--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/Cauchy_Integral_Formula.thy Mon Dec 02 17:51:54 2019 +0100
@@ -0,0 +1,2787 @@
+section \<open>Cauchy's Integral Formula\<close>
+theory Cauchy_Integral_Formula
+ imports Winding_Numbers
+begin
+
+subsection\<open>Proof\<close>
+
+lemma Cauchy_integral_formula_weak:
+ assumes s: "convex s" and "finite k" and conf: "continuous_on s f"
+ and fcd: "(\<And>x. x \<in> interior s - k \<Longrightarrow> f field_differentiable at x)"
+ and z: "z \<in> interior s - k" and vpg: "valid_path \<gamma>"
+ and pasz: "path_image \<gamma> \<subseteq> s - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+proof -
+ obtain f' where f': "(f has_field_derivative f') (at z)"
+ using fcd [OF z] by (auto simp: field_differentiable_def)
+ have pas: "path_image \<gamma> \<subseteq> s" and znotin: "z \<notin> path_image \<gamma>" using pasz by blast+
+ have c: "continuous (at x within s) (\<lambda>w. if w = z then f' else (f w - f z) / (w - z))" if "x \<in> s" for x
+ proof (cases "x = z")
+ case True then show ?thesis
+ apply (simp add: continuous_within)
+ apply (rule Lim_transform_away_within [of _ "z+1" _ "\<lambda>w::complex. (f w - f z)/(w - z)"])
+ using has_field_derivative_at_within has_field_derivative_iff f'
+ apply (fastforce simp add:)+
+ done
+ next
+ case False
+ then have dxz: "dist x z > 0" by auto
+ have cf: "continuous (at x within s) f"
+ using conf continuous_on_eq_continuous_within that by blast
+ have "continuous (at x within s) (\<lambda>w. (f w - f z) / (w - z))"
+ by (rule cf continuous_intros | simp add: False)+
+ then show ?thesis
+ apply (rule continuous_transform_within [OF _ dxz that, of "\<lambda>w::complex. (f w - f z)/(w - z)"])
+ apply (force simp: dist_commute)
+ done
+ qed
+ have fink': "finite (insert z k)" using \<open>finite k\<close> by blast
+ have *: "((\<lambda>w. if w = z then f' else (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
+ apply (rule Cauchy_theorem_convex [OF _ s fink' _ vpg pas loop])
+ using c apply (force simp: continuous_on_eq_continuous_within)
+ apply (rename_tac w)
+ apply (rule_tac d="dist w z" and f = "\<lambda>w. (f w - f z)/(w - z)" in field_differentiable_transform_within)
+ apply (simp_all add: dist_pos_lt dist_commute)
+ apply (metis less_irrefl)
+ apply (rule derivative_intros fcd | simp)+
+ done
+ show ?thesis
+ apply (rule has_contour_integral_eq)
+ using znotin has_contour_integral_add [OF has_contour_integral_lmul [OF has_contour_integral_winding_number [OF vpg znotin], of "f z"] *]
+ apply (auto simp: ac_simps divide_simps)
+ done
+qed
+
+theorem Cauchy_integral_formula_convex_simple:
+ "\<lbrakk>convex s; f holomorphic_on s; z \<in> interior s; valid_path \<gamma>; path_image \<gamma> \<subseteq> s - {z};
+ pathfinish \<gamma> = pathstart \<gamma>\<rbrakk>
+ \<Longrightarrow> ((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+ apply (rule Cauchy_integral_formula_weak [where k = "{}"])
+ using holomorphic_on_imp_continuous_on
+ by auto (metis at_within_interior holomorphic_on_def interiorE subsetCE)
+
+text\<open> Hence the Cauchy formula for points inside a circle.\<close>
+
+theorem Cauchy_integral_circlepath:
+ assumes contf: "continuous_on (cball z r) f" and holf: "f holomorphic_on (ball z r)" and wz: "norm(w - z) < r"
+ shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w))
+ (circlepath z r)"
+proof -
+ have "r > 0"
+ using assms le_less_trans norm_ge_zero by blast
+ have "((\<lambda>u. f u / (u - w)) has_contour_integral (2 * pi) * \<i> * winding_number (circlepath z r) w * f w)
+ (circlepath z r)"
+ proof (rule Cauchy_integral_formula_weak [where s = "cball z r" and k = "{}"])
+ show "\<And>x. x \<in> interior (cball z r) - {} \<Longrightarrow>
+ f field_differentiable at x"
+ using holf holomorphic_on_imp_differentiable_at by auto
+ have "w \<notin> sphere z r"
+ by simp (metis dist_commute dist_norm not_le order_refl wz)
+ then show "path_image (circlepath z r) \<subseteq> cball z r - {w}"
+ using \<open>r > 0\<close> by (auto simp add: cball_def sphere_def)
+ qed (use wz in \<open>simp_all add: dist_norm norm_minus_commute contf\<close>)
+ then show ?thesis
+ by (simp add: winding_number_circlepath assms)
+qed
+
+corollary\<^marker>\<open>tag unimportant\<close> Cauchy_integral_circlepath_simple:
+ assumes "f holomorphic_on cball z r" "norm(w - z) < r"
+ shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w))
+ (circlepath z r)"
+using assms by (force simp: holomorphic_on_imp_continuous_on holomorphic_on_subset Cauchy_integral_circlepath)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>General stepping result for derivative formulas\<close>
+
+lemma Cauchy_next_derivative:
+ assumes "continuous_on (path_image \<gamma>) f'"
+ and leB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
+ and int: "\<And>w. w \<in> s - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f' u / (u - w)^k) has_contour_integral f w) \<gamma>"
+ and k: "k \<noteq> 0"
+ and "open s"
+ and \<gamma>: "valid_path \<gamma>"
+ and w: "w \<in> s - path_image \<gamma>"
+ shows "(\<lambda>u. f' u / (u - w)^(Suc k)) contour_integrable_on \<gamma>"
+ and "(f has_field_derivative (k * contour_integral \<gamma> (\<lambda>u. f' u/(u - w)^(Suc k))))
+ (at w)" (is "?thes2")
+proof -
+ have "open (s - path_image \<gamma>)" using \<open>open s\<close> closed_valid_path_image \<gamma> by blast
+ then obtain d where "d>0" and d: "ball w d \<subseteq> s - path_image \<gamma>" using w
+ using open_contains_ball by blast
+ have [simp]: "\<And>n. cmod (1 + of_nat n) = 1 + of_nat n"
+ by (metis norm_of_nat of_nat_Suc)
+ have cint: "\<And>x. \<lbrakk>x \<noteq> w; cmod (x - w) < d\<rbrakk>
+ \<Longrightarrow> (\<lambda>z. (f' z / (z - x) ^ k - f' z / (z - w) ^ k) / (x * k - w * k)) contour_integrable_on \<gamma>"
+ apply (rule contour_integrable_div [OF contour_integrable_diff])
+ using int w d
+ by (force simp: dist_norm norm_minus_commute intro!: has_contour_integral_integrable)+
+ have 1: "\<forall>\<^sub>F n in at w. (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k)
+ contour_integrable_on \<gamma>"
+ unfolding eventually_at
+ apply (rule_tac x=d in exI)
+ apply (simp add: \<open>d > 0\<close> dist_norm field_simps cint)
+ done
+ have bim_g: "bounded (image f' (path_image \<gamma>))"
+ by (simp add: compact_imp_bounded compact_continuous_image compact_valid_path_image assms)
+ then obtain C where "C > 0" and C: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cmod (f' (\<gamma> x)) \<le> C"
+ by (force simp: bounded_pos path_image_def)
+ have twom: "\<forall>\<^sub>F n in at w.
+ \<forall>x\<in>path_image \<gamma>.
+ cmod ((inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k - inverse (x - w) ^ Suc k) < e"
+ if "0 < e" for e
+ proof -
+ have *: "cmod ((inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k) - inverse (x - w) ^ Suc k) < e"
+ if x: "x \<in> path_image \<gamma>" and "u \<noteq> w" and uwd: "cmod (u - w) < d/2"
+ and uw_less: "cmod (u - w) < e * (d/2) ^ (k+2) / (1 + real k)"
+ for u x
+ proof -
+ define ff where [abs_def]:
+ "ff n w =
+ (if n = 0 then inverse(x - w)^k
+ else if n = 1 then k / (x - w)^(Suc k)
+ else (k * of_real(Suc k)) / (x - w)^(k + 2))" for n :: nat and w
+ have km1: "\<And>z::complex. z \<noteq> 0 \<Longrightarrow> z ^ (k - Suc 0) = z ^ k / z"
+ by (simp add: field_simps) (metis Suc_pred \<open>k \<noteq> 0\<close> neq0_conv power_Suc)
+ have ff1: "(ff i has_field_derivative ff (Suc i) z) (at z within ball w (d/2))"
+ if "z \<in> ball w (d/2)" "i \<le> 1" for i z
+ proof -
+ have "z \<notin> path_image \<gamma>"
+ using \<open>x \<in> path_image \<gamma>\<close> d that ball_divide_subset_numeral by blast
+ then have xz[simp]: "x \<noteq> z" using \<open>x \<in> path_image \<gamma>\<close> by blast
+ then have neq: "x * x + z * z \<noteq> x * (z * 2)"
+ by (blast intro: dest!: sum_sqs_eq)
+ with xz have "\<And>v. v \<noteq> 0 \<Longrightarrow> (x * x + z * z) * v \<noteq> (x * (z * 2) * v)" by auto
+ then have neqq: "\<And>v. v \<noteq> 0 \<Longrightarrow> x * (x * v) + z * (z * v) \<noteq> x * (z * (2 * v))"
+ by (simp add: algebra_simps)
+ show ?thesis using \<open>i \<le> 1\<close>
+ apply (simp add: ff_def dist_norm Nat.le_Suc_eq km1, safe)
+ apply (rule derivative_eq_intros | simp add: km1 | simp add: field_simps neq neqq)+
+ done
+ qed
+ { fix a::real and b::real assume ab: "a > 0" "b > 0"
+ then have "k * (1 + real k) * (1 / a) \<le> k * (1 + real k) * (4 / b) \<longleftrightarrow> b \<le> 4 * a"
+ by (subst mult_le_cancel_left_pos)
+ (use \<open>k \<noteq> 0\<close> in \<open>auto simp: divide_simps\<close>)
+ with ab have "real k * (1 + real k) / a \<le> (real k * 4 + real k * real k * 4) / b \<longleftrightarrow> b \<le> 4 * a"
+ by (simp add: field_simps)
+ } note canc = this
+ have ff2: "cmod (ff (Suc 1) v) \<le> real (k * (k + 1)) / (d/2) ^ (k + 2)"
+ if "v \<in> ball w (d/2)" for v
+ proof -
+ have lessd: "\<And>z. cmod (\<gamma> z - v) < d/2 \<Longrightarrow> cmod (w - \<gamma> z) < d"
+ by (metis that norm_minus_commute norm_triangle_half_r dist_norm mem_ball)
+ have "d/2 \<le> cmod (x - v)" using d x that
+ using lessd d x
+ by (auto simp add: dist_norm path_image_def ball_def not_less [symmetric] del: divide_const_simps)
+ then have "d \<le> cmod (x - v) * 2"
+ by (simp add: field_split_simps)
+ then have dpow_le: "d ^ (k+2) \<le> (cmod (x - v) * 2) ^ (k+2)"
+ using \<open>0 < d\<close> order_less_imp_le power_mono by blast
+ have "x \<noteq> v" using that
+ using \<open>x \<in> path_image \<gamma>\<close> ball_divide_subset_numeral d by fastforce
+ then show ?thesis
+ using \<open>d > 0\<close> apply (simp add: ff_def norm_mult norm_divide norm_power dist_norm canc)
+ using dpow_le apply (simp add: field_split_simps)
+ done
+ qed
+ have ub: "u \<in> ball w (d/2)"
+ using uwd by (simp add: dist_commute dist_norm)
+ have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
+ \<le> (real k * 4 + real k * real k * 4) * (cmod (u - w) * cmod (u - w)) / (d * (d * (d/2) ^ k))"
+ using complex_Taylor [OF _ ff1 ff2 _ ub, of w, simplified]
+ by (simp add: ff_def \<open>0 < d\<close>)
+ then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
+ \<le> (cmod (u - w) * real k) * (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)"
+ by (simp add: field_simps)
+ then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
+ / (cmod (u - w) * real k)
+ \<le> (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)"
+ using \<open>k \<noteq> 0\<close> \<open>u \<noteq> w\<close> by (simp add: mult_ac zero_less_mult_iff pos_divide_le_eq)
+ also have "\<dots> < e"
+ using uw_less \<open>0 < d\<close> by (simp add: mult_ac divide_simps)
+ finally have e: "cmod (inverse (x-u)^k - (inverse (x-w)^k + of_nat k * (u-w) / ((x-w) * (x-w)^k)))
+ / cmod ((u - w) * real k) < e"
+ by (simp add: norm_mult)
+ have "x \<noteq> u"
+ using uwd \<open>0 < d\<close> x d by (force simp: dist_norm ball_def norm_minus_commute)
+ show ?thesis
+ apply (rule le_less_trans [OF _ e])
+ using \<open>k \<noteq> 0\<close> \<open>x \<noteq> u\<close> \<open>u \<noteq> w\<close>
+ apply (simp add: field_simps norm_divide [symmetric])
+ done
+ qed
+ show ?thesis
+ unfolding eventually_at
+ apply (rule_tac x = "min (d/2) ((e*(d/2)^(k + 2))/(Suc k))" in exI)
+ apply (force simp: \<open>d > 0\<close> dist_norm that simp del: power_Suc intro: *)
+ done
+ qed
+ have 2: "uniform_limit (path_image \<gamma>) (\<lambda>n x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) (\<lambda>x. f' x / (x - w) ^ Suc k) (at w)"
+ unfolding uniform_limit_iff dist_norm
+ proof clarify
+ fix e::real
+ assume "0 < e"
+ have *: "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+ f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) < e"
+ if ec: "cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+ inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k) < e / C"
+ and x: "0 \<le> x" "x \<le> 1"
+ for u x
+ proof (cases "(f' (\<gamma> x)) = 0")
+ case True then show ?thesis by (simp add: \<open>0 < e\<close>)
+ next
+ case False
+ have "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+ f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) =
+ cmod (f' (\<gamma> x) * ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+ inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k))"
+ by (simp add: field_simps)
+ also have "\<dots> = cmod (f' (\<gamma> x)) *
+ cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+ inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k)"
+ by (simp add: norm_mult)
+ also have "\<dots> < cmod (f' (\<gamma> x)) * (e/C)"
+ using False mult_strict_left_mono [OF ec] by force
+ also have "\<dots> \<le> e" using C
+ by (metis False \<open>0 < e\<close> frac_le less_eq_real_def mult.commute pos_le_divide_eq x zero_less_norm_iff)
+ finally show ?thesis .
+ qed
+ show "\<forall>\<^sub>F n in at w.
+ \<forall>x\<in>path_image \<gamma>.
+ cmod (f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k - f' x / (x - w) ^ Suc k) < e"
+ using twom [OF divide_pos_pos [OF \<open>0 < e\<close> \<open>C > 0\<close>]] unfolding path_image_def
+ by (force intro: * elim: eventually_mono)
+ qed
+ show "(\<lambda>u. f' u / (u - w) ^ (Suc k)) contour_integrable_on \<gamma>"
+ by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
+ have *: "(\<lambda>n. contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k))
+ \<midarrow>w\<rightarrow> contour_integral \<gamma> (\<lambda>u. f' u / (u - w) ^ (Suc k))"
+ by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
+ have **: "contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k)) =
+ (f u - f w) / (u - w) / k"
+ if "dist u w < d" for u
+ proof -
+ have u: "u \<in> s - path_image \<gamma>"
+ by (metis subsetD d dist_commute mem_ball that)
+ show ?thesis
+ apply (rule contour_integral_unique)
+ apply (simp add: diff_divide_distrib algebra_simps)
+ apply (intro has_contour_integral_diff has_contour_integral_div)
+ using u w apply (simp_all add: field_simps int)
+ done
+ qed
+ show ?thes2
+ apply (simp add: has_field_derivative_iff del: power_Suc)
+ apply (rule Lim_transform_within [OF tendsto_mult_left [OF *] \<open>0 < d\<close> ])
+ apply (simp add: \<open>k \<noteq> 0\<close> **)
+ done
+qed
+
+lemma Cauchy_next_derivative_circlepath:
+ assumes contf: "continuous_on (path_image (circlepath z r)) f"
+ and int: "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>u. f u / (u - w)^k) has_contour_integral g w) (circlepath z r)"
+ and k: "k \<noteq> 0"
+ and w: "w \<in> ball z r"
+ shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
+ (is "?thes1")
+ and "(g has_field_derivative (k * contour_integral (circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)))) (at w)"
+ (is "?thes2")
+proof -
+ have "r > 0" using w
+ using ball_eq_empty by fastforce
+ have wim: "w \<in> ball z r - path_image (circlepath z r)"
+ using w by (auto simp: dist_norm)
+ show ?thes1 ?thes2
+ by (rule Cauchy_next_derivative [OF contf _ int k open_ball valid_path_circlepath wim, where B = "2 * pi * \<bar>r\<bar>"];
+ auto simp: vector_derivative_circlepath norm_mult)+
+qed
+
+
+text\<open> In particular, the first derivative formula.\<close>
+
+lemma Cauchy_derivative_integral_circlepath:
+ assumes contf: "continuous_on (cball z r) f"
+ and holf: "f holomorphic_on ball z r"
+ and w: "w \<in> ball z r"
+ shows "(\<lambda>u. f u/(u - w)^2) contour_integrable_on (circlepath z r)"
+ (is "?thes1")
+ and "(f has_field_derivative (1 / (2 * of_real pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u / (u - w)^2))) (at w)"
+ (is "?thes2")
+proof -
+ have [simp]: "r \<ge> 0" using w
+ using ball_eq_empty by fastforce
+ have f: "continuous_on (path_image (circlepath z r)) f"
+ by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def)
+ have int: "\<And>w. dist z w < r \<Longrightarrow>
+ ((\<lambda>u. f u / (u - w)) has_contour_integral (\<lambda>x. 2 * of_real pi * \<i> * f x) w) (circlepath z r)"
+ by (rule Cauchy_integral_circlepath [OF contf holf]) (simp add: dist_norm norm_minus_commute)
+ show ?thes1
+ apply (simp add: power2_eq_square)
+ apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1, simplified])
+ apply (blast intro: int)
+ done
+ have "((\<lambda>x. 2 * of_real pi * \<i> * f x) has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2)) (at w)"
+ apply (simp add: power2_eq_square)
+ apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1 and g = "\<lambda>x. 2 * of_real pi * \<i> * f x", simplified])
+ apply (blast intro: int)
+ done
+ then have fder: "(f has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2) / (2 * of_real pi * \<i>)) (at w)"
+ by (rule DERIV_cdivide [where f = "\<lambda>x. 2 * of_real pi * \<i> * f x" and c = "2 * of_real pi * \<i>", simplified])
+ show ?thes2
+ by simp (rule fder)
+qed
+
+subsection\<open>Existence of all higher derivatives\<close>
+
+proposition derivative_is_holomorphic:
+ assumes "open S"
+ and fder: "\<And>z. z \<in> S \<Longrightarrow> (f has_field_derivative f' z) (at z)"
+ shows "f' holomorphic_on S"
+proof -
+ have *: "\<exists>h. (f' has_field_derivative h) (at z)" if "z \<in> S" for z
+ proof -
+ obtain r where "r > 0" and r: "cball z r \<subseteq> S"
+ using open_contains_cball \<open>z \<in> S\<close> \<open>open S\<close> by blast
+ then have holf_cball: "f holomorphic_on cball z r"
+ apply (simp add: holomorphic_on_def)
+ using field_differentiable_at_within field_differentiable_def fder by blast
+ then have "continuous_on (path_image (circlepath z r)) f"
+ using \<open>r > 0\<close> by (force elim: holomorphic_on_subset [THEN holomorphic_on_imp_continuous_on])
+ then have contfpi: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1/(2 * of_real pi*\<i>) * f x)"
+ by (auto intro: continuous_intros)+
+ have contf_cball: "continuous_on (cball z r) f" using holf_cball
+ by (simp add: holomorphic_on_imp_continuous_on holomorphic_on_subset)
+ have holf_ball: "f holomorphic_on ball z r" using holf_cball
+ using ball_subset_cball holomorphic_on_subset by blast
+ { fix w assume w: "w \<in> ball z r"
+ have intf: "(\<lambda>u. f u / (u - w)\<^sup>2) contour_integrable_on circlepath z r"
+ by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
+ have fder': "(f has_field_derivative 1 / (2 * of_real pi * \<i>) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2))
+ (at w)"
+ by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
+ have f'_eq: "f' w = contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>)"
+ using fder' ball_subset_cball r w by (force intro: DERIV_unique [OF fder])
+ have "((\<lambda>u. f u / (u - w)\<^sup>2 / (2 * of_real pi * \<i>)) has_contour_integral
+ contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
+ (circlepath z r)"
+ by (rule has_contour_integral_div [OF has_contour_integral_integral [OF intf]])
+ then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral
+ contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
+ (circlepath z r)"
+ by (simp add: algebra_simps)
+ then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral f' w) (circlepath z r)"
+ by (simp add: f'_eq)
+ } note * = this
+ show ?thesis
+ apply (rule exI)
+ apply (rule Cauchy_next_derivative_circlepath [OF contfpi, of 2 f', simplified])
+ apply (simp_all add: \<open>0 < r\<close> * dist_norm)
+ done
+ qed
+ show ?thesis
+ by (simp add: holomorphic_on_open [OF \<open>open S\<close>] *)
+qed
+
+lemma holomorphic_deriv [holomorphic_intros]:
+ "\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv f) holomorphic_on S"
+by (metis DERIV_deriv_iff_field_differentiable at_within_open derivative_is_holomorphic holomorphic_on_def)
+
+lemma analytic_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv f) analytic_on S"
+ using analytic_on_holomorphic holomorphic_deriv by auto
+
+lemma holomorphic_higher_deriv [holomorphic_intros]: "\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv ^^ n) f holomorphic_on S"
+ by (induction n) (auto simp: holomorphic_deriv)
+
+lemma analytic_higher_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv ^^ n) f analytic_on S"
+ unfolding analytic_on_def using holomorphic_higher_deriv by blast
+
+lemma has_field_derivative_higher_deriv:
+ "\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk>
+ \<Longrightarrow> ((deriv ^^ n) f has_field_derivative (deriv ^^ (Suc n)) f x) (at x)"
+by (metis (no_types, hide_lams) DERIV_deriv_iff_field_differentiable at_within_open comp_apply
+ funpow.simps(2) holomorphic_higher_deriv holomorphic_on_def)
+
+lemma valid_path_compose_holomorphic:
+ assumes "valid_path g" and holo:"f holomorphic_on S" and "open S" "path_image g \<subseteq> S"
+ shows "valid_path (f \<circ> g)"
+proof (rule valid_path_compose[OF \<open>valid_path g\<close>])
+ fix x assume "x \<in> path_image g"
+ then show "f field_differentiable at x"
+ using analytic_on_imp_differentiable_at analytic_on_open assms holo by blast
+next
+ have "deriv f holomorphic_on S"
+ using holomorphic_deriv holo \<open>open S\<close> by auto
+ then show "continuous_on (path_image g) (deriv f)"
+ using assms(4) holomorphic_on_imp_continuous_on holomorphic_on_subset by auto
+qed
+
+subsection\<open>Morera's theorem\<close>
+
+lemma Morera_local_triangle_ball:
+ assumes "\<And>z. z \<in> S
+ \<Longrightarrow> \<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and>
+ (\<forall>b c. closed_segment b c \<subseteq> ball a e
+ \<longrightarrow> contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0)"
+ shows "f analytic_on S"
+proof -
+ { fix z assume "z \<in> S"
+ with assms obtain e a where
+ "0 < e" and z: "z \<in> ball a e" and contf: "continuous_on (ball a e) f"
+ and 0: "\<And>b c. closed_segment b c \<subseteq> ball a e
+ \<Longrightarrow> contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+ by blast
+ have az: "dist a z < e" using mem_ball z by blast
+ have sb_ball: "ball z (e - dist a z) \<subseteq> ball a e"
+ by (simp add: dist_commute ball_subset_ball_iff)
+ have "\<exists>e>0. f holomorphic_on ball z e"
+ proof (intro exI conjI)
+ have sub_ball: "\<And>y. dist a y < e \<Longrightarrow> closed_segment a y \<subseteq> ball a e"
+ by (meson \<open>0 < e\<close> centre_in_ball convex_ball convex_contains_segment mem_ball)
+ show "f holomorphic_on ball z (e - dist a z)"
+ apply (rule holomorphic_on_subset [OF _ sb_ball])
+ apply (rule derivative_is_holomorphic[OF open_ball])
+ apply (rule triangle_contour_integrals_starlike_primitive [OF contf _ open_ball, of a])
+ apply (simp_all add: 0 \<open>0 < e\<close> sub_ball)
+ done
+ qed (simp add: az)
+ }
+ then show ?thesis
+ by (simp add: analytic_on_def)
+qed
+
+lemma Morera_local_triangle:
+ assumes "\<And>z. z \<in> S
+ \<Longrightarrow> \<exists>t. open t \<and> z \<in> t \<and> continuous_on t f \<and>
+ (\<forall>a b c. convex hull {a,b,c} \<subseteq> t
+ \<longrightarrow> contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0)"
+ shows "f analytic_on S"
+proof -
+ { fix z assume "z \<in> S"
+ with assms obtain t where
+ "open t" and z: "z \<in> t" and contf: "continuous_on t f"
+ and 0: "\<And>a b c. convex hull {a,b,c} \<subseteq> t
+ \<Longrightarrow> contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+ by force
+ then obtain e where "e>0" and e: "ball z e \<subseteq> t"
+ using open_contains_ball by blast
+ have [simp]: "continuous_on (ball z e) f" using contf
+ using continuous_on_subset e by blast
+ have eq0: "\<And>b c. closed_segment b c \<subseteq> ball z e \<Longrightarrow>
+ contour_integral (linepath z b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c z) f = 0"
+ by (meson 0 z \<open>0 < e\<close> centre_in_ball closed_segment_subset convex_ball dual_order.trans e starlike_convex_subset)
+ have "\<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and>
+ (\<forall>b c. closed_segment b c \<subseteq> ball a e \<longrightarrow>
+ contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0)"
+ using \<open>e > 0\<close> eq0 by force
+ }
+ then show ?thesis
+ by (simp add: Morera_local_triangle_ball)
+qed
+
+proposition Morera_triangle:
+ "\<lbrakk>continuous_on S f; open S;
+ \<And>a b c. convex hull {a,b,c} \<subseteq> S
+ \<longrightarrow> contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0\<rbrakk>
+ \<Longrightarrow> f analytic_on S"
+ using Morera_local_triangle by blast
+
+subsection\<open>Combining theorems for higher derivatives including Leibniz rule\<close>
+
+lemma higher_deriv_linear [simp]:
+ "(deriv ^^ n) (\<lambda>w. c*w) = (\<lambda>z. if n = 0 then c*z else if n = 1 then c else 0)"
+ by (induction n) auto
+
+lemma higher_deriv_const [simp]: "(deriv ^^ n) (\<lambda>w. c) = (\<lambda>w. if n=0 then c else 0)"
+ by (induction n) auto
+
+lemma higher_deriv_ident [simp]:
+ "(deriv ^^ n) (\<lambda>w. w) z = (if n = 0 then z else if n = 1 then 1 else 0)"
+ apply (induction n, simp)
+ apply (metis higher_deriv_linear lambda_one)
+ done
+
+lemma higher_deriv_id [simp]:
+ "(deriv ^^ n) id z = (if n = 0 then z else if n = 1 then 1 else 0)"
+ by (simp add: id_def)
+
+lemma has_complex_derivative_funpow_1:
+ "\<lbrakk>(f has_field_derivative 1) (at z); f z = z\<rbrakk> \<Longrightarrow> (f^^n has_field_derivative 1) (at z)"
+ apply (induction n, auto)
+ apply (simp add: id_def)
+ by (metis DERIV_chain comp_funpow comp_id funpow_swap1 mult.right_neutral)
+
+lemma higher_deriv_uminus:
+ assumes "f holomorphic_on S" "open S" and z: "z \<in> S"
+ shows "(deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)"
+using z
+proof (induction n arbitrary: z)
+ case 0 then show ?case by simp
+next
+ case (Suc n z)
+ have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
+ using Suc.prems assms has_field_derivative_higher_deriv by auto
+ have "((deriv ^^ n) (\<lambda>w. - f w) has_field_derivative - deriv ((deriv ^^ n) f) z) (at z)"
+ apply (rule has_field_derivative_transform_within_open [of "\<lambda>w. -((deriv ^^ n) f w)"])
+ apply (rule derivative_eq_intros | rule * refl assms)+
+ apply (auto simp add: Suc)
+ done
+ then show ?case
+ by (simp add: DERIV_imp_deriv)
+qed
+
+lemma higher_deriv_add:
+ fixes z::complex
+ assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
+ shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
+using z
+proof (induction n arbitrary: z)
+ case 0 then show ?case by simp
+next
+ case (Suc n z)
+ have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
+ "((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
+ using Suc.prems assms has_field_derivative_higher_deriv by auto
+ have "((deriv ^^ n) (\<lambda>w. f w + g w) has_field_derivative
+ deriv ((deriv ^^ n) f) z + deriv ((deriv ^^ n) g) z) (at z)"
+ apply (rule has_field_derivative_transform_within_open [of "\<lambda>w. (deriv ^^ n) f w + (deriv ^^ n) g w"])
+ apply (rule derivative_eq_intros | rule * refl assms)+
+ apply (auto simp add: Suc)
+ done
+ then show ?case
+ by (simp add: DERIV_imp_deriv)
+qed
+
+lemma higher_deriv_diff:
+ fixes z::complex
+ assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
+ shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
+ apply (simp only: Groups.group_add_class.diff_conv_add_uminus higher_deriv_add)
+ apply (subst higher_deriv_add)
+ using assms holomorphic_on_minus apply (auto simp: higher_deriv_uminus)
+ done
+
+lemma bb: "Suc n choose k = (n choose k) + (if k = 0 then 0 else (n choose (k - 1)))"
+ by (cases k) simp_all
+
+lemma higher_deriv_mult:
+ fixes z::complex
+ assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
+ shows "(deriv ^^ n) (\<lambda>w. f w * g w) z =
+ (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
+using z
+proof (induction n arbitrary: z)
+ case 0 then show ?case by simp
+next
+ case (Suc n z)
+ have *: "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
+ "\<And>n. ((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
+ using Suc.prems assms has_field_derivative_higher_deriv by auto
+ have sumeq: "(\<Sum>i = 0..n.
+ of_nat (n choose i) * (deriv ((deriv ^^ i) f) z * (deriv ^^ (n - i)) g z + deriv ((deriv ^^ (n - i)) g) z * (deriv ^^ i) f z)) =
+ g z * deriv ((deriv ^^ n) f) z + (\<Sum>i = 0..n. (deriv ^^ i) f z * (of_nat (Suc n choose i) * (deriv ^^ (Suc n - i)) g z))"
+ apply (simp add: bb algebra_simps sum.distrib)
+ apply (subst (4) sum_Suc_reindex)
+ apply (auto simp: algebra_simps Suc_diff_le intro: sum.cong)
+ done
+ have "((deriv ^^ n) (\<lambda>w. f w * g w) has_field_derivative
+ (\<Sum>i = 0..Suc n. (Suc n choose i) * (deriv ^^ i) f z * (deriv ^^ (Suc n - i)) g z))
+ (at z)"
+ apply (rule has_field_derivative_transform_within_open
+ [of "\<lambda>w. (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f w * (deriv ^^ (n - i)) g w)"])
+ apply (simp add: algebra_simps)
+ apply (rule DERIV_cong [OF DERIV_sum])
+ apply (rule DERIV_cmult)
+ apply (auto intro: DERIV_mult * sumeq \<open>open S\<close> Suc.prems Suc.IH [symmetric])
+ done
+ then show ?case
+ unfolding funpow.simps o_apply
+ by (simp add: DERIV_imp_deriv)
+qed
+
+lemma higher_deriv_transform_within_open:
+ fixes z::complex
+ assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
+ and fg: "\<And>w. w \<in> S \<Longrightarrow> f w = g w"
+ shows "(deriv ^^ i) f z = (deriv ^^ i) g z"
+using z
+by (induction i arbitrary: z)
+ (auto simp: fg intro: complex_derivative_transform_within_open holomorphic_higher_deriv assms)
+
+lemma higher_deriv_compose_linear:
+ fixes z::complex
+ assumes f: "f holomorphic_on T" and S: "open S" and T: "open T" and z: "z \<in> S"
+ and fg: "\<And>w. w \<in> S \<Longrightarrow> u * w \<in> T"
+ shows "(deriv ^^ n) (\<lambda>w. f (u * w)) z = u^n * (deriv ^^ n) f (u * z)"
+using z
+proof (induction n arbitrary: z)
+ case 0 then show ?case by simp
+next
+ case (Suc n z)
+ have holo0: "f holomorphic_on (*) u ` S"
+ by (meson fg f holomorphic_on_subset image_subset_iff)
+ have holo2: "(deriv ^^ n) f holomorphic_on (*) u ` S"
+ by (meson f fg holomorphic_higher_deriv holomorphic_on_subset image_subset_iff T)
+ have holo3: "(\<lambda>z. u ^ n * (deriv ^^ n) f (u * z)) holomorphic_on S"
+ by (intro holo2 holomorphic_on_compose [where g="(deriv ^^ n) f", unfolded o_def] holomorphic_intros)
+ have holo1: "(\<lambda>w. f (u * w)) holomorphic_on S"
+ apply (rule holomorphic_on_compose [where g=f, unfolded o_def])
+ apply (rule holo0 holomorphic_intros)+
+ done
+ have "deriv ((deriv ^^ n) (\<lambda>w. f (u * w))) z = deriv (\<lambda>z. u^n * (deriv ^^ n) f (u*z)) z"
+ apply (rule complex_derivative_transform_within_open [OF _ holo3 S Suc.prems])
+ apply (rule holomorphic_higher_deriv [OF holo1 S])
+ apply (simp add: Suc.IH)
+ done
+ also have "\<dots> = u^n * deriv (\<lambda>z. (deriv ^^ n) f (u * z)) z"
+ apply (rule deriv_cmult)
+ apply (rule analytic_on_imp_differentiable_at [OF _ Suc.prems])
+ apply (rule analytic_on_compose_gen [where g="(deriv ^^ n) f" and T=T, unfolded o_def])
+ apply (simp)
+ apply (simp add: analytic_on_open f holomorphic_higher_deriv T)
+ apply (blast intro: fg)
+ done
+ also have "\<dots> = u * u ^ n * deriv ((deriv ^^ n) f) (u * z)"
+ apply (subst deriv_chain [where g = "(deriv ^^ n) f" and f = "(*) u", unfolded o_def])
+ apply (rule derivative_intros)
+ using Suc.prems field_differentiable_def f fg has_field_derivative_higher_deriv T apply blast
+ apply (simp)
+ done
+ finally show ?case
+ by simp
+qed
+
+lemma higher_deriv_add_at:
+ assumes "f analytic_on {z}" "g analytic_on {z}"
+ shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
+proof -
+ have "f analytic_on {z} \<and> g analytic_on {z}"
+ using assms by blast
+ with higher_deriv_add show ?thesis
+ by (auto simp: analytic_at_two)
+qed
+
+lemma higher_deriv_diff_at:
+ assumes "f analytic_on {z}" "g analytic_on {z}"
+ shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
+proof -
+ have "f analytic_on {z} \<and> g analytic_on {z}"
+ using assms by blast
+ with higher_deriv_diff show ?thesis
+ by (auto simp: analytic_at_two)
+qed
+
+lemma higher_deriv_uminus_at:
+ "f analytic_on {z} \<Longrightarrow> (deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)"
+ using higher_deriv_uminus
+ by (auto simp: analytic_at)
+
+lemma higher_deriv_mult_at:
+ assumes "f analytic_on {z}" "g analytic_on {z}"
+ shows "(deriv ^^ n) (\<lambda>w. f w * g w) z =
+ (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
+proof -
+ have "f analytic_on {z} \<and> g analytic_on {z}"
+ using assms by blast
+ with higher_deriv_mult show ?thesis
+ by (auto simp: analytic_at_two)
+qed
+
+
+text\<open> Nonexistence of isolated singularities and a stronger integral formula.\<close>
+
+proposition no_isolated_singularity:
+ fixes z::complex
+ assumes f: "continuous_on S f" and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K"
+ shows "f holomorphic_on S"
+proof -
+ { fix z
+ assume "z \<in> S" and cdf: "\<And>x. x \<in> S - K \<Longrightarrow> f field_differentiable at x"
+ have "f field_differentiable at z"
+ proof (cases "z \<in> K")
+ case False then show ?thesis by (blast intro: cdf \<open>z \<in> S\<close>)
+ next
+ case True
+ with finite_set_avoid [OF K, of z]
+ obtain d where "d>0" and d: "\<And>x. \<lbrakk>x\<in>K; x \<noteq> z\<rbrakk> \<Longrightarrow> d \<le> dist z x"
+ by blast
+ obtain e where "e>0" and e: "ball z e \<subseteq> S"
+ using S \<open>z \<in> S\<close> by (force simp: open_contains_ball)
+ have fde: "continuous_on (ball z (min d e)) f"
+ by (metis Int_iff ball_min_Int continuous_on_subset e f subsetI)
+ have cont: "{a,b,c} \<subseteq> ball z (min d e) \<Longrightarrow> continuous_on (convex hull {a, b, c}) f" for a b c
+ by (simp add: hull_minimal continuous_on_subset [OF fde])
+ have fd: "\<lbrakk>{a,b,c} \<subseteq> ball z (min d e); x \<in> interior (convex hull {a, b, c}) - K\<rbrakk>
+ \<Longrightarrow> f field_differentiable at x" for a b c x
+ by (metis cdf Diff_iff Int_iff ball_min_Int subsetD convex_ball e interior_mono interior_subset subset_hull)
+ obtain g where "\<And>w. w \<in> ball z (min d e) \<Longrightarrow> (g has_field_derivative f w) (at w within ball z (min d e))"
+ apply (rule contour_integral_convex_primitive
+ [OF convex_ball fde Cauchy_theorem_triangle_cofinite [OF _ K]])
+ using cont fd by auto
+ then have "f holomorphic_on ball z (min d e)"
+ by (metis open_ball at_within_open derivative_is_holomorphic)
+ then show ?thesis
+ unfolding holomorphic_on_def
+ by (metis open_ball \<open>0 < d\<close> \<open>0 < e\<close> at_within_open centre_in_ball min_less_iff_conj)
+ qed
+ }
+ with holf S K show ?thesis
+ by (simp add: holomorphic_on_open open_Diff finite_imp_closed field_differentiable_def [symmetric])
+qed
+
+lemma no_isolated_singularity':
+ fixes z::complex
+ assumes f: "\<And>z. z \<in> K \<Longrightarrow> (f \<longlongrightarrow> f z) (at z within S)"
+ and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K"
+ shows "f holomorphic_on S"
+proof (rule no_isolated_singularity[OF _ assms(2-)])
+ show "continuous_on S f" unfolding continuous_on_def
+ proof
+ fix z assume z: "z \<in> S"
+ show "(f \<longlongrightarrow> f z) (at z within S)"
+ proof (cases "z \<in> K")
+ case False
+ from holf have "continuous_on (S - K) f"
+ by (rule holomorphic_on_imp_continuous_on)
+ with z False have "(f \<longlongrightarrow> f z) (at z within (S - K))"
+ by (simp add: continuous_on_def)
+ also from z K S False have "at z within (S - K) = at z within S"
+ by (subst (1 2) at_within_open) (auto intro: finite_imp_closed)
+ finally show "(f \<longlongrightarrow> f z) (at z within S)" .
+ qed (insert assms z, simp_all)
+ qed
+qed
+
+proposition Cauchy_integral_formula_convex:
+ assumes S: "convex S" and K: "finite K" and contf: "continuous_on S f"
+ and fcd: "(\<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x)"
+ and z: "z \<in> interior S" and vpg: "valid_path \<gamma>"
+ and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+proof -
+ have *: "\<And>x. x \<in> interior S \<Longrightarrow> f field_differentiable at x"
+ unfolding holomorphic_on_open [symmetric] field_differentiable_def
+ using no_isolated_singularity [where S = "interior S"]
+ by (meson K contf continuous_at_imp_continuous_on continuous_on_interior fcd
+ field_differentiable_at_within field_differentiable_def holomorphic_onI
+ holomorphic_on_imp_differentiable_at open_interior)
+ show ?thesis
+ by (rule Cauchy_integral_formula_weak [OF S finite.emptyI contf]) (use * assms in auto)
+qed
+
+text\<open> Formula for higher derivatives.\<close>
+
+lemma Cauchy_has_contour_integral_higher_derivative_circlepath:
+ assumes contf: "continuous_on (cball z r) f"
+ and holf: "f holomorphic_on ball z r"
+ and w: "w \<in> ball z r"
+ shows "((\<lambda>u. f u / (u - w) ^ (Suc k)) has_contour_integral ((2 * pi * \<i>) / (fact k) * (deriv ^^ k) f w))
+ (circlepath z r)"
+using w
+proof (induction k arbitrary: w)
+ case 0 then show ?case
+ using assms by (auto simp: Cauchy_integral_circlepath dist_commute dist_norm)
+next
+ case (Suc k)
+ have [simp]: "r > 0" using w
+ using ball_eq_empty by fastforce
+ have f: "continuous_on (path_image (circlepath z r)) f"
+ by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def less_imp_le)
+ obtain X where X: "((\<lambda>u. f u / (u - w) ^ Suc (Suc k)) has_contour_integral X) (circlepath z r)"
+ using Cauchy_next_derivative_circlepath(1) [OF f Suc.IH _ Suc.prems]
+ by (auto simp: contour_integrable_on_def)
+ then have con: "contour_integral (circlepath z r) ((\<lambda>u. f u / (u - w) ^ Suc (Suc k))) = X"
+ by (rule contour_integral_unique)
+ have "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) w) (at w)"
+ using Suc.prems assms has_field_derivative_higher_deriv by auto
+ then have dnf_diff: "\<And>n. (deriv ^^ n) f field_differentiable (at w)"
+ by (force simp: field_differentiable_def)
+ have "deriv (\<lambda>w. complex_of_real (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w) w =
+ of_nat (Suc k) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w) ^ Suc (Suc k))"
+ by (force intro!: DERIV_imp_deriv Cauchy_next_derivative_circlepath [OF f Suc.IH _ Suc.prems])
+ also have "\<dots> = of_nat (Suc k) * X"
+ by (simp only: con)
+ finally have "deriv (\<lambda>w. ((2 * pi) * \<i> / (fact k)) * (deriv ^^ k) f w) w = of_nat (Suc k) * X" .
+ then have "((2 * pi) * \<i> / (fact k)) * deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X"
+ by (metis deriv_cmult dnf_diff)
+ then have "deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X / ((2 * pi) * \<i> / (fact k))"
+ by (simp add: field_simps)
+ then show ?case
+ using of_nat_eq_0_iff X by fastforce
+qed
+
+lemma Cauchy_higher_derivative_integral_circlepath:
+ assumes contf: "continuous_on (cball z r) f"
+ and holf: "f holomorphic_on ball z r"
+ and w: "w \<in> ball z r"
+ shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
+ (is "?thes1")
+ and "(deriv ^^ k) f w = (fact k) / (2 * pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k))"
+ (is "?thes2")
+proof -
+ have *: "((\<lambda>u. f u / (u - w) ^ Suc k) has_contour_integral (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w)
+ (circlepath z r)"
+ using Cauchy_has_contour_integral_higher_derivative_circlepath [OF assms]
+ by simp
+ show ?thes1 using *
+ using contour_integrable_on_def by blast
+ show ?thes2
+ unfolding contour_integral_unique [OF *] by (simp add: field_split_simps)
+qed
+
+corollary Cauchy_contour_integral_circlepath:
+ assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r"
+ shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)) = (2 * pi * \<i>) * (deriv ^^ k) f w / (fact k)"
+by (simp add: Cauchy_higher_derivative_integral_circlepath [OF assms])
+
+lemma Cauchy_contour_integral_circlepath_2:
+ assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r"
+ shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^2) = (2 * pi * \<i>) * deriv f w"
+ using Cauchy_contour_integral_circlepath [OF assms, of 1]
+ by (simp add: power2_eq_square)
+
+
+subsection\<open>A holomorphic function is analytic, i.e. has local power series\<close>
+
+theorem holomorphic_power_series:
+ assumes holf: "f holomorphic_on ball z r"
+ and w: "w \<in> ball z r"
+ shows "((\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
+proof -
+ \<comment> \<open>Replacing \<^term>\<open>r\<close> and the original (weak) premises with stronger ones\<close>
+ obtain r where "r > 0" and holfc: "f holomorphic_on cball z r" and w: "w \<in> ball z r"
+ proof
+ have "cball z ((r + dist w z) / 2) \<subseteq> ball z r"
+ using w by (simp add: dist_commute field_sum_of_halves subset_eq)
+ then show "f holomorphic_on cball z ((r + dist w z) / 2)"
+ by (rule holomorphic_on_subset [OF holf])
+ have "r > 0"
+ using w by clarsimp (metis dist_norm le_less_trans norm_ge_zero)
+ then show "0 < (r + dist w z) / 2"
+ by simp (use zero_le_dist [of w z] in linarith)
+ qed (use w in \<open>auto simp: dist_commute\<close>)
+ then have holf: "f holomorphic_on ball z r"
+ using ball_subset_cball holomorphic_on_subset by blast
+ have contf: "continuous_on (cball z r) f"
+ by (simp add: holfc holomorphic_on_imp_continuous_on)
+ have cint: "\<And>k. (\<lambda>u. f u / (u - z) ^ Suc k) contour_integrable_on circlepath z r"
+ by (rule Cauchy_higher_derivative_integral_circlepath [OF contf holf]) (simp add: \<open>0 < r\<close>)
+ obtain B where "0 < B" and B: "\<And>u. u \<in> cball z r \<Longrightarrow> norm(f u) \<le> B"
+ by (metis (no_types) bounded_pos compact_cball compact_continuous_image compact_imp_bounded contf image_eqI)
+ obtain k where k: "0 < k" "k \<le> r" and wz_eq: "norm(w - z) = r - k"
+ and kle: "\<And>u. norm(u - z) = r \<Longrightarrow> k \<le> norm(u - w)"
+ proof
+ show "\<And>u. cmod (u - z) = r \<Longrightarrow> r - dist z w \<le> cmod (u - w)"
+ by (metis add_diff_eq diff_add_cancel dist_norm norm_diff_ineq)
+ qed (use w in \<open>auto simp: dist_norm norm_minus_commute\<close>)
+ have ul: "uniform_limit (sphere z r) (\<lambda>n x. (\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k))) (\<lambda>x. f x / (x - w)) sequentially"
+ unfolding uniform_limit_iff dist_norm
+ proof clarify
+ fix e::real
+ assume "0 < e"
+ have rr: "0 \<le> (r - k) / r" "(r - k) / r < 1" using k by auto
+ obtain n where n: "((r - k) / r) ^ n < e / B * k"
+ using real_arch_pow_inv [of "e/B*k" "(r - k)/r"] \<open>0 < e\<close> \<open>0 < B\<close> k by force
+ have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) - f u / (u - w)) < e"
+ if "n \<le> N" and r: "r = dist z u" for N u
+ proof -
+ have N: "((r - k) / r) ^ N < e / B * k"
+ apply (rule le_less_trans [OF power_decreasing n])
+ using \<open>n \<le> N\<close> k by auto
+ have u [simp]: "(u \<noteq> z) \<and> (u \<noteq> w)"
+ using \<open>0 < r\<close> r w by auto
+ have wzu_not1: "(w - z) / (u - z) \<noteq> 1"
+ by (metis (no_types) dist_norm divide_eq_1_iff less_irrefl mem_ball norm_minus_commute r w)
+ have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) * (u - w) - f u)
+ = norm ((\<Sum>k<N. (((w - z) / (u - z)) ^ k)) * f u * (u - w) / (u - z) - f u)"
+ unfolding sum_distrib_right sum_divide_distrib power_divide by (simp add: algebra_simps)
+ also have "\<dots> = norm ((((w - z) / (u - z)) ^ N - 1) * (u - w) / (((w - z) / (u - z) - 1) * (u - z)) - 1) * norm (f u)"
+ using \<open>0 < B\<close>
+ apply (auto simp: geometric_sum [OF wzu_not1])
+ apply (simp add: field_simps norm_mult [symmetric])
+ done
+ also have "\<dots> = norm ((u-z) ^ N * (w - u) - ((w - z) ^ N - (u-z) ^ N) * (u-w)) / (r ^ N * norm (u-w)) * norm (f u)"
+ using \<open>0 < r\<close> r by (simp add: divide_simps norm_mult norm_divide norm_power dist_norm norm_minus_commute)
+ also have "\<dots> = norm ((w - z) ^ N * (w - u)) / (r ^ N * norm (u - w)) * norm (f u)"
+ by (simp add: algebra_simps)
+ also have "\<dots> = norm (w - z) ^ N * norm (f u) / r ^ N"
+ by (simp add: norm_mult norm_power norm_minus_commute)
+ also have "\<dots> \<le> (((r - k)/r)^N) * B"
+ using \<open>0 < r\<close> w k
+ apply (simp add: divide_simps)
+ apply (rule mult_mono [OF power_mono])
+ apply (auto simp: norm_divide wz_eq norm_power dist_norm norm_minus_commute B r)
+ done
+ also have "\<dots> < e * k"
+ using \<open>0 < B\<close> N by (simp add: divide_simps)
+ also have "\<dots> \<le> e * norm (u - w)"
+ using r kle \<open>0 < e\<close> by (simp add: dist_commute dist_norm)
+ finally show ?thesis
+ by (simp add: field_split_simps norm_divide del: power_Suc)
+ qed
+ with \<open>0 < r\<close> show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>sphere z r.
+ norm ((\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k)) - f x / (x - w)) < e"
+ by (auto simp: mult_ac less_imp_le eventually_sequentially Ball_def)
+ qed
+ have eq: "\<forall>\<^sub>F x in sequentially.
+ contour_integral (circlepath z r) (\<lambda>u. \<Sum>k<x. (w - z) ^ k * (f u / (u - z) ^ Suc k)) =
+ (\<Sum>k<x. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z) ^ k)"
+ apply (rule eventuallyI)
+ apply (subst contour_integral_sum, simp)
+ using contour_integrable_lmul [OF cint, of "(w - z) ^ a" for a] apply (simp add: field_simps)
+ apply (simp only: contour_integral_lmul cint algebra_simps)
+ done
+ have cic: "\<And>u. (\<lambda>y. \<Sum>k<u. (w - z) ^ k * (f y / (y - z) ^ Suc k)) contour_integrable_on circlepath z r"
+ apply (intro contour_integrable_sum contour_integrable_lmul, simp)
+ using \<open>0 < r\<close> by (force intro!: Cauchy_higher_derivative_integral_circlepath [OF contf holf])
+ have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k)
+ sums contour_integral (circlepath z r) (\<lambda>u. f u/(u - w))"
+ unfolding sums_def
+ apply (intro Lim_transform_eventually [OF _ eq] contour_integral_uniform_limit_circlepath [OF eventuallyI ul] cic)
+ using \<open>0 < r\<close> apply auto
+ done
+ then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k)
+ sums (2 * of_real pi * \<i> * f w)"
+ using w by (auto simp: dist_commute dist_norm contour_integral_unique [OF Cauchy_integral_circlepath_simple [OF holfc]])
+ then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z)^k / (\<i> * (of_real pi * 2)))
+ sums ((2 * of_real pi * \<i> * f w) / (\<i> * (complex_of_real pi * 2)))"
+ by (rule sums_divide)
+ then have "(\<lambda>n. (w - z) ^ n * contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc n) / (\<i> * (of_real pi * 2)))
+ sums f w"
+ by (simp add: field_simps)
+ then show ?thesis
+ by (simp add: field_simps \<open>0 < r\<close> Cauchy_higher_derivative_integral_circlepath [OF contf holf])
+qed
+
+subsection\<open>The Liouville theorem and the Fundamental Theorem of Algebra\<close>
+
+text\<open> These weak Liouville versions don't even need the derivative formula.\<close>
+
+lemma Liouville_weak_0:
+ assumes holf: "f holomorphic_on UNIV" and inf: "(f \<longlongrightarrow> 0) at_infinity"
+ shows "f z = 0"
+proof (rule ccontr)
+ assume fz: "f z \<noteq> 0"
+ with inf [unfolded Lim_at_infinity, rule_format, of "norm(f z)/2"]
+ obtain B where B: "\<And>x. B \<le> cmod x \<Longrightarrow> norm (f x) * 2 < cmod (f z)"
+ by (auto simp: dist_norm)
+ define R where "R = 1 + \<bar>B\<bar> + norm z"
+ have "R > 0" unfolding R_def
+ proof -
+ have "0 \<le> cmod z + \<bar>B\<bar>"
+ by (metis (full_types) add_nonneg_nonneg norm_ge_zero real_norm_def)
+ then show "0 < 1 + \<bar>B\<bar> + cmod z"
+ by linarith
+ qed
+ have *: "((\<lambda>u. f u / (u - z)) has_contour_integral 2 * complex_of_real pi * \<i> * f z) (circlepath z R)"
+ apply (rule Cauchy_integral_circlepath)
+ using \<open>R > 0\<close> apply (auto intro: holomorphic_on_subset [OF holf] holomorphic_on_imp_continuous_on)+
+ done
+ have "cmod (x - z) = R \<Longrightarrow> cmod (f x) * 2 < cmod (f z)" for x
+ unfolding R_def
+ by (rule B) (use norm_triangle_ineq4 [of x z] in auto)
+ with \<open>R > 0\<close> fz show False
+ using has_contour_integral_bound_circlepath [OF *, of "norm(f z)/2/R"]
+ by (auto simp: less_imp_le norm_mult norm_divide field_split_simps)
+qed
+
+proposition Liouville_weak:
+ assumes "f holomorphic_on UNIV" and "(f \<longlongrightarrow> l) at_infinity"
+ shows "f z = l"
+ using Liouville_weak_0 [of "\<lambda>z. f z - l"]
+ by (simp add: assms holomorphic_on_diff LIM_zero)
+
+proposition Liouville_weak_inverse:
+ assumes "f holomorphic_on UNIV" and unbounded: "\<And>B. eventually (\<lambda>x. norm (f x) \<ge> B) at_infinity"
+ obtains z where "f z = 0"
+proof -
+ { assume f: "\<And>z. f z \<noteq> 0"
+ have 1: "(\<lambda>x. 1 / f x) holomorphic_on UNIV"
+ by (simp add: holomorphic_on_divide assms f)
+ have 2: "((\<lambda>x. 1 / f x) \<longlongrightarrow> 0) at_infinity"
+ apply (rule tendstoI [OF eventually_mono])
+ apply (rule_tac B="2/e" in unbounded)
+ apply (simp add: dist_norm norm_divide field_split_simps)
+ done
+ have False
+ using Liouville_weak_0 [OF 1 2] f by simp
+ }
+ then show ?thesis
+ using that by blast
+qed
+
+text\<open> In particular we get the Fundamental Theorem of Algebra.\<close>
+
+theorem fundamental_theorem_of_algebra:
+ fixes a :: "nat \<Rightarrow> complex"
+ assumes "a 0 = 0 \<or> (\<exists>i \<in> {1..n}. a i \<noteq> 0)"
+ obtains z where "(\<Sum>i\<le>n. a i * z^i) = 0"
+using assms
+proof (elim disjE bexE)
+ assume "a 0 = 0" then show ?thesis
+ by (auto simp: that [of 0])
+next
+ fix i
+ assume i: "i \<in> {1..n}" and nz: "a i \<noteq> 0"
+ have 1: "(\<lambda>z. \<Sum>i\<le>n. a i * z^i) holomorphic_on UNIV"
+ by (rule holomorphic_intros)+
+ show thesis
+ proof (rule Liouville_weak_inverse [OF 1])
+ show "\<forall>\<^sub>F x in at_infinity. B \<le> cmod (\<Sum>i\<le>n. a i * x ^ i)" for B
+ using i nz by (intro polyfun_extremal exI[of _ i]) auto
+ qed (use that in auto)
+qed
+
+subsection\<open>Weierstrass convergence theorem\<close>
+
+lemma holomorphic_uniform_limit:
+ assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and> (f n) holomorphic_on ball z r) F"
+ and ulim: "uniform_limit (cball z r) f g F"
+ and F: "\<not> trivial_limit F"
+ obtains "continuous_on (cball z r) g" "g holomorphic_on ball z r"
+proof (cases r "0::real" rule: linorder_cases)
+ case less then show ?thesis by (force simp: ball_empty less_imp_le continuous_on_def holomorphic_on_def intro: that)
+next
+ case equal then show ?thesis
+ by (force simp: holomorphic_on_def intro: that)
+next
+ case greater
+ have contg: "continuous_on (cball z r) g"
+ using cont uniform_limit_theorem [OF eventually_mono ulim F] by blast
+ have "path_image (circlepath z r) \<subseteq> cball z r"
+ using \<open>0 < r\<close> by auto
+ then have 1: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1 / (2 * complex_of_real pi * \<i>) * g x)"
+ by (intro continuous_intros continuous_on_subset [OF contg])
+ have 2: "((\<lambda>u. 1 / (2 * of_real pi * \<i>) * g u / (u - w) ^ 1) has_contour_integral g w) (circlepath z r)"
+ if w: "w \<in> ball z r" for w
+ proof -
+ define d where "d = (r - norm(w - z))"
+ have "0 < d" "d \<le> r" using w by (auto simp: norm_minus_commute d_def dist_norm)
+ have dle: "\<And>u. cmod (z - u) = r \<Longrightarrow> d \<le> cmod (u - w)"
+ unfolding d_def by (metis add_diff_eq diff_add_cancel norm_diff_ineq norm_minus_commute)
+ have ev_int: "\<forall>\<^sub>F n in F. (\<lambda>u. f n u / (u - w)) contour_integrable_on circlepath z r"
+ apply (rule eventually_mono [OF cont])
+ using w
+ apply (auto intro: Cauchy_higher_derivative_integral_circlepath [where k=0, simplified])
+ done
+ have ul_less: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)) (\<lambda>x. g x / (x - w)) F"
+ using greater \<open>0 < d\<close>
+ apply (clarsimp simp add: uniform_limit_iff dist_norm norm_divide diff_divide_distrib [symmetric] divide_simps)
+ apply (rule_tac e1="e * d" in eventually_mono [OF uniform_limitD [OF ulim]])
+ apply (force simp: dist_norm intro: dle mult_left_mono less_le_trans)+
+ done
+ have g_cint: "(\<lambda>u. g u/(u - w)) contour_integrable_on circlepath z r"
+ by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>])
+ have cif_tends_cig: "((\<lambda>n. contour_integral(circlepath z r) (\<lambda>u. f n u / (u - w))) \<longlongrightarrow> contour_integral(circlepath z r) (\<lambda>u. g u/(u - w))) F"
+ by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>])
+ have f_tends_cig: "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> contour_integral (circlepath z r) (\<lambda>u. g u / (u - w))) F"
+ proof (rule Lim_transform_eventually)
+ show "\<forall>\<^sub>F x in F. contour_integral (circlepath z r) (\<lambda>u. f x u / (u - w))
+ = 2 * of_real pi * \<i> * f x w"
+ apply (rule eventually_mono [OF cont contour_integral_unique [OF Cauchy_integral_circlepath]])
+ using w\<open>0 < d\<close> d_def by auto
+ qed (auto simp: cif_tends_cig)
+ have "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in F. dist (f n w) (g w) < e"
+ by (rule eventually_mono [OF uniform_limitD [OF ulim]]) (use w in auto)
+ then have "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> 2 * of_real pi * \<i> * g w) F"
+ by (rule tendsto_mult_left [OF tendstoI])
+ then have "((\<lambda>u. g u / (u - w)) has_contour_integral 2 * of_real pi * \<i> * g w) (circlepath z r)"
+ using has_contour_integral_integral [OF g_cint] tendsto_unique [OF F f_tends_cig] w
+ by fastforce
+ then have "((\<lambda>u. g u / (2 * of_real pi * \<i> * (u - w))) has_contour_integral g w) (circlepath z r)"
+ using has_contour_integral_div [where c = "2 * of_real pi * \<i>"]
+ by (force simp: field_simps)
+ then show ?thesis
+ by (simp add: dist_norm)
+ qed
+ show ?thesis
+ using Cauchy_next_derivative_circlepath(2) [OF 1 2, simplified]
+ by (fastforce simp add: holomorphic_on_open contg intro: that)
+qed
+
+
+text\<open> Version showing that the limit is the limit of the derivatives.\<close>
+
+proposition has_complex_derivative_uniform_limit:
+ fixes z::complex
+ assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and>
+ (\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))) F"
+ and ulim: "uniform_limit (cball z r) f g F"
+ and F: "\<not> trivial_limit F" and "0 < r"
+ obtains g' where
+ "continuous_on (cball z r) g"
+ "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
+proof -
+ let ?conint = "contour_integral (circlepath z r)"
+ have g: "continuous_on (cball z r) g" "g holomorphic_on ball z r"
+ by (rule holomorphic_uniform_limit [OF eventually_mono [OF cont] ulim F];
+ auto simp: holomorphic_on_open field_differentiable_def)+
+ then obtain g' where g': "\<And>x. x \<in> ball z r \<Longrightarrow> (g has_field_derivative g' x) (at x)"
+ using DERIV_deriv_iff_has_field_derivative
+ by (fastforce simp add: holomorphic_on_open)
+ then have derg: "\<And>x. x \<in> ball z r \<Longrightarrow> deriv g x = g' x"
+ by (simp add: DERIV_imp_deriv)
+ have tends_f'n_g': "((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" if w: "w \<in> ball z r" for w
+ proof -
+ have eq_f': "?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2) = (f' n w - g' w) * (2 * of_real pi * \<i>)"
+ if cont_fn: "continuous_on (cball z r) (f n)"
+ and fnd: "\<And>w. w \<in> ball z r \<Longrightarrow> (f n has_field_derivative f' n w) (at w)" for n
+ proof -
+ have hol_fn: "f n holomorphic_on ball z r"
+ using fnd by (force simp: holomorphic_on_open)
+ have "(f n has_field_derivative 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)) (at w)"
+ by (rule Cauchy_derivative_integral_circlepath [OF cont_fn hol_fn w])
+ then have f': "f' n w = 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)"
+ using DERIV_unique [OF fnd] w by blast
+ show ?thesis
+ by (simp add: f' Cauchy_contour_integral_circlepath_2 [OF g w] derg [OF w] field_split_simps)
+ qed
+ define d where "d = (r - norm(w - z))^2"
+ have "d > 0"
+ using w by (simp add: dist_commute dist_norm d_def)
+ have dle: "d \<le> cmod ((y - w)\<^sup>2)" if "r = cmod (z - y)" for y
+ proof -
+ have "w \<in> ball z (cmod (z - y))"
+ using that w by fastforce
+ then have "cmod (w - z) \<le> cmod (z - y)"
+ by (simp add: dist_complex_def norm_minus_commute)
+ moreover have "cmod (z - y) - cmod (w - z) \<le> cmod (y - w)"
+ by (metis diff_add_cancel diff_add_eq_diff_diff_swap norm_minus_commute norm_triangle_ineq2)
+ ultimately show ?thesis
+ using that by (simp add: d_def norm_power power_mono)
+ qed
+ have 1: "\<forall>\<^sub>F n in F. (\<lambda>x. f n x / (x - w)\<^sup>2) contour_integrable_on circlepath z r"
+ by (force simp: holomorphic_on_open intro: w Cauchy_derivative_integral_circlepath eventually_mono [OF cont])
+ have 2: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)\<^sup>2) (\<lambda>x. g x / (x - w)\<^sup>2) F"
+ unfolding uniform_limit_iff
+ proof clarify
+ fix e::real
+ assume "0 < e"
+ with \<open>r > 0\<close> show "\<forall>\<^sub>F n in F. \<forall>x\<in>sphere z r. dist (f n x / (x - w)\<^sup>2) (g x / (x - w)\<^sup>2) < e"
+ apply (simp add: norm_divide field_split_simps sphere_def dist_norm)
+ apply (rule eventually_mono [OF uniform_limitD [OF ulim], of "e*d"])
+ apply (simp add: \<open>0 < d\<close>)
+ apply (force simp: dist_norm dle intro: less_le_trans)
+ done
+ qed
+ have "((\<lambda>n. contour_integral (circlepath z r) (\<lambda>x. f n x / (x - w)\<^sup>2))
+ \<longlongrightarrow> contour_integral (circlepath z r) ((\<lambda>x. g x / (x - w)\<^sup>2))) F"
+ by (rule contour_integral_uniform_limit_circlepath [OF 1 2 F \<open>0 < r\<close>])
+ then have tendsto_0: "((\<lambda>n. 1 / (2 * of_real pi * \<i>) * (?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2))) \<longlongrightarrow> 0) F"
+ using Lim_null by (force intro!: tendsto_mult_right_zero)
+ have "((\<lambda>n. f' n w - g' w) \<longlongrightarrow> 0) F"
+ apply (rule Lim_transform_eventually [OF tendsto_0])
+ apply (force simp: divide_simps intro: eq_f' eventually_mono [OF cont])
+ done
+ then show ?thesis using Lim_null by blast
+ qed
+ obtain g' where "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
+ by (blast intro: tends_f'n_g' g')
+ then show ?thesis using g
+ using that by blast
+qed
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Some more simple/convenient versions for applications\<close>
+
+lemma holomorphic_uniform_sequence:
+ assumes S: "open S"
+ and hol_fn: "\<And>n. (f n) holomorphic_on S"
+ and ulim_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially"
+ shows "g holomorphic_on S"
+proof -
+ have "\<exists>f'. (g has_field_derivative f') (at z)" if "z \<in> S" for z
+ proof -
+ obtain r where "0 < r" and r: "cball z r \<subseteq> S"
+ and ul: "uniform_limit (cball z r) f g sequentially"
+ using ulim_g [OF \<open>z \<in> S\<close>] by blast
+ have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and> f n holomorphic_on ball z r"
+ proof (intro eventuallyI conjI)
+ show "continuous_on (cball z r) (f x)" for x
+ using hol_fn holomorphic_on_imp_continuous_on holomorphic_on_subset r by blast
+ show "f x holomorphic_on ball z r" for x
+ by (metis hol_fn holomorphic_on_subset interior_cball interior_subset r)
+ qed
+ show ?thesis
+ apply (rule holomorphic_uniform_limit [OF *])
+ using \<open>0 < r\<close> centre_in_ball ul
+ apply (auto simp: holomorphic_on_open)
+ done
+ qed
+ with S show ?thesis
+ by (simp add: holomorphic_on_open)
+qed
+
+lemma has_complex_derivative_uniform_sequence:
+ fixes S :: "complex set"
+ assumes S: "open S"
+ and hfd: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_field_derivative f' n x) (at x)"
+ and ulim_g: "\<And>x. x \<in> S
+ \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially"
+ shows "\<exists>g'. \<forall>x \<in> S. (g has_field_derivative g' x) (at x) \<and> ((\<lambda>n. f' n x) \<longlongrightarrow> g' x) sequentially"
+proof -
+ have y: "\<exists>y. (g has_field_derivative y) (at z) \<and> (\<lambda>n. f' n z) \<longlonglongrightarrow> y" if "z \<in> S" for z
+ proof -
+ obtain r where "0 < r" and r: "cball z r \<subseteq> S"
+ and ul: "uniform_limit (cball z r) f g sequentially"
+ using ulim_g [OF \<open>z \<in> S\<close>] by blast
+ have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and>
+ (\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))"
+ proof (intro eventuallyI conjI ballI)
+ show "continuous_on (cball z r) (f x)" for x
+ by (meson S continuous_on_subset hfd holomorphic_on_imp_continuous_on holomorphic_on_open r)
+ show "w \<in> ball z r \<Longrightarrow> (f x has_field_derivative f' x w) (at w)" for w x
+ using ball_subset_cball hfd r by blast
+ qed
+ show ?thesis
+ by (rule has_complex_derivative_uniform_limit [OF *, of g]) (use \<open>0 < r\<close> ul in \<open>force+\<close>)
+ qed
+ show ?thesis
+ by (rule bchoice) (blast intro: y)
+qed
+
+subsection\<open>On analytic functions defined by a series\<close>
+
+lemma series_and_derivative_comparison:
+ fixes S :: "complex set"
+ assumes S: "open S"
+ and h: "summable h"
+ and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+ and to_g: "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. norm (f n x) \<le> h n"
+ obtains g g' where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+proof -
+ obtain g where g: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
+ using Weierstrass_m_test_ev [OF to_g h] by force
+ have *: "\<exists>d>0. cball x d \<subseteq> S \<and> uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
+ if "x \<in> S" for x
+ proof -
+ obtain d where "d>0" and d: "cball x d \<subseteq> S"
+ using open_contains_cball [of "S"] \<open>x \<in> S\<close> S by blast
+ show ?thesis
+ proof (intro conjI exI)
+ show "uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
+ using d g uniform_limit_on_subset by (force simp: dist_norm eventually_sequentially)
+ qed (use \<open>d > 0\<close> d in auto)
+ qed
+ have "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i x) \<longlonglongrightarrow> g x"
+ by (metis tendsto_uniform_limitI [OF g])
+ moreover have "\<exists>g'. \<forall>x\<in>S. (g has_field_derivative g' x) (at x) \<and> (\<lambda>n. \<Sum>i<n. f' i x) \<longlonglongrightarrow> g' x"
+ by (rule has_complex_derivative_uniform_sequence [OF S]) (auto intro: * hfd DERIV_sum)+
+ ultimately show ?thesis
+ by (metis sums_def that)
+qed
+
+text\<open>A version where we only have local uniform/comparative convergence.\<close>
+
+lemma series_and_derivative_comparison_local:
+ fixes S :: "complex set"
+ assumes S: "open S"
+ and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+ and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. norm (f n y) \<le> h n)"
+ shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+proof -
+ have "\<exists>y. (\<lambda>n. f n z) sums (\<Sum>n. f n z) \<and> (\<lambda>n. f' n z) sums y \<and> ((\<lambda>x. \<Sum>n. f n x) has_field_derivative y) (at z)"
+ if "z \<in> S" for z
+ proof -
+ obtain d h where "0 < d" "summable h" and le_h: "\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball z d \<inter> S. norm (f n y) \<le> h n"
+ using to_g \<open>z \<in> S\<close> by meson
+ then obtain r where "r>0" and r: "ball z r \<subseteq> ball z d \<inter> S" using \<open>z \<in> S\<close> S
+ by (metis Int_iff open_ball centre_in_ball open_Int open_contains_ball_eq)
+ have 1: "open (ball z d \<inter> S)"
+ by (simp add: open_Int S)
+ have 2: "\<And>n x. x \<in> ball z d \<inter> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+ by (auto simp: hfd)
+ obtain g g' where gg': "\<forall>x \<in> ball z d \<inter> S. ((\<lambda>n. f n x) sums g x) \<and>
+ ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+ by (auto intro: le_h series_and_derivative_comparison [OF 1 \<open>summable h\<close> hfd])
+ then have "(\<lambda>n. f' n z) sums g' z"
+ by (meson \<open>0 < r\<close> centre_in_ball contra_subsetD r)
+ moreover have "(\<lambda>n. f n z) sums (\<Sum>n. f n z)"
+ using summable_sums centre_in_ball \<open>0 < d\<close> \<open>summable h\<close> le_h
+ by (metis (full_types) Int_iff gg' summable_def that)
+ moreover have "((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' z) (at z)"
+ proof (rule has_field_derivative_transform_within)
+ show "\<And>x. dist x z < r \<Longrightarrow> g x = (\<Sum>n. f n x)"
+ by (metis subsetD dist_commute gg' mem_ball r sums_unique)
+ qed (use \<open>0 < r\<close> gg' \<open>z \<in> S\<close> \<open>0 < d\<close> in auto)
+ ultimately show ?thesis by auto
+ qed
+ then show ?thesis
+ by (rule_tac x="\<lambda>x. suminf (\<lambda>n. f n x)" in exI) meson
+qed
+
+
+text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close>
+
+lemma series_and_derivative_comparison_complex:
+ fixes S :: "complex set"
+ assumes S: "open S"
+ and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+ and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))"
+ shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+apply (rule series_and_derivative_comparison_local [OF S hfd], assumption)
+apply (rule ex_forward [OF to_g], assumption)
+apply (erule exE)
+apply (rule_tac x="Re \<circ> h" in exI)
+apply (force simp: summable_Re o_def nonneg_Reals_cmod_eq_Re image_subset_iff)
+done
+
+text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close>
+lemma series_differentiable_comparison_complex:
+ fixes S :: "complex set"
+ assumes S: "open S"
+ and hfd: "\<And>n x. x \<in> S \<Longrightarrow> f n field_differentiable (at x)"
+ and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))"
+ obtains g where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> g field_differentiable (at x)"
+proof -
+ have hfd': "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative deriv (f n) x) (at x)"
+ using hfd field_differentiable_derivI by blast
+ have "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. deriv (f n) x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+ by (metis series_and_derivative_comparison_complex [OF S hfd' to_g])
+ then show ?thesis
+ using field_differentiable_def that by blast
+qed
+
+text\<open>In particular, a power series is analytic inside circle of convergence.\<close>
+
+lemma power_series_and_derivative_0:
+ fixes a :: "nat \<Rightarrow> complex" and r::real
+ assumes "summable (\<lambda>n. a n * r^n)"
+ shows "\<exists>g g'. \<forall>z. cmod z < r \<longrightarrow>
+ ((\<lambda>n. a n * z^n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * z^(n - 1)) sums g' z) \<and> (g has_field_derivative g' z) (at z)"
+proof (cases "0 < r")
+ case True
+ have der: "\<And>n z. ((\<lambda>x. a n * x ^ n) has_field_derivative of_nat n * a n * z ^ (n - 1)) (at z)"
+ by (rule derivative_eq_intros | simp)+
+ have y_le: "\<lbrakk>cmod (z - y) * 2 < r - cmod z\<rbrakk> \<Longrightarrow> cmod y \<le> cmod (of_real r + of_real (cmod z)) / 2" for z y
+ using \<open>r > 0\<close>
+ apply (auto simp: algebra_simps norm_mult norm_divide norm_power simp flip: of_real_add)
+ using norm_triangle_ineq2 [of y z]
+ apply (simp only: diff_le_eq norm_minus_commute mult_2)
+ done
+ have "summable (\<lambda>n. a n * complex_of_real r ^ n)"
+ using assms \<open>r > 0\<close> by simp
+ moreover have "\<And>z. cmod z < r \<Longrightarrow> cmod ((of_real r + of_real (cmod z)) / 2) < cmod (of_real r)"
+ using \<open>r > 0\<close>
+ by (simp flip: of_real_add)
+ ultimately have sum: "\<And>z. cmod z < r \<Longrightarrow> summable (\<lambda>n. of_real (cmod (a n)) * ((of_real r + complex_of_real (cmod z)) / 2) ^ n)"
+ by (rule power_series_conv_imp_absconv_weak)
+ have "\<exists>g g'. \<forall>z \<in> ball 0 r. (\<lambda>n. (a n) * z ^ n) sums g z \<and>
+ (\<lambda>n. of_nat n * (a n) * z ^ (n - 1)) sums g' z \<and> (g has_field_derivative g' z) (at z)"
+ apply (rule series_and_derivative_comparison_complex [OF open_ball der])
+ apply (rule_tac x="(r - norm z)/2" in exI)
+ apply (rule_tac x="\<lambda>n. of_real(norm(a n)*((r + norm z)/2)^n)" in exI)
+ using \<open>r > 0\<close>
+ apply (auto simp: sum eventually_sequentially norm_mult norm_power dist_norm intro!: mult_left_mono power_mono y_le)
+ done
+ then show ?thesis
+ by (simp add: ball_def)
+next
+ case False then show ?thesis
+ apply (simp add: not_less)
+ using less_le_trans norm_not_less_zero by blast
+qed
+
+proposition\<^marker>\<open>tag unimportant\<close> power_series_and_derivative:
+ fixes a :: "nat \<Rightarrow> complex" and r::real
+ assumes "summable (\<lambda>n. a n * r^n)"
+ obtains g g' where "\<forall>z \<in> ball w r.
+ ((\<lambda>n. a n * (z - w) ^ n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * (z - w) ^ (n - 1)) sums g' z) \<and>
+ (g has_field_derivative g' z) (at z)"
+ using power_series_and_derivative_0 [OF assms]
+ apply clarify
+ apply (rule_tac g="(\<lambda>z. g(z - w))" in that)
+ using DERIV_shift [where z="-w"]
+ apply (auto simp: norm_minus_commute Ball_def dist_norm)
+ done
+
+proposition\<^marker>\<open>tag unimportant\<close> power_series_holomorphic:
+ assumes "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>n. a n*(w - z)^n) sums f w)"
+ shows "f holomorphic_on ball z r"
+proof -
+ have "\<exists>f'. (f has_field_derivative f') (at w)" if w: "dist z w < r" for w
+ proof -
+ have inb: "z + complex_of_real ((dist z w + r) / 2) \<in> ball z r"
+ proof -
+ have wz: "cmod (w - z) < r" using w
+ by (auto simp: field_split_simps dist_norm norm_minus_commute)
+ then have "0 \<le> r"
+ by (meson less_eq_real_def norm_ge_zero order_trans)
+ show ?thesis
+ using w by (simp add: dist_norm \<open>0\<le>r\<close> flip: of_real_add)
+ qed
+ have sum: "summable (\<lambda>n. a n * of_real (((cmod (z - w) + r) / 2) ^ n))"
+ using assms [OF inb] by (force simp: summable_def dist_norm)
+ obtain g g' where gg': "\<And>u. u \<in> ball z ((cmod (z - w) + r) / 2) \<Longrightarrow>
+ (\<lambda>n. a n * (u - z) ^ n) sums g u \<and>
+ (\<lambda>n. of_nat n * a n * (u - z) ^ (n - 1)) sums g' u \<and> (g has_field_derivative g' u) (at u)"
+ by (rule power_series_and_derivative [OF sum, of z]) fastforce
+ have [simp]: "g u = f u" if "cmod (u - w) < (r - cmod (z - w)) / 2" for u
+ proof -
+ have less: "cmod (z - u) * 2 < cmod (z - w) + r"
+ using that dist_triangle2 [of z u w]
+ by (simp add: dist_norm [symmetric] algebra_simps)
+ show ?thesis
+ apply (rule sums_unique2 [of "\<lambda>n. a n*(u - z)^n"])
+ using gg' [of u] less w
+ apply (auto simp: assms dist_norm)
+ done
+ qed
+ have "(f has_field_derivative g' w) (at w)"
+ by (rule has_field_derivative_transform_within [where d="(r - norm(z - w))/2"])
+ (use w gg' [of w] in \<open>(force simp: dist_norm)+\<close>)
+ then show ?thesis ..
+ qed
+ then show ?thesis by (simp add: holomorphic_on_open)
+qed
+
+corollary holomorphic_iff_power_series:
+ "f holomorphic_on ball z r \<longleftrightarrow>
+ (\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
+ apply (intro iffI ballI holomorphic_power_series, assumption+)
+ apply (force intro: power_series_holomorphic [where a = "\<lambda>n. (deriv ^^ n) f z / (fact n)"])
+ done
+
+lemma power_series_analytic:
+ "(\<And>w. w \<in> ball z r \<Longrightarrow> (\<lambda>n. a n*(w - z)^n) sums f w) \<Longrightarrow> f analytic_on ball z r"
+ by (force simp: analytic_on_open intro!: power_series_holomorphic)
+
+lemma analytic_iff_power_series:
+ "f analytic_on ball z r \<longleftrightarrow>
+ (\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
+ by (simp add: analytic_on_open holomorphic_iff_power_series)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Equality between holomorphic functions, on open ball then connected set\<close>
+
+lemma holomorphic_fun_eq_on_ball:
+ "\<lbrakk>f holomorphic_on ball z r; g holomorphic_on ball z r;
+ w \<in> ball z r;
+ \<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z\<rbrakk>
+ \<Longrightarrow> f w = g w"
+ apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
+ apply (auto simp: holomorphic_iff_power_series)
+ done
+
+lemma holomorphic_fun_eq_0_on_ball:
+ "\<lbrakk>f holomorphic_on ball z r; w \<in> ball z r;
+ \<And>n. (deriv ^^ n) f z = 0\<rbrakk>
+ \<Longrightarrow> f w = 0"
+ apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
+ apply (auto simp: holomorphic_iff_power_series)
+ done
+
+lemma holomorphic_fun_eq_0_on_connected:
+ assumes holf: "f holomorphic_on S" and "open S"
+ and cons: "connected S"
+ and der: "\<And>n. (deriv ^^ n) f z = 0"
+ and "z \<in> S" "w \<in> S"
+ shows "f w = 0"
+proof -
+ have *: "ball x e \<subseteq> (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
+ if "\<forall>u. (deriv ^^ u) f x = 0" "ball x e \<subseteq> S" for x e
+ proof -
+ have "\<And>x' n. dist x x' < e \<Longrightarrow> (deriv ^^ n) f x' = 0"
+ apply (rule holomorphic_fun_eq_0_on_ball [OF holomorphic_higher_deriv])
+ apply (rule holomorphic_on_subset [OF holf])
+ using that apply simp_all
+ by (metis funpow_add o_apply)
+ with that show ?thesis by auto
+ qed
+ have 1: "openin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
+ apply (rule open_subset, force)
+ using \<open>open S\<close>
+ apply (simp add: open_contains_ball Ball_def)
+ apply (erule all_forward)
+ using "*" by auto blast+
+ have 2: "closedin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
+ using assms
+ by (auto intro: continuous_closedin_preimage_constant holomorphic_on_imp_continuous_on holomorphic_higher_deriv)
+ obtain e where "e>0" and e: "ball w e \<subseteq> S" using openE [OF \<open>open S\<close> \<open>w \<in> S\<close>] .
+ then have holfb: "f holomorphic_on ball w e"
+ using holf holomorphic_on_subset by blast
+ have 3: "(\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}) = S \<Longrightarrow> f w = 0"
+ using \<open>e>0\<close> e by (force intro: holomorphic_fun_eq_0_on_ball [OF holfb])
+ show ?thesis
+ using cons der \<open>z \<in> S\<close>
+ apply (simp add: connected_clopen)
+ apply (drule_tac x="\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}" in spec)
+ apply (auto simp: 1 2 3)
+ done
+qed
+
+lemma holomorphic_fun_eq_on_connected:
+ assumes "f holomorphic_on S" "g holomorphic_on S" and "open S" "connected S"
+ and "\<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z"
+ and "z \<in> S" "w \<in> S"
+ shows "f w = g w"
+proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>x. f x - g x" S z, simplified])
+ show "(\<lambda>x. f x - g x) holomorphic_on S"
+ by (intro assms holomorphic_intros)
+ show "\<And>n. (deriv ^^ n) (\<lambda>x. f x - g x) z = 0"
+ using assms higher_deriv_diff by auto
+qed (use assms in auto)
+
+lemma holomorphic_fun_eq_const_on_connected:
+ assumes holf: "f holomorphic_on S" and "open S"
+ and cons: "connected S"
+ and der: "\<And>n. 0 < n \<Longrightarrow> (deriv ^^ n) f z = 0"
+ and "z \<in> S" "w \<in> S"
+ shows "f w = f z"
+proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>w. f w - f z" S z, simplified])
+ show "(\<lambda>w. f w - f z) holomorphic_on S"
+ by (intro assms holomorphic_intros)
+ show "\<And>n. (deriv ^^ n) (\<lambda>w. f w - f z) z = 0"
+ by (subst higher_deriv_diff) (use assms in \<open>auto intro: holomorphic_intros\<close>)
+qed (use assms in auto)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Some basic lemmas about poles/singularities\<close>
+
+lemma pole_lemma:
+ assumes holf: "f holomorphic_on S" and a: "a \<in> interior S"
+ shows "(\<lambda>z. if z = a then deriv f a
+ else (f z - f a) / (z - a)) holomorphic_on S" (is "?F holomorphic_on S")
+proof -
+ have F1: "?F field_differentiable (at u within S)" if "u \<in> S" "u \<noteq> a" for u
+ proof -
+ have fcd: "f field_differentiable at u within S"
+ using holf holomorphic_on_def by (simp add: \<open>u \<in> S\<close>)
+ have cd: "(\<lambda>z. (f z - f a) / (z - a)) field_differentiable at u within S"
+ by (rule fcd derivative_intros | simp add: that)+
+ have "0 < dist a u" using that dist_nz by blast
+ then show ?thesis
+ by (rule field_differentiable_transform_within [OF _ _ _ cd]) (auto simp: \<open>u \<in> S\<close>)
+ qed
+ have F2: "?F field_differentiable at a" if "0 < e" "ball a e \<subseteq> S" for e
+ proof -
+ have holfb: "f holomorphic_on ball a e"
+ by (rule holomorphic_on_subset [OF holf \<open>ball a e \<subseteq> S\<close>])
+ have 2: "?F holomorphic_on ball a e - {a}"
+ apply (simp add: holomorphic_on_def flip: field_differentiable_def)
+ using mem_ball that
+ apply (auto intro: F1 field_differentiable_within_subset)
+ done
+ have "isCont (\<lambda>z. if z = a then deriv f a else (f z - f a) / (z - a)) x"
+ if "dist a x < e" for x
+ proof (cases "x=a")
+ case True
+ then have "f field_differentiable at a"
+ using holfb \<open>0 < e\<close> holomorphic_on_imp_differentiable_at by auto
+ with True show ?thesis
+ by (auto simp: continuous_at has_field_derivative_iff simp flip: DERIV_deriv_iff_field_differentiable
+ elim: rev_iffD1 [OF _ LIM_equal])
+ next
+ case False with 2 that show ?thesis
+ by (force simp: holomorphic_on_open open_Diff field_differentiable_def [symmetric] field_differentiable_imp_continuous_at)
+ qed
+ then have 1: "continuous_on (ball a e) ?F"
+ by (clarsimp simp: continuous_on_eq_continuous_at)
+ have "?F holomorphic_on ball a e"
+ by (auto intro: no_isolated_singularity [OF 1 2])
+ with that show ?thesis
+ by (simp add: holomorphic_on_open field_differentiable_def [symmetric]
+ field_differentiable_at_within)
+ qed
+ show ?thesis
+ proof
+ fix x assume "x \<in> S" show "?F field_differentiable at x within S"
+ proof (cases "x=a")
+ case True then show ?thesis
+ using a by (auto simp: mem_interior intro: field_differentiable_at_within F2)
+ next
+ case False with F1 \<open>x \<in> S\<close>
+ show ?thesis by blast
+ qed
+ qed
+qed
+
+lemma pole_theorem:
+ assumes holg: "g holomorphic_on S" and a: "a \<in> interior S"
+ and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
+ shows "(\<lambda>z. if z = a then deriv g a
+ else f z - g a/(z - a)) holomorphic_on S"
+ using pole_lemma [OF holg a]
+ by (rule holomorphic_transform) (simp add: eq field_split_simps)
+
+lemma pole_lemma_open:
+ assumes "f holomorphic_on S" "open S"
+ shows "(\<lambda>z. if z = a then deriv f a else (f z - f a)/(z - a)) holomorphic_on S"
+proof (cases "a \<in> S")
+ case True with assms interior_eq pole_lemma
+ show ?thesis by fastforce
+next
+ case False with assms show ?thesis
+ apply (simp add: holomorphic_on_def field_differentiable_def [symmetric], clarify)
+ apply (rule field_differentiable_transform_within [where f = "\<lambda>z. (f z - f a)/(z - a)" and d = 1])
+ apply (rule derivative_intros | force)+
+ done
+qed
+
+lemma pole_theorem_open:
+ assumes holg: "g holomorphic_on S" and S: "open S"
+ and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
+ shows "(\<lambda>z. if z = a then deriv g a
+ else f z - g a/(z - a)) holomorphic_on S"
+ using pole_lemma_open [OF holg S]
+ by (rule holomorphic_transform) (auto simp: eq divide_simps)
+
+lemma pole_theorem_0:
+ assumes holg: "g holomorphic_on S" and a: "a \<in> interior S"
+ and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
+ and [simp]: "f a = deriv g a" "g a = 0"
+ shows "f holomorphic_on S"
+ using pole_theorem [OF holg a eq]
+ by (rule holomorphic_transform) (auto simp: eq field_split_simps)
+
+lemma pole_theorem_open_0:
+ assumes holg: "g holomorphic_on S" and S: "open S"
+ and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
+ and [simp]: "f a = deriv g a" "g a = 0"
+ shows "f holomorphic_on S"
+ using pole_theorem_open [OF holg S eq]
+ by (rule holomorphic_transform) (auto simp: eq field_split_simps)
+
+lemma pole_theorem_analytic:
+ assumes g: "g analytic_on S"
+ and eq: "\<And>z. z \<in> S
+ \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)"
+ shows "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S" (is "?F analytic_on S")
+ unfolding analytic_on_def
+proof
+ fix x
+ assume "x \<in> S"
+ with g obtain e where "0 < e" and e: "g holomorphic_on ball x e"
+ by (auto simp add: analytic_on_def)
+ obtain d where "0 < d" and d: "\<And>w. w \<in> ball x d - {a} \<Longrightarrow> g w = (w - a) * f w"
+ using \<open>x \<in> S\<close> eq by blast
+ have "?F holomorphic_on ball x (min d e)"
+ using d e \<open>x \<in> S\<close> by (fastforce simp: holomorphic_on_subset subset_ball intro!: pole_theorem_open)
+ then show "\<exists>e>0. ?F holomorphic_on ball x e"
+ using \<open>0 < d\<close> \<open>0 < e\<close> not_le by fastforce
+qed
+
+lemma pole_theorem_analytic_0:
+ assumes g: "g analytic_on S"
+ and eq: "\<And>z. z \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)"
+ and [simp]: "f a = deriv g a" "g a = 0"
+ shows "f analytic_on S"
+proof -
+ have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f"
+ by auto
+ show ?thesis
+ using pole_theorem_analytic [OF g eq] by simp
+qed
+
+lemma pole_theorem_analytic_open_superset:
+ assumes g: "g analytic_on S" and "S \<subseteq> T" "open T"
+ and eq: "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z"
+ shows "(\<lambda>z. if z = a then deriv g a
+ else f z - g a/(z - a)) analytic_on S"
+proof (rule pole_theorem_analytic [OF g])
+ fix z
+ assume "z \<in> S"
+ then obtain e where "0 < e" and e: "ball z e \<subseteq> T"
+ using assms openE by blast
+ then show "\<exists>d>0. \<forall>w\<in>ball z d - {a}. g w = (w - a) * f w"
+ using eq by auto
+qed
+
+lemma pole_theorem_analytic_open_superset_0:
+ assumes g: "g analytic_on S" "S \<subseteq> T" "open T" "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z"
+ and [simp]: "f a = deriv g a" "g a = 0"
+ shows "f analytic_on S"
+proof -
+ have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f"
+ by auto
+ have "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S"
+ by (rule pole_theorem_analytic_open_superset [OF g])
+ then show ?thesis by simp
+qed
+
+
+subsection\<open>General, homology form of Cauchy's theorem\<close>
+
+text\<open>Proof is based on Dixon's, as presented in Lang's "Complex Analysis" book (page 147).\<close>
+
+lemma contour_integral_continuous_on_linepath_2D:
+ assumes "open U" and cont_dw: "\<And>w. w \<in> U \<Longrightarrow> F w contour_integrable_on (linepath a b)"
+ and cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). F x y)"
+ and abu: "closed_segment a b \<subseteq> U"
+ shows "continuous_on U (\<lambda>w. contour_integral (linepath a b) (F w))"
+proof -
+ have *: "\<exists>d>0. \<forall>x'\<in>U. dist x' w < d \<longrightarrow>
+ dist (contour_integral (linepath a b) (F x'))
+ (contour_integral (linepath a b) (F w)) \<le> \<epsilon>"
+ if "w \<in> U" "0 < \<epsilon>" "a \<noteq> b" for w \<epsilon>
+ proof -
+ obtain \<delta> where "\<delta>>0" and \<delta>: "cball w \<delta> \<subseteq> U" using open_contains_cball \<open>open U\<close> \<open>w \<in> U\<close> by force
+ let ?TZ = "cball w \<delta> \<times> closed_segment a b"
+ have "uniformly_continuous_on ?TZ (\<lambda>(x,y). F x y)"
+ proof (rule compact_uniformly_continuous)
+ show "continuous_on ?TZ (\<lambda>(x,y). F x y)"
+ by (rule continuous_on_subset[OF cond_uu]) (use SigmaE \<delta> abu in blast)
+ show "compact ?TZ"
+ by (simp add: compact_Times)
+ qed
+ then obtain \<eta> where "\<eta>>0"
+ and \<eta>: "\<And>x x'. \<lbrakk>x\<in>?TZ; x'\<in>?TZ; dist x' x < \<eta>\<rbrakk> \<Longrightarrow>
+ dist ((\<lambda>(x,y). F x y) x') ((\<lambda>(x,y). F x y) x) < \<epsilon>/norm(b - a)"
+ apply (rule uniformly_continuous_onE [where e = "\<epsilon>/norm(b - a)"])
+ using \<open>0 < \<epsilon>\<close> \<open>a \<noteq> b\<close> by auto
+ have \<eta>: "\<lbrakk>norm (w - x1) \<le> \<delta>; x2 \<in> closed_segment a b;
+ norm (w - x1') \<le> \<delta>; x2' \<in> closed_segment a b; norm ((x1', x2') - (x1, x2)) < \<eta>\<rbrakk>
+ \<Longrightarrow> norm (F x1' x2' - F x1 x2) \<le> \<epsilon> / cmod (b - a)"
+ for x1 x2 x1' x2'
+ using \<eta> [of "(x1,x2)" "(x1',x2')"] by (force simp: dist_norm)
+ have le_ee: "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>"
+ if "x' \<in> U" "cmod (x' - w) < \<delta>" "cmod (x' - w) < \<eta>" for x'
+ proof -
+ have "(\<lambda>x. F x' x - F w x) contour_integrable_on linepath a b"
+ by (simp add: \<open>w \<in> U\<close> cont_dw contour_integrable_diff that)
+ then have "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>/norm(b - a) * norm(b - a)"
+ apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_integral _ \<eta>])
+ using \<open>0 < \<epsilon>\<close> \<open>0 < \<delta>\<close> that apply (auto simp: norm_minus_commute)
+ done
+ also have "\<dots> = \<epsilon>" using \<open>a \<noteq> b\<close> by simp
+ finally show ?thesis .
+ qed
+ show ?thesis
+ apply (rule_tac x="min \<delta> \<eta>" in exI)
+ using \<open>0 < \<delta>\<close> \<open>0 < \<eta>\<close>
+ apply (auto simp: dist_norm contour_integral_diff [OF cont_dw cont_dw, symmetric] \<open>w \<in> U\<close> intro: le_ee)
+ done
+ qed
+ show ?thesis
+ proof (cases "a=b")
+ case True
+ then show ?thesis by simp
+ next
+ case False
+ show ?thesis
+ by (rule continuous_onI) (use False in \<open>auto intro: *\<close>)
+ qed
+qed
+
+text\<open>This version has \<^term>\<open>polynomial_function \<gamma>\<close> as an additional assumption.\<close>
+lemma Cauchy_integral_formula_global_weak:
+ assumes "open U" and holf: "f holomorphic_on U"
+ and z: "z \<in> U" and \<gamma>: "polynomial_function \<gamma>"
+ and pasz: "path_image \<gamma> \<subseteq> U - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ and zero: "\<And>w. w \<notin> U \<Longrightarrow> winding_number \<gamma> w = 0"
+ shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+proof -
+ obtain \<gamma>' where pf\<gamma>': "polynomial_function \<gamma>'" and \<gamma>': "\<And>x. (\<gamma> has_vector_derivative (\<gamma>' x)) (at x)"
+ using has_vector_derivative_polynomial_function [OF \<gamma>] by blast
+ then have "bounded(path_image \<gamma>')"
+ by (simp add: path_image_def compact_imp_bounded compact_continuous_image continuous_on_polymonial_function)
+ then obtain B where "B>0" and B: "\<And>x. x \<in> path_image \<gamma>' \<Longrightarrow> norm x \<le> B"
+ using bounded_pos by force
+ define d where [abs_def]: "d z w = (if w = z then deriv f z else (f w - f z)/(w - z))" for z w
+ define v where "v = {w. w \<notin> path_image \<gamma> \<and> winding_number \<gamma> w = 0}"
+ have "path \<gamma>" "valid_path \<gamma>" using \<gamma>
+ by (auto simp: path_polynomial_function valid_path_polynomial_function)
+ then have ov: "open v"
+ by (simp add: v_def open_winding_number_levelsets loop)
+ have uv_Un: "U \<union> v = UNIV"
+ using pasz zero by (auto simp: v_def)
+ have conf: "continuous_on U f"
+ by (metis holf holomorphic_on_imp_continuous_on)
+ have hol_d: "(d y) holomorphic_on U" if "y \<in> U" for y
+ proof -
+ have *: "(\<lambda>c. if c = y then deriv f y else (f c - f y) / (c - y)) holomorphic_on U"
+ by (simp add: holf pole_lemma_open \<open>open U\<close>)
+ then have "isCont (\<lambda>x. if x = y then deriv f y else (f x - f y) / (x - y)) y"
+ using at_within_open field_differentiable_imp_continuous_at holomorphic_on_def that \<open>open U\<close> by fastforce
+ then have "continuous_on U (d y)"
+ apply (simp add: d_def continuous_on_eq_continuous_at \<open>open U\<close>, clarify)
+ using * holomorphic_on_def
+ by (meson field_differentiable_within_open field_differentiable_imp_continuous_at \<open>open U\<close>)
+ moreover have "d y holomorphic_on U - {y}"
+ proof -
+ have "\<And>w. w \<in> U - {y} \<Longrightarrow>
+ (\<lambda>w. if w = y then deriv f y else (f w - f y) / (w - y)) field_differentiable at w"
+ apply (rule_tac d="dist w y" and f = "\<lambda>w. (f w - f y)/(w - y)" in field_differentiable_transform_within)
+ apply (auto simp: dist_pos_lt dist_commute intro!: derivative_intros)
+ using \<open>open U\<close> holf holomorphic_on_imp_differentiable_at by blast
+ then show ?thesis
+ unfolding field_differentiable_def by (simp add: d_def holomorphic_on_open \<open>open U\<close> open_delete)
+ qed
+ ultimately show ?thesis
+ by (rule no_isolated_singularity) (auto simp: \<open>open U\<close>)
+ qed
+ have cint_fxy: "(\<lambda>x. (f x - f y) / (x - y)) contour_integrable_on \<gamma>" if "y \<notin> path_image \<gamma>" for y
+ proof (rule contour_integrable_holomorphic_simple [where S = "U-{y}"])
+ show "(\<lambda>x. (f x - f y) / (x - y)) holomorphic_on U - {y}"
+ by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
+ show "path_image \<gamma> \<subseteq> U - {y}"
+ using pasz that by blast
+ qed (auto simp: \<open>open U\<close> open_delete \<open>valid_path \<gamma>\<close>)
+ define h where
+ "h z = (if z \<in> U then contour_integral \<gamma> (d z) else contour_integral \<gamma> (\<lambda>w. f w/(w - z)))" for z
+ have U: "((d z) has_contour_integral h z) \<gamma>" if "z \<in> U" for z
+ proof -
+ have "d z holomorphic_on U"
+ by (simp add: hol_d that)
+ with that show ?thesis
+ apply (simp add: h_def)
+ by (meson Diff_subset \<open>open U\<close> \<open>valid_path \<gamma>\<close> contour_integrable_holomorphic_simple has_contour_integral_integral pasz subset_trans)
+ qed
+ have V: "((\<lambda>w. f w / (w - z)) has_contour_integral h z) \<gamma>" if z: "z \<in> v" for z
+ proof -
+ have 0: "0 = (f z) * 2 * of_real (2 * pi) * \<i> * winding_number \<gamma> z"
+ using v_def z by auto
+ then have "((\<lambda>x. 1 / (x - z)) has_contour_integral 0) \<gamma>"
+ using z v_def has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close>] by fastforce
+ then have "((\<lambda>x. f z * (1 / (x - z))) has_contour_integral 0) \<gamma>"
+ using has_contour_integral_lmul by fastforce
+ then have "((\<lambda>x. f z / (x - z)) has_contour_integral 0) \<gamma>"
+ by (simp add: field_split_simps)
+ moreover have "((\<lambda>x. (f x - f z) / (x - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
+ using z
+ apply (auto simp: v_def)
+ apply (metis (no_types, lifting) contour_integrable_eq d_def has_contour_integral_eq has_contour_integral_integral cint_fxy)
+ done
+ ultimately have *: "((\<lambda>x. f z / (x - z) + (f x - f z) / (x - z)) has_contour_integral (0 + contour_integral \<gamma> (d z))) \<gamma>"
+ by (rule has_contour_integral_add)
+ have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
+ if "z \<in> U"
+ using * by (auto simp: divide_simps has_contour_integral_eq)
+ moreover have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. f w / (w - z))) \<gamma>"
+ if "z \<notin> U"
+ apply (rule has_contour_integral_integral [OF contour_integrable_holomorphic_simple [where S=U]])
+ using U pasz \<open>valid_path \<gamma>\<close> that
+ apply (auto intro: holomorphic_on_imp_continuous_on hol_d)
+ apply (rule continuous_intros conf holomorphic_intros holf assms | force)+
+ done
+ ultimately show ?thesis
+ using z by (simp add: h_def)
+ qed
+ have znot: "z \<notin> path_image \<gamma>"
+ using pasz by blast
+ obtain d0 where "d0>0" and d0: "\<And>x y. x \<in> path_image \<gamma> \<Longrightarrow> y \<in> - U \<Longrightarrow> d0 \<le> dist x y"
+ using separate_compact_closed [of "path_image \<gamma>" "-U"] pasz \<open>open U\<close> \<open>path \<gamma>\<close> compact_path_image
+ by blast
+ obtain dd where "0 < dd" and dd: "{y + k | y k. y \<in> path_image \<gamma> \<and> k \<in> ball 0 dd} \<subseteq> U"
+ apply (rule that [of "d0/2"])
+ using \<open>0 < d0\<close>
+ apply (auto simp: dist_norm dest: d0)
+ done
+ have "\<And>x x'. \<lbrakk>x \<in> path_image \<gamma>; dist x x' * 2 < dd\<rbrakk> \<Longrightarrow> \<exists>y k. x' = y + k \<and> y \<in> path_image \<gamma> \<and> dist 0 k * 2 \<le> dd"
+ apply (rule_tac x=x in exI)
+ apply (rule_tac x="x'-x" in exI)
+ apply (force simp: dist_norm)
+ done
+ then have 1: "path_image \<gamma> \<subseteq> interior {y + k |y k. y \<in> path_image \<gamma> \<and> k \<in> cball 0 (dd / 2)}"
+ apply (clarsimp simp add: mem_interior)
+ using \<open>0 < dd\<close>
+ apply (rule_tac x="dd/2" in exI, auto)
+ done
+ obtain T where "compact T" and subt: "path_image \<gamma> \<subseteq> interior T" and T: "T \<subseteq> U"
+ apply (rule that [OF _ 1])
+ apply (fastforce simp add: \<open>valid_path \<gamma>\<close> compact_valid_path_image intro!: compact_sums)
+ apply (rule order_trans [OF _ dd])
+ using \<open>0 < dd\<close> by fastforce
+ obtain L where "L>0"
+ and L: "\<And>f B. \<lbrakk>f holomorphic_on interior T; \<And>z. z\<in>interior T \<Longrightarrow> cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
+ cmod (contour_integral \<gamma> f) \<le> L * B"
+ using contour_integral_bound_exists [OF open_interior \<open>valid_path \<gamma>\<close> subt]
+ by blast
+ have "bounded(f ` T)"
+ by (meson \<open>compact T\<close> compact_continuous_image compact_imp_bounded conf continuous_on_subset T)
+ then obtain D where "D>0" and D: "\<And>x. x \<in> T \<Longrightarrow> norm (f x) \<le> D"
+ by (auto simp: bounded_pos)
+ obtain C where "C>0" and C: "\<And>x. x \<in> T \<Longrightarrow> norm x \<le> C"
+ using \<open>compact T\<close> bounded_pos compact_imp_bounded by force
+ have "dist (h y) 0 \<le> e" if "0 < e" and le: "D * L / e + C \<le> cmod y" for e y
+ proof -
+ have "D * L / e > 0" using \<open>D>0\<close> \<open>L>0\<close> \<open>e>0\<close> by simp
+ with le have ybig: "norm y > C" by force
+ with C have "y \<notin> T" by force
+ then have ynot: "y \<notin> path_image \<gamma>"
+ using subt interior_subset by blast
+ have [simp]: "winding_number \<gamma> y = 0"
+ apply (rule winding_number_zero_outside [of _ "cball 0 C"])
+ using ybig interior_subset subt
+ apply (force simp: loop \<open>path \<gamma>\<close> dist_norm intro!: C)+
+ done
+ have [simp]: "h y = contour_integral \<gamma> (\<lambda>w. f w/(w - y))"
+ by (rule contour_integral_unique [symmetric]) (simp add: v_def ynot V)
+ have holint: "(\<lambda>w. f w / (w - y)) holomorphic_on interior T"
+ apply (rule holomorphic_on_divide)
+ using holf holomorphic_on_subset interior_subset T apply blast
+ apply (rule holomorphic_intros)+
+ using \<open>y \<notin> T\<close> interior_subset by auto
+ have leD: "cmod (f z / (z - y)) \<le> D * (e / L / D)" if z: "z \<in> interior T" for z
+ proof -
+ have "D * L / e + cmod z \<le> cmod y"
+ using le C [of z] z using interior_subset by force
+ then have DL2: "D * L / e \<le> cmod (z - y)"
+ using norm_triangle_ineq2 [of y z] by (simp add: norm_minus_commute)
+ have "cmod (f z / (z - y)) = cmod (f z) * inverse (cmod (z - y))"
+ by (simp add: norm_mult norm_inverse Fields.field_class.field_divide_inverse)
+ also have "\<dots> \<le> D * (e / L / D)"
+ apply (rule mult_mono)
+ using that D interior_subset apply blast
+ using \<open>L>0\<close> \<open>e>0\<close> \<open>D>0\<close> DL2
+ apply (auto simp: norm_divide field_split_simps)
+ done
+ finally show ?thesis .
+ qed
+ have "dist (h y) 0 = cmod (contour_integral \<gamma> (\<lambda>w. f w / (w - y)))"
+ by (simp add: dist_norm)
+ also have "\<dots> \<le> L * (D * (e / L / D))"
+ by (rule L [OF holint leD])
+ also have "\<dots> = e"
+ using \<open>L>0\<close> \<open>0 < D\<close> by auto
+ finally show ?thesis .
+ qed
+ then have "(h \<longlongrightarrow> 0) at_infinity"
+ by (meson Lim_at_infinityI)
+ moreover have "h holomorphic_on UNIV"
+ proof -
+ have con_ff: "continuous (at (x,z)) (\<lambda>(x,y). (f y - f x) / (y - x))"
+ if "x \<in> U" "z \<in> U" "x \<noteq> z" for x z
+ using that conf
+ apply (simp add: split_def continuous_on_eq_continuous_at \<open>open U\<close>)
+ apply (simp | rule continuous_intros continuous_within_compose2 [where g=f])+
+ done
+ have con_fstsnd: "continuous_on UNIV (\<lambda>x. (fst x - snd x) ::complex)"
+ by (rule continuous_intros)+
+ have open_uu_Id: "open (U \<times> U - Id)"
+ apply (rule open_Diff)
+ apply (simp add: open_Times \<open>open U\<close>)
+ using continuous_closed_preimage_constant [OF con_fstsnd closed_UNIV, of 0]
+ apply (auto simp: Id_fstsnd_eq algebra_simps)
+ done
+ have con_derf: "continuous (at z) (deriv f)" if "z \<in> U" for z
+ apply (rule continuous_on_interior [of U])
+ apply (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on \<open>open U\<close>)
+ by (simp add: interior_open that \<open>open U\<close>)
+ have tendsto_f': "((\<lambda>(x,y). if y = x then deriv f (x)
+ else (f (y) - f (x)) / (y - x)) \<longlongrightarrow> deriv f x)
+ (at (x, x) within U \<times> U)" if "x \<in> U" for x
+ proof (rule Lim_withinI)
+ fix e::real assume "0 < e"
+ obtain k1 where "k1>0" and k1: "\<And>x'. norm (x' - x) \<le> k1 \<Longrightarrow> norm (deriv f x' - deriv f x) < e"
+ using \<open>0 < e\<close> continuous_within_E [OF con_derf [OF \<open>x \<in> U\<close>]]
+ by (metis UNIV_I dist_norm)
+ obtain k2 where "k2>0" and k2: "ball x k2 \<subseteq> U"
+ by (blast intro: openE [OF \<open>open U\<close>] \<open>x \<in> U\<close>)
+ have neq: "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e"
+ if "z' \<noteq> x'" and less_k1: "norm (x'-x, z'-x) < k1" and less_k2: "norm (x'-x, z'-x) < k2"
+ for x' z'
+ proof -
+ have cs_less: "w \<in> closed_segment x' z' \<Longrightarrow> cmod (w - x) \<le> norm (x'-x, z'-x)" for w
+ apply (drule segment_furthest_le [where y=x])
+ by (metis (no_types) dist_commute dist_norm norm_fst_le norm_snd_le order_trans)
+ have derf_le: "w \<in> closed_segment x' z' \<Longrightarrow> z' \<noteq> x' \<Longrightarrow> cmod (deriv f w - deriv f x) \<le> e" for w
+ by (blast intro: cs_less less_k1 k1 [unfolded divide_const_simps dist_norm] less_imp_le le_less_trans)
+ have f_has_der: "\<And>x. x \<in> U \<Longrightarrow> (f has_field_derivative deriv f x) (at x within U)"
+ by (metis DERIV_deriv_iff_field_differentiable at_within_open holf holomorphic_on_def \<open>open U\<close>)
+ have "closed_segment x' z' \<subseteq> U"
+ by (rule order_trans [OF _ k2]) (simp add: cs_less le_less_trans [OF _ less_k2] dist_complex_def norm_minus_commute subset_iff)
+ then have cint_derf: "(deriv f has_contour_integral f z' - f x') (linepath x' z')"
+ using contour_integral_primitive [OF f_has_der valid_path_linepath] pasz by simp
+ then have *: "((\<lambda>x. deriv f x / (z' - x')) has_contour_integral (f z' - f x') / (z' - x')) (linepath x' z')"
+ by (rule has_contour_integral_div)
+ have "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e/norm(z' - x') * norm(z' - x')"
+ apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_diff [OF *]])
+ using has_contour_integral_div [where c = "z' - x'", OF has_contour_integral_const_linepath [of "deriv f x" z' x']]
+ \<open>e > 0\<close> \<open>z' \<noteq> x'\<close>
+ apply (auto simp: norm_divide divide_simps derf_le)
+ done
+ also have "\<dots> \<le> e" using \<open>0 < e\<close> by simp
+ finally show ?thesis .
+ qed
+ show "\<exists>d>0. \<forall>xa\<in>U \<times> U.
+ 0 < dist xa (x, x) \<and> dist xa (x, x) < d \<longrightarrow>
+ dist (case xa of (x, y) \<Rightarrow> if y = x then deriv f x else (f y - f x) / (y - x)) (deriv f x) \<le> e"
+ apply (rule_tac x="min k1 k2" in exI)
+ using \<open>k1>0\<close> \<open>k2>0\<close> \<open>e>0\<close>
+ apply (force simp: dist_norm neq intro: dual_order.strict_trans2 k1 less_imp_le norm_fst_le)
+ done
+ qed
+ have con_pa_f: "continuous_on (path_image \<gamma>) f"
+ by (meson holf holomorphic_on_imp_continuous_on holomorphic_on_subset interior_subset subt T)
+ have le_B: "\<And>T. T \<in> {0..1} \<Longrightarrow> cmod (vector_derivative \<gamma> (at T)) \<le> B"
+ apply (rule B)
+ using \<gamma>' using path_image_def vector_derivative_at by fastforce
+ have f_has_cint: "\<And>w. w \<in> v - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f u / (u - w) ^ 1) has_contour_integral h w) \<gamma>"
+ by (simp add: V)
+ have cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). d x y)"
+ apply (simp add: continuous_on_eq_continuous_within d_def continuous_within tendsto_f')
+ apply (simp add: tendsto_within_open_NO_MATCH open_Times \<open>open U\<close>, clarify)
+ apply (rule Lim_transform_within_open [OF _ open_uu_Id, where f = "(\<lambda>(x,y). (f y - f x) / (y - x))"])
+ using con_ff
+ apply (auto simp: continuous_within)
+ done
+ have hol_dw: "(\<lambda>z. d z w) holomorphic_on U" if "w \<in> U" for w
+ proof -
+ have "continuous_on U ((\<lambda>(x,y). d x y) \<circ> (\<lambda>z. (w,z)))"
+ by (rule continuous_on_compose continuous_intros continuous_on_subset [OF cond_uu] | force intro: that)+
+ then have *: "continuous_on U (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z))"
+ by (rule rev_iffD1 [OF _ continuous_on_cong [OF refl]]) (simp add: d_def field_simps)
+ have **: "\<And>x. \<lbrakk>x \<in> U; x \<noteq> w\<rbrakk> \<Longrightarrow> (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z)) field_differentiable at x"
+ apply (rule_tac f = "\<lambda>x. (f w - f x)/(w - x)" and d = "dist x w" in field_differentiable_transform_within)
+ apply (rule \<open>open U\<close> derivative_intros holomorphic_on_imp_differentiable_at [OF holf] | force simp: dist_commute)+
+ done
+ show ?thesis
+ unfolding d_def
+ apply (rule no_isolated_singularity [OF * _ \<open>open U\<close>, where K = "{w}"])
+ apply (auto simp: field_differentiable_def [symmetric] holomorphic_on_open open_Diff \<open>open U\<close> **)
+ done
+ qed
+ { fix a b
+ assume abu: "closed_segment a b \<subseteq> U"
+ then have "\<And>w. w \<in> U \<Longrightarrow> (\<lambda>z. d z w) contour_integrable_on (linepath a b)"
+ by (metis hol_dw continuous_on_subset contour_integrable_continuous_linepath holomorphic_on_imp_continuous_on)
+ then have cont_cint_d: "continuous_on U (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
+ apply (rule contour_integral_continuous_on_linepath_2D [OF \<open>open U\<close> _ _ abu])
+ apply (auto intro: continuous_on_swap_args cond_uu)
+ done
+ have cont_cint_d\<gamma>: "continuous_on {0..1} ((\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) \<circ> \<gamma>)"
+ proof (rule continuous_on_compose)
+ show "continuous_on {0..1} \<gamma>"
+ using \<open>path \<gamma>\<close> path_def by blast
+ show "continuous_on (\<gamma> ` {0..1}) (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
+ using pasz unfolding path_image_def
+ by (auto intro!: continuous_on_subset [OF cont_cint_d])
+ qed
+ have cint_cint: "(\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) contour_integrable_on \<gamma>"
+ apply (simp add: contour_integrable_on)
+ apply (rule integrable_continuous_real)
+ apply (rule continuous_on_mult [OF cont_cint_d\<gamma> [unfolded o_def]])
+ using pf\<gamma>'
+ by (simp add: continuous_on_polymonial_function vector_derivative_at [OF \<gamma>'])
+ have "contour_integral (linepath a b) h = contour_integral (linepath a b) (\<lambda>z. contour_integral \<gamma> (d z))"
+ using abu by (force simp: h_def intro: contour_integral_eq)
+ also have "\<dots> = contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
+ apply (rule contour_integral_swap)
+ apply (rule continuous_on_subset [OF cond_uu])
+ using abu pasz \<open>valid_path \<gamma>\<close>
+ apply (auto intro!: continuous_intros)
+ by (metis \<gamma>' continuous_on_eq path_def path_polynomial_function pf\<gamma>' vector_derivative_at)
+ finally have cint_h_eq:
+ "contour_integral (linepath a b) h =
+ contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))" .
+ note cint_cint cint_h_eq
+ } note cint_h = this
+ have conthu: "continuous_on U h"
+ proof (simp add: continuous_on_sequentially, clarify)
+ fix a x
+ assume x: "x \<in> U" and au: "\<forall>n. a n \<in> U" and ax: "a \<longlonglongrightarrow> x"
+ then have A1: "\<forall>\<^sub>F n in sequentially. d (a n) contour_integrable_on \<gamma>"
+ by (meson U contour_integrable_on_def eventuallyI)
+ obtain dd where "dd>0" and dd: "cball x dd \<subseteq> U" using open_contains_cball \<open>open U\<close> x by force
+ have A2: "uniform_limit (path_image \<gamma>) (\<lambda>n. d (a n)) (d x) sequentially"
+ unfolding uniform_limit_iff dist_norm
+ proof clarify
+ fix ee::real
+ assume "0 < ee"
+ show "\<forall>\<^sub>F n in sequentially. \<forall>\<xi>\<in>path_image \<gamma>. cmod (d (a n) \<xi> - d x \<xi>) < ee"
+ proof -
+ let ?ddpa = "{(w,z) |w z. w \<in> cball x dd \<and> z \<in> path_image \<gamma>}"
+ have "uniformly_continuous_on ?ddpa (\<lambda>(x,y). d x y)"
+ apply (rule compact_uniformly_continuous [OF continuous_on_subset[OF cond_uu]])
+ using dd pasz \<open>valid_path \<gamma>\<close>
+ apply (auto simp: compact_Times compact_valid_path_image simp del: mem_cball)
+ done
+ then obtain kk where "kk>0"
+ and kk: "\<And>x x'. \<lbrakk>x \<in> ?ddpa; x' \<in> ?ddpa; dist x' x < kk\<rbrakk> \<Longrightarrow>
+ dist ((\<lambda>(x,y). d x y) x') ((\<lambda>(x,y). d x y) x) < ee"
+ by (rule uniformly_continuous_onE [where e = ee]) (use \<open>0 < ee\<close> in auto)
+ have kk: "\<lbrakk>norm (w - x) \<le> dd; z \<in> path_image \<gamma>; norm ((w, z) - (x, z)) < kk\<rbrakk> \<Longrightarrow> norm (d w z - d x z) < ee"
+ for w z
+ using \<open>dd>0\<close> kk [of "(x,z)" "(w,z)"] by (force simp: norm_minus_commute dist_norm)
+ show ?thesis
+ using ax unfolding lim_sequentially eventually_sequentially
+ apply (drule_tac x="min dd kk" in spec)
+ using \<open>dd > 0\<close> \<open>kk > 0\<close>
+ apply (fastforce simp: kk dist_norm)
+ done
+ qed
+ qed
+ have "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> contour_integral \<gamma> (d x)"
+ by (rule contour_integral_uniform_limit [OF A1 A2 le_B]) (auto simp: \<open>valid_path \<gamma>\<close>)
+ then have tendsto_hx: "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> h x"
+ by (simp add: h_def x)
+ then show "(h \<circ> a) \<longlonglongrightarrow> h x"
+ by (simp add: h_def x au o_def)
+ qed
+ show ?thesis
+ proof (simp add: holomorphic_on_open field_differentiable_def [symmetric], clarify)
+ fix z0
+ consider "z0 \<in> v" | "z0 \<in> U" using uv_Un by blast
+ then show "h field_differentiable at z0"
+ proof cases
+ assume "z0 \<in> v" then show ?thesis
+ using Cauchy_next_derivative [OF con_pa_f le_B f_has_cint _ ov] V f_has_cint \<open>valid_path \<gamma>\<close>
+ by (auto simp: field_differentiable_def v_def)
+ next
+ assume "z0 \<in> U" then
+ obtain e where "e>0" and e: "ball z0 e \<subseteq> U" by (blast intro: openE [OF \<open>open U\<close>])
+ have *: "contour_integral (linepath a b) h + contour_integral (linepath b c) h + contour_integral (linepath c a) h = 0"
+ if abc_subset: "convex hull {a, b, c} \<subseteq> ball z0 e" for a b c
+ proof -
+ have *: "\<And>x1 x2 z. z \<in> U \<Longrightarrow> closed_segment x1 x2 \<subseteq> U \<Longrightarrow> (\<lambda>w. d w z) contour_integrable_on linepath x1 x2"
+ using hol_dw holomorphic_on_imp_continuous_on \<open>open U\<close>
+ by (auto intro!: contour_integrable_holomorphic_simple)
+ have abc: "closed_segment a b \<subseteq> U" "closed_segment b c \<subseteq> U" "closed_segment c a \<subseteq> U"
+ using that e segments_subset_convex_hull by fastforce+
+ have eq0: "\<And>w. w \<in> U \<Longrightarrow> contour_integral (linepath a b +++ linepath b c +++ linepath c a) (\<lambda>z. d z w) = 0"
+ apply (rule contour_integral_unique [OF Cauchy_theorem_triangle])
+ apply (rule holomorphic_on_subset [OF hol_dw])
+ using e abc_subset by auto
+ have "contour_integral \<gamma>
+ (\<lambda>x. contour_integral (linepath a b) (\<lambda>z. d z x) +
+ (contour_integral (linepath b c) (\<lambda>z. d z x) +
+ contour_integral (linepath c a) (\<lambda>z. d z x))) = 0"
+ apply (rule contour_integral_eq_0)
+ using abc pasz U
+ apply (subst contour_integral_join [symmetric], auto intro: eq0 *)+
+ done
+ then show ?thesis
+ by (simp add: cint_h abc contour_integrable_add contour_integral_add [symmetric] add_ac)
+ qed
+ show ?thesis
+ using e \<open>e > 0\<close>
+ by (auto intro!: holomorphic_on_imp_differentiable_at [OF _ open_ball] analytic_imp_holomorphic
+ Morera_triangle continuous_on_subset [OF conthu] *)
+ qed
+ qed
+ qed
+ ultimately have [simp]: "h z = 0" for z
+ by (meson Liouville_weak)
+ have "((\<lambda>w. 1 / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z) \<gamma>"
+ by (rule has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close> znot])
+ then have "((\<lambda>w. f z * (1 / (w - z))) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
+ by (metis mult.commute has_contour_integral_lmul)
+ then have 1: "((\<lambda>w. f z / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
+ by (simp add: field_split_simps)
+ moreover have 2: "((\<lambda>w. (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
+ using U [OF z] pasz d_def by (force elim: has_contour_integral_eq [where g = "\<lambda>w. (f w - f z)/(w - z)"])
+ show ?thesis
+ using has_contour_integral_add [OF 1 2] by (simp add: diff_divide_distrib)
+qed
+
+theorem Cauchy_integral_formula_global:
+ assumes S: "open S" and holf: "f holomorphic_on S"
+ and z: "z \<in> S" and vpg: "valid_path \<gamma>"
+ and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ and zero: "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0"
+ shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+proof -
+ have "path \<gamma>" using vpg by (blast intro: valid_path_imp_path)
+ have hols: "(\<lambda>w. f w / (w - z)) holomorphic_on S - {z}" "(\<lambda>w. 1 / (w - z)) holomorphic_on S - {z}"
+ by (rule holomorphic_intros holomorphic_on_subset [OF holf] | force)+
+ then have cint_fw: "(\<lambda>w. f w / (w - z)) contour_integrable_on \<gamma>"
+ by (meson contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on open_delete S vpg pasz)
+ obtain d where "d>0"
+ and d: "\<And>g h. \<lbrakk>valid_path g; valid_path h; \<forall>t\<in>{0..1}. cmod (g t - \<gamma> t) < d \<and> cmod (h t - \<gamma> t) < d;
+ pathstart h = pathstart g \<and> pathfinish h = pathfinish g\<rbrakk>
+ \<Longrightarrow> path_image h \<subseteq> S - {z} \<and> (\<forall>f. f holomorphic_on S - {z} \<longrightarrow> contour_integral h f = contour_integral g f)"
+ using contour_integral_nearby_ends [OF _ \<open>path \<gamma>\<close> pasz] S by (simp add: open_Diff) metis
+ obtain p where polyp: "polynomial_function p"
+ and ps: "pathstart p = pathstart \<gamma>" and pf: "pathfinish p = pathfinish \<gamma>" and led: "\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < d"
+ using path_approx_polynomial_function [OF \<open>path \<gamma>\<close> \<open>d > 0\<close>] by blast
+ then have ploop: "pathfinish p = pathstart p" using loop by auto
+ have vpp: "valid_path p" using polyp valid_path_polynomial_function by blast
+ have [simp]: "z \<notin> path_image \<gamma>" using pasz by blast
+ have paps: "path_image p \<subseteq> S - {z}" and cint_eq: "(\<And>f. f holomorphic_on S - {z} \<Longrightarrow> contour_integral p f = contour_integral \<gamma> f)"
+ using pf ps led d [OF vpg vpp] \<open>d > 0\<close> by auto
+ have wn_eq: "winding_number p z = winding_number \<gamma> z"
+ using vpp paps
+ by (simp add: subset_Diff_insert vpg valid_path_polynomial_function winding_number_valid_path cint_eq hols)
+ have "winding_number p w = winding_number \<gamma> w" if "w \<notin> S" for w
+ proof -
+ have hol: "(\<lambda>v. 1 / (v - w)) holomorphic_on S - {z}"
+ using that by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
+ have "w \<notin> path_image p" "w \<notin> path_image \<gamma>" using paps pasz that by auto
+ then show ?thesis
+ using vpp vpg by (simp add: subset_Diff_insert valid_path_polynomial_function winding_number_valid_path cint_eq [OF hol])
+ qed
+ then have wn0: "\<And>w. w \<notin> S \<Longrightarrow> winding_number p w = 0"
+ by (simp add: zero)
+ show ?thesis
+ using Cauchy_integral_formula_global_weak [OF S holf z polyp paps ploop wn0] hols
+ by (metis wn_eq cint_eq has_contour_integral_eqpath cint_fw cint_eq)
+qed
+
+theorem Cauchy_theorem_global:
+ assumes S: "open S" and holf: "f holomorphic_on S"
+ and vpg: "valid_path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ and pas: "path_image \<gamma> \<subseteq> S"
+ and zero: "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0"
+ shows "(f has_contour_integral 0) \<gamma>"
+proof -
+ obtain z where "z \<in> S" and znot: "z \<notin> path_image \<gamma>"
+ proof -
+ have "compact (path_image \<gamma>)"
+ using compact_valid_path_image vpg by blast
+ then have "path_image \<gamma> \<noteq> S"
+ by (metis (no_types) compact_open path_image_nonempty S)
+ with pas show ?thesis by (blast intro: that)
+ qed
+ then have pasz: "path_image \<gamma> \<subseteq> S - {z}" using pas by blast
+ have hol: "(\<lambda>w. (w - z) * f w) holomorphic_on S"
+ by (rule holomorphic_intros holf)+
+ show ?thesis
+ using Cauchy_integral_formula_global [OF S hol \<open>z \<in> S\<close> vpg pasz loop zero]
+ by (auto simp: znot elim!: has_contour_integral_eq)
+qed
+
+corollary Cauchy_theorem_global_outside:
+ assumes "open S" "f holomorphic_on S" "valid_path \<gamma>" "pathfinish \<gamma> = pathstart \<gamma>" "path_image \<gamma> \<subseteq> S"
+ "\<And>w. w \<notin> S \<Longrightarrow> w \<in> outside(path_image \<gamma>)"
+ shows "(f has_contour_integral 0) \<gamma>"
+by (metis Cauchy_theorem_global assms winding_number_zero_in_outside valid_path_imp_path)
+
+lemma simply_connected_imp_winding_number_zero:
+ assumes "simply_connected S" "path g"
+ "path_image g \<subseteq> S" "pathfinish g = pathstart g" "z \<notin> S"
+ shows "winding_number g z = 0"
+proof -
+ have hom: "homotopic_loops S g (linepath (pathstart g) (pathstart g))"
+ by (meson assms homotopic_paths_imp_homotopic_loops pathfinish_linepath simply_connected_eq_contractible_path)
+ then have "homotopic_paths (- {z}) g (linepath (pathstart g) (pathstart g))"
+ by (meson \<open>z \<notin> S\<close> homotopic_loops_imp_homotopic_paths_null homotopic_paths_subset subset_Compl_singleton)
+ then have "winding_number g z = winding_number(linepath (pathstart g) (pathstart g)) z"
+ by (rule winding_number_homotopic_paths)
+ also have "\<dots> = 0"
+ using assms by (force intro: winding_number_trivial)
+ finally show ?thesis .
+qed
+
+lemma Cauchy_theorem_simply_connected:
+ assumes "open S" "simply_connected S" "f holomorphic_on S" "valid_path g"
+ "path_image g \<subseteq> S" "pathfinish g = pathstart g"
+ shows "(f has_contour_integral 0) g"
+using assms
+apply (simp add: simply_connected_eq_contractible_path)
+apply (auto intro!: Cauchy_theorem_null_homotopic [where a = "pathstart g"]
+ homotopic_paths_imp_homotopic_loops)
+using valid_path_imp_path by blast
+
+proposition\<^marker>\<open>tag unimportant\<close> holomorphic_logarithm_exists:
+ assumes A: "convex A" "open A"
+ and f: "f holomorphic_on A" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0"
+ and z0: "z0 \<in> A"
+ obtains g where "g holomorphic_on A" and "\<And>x. x \<in> A \<Longrightarrow> exp (g x) = f x"
+proof -
+ note f' = holomorphic_derivI [OF f(1) A(2)]
+ obtain g where g: "\<And>x. x \<in> A \<Longrightarrow> (g has_field_derivative deriv f x / f x) (at x)"
+ proof (rule holomorphic_convex_primitive' [OF A])
+ show "(\<lambda>x. deriv f x / f x) holomorphic_on A"
+ by (intro holomorphic_intros f A)
+ qed (auto simp: A at_within_open[of _ A])
+ define h where "h = (\<lambda>x. -g z0 + ln (f z0) + g x)"
+ from g and A have g_holo: "g holomorphic_on A"
+ by (auto simp: holomorphic_on_def at_within_open[of _ A] field_differentiable_def)
+ hence h_holo: "h holomorphic_on A"
+ by (auto simp: h_def intro!: holomorphic_intros)
+ have "\<exists>c. \<forall>x\<in>A. f x / exp (h x) - 1 = c"
+ proof (rule has_field_derivative_zero_constant, goal_cases)
+ case (2 x)
+ note [simp] = at_within_open[OF _ \<open>open A\<close>]
+ from 2 and z0 and f show ?case
+ by (auto simp: h_def exp_diff field_simps intro!: derivative_eq_intros g f')
+ qed fact+
+ then obtain c where c: "\<And>x. x \<in> A \<Longrightarrow> f x / exp (h x) - 1 = c"
+ by blast
+ from c[OF z0] and z0 and f have "c = 0"
+ by (simp add: h_def)
+ with c have "\<And>x. x \<in> A \<Longrightarrow> exp (h x) = f x" by simp
+ from that[OF h_holo this] show ?thesis .
+qed
+
+
+(* FIXME mv to Cauchy_Integral_Theorem.thy *)
+subsection\<open>Cauchy's inequality and more versions of Liouville\<close>
+
+lemma Cauchy_higher_deriv_bound:
+ assumes holf: "f holomorphic_on (ball z r)"
+ and contf: "continuous_on (cball z r) f"
+ and fin : "\<And>w. w \<in> ball z r \<Longrightarrow> f w \<in> ball y B0"
+ and "0 < r" and "0 < n"
+ shows "norm ((deriv ^^ n) f z) \<le> (fact n) * B0 / r^n"
+proof -
+ have "0 < B0" using \<open>0 < r\<close> fin [of z]
+ by (metis ball_eq_empty ex_in_conv fin not_less)
+ have le_B0: "\<And>w. cmod (w - z) \<le> r \<Longrightarrow> cmod (f w - y) \<le> B0"
+ apply (rule continuous_on_closure_norm_le [of "ball z r" "\<lambda>w. f w - y"])
+ apply (auto simp: \<open>0 < r\<close> dist_norm norm_minus_commute)
+ apply (rule continuous_intros contf)+
+ using fin apply (simp add: dist_commute dist_norm less_eq_real_def)
+ done
+ have "(deriv ^^ n) f z = (deriv ^^ n) (\<lambda>w. f w) z - (deriv ^^ n) (\<lambda>w. y) z"
+ using \<open>0 < n\<close> by simp
+ also have "... = (deriv ^^ n) (\<lambda>w. f w - y) z"
+ by (rule higher_deriv_diff [OF holf, symmetric]) (auto simp: \<open>0 < r\<close>)
+ finally have "(deriv ^^ n) f z = (deriv ^^ n) (\<lambda>w. f w - y) z" .
+ have contf': "continuous_on (cball z r) (\<lambda>u. f u - y)"
+ by (rule contf continuous_intros)+
+ have holf': "(\<lambda>u. (f u - y)) holomorphic_on (ball z r)"
+ by (simp add: holf holomorphic_on_diff)
+ define a where "a = (2 * pi)/(fact n)"
+ have "0 < a" by (simp add: a_def)
+ have "B0/r^(Suc n)*2 * pi * r = a*((fact n)*B0/r^n)"
+ using \<open>0 < r\<close> by (simp add: a_def field_split_simps)
+ have der_dif: "(deriv ^^ n) (\<lambda>w. f w - y) z = (deriv ^^ n) f z"
+ using \<open>0 < r\<close> \<open>0 < n\<close>
+ by (auto simp: higher_deriv_diff [OF holf holomorphic_on_const])
+ have "norm ((2 * of_real pi * \<i>)/(fact n) * (deriv ^^ n) (\<lambda>w. f w - y) z)
+ \<le> (B0/r^(Suc n)) * (2 * pi * r)"
+ apply (rule has_contour_integral_bound_circlepath [of "(\<lambda>u. (f u - y)/(u - z)^(Suc n))" _ z])
+ using Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf' holf']
+ using \<open>0 < B0\<close> \<open>0 < r\<close>
+ apply (auto simp: norm_divide norm_mult norm_power divide_simps le_B0)
+ done
+ then show ?thesis
+ using \<open>0 < r\<close>
+ by (auto simp: norm_divide norm_mult norm_power field_simps der_dif le_B0)
+qed
+
+lemma Cauchy_inequality:
+ assumes holf: "f holomorphic_on (ball \<xi> r)"
+ and contf: "continuous_on (cball \<xi> r) f"
+ and "0 < r"
+ and nof: "\<And>x. norm(\<xi>-x) = r \<Longrightarrow> norm(f x) \<le> B"
+ shows "norm ((deriv ^^ n) f \<xi>) \<le> (fact n) * B / r^n"
+proof -
+ obtain x where "norm (\<xi>-x) = r"
+ by (metis abs_of_nonneg add_diff_cancel_left' \<open>0 < r\<close> diff_add_cancel
+ dual_order.strict_implies_order norm_of_real)
+ then have "0 \<le> B"
+ by (metis nof norm_not_less_zero not_le order_trans)
+ have "((\<lambda>u. f u / (u - \<xi>) ^ Suc n) has_contour_integral (2 * pi) * \<i> / fact n * (deriv ^^ n) f \<xi>)
+ (circlepath \<xi> r)"
+ apply (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf])
+ using \<open>0 < r\<close> by simp
+ then have "norm ((2 * pi * \<i>)/(fact n) * (deriv ^^ n) f \<xi>) \<le> (B / r^(Suc n)) * (2 * pi * r)"
+ apply (rule has_contour_integral_bound_circlepath)
+ using \<open>0 \<le> B\<close> \<open>0 < r\<close>
+ apply (simp_all add: norm_divide norm_power nof frac_le norm_minus_commute del: power_Suc)
+ done
+ then show ?thesis using \<open>0 < r\<close>
+ by (simp add: norm_divide norm_mult field_simps)
+qed
+
+lemma Liouville_polynomial:
+ assumes holf: "f holomorphic_on UNIV"
+ and nof: "\<And>z. A \<le> norm z \<Longrightarrow> norm(f z) \<le> B * norm z ^ n"
+ shows "f \<xi> = (\<Sum>k\<le>n. (deriv^^k) f 0 / fact k * \<xi> ^ k)"
+proof (cases rule: le_less_linear [THEN disjE])
+ assume "B \<le> 0"
+ then have "\<And>z. A \<le> norm z \<Longrightarrow> norm(f z) = 0"
+ by (metis nof less_le_trans zero_less_mult_iff neqE norm_not_less_zero norm_power not_le)
+ then have f0: "(f \<longlongrightarrow> 0) at_infinity"
+ using Lim_at_infinity by force
+ then have [simp]: "f = (\<lambda>w. 0)"
+ using Liouville_weak [OF holf, of 0]
+ by (simp add: eventually_at_infinity f0) meson
+ show ?thesis by simp
+next
+ assume "0 < B"
+ have "((\<lambda>k. (deriv ^^ k) f 0 / (fact k) * (\<xi> - 0)^k) sums f \<xi>)"
+ apply (rule holomorphic_power_series [where r = "norm \<xi> + 1"])
+ using holf holomorphic_on_subset apply auto
+ done
+ then have sumsf: "((\<lambda>k. (deriv ^^ k) f 0 / (fact k) * \<xi>^k) sums f \<xi>)" by simp
+ have "(deriv ^^ k) f 0 / fact k * \<xi> ^ k = 0" if "k>n" for k
+ proof (cases "(deriv ^^ k) f 0 = 0")
+ case True then show ?thesis by simp
+ next
+ case False
+ define w where "w = complex_of_real (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
+ have "1 \<le> abs (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
+ using \<open>0 < B\<close> by simp
+ then have wge1: "1 \<le> norm w"
+ by (metis norm_of_real w_def)
+ then have "w \<noteq> 0" by auto
+ have kB: "0 < fact k * B"
+ using \<open>0 < B\<close> by simp
+ then have "0 \<le> fact k * B / cmod ((deriv ^^ k) f 0)"
+ by simp
+ then have wgeA: "A \<le> cmod w"
+ by (simp only: w_def norm_of_real)
+ have "fact k * B / cmod ((deriv ^^ k) f 0) < abs (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
+ using \<open>0 < B\<close> by simp
+ then have wge: "fact k * B / cmod ((deriv ^^ k) f 0) < norm w"
+ by (metis norm_of_real w_def)
+ then have "fact k * B / norm w < cmod ((deriv ^^ k) f 0)"
+ using False by (simp add: field_split_simps mult.commute split: if_split_asm)
+ also have "... \<le> fact k * (B * norm w ^ n) / norm w ^ k"
+ apply (rule Cauchy_inequality)
+ using holf holomorphic_on_subset apply force
+ using holf holomorphic_on_imp_continuous_on holomorphic_on_subset apply blast
+ using \<open>w \<noteq> 0\<close> apply simp
+ by (metis nof wgeA dist_0_norm dist_norm)
+ also have "... = fact k * (B * 1 / cmod w ^ (k-n))"
+ apply (simp only: mult_cancel_left times_divide_eq_right [symmetric])
+ using \<open>k>n\<close> \<open>w \<noteq> 0\<close> \<open>0 < B\<close> apply (simp add: field_split_simps semiring_normalization_rules)
+ done
+ also have "... = fact k * B / cmod w ^ (k-n)"
+ by simp
+ finally have "fact k * B / cmod w < fact k * B / cmod w ^ (k - n)" .
+ then have "1 / cmod w < 1 / cmod w ^ (k - n)"
+ by (metis kB divide_inverse inverse_eq_divide mult_less_cancel_left_pos)
+ then have "cmod w ^ (k - n) < cmod w"
+ by (metis frac_le le_less_trans norm_ge_zero norm_one not_less order_refl wge1 zero_less_one)
+ with self_le_power [OF wge1] have False
+ by (meson diff_is_0_eq not_gr0 not_le that)
+ then show ?thesis by blast
+ qed
+ then have "(deriv ^^ (k + Suc n)) f 0 / fact (k + Suc n) * \<xi> ^ (k + Suc n) = 0" for k
+ using not_less_eq by blast
+ then have "(\<lambda>i. (deriv ^^ (i + Suc n)) f 0 / fact (i + Suc n) * \<xi> ^ (i + Suc n)) sums 0"
+ by (rule sums_0)
+ with sums_split_initial_segment [OF sumsf, where n = "Suc n"]
+ show ?thesis
+ using atLeast0AtMost lessThan_Suc_atMost sums_unique2 by fastforce
+qed
+
+text\<open>Every bounded entire function is a constant function.\<close>
+theorem Liouville_theorem:
+ assumes holf: "f holomorphic_on UNIV"
+ and bf: "bounded (range f)"
+ obtains c where "\<And>z. f z = c"
+proof -
+ obtain B where "\<And>z. cmod (f z) \<le> B"
+ by (meson bf bounded_pos rangeI)
+ then show ?thesis
+ using Liouville_polynomial [OF holf, of 0 B 0, simplified] that by blast
+qed
+
+text\<open>A holomorphic function f has only isolated zeros unless f is 0.\<close>
+
+lemma powser_0_nonzero:
+ fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
+ assumes r: "0 < r"
+ and sm: "\<And>x. norm (x - \<xi>) < r \<Longrightarrow> (\<lambda>n. a n * (x - \<xi>) ^ n) sums (f x)"
+ and [simp]: "f \<xi> = 0"
+ and m0: "a m \<noteq> 0" and "m>0"
+ obtains s where "0 < s" and "\<And>z. z \<in> cball \<xi> s - {\<xi>} \<Longrightarrow> f z \<noteq> 0"
+proof -
+ have "r \<le> conv_radius a"
+ using sm sums_summable by (auto simp: le_conv_radius_iff [where \<xi>=\<xi>])
+ obtain m where am: "a m \<noteq> 0" and az [simp]: "(\<And>n. n<m \<Longrightarrow> a n = 0)"
+ apply (rule_tac m = "LEAST n. a n \<noteq> 0" in that)
+ using m0
+ apply (rule LeastI2)
+ apply (fastforce intro: dest!: not_less_Least)+
+ done
+ define b where "b i = a (i+m) / a m" for i
+ define g where "g x = suminf (\<lambda>i. b i * (x - \<xi>) ^ i)" for x
+ have [simp]: "b 0 = 1"
+ by (simp add: am b_def)
+ { fix x::'a
+ assume "norm (x - \<xi>) < r"
+ then have "(\<lambda>n. (a m * (x - \<xi>)^m) * (b n * (x - \<xi>)^n)) sums (f x)"
+ using am az sm sums_zero_iff_shift [of m "(\<lambda>n. a n * (x - \<xi>) ^ n)" "f x"]
+ by (simp add: b_def monoid_mult_class.power_add algebra_simps)
+ then have "x \<noteq> \<xi> \<Longrightarrow> (\<lambda>n. b n * (x - \<xi>)^n) sums (f x / (a m * (x - \<xi>)^m))"
+ using am by (simp add: sums_mult_D)
+ } note bsums = this
+ then have "norm (x - \<xi>) < r \<Longrightarrow> summable (\<lambda>n. b n * (x - \<xi>)^n)" for x
+ using sums_summable by (cases "x=\<xi>") auto
+ then have "r \<le> conv_radius b"
+ by (simp add: le_conv_radius_iff [where \<xi>=\<xi>])
+ then have "r/2 < conv_radius b"
+ using not_le order_trans r by fastforce
+ then have "continuous_on (cball \<xi> (r/2)) g"
+ using powser_continuous_suminf [of "r/2" b \<xi>] by (simp add: g_def)
+ then obtain s where "s>0" "\<And>x. \<lbrakk>norm (x - \<xi>) \<le> s; norm (x - \<xi>) \<le> r/2\<rbrakk> \<Longrightarrow> dist (g x) (g \<xi>) < 1/2"
+ apply (rule continuous_onE [where x=\<xi> and e = "1/2"])
+ using r apply (auto simp: norm_minus_commute dist_norm)
+ done
+ moreover have "g \<xi> = 1"
+ by (simp add: g_def)
+ ultimately have gnz: "\<And>x. \<lbrakk>norm (x - \<xi>) \<le> s; norm (x - \<xi>) \<le> r/2\<rbrakk> \<Longrightarrow> (g x) \<noteq> 0"
+ by fastforce
+ have "f x \<noteq> 0" if "x \<noteq> \<xi>" "norm (x - \<xi>) \<le> s" "norm (x - \<xi>) \<le> r/2" for x
+ using bsums [of x] that gnz [of x]
+ apply (auto simp: g_def)
+ using r sums_iff by fastforce
+ then show ?thesis
+ apply (rule_tac s="min s (r/2)" in that)
+ using \<open>0 < r\<close> \<open>0 < s\<close> by (auto simp: dist_commute dist_norm)
+qed
+
+subsection \<open>Complex functions and power series\<close>
+
+text \<open>
+ The following defines the power series expansion of a complex function at a given point
+ (assuming that it is analytic at that point).
+\<close>
+definition\<^marker>\<open>tag important\<close> fps_expansion :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex fps" where
+ "fps_expansion f z0 = Abs_fps (\<lambda>n. (deriv ^^ n) f z0 / fact n)"
+
+lemma
+ fixes r :: ereal
+ assumes "f holomorphic_on eball z0 r"
+ shows conv_radius_fps_expansion: "fps_conv_radius (fps_expansion f z0) \<ge> r"
+ and eval_fps_expansion: "\<And>z. z \<in> eball z0 r \<Longrightarrow> eval_fps (fps_expansion f z0) (z - z0) = f z"
+ and eval_fps_expansion': "\<And>z. norm z < r \<Longrightarrow> eval_fps (fps_expansion f z0) z = f (z0 + z)"
+proof -
+ have "(\<lambda>n. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z"
+ if "z \<in> ball z0 r'" "ereal r' < r" for z r'
+ proof -
+ from that(2) have "ereal r' \<le> r" by simp
+ from assms(1) and this have "f holomorphic_on ball z0 r'"
+ by (rule holomorphic_on_subset[OF _ ball_eball_mono])
+ from holomorphic_power_series [OF this that(1)]
+ show ?thesis by (simp add: fps_expansion_def)
+ qed
+ hence *: "(\<lambda>n. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z"
+ if "z \<in> eball z0 r" for z
+ using that by (subst (asm) eball_conv_UNION_balls) blast
+ show "fps_conv_radius (fps_expansion f z0) \<ge> r" unfolding fps_conv_radius_def
+ proof (rule conv_radius_geI_ex)
+ fix r' :: real assume r': "r' > 0" "ereal r' < r"
+ thus "\<exists>z. norm z = r' \<and> summable (\<lambda>n. fps_nth (fps_expansion f z0) n * z ^ n)"
+ using *[of "z0 + of_real r'"]
+ by (intro exI[of _ "of_real r'"]) (auto simp: summable_def dist_norm)
+ qed
+ show "eval_fps (fps_expansion f z0) (z - z0) = f z" if "z \<in> eball z0 r" for z
+ using *[OF that] by (simp add: eval_fps_def sums_iff)
+ show "eval_fps (fps_expansion f z0) z = f (z0 + z)" if "ereal (norm z) < r" for z
+ using *[of "z0 + z"] and that by (simp add: eval_fps_def sums_iff dist_norm)
+qed
+
+
+text \<open>
+ We can now show several more facts about power series expansions (at least in the complex case)
+ with relative ease that would have been trickier without complex analysis.
+\<close>
+lemma
+ fixes f :: "complex fps" and r :: ereal
+ assumes "\<And>z. ereal (norm z) < r \<Longrightarrow> eval_fps f z \<noteq> 0"
+ shows fps_conv_radius_inverse: "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f)"
+ and eval_fps_inverse: "\<And>z. ereal (norm z) < fps_conv_radius f \<Longrightarrow> ereal (norm z) < r \<Longrightarrow>
+ eval_fps (inverse f) z = inverse (eval_fps f z)"
+proof -
+ define R where "R = min (fps_conv_radius f) r"
+ have *: "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f) \<and>
+ (\<forall>z\<in>eball 0 (min (fps_conv_radius f) r). eval_fps (inverse f) z = inverse (eval_fps f z))"
+ proof (cases "min r (fps_conv_radius f) > 0")
+ case True
+ define f' where "f' = fps_expansion (\<lambda>z. inverse (eval_fps f z)) 0"
+ have holo: "(\<lambda>z. inverse (eval_fps f z)) holomorphic_on eball 0 (min r (fps_conv_radius f))"
+ using assms by (intro holomorphic_intros) auto
+ from holo have radius: "fps_conv_radius f' \<ge> min r (fps_conv_radius f)"
+ unfolding f'_def by (rule conv_radius_fps_expansion)
+ have eval_f': "eval_fps f' z = inverse (eval_fps f z)"
+ if "norm z < fps_conv_radius f" "norm z < r" for z
+ using that unfolding f'_def by (subst eval_fps_expansion'[OF holo]) auto
+
+ have "f * f' = 1"
+ proof (rule eval_fps_eqD)
+ from radius and True have "0 < min (fps_conv_radius f) (fps_conv_radius f')"
+ by (auto simp: min_def split: if_splits)
+ also have "\<dots> \<le> fps_conv_radius (f * f')" by (rule fps_conv_radius_mult)
+ finally show "\<dots> > 0" .
+ next
+ from True have "R > 0" by (auto simp: R_def)
+ hence "eventually (\<lambda>z. z \<in> eball 0 R) (nhds 0)"
+ by (intro eventually_nhds_in_open) (auto simp: zero_ereal_def)
+ thus "eventually (\<lambda>z. eval_fps (f * f') z = eval_fps 1 z) (nhds 0)"
+ proof eventually_elim
+ case (elim z)
+ hence "eval_fps (f * f') z = eval_fps f z * eval_fps f' z"
+ using radius by (intro eval_fps_mult)
+ (auto simp: R_def min_def split: if_splits intro: less_trans)
+ also have "eval_fps f' z = inverse (eval_fps f z)"
+ using elim by (intro eval_f') (auto simp: R_def)
+ also from elim have "eval_fps f z \<noteq> 0"
+ by (intro assms) (auto simp: R_def)
+ hence "eval_fps f z * inverse (eval_fps f z) = eval_fps 1 z"
+ by simp
+ finally show "eval_fps (f * f') z = eval_fps 1 z" .
+ qed
+ qed simp_all
+ hence "f' = inverse f"
+ by (intro fps_inverse_unique [symmetric]) (simp_all add: mult_ac)
+ with eval_f' and radius show ?thesis by simp
+ next
+ case False
+ hence *: "eball 0 R = {}"
+ by (intro eball_empty) (auto simp: R_def min_def split: if_splits)
+ show ?thesis
+ proof safe
+ from False have "min r (fps_conv_radius f) \<le> 0"
+ by (simp add: min_def)
+ also have "0 \<le> fps_conv_radius (inverse f)"
+ by (simp add: fps_conv_radius_def conv_radius_nonneg)
+ finally show "min r (fps_conv_radius f) \<le> \<dots>" .
+ qed (unfold * [unfolded R_def], auto)
+ qed
+
+ from * show "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f)" by blast
+ from * show "eval_fps (inverse f) z = inverse (eval_fps f z)"
+ if "ereal (norm z) < fps_conv_radius f" "ereal (norm z) < r" for z
+ using that by auto
+qed
+
+lemma
+ fixes f g :: "complex fps" and r :: ereal
+ defines "R \<equiv> Min {r, fps_conv_radius f, fps_conv_radius g}"
+ assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0"
+ assumes nz: "\<And>z. z \<in> eball 0 r \<Longrightarrow> eval_fps g z \<noteq> 0"
+ shows fps_conv_radius_divide': "fps_conv_radius (f / g) \<ge> R"
+ and eval_fps_divide':
+ "ereal (norm z) < R \<Longrightarrow> eval_fps (f / g) z = eval_fps f z / eval_fps g z"
+proof -
+ from nz[of 0] and \<open>r > 0\<close> have nz': "fps_nth g 0 \<noteq> 0"
+ by (auto simp: eval_fps_at_0 zero_ereal_def)
+ have "R \<le> min r (fps_conv_radius g)"
+ by (auto simp: R_def intro: min.coboundedI2)
+ also have "min r (fps_conv_radius g) \<le> fps_conv_radius (inverse g)"
+ by (intro fps_conv_radius_inverse assms) (auto simp: zero_ereal_def)
+ finally have radius: "fps_conv_radius (inverse g) \<ge> R" .
+ have "R \<le> min (fps_conv_radius f) (fps_conv_radius (inverse g))"
+ by (intro radius min.boundedI) (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
+ also have "\<dots> \<le> fps_conv_radius (f * inverse g)"
+ by (rule fps_conv_radius_mult)
+ also have "f * inverse g = f / g"
+ by (intro fps_divide_unit [symmetric] nz')
+ finally show "fps_conv_radius (f / g) \<ge> R" .
+
+ assume z: "ereal (norm z) < R"
+ have "eval_fps (f * inverse g) z = eval_fps f z * eval_fps (inverse g) z"
+ using radius by (intro eval_fps_mult less_le_trans[OF z])
+ (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
+ also have "eval_fps (inverse g) z = inverse (eval_fps g z)" using \<open>r > 0\<close>
+ by (intro eval_fps_inverse[where r = r] less_le_trans[OF z] nz)
+ (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
+ also have "f * inverse g = f / g" by fact
+ finally show "eval_fps (f / g) z = eval_fps f z / eval_fps g z" by (simp add: field_split_simps)
+qed
+
+lemma
+ fixes f g :: "complex fps" and r :: ereal
+ defines "R \<equiv> Min {r, fps_conv_radius f, fps_conv_radius g}"
+ assumes "subdegree g \<le> subdegree f"
+ assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0"
+ assumes "\<And>z. z \<in> eball 0 r \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> eval_fps g z \<noteq> 0"
+ shows fps_conv_radius_divide: "fps_conv_radius (f / g) \<ge> R"
+ and eval_fps_divide:
+ "ereal (norm z) < R \<Longrightarrow> c = fps_nth f (subdegree g) / fps_nth g (subdegree g) \<Longrightarrow>
+ eval_fps (f / g) z = (if z = 0 then c else eval_fps f z / eval_fps g z)"
+proof -
+ define f' g' where "f' = fps_shift (subdegree g) f" and "g' = fps_shift (subdegree g) g"
+ have f_eq: "f = f' * fps_X ^ subdegree g" and g_eq: "g = g' * fps_X ^ subdegree g"
+ unfolding f'_def g'_def by (rule subdegree_decompose' le_refl | fact)+
+ have subdegree: "subdegree f' = subdegree f - subdegree g" "subdegree g' = 0"
+ using assms(2) by (simp_all add: f'_def g'_def)
+ have [simp]: "fps_conv_radius f' = fps_conv_radius f" "fps_conv_radius g' = fps_conv_radius g"
+ by (simp_all add: f'_def g'_def)
+ have [simp]: "fps_nth f' 0 = fps_nth f (subdegree g)"
+ "fps_nth g' 0 = fps_nth g (subdegree g)" by (simp_all add: f'_def g'_def)
+ have g_nz: "g \<noteq> 0"
+ proof -
+ define z :: complex where "z = (if r = \<infinity> then 1 else of_real (real_of_ereal r / 2))"
+ from \<open>r > 0\<close> have "z \<in> eball 0 r"
+ by (cases r) (auto simp: z_def eball_def)
+ moreover have "z \<noteq> 0" using \<open>r > 0\<close>
+ by (cases r) (auto simp: z_def)
+ ultimately have "eval_fps g z \<noteq> 0" by (rule assms(6))
+ thus "g \<noteq> 0" by auto
+ qed
+ have fg: "f / g = f' * inverse g'"
+ by (subst f_eq, subst (2) g_eq) (insert g_nz, simp add: fps_divide_unit)
+
+ have g'_nz: "eval_fps g' z \<noteq> 0" if z: "norm z < min r (fps_conv_radius g)" for z
+ proof (cases "z = 0")
+ case False
+ with assms and z have "eval_fps g z \<noteq> 0" by auto
+ also from z have "eval_fps g z = eval_fps g' z * z ^ subdegree g"
+ by (subst g_eq) (auto simp: eval_fps_mult)
+ finally show ?thesis by auto
+ qed (insert \<open>g \<noteq> 0\<close>, auto simp: g'_def eval_fps_at_0)
+
+ have "R \<le> min (min r (fps_conv_radius g)) (fps_conv_radius g')"
+ by (auto simp: R_def min.coboundedI1 min.coboundedI2)
+ also have "\<dots> \<le> fps_conv_radius (inverse g')"
+ using g'_nz by (rule fps_conv_radius_inverse)
+ finally have conv_radius_inv: "R \<le> fps_conv_radius (inverse g')" .
+ hence "R \<le> fps_conv_radius (f' * inverse g')"
+ by (intro order.trans[OF _ fps_conv_radius_mult])
+ (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
+ thus "fps_conv_radius (f / g) \<ge> R" by (simp add: fg)
+
+ fix z c :: complex assume z: "ereal (norm z) < R"
+ assume c: "c = fps_nth f (subdegree g) / fps_nth g (subdegree g)"
+ show "eval_fps (f / g) z = (if z = 0 then c else eval_fps f z / eval_fps g z)"
+ proof (cases "z = 0")
+ case False
+ from z and conv_radius_inv have "ereal (norm z) < fps_conv_radius (inverse g')"
+ by simp
+ with z have "eval_fps (f / g) z = eval_fps f' z * eval_fps (inverse g') z"
+ unfolding fg by (subst eval_fps_mult) (auto simp: R_def)
+ also have "eval_fps (inverse g') z = inverse (eval_fps g' z)"
+ using z by (intro eval_fps_inverse[of "min r (fps_conv_radius g')"] g'_nz) (auto simp: R_def)
+ also have "eval_fps f' z * \<dots> = eval_fps f z / eval_fps g z"
+ using z False assms(2) by (simp add: f'_def g'_def eval_fps_shift R_def)
+ finally show ?thesis using False by simp
+ qed (simp_all add: eval_fps_at_0 fg field_simps c)
+qed
+
+lemma has_fps_expansion_fps_expansion [intro]:
+ assumes "open A" "0 \<in> A" "f holomorphic_on A"
+ shows "f has_fps_expansion fps_expansion f 0"
+proof -
+ from assms(1,2) obtain r where r: "r > 0 " "ball 0 r \<subseteq> A"
+ by (auto simp: open_contains_ball)
+ have holo: "f holomorphic_on eball 0 (ereal r)"
+ using r(2) and assms(3) by auto
+ from r(1) have "0 < ereal r" by simp
+ also have "r \<le> fps_conv_radius (fps_expansion f 0)"
+ using holo by (intro conv_radius_fps_expansion) auto
+ finally have "\<dots> > 0" .
+ moreover have "eventually (\<lambda>z. z \<in> ball 0 r) (nhds 0)"
+ using r(1) by (intro eventually_nhds_in_open) auto
+ hence "eventually (\<lambda>z. eval_fps (fps_expansion f 0) z = f z) (nhds 0)"
+ by eventually_elim (subst eval_fps_expansion'[OF holo], auto)
+ ultimately show ?thesis using r(1) by (auto simp: has_fps_expansion_def)
+qed
+
+lemma fps_conv_radius_tan:
+ fixes c :: complex
+ assumes "c \<noteq> 0"
+ shows "fps_conv_radius (fps_tan c) \<ge> pi / (2 * norm c)"
+proof -
+ have "fps_conv_radius (fps_tan c) \<ge>
+ Min {pi / (2 * norm c), fps_conv_radius (fps_sin c), fps_conv_radius (fps_cos c)}"
+ unfolding fps_tan_def
+ proof (rule fps_conv_radius_divide)
+ fix z :: complex assume "z \<in> eball 0 (pi / (2 * norm c))"
+ with cos_eq_zero_imp_norm_ge[of "c*z"] assms
+ show "eval_fps (fps_cos c) z \<noteq> 0" by (auto simp: norm_mult field_simps)
+ qed (insert assms, auto)
+ thus ?thesis by (simp add: min_def)
+qed
+
+lemma eval_fps_tan:
+ fixes c :: complex
+ assumes "norm z < pi / (2 * norm c)"
+ shows "eval_fps (fps_tan c) z = tan (c * z)"
+proof (cases "c = 0")
+ case False
+ show ?thesis unfolding fps_tan_def
+ proof (subst eval_fps_divide'[where r = "pi / (2 * norm c)"])
+ fix z :: complex assume "z \<in> eball 0 (pi / (2 * norm c))"
+ with cos_eq_zero_imp_norm_ge[of "c*z"] assms
+ show "eval_fps (fps_cos c) z \<noteq> 0" using False by (auto simp: norm_mult field_simps)
+ qed (insert False assms, auto simp: field_simps tan_def)
+ qed simp_all
+
+end
--- a/src/HOL/Complex_Analysis/Cauchy_Integral_Theorem.thy Mon Dec 02 22:40:16 2019 -0500
+++ b/src/HOL/Complex_Analysis/Cauchy_Integral_Theorem.thy Mon Dec 02 17:51:54 2019 +0100
@@ -5,6 +5,7 @@
theory Cauchy_Integral_Theorem
imports
"HOL-Analysis.Analysis"
+ Contour_Integration
begin
lemma leibniz_rule_holomorphic:
@@ -38,1005 +39,6 @@
shows "(\<lambda>x. f x powr g x :: complex) \<in> measurable M borel"
using assms by (simp add: powr_def)
-subsection\<open>Contour Integrals along a path\<close>
-
-text\<open>This definition is for complex numbers only, and does not generalise to line integrals in a vector field\<close>
-
-text\<open>piecewise differentiable function on [0,1]\<close>
-
-definition\<^marker>\<open>tag important\<close> has_contour_integral :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool"
- (infixr "has'_contour'_integral" 50)
- where "(f has_contour_integral i) g \<equiv>
- ((\<lambda>x. f(g x) * vector_derivative g (at x within {0..1}))
- has_integral i) {0..1}"
-
-definition\<^marker>\<open>tag important\<close> contour_integrable_on
- (infixr "contour'_integrable'_on" 50)
- where "f contour_integrable_on g \<equiv> \<exists>i. (f has_contour_integral i) g"
-
-definition\<^marker>\<open>tag important\<close> contour_integral
- where "contour_integral g f \<equiv> SOME i. (f has_contour_integral i) g \<or> \<not> f contour_integrable_on g \<and> i=0"
-
-lemma not_integrable_contour_integral: "\<not> f contour_integrable_on g \<Longrightarrow> contour_integral g f = 0"
- unfolding contour_integrable_on_def contour_integral_def by blast
-
-lemma contour_integral_unique: "(f has_contour_integral i) g \<Longrightarrow> contour_integral g f = i"
- apply (simp add: contour_integral_def has_contour_integral_def contour_integrable_on_def)
- using has_integral_unique by blast
-
-lemma has_contour_integral_eqpath:
- "\<lbrakk>(f has_contour_integral y) p; f contour_integrable_on \<gamma>;
- contour_integral p f = contour_integral \<gamma> f\<rbrakk>
- \<Longrightarrow> (f has_contour_integral y) \<gamma>"
-using contour_integrable_on_def contour_integral_unique by auto
-
-lemma has_contour_integral_integral:
- "f contour_integrable_on i \<Longrightarrow> (f has_contour_integral (contour_integral i f)) i"
- by (metis contour_integral_unique contour_integrable_on_def)
-
-lemma has_contour_integral_unique:
- "(f has_contour_integral i) g \<Longrightarrow> (f has_contour_integral j) g \<Longrightarrow> i = j"
- using has_integral_unique
- by (auto simp: has_contour_integral_def)
-
-lemma has_contour_integral_integrable: "(f has_contour_integral i) g \<Longrightarrow> f contour_integrable_on g"
- using contour_integrable_on_def by blast
-
-text\<open>Show that we can forget about the localized derivative.\<close>
-
-lemma has_integral_localized_vector_derivative:
- "((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} \<longleftrightarrow>
- ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}"
-proof -
- have *: "{a..b} - {a,b} = interior {a..b}"
- by (simp add: atLeastAtMost_diff_ends)
- show ?thesis
- apply (rule has_integral_spike_eq [of "{a,b}"])
- apply (auto simp: at_within_interior [of _ "{a..b}"])
- done
-qed
-
-lemma integrable_on_localized_vector_derivative:
- "(\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b} \<longleftrightarrow>
- (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}"
- by (simp add: integrable_on_def has_integral_localized_vector_derivative)
-
-lemma has_contour_integral:
- "(f has_contour_integral i) g \<longleftrightarrow>
- ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
- by (simp add: has_integral_localized_vector_derivative has_contour_integral_def)
-
-lemma contour_integrable_on:
- "f contour_integrable_on g \<longleftrightarrow>
- (\<lambda>t. f(g t) * vector_derivative g (at t)) integrable_on {0..1}"
- by (simp add: has_contour_integral integrable_on_def contour_integrable_on_def)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Reversing a path\<close>
-
-
-
-lemma has_contour_integral_reversepath:
- assumes "valid_path g" and f: "(f has_contour_integral i) g"
- shows "(f has_contour_integral (-i)) (reversepath g)"
-proof -
- { fix S x
- assume xs: "g C1_differentiable_on ({0..1} - S)" "x \<notin> (-) 1 ` S" "0 \<le> x" "x \<le> 1"
- have "vector_derivative (\<lambda>x. g (1 - x)) (at x within {0..1}) =
- - vector_derivative g (at (1 - x) within {0..1})"
- proof -
- obtain f' where f': "(g has_vector_derivative f') (at (1 - x))"
- using xs
- by (force simp: has_vector_derivative_def C1_differentiable_on_def)
- have "(g \<circ> (\<lambda>x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)"
- by (intro vector_diff_chain_within has_vector_derivative_at_within [OF f'] derivative_eq_intros | simp)+
- then have mf': "((\<lambda>x. g (1 - x)) has_vector_derivative -f') (at x)"
- by (simp add: o_def)
- show ?thesis
- using xs
- by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f'])
- qed
- } note * = this
- obtain S where S: "continuous_on {0..1} g" "finite S" "g C1_differentiable_on {0..1} - S"
- using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
- have "((\<lambda>x. - (f (g (1 - x)) * vector_derivative g (at (1 - x) within {0..1}))) has_integral -i)
- {0..1}"
- using has_integral_affinity01 [where m= "-1" and c=1, OF f [unfolded has_contour_integral_def]]
- by (simp add: has_integral_neg)
- then show ?thesis
- using S
- apply (clarsimp simp: reversepath_def has_contour_integral_def)
- apply (rule_tac S = "(\<lambda>x. 1 - x) ` S" in has_integral_spike_finite)
- apply (auto simp: *)
- done
-qed
-
-lemma contour_integrable_reversepath:
- "valid_path g \<Longrightarrow> f contour_integrable_on g \<Longrightarrow> f contour_integrable_on (reversepath g)"
- using has_contour_integral_reversepath contour_integrable_on_def by blast
-
-lemma contour_integrable_reversepath_eq:
- "valid_path g \<Longrightarrow> (f contour_integrable_on (reversepath g) \<longleftrightarrow> f contour_integrable_on g)"
- using contour_integrable_reversepath valid_path_reversepath by fastforce
-
-lemma contour_integral_reversepath:
- assumes "valid_path g"
- shows "contour_integral (reversepath g) f = - (contour_integral g f)"
-proof (cases "f contour_integrable_on g")
- case True then show ?thesis
- by (simp add: assms contour_integral_unique has_contour_integral_integral has_contour_integral_reversepath)
-next
- case False then have "\<not> f contour_integrable_on (reversepath g)"
- by (simp add: assms contour_integrable_reversepath_eq)
- with False show ?thesis by (simp add: not_integrable_contour_integral)
-qed
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Joining two paths together\<close>
-
-lemma has_contour_integral_join:
- assumes "(f has_contour_integral i1) g1" "(f has_contour_integral i2) g2"
- "valid_path g1" "valid_path g2"
- shows "(f has_contour_integral (i1 + i2)) (g1 +++ g2)"
-proof -
- obtain s1 s2
- where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
- and s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
- using assms
- by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- have 1: "((\<lambda>x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}"
- and 2: "((\<lambda>x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}"
- using assms
- by (auto simp: has_contour_integral)
- have i1: "((\<lambda>x. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}"
- and i2: "((\<lambda>x. (2*f (g2 (2*x - 1))) * vector_derivative g2 (at (2*x - 1))) has_integral i2) {1/2..1}"
- using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]]
- has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]]
- by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac)
- have g1: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
- vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
- 2 *\<^sub>R vector_derivative g1 (at (z*2))" for z
- apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>z - 1/2\<bar>"]])
- apply (simp_all add: dist_real_def abs_if split: if_split_asm)
- apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def])
- apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
- using s1
- apply (auto simp: algebra_simps vector_derivative_works)
- done
- have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
- vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
- 2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))" for z
- apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2 (2*x - 1))" and d = "\<bar>z - 1/2\<bar>"]])
- apply (simp_all add: dist_real_def abs_if split: if_split_asm)
- apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def])
- apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
- using s2
- apply (auto simp: algebra_simps vector_derivative_works)
- done
- have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}"
- apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) ((*)2 -` s1)"])
- using s1
- apply (force intro: finite_vimageI [where h = "(*)2"] inj_onI)
- apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
- done
- moreover have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}"
- apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\<lambda>x. 2*x-1) -` s2)"])
- using s2
- apply (force intro: finite_vimageI [where h = "\<lambda>x. 2*x-1"] inj_onI)
- apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2)
- done
- ultimately
- show ?thesis
- apply (simp add: has_contour_integral)
- apply (rule has_integral_combine [where c = "1/2"], auto)
- done
-qed
-
-lemma contour_integrable_joinI:
- assumes "f contour_integrable_on g1" "f contour_integrable_on g2"
- "valid_path g1" "valid_path g2"
- shows "f contour_integrable_on (g1 +++ g2)"
- using assms
- by (meson has_contour_integral_join contour_integrable_on_def)
-
-lemma contour_integrable_joinD1:
- assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g1"
- shows "f contour_integrable_on g1"
-proof -
- obtain s1
- where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
- using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- have "(\<lambda>x. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
- using assms
- apply (auto simp: contour_integrable_on)
- apply (drule integrable_on_subcbox [where a=0 and b="1/2"])
- apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified])
- done
- then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2))/2) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
- by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
- have g1: "\<lbrakk>0 < z; z < 1; z \<notin> s1\<rbrakk> \<Longrightarrow>
- vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) =
- 2 *\<^sub>R vector_derivative g1 (at z)" for z
- apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>(z-1)/2\<bar>"]])
- apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
- apply (rule vector_diff_chain_at [of "\<lambda>x. x*2" 2 _ g1, simplified o_def])
- using s1
- apply (auto simp: vector_derivative_works has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
- done
- show ?thesis
- using s1
- apply (auto simp: contour_integrable_on)
- apply (rule integrable_spike_finite [of "{0,1} \<union> s1", OF _ _ *])
- apply (auto simp: joinpaths_def scaleR_conv_of_real g1)
- done
-qed
-
-lemma contour_integrable_joinD2:
- assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g2"
- shows "f contour_integrable_on g2"
-proof -
- obtain s2
- where s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
- using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- have "(\<lambda>x. f ((g1 +++ g2) (x/2 + 1/2)) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) integrable_on {0..1}"
- using assms
- apply (auto simp: contour_integrable_on)
- apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto)
- apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified])
- apply (simp add: image_affinity_atLeastAtMost_diff)
- done
- then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2 + 1/2))/2) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2)))
- integrable_on {0..1}"
- by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
- have g2: "\<lbrakk>0 < z; z < 1; z \<notin> s2\<rbrakk> \<Longrightarrow>
- vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2+1/2)) =
- 2 *\<^sub>R vector_derivative g2 (at z)" for z
- apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2(2*x-1))" and d = "\<bar>z/2\<bar>"]])
- apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
- apply (rule vector_diff_chain_at [of "\<lambda>x. x*2-1" 2 _ g2, simplified o_def])
- using s2
- apply (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left
- vector_derivative_works add_divide_distrib)
- done
- show ?thesis
- using s2
- apply (auto simp: contour_integrable_on)
- apply (rule integrable_spike_finite [of "{0,1} \<union> s2", OF _ _ *])
- apply (auto simp: joinpaths_def scaleR_conv_of_real g2)
- done
-qed
-
-lemma contour_integrable_join [simp]:
- shows
- "\<lbrakk>valid_path g1; valid_path g2\<rbrakk>
- \<Longrightarrow> f contour_integrable_on (g1 +++ g2) \<longleftrightarrow> f contour_integrable_on g1 \<and> f contour_integrable_on g2"
-using contour_integrable_joinD1 contour_integrable_joinD2 contour_integrable_joinI by blast
-
-lemma contour_integral_join [simp]:
- shows
- "\<lbrakk>f contour_integrable_on g1; f contour_integrable_on g2; valid_path g1; valid_path g2\<rbrakk>
- \<Longrightarrow> contour_integral (g1 +++ g2) f = contour_integral g1 f + contour_integral g2 f"
- by (simp add: has_contour_integral_integral has_contour_integral_join contour_integral_unique)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Shifting the starting point of a (closed) path\<close>
-
-lemma has_contour_integral_shiftpath:
- assumes f: "(f has_contour_integral i) g" "valid_path g"
- and a: "a \<in> {0..1}"
- shows "(f has_contour_integral i) (shiftpath a g)"
-proof -
- obtain s
- where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
- using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- have *: "((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
- using assms by (auto simp: has_contour_integral)
- then have i: "i = integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)) +
- integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))"
- apply (rule has_integral_unique)
- apply (subst add.commute)
- apply (subst Henstock_Kurzweil_Integration.integral_combine)
- using assms * integral_unique by auto
- { fix x
- have "0 \<le> x \<Longrightarrow> x + a < 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a) ` s \<Longrightarrow>
- vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))"
- unfolding shiftpath_def
- apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x))" and d = "dist(1-a) x"]])
- apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
- apply (rule vector_diff_chain_at [of "\<lambda>x. x+a" 1 _ g, simplified o_def scaleR_one])
- apply (intro derivative_eq_intros | simp)+
- using g
- apply (drule_tac x="x+a" in bspec)
- using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
- done
- } note vd1 = this
- { fix x
- have "1 < x + a \<Longrightarrow> x \<le> 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a + 1) ` s \<Longrightarrow>
- vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))"
- unfolding shiftpath_def
- apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x-1))" and d = "dist (1-a) x"]])
- apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
- apply (rule vector_diff_chain_at [of "\<lambda>x. x+a-1" 1 _ g, simplified o_def scaleR_one])
- apply (intro derivative_eq_intros | simp)+
- using g
- apply (drule_tac x="x+a-1" in bspec)
- using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
- done
- } note vd2 = this
- have va1: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})"
- using * a by (fastforce intro: integrable_subinterval_real)
- have v0a: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})"
- apply (rule integrable_subinterval_real)
- using * a by auto
- have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
- has_integral integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x))) {0..1 - a}"
- apply (rule has_integral_spike_finite
- [where S = "{1-a} \<union> (\<lambda>x. x-a) ` s" and f = "\<lambda>x. f(g(a+x)) * vector_derivative g (at(a+x))"])
- using s apply blast
- using a apply (auto simp: algebra_simps vd1)
- apply (force simp: shiftpath_def add.commute)
- using has_integral_affinity [where m=1 and c=a, simplified, OF integrable_integral [OF va1]]
- apply (simp add: image_affinity_atLeastAtMost_diff [where m=1 and c=a, simplified] add.commute)
- done
- moreover
- have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
- has_integral integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))) {1 - a..1}"
- apply (rule has_integral_spike_finite
- [where S = "{1-a} \<union> (\<lambda>x. x-a+1) ` s" and f = "\<lambda>x. f(g(a+x-1)) * vector_derivative g (at(a+x-1))"])
- using s apply blast
- using a apply (auto simp: algebra_simps vd2)
- apply (force simp: shiftpath_def add.commute)
- using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]]
- apply (simp add: image_affinity_atLeastAtMost [where m=1 and c="1-a", simplified])
- apply (simp add: algebra_simps)
- done
- ultimately show ?thesis
- using a
- by (auto simp: i has_contour_integral intro: has_integral_combine [where c = "1-a"])
-qed
-
-lemma has_contour_integral_shiftpath_D:
- assumes "(f has_contour_integral i) (shiftpath a g)"
- "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
- shows "(f has_contour_integral i) g"
-proof -
- obtain s
- where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
- using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- { fix x
- assume x: "0 < x" "x < 1" "x \<notin> s"
- then have gx: "g differentiable at x"
- using g by auto
- have "vector_derivative g (at x within {0..1}) =
- vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})"
- apply (rule vector_derivative_at_within_ivl
- [OF has_vector_derivative_transform_within_open
- [where f = "(shiftpath (1 - a) (shiftpath a g))" and S = "{0<..<1}-s"]])
- using s g assms x
- apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath
- at_within_interior [of _ "{0..1}"] vector_derivative_works [symmetric])
- apply (rule differentiable_transform_within [OF gx, of "min x (1-x)"])
- apply (auto simp: dist_real_def shiftpath_shiftpath abs_if split: if_split_asm)
- done
- } note vd = this
- have fi: "(f has_contour_integral i) (shiftpath (1 - a) (shiftpath a g))"
- using assms by (auto intro!: has_contour_integral_shiftpath)
- show ?thesis
- apply (simp add: has_contour_integral_def)
- apply (rule has_integral_spike_finite [of "{0,1} \<union> s", OF _ _ fi [unfolded has_contour_integral_def]])
- using s assms vd
- apply (auto simp: Path_Connected.shiftpath_shiftpath)
- done
-qed
-
-lemma has_contour_integral_shiftpath_eq:
- assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
- shows "(f has_contour_integral i) (shiftpath a g) \<longleftrightarrow> (f has_contour_integral i) g"
- using assms has_contour_integral_shiftpath has_contour_integral_shiftpath_D by blast
-
-lemma contour_integrable_on_shiftpath_eq:
- assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
- shows "f contour_integrable_on (shiftpath a g) \<longleftrightarrow> f contour_integrable_on g"
-using assms contour_integrable_on_def has_contour_integral_shiftpath_eq by auto
-
-lemma contour_integral_shiftpath:
- assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
- shows "contour_integral (shiftpath a g) f = contour_integral g f"
- using assms
- by (simp add: contour_integral_def contour_integrable_on_def has_contour_integral_shiftpath_eq)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>More about straight-line paths\<close>
-
-lemma has_contour_integral_linepath:
- shows "(f has_contour_integral i) (linepath a b) \<longleftrightarrow>
- ((\<lambda>x. f(linepath a b x) * (b - a)) has_integral i) {0..1}"
- by (simp add: has_contour_integral)
-
-lemma has_contour_integral_trivial [iff]: "(f has_contour_integral 0) (linepath a a)"
- by (simp add: has_contour_integral_linepath)
-
-lemma has_contour_integral_trivial_iff [simp]: "(f has_contour_integral i) (linepath a a) \<longleftrightarrow> i=0"
- using has_contour_integral_unique by blast
-
-lemma contour_integral_trivial [simp]: "contour_integral (linepath a a) f = 0"
- using has_contour_integral_trivial contour_integral_unique by blast
-
-
-subsection\<open>Relation to subpath construction\<close>
-
-lemma has_contour_integral_subpath_refl [iff]: "(f has_contour_integral 0) (subpath u u g)"
- by (simp add: has_contour_integral subpath_def)
-
-lemma contour_integrable_subpath_refl [iff]: "f contour_integrable_on (subpath u u g)"
- using has_contour_integral_subpath_refl contour_integrable_on_def by blast
-
-lemma contour_integral_subpath_refl [simp]: "contour_integral (subpath u u g) f = 0"
- by (simp add: contour_integral_unique)
-
-lemma has_contour_integral_subpath:
- assumes f: "f contour_integrable_on g" and g: "valid_path g"
- and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
- shows "(f has_contour_integral integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x)))
- (subpath u v g)"
-proof (cases "v=u")
- case True
- then show ?thesis
- using f by (simp add: contour_integrable_on_def subpath_def has_contour_integral)
-next
- case False
- obtain s where s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x" and fs: "finite s"
- using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast
- have *: "((\<lambda>x. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u)))
- has_integral (1 / (v - u)) * integral {u..v} (\<lambda>t. f (g t) * vector_derivative g (at t)))
- {0..1}"
- using f uv
- apply (simp add: contour_integrable_on subpath_def has_contour_integral)
- apply (drule integrable_on_subcbox [where a=u and b=v, simplified])
- apply (simp_all add: has_integral_integral)
- apply (drule has_integral_affinity [where m="v-u" and c=u, simplified])
- apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real)
- apply (simp add: divide_simps False)
- done
- { fix x
- have "x \<in> {0..1} \<Longrightarrow>
- x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s \<Longrightarrow>
- vector_derivative (\<lambda>x. g ((v-u) * x + u)) (at x) = (v-u) *\<^sub>R vector_derivative g (at ((v-u) * x + u))"
- apply (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]])
- apply (intro derivative_eq_intros | simp)+
- apply (cut_tac s [of "(v - u) * x + u"])
- using uv mult_left_le [of x "v-u"]
- apply (auto simp: vector_derivative_works)
- done
- } note vd = this
- show ?thesis
- apply (cut_tac has_integral_cmul [OF *, where c = "v-u"])
- using fs assms
- apply (simp add: False subpath_def has_contour_integral)
- apply (rule_tac S = "(\<lambda>t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite)
- apply (auto simp: inj_on_def False finite_vimageI vd scaleR_conv_of_real)
- done
-qed
-
-lemma contour_integrable_subpath:
- assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
- shows "f contour_integrable_on (subpath u v g)"
- apply (cases u v rule: linorder_class.le_cases)
- apply (metis contour_integrable_on_def has_contour_integral_subpath [OF assms])
- apply (subst reversepath_subpath [symmetric])
- apply (rule contour_integrable_reversepath)
- using assms apply (blast intro: valid_path_subpath)
- apply (simp add: contour_integrable_on_def)
- using assms apply (blast intro: has_contour_integral_subpath)
- done
-
-lemma has_integral_contour_integral_subpath:
- assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
- shows "(((\<lambda>x. f(g x) * vector_derivative g (at x)))
- has_integral contour_integral (subpath u v g) f) {u..v}"
- using assms
- apply (auto simp: has_integral_integrable_integral)
- apply (rule integrable_on_subcbox [where a=u and b=v and S = "{0..1}", simplified])
- apply (auto simp: contour_integral_unique [OF has_contour_integral_subpath] contour_integrable_on)
- done
-
-lemma contour_integral_subcontour_integral:
- assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
- shows "contour_integral (subpath u v g) f =
- integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x))"
- using assms has_contour_integral_subpath contour_integral_unique by blast
-
-lemma contour_integral_subpath_combine_less:
- assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
- "u<v" "v<w"
- shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
- contour_integral (subpath u w g) f"
- using assms apply (auto simp: contour_integral_subcontour_integral)
- apply (rule Henstock_Kurzweil_Integration.integral_combine, auto)
- apply (rule integrable_on_subcbox [where a=u and b=w and S = "{0..1}", simplified])
- apply (auto simp: contour_integrable_on)
- done
-
-lemma contour_integral_subpath_combine:
- assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
- shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
- contour_integral (subpath u w g) f"
-proof (cases "u\<noteq>v \<and> v\<noteq>w \<and> u\<noteq>w")
- case True
- have *: "subpath v u g = reversepath(subpath u v g) \<and>
- subpath w u g = reversepath(subpath u w g) \<and>
- subpath w v g = reversepath(subpath v w g)"
- by (auto simp: reversepath_subpath)
- have "u < v \<and> v < w \<or>
- u < w \<and> w < v \<or>
- v < u \<and> u < w \<or>
- v < w \<and> w < u \<or>
- w < u \<and> u < v \<or>
- w < v \<and> v < u"
- using True assms by linarith
- with assms show ?thesis
- using contour_integral_subpath_combine_less [of f g u v w]
- contour_integral_subpath_combine_less [of f g u w v]
- contour_integral_subpath_combine_less [of f g v u w]
- contour_integral_subpath_combine_less [of f g v w u]
- contour_integral_subpath_combine_less [of f g w u v]
- contour_integral_subpath_combine_less [of f g w v u]
- apply simp
- apply (elim disjE)
- apply (auto simp: * contour_integral_reversepath contour_integrable_subpath
- valid_path_subpath algebra_simps)
- done
-next
- case False
- then show ?thesis
- apply (auto)
- using assms
- by (metis eq_neg_iff_add_eq_0 contour_integral_reversepath reversepath_subpath valid_path_subpath)
-qed
-
-lemma contour_integral_integral:
- "contour_integral g f = integral {0..1} (\<lambda>x. f (g x) * vector_derivative g (at x))"
- by (simp add: contour_integral_def integral_def has_contour_integral contour_integrable_on)
-
-lemma contour_integral_cong:
- assumes "g = g'" "\<And>x. x \<in> path_image g \<Longrightarrow> f x = f' x"
- shows "contour_integral g f = contour_integral g' f'"
- unfolding contour_integral_integral using assms
- by (intro integral_cong) (auto simp: path_image_def)
-
-
-text \<open>Contour integral along a segment on the real axis\<close>
-
-lemma has_contour_integral_linepath_Reals_iff:
- fixes a b :: complex and f :: "complex \<Rightarrow> complex"
- assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
- shows "(f has_contour_integral I) (linepath a b) \<longleftrightarrow>
- ((\<lambda>x. f (of_real x)) has_integral I) {Re a..Re b}"
-proof -
- from assms have [simp]: "of_real (Re a) = a" "of_real (Re b) = b"
- by (simp_all add: complex_eq_iff)
- from assms have "a \<noteq> b" by auto
- have "((\<lambda>x. f (of_real x)) has_integral I) (cbox (Re a) (Re b)) \<longleftrightarrow>
- ((\<lambda>x. f (a + b * of_real x - a * of_real x)) has_integral I /\<^sub>R (Re b - Re a)) {0..1}"
- by (subst has_integral_affinity_iff [of "Re b - Re a" _ "Re a", symmetric])
- (insert assms, simp_all add: field_simps scaleR_conv_of_real)
- also have "(\<lambda>x. f (a + b * of_real x - a * of_real x)) =
- (\<lambda>x. (f (a + b * of_real x - a * of_real x) * (b - a)) /\<^sub>R (Re b - Re a))"
- using \<open>a \<noteq> b\<close> by (auto simp: field_simps fun_eq_iff scaleR_conv_of_real)
- also have "(\<dots> has_integral I /\<^sub>R (Re b - Re a)) {0..1} \<longleftrightarrow>
- ((\<lambda>x. f (linepath a b x) * (b - a)) has_integral I) {0..1}" using assms
- by (subst has_integral_cmul_iff) (auto simp: linepath_def scaleR_conv_of_real algebra_simps)
- also have "\<dots> \<longleftrightarrow> (f has_contour_integral I) (linepath a b)" unfolding has_contour_integral_def
- by (intro has_integral_cong) (simp add: vector_derivative_linepath_within)
- finally show ?thesis by simp
-qed
-
-lemma contour_integrable_linepath_Reals_iff:
- fixes a b :: complex and f :: "complex \<Rightarrow> complex"
- assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
- shows "(f contour_integrable_on linepath a b) \<longleftrightarrow>
- (\<lambda>x. f (of_real x)) integrable_on {Re a..Re b}"
- using has_contour_integral_linepath_Reals_iff[OF assms, of f]
- by (auto simp: contour_integrable_on_def integrable_on_def)
-
-lemma contour_integral_linepath_Reals_eq:
- fixes a b :: complex and f :: "complex \<Rightarrow> complex"
- assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
- shows "contour_integral (linepath a b) f = integral {Re a..Re b} (\<lambda>x. f (of_real x))"
-proof (cases "f contour_integrable_on linepath a b")
- case True
- thus ?thesis using has_contour_integral_linepath_Reals_iff[OF assms, of f]
- using has_contour_integral_integral has_contour_integral_unique by blast
-next
- case False
- thus ?thesis using contour_integrable_linepath_Reals_iff[OF assms, of f]
- by (simp add: not_integrable_contour_integral not_integrable_integral)
-qed
-
-
-
-text\<open>Cauchy's theorem where there's a primitive\<close>
-
-lemma contour_integral_primitive_lemma:
- fixes f :: "complex \<Rightarrow> complex" and g :: "real \<Rightarrow> complex"
- assumes "a \<le> b"
- and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
- and "g piecewise_differentiable_on {a..b}" "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
- shows "((\<lambda>x. f'(g x) * vector_derivative g (at x within {a..b}))
- has_integral (f(g b) - f(g a))) {a..b}"
-proof -
- obtain k where k: "finite k" "\<forall>x\<in>{a..b} - k. g differentiable (at x within {a..b})" and cg: "continuous_on {a..b} g"
- using assms by (auto simp: piecewise_differentiable_on_def)
- have cfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
- apply (rule continuous_on_compose [OF cg, unfolded o_def])
- using assms
- apply (metis field_differentiable_def field_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
- done
- { fix x::real
- assume a: "a < x" and b: "x < b" and xk: "x \<notin> k"
- then have "g differentiable at x within {a..b}"
- using k by (simp add: differentiable_at_withinI)
- then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})"
- by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real)
- then have gdiff: "(g has_derivative (\<lambda>u. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})"
- by (simp add: has_vector_derivative_def scaleR_conv_of_real)
- have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})"
- using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
- then have fdiff: "(f has_derivative (*) (f' (g x))) (at (g x) within g ` {a..b})"
- by (simp add: has_field_derivative_def)
- have "((\<lambda>x. f (g x)) has_vector_derivative f' (g x) * vector_derivative g (at x within {a..b})) (at x within {a..b})"
- using diff_chain_within [OF gdiff fdiff]
- by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac)
- } note * = this
- show ?thesis
- apply (rule fundamental_theorem_of_calculus_interior_strong)
- using k assms cfg *
- apply (auto simp: at_within_Icc_at)
- done
-qed
-
-lemma contour_integral_primitive:
- assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
- and "valid_path g" "path_image g \<subseteq> s"
- shows "(f' has_contour_integral (f(pathfinish g) - f(pathstart g))) g"
- using assms
- apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_contour_integral_def)
- apply (auto intro!: piecewise_C1_imp_differentiable contour_integral_primitive_lemma [of 0 1 s])
- done
-
-corollary Cauchy_theorem_primitive:
- assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
- and "valid_path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g"
- shows "(f' has_contour_integral 0) g"
- using assms
- by (metis diff_self contour_integral_primitive)
-
-text\<open>Existence of path integral for continuous function\<close>
-lemma contour_integrable_continuous_linepath:
- assumes "continuous_on (closed_segment a b) f"
- shows "f contour_integrable_on (linepath a b)"
-proof -
- have "continuous_on {0..1} ((\<lambda>x. f x * (b - a)) \<circ> linepath a b)"
- apply (rule continuous_on_compose [OF continuous_on_linepath], simp add: linepath_image_01)
- apply (rule continuous_intros | simp add: assms)+
- done
- then show ?thesis
- apply (simp add: contour_integrable_on_def has_contour_integral_def integrable_on_def [symmetric])
- apply (rule integrable_continuous [of 0 "1::real", simplified])
- apply (rule continuous_on_eq [where f = "\<lambda>x. f(linepath a b x)*(b - a)"])
- apply (auto simp: vector_derivative_linepath_within)
- done
-qed
-
-lemma has_field_der_id: "((\<lambda>x. x\<^sup>2 / 2) has_field_derivative x) (at x)"
- by (rule has_derivative_imp_has_field_derivative)
- (rule derivative_intros | simp)+
-
-lemma contour_integral_id [simp]: "contour_integral (linepath a b) (\<lambda>y. y) = (b^2 - a^2)/2"
- apply (rule contour_integral_unique)
- using contour_integral_primitive [of UNIV "\<lambda>x. x^2/2" "\<lambda>x. x" "linepath a b"]
- apply (auto simp: field_simps has_field_der_id)
- done
-
-lemma contour_integrable_on_const [iff]: "(\<lambda>x. c) contour_integrable_on (linepath a b)"
- by (simp add: contour_integrable_continuous_linepath)
-
-lemma contour_integrable_on_id [iff]: "(\<lambda>x. x) contour_integrable_on (linepath a b)"
- by (simp add: contour_integrable_continuous_linepath)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetical combining theorems\<close>
-
-lemma has_contour_integral_neg:
- "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. -(f x)) has_contour_integral (-i)) g"
- by (simp add: has_integral_neg has_contour_integral_def)
-
-lemma has_contour_integral_add:
- "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
- \<Longrightarrow> ((\<lambda>x. f1 x + f2 x) has_contour_integral (i1 + i2)) g"
- by (simp add: has_integral_add has_contour_integral_def algebra_simps)
-
-lemma has_contour_integral_diff:
- "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
- \<Longrightarrow> ((\<lambda>x. f1 x - f2 x) has_contour_integral (i1 - i2)) g"
- by (simp add: has_integral_diff has_contour_integral_def algebra_simps)
-
-lemma has_contour_integral_lmul:
- "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. c * (f x)) has_contour_integral (c*i)) g"
-apply (simp add: has_contour_integral_def)
-apply (drule has_integral_mult_right)
-apply (simp add: algebra_simps)
-done
-
-lemma has_contour_integral_rmul:
- "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. (f x) * c) has_contour_integral (i*c)) g"
-apply (drule has_contour_integral_lmul)
-apply (simp add: mult.commute)
-done
-
-lemma has_contour_integral_div:
- "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. f x/c) has_contour_integral (i/c)) g"
- by (simp add: field_class.field_divide_inverse) (metis has_contour_integral_rmul)
-
-lemma has_contour_integral_eq:
- "\<lbrakk>(f has_contour_integral y) p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> (g has_contour_integral y) p"
-apply (simp add: path_image_def has_contour_integral_def)
-by (metis (no_types, lifting) image_eqI has_integral_eq)
-
-lemma has_contour_integral_bound_linepath:
- assumes "(f has_contour_integral i) (linepath a b)"
- "0 \<le> B" "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B"
- shows "norm i \<le> B * norm(b - a)"
-proof -
- { fix x::real
- assume x: "0 \<le> x" "x \<le> 1"
- have "norm (f (linepath a b x)) *
- norm (vector_derivative (linepath a b) (at x within {0..1})) \<le> B * norm (b - a)"
- by (auto intro: mult_mono simp: assms linepath_in_path of_real_linepath vector_derivative_linepath_within x)
- } note * = this
- have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))"
- apply (rule has_integral_bound
- [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
- using assms * unfolding has_contour_integral_def
- apply (auto simp: norm_mult)
- done
- then show ?thesis
- by (auto simp: content_real)
-qed
-
-(*UNUSED
-lemma has_contour_integral_bound_linepath_strong:
- fixes a :: real and f :: "complex \<Rightarrow> real"
- assumes "(f has_contour_integral i) (linepath a b)"
- "finite k"
- "0 \<le> B" "\<And>x::real. x \<in> closed_segment a b - k \<Longrightarrow> norm(f x) \<le> B"
- shows "norm i \<le> B*norm(b - a)"
-*)
-
-lemma has_contour_integral_const_linepath: "((\<lambda>x. c) has_contour_integral c*(b - a))(linepath a b)"
- unfolding has_contour_integral_linepath
- by (metis content_real diff_0_right has_integral_const_real lambda_one of_real_1 scaleR_conv_of_real zero_le_one)
-
-lemma has_contour_integral_0: "((\<lambda>x. 0) has_contour_integral 0) g"
- by (simp add: has_contour_integral_def)
-
-lemma has_contour_integral_is_0:
- "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> (f has_contour_integral 0) g"
- by (rule has_contour_integral_eq [OF has_contour_integral_0]) auto
-
-lemma has_contour_integral_sum:
- "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a has_contour_integral i a) p\<rbrakk>
- \<Longrightarrow> ((\<lambda>x. sum (\<lambda>a. f a x) s) has_contour_integral sum i s) p"
- by (induction s rule: finite_induct) (auto simp: has_contour_integral_0 has_contour_integral_add)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Operations on path integrals\<close>
-
-lemma contour_integral_const_linepath [simp]: "contour_integral (linepath a b) (\<lambda>x. c) = c*(b - a)"
- by (rule contour_integral_unique [OF has_contour_integral_const_linepath])
-
-lemma contour_integral_neg:
- "f contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. -(f x)) = -(contour_integral g f)"
- by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_neg)
-
-lemma contour_integral_add:
- "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x + f2 x) =
- contour_integral g f1 + contour_integral g f2"
- by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_add)
-
-lemma contour_integral_diff:
- "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x - f2 x) =
- contour_integral g f1 - contour_integral g f2"
- by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_diff)
-
-lemma contour_integral_lmul:
- shows "f contour_integrable_on g
- \<Longrightarrow> contour_integral g (\<lambda>x. c * f x) = c*contour_integral g f"
- by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_lmul)
-
-lemma contour_integral_rmul:
- shows "f contour_integrable_on g
- \<Longrightarrow> contour_integral g (\<lambda>x. f x * c) = contour_integral g f * c"
- by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_rmul)
-
-lemma contour_integral_div:
- shows "f contour_integrable_on g
- \<Longrightarrow> contour_integral g (\<lambda>x. f x / c) = contour_integral g f / c"
- by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_div)
-
-lemma contour_integral_eq:
- "(\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x) \<Longrightarrow> contour_integral p f = contour_integral p g"
- apply (simp add: contour_integral_def)
- using has_contour_integral_eq
- by (metis contour_integral_unique has_contour_integral_integrable has_contour_integral_integral)
-
-lemma contour_integral_eq_0:
- "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> contour_integral g f = 0"
- by (simp add: has_contour_integral_is_0 contour_integral_unique)
-
-lemma contour_integral_bound_linepath:
- shows
- "\<lbrakk>f contour_integrable_on (linepath a b);
- 0 \<le> B; \<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
- \<Longrightarrow> norm(contour_integral (linepath a b) f) \<le> B*norm(b - a)"
- apply (rule has_contour_integral_bound_linepath [of f])
- apply (auto simp: has_contour_integral_integral)
- done
-
-lemma contour_integral_0 [simp]: "contour_integral g (\<lambda>x. 0) = 0"
- by (simp add: contour_integral_unique has_contour_integral_0)
-
-lemma contour_integral_sum:
- "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
- \<Longrightarrow> contour_integral p (\<lambda>x. sum (\<lambda>a. f a x) s) = sum (\<lambda>a. contour_integral p (f a)) s"
- by (auto simp: contour_integral_unique has_contour_integral_sum has_contour_integral_integral)
-
-lemma contour_integrable_eq:
- "\<lbrakk>f contour_integrable_on p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g contour_integrable_on p"
- unfolding contour_integrable_on_def
- by (metis has_contour_integral_eq)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetic theorems for path integrability\<close>
-
-lemma contour_integrable_neg:
- "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. -(f x)) contour_integrable_on g"
- using has_contour_integral_neg contour_integrable_on_def by blast
-
-lemma contour_integrable_add:
- "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x + f2 x) contour_integrable_on g"
- using has_contour_integral_add contour_integrable_on_def
- by fastforce
-
-lemma contour_integrable_diff:
- "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x - f2 x) contour_integrable_on g"
- using has_contour_integral_diff contour_integrable_on_def
- by fastforce
-
-lemma contour_integrable_lmul:
- "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. c * f x) contour_integrable_on g"
- using has_contour_integral_lmul contour_integrable_on_def
- by fastforce
-
-lemma contour_integrable_rmul:
- "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x * c) contour_integrable_on g"
- using has_contour_integral_rmul contour_integrable_on_def
- by fastforce
-
-lemma contour_integrable_div:
- "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x / c) contour_integrable_on g"
- using has_contour_integral_div contour_integrable_on_def
- by fastforce
-
-lemma contour_integrable_sum:
- "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
- \<Longrightarrow> (\<lambda>x. sum (\<lambda>a. f a x) s) contour_integrable_on p"
- unfolding contour_integrable_on_def
- by (metis has_contour_integral_sum)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Reversing a path integral\<close>
-
-lemma has_contour_integral_reverse_linepath:
- "(f has_contour_integral i) (linepath a b)
- \<Longrightarrow> (f has_contour_integral (-i)) (linepath b a)"
- using has_contour_integral_reversepath valid_path_linepath by fastforce
-
-lemma contour_integral_reverse_linepath:
- "continuous_on (closed_segment a b) f
- \<Longrightarrow> contour_integral (linepath a b) f = - (contour_integral(linepath b a) f)"
-apply (rule contour_integral_unique)
-apply (rule has_contour_integral_reverse_linepath)
-by (simp add: closed_segment_commute contour_integrable_continuous_linepath has_contour_integral_integral)
-
-
-(* Splitting a path integral in a flat way.*)
-
-lemma has_contour_integral_split:
- assumes f: "(f has_contour_integral i) (linepath a c)" "(f has_contour_integral j) (linepath c b)"
- and k: "0 \<le> k" "k \<le> 1"
- and c: "c - a = k *\<^sub>R (b - a)"
- shows "(f has_contour_integral (i + j)) (linepath a b)"
-proof (cases "k = 0 \<or> k = 1")
- case True
- then show ?thesis
- using assms by auto
-next
- case False
- then have k: "0 < k" "k < 1" "complex_of_real k \<noteq> 1"
- using assms by auto
- have c': "c = k *\<^sub>R (b - a) + a"
- by (metis diff_add_cancel c)
- have bc: "(b - c) = (1 - k) *\<^sub>R (b - a)"
- by (simp add: algebra_simps c')
- { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}"
- have **: "\<And>x. ((k - x) / k) *\<^sub>R a + (x / k) *\<^sub>R c = (1 - x) *\<^sub>R a + x *\<^sub>R b"
- using False apply (simp add: c' algebra_simps)
- apply (simp add: real_vector.scale_left_distrib [symmetric] field_split_simps)
- done
- have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}"
- using k has_integral_affinity01 [OF *, of "inverse k" "0"]
- apply (simp add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost ** c)
- apply (auto dest: has_integral_cmul [where c = "inverse k"])
- done
- } note fi = this
- { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}"
- have **: "\<And>x. (((1 - x) / (1 - k)) *\<^sub>R c + ((x - k) / (1 - k)) *\<^sub>R b) = ((1 - x) *\<^sub>R a + x *\<^sub>R b)"
- using k
- apply (simp add: c' field_simps)
- apply (simp add: scaleR_conv_of_real divide_simps)
- apply (simp add: field_simps)
- done
- have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}"
- using k has_integral_affinity01 [OF *, of "inverse(1 - k)" "-(k/(1 - k))"]
- apply (simp add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc)
- apply (auto dest: has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"])
- done
- } note fj = this
- show ?thesis
- using f k
- apply (simp add: has_contour_integral_linepath)
- apply (simp add: linepath_def)
- apply (rule has_integral_combine [OF _ _ fi fj], simp_all)
- done
-qed
-
-lemma continuous_on_closed_segment_transform:
- assumes f: "continuous_on (closed_segment a b) f"
- and k: "0 \<le> k" "k \<le> 1"
- and c: "c - a = k *\<^sub>R (b - a)"
- shows "continuous_on (closed_segment a c) f"
-proof -
- have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
- using c by (simp add: algebra_simps)
- have "closed_segment a c \<subseteq> closed_segment a b"
- by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment)
- then show "continuous_on (closed_segment a c) f"
- by (rule continuous_on_subset [OF f])
-qed
-
-lemma contour_integral_split:
- assumes f: "continuous_on (closed_segment a b) f"
- and k: "0 \<le> k" "k \<le> 1"
- and c: "c - a = k *\<^sub>R (b - a)"
- shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
-proof -
- have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
- using c by (simp add: algebra_simps)
- have "closed_segment a c \<subseteq> closed_segment a b"
- by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment)
- moreover have "closed_segment c b \<subseteq> closed_segment a b"
- by (metis c' ends_in_segment(2) in_segment(1) k subset_closed_segment)
- ultimately
- have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f"
- by (auto intro: continuous_on_subset [OF f])
- show ?thesis
- by (rule contour_integral_unique) (meson "*" c contour_integrable_continuous_linepath has_contour_integral_integral has_contour_integral_split k)
-qed
-
-lemma contour_integral_split_linepath:
- assumes f: "continuous_on (closed_segment a b) f"
- and c: "c \<in> closed_segment a b"
- shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
- using c by (auto simp: closed_segment_def algebra_simps intro!: contour_integral_split [OF f])
-
text\<open>The special case of midpoints used in the main quadrisection\<close>
lemma has_contour_integral_midpoint:
@@ -1056,7 +58,6 @@
apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
done
-
text\<open>A couple of special case lemmas that are useful below\<close>
lemma triangle_linear_has_chain_integral:
@@ -1081,87 +82,6 @@
apply (simp add: valid_path_join)
done
-subsection\<open>Reversing the order in a double path integral\<close>
-
-text\<open>The condition is stronger than needed but it's often true in typical situations\<close>
-
-lemma fst_im_cbox [simp]: "cbox c d \<noteq> {} \<Longrightarrow> (fst ` cbox (a,c) (b,d)) = cbox a b"
- by (auto simp: cbox_Pair_eq)
-
-lemma snd_im_cbox [simp]: "cbox a b \<noteq> {} \<Longrightarrow> (snd ` cbox (a,c) (b,d)) = cbox c d"
- by (auto simp: cbox_Pair_eq)
-
-proposition contour_integral_swap:
- assumes fcon: "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1,y2). f y1 y2)"
- and vp: "valid_path g" "valid_path h"
- and gvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))"
- and hvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))"
- shows "contour_integral g (\<lambda>w. contour_integral h (f w)) =
- contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
-proof -
- have gcon: "continuous_on {0..1} g" and hcon: "continuous_on {0..1} h"
- using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
- have fgh1: "\<And>x. (\<lambda>t. f (g x) (h t)) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>t. (g x, h t))"
- by (rule ext) simp
- have fgh2: "\<And>x. (\<lambda>t. f (g t) (h x)) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>t. (g t, h x))"
- by (rule ext) simp
- have fcon_im1: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g x, h t)) ` {0..1}) (\<lambda>(x, y). f x y)"
- by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
- have fcon_im2: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g t, h x)) ` {0..1}) (\<lambda>(x, y). f x y)"
- by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
- have "\<And>y. y \<in> {0..1} \<Longrightarrow> continuous_on {0..1} (\<lambda>x. f (g x) (h y))"
- by (subst fgh2) (rule fcon_im2 gcon continuous_intros | simp)+
- then have vdg: "\<And>y. y \<in> {0..1} \<Longrightarrow> (\<lambda>x. f (g x) (h y) * vector_derivative g (at x)) integrable_on {0..1}"
- using continuous_on_mult gvcon integrable_continuous_real by blast
- have "(\<lambda>z. vector_derivative g (at (fst z))) = (\<lambda>x. vector_derivative g (at x)) \<circ> fst"
- by auto
- then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\<lambda>x. vector_derivative g (at (fst x)))"
- apply (rule ssubst)
- apply (rule continuous_intros | simp add: gvcon)+
- done
- have "(\<lambda>z. vector_derivative h (at (snd z))) = (\<lambda>x. vector_derivative h (at x)) \<circ> snd"
- by auto
- then have hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\<lambda>x. vector_derivative h (at (snd x)))"
- apply (rule ssubst)
- apply (rule continuous_intros | simp add: hvcon)+
- done
- have "(\<lambda>x. f (g (fst x)) (h (snd x))) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>w. ((g \<circ> fst) w, (h \<circ> snd) w))"
- by auto
- then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))"
- apply (rule ssubst)
- apply (rule gcon hcon continuous_intros | simp)+
- apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon])
- done
- have "integral {0..1} (\<lambda>x. contour_integral h (f (g x)) * vector_derivative g (at x)) =
- integral {0..1} (\<lambda>x. contour_integral h (\<lambda>y. f (g x) y * vector_derivative g (at x)))"
- proof (rule integral_cong [OF contour_integral_rmul [symmetric]])
- show "\<And>x. x \<in> {0..1} \<Longrightarrow> f (g x) contour_integrable_on h"
- unfolding contour_integrable_on
- apply (rule integrable_continuous_real)
- apply (rule continuous_on_mult [OF _ hvcon])
- apply (subst fgh1)
- apply (rule fcon_im1 hcon continuous_intros | simp)+
- done
- qed
- also have "\<dots> = integral {0..1}
- (\<lambda>y. contour_integral g (\<lambda>x. f x (h y) * vector_derivative h (at y)))"
- unfolding contour_integral_integral
- apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified])
- apply (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+
- unfolding integral_mult_left [symmetric]
- apply (simp only: mult_ac)
- done
- also have "\<dots> = contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
- unfolding contour_integral_integral
- apply (rule integral_cong)
- unfolding integral_mult_left [symmetric]
- apply (simp add: algebra_simps)
- done
- finally show ?thesis
- by (simp add: contour_integral_integral)
-qed
-
-
subsection\<^marker>\<open>tag unimportant\<close> \<open>The key quadrisection step\<close>
lemma norm_sum_half:
@@ -2431,21 +1351,6 @@
using contour_integral_nearby [OF assms, where atends=False]
unfolding linked_paths_def by simp_all
-lemma C1_differentiable_polynomial_function:
- fixes p :: "real \<Rightarrow> 'a::euclidean_space"
- shows "polynomial_function p \<Longrightarrow> p C1_differentiable_on S"
- by (metis continuous_on_polymonial_function C1_differentiable_on_def has_vector_derivative_polynomial_function)
-
-lemma valid_path_polynomial_function:
- fixes p :: "real \<Rightarrow> 'a::euclidean_space"
- shows "polynomial_function p \<Longrightarrow> valid_path p"
-by (force simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_polymonial_function C1_differentiable_polynomial_function)
-
-lemma valid_path_subpath_trivial [simp]:
- fixes g :: "real \<Rightarrow> 'a::euclidean_space"
- shows "z \<noteq> g x \<Longrightarrow> valid_path (subpath x x g)"
- by (simp add: subpath_def valid_path_polynomial_function)
-
lemma contour_integral_bound_exists:
assumes S: "open S"
and g: "valid_path g"
@@ -2497,1188 +1402,6 @@
by (intro exI[of _ L]) auto
qed
-text\<open>We can treat even non-rectifiable paths as having a "length" for bounds on analytic functions in open sets.\<close>
-
-subsection \<open>Winding Numbers\<close>
-
-definition\<^marker>\<open>tag important\<close> winding_number_prop :: "[real \<Rightarrow> complex, complex, real, real \<Rightarrow> complex, complex] \<Rightarrow> bool" where
- "winding_number_prop \<gamma> z e p n \<equiv>
- valid_path p \<and> z \<notin> path_image p \<and>
- pathstart p = pathstart \<gamma> \<and>
- pathfinish p = pathfinish \<gamma> \<and>
- (\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and>
- contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
-
-definition\<^marker>\<open>tag important\<close> winding_number:: "[real \<Rightarrow> complex, complex] \<Rightarrow> complex" where
- "winding_number \<gamma> z \<equiv> SOME n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n"
-
-
-lemma winding_number:
- assumes "path \<gamma>" "z \<notin> path_image \<gamma>" "0 < e"
- shows "\<exists>p. winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
-proof -
- have "path_image \<gamma> \<subseteq> UNIV - {z}"
- using assms by blast
- then obtain d
- where d: "d>0"
- and pi_eq: "\<And>h1 h2. valid_path h1 \<and> valid_path h2 \<and>
- (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d) \<and>
- pathstart h2 = pathstart h1 \<and> pathfinish h2 = pathfinish h1 \<longrightarrow>
- path_image h1 \<subseteq> UNIV - {z} \<and> path_image h2 \<subseteq> UNIV - {z} \<and>
- (\<forall>f. f holomorphic_on UNIV - {z} \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
- using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
- then obtain h where h: "polynomial_function h \<and> pathstart h = pathstart \<gamma> \<and> pathfinish h = pathfinish \<gamma> \<and>
- (\<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)"
- using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "d/2"] d by auto
- define nn where "nn = 1/(2* pi*\<i>) * contour_integral h (\<lambda>w. 1/(w - z))"
- have "\<exists>n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n"
- proof (rule_tac x=nn in exI, clarify)
- fix e::real
- assume e: "e>0"
- obtain p where p: "polynomial_function p \<and>
- pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < min e (d/2))"
- using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "min e (d/2)"] d \<open>0<e\<close> by auto
- have "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
- by (auto simp: intro!: holomorphic_intros)
- then show "\<exists>p. winding_number_prop \<gamma> z e p nn"
- apply (rule_tac x=p in exI)
- using pi_eq [of h p] h p d
- apply (auto simp: valid_path_polynomial_function norm_minus_commute nn_def winding_number_prop_def)
- done
- qed
- then show ?thesis
- unfolding winding_number_def by (rule someI2_ex) (blast intro: \<open>0<e\<close>)
-qed
-
-lemma winding_number_unique:
- assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
- and pi: "\<And>e. e>0 \<Longrightarrow> \<exists>p. winding_number_prop \<gamma> z e p n"
- shows "winding_number \<gamma> z = n"
-proof -
- have "path_image \<gamma> \<subseteq> UNIV - {z}"
- using assms by blast
- then obtain e
- where e: "e>0"
- and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
- (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
- pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
- contour_integral h2 f = contour_integral h1 f"
- using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
- obtain p where p: "winding_number_prop \<gamma> z e p n"
- using pi [OF e] by blast
- obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)"
- using winding_number [OF \<gamma> e] by blast
- have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
- using p by (auto simp: winding_number_prop_def)
- also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))"
- proof (rule pi_eq)
- show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
- by (auto intro!: holomorphic_intros)
- qed (use p q in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
- also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
- using q by (auto simp: winding_number_prop_def)
- finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
- then show ?thesis
- by simp
-qed
-
-(*NB not winding_number_prop here due to the loop in p*)
-lemma winding_number_unique_loop:
- assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
- and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- and pi:
- "\<And>e. e>0 \<Longrightarrow> \<exists>p. valid_path p \<and> z \<notin> path_image p \<and>
- pathfinish p = pathstart p \<and>
- (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
- contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
- shows "winding_number \<gamma> z = n"
-proof -
- have "path_image \<gamma> \<subseteq> UNIV - {z}"
- using assms by blast
- then obtain e
- where e: "e>0"
- and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
- (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
- pathfinish h1 = pathstart h1; pathfinish h2 = pathstart h2; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
- contour_integral h2 f = contour_integral h1 f"
- using contour_integral_nearby_loops [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
- obtain p where p:
- "valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and>
- (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
- contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
- using pi [OF e] by blast
- obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)"
- using winding_number [OF \<gamma> e] by blast
- have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
- using p by auto
- also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))"
- proof (rule pi_eq)
- show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
- by (auto intro!: holomorphic_intros)
- qed (use p q loop in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
- also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
- using q by (auto simp: winding_number_prop_def)
- finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
- then show ?thesis
- by simp
-qed
-
-proposition winding_number_valid_path:
- assumes "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
- shows "winding_number \<gamma> z = 1/(2*pi*\<i>) * contour_integral \<gamma> (\<lambda>w. 1/(w - z))"
- by (rule winding_number_unique)
- (use assms in \<open>auto simp: valid_path_imp_path winding_number_prop_def\<close>)
-
-proposition has_contour_integral_winding_number:
- assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
- shows "((\<lambda>w. 1/(w - z)) has_contour_integral (2*pi*\<i>*winding_number \<gamma> z)) \<gamma>"
-by (simp add: winding_number_valid_path has_contour_integral_integral contour_integrable_inversediff assms)
-
-lemma winding_number_trivial [simp]: "z \<noteq> a \<Longrightarrow> winding_number(linepath a a) z = 0"
- by (simp add: winding_number_valid_path)
-
-lemma winding_number_subpath_trivial [simp]: "z \<noteq> g x \<Longrightarrow> winding_number (subpath x x g) z = 0"
- by (simp add: path_image_subpath winding_number_valid_path)
-
-lemma winding_number_join:
- assumes \<gamma>1: "path \<gamma>1" "z \<notin> path_image \<gamma>1"
- and \<gamma>2: "path \<gamma>2" "z \<notin> path_image \<gamma>2"
- and "pathfinish \<gamma>1 = pathstart \<gamma>2"
- shows "winding_number(\<gamma>1 +++ \<gamma>2) z = winding_number \<gamma>1 z + winding_number \<gamma>2 z"
-proof (rule winding_number_unique)
- show "\<exists>p. winding_number_prop (\<gamma>1 +++ \<gamma>2) z e p
- (winding_number \<gamma>1 z + winding_number \<gamma>2 z)" if "e > 0" for e
- proof -
- obtain p1 where "winding_number_prop \<gamma>1 z e p1 (winding_number \<gamma>1 z)"
- using \<open>0 < e\<close> \<gamma>1 winding_number by blast
- moreover
- obtain p2 where "winding_number_prop \<gamma>2 z e p2 (winding_number \<gamma>2 z)"
- using \<open>0 < e\<close> \<gamma>2 winding_number by blast
- ultimately
- have "winding_number_prop (\<gamma>1+++\<gamma>2) z e (p1+++p2) (winding_number \<gamma>1 z + winding_number \<gamma>2 z)"
- using assms
- apply (simp add: winding_number_prop_def not_in_path_image_join contour_integrable_inversediff algebra_simps)
- apply (auto simp: joinpaths_def)
- done
- then show ?thesis
- by blast
- qed
-qed (use assms in \<open>auto simp: not_in_path_image_join\<close>)
-
-lemma winding_number_reversepath:
- assumes "path \<gamma>" "z \<notin> path_image \<gamma>"
- shows "winding_number(reversepath \<gamma>) z = - (winding_number \<gamma> z)"
-proof (rule winding_number_unique)
- show "\<exists>p. winding_number_prop (reversepath \<gamma>) z e p (- winding_number \<gamma> z)" if "e > 0" for e
- proof -
- obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
- using \<open>0 < e\<close> assms winding_number by blast
- then have "winding_number_prop (reversepath \<gamma>) z e (reversepath p) (- winding_number \<gamma> z)"
- using assms
- apply (simp add: winding_number_prop_def contour_integral_reversepath contour_integrable_inversediff valid_path_imp_reverse)
- apply (auto simp: reversepath_def)
- done
- then show ?thesis
- by blast
- qed
-qed (use assms in auto)
-
-lemma winding_number_shiftpath:
- assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
- and "pathfinish \<gamma> = pathstart \<gamma>" "a \<in> {0..1}"
- shows "winding_number(shiftpath a \<gamma>) z = winding_number \<gamma> z"
-proof (rule winding_number_unique_loop)
- show "\<exists>p. valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and>
- (\<forall>t\<in>{0..1}. cmod (shiftpath a \<gamma> t - p t) < e) \<and>
- contour_integral p (\<lambda>w. 1 / (w - z)) =
- complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
- if "e > 0" for e
- proof -
- obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
- using \<open>0 < e\<close> assms winding_number by blast
- then show ?thesis
- apply (rule_tac x="shiftpath a p" in exI)
- using assms that
- apply (auto simp: winding_number_prop_def path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath contour_integral_shiftpath)
- apply (simp add: shiftpath_def)
- done
- qed
-qed (use assms in \<open>auto simp: path_shiftpath path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath\<close>)
-
-lemma winding_number_split_linepath:
- assumes "c \<in> closed_segment a b" "z \<notin> closed_segment a b"
- shows "winding_number(linepath a b) z = winding_number(linepath a c) z + winding_number(linepath c b) z"
-proof -
- have "z \<notin> closed_segment a c" "z \<notin> closed_segment c b"
- using assms by (meson convex_contains_segment convex_segment ends_in_segment subsetCE)+
- then show ?thesis
- using assms
- by (simp add: winding_number_valid_path contour_integral_split_linepath [symmetric] continuous_on_inversediff field_simps)
-qed
-
-lemma winding_number_cong:
- "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> p t = q t) \<Longrightarrow> winding_number p z = winding_number q z"
- by (simp add: winding_number_def winding_number_prop_def pathstart_def pathfinish_def)
-
-lemma winding_number_constI:
- assumes "c\<noteq>z" "\<And>t. \<lbrakk>0\<le>t; t\<le>1\<rbrakk> \<Longrightarrow> g t = c"
- shows "winding_number g z = 0"
-proof -
- have "winding_number g z = winding_number (linepath c c) z"
- apply (rule winding_number_cong)
- using assms unfolding linepath_def by auto
- moreover have "winding_number (linepath c c) z =0"
- apply (rule winding_number_trivial)
- using assms by auto
- ultimately show ?thesis by auto
-qed
-
-lemma winding_number_offset: "winding_number p z = winding_number (\<lambda>w. p w - z) 0"
- unfolding winding_number_def
-proof (intro ext arg_cong [where f = Eps] arg_cong [where f = All] imp_cong refl, safe)
- fix n e g
- assume "0 < e" and g: "winding_number_prop p z e g n"
- then show "\<exists>r. winding_number_prop (\<lambda>w. p w - z) 0 e r n"
- by (rule_tac x="\<lambda>t. g t - z" in exI)
- (force simp: winding_number_prop_def contour_integral_integral valid_path_def path_defs
- vector_derivative_def has_vector_derivative_diff_const piecewise_C1_differentiable_diff C1_differentiable_imp_piecewise)
-next
- fix n e g
- assume "0 < e" and g: "winding_number_prop (\<lambda>w. p w - z) 0 e g n"
- then show "\<exists>r. winding_number_prop p z e r n"
- apply (rule_tac x="\<lambda>t. g t + z" in exI)
- apply (simp add: winding_number_prop_def contour_integral_integral valid_path_def path_defs
- piecewise_C1_differentiable_add vector_derivative_def has_vector_derivative_add_const C1_differentiable_imp_piecewise)
- apply (force simp: algebra_simps)
- done
-qed
-
-subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Some lemmas about negating a path\<close>
-
-lemma valid_path_negatepath: "valid_path \<gamma> \<Longrightarrow> valid_path (uminus \<circ> \<gamma>)"
- unfolding o_def using piecewise_C1_differentiable_neg valid_path_def by blast
-
-lemma has_contour_integral_negatepath:
- assumes \<gamma>: "valid_path \<gamma>" and cint: "((\<lambda>z. f (- z)) has_contour_integral - i) \<gamma>"
- shows "(f has_contour_integral i) (uminus \<circ> \<gamma>)"
-proof -
- obtain S where cont: "continuous_on {0..1} \<gamma>" and "finite S" and diff: "\<gamma> C1_differentiable_on {0..1} - S"
- using \<gamma> by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
- have "((\<lambda>x. - (f (- \<gamma> x) * vector_derivative \<gamma> (at x within {0..1}))) has_integral i) {0..1}"
- using cint by (auto simp: has_contour_integral_def dest: has_integral_neg)
- then
- have "((\<lambda>x. f (- \<gamma> x) * vector_derivative (uminus \<circ> \<gamma>) (at x within {0..1})) has_integral i) {0..1}"
- proof (rule rev_iffD1 [OF _ has_integral_spike_eq])
- show "negligible S"
- by (simp add: \<open>finite S\<close> negligible_finite)
- show "f (- \<gamma> x) * vector_derivative (uminus \<circ> \<gamma>) (at x within {0..1}) =
- - (f (- \<gamma> x) * vector_derivative \<gamma> (at x within {0..1}))"
- if "x \<in> {0..1} - S" for x
- proof -
- have "vector_derivative (uminus \<circ> \<gamma>) (at x within cbox 0 1) = - vector_derivative \<gamma> (at x within cbox 0 1)"
- proof (rule vector_derivative_within_cbox)
- show "(uminus \<circ> \<gamma> has_vector_derivative - vector_derivative \<gamma> (at x within cbox 0 1)) (at x within cbox 0 1)"
- using that unfolding o_def
- by (metis C1_differentiable_on_eq UNIV_I diff differentiable_subset has_vector_derivative_minus subsetI that vector_derivative_works)
- qed (use that in auto)
- then show ?thesis
- by simp
- qed
- qed
- then show ?thesis by (simp add: has_contour_integral_def)
-qed
-
-lemma winding_number_negatepath:
- assumes \<gamma>: "valid_path \<gamma>" and 0: "0 \<notin> path_image \<gamma>"
- shows "winding_number(uminus \<circ> \<gamma>) 0 = winding_number \<gamma> 0"
-proof -
- have "(/) 1 contour_integrable_on \<gamma>"
- using "0" \<gamma> contour_integrable_inversediff by fastforce
- then have "((\<lambda>z. 1/z) has_contour_integral contour_integral \<gamma> ((/) 1)) \<gamma>"
- by (rule has_contour_integral_integral)
- then have "((\<lambda>z. 1 / - z) has_contour_integral - contour_integral \<gamma> ((/) 1)) \<gamma>"
- using has_contour_integral_neg by auto
- then show ?thesis
- using assms
- apply (simp add: winding_number_valid_path valid_path_negatepath image_def path_defs)
- apply (simp add: contour_integral_unique has_contour_integral_negatepath)
- done
-qed
-
-lemma contour_integrable_negatepath:
- assumes \<gamma>: "valid_path \<gamma>" and pi: "(\<lambda>z. f (- z)) contour_integrable_on \<gamma>"
- shows "f contour_integrable_on (uminus \<circ> \<gamma>)"
- by (metis \<gamma> add.inverse_inverse contour_integrable_on_def has_contour_integral_negatepath pi)
-
-(* A combined theorem deducing several things piecewise.*)
-lemma winding_number_join_pos_combined:
- "\<lbrakk>valid_path \<gamma>1; z \<notin> path_image \<gamma>1; 0 < Re(winding_number \<gamma>1 z);
- valid_path \<gamma>2; z \<notin> path_image \<gamma>2; 0 < Re(winding_number \<gamma>2 z); pathfinish \<gamma>1 = pathstart \<gamma>2\<rbrakk>
- \<Longrightarrow> valid_path(\<gamma>1 +++ \<gamma>2) \<and> z \<notin> path_image(\<gamma>1 +++ \<gamma>2) \<and> 0 < Re(winding_number(\<gamma>1 +++ \<gamma>2) z)"
- by (simp add: valid_path_join path_image_join winding_number_join valid_path_imp_path)
-
-
-subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Useful sufficient conditions for the winding number to be positive\<close>
-
-lemma Re_winding_number:
- "\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>\<rbrakk>
- \<Longrightarrow> Re(winding_number \<gamma> z) = Im(contour_integral \<gamma> (\<lambda>w. 1/(w - z))) / (2*pi)"
-by (simp add: winding_number_valid_path field_simps Re_divide power2_eq_square)
-
-lemma winding_number_pos_le:
- assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
- and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> 0 \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
- shows "0 \<le> Re(winding_number \<gamma> z)"
-proof -
- have ge0: "0 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" if x: "0 < x" "x < 1" for x
- using ge by (simp add: Complex.Im_divide algebra_simps x)
- let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)"
- let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))"
- have hi: "(?vd has_integral ?int z) (cbox 0 1)"
- unfolding box_real
- apply (subst has_contour_integral [symmetric])
- using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral)
- have "0 \<le> Im (?int z)"
- proof (rule has_integral_component_nonneg [of \<i>, simplified])
- show "\<And>x. x \<in> cbox 0 1 \<Longrightarrow> 0 \<le> Im (if 0 < x \<and> x < 1 then ?vd x else 0)"
- by (force simp: ge0)
- show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else 0) has_integral ?int z) (cbox 0 1)"
- by (rule has_integral_spike_interior [OF hi]) simp
- qed
- then show ?thesis
- by (simp add: Re_winding_number [OF \<gamma>] field_simps)
-qed
-
-lemma winding_number_pos_lt_lemma:
- assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
- and e: "0 < e"
- and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
- shows "0 < Re(winding_number \<gamma> z)"
-proof -
- let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)"
- let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))"
- have hi: "(?vd has_integral ?int z) (cbox 0 1)"
- unfolding box_real
- apply (subst has_contour_integral [symmetric])
- using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral)
- have "e \<le> Im (contour_integral \<gamma> (\<lambda>w. 1 / (w - z)))"
- proof (rule has_integral_component_le [of \<i> "\<lambda>x. \<i>*e" "\<i>*e" "{0..1}", simplified])
- show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else \<i> * complex_of_real e) has_integral ?int z) {0..1}"
- by (rule has_integral_spike_interior [OF hi, simplified box_real]) (use e in simp)
- show "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow>
- e \<le> Im (if 0 < x \<and> x < 1 then ?vd x else \<i> * complex_of_real e)"
- by (simp add: ge)
- qed (use has_integral_const_real [of _ 0 1] in auto)
- with e show ?thesis
- by (simp add: Re_winding_number [OF \<gamma>] field_simps)
-qed
-
-lemma winding_number_pos_lt:
- assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
- and e: "0 < e"
- and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
- shows "0 < Re (winding_number \<gamma> z)"
-proof -
- have bm: "bounded ((\<lambda>w. w - z) ` (path_image \<gamma>))"
- using bounded_translation [of _ "-z"] \<gamma> by (simp add: bounded_valid_path_image)
- then obtain B where B: "B > 0" and Bno: "\<And>x. x \<in> (\<lambda>w. w - z) ` (path_image \<gamma>) \<Longrightarrow> norm x \<le> B"
- using bounded_pos [THEN iffD1, OF bm] by blast
- { fix x::real assume x: "0 < x" "x < 1"
- then have B2: "cmod (\<gamma> x - z)^2 \<le> B^2" using Bno [of "\<gamma> x - z"]
- by (simp add: path_image_def power2_eq_square mult_mono')
- with x have "\<gamma> x \<noteq> z" using \<gamma>
- using path_image_def by fastforce
- then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z)) / (cmod (\<gamma> x - z))\<^sup>2"
- using B ge [OF x] B2 e
- apply (rule_tac y="e / (cmod (\<gamma> x - z))\<^sup>2" in order_trans)
- apply (auto simp: divide_left_mono divide_right_mono)
- done
- then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
- by (simp add: complex_div_cnj [of _ "\<gamma> x - z" for x] del: complex_cnj_diff times_complex.sel)
- } note * = this
- show ?thesis
- using e B by (simp add: * winding_number_pos_lt_lemma [OF \<gamma>, of "e/B^2"])
-qed
-
-subsection\<open>The winding number is an integer\<close>
-
-text\<open>Proof from the book Complex Analysis by Lars V. Ahlfors, Chapter 4, section 2.1,
- Also on page 134 of Serge Lang's book with the name title, etc.\<close>
-
-lemma exp_fg:
- fixes z::complex
- assumes g: "(g has_vector_derivative g') (at x within s)"
- and f: "(f has_vector_derivative (g' / (g x - z))) (at x within s)"
- and z: "g x \<noteq> z"
- shows "((\<lambda>x. exp(-f x) * (g x - z)) has_vector_derivative 0) (at x within s)"
-proof -
- have *: "(exp \<circ> (\<lambda>x. (- f x)) has_vector_derivative - (g' / (g x - z)) * exp (- f x)) (at x within s)"
- using assms unfolding has_vector_derivative_def scaleR_conv_of_real
- by (auto intro!: derivative_eq_intros)
- show ?thesis
- apply (rule has_vector_derivative_eq_rhs)
- using z
- apply (auto intro!: derivative_eq_intros * [unfolded o_def] g)
- done
-qed
-
-lemma winding_number_exp_integral:
- fixes z::complex
- assumes \<gamma>: "\<gamma> piecewise_C1_differentiable_on {a..b}"
- and ab: "a \<le> b"
- and z: "z \<notin> \<gamma> ` {a..b}"
- shows "(\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)) integrable_on {a..b}"
- (is "?thesis1")
- "exp (- (integral {a..b} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))) * (\<gamma> b - z) = \<gamma> a - z"
- (is "?thesis2")
-proof -
- let ?D\<gamma> = "\<lambda>x. vector_derivative \<gamma> (at x)"
- have [simp]: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<gamma> x \<noteq> z"
- using z by force
- have cong: "continuous_on {a..b} \<gamma>"
- using \<gamma> by (simp add: piecewise_C1_differentiable_on_def)
- obtain k where fink: "finite k" and g_C1_diff: "\<gamma> C1_differentiable_on ({a..b} - k)"
- using \<gamma> by (force simp: piecewise_C1_differentiable_on_def)
- have \<circ>: "open ({a<..<b} - k)"
- using \<open>finite k\<close> by (simp add: finite_imp_closed open_Diff)
- moreover have "{a<..<b} - k \<subseteq> {a..b} - k"
- by force
- ultimately have g_diff_at: "\<And>x. \<lbrakk>x \<notin> k; x \<in> {a<..<b}\<rbrakk> \<Longrightarrow> \<gamma> differentiable at x"
- by (metis Diff_iff differentiable_on_subset C1_diff_imp_diff [OF g_C1_diff] differentiable_on_def at_within_open)
- { fix w
- assume "w \<noteq> z"
- have "continuous_on (ball w (cmod (w - z))) (\<lambda>w. 1 / (w - z))"
- by (auto simp: dist_norm intro!: continuous_intros)
- moreover have "\<And>x. cmod (w - x) < cmod (w - z) \<Longrightarrow> \<exists>f'. ((\<lambda>w. 1 / (w - z)) has_field_derivative f') (at x)"
- by (auto simp: intro!: derivative_eq_intros)
- ultimately have "\<exists>h. \<forall>y. norm(y - w) < norm(w - z) \<longrightarrow> (h has_field_derivative 1/(y - z)) (at y)"
- using holomorphic_convex_primitive [of "ball w (norm(w - z))" "{}" "\<lambda>w. 1/(w - z)"]
- by (force simp: field_differentiable_def Ball_def dist_norm at_within_open_NO_MATCH norm_minus_commute)
- }
- then obtain h where h: "\<And>w y. w \<noteq> z \<Longrightarrow> norm(y - w) < norm(w - z) \<Longrightarrow> (h w has_field_derivative 1/(y - z)) (at y)"
- by meson
- have exy: "\<exists>y. ((\<lambda>x. inverse (\<gamma> x - z) * ?D\<gamma> x) has_integral y) {a..b}"
- unfolding integrable_on_def [symmetric]
- proof (rule contour_integral_local_primitive_any [OF piecewise_C1_imp_differentiable [OF \<gamma>]])
- show "\<exists>d h. 0 < d \<and>
- (\<forall>y. cmod (y - w) < d \<longrightarrow> (h has_field_derivative inverse (y - z))(at y within - {z}))"
- if "w \<in> - {z}" for w
- apply (rule_tac x="norm(w - z)" in exI)
- using that inverse_eq_divide has_field_derivative_at_within h
- by (metis Compl_insert DiffD2 insertCI right_minus_eq zero_less_norm_iff)
- qed simp
- have vg_int: "(\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)) integrable_on {a..b}"
- unfolding box_real [symmetric] divide_inverse_commute
- by (auto intro!: exy integrable_subinterval simp add: integrable_on_def ab)
- with ab show ?thesis1
- by (simp add: divide_inverse_commute integral_def integrable_on_def)
- { fix t
- assume t: "t \<in> {a..b}"
- have cball: "continuous_on (ball (\<gamma> t) (dist (\<gamma> t) z)) (\<lambda>x. inverse (x - z))"
- using z by (auto intro!: continuous_intros simp: dist_norm)
- have icd: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow> (\<lambda>w. inverse (w - z)) field_differentiable at x"
- unfolding field_differentiable_def by (force simp: intro!: derivative_eq_intros)
- obtain h where h: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow>
- (h has_field_derivative inverse (x - z)) (at x within {y. cmod (\<gamma> t - y) < cmod (\<gamma> t - z)})"
- using holomorphic_convex_primitive [where f = "\<lambda>w. inverse(w - z)", OF convex_ball finite.emptyI cball icd]
- by simp (auto simp: ball_def dist_norm that)
- { fix x D
- assume x: "x \<notin> k" "a < x" "x < b"
- then have "x \<in> interior ({a..b} - k)"
- using open_subset_interior [OF \<circ>] by fastforce
- then have con: "isCont ?D\<gamma> x"
- using g_C1_diff x by (auto simp: C1_differentiable_on_eq intro: continuous_on_interior)
- then have con_vd: "continuous (at x within {a..b}) (\<lambda>x. ?D\<gamma> x)"
- by (rule continuous_at_imp_continuous_within)
- have gdx: "\<gamma> differentiable at x"
- using x by (simp add: g_diff_at)
- have "\<And>d. \<lbrakk>x \<notin> k; a < x; x < b;
- (\<gamma> has_vector_derivative d) (at x); a \<le> t; t \<le> b\<rbrakk>
- \<Longrightarrow> ((\<lambda>x. integral {a..x}
- (\<lambda>x. ?D\<gamma> x /
- (\<gamma> x - z))) has_vector_derivative
- d / (\<gamma> x - z))
- (at x within {a..b})"
- apply (rule has_vector_derivative_eq_rhs)
- apply (rule integral_has_vector_derivative_continuous_at [where S = "{}", simplified])
- apply (rule con_vd continuous_intros cong vg_int | simp add: continuous_at_imp_continuous_within has_vector_derivative_continuous vector_derivative_at)+
- done
- then have "((\<lambda>c. exp (- integral {a..c} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) * (\<gamma> c - z)) has_derivative (\<lambda>h. 0))
- (at x within {a..b})"
- using x gdx t
- apply (clarsimp simp add: differentiable_iff_scaleR)
- apply (rule exp_fg [unfolded has_vector_derivative_def, simplified], blast intro: has_derivative_at_withinI)
- apply (simp_all add: has_vector_derivative_def [symmetric])
- done
- } note * = this
- have "exp (- (integral {a..t} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)))) * (\<gamma> t - z) =\<gamma> a - z"
- apply (rule has_derivative_zero_unique_strong_interval [of "{a,b} \<union> k" a b])
- using t
- apply (auto intro!: * continuous_intros fink cong indefinite_integral_continuous_1 [OF vg_int] simp add: ab)+
- done
- }
- with ab show ?thesis2
- by (simp add: divide_inverse_commute integral_def)
-qed
-
-lemma winding_number_exp_2pi:
- "\<lbrakk>path p; z \<notin> path_image p\<rbrakk>
- \<Longrightarrow> pathfinish p - z = exp (2 * pi * \<i> * winding_number p z) * (pathstart p - z)"
-using winding_number [of p z 1] unfolding valid_path_def path_image_def pathstart_def pathfinish_def winding_number_prop_def
- by (force dest: winding_number_exp_integral(2) [of _ 0 1 z] simp: field_simps contour_integral_integral exp_minus)
-
-lemma integer_winding_number_eq:
- assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
- shows "winding_number \<gamma> z \<in> \<int> \<longleftrightarrow> pathfinish \<gamma> = pathstart \<gamma>"
-proof -
- obtain p where p: "valid_path p" "z \<notin> path_image p"
- "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
- and eq: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
- using winding_number [OF assms, of 1] unfolding winding_number_prop_def by auto
- then have wneq: "winding_number \<gamma> z = winding_number p z"
- using eq winding_number_valid_path by force
- have iff: "(winding_number \<gamma> z \<in> \<int>) \<longleftrightarrow> (exp (contour_integral p (\<lambda>w. 1 / (w - z))) = 1)"
- using eq by (simp add: exp_eq_1 complex_is_Int_iff)
- have "exp (contour_integral p (\<lambda>w. 1 / (w - z))) = (\<gamma> 1 - z) / (\<gamma> 0 - z)"
- using p winding_number_exp_integral(2) [of p 0 1 z]
- apply (simp add: valid_path_def path_defs contour_integral_integral exp_minus field_split_simps)
- by (metis path_image_def pathstart_def pathstart_in_path_image)
- then have "winding_number p z \<in> \<int> \<longleftrightarrow> pathfinish p = pathstart p"
- using p wneq iff by (auto simp: path_defs)
- then show ?thesis using p eq
- by (auto simp: winding_number_valid_path)
-qed
-
-theorem integer_winding_number:
- "\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> \<Longrightarrow> winding_number \<gamma> z \<in> \<int>"
-by (metis integer_winding_number_eq)
-
-
-text\<open>If the winding number's magnitude is at least one, then the path must contain points in every direction.*)
- We can thus bound the winding number of a path that doesn't intersect a given ray. \<close>
-
-lemma winding_number_pos_meets:
- fixes z::complex
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and 1: "Re (winding_number \<gamma> z) \<ge> 1"
- and w: "w \<noteq> z"
- shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
-proof -
- have [simp]: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> \<gamma> x \<noteq> z"
- using z by (auto simp: path_image_def)
- have [simp]: "z \<notin> \<gamma> ` {0..1}"
- using path_image_def z by auto
- have gpd: "\<gamma> piecewise_C1_differentiable_on {0..1}"
- using \<gamma> valid_path_def by blast
- define r where "r = (w - z) / (\<gamma> 0 - z)"
- have [simp]: "r \<noteq> 0"
- using w z by (auto simp: r_def)
- have cont: "continuous_on {0..1}
- (\<lambda>x. Im (integral {0..x} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))))"
- by (intro continuous_intros indefinite_integral_continuous_1 winding_number_exp_integral [OF gpd]; simp)
- have "Arg2pi r \<le> 2*pi"
- by (simp add: Arg2pi less_eq_real_def)
- also have "\<dots> \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))"
- using 1
- apply (simp add: winding_number_valid_path [OF \<gamma> z] contour_integral_integral)
- apply (simp add: Complex.Re_divide field_simps power2_eq_square)
- done
- finally have "Arg2pi r \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))" .
- then have "\<exists>t. t \<in> {0..1} \<and> Im(integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r"
- by (simp add: Arg2pi_ge_0 cont IVT')
- then obtain t where t: "t \<in> {0..1}"
- and eqArg: "Im (integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r"
- by blast
- define i where "i = integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
- have iArg: "Arg2pi r = Im i"
- using eqArg by (simp add: i_def)
- have gpdt: "\<gamma> piecewise_C1_differentiable_on {0..t}"
- by (metis atLeastAtMost_iff atLeastatMost_subset_iff order_refl piecewise_C1_differentiable_on_subset gpd t)
- have "exp (- i) * (\<gamma> t - z) = \<gamma> 0 - z"
- unfolding i_def
- apply (rule winding_number_exp_integral [OF gpdt])
- using t z unfolding path_image_def by force+
- then have *: "\<gamma> t - z = exp i * (\<gamma> 0 - z)"
- by (simp add: exp_minus field_simps)
- then have "(w - z) = r * (\<gamma> 0 - z)"
- by (simp add: r_def)
- then have "z + complex_of_real (exp (Re i)) * (w - z) / complex_of_real (cmod r) = \<gamma> t"
- apply simp
- apply (subst Complex_Transcendental.Arg2pi_eq [of r])
- apply (simp add: iArg)
- using * apply (simp add: exp_eq_polar field_simps)
- done
- with t show ?thesis
- by (rule_tac x="exp(Re i) / norm r" in exI) (auto simp: path_image_def)
-qed
-
-lemma winding_number_big_meets:
- fixes z::complex
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "\<bar>Re (winding_number \<gamma> z)\<bar> \<ge> 1"
- and w: "w \<noteq> z"
- shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
-proof -
- { assume "Re (winding_number \<gamma> z) \<le> - 1"
- then have "Re (winding_number (reversepath \<gamma>) z) \<ge> 1"
- by (simp add: \<gamma> valid_path_imp_path winding_number_reversepath z)
- moreover have "valid_path (reversepath \<gamma>)"
- using \<gamma> valid_path_imp_reverse by auto
- moreover have "z \<notin> path_image (reversepath \<gamma>)"
- by (simp add: z)
- ultimately have "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image (reversepath \<gamma>)"
- using winding_number_pos_meets w by blast
- then have ?thesis
- by simp
- }
- then show ?thesis
- using assms
- by (simp add: abs_if winding_number_pos_meets split: if_split_asm)
-qed
-
-lemma winding_number_less_1:
- fixes z::complex
- shows
- "\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>; w \<noteq> z;
- \<And>a::real. 0 < a \<Longrightarrow> z + a*(w - z) \<notin> path_image \<gamma>\<rbrakk>
- \<Longrightarrow> Re(winding_number \<gamma> z) < 1"
- by (auto simp: not_less dest: winding_number_big_meets)
-
-text\<open>One way of proving that WN=1 for a loop.\<close>
-lemma winding_number_eq_1:
- fixes z::complex
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- and 0: "0 < Re(winding_number \<gamma> z)" and 2: "Re(winding_number \<gamma> z) < 2"
- shows "winding_number \<gamma> z = 1"
-proof -
- have "winding_number \<gamma> z \<in> Ints"
- by (simp add: \<gamma> integer_winding_number loop valid_path_imp_path z)
- then show ?thesis
- using 0 2 by (auto simp: Ints_def)
-qed
-
-subsection\<open>Continuity of winding number and invariance on connected sets\<close>
-
-lemma continuous_at_winding_number:
- fixes z::complex
- assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
- shows "continuous (at z) (winding_number \<gamma>)"
-proof -
- obtain e where "e>0" and cbg: "cball z e \<subseteq> - path_image \<gamma>"
- using open_contains_cball [of "- path_image \<gamma>"] z
- by (force simp: closed_def [symmetric] closed_path_image [OF \<gamma>])
- then have ppag: "path_image \<gamma> \<subseteq> - cball z (e/2)"
- by (force simp: cball_def dist_norm)
- have oc: "open (- cball z (e / 2))"
- by (simp add: closed_def [symmetric])
- obtain d where "d>0" and pi_eq:
- "\<And>h1 h2. \<lbrakk>valid_path h1; valid_path h2;
- (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d);
- pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1\<rbrakk>
- \<Longrightarrow>
- path_image h1 \<subseteq> - cball z (e / 2) \<and>
- path_image h2 \<subseteq> - cball z (e / 2) \<and>
- (\<forall>f. f holomorphic_on - cball z (e / 2) \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
- using contour_integral_nearby_ends [OF oc \<gamma> ppag] by metis
- obtain p where p: "valid_path p" "z \<notin> path_image p"
- "pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma>"
- and pg: "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (\<gamma> t - p t) < min d e / 2"
- and pi: "contour_integral p (\<lambda>x. 1 / (x - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
- using winding_number [OF \<gamma> z, of "min d e / 2"] \<open>d>0\<close> \<open>e>0\<close> by (auto simp: winding_number_prop_def)
- { fix w
- assume d2: "cmod (w - z) < d/2" and e2: "cmod (w - z) < e/2"
- then have wnotp: "w \<notin> path_image p"
- using cbg \<open>d>0\<close> \<open>e>0\<close>
- apply (simp add: path_image_def cball_def dist_norm, clarify)
- apply (frule pg)
- apply (drule_tac c="\<gamma> x" in subsetD)
- apply (auto simp: less_eq_real_def norm_minus_commute norm_triangle_half_l)
- done
- have wnotg: "w \<notin> path_image \<gamma>"
- using cbg e2 \<open>e>0\<close> by (force simp: dist_norm norm_minus_commute)
- { fix k::real
- assume k: "k>0"
- then obtain q where q: "valid_path q" "w \<notin> path_image q"
- "pathstart q = pathstart \<gamma> \<and> pathfinish q = pathfinish \<gamma>"
- and qg: "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (\<gamma> t - q t) < min k (min d e) / 2"
- and qi: "contour_integral q (\<lambda>u. 1 / (u - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
- using winding_number [OF \<gamma> wnotg, of "min k (min d e) / 2"] \<open>d>0\<close> \<open>e>0\<close> k
- by (force simp: min_divide_distrib_right winding_number_prop_def)
- have "contour_integral p (\<lambda>u. 1 / (u - w)) = contour_integral q (\<lambda>u. 1 / (u - w))"
- apply (rule pi_eq [OF \<open>valid_path q\<close> \<open>valid_path p\<close>, THEN conjunct2, THEN conjunct2, rule_format])
- apply (frule pg)
- apply (frule qg)
- using p q \<open>d>0\<close> e2
- apply (auto simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
- done
- then have "contour_integral p (\<lambda>x. 1 / (x - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
- by (simp add: pi qi)
- } note pip = this
- have "path p"
- using p by (simp add: valid_path_imp_path)
- then have "winding_number p w = winding_number \<gamma> w"
- apply (rule winding_number_unique [OF _ wnotp])
- apply (rule_tac x=p in exI)
- apply (simp add: p wnotp min_divide_distrib_right pip winding_number_prop_def)
- done
- } note wnwn = this
- obtain pe where "pe>0" and cbp: "cball z (3 / 4 * pe) \<subseteq> - path_image p"
- using p open_contains_cball [of "- path_image p"]
- by (force simp: closed_def [symmetric] closed_path_image [OF valid_path_imp_path])
- obtain L
- where "L>0"
- and L: "\<And>f B. \<lbrakk>f holomorphic_on - cball z (3 / 4 * pe);
- \<forall>z \<in> - cball z (3 / 4 * pe). cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
- cmod (contour_integral p f) \<le> L * B"
- using contour_integral_bound_exists [of "- cball z (3/4*pe)" p] cbp \<open>valid_path p\<close> by blast
- { fix e::real and w::complex
- assume e: "0 < e" and w: "cmod (w - z) < pe/4" "cmod (w - z) < e * pe\<^sup>2 / (8 * L)"
- then have [simp]: "w \<notin> path_image p"
- using cbp p(2) \<open>0 < pe\<close>
- by (force simp: dist_norm norm_minus_commute path_image_def cball_def)
- have [simp]: "contour_integral p (\<lambda>x. 1/(x - w)) - contour_integral p (\<lambda>x. 1/(x - z)) =
- contour_integral p (\<lambda>x. 1/(x - w) - 1/(x - z))"
- by (simp add: p contour_integrable_inversediff contour_integral_diff)
- { fix x
- assume pe: "3/4 * pe < cmod (z - x)"
- have "cmod (w - x) < pe/4 + cmod (z - x)"
- by (meson add_less_cancel_right norm_diff_triangle_le order_refl order_trans_rules(21) w(1))
- then have wx: "cmod (w - x) < 4/3 * cmod (z - x)" using pe by simp
- have "cmod (z - x) \<le> cmod (z - w) + cmod (w - x)"
- using norm_diff_triangle_le by blast
- also have "\<dots> < pe/4 + cmod (w - x)"
- using w by (simp add: norm_minus_commute)
- finally have "pe/2 < cmod (w - x)"
- using pe by auto
- then have "(pe/2)^2 < cmod (w - x) ^ 2"
- apply (rule power_strict_mono)
- using \<open>pe>0\<close> by auto
- then have pe2: "pe^2 < 4 * cmod (w - x) ^ 2"
- by (simp add: power_divide)
- have "8 * L * cmod (w - z) < e * pe\<^sup>2"
- using w \<open>L>0\<close> by (simp add: field_simps)
- also have "\<dots> < e * 4 * cmod (w - x) * cmod (w - x)"
- using pe2 \<open>e>0\<close> by (simp add: power2_eq_square)
- also have "\<dots> < e * 4 * cmod (w - x) * (4/3 * cmod (z - x))"
- using wx
- apply (rule mult_strict_left_mono)
- using pe2 e not_less_iff_gr_or_eq by fastforce
- finally have "L * cmod (w - z) < 2/3 * e * cmod (w - x) * cmod (z - x)"
- by simp
- also have "\<dots> \<le> e * cmod (w - x) * cmod (z - x)"
- using e by simp
- finally have Lwz: "L * cmod (w - z) < e * cmod (w - x) * cmod (z - x)" .
- have "L * cmod (1 / (x - w) - 1 / (x - z)) \<le> e"
- apply (cases "x=z \<or> x=w")
- using pe \<open>pe>0\<close> w \<open>L>0\<close>
- apply (force simp: norm_minus_commute)
- using wx w(2) \<open>L>0\<close> pe pe2 Lwz
- apply (auto simp: divide_simps mult_less_0_iff norm_minus_commute norm_divide norm_mult power2_eq_square)
- done
- } note L_cmod_le = this
- have *: "cmod (contour_integral p (\<lambda>x. 1 / (x - w) - 1 / (x - z))) \<le> L * (e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2)"
- apply (rule L)
- using \<open>pe>0\<close> w
- apply (force simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
- using \<open>pe>0\<close> w \<open>L>0\<close>
- apply (auto simp: cball_def dist_norm field_simps L_cmod_le simp del: less_divide_eq_numeral1 le_divide_eq_numeral1)
- done
- have "cmod (contour_integral p (\<lambda>x. 1 / (x - w)) - contour_integral p (\<lambda>x. 1 / (x - z))) < 2*e"
- apply simp
- apply (rule le_less_trans [OF *])
- using \<open>L>0\<close> e
- apply (force simp: field_simps)
- done
- then have "cmod (winding_number p w - winding_number p z) < e"
- using pi_ge_two e
- by (force simp: winding_number_valid_path p field_simps norm_divide norm_mult intro: less_le_trans)
- } note cmod_wn_diff = this
- then have "isCont (winding_number p) z"
- apply (simp add: continuous_at_eps_delta, clarify)
- apply (rule_tac x="min (pe/4) (e/2*pe^2/L/4)" in exI)
- using \<open>pe>0\<close> \<open>L>0\<close>
- apply (simp add: dist_norm cmod_wn_diff)
- done
- then show ?thesis
- apply (rule continuous_transform_within [where d = "min d e / 2"])
- apply (auto simp: \<open>d>0\<close> \<open>e>0\<close> dist_norm wnwn)
- done
-qed
-
-corollary continuous_on_winding_number:
- "path \<gamma> \<Longrightarrow> continuous_on (- path_image \<gamma>) (\<lambda>w. winding_number \<gamma> w)"
- by (simp add: continuous_at_imp_continuous_on continuous_at_winding_number)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number is constant on a connected region\<close>
-
-lemma winding_number_constant:
- assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and cs: "connected S" and sg: "S \<inter> path_image \<gamma> = {}"
- shows "winding_number \<gamma> constant_on S"
-proof -
- have *: "1 \<le> cmod (winding_number \<gamma> y - winding_number \<gamma> z)"
- if ne: "winding_number \<gamma> y \<noteq> winding_number \<gamma> z" and "y \<in> S" "z \<in> S" for y z
- proof -
- have "winding_number \<gamma> y \<in> \<int>" "winding_number \<gamma> z \<in> \<int>"
- using that integer_winding_number [OF \<gamma> loop] sg \<open>y \<in> S\<close> by auto
- with ne show ?thesis
- by (auto simp: Ints_def simp flip: of_int_diff)
- qed
- have cont: "continuous_on S (\<lambda>w. winding_number \<gamma> w)"
- using continuous_on_winding_number [OF \<gamma>] sg
- by (meson continuous_on_subset disjoint_eq_subset_Compl)
- show ?thesis
- using "*" zero_less_one
- by (blast intro: continuous_discrete_range_constant [OF cs cont])
-qed
-
-lemma winding_number_eq:
- "\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; w \<in> S; z \<in> S; connected S; S \<inter> path_image \<gamma> = {}\<rbrakk>
- \<Longrightarrow> winding_number \<gamma> w = winding_number \<gamma> z"
- using winding_number_constant by (metis constant_on_def)
-
-lemma open_winding_number_levelsets:
- assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- shows "open {z. z \<notin> path_image \<gamma> \<and> winding_number \<gamma> z = k}"
-proof -
- have opn: "open (- path_image \<gamma>)"
- by (simp add: closed_path_image \<gamma> open_Compl)
- { fix z assume z: "z \<notin> path_image \<gamma>" and k: "k = winding_number \<gamma> z"
- obtain e where e: "e>0" "ball z e \<subseteq> - path_image \<gamma>"
- using open_contains_ball [of "- path_image \<gamma>"] opn z
- by blast
- have "\<exists>e>0. \<forall>y. dist y z < e \<longrightarrow> y \<notin> path_image \<gamma> \<and> winding_number \<gamma> y = winding_number \<gamma> z"
- apply (rule_tac x=e in exI)
- using e apply (simp add: dist_norm ball_def norm_minus_commute)
- apply (auto simp: dist_norm norm_minus_commute intro!: winding_number_eq [OF assms, where S = "ball z e"])
- done
- } then
- show ?thesis
- by (auto simp: open_dist)
-qed
-
-subsection\<open>Winding number is zero "outside" a curve\<close>
-
-proposition winding_number_zero_in_outside:
- assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and z: "z \<in> outside (path_image \<gamma>)"
- shows "winding_number \<gamma> z = 0"
-proof -
- obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
- using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
- obtain w::complex where w: "w \<notin> ball 0 (B + 1)"
- by (metis abs_of_nonneg le_less less_irrefl mem_ball_0 norm_of_real)
- have "- ball 0 (B + 1) \<subseteq> outside (path_image \<gamma>)"
- apply (rule outside_subset_convex)
- using B subset_ball by auto
- then have wout: "w \<in> outside (path_image \<gamma>)"
- using w by blast
- moreover have "winding_number \<gamma> constant_on outside (path_image \<gamma>)"
- using winding_number_constant [OF \<gamma> loop, of "outside(path_image \<gamma>)"] connected_outside
- by (metis DIM_complex bounded_path_image dual_order.refl \<gamma> outside_no_overlap)
- ultimately have "winding_number \<gamma> z = winding_number \<gamma> w"
- by (metis (no_types, hide_lams) constant_on_def z)
- also have "\<dots> = 0"
- proof -
- have wnot: "w \<notin> path_image \<gamma>" using wout by (simp add: outside_def)
- { fix e::real assume "0<e"
- obtain p where p: "polynomial_function p" "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
- and pg1: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < 1)"
- and pge: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < e)"
- using path_approx_polynomial_function [OF \<gamma>, of "min 1 e"] \<open>e>0\<close> by force
- have pip: "path_image p \<subseteq> ball 0 (B + 1)"
- using B
- apply (clarsimp simp add: path_image_def dist_norm ball_def)
- apply (frule (1) pg1)
- apply (fastforce dest: norm_add_less)
- done
- then have "w \<notin> path_image p" using w by blast
- then have "\<exists>p. valid_path p \<and> w \<notin> path_image p \<and>
- pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and>
- (\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0"
- apply (rule_tac x=p in exI)
- apply (simp add: p valid_path_polynomial_function)
- apply (intro conjI)
- using pge apply (simp add: norm_minus_commute)
- apply (rule contour_integral_unique [OF Cauchy_theorem_convex_simple [OF _ convex_ball [of 0 "B+1"]]])
- apply (rule holomorphic_intros | simp add: dist_norm)+
- using mem_ball_0 w apply blast
- using p apply (simp_all add: valid_path_polynomial_function loop pip)
- done
- }
- then show ?thesis
- by (auto intro: winding_number_unique [OF \<gamma>] simp add: winding_number_prop_def wnot)
- qed
- finally show ?thesis .
-qed
-
-corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_const: "a \<noteq> z \<Longrightarrow> winding_number (\<lambda>t. a) z = 0"
- by (rule winding_number_zero_in_outside)
- (auto simp: pathfinish_def pathstart_def path_polynomial_function)
-
-corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_outside:
- "\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> s; path_image \<gamma> \<subseteq> s\<rbrakk> \<Longrightarrow> winding_number \<gamma> z = 0"
- by (meson convex_in_outside outside_mono subsetCE winding_number_zero_in_outside)
-
-lemma winding_number_zero_at_infinity:
- assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- shows "\<exists>B. \<forall>z. B \<le> norm z \<longrightarrow> winding_number \<gamma> z = 0"
-proof -
- obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
- using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
- then show ?thesis
- apply (rule_tac x="B+1" in exI, clarify)
- apply (rule winding_number_zero_outside [OF \<gamma> convex_cball [of 0 B] loop])
- apply (meson less_add_one mem_cball_0 not_le order_trans)
- using ball_subset_cball by blast
-qed
-
-lemma winding_number_zero_point:
- "\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; open s; path_image \<gamma> \<subseteq> s\<rbrakk>
- \<Longrightarrow> \<exists>z. z \<in> s \<and> winding_number \<gamma> z = 0"
- using outside_compact_in_open [of "path_image \<gamma>" s] path_image_nonempty winding_number_zero_in_outside
- by (fastforce simp add: compact_path_image)
-
-
-text\<open>If a path winds round a set, it winds rounds its inside.\<close>
-lemma winding_number_around_inside:
- assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- and cls: "closed s" and cos: "connected s" and s_disj: "s \<inter> path_image \<gamma> = {}"
- and z: "z \<in> s" and wn_nz: "winding_number \<gamma> z \<noteq> 0" and w: "w \<in> s \<union> inside s"
- shows "winding_number \<gamma> w = winding_number \<gamma> z"
-proof -
- have ssb: "s \<subseteq> inside(path_image \<gamma>)"
- proof
- fix x :: complex
- assume "x \<in> s"
- hence "x \<notin> path_image \<gamma>"
- by (meson disjoint_iff_not_equal s_disj)
- thus "x \<in> inside (path_image \<gamma>)"
- using \<open>x \<in> s\<close> by (metis (no_types) ComplI UnE cos \<gamma> loop s_disj union_with_outside winding_number_eq winding_number_zero_in_outside wn_nz z)
-qed
- show ?thesis
- apply (rule winding_number_eq [OF \<gamma> loop w])
- using z apply blast
- apply (simp add: cls connected_with_inside cos)
- apply (simp add: Int_Un_distrib2 s_disj, safe)
- by (meson ssb inside_inside_compact_connected [OF cls, of "path_image \<gamma>"] compact_path_image connected_path_image contra_subsetD disjoint_iff_not_equal \<gamma> inside_no_overlap)
- qed
-
-
-text\<open>Bounding a WN by 1/2 for a path and point in opposite halfspaces.\<close>
-lemma winding_number_subpath_continuous:
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
- shows "continuous_on {0..1} (\<lambda>x. winding_number(subpath 0 x \<gamma>) z)"
-proof -
- have *: "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
- winding_number (subpath 0 x \<gamma>) z"
- if x: "0 \<le> x" "x \<le> 1" for x
- proof -
- have "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
- 1 / (2*pi*\<i>) * contour_integral (subpath 0 x \<gamma>) (\<lambda>w. 1/(w - z))"
- using assms x
- apply (simp add: contour_integral_subcontour_integral [OF contour_integrable_inversediff])
- done
- also have "\<dots> = winding_number (subpath 0 x \<gamma>) z"
- apply (subst winding_number_valid_path)
- using assms x
- apply (simp_all add: path_image_subpath valid_path_subpath)
- by (force simp: path_image_def)
- finally show ?thesis .
- qed
- show ?thesis
- apply (rule continuous_on_eq
- [where f = "\<lambda>x. 1 / (2*pi*\<i>) *
- integral {0..x} (\<lambda>t. 1/(\<gamma> t - z) * vector_derivative \<gamma> (at t))"])
- apply (rule continuous_intros)+
- apply (rule indefinite_integral_continuous_1)
- apply (rule contour_integrable_inversediff [OF assms, unfolded contour_integrable_on])
- using assms
- apply (simp add: *)
- done
-qed
-
-lemma winding_number_ivt_pos:
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> Re(winding_number \<gamma> z)"
- shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
- apply (rule ivt_increasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right], simp)
- apply (rule winding_number_subpath_continuous [OF \<gamma> z])
- using assms
- apply (auto simp: path_image_def image_def)
- done
-
-lemma winding_number_ivt_neg:
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "Re(winding_number \<gamma> z) \<le> w" "w \<le> 0"
- shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
- apply (rule ivt_decreasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right], simp)
- apply (rule winding_number_subpath_continuous [OF \<gamma> z])
- using assms
- apply (auto simp: path_image_def image_def)
- done
-
-lemma winding_number_ivt_abs:
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> \<bar>Re(winding_number \<gamma> z)\<bar>"
- shows "\<exists>t \<in> {0..1}. \<bar>Re (winding_number (subpath 0 t \<gamma>) z)\<bar> = w"
- using assms winding_number_ivt_pos [of \<gamma> z w] winding_number_ivt_neg [of \<gamma> z "-w"]
- by force
-
-lemma winding_number_lt_half_lemma:
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and az: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
- shows "Re(winding_number \<gamma> z) < 1/2"
-proof -
- { assume "Re(winding_number \<gamma> z) \<ge> 1/2"
- then obtain t::real where t: "0 \<le> t" "t \<le> 1" and sub12: "Re (winding_number (subpath 0 t \<gamma>) z) = 1/2"
- using winding_number_ivt_pos [OF \<gamma> z, of "1/2"] by auto
- have gt: "\<gamma> t - z = - (of_real (exp (- (2 * pi * Im (winding_number (subpath 0 t \<gamma>) z)))) * (\<gamma> 0 - z))"
- using winding_number_exp_2pi [of "subpath 0 t \<gamma>" z]
- apply (simp add: t \<gamma> valid_path_imp_path)
- using closed_segment_eq_real_ivl path_image_def t z by (fastforce simp: path_image_subpath Euler sub12)
- have "b < a \<bullet> \<gamma> 0"
- proof -
- have "\<gamma> 0 \<in> {c. b < a \<bullet> c}"
- by (metis (no_types) pag atLeastAtMost_iff image_subset_iff order_refl path_image_def zero_le_one)
- thus ?thesis
- by blast
- qed
- moreover have "b < a \<bullet> \<gamma> t"
- proof -
- have "\<gamma> t \<in> {c. b < a \<bullet> c}"
- by (metis (no_types) pag atLeastAtMost_iff image_subset_iff path_image_def t)
- thus ?thesis
- by blast
- qed
- ultimately have "0 < a \<bullet> (\<gamma> 0 - z)" "0 < a \<bullet> (\<gamma> t - z)" using az
- by (simp add: inner_diff_right)+
- then have False
- by (simp add: gt inner_mult_right mult_less_0_iff)
- }
- then show ?thesis by force
-qed
-
-lemma winding_number_lt_half:
- assumes "valid_path \<gamma>" "a \<bullet> z \<le> b" "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
- shows "\<bar>Re (winding_number \<gamma> z)\<bar> < 1/2"
-proof -
- have "z \<notin> path_image \<gamma>" using assms by auto
- with assms show ?thesis
- apply (simp add: winding_number_lt_half_lemma abs_if del: less_divide_eq_numeral1)
- apply (metis complex_inner_1_right winding_number_lt_half_lemma [OF valid_path_imp_reverse, of \<gamma> z a b]
- winding_number_reversepath valid_path_imp_path inner_minus_left path_image_reversepath)
- done
-qed
-
-lemma winding_number_le_half:
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
- and anz: "a \<noteq> 0" and azb: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w \<ge> b}"
- shows "\<bar>Re (winding_number \<gamma> z)\<bar> \<le> 1/2"
-proof -
- { assume wnz_12: "\<bar>Re (winding_number \<gamma> z)\<bar> > 1/2"
- have "isCont (winding_number \<gamma>) z"
- by (metis continuous_at_winding_number valid_path_imp_path \<gamma> z)
- then obtain d where "d>0" and d: "\<And>x'. dist x' z < d \<Longrightarrow> dist (winding_number \<gamma> x') (winding_number \<gamma> z) < \<bar>Re(winding_number \<gamma> z)\<bar> - 1/2"
- using continuous_at_eps_delta wnz_12 diff_gt_0_iff_gt by blast
- define z' where "z' = z - (d / (2 * cmod a)) *\<^sub>R a"
- have *: "a \<bullet> z' \<le> b - d / 3 * cmod a"
- unfolding z'_def inner_mult_right' divide_inverse
- apply (simp add: field_split_simps algebra_simps dot_square_norm power2_eq_square anz)
- apply (metis \<open>0 < d\<close> add_increasing azb less_eq_real_def mult_nonneg_nonneg mult_right_mono norm_ge_zero norm_numeral)
- done
- have "cmod (winding_number \<gamma> z' - winding_number \<gamma> z) < \<bar>Re (winding_number \<gamma> z)\<bar> - 1/2"
- using d [of z'] anz \<open>d>0\<close> by (simp add: dist_norm z'_def)
- then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - cmod (winding_number \<gamma> z' - winding_number \<gamma> z)"
- by simp
- then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - \<bar>Re (winding_number \<gamma> z') - Re (winding_number \<gamma> z)\<bar>"
- using abs_Re_le_cmod [of "winding_number \<gamma> z' - winding_number \<gamma> z"] by simp
- then have wnz_12': "\<bar>Re (winding_number \<gamma> z')\<bar> > 1/2"
- by linarith
- moreover have "\<bar>Re (winding_number \<gamma> z')\<bar> < 1/2"
- apply (rule winding_number_lt_half [OF \<gamma> *])
- using azb \<open>d>0\<close> pag
- apply (auto simp: add_strict_increasing anz field_split_simps dest!: subsetD)
- done
- ultimately have False
- by simp
- }
- then show ?thesis by force
-qed
-
-lemma winding_number_lt_half_linepath: "z \<notin> closed_segment a b \<Longrightarrow> \<bar>Re (winding_number (linepath a b) z)\<bar> < 1/2"
- using separating_hyperplane_closed_point [of "closed_segment a b" z]
- apply auto
- apply (simp add: closed_segment_def)
- apply (drule less_imp_le)
- apply (frule winding_number_lt_half [OF valid_path_linepath [of a b]])
- apply (auto simp: segment)
- done
-
-
-text\<open> Positivity of WN for a linepath.\<close>
-lemma winding_number_linepath_pos_lt:
- assumes "0 < Im ((b - a) * cnj (b - z))"
- shows "0 < Re(winding_number(linepath a b) z)"
-proof -
- have z: "z \<notin> path_image (linepath a b)"
- using assms
- by (simp add: closed_segment_def) (force simp: algebra_simps)
- show ?thesis
- apply (rule winding_number_pos_lt [OF valid_path_linepath z assms])
- apply (simp add: linepath_def algebra_simps)
- done
-qed
-
-
-subsection\<open>Cauchy's integral formula, again for a convex enclosing set\<close>
-
-lemma Cauchy_integral_formula_weak:
- assumes s: "convex s" and "finite k" and conf: "continuous_on s f"
- and fcd: "(\<And>x. x \<in> interior s - k \<Longrightarrow> f field_differentiable at x)"
- and z: "z \<in> interior s - k" and vpg: "valid_path \<gamma>"
- and pasz: "path_image \<gamma> \<subseteq> s - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
-proof -
- obtain f' where f': "(f has_field_derivative f') (at z)"
- using fcd [OF z] by (auto simp: field_differentiable_def)
- have pas: "path_image \<gamma> \<subseteq> s" and znotin: "z \<notin> path_image \<gamma>" using pasz by blast+
- have c: "continuous (at x within s) (\<lambda>w. if w = z then f' else (f w - f z) / (w - z))" if "x \<in> s" for x
- proof (cases "x = z")
- case True then show ?thesis
- apply (simp add: continuous_within)
- apply (rule Lim_transform_away_within [of _ "z+1" _ "\<lambda>w::complex. (f w - f z)/(w - z)"])
- using has_field_derivative_at_within has_field_derivative_iff f'
- apply (fastforce simp add:)+
- done
- next
- case False
- then have dxz: "dist x z > 0" by auto
- have cf: "continuous (at x within s) f"
- using conf continuous_on_eq_continuous_within that by blast
- have "continuous (at x within s) (\<lambda>w. (f w - f z) / (w - z))"
- by (rule cf continuous_intros | simp add: False)+
- then show ?thesis
- apply (rule continuous_transform_within [OF _ dxz that, of "\<lambda>w::complex. (f w - f z)/(w - z)"])
- apply (force simp: dist_commute)
- done
- qed
- have fink': "finite (insert z k)" using \<open>finite k\<close> by blast
- have *: "((\<lambda>w. if w = z then f' else (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
- apply (rule Cauchy_theorem_convex [OF _ s fink' _ vpg pas loop])
- using c apply (force simp: continuous_on_eq_continuous_within)
- apply (rename_tac w)
- apply (rule_tac d="dist w z" and f = "\<lambda>w. (f w - f z)/(w - z)" in field_differentiable_transform_within)
- apply (simp_all add: dist_pos_lt dist_commute)
- apply (metis less_irrefl)
- apply (rule derivative_intros fcd | simp)+
- done
- show ?thesis
- apply (rule has_contour_integral_eq)
- using znotin has_contour_integral_add [OF has_contour_integral_lmul [OF has_contour_integral_winding_number [OF vpg znotin], of "f z"] *]
- apply (auto simp: ac_simps divide_simps)
- done
-qed
-
-theorem Cauchy_integral_formula_convex_simple:
- "\<lbrakk>convex s; f holomorphic_on s; z \<in> interior s; valid_path \<gamma>; path_image \<gamma> \<subseteq> s - {z};
- pathfinish \<gamma> = pathstart \<gamma>\<rbrakk>
- \<Longrightarrow> ((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
- apply (rule Cauchy_integral_formula_weak [where k = "{}"])
- using holomorphic_on_imp_continuous_on
- by auto (metis at_within_interior holomorphic_on_def interiorE subsetCE)
subsection\<open>Homotopy forms of Cauchy's theorem\<close>
@@ -3870,3290 +1593,4 @@
apply (blast dest: holomorphic_on_imp_continuous_on homotopic_loops_imp_subset)
by (simp add: Cauchy_theorem_homotopic_loops)
-subsection\<^marker>\<open>tag unimportant\<close> \<open>More winding number properties\<close>
-
-text\<open>including the fact that it's +-1 inside a simple closed curve.\<close>
-
-lemma winding_number_homotopic_paths:
- assumes "homotopic_paths (-{z}) g h"
- shows "winding_number g z = winding_number h z"
-proof -
- have "path g" "path h" using homotopic_paths_imp_path [OF assms] by auto
- moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
- using homotopic_paths_imp_subset [OF assms] by auto
- ultimately obtain d e where "d > 0" "e > 0"
- and d: "\<And>p. \<lbrakk>path p; pathstart p = pathstart g; pathfinish p = pathfinish g; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
- \<Longrightarrow> homotopic_paths (-{z}) g p"
- and e: "\<And>q. \<lbrakk>path q; pathstart q = pathstart h; pathfinish q = pathfinish h; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
- \<Longrightarrow> homotopic_paths (-{z}) h q"
- using homotopic_nearby_paths [of g "-{z}"] homotopic_nearby_paths [of h "-{z}"] by force
- obtain p where p:
- "valid_path p" "z \<notin> path_image p"
- "pathstart p = pathstart g" "pathfinish p = pathfinish g"
- and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
- and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
- using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast
- obtain q where q:
- "valid_path q" "z \<notin> path_image q"
- "pathstart q = pathstart h" "pathfinish q = pathfinish h"
- and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
- and paq: "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
- using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast
- have "homotopic_paths (- {z}) g p"
- by (simp add: d p valid_path_imp_path norm_minus_commute gp_less)
- moreover have "homotopic_paths (- {z}) h q"
- by (simp add: e q valid_path_imp_path norm_minus_commute hq_less)
- ultimately have "homotopic_paths (- {z}) p q"
- by (blast intro: homotopic_paths_trans homotopic_paths_sym assms)
- then have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
- by (rule Cauchy_theorem_homotopic_paths) (auto intro!: holomorphic_intros simp: p q)
- then show ?thesis
- by (simp add: pap paq)
-qed
-
-lemma winding_number_homotopic_loops:
- assumes "homotopic_loops (-{z}) g h"
- shows "winding_number g z = winding_number h z"
-proof -
- have "path g" "path h" using homotopic_loops_imp_path [OF assms] by auto
- moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
- using homotopic_loops_imp_subset [OF assms] by auto
- moreover have gloop: "pathfinish g = pathstart g" and hloop: "pathfinish h = pathstart h"
- using homotopic_loops_imp_loop [OF assms] by auto
- ultimately obtain d e where "d > 0" "e > 0"
- and d: "\<And>p. \<lbrakk>path p; pathfinish p = pathstart p; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
- \<Longrightarrow> homotopic_loops (-{z}) g p"
- and e: "\<And>q. \<lbrakk>path q; pathfinish q = pathstart q; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
- \<Longrightarrow> homotopic_loops (-{z}) h q"
- using homotopic_nearby_loops [of g "-{z}"] homotopic_nearby_loops [of h "-{z}"] by force
- obtain p where p:
- "valid_path p" "z \<notin> path_image p"
- "pathstart p = pathstart g" "pathfinish p = pathfinish g"
- and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
- and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
- using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast
- obtain q where q:
- "valid_path q" "z \<notin> path_image q"
- "pathstart q = pathstart h" "pathfinish q = pathfinish h"
- and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
- and paq: "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
- using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast
- have gp: "homotopic_loops (- {z}) g p"
- by (simp add: gloop d gp_less norm_minus_commute p valid_path_imp_path)
- have hq: "homotopic_loops (- {z}) h q"
- by (simp add: e hloop hq_less norm_minus_commute q valid_path_imp_path)
- have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
- proof (rule Cauchy_theorem_homotopic_loops)
- show "homotopic_loops (- {z}) p q"
- by (blast intro: homotopic_loops_trans homotopic_loops_sym gp hq assms)
- qed (auto intro!: holomorphic_intros simp: p q)
- then show ?thesis
- by (simp add: pap paq)
-qed
-
-lemma winding_number_paths_linear_eq:
- "\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g;
- \<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
- \<Longrightarrow> winding_number h z = winding_number g z"
- by (blast intro: sym homotopic_paths_linear winding_number_homotopic_paths)
-
-lemma winding_number_loops_linear_eq:
- "\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h;
- \<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
- \<Longrightarrow> winding_number h z = winding_number g z"
- by (blast intro: sym homotopic_loops_linear winding_number_homotopic_loops)
-
-lemma winding_number_nearby_paths_eq:
- "\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g;
- \<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
- \<Longrightarrow> winding_number h z = winding_number g z"
- by (metis segment_bound(2) norm_minus_commute not_le winding_number_paths_linear_eq)
-
-lemma winding_number_nearby_loops_eq:
- "\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h;
- \<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
- \<Longrightarrow> winding_number h z = winding_number g z"
- by (metis segment_bound(2) norm_minus_commute not_le winding_number_loops_linear_eq)
-
-
-lemma winding_number_subpath_combine:
- "\<lbrakk>path g; z \<notin> path_image g;
- u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
- \<Longrightarrow> winding_number (subpath u v g) z + winding_number (subpath v w g) z =
- winding_number (subpath u w g) z"
-apply (rule trans [OF winding_number_join [THEN sym]
- winding_number_homotopic_paths [OF homotopic_join_subpaths]])
- using path_image_subpath_subset by auto
-
-subsection\<open>Partial circle path\<close>
-
-definition\<^marker>\<open>tag important\<close> part_circlepath :: "[complex, real, real, real, real] \<Rightarrow> complex"
- where "part_circlepath z r s t \<equiv> \<lambda>x. z + of_real r * exp (\<i> * of_real (linepath s t x))"
-
-lemma pathstart_part_circlepath [simp]:
- "pathstart(part_circlepath z r s t) = z + r*exp(\<i> * s)"
-by (metis part_circlepath_def pathstart_def pathstart_linepath)
-
-lemma pathfinish_part_circlepath [simp]:
- "pathfinish(part_circlepath z r s t) = z + r*exp(\<i>*t)"
-by (metis part_circlepath_def pathfinish_def pathfinish_linepath)
-
-lemma reversepath_part_circlepath[simp]:
- "reversepath (part_circlepath z r s t) = part_circlepath z r t s"
- unfolding part_circlepath_def reversepath_def linepath_def
- by (auto simp:algebra_simps)
-
-lemma has_vector_derivative_part_circlepath [derivative_intros]:
- "((part_circlepath z r s t) has_vector_derivative
- (\<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)))
- (at x within X)"
- apply (simp add: part_circlepath_def linepath_def scaleR_conv_of_real)
- apply (rule has_vector_derivative_real_field)
- apply (rule derivative_eq_intros | simp)+
- done
-
-lemma differentiable_part_circlepath:
- "part_circlepath c r a b differentiable at x within A"
- using has_vector_derivative_part_circlepath[of c r a b x A] differentiableI_vector by blast
-
-lemma vector_derivative_part_circlepath:
- "vector_derivative (part_circlepath z r s t) (at x) =
- \<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)"
- using has_vector_derivative_part_circlepath vector_derivative_at by blast
-
-lemma vector_derivative_part_circlepath01:
- "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
- \<Longrightarrow> vector_derivative (part_circlepath z r s t) (at x within {0..1}) =
- \<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)"
- using has_vector_derivative_part_circlepath
- by (auto simp: vector_derivative_at_within_ivl)
-
-lemma valid_path_part_circlepath [simp]: "valid_path (part_circlepath z r s t)"
- apply (simp add: valid_path_def)
- apply (rule C1_differentiable_imp_piecewise)
- apply (auto simp: C1_differentiable_on_eq vector_derivative_works vector_derivative_part_circlepath has_vector_derivative_part_circlepath
- intro!: continuous_intros)
- done
-
-lemma path_part_circlepath [simp]: "path (part_circlepath z r s t)"
- by (simp add: valid_path_imp_path)
-
-proposition path_image_part_circlepath:
- assumes "s \<le> t"
- shows "path_image (part_circlepath z r s t) = {z + r * exp(\<i> * of_real x) | x. s \<le> x \<and> x \<le> t}"
-proof -
- { fix z::real
- assume "0 \<le> z" "z \<le> 1"
- with \<open>s \<le> t\<close> have "\<exists>x. (exp (\<i> * linepath s t z) = exp (\<i> * of_real x)) \<and> s \<le> x \<and> x \<le> t"
- apply (rule_tac x="(1 - z) * s + z * t" in exI)
- apply (simp add: linepath_def scaleR_conv_of_real algebra_simps)
- apply (rule conjI)
- using mult_right_mono apply blast
- using affine_ineq by (metis "mult.commute")
- }
- moreover
- { fix z
- assume "s \<le> z" "z \<le> t"
- then have "z + of_real r * exp (\<i> * of_real z) \<in> (\<lambda>x. z + of_real r * exp (\<i> * linepath s t x)) ` {0..1}"
- apply (rule_tac x="(z - s)/(t - s)" in image_eqI)
- apply (simp add: linepath_def scaleR_conv_of_real divide_simps exp_eq)
- apply (auto simp: field_split_simps)
- done
- }
- ultimately show ?thesis
- by (fastforce simp add: path_image_def part_circlepath_def)
-qed
-
-lemma path_image_part_circlepath':
- "path_image (part_circlepath z r s t) = (\<lambda>x. z + r * cis x) ` closed_segment s t"
-proof -
- have "path_image (part_circlepath z r s t) =
- (\<lambda>x. z + r * exp(\<i> * of_real x)) ` linepath s t ` {0..1}"
- by (simp add: image_image path_image_def part_circlepath_def)
- also have "linepath s t ` {0..1} = closed_segment s t"
- by (rule linepath_image_01)
- finally show ?thesis by (simp add: cis_conv_exp)
-qed
-
-lemma path_image_part_circlepath_subset:
- "\<lbrakk>s \<le> t; 0 \<le> r\<rbrakk> \<Longrightarrow> path_image(part_circlepath z r s t) \<subseteq> sphere z r"
-by (auto simp: path_image_part_circlepath sphere_def dist_norm algebra_simps norm_mult)
-
-lemma in_path_image_part_circlepath:
- assumes "w \<in> path_image(part_circlepath z r s t)" "s \<le> t" "0 \<le> r"
- shows "norm(w - z) = r"
-proof -
- have "w \<in> {c. dist z c = r}"
- by (metis (no_types) path_image_part_circlepath_subset sphere_def subset_eq assms)
- thus ?thesis
- by (simp add: dist_norm norm_minus_commute)
-qed
-
-lemma path_image_part_circlepath_subset':
- assumes "r \<ge> 0"
- shows "path_image (part_circlepath z r s t) \<subseteq> sphere z r"
-proof (cases "s \<le> t")
- case True
- thus ?thesis using path_image_part_circlepath_subset[of s t r z] assms by simp
-next
- case False
- thus ?thesis using path_image_part_circlepath_subset[of t s r z] assms
- by (subst reversepath_part_circlepath [symmetric], subst path_image_reversepath) simp_all
-qed
-
-lemma part_circlepath_cnj: "cnj (part_circlepath c r a b x) = part_circlepath (cnj c) r (-a) (-b) x"
- by (simp add: part_circlepath_def exp_cnj linepath_def algebra_simps)
-
-lemma contour_integral_bound_part_circlepath:
- assumes "f contour_integrable_on part_circlepath c r a b"
- assumes "B \<ge> 0" "r \<ge> 0" "\<And>x. x \<in> path_image (part_circlepath c r a b) \<Longrightarrow> norm (f x) \<le> B"
- shows "norm (contour_integral (part_circlepath c r a b) f) \<le> B * r * \<bar>b - a\<bar>"
-proof -
- let ?I = "integral {0..1} (\<lambda>x. f (part_circlepath c r a b x) * \<i> * of_real (r * (b - a)) *
- exp (\<i> * linepath a b x))"
- have "norm ?I \<le> integral {0..1} (\<lambda>x::real. B * 1 * (r * \<bar>b - a\<bar>) * 1)"
- proof (rule integral_norm_bound_integral, goal_cases)
- case 1
- with assms(1) show ?case
- by (simp add: contour_integrable_on vector_derivative_part_circlepath mult_ac)
- next
- case (3 x)
- with assms(2-) show ?case unfolding norm_mult norm_of_real abs_mult
- by (intro mult_mono) (auto simp: path_image_def)
- qed auto
- also have "?I = contour_integral (part_circlepath c r a b) f"
- by (simp add: contour_integral_integral vector_derivative_part_circlepath mult_ac)
- finally show ?thesis by simp
-qed
-
-lemma has_contour_integral_part_circlepath_iff:
- assumes "a < b"
- shows "(f has_contour_integral I) (part_circlepath c r a b) \<longleftrightarrow>
- ((\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) has_integral I) {a..b}"
-proof -
- have "(f has_contour_integral I) (part_circlepath c r a b) \<longleftrightarrow>
- ((\<lambda>x. f (part_circlepath c r a b x) * vector_derivative (part_circlepath c r a b)
- (at x within {0..1})) has_integral I) {0..1}"
- unfolding has_contour_integral_def ..
- also have "\<dots> \<longleftrightarrow> ((\<lambda>x. f (part_circlepath c r a b x) * r * (b - a) * \<i> *
- cis (linepath a b x)) has_integral I) {0..1}"
- by (intro has_integral_cong, subst vector_derivative_part_circlepath01)
- (simp_all add: cis_conv_exp)
- also have "\<dots> \<longleftrightarrow> ((\<lambda>x. f (c + r * exp (\<i> * linepath (of_real a) (of_real b) x)) *
- r * \<i> * exp (\<i> * linepath (of_real a) (of_real b) x) *
- vector_derivative (linepath (of_real a) (of_real b))
- (at x within {0..1})) has_integral I) {0..1}"
- by (intro has_integral_cong, subst vector_derivative_linepath_within)
- (auto simp: part_circlepath_def cis_conv_exp of_real_linepath [symmetric])
- also have "\<dots> \<longleftrightarrow> ((\<lambda>z. f (c + r * exp (\<i> * z)) * r * \<i> * exp (\<i> * z)) has_contour_integral I)
- (linepath (of_real a) (of_real b))"
- by (simp add: has_contour_integral_def)
- also have "\<dots> \<longleftrightarrow> ((\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) has_integral I) {a..b}" using assms
- by (subst has_contour_integral_linepath_Reals_iff) (simp_all add: cis_conv_exp)
- finally show ?thesis .
-qed
-
-lemma contour_integrable_part_circlepath_iff:
- assumes "a < b"
- shows "f contour_integrable_on (part_circlepath c r a b) \<longleftrightarrow>
- (\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}"
- using assms by (auto simp: contour_integrable_on_def integrable_on_def
- has_contour_integral_part_circlepath_iff)
-
-lemma contour_integral_part_circlepath_eq:
- assumes "a < b"
- shows "contour_integral (part_circlepath c r a b) f =
- integral {a..b} (\<lambda>t. f (c + r * cis t) * r * \<i> * cis t)"
-proof (cases "f contour_integrable_on part_circlepath c r a b")
- case True
- hence "(\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}"
- using assms by (simp add: contour_integrable_part_circlepath_iff)
- with True show ?thesis
- using has_contour_integral_part_circlepath_iff[OF assms]
- contour_integral_unique has_integral_integrable_integral by blast
-next
- case False
- hence "\<not>(\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}"
- using assms by (simp add: contour_integrable_part_circlepath_iff)
- with False show ?thesis
- by (simp add: not_integrable_contour_integral not_integrable_integral)
-qed
-
-lemma contour_integral_part_circlepath_reverse:
- "contour_integral (part_circlepath c r a b) f = -contour_integral (part_circlepath c r b a) f"
- by (subst reversepath_part_circlepath [symmetric], subst contour_integral_reversepath) simp_all
-
-lemma contour_integral_part_circlepath_reverse':
- "b < a \<Longrightarrow> contour_integral (part_circlepath c r a b) f =
- -contour_integral (part_circlepath c r b a) f"
- by (rule contour_integral_part_circlepath_reverse)
-
-lemma finite_bounded_log: "finite {z::complex. norm z \<le> b \<and> exp z = w}"
-proof (cases "w = 0")
- case True then show ?thesis by auto
-next
- case False
- have *: "finite {x. cmod (complex_of_real (2 * real_of_int x * pi) * \<i>) \<le> b + cmod (Ln w)}"
- apply (simp add: norm_mult finite_int_iff_bounded_le)
- apply (rule_tac x="\<lfloor>(b + cmod (Ln w)) / (2*pi)\<rfloor>" in exI)
- apply (auto simp: field_split_simps le_floor_iff)
- done
- have [simp]: "\<And>P f. {z. P z \<and> (\<exists>n. z = f n)} = f ` {n. P (f n)}"
- by blast
- show ?thesis
- apply (subst exp_Ln [OF False, symmetric])
- apply (simp add: exp_eq)
- using norm_add_leD apply (fastforce intro: finite_subset [OF _ *])
- done
-qed
-
-lemma finite_bounded_log2:
- fixes a::complex
- assumes "a \<noteq> 0"
- shows "finite {z. norm z \<le> b \<and> exp(a*z) = w}"
-proof -
- have *: "finite ((\<lambda>z. z / a) ` {z. cmod z \<le> b * cmod a \<and> exp z = w})"
- by (rule finite_imageI [OF finite_bounded_log])
- show ?thesis
- by (rule finite_subset [OF _ *]) (force simp: assms norm_mult)
-qed
-
-lemma has_contour_integral_bound_part_circlepath_strong:
- assumes fi: "(f has_contour_integral i) (part_circlepath z r s t)"
- and "finite k" and le: "0 \<le> B" "0 < r" "s \<le> t"
- and B: "\<And>x. x \<in> path_image(part_circlepath z r s t) - k \<Longrightarrow> norm(f x) \<le> B"
- shows "cmod i \<le> B * r * (t - s)"
-proof -
- consider "s = t" | "s < t" using \<open>s \<le> t\<close> by linarith
- then show ?thesis
- proof cases
- case 1 with fi [unfolded has_contour_integral]
- have "i = 0" by (simp add: vector_derivative_part_circlepath)
- with assms show ?thesis by simp
- next
- case 2
- have [simp]: "\<bar>r\<bar> = r" using \<open>r > 0\<close> by linarith
- have [simp]: "cmod (complex_of_real t - complex_of_real s) = t-s"
- by (metis "2" abs_of_pos diff_gt_0_iff_gt norm_of_real of_real_diff)
- have "finite (part_circlepath z r s t -` {y} \<inter> {0..1})" if "y \<in> k" for y
- proof -
- define w where "w = (y - z)/of_real r / exp(\<i> * of_real s)"
- have fin: "finite (of_real -` {z. cmod z \<le> 1 \<and> exp (\<i> * complex_of_real (t - s) * z) = w})"
- apply (rule finite_vimageI [OF finite_bounded_log2])
- using \<open>s < t\<close> apply (auto simp: inj_of_real)
- done
- show ?thesis
- apply (simp add: part_circlepath_def linepath_def vimage_def)
- apply (rule finite_subset [OF _ fin])
- using le
- apply (auto simp: w_def algebra_simps scaleR_conv_of_real exp_add exp_diff)
- done
- qed
- then have fin01: "finite ((part_circlepath z r s t) -` k \<inter> {0..1})"
- by (rule finite_finite_vimage_IntI [OF \<open>finite k\<close>])
- have **: "((\<lambda>x. if (part_circlepath z r s t x) \<in> k then 0
- else f(part_circlepath z r s t x) *
- vector_derivative (part_circlepath z r s t) (at x)) has_integral i) {0..1}"
- by (rule has_integral_spike [OF negligible_finite [OF fin01]]) (use fi has_contour_integral in auto)
- have *: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1; part_circlepath z r s t x \<notin> k\<rbrakk> \<Longrightarrow> cmod (f (part_circlepath z r s t x)) \<le> B"
- by (auto intro!: B [unfolded path_image_def image_def, simplified])
- show ?thesis
- apply (rule has_integral_bound [where 'a=real, simplified, OF _ **, simplified])
- using assms apply force
- apply (simp add: norm_mult vector_derivative_part_circlepath)
- using le * "2" \<open>r > 0\<close> by auto
- qed
-qed
-
-lemma has_contour_integral_bound_part_circlepath:
- "\<lbrakk>(f has_contour_integral i) (part_circlepath z r s t);
- 0 \<le> B; 0 < r; s \<le> t;
- \<And>x. x \<in> path_image(part_circlepath z r s t) \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
- \<Longrightarrow> norm i \<le> B*r*(t - s)"
- by (auto intro: has_contour_integral_bound_part_circlepath_strong)
-
-lemma contour_integrable_continuous_part_circlepath:
- "continuous_on (path_image (part_circlepath z r s t)) f
- \<Longrightarrow> f contour_integrable_on (part_circlepath z r s t)"
- apply (simp add: contour_integrable_on has_contour_integral_def vector_derivative_part_circlepath path_image_def)
- apply (rule integrable_continuous_real)
- apply (fast intro: path_part_circlepath [unfolded path_def] continuous_intros continuous_on_compose2 [where g=f, OF _ _ order_refl])
- done
-
-proposition winding_number_part_circlepath_pos_less:
- assumes "s < t" and no: "norm(w - z) < r"
- shows "0 < Re (winding_number(part_circlepath z r s t) w)"
-proof -
- have "0 < r" by (meson no norm_not_less_zero not_le order.strict_trans2)
- note valid_path_part_circlepath
- moreover have " w \<notin> path_image (part_circlepath z r s t)"
- using assms by (auto simp: path_image_def image_def part_circlepath_def norm_mult linepath_def)
- moreover have "0 < r * (t - s) * (r - cmod (w - z))"
- using assms by (metis \<open>0 < r\<close> diff_gt_0_iff_gt mult_pos_pos)
- ultimately show ?thesis
- apply (rule winding_number_pos_lt [where e = "r*(t - s)*(r - norm(w - z))"])
- apply (simp add: vector_derivative_part_circlepath right_diff_distrib [symmetric] mult_ac)
- apply (rule mult_left_mono)+
- using Re_Im_le_cmod [of "w-z" "linepath s t x" for x]
- apply (simp add: exp_Euler cos_of_real sin_of_real part_circlepath_def algebra_simps cos_squared_eq [unfolded power2_eq_square])
- using assms \<open>0 < r\<close> by auto
-qed
-
-lemma simple_path_part_circlepath:
- "simple_path(part_circlepath z r s t) \<longleftrightarrow> (r \<noteq> 0 \<and> s \<noteq> t \<and> \<bar>s - t\<bar> \<le> 2*pi)"
-proof (cases "r = 0 \<or> s = t")
- case True
- then show ?thesis
- unfolding part_circlepath_def simple_path_def
- by (rule disjE) (force intro: bexI [where x = "1/4"] bexI [where x = "1/3"])+
-next
- case False then have "r \<noteq> 0" "s \<noteq> t" by auto
- have *: "\<And>x y z s t. \<i>*((1 - x) * s + x * t) = \<i>*(((1 - y) * s + y * t)) + z \<longleftrightarrow> \<i>*(x - y) * (t - s) = z"
- by (simp add: algebra_simps)
- have abs01: "\<And>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1
- \<Longrightarrow> (x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0 \<longleftrightarrow> \<bar>x - y\<bar> \<in> {0,1})"
- by auto
- have **: "\<And>x y. (\<exists>n. (complex_of_real x - of_real y) * (of_real t - of_real s) = 2 * (of_int n * of_real pi)) \<longleftrightarrow>
- (\<exists>n. \<bar>x - y\<bar> * (t - s) = 2 * (of_int n * pi))"
- by (force simp: algebra_simps abs_if dest: arg_cong [where f=Re] arg_cong [where f=complex_of_real]
- intro: exI [where x = "-n" for n])
- have 1: "\<bar>s - t\<bar> \<le> 2 * pi"
- if "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow> (\<exists>n. x * (t - s) = 2 * (real_of_int n * pi)) \<longrightarrow> x = 0 \<or> x = 1"
- proof (rule ccontr)
- assume "\<not> \<bar>s - t\<bar> \<le> 2 * pi"
- then have *: "\<And>n. t - s \<noteq> of_int n * \<bar>s - t\<bar>"
- using False that [of "2*pi / \<bar>t - s\<bar>"]
- by (simp add: abs_minus_commute divide_simps)
- show False
- using * [of 1] * [of "-1"] by auto
- qed
- have 2: "\<bar>s - t\<bar> = \<bar>2 * (real_of_int n * pi) / x\<bar>" if "x \<noteq> 0" "x * (t - s) = 2 * (real_of_int n * pi)" for x n
- proof -
- have "t-s = 2 * (real_of_int n * pi)/x"
- using that by (simp add: field_simps)
- then show ?thesis by (metis abs_minus_commute)
- qed
- have abs_away: "\<And>P. (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. P \<bar>x - y\<bar>) \<longleftrightarrow> (\<forall>x::real. 0 \<le> x \<and> x \<le> 1 \<longrightarrow> P x)"
- by force
- show ?thesis using False
- apply (simp add: simple_path_def)
- apply (simp add: part_circlepath_def linepath_def exp_eq * ** abs01 del: Set.insert_iff)
- apply (subst abs_away)
- apply (auto simp: 1)
- apply (rule ccontr)
- apply (auto simp: 2 field_split_simps abs_mult dest: of_int_leD)
- done
-qed
-
-lemma arc_part_circlepath:
- assumes "r \<noteq> 0" "s \<noteq> t" "\<bar>s - t\<bar> < 2*pi"
- shows "arc (part_circlepath z r s t)"
-proof -
- have *: "x = y" if eq: "\<i> * (linepath s t x) = \<i> * (linepath s t y) + 2 * of_int n * complex_of_real pi * \<i>"
- and x: "x \<in> {0..1}" and y: "y \<in> {0..1}" for x y n
- proof (rule ccontr)
- assume "x \<noteq> y"
- have "(linepath s t x) = (linepath s t y) + 2 * of_int n * complex_of_real pi"
- by (metis add_divide_eq_iff complex_i_not_zero mult.commute nonzero_mult_div_cancel_left eq)
- then have "s*y + t*x = s*x + (t*y + of_int n * (pi * 2))"
- by (force simp: algebra_simps linepath_def dest: arg_cong [where f=Re])
- with \<open>x \<noteq> y\<close> have st: "s-t = (of_int n * (pi * 2) / (y-x))"
- by (force simp: field_simps)
- have "\<bar>real_of_int n\<bar> < \<bar>y - x\<bar>"
- using assms \<open>x \<noteq> y\<close> by (simp add: st abs_mult field_simps)
- then show False
- using assms x y st by (auto dest: of_int_lessD)
- qed
- show ?thesis
- using assms
- apply (simp add: arc_def)
- apply (simp add: part_circlepath_def inj_on_def exp_eq)
- apply (blast intro: *)
- done
-qed
-
-subsection\<open>Special case of one complete circle\<close>
-
-definition\<^marker>\<open>tag important\<close> circlepath :: "[complex, real, real] \<Rightarrow> complex"
- where "circlepath z r \<equiv> part_circlepath z r 0 (2*pi)"
-
-lemma circlepath: "circlepath z r = (\<lambda>x. z + r * exp(2 * of_real pi * \<i> * of_real x))"
- by (simp add: circlepath_def part_circlepath_def linepath_def algebra_simps)
-
-lemma pathstart_circlepath [simp]: "pathstart (circlepath z r) = z + r"
- by (simp add: circlepath_def)
-
-lemma pathfinish_circlepath [simp]: "pathfinish (circlepath z r) = z + r"
- by (simp add: circlepath_def) (metis exp_two_pi_i mult.commute)
-
-lemma circlepath_minus: "circlepath z (-r) x = circlepath z r (x + 1/2)"
-proof -
- have "z + of_real r * exp (2 * pi * \<i> * (x + 1/2)) =
- z + of_real r * exp (2 * pi * \<i> * x + pi * \<i>)"
- by (simp add: divide_simps) (simp add: algebra_simps)
- also have "\<dots> = z - r * exp (2 * pi * \<i> * x)"
- by (simp add: exp_add)
- finally show ?thesis
- by (simp add: circlepath path_image_def sphere_def dist_norm)
-qed
-
-lemma circlepath_add1: "circlepath z r (x+1) = circlepath z r x"
- using circlepath_minus [of z r "x+1/2"] circlepath_minus [of z "-r" x]
- by (simp add: add.commute)
-
-lemma circlepath_add_half: "circlepath z r (x + 1/2) = circlepath z r (x - 1/2)"
- using circlepath_add1 [of z r "x-1/2"]
- by (simp add: add.commute)
-
-lemma path_image_circlepath_minus_subset:
- "path_image (circlepath z (-r)) \<subseteq> path_image (circlepath z r)"
- apply (simp add: path_image_def image_def circlepath_minus, clarify)
- apply (case_tac "xa \<le> 1/2", force)
- apply (force simp: circlepath_add_half)+
- done
-
-lemma path_image_circlepath_minus: "path_image (circlepath z (-r)) = path_image (circlepath z r)"
- using path_image_circlepath_minus_subset by fastforce
-
-lemma has_vector_derivative_circlepath [derivative_intros]:
- "((circlepath z r) has_vector_derivative (2 * pi * \<i> * r * exp (2 * of_real pi * \<i> * of_real x)))
- (at x within X)"
- apply (simp add: circlepath_def scaleR_conv_of_real)
- apply (rule derivative_eq_intros)
- apply (simp add: algebra_simps)
- done
-
-lemma vector_derivative_circlepath:
- "vector_derivative (circlepath z r) (at x) =
- 2 * pi * \<i> * r * exp(2 * of_real pi * \<i> * x)"
-using has_vector_derivative_circlepath vector_derivative_at by blast
-
-lemma vector_derivative_circlepath01:
- "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
- \<Longrightarrow> vector_derivative (circlepath z r) (at x within {0..1}) =
- 2 * pi * \<i> * r * exp(2 * of_real pi * \<i> * x)"
- using has_vector_derivative_circlepath
- by (auto simp: vector_derivative_at_within_ivl)
-
-lemma valid_path_circlepath [simp]: "valid_path (circlepath z r)"
- by (simp add: circlepath_def)
-
-lemma path_circlepath [simp]: "path (circlepath z r)"
- by (simp add: valid_path_imp_path)
-
-lemma path_image_circlepath_nonneg:
- assumes "0 \<le> r" shows "path_image (circlepath z r) = sphere z r"
-proof -
- have *: "x \<in> (\<lambda>u. z + (cmod (x - z)) * exp (\<i> * (of_real u * (of_real pi * 2)))) ` {0..1}" for x
- proof (cases "x = z")
- case True then show ?thesis by force
- next
- case False
- define w where "w = x - z"
- then have "w \<noteq> 0" by (simp add: False)
- have **: "\<And>t. \<lbrakk>Re w = cos t * cmod w; Im w = sin t * cmod w\<rbrakk> \<Longrightarrow> w = of_real (cmod w) * exp (\<i> * t)"
- using cis_conv_exp complex_eq_iff by auto
- show ?thesis
- apply (rule sincos_total_2pi [of "Re(w/of_real(norm w))" "Im(w/of_real(norm w))"])
- apply (simp add: divide_simps \<open>w \<noteq> 0\<close> cmod_power2 [symmetric])
- apply (rule_tac x="t / (2*pi)" in image_eqI)
- apply (simp add: field_simps \<open>w \<noteq> 0\<close>)
- using False **
- apply (auto simp: w_def)
- done
- qed
- show ?thesis
- unfolding circlepath path_image_def sphere_def dist_norm
- by (force simp: assms algebra_simps norm_mult norm_minus_commute intro: *)
-qed
-
-lemma path_image_circlepath [simp]:
- "path_image (circlepath z r) = sphere z \<bar>r\<bar>"
- using path_image_circlepath_minus
- by (force simp: path_image_circlepath_nonneg abs_if)
-
-lemma has_contour_integral_bound_circlepath_strong:
- "\<lbrakk>(f has_contour_integral i) (circlepath z r);
- finite k; 0 \<le> B; 0 < r;
- \<And>x. \<lbrakk>norm(x - z) = r; x \<notin> k\<rbrakk> \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
- \<Longrightarrow> norm i \<le> B*(2*pi*r)"
- unfolding circlepath_def
- by (auto simp: algebra_simps in_path_image_part_circlepath dest!: has_contour_integral_bound_part_circlepath_strong)
-
-lemma has_contour_integral_bound_circlepath:
- "\<lbrakk>(f has_contour_integral i) (circlepath z r);
- 0 \<le> B; 0 < r; \<And>x. norm(x - z) = r \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
- \<Longrightarrow> norm i \<le> B*(2*pi*r)"
- by (auto intro: has_contour_integral_bound_circlepath_strong)
-
-lemma contour_integrable_continuous_circlepath:
- "continuous_on (path_image (circlepath z r)) f
- \<Longrightarrow> f contour_integrable_on (circlepath z r)"
- by (simp add: circlepath_def contour_integrable_continuous_part_circlepath)
-
-lemma simple_path_circlepath: "simple_path(circlepath z r) \<longleftrightarrow> (r \<noteq> 0)"
- by (simp add: circlepath_def simple_path_part_circlepath)
-
-lemma notin_path_image_circlepath [simp]: "cmod (w - z) < r \<Longrightarrow> w \<notin> path_image (circlepath z r)"
- by (simp add: sphere_def dist_norm norm_minus_commute)
-
-lemma contour_integral_circlepath:
- assumes "r > 0"
- shows "contour_integral (circlepath z r) (\<lambda>w. 1 / (w - z)) = 2 * complex_of_real pi * \<i>"
-proof (rule contour_integral_unique)
- show "((\<lambda>w. 1 / (w - z)) has_contour_integral 2 * complex_of_real pi * \<i>) (circlepath z r)"
- unfolding has_contour_integral_def using assms
- apply (subst has_integral_cong)
- apply (simp add: vector_derivative_circlepath01)
- using has_integral_const_real [of _ 0 1] apply (force simp: circlepath)
- done
-qed
-
-lemma winding_number_circlepath_centre: "0 < r \<Longrightarrow> winding_number (circlepath z r) z = 1"
- apply (rule winding_number_unique_loop)
- apply (simp_all add: sphere_def valid_path_imp_path)
- apply (rule_tac x="circlepath z r" in exI)
- apply (simp add: sphere_def contour_integral_circlepath)
- done
-
-proposition winding_number_circlepath:
- assumes "norm(w - z) < r" shows "winding_number(circlepath z r) w = 1"
-proof (cases "w = z")
- case True then show ?thesis
- using assms winding_number_circlepath_centre by auto
-next
- case False
- have [simp]: "r > 0"
- using assms le_less_trans norm_ge_zero by blast
- define r' where "r' = norm(w - z)"
- have "r' < r"
- by (simp add: assms r'_def)
- have disjo: "cball z r' \<inter> sphere z r = {}"
- using \<open>r' < r\<close> by (force simp: cball_def sphere_def)
- have "winding_number(circlepath z r) w = winding_number(circlepath z r) z"
- proof (rule winding_number_around_inside [where s = "cball z r'"])
- show "winding_number (circlepath z r) z \<noteq> 0"
- by (simp add: winding_number_circlepath_centre)
- show "cball z r' \<inter> path_image (circlepath z r) = {}"
- by (simp add: disjo less_eq_real_def)
- qed (auto simp: r'_def dist_norm norm_minus_commute)
- also have "\<dots> = 1"
- by (simp add: winding_number_circlepath_centre)
- finally show ?thesis .
-qed
-
-
-text\<open> Hence the Cauchy formula for points inside a circle.\<close>
-
-theorem Cauchy_integral_circlepath:
- assumes contf: "continuous_on (cball z r) f" and holf: "f holomorphic_on (ball z r)" and wz: "norm(w - z) < r"
- shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w))
- (circlepath z r)"
-proof -
- have "r > 0"
- using assms le_less_trans norm_ge_zero by blast
- have "((\<lambda>u. f u / (u - w)) has_contour_integral (2 * pi) * \<i> * winding_number (circlepath z r) w * f w)
- (circlepath z r)"
- proof (rule Cauchy_integral_formula_weak [where s = "cball z r" and k = "{}"])
- show "\<And>x. x \<in> interior (cball z r) - {} \<Longrightarrow>
- f field_differentiable at x"
- using holf holomorphic_on_imp_differentiable_at by auto
- have "w \<notin> sphere z r"
- by simp (metis dist_commute dist_norm not_le order_refl wz)
- then show "path_image (circlepath z r) \<subseteq> cball z r - {w}"
- using \<open>r > 0\<close> by (auto simp add: cball_def sphere_def)
- qed (use wz in \<open>simp_all add: dist_norm norm_minus_commute contf\<close>)
- then show ?thesis
- by (simp add: winding_number_circlepath assms)
-qed
-
-corollary\<^marker>\<open>tag unimportant\<close> Cauchy_integral_circlepath_simple:
- assumes "f holomorphic_on cball z r" "norm(w - z) < r"
- shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w))
- (circlepath z r)"
-using assms by (force simp: holomorphic_on_imp_continuous_on holomorphic_on_subset Cauchy_integral_circlepath)
-
-
-lemma no_bounded_connected_component_imp_winding_number_zero:
- assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
- and nb: "\<And>z. bounded (connected_component_set (- s) z) \<longrightarrow> z \<in> s"
- shows "winding_number g z = 0"
-apply (rule winding_number_zero_in_outside)
-apply (simp_all add: assms)
-by (metis nb [of z] \<open>path_image g \<subseteq> s\<close> \<open>z \<notin> s\<close> contra_subsetD mem_Collect_eq outside outside_mono)
-
-lemma no_bounded_path_component_imp_winding_number_zero:
- assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
- and nb: "\<And>z. bounded (path_component_set (- s) z) \<longrightarrow> z \<in> s"
- shows "winding_number g z = 0"
-apply (rule no_bounded_connected_component_imp_winding_number_zero [OF g])
-by (simp add: bounded_subset nb path_component_subset_connected_component)
-
-
-subsection\<open> Uniform convergence of path integral\<close>
-
-text\<open>Uniform convergence when the derivative of the path is bounded, and in particular for the special case of a circle.\<close>
-
-proposition contour_integral_uniform_limit:
- assumes ev_fint: "eventually (\<lambda>n::'a. (f n) contour_integrable_on \<gamma>) F"
- and ul_f: "uniform_limit (path_image \<gamma>) f l F"
- and noleB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
- and \<gamma>: "valid_path \<gamma>"
- and [simp]: "\<not> trivial_limit F"
- shows "l contour_integrable_on \<gamma>" "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
-proof -
- have "0 \<le> B" by (meson noleB [of 0] atLeastAtMost_iff norm_ge_zero order_refl order_trans zero_le_one)
- { fix e::real
- assume "0 < e"
- then have "0 < e / (\<bar>B\<bar> + 1)" by simp
- then have "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. cmod (f n x - l x) < e / (\<bar>B\<bar> + 1)"
- using ul_f [unfolded uniform_limit_iff dist_norm] by auto
- with ev_fint
- obtain a where fga: "\<And>x. x \<in> {0..1} \<Longrightarrow> cmod (f a (\<gamma> x) - l (\<gamma> x)) < e / (\<bar>B\<bar> + 1)"
- and inta: "(\<lambda>t. f a (\<gamma> t) * vector_derivative \<gamma> (at t)) integrable_on {0..1}"
- using eventually_happens [OF eventually_conj]
- by (fastforce simp: contour_integrable_on path_image_def)
- have Ble: "B * e / (\<bar>B\<bar> + 1) \<le> e"
- using \<open>0 \<le> B\<close> \<open>0 < e\<close> by (simp add: field_split_simps)
- have "\<exists>h. (\<forall>x\<in>{0..1}. cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - h x) \<le> e) \<and> h integrable_on {0..1}"
- proof (intro exI conjI ballI)
- show "cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - f a (\<gamma> x) * vector_derivative \<gamma> (at x)) \<le> e"
- if "x \<in> {0..1}" for x
- apply (rule order_trans [OF _ Ble])
- using noleB [OF that] fga [OF that] \<open>0 \<le> B\<close> \<open>0 < e\<close>
- apply (simp add: norm_mult left_diff_distrib [symmetric] norm_minus_commute divide_simps)
- apply (fastforce simp: mult_ac dest: mult_mono [OF less_imp_le])
- done
- qed (rule inta)
- }
- then show lintg: "l contour_integrable_on \<gamma>"
- unfolding contour_integrable_on by (metis (mono_tags, lifting)integrable_uniform_limit_real)
- { fix e::real
- define B' where "B' = B + 1"
- have B': "B' > 0" "B' > B" using \<open>0 \<le> B\<close> by (auto simp: B'_def)
- assume "0 < e"
- then have ev_no': "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. 2 * cmod (f n x - l x) < e / B'"
- using ul_f [unfolded uniform_limit_iff dist_norm, rule_format, of "e / B' / 2"] B'
- by (simp add: field_simps)
- have ie: "integral {0..1::real} (\<lambda>x. e / 2) < e" using \<open>0 < e\<close> by simp
- have *: "cmod (f x (\<gamma> t) * vector_derivative \<gamma> (at t) - l (\<gamma> t) * vector_derivative \<gamma> (at t)) \<le> e / 2"
- if t: "t\<in>{0..1}" and leB': "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) < e / B'" for x t
- proof -
- have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) \<le> e * (B/ B')"
- using mult_mono [OF less_imp_le [OF leB'] noleB] B' \<open>0 < e\<close> t by auto
- also have "\<dots> < e"
- by (simp add: B' \<open>0 < e\<close> mult_imp_div_pos_less)
- finally have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) < e" .
- then show ?thesis
- by (simp add: left_diff_distrib [symmetric] norm_mult)
- qed
- have le_e: "\<And>x. \<lbrakk>\<forall>xa\<in>{0..1}. 2 * cmod (f x (\<gamma> xa) - l (\<gamma> xa)) < e / B'; f x contour_integrable_on \<gamma>\<rbrakk>
- \<Longrightarrow> cmod (integral {0..1}
- (\<lambda>u. f x (\<gamma> u) * vector_derivative \<gamma> (at u) - l (\<gamma> u) * vector_derivative \<gamma> (at u))) < e"
- apply (rule le_less_trans [OF integral_norm_bound_integral ie])
- apply (simp add: lintg integrable_diff contour_integrable_on [symmetric])
- apply (blast intro: *)+
- done
- have "\<forall>\<^sub>F x in F. dist (contour_integral \<gamma> (f x)) (contour_integral \<gamma> l) < e"
- apply (rule eventually_mono [OF eventually_conj [OF ev_no' ev_fint]])
- apply (simp add: dist_norm contour_integrable_on path_image_def contour_integral_integral)
- apply (simp add: lintg integral_diff [symmetric] contour_integrable_on [symmetric] le_e)
- done
- }
- then show "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
- by (rule tendstoI)
-qed
-
-corollary\<^marker>\<open>tag unimportant\<close> contour_integral_uniform_limit_circlepath:
- assumes "\<forall>\<^sub>F n::'a in F. (f n) contour_integrable_on (circlepath z r)"
- and "uniform_limit (sphere z r) f l F"
- and "\<not> trivial_limit F" "0 < r"
- shows "l contour_integrable_on (circlepath z r)"
- "((\<lambda>n. contour_integral (circlepath z r) (f n)) \<longlongrightarrow> contour_integral (circlepath z r) l) F"
- using assms by (auto simp: vector_derivative_circlepath norm_mult intro!: contour_integral_uniform_limit)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>General stepping result for derivative formulas\<close>
-
-lemma Cauchy_next_derivative:
- assumes "continuous_on (path_image \<gamma>) f'"
- and leB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
- and int: "\<And>w. w \<in> s - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f' u / (u - w)^k) has_contour_integral f w) \<gamma>"
- and k: "k \<noteq> 0"
- and "open s"
- and \<gamma>: "valid_path \<gamma>"
- and w: "w \<in> s - path_image \<gamma>"
- shows "(\<lambda>u. f' u / (u - w)^(Suc k)) contour_integrable_on \<gamma>"
- and "(f has_field_derivative (k * contour_integral \<gamma> (\<lambda>u. f' u/(u - w)^(Suc k))))
- (at w)" (is "?thes2")
-proof -
- have "open (s - path_image \<gamma>)" using \<open>open s\<close> closed_valid_path_image \<gamma> by blast
- then obtain d where "d>0" and d: "ball w d \<subseteq> s - path_image \<gamma>" using w
- using open_contains_ball by blast
- have [simp]: "\<And>n. cmod (1 + of_nat n) = 1 + of_nat n"
- by (metis norm_of_nat of_nat_Suc)
- have cint: "\<And>x. \<lbrakk>x \<noteq> w; cmod (x - w) < d\<rbrakk>
- \<Longrightarrow> (\<lambda>z. (f' z / (z - x) ^ k - f' z / (z - w) ^ k) / (x * k - w * k)) contour_integrable_on \<gamma>"
- apply (rule contour_integrable_div [OF contour_integrable_diff])
- using int w d
- by (force simp: dist_norm norm_minus_commute intro!: has_contour_integral_integrable)+
- have 1: "\<forall>\<^sub>F n in at w. (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k)
- contour_integrable_on \<gamma>"
- unfolding eventually_at
- apply (rule_tac x=d in exI)
- apply (simp add: \<open>d > 0\<close> dist_norm field_simps cint)
- done
- have bim_g: "bounded (image f' (path_image \<gamma>))"
- by (simp add: compact_imp_bounded compact_continuous_image compact_valid_path_image assms)
- then obtain C where "C > 0" and C: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cmod (f' (\<gamma> x)) \<le> C"
- by (force simp: bounded_pos path_image_def)
- have twom: "\<forall>\<^sub>F n in at w.
- \<forall>x\<in>path_image \<gamma>.
- cmod ((inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k - inverse (x - w) ^ Suc k) < e"
- if "0 < e" for e
- proof -
- have *: "cmod ((inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k) - inverse (x - w) ^ Suc k) < e"
- if x: "x \<in> path_image \<gamma>" and "u \<noteq> w" and uwd: "cmod (u - w) < d/2"
- and uw_less: "cmod (u - w) < e * (d/2) ^ (k+2) / (1 + real k)"
- for u x
- proof -
- define ff where [abs_def]:
- "ff n w =
- (if n = 0 then inverse(x - w)^k
- else if n = 1 then k / (x - w)^(Suc k)
- else (k * of_real(Suc k)) / (x - w)^(k + 2))" for n :: nat and w
- have km1: "\<And>z::complex. z \<noteq> 0 \<Longrightarrow> z ^ (k - Suc 0) = z ^ k / z"
- by (simp add: field_simps) (metis Suc_pred \<open>k \<noteq> 0\<close> neq0_conv power_Suc)
- have ff1: "(ff i has_field_derivative ff (Suc i) z) (at z within ball w (d/2))"
- if "z \<in> ball w (d/2)" "i \<le> 1" for i z
- proof -
- have "z \<notin> path_image \<gamma>"
- using \<open>x \<in> path_image \<gamma>\<close> d that ball_divide_subset_numeral by blast
- then have xz[simp]: "x \<noteq> z" using \<open>x \<in> path_image \<gamma>\<close> by blast
- then have neq: "x * x + z * z \<noteq> x * (z * 2)"
- by (blast intro: dest!: sum_sqs_eq)
- with xz have "\<And>v. v \<noteq> 0 \<Longrightarrow> (x * x + z * z) * v \<noteq> (x * (z * 2) * v)" by auto
- then have neqq: "\<And>v. v \<noteq> 0 \<Longrightarrow> x * (x * v) + z * (z * v) \<noteq> x * (z * (2 * v))"
- by (simp add: algebra_simps)
- show ?thesis using \<open>i \<le> 1\<close>
- apply (simp add: ff_def dist_norm Nat.le_Suc_eq km1, safe)
- apply (rule derivative_eq_intros | simp add: km1 | simp add: field_simps neq neqq)+
- done
- qed
- { fix a::real and b::real assume ab: "a > 0" "b > 0"
- then have "k * (1 + real k) * (1 / a) \<le> k * (1 + real k) * (4 / b) \<longleftrightarrow> b \<le> 4 * a"
- by (subst mult_le_cancel_left_pos)
- (use \<open>k \<noteq> 0\<close> in \<open>auto simp: divide_simps\<close>)
- with ab have "real k * (1 + real k) / a \<le> (real k * 4 + real k * real k * 4) / b \<longleftrightarrow> b \<le> 4 * a"
- by (simp add: field_simps)
- } note canc = this
- have ff2: "cmod (ff (Suc 1) v) \<le> real (k * (k + 1)) / (d/2) ^ (k + 2)"
- if "v \<in> ball w (d/2)" for v
- proof -
- have lessd: "\<And>z. cmod (\<gamma> z - v) < d/2 \<Longrightarrow> cmod (w - \<gamma> z) < d"
- by (metis that norm_minus_commute norm_triangle_half_r dist_norm mem_ball)
- have "d/2 \<le> cmod (x - v)" using d x that
- using lessd d x
- by (auto simp add: dist_norm path_image_def ball_def not_less [symmetric] del: divide_const_simps)
- then have "d \<le> cmod (x - v) * 2"
- by (simp add: field_split_simps)
- then have dpow_le: "d ^ (k+2) \<le> (cmod (x - v) * 2) ^ (k+2)"
- using \<open>0 < d\<close> order_less_imp_le power_mono by blast
- have "x \<noteq> v" using that
- using \<open>x \<in> path_image \<gamma>\<close> ball_divide_subset_numeral d by fastforce
- then show ?thesis
- using \<open>d > 0\<close> apply (simp add: ff_def norm_mult norm_divide norm_power dist_norm canc)
- using dpow_le apply (simp add: field_split_simps)
- done
- qed
- have ub: "u \<in> ball w (d/2)"
- using uwd by (simp add: dist_commute dist_norm)
- have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
- \<le> (real k * 4 + real k * real k * 4) * (cmod (u - w) * cmod (u - w)) / (d * (d * (d/2) ^ k))"
- using complex_Taylor [OF _ ff1 ff2 _ ub, of w, simplified]
- by (simp add: ff_def \<open>0 < d\<close>)
- then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
- \<le> (cmod (u - w) * real k) * (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)"
- by (simp add: field_simps)
- then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
- / (cmod (u - w) * real k)
- \<le> (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)"
- using \<open>k \<noteq> 0\<close> \<open>u \<noteq> w\<close> by (simp add: mult_ac zero_less_mult_iff pos_divide_le_eq)
- also have "\<dots> < e"
- using uw_less \<open>0 < d\<close> by (simp add: mult_ac divide_simps)
- finally have e: "cmod (inverse (x-u)^k - (inverse (x-w)^k + of_nat k * (u-w) / ((x-w) * (x-w)^k)))
- / cmod ((u - w) * real k) < e"
- by (simp add: norm_mult)
- have "x \<noteq> u"
- using uwd \<open>0 < d\<close> x d by (force simp: dist_norm ball_def norm_minus_commute)
- show ?thesis
- apply (rule le_less_trans [OF _ e])
- using \<open>k \<noteq> 0\<close> \<open>x \<noteq> u\<close> \<open>u \<noteq> w\<close>
- apply (simp add: field_simps norm_divide [symmetric])
- done
- qed
- show ?thesis
- unfolding eventually_at
- apply (rule_tac x = "min (d/2) ((e*(d/2)^(k + 2))/(Suc k))" in exI)
- apply (force simp: \<open>d > 0\<close> dist_norm that simp del: power_Suc intro: *)
- done
- qed
- have 2: "uniform_limit (path_image \<gamma>) (\<lambda>n x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) (\<lambda>x. f' x / (x - w) ^ Suc k) (at w)"
- unfolding uniform_limit_iff dist_norm
- proof clarify
- fix e::real
- assume "0 < e"
- have *: "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
- f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) < e"
- if ec: "cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
- inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k) < e / C"
- and x: "0 \<le> x" "x \<le> 1"
- for u x
- proof (cases "(f' (\<gamma> x)) = 0")
- case True then show ?thesis by (simp add: \<open>0 < e\<close>)
- next
- case False
- have "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
- f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) =
- cmod (f' (\<gamma> x) * ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
- inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k))"
- by (simp add: field_simps)
- also have "\<dots> = cmod (f' (\<gamma> x)) *
- cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
- inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k)"
- by (simp add: norm_mult)
- also have "\<dots> < cmod (f' (\<gamma> x)) * (e/C)"
- using False mult_strict_left_mono [OF ec] by force
- also have "\<dots> \<le> e" using C
- by (metis False \<open>0 < e\<close> frac_le less_eq_real_def mult.commute pos_le_divide_eq x zero_less_norm_iff)
- finally show ?thesis .
- qed
- show "\<forall>\<^sub>F n in at w.
- \<forall>x\<in>path_image \<gamma>.
- cmod (f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k - f' x / (x - w) ^ Suc k) < e"
- using twom [OF divide_pos_pos [OF \<open>0 < e\<close> \<open>C > 0\<close>]] unfolding path_image_def
- by (force intro: * elim: eventually_mono)
- qed
- show "(\<lambda>u. f' u / (u - w) ^ (Suc k)) contour_integrable_on \<gamma>"
- by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
- have *: "(\<lambda>n. contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k))
- \<midarrow>w\<rightarrow> contour_integral \<gamma> (\<lambda>u. f' u / (u - w) ^ (Suc k))"
- by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
- have **: "contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k)) =
- (f u - f w) / (u - w) / k"
- if "dist u w < d" for u
- proof -
- have u: "u \<in> s - path_image \<gamma>"
- by (metis subsetD d dist_commute mem_ball that)
- show ?thesis
- apply (rule contour_integral_unique)
- apply (simp add: diff_divide_distrib algebra_simps)
- apply (intro has_contour_integral_diff has_contour_integral_div)
- using u w apply (simp_all add: field_simps int)
- done
- qed
- show ?thes2
- apply (simp add: has_field_derivative_iff del: power_Suc)
- apply (rule Lim_transform_within [OF tendsto_mult_left [OF *] \<open>0 < d\<close> ])
- apply (simp add: \<open>k \<noteq> 0\<close> **)
- done
-qed
-
-lemma Cauchy_next_derivative_circlepath:
- assumes contf: "continuous_on (path_image (circlepath z r)) f"
- and int: "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>u. f u / (u - w)^k) has_contour_integral g w) (circlepath z r)"
- and k: "k \<noteq> 0"
- and w: "w \<in> ball z r"
- shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
- (is "?thes1")
- and "(g has_field_derivative (k * contour_integral (circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)))) (at w)"
- (is "?thes2")
-proof -
- have "r > 0" using w
- using ball_eq_empty by fastforce
- have wim: "w \<in> ball z r - path_image (circlepath z r)"
- using w by (auto simp: dist_norm)
- show ?thes1 ?thes2
- by (rule Cauchy_next_derivative [OF contf _ int k open_ball valid_path_circlepath wim, where B = "2 * pi * \<bar>r\<bar>"];
- auto simp: vector_derivative_circlepath norm_mult)+
-qed
-
-
-text\<open> In particular, the first derivative formula.\<close>
-
-lemma Cauchy_derivative_integral_circlepath:
- assumes contf: "continuous_on (cball z r) f"
- and holf: "f holomorphic_on ball z r"
- and w: "w \<in> ball z r"
- shows "(\<lambda>u. f u/(u - w)^2) contour_integrable_on (circlepath z r)"
- (is "?thes1")
- and "(f has_field_derivative (1 / (2 * of_real pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u / (u - w)^2))) (at w)"
- (is "?thes2")
-proof -
- have [simp]: "r \<ge> 0" using w
- using ball_eq_empty by fastforce
- have f: "continuous_on (path_image (circlepath z r)) f"
- by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def)
- have int: "\<And>w. dist z w < r \<Longrightarrow>
- ((\<lambda>u. f u / (u - w)) has_contour_integral (\<lambda>x. 2 * of_real pi * \<i> * f x) w) (circlepath z r)"
- by (rule Cauchy_integral_circlepath [OF contf holf]) (simp add: dist_norm norm_minus_commute)
- show ?thes1
- apply (simp add: power2_eq_square)
- apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1, simplified])
- apply (blast intro: int)
- done
- have "((\<lambda>x. 2 * of_real pi * \<i> * f x) has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2)) (at w)"
- apply (simp add: power2_eq_square)
- apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1 and g = "\<lambda>x. 2 * of_real pi * \<i> * f x", simplified])
- apply (blast intro: int)
- done
- then have fder: "(f has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2) / (2 * of_real pi * \<i>)) (at w)"
- by (rule DERIV_cdivide [where f = "\<lambda>x. 2 * of_real pi * \<i> * f x" and c = "2 * of_real pi * \<i>", simplified])
- show ?thes2
- by simp (rule fder)
-qed
-
-subsection\<open>Existence of all higher derivatives\<close>
-
-proposition derivative_is_holomorphic:
- assumes "open S"
- and fder: "\<And>z. z \<in> S \<Longrightarrow> (f has_field_derivative f' z) (at z)"
- shows "f' holomorphic_on S"
-proof -
- have *: "\<exists>h. (f' has_field_derivative h) (at z)" if "z \<in> S" for z
- proof -
- obtain r where "r > 0" and r: "cball z r \<subseteq> S"
- using open_contains_cball \<open>z \<in> S\<close> \<open>open S\<close> by blast
- then have holf_cball: "f holomorphic_on cball z r"
- apply (simp add: holomorphic_on_def)
- using field_differentiable_at_within field_differentiable_def fder by blast
- then have "continuous_on (path_image (circlepath z r)) f"
- using \<open>r > 0\<close> by (force elim: holomorphic_on_subset [THEN holomorphic_on_imp_continuous_on])
- then have contfpi: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1/(2 * of_real pi*\<i>) * f x)"
- by (auto intro: continuous_intros)+
- have contf_cball: "continuous_on (cball z r) f" using holf_cball
- by (simp add: holomorphic_on_imp_continuous_on holomorphic_on_subset)
- have holf_ball: "f holomorphic_on ball z r" using holf_cball
- using ball_subset_cball holomorphic_on_subset by blast
- { fix w assume w: "w \<in> ball z r"
- have intf: "(\<lambda>u. f u / (u - w)\<^sup>2) contour_integrable_on circlepath z r"
- by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
- have fder': "(f has_field_derivative 1 / (2 * of_real pi * \<i>) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2))
- (at w)"
- by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
- have f'_eq: "f' w = contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>)"
- using fder' ball_subset_cball r w by (force intro: DERIV_unique [OF fder])
- have "((\<lambda>u. f u / (u - w)\<^sup>2 / (2 * of_real pi * \<i>)) has_contour_integral
- contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
- (circlepath z r)"
- by (rule has_contour_integral_div [OF has_contour_integral_integral [OF intf]])
- then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral
- contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
- (circlepath z r)"
- by (simp add: algebra_simps)
- then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral f' w) (circlepath z r)"
- by (simp add: f'_eq)
- } note * = this
- show ?thesis
- apply (rule exI)
- apply (rule Cauchy_next_derivative_circlepath [OF contfpi, of 2 f', simplified])
- apply (simp_all add: \<open>0 < r\<close> * dist_norm)
- done
- qed
- show ?thesis
- by (simp add: holomorphic_on_open [OF \<open>open S\<close>] *)
-qed
-
-lemma holomorphic_deriv [holomorphic_intros]:
- "\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv f) holomorphic_on S"
-by (metis DERIV_deriv_iff_field_differentiable at_within_open derivative_is_holomorphic holomorphic_on_def)
-
-lemma analytic_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv f) analytic_on S"
- using analytic_on_holomorphic holomorphic_deriv by auto
-
-lemma holomorphic_higher_deriv [holomorphic_intros]: "\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv ^^ n) f holomorphic_on S"
- by (induction n) (auto simp: holomorphic_deriv)
-
-lemma analytic_higher_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv ^^ n) f analytic_on S"
- unfolding analytic_on_def using holomorphic_higher_deriv by blast
-
-lemma has_field_derivative_higher_deriv:
- "\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk>
- \<Longrightarrow> ((deriv ^^ n) f has_field_derivative (deriv ^^ (Suc n)) f x) (at x)"
-by (metis (no_types, hide_lams) DERIV_deriv_iff_field_differentiable at_within_open comp_apply
- funpow.simps(2) holomorphic_higher_deriv holomorphic_on_def)
-
-lemma valid_path_compose_holomorphic:
- assumes "valid_path g" and holo:"f holomorphic_on S" and "open S" "path_image g \<subseteq> S"
- shows "valid_path (f \<circ> g)"
-proof (rule valid_path_compose[OF \<open>valid_path g\<close>])
- fix x assume "x \<in> path_image g"
- then show "f field_differentiable at x"
- using analytic_on_imp_differentiable_at analytic_on_open assms holo by blast
-next
- have "deriv f holomorphic_on S"
- using holomorphic_deriv holo \<open>open S\<close> by auto
- then show "continuous_on (path_image g) (deriv f)"
- using assms(4) holomorphic_on_imp_continuous_on holomorphic_on_subset by auto
-qed
-
-
-subsection\<open>Morera's theorem\<close>
-
-lemma Morera_local_triangle_ball:
- assumes "\<And>z. z \<in> S
- \<Longrightarrow> \<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and>
- (\<forall>b c. closed_segment b c \<subseteq> ball a e
- \<longrightarrow> contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0)"
- shows "f analytic_on S"
-proof -
- { fix z assume "z \<in> S"
- with assms obtain e a where
- "0 < e" and z: "z \<in> ball a e" and contf: "continuous_on (ball a e) f"
- and 0: "\<And>b c. closed_segment b c \<subseteq> ball a e
- \<Longrightarrow> contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0"
- by blast
- have az: "dist a z < e" using mem_ball z by blast
- have sb_ball: "ball z (e - dist a z) \<subseteq> ball a e"
- by (simp add: dist_commute ball_subset_ball_iff)
- have "\<exists>e>0. f holomorphic_on ball z e"
- proof (intro exI conjI)
- have sub_ball: "\<And>y. dist a y < e \<Longrightarrow> closed_segment a y \<subseteq> ball a e"
- by (meson \<open>0 < e\<close> centre_in_ball convex_ball convex_contains_segment mem_ball)
- show "f holomorphic_on ball z (e - dist a z)"
- apply (rule holomorphic_on_subset [OF _ sb_ball])
- apply (rule derivative_is_holomorphic[OF open_ball])
- apply (rule triangle_contour_integrals_starlike_primitive [OF contf _ open_ball, of a])
- apply (simp_all add: 0 \<open>0 < e\<close> sub_ball)
- done
- qed (simp add: az)
- }
- then show ?thesis
- by (simp add: analytic_on_def)
-qed
-
-lemma Morera_local_triangle:
- assumes "\<And>z. z \<in> S
- \<Longrightarrow> \<exists>t. open t \<and> z \<in> t \<and> continuous_on t f \<and>
- (\<forall>a b c. convex hull {a,b,c} \<subseteq> t
- \<longrightarrow> contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0)"
- shows "f analytic_on S"
-proof -
- { fix z assume "z \<in> S"
- with assms obtain t where
- "open t" and z: "z \<in> t" and contf: "continuous_on t f"
- and 0: "\<And>a b c. convex hull {a,b,c} \<subseteq> t
- \<Longrightarrow> contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0"
- by force
- then obtain e where "e>0" and e: "ball z e \<subseteq> t"
- using open_contains_ball by blast
- have [simp]: "continuous_on (ball z e) f" using contf
- using continuous_on_subset e by blast
- have eq0: "\<And>b c. closed_segment b c \<subseteq> ball z e \<Longrightarrow>
- contour_integral (linepath z b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c z) f = 0"
- by (meson 0 z \<open>0 < e\<close> centre_in_ball closed_segment_subset convex_ball dual_order.trans e starlike_convex_subset)
- have "\<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and>
- (\<forall>b c. closed_segment b c \<subseteq> ball a e \<longrightarrow>
- contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0)"
- using \<open>e > 0\<close> eq0 by force
- }
- then show ?thesis
- by (simp add: Morera_local_triangle_ball)
-qed
-
-proposition Morera_triangle:
- "\<lbrakk>continuous_on S f; open S;
- \<And>a b c. convex hull {a,b,c} \<subseteq> S
- \<longrightarrow> contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0\<rbrakk>
- \<Longrightarrow> f analytic_on S"
- using Morera_local_triangle by blast
-
-subsection\<open>Combining theorems for higher derivatives including Leibniz rule\<close>
-
-lemma higher_deriv_linear [simp]:
- "(deriv ^^ n) (\<lambda>w. c*w) = (\<lambda>z. if n = 0 then c*z else if n = 1 then c else 0)"
- by (induction n) auto
-
-lemma higher_deriv_const [simp]: "(deriv ^^ n) (\<lambda>w. c) = (\<lambda>w. if n=0 then c else 0)"
- by (induction n) auto
-
-lemma higher_deriv_ident [simp]:
- "(deriv ^^ n) (\<lambda>w. w) z = (if n = 0 then z else if n = 1 then 1 else 0)"
- apply (induction n, simp)
- apply (metis higher_deriv_linear lambda_one)
- done
-
-lemma higher_deriv_id [simp]:
- "(deriv ^^ n) id z = (if n = 0 then z else if n = 1 then 1 else 0)"
- by (simp add: id_def)
-
-lemma has_complex_derivative_funpow_1:
- "\<lbrakk>(f has_field_derivative 1) (at z); f z = z\<rbrakk> \<Longrightarrow> (f^^n has_field_derivative 1) (at z)"
- apply (induction n, auto)
- apply (simp add: id_def)
- by (metis DERIV_chain comp_funpow comp_id funpow_swap1 mult.right_neutral)
-
-lemma higher_deriv_uminus:
- assumes "f holomorphic_on S" "open S" and z: "z \<in> S"
- shows "(deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)"
-using z
-proof (induction n arbitrary: z)
- case 0 then show ?case by simp
-next
- case (Suc n z)
- have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
- using Suc.prems assms has_field_derivative_higher_deriv by auto
- have "((deriv ^^ n) (\<lambda>w. - f w) has_field_derivative - deriv ((deriv ^^ n) f) z) (at z)"
- apply (rule has_field_derivative_transform_within_open [of "\<lambda>w. -((deriv ^^ n) f w)"])
- apply (rule derivative_eq_intros | rule * refl assms)+
- apply (auto simp add: Suc)
- done
- then show ?case
- by (simp add: DERIV_imp_deriv)
-qed
-
-lemma higher_deriv_add:
- fixes z::complex
- assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
- shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
-using z
-proof (induction n arbitrary: z)
- case 0 then show ?case by simp
-next
- case (Suc n z)
- have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
- "((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
- using Suc.prems assms has_field_derivative_higher_deriv by auto
- have "((deriv ^^ n) (\<lambda>w. f w + g w) has_field_derivative
- deriv ((deriv ^^ n) f) z + deriv ((deriv ^^ n) g) z) (at z)"
- apply (rule has_field_derivative_transform_within_open [of "\<lambda>w. (deriv ^^ n) f w + (deriv ^^ n) g w"])
- apply (rule derivative_eq_intros | rule * refl assms)+
- apply (auto simp add: Suc)
- done
- then show ?case
- by (simp add: DERIV_imp_deriv)
-qed
-
-lemma higher_deriv_diff:
- fixes z::complex
- assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
- shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
- apply (simp only: Groups.group_add_class.diff_conv_add_uminus higher_deriv_add)
- apply (subst higher_deriv_add)
- using assms holomorphic_on_minus apply (auto simp: higher_deriv_uminus)
- done
-
-lemma bb: "Suc n choose k = (n choose k) + (if k = 0 then 0 else (n choose (k - 1)))"
- by (cases k) simp_all
-
-lemma higher_deriv_mult:
- fixes z::complex
- assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
- shows "(deriv ^^ n) (\<lambda>w. f w * g w) z =
- (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
-using z
-proof (induction n arbitrary: z)
- case 0 then show ?case by simp
-next
- case (Suc n z)
- have *: "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
- "\<And>n. ((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
- using Suc.prems assms has_field_derivative_higher_deriv by auto
- have sumeq: "(\<Sum>i = 0..n.
- of_nat (n choose i) * (deriv ((deriv ^^ i) f) z * (deriv ^^ (n - i)) g z + deriv ((deriv ^^ (n - i)) g) z * (deriv ^^ i) f z)) =
- g z * deriv ((deriv ^^ n) f) z + (\<Sum>i = 0..n. (deriv ^^ i) f z * (of_nat (Suc n choose i) * (deriv ^^ (Suc n - i)) g z))"
- apply (simp add: bb algebra_simps sum.distrib)
- apply (subst (4) sum_Suc_reindex)
- apply (auto simp: algebra_simps Suc_diff_le intro: sum.cong)
- done
- have "((deriv ^^ n) (\<lambda>w. f w * g w) has_field_derivative
- (\<Sum>i = 0..Suc n. (Suc n choose i) * (deriv ^^ i) f z * (deriv ^^ (Suc n - i)) g z))
- (at z)"
- apply (rule has_field_derivative_transform_within_open
- [of "\<lambda>w. (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f w * (deriv ^^ (n - i)) g w)"])
- apply (simp add: algebra_simps)
- apply (rule DERIV_cong [OF DERIV_sum])
- apply (rule DERIV_cmult)
- apply (auto intro: DERIV_mult * sumeq \<open>open S\<close> Suc.prems Suc.IH [symmetric])
- done
- then show ?case
- unfolding funpow.simps o_apply
- by (simp add: DERIV_imp_deriv)
-qed
-
-lemma higher_deriv_transform_within_open:
- fixes z::complex
- assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
- and fg: "\<And>w. w \<in> S \<Longrightarrow> f w = g w"
- shows "(deriv ^^ i) f z = (deriv ^^ i) g z"
-using z
-by (induction i arbitrary: z)
- (auto simp: fg intro: complex_derivative_transform_within_open holomorphic_higher_deriv assms)
-
-lemma higher_deriv_compose_linear:
- fixes z::complex
- assumes f: "f holomorphic_on T" and S: "open S" and T: "open T" and z: "z \<in> S"
- and fg: "\<And>w. w \<in> S \<Longrightarrow> u * w \<in> T"
- shows "(deriv ^^ n) (\<lambda>w. f (u * w)) z = u^n * (deriv ^^ n) f (u * z)"
-using z
-proof (induction n arbitrary: z)
- case 0 then show ?case by simp
-next
- case (Suc n z)
- have holo0: "f holomorphic_on (*) u ` S"
- by (meson fg f holomorphic_on_subset image_subset_iff)
- have holo2: "(deriv ^^ n) f holomorphic_on (*) u ` S"
- by (meson f fg holomorphic_higher_deriv holomorphic_on_subset image_subset_iff T)
- have holo3: "(\<lambda>z. u ^ n * (deriv ^^ n) f (u * z)) holomorphic_on S"
- by (intro holo2 holomorphic_on_compose [where g="(deriv ^^ n) f", unfolded o_def] holomorphic_intros)
- have holo1: "(\<lambda>w. f (u * w)) holomorphic_on S"
- apply (rule holomorphic_on_compose [where g=f, unfolded o_def])
- apply (rule holo0 holomorphic_intros)+
- done
- have "deriv ((deriv ^^ n) (\<lambda>w. f (u * w))) z = deriv (\<lambda>z. u^n * (deriv ^^ n) f (u*z)) z"
- apply (rule complex_derivative_transform_within_open [OF _ holo3 S Suc.prems])
- apply (rule holomorphic_higher_deriv [OF holo1 S])
- apply (simp add: Suc.IH)
- done
- also have "\<dots> = u^n * deriv (\<lambda>z. (deriv ^^ n) f (u * z)) z"
- apply (rule deriv_cmult)
- apply (rule analytic_on_imp_differentiable_at [OF _ Suc.prems])
- apply (rule analytic_on_compose_gen [where g="(deriv ^^ n) f" and T=T, unfolded o_def])
- apply (simp)
- apply (simp add: analytic_on_open f holomorphic_higher_deriv T)
- apply (blast intro: fg)
- done
- also have "\<dots> = u * u ^ n * deriv ((deriv ^^ n) f) (u * z)"
- apply (subst deriv_chain [where g = "(deriv ^^ n) f" and f = "(*) u", unfolded o_def])
- apply (rule derivative_intros)
- using Suc.prems field_differentiable_def f fg has_field_derivative_higher_deriv T apply blast
- apply (simp)
- done
- finally show ?case
- by simp
-qed
-
-lemma higher_deriv_add_at:
- assumes "f analytic_on {z}" "g analytic_on {z}"
- shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
-proof -
- have "f analytic_on {z} \<and> g analytic_on {z}"
- using assms by blast
- with higher_deriv_add show ?thesis
- by (auto simp: analytic_at_two)
-qed
-
-lemma higher_deriv_diff_at:
- assumes "f analytic_on {z}" "g analytic_on {z}"
- shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
-proof -
- have "f analytic_on {z} \<and> g analytic_on {z}"
- using assms by blast
- with higher_deriv_diff show ?thesis
- by (auto simp: analytic_at_two)
-qed
-
-lemma higher_deriv_uminus_at:
- "f analytic_on {z} \<Longrightarrow> (deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)"
- using higher_deriv_uminus
- by (auto simp: analytic_at)
-
-lemma higher_deriv_mult_at:
- assumes "f analytic_on {z}" "g analytic_on {z}"
- shows "(deriv ^^ n) (\<lambda>w. f w * g w) z =
- (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
-proof -
- have "f analytic_on {z} \<and> g analytic_on {z}"
- using assms by blast
- with higher_deriv_mult show ?thesis
- by (auto simp: analytic_at_two)
-qed
-
-
-text\<open> Nonexistence of isolated singularities and a stronger integral formula.\<close>
-
-proposition no_isolated_singularity:
- fixes z::complex
- assumes f: "continuous_on S f" and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K"
- shows "f holomorphic_on S"
-proof -
- { fix z
- assume "z \<in> S" and cdf: "\<And>x. x \<in> S - K \<Longrightarrow> f field_differentiable at x"
- have "f field_differentiable at z"
- proof (cases "z \<in> K")
- case False then show ?thesis by (blast intro: cdf \<open>z \<in> S\<close>)
- next
- case True
- with finite_set_avoid [OF K, of z]
- obtain d where "d>0" and d: "\<And>x. \<lbrakk>x\<in>K; x \<noteq> z\<rbrakk> \<Longrightarrow> d \<le> dist z x"
- by blast
- obtain e where "e>0" and e: "ball z e \<subseteq> S"
- using S \<open>z \<in> S\<close> by (force simp: open_contains_ball)
- have fde: "continuous_on (ball z (min d e)) f"
- by (metis Int_iff ball_min_Int continuous_on_subset e f subsetI)
- have cont: "{a,b,c} \<subseteq> ball z (min d e) \<Longrightarrow> continuous_on (convex hull {a, b, c}) f" for a b c
- by (simp add: hull_minimal continuous_on_subset [OF fde])
- have fd: "\<lbrakk>{a,b,c} \<subseteq> ball z (min d e); x \<in> interior (convex hull {a, b, c}) - K\<rbrakk>
- \<Longrightarrow> f field_differentiable at x" for a b c x
- by (metis cdf Diff_iff Int_iff ball_min_Int subsetD convex_ball e interior_mono interior_subset subset_hull)
- obtain g where "\<And>w. w \<in> ball z (min d e) \<Longrightarrow> (g has_field_derivative f w) (at w within ball z (min d e))"
- apply (rule contour_integral_convex_primitive
- [OF convex_ball fde Cauchy_theorem_triangle_cofinite [OF _ K]])
- using cont fd by auto
- then have "f holomorphic_on ball z (min d e)"
- by (metis open_ball at_within_open derivative_is_holomorphic)
- then show ?thesis
- unfolding holomorphic_on_def
- by (metis open_ball \<open>0 < d\<close> \<open>0 < e\<close> at_within_open centre_in_ball min_less_iff_conj)
- qed
- }
- with holf S K show ?thesis
- by (simp add: holomorphic_on_open open_Diff finite_imp_closed field_differentiable_def [symmetric])
-qed
-
-lemma no_isolated_singularity':
- fixes z::complex
- assumes f: "\<And>z. z \<in> K \<Longrightarrow> (f \<longlongrightarrow> f z) (at z within S)"
- and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K"
- shows "f holomorphic_on S"
-proof (rule no_isolated_singularity[OF _ assms(2-)])
- show "continuous_on S f" unfolding continuous_on_def
- proof
- fix z assume z: "z \<in> S"
- show "(f \<longlongrightarrow> f z) (at z within S)"
- proof (cases "z \<in> K")
- case False
- from holf have "continuous_on (S - K) f"
- by (rule holomorphic_on_imp_continuous_on)
- with z False have "(f \<longlongrightarrow> f z) (at z within (S - K))"
- by (simp add: continuous_on_def)
- also from z K S False have "at z within (S - K) = at z within S"
- by (subst (1 2) at_within_open) (auto intro: finite_imp_closed)
- finally show "(f \<longlongrightarrow> f z) (at z within S)" .
- qed (insert assms z, simp_all)
- qed
-qed
-
-proposition Cauchy_integral_formula_convex:
- assumes S: "convex S" and K: "finite K" and contf: "continuous_on S f"
- and fcd: "(\<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x)"
- and z: "z \<in> interior S" and vpg: "valid_path \<gamma>"
- and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
-proof -
- have *: "\<And>x. x \<in> interior S \<Longrightarrow> f field_differentiable at x"
- unfolding holomorphic_on_open [symmetric] field_differentiable_def
- using no_isolated_singularity [where S = "interior S"]
- by (meson K contf continuous_at_imp_continuous_on continuous_on_interior fcd
- field_differentiable_at_within field_differentiable_def holomorphic_onI
- holomorphic_on_imp_differentiable_at open_interior)
- show ?thesis
- by (rule Cauchy_integral_formula_weak [OF S finite.emptyI contf]) (use * assms in auto)
-qed
-
-text\<open> Formula for higher derivatives.\<close>
-
-lemma Cauchy_has_contour_integral_higher_derivative_circlepath:
- assumes contf: "continuous_on (cball z r) f"
- and holf: "f holomorphic_on ball z r"
- and w: "w \<in> ball z r"
- shows "((\<lambda>u. f u / (u - w) ^ (Suc k)) has_contour_integral ((2 * pi * \<i>) / (fact k) * (deriv ^^ k) f w))
- (circlepath z r)"
-using w
-proof (induction k arbitrary: w)
- case 0 then show ?case
- using assms by (auto simp: Cauchy_integral_circlepath dist_commute dist_norm)
-next
- case (Suc k)
- have [simp]: "r > 0" using w
- using ball_eq_empty by fastforce
- have f: "continuous_on (path_image (circlepath z r)) f"
- by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def less_imp_le)
- obtain X where X: "((\<lambda>u. f u / (u - w) ^ Suc (Suc k)) has_contour_integral X) (circlepath z r)"
- using Cauchy_next_derivative_circlepath(1) [OF f Suc.IH _ Suc.prems]
- by (auto simp: contour_integrable_on_def)
- then have con: "contour_integral (circlepath z r) ((\<lambda>u. f u / (u - w) ^ Suc (Suc k))) = X"
- by (rule contour_integral_unique)
- have "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) w) (at w)"
- using Suc.prems assms has_field_derivative_higher_deriv by auto
- then have dnf_diff: "\<And>n. (deriv ^^ n) f field_differentiable (at w)"
- by (force simp: field_differentiable_def)
- have "deriv (\<lambda>w. complex_of_real (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w) w =
- of_nat (Suc k) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w) ^ Suc (Suc k))"
- by (force intro!: DERIV_imp_deriv Cauchy_next_derivative_circlepath [OF f Suc.IH _ Suc.prems])
- also have "\<dots> = of_nat (Suc k) * X"
- by (simp only: con)
- finally have "deriv (\<lambda>w. ((2 * pi) * \<i> / (fact k)) * (deriv ^^ k) f w) w = of_nat (Suc k) * X" .
- then have "((2 * pi) * \<i> / (fact k)) * deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X"
- by (metis deriv_cmult dnf_diff)
- then have "deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X / ((2 * pi) * \<i> / (fact k))"
- by (simp add: field_simps)
- then show ?case
- using of_nat_eq_0_iff X by fastforce
-qed
-
-lemma Cauchy_higher_derivative_integral_circlepath:
- assumes contf: "continuous_on (cball z r) f"
- and holf: "f holomorphic_on ball z r"
- and w: "w \<in> ball z r"
- shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
- (is "?thes1")
- and "(deriv ^^ k) f w = (fact k) / (2 * pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k))"
- (is "?thes2")
-proof -
- have *: "((\<lambda>u. f u / (u - w) ^ Suc k) has_contour_integral (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w)
- (circlepath z r)"
- using Cauchy_has_contour_integral_higher_derivative_circlepath [OF assms]
- by simp
- show ?thes1 using *
- using contour_integrable_on_def by blast
- show ?thes2
- unfolding contour_integral_unique [OF *] by (simp add: field_split_simps)
-qed
-
-corollary Cauchy_contour_integral_circlepath:
- assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r"
- shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)) = (2 * pi * \<i>) * (deriv ^^ k) f w / (fact k)"
-by (simp add: Cauchy_higher_derivative_integral_circlepath [OF assms])
-
-lemma Cauchy_contour_integral_circlepath_2:
- assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r"
- shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^2) = (2 * pi * \<i>) * deriv f w"
- using Cauchy_contour_integral_circlepath [OF assms, of 1]
- by (simp add: power2_eq_square)
-
-
-subsection\<open>A holomorphic function is analytic, i.e. has local power series\<close>
-
-theorem holomorphic_power_series:
- assumes holf: "f holomorphic_on ball z r"
- and w: "w \<in> ball z r"
- shows "((\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
-proof -
- \<comment> \<open>Replacing \<^term>\<open>r\<close> and the original (weak) premises with stronger ones\<close>
- obtain r where "r > 0" and holfc: "f holomorphic_on cball z r" and w: "w \<in> ball z r"
- proof
- have "cball z ((r + dist w z) / 2) \<subseteq> ball z r"
- using w by (simp add: dist_commute field_sum_of_halves subset_eq)
- then show "f holomorphic_on cball z ((r + dist w z) / 2)"
- by (rule holomorphic_on_subset [OF holf])
- have "r > 0"
- using w by clarsimp (metis dist_norm le_less_trans norm_ge_zero)
- then show "0 < (r + dist w z) / 2"
- by simp (use zero_le_dist [of w z] in linarith)
- qed (use w in \<open>auto simp: dist_commute\<close>)
- then have holf: "f holomorphic_on ball z r"
- using ball_subset_cball holomorphic_on_subset by blast
- have contf: "continuous_on (cball z r) f"
- by (simp add: holfc holomorphic_on_imp_continuous_on)
- have cint: "\<And>k. (\<lambda>u. f u / (u - z) ^ Suc k) contour_integrable_on circlepath z r"
- by (rule Cauchy_higher_derivative_integral_circlepath [OF contf holf]) (simp add: \<open>0 < r\<close>)
- obtain B where "0 < B" and B: "\<And>u. u \<in> cball z r \<Longrightarrow> norm(f u) \<le> B"
- by (metis (no_types) bounded_pos compact_cball compact_continuous_image compact_imp_bounded contf image_eqI)
- obtain k where k: "0 < k" "k \<le> r" and wz_eq: "norm(w - z) = r - k"
- and kle: "\<And>u. norm(u - z) = r \<Longrightarrow> k \<le> norm(u - w)"
- proof
- show "\<And>u. cmod (u - z) = r \<Longrightarrow> r - dist z w \<le> cmod (u - w)"
- by (metis add_diff_eq diff_add_cancel dist_norm norm_diff_ineq)
- qed (use w in \<open>auto simp: dist_norm norm_minus_commute\<close>)
- have ul: "uniform_limit (sphere z r) (\<lambda>n x. (\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k))) (\<lambda>x. f x / (x - w)) sequentially"
- unfolding uniform_limit_iff dist_norm
- proof clarify
- fix e::real
- assume "0 < e"
- have rr: "0 \<le> (r - k) / r" "(r - k) / r < 1" using k by auto
- obtain n where n: "((r - k) / r) ^ n < e / B * k"
- using real_arch_pow_inv [of "e/B*k" "(r - k)/r"] \<open>0 < e\<close> \<open>0 < B\<close> k by force
- have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) - f u / (u - w)) < e"
- if "n \<le> N" and r: "r = dist z u" for N u
- proof -
- have N: "((r - k) / r) ^ N < e / B * k"
- apply (rule le_less_trans [OF power_decreasing n])
- using \<open>n \<le> N\<close> k by auto
- have u [simp]: "(u \<noteq> z) \<and> (u \<noteq> w)"
- using \<open>0 < r\<close> r w by auto
- have wzu_not1: "(w - z) / (u - z) \<noteq> 1"
- by (metis (no_types) dist_norm divide_eq_1_iff less_irrefl mem_ball norm_minus_commute r w)
- have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) * (u - w) - f u)
- = norm ((\<Sum>k<N. (((w - z) / (u - z)) ^ k)) * f u * (u - w) / (u - z) - f u)"
- unfolding sum_distrib_right sum_divide_distrib power_divide by (simp add: algebra_simps)
- also have "\<dots> = norm ((((w - z) / (u - z)) ^ N - 1) * (u - w) / (((w - z) / (u - z) - 1) * (u - z)) - 1) * norm (f u)"
- using \<open>0 < B\<close>
- apply (auto simp: geometric_sum [OF wzu_not1])
- apply (simp add: field_simps norm_mult [symmetric])
- done
- also have "\<dots> = norm ((u-z) ^ N * (w - u) - ((w - z) ^ N - (u-z) ^ N) * (u-w)) / (r ^ N * norm (u-w)) * norm (f u)"
- using \<open>0 < r\<close> r by (simp add: divide_simps norm_mult norm_divide norm_power dist_norm norm_minus_commute)
- also have "\<dots> = norm ((w - z) ^ N * (w - u)) / (r ^ N * norm (u - w)) * norm (f u)"
- by (simp add: algebra_simps)
- also have "\<dots> = norm (w - z) ^ N * norm (f u) / r ^ N"
- by (simp add: norm_mult norm_power norm_minus_commute)
- also have "\<dots> \<le> (((r - k)/r)^N) * B"
- using \<open>0 < r\<close> w k
- apply (simp add: divide_simps)
- apply (rule mult_mono [OF power_mono])
- apply (auto simp: norm_divide wz_eq norm_power dist_norm norm_minus_commute B r)
- done
- also have "\<dots> < e * k"
- using \<open>0 < B\<close> N by (simp add: divide_simps)
- also have "\<dots> \<le> e * norm (u - w)"
- using r kle \<open>0 < e\<close> by (simp add: dist_commute dist_norm)
- finally show ?thesis
- by (simp add: field_split_simps norm_divide del: power_Suc)
- qed
- with \<open>0 < r\<close> show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>sphere z r.
- norm ((\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k)) - f x / (x - w)) < e"
- by (auto simp: mult_ac less_imp_le eventually_sequentially Ball_def)
- qed
- have eq: "\<forall>\<^sub>F x in sequentially.
- contour_integral (circlepath z r) (\<lambda>u. \<Sum>k<x. (w - z) ^ k * (f u / (u - z) ^ Suc k)) =
- (\<Sum>k<x. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z) ^ k)"
- apply (rule eventuallyI)
- apply (subst contour_integral_sum, simp)
- using contour_integrable_lmul [OF cint, of "(w - z) ^ a" for a] apply (simp add: field_simps)
- apply (simp only: contour_integral_lmul cint algebra_simps)
- done
- have cic: "\<And>u. (\<lambda>y. \<Sum>k<u. (w - z) ^ k * (f y / (y - z) ^ Suc k)) contour_integrable_on circlepath z r"
- apply (intro contour_integrable_sum contour_integrable_lmul, simp)
- using \<open>0 < r\<close> by (force intro!: Cauchy_higher_derivative_integral_circlepath [OF contf holf])
- have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k)
- sums contour_integral (circlepath z r) (\<lambda>u. f u/(u - w))"
- unfolding sums_def
- apply (intro Lim_transform_eventually [OF _ eq] contour_integral_uniform_limit_circlepath [OF eventuallyI ul] cic)
- using \<open>0 < r\<close> apply auto
- done
- then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k)
- sums (2 * of_real pi * \<i> * f w)"
- using w by (auto simp: dist_commute dist_norm contour_integral_unique [OF Cauchy_integral_circlepath_simple [OF holfc]])
- then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z)^k / (\<i> * (of_real pi * 2)))
- sums ((2 * of_real pi * \<i> * f w) / (\<i> * (complex_of_real pi * 2)))"
- by (rule sums_divide)
- then have "(\<lambda>n. (w - z) ^ n * contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc n) / (\<i> * (of_real pi * 2)))
- sums f w"
- by (simp add: field_simps)
- then show ?thesis
- by (simp add: field_simps \<open>0 < r\<close> Cauchy_higher_derivative_integral_circlepath [OF contf holf])
-qed
-
-
-subsection\<open>The Liouville theorem and the Fundamental Theorem of Algebra\<close>
-
-text\<open> These weak Liouville versions don't even need the derivative formula.\<close>
-
-lemma Liouville_weak_0:
- assumes holf: "f holomorphic_on UNIV" and inf: "(f \<longlongrightarrow> 0) at_infinity"
- shows "f z = 0"
-proof (rule ccontr)
- assume fz: "f z \<noteq> 0"
- with inf [unfolded Lim_at_infinity, rule_format, of "norm(f z)/2"]
- obtain B where B: "\<And>x. B \<le> cmod x \<Longrightarrow> norm (f x) * 2 < cmod (f z)"
- by (auto simp: dist_norm)
- define R where "R = 1 + \<bar>B\<bar> + norm z"
- have "R > 0" unfolding R_def
- proof -
- have "0 \<le> cmod z + \<bar>B\<bar>"
- by (metis (full_types) add_nonneg_nonneg norm_ge_zero real_norm_def)
- then show "0 < 1 + \<bar>B\<bar> + cmod z"
- by linarith
- qed
- have *: "((\<lambda>u. f u / (u - z)) has_contour_integral 2 * complex_of_real pi * \<i> * f z) (circlepath z R)"
- apply (rule Cauchy_integral_circlepath)
- using \<open>R > 0\<close> apply (auto intro: holomorphic_on_subset [OF holf] holomorphic_on_imp_continuous_on)+
- done
- have "cmod (x - z) = R \<Longrightarrow> cmod (f x) * 2 < cmod (f z)" for x
- unfolding R_def
- by (rule B) (use norm_triangle_ineq4 [of x z] in auto)
- with \<open>R > 0\<close> fz show False
- using has_contour_integral_bound_circlepath [OF *, of "norm(f z)/2/R"]
- by (auto simp: less_imp_le norm_mult norm_divide field_split_simps)
-qed
-
-proposition Liouville_weak:
- assumes "f holomorphic_on UNIV" and "(f \<longlongrightarrow> l) at_infinity"
- shows "f z = l"
- using Liouville_weak_0 [of "\<lambda>z. f z - l"]
- by (simp add: assms holomorphic_on_diff LIM_zero)
-
-proposition Liouville_weak_inverse:
- assumes "f holomorphic_on UNIV" and unbounded: "\<And>B. eventually (\<lambda>x. norm (f x) \<ge> B) at_infinity"
- obtains z where "f z = 0"
-proof -
- { assume f: "\<And>z. f z \<noteq> 0"
- have 1: "(\<lambda>x. 1 / f x) holomorphic_on UNIV"
- by (simp add: holomorphic_on_divide assms f)
- have 2: "((\<lambda>x. 1 / f x) \<longlongrightarrow> 0) at_infinity"
- apply (rule tendstoI [OF eventually_mono])
- apply (rule_tac B="2/e" in unbounded)
- apply (simp add: dist_norm norm_divide field_split_simps)
- done
- have False
- using Liouville_weak_0 [OF 1 2] f by simp
- }
- then show ?thesis
- using that by blast
-qed
-
-text\<open> In particular we get the Fundamental Theorem of Algebra.\<close>
-
-theorem fundamental_theorem_of_algebra:
- fixes a :: "nat \<Rightarrow> complex"
- assumes "a 0 = 0 \<or> (\<exists>i \<in> {1..n}. a i \<noteq> 0)"
- obtains z where "(\<Sum>i\<le>n. a i * z^i) = 0"
-using assms
-proof (elim disjE bexE)
- assume "a 0 = 0" then show ?thesis
- by (auto simp: that [of 0])
-next
- fix i
- assume i: "i \<in> {1..n}" and nz: "a i \<noteq> 0"
- have 1: "(\<lambda>z. \<Sum>i\<le>n. a i * z^i) holomorphic_on UNIV"
- by (rule holomorphic_intros)+
- show thesis
- proof (rule Liouville_weak_inverse [OF 1])
- show "\<forall>\<^sub>F x in at_infinity. B \<le> cmod (\<Sum>i\<le>n. a i * x ^ i)" for B
- using i nz by (intro polyfun_extremal exI[of _ i]) auto
- qed (use that in auto)
-qed
-
-subsection\<open>Weierstrass convergence theorem\<close>
-
-lemma holomorphic_uniform_limit:
- assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and> (f n) holomorphic_on ball z r) F"
- and ulim: "uniform_limit (cball z r) f g F"
- and F: "\<not> trivial_limit F"
- obtains "continuous_on (cball z r) g" "g holomorphic_on ball z r"
-proof (cases r "0::real" rule: linorder_cases)
- case less then show ?thesis by (force simp: ball_empty less_imp_le continuous_on_def holomorphic_on_def intro: that)
-next
- case equal then show ?thesis
- by (force simp: holomorphic_on_def intro: that)
-next
- case greater
- have contg: "continuous_on (cball z r) g"
- using cont uniform_limit_theorem [OF eventually_mono ulim F] by blast
- have "path_image (circlepath z r) \<subseteq> cball z r"
- using \<open>0 < r\<close> by auto
- then have 1: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1 / (2 * complex_of_real pi * \<i>) * g x)"
- by (intro continuous_intros continuous_on_subset [OF contg])
- have 2: "((\<lambda>u. 1 / (2 * of_real pi * \<i>) * g u / (u - w) ^ 1) has_contour_integral g w) (circlepath z r)"
- if w: "w \<in> ball z r" for w
- proof -
- define d where "d = (r - norm(w - z))"
- have "0 < d" "d \<le> r" using w by (auto simp: norm_minus_commute d_def dist_norm)
- have dle: "\<And>u. cmod (z - u) = r \<Longrightarrow> d \<le> cmod (u - w)"
- unfolding d_def by (metis add_diff_eq diff_add_cancel norm_diff_ineq norm_minus_commute)
- have ev_int: "\<forall>\<^sub>F n in F. (\<lambda>u. f n u / (u - w)) contour_integrable_on circlepath z r"
- apply (rule eventually_mono [OF cont])
- using w
- apply (auto intro: Cauchy_higher_derivative_integral_circlepath [where k=0, simplified])
- done
- have ul_less: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)) (\<lambda>x. g x / (x - w)) F"
- using greater \<open>0 < d\<close>
- apply (clarsimp simp add: uniform_limit_iff dist_norm norm_divide diff_divide_distrib [symmetric] divide_simps)
- apply (rule_tac e1="e * d" in eventually_mono [OF uniform_limitD [OF ulim]])
- apply (force simp: dist_norm intro: dle mult_left_mono less_le_trans)+
- done
- have g_cint: "(\<lambda>u. g u/(u - w)) contour_integrable_on circlepath z r"
- by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>])
- have cif_tends_cig: "((\<lambda>n. contour_integral(circlepath z r) (\<lambda>u. f n u / (u - w))) \<longlongrightarrow> contour_integral(circlepath z r) (\<lambda>u. g u/(u - w))) F"
- by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>])
- have f_tends_cig: "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> contour_integral (circlepath z r) (\<lambda>u. g u / (u - w))) F"
- proof (rule Lim_transform_eventually)
- show "\<forall>\<^sub>F x in F. contour_integral (circlepath z r) (\<lambda>u. f x u / (u - w))
- = 2 * of_real pi * \<i> * f x w"
- apply (rule eventually_mono [OF cont contour_integral_unique [OF Cauchy_integral_circlepath]])
- using w\<open>0 < d\<close> d_def by auto
- qed (auto simp: cif_tends_cig)
- have "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in F. dist (f n w) (g w) < e"
- by (rule eventually_mono [OF uniform_limitD [OF ulim]]) (use w in auto)
- then have "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> 2 * of_real pi * \<i> * g w) F"
- by (rule tendsto_mult_left [OF tendstoI])
- then have "((\<lambda>u. g u / (u - w)) has_contour_integral 2 * of_real pi * \<i> * g w) (circlepath z r)"
- using has_contour_integral_integral [OF g_cint] tendsto_unique [OF F f_tends_cig] w
- by fastforce
- then have "((\<lambda>u. g u / (2 * of_real pi * \<i> * (u - w))) has_contour_integral g w) (circlepath z r)"
- using has_contour_integral_div [where c = "2 * of_real pi * \<i>"]
- by (force simp: field_simps)
- then show ?thesis
- by (simp add: dist_norm)
- qed
- show ?thesis
- using Cauchy_next_derivative_circlepath(2) [OF 1 2, simplified]
- by (fastforce simp add: holomorphic_on_open contg intro: that)
-qed
-
-
-text\<open> Version showing that the limit is the limit of the derivatives.\<close>
-
-proposition has_complex_derivative_uniform_limit:
- fixes z::complex
- assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and>
- (\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))) F"
- and ulim: "uniform_limit (cball z r) f g F"
- and F: "\<not> trivial_limit F" and "0 < r"
- obtains g' where
- "continuous_on (cball z r) g"
- "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
-proof -
- let ?conint = "contour_integral (circlepath z r)"
- have g: "continuous_on (cball z r) g" "g holomorphic_on ball z r"
- by (rule holomorphic_uniform_limit [OF eventually_mono [OF cont] ulim F];
- auto simp: holomorphic_on_open field_differentiable_def)+
- then obtain g' where g': "\<And>x. x \<in> ball z r \<Longrightarrow> (g has_field_derivative g' x) (at x)"
- using DERIV_deriv_iff_has_field_derivative
- by (fastforce simp add: holomorphic_on_open)
- then have derg: "\<And>x. x \<in> ball z r \<Longrightarrow> deriv g x = g' x"
- by (simp add: DERIV_imp_deriv)
- have tends_f'n_g': "((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" if w: "w \<in> ball z r" for w
- proof -
- have eq_f': "?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2) = (f' n w - g' w) * (2 * of_real pi * \<i>)"
- if cont_fn: "continuous_on (cball z r) (f n)"
- and fnd: "\<And>w. w \<in> ball z r \<Longrightarrow> (f n has_field_derivative f' n w) (at w)" for n
- proof -
- have hol_fn: "f n holomorphic_on ball z r"
- using fnd by (force simp: holomorphic_on_open)
- have "(f n has_field_derivative 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)) (at w)"
- by (rule Cauchy_derivative_integral_circlepath [OF cont_fn hol_fn w])
- then have f': "f' n w = 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)"
- using DERIV_unique [OF fnd] w by blast
- show ?thesis
- by (simp add: f' Cauchy_contour_integral_circlepath_2 [OF g w] derg [OF w] field_split_simps)
- qed
- define d where "d = (r - norm(w - z))^2"
- have "d > 0"
- using w by (simp add: dist_commute dist_norm d_def)
- have dle: "d \<le> cmod ((y - w)\<^sup>2)" if "r = cmod (z - y)" for y
- proof -
- have "w \<in> ball z (cmod (z - y))"
- using that w by fastforce
- then have "cmod (w - z) \<le> cmod (z - y)"
- by (simp add: dist_complex_def norm_minus_commute)
- moreover have "cmod (z - y) - cmod (w - z) \<le> cmod (y - w)"
- by (metis diff_add_cancel diff_add_eq_diff_diff_swap norm_minus_commute norm_triangle_ineq2)
- ultimately show ?thesis
- using that by (simp add: d_def norm_power power_mono)
- qed
- have 1: "\<forall>\<^sub>F n in F. (\<lambda>x. f n x / (x - w)\<^sup>2) contour_integrable_on circlepath z r"
- by (force simp: holomorphic_on_open intro: w Cauchy_derivative_integral_circlepath eventually_mono [OF cont])
- have 2: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)\<^sup>2) (\<lambda>x. g x / (x - w)\<^sup>2) F"
- unfolding uniform_limit_iff
- proof clarify
- fix e::real
- assume "0 < e"
- with \<open>r > 0\<close> show "\<forall>\<^sub>F n in F. \<forall>x\<in>sphere z r. dist (f n x / (x - w)\<^sup>2) (g x / (x - w)\<^sup>2) < e"
- apply (simp add: norm_divide field_split_simps sphere_def dist_norm)
- apply (rule eventually_mono [OF uniform_limitD [OF ulim], of "e*d"])
- apply (simp add: \<open>0 < d\<close>)
- apply (force simp: dist_norm dle intro: less_le_trans)
- done
- qed
- have "((\<lambda>n. contour_integral (circlepath z r) (\<lambda>x. f n x / (x - w)\<^sup>2))
- \<longlongrightarrow> contour_integral (circlepath z r) ((\<lambda>x. g x / (x - w)\<^sup>2))) F"
- by (rule contour_integral_uniform_limit_circlepath [OF 1 2 F \<open>0 < r\<close>])
- then have tendsto_0: "((\<lambda>n. 1 / (2 * of_real pi * \<i>) * (?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2))) \<longlongrightarrow> 0) F"
- using Lim_null by (force intro!: tendsto_mult_right_zero)
- have "((\<lambda>n. f' n w - g' w) \<longlongrightarrow> 0) F"
- apply (rule Lim_transform_eventually [OF tendsto_0])
- apply (force simp: divide_simps intro: eq_f' eventually_mono [OF cont])
- done
- then show ?thesis using Lim_null by blast
- qed
- obtain g' where "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
- by (blast intro: tends_f'n_g' g')
- then show ?thesis using g
- using that by blast
-qed
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Some more simple/convenient versions for applications\<close>
-
-lemma holomorphic_uniform_sequence:
- assumes S: "open S"
- and hol_fn: "\<And>n. (f n) holomorphic_on S"
- and ulim_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially"
- shows "g holomorphic_on S"
-proof -
- have "\<exists>f'. (g has_field_derivative f') (at z)" if "z \<in> S" for z
- proof -
- obtain r where "0 < r" and r: "cball z r \<subseteq> S"
- and ul: "uniform_limit (cball z r) f g sequentially"
- using ulim_g [OF \<open>z \<in> S\<close>] by blast
- have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and> f n holomorphic_on ball z r"
- proof (intro eventuallyI conjI)
- show "continuous_on (cball z r) (f x)" for x
- using hol_fn holomorphic_on_imp_continuous_on holomorphic_on_subset r by blast
- show "f x holomorphic_on ball z r" for x
- by (metis hol_fn holomorphic_on_subset interior_cball interior_subset r)
- qed
- show ?thesis
- apply (rule holomorphic_uniform_limit [OF *])
- using \<open>0 < r\<close> centre_in_ball ul
- apply (auto simp: holomorphic_on_open)
- done
- qed
- with S show ?thesis
- by (simp add: holomorphic_on_open)
-qed
-
-lemma has_complex_derivative_uniform_sequence:
- fixes S :: "complex set"
- assumes S: "open S"
- and hfd: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_field_derivative f' n x) (at x)"
- and ulim_g: "\<And>x. x \<in> S
- \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially"
- shows "\<exists>g'. \<forall>x \<in> S. (g has_field_derivative g' x) (at x) \<and> ((\<lambda>n. f' n x) \<longlongrightarrow> g' x) sequentially"
-proof -
- have y: "\<exists>y. (g has_field_derivative y) (at z) \<and> (\<lambda>n. f' n z) \<longlonglongrightarrow> y" if "z \<in> S" for z
- proof -
- obtain r where "0 < r" and r: "cball z r \<subseteq> S"
- and ul: "uniform_limit (cball z r) f g sequentially"
- using ulim_g [OF \<open>z \<in> S\<close>] by blast
- have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and>
- (\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))"
- proof (intro eventuallyI conjI ballI)
- show "continuous_on (cball z r) (f x)" for x
- by (meson S continuous_on_subset hfd holomorphic_on_imp_continuous_on holomorphic_on_open r)
- show "w \<in> ball z r \<Longrightarrow> (f x has_field_derivative f' x w) (at w)" for w x
- using ball_subset_cball hfd r by blast
- qed
- show ?thesis
- by (rule has_complex_derivative_uniform_limit [OF *, of g]) (use \<open>0 < r\<close> ul in \<open>force+\<close>)
- qed
- show ?thesis
- by (rule bchoice) (blast intro: y)
-qed
-
-subsection\<open>On analytic functions defined by a series\<close>
-
-lemma series_and_derivative_comparison:
- fixes S :: "complex set"
- assumes S: "open S"
- and h: "summable h"
- and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
- and to_g: "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. norm (f n x) \<le> h n"
- obtains g g' where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
-proof -
- obtain g where g: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
- using Weierstrass_m_test_ev [OF to_g h] by force
- have *: "\<exists>d>0. cball x d \<subseteq> S \<and> uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
- if "x \<in> S" for x
- proof -
- obtain d where "d>0" and d: "cball x d \<subseteq> S"
- using open_contains_cball [of "S"] \<open>x \<in> S\<close> S by blast
- show ?thesis
- proof (intro conjI exI)
- show "uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
- using d g uniform_limit_on_subset by (force simp: dist_norm eventually_sequentially)
- qed (use \<open>d > 0\<close> d in auto)
- qed
- have "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i x) \<longlonglongrightarrow> g x"
- by (metis tendsto_uniform_limitI [OF g])
- moreover have "\<exists>g'. \<forall>x\<in>S. (g has_field_derivative g' x) (at x) \<and> (\<lambda>n. \<Sum>i<n. f' i x) \<longlonglongrightarrow> g' x"
- by (rule has_complex_derivative_uniform_sequence [OF S]) (auto intro: * hfd DERIV_sum)+
- ultimately show ?thesis
- by (metis sums_def that)
-qed
-
-text\<open>A version where we only have local uniform/comparative convergence.\<close>
-
-lemma series_and_derivative_comparison_local:
- fixes S :: "complex set"
- assumes S: "open S"
- and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
- and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. norm (f n y) \<le> h n)"
- shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
-proof -
- have "\<exists>y. (\<lambda>n. f n z) sums (\<Sum>n. f n z) \<and> (\<lambda>n. f' n z) sums y \<and> ((\<lambda>x. \<Sum>n. f n x) has_field_derivative y) (at z)"
- if "z \<in> S" for z
- proof -
- obtain d h where "0 < d" "summable h" and le_h: "\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball z d \<inter> S. norm (f n y) \<le> h n"
- using to_g \<open>z \<in> S\<close> by meson
- then obtain r where "r>0" and r: "ball z r \<subseteq> ball z d \<inter> S" using \<open>z \<in> S\<close> S
- by (metis Int_iff open_ball centre_in_ball open_Int open_contains_ball_eq)
- have 1: "open (ball z d \<inter> S)"
- by (simp add: open_Int S)
- have 2: "\<And>n x. x \<in> ball z d \<inter> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
- by (auto simp: hfd)
- obtain g g' where gg': "\<forall>x \<in> ball z d \<inter> S. ((\<lambda>n. f n x) sums g x) \<and>
- ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
- by (auto intro: le_h series_and_derivative_comparison [OF 1 \<open>summable h\<close> hfd])
- then have "(\<lambda>n. f' n z) sums g' z"
- by (meson \<open>0 < r\<close> centre_in_ball contra_subsetD r)
- moreover have "(\<lambda>n. f n z) sums (\<Sum>n. f n z)"
- using summable_sums centre_in_ball \<open>0 < d\<close> \<open>summable h\<close> le_h
- by (metis (full_types) Int_iff gg' summable_def that)
- moreover have "((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' z) (at z)"
- proof (rule has_field_derivative_transform_within)
- show "\<And>x. dist x z < r \<Longrightarrow> g x = (\<Sum>n. f n x)"
- by (metis subsetD dist_commute gg' mem_ball r sums_unique)
- qed (use \<open>0 < r\<close> gg' \<open>z \<in> S\<close> \<open>0 < d\<close> in auto)
- ultimately show ?thesis by auto
- qed
- then show ?thesis
- by (rule_tac x="\<lambda>x. suminf (\<lambda>n. f n x)" in exI) meson
-qed
-
-
-text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close>
-
-lemma series_and_derivative_comparison_complex:
- fixes S :: "complex set"
- assumes S: "open S"
- and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
- and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))"
- shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
-apply (rule series_and_derivative_comparison_local [OF S hfd], assumption)
-apply (rule ex_forward [OF to_g], assumption)
-apply (erule exE)
-apply (rule_tac x="Re \<circ> h" in exI)
-apply (force simp: summable_Re o_def nonneg_Reals_cmod_eq_Re image_subset_iff)
-done
-
-text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close>
-lemma series_differentiable_comparison_complex:
- fixes S :: "complex set"
- assumes S: "open S"
- and hfd: "\<And>n x. x \<in> S \<Longrightarrow> f n field_differentiable (at x)"
- and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))"
- obtains g where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> g field_differentiable (at x)"
-proof -
- have hfd': "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative deriv (f n) x) (at x)"
- using hfd field_differentiable_derivI by blast
- have "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. deriv (f n) x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
- by (metis series_and_derivative_comparison_complex [OF S hfd' to_g])
- then show ?thesis
- using field_differentiable_def that by blast
-qed
-
-text\<open>In particular, a power series is analytic inside circle of convergence.\<close>
-
-lemma power_series_and_derivative_0:
- fixes a :: "nat \<Rightarrow> complex" and r::real
- assumes "summable (\<lambda>n. a n * r^n)"
- shows "\<exists>g g'. \<forall>z. cmod z < r \<longrightarrow>
- ((\<lambda>n. a n * z^n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * z^(n - 1)) sums g' z) \<and> (g has_field_derivative g' z) (at z)"
-proof (cases "0 < r")
- case True
- have der: "\<And>n z. ((\<lambda>x. a n * x ^ n) has_field_derivative of_nat n * a n * z ^ (n - 1)) (at z)"
- by (rule derivative_eq_intros | simp)+
- have y_le: "\<lbrakk>cmod (z - y) * 2 < r - cmod z\<rbrakk> \<Longrightarrow> cmod y \<le> cmod (of_real r + of_real (cmod z)) / 2" for z y
- using \<open>r > 0\<close>
- apply (auto simp: algebra_simps norm_mult norm_divide norm_power simp flip: of_real_add)
- using norm_triangle_ineq2 [of y z]
- apply (simp only: diff_le_eq norm_minus_commute mult_2)
- done
- have "summable (\<lambda>n. a n * complex_of_real r ^ n)"
- using assms \<open>r > 0\<close> by simp
- moreover have "\<And>z. cmod z < r \<Longrightarrow> cmod ((of_real r + of_real (cmod z)) / 2) < cmod (of_real r)"
- using \<open>r > 0\<close>
- by (simp flip: of_real_add)
- ultimately have sum: "\<And>z. cmod z < r \<Longrightarrow> summable (\<lambda>n. of_real (cmod (a n)) * ((of_real r + complex_of_real (cmod z)) / 2) ^ n)"
- by (rule power_series_conv_imp_absconv_weak)
- have "\<exists>g g'. \<forall>z \<in> ball 0 r. (\<lambda>n. (a n) * z ^ n) sums g z \<and>
- (\<lambda>n. of_nat n * (a n) * z ^ (n - 1)) sums g' z \<and> (g has_field_derivative g' z) (at z)"
- apply (rule series_and_derivative_comparison_complex [OF open_ball der])
- apply (rule_tac x="(r - norm z)/2" in exI)
- apply (rule_tac x="\<lambda>n. of_real(norm(a n)*((r + norm z)/2)^n)" in exI)
- using \<open>r > 0\<close>
- apply (auto simp: sum eventually_sequentially norm_mult norm_power dist_norm intro!: mult_left_mono power_mono y_le)
- done
- then show ?thesis
- by (simp add: ball_def)
-next
- case False then show ?thesis
- apply (simp add: not_less)
- using less_le_trans norm_not_less_zero by blast
-qed
-
-proposition\<^marker>\<open>tag unimportant\<close> power_series_and_derivative:
- fixes a :: "nat \<Rightarrow> complex" and r::real
- assumes "summable (\<lambda>n. a n * r^n)"
- obtains g g' where "\<forall>z \<in> ball w r.
- ((\<lambda>n. a n * (z - w) ^ n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * (z - w) ^ (n - 1)) sums g' z) \<and>
- (g has_field_derivative g' z) (at z)"
- using power_series_and_derivative_0 [OF assms]
- apply clarify
- apply (rule_tac g="(\<lambda>z. g(z - w))" in that)
- using DERIV_shift [where z="-w"]
- apply (auto simp: norm_minus_commute Ball_def dist_norm)
- done
-
-proposition\<^marker>\<open>tag unimportant\<close> power_series_holomorphic:
- assumes "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>n. a n*(w - z)^n) sums f w)"
- shows "f holomorphic_on ball z r"
-proof -
- have "\<exists>f'. (f has_field_derivative f') (at w)" if w: "dist z w < r" for w
- proof -
- have inb: "z + complex_of_real ((dist z w + r) / 2) \<in> ball z r"
- proof -
- have wz: "cmod (w - z) < r" using w
- by (auto simp: field_split_simps dist_norm norm_minus_commute)
- then have "0 \<le> r"
- by (meson less_eq_real_def norm_ge_zero order_trans)
- show ?thesis
- using w by (simp add: dist_norm \<open>0\<le>r\<close> flip: of_real_add)
- qed
- have sum: "summable (\<lambda>n. a n * of_real (((cmod (z - w) + r) / 2) ^ n))"
- using assms [OF inb] by (force simp: summable_def dist_norm)
- obtain g g' where gg': "\<And>u. u \<in> ball z ((cmod (z - w) + r) / 2) \<Longrightarrow>
- (\<lambda>n. a n * (u - z) ^ n) sums g u \<and>
- (\<lambda>n. of_nat n * a n * (u - z) ^ (n - 1)) sums g' u \<and> (g has_field_derivative g' u) (at u)"
- by (rule power_series_and_derivative [OF sum, of z]) fastforce
- have [simp]: "g u = f u" if "cmod (u - w) < (r - cmod (z - w)) / 2" for u
- proof -
- have less: "cmod (z - u) * 2 < cmod (z - w) + r"
- using that dist_triangle2 [of z u w]
- by (simp add: dist_norm [symmetric] algebra_simps)
- show ?thesis
- apply (rule sums_unique2 [of "\<lambda>n. a n*(u - z)^n"])
- using gg' [of u] less w
- apply (auto simp: assms dist_norm)
- done
- qed
- have "(f has_field_derivative g' w) (at w)"
- by (rule has_field_derivative_transform_within [where d="(r - norm(z - w))/2"])
- (use w gg' [of w] in \<open>(force simp: dist_norm)+\<close>)
- then show ?thesis ..
- qed
- then show ?thesis by (simp add: holomorphic_on_open)
-qed
-
-corollary holomorphic_iff_power_series:
- "f holomorphic_on ball z r \<longleftrightarrow>
- (\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
- apply (intro iffI ballI holomorphic_power_series, assumption+)
- apply (force intro: power_series_holomorphic [where a = "\<lambda>n. (deriv ^^ n) f z / (fact n)"])
- done
-
-lemma power_series_analytic:
- "(\<And>w. w \<in> ball z r \<Longrightarrow> (\<lambda>n. a n*(w - z)^n) sums f w) \<Longrightarrow> f analytic_on ball z r"
- by (force simp: analytic_on_open intro!: power_series_holomorphic)
-
-lemma analytic_iff_power_series:
- "f analytic_on ball z r \<longleftrightarrow>
- (\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
- by (simp add: analytic_on_open holomorphic_iff_power_series)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Equality between holomorphic functions, on open ball then connected set\<close>
-
-lemma holomorphic_fun_eq_on_ball:
- "\<lbrakk>f holomorphic_on ball z r; g holomorphic_on ball z r;
- w \<in> ball z r;
- \<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z\<rbrakk>
- \<Longrightarrow> f w = g w"
- apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
- apply (auto simp: holomorphic_iff_power_series)
- done
-
-lemma holomorphic_fun_eq_0_on_ball:
- "\<lbrakk>f holomorphic_on ball z r; w \<in> ball z r;
- \<And>n. (deriv ^^ n) f z = 0\<rbrakk>
- \<Longrightarrow> f w = 0"
- apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
- apply (auto simp: holomorphic_iff_power_series)
- done
-
-lemma holomorphic_fun_eq_0_on_connected:
- assumes holf: "f holomorphic_on S" and "open S"
- and cons: "connected S"
- and der: "\<And>n. (deriv ^^ n) f z = 0"
- and "z \<in> S" "w \<in> S"
- shows "f w = 0"
-proof -
- have *: "ball x e \<subseteq> (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
- if "\<forall>u. (deriv ^^ u) f x = 0" "ball x e \<subseteq> S" for x e
- proof -
- have "\<And>x' n. dist x x' < e \<Longrightarrow> (deriv ^^ n) f x' = 0"
- apply (rule holomorphic_fun_eq_0_on_ball [OF holomorphic_higher_deriv])
- apply (rule holomorphic_on_subset [OF holf])
- using that apply simp_all
- by (metis funpow_add o_apply)
- with that show ?thesis by auto
- qed
- have 1: "openin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
- apply (rule open_subset, force)
- using \<open>open S\<close>
- apply (simp add: open_contains_ball Ball_def)
- apply (erule all_forward)
- using "*" by auto blast+
- have 2: "closedin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
- using assms
- by (auto intro: continuous_closedin_preimage_constant holomorphic_on_imp_continuous_on holomorphic_higher_deriv)
- obtain e where "e>0" and e: "ball w e \<subseteq> S" using openE [OF \<open>open S\<close> \<open>w \<in> S\<close>] .
- then have holfb: "f holomorphic_on ball w e"
- using holf holomorphic_on_subset by blast
- have 3: "(\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}) = S \<Longrightarrow> f w = 0"
- using \<open>e>0\<close> e by (force intro: holomorphic_fun_eq_0_on_ball [OF holfb])
- show ?thesis
- using cons der \<open>z \<in> S\<close>
- apply (simp add: connected_clopen)
- apply (drule_tac x="\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}" in spec)
- apply (auto simp: 1 2 3)
- done
-qed
-
-lemma holomorphic_fun_eq_on_connected:
- assumes "f holomorphic_on S" "g holomorphic_on S" and "open S" "connected S"
- and "\<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z"
- and "z \<in> S" "w \<in> S"
- shows "f w = g w"
-proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>x. f x - g x" S z, simplified])
- show "(\<lambda>x. f x - g x) holomorphic_on S"
- by (intro assms holomorphic_intros)
- show "\<And>n. (deriv ^^ n) (\<lambda>x. f x - g x) z = 0"
- using assms higher_deriv_diff by auto
-qed (use assms in auto)
-
-lemma holomorphic_fun_eq_const_on_connected:
- assumes holf: "f holomorphic_on S" and "open S"
- and cons: "connected S"
- and der: "\<And>n. 0 < n \<Longrightarrow> (deriv ^^ n) f z = 0"
- and "z \<in> S" "w \<in> S"
- shows "f w = f z"
-proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>w. f w - f z" S z, simplified])
- show "(\<lambda>w. f w - f z) holomorphic_on S"
- by (intro assms holomorphic_intros)
- show "\<And>n. (deriv ^^ n) (\<lambda>w. f w - f z) z = 0"
- by (subst higher_deriv_diff) (use assms in \<open>auto intro: holomorphic_intros\<close>)
-qed (use assms in auto)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Some basic lemmas about poles/singularities\<close>
-
-lemma pole_lemma:
- assumes holf: "f holomorphic_on S" and a: "a \<in> interior S"
- shows "(\<lambda>z. if z = a then deriv f a
- else (f z - f a) / (z - a)) holomorphic_on S" (is "?F holomorphic_on S")
-proof -
- have F1: "?F field_differentiable (at u within S)" if "u \<in> S" "u \<noteq> a" for u
- proof -
- have fcd: "f field_differentiable at u within S"
- using holf holomorphic_on_def by (simp add: \<open>u \<in> S\<close>)
- have cd: "(\<lambda>z. (f z - f a) / (z - a)) field_differentiable at u within S"
- by (rule fcd derivative_intros | simp add: that)+
- have "0 < dist a u" using that dist_nz by blast
- then show ?thesis
- by (rule field_differentiable_transform_within [OF _ _ _ cd]) (auto simp: \<open>u \<in> S\<close>)
- qed
- have F2: "?F field_differentiable at a" if "0 < e" "ball a e \<subseteq> S" for e
- proof -
- have holfb: "f holomorphic_on ball a e"
- by (rule holomorphic_on_subset [OF holf \<open>ball a e \<subseteq> S\<close>])
- have 2: "?F holomorphic_on ball a e - {a}"
- apply (simp add: holomorphic_on_def flip: field_differentiable_def)
- using mem_ball that
- apply (auto intro: F1 field_differentiable_within_subset)
- done
- have "isCont (\<lambda>z. if z = a then deriv f a else (f z - f a) / (z - a)) x"
- if "dist a x < e" for x
- proof (cases "x=a")
- case True
- then have "f field_differentiable at a"
- using holfb \<open>0 < e\<close> holomorphic_on_imp_differentiable_at by auto
- with True show ?thesis
- by (auto simp: continuous_at has_field_derivative_iff simp flip: DERIV_deriv_iff_field_differentiable
- elim: rev_iffD1 [OF _ LIM_equal])
- next
- case False with 2 that show ?thesis
- by (force simp: holomorphic_on_open open_Diff field_differentiable_def [symmetric] field_differentiable_imp_continuous_at)
- qed
- then have 1: "continuous_on (ball a e) ?F"
- by (clarsimp simp: continuous_on_eq_continuous_at)
- have "?F holomorphic_on ball a e"
- by (auto intro: no_isolated_singularity [OF 1 2])
- with that show ?thesis
- by (simp add: holomorphic_on_open field_differentiable_def [symmetric]
- field_differentiable_at_within)
- qed
- show ?thesis
- proof
- fix x assume "x \<in> S" show "?F field_differentiable at x within S"
- proof (cases "x=a")
- case True then show ?thesis
- using a by (auto simp: mem_interior intro: field_differentiable_at_within F2)
- next
- case False with F1 \<open>x \<in> S\<close>
- show ?thesis by blast
- qed
- qed
-qed
-
-lemma pole_theorem:
- assumes holg: "g holomorphic_on S" and a: "a \<in> interior S"
- and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
- shows "(\<lambda>z. if z = a then deriv g a
- else f z - g a/(z - a)) holomorphic_on S"
- using pole_lemma [OF holg a]
- by (rule holomorphic_transform) (simp add: eq field_split_simps)
-
-lemma pole_lemma_open:
- assumes "f holomorphic_on S" "open S"
- shows "(\<lambda>z. if z = a then deriv f a else (f z - f a)/(z - a)) holomorphic_on S"
-proof (cases "a \<in> S")
- case True with assms interior_eq pole_lemma
- show ?thesis by fastforce
-next
- case False with assms show ?thesis
- apply (simp add: holomorphic_on_def field_differentiable_def [symmetric], clarify)
- apply (rule field_differentiable_transform_within [where f = "\<lambda>z. (f z - f a)/(z - a)" and d = 1])
- apply (rule derivative_intros | force)+
- done
-qed
-
-lemma pole_theorem_open:
- assumes holg: "g holomorphic_on S" and S: "open S"
- and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
- shows "(\<lambda>z. if z = a then deriv g a
- else f z - g a/(z - a)) holomorphic_on S"
- using pole_lemma_open [OF holg S]
- by (rule holomorphic_transform) (auto simp: eq divide_simps)
-
-lemma pole_theorem_0:
- assumes holg: "g holomorphic_on S" and a: "a \<in> interior S"
- and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
- and [simp]: "f a = deriv g a" "g a = 0"
- shows "f holomorphic_on S"
- using pole_theorem [OF holg a eq]
- by (rule holomorphic_transform) (auto simp: eq field_split_simps)
-
-lemma pole_theorem_open_0:
- assumes holg: "g holomorphic_on S" and S: "open S"
- and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
- and [simp]: "f a = deriv g a" "g a = 0"
- shows "f holomorphic_on S"
- using pole_theorem_open [OF holg S eq]
- by (rule holomorphic_transform) (auto simp: eq field_split_simps)
-
-lemma pole_theorem_analytic:
- assumes g: "g analytic_on S"
- and eq: "\<And>z. z \<in> S
- \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)"
- shows "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S" (is "?F analytic_on S")
- unfolding analytic_on_def
-proof
- fix x
- assume "x \<in> S"
- with g obtain e where "0 < e" and e: "g holomorphic_on ball x e"
- by (auto simp add: analytic_on_def)
- obtain d where "0 < d" and d: "\<And>w. w \<in> ball x d - {a} \<Longrightarrow> g w = (w - a) * f w"
- using \<open>x \<in> S\<close> eq by blast
- have "?F holomorphic_on ball x (min d e)"
- using d e \<open>x \<in> S\<close> by (fastforce simp: holomorphic_on_subset subset_ball intro!: pole_theorem_open)
- then show "\<exists>e>0. ?F holomorphic_on ball x e"
- using \<open>0 < d\<close> \<open>0 < e\<close> not_le by fastforce
-qed
-
-lemma pole_theorem_analytic_0:
- assumes g: "g analytic_on S"
- and eq: "\<And>z. z \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)"
- and [simp]: "f a = deriv g a" "g a = 0"
- shows "f analytic_on S"
-proof -
- have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f"
- by auto
- show ?thesis
- using pole_theorem_analytic [OF g eq] by simp
-qed
-
-lemma pole_theorem_analytic_open_superset:
- assumes g: "g analytic_on S" and "S \<subseteq> T" "open T"
- and eq: "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z"
- shows "(\<lambda>z. if z = a then deriv g a
- else f z - g a/(z - a)) analytic_on S"
-proof (rule pole_theorem_analytic [OF g])
- fix z
- assume "z \<in> S"
- then obtain e where "0 < e" and e: "ball z e \<subseteq> T"
- using assms openE by blast
- then show "\<exists>d>0. \<forall>w\<in>ball z d - {a}. g w = (w - a) * f w"
- using eq by auto
-qed
-
-lemma pole_theorem_analytic_open_superset_0:
- assumes g: "g analytic_on S" "S \<subseteq> T" "open T" "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z"
- and [simp]: "f a = deriv g a" "g a = 0"
- shows "f analytic_on S"
-proof -
- have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f"
- by auto
- have "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S"
- by (rule pole_theorem_analytic_open_superset [OF g])
- then show ?thesis by simp
-qed
-
-
-subsection\<open>General, homology form of Cauchy's theorem\<close>
-
-text\<open>Proof is based on Dixon's, as presented in Lang's "Complex Analysis" book (page 147).\<close>
-
-lemma contour_integral_continuous_on_linepath_2D:
- assumes "open U" and cont_dw: "\<And>w. w \<in> U \<Longrightarrow> F w contour_integrable_on (linepath a b)"
- and cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). F x y)"
- and abu: "closed_segment a b \<subseteq> U"
- shows "continuous_on U (\<lambda>w. contour_integral (linepath a b) (F w))"
-proof -
- have *: "\<exists>d>0. \<forall>x'\<in>U. dist x' w < d \<longrightarrow>
- dist (contour_integral (linepath a b) (F x'))
- (contour_integral (linepath a b) (F w)) \<le> \<epsilon>"
- if "w \<in> U" "0 < \<epsilon>" "a \<noteq> b" for w \<epsilon>
- proof -
- obtain \<delta> where "\<delta>>0" and \<delta>: "cball w \<delta> \<subseteq> U" using open_contains_cball \<open>open U\<close> \<open>w \<in> U\<close> by force
- let ?TZ = "cball w \<delta> \<times> closed_segment a b"
- have "uniformly_continuous_on ?TZ (\<lambda>(x,y). F x y)"
- proof (rule compact_uniformly_continuous)
- show "continuous_on ?TZ (\<lambda>(x,y). F x y)"
- by (rule continuous_on_subset[OF cond_uu]) (use SigmaE \<delta> abu in blast)
- show "compact ?TZ"
- by (simp add: compact_Times)
- qed
- then obtain \<eta> where "\<eta>>0"
- and \<eta>: "\<And>x x'. \<lbrakk>x\<in>?TZ; x'\<in>?TZ; dist x' x < \<eta>\<rbrakk> \<Longrightarrow>
- dist ((\<lambda>(x,y). F x y) x') ((\<lambda>(x,y). F x y) x) < \<epsilon>/norm(b - a)"
- apply (rule uniformly_continuous_onE [where e = "\<epsilon>/norm(b - a)"])
- using \<open>0 < \<epsilon>\<close> \<open>a \<noteq> b\<close> by auto
- have \<eta>: "\<lbrakk>norm (w - x1) \<le> \<delta>; x2 \<in> closed_segment a b;
- norm (w - x1') \<le> \<delta>; x2' \<in> closed_segment a b; norm ((x1', x2') - (x1, x2)) < \<eta>\<rbrakk>
- \<Longrightarrow> norm (F x1' x2' - F x1 x2) \<le> \<epsilon> / cmod (b - a)"
- for x1 x2 x1' x2'
- using \<eta> [of "(x1,x2)" "(x1',x2')"] by (force simp: dist_norm)
- have le_ee: "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>"
- if "x' \<in> U" "cmod (x' - w) < \<delta>" "cmod (x' - w) < \<eta>" for x'
- proof -
- have "(\<lambda>x. F x' x - F w x) contour_integrable_on linepath a b"
- by (simp add: \<open>w \<in> U\<close> cont_dw contour_integrable_diff that)
- then have "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>/norm(b - a) * norm(b - a)"
- apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_integral _ \<eta>])
- using \<open>0 < \<epsilon>\<close> \<open>0 < \<delta>\<close> that apply (auto simp: norm_minus_commute)
- done
- also have "\<dots> = \<epsilon>" using \<open>a \<noteq> b\<close> by simp
- finally show ?thesis .
- qed
- show ?thesis
- apply (rule_tac x="min \<delta> \<eta>" in exI)
- using \<open>0 < \<delta>\<close> \<open>0 < \<eta>\<close>
- apply (auto simp: dist_norm contour_integral_diff [OF cont_dw cont_dw, symmetric] \<open>w \<in> U\<close> intro: le_ee)
- done
- qed
- show ?thesis
- proof (cases "a=b")
- case True
- then show ?thesis by simp
- next
- case False
- show ?thesis
- by (rule continuous_onI) (use False in \<open>auto intro: *\<close>)
- qed
-qed
-
-text\<open>This version has \<^term>\<open>polynomial_function \<gamma>\<close> as an additional assumption.\<close>
-lemma Cauchy_integral_formula_global_weak:
- assumes "open U" and holf: "f holomorphic_on U"
- and z: "z \<in> U" and \<gamma>: "polynomial_function \<gamma>"
- and pasz: "path_image \<gamma> \<subseteq> U - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- and zero: "\<And>w. w \<notin> U \<Longrightarrow> winding_number \<gamma> w = 0"
- shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
-proof -
- obtain \<gamma>' where pf\<gamma>': "polynomial_function \<gamma>'" and \<gamma>': "\<And>x. (\<gamma> has_vector_derivative (\<gamma>' x)) (at x)"
- using has_vector_derivative_polynomial_function [OF \<gamma>] by blast
- then have "bounded(path_image \<gamma>')"
- by (simp add: path_image_def compact_imp_bounded compact_continuous_image continuous_on_polymonial_function)
- then obtain B where "B>0" and B: "\<And>x. x \<in> path_image \<gamma>' \<Longrightarrow> norm x \<le> B"
- using bounded_pos by force
- define d where [abs_def]: "d z w = (if w = z then deriv f z else (f w - f z)/(w - z))" for z w
- define v where "v = {w. w \<notin> path_image \<gamma> \<and> winding_number \<gamma> w = 0}"
- have "path \<gamma>" "valid_path \<gamma>" using \<gamma>
- by (auto simp: path_polynomial_function valid_path_polynomial_function)
- then have ov: "open v"
- by (simp add: v_def open_winding_number_levelsets loop)
- have uv_Un: "U \<union> v = UNIV"
- using pasz zero by (auto simp: v_def)
- have conf: "continuous_on U f"
- by (metis holf holomorphic_on_imp_continuous_on)
- have hol_d: "(d y) holomorphic_on U" if "y \<in> U" for y
- proof -
- have *: "(\<lambda>c. if c = y then deriv f y else (f c - f y) / (c - y)) holomorphic_on U"
- by (simp add: holf pole_lemma_open \<open>open U\<close>)
- then have "isCont (\<lambda>x. if x = y then deriv f y else (f x - f y) / (x - y)) y"
- using at_within_open field_differentiable_imp_continuous_at holomorphic_on_def that \<open>open U\<close> by fastforce
- then have "continuous_on U (d y)"
- apply (simp add: d_def continuous_on_eq_continuous_at \<open>open U\<close>, clarify)
- using * holomorphic_on_def
- by (meson field_differentiable_within_open field_differentiable_imp_continuous_at \<open>open U\<close>)
- moreover have "d y holomorphic_on U - {y}"
- proof -
- have "\<And>w. w \<in> U - {y} \<Longrightarrow>
- (\<lambda>w. if w = y then deriv f y else (f w - f y) / (w - y)) field_differentiable at w"
- apply (rule_tac d="dist w y" and f = "\<lambda>w. (f w - f y)/(w - y)" in field_differentiable_transform_within)
- apply (auto simp: dist_pos_lt dist_commute intro!: derivative_intros)
- using \<open>open U\<close> holf holomorphic_on_imp_differentiable_at by blast
- then show ?thesis
- unfolding field_differentiable_def by (simp add: d_def holomorphic_on_open \<open>open U\<close> open_delete)
- qed
- ultimately show ?thesis
- by (rule no_isolated_singularity) (auto simp: \<open>open U\<close>)
- qed
- have cint_fxy: "(\<lambda>x. (f x - f y) / (x - y)) contour_integrable_on \<gamma>" if "y \<notin> path_image \<gamma>" for y
- proof (rule contour_integrable_holomorphic_simple [where S = "U-{y}"])
- show "(\<lambda>x. (f x - f y) / (x - y)) holomorphic_on U - {y}"
- by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
- show "path_image \<gamma> \<subseteq> U - {y}"
- using pasz that by blast
- qed (auto simp: \<open>open U\<close> open_delete \<open>valid_path \<gamma>\<close>)
- define h where
- "h z = (if z \<in> U then contour_integral \<gamma> (d z) else contour_integral \<gamma> (\<lambda>w. f w/(w - z)))" for z
- have U: "((d z) has_contour_integral h z) \<gamma>" if "z \<in> U" for z
- proof -
- have "d z holomorphic_on U"
- by (simp add: hol_d that)
- with that show ?thesis
- apply (simp add: h_def)
- by (meson Diff_subset \<open>open U\<close> \<open>valid_path \<gamma>\<close> contour_integrable_holomorphic_simple has_contour_integral_integral pasz subset_trans)
- qed
- have V: "((\<lambda>w. f w / (w - z)) has_contour_integral h z) \<gamma>" if z: "z \<in> v" for z
- proof -
- have 0: "0 = (f z) * 2 * of_real (2 * pi) * \<i> * winding_number \<gamma> z"
- using v_def z by auto
- then have "((\<lambda>x. 1 / (x - z)) has_contour_integral 0) \<gamma>"
- using z v_def has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close>] by fastforce
- then have "((\<lambda>x. f z * (1 / (x - z))) has_contour_integral 0) \<gamma>"
- using has_contour_integral_lmul by fastforce
- then have "((\<lambda>x. f z / (x - z)) has_contour_integral 0) \<gamma>"
- by (simp add: field_split_simps)
- moreover have "((\<lambda>x. (f x - f z) / (x - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
- using z
- apply (auto simp: v_def)
- apply (metis (no_types, lifting) contour_integrable_eq d_def has_contour_integral_eq has_contour_integral_integral cint_fxy)
- done
- ultimately have *: "((\<lambda>x. f z / (x - z) + (f x - f z) / (x - z)) has_contour_integral (0 + contour_integral \<gamma> (d z))) \<gamma>"
- by (rule has_contour_integral_add)
- have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
- if "z \<in> U"
- using * by (auto simp: divide_simps has_contour_integral_eq)
- moreover have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. f w / (w - z))) \<gamma>"
- if "z \<notin> U"
- apply (rule has_contour_integral_integral [OF contour_integrable_holomorphic_simple [where S=U]])
- using U pasz \<open>valid_path \<gamma>\<close> that
- apply (auto intro: holomorphic_on_imp_continuous_on hol_d)
- apply (rule continuous_intros conf holomorphic_intros holf assms | force)+
- done
- ultimately show ?thesis
- using z by (simp add: h_def)
- qed
- have znot: "z \<notin> path_image \<gamma>"
- using pasz by blast
- obtain d0 where "d0>0" and d0: "\<And>x y. x \<in> path_image \<gamma> \<Longrightarrow> y \<in> - U \<Longrightarrow> d0 \<le> dist x y"
- using separate_compact_closed [of "path_image \<gamma>" "-U"] pasz \<open>open U\<close> \<open>path \<gamma>\<close> compact_path_image
- by blast
- obtain dd where "0 < dd" and dd: "{y + k | y k. y \<in> path_image \<gamma> \<and> k \<in> ball 0 dd} \<subseteq> U"
- apply (rule that [of "d0/2"])
- using \<open>0 < d0\<close>
- apply (auto simp: dist_norm dest: d0)
- done
- have "\<And>x x'. \<lbrakk>x \<in> path_image \<gamma>; dist x x' * 2 < dd\<rbrakk> \<Longrightarrow> \<exists>y k. x' = y + k \<and> y \<in> path_image \<gamma> \<and> dist 0 k * 2 \<le> dd"
- apply (rule_tac x=x in exI)
- apply (rule_tac x="x'-x" in exI)
- apply (force simp: dist_norm)
- done
- then have 1: "path_image \<gamma> \<subseteq> interior {y + k |y k. y \<in> path_image \<gamma> \<and> k \<in> cball 0 (dd / 2)}"
- apply (clarsimp simp add: mem_interior)
- using \<open>0 < dd\<close>
- apply (rule_tac x="dd/2" in exI, auto)
- done
- obtain T where "compact T" and subt: "path_image \<gamma> \<subseteq> interior T" and T: "T \<subseteq> U"
- apply (rule that [OF _ 1])
- apply (fastforce simp add: \<open>valid_path \<gamma>\<close> compact_valid_path_image intro!: compact_sums)
- apply (rule order_trans [OF _ dd])
- using \<open>0 < dd\<close> by fastforce
- obtain L where "L>0"
- and L: "\<And>f B. \<lbrakk>f holomorphic_on interior T; \<And>z. z\<in>interior T \<Longrightarrow> cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
- cmod (contour_integral \<gamma> f) \<le> L * B"
- using contour_integral_bound_exists [OF open_interior \<open>valid_path \<gamma>\<close> subt]
- by blast
- have "bounded(f ` T)"
- by (meson \<open>compact T\<close> compact_continuous_image compact_imp_bounded conf continuous_on_subset T)
- then obtain D where "D>0" and D: "\<And>x. x \<in> T \<Longrightarrow> norm (f x) \<le> D"
- by (auto simp: bounded_pos)
- obtain C where "C>0" and C: "\<And>x. x \<in> T \<Longrightarrow> norm x \<le> C"
- using \<open>compact T\<close> bounded_pos compact_imp_bounded by force
- have "dist (h y) 0 \<le> e" if "0 < e" and le: "D * L / e + C \<le> cmod y" for e y
- proof -
- have "D * L / e > 0" using \<open>D>0\<close> \<open>L>0\<close> \<open>e>0\<close> by simp
- with le have ybig: "norm y > C" by force
- with C have "y \<notin> T" by force
- then have ynot: "y \<notin> path_image \<gamma>"
- using subt interior_subset by blast
- have [simp]: "winding_number \<gamma> y = 0"
- apply (rule winding_number_zero_outside [of _ "cball 0 C"])
- using ybig interior_subset subt
- apply (force simp: loop \<open>path \<gamma>\<close> dist_norm intro!: C)+
- done
- have [simp]: "h y = contour_integral \<gamma> (\<lambda>w. f w/(w - y))"
- by (rule contour_integral_unique [symmetric]) (simp add: v_def ynot V)
- have holint: "(\<lambda>w. f w / (w - y)) holomorphic_on interior T"
- apply (rule holomorphic_on_divide)
- using holf holomorphic_on_subset interior_subset T apply blast
- apply (rule holomorphic_intros)+
- using \<open>y \<notin> T\<close> interior_subset by auto
- have leD: "cmod (f z / (z - y)) \<le> D * (e / L / D)" if z: "z \<in> interior T" for z
- proof -
- have "D * L / e + cmod z \<le> cmod y"
- using le C [of z] z using interior_subset by force
- then have DL2: "D * L / e \<le> cmod (z - y)"
- using norm_triangle_ineq2 [of y z] by (simp add: norm_minus_commute)
- have "cmod (f z / (z - y)) = cmod (f z) * inverse (cmod (z - y))"
- by (simp add: norm_mult norm_inverse Fields.field_class.field_divide_inverse)
- also have "\<dots> \<le> D * (e / L / D)"
- apply (rule mult_mono)
- using that D interior_subset apply blast
- using \<open>L>0\<close> \<open>e>0\<close> \<open>D>0\<close> DL2
- apply (auto simp: norm_divide field_split_simps)
- done
- finally show ?thesis .
- qed
- have "dist (h y) 0 = cmod (contour_integral \<gamma> (\<lambda>w. f w / (w - y)))"
- by (simp add: dist_norm)
- also have "\<dots> \<le> L * (D * (e / L / D))"
- by (rule L [OF holint leD])
- also have "\<dots> = e"
- using \<open>L>0\<close> \<open>0 < D\<close> by auto
- finally show ?thesis .
- qed
- then have "(h \<longlongrightarrow> 0) at_infinity"
- by (meson Lim_at_infinityI)
- moreover have "h holomorphic_on UNIV"
- proof -
- have con_ff: "continuous (at (x,z)) (\<lambda>(x,y). (f y - f x) / (y - x))"
- if "x \<in> U" "z \<in> U" "x \<noteq> z" for x z
- using that conf
- apply (simp add: split_def continuous_on_eq_continuous_at \<open>open U\<close>)
- apply (simp | rule continuous_intros continuous_within_compose2 [where g=f])+
- done
- have con_fstsnd: "continuous_on UNIV (\<lambda>x. (fst x - snd x) ::complex)"
- by (rule continuous_intros)+
- have open_uu_Id: "open (U \<times> U - Id)"
- apply (rule open_Diff)
- apply (simp add: open_Times \<open>open U\<close>)
- using continuous_closed_preimage_constant [OF con_fstsnd closed_UNIV, of 0]
- apply (auto simp: Id_fstsnd_eq algebra_simps)
- done
- have con_derf: "continuous (at z) (deriv f)" if "z \<in> U" for z
- apply (rule continuous_on_interior [of U])
- apply (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on \<open>open U\<close>)
- by (simp add: interior_open that \<open>open U\<close>)
- have tendsto_f': "((\<lambda>(x,y). if y = x then deriv f (x)
- else (f (y) - f (x)) / (y - x)) \<longlongrightarrow> deriv f x)
- (at (x, x) within U \<times> U)" if "x \<in> U" for x
- proof (rule Lim_withinI)
- fix e::real assume "0 < e"
- obtain k1 where "k1>0" and k1: "\<And>x'. norm (x' - x) \<le> k1 \<Longrightarrow> norm (deriv f x' - deriv f x) < e"
- using \<open>0 < e\<close> continuous_within_E [OF con_derf [OF \<open>x \<in> U\<close>]]
- by (metis UNIV_I dist_norm)
- obtain k2 where "k2>0" and k2: "ball x k2 \<subseteq> U"
- by (blast intro: openE [OF \<open>open U\<close>] \<open>x \<in> U\<close>)
- have neq: "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e"
- if "z' \<noteq> x'" and less_k1: "norm (x'-x, z'-x) < k1" and less_k2: "norm (x'-x, z'-x) < k2"
- for x' z'
- proof -
- have cs_less: "w \<in> closed_segment x' z' \<Longrightarrow> cmod (w - x) \<le> norm (x'-x, z'-x)" for w
- apply (drule segment_furthest_le [where y=x])
- by (metis (no_types) dist_commute dist_norm norm_fst_le norm_snd_le order_trans)
- have derf_le: "w \<in> closed_segment x' z' \<Longrightarrow> z' \<noteq> x' \<Longrightarrow> cmod (deriv f w - deriv f x) \<le> e" for w
- by (blast intro: cs_less less_k1 k1 [unfolded divide_const_simps dist_norm] less_imp_le le_less_trans)
- have f_has_der: "\<And>x. x \<in> U \<Longrightarrow> (f has_field_derivative deriv f x) (at x within U)"
- by (metis DERIV_deriv_iff_field_differentiable at_within_open holf holomorphic_on_def \<open>open U\<close>)
- have "closed_segment x' z' \<subseteq> U"
- by (rule order_trans [OF _ k2]) (simp add: cs_less le_less_trans [OF _ less_k2] dist_complex_def norm_minus_commute subset_iff)
- then have cint_derf: "(deriv f has_contour_integral f z' - f x') (linepath x' z')"
- using contour_integral_primitive [OF f_has_der valid_path_linepath] pasz by simp
- then have *: "((\<lambda>x. deriv f x / (z' - x')) has_contour_integral (f z' - f x') / (z' - x')) (linepath x' z')"
- by (rule has_contour_integral_div)
- have "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e/norm(z' - x') * norm(z' - x')"
- apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_diff [OF *]])
- using has_contour_integral_div [where c = "z' - x'", OF has_contour_integral_const_linepath [of "deriv f x" z' x']]
- \<open>e > 0\<close> \<open>z' \<noteq> x'\<close>
- apply (auto simp: norm_divide divide_simps derf_le)
- done
- also have "\<dots> \<le> e" using \<open>0 < e\<close> by simp
- finally show ?thesis .
- qed
- show "\<exists>d>0. \<forall>xa\<in>U \<times> U.
- 0 < dist xa (x, x) \<and> dist xa (x, x) < d \<longrightarrow>
- dist (case xa of (x, y) \<Rightarrow> if y = x then deriv f x else (f y - f x) / (y - x)) (deriv f x) \<le> e"
- apply (rule_tac x="min k1 k2" in exI)
- using \<open>k1>0\<close> \<open>k2>0\<close> \<open>e>0\<close>
- apply (force simp: dist_norm neq intro: dual_order.strict_trans2 k1 less_imp_le norm_fst_le)
- done
- qed
- have con_pa_f: "continuous_on (path_image \<gamma>) f"
- by (meson holf holomorphic_on_imp_continuous_on holomorphic_on_subset interior_subset subt T)
- have le_B: "\<And>T. T \<in> {0..1} \<Longrightarrow> cmod (vector_derivative \<gamma> (at T)) \<le> B"
- apply (rule B)
- using \<gamma>' using path_image_def vector_derivative_at by fastforce
- have f_has_cint: "\<And>w. w \<in> v - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f u / (u - w) ^ 1) has_contour_integral h w) \<gamma>"
- by (simp add: V)
- have cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). d x y)"
- apply (simp add: continuous_on_eq_continuous_within d_def continuous_within tendsto_f')
- apply (simp add: tendsto_within_open_NO_MATCH open_Times \<open>open U\<close>, clarify)
- apply (rule Lim_transform_within_open [OF _ open_uu_Id, where f = "(\<lambda>(x,y). (f y - f x) / (y - x))"])
- using con_ff
- apply (auto simp: continuous_within)
- done
- have hol_dw: "(\<lambda>z. d z w) holomorphic_on U" if "w \<in> U" for w
- proof -
- have "continuous_on U ((\<lambda>(x,y). d x y) \<circ> (\<lambda>z. (w,z)))"
- by (rule continuous_on_compose continuous_intros continuous_on_subset [OF cond_uu] | force intro: that)+
- then have *: "continuous_on U (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z))"
- by (rule rev_iffD1 [OF _ continuous_on_cong [OF refl]]) (simp add: d_def field_simps)
- have **: "\<And>x. \<lbrakk>x \<in> U; x \<noteq> w\<rbrakk> \<Longrightarrow> (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z)) field_differentiable at x"
- apply (rule_tac f = "\<lambda>x. (f w - f x)/(w - x)" and d = "dist x w" in field_differentiable_transform_within)
- apply (rule \<open>open U\<close> derivative_intros holomorphic_on_imp_differentiable_at [OF holf] | force simp: dist_commute)+
- done
- show ?thesis
- unfolding d_def
- apply (rule no_isolated_singularity [OF * _ \<open>open U\<close>, where K = "{w}"])
- apply (auto simp: field_differentiable_def [symmetric] holomorphic_on_open open_Diff \<open>open U\<close> **)
- done
- qed
- { fix a b
- assume abu: "closed_segment a b \<subseteq> U"
- then have "\<And>w. w \<in> U \<Longrightarrow> (\<lambda>z. d z w) contour_integrable_on (linepath a b)"
- by (metis hol_dw continuous_on_subset contour_integrable_continuous_linepath holomorphic_on_imp_continuous_on)
- then have cont_cint_d: "continuous_on U (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
- apply (rule contour_integral_continuous_on_linepath_2D [OF \<open>open U\<close> _ _ abu])
- apply (auto intro: continuous_on_swap_args cond_uu)
- done
- have cont_cint_d\<gamma>: "continuous_on {0..1} ((\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) \<circ> \<gamma>)"
- proof (rule continuous_on_compose)
- show "continuous_on {0..1} \<gamma>"
- using \<open>path \<gamma>\<close> path_def by blast
- show "continuous_on (\<gamma> ` {0..1}) (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
- using pasz unfolding path_image_def
- by (auto intro!: continuous_on_subset [OF cont_cint_d])
- qed
- have cint_cint: "(\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) contour_integrable_on \<gamma>"
- apply (simp add: contour_integrable_on)
- apply (rule integrable_continuous_real)
- apply (rule continuous_on_mult [OF cont_cint_d\<gamma> [unfolded o_def]])
- using pf\<gamma>'
- by (simp add: continuous_on_polymonial_function vector_derivative_at [OF \<gamma>'])
- have "contour_integral (linepath a b) h = contour_integral (linepath a b) (\<lambda>z. contour_integral \<gamma> (d z))"
- using abu by (force simp: h_def intro: contour_integral_eq)
- also have "\<dots> = contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
- apply (rule contour_integral_swap)
- apply (rule continuous_on_subset [OF cond_uu])
- using abu pasz \<open>valid_path \<gamma>\<close>
- apply (auto intro!: continuous_intros)
- by (metis \<gamma>' continuous_on_eq path_def path_polynomial_function pf\<gamma>' vector_derivative_at)
- finally have cint_h_eq:
- "contour_integral (linepath a b) h =
- contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))" .
- note cint_cint cint_h_eq
- } note cint_h = this
- have conthu: "continuous_on U h"
- proof (simp add: continuous_on_sequentially, clarify)
- fix a x
- assume x: "x \<in> U" and au: "\<forall>n. a n \<in> U" and ax: "a \<longlonglongrightarrow> x"
- then have A1: "\<forall>\<^sub>F n in sequentially. d (a n) contour_integrable_on \<gamma>"
- by (meson U contour_integrable_on_def eventuallyI)
- obtain dd where "dd>0" and dd: "cball x dd \<subseteq> U" using open_contains_cball \<open>open U\<close> x by force
- have A2: "uniform_limit (path_image \<gamma>) (\<lambda>n. d (a n)) (d x) sequentially"
- unfolding uniform_limit_iff dist_norm
- proof clarify
- fix ee::real
- assume "0 < ee"
- show "\<forall>\<^sub>F n in sequentially. \<forall>\<xi>\<in>path_image \<gamma>. cmod (d (a n) \<xi> - d x \<xi>) < ee"
- proof -
- let ?ddpa = "{(w,z) |w z. w \<in> cball x dd \<and> z \<in> path_image \<gamma>}"
- have "uniformly_continuous_on ?ddpa (\<lambda>(x,y). d x y)"
- apply (rule compact_uniformly_continuous [OF continuous_on_subset[OF cond_uu]])
- using dd pasz \<open>valid_path \<gamma>\<close>
- apply (auto simp: compact_Times compact_valid_path_image simp del: mem_cball)
- done
- then obtain kk where "kk>0"
- and kk: "\<And>x x'. \<lbrakk>x \<in> ?ddpa; x' \<in> ?ddpa; dist x' x < kk\<rbrakk> \<Longrightarrow>
- dist ((\<lambda>(x,y). d x y) x') ((\<lambda>(x,y). d x y) x) < ee"
- by (rule uniformly_continuous_onE [where e = ee]) (use \<open>0 < ee\<close> in auto)
- have kk: "\<lbrakk>norm (w - x) \<le> dd; z \<in> path_image \<gamma>; norm ((w, z) - (x, z)) < kk\<rbrakk> \<Longrightarrow> norm (d w z - d x z) < ee"
- for w z
- using \<open>dd>0\<close> kk [of "(x,z)" "(w,z)"] by (force simp: norm_minus_commute dist_norm)
- show ?thesis
- using ax unfolding lim_sequentially eventually_sequentially
- apply (drule_tac x="min dd kk" in spec)
- using \<open>dd > 0\<close> \<open>kk > 0\<close>
- apply (fastforce simp: kk dist_norm)
- done
- qed
- qed
- have "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> contour_integral \<gamma> (d x)"
- by (rule contour_integral_uniform_limit [OF A1 A2 le_B]) (auto simp: \<open>valid_path \<gamma>\<close>)
- then have tendsto_hx: "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> h x"
- by (simp add: h_def x)
- then show "(h \<circ> a) \<longlonglongrightarrow> h x"
- by (simp add: h_def x au o_def)
- qed
- show ?thesis
- proof (simp add: holomorphic_on_open field_differentiable_def [symmetric], clarify)
- fix z0
- consider "z0 \<in> v" | "z0 \<in> U" using uv_Un by blast
- then show "h field_differentiable at z0"
- proof cases
- assume "z0 \<in> v" then show ?thesis
- using Cauchy_next_derivative [OF con_pa_f le_B f_has_cint _ ov] V f_has_cint \<open>valid_path \<gamma>\<close>
- by (auto simp: field_differentiable_def v_def)
- next
- assume "z0 \<in> U" then
- obtain e where "e>0" and e: "ball z0 e \<subseteq> U" by (blast intro: openE [OF \<open>open U\<close>])
- have *: "contour_integral (linepath a b) h + contour_integral (linepath b c) h + contour_integral (linepath c a) h = 0"
- if abc_subset: "convex hull {a, b, c} \<subseteq> ball z0 e" for a b c
- proof -
- have *: "\<And>x1 x2 z. z \<in> U \<Longrightarrow> closed_segment x1 x2 \<subseteq> U \<Longrightarrow> (\<lambda>w. d w z) contour_integrable_on linepath x1 x2"
- using hol_dw holomorphic_on_imp_continuous_on \<open>open U\<close>
- by (auto intro!: contour_integrable_holomorphic_simple)
- have abc: "closed_segment a b \<subseteq> U" "closed_segment b c \<subseteq> U" "closed_segment c a \<subseteq> U"
- using that e segments_subset_convex_hull by fastforce+
- have eq0: "\<And>w. w \<in> U \<Longrightarrow> contour_integral (linepath a b +++ linepath b c +++ linepath c a) (\<lambda>z. d z w) = 0"
- apply (rule contour_integral_unique [OF Cauchy_theorem_triangle])
- apply (rule holomorphic_on_subset [OF hol_dw])
- using e abc_subset by auto
- have "contour_integral \<gamma>
- (\<lambda>x. contour_integral (linepath a b) (\<lambda>z. d z x) +
- (contour_integral (linepath b c) (\<lambda>z. d z x) +
- contour_integral (linepath c a) (\<lambda>z. d z x))) = 0"
- apply (rule contour_integral_eq_0)
- using abc pasz U
- apply (subst contour_integral_join [symmetric], auto intro: eq0 *)+
- done
- then show ?thesis
- by (simp add: cint_h abc contour_integrable_add contour_integral_add [symmetric] add_ac)
- qed
- show ?thesis
- using e \<open>e > 0\<close>
- by (auto intro!: holomorphic_on_imp_differentiable_at [OF _ open_ball] analytic_imp_holomorphic
- Morera_triangle continuous_on_subset [OF conthu] *)
- qed
- qed
- qed
- ultimately have [simp]: "h z = 0" for z
- by (meson Liouville_weak)
- have "((\<lambda>w. 1 / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z) \<gamma>"
- by (rule has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close> znot])
- then have "((\<lambda>w. f z * (1 / (w - z))) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
- by (metis mult.commute has_contour_integral_lmul)
- then have 1: "((\<lambda>w. f z / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
- by (simp add: field_split_simps)
- moreover have 2: "((\<lambda>w. (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
- using U [OF z] pasz d_def by (force elim: has_contour_integral_eq [where g = "\<lambda>w. (f w - f z)/(w - z)"])
- show ?thesis
- using has_contour_integral_add [OF 1 2] by (simp add: diff_divide_distrib)
-qed
-
-theorem Cauchy_integral_formula_global:
- assumes S: "open S" and holf: "f holomorphic_on S"
- and z: "z \<in> S" and vpg: "valid_path \<gamma>"
- and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- and zero: "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0"
- shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
-proof -
- have "path \<gamma>" using vpg by (blast intro: valid_path_imp_path)
- have hols: "(\<lambda>w. f w / (w - z)) holomorphic_on S - {z}" "(\<lambda>w. 1 / (w - z)) holomorphic_on S - {z}"
- by (rule holomorphic_intros holomorphic_on_subset [OF holf] | force)+
- then have cint_fw: "(\<lambda>w. f w / (w - z)) contour_integrable_on \<gamma>"
- by (meson contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on open_delete S vpg pasz)
- obtain d where "d>0"
- and d: "\<And>g h. \<lbrakk>valid_path g; valid_path h; \<forall>t\<in>{0..1}. cmod (g t - \<gamma> t) < d \<and> cmod (h t - \<gamma> t) < d;
- pathstart h = pathstart g \<and> pathfinish h = pathfinish g\<rbrakk>
- \<Longrightarrow> path_image h \<subseteq> S - {z} \<and> (\<forall>f. f holomorphic_on S - {z} \<longrightarrow> contour_integral h f = contour_integral g f)"
- using contour_integral_nearby_ends [OF _ \<open>path \<gamma>\<close> pasz] S by (simp add: open_Diff) metis
- obtain p where polyp: "polynomial_function p"
- and ps: "pathstart p = pathstart \<gamma>" and pf: "pathfinish p = pathfinish \<gamma>" and led: "\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < d"
- using path_approx_polynomial_function [OF \<open>path \<gamma>\<close> \<open>d > 0\<close>] by blast
- then have ploop: "pathfinish p = pathstart p" using loop by auto
- have vpp: "valid_path p" using polyp valid_path_polynomial_function by blast
- have [simp]: "z \<notin> path_image \<gamma>" using pasz by blast
- have paps: "path_image p \<subseteq> S - {z}" and cint_eq: "(\<And>f. f holomorphic_on S - {z} \<Longrightarrow> contour_integral p f = contour_integral \<gamma> f)"
- using pf ps led d [OF vpg vpp] \<open>d > 0\<close> by auto
- have wn_eq: "winding_number p z = winding_number \<gamma> z"
- using vpp paps
- by (simp add: subset_Diff_insert vpg valid_path_polynomial_function winding_number_valid_path cint_eq hols)
- have "winding_number p w = winding_number \<gamma> w" if "w \<notin> S" for w
- proof -
- have hol: "(\<lambda>v. 1 / (v - w)) holomorphic_on S - {z}"
- using that by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
- have "w \<notin> path_image p" "w \<notin> path_image \<gamma>" using paps pasz that by auto
- then show ?thesis
- using vpp vpg by (simp add: subset_Diff_insert valid_path_polynomial_function winding_number_valid_path cint_eq [OF hol])
- qed
- then have wn0: "\<And>w. w \<notin> S \<Longrightarrow> winding_number p w = 0"
- by (simp add: zero)
- show ?thesis
- using Cauchy_integral_formula_global_weak [OF S holf z polyp paps ploop wn0] hols
- by (metis wn_eq cint_eq has_contour_integral_eqpath cint_fw cint_eq)
-qed
-
-theorem Cauchy_theorem_global:
- assumes S: "open S" and holf: "f holomorphic_on S"
- and vpg: "valid_path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- and pas: "path_image \<gamma> \<subseteq> S"
- and zero: "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0"
- shows "(f has_contour_integral 0) \<gamma>"
-proof -
- obtain z where "z \<in> S" and znot: "z \<notin> path_image \<gamma>"
- proof -
- have "compact (path_image \<gamma>)"
- using compact_valid_path_image vpg by blast
- then have "path_image \<gamma> \<noteq> S"
- by (metis (no_types) compact_open path_image_nonempty S)
- with pas show ?thesis by (blast intro: that)
- qed
- then have pasz: "path_image \<gamma> \<subseteq> S - {z}" using pas by blast
- have hol: "(\<lambda>w. (w - z) * f w) holomorphic_on S"
- by (rule holomorphic_intros holf)+
- show ?thesis
- using Cauchy_integral_formula_global [OF S hol \<open>z \<in> S\<close> vpg pasz loop zero]
- by (auto simp: znot elim!: has_contour_integral_eq)
-qed
-
-corollary Cauchy_theorem_global_outside:
- assumes "open S" "f holomorphic_on S" "valid_path \<gamma>" "pathfinish \<gamma> = pathstart \<gamma>" "path_image \<gamma> \<subseteq> S"
- "\<And>w. w \<notin> S \<Longrightarrow> w \<in> outside(path_image \<gamma>)"
- shows "(f has_contour_integral 0) \<gamma>"
-by (metis Cauchy_theorem_global assms winding_number_zero_in_outside valid_path_imp_path)
-
-lemma simply_connected_imp_winding_number_zero:
- assumes "simply_connected S" "path g"
- "path_image g \<subseteq> S" "pathfinish g = pathstart g" "z \<notin> S"
- shows "winding_number g z = 0"
-proof -
- have hom: "homotopic_loops S g (linepath (pathstart g) (pathstart g))"
- by (meson assms homotopic_paths_imp_homotopic_loops pathfinish_linepath simply_connected_eq_contractible_path)
- then have "homotopic_paths (- {z}) g (linepath (pathstart g) (pathstart g))"
- by (meson \<open>z \<notin> S\<close> homotopic_loops_imp_homotopic_paths_null homotopic_paths_subset subset_Compl_singleton)
- then have "winding_number g z = winding_number(linepath (pathstart g) (pathstart g)) z"
- by (rule winding_number_homotopic_paths)
- also have "\<dots> = 0"
- using assms by (force intro: winding_number_trivial)
- finally show ?thesis .
-qed
-
-lemma Cauchy_theorem_simply_connected:
- assumes "open S" "simply_connected S" "f holomorphic_on S" "valid_path g"
- "path_image g \<subseteq> S" "pathfinish g = pathstart g"
- shows "(f has_contour_integral 0) g"
-using assms
-apply (simp add: simply_connected_eq_contractible_path)
-apply (auto intro!: Cauchy_theorem_null_homotopic [where a = "pathstart g"]
- homotopic_paths_imp_homotopic_loops)
-using valid_path_imp_path by blast
-
-proposition\<^marker>\<open>tag unimportant\<close> holomorphic_logarithm_exists:
- assumes A: "convex A" "open A"
- and f: "f holomorphic_on A" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0"
- and z0: "z0 \<in> A"
- obtains g where "g holomorphic_on A" and "\<And>x. x \<in> A \<Longrightarrow> exp (g x) = f x"
-proof -
- note f' = holomorphic_derivI [OF f(1) A(2)]
- obtain g where g: "\<And>x. x \<in> A \<Longrightarrow> (g has_field_derivative deriv f x / f x) (at x)"
- proof (rule holomorphic_convex_primitive' [OF A])
- show "(\<lambda>x. deriv f x / f x) holomorphic_on A"
- by (intro holomorphic_intros f A)
- qed (auto simp: A at_within_open[of _ A])
- define h where "h = (\<lambda>x. -g z0 + ln (f z0) + g x)"
- from g and A have g_holo: "g holomorphic_on A"
- by (auto simp: holomorphic_on_def at_within_open[of _ A] field_differentiable_def)
- hence h_holo: "h holomorphic_on A"
- by (auto simp: h_def intro!: holomorphic_intros)
- have "\<exists>c. \<forall>x\<in>A. f x / exp (h x) - 1 = c"
- proof (rule has_field_derivative_zero_constant, goal_cases)
- case (2 x)
- note [simp] = at_within_open[OF _ \<open>open A\<close>]
- from 2 and z0 and f show ?case
- by (auto simp: h_def exp_diff field_simps intro!: derivative_eq_intros g f')
- qed fact+
- then obtain c where c: "\<And>x. x \<in> A \<Longrightarrow> f x / exp (h x) - 1 = c"
- by blast
- from c[OF z0] and z0 and f have "c = 0"
- by (simp add: h_def)
- with c have "\<And>x. x \<in> A \<Longrightarrow> exp (h x) = f x" by simp
- from that[OF h_holo this] show ?thesis .
-qed
-
-subsection \<open>Complex functions and power series\<close>
-
-text \<open>
- The following defines the power series expansion of a complex function at a given point
- (assuming that it is analytic at that point).
-\<close>
-definition\<^marker>\<open>tag important\<close> fps_expansion :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex fps" where
- "fps_expansion f z0 = Abs_fps (\<lambda>n. (deriv ^^ n) f z0 / fact n)"
-
-lemma
- fixes r :: ereal
- assumes "f holomorphic_on eball z0 r"
- shows conv_radius_fps_expansion: "fps_conv_radius (fps_expansion f z0) \<ge> r"
- and eval_fps_expansion: "\<And>z. z \<in> eball z0 r \<Longrightarrow> eval_fps (fps_expansion f z0) (z - z0) = f z"
- and eval_fps_expansion': "\<And>z. norm z < r \<Longrightarrow> eval_fps (fps_expansion f z0) z = f (z0 + z)"
-proof -
- have "(\<lambda>n. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z"
- if "z \<in> ball z0 r'" "ereal r' < r" for z r'
- proof -
- from that(2) have "ereal r' \<le> r" by simp
- from assms(1) and this have "f holomorphic_on ball z0 r'"
- by (rule holomorphic_on_subset[OF _ ball_eball_mono])
- from holomorphic_power_series [OF this that(1)]
- show ?thesis by (simp add: fps_expansion_def)
- qed
- hence *: "(\<lambda>n. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z"
- if "z \<in> eball z0 r" for z
- using that by (subst (asm) eball_conv_UNION_balls) blast
- show "fps_conv_radius (fps_expansion f z0) \<ge> r" unfolding fps_conv_radius_def
- proof (rule conv_radius_geI_ex)
- fix r' :: real assume r': "r' > 0" "ereal r' < r"
- thus "\<exists>z. norm z = r' \<and> summable (\<lambda>n. fps_nth (fps_expansion f z0) n * z ^ n)"
- using *[of "z0 + of_real r'"]
- by (intro exI[of _ "of_real r'"]) (auto simp: summable_def dist_norm)
- qed
- show "eval_fps (fps_expansion f z0) (z - z0) = f z" if "z \<in> eball z0 r" for z
- using *[OF that] by (simp add: eval_fps_def sums_iff)
- show "eval_fps (fps_expansion f z0) z = f (z0 + z)" if "ereal (norm z) < r" for z
- using *[of "z0 + z"] and that by (simp add: eval_fps_def sums_iff dist_norm)
-qed
-
-
-text \<open>
- We can now show several more facts about power series expansions (at least in the complex case)
- with relative ease that would have been trickier without complex analysis.
-\<close>
-lemma
- fixes f :: "complex fps" and r :: ereal
- assumes "\<And>z. ereal (norm z) < r \<Longrightarrow> eval_fps f z \<noteq> 0"
- shows fps_conv_radius_inverse: "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f)"
- and eval_fps_inverse: "\<And>z. ereal (norm z) < fps_conv_radius f \<Longrightarrow> ereal (norm z) < r \<Longrightarrow>
- eval_fps (inverse f) z = inverse (eval_fps f z)"
-proof -
- define R where "R = min (fps_conv_radius f) r"
- have *: "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f) \<and>
- (\<forall>z\<in>eball 0 (min (fps_conv_radius f) r). eval_fps (inverse f) z = inverse (eval_fps f z))"
- proof (cases "min r (fps_conv_radius f) > 0")
- case True
- define f' where "f' = fps_expansion (\<lambda>z. inverse (eval_fps f z)) 0"
- have holo: "(\<lambda>z. inverse (eval_fps f z)) holomorphic_on eball 0 (min r (fps_conv_radius f))"
- using assms by (intro holomorphic_intros) auto
- from holo have radius: "fps_conv_radius f' \<ge> min r (fps_conv_radius f)"
- unfolding f'_def by (rule conv_radius_fps_expansion)
- have eval_f': "eval_fps f' z = inverse (eval_fps f z)"
- if "norm z < fps_conv_radius f" "norm z < r" for z
- using that unfolding f'_def by (subst eval_fps_expansion'[OF holo]) auto
-
- have "f * f' = 1"
- proof (rule eval_fps_eqD)
- from radius and True have "0 < min (fps_conv_radius f) (fps_conv_radius f')"
- by (auto simp: min_def split: if_splits)
- also have "\<dots> \<le> fps_conv_radius (f * f')" by (rule fps_conv_radius_mult)
- finally show "\<dots> > 0" .
- next
- from True have "R > 0" by (auto simp: R_def)
- hence "eventually (\<lambda>z. z \<in> eball 0 R) (nhds 0)"
- by (intro eventually_nhds_in_open) (auto simp: zero_ereal_def)
- thus "eventually (\<lambda>z. eval_fps (f * f') z = eval_fps 1 z) (nhds 0)"
- proof eventually_elim
- case (elim z)
- hence "eval_fps (f * f') z = eval_fps f z * eval_fps f' z"
- using radius by (intro eval_fps_mult)
- (auto simp: R_def min_def split: if_splits intro: less_trans)
- also have "eval_fps f' z = inverse (eval_fps f z)"
- using elim by (intro eval_f') (auto simp: R_def)
- also from elim have "eval_fps f z \<noteq> 0"
- by (intro assms) (auto simp: R_def)
- hence "eval_fps f z * inverse (eval_fps f z) = eval_fps 1 z"
- by simp
- finally show "eval_fps (f * f') z = eval_fps 1 z" .
- qed
- qed simp_all
- hence "f' = inverse f"
- by (intro fps_inverse_unique [symmetric]) (simp_all add: mult_ac)
- with eval_f' and radius show ?thesis by simp
- next
- case False
- hence *: "eball 0 R = {}"
- by (intro eball_empty) (auto simp: R_def min_def split: if_splits)
- show ?thesis
- proof safe
- from False have "min r (fps_conv_radius f) \<le> 0"
- by (simp add: min_def)
- also have "0 \<le> fps_conv_radius (inverse f)"
- by (simp add: fps_conv_radius_def conv_radius_nonneg)
- finally show "min r (fps_conv_radius f) \<le> \<dots>" .
- qed (unfold * [unfolded R_def], auto)
- qed
-
- from * show "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f)" by blast
- from * show "eval_fps (inverse f) z = inverse (eval_fps f z)"
- if "ereal (norm z) < fps_conv_radius f" "ereal (norm z) < r" for z
- using that by auto
-qed
-
-lemma
- fixes f g :: "complex fps" and r :: ereal
- defines "R \<equiv> Min {r, fps_conv_radius f, fps_conv_radius g}"
- assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0"
- assumes nz: "\<And>z. z \<in> eball 0 r \<Longrightarrow> eval_fps g z \<noteq> 0"
- shows fps_conv_radius_divide': "fps_conv_radius (f / g) \<ge> R"
- and eval_fps_divide':
- "ereal (norm z) < R \<Longrightarrow> eval_fps (f / g) z = eval_fps f z / eval_fps g z"
-proof -
- from nz[of 0] and \<open>r > 0\<close> have nz': "fps_nth g 0 \<noteq> 0"
- by (auto simp: eval_fps_at_0 zero_ereal_def)
- have "R \<le> min r (fps_conv_radius g)"
- by (auto simp: R_def intro: min.coboundedI2)
- also have "min r (fps_conv_radius g) \<le> fps_conv_radius (inverse g)"
- by (intro fps_conv_radius_inverse assms) (auto simp: zero_ereal_def)
- finally have radius: "fps_conv_radius (inverse g) \<ge> R" .
- have "R \<le> min (fps_conv_radius f) (fps_conv_radius (inverse g))"
- by (intro radius min.boundedI) (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
- also have "\<dots> \<le> fps_conv_radius (f * inverse g)"
- by (rule fps_conv_radius_mult)
- also have "f * inverse g = f / g"
- by (intro fps_divide_unit [symmetric] nz')
- finally show "fps_conv_radius (f / g) \<ge> R" .
-
- assume z: "ereal (norm z) < R"
- have "eval_fps (f * inverse g) z = eval_fps f z * eval_fps (inverse g) z"
- using radius by (intro eval_fps_mult less_le_trans[OF z])
- (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
- also have "eval_fps (inverse g) z = inverse (eval_fps g z)" using \<open>r > 0\<close>
- by (intro eval_fps_inverse[where r = r] less_le_trans[OF z] nz)
- (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
- also have "f * inverse g = f / g" by fact
- finally show "eval_fps (f / g) z = eval_fps f z / eval_fps g z" by (simp add: field_split_simps)
-qed
-
-lemma
- fixes f g :: "complex fps" and r :: ereal
- defines "R \<equiv> Min {r, fps_conv_radius f, fps_conv_radius g}"
- assumes "subdegree g \<le> subdegree f"
- assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0"
- assumes "\<And>z. z \<in> eball 0 r \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> eval_fps g z \<noteq> 0"
- shows fps_conv_radius_divide: "fps_conv_radius (f / g) \<ge> R"
- and eval_fps_divide:
- "ereal (norm z) < R \<Longrightarrow> c = fps_nth f (subdegree g) / fps_nth g (subdegree g) \<Longrightarrow>
- eval_fps (f / g) z = (if z = 0 then c else eval_fps f z / eval_fps g z)"
-proof -
- define f' g' where "f' = fps_shift (subdegree g) f" and "g' = fps_shift (subdegree g) g"
- have f_eq: "f = f' * fps_X ^ subdegree g" and g_eq: "g = g' * fps_X ^ subdegree g"
- unfolding f'_def g'_def by (rule subdegree_decompose' le_refl | fact)+
- have subdegree: "subdegree f' = subdegree f - subdegree g" "subdegree g' = 0"
- using assms(2) by (simp_all add: f'_def g'_def)
- have [simp]: "fps_conv_radius f' = fps_conv_radius f" "fps_conv_radius g' = fps_conv_radius g"
- by (simp_all add: f'_def g'_def)
- have [simp]: "fps_nth f' 0 = fps_nth f (subdegree g)"
- "fps_nth g' 0 = fps_nth g (subdegree g)" by (simp_all add: f'_def g'_def)
- have g_nz: "g \<noteq> 0"
- proof -
- define z :: complex where "z = (if r = \<infinity> then 1 else of_real (real_of_ereal r / 2))"
- from \<open>r > 0\<close> have "z \<in> eball 0 r"
- by (cases r) (auto simp: z_def eball_def)
- moreover have "z \<noteq> 0" using \<open>r > 0\<close>
- by (cases r) (auto simp: z_def)
- ultimately have "eval_fps g z \<noteq> 0" by (rule assms(6))
- thus "g \<noteq> 0" by auto
- qed
- have fg: "f / g = f' * inverse g'"
- by (subst f_eq, subst (2) g_eq) (insert g_nz, simp add: fps_divide_unit)
-
- have g'_nz: "eval_fps g' z \<noteq> 0" if z: "norm z < min r (fps_conv_radius g)" for z
- proof (cases "z = 0")
- case False
- with assms and z have "eval_fps g z \<noteq> 0" by auto
- also from z have "eval_fps g z = eval_fps g' z * z ^ subdegree g"
- by (subst g_eq) (auto simp: eval_fps_mult)
- finally show ?thesis by auto
- qed (insert \<open>g \<noteq> 0\<close>, auto simp: g'_def eval_fps_at_0)
-
- have "R \<le> min (min r (fps_conv_radius g)) (fps_conv_radius g')"
- by (auto simp: R_def min.coboundedI1 min.coboundedI2)
- also have "\<dots> \<le> fps_conv_radius (inverse g')"
- using g'_nz by (rule fps_conv_radius_inverse)
- finally have conv_radius_inv: "R \<le> fps_conv_radius (inverse g')" .
- hence "R \<le> fps_conv_radius (f' * inverse g')"
- by (intro order.trans[OF _ fps_conv_radius_mult])
- (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
- thus "fps_conv_radius (f / g) \<ge> R" by (simp add: fg)
-
- fix z c :: complex assume z: "ereal (norm z) < R"
- assume c: "c = fps_nth f (subdegree g) / fps_nth g (subdegree g)"
- show "eval_fps (f / g) z = (if z = 0 then c else eval_fps f z / eval_fps g z)"
- proof (cases "z = 0")
- case False
- from z and conv_radius_inv have "ereal (norm z) < fps_conv_radius (inverse g')"
- by simp
- with z have "eval_fps (f / g) z = eval_fps f' z * eval_fps (inverse g') z"
- unfolding fg by (subst eval_fps_mult) (auto simp: R_def)
- also have "eval_fps (inverse g') z = inverse (eval_fps g' z)"
- using z by (intro eval_fps_inverse[of "min r (fps_conv_radius g')"] g'_nz) (auto simp: R_def)
- also have "eval_fps f' z * \<dots> = eval_fps f z / eval_fps g z"
- using z False assms(2) by (simp add: f'_def g'_def eval_fps_shift R_def)
- finally show ?thesis using False by simp
- qed (simp_all add: eval_fps_at_0 fg field_simps c)
-qed
-
-lemma has_fps_expansion_fps_expansion [intro]:
- assumes "open A" "0 \<in> A" "f holomorphic_on A"
- shows "f has_fps_expansion fps_expansion f 0"
-proof -
- from assms(1,2) obtain r where r: "r > 0 " "ball 0 r \<subseteq> A"
- by (auto simp: open_contains_ball)
- have holo: "f holomorphic_on eball 0 (ereal r)"
- using r(2) and assms(3) by auto
- from r(1) have "0 < ereal r" by simp
- also have "r \<le> fps_conv_radius (fps_expansion f 0)"
- using holo by (intro conv_radius_fps_expansion) auto
- finally have "\<dots> > 0" .
- moreover have "eventually (\<lambda>z. z \<in> ball 0 r) (nhds 0)"
- using r(1) by (intro eventually_nhds_in_open) auto
- hence "eventually (\<lambda>z. eval_fps (fps_expansion f 0) z = f z) (nhds 0)"
- by eventually_elim (subst eval_fps_expansion'[OF holo], auto)
- ultimately show ?thesis using r(1) by (auto simp: has_fps_expansion_def)
-qed
-
-lemma fps_conv_radius_tan:
- fixes c :: complex
- assumes "c \<noteq> 0"
- shows "fps_conv_radius (fps_tan c) \<ge> pi / (2 * norm c)"
-proof -
- have "fps_conv_radius (fps_tan c) \<ge>
- Min {pi / (2 * norm c), fps_conv_radius (fps_sin c), fps_conv_radius (fps_cos c)}"
- unfolding fps_tan_def
- proof (rule fps_conv_radius_divide)
- fix z :: complex assume "z \<in> eball 0 (pi / (2 * norm c))"
- with cos_eq_zero_imp_norm_ge[of "c*z"] assms
- show "eval_fps (fps_cos c) z \<noteq> 0" by (auto simp: norm_mult field_simps)
- qed (insert assms, auto)
- thus ?thesis by (simp add: min_def)
-qed
-
-lemma eval_fps_tan:
- fixes c :: complex
- assumes "norm z < pi / (2 * norm c)"
- shows "eval_fps (fps_tan c) z = tan (c * z)"
-proof (cases "c = 0")
- case False
- show ?thesis unfolding fps_tan_def
- proof (subst eval_fps_divide'[where r = "pi / (2 * norm c)"])
- fix z :: complex assume "z \<in> eball 0 (pi / (2 * norm c))"
- with cos_eq_zero_imp_norm_ge[of "c*z"] assms
- show "eval_fps (fps_cos c) z \<noteq> 0" using False by (auto simp: norm_mult field_simps)
- qed (insert False assms, auto simp: field_simps tan_def)
-qed simp_all
-
-end
+end
\ No newline at end of file
--- a/src/HOL/Complex_Analysis/Complex_Analysis.thy Mon Dec 02 22:40:16 2019 -0500
+++ b/src/HOL/Complex_Analysis/Complex_Analysis.thy Mon Dec 02 17:51:54 2019 +0100
@@ -1,6 +1,7 @@
theory Complex_Analysis
- imports
- Winding_Numbers
+imports
+ Residue_Theorem
+ Riemann_Mapping
begin
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/Complex_Residues.thy Mon Dec 02 17:51:54 2019 +0100
@@ -0,0 +1,545 @@
+theory Complex_Residues
+ imports Complex_Singularities
+begin
+
+subsection \<open>Definition of residues\<close>
+
+text\<open>Wenda Li and LC Paulson (2016). A Formal Proof of Cauchy's Residue Theorem.
+ Interactive Theorem Proving\<close>
+
+definition\<^marker>\<open>tag important\<close> residue :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex" where
+ "residue f z = (SOME int. \<exists>e>0. \<forall>\<epsilon>>0. \<epsilon><e
+ \<longrightarrow> (f has_contour_integral 2*pi* \<i> *int) (circlepath z \<epsilon>))"
+
+lemma Eps_cong:
+ assumes "\<And>x. P x = Q x"
+ shows "Eps P = Eps Q"
+ using ext[of P Q, OF assms] by simp
+
+lemma residue_cong:
+ assumes eq: "eventually (\<lambda>z. f z = g z) (at z)" and "z = z'"
+ shows "residue f z = residue g z'"
+proof -
+ from assms have eq': "eventually (\<lambda>z. g z = f z) (at z)"
+ by (simp add: eq_commute)
+ let ?P = "\<lambda>f c e. (\<forall>\<epsilon>>0. \<epsilon> < e \<longrightarrow>
+ (f has_contour_integral of_real (2 * pi) * \<i> * c) (circlepath z \<epsilon>))"
+ have "residue f z = residue g z" unfolding residue_def
+ proof (rule Eps_cong)
+ fix c :: complex
+ have "\<exists>e>0. ?P g c e"
+ if "\<exists>e>0. ?P f c e" and "eventually (\<lambda>z. f z = g z) (at z)" for f g
+ proof -
+ from that(1) obtain e where e: "e > 0" "?P f c e"
+ by blast
+ from that(2) obtain e' where e': "e' > 0" "\<And>z'. z' \<noteq> z \<Longrightarrow> dist z' z < e' \<Longrightarrow> f z' = g z'"
+ unfolding eventually_at by blast
+ have "?P g c (min e e')"
+ proof (intro allI exI impI, goal_cases)
+ case (1 \<epsilon>)
+ hence "(f has_contour_integral of_real (2 * pi) * \<i> * c) (circlepath z \<epsilon>)"
+ using e(2) by auto
+ thus ?case
+ proof (rule has_contour_integral_eq)
+ fix z' assume "z' \<in> path_image (circlepath z \<epsilon>)"
+ hence "dist z' z < e'" and "z' \<noteq> z"
+ using 1 by (auto simp: dist_commute)
+ with e'(2)[of z'] show "f z' = g z'" by simp
+ qed
+ qed
+ moreover from e and e' have "min e e' > 0" by auto
+ ultimately show ?thesis by blast
+ qed
+ from this[OF _ eq] and this[OF _ eq']
+ show "(\<exists>e>0. ?P f c e) \<longleftrightarrow> (\<exists>e>0. ?P g c e)"
+ by blast
+ qed
+ with assms show ?thesis by simp
+qed
+
+lemma contour_integral_circlepath_eq:
+ assumes "open s" and f_holo:"f holomorphic_on (s-{z})" and "0<e1" "e1\<le>e2"
+ and e2_cball:"cball z e2 \<subseteq> s"
+ shows
+ "f contour_integrable_on circlepath z e1"
+ "f contour_integrable_on circlepath z e2"
+ "contour_integral (circlepath z e2) f = contour_integral (circlepath z e1) f"
+proof -
+ define l where "l \<equiv> linepath (z+e2) (z+e1)"
+ have [simp]:"valid_path l" "pathstart l=z+e2" "pathfinish l=z+e1" unfolding l_def by auto
+ have "e2>0" using \<open>e1>0\<close> \<open>e1\<le>e2\<close> by auto
+ have zl_img:"z\<notin>path_image l"
+ proof
+ assume "z \<in> path_image l"
+ then have "e2 \<le> cmod (e2 - e1)"
+ using segment_furthest_le[of z "z+e2" "z+e1" "z+e2",simplified] \<open>e1>0\<close> \<open>e2>0\<close> unfolding l_def
+ by (auto simp add:closed_segment_commute)
+ thus False using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
+ apply (subst (asm) norm_of_real)
+ by auto
+ qed
+ define g where "g \<equiv> circlepath z e2 +++ l +++ reversepath (circlepath z e1) +++ reversepath l"
+ show [simp]: "f contour_integrable_on circlepath z e2" "f contour_integrable_on (circlepath z e1)"
+ proof -
+ show "f contour_integrable_on circlepath z e2"
+ apply (intro contour_integrable_continuous_circlepath[OF
+ continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
+ using \<open>e2>0\<close> e2_cball by auto
+ show "f contour_integrable_on (circlepath z e1)"
+ apply (intro contour_integrable_continuous_circlepath[OF
+ continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
+ using \<open>e1>0\<close> \<open>e1\<le>e2\<close> e2_cball by auto
+ qed
+ have [simp]:"f contour_integrable_on l"
+ proof -
+ have "closed_segment (z + e2) (z + e1) \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
+ by (intro closed_segment_subset,auto simp add:dist_norm)
+ hence "closed_segment (z + e2) (z + e1) \<subseteq> s - {z}" using zl_img e2_cball unfolding l_def
+ by auto
+ then show "f contour_integrable_on l" unfolding l_def
+ apply (intro contour_integrable_continuous_linepath[OF
+ continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
+ by auto
+ qed
+ let ?ig="\<lambda>g. contour_integral g f"
+ have "(f has_contour_integral 0) g"
+ proof (rule Cauchy_theorem_global[OF _ f_holo])
+ show "open (s - {z})" using \<open>open s\<close> by auto
+ show "valid_path g" unfolding g_def l_def by auto
+ show "pathfinish g = pathstart g" unfolding g_def l_def by auto
+ next
+ have path_img:"path_image g \<subseteq> cball z e2"
+ proof -
+ have "closed_segment (z + e2) (z + e1) \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
+ by (intro closed_segment_subset,auto simp add:dist_norm)
+ moreover have "sphere z \<bar>e1\<bar> \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1\<le>e2\<close> \<open>e1>0\<close> by auto
+ ultimately show ?thesis unfolding g_def l_def using \<open>e2>0\<close>
+ by (simp add: path_image_join closed_segment_commute)
+ qed
+ show "path_image g \<subseteq> s - {z}"
+ proof -
+ have "z\<notin>path_image g" using zl_img
+ unfolding g_def l_def by (auto simp add: path_image_join closed_segment_commute)
+ moreover note \<open>cball z e2 \<subseteq> s\<close> and path_img
+ ultimately show ?thesis by auto
+ qed
+ show "winding_number g w = 0" when"w \<notin> s - {z}" for w
+ proof -
+ have "winding_number g w = 0" when "w\<notin>s" using that e2_cball
+ apply (intro winding_number_zero_outside[OF _ _ _ _ path_img])
+ by (auto simp add:g_def l_def)
+ moreover have "winding_number g z=0"
+ proof -
+ let ?Wz="\<lambda>g. winding_number g z"
+ have "?Wz g = ?Wz (circlepath z e2) + ?Wz l + ?Wz (reversepath (circlepath z e1))
+ + ?Wz (reversepath l)"
+ using \<open>e2>0\<close> \<open>e1>0\<close> zl_img unfolding g_def l_def
+ by (subst winding_number_join,auto simp add:path_image_join closed_segment_commute)+
+ also have "... = ?Wz (circlepath z e2) + ?Wz (reversepath (circlepath z e1))"
+ using zl_img
+ apply (subst (2) winding_number_reversepath)
+ by (auto simp add:l_def closed_segment_commute)
+ also have "... = 0"
+ proof -
+ have "?Wz (circlepath z e2) = 1" using \<open>e2>0\<close>
+ by (auto intro: winding_number_circlepath_centre)
+ moreover have "?Wz (reversepath (circlepath z e1)) = -1" using \<open>e1>0\<close>
+ apply (subst winding_number_reversepath)
+ by (auto intro: winding_number_circlepath_centre)
+ ultimately show ?thesis by auto
+ qed
+ finally show ?thesis .
+ qed
+ ultimately show ?thesis using that by auto
+ qed
+ qed
+ then have "0 = ?ig g" using contour_integral_unique by simp
+ also have "... = ?ig (circlepath z e2) + ?ig l + ?ig (reversepath (circlepath z e1))
+ + ?ig (reversepath l)"
+ unfolding g_def
+ by (auto simp add:contour_integrable_reversepath_eq)
+ also have "... = ?ig (circlepath z e2) - ?ig (circlepath z e1)"
+ by (auto simp add:contour_integral_reversepath)
+ finally show "contour_integral (circlepath z e2) f = contour_integral (circlepath z e1) f"
+ by simp
+qed
+
+lemma base_residue:
+ assumes "open s" "z\<in>s" "r>0" and f_holo:"f holomorphic_on (s - {z})"
+ and r_cball:"cball z r \<subseteq> s"
+ shows "(f has_contour_integral 2 * pi * \<i> * (residue f z)) (circlepath z r)"
+proof -
+ obtain e where "e>0" and e_cball:"cball z e \<subseteq> s"
+ using open_contains_cball[of s] \<open>open s\<close> \<open>z\<in>s\<close> by auto
+ define c where "c \<equiv> 2 * pi * \<i>"
+ define i where "i \<equiv> contour_integral (circlepath z e) f / c"
+ have "(f has_contour_integral c*i) (circlepath z \<epsilon>)" when "\<epsilon>>0" "\<epsilon><e" for \<epsilon>
+ proof -
+ have "contour_integral (circlepath z e) f = contour_integral (circlepath z \<epsilon>) f"
+ "f contour_integrable_on circlepath z \<epsilon>"
+ "f contour_integrable_on circlepath z e"
+ using \<open>\<epsilon><e\<close>
+ by (intro contour_integral_circlepath_eq[OF \<open>open s\<close> f_holo \<open>\<epsilon>>0\<close> _ e_cball],auto)+
+ then show ?thesis unfolding i_def c_def
+ by (auto intro:has_contour_integral_integral)
+ qed
+ then have "\<exists>e>0. \<forall>\<epsilon>>0. \<epsilon><e \<longrightarrow> (f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
+ unfolding residue_def c_def
+ apply (rule_tac someI[of _ i],intro exI[where x=e])
+ by (auto simp add:\<open>e>0\<close> c_def)
+ then obtain e' where "e'>0"
+ and e'_def:"\<forall>\<epsilon>>0. \<epsilon><e' \<longrightarrow> (f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
+ by auto
+ let ?int="\<lambda>e. contour_integral (circlepath z e) f"
+ define \<epsilon> where "\<epsilon> \<equiv> Min {r,e'} / 2"
+ have "\<epsilon>>0" "\<epsilon>\<le>r" "\<epsilon><e'" using \<open>r>0\<close> \<open>e'>0\<close> unfolding \<epsilon>_def by auto
+ have "(f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
+ using e'_def[rule_format,OF \<open>\<epsilon>>0\<close> \<open>\<epsilon><e'\<close>] .
+ then show ?thesis unfolding c_def
+ using contour_integral_circlepath_eq[OF \<open>open s\<close> f_holo \<open>\<epsilon>>0\<close> \<open>\<epsilon>\<le>r\<close> r_cball]
+ by (auto elim: has_contour_integral_eqpath[of _ _ "circlepath z \<epsilon>" "circlepath z r"])
+qed
+
+lemma residue_holo:
+ assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s"
+ shows "residue f z = 0"
+proof -
+ define c where "c \<equiv> 2 * pi * \<i>"
+ obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
+ using open_contains_cball_eq by blast
+ have "(f has_contour_integral c*residue f z) (circlepath z e)"
+ using f_holo
+ by (auto intro: base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
+ moreover have "(f has_contour_integral 0) (circlepath z e)"
+ using f_holo e_cball \<open>e>0\<close>
+ by (auto intro: Cauchy_theorem_convex_simple[of _ "cball z e"])
+ ultimately have "c*residue f z =0"
+ using has_contour_integral_unique by blast
+ thus ?thesis unfolding c_def by auto
+qed
+
+lemma residue_const:"residue (\<lambda>_. c) z = 0"
+ by (intro residue_holo[of "UNIV::complex set"],auto intro:holomorphic_intros)
+
+lemma residue_add:
+ assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
+ and g_holo:"g holomorphic_on s - {z}"
+ shows "residue (\<lambda>z. f z + g z) z= residue f z + residue g z"
+proof -
+ define c where "c \<equiv> 2 * pi * \<i>"
+ define fg where "fg \<equiv> (\<lambda>z. f z+g z)"
+ obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
+ using open_contains_cball_eq by blast
+ have "(fg has_contour_integral c * residue fg z) (circlepath z e)"
+ unfolding fg_def using f_holo g_holo
+ apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
+ by (auto intro:holomorphic_intros)
+ moreover have "(fg has_contour_integral c*residue f z + c* residue g z) (circlepath z e)"
+ unfolding fg_def using f_holo g_holo
+ by (auto intro: has_contour_integral_add base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
+ ultimately have "c*(residue f z + residue g z) = c * residue fg z"
+ using has_contour_integral_unique by (auto simp add:distrib_left)
+ thus ?thesis unfolding fg_def
+ by (auto simp add:c_def)
+qed
+
+lemma residue_lmul:
+ assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
+ shows "residue (\<lambda>z. c * (f z)) z= c * residue f z"
+proof (cases "c=0")
+ case True
+ thus ?thesis using residue_const by auto
+next
+ case False
+ define c' where "c' \<equiv> 2 * pi * \<i>"
+ define f' where "f' \<equiv> (\<lambda>z. c * (f z))"
+ obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
+ using open_contains_cball_eq by blast
+ have "(f' has_contour_integral c' * residue f' z) (circlepath z e)"
+ unfolding f'_def using f_holo
+ apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c'_def])
+ by (auto intro:holomorphic_intros)
+ moreover have "(f' has_contour_integral c * (c' * residue f z)) (circlepath z e)"
+ unfolding f'_def using f_holo
+ by (auto intro: has_contour_integral_lmul
+ base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c'_def])
+ ultimately have "c' * residue f' z = c * (c' * residue f z)"
+ using has_contour_integral_unique by auto
+ thus ?thesis unfolding f'_def c'_def using False
+ by (auto simp add:field_simps)
+qed
+
+lemma residue_rmul:
+ assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
+ shows "residue (\<lambda>z. (f z) * c) z= residue f z * c"
+using residue_lmul[OF assms,of c] by (auto simp add:algebra_simps)
+
+lemma residue_div:
+ assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
+ shows "residue (\<lambda>z. (f z) / c) z= residue f z / c "
+using residue_lmul[OF assms,of "1/c"] by (auto simp add:algebra_simps)
+
+lemma residue_neg:
+ assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
+ shows "residue (\<lambda>z. - (f z)) z= - residue f z"
+using residue_lmul[OF assms,of "-1"] by auto
+
+lemma residue_diff:
+ assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
+ and g_holo:"g holomorphic_on s - {z}"
+ shows "residue (\<lambda>z. f z - g z) z= residue f z - residue g z"
+using residue_add[OF assms(1,2,3),of "\<lambda>z. - g z"] residue_neg[OF assms(1,2,4)]
+by (auto intro:holomorphic_intros g_holo)
+
+lemma residue_simple:
+ assumes "open s" "z\<in>s" and f_holo:"f holomorphic_on s"
+ shows "residue (\<lambda>w. f w / (w - z)) z = f z"
+proof -
+ define c where "c \<equiv> 2 * pi * \<i>"
+ define f' where "f' \<equiv> \<lambda>w. f w / (w - z)"
+ obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
+ using open_contains_cball_eq by blast
+ have "(f' has_contour_integral c * f z) (circlepath z e)"
+ unfolding f'_def c_def using \<open>e>0\<close> f_holo e_cball
+ by (auto intro!: Cauchy_integral_circlepath_simple holomorphic_intros)
+ moreover have "(f' has_contour_integral c * residue f' z) (circlepath z e)"
+ unfolding f'_def using f_holo
+ apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
+ by (auto intro!:holomorphic_intros)
+ ultimately have "c * f z = c * residue f' z"
+ using has_contour_integral_unique by blast
+ thus ?thesis unfolding c_def f'_def by auto
+qed
+
+lemma residue_simple':
+ assumes s: "open s" "z \<in> s" and holo: "f holomorphic_on (s - {z})"
+ and lim: "((\<lambda>w. f w * (w - z)) \<longlongrightarrow> c) (at z)"
+ shows "residue f z = c"
+proof -
+ define g where "g = (\<lambda>w. if w = z then c else f w * (w - z))"
+ from holo have "(\<lambda>w. f w * (w - z)) holomorphic_on (s - {z})" (is "?P")
+ by (force intro: holomorphic_intros)
+ also have "?P \<longleftrightarrow> g holomorphic_on (s - {z})"
+ by (intro holomorphic_cong refl) (simp_all add: g_def)
+ finally have *: "g holomorphic_on (s - {z})" .
+
+ note lim
+ also have "(\<lambda>w. f w * (w - z)) \<midarrow>z\<rightarrow> c \<longleftrightarrow> g \<midarrow>z\<rightarrow> g z"
+ by (intro filterlim_cong refl) (simp_all add: g_def [abs_def] eventually_at_filter)
+ finally have **: "g \<midarrow>z\<rightarrow> g z" .
+
+ have g_holo: "g holomorphic_on s"
+ by (rule no_isolated_singularity'[where K = "{z}"])
+ (insert assms * **, simp_all add: at_within_open_NO_MATCH)
+ from s and this have "residue (\<lambda>w. g w / (w - z)) z = g z"
+ by (rule residue_simple)
+ also have "\<forall>\<^sub>F za in at z. g za / (za - z) = f za"
+ unfolding eventually_at by (auto intro!: exI[of _ 1] simp: field_simps g_def)
+ hence "residue (\<lambda>w. g w / (w - z)) z = residue f z"
+ by (intro residue_cong refl)
+ finally show ?thesis
+ by (simp add: g_def)
+qed
+
+lemma residue_holomorphic_over_power:
+ assumes "open A" "z0 \<in> A" "f holomorphic_on A"
+ shows "residue (\<lambda>z. f z / (z - z0) ^ Suc n) z0 = (deriv ^^ n) f z0 / fact n"
+proof -
+ let ?f = "\<lambda>z. f z / (z - z0) ^ Suc n"
+ from assms(1,2) obtain r where r: "r > 0" "cball z0 r \<subseteq> A"
+ by (auto simp: open_contains_cball)
+ have "(?f has_contour_integral 2 * pi * \<i> * residue ?f z0) (circlepath z0 r)"
+ using r assms by (intro base_residue[of A]) (auto intro!: holomorphic_intros)
+ moreover have "(?f has_contour_integral 2 * pi * \<i> / fact n * (deriv ^^ n) f z0) (circlepath z0 r)"
+ using assms r
+ by (intro Cauchy_has_contour_integral_higher_derivative_circlepath)
+ (auto intro!: holomorphic_on_subset[OF assms(3)] holomorphic_on_imp_continuous_on)
+ ultimately have "2 * pi * \<i> * residue ?f z0 = 2 * pi * \<i> / fact n * (deriv ^^ n) f z0"
+ by (rule has_contour_integral_unique)
+ thus ?thesis by (simp add: field_simps)
+qed
+
+lemma residue_holomorphic_over_power':
+ assumes "open A" "0 \<in> A" "f holomorphic_on A"
+ shows "residue (\<lambda>z. f z / z ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
+ using residue_holomorphic_over_power[OF assms] by simp
+
+theorem residue_fps_expansion_over_power_at_0:
+ assumes "f has_fps_expansion F"
+ shows "residue (\<lambda>z. f z / z ^ Suc n) 0 = fps_nth F n"
+proof -
+ from has_fps_expansion_imp_holomorphic[OF assms] guess s . note s = this
+ have "residue (\<lambda>z. f z / (z - 0) ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
+ using assms s unfolding has_fps_expansion_def
+ by (intro residue_holomorphic_over_power[of s]) (auto simp: zero_ereal_def)
+ also from assms have "\<dots> = fps_nth F n"
+ by (subst fps_nth_fps_expansion) auto
+ finally show ?thesis by simp
+qed
+
+lemma residue_pole_order:
+ fixes f::"complex \<Rightarrow> complex" and z::complex
+ defines "n \<equiv> nat (- zorder f z)" and "h \<equiv> zor_poly f z"
+ assumes f_iso:"isolated_singularity_at f z"
+ and pole:"is_pole f z"
+ shows "residue f z = ((deriv ^^ (n - 1)) h z / fact (n-1))"
+proof -
+ define g where "g \<equiv> \<lambda>x. if x=z then 0 else inverse (f x)"
+ obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}"
+ using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast
+ obtain r where "0 < n" "0 < r" and r_cball:"cball z r \<subseteq> ball z e" and h_holo: "h holomorphic_on cball z r"
+ and h_divide:"(\<forall>w\<in>cball z r. (w\<noteq>z \<longrightarrow> f w = h w / (w - z) ^ n) \<and> h w \<noteq> 0)"
+ proof -
+ obtain r where r:"zorder f z < 0" "h z \<noteq> 0" "r>0" "cball z r \<subseteq> ball z e" "h holomorphic_on cball z r"
+ "(\<forall>w\<in>cball z r - {z}. f w = h w / (w - z) ^ n \<and> h w \<noteq> 0)"
+ using zorder_exist_pole[OF f_holo,simplified,OF \<open>is_pole f z\<close>,folded n_def h_def] by auto
+ have "n>0" using \<open>zorder f z < 0\<close> unfolding n_def by simp
+ moreover have "(\<forall>w\<in>cball z r. (w\<noteq>z \<longrightarrow> f w = h w / (w - z) ^ n) \<and> h w \<noteq> 0)"
+ using \<open>h z\<noteq>0\<close> r(6) by blast
+ ultimately show ?thesis using r(3,4,5) that by blast
+ qed
+ have r_nonzero:"\<And>w. w \<in> ball z r - {z} \<Longrightarrow> f w \<noteq> 0"
+ using h_divide by simp
+ define c where "c \<equiv> 2 * pi * \<i>"
+ define der_f where "der_f \<equiv> ((deriv ^^ (n - 1)) h z / fact (n-1))"
+ define h' where "h' \<equiv> \<lambda>u. h u / (u - z) ^ n"
+ have "(h' has_contour_integral c / fact (n - 1) * (deriv ^^ (n - 1)) h z) (circlepath z r)"
+ unfolding h'_def
+ proof (rule Cauchy_has_contour_integral_higher_derivative_circlepath[of z r h z "n-1",
+ folded c_def Suc_pred'[OF \<open>n>0\<close>]])
+ show "continuous_on (cball z r) h" using holomorphic_on_imp_continuous_on h_holo by simp
+ show "h holomorphic_on ball z r" using h_holo by auto
+ show " z \<in> ball z r" using \<open>r>0\<close> by auto
+ qed
+ then have "(h' has_contour_integral c * der_f) (circlepath z r)" unfolding der_f_def by auto
+ then have "(f has_contour_integral c * der_f) (circlepath z r)"
+ proof (elim has_contour_integral_eq)
+ fix x assume "x \<in> path_image (circlepath z r)"
+ hence "x\<in>cball z r - {z}" using \<open>r>0\<close> by auto
+ then show "h' x = f x" using h_divide unfolding h'_def by auto
+ qed
+ moreover have "(f has_contour_integral c * residue f z) (circlepath z r)"
+ using base_residue[of \<open>ball z e\<close> z,simplified,OF \<open>r>0\<close> f_holo r_cball,folded c_def]
+ unfolding c_def by simp
+ ultimately have "c * der_f = c * residue f z" using has_contour_integral_unique by blast
+ hence "der_f = residue f z" unfolding c_def by auto
+ thus ?thesis unfolding der_f_def by auto
+qed
+
+lemma residue_simple_pole:
+ assumes "isolated_singularity_at f z0"
+ assumes "is_pole f z0" "zorder f z0 = - 1"
+ shows "residue f z0 = zor_poly f z0 z0"
+ using assms by (subst residue_pole_order) simp_all
+
+lemma residue_simple_pole_limit:
+ assumes "isolated_singularity_at f z0"
+ assumes "is_pole f z0" "zorder f z0 = - 1"
+ assumes "((\<lambda>x. f (g x) * (g x - z0)) \<longlongrightarrow> c) F"
+ assumes "filterlim g (at z0) F" "F \<noteq> bot"
+ shows "residue f z0 = c"
+proof -
+ have "residue f z0 = zor_poly f z0 z0"
+ by (rule residue_simple_pole assms)+
+ also have "\<dots> = c"
+ apply (rule zor_poly_pole_eqI)
+ using assms by auto
+ finally show ?thesis .
+qed
+
+lemma
+ assumes f_holo:"f holomorphic_on s" and g_holo:"g holomorphic_on s"
+ and "open s" "connected s" "z \<in> s"
+ assumes g_deriv:"(g has_field_derivative g') (at z)"
+ assumes "f z \<noteq> 0" "g z = 0" "g' \<noteq> 0"
+ shows porder_simple_pole_deriv: "zorder (\<lambda>w. f w / g w) z = - 1"
+ and residue_simple_pole_deriv: "residue (\<lambda>w. f w / g w) z = f z / g'"
+proof -
+ have [simp]:"isolated_singularity_at f z" "isolated_singularity_at g z"
+ using isolated_singularity_at_holomorphic[OF _ \<open>open s\<close> \<open>z\<in>s\<close>] f_holo g_holo
+ by (meson Diff_subset holomorphic_on_subset)+
+ have [simp]:"not_essential f z" "not_essential g z"
+ unfolding not_essential_def using f_holo g_holo assms(3,5)
+ by (meson continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on)+
+ have g_nconst:"\<exists>\<^sub>F w in at z. g w \<noteq>0 "
+ proof (rule ccontr)
+ assume "\<not> (\<exists>\<^sub>F w in at z. g w \<noteq> 0)"
+ then have "\<forall>\<^sub>F w in nhds z. g w = 0"
+ unfolding eventually_at eventually_nhds frequently_at using \<open>g z = 0\<close>
+ by (metis open_ball UNIV_I centre_in_ball dist_commute mem_ball)
+ then have "deriv g z = deriv (\<lambda>_. 0) z"
+ by (intro deriv_cong_ev) auto
+ then have "deriv g z = 0" by auto
+ then have "g' = 0" using g_deriv DERIV_imp_deriv by blast
+ then show False using \<open>g'\<noteq>0\<close> by auto
+ qed
+
+ have "zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z"
+ proof -
+ have "\<forall>\<^sub>F w in at z. f w \<noteq>0 \<and> w\<in>s"
+ apply (rule non_zero_neighbour_alt)
+ using assms by auto
+ with g_nconst have "\<exists>\<^sub>F w in at z. f w * g w \<noteq> 0"
+ by (elim frequently_rev_mp eventually_rev_mp,auto)
+ then show ?thesis using zorder_divide[of f z g] by auto
+ qed
+ moreover have "zorder f z=0"
+ apply (rule zorder_zero_eqI[OF f_holo \<open>open s\<close> \<open>z\<in>s\<close>])
+ using \<open>f z\<noteq>0\<close> by auto
+ moreover have "zorder g z=1"
+ apply (rule zorder_zero_eqI[OF g_holo \<open>open s\<close> \<open>z\<in>s\<close>])
+ subgoal using assms(8) by auto
+ subgoal using DERIV_imp_deriv assms(9) g_deriv by auto
+ subgoal by simp
+ done
+ ultimately show "zorder (\<lambda>w. f w / g w) z = - 1" by auto
+
+ show "residue (\<lambda>w. f w / g w) z = f z / g'"
+ proof (rule residue_simple_pole_limit[where g=id and F="at z",simplified])
+ show "zorder (\<lambda>w. f w / g w) z = - 1" by fact
+ show "isolated_singularity_at (\<lambda>w. f w / g w) z"
+ by (auto intro: singularity_intros)
+ show "is_pole (\<lambda>w. f w / g w) z"
+ proof (rule is_pole_divide)
+ have "\<forall>\<^sub>F x in at z. g x \<noteq> 0"
+ apply (rule non_zero_neighbour)
+ using g_nconst by auto
+ moreover have "g \<midarrow>z\<rightarrow> 0"
+ using DERIV_isCont assms(8) continuous_at g_deriv by force
+ ultimately show "filterlim g (at 0) (at z)" unfolding filterlim_at by simp
+ show "isCont f z"
+ using assms(3,5) continuous_on_eq_continuous_at f_holo holomorphic_on_imp_continuous_on
+ by auto
+ show "f z \<noteq> 0" by fact
+ qed
+ show "filterlim id (at z) (at z)" by (simp add: filterlim_iff)
+ have "((\<lambda>w. (f w * (w - z)) / g w) \<longlongrightarrow> f z / g') (at z)"
+ proof (rule lhopital_complex_simple)
+ show "((\<lambda>w. f w * (w - z)) has_field_derivative f z) (at z)"
+ using assms by (auto intro!: derivative_eq_intros holomorphic_derivI[OF f_holo])
+ show "(g has_field_derivative g') (at z)" by fact
+ qed (insert assms, auto)
+ then show "((\<lambda>w. (f w / g w) * (w - z)) \<longlongrightarrow> f z / g') (at z)"
+ by (simp add: field_split_simps)
+ qed
+qed
+
+
+subsection \<open>Poles and residues of some well-known functions\<close>
+
+(* TODO: add more material here for other functions *)
+lemma is_pole_Gamma: "is_pole Gamma (-of_nat n)"
+ unfolding is_pole_def using Gamma_poles .
+
+lemma Gamma_residue:
+ "residue Gamma (-of_nat n) = (-1) ^ n / fact n"
+proof (rule residue_simple')
+ show "open (- (\<int>\<^sub>\<le>\<^sub>0 - {-of_nat n}) :: complex set)"
+ by (intro open_Compl closed_subset_Ints) auto
+ show "Gamma holomorphic_on (- (\<int>\<^sub>\<le>\<^sub>0 - {-of_nat n}) - {- of_nat n})"
+ by (rule holomorphic_Gamma) auto
+ show "(\<lambda>w. Gamma w * (w - (-of_nat n))) \<midarrow>(-of_nat n)\<rightarrow> (- 1) ^ n / fact n"
+ using Gamma_residues[of n] by simp
+qed auto
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/Complex_Singularities.thy Mon Dec 02 17:51:54 2019 +0100
@@ -0,0 +1,1534 @@
+theory Complex_Singularities
+ imports Conformal_Mappings
+begin
+
+subsection \<open>Non-essential singular points\<close>
+
+definition\<^marker>\<open>tag important\<close> is_pole ::
+ "('a::topological_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> bool" where
+ "is_pole f a = (LIM x (at a). f x :> at_infinity)"
+
+lemma is_pole_cong:
+ assumes "eventually (\<lambda>x. f x = g x) (at a)" "a=b"
+ shows "is_pole f a \<longleftrightarrow> is_pole g b"
+ unfolding is_pole_def using assms by (intro filterlim_cong,auto)
+
+lemma is_pole_transform:
+ assumes "is_pole f a" "eventually (\<lambda>x. f x = g x) (at a)" "a=b"
+ shows "is_pole g b"
+ using is_pole_cong assms by auto
+
+lemma is_pole_tendsto:
+ fixes f::"('a::topological_space \<Rightarrow> 'b::real_normed_div_algebra)"
+ shows "is_pole f x \<Longrightarrow> ((inverse o f) \<longlongrightarrow> 0) (at x)"
+unfolding is_pole_def
+by (auto simp add:filterlim_inverse_at_iff[symmetric] comp_def filterlim_at)
+
+lemma is_pole_inverse_holomorphic:
+ assumes "open s"
+ and f_holo:"f holomorphic_on (s-{z})"
+ and pole:"is_pole f z"
+ and non_z:"\<forall>x\<in>s-{z}. f x\<noteq>0"
+ shows "(\<lambda>x. if x=z then 0 else inverse (f x)) holomorphic_on s"
+proof -
+ define g where "g \<equiv> \<lambda>x. if x=z then 0 else inverse (f x)"
+ have "isCont g z" unfolding isCont_def using is_pole_tendsto[OF pole]
+ apply (subst Lim_cong_at[where b=z and y=0 and g="inverse \<circ> f"])
+ by (simp_all add:g_def)
+ moreover have "continuous_on (s-{z}) f" using f_holo holomorphic_on_imp_continuous_on by auto
+ hence "continuous_on (s-{z}) (inverse o f)" unfolding comp_def
+ by (auto elim!:continuous_on_inverse simp add:non_z)
+ hence "continuous_on (s-{z}) g" unfolding g_def
+ apply (subst continuous_on_cong[where t="s-{z}" and g="inverse o f"])
+ by auto
+ ultimately have "continuous_on s g" using open_delete[OF \<open>open s\<close>] \<open>open s\<close>
+ by (auto simp add:continuous_on_eq_continuous_at)
+ moreover have "(inverse o f) holomorphic_on (s-{z})"
+ unfolding comp_def using f_holo
+ by (auto elim!:holomorphic_on_inverse simp add:non_z)
+ hence "g holomorphic_on (s-{z})"
+ apply (subst holomorphic_cong[where t="s-{z}" and g="inverse o f"])
+ by (auto simp add:g_def)
+ ultimately show ?thesis unfolding g_def using \<open>open s\<close>
+ by (auto elim!: no_isolated_singularity)
+qed
+
+lemma not_is_pole_holomorphic:
+ assumes "open A" "x \<in> A" "f holomorphic_on A"
+ shows "\<not>is_pole f x"
+proof -
+ have "continuous_on A f" by (intro holomorphic_on_imp_continuous_on) fact
+ with assms have "isCont f x" by (simp add: continuous_on_eq_continuous_at)
+ hence "f \<midarrow>x\<rightarrow> f x" by (simp add: isCont_def)
+ thus "\<not>is_pole f x" unfolding is_pole_def
+ using not_tendsto_and_filterlim_at_infinity[of "at x" f "f x"] by auto
+qed
+
+lemma is_pole_inverse_power: "n > 0 \<Longrightarrow> is_pole (\<lambda>z::complex. 1 / (z - a) ^ n) a"
+ unfolding is_pole_def inverse_eq_divide [symmetric]
+ by (intro filterlim_compose[OF filterlim_inverse_at_infinity] tendsto_intros)
+ (auto simp: filterlim_at eventually_at intro!: exI[of _ 1] tendsto_eq_intros)
+
+lemma is_pole_inverse: "is_pole (\<lambda>z::complex. 1 / (z - a)) a"
+ using is_pole_inverse_power[of 1 a] by simp
+
+lemma is_pole_divide:
+ fixes f :: "'a :: t2_space \<Rightarrow> 'b :: real_normed_field"
+ assumes "isCont f z" "filterlim g (at 0) (at z)" "f z \<noteq> 0"
+ shows "is_pole (\<lambda>z. f z / g z) z"
+proof -
+ have "filterlim (\<lambda>z. f z * inverse (g z)) at_infinity (at z)"
+ by (intro tendsto_mult_filterlim_at_infinity[of _ "f z"]
+ filterlim_compose[OF filterlim_inverse_at_infinity])+
+ (insert assms, auto simp: isCont_def)
+ thus ?thesis by (simp add: field_split_simps is_pole_def)
+qed
+
+lemma is_pole_basic:
+ assumes "f holomorphic_on A" "open A" "z \<in> A" "f z \<noteq> 0" "n > 0"
+ shows "is_pole (\<lambda>w. f w / (w - z) ^ n) z"
+proof (rule is_pole_divide)
+ have "continuous_on A f" by (rule holomorphic_on_imp_continuous_on) fact
+ with assms show "isCont f z" by (auto simp: continuous_on_eq_continuous_at)
+ have "filterlim (\<lambda>w. (w - z) ^ n) (nhds 0) (at z)"
+ using assms by (auto intro!: tendsto_eq_intros)
+ thus "filterlim (\<lambda>w. (w - z) ^ n) (at 0) (at z)"
+ by (intro filterlim_atI tendsto_eq_intros)
+ (insert assms, auto simp: eventually_at_filter)
+qed fact+
+
+lemma is_pole_basic':
+ assumes "f holomorphic_on A" "open A" "0 \<in> A" "f 0 \<noteq> 0" "n > 0"
+ shows "is_pole (\<lambda>w. f w / w ^ n) 0"
+ using is_pole_basic[of f A 0] assms by simp
+
+text \<open>The proposition
+ \<^term>\<open>\<exists>x. ((f::complex\<Rightarrow>complex) \<longlongrightarrow> x) (at z) \<or> is_pole f z\<close>
+can be interpreted as the complex function \<^term>\<open>f\<close> has a non-essential singularity at \<^term>\<open>z\<close>
+(i.e. the singularity is either removable or a pole).\<close>
+definition not_essential::"[complex \<Rightarrow> complex, complex] \<Rightarrow> bool" where
+ "not_essential f z = (\<exists>x. f\<midarrow>z\<rightarrow>x \<or> is_pole f z)"
+
+definition isolated_singularity_at::"[complex \<Rightarrow> complex, complex] \<Rightarrow> bool" where
+ "isolated_singularity_at f z = (\<exists>r>0. f analytic_on ball z r-{z})"
+
+named_theorems singularity_intros "introduction rules for singularities"
+
+lemma holomorphic_factor_unique:
+ fixes f::"complex \<Rightarrow> complex" and z::complex and r::real and m n::int
+ assumes "r>0" "g z\<noteq>0" "h z\<noteq>0"
+ and asm:"\<forall>w\<in>ball z r-{z}. f w = g w * (w-z) powr n \<and> g w\<noteq>0 \<and> f w = h w * (w - z) powr m \<and> h w\<noteq>0"
+ and g_holo:"g holomorphic_on ball z r" and h_holo:"h holomorphic_on ball z r"
+ shows "n=m"
+proof -
+ have [simp]:"at z within ball z r \<noteq> bot" using \<open>r>0\<close>
+ by (auto simp add:at_within_ball_bot_iff)
+ have False when "n>m"
+ proof -
+ have "(h \<longlongrightarrow> 0) (at z within ball z r)"
+ proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powr (n - m) * g w"])
+ have "\<forall>w\<in>ball z r-{z}. h w = (w-z)powr(n-m) * g w"
+ using \<open>n>m\<close> asm \<open>r>0\<close>
+ apply (auto simp add:field_simps powr_diff)
+ by force
+ then show "\<lbrakk>x' \<in> ball z r; 0 < dist x' z;dist x' z < r\<rbrakk>
+ \<Longrightarrow> (x' - z) powr (n - m) * g x' = h x'" for x' by auto
+ next
+ define F where "F \<equiv> at z within ball z r"
+ define f' where "f' \<equiv> \<lambda>x. (x - z) powr (n-m)"
+ have "f' z=0" using \<open>n>m\<close> unfolding f'_def by auto
+ moreover have "continuous F f'" unfolding f'_def F_def continuous_def
+ apply (subst Lim_ident_at)
+ using \<open>n>m\<close> by (auto intro!:tendsto_powr_complex_0 tendsto_eq_intros)
+ ultimately have "(f' \<longlongrightarrow> 0) F" unfolding F_def
+ by (simp add: continuous_within)
+ moreover have "(g \<longlongrightarrow> g z) F"
+ using holomorphic_on_imp_continuous_on[OF g_holo,unfolded continuous_on_def] \<open>r>0\<close>
+ unfolding F_def by auto
+ ultimately show " ((\<lambda>w. f' w * g w) \<longlongrightarrow> 0) F" using tendsto_mult by fastforce
+ qed
+ moreover have "(h \<longlongrightarrow> h z) (at z within ball z r)"
+ using holomorphic_on_imp_continuous_on[OF h_holo]
+ by (auto simp add:continuous_on_def \<open>r>0\<close>)
+ ultimately have "h z=0" by (auto intro!: tendsto_unique)
+ thus False using \<open>h z\<noteq>0\<close> by auto
+ qed
+ moreover have False when "m>n"
+ proof -
+ have "(g \<longlongrightarrow> 0) (at z within ball z r)"
+ proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powr (m - n) * h w"])
+ have "\<forall>w\<in>ball z r -{z}. g w = (w-z) powr (m-n) * h w" using \<open>m>n\<close> asm
+ apply (auto simp add:field_simps powr_diff)
+ by force
+ then show "\<lbrakk>x' \<in> ball z r; 0 < dist x' z;dist x' z < r\<rbrakk>
+ \<Longrightarrow> (x' - z) powr (m - n) * h x' = g x'" for x' by auto
+ next
+ define F where "F \<equiv> at z within ball z r"
+ define f' where "f' \<equiv>\<lambda>x. (x - z) powr (m-n)"
+ have "f' z=0" using \<open>m>n\<close> unfolding f'_def by auto
+ moreover have "continuous F f'" unfolding f'_def F_def continuous_def
+ apply (subst Lim_ident_at)
+ using \<open>m>n\<close> by (auto intro!:tendsto_powr_complex_0 tendsto_eq_intros)
+ ultimately have "(f' \<longlongrightarrow> 0) F" unfolding F_def
+ by (simp add: continuous_within)
+ moreover have "(h \<longlongrightarrow> h z) F"
+ using holomorphic_on_imp_continuous_on[OF h_holo,unfolded continuous_on_def] \<open>r>0\<close>
+ unfolding F_def by auto
+ ultimately show " ((\<lambda>w. f' w * h w) \<longlongrightarrow> 0) F" using tendsto_mult by fastforce
+ qed
+ moreover have "(g \<longlongrightarrow> g z) (at z within ball z r)"
+ using holomorphic_on_imp_continuous_on[OF g_holo]
+ by (auto simp add:continuous_on_def \<open>r>0\<close>)
+ ultimately have "g z=0" by (auto intro!: tendsto_unique)
+ thus False using \<open>g z\<noteq>0\<close> by auto
+ qed
+ ultimately show "n=m" by fastforce
+qed
+
+lemma holomorphic_factor_puncture:
+ assumes f_iso:"isolated_singularity_at f z"
+ and "not_essential f z" \<comment> \<open>\<^term>\<open>f\<close> has either a removable singularity or a pole at \<^term>\<open>z\<close>\<close>
+ and non_zero:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" \<comment> \<open>\<^term>\<open>f\<close> will not be constantly zero in a neighbour of \<^term>\<open>z\<close>\<close>
+ shows "\<exists>!n::int. \<exists>g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
+ \<and> (\<forall>w\<in>cball z r-{z}. f w = g w * (w-z) powr n \<and> g w\<noteq>0)"
+proof -
+ define P where "P = (\<lambda>f n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
+ have imp_unique:"\<exists>!n::int. \<exists>g r. P f n g r" when "\<exists>n g r. P f n g r"
+ proof (rule ex_ex1I[OF that])
+ fix n1 n2 :: int
+ assume g1_asm:"\<exists>g1 r1. P f n1 g1 r1" and g2_asm:"\<exists>g2 r2. P f n2 g2 r2"
+ define fac where "fac \<equiv> \<lambda>n g r. \<forall>w\<in>cball z r-{z}. f w = g w * (w - z) powr (of_int n) \<and> g w \<noteq> 0"
+ obtain g1 r1 where "0 < r1" and g1_holo: "g1 holomorphic_on cball z r1" and "g1 z\<noteq>0"
+ and "fac n1 g1 r1" using g1_asm unfolding P_def fac_def by auto
+ obtain g2 r2 where "0 < r2" and g2_holo: "g2 holomorphic_on cball z r2" and "g2 z\<noteq>0"
+ and "fac n2 g2 r2" using g2_asm unfolding P_def fac_def by auto
+ define r where "r \<equiv> min r1 r2"
+ have "r>0" using \<open>r1>0\<close> \<open>r2>0\<close> unfolding r_def by auto
+ moreover have "\<forall>w\<in>ball z r-{z}. f w = g1 w * (w-z) powr n1 \<and> g1 w\<noteq>0
+ \<and> f w = g2 w * (w - z) powr n2 \<and> g2 w\<noteq>0"
+ using \<open>fac n1 g1 r1\<close> \<open>fac n2 g2 r2\<close> unfolding fac_def r_def
+ by fastforce
+ ultimately show "n1=n2" using g1_holo g2_holo \<open>g1 z\<noteq>0\<close> \<open>g2 z\<noteq>0\<close>
+ apply (elim holomorphic_factor_unique)
+ by (auto simp add:r_def)
+ qed
+
+ have P_exist:"\<exists> n g r. P h n g r" when
+ "\<exists>z'. (h \<longlongrightarrow> z') (at z)" "isolated_singularity_at h z" "\<exists>\<^sub>Fw in (at z). h w\<noteq>0"
+ for h
+ proof -
+ from that(2) obtain r where "r>0" "h analytic_on ball z r - {z}"
+ unfolding isolated_singularity_at_def by auto
+ obtain z' where "(h \<longlongrightarrow> z') (at z)" using \<open>\<exists>z'. (h \<longlongrightarrow> z') (at z)\<close> by auto
+ define h' where "h'=(\<lambda>x. if x=z then z' else h x)"
+ have "h' holomorphic_on ball z r"
+ apply (rule no_isolated_singularity'[of "{z}"])
+ subgoal by (metis LIM_equal Lim_at_imp_Lim_at_within \<open>h \<midarrow>z\<rightarrow> z'\<close> empty_iff h'_def insert_iff)
+ subgoal using \<open>h analytic_on ball z r - {z}\<close> analytic_imp_holomorphic h'_def holomorphic_transform
+ by fastforce
+ by auto
+ have ?thesis when "z'=0"
+ proof -
+ have "h' z=0" using that unfolding h'_def by auto
+ moreover have "\<not> h' constant_on ball z r"
+ using \<open>\<exists>\<^sub>Fw in (at z). h w\<noteq>0\<close> unfolding constant_on_def frequently_def eventually_at h'_def
+ apply simp
+ by (metis \<open>0 < r\<close> centre_in_ball dist_commute mem_ball that)
+ moreover note \<open>h' holomorphic_on ball z r\<close>
+ ultimately obtain g r1 n where "0 < n" "0 < r1" "ball z r1 \<subseteq> ball z r" and
+ g:"g holomorphic_on ball z r1"
+ "\<And>w. w \<in> ball z r1 \<Longrightarrow> h' w = (w - z) ^ n * g w"
+ "\<And>w. w \<in> ball z r1 \<Longrightarrow> g w \<noteq> 0"
+ using holomorphic_factor_zero_nonconstant[of _ "ball z r" z thesis,simplified,
+ OF \<open>h' holomorphic_on ball z r\<close> \<open>r>0\<close> \<open>h' z=0\<close> \<open>\<not> h' constant_on ball z r\<close>]
+ by (auto simp add:dist_commute)
+ define rr where "rr=r1/2"
+ have "P h' n g rr"
+ unfolding P_def rr_def
+ using \<open>n>0\<close> \<open>r1>0\<close> g by (auto simp add:powr_nat)
+ then have "P h n g rr"
+ unfolding h'_def P_def by auto
+ then show ?thesis unfolding P_def by blast
+ qed
+ moreover have ?thesis when "z'\<noteq>0"
+ proof -
+ have "h' z\<noteq>0" using that unfolding h'_def by auto
+ obtain r1 where "r1>0" "cball z r1 \<subseteq> ball z r" "\<forall>x\<in>cball z r1. h' x\<noteq>0"
+ proof -
+ have "isCont h' z" "h' z\<noteq>0"
+ by (auto simp add: Lim_cong_within \<open>h \<midarrow>z\<rightarrow> z'\<close> \<open>z'\<noteq>0\<close> continuous_at h'_def)
+ then obtain r2 where r2:"r2>0" "\<forall>x\<in>ball z r2. h' x\<noteq>0"
+ using continuous_at_avoid[of z h' 0 ] unfolding ball_def by auto
+ define r1 where "r1=min r2 r / 2"
+ have "0 < r1" "cball z r1 \<subseteq> ball z r"
+ using \<open>r2>0\<close> \<open>r>0\<close> unfolding r1_def by auto
+ moreover have "\<forall>x\<in>cball z r1. h' x \<noteq> 0"
+ using r2 unfolding r1_def by simp
+ ultimately show ?thesis using that by auto
+ qed
+ then have "P h' 0 h' r1" using \<open>h' holomorphic_on ball z r\<close> unfolding P_def by auto
+ then have "P h 0 h' r1" unfolding P_def h'_def by auto
+ then show ?thesis unfolding P_def by blast
+ qed
+ ultimately show ?thesis by auto
+ qed
+
+ have ?thesis when "\<exists>x. (f \<longlongrightarrow> x) (at z)"
+ apply (rule_tac imp_unique[unfolded P_def])
+ using P_exist[OF that(1) f_iso non_zero] unfolding P_def .
+ moreover have ?thesis when "is_pole f z"
+ proof (rule imp_unique[unfolded P_def])
+ obtain e where [simp]:"e>0" and e_holo:"f holomorphic_on ball z e - {z}" and e_nz: "\<forall>x\<in>ball z e-{z}. f x\<noteq>0"
+ proof -
+ have "\<forall>\<^sub>F z in at z. f z \<noteq> 0"
+ using \<open>is_pole f z\<close> filterlim_at_infinity_imp_eventually_ne unfolding is_pole_def
+ by auto
+ then obtain e1 where e1:"e1>0" "\<forall>x\<in>ball z e1-{z}. f x\<noteq>0"
+ using that eventually_at[of "\<lambda>x. f x\<noteq>0" z UNIV,simplified] by (auto simp add:dist_commute)
+ obtain e2 where e2:"e2>0" "f holomorphic_on ball z e2 - {z}"
+ using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by auto
+ define e where "e=min e1 e2"
+ show ?thesis
+ apply (rule that[of e])
+ using e1 e2 unfolding e_def by auto
+ qed
+
+ define h where "h \<equiv> \<lambda>x. inverse (f x)"
+
+ have "\<exists>n g r. P h n g r"
+ proof -
+ have "h \<midarrow>z\<rightarrow> 0"
+ using Lim_transform_within_open assms(2) h_def is_pole_tendsto that by fastforce
+ moreover have "\<exists>\<^sub>Fw in (at z). h w\<noteq>0"
+ using non_zero
+ apply (elim frequently_rev_mp)
+ unfolding h_def eventually_at by (auto intro:exI[where x=1])
+ moreover have "isolated_singularity_at h z"
+ unfolding isolated_singularity_at_def h_def
+ apply (rule exI[where x=e])
+ using e_holo e_nz \<open>e>0\<close> by (metis open_ball analytic_on_open
+ holomorphic_on_inverse open_delete)
+ ultimately show ?thesis
+ using P_exist[of h] by auto
+ qed
+ then obtain n g r
+ where "0 < r" and
+ g_holo:"g holomorphic_on cball z r" and "g z\<noteq>0" and
+ g_fac:"(\<forall>w\<in>cball z r-{z}. h w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ unfolding P_def by auto
+ have "P f (-n) (inverse o g) r"
+ proof -
+ have "f w = inverse (g w) * (w - z) powr of_int (- n)" when "w\<in>cball z r - {z}" for w
+ using g_fac[rule_format,of w] that unfolding h_def
+ apply (auto simp add:powr_minus )
+ by (metis inverse_inverse_eq inverse_mult_distrib)
+ then show ?thesis
+ unfolding P_def comp_def
+ using \<open>r>0\<close> g_holo g_fac \<open>g z\<noteq>0\<close> by (auto intro:holomorphic_intros)
+ qed
+ then show "\<exists>x g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z \<noteq> 0
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int x \<and> g w \<noteq> 0)"
+ unfolding P_def by blast
+ qed
+ ultimately show ?thesis using \<open>not_essential f z\<close> unfolding not_essential_def by presburger
+qed
+
+lemma not_essential_transform:
+ assumes "not_essential g z"
+ assumes "\<forall>\<^sub>F w in (at z). g w = f w"
+ shows "not_essential f z"
+ using assms unfolding not_essential_def
+ by (simp add: filterlim_cong is_pole_cong)
+
+lemma isolated_singularity_at_transform:
+ assumes "isolated_singularity_at g z"
+ assumes "\<forall>\<^sub>F w in (at z). g w = f w"
+ shows "isolated_singularity_at f z"
+proof -
+ obtain r1 where "r1>0" and r1:"g analytic_on ball z r1 - {z}"
+ using assms(1) unfolding isolated_singularity_at_def by auto
+ obtain r2 where "r2>0" and r2:" \<forall>x. x \<noteq> z \<and> dist x z < r2 \<longrightarrow> g x = f x"
+ using assms(2) unfolding eventually_at by auto
+ define r3 where "r3=min r1 r2"
+ have "r3>0" unfolding r3_def using \<open>r1>0\<close> \<open>r2>0\<close> by auto
+ moreover have "f analytic_on ball z r3 - {z}"
+ proof -
+ have "g holomorphic_on ball z r3 - {z}"
+ using r1 unfolding r3_def by (subst (asm) analytic_on_open,auto)
+ then have "f holomorphic_on ball z r3 - {z}"
+ using r2 unfolding r3_def
+ by (auto simp add:dist_commute elim!:holomorphic_transform)
+ then show ?thesis by (subst analytic_on_open,auto)
+ qed
+ ultimately show ?thesis unfolding isolated_singularity_at_def by auto
+qed
+
+lemma not_essential_powr[singularity_intros]:
+ assumes "LIM w (at z). f w :> (at x)"
+ shows "not_essential (\<lambda>w. (f w) powr (of_int n)) z"
+proof -
+ define fp where "fp=(\<lambda>w. (f w) powr (of_int n))"
+ have ?thesis when "n>0"
+ proof -
+ have "(\<lambda>w. (f w) ^ (nat n)) \<midarrow>z\<rightarrow> x ^ nat n"
+ using that assms unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
+ then have "fp \<midarrow>z\<rightarrow> x ^ nat n" unfolding fp_def
+ apply (elim Lim_transform_within[where d=1],simp)
+ by (metis less_le powr_0 powr_of_int that zero_less_nat_eq zero_power)
+ then show ?thesis unfolding not_essential_def fp_def by auto
+ qed
+ moreover have ?thesis when "n=0"
+ proof -
+ have "fp \<midarrow>z\<rightarrow> 1 "
+ apply (subst tendsto_cong[where g="\<lambda>_.1"])
+ using that filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def by auto
+ then show ?thesis unfolding fp_def not_essential_def by auto
+ qed
+ moreover have ?thesis when "n<0"
+ proof (cases "x=0")
+ case True
+ have "LIM w (at z). inverse ((f w) ^ (nat (-n))) :> at_infinity"
+ apply (subst filterlim_inverse_at_iff[symmetric],simp)
+ apply (rule filterlim_atI)
+ subgoal using assms True that unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
+ subgoal using filterlim_at_within_not_equal[OF assms,of 0]
+ by (eventually_elim,insert that,auto)
+ done
+ then have "LIM w (at z). fp w :> at_infinity"
+ proof (elim filterlim_mono_eventually)
+ show "\<forall>\<^sub>F x in at z. inverse (f x ^ nat (- n)) = fp x"
+ using filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def
+ apply eventually_elim
+ using powr_of_int that by auto
+ qed auto
+ then show ?thesis unfolding fp_def not_essential_def is_pole_def by auto
+ next
+ case False
+ let ?xx= "inverse (x ^ (nat (-n)))"
+ have "(\<lambda>w. inverse ((f w) ^ (nat (-n)))) \<midarrow>z\<rightarrow>?xx"
+ using assms False unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
+ then have "fp \<midarrow>z\<rightarrow>?xx"
+ apply (elim Lim_transform_within[where d=1],simp)
+ unfolding fp_def by (metis inverse_zero nat_mono_iff nat_zero_as_int neg_0_less_iff_less
+ not_le power_eq_0_iff powr_0 powr_of_int that)
+ then show ?thesis unfolding fp_def not_essential_def by auto
+ qed
+ ultimately show ?thesis by linarith
+qed
+
+lemma isolated_singularity_at_powr[singularity_intros]:
+ assumes "isolated_singularity_at f z" "\<forall>\<^sub>F w in (at z). f w\<noteq>0"
+ shows "isolated_singularity_at (\<lambda>w. (f w) powr (of_int n)) z"
+proof -
+ obtain r1 where "r1>0" "f analytic_on ball z r1 - {z}"
+ using assms(1) unfolding isolated_singularity_at_def by auto
+ then have r1:"f holomorphic_on ball z r1 - {z}"
+ using analytic_on_open[of "ball z r1-{z}" f] by blast
+ obtain r2 where "r2>0" and r2:"\<forall>w. w \<noteq> z \<and> dist w z < r2 \<longrightarrow> f w \<noteq> 0"
+ using assms(2) unfolding eventually_at by auto
+ define r3 where "r3=min r1 r2"
+ have "(\<lambda>w. (f w) powr of_int n) holomorphic_on ball z r3 - {z}"
+ apply (rule holomorphic_on_powr_of_int)
+ subgoal unfolding r3_def using r1 by auto
+ subgoal unfolding r3_def using r2 by (auto simp add:dist_commute)
+ done
+ moreover have "r3>0" unfolding r3_def using \<open>0 < r1\<close> \<open>0 < r2\<close> by linarith
+ ultimately show ?thesis unfolding isolated_singularity_at_def
+ apply (subst (asm) analytic_on_open[symmetric])
+ by auto
+qed
+
+lemma non_zero_neighbour:
+ assumes f_iso:"isolated_singularity_at f z"
+ and f_ness:"not_essential f z"
+ and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
+ shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0"
+proof -
+ obtain fn fp fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
+ and fr: "fp holomorphic_on cball z fr"
+ "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
+ using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
+ have "f w \<noteq> 0" when " w \<noteq> z" "dist w z < fr" for w
+ proof -
+ have "f w = fp w * (w - z) powr of_int fn" "fp w \<noteq> 0"
+ using fr(2)[rule_format, of w] using that by (auto simp add:dist_commute)
+ moreover have "(w - z) powr of_int fn \<noteq>0"
+ unfolding powr_eq_0_iff using \<open>w\<noteq>z\<close> by auto
+ ultimately show ?thesis by auto
+ qed
+ then show ?thesis using \<open>fr>0\<close> unfolding eventually_at by auto
+qed
+
+lemma non_zero_neighbour_pole:
+ assumes "is_pole f z"
+ shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0"
+ using assms filterlim_at_infinity_imp_eventually_ne[of f "at z" 0]
+ unfolding is_pole_def by auto
+
+lemma non_zero_neighbour_alt:
+ assumes holo: "f holomorphic_on S"
+ and "open S" "connected S" "z \<in> S" "\<beta> \<in> S" "f \<beta> \<noteq> 0"
+ shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0 \<and> w\<in>S"
+proof (cases "f z = 0")
+ case True
+ from isolated_zeros[OF holo \<open>open S\<close> \<open>connected S\<close> \<open>z \<in> S\<close> True \<open>\<beta> \<in> S\<close> \<open>f \<beta> \<noteq> 0\<close>]
+ obtain r where "0 < r" "ball z r \<subseteq> S" "\<forall>w \<in> ball z r - {z}.f w \<noteq> 0" by metis
+ then show ?thesis unfolding eventually_at
+ apply (rule_tac x=r in exI)
+ by (auto simp add:dist_commute)
+next
+ case False
+ obtain r1 where r1:"r1>0" "\<forall>y. dist z y < r1 \<longrightarrow> f y \<noteq> 0"
+ using continuous_at_avoid[of z f, OF _ False] assms(2,4) continuous_on_eq_continuous_at
+ holo holomorphic_on_imp_continuous_on by blast
+ obtain r2 where r2:"r2>0" "ball z r2 \<subseteq> S"
+ using assms(2) assms(4) openE by blast
+ show ?thesis unfolding eventually_at
+ apply (rule_tac x="min r1 r2" in exI)
+ using r1 r2 by (auto simp add:dist_commute)
+qed
+
+lemma not_essential_times[singularity_intros]:
+ assumes f_ness:"not_essential f z" and g_ness:"not_essential g z"
+ assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
+ shows "not_essential (\<lambda>w. f w * g w) z"
+proof -
+ define fg where "fg = (\<lambda>w. f w * g w)"
+ have ?thesis when "\<not> ((\<exists>\<^sub>Fw in (at z). f w\<noteq>0) \<and> (\<exists>\<^sub>Fw in (at z). g w\<noteq>0))"
+ proof -
+ have "\<forall>\<^sub>Fw in (at z). fg w=0"
+ using that[unfolded frequently_def, simplified] unfolding fg_def
+ by (auto elim: eventually_rev_mp)
+ from tendsto_cong[OF this] have "fg \<midarrow>z\<rightarrow>0" by auto
+ then show ?thesis unfolding not_essential_def fg_def by auto
+ qed
+ moreover have ?thesis when f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" and g_nconst:"\<exists>\<^sub>Fw in (at z). g w\<noteq>0"
+ proof -
+ obtain fn fp fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
+ and fr: "fp holomorphic_on cball z fr"
+ "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
+ using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
+ obtain gn gp gr where [simp]:"gp z \<noteq> 0" and "gr > 0"
+ and gr: "gp holomorphic_on cball z gr"
+ "\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powr of_int gn \<and> gp w \<noteq> 0"
+ using holomorphic_factor_puncture[OF g_iso g_ness g_nconst,THEN ex1_implies_ex] by auto
+
+ define r1 where "r1=(min fr gr)"
+ have "r1>0" unfolding r1_def using \<open>fr>0\<close> \<open>gr>0\<close> by auto
+ have fg_times:"fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" and fgp_nz:"fp w*gp w\<noteq>0"
+ when "w\<in>ball z r1 - {z}" for w
+ proof -
+ have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
+ using fr(2)[rule_format,of w] that unfolding r1_def by auto
+ moreover have "g w = gp w * (w - z) powr of_int gn" "gp w \<noteq> 0"
+ using gr(2)[rule_format, of w] that unfolding r1_def by auto
+ ultimately show "fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" "fp w*gp w\<noteq>0"
+ unfolding fg_def by (auto simp add:powr_add)
+ qed
+
+ have [intro]: "fp \<midarrow>z\<rightarrow>fp z" "gp \<midarrow>z\<rightarrow>gp z"
+ using fr(1) \<open>fr>0\<close> gr(1) \<open>gr>0\<close>
+ by (meson open_ball ball_subset_cball centre_in_ball
+ continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on
+ holomorphic_on_subset)+
+ have ?thesis when "fn+gn>0"
+ proof -
+ have "(\<lambda>w. (fp w * gp w) * (w - z) ^ (nat (fn+gn))) \<midarrow>z\<rightarrow>0"
+ using that by (auto intro!:tendsto_eq_intros)
+ then have "fg \<midarrow>z\<rightarrow> 0"
+ apply (elim Lim_transform_within[OF _ \<open>r1>0\<close>])
+ by (metis (no_types, hide_lams) Diff_iff cball_trivial dist_commute dist_self
+ eq_iff_diff_eq_0 fg_times less_le linorder_not_le mem_ball mem_cball powr_of_int
+ that)
+ then show ?thesis unfolding not_essential_def fg_def by auto
+ qed
+ moreover have ?thesis when "fn+gn=0"
+ proof -
+ have "(\<lambda>w. fp w * gp w) \<midarrow>z\<rightarrow>fp z*gp z"
+ using that by (auto intro!:tendsto_eq_intros)
+ then have "fg \<midarrow>z\<rightarrow> fp z*gp z"
+ apply (elim Lim_transform_within[OF _ \<open>r1>0\<close>])
+ apply (subst fg_times)
+ by (auto simp add:dist_commute that)
+ then show ?thesis unfolding not_essential_def fg_def by auto
+ qed
+ moreover have ?thesis when "fn+gn<0"
+ proof -
+ have "LIM w (at z). fp w * gp w / (w-z)^nat (-(fn+gn)) :> at_infinity"
+ apply (rule filterlim_divide_at_infinity)
+ apply (insert that, auto intro!:tendsto_eq_intros filterlim_atI)
+ using eventually_at_topological by blast
+ then have "is_pole fg z" unfolding is_pole_def
+ apply (elim filterlim_transform_within[OF _ _ \<open>r1>0\<close>],simp)
+ apply (subst fg_times,simp add:dist_commute)
+ apply (subst powr_of_int)
+ using that by (auto simp add:field_split_simps)
+ then show ?thesis unfolding not_essential_def fg_def by auto
+ qed
+ ultimately show ?thesis unfolding not_essential_def fg_def by fastforce
+ qed
+ ultimately show ?thesis by auto
+qed
+
+lemma not_essential_inverse[singularity_intros]:
+ assumes f_ness:"not_essential f z"
+ assumes f_iso:"isolated_singularity_at f z"
+ shows "not_essential (\<lambda>w. inverse (f w)) z"
+proof -
+ define vf where "vf = (\<lambda>w. inverse (f w))"
+ have ?thesis when "\<not>(\<exists>\<^sub>Fw in (at z). f w\<noteq>0)"
+ proof -
+ have "\<forall>\<^sub>Fw in (at z). f w=0"
+ using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp)
+ then have "\<forall>\<^sub>Fw in (at z). vf w=0"
+ unfolding vf_def by auto
+ from tendsto_cong[OF this] have "vf \<midarrow>z\<rightarrow>0" unfolding vf_def by auto
+ then show ?thesis unfolding not_essential_def vf_def by auto
+ qed
+ moreover have ?thesis when "is_pole f z"
+ proof -
+ have "vf \<midarrow>z\<rightarrow>0"
+ using that filterlim_at filterlim_inverse_at_iff unfolding is_pole_def vf_def by blast
+ then show ?thesis unfolding not_essential_def vf_def by auto
+ qed
+ moreover have ?thesis when "\<exists>x. f\<midarrow>z\<rightarrow>x " and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
+ proof -
+ from that obtain fz where fz:"f\<midarrow>z\<rightarrow>fz" by auto
+ have ?thesis when "fz=0"
+ proof -
+ have "(\<lambda>w. inverse (vf w)) \<midarrow>z\<rightarrow>0"
+ using fz that unfolding vf_def by auto
+ moreover have "\<forall>\<^sub>F w in at z. inverse (vf w) \<noteq> 0"
+ using non_zero_neighbour[OF f_iso f_ness f_nconst]
+ unfolding vf_def by auto
+ ultimately have "is_pole vf z"
+ using filterlim_inverse_at_iff[of vf "at z"] unfolding filterlim_at is_pole_def by auto
+ then show ?thesis unfolding not_essential_def vf_def by auto
+ qed
+ moreover have ?thesis when "fz\<noteq>0"
+ proof -
+ have "vf \<midarrow>z\<rightarrow>inverse fz"
+ using fz that unfolding vf_def by (auto intro:tendsto_eq_intros)
+ then show ?thesis unfolding not_essential_def vf_def by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ ultimately show ?thesis using f_ness unfolding not_essential_def by auto
+qed
+
+lemma isolated_singularity_at_inverse[singularity_intros]:
+ assumes f_iso:"isolated_singularity_at f z"
+ and f_ness:"not_essential f z"
+ shows "isolated_singularity_at (\<lambda>w. inverse (f w)) z"
+proof -
+ define vf where "vf = (\<lambda>w. inverse (f w))"
+ have ?thesis when "\<not>(\<exists>\<^sub>Fw in (at z). f w\<noteq>0)"
+ proof -
+ have "\<forall>\<^sub>Fw in (at z). f w=0"
+ using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp)
+ then have "\<forall>\<^sub>Fw in (at z). vf w=0"
+ unfolding vf_def by auto
+ then obtain d1 where "d1>0" and d1:"\<forall>x. x \<noteq> z \<and> dist x z < d1 \<longrightarrow> vf x = 0"
+ unfolding eventually_at by auto
+ then have "vf holomorphic_on ball z d1-{z}"
+ apply (rule_tac holomorphic_transform[of "\<lambda>_. 0"])
+ by (auto simp add:dist_commute)
+ then have "vf analytic_on ball z d1 - {z}"
+ by (simp add: analytic_on_open open_delete)
+ then show ?thesis using \<open>d1>0\<close> unfolding isolated_singularity_at_def vf_def by auto
+ qed
+ moreover have ?thesis when f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
+ proof -
+ have "\<forall>\<^sub>F w in at z. f w \<noteq> 0" using non_zero_neighbour[OF f_iso f_ness f_nconst] .
+ then obtain d1 where d1:"d1>0" "\<forall>x. x \<noteq> z \<and> dist x z < d1 \<longrightarrow> f x \<noteq> 0"
+ unfolding eventually_at by auto
+ obtain d2 where "d2>0" and d2:"f analytic_on ball z d2 - {z}"
+ using f_iso unfolding isolated_singularity_at_def by auto
+ define d3 where "d3=min d1 d2"
+ have "d3>0" unfolding d3_def using \<open>d1>0\<close> \<open>d2>0\<close> by auto
+ moreover have "vf analytic_on ball z d3 - {z}"
+ unfolding vf_def
+ apply (rule analytic_on_inverse)
+ subgoal using d2 unfolding d3_def by (elim analytic_on_subset) auto
+ subgoal for w using d1 unfolding d3_def by (auto simp add:dist_commute)
+ done
+ ultimately show ?thesis unfolding isolated_singularity_at_def vf_def by auto
+ qed
+ ultimately show ?thesis by auto
+qed
+
+lemma not_essential_divide[singularity_intros]:
+ assumes f_ness:"not_essential f z" and g_ness:"not_essential g z"
+ assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
+ shows "not_essential (\<lambda>w. f w / g w) z"
+proof -
+ have "not_essential (\<lambda>w. f w * inverse (g w)) z"
+ apply (rule not_essential_times[where g="\<lambda>w. inverse (g w)"])
+ using assms by (auto intro: isolated_singularity_at_inverse not_essential_inverse)
+ then show ?thesis by (simp add:field_simps)
+qed
+
+lemma
+ assumes f_iso:"isolated_singularity_at f z"
+ and g_iso:"isolated_singularity_at g z"
+ shows isolated_singularity_at_times[singularity_intros]:
+ "isolated_singularity_at (\<lambda>w. f w * g w) z" and
+ isolated_singularity_at_add[singularity_intros]:
+ "isolated_singularity_at (\<lambda>w. f w + g w) z"
+proof -
+ obtain d1 d2 where "d1>0" "d2>0"
+ and d1:"f analytic_on ball z d1 - {z}" and d2:"g analytic_on ball z d2 - {z}"
+ using f_iso g_iso unfolding isolated_singularity_at_def by auto
+ define d3 where "d3=min d1 d2"
+ have "d3>0" unfolding d3_def using \<open>d1>0\<close> \<open>d2>0\<close> by auto
+
+ have "(\<lambda>w. f w * g w) analytic_on ball z d3 - {z}"
+ apply (rule analytic_on_mult)
+ using d1 d2 unfolding d3_def by (auto elim:analytic_on_subset)
+ then show "isolated_singularity_at (\<lambda>w. f w * g w) z"
+ using \<open>d3>0\<close> unfolding isolated_singularity_at_def by auto
+ have "(\<lambda>w. f w + g w) analytic_on ball z d3 - {z}"
+ apply (rule analytic_on_add)
+ using d1 d2 unfolding d3_def by (auto elim:analytic_on_subset)
+ then show "isolated_singularity_at (\<lambda>w. f w + g w) z"
+ using \<open>d3>0\<close> unfolding isolated_singularity_at_def by auto
+qed
+
+lemma isolated_singularity_at_uminus[singularity_intros]:
+ assumes f_iso:"isolated_singularity_at f z"
+ shows "isolated_singularity_at (\<lambda>w. - f w) z"
+ using assms unfolding isolated_singularity_at_def using analytic_on_neg by blast
+
+lemma isolated_singularity_at_id[singularity_intros]:
+ "isolated_singularity_at (\<lambda>w. w) z"
+ unfolding isolated_singularity_at_def by (simp add: gt_ex)
+
+lemma isolated_singularity_at_minus[singularity_intros]:
+ assumes f_iso:"isolated_singularity_at f z"
+ and g_iso:"isolated_singularity_at g z"
+ shows "isolated_singularity_at (\<lambda>w. f w - g w) z"
+ using isolated_singularity_at_uminus[THEN isolated_singularity_at_add[OF f_iso,of "\<lambda>w. - g w"]
+ ,OF g_iso] by simp
+
+lemma isolated_singularity_at_divide[singularity_intros]:
+ assumes f_iso:"isolated_singularity_at f z"
+ and g_iso:"isolated_singularity_at g z"
+ and g_ness:"not_essential g z"
+ shows "isolated_singularity_at (\<lambda>w. f w / g w) z"
+ using isolated_singularity_at_inverse[THEN isolated_singularity_at_times[OF f_iso,
+ of "\<lambda>w. inverse (g w)"],OF g_iso g_ness] by (simp add:field_simps)
+
+lemma isolated_singularity_at_const[singularity_intros]:
+ "isolated_singularity_at (\<lambda>w. c) z"
+ unfolding isolated_singularity_at_def by (simp add: gt_ex)
+
+lemma isolated_singularity_at_holomorphic:
+ assumes "f holomorphic_on s-{z}" "open s" "z\<in>s"
+ shows "isolated_singularity_at f z"
+ using assms unfolding isolated_singularity_at_def
+ by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)
+
+subsubsection \<open>The order of non-essential singularities (i.e. removable singularities or poles)\<close>
+
+
+definition\<^marker>\<open>tag important\<close> zorder :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> int" where
+ "zorder f z = (THE n. (\<exists>h r. r>0 \<and> h holomorphic_on cball z r \<and> h z\<noteq>0
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w-z) powr (of_int n)
+ \<and> h w \<noteq>0)))"
+
+definition\<^marker>\<open>tag important\<close> zor_poly
+ ::"[complex \<Rightarrow> complex, complex] \<Rightarrow> complex \<Rightarrow> complex" where
+ "zor_poly f z = (SOME h. \<exists>r. r > 0 \<and> h holomorphic_on cball z r \<and> h z \<noteq> 0
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w - z) powr (zorder f z)
+ \<and> h w \<noteq>0))"
+
+lemma zorder_exist:
+ fixes f::"complex \<Rightarrow> complex" and z::complex
+ defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z"
+ assumes f_iso:"isolated_singularity_at f z"
+ and f_ness:"not_essential f z"
+ and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
+ shows "g z\<noteq>0 \<and> (\<exists>r. r>0 \<and> g holomorphic_on cball z r
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr n \<and> g w \<noteq>0))"
+proof -
+ define P where "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
+ have "\<exists>!n. \<exists>g r. P n g r"
+ using holomorphic_factor_puncture[OF assms(3-)] unfolding P_def by auto
+ then have "\<exists>g r. P n g r"
+ unfolding n_def P_def zorder_def
+ by (drule_tac theI',argo)
+ then have "\<exists>r. P n g r"
+ unfolding P_def zor_poly_def g_def n_def
+ by (drule_tac someI_ex,argo)
+ then obtain r1 where "P n g r1" by auto
+ then show ?thesis unfolding P_def by auto
+qed
+
+lemma
+ fixes f::"complex \<Rightarrow> complex" and z::complex
+ assumes f_iso:"isolated_singularity_at f z"
+ and f_ness:"not_essential f z"
+ and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
+ shows zorder_inverse: "zorder (\<lambda>w. inverse (f w)) z = - zorder f z"
+ and zor_poly_inverse: "\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. inverse (f w)) z w
+ = inverse (zor_poly f z w)"
+proof -
+ define vf where "vf = (\<lambda>w. inverse (f w))"
+ define fn vfn where
+ "fn = zorder f z" and "vfn = zorder vf z"
+ define fp vfp where
+ "fp = zor_poly f z" and "vfp = zor_poly vf z"
+
+ obtain fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
+ and fr: "fp holomorphic_on cball z fr"
+ "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
+ using zorder_exist[OF f_iso f_ness f_nconst,folded fn_def fp_def]
+ by auto
+ have fr_inverse: "vf w = (inverse (fp w)) * (w - z) powr (of_int (-fn))"
+ and fr_nz: "inverse (fp w)\<noteq>0"
+ when "w\<in>ball z fr - {z}" for w
+ proof -
+ have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
+ using fr(2)[rule_format,of w] that by auto
+ then show "vf w = (inverse (fp w)) * (w - z) powr (of_int (-fn))" "inverse (fp w)\<noteq>0"
+ unfolding vf_def by (auto simp add:powr_minus)
+ qed
+ obtain vfr where [simp]:"vfp z \<noteq> 0" and "vfr>0" and vfr:"vfp holomorphic_on cball z vfr"
+ "(\<forall>w\<in>cball z vfr - {z}. vf w = vfp w * (w - z) powr of_int vfn \<and> vfp w \<noteq> 0)"
+ proof -
+ have "isolated_singularity_at vf z"
+ using isolated_singularity_at_inverse[OF f_iso f_ness] unfolding vf_def .
+ moreover have "not_essential vf z"
+ using not_essential_inverse[OF f_ness f_iso] unfolding vf_def .
+ moreover have "\<exists>\<^sub>F w in at z. vf w \<noteq> 0"
+ using f_nconst unfolding vf_def by (auto elim:frequently_elim1)
+ ultimately show ?thesis using zorder_exist[of vf z, folded vfn_def vfp_def] that by auto
+ qed
+
+
+ define r1 where "r1 = min fr vfr"
+ have "r1>0" using \<open>fr>0\<close> \<open>vfr>0\<close> unfolding r1_def by simp
+ show "vfn = - fn"
+ apply (rule holomorphic_factor_unique[of r1 vfp z "\<lambda>w. inverse (fp w)" vf])
+ subgoal using \<open>r1>0\<close> by simp
+ subgoal by simp
+ subgoal by simp
+ subgoal
+ proof (rule ballI)
+ fix w assume "w \<in> ball z r1 - {z}"
+ then have "w \<in> ball z fr - {z}" "w \<in> cball z vfr - {z}" unfolding r1_def by auto
+ from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)]
+ show "vf w = vfp w * (w - z) powr of_int vfn \<and> vfp w \<noteq> 0
+ \<and> vf w = inverse (fp w) * (w - z) powr of_int (- fn) \<and> inverse (fp w) \<noteq> 0" by auto
+ qed
+ subgoal using vfr(1) unfolding r1_def by (auto intro!:holomorphic_intros)
+ subgoal using fr unfolding r1_def by (auto intro!:holomorphic_intros)
+ done
+
+ have "vfp w = inverse (fp w)" when "w\<in>ball z r1-{z}" for w
+ proof -
+ have "w \<in> ball z fr - {z}" "w \<in> cball z vfr - {z}" "w\<noteq>z" using that unfolding r1_def by auto
+ from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)] \<open>vfn = - fn\<close> \<open>w\<noteq>z\<close>
+ show ?thesis by auto
+ qed
+ then show "\<forall>\<^sub>Fw in (at z). vfp w = inverse (fp w)"
+ unfolding eventually_at using \<open>r1>0\<close>
+ apply (rule_tac x=r1 in exI)
+ by (auto simp add:dist_commute)
+qed
+
+lemma
+ fixes f g::"complex \<Rightarrow> complex" and z::complex
+ assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
+ and f_ness:"not_essential f z" and g_ness:"not_essential g z"
+ and fg_nconst: "\<exists>\<^sub>Fw in (at z). f w * g w\<noteq> 0"
+ shows zorder_times:"zorder (\<lambda>w. f w * g w) z = zorder f z + zorder g z" and
+ zor_poly_times:"\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w * g w) z w
+ = zor_poly f z w *zor_poly g z w"
+proof -
+ define fg where "fg = (\<lambda>w. f w * g w)"
+ define fn gn fgn where
+ "fn = zorder f z" and "gn = zorder g z" and "fgn = zorder fg z"
+ define fp gp fgp where
+ "fp = zor_poly f z" and "gp = zor_poly g z" and "fgp = zor_poly fg z"
+ have f_nconst:"\<exists>\<^sub>Fw in (at z). f w \<noteq> 0" and g_nconst:"\<exists>\<^sub>Fw in (at z).g w\<noteq> 0"
+ using fg_nconst by (auto elim!:frequently_elim1)
+ obtain fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
+ and fr: "fp holomorphic_on cball z fr"
+ "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
+ using zorder_exist[OF f_iso f_ness f_nconst,folded fp_def fn_def] by auto
+ obtain gr where [simp]:"gp z \<noteq> 0" and "gr > 0"
+ and gr: "gp holomorphic_on cball z gr"
+ "\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powr of_int gn \<and> gp w \<noteq> 0"
+ using zorder_exist[OF g_iso g_ness g_nconst,folded gn_def gp_def] by auto
+ define r1 where "r1=min fr gr"
+ have "r1>0" unfolding r1_def using \<open>fr>0\<close> \<open>gr>0\<close> by auto
+ have fg_times:"fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" and fgp_nz:"fp w*gp w\<noteq>0"
+ when "w\<in>ball z r1 - {z}" for w
+ proof -
+ have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
+ using fr(2)[rule_format,of w] that unfolding r1_def by auto
+ moreover have "g w = gp w * (w - z) powr of_int gn" "gp w \<noteq> 0"
+ using gr(2)[rule_format, of w] that unfolding r1_def by auto
+ ultimately show "fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" "fp w*gp w\<noteq>0"
+ unfolding fg_def by (auto simp add:powr_add)
+ qed
+
+ obtain fgr where [simp]:"fgp z \<noteq> 0" and "fgr > 0"
+ and fgr: "fgp holomorphic_on cball z fgr"
+ "\<forall>w\<in>cball z fgr - {z}. fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0"
+ proof -
+ have "fgp z \<noteq> 0 \<and> (\<exists>r>0. fgp holomorphic_on cball z r
+ \<and> (\<forall>w\<in>cball z r - {z}. fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0))"
+ apply (rule zorder_exist[of fg z, folded fgn_def fgp_def])
+ subgoal unfolding fg_def using isolated_singularity_at_times[OF f_iso g_iso] .
+ subgoal unfolding fg_def using not_essential_times[OF f_ness g_ness f_iso g_iso] .
+ subgoal unfolding fg_def using fg_nconst .
+ done
+ then show ?thesis using that by blast
+ qed
+ define r2 where "r2 = min fgr r1"
+ have "r2>0" using \<open>r1>0\<close> \<open>fgr>0\<close> unfolding r2_def by simp
+ show "fgn = fn + gn "
+ apply (rule holomorphic_factor_unique[of r2 fgp z "\<lambda>w. fp w * gp w" fg])
+ subgoal using \<open>r2>0\<close> by simp
+ subgoal by simp
+ subgoal by simp
+ subgoal
+ proof (rule ballI)
+ fix w assume "w \<in> ball z r2 - {z}"
+ then have "w \<in> ball z r1 - {z}" "w \<in> cball z fgr - {z}" unfolding r2_def by auto
+ from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)]
+ show "fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0
+ \<and> fg w = fp w * gp w * (w - z) powr of_int (fn + gn) \<and> fp w * gp w \<noteq> 0" by auto
+ qed
+ subgoal using fgr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
+ subgoal using fr(1) gr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
+ done
+
+ have "fgp w = fp w *gp w" when "w\<in>ball z r2-{z}" for w
+ proof -
+ have "w \<in> ball z r1 - {z}" "w \<in> cball z fgr - {z}" "w\<noteq>z" using that unfolding r2_def by auto
+ from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)] \<open>fgn = fn + gn\<close> \<open>w\<noteq>z\<close>
+ show ?thesis by auto
+ qed
+ then show "\<forall>\<^sub>Fw in (at z). fgp w = fp w * gp w"
+ using \<open>r2>0\<close> unfolding eventually_at by (auto simp add:dist_commute)
+qed
+
+lemma
+ fixes f g::"complex \<Rightarrow> complex" and z::complex
+ assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
+ and f_ness:"not_essential f z" and g_ness:"not_essential g z"
+ and fg_nconst: "\<exists>\<^sub>Fw in (at z). f w * g w\<noteq> 0"
+ shows zorder_divide:"zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z" and
+ zor_poly_divide:"\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w / g w) z w
+ = zor_poly f z w / zor_poly g z w"
+proof -
+ have f_nconst:"\<exists>\<^sub>Fw in (at z). f w \<noteq> 0" and g_nconst:"\<exists>\<^sub>Fw in (at z).g w\<noteq> 0"
+ using fg_nconst by (auto elim!:frequently_elim1)
+ define vg where "vg=(\<lambda>w. inverse (g w))"
+ have "zorder (\<lambda>w. f w * vg w) z = zorder f z + zorder vg z"
+ apply (rule zorder_times[OF f_iso _ f_ness,of vg])
+ subgoal unfolding vg_def using isolated_singularity_at_inverse[OF g_iso g_ness] .
+ subgoal unfolding vg_def using not_essential_inverse[OF g_ness g_iso] .
+ subgoal unfolding vg_def using fg_nconst by (auto elim!:frequently_elim1)
+ done
+ then show "zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z"
+ using zorder_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def
+ by (auto simp add:field_simps)
+
+ have "\<forall>\<^sub>F w in at z. zor_poly (\<lambda>w. f w * vg w) z w = zor_poly f z w * zor_poly vg z w"
+ apply (rule zor_poly_times[OF f_iso _ f_ness,of vg])
+ subgoal unfolding vg_def using isolated_singularity_at_inverse[OF g_iso g_ness] .
+ subgoal unfolding vg_def using not_essential_inverse[OF g_ness g_iso] .
+ subgoal unfolding vg_def using fg_nconst by (auto elim!:frequently_elim1)
+ done
+ then show "\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w / g w) z w = zor_poly f z w / zor_poly g z w"
+ using zor_poly_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def
+ apply eventually_elim
+ by (auto simp add:field_simps)
+qed
+
+lemma zorder_exist_zero:
+ fixes f::"complex \<Rightarrow> complex" and z::complex
+ defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z"
+ assumes holo: "f holomorphic_on s" and
+ "open s" "connected s" "z\<in>s"
+ and non_const: "\<exists>w\<in>s. f w \<noteq> 0"
+ shows "(if f z=0 then n > 0 else n=0) \<and> (\<exists>r. r>0 \<and> cball z r \<subseteq> s \<and> g holomorphic_on cball z r
+ \<and> (\<forall>w\<in>cball z r. f w = g w * (w-z) ^ nat n \<and> g w \<noteq>0))"
+proof -
+ obtain r where "g z \<noteq> 0" and r: "r>0" "cball z r \<subseteq> s" "g holomorphic_on cball z r"
+ "(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ proof -
+ have "g z \<noteq> 0 \<and> (\<exists>r>0. g holomorphic_on cball z r
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0))"
+ proof (rule zorder_exist[of f z,folded g_def n_def])
+ show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
+ using holo assms(4,6)
+ by (meson Diff_subset open_ball analytic_on_holomorphic holomorphic_on_subset openE)
+ show "not_essential f z" unfolding not_essential_def
+ using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on
+ by fastforce
+ have "\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w\<in>s"
+ proof -
+ obtain w where "w\<in>s" "f w\<noteq>0" using non_const by auto
+ then show ?thesis
+ by (rule non_zero_neighbour_alt[OF holo \<open>open s\<close> \<open>connected s\<close> \<open>z\<in>s\<close>])
+ qed
+ then show "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
+ apply (elim eventually_frequentlyE)
+ by auto
+ qed
+ then obtain r1 where "g z \<noteq> 0" "r1>0" and r1:"g holomorphic_on cball z r1"
+ "(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ by auto
+ obtain r2 where r2: "r2>0" "cball z r2 \<subseteq> s"
+ using assms(4,6) open_contains_cball_eq by blast
+ define r3 where "r3=min r1 r2"
+ have "r3>0" "cball z r3 \<subseteq> s" using \<open>r1>0\<close> r2 unfolding r3_def by auto
+ moreover have "g holomorphic_on cball z r3"
+ using r1(1) unfolding r3_def by auto
+ moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ using r1(2) unfolding r3_def by auto
+ ultimately show ?thesis using that[of r3] \<open>g z\<noteq>0\<close> by auto
+ qed
+
+ have if_0:"if f z=0 then n > 0 else n=0"
+ proof -
+ have "f\<midarrow> z \<rightarrow> f z"
+ by (metis assms(4,6,7) at_within_open continuous_on holo holomorphic_on_imp_continuous_on)
+ then have "(\<lambda>w. g w * (w - z) powr of_int n) \<midarrow>z\<rightarrow> f z"
+ apply (elim Lim_transform_within_open[where s="ball z r"])
+ using r by auto
+ moreover have "g \<midarrow>z\<rightarrow>g z"
+ by (metis (mono_tags, lifting) open_ball at_within_open_subset
+ ball_subset_cball centre_in_ball continuous_on holomorphic_on_imp_continuous_on r(1,3) subsetCE)
+ ultimately have "(\<lambda>w. (g w * (w - z) powr of_int n) / g w) \<midarrow>z\<rightarrow> f z/g z"
+ apply (rule_tac tendsto_divide)
+ using \<open>g z\<noteq>0\<close> by auto
+ then have powr_tendsto:"(\<lambda>w. (w - z) powr of_int n) \<midarrow>z\<rightarrow> f z/g z"
+ apply (elim Lim_transform_within_open[where s="ball z r"])
+ using r by auto
+
+ have ?thesis when "n\<ge>0" "f z=0"
+ proof -
+ have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> f z/g z"
+ using powr_tendsto
+ apply (elim Lim_transform_within[where d=r])
+ by (auto simp add: powr_of_int \<open>n\<ge>0\<close> \<open>r>0\<close>)
+ then have *:"(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 0" using \<open>f z=0\<close> by simp
+ moreover have False when "n=0"
+ proof -
+ have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 1"
+ using \<open>n=0\<close> by auto
+ then show False using * using LIM_unique zero_neq_one by blast
+ qed
+ ultimately show ?thesis using that by fastforce
+ qed
+ moreover have ?thesis when "n\<ge>0" "f z\<noteq>0"
+ proof -
+ have False when "n>0"
+ proof -
+ have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> f z/g z"
+ using powr_tendsto
+ apply (elim Lim_transform_within[where d=r])
+ by (auto simp add: powr_of_int \<open>n\<ge>0\<close> \<open>r>0\<close>)
+ moreover have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 0"
+ using \<open>n>0\<close> by (auto intro!:tendsto_eq_intros)
+ ultimately show False using \<open>f z\<noteq>0\<close> \<open>g z\<noteq>0\<close> using LIM_unique divide_eq_0_iff by blast
+ qed
+ then show ?thesis using that by force
+ qed
+ moreover have False when "n<0"
+ proof -
+ have "(\<lambda>w. inverse ((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> f z/g z"
+ "(\<lambda>w.((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> 0"
+ subgoal using powr_tendsto powr_of_int that
+ by (elim Lim_transform_within_open[where s=UNIV],auto)
+ subgoal using that by (auto intro!:tendsto_eq_intros)
+ done
+ from tendsto_mult[OF this,simplified]
+ have "(\<lambda>x. inverse ((x - z) ^ nat (- n)) * (x - z) ^ nat (- n)) \<midarrow>z\<rightarrow> 0" .
+ then have "(\<lambda>x. 1::complex) \<midarrow>z\<rightarrow> 0"
+ by (elim Lim_transform_within_open[where s=UNIV],auto)
+ then show False using LIM_const_eq by fastforce
+ qed
+ ultimately show ?thesis by fastforce
+ qed
+ moreover have "f w = g w * (w-z) ^ nat n \<and> g w \<noteq>0" when "w\<in>cball z r" for w
+ proof (cases "w=z")
+ case True
+ then have "f \<midarrow>z\<rightarrow>f w"
+ using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on by fastforce
+ then have "(\<lambda>w. g w * (w-z) ^ nat n) \<midarrow>z\<rightarrow>f w"
+ proof (elim Lim_transform_within[OF _ \<open>r>0\<close>])
+ fix x assume "0 < dist x z" "dist x z < r"
+ then have "x \<in> cball z r - {z}" "x\<noteq>z"
+ unfolding cball_def by (auto simp add: dist_commute)
+ then have "f x = g x * (x - z) powr of_int n"
+ using r(4)[rule_format,of x] by simp
+ also have "... = g x * (x - z) ^ nat n"
+ apply (subst powr_of_int)
+ using if_0 \<open>x\<noteq>z\<close> by (auto split:if_splits)
+ finally show "f x = g x * (x - z) ^ nat n" .
+ qed
+ moreover have "(\<lambda>w. g w * (w-z) ^ nat n) \<midarrow>z\<rightarrow> g w * (w-z) ^ nat n"
+ using True apply (auto intro!:tendsto_eq_intros)
+ by (metis open_ball at_within_open_subset ball_subset_cball centre_in_ball
+ continuous_on holomorphic_on_imp_continuous_on r(1) r(3) that)
+ ultimately have "f w = g w * (w-z) ^ nat n" using LIM_unique by blast
+ then show ?thesis using \<open>g z\<noteq>0\<close> True by auto
+ next
+ case False
+ then have "f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0"
+ using r(4) that by auto
+ then show ?thesis using False if_0 powr_of_int by (auto split:if_splits)
+ qed
+ ultimately show ?thesis using r by auto
+qed
+
+lemma zorder_exist_pole:
+ fixes f::"complex \<Rightarrow> complex" and z::complex
+ defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z"
+ assumes holo: "f holomorphic_on s-{z}" and
+ "open s" "z\<in>s"
+ and "is_pole f z"
+ shows "n < 0 \<and> g z\<noteq>0 \<and> (\<exists>r. r>0 \<and> cball z r \<subseteq> s \<and> g holomorphic_on cball z r
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w / (w-z) ^ nat (- n) \<and> g w \<noteq>0))"
+proof -
+ obtain r where "g z \<noteq> 0" and r: "r>0" "cball z r \<subseteq> s" "g holomorphic_on cball z r"
+ "(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ proof -
+ have "g z \<noteq> 0 \<and> (\<exists>r>0. g holomorphic_on cball z r
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0))"
+ proof (rule zorder_exist[of f z,folded g_def n_def])
+ show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
+ using holo assms(4,5)
+ by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)
+ show "not_essential f z" unfolding not_essential_def
+ using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on
+ by fastforce
+ from non_zero_neighbour_pole[OF \<open>is_pole f z\<close>] show "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
+ apply (elim eventually_frequentlyE)
+ by auto
+ qed
+ then obtain r1 where "g z \<noteq> 0" "r1>0" and r1:"g holomorphic_on cball z r1"
+ "(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ by auto
+ obtain r2 where r2: "r2>0" "cball z r2 \<subseteq> s"
+ using assms(4,5) open_contains_cball_eq by metis
+ define r3 where "r3=min r1 r2"
+ have "r3>0" "cball z r3 \<subseteq> s" using \<open>r1>0\<close> r2 unfolding r3_def by auto
+ moreover have "g holomorphic_on cball z r3"
+ using r1(1) unfolding r3_def by auto
+ moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ using r1(2) unfolding r3_def by auto
+ ultimately show ?thesis using that[of r3] \<open>g z\<noteq>0\<close> by auto
+ qed
+
+ have "n<0"
+ proof (rule ccontr)
+ assume " \<not> n < 0"
+ define c where "c=(if n=0 then g z else 0)"
+ have [simp]:"g \<midarrow>z\<rightarrow> g z"
+ by (metis open_ball at_within_open ball_subset_cball centre_in_ball
+ continuous_on holomorphic_on_imp_continuous_on holomorphic_on_subset r(1) r(3) )
+ have "\<forall>\<^sub>F x in at z. f x = g x * (x - z) ^ nat n"
+ unfolding eventually_at_topological
+ apply (rule_tac exI[where x="ball z r"])
+ using r powr_of_int \<open>\<not> n < 0\<close> by auto
+ moreover have "(\<lambda>x. g x * (x - z) ^ nat n) \<midarrow>z\<rightarrow>c"
+ proof (cases "n=0")
+ case True
+ then show ?thesis unfolding c_def by simp
+ next
+ case False
+ then have "(\<lambda>x. (x - z) ^ nat n) \<midarrow>z\<rightarrow> 0" using \<open>\<not> n < 0\<close>
+ by (auto intro!:tendsto_eq_intros)
+ from tendsto_mult[OF _ this,of g "g z",simplified]
+ show ?thesis unfolding c_def using False by simp
+ qed
+ ultimately have "f \<midarrow>z\<rightarrow>c" using tendsto_cong by fast
+ then show False using \<open>is_pole f z\<close> at_neq_bot not_tendsto_and_filterlim_at_infinity
+ unfolding is_pole_def by blast
+ qed
+ moreover have "\<forall>w\<in>cball z r - {z}. f w = g w / (w-z) ^ nat (- n) \<and> g w \<noteq>0"
+ using r(4) \<open>n<0\<close> powr_of_int
+ by (metis Diff_iff divide_inverse eq_iff_diff_eq_0 insert_iff linorder_not_le)
+ ultimately show ?thesis using r(1-3) \<open>g z\<noteq>0\<close> by auto
+qed
+
+lemma zorder_eqI:
+ assumes "open s" "z \<in> s" "g holomorphic_on s" "g z \<noteq> 0"
+ assumes fg_eq:"\<And>w. \<lbrakk>w \<in> s;w\<noteq>z\<rbrakk> \<Longrightarrow> f w = g w * (w - z) powr n"
+ shows "zorder f z = n"
+proof -
+ have "continuous_on s g" by (rule holomorphic_on_imp_continuous_on) fact
+ moreover have "open (-{0::complex})" by auto
+ ultimately have "open ((g -` (-{0})) \<inter> s)"
+ unfolding continuous_on_open_vimage[OF \<open>open s\<close>] by blast
+ moreover from assms have "z \<in> (g -` (-{0})) \<inter> s" by auto
+ ultimately obtain r where r: "r > 0" "cball z r \<subseteq> s \<inter> (g -` (-{0}))"
+ unfolding open_contains_cball by blast
+
+ let ?gg= "(\<lambda>w. g w * (w - z) powr n)"
+ define P where "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
+ have "P n g r"
+ unfolding P_def using r assms(3,4,5) by auto
+ then have "\<exists>g r. P n g r" by auto
+ moreover have unique: "\<exists>!n. \<exists>g r. P n g r" unfolding P_def
+ proof (rule holomorphic_factor_puncture)
+ have "ball z r-{z} \<subseteq> s" using r using ball_subset_cball by blast
+ then have "?gg holomorphic_on ball z r-{z}"
+ using \<open>g holomorphic_on s\<close> r by (auto intro!: holomorphic_intros)
+ then have "f holomorphic_on ball z r - {z}"
+ apply (elim holomorphic_transform)
+ using fg_eq \<open>ball z r-{z} \<subseteq> s\<close> by auto
+ then show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
+ using analytic_on_open open_delete r(1) by blast
+ next
+ have "not_essential ?gg z"
+ proof (intro singularity_intros)
+ show "not_essential g z"
+ by (meson \<open>continuous_on s g\<close> assms(1) assms(2) continuous_on_eq_continuous_at
+ isCont_def not_essential_def)
+ show " \<forall>\<^sub>F w in at z. w - z \<noteq> 0" by (simp add: eventually_at_filter)
+ then show "LIM w at z. w - z :> at 0"
+ unfolding filterlim_at by (auto intro:tendsto_eq_intros)
+ show "isolated_singularity_at g z"
+ by (meson Diff_subset open_ball analytic_on_holomorphic
+ assms(1,2,3) holomorphic_on_subset isolated_singularity_at_def openE)
+ qed
+ then show "not_essential f z"
+ apply (elim not_essential_transform)
+ unfolding eventually_at using assms(1,2) assms(5)[symmetric]
+ by (metis dist_commute mem_ball openE subsetCE)
+ show "\<exists>\<^sub>F w in at z. f w \<noteq> 0" unfolding frequently_at
+ proof (rule,rule)
+ fix d::real assume "0 < d"
+ define z' where "z'=z+min d r / 2"
+ have "z' \<noteq> z" " dist z' z < d "
+ unfolding z'_def using \<open>d>0\<close> \<open>r>0\<close>
+ by (auto simp add:dist_norm)
+ moreover have "f z' \<noteq> 0"
+ proof (subst fg_eq[OF _ \<open>z'\<noteq>z\<close>])
+ have "z' \<in> cball z r" unfolding z'_def using \<open>r>0\<close> \<open>d>0\<close> by (auto simp add:dist_norm)
+ then show " z' \<in> s" using r(2) by blast
+ show "g z' * (z' - z) powr of_int n \<noteq> 0"
+ using P_def \<open>P n g r\<close> \<open>z' \<in> cball z r\<close> calculation(1) by auto
+ qed
+ ultimately show "\<exists>x\<in>UNIV. x \<noteq> z \<and> dist x z < d \<and> f x \<noteq> 0" by auto
+ qed
+ qed
+ ultimately have "(THE n. \<exists>g r. P n g r) = n"
+ by (rule_tac the1_equality)
+ then show ?thesis unfolding zorder_def P_def by blast
+qed
+
+lemma simple_zeroI:
+ assumes "open s" "z \<in> s" "g holomorphic_on s" "g z \<noteq> 0"
+ assumes "\<And>w. w \<in> s \<Longrightarrow> f w = g w * (w - z)"
+ shows "zorder f z = 1"
+ using assms(1-4) by (rule zorder_eqI) (use assms(5) in auto)
+
+lemma higher_deriv_power:
+ shows "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) w =
+ pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)"
+proof (induction j arbitrary: w)
+ case 0
+ thus ?case by auto
+next
+ case (Suc j w)
+ have "(deriv ^^ Suc j) (\<lambda>w. (w - z) ^ n) w = deriv ((deriv ^^ j) (\<lambda>w. (w - z) ^ n)) w"
+ by simp
+ also have "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) =
+ (\<lambda>w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))"
+ using Suc by (intro Suc.IH ext)
+ also {
+ have "(\<dots> has_field_derivative of_nat (n - j) *
+ pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - Suc j)) (at w)"
+ using Suc.prems by (auto intro!: derivative_eq_intros)
+ also have "of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j =
+ pochhammer (of_nat (Suc n - Suc j)) (Suc j)"
+ by (cases "Suc j \<le> n", subst pochhammer_rec)
+ (insert Suc.prems, simp_all add: algebra_simps Suc_diff_le pochhammer_0_left)
+ finally have "deriv (\<lambda>w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) w =
+ \<dots> * (w - z) ^ (n - Suc j)"
+ by (rule DERIV_imp_deriv)
+ }
+ finally show ?case .
+qed
+
+lemma zorder_zero_eqI:
+ assumes f_holo:"f holomorphic_on s" and "open s" "z \<in> s"
+ assumes zero: "\<And>i. i < nat n \<Longrightarrow> (deriv ^^ i) f z = 0"
+ assumes nz: "(deriv ^^ nat n) f z \<noteq> 0" and "n\<ge>0"
+ shows "zorder f z = n"
+proof -
+ obtain r where [simp]:"r>0" and "ball z r \<subseteq> s"
+ using \<open>open s\<close> \<open>z\<in>s\<close> openE by blast
+ have nz':"\<exists>w\<in>ball z r. f w \<noteq> 0"
+ proof (rule ccontr)
+ assume "\<not> (\<exists>w\<in>ball z r. f w \<noteq> 0)"
+ then have "eventually (\<lambda>u. f u = 0) (nhds z)"
+ using \<open>r>0\<close> unfolding eventually_nhds
+ apply (rule_tac x="ball z r" in exI)
+ by auto
+ then have "(deriv ^^ nat n) f z = (deriv ^^ nat n) (\<lambda>_. 0) z"
+ by (intro higher_deriv_cong_ev) auto
+ also have "(deriv ^^ nat n) (\<lambda>_. 0) z = 0"
+ by (induction n) simp_all
+ finally show False using nz by contradiction
+ qed
+
+ define zn g where "zn = zorder f z" and "g = zor_poly f z"
+ obtain e where e_if:"if f z = 0 then 0 < zn else zn = 0" and
+ [simp]:"e>0" and "cball z e \<subseteq> ball z r" and
+ g_holo:"g holomorphic_on cball z e" and
+ e_fac:"(\<forall>w\<in>cball z e. f w = g w * (w - z) ^ nat zn \<and> g w \<noteq> 0)"
+ proof -
+ have "f holomorphic_on ball z r"
+ using f_holo \<open>ball z r \<subseteq> s\<close> by auto
+ from that zorder_exist_zero[of f "ball z r" z,simplified,OF this nz',folded zn_def g_def]
+ show ?thesis by blast
+ qed
+ from this(1,2,5) have "zn\<ge>0" "g z\<noteq>0"
+ subgoal by (auto split:if_splits)
+ subgoal using \<open>0 < e\<close> ball_subset_cball centre_in_ball e_fac by blast
+ done
+
+ define A where "A = (\<lambda>i. of_nat (i choose (nat zn)) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z)"
+ have deriv_A:"(deriv ^^ i) f z = (if zn \<le> int i then A i else 0)" for i
+ proof -
+ have "eventually (\<lambda>w. w \<in> ball z e) (nhds z)"
+ using \<open>cball z e \<subseteq> ball z r\<close> \<open>e>0\<close> by (intro eventually_nhds_in_open) auto
+ hence "eventually (\<lambda>w. f w = (w - z) ^ (nat zn) * g w) (nhds z)"
+ apply eventually_elim
+ by (use e_fac in auto)
+ hence "(deriv ^^ i) f z = (deriv ^^ i) (\<lambda>w. (w - z) ^ nat zn * g w) z"
+ by (intro higher_deriv_cong_ev) auto
+ also have "\<dots> = (\<Sum>j=0..i. of_nat (i choose j) *
+ (deriv ^^ j) (\<lambda>w. (w - z) ^ nat zn) z * (deriv ^^ (i - j)) g z)"
+ using g_holo \<open>e>0\<close>
+ by (intro higher_deriv_mult[of _ "ball z e"]) (auto intro!: holomorphic_intros)
+ also have "\<dots> = (\<Sum>j=0..i. if j = nat zn then
+ of_nat (i choose nat zn) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z else 0)"
+ proof (intro sum.cong refl, goal_cases)
+ case (1 j)
+ have "(deriv ^^ j) (\<lambda>w. (w - z) ^ nat zn) z =
+ pochhammer (of_nat (Suc (nat zn) - j)) j * 0 ^ (nat zn - j)"
+ by (subst higher_deriv_power) auto
+ also have "\<dots> = (if j = nat zn then fact j else 0)"
+ by (auto simp: not_less pochhammer_0_left pochhammer_fact)
+ also have "of_nat (i choose j) * \<dots> * (deriv ^^ (i - j)) g z =
+ (if j = nat zn then of_nat (i choose (nat zn)) * fact (nat zn)
+ * (deriv ^^ (i - nat zn)) g z else 0)"
+ by simp
+ finally show ?case .
+ qed
+ also have "\<dots> = (if i \<ge> zn then A i else 0)"
+ by (auto simp: A_def)
+ finally show "(deriv ^^ i) f z = \<dots>" .
+ qed
+
+ have False when "n<zn"
+ proof -
+ have "(deriv ^^ nat n) f z = 0"
+ using deriv_A[of "nat n"] that \<open>n\<ge>0\<close> by auto
+ with nz show False by auto
+ qed
+ moreover have "n\<le>zn"
+ proof -
+ have "g z \<noteq> 0" using e_fac[rule_format,of z] \<open>e>0\<close> by simp
+ then have "(deriv ^^ nat zn) f z \<noteq> 0"
+ using deriv_A[of "nat zn"] by(auto simp add:A_def)
+ then have "nat zn \<ge> nat n" using zero[of "nat zn"] by linarith
+ moreover have "zn\<ge>0" using e_if by (auto split:if_splits)
+ ultimately show ?thesis using nat_le_eq_zle by blast
+ qed
+ ultimately show ?thesis unfolding zn_def by fastforce
+qed
+
+lemma
+ assumes "eventually (\<lambda>z. f z = g z) (at z)" "z = z'"
+ shows zorder_cong:"zorder f z = zorder g z'" and zor_poly_cong:"zor_poly f z = zor_poly g z'"
+proof -
+ define P where "P = (\<lambda>ff n h r. 0 < r \<and> h holomorphic_on cball z r \<and> h z\<noteq>0
+ \<and> (\<forall>w\<in>cball z r - {z}. ff w = h w * (w-z) powr (of_int n) \<and> h w\<noteq>0))"
+ have "(\<exists>r. P f n h r) = (\<exists>r. P g n h r)" for n h
+ proof -
+ have *: "\<exists>r. P g n h r" if "\<exists>r. P f n h r" and "eventually (\<lambda>x. f x = g x) (at z)" for f g
+ proof -
+ from that(1) obtain r1 where r1_P:"P f n h r1" by auto
+ from that(2) obtain r2 where "r2>0" and r2_dist:"\<forall>x. x \<noteq> z \<and> dist x z \<le> r2 \<longrightarrow> f x = g x"
+ unfolding eventually_at_le by auto
+ define r where "r=min r1 r2"
+ have "r>0" "h z\<noteq>0" using r1_P \<open>r2>0\<close> unfolding r_def P_def by auto
+ moreover have "h holomorphic_on cball z r"
+ using r1_P unfolding P_def r_def by auto
+ moreover have "g w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0" when "w\<in>cball z r - {z}" for w
+ proof -
+ have "f w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0"
+ using r1_P that unfolding P_def r_def by auto
+ moreover have "f w=g w" using r2_dist[rule_format,of w] that unfolding r_def
+ by (simp add: dist_commute)
+ ultimately show ?thesis by simp
+ qed
+ ultimately show ?thesis unfolding P_def by auto
+ qed
+ from assms have eq': "eventually (\<lambda>z. g z = f z) (at z)"
+ by (simp add: eq_commute)
+ show ?thesis
+ by (rule iffI[OF *[OF _ assms(1)] *[OF _ eq']])
+ qed
+ then show "zorder f z = zorder g z'" "zor_poly f z = zor_poly g z'"
+ using \<open>z=z'\<close> unfolding P_def zorder_def zor_poly_def by auto
+qed
+
+lemma zorder_nonzero_div_power:
+ assumes "open s" "z \<in> s" "f holomorphic_on s" "f z \<noteq> 0" "n > 0"
+ shows "zorder (\<lambda>w. f w / (w - z) ^ n) z = - n"
+ apply (rule zorder_eqI[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>f holomorphic_on s\<close> \<open>f z\<noteq>0\<close>])
+ apply (subst powr_of_int)
+ using \<open>n>0\<close> by (auto simp add:field_simps)
+
+lemma zor_poly_eq:
+ assumes "isolated_singularity_at f z" "not_essential f z" "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
+ shows "eventually (\<lambda>w. zor_poly f z w = f w * (w - z) powr - zorder f z) (at z)"
+proof -
+ obtain r where r:"r>0"
+ "(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w * (w - z) powr of_int (zorder f z))"
+ using zorder_exist[OF assms] by blast
+ then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w * (w - z) powr - zorder f z"
+ by (auto simp: field_simps powr_minus)
+ have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
+ using r eventually_at_ball'[of r z UNIV] by auto
+ thus ?thesis by eventually_elim (insert *, auto)
+qed
+
+lemma zor_poly_zero_eq:
+ assumes "f holomorphic_on s" "open s" "connected s" "z \<in> s" "\<exists>w\<in>s. f w \<noteq> 0"
+ shows "eventually (\<lambda>w. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)) (at z)"
+proof -
+ obtain r where r:"r>0"
+ "(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w * (w - z) ^ nat (zorder f z))"
+ using zorder_exist_zero[OF assms] by auto
+ then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)"
+ by (auto simp: field_simps powr_minus)
+ have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
+ using r eventually_at_ball'[of r z UNIV] by auto
+ thus ?thesis by eventually_elim (insert *, auto)
+qed
+
+lemma zor_poly_pole_eq:
+ assumes f_iso:"isolated_singularity_at f z" "is_pole f z"
+ shows "eventually (\<lambda>w. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)) (at z)"
+proof -
+ obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}"
+ using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast
+ obtain r where r:"r>0"
+ "(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w / (w - z) ^ nat (- zorder f z))"
+ using zorder_exist_pole[OF f_holo,simplified,OF \<open>is_pole f z\<close>] by auto
+ then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)"
+ by (auto simp: field_simps)
+ have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
+ using r eventually_at_ball'[of r z UNIV] by auto
+ thus ?thesis by eventually_elim (insert *, auto)
+qed
+
+lemma zor_poly_eqI:
+ fixes f :: "complex \<Rightarrow> complex" and z0 :: complex
+ defines "n \<equiv> zorder f z0"
+ assumes "isolated_singularity_at f z0" "not_essential f z0" "\<exists>\<^sub>F w in at z0. f w \<noteq> 0"
+ assumes lim: "((\<lambda>x. f (g x) * (g x - z0) powr - n) \<longlongrightarrow> c) F"
+ assumes g: "filterlim g (at z0) F" and "F \<noteq> bot"
+ shows "zor_poly f z0 z0 = c"
+proof -
+ from zorder_exist[OF assms(2-4)] obtain r where
+ r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r"
+ "\<And>w. w \<in> cball z0 r - {z0} \<Longrightarrow> f w = zor_poly f z0 w * (w - z0) powr n"
+ unfolding n_def by blast
+ from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)"
+ using eventually_at_ball'[of r z0 UNIV] by auto
+ hence "eventually (\<lambda>w. zor_poly f z0 w = f w * (w - z0) powr - n) (at z0)"
+ by eventually_elim (insert r, auto simp: field_simps powr_minus)
+ moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
+ using r by (intro holomorphic_on_imp_continuous_on) auto
+ with r(1,2) have "isCont (zor_poly f z0) z0"
+ by (auto simp: continuous_on_eq_continuous_at)
+ hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
+ unfolding isCont_def .
+ ultimately have "((\<lambda>w. f w * (w - z0) powr - n) \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
+ by (blast intro: Lim_transform_eventually)
+ hence "((\<lambda>x. f (g x) * (g x - z0) powr - n) \<longlongrightarrow> zor_poly f z0 z0) F"
+ by (rule filterlim_compose[OF _ g])
+ from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis .
+qed
+
+lemma zor_poly_zero_eqI:
+ fixes f :: "complex \<Rightarrow> complex" and z0 :: complex
+ defines "n \<equiv> zorder f z0"
+ assumes "f holomorphic_on A" "open A" "connected A" "z0 \<in> A" "\<exists>z\<in>A. f z \<noteq> 0"
+ assumes lim: "((\<lambda>x. f (g x) / (g x - z0) ^ nat n) \<longlongrightarrow> c) F"
+ assumes g: "filterlim g (at z0) F" and "F \<noteq> bot"
+ shows "zor_poly f z0 z0 = c"
+proof -
+ from zorder_exist_zero[OF assms(2-6)] obtain r where
+ r: "r > 0" "cball z0 r \<subseteq> A" "zor_poly f z0 holomorphic_on cball z0 r"
+ "\<And>w. w \<in> cball z0 r \<Longrightarrow> f w = zor_poly f z0 w * (w - z0) ^ nat n"
+ unfolding n_def by blast
+ from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)"
+ using eventually_at_ball'[of r z0 UNIV] by auto
+ hence "eventually (\<lambda>w. zor_poly f z0 w = f w / (w - z0) ^ nat n) (at z0)"
+ by eventually_elim (insert r, auto simp: field_simps)
+ moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
+ using r by (intro holomorphic_on_imp_continuous_on) auto
+ with r(1,2) have "isCont (zor_poly f z0) z0"
+ by (auto simp: continuous_on_eq_continuous_at)
+ hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
+ unfolding isCont_def .
+ ultimately have "((\<lambda>w. f w / (w - z0) ^ nat n) \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
+ by (blast intro: Lim_transform_eventually)
+ hence "((\<lambda>x. f (g x) / (g x - z0) ^ nat n) \<longlongrightarrow> zor_poly f z0 z0) F"
+ by (rule filterlim_compose[OF _ g])
+ from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis .
+qed
+
+lemma zor_poly_pole_eqI:
+ fixes f :: "complex \<Rightarrow> complex" and z0 :: complex
+ defines "n \<equiv> zorder f z0"
+ assumes f_iso:"isolated_singularity_at f z0" and "is_pole f z0"
+ assumes lim: "((\<lambda>x. f (g x) * (g x - z0) ^ nat (-n)) \<longlongrightarrow> c) F"
+ assumes g: "filterlim g (at z0) F" and "F \<noteq> bot"
+ shows "zor_poly f z0 z0 = c"
+proof -
+ obtain r where r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r"
+ proof -
+ have "\<exists>\<^sub>F w in at z0. f w \<noteq> 0"
+ using non_zero_neighbour_pole[OF \<open>is_pole f z0\<close>] by (auto elim:eventually_frequentlyE)
+ moreover have "not_essential f z0" unfolding not_essential_def using \<open>is_pole f z0\<close> by simp
+ ultimately show ?thesis using that zorder_exist[OF f_iso,folded n_def] by auto
+ qed
+ from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)"
+ using eventually_at_ball'[of r z0 UNIV] by auto
+ have "eventually (\<lambda>w. zor_poly f z0 w = f w * (w - z0) ^ nat (-n)) (at z0)"
+ using zor_poly_pole_eq[OF f_iso \<open>is_pole f z0\<close>] unfolding n_def .
+ moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
+ using r by (intro holomorphic_on_imp_continuous_on) auto
+ with r(1,2) have "isCont (zor_poly f z0) z0"
+ by (auto simp: continuous_on_eq_continuous_at)
+ hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
+ unfolding isCont_def .
+ ultimately have "((\<lambda>w. f w * (w - z0) ^ nat (-n)) \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
+ by (blast intro: Lim_transform_eventually)
+ hence "((\<lambda>x. f (g x) * (g x - z0) ^ nat (-n)) \<longlongrightarrow> zor_poly f z0 z0) F"
+ by (rule filterlim_compose[OF _ g])
+ from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis .
+qed
+
+end
\ No newline at end of file
--- a/src/HOL/Complex_Analysis/Conformal_Mappings.thy Mon Dec 02 22:40:16 2019 -0500
+++ b/src/HOL/Complex_Analysis/Conformal_Mappings.thy Mon Dec 02 17:51:54 2019 +0100
@@ -5,222 +5,10 @@
text\<open>Also Cauchy's residue theorem by Wenda Li (2016)\<close>
theory Conformal_Mappings
-imports Cauchy_Integral_Theorem
+imports Cauchy_Integral_Formula
begin
-(* FIXME mv to Cauchy_Integral_Theorem.thy *)
-subsection\<open>Cauchy's inequality and more versions of Liouville\<close>
-
-lemma Cauchy_higher_deriv_bound:
- assumes holf: "f holomorphic_on (ball z r)"
- and contf: "continuous_on (cball z r) f"
- and fin : "\<And>w. w \<in> ball z r \<Longrightarrow> f w \<in> ball y B0"
- and "0 < r" and "0 < n"
- shows "norm ((deriv ^^ n) f z) \<le> (fact n) * B0 / r^n"
-proof -
- have "0 < B0" using \<open>0 < r\<close> fin [of z]
- by (metis ball_eq_empty ex_in_conv fin not_less)
- have le_B0: "\<And>w. cmod (w - z) \<le> r \<Longrightarrow> cmod (f w - y) \<le> B0"
- apply (rule continuous_on_closure_norm_le [of "ball z r" "\<lambda>w. f w - y"])
- apply (auto simp: \<open>0 < r\<close> dist_norm norm_minus_commute)
- apply (rule continuous_intros contf)+
- using fin apply (simp add: dist_commute dist_norm less_eq_real_def)
- done
- have "(deriv ^^ n) f z = (deriv ^^ n) (\<lambda>w. f w) z - (deriv ^^ n) (\<lambda>w. y) z"
- using \<open>0 < n\<close> by simp
- also have "... = (deriv ^^ n) (\<lambda>w. f w - y) z"
- by (rule higher_deriv_diff [OF holf, symmetric]) (auto simp: \<open>0 < r\<close>)
- finally have "(deriv ^^ n) f z = (deriv ^^ n) (\<lambda>w. f w - y) z" .
- have contf': "continuous_on (cball z r) (\<lambda>u. f u - y)"
- by (rule contf continuous_intros)+
- have holf': "(\<lambda>u. (f u - y)) holomorphic_on (ball z r)"
- by (simp add: holf holomorphic_on_diff)
- define a where "a = (2 * pi)/(fact n)"
- have "0 < a" by (simp add: a_def)
- have "B0/r^(Suc n)*2 * pi * r = a*((fact n)*B0/r^n)"
- using \<open>0 < r\<close> by (simp add: a_def field_split_simps)
- have der_dif: "(deriv ^^ n) (\<lambda>w. f w - y) z = (deriv ^^ n) f z"
- using \<open>0 < r\<close> \<open>0 < n\<close>
- by (auto simp: higher_deriv_diff [OF holf holomorphic_on_const])
- have "norm ((2 * of_real pi * \<i>)/(fact n) * (deriv ^^ n) (\<lambda>w. f w - y) z)
- \<le> (B0/r^(Suc n)) * (2 * pi * r)"
- apply (rule has_contour_integral_bound_circlepath [of "(\<lambda>u. (f u - y)/(u - z)^(Suc n))" _ z])
- using Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf' holf']
- using \<open>0 < B0\<close> \<open>0 < r\<close>
- apply (auto simp: norm_divide norm_mult norm_power divide_simps le_B0)
- done
- then show ?thesis
- using \<open>0 < r\<close>
- by (auto simp: norm_divide norm_mult norm_power field_simps der_dif le_B0)
-qed
-
-lemma Cauchy_inequality:
- assumes holf: "f holomorphic_on (ball \<xi> r)"
- and contf: "continuous_on (cball \<xi> r) f"
- and "0 < r"
- and nof: "\<And>x. norm(\<xi>-x) = r \<Longrightarrow> norm(f x) \<le> B"
- shows "norm ((deriv ^^ n) f \<xi>) \<le> (fact n) * B / r^n"
-proof -
- obtain x where "norm (\<xi>-x) = r"
- by (metis abs_of_nonneg add_diff_cancel_left' \<open>0 < r\<close> diff_add_cancel
- dual_order.strict_implies_order norm_of_real)
- then have "0 \<le> B"
- by (metis nof norm_not_less_zero not_le order_trans)
- have "((\<lambda>u. f u / (u - \<xi>) ^ Suc n) has_contour_integral (2 * pi) * \<i> / fact n * (deriv ^^ n) f \<xi>)
- (circlepath \<xi> r)"
- apply (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf])
- using \<open>0 < r\<close> by simp
- then have "norm ((2 * pi * \<i>)/(fact n) * (deriv ^^ n) f \<xi>) \<le> (B / r^(Suc n)) * (2 * pi * r)"
- apply (rule has_contour_integral_bound_circlepath)
- using \<open>0 \<le> B\<close> \<open>0 < r\<close>
- apply (simp_all add: norm_divide norm_power nof frac_le norm_minus_commute del: power_Suc)
- done
- then show ?thesis using \<open>0 < r\<close>
- by (simp add: norm_divide norm_mult field_simps)
-qed
-
-lemma Liouville_polynomial:
- assumes holf: "f holomorphic_on UNIV"
- and nof: "\<And>z. A \<le> norm z \<Longrightarrow> norm(f z) \<le> B * norm z ^ n"
- shows "f \<xi> = (\<Sum>k\<le>n. (deriv^^k) f 0 / fact k * \<xi> ^ k)"
-proof (cases rule: le_less_linear [THEN disjE])
- assume "B \<le> 0"
- then have "\<And>z. A \<le> norm z \<Longrightarrow> norm(f z) = 0"
- by (metis nof less_le_trans zero_less_mult_iff neqE norm_not_less_zero norm_power not_le)
- then have f0: "(f \<longlongrightarrow> 0) at_infinity"
- using Lim_at_infinity by force
- then have [simp]: "f = (\<lambda>w. 0)"
- using Liouville_weak [OF holf, of 0]
- by (simp add: eventually_at_infinity f0) meson
- show ?thesis by simp
-next
- assume "0 < B"
- have "((\<lambda>k. (deriv ^^ k) f 0 / (fact k) * (\<xi> - 0)^k) sums f \<xi>)"
- apply (rule holomorphic_power_series [where r = "norm \<xi> + 1"])
- using holf holomorphic_on_subset apply auto
- done
- then have sumsf: "((\<lambda>k. (deriv ^^ k) f 0 / (fact k) * \<xi>^k) sums f \<xi>)" by simp
- have "(deriv ^^ k) f 0 / fact k * \<xi> ^ k = 0" if "k>n" for k
- proof (cases "(deriv ^^ k) f 0 = 0")
- case True then show ?thesis by simp
- next
- case False
- define w where "w = complex_of_real (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
- have "1 \<le> abs (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
- using \<open>0 < B\<close> by simp
- then have wge1: "1 \<le> norm w"
- by (metis norm_of_real w_def)
- then have "w \<noteq> 0" by auto
- have kB: "0 < fact k * B"
- using \<open>0 < B\<close> by simp
- then have "0 \<le> fact k * B / cmod ((deriv ^^ k) f 0)"
- by simp
- then have wgeA: "A \<le> cmod w"
- by (simp only: w_def norm_of_real)
- have "fact k * B / cmod ((deriv ^^ k) f 0) < abs (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
- using \<open>0 < B\<close> by simp
- then have wge: "fact k * B / cmod ((deriv ^^ k) f 0) < norm w"
- by (metis norm_of_real w_def)
- then have "fact k * B / norm w < cmod ((deriv ^^ k) f 0)"
- using False by (simp add: field_split_simps mult.commute split: if_split_asm)
- also have "... \<le> fact k * (B * norm w ^ n) / norm w ^ k"
- apply (rule Cauchy_inequality)
- using holf holomorphic_on_subset apply force
- using holf holomorphic_on_imp_continuous_on holomorphic_on_subset apply blast
- using \<open>w \<noteq> 0\<close> apply simp
- by (metis nof wgeA dist_0_norm dist_norm)
- also have "... = fact k * (B * 1 / cmod w ^ (k-n))"
- apply (simp only: mult_cancel_left times_divide_eq_right [symmetric])
- using \<open>k>n\<close> \<open>w \<noteq> 0\<close> \<open>0 < B\<close> apply (simp add: field_split_simps semiring_normalization_rules)
- done
- also have "... = fact k * B / cmod w ^ (k-n)"
- by simp
- finally have "fact k * B / cmod w < fact k * B / cmod w ^ (k - n)" .
- then have "1 / cmod w < 1 / cmod w ^ (k - n)"
- by (metis kB divide_inverse inverse_eq_divide mult_less_cancel_left_pos)
- then have "cmod w ^ (k - n) < cmod w"
- by (metis frac_le le_less_trans norm_ge_zero norm_one not_less order_refl wge1 zero_less_one)
- with self_le_power [OF wge1] have False
- by (meson diff_is_0_eq not_gr0 not_le that)
- then show ?thesis by blast
- qed
- then have "(deriv ^^ (k + Suc n)) f 0 / fact (k + Suc n) * \<xi> ^ (k + Suc n) = 0" for k
- using not_less_eq by blast
- then have "(\<lambda>i. (deriv ^^ (i + Suc n)) f 0 / fact (i + Suc n) * \<xi> ^ (i + Suc n)) sums 0"
- by (rule sums_0)
- with sums_split_initial_segment [OF sumsf, where n = "Suc n"]
- show ?thesis
- using atLeast0AtMost lessThan_Suc_atMost sums_unique2 by fastforce
-qed
-
-text\<open>Every bounded entire function is a constant function.\<close>
-theorem Liouville_theorem:
- assumes holf: "f holomorphic_on UNIV"
- and bf: "bounded (range f)"
- obtains c where "\<And>z. f z = c"
-proof -
- obtain B where "\<And>z. cmod (f z) \<le> B"
- by (meson bf bounded_pos rangeI)
- then show ?thesis
- using Liouville_polynomial [OF holf, of 0 B 0, simplified] that by blast
-qed
-
-text\<open>A holomorphic function f has only isolated zeros unless f is 0.\<close>
-
-lemma powser_0_nonzero:
- fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
- assumes r: "0 < r"
- and sm: "\<And>x. norm (x - \<xi>) < r \<Longrightarrow> (\<lambda>n. a n * (x - \<xi>) ^ n) sums (f x)"
- and [simp]: "f \<xi> = 0"
- and m0: "a m \<noteq> 0" and "m>0"
- obtains s where "0 < s" and "\<And>z. z \<in> cball \<xi> s - {\<xi>} \<Longrightarrow> f z \<noteq> 0"
-proof -
- have "r \<le> conv_radius a"
- using sm sums_summable by (auto simp: le_conv_radius_iff [where \<xi>=\<xi>])
- obtain m where am: "a m \<noteq> 0" and az [simp]: "(\<And>n. n<m \<Longrightarrow> a n = 0)"
- apply (rule_tac m = "LEAST n. a n \<noteq> 0" in that)
- using m0
- apply (rule LeastI2)
- apply (fastforce intro: dest!: not_less_Least)+
- done
- define b where "b i = a (i+m) / a m" for i
- define g where "g x = suminf (\<lambda>i. b i * (x - \<xi>) ^ i)" for x
- have [simp]: "b 0 = 1"
- by (simp add: am b_def)
- { fix x::'a
- assume "norm (x - \<xi>) < r"
- then have "(\<lambda>n. (a m * (x - \<xi>)^m) * (b n * (x - \<xi>)^n)) sums (f x)"
- using am az sm sums_zero_iff_shift [of m "(\<lambda>n. a n * (x - \<xi>) ^ n)" "f x"]
- by (simp add: b_def monoid_mult_class.power_add algebra_simps)
- then have "x \<noteq> \<xi> \<Longrightarrow> (\<lambda>n. b n * (x - \<xi>)^n) sums (f x / (a m * (x - \<xi>)^m))"
- using am by (simp add: sums_mult_D)
- } note bsums = this
- then have "norm (x - \<xi>) < r \<Longrightarrow> summable (\<lambda>n. b n * (x - \<xi>)^n)" for x
- using sums_summable by (cases "x=\<xi>") auto
- then have "r \<le> conv_radius b"
- by (simp add: le_conv_radius_iff [where \<xi>=\<xi>])
- then have "r/2 < conv_radius b"
- using not_le order_trans r by fastforce
- then have "continuous_on (cball \<xi> (r/2)) g"
- using powser_continuous_suminf [of "r/2" b \<xi>] by (simp add: g_def)
- then obtain s where "s>0" "\<And>x. \<lbrakk>norm (x - \<xi>) \<le> s; norm (x - \<xi>) \<le> r/2\<rbrakk> \<Longrightarrow> dist (g x) (g \<xi>) < 1/2"
- apply (rule continuous_onE [where x=\<xi> and e = "1/2"])
- using r apply (auto simp: norm_minus_commute dist_norm)
- done
- moreover have "g \<xi> = 1"
- by (simp add: g_def)
- ultimately have gnz: "\<And>x. \<lbrakk>norm (x - \<xi>) \<le> s; norm (x - \<xi>) \<le> r/2\<rbrakk> \<Longrightarrow> (g x) \<noteq> 0"
- by fastforce
- have "f x \<noteq> 0" if "x \<noteq> \<xi>" "norm (x - \<xi>) \<le> s" "norm (x - \<xi>) \<le> r/2" for x
- using bsums [of x] that gnz [of x]
- apply (auto simp: g_def)
- using r sums_iff by fastforce
- then show ?thesis
- apply (rule_tac s="min s (r/2)" in that)
- using \<open>0 < r\<close> \<open>0 < s\<close> by (auto simp: dist_commute dist_norm)
-qed
-
subsection \<open>Analytic continuation\<close>
proposition isolated_zeros:
@@ -2173,2944 +1961,4 @@
qed
qed
-subsection \<open>Cauchy's residue theorem\<close>
-
-text\<open>Wenda Li and LC Paulson (2016). A Formal Proof of Cauchy's Residue Theorem.
- Interactive Theorem Proving\<close>
-
-definition\<^marker>\<open>tag important\<close> residue :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex" where
- "residue f z = (SOME int. \<exists>e>0. \<forall>\<epsilon>>0. \<epsilon><e
- \<longrightarrow> (f has_contour_integral 2*pi* \<i> *int) (circlepath z \<epsilon>))"
-
-lemma Eps_cong:
- assumes "\<And>x. P x = Q x"
- shows "Eps P = Eps Q"
- using ext[of P Q, OF assms] by simp
-
-lemma residue_cong:
- assumes eq: "eventually (\<lambda>z. f z = g z) (at z)" and "z = z'"
- shows "residue f z = residue g z'"
-proof -
- from assms have eq': "eventually (\<lambda>z. g z = f z) (at z)"
- by (simp add: eq_commute)
- let ?P = "\<lambda>f c e. (\<forall>\<epsilon>>0. \<epsilon> < e \<longrightarrow>
- (f has_contour_integral of_real (2 * pi) * \<i> * c) (circlepath z \<epsilon>))"
- have "residue f z = residue g z" unfolding residue_def
- proof (rule Eps_cong)
- fix c :: complex
- have "\<exists>e>0. ?P g c e"
- if "\<exists>e>0. ?P f c e" and "eventually (\<lambda>z. f z = g z) (at z)" for f g
- proof -
- from that(1) obtain e where e: "e > 0" "?P f c e"
- by blast
- from that(2) obtain e' where e': "e' > 0" "\<And>z'. z' \<noteq> z \<Longrightarrow> dist z' z < e' \<Longrightarrow> f z' = g z'"
- unfolding eventually_at by blast
- have "?P g c (min e e')"
- proof (intro allI exI impI, goal_cases)
- case (1 \<epsilon>)
- hence "(f has_contour_integral of_real (2 * pi) * \<i> * c) (circlepath z \<epsilon>)"
- using e(2) by auto
- thus ?case
- proof (rule has_contour_integral_eq)
- fix z' assume "z' \<in> path_image (circlepath z \<epsilon>)"
- hence "dist z' z < e'" and "z' \<noteq> z"
- using 1 by (auto simp: dist_commute)
- with e'(2)[of z'] show "f z' = g z'" by simp
- qed
- qed
- moreover from e and e' have "min e e' > 0" by auto
- ultimately show ?thesis by blast
- qed
- from this[OF _ eq] and this[OF _ eq']
- show "(\<exists>e>0. ?P f c e) \<longleftrightarrow> (\<exists>e>0. ?P g c e)"
- by blast
- qed
- with assms show ?thesis by simp
-qed
-
-lemma contour_integral_circlepath_eq:
- assumes "open s" and f_holo:"f holomorphic_on (s-{z})" and "0<e1" "e1\<le>e2"
- and e2_cball:"cball z e2 \<subseteq> s"
- shows
- "f contour_integrable_on circlepath z e1"
- "f contour_integrable_on circlepath z e2"
- "contour_integral (circlepath z e2) f = contour_integral (circlepath z e1) f"
-proof -
- define l where "l \<equiv> linepath (z+e2) (z+e1)"
- have [simp]:"valid_path l" "pathstart l=z+e2" "pathfinish l=z+e1" unfolding l_def by auto
- have "e2>0" using \<open>e1>0\<close> \<open>e1\<le>e2\<close> by auto
- have zl_img:"z\<notin>path_image l"
- proof
- assume "z \<in> path_image l"
- then have "e2 \<le> cmod (e2 - e1)"
- using segment_furthest_le[of z "z+e2" "z+e1" "z+e2",simplified] \<open>e1>0\<close> \<open>e2>0\<close> unfolding l_def
- by (auto simp add:closed_segment_commute)
- thus False using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
- apply (subst (asm) norm_of_real)
- by auto
- qed
- define g where "g \<equiv> circlepath z e2 +++ l +++ reversepath (circlepath z e1) +++ reversepath l"
- show [simp]: "f contour_integrable_on circlepath z e2" "f contour_integrable_on (circlepath z e1)"
- proof -
- show "f contour_integrable_on circlepath z e2"
- apply (intro contour_integrable_continuous_circlepath[OF
- continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
- using \<open>e2>0\<close> e2_cball by auto
- show "f contour_integrable_on (circlepath z e1)"
- apply (intro contour_integrable_continuous_circlepath[OF
- continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
- using \<open>e1>0\<close> \<open>e1\<le>e2\<close> e2_cball by auto
- qed
- have [simp]:"f contour_integrable_on l"
- proof -
- have "closed_segment (z + e2) (z + e1) \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
- by (intro closed_segment_subset,auto simp add:dist_norm)
- hence "closed_segment (z + e2) (z + e1) \<subseteq> s - {z}" using zl_img e2_cball unfolding l_def
- by auto
- then show "f contour_integrable_on l" unfolding l_def
- apply (intro contour_integrable_continuous_linepath[OF
- continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
- by auto
- qed
- let ?ig="\<lambda>g. contour_integral g f"
- have "(f has_contour_integral 0) g"
- proof (rule Cauchy_theorem_global[OF _ f_holo])
- show "open (s - {z})" using \<open>open s\<close> by auto
- show "valid_path g" unfolding g_def l_def by auto
- show "pathfinish g = pathstart g" unfolding g_def l_def by auto
- next
- have path_img:"path_image g \<subseteq> cball z e2"
- proof -
- have "closed_segment (z + e2) (z + e1) \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
- by (intro closed_segment_subset,auto simp add:dist_norm)
- moreover have "sphere z \<bar>e1\<bar> \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1\<le>e2\<close> \<open>e1>0\<close> by auto
- ultimately show ?thesis unfolding g_def l_def using \<open>e2>0\<close>
- by (simp add: path_image_join closed_segment_commute)
- qed
- show "path_image g \<subseteq> s - {z}"
- proof -
- have "z\<notin>path_image g" using zl_img
- unfolding g_def l_def by (auto simp add: path_image_join closed_segment_commute)
- moreover note \<open>cball z e2 \<subseteq> s\<close> and path_img
- ultimately show ?thesis by auto
- qed
- show "winding_number g w = 0" when"w \<notin> s - {z}" for w
- proof -
- have "winding_number g w = 0" when "w\<notin>s" using that e2_cball
- apply (intro winding_number_zero_outside[OF _ _ _ _ path_img])
- by (auto simp add:g_def l_def)
- moreover have "winding_number g z=0"
- proof -
- let ?Wz="\<lambda>g. winding_number g z"
- have "?Wz g = ?Wz (circlepath z e2) + ?Wz l + ?Wz (reversepath (circlepath z e1))
- + ?Wz (reversepath l)"
- using \<open>e2>0\<close> \<open>e1>0\<close> zl_img unfolding g_def l_def
- by (subst winding_number_join,auto simp add:path_image_join closed_segment_commute)+
- also have "... = ?Wz (circlepath z e2) + ?Wz (reversepath (circlepath z e1))"
- using zl_img
- apply (subst (2) winding_number_reversepath)
- by (auto simp add:l_def closed_segment_commute)
- also have "... = 0"
- proof -
- have "?Wz (circlepath z e2) = 1" using \<open>e2>0\<close>
- by (auto intro: winding_number_circlepath_centre)
- moreover have "?Wz (reversepath (circlepath z e1)) = -1" using \<open>e1>0\<close>
- apply (subst winding_number_reversepath)
- by (auto intro: winding_number_circlepath_centre)
- ultimately show ?thesis by auto
- qed
- finally show ?thesis .
- qed
- ultimately show ?thesis using that by auto
- qed
- qed
- then have "0 = ?ig g" using contour_integral_unique by simp
- also have "... = ?ig (circlepath z e2) + ?ig l + ?ig (reversepath (circlepath z e1))
- + ?ig (reversepath l)"
- unfolding g_def
- by (auto simp add:contour_integrable_reversepath_eq)
- also have "... = ?ig (circlepath z e2) - ?ig (circlepath z e1)"
- by (auto simp add:contour_integral_reversepath)
- finally show "contour_integral (circlepath z e2) f = contour_integral (circlepath z e1) f"
- by simp
-qed
-
-lemma base_residue:
- assumes "open s" "z\<in>s" "r>0" and f_holo:"f holomorphic_on (s - {z})"
- and r_cball:"cball z r \<subseteq> s"
- shows "(f has_contour_integral 2 * pi * \<i> * (residue f z)) (circlepath z r)"
-proof -
- obtain e where "e>0" and e_cball:"cball z e \<subseteq> s"
- using open_contains_cball[of s] \<open>open s\<close> \<open>z\<in>s\<close> by auto
- define c where "c \<equiv> 2 * pi * \<i>"
- define i where "i \<equiv> contour_integral (circlepath z e) f / c"
- have "(f has_contour_integral c*i) (circlepath z \<epsilon>)" when "\<epsilon>>0" "\<epsilon><e" for \<epsilon>
- proof -
- have "contour_integral (circlepath z e) f = contour_integral (circlepath z \<epsilon>) f"
- "f contour_integrable_on circlepath z \<epsilon>"
- "f contour_integrable_on circlepath z e"
- using \<open>\<epsilon><e\<close>
- by (intro contour_integral_circlepath_eq[OF \<open>open s\<close> f_holo \<open>\<epsilon>>0\<close> _ e_cball],auto)+
- then show ?thesis unfolding i_def c_def
- by (auto intro:has_contour_integral_integral)
- qed
- then have "\<exists>e>0. \<forall>\<epsilon>>0. \<epsilon><e \<longrightarrow> (f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
- unfolding residue_def c_def
- apply (rule_tac someI[of _ i],intro exI[where x=e])
- by (auto simp add:\<open>e>0\<close> c_def)
- then obtain e' where "e'>0"
- and e'_def:"\<forall>\<epsilon>>0. \<epsilon><e' \<longrightarrow> (f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
- by auto
- let ?int="\<lambda>e. contour_integral (circlepath z e) f"
- define \<epsilon> where "\<epsilon> \<equiv> Min {r,e'} / 2"
- have "\<epsilon>>0" "\<epsilon>\<le>r" "\<epsilon><e'" using \<open>r>0\<close> \<open>e'>0\<close> unfolding \<epsilon>_def by auto
- have "(f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
- using e'_def[rule_format,OF \<open>\<epsilon>>0\<close> \<open>\<epsilon><e'\<close>] .
- then show ?thesis unfolding c_def
- using contour_integral_circlepath_eq[OF \<open>open s\<close> f_holo \<open>\<epsilon>>0\<close> \<open>\<epsilon>\<le>r\<close> r_cball]
- by (auto elim: has_contour_integral_eqpath[of _ _ "circlepath z \<epsilon>" "circlepath z r"])
-qed
-
-lemma residue_holo:
- assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s"
- shows "residue f z = 0"
-proof -
- define c where "c \<equiv> 2 * pi * \<i>"
- obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
- using open_contains_cball_eq by blast
- have "(f has_contour_integral c*residue f z) (circlepath z e)"
- using f_holo
- by (auto intro: base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
- moreover have "(f has_contour_integral 0) (circlepath z e)"
- using f_holo e_cball \<open>e>0\<close>
- by (auto intro: Cauchy_theorem_convex_simple[of _ "cball z e"])
- ultimately have "c*residue f z =0"
- using has_contour_integral_unique by blast
- thus ?thesis unfolding c_def by auto
-qed
-
-lemma residue_const:"residue (\<lambda>_. c) z = 0"
- by (intro residue_holo[of "UNIV::complex set"],auto intro:holomorphic_intros)
-
-lemma residue_add:
- assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
- and g_holo:"g holomorphic_on s - {z}"
- shows "residue (\<lambda>z. f z + g z) z= residue f z + residue g z"
-proof -
- define c where "c \<equiv> 2 * pi * \<i>"
- define fg where "fg \<equiv> (\<lambda>z. f z+g z)"
- obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
- using open_contains_cball_eq by blast
- have "(fg has_contour_integral c * residue fg z) (circlepath z e)"
- unfolding fg_def using f_holo g_holo
- apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
- by (auto intro:holomorphic_intros)
- moreover have "(fg has_contour_integral c*residue f z + c* residue g z) (circlepath z e)"
- unfolding fg_def using f_holo g_holo
- by (auto intro: has_contour_integral_add base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
- ultimately have "c*(residue f z + residue g z) = c * residue fg z"
- using has_contour_integral_unique by (auto simp add:distrib_left)
- thus ?thesis unfolding fg_def
- by (auto simp add:c_def)
-qed
-
-lemma residue_lmul:
- assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
- shows "residue (\<lambda>z. c * (f z)) z= c * residue f z"
-proof (cases "c=0")
- case True
- thus ?thesis using residue_const by auto
-next
- case False
- define c' where "c' \<equiv> 2 * pi * \<i>"
- define f' where "f' \<equiv> (\<lambda>z. c * (f z))"
- obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
- using open_contains_cball_eq by blast
- have "(f' has_contour_integral c' * residue f' z) (circlepath z e)"
- unfolding f'_def using f_holo
- apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c'_def])
- by (auto intro:holomorphic_intros)
- moreover have "(f' has_contour_integral c * (c' * residue f z)) (circlepath z e)"
- unfolding f'_def using f_holo
- by (auto intro: has_contour_integral_lmul
- base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c'_def])
- ultimately have "c' * residue f' z = c * (c' * residue f z)"
- using has_contour_integral_unique by auto
- thus ?thesis unfolding f'_def c'_def using False
- by (auto simp add:field_simps)
-qed
-
-lemma residue_rmul:
- assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
- shows "residue (\<lambda>z. (f z) * c) z= residue f z * c"
-using residue_lmul[OF assms,of c] by (auto simp add:algebra_simps)
-
-lemma residue_div:
- assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
- shows "residue (\<lambda>z. (f z) / c) z= residue f z / c "
-using residue_lmul[OF assms,of "1/c"] by (auto simp add:algebra_simps)
-
-lemma residue_neg:
- assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
- shows "residue (\<lambda>z. - (f z)) z= - residue f z"
-using residue_lmul[OF assms,of "-1"] by auto
-
-lemma residue_diff:
- assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
- and g_holo:"g holomorphic_on s - {z}"
- shows "residue (\<lambda>z. f z - g z) z= residue f z - residue g z"
-using residue_add[OF assms(1,2,3),of "\<lambda>z. - g z"] residue_neg[OF assms(1,2,4)]
-by (auto intro:holomorphic_intros g_holo)
-
-lemma residue_simple:
- assumes "open s" "z\<in>s" and f_holo:"f holomorphic_on s"
- shows "residue (\<lambda>w. f w / (w - z)) z = f z"
-proof -
- define c where "c \<equiv> 2 * pi * \<i>"
- define f' where "f' \<equiv> \<lambda>w. f w / (w - z)"
- obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
- using open_contains_cball_eq by blast
- have "(f' has_contour_integral c * f z) (circlepath z e)"
- unfolding f'_def c_def using \<open>e>0\<close> f_holo e_cball
- by (auto intro!: Cauchy_integral_circlepath_simple holomorphic_intros)
- moreover have "(f' has_contour_integral c * residue f' z) (circlepath z e)"
- unfolding f'_def using f_holo
- apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
- by (auto intro!:holomorphic_intros)
- ultimately have "c * f z = c * residue f' z"
- using has_contour_integral_unique by blast
- thus ?thesis unfolding c_def f'_def by auto
-qed
-
-lemma residue_simple':
- assumes s: "open s" "z \<in> s" and holo: "f holomorphic_on (s - {z})"
- and lim: "((\<lambda>w. f w * (w - z)) \<longlongrightarrow> c) (at z)"
- shows "residue f z = c"
-proof -
- define g where "g = (\<lambda>w. if w = z then c else f w * (w - z))"
- from holo have "(\<lambda>w. f w * (w - z)) holomorphic_on (s - {z})" (is "?P")
- by (force intro: holomorphic_intros)
- also have "?P \<longleftrightarrow> g holomorphic_on (s - {z})"
- by (intro holomorphic_cong refl) (simp_all add: g_def)
- finally have *: "g holomorphic_on (s - {z})" .
-
- note lim
- also have "(\<lambda>w. f w * (w - z)) \<midarrow>z\<rightarrow> c \<longleftrightarrow> g \<midarrow>z\<rightarrow> g z"
- by (intro filterlim_cong refl) (simp_all add: g_def [abs_def] eventually_at_filter)
- finally have **: "g \<midarrow>z\<rightarrow> g z" .
-
- have g_holo: "g holomorphic_on s"
- by (rule no_isolated_singularity'[where K = "{z}"])
- (insert assms * **, simp_all add: at_within_open_NO_MATCH)
- from s and this have "residue (\<lambda>w. g w / (w - z)) z = g z"
- by (rule residue_simple)
- also have "\<forall>\<^sub>F za in at z. g za / (za - z) = f za"
- unfolding eventually_at by (auto intro!: exI[of _ 1] simp: field_simps g_def)
- hence "residue (\<lambda>w. g w / (w - z)) z = residue f z"
- by (intro residue_cong refl)
- finally show ?thesis
- by (simp add: g_def)
-qed
-
-lemma residue_holomorphic_over_power:
- assumes "open A" "z0 \<in> A" "f holomorphic_on A"
- shows "residue (\<lambda>z. f z / (z - z0) ^ Suc n) z0 = (deriv ^^ n) f z0 / fact n"
-proof -
- let ?f = "\<lambda>z. f z / (z - z0) ^ Suc n"
- from assms(1,2) obtain r where r: "r > 0" "cball z0 r \<subseteq> A"
- by (auto simp: open_contains_cball)
- have "(?f has_contour_integral 2 * pi * \<i> * residue ?f z0) (circlepath z0 r)"
- using r assms by (intro base_residue[of A]) (auto intro!: holomorphic_intros)
- moreover have "(?f has_contour_integral 2 * pi * \<i> / fact n * (deriv ^^ n) f z0) (circlepath z0 r)"
- using assms r
- by (intro Cauchy_has_contour_integral_higher_derivative_circlepath)
- (auto intro!: holomorphic_on_subset[OF assms(3)] holomorphic_on_imp_continuous_on)
- ultimately have "2 * pi * \<i> * residue ?f z0 = 2 * pi * \<i> / fact n * (deriv ^^ n) f z0"
- by (rule has_contour_integral_unique)
- thus ?thesis by (simp add: field_simps)
-qed
-
-lemma residue_holomorphic_over_power':
- assumes "open A" "0 \<in> A" "f holomorphic_on A"
- shows "residue (\<lambda>z. f z / z ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
- using residue_holomorphic_over_power[OF assms] by simp
-
-theorem residue_fps_expansion_over_power_at_0:
- assumes "f has_fps_expansion F"
- shows "residue (\<lambda>z. f z / z ^ Suc n) 0 = fps_nth F n"
-proof -
- from has_fps_expansion_imp_holomorphic[OF assms] guess s . note s = this
- have "residue (\<lambda>z. f z / (z - 0) ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
- using assms s unfolding has_fps_expansion_def
- by (intro residue_holomorphic_over_power[of s]) (auto simp: zero_ereal_def)
- also from assms have "\<dots> = fps_nth F n"
- by (subst fps_nth_fps_expansion) auto
- finally show ?thesis by simp
-qed
-
-lemma get_integrable_path:
- assumes "open s" "connected (s-pts)" "finite pts" "f holomorphic_on (s-pts) " "a\<in>s-pts" "b\<in>s-pts"
- obtains g where "valid_path g" "pathstart g = a" "pathfinish g = b"
- "path_image g \<subseteq> s-pts" "f contour_integrable_on g" using assms
-proof (induct arbitrary:s thesis a rule:finite_induct[OF \<open>finite pts\<close>])
- case 1
- obtain g where "valid_path g" "path_image g \<subseteq> s" "pathstart g = a" "pathfinish g = b"
- using connected_open_polynomial_connected[OF \<open>open s\<close>,of a b ] \<open>connected (s - {})\<close>
- valid_path_polynomial_function "1.prems"(6) "1.prems"(7) by auto
- moreover have "f contour_integrable_on g"
- using contour_integrable_holomorphic_simple[OF _ \<open>open s\<close> \<open>valid_path g\<close> \<open>path_image g \<subseteq> s\<close>,of f]
- \<open>f holomorphic_on s - {}\<close>
- by auto
- ultimately show ?case using "1"(1)[of g] by auto
-next
- case idt:(2 p pts)
- obtain e where "e>0" and e:"\<forall>w\<in>ball a e. w \<in> s \<and> (w \<noteq> a \<longrightarrow> w \<notin> insert p pts)"
- using finite_ball_avoid[OF \<open>open s\<close> \<open>finite (insert p pts)\<close>, of a]
- \<open>a \<in> s - insert p pts\<close>
- by auto
- define a' where "a' \<equiv> a+e/2"
- have "a'\<in>s-{p} -pts" using e[rule_format,of "a+e/2"] \<open>e>0\<close>
- by (auto simp add:dist_complex_def a'_def)
- then obtain g' where g'[simp]:"valid_path g'" "pathstart g' = a'" "pathfinish g' = b"
- "path_image g' \<subseteq> s - {p} - pts" "f contour_integrable_on g'"
- using idt.hyps(3)[of a' "s-{p}"] idt.prems idt.hyps(1)
- by (metis Diff_insert2 open_delete)
- define g where "g \<equiv> linepath a a' +++ g'"
- have "valid_path g" unfolding g_def by (auto intro: valid_path_join)
- moreover have "pathstart g = a" and "pathfinish g = b" unfolding g_def by auto
- moreover have "path_image g \<subseteq> s - insert p pts" unfolding g_def
- proof (rule subset_path_image_join)
- have "closed_segment a a' \<subseteq> ball a e" using \<open>e>0\<close>
- by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
- then show "path_image (linepath a a') \<subseteq> s - insert p pts" using e idt(9)
- by auto
- next
- show "path_image g' \<subseteq> s - insert p pts" using g'(4) by blast
- qed
- moreover have "f contour_integrable_on g"
- proof -
- have "closed_segment a a' \<subseteq> ball a e" using \<open>e>0\<close>
- by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
- then have "continuous_on (closed_segment a a') f"
- using e idt.prems(6) holomorphic_on_imp_continuous_on[OF idt.prems(5)]
- apply (elim continuous_on_subset)
- by auto
- then have "f contour_integrable_on linepath a a'"
- using contour_integrable_continuous_linepath by auto
- then show ?thesis unfolding g_def
- apply (rule contour_integrable_joinI)
- by (auto simp add: \<open>e>0\<close>)
- qed
- ultimately show ?case using idt.prems(1)[of g] by auto
-qed
-
-lemma Cauchy_theorem_aux:
- assumes "open s" "connected (s-pts)" "finite pts" "pts \<subseteq> s" "f holomorphic_on s-pts"
- "valid_path g" "pathfinish g = pathstart g" "path_image g \<subseteq> s-pts"
- "\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0"
- "\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
- shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
- using assms
-proof (induct arbitrary:s g rule:finite_induct[OF \<open>finite pts\<close>])
- case 1
- then show ?case by (simp add: Cauchy_theorem_global contour_integral_unique)
-next
- case (2 p pts)
- note fin[simp] = \<open>finite (insert p pts)\<close>
- and connected = \<open>connected (s - insert p pts)\<close>
- and valid[simp] = \<open>valid_path g\<close>
- and g_loop[simp] = \<open>pathfinish g = pathstart g\<close>
- and holo[simp]= \<open>f holomorphic_on s - insert p pts\<close>
- and path_img = \<open>path_image g \<subseteq> s - insert p pts\<close>
- and winding = \<open>\<forall>z. z \<notin> s \<longrightarrow> winding_number g z = 0\<close>
- and h = \<open>\<forall>pa\<in>s. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s \<and> (w \<noteq> pa \<longrightarrow> w \<notin> insert p pts))\<close>
- have "h p>0" and "p\<in>s"
- and h_p: "\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> insert p pts)"
- using h \<open>insert p pts \<subseteq> s\<close> by auto
- obtain pg where pg[simp]: "valid_path pg" "pathstart pg = pathstart g" "pathfinish pg=p+h p"
- "path_image pg \<subseteq> s-insert p pts" "f contour_integrable_on pg"
- proof -
- have "p + h p\<in>cball p (h p)" using h[rule_format,of p]
- by (simp add: \<open>p \<in> s\<close> dist_norm)
- then have "p + h p \<in> s - insert p pts" using h[rule_format,of p] \<open>insert p pts \<subseteq> s\<close>
- by fastforce
- moreover have "pathstart g \<in> s - insert p pts " using path_img by auto
- ultimately show ?thesis
- using get_integrable_path[OF \<open>open s\<close> connected fin holo,of "pathstart g" "p+h p"] that
- by blast
- qed
- obtain n::int where "n=winding_number g p"
- using integer_winding_number[OF _ g_loop,of p] valid path_img
- by (metis DiffD2 Ints_cases insertI1 subset_eq valid_path_imp_path)
- define p_circ where "p_circ \<equiv> circlepath p (h p)"
- define p_circ_pt where "p_circ_pt \<equiv> linepath (p+h p) (p+h p)"
- define n_circ where "n_circ \<equiv> \<lambda>n. ((+++) p_circ ^^ n) p_circ_pt"
- define cp where "cp \<equiv> if n\<ge>0 then reversepath (n_circ (nat n)) else n_circ (nat (- n))"
- have n_circ:"valid_path (n_circ k)"
- "winding_number (n_circ k) p = k"
- "pathstart (n_circ k) = p + h p" "pathfinish (n_circ k) = p + h p"
- "path_image (n_circ k) = (if k=0 then {p + h p} else sphere p (h p))"
- "p \<notin> path_image (n_circ k)"
- "\<And>p'. p'\<notin>s - pts \<Longrightarrow> winding_number (n_circ k) p'=0 \<and> p'\<notin>path_image (n_circ k)"
- "f contour_integrable_on (n_circ k)"
- "contour_integral (n_circ k) f = k * contour_integral p_circ f"
- for k
- proof (induct k)
- case 0
- show "valid_path (n_circ 0)"
- and "path_image (n_circ 0) = (if 0=0 then {p + h p} else sphere p (h p))"
- and "winding_number (n_circ 0) p = of_nat 0"
- and "pathstart (n_circ 0) = p + h p"
- and "pathfinish (n_circ 0) = p + h p"
- and "p \<notin> path_image (n_circ 0)"
- unfolding n_circ_def p_circ_pt_def using \<open>h p > 0\<close>
- by (auto simp add: dist_norm)
- show "winding_number (n_circ 0) p'=0 \<and> p'\<notin>path_image (n_circ 0)" when "p'\<notin>s- pts" for p'
- unfolding n_circ_def p_circ_pt_def
- apply (auto intro!:winding_number_trivial)
- by (metis Diff_iff pathfinish_in_path_image pg(3) pg(4) subsetCE subset_insertI that)+
- show "f contour_integrable_on (n_circ 0)"
- unfolding n_circ_def p_circ_pt_def
- by (auto intro!:contour_integrable_continuous_linepath simp add:continuous_on_sing)
- show "contour_integral (n_circ 0) f = of_nat 0 * contour_integral p_circ f"
- unfolding n_circ_def p_circ_pt_def by auto
- next
- case (Suc k)
- have n_Suc:"n_circ (Suc k) = p_circ +++ n_circ k" unfolding n_circ_def by auto
- have pcirc:"p \<notin> path_image p_circ" "valid_path p_circ" "pathfinish p_circ = pathstart (n_circ k)"
- using Suc(3) unfolding p_circ_def using \<open>h p > 0\<close> by (auto simp add: p_circ_def)
- have pcirc_image:"path_image p_circ \<subseteq> s - insert p pts"
- proof -
- have "path_image p_circ \<subseteq> cball p (h p)" using \<open>0 < h p\<close> p_circ_def by auto
- then show ?thesis using h_p pcirc(1) by auto
- qed
- have pcirc_integrable:"f contour_integrable_on p_circ"
- by (auto simp add:p_circ_def intro!: pcirc_image[unfolded p_circ_def]
- contour_integrable_continuous_circlepath holomorphic_on_imp_continuous_on
- holomorphic_on_subset[OF holo])
- show "valid_path (n_circ (Suc k))"
- using valid_path_join[OF pcirc(2) Suc(1) pcirc(3)] unfolding n_circ_def by auto
- show "path_image (n_circ (Suc k))
- = (if Suc k = 0 then {p + complex_of_real (h p)} else sphere p (h p))"
- proof -
- have "path_image p_circ = sphere p (h p)"
- unfolding p_circ_def using \<open>0 < h p\<close> by auto
- then show ?thesis unfolding n_Suc using Suc.hyps(5) \<open>h p>0\<close>
- by (auto simp add: path_image_join[OF pcirc(3)] dist_norm)
- qed
- then show "p \<notin> path_image (n_circ (Suc k))" using \<open>h p>0\<close> by auto
- show "winding_number (n_circ (Suc k)) p = of_nat (Suc k)"
- proof -
- have "winding_number p_circ p = 1"
- by (simp add: \<open>h p > 0\<close> p_circ_def winding_number_circlepath_centre)
- moreover have "p \<notin> path_image (n_circ k)" using Suc(5) \<open>h p>0\<close> by auto
- then have "winding_number (p_circ +++ n_circ k) p
- = winding_number p_circ p + winding_number (n_circ k) p"
- using valid_path_imp_path Suc.hyps(1) Suc.hyps(2) pcirc
- apply (intro winding_number_join)
- by auto
- ultimately show ?thesis using Suc(2) unfolding n_circ_def
- by auto
- qed
- show "pathstart (n_circ (Suc k)) = p + h p"
- by (simp add: n_circ_def p_circ_def)
- show "pathfinish (n_circ (Suc k)) = p + h p"
- using Suc(4) unfolding n_circ_def by auto
- show "winding_number (n_circ (Suc k)) p'=0 \<and> p'\<notin>path_image (n_circ (Suc k))" when "p'\<notin>s-pts" for p'
- proof -
- have " p' \<notin> path_image p_circ" using \<open>p \<in> s\<close> h p_circ_def that using pcirc_image by blast
- moreover have "p' \<notin> path_image (n_circ k)"
- using Suc.hyps(7) that by blast
- moreover have "winding_number p_circ p' = 0"
- proof -
- have "path_image p_circ \<subseteq> cball p (h p)"
- using h unfolding p_circ_def using \<open>p \<in> s\<close> by fastforce
- moreover have "p'\<notin>cball p (h p)" using \<open>p \<in> s\<close> h that "2.hyps"(2) by fastforce
- ultimately show ?thesis unfolding p_circ_def
- apply (intro winding_number_zero_outside)
- by auto
- qed
- ultimately show ?thesis
- unfolding n_Suc
- apply (subst winding_number_join)
- by (auto simp: valid_path_imp_path pcirc Suc that not_in_path_image_join Suc.hyps(7)[OF that])
- qed
- show "f contour_integrable_on (n_circ (Suc k))"
- unfolding n_Suc
- by (rule contour_integrable_joinI[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)])
- show "contour_integral (n_circ (Suc k)) f = (Suc k) * contour_integral p_circ f"
- unfolding n_Suc
- by (auto simp add:contour_integral_join[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)]
- Suc(9) algebra_simps)
- qed
- have cp[simp]:"pathstart cp = p + h p" "pathfinish cp = p + h p"
- "valid_path cp" "path_image cp \<subseteq> s - insert p pts"
- "winding_number cp p = - n"
- "\<And>p'. p'\<notin>s - pts \<Longrightarrow> winding_number cp p'=0 \<and> p' \<notin> path_image cp"
- "f contour_integrable_on cp"
- "contour_integral cp f = - n * contour_integral p_circ f"
- proof -
- show "pathstart cp = p + h p" and "pathfinish cp = p + h p" and "valid_path cp"
- using n_circ unfolding cp_def by auto
- next
- have "sphere p (h p) \<subseteq> s - insert p pts"
- using h[rule_format,of p] \<open>insert p pts \<subseteq> s\<close> by force
- moreover have "p + complex_of_real (h p) \<in> s - insert p pts"
- using pg(3) pg(4) by (metis pathfinish_in_path_image subsetCE)
- ultimately show "path_image cp \<subseteq> s - insert p pts" unfolding cp_def
- using n_circ(5) by auto
- next
- show "winding_number cp p = - n"
- unfolding cp_def using winding_number_reversepath n_circ \<open>h p>0\<close>
- by (auto simp: valid_path_imp_path)
- next
- show "winding_number cp p'=0 \<and> p' \<notin> path_image cp" when "p'\<notin>s - pts" for p'
- unfolding cp_def
- apply (auto)
- apply (subst winding_number_reversepath)
- by (auto simp add: valid_path_imp_path n_circ(7)[OF that] n_circ(1))
- next
- show "f contour_integrable_on cp" unfolding cp_def
- using contour_integrable_reversepath_eq n_circ(1,8) by auto
- next
- show "contour_integral cp f = - n * contour_integral p_circ f"
- unfolding cp_def using contour_integral_reversepath[OF n_circ(1)] n_circ(9)
- by auto
- qed
- define g' where "g' \<equiv> g +++ pg +++ cp +++ (reversepath pg)"
- have "contour_integral g' f = (\<Sum>p\<in>pts. winding_number g' p * contour_integral (circlepath p (h p)) f)"
- proof (rule "2.hyps"(3)[of "s-{p}" "g'",OF _ _ \<open>finite pts\<close> ])
- show "connected (s - {p} - pts)" using connected by (metis Diff_insert2)
- show "open (s - {p})" using \<open>open s\<close> by auto
- show " pts \<subseteq> s - {p}" using \<open>insert p pts \<subseteq> s\<close> \<open> p \<notin> pts\<close> by blast
- show "f holomorphic_on s - {p} - pts" using holo \<open>p \<notin> pts\<close> by (metis Diff_insert2)
- show "valid_path g'"
- unfolding g'_def cp_def using n_circ valid pg g_loop
- by (auto intro!:valid_path_join )
- show "pathfinish g' = pathstart g'"
- unfolding g'_def cp_def using pg(2) by simp
- show "path_image g' \<subseteq> s - {p} - pts"
- proof -
- define s' where "s' \<equiv> s - {p} - pts"
- have s':"s' = s-insert p pts " unfolding s'_def by auto
- then show ?thesis using path_img pg(4) cp(4)
- unfolding g'_def
- apply (fold s'_def s')
- apply (intro subset_path_image_join)
- by auto
- qed
- note path_join_imp[simp]
- show "\<forall>z. z \<notin> s - {p} \<longrightarrow> winding_number g' z = 0"
- proof clarify
- fix z assume z:"z\<notin>s - {p}"
- have "winding_number (g +++ pg +++ cp +++ reversepath pg) z = winding_number g z
- + winding_number (pg +++ cp +++ (reversepath pg)) z"
- proof (rule winding_number_join)
- show "path g" using \<open>valid_path g\<close> by (simp add: valid_path_imp_path)
- show "z \<notin> path_image g" using z path_img by auto
- show "path (pg +++ cp +++ reversepath pg)" using pg(3) cp
- by (simp add: valid_path_imp_path)
- next
- have "path_image (pg +++ cp +++ reversepath pg) \<subseteq> s - insert p pts"
- using pg(4) cp(4) by (auto simp:subset_path_image_join)
- then show "z \<notin> path_image (pg +++ cp +++ reversepath pg)" using z by auto
- next
- show "pathfinish g = pathstart (pg +++ cp +++ reversepath pg)" using g_loop by auto
- qed
- also have "... = winding_number g z + (winding_number pg z
- + winding_number (cp +++ (reversepath pg)) z)"
- proof (subst add_left_cancel,rule winding_number_join)
- show "path pg" and "path (cp +++ reversepath pg)"
- and "pathfinish pg = pathstart (cp +++ reversepath pg)"
- by (auto simp add: valid_path_imp_path)
- show "z \<notin> path_image pg" using pg(4) z by blast
- show "z \<notin> path_image (cp +++ reversepath pg)" using z
- by (metis Diff_iff \<open>z \<notin> path_image pg\<close> contra_subsetD cp(4) insertI1
- not_in_path_image_join path_image_reversepath singletonD)
- qed
- also have "... = winding_number g z + (winding_number pg z
- + (winding_number cp z + winding_number (reversepath pg) z))"
- apply (auto intro!:winding_number_join simp: valid_path_imp_path)
- apply (metis Diff_iff contra_subsetD cp(4) insertI1 singletonD z)
- by (metis Diff_insert2 Diff_subset contra_subsetD pg(4) z)
- also have "... = winding_number g z + winding_number cp z"
- apply (subst winding_number_reversepath)
- apply (auto simp: valid_path_imp_path)
- by (metis Diff_iff contra_subsetD insertI1 pg(4) singletonD z)
- finally have "winding_number g' z = winding_number g z + winding_number cp z"
- unfolding g'_def .
- moreover have "winding_number g z + winding_number cp z = 0"
- using winding z \<open>n=winding_number g p\<close> by auto
- ultimately show "winding_number g' z = 0" unfolding g'_def by auto
- qed
- show "\<forall>pa\<in>s - {p}. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s - {p} \<and> (w \<noteq> pa \<longrightarrow> w \<notin> pts))"
- using h by fastforce
- qed
- moreover have "contour_integral g' f = contour_integral g f
- - winding_number g p * contour_integral p_circ f"
- proof -
- have "contour_integral g' f = contour_integral g f
- + contour_integral (pg +++ cp +++ reversepath pg) f"
- unfolding g'_def
- apply (subst contour_integral_join)
- by (auto simp add:open_Diff[OF \<open>open s\<close>,OF finite_imp_closed[OF fin]]
- intro!: contour_integrable_holomorphic_simple[OF holo _ _ path_img]
- contour_integrable_reversepath)
- also have "... = contour_integral g f + contour_integral pg f
- + contour_integral (cp +++ reversepath pg) f"
- apply (subst contour_integral_join)
- by (auto simp add:contour_integrable_reversepath)
- also have "... = contour_integral g f + contour_integral pg f
- + contour_integral cp f + contour_integral (reversepath pg) f"
- apply (subst contour_integral_join)
- by (auto simp add:contour_integrable_reversepath)
- also have "... = contour_integral g f + contour_integral cp f"
- using contour_integral_reversepath
- by (auto simp add:contour_integrable_reversepath)
- also have "... = contour_integral g f - winding_number g p * contour_integral p_circ f"
- using \<open>n=winding_number g p\<close> by auto
- finally show ?thesis .
- qed
- moreover have "winding_number g' p' = winding_number g p'" when "p'\<in>pts" for p'
- proof -
- have [simp]: "p' \<notin> path_image g" "p' \<notin> path_image pg" "p'\<notin>path_image cp"
- using "2.prems"(8) that
- apply blast
- apply (metis Diff_iff Diff_insert2 contra_subsetD pg(4) that)
- by (meson DiffD2 cp(4) rev_subsetD subset_insertI that)
- have "winding_number g' p' = winding_number g p'
- + winding_number (pg +++ cp +++ reversepath pg) p'" unfolding g'_def
- apply (subst winding_number_join)
- apply (simp_all add: valid_path_imp_path)
- apply (intro not_in_path_image_join)
- by auto
- also have "... = winding_number g p' + winding_number pg p'
- + winding_number (cp +++ reversepath pg) p'"
- apply (subst winding_number_join)
- apply (simp_all add: valid_path_imp_path)
- apply (intro not_in_path_image_join)
- by auto
- also have "... = winding_number g p' + winding_number pg p'+ winding_number cp p'
- + winding_number (reversepath pg) p'"
- apply (subst winding_number_join)
- by (simp_all add: valid_path_imp_path)
- also have "... = winding_number g p' + winding_number cp p'"
- apply (subst winding_number_reversepath)
- by (simp_all add: valid_path_imp_path)
- also have "... = winding_number g p'" using that by auto
- finally show ?thesis .
- qed
- ultimately show ?case unfolding p_circ_def
- apply (subst (asm) sum.cong[OF refl,
- of pts _ "\<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"])
- by (auto simp add:sum.insert[OF \<open>finite pts\<close> \<open>p\<notin>pts\<close>] algebra_simps)
-qed
-
-lemma Cauchy_theorem_singularities:
- assumes "open s" "connected s" "finite pts" and
- holo:"f holomorphic_on s-pts" and
- "valid_path g" and
- loop:"pathfinish g = pathstart g" and
- "path_image g \<subseteq> s-pts" and
- homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0" and
- avoid:"\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
- shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
- (is "?L=?R")
-proof -
- define circ where "circ \<equiv> \<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"
- define pts1 where "pts1 \<equiv> pts \<inter> s"
- define pts2 where "pts2 \<equiv> pts - pts1"
- have "pts=pts1 \<union> pts2" "pts1 \<inter> pts2 = {}" "pts2 \<inter> s={}" "pts1\<subseteq>s"
- unfolding pts1_def pts2_def by auto
- have "contour_integral g f = (\<Sum>p\<in>pts1. circ p)" unfolding circ_def
- proof (rule Cauchy_theorem_aux[OF \<open>open s\<close> _ _ \<open>pts1\<subseteq>s\<close> _ \<open>valid_path g\<close> loop _ homo])
- have "finite pts1" unfolding pts1_def using \<open>finite pts\<close> by auto
- then show "connected (s - pts1)"
- using \<open>open s\<close> \<open>connected s\<close> connected_open_delete_finite[of s] by auto
- next
- show "finite pts1" using \<open>pts = pts1 \<union> pts2\<close> assms(3) by auto
- show "f holomorphic_on s - pts1" by (metis Diff_Int2 Int_absorb holo pts1_def)
- show "path_image g \<subseteq> s - pts1" using assms(7) pts1_def by auto
- show "\<forall>p\<in>s. 0 < h p \<and> (\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pts1))"
- by (simp add: avoid pts1_def)
- qed
- moreover have "sum circ pts2=0"
- proof -
- have "winding_number g p=0" when "p\<in>pts2" for p
- using \<open>pts2 \<inter> s={}\<close> that homo[rule_format,of p] by auto
- thus ?thesis unfolding circ_def
- apply (intro sum.neutral)
- by auto
- qed
- moreover have "?R=sum circ pts1 + sum circ pts2"
- unfolding circ_def
- using sum.union_disjoint[OF _ _ \<open>pts1 \<inter> pts2 = {}\<close>] \<open>finite pts\<close> \<open>pts=pts1 \<union> pts2\<close>
- by blast
- ultimately show ?thesis
- apply (fold circ_def)
- by auto
-qed
-
-theorem Residue_theorem:
- fixes s pts::"complex set" and f::"complex \<Rightarrow> complex"
- and g::"real \<Rightarrow> complex"
- assumes "open s" "connected s" "finite pts" and
- holo:"f holomorphic_on s-pts" and
- "valid_path g" and
- loop:"pathfinish g = pathstart g" and
- "path_image g \<subseteq> s-pts" and
- homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0"
- shows "contour_integral g f = 2 * pi * \<i> *(\<Sum>p\<in>pts. winding_number g p * residue f p)"
-proof -
- define c where "c \<equiv> 2 * pi * \<i>"
- obtain h where avoid:"\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
- using finite_cball_avoid[OF \<open>open s\<close> \<open>finite pts\<close>] by metis
- have "contour_integral g f
- = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
- using Cauchy_theorem_singularities[OF assms avoid] .
- also have "... = (\<Sum>p\<in>pts. c * winding_number g p * residue f p)"
- proof (intro sum.cong)
- show "pts = pts" by simp
- next
- fix x assume "x \<in> pts"
- show "winding_number g x * contour_integral (circlepath x (h x)) f
- = c * winding_number g x * residue f x"
- proof (cases "x\<in>s")
- case False
- then have "winding_number g x=0" using homo by auto
- thus ?thesis by auto
- next
- case True
- have "contour_integral (circlepath x (h x)) f = c* residue f x"
- using \<open>x\<in>pts\<close> \<open>finite pts\<close> avoid[rule_format,OF True]
- apply (intro base_residue[of "s-(pts-{x})",THEN contour_integral_unique,folded c_def])
- by (auto intro:holomorphic_on_subset[OF holo] open_Diff[OF \<open>open s\<close> finite_imp_closed])
- then show ?thesis by auto
- qed
- qed
- also have "... = c * (\<Sum>p\<in>pts. winding_number g p * residue f p)"
- by (simp add: sum_distrib_left algebra_simps)
- finally show ?thesis unfolding c_def .
-qed
-
-subsection \<open>Non-essential singular points\<close>
-
-definition\<^marker>\<open>tag important\<close> is_pole ::
- "('a::topological_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> bool" where
- "is_pole f a = (LIM x (at a). f x :> at_infinity)"
-
-lemma is_pole_cong:
- assumes "eventually (\<lambda>x. f x = g x) (at a)" "a=b"
- shows "is_pole f a \<longleftrightarrow> is_pole g b"
- unfolding is_pole_def using assms by (intro filterlim_cong,auto)
-
-lemma is_pole_transform:
- assumes "is_pole f a" "eventually (\<lambda>x. f x = g x) (at a)" "a=b"
- shows "is_pole g b"
- using is_pole_cong assms by auto
-
-lemma is_pole_tendsto:
- fixes f::"('a::topological_space \<Rightarrow> 'b::real_normed_div_algebra)"
- shows "is_pole f x \<Longrightarrow> ((inverse o f) \<longlongrightarrow> 0) (at x)"
-unfolding is_pole_def
-by (auto simp add:filterlim_inverse_at_iff[symmetric] comp_def filterlim_at)
-
-lemma is_pole_inverse_holomorphic:
- assumes "open s"
- and f_holo:"f holomorphic_on (s-{z})"
- and pole:"is_pole f z"
- and non_z:"\<forall>x\<in>s-{z}. f x\<noteq>0"
- shows "(\<lambda>x. if x=z then 0 else inverse (f x)) holomorphic_on s"
-proof -
- define g where "g \<equiv> \<lambda>x. if x=z then 0 else inverse (f x)"
- have "isCont g z" unfolding isCont_def using is_pole_tendsto[OF pole]
- apply (subst Lim_cong_at[where b=z and y=0 and g="inverse \<circ> f"])
- by (simp_all add:g_def)
- moreover have "continuous_on (s-{z}) f" using f_holo holomorphic_on_imp_continuous_on by auto
- hence "continuous_on (s-{z}) (inverse o f)" unfolding comp_def
- by (auto elim!:continuous_on_inverse simp add:non_z)
- hence "continuous_on (s-{z}) g" unfolding g_def
- apply (subst continuous_on_cong[where t="s-{z}" and g="inverse o f"])
- by auto
- ultimately have "continuous_on s g" using open_delete[OF \<open>open s\<close>] \<open>open s\<close>
- by (auto simp add:continuous_on_eq_continuous_at)
- moreover have "(inverse o f) holomorphic_on (s-{z})"
- unfolding comp_def using f_holo
- by (auto elim!:holomorphic_on_inverse simp add:non_z)
- hence "g holomorphic_on (s-{z})"
- apply (subst holomorphic_cong[where t="s-{z}" and g="inverse o f"])
- by (auto simp add:g_def)
- ultimately show ?thesis unfolding g_def using \<open>open s\<close>
- by (auto elim!: no_isolated_singularity)
-qed
-
-lemma not_is_pole_holomorphic:
- assumes "open A" "x \<in> A" "f holomorphic_on A"
- shows "\<not>is_pole f x"
-proof -
- have "continuous_on A f" by (intro holomorphic_on_imp_continuous_on) fact
- with assms have "isCont f x" by (simp add: continuous_on_eq_continuous_at)
- hence "f \<midarrow>x\<rightarrow> f x" by (simp add: isCont_def)
- thus "\<not>is_pole f x" unfolding is_pole_def
- using not_tendsto_and_filterlim_at_infinity[of "at x" f "f x"] by auto
-qed
-
-lemma is_pole_inverse_power: "n > 0 \<Longrightarrow> is_pole (\<lambda>z::complex. 1 / (z - a) ^ n) a"
- unfolding is_pole_def inverse_eq_divide [symmetric]
- by (intro filterlim_compose[OF filterlim_inverse_at_infinity] tendsto_intros)
- (auto simp: filterlim_at eventually_at intro!: exI[of _ 1] tendsto_eq_intros)
-
-lemma is_pole_inverse: "is_pole (\<lambda>z::complex. 1 / (z - a)) a"
- using is_pole_inverse_power[of 1 a] by simp
-
-lemma is_pole_divide:
- fixes f :: "'a :: t2_space \<Rightarrow> 'b :: real_normed_field"
- assumes "isCont f z" "filterlim g (at 0) (at z)" "f z \<noteq> 0"
- shows "is_pole (\<lambda>z. f z / g z) z"
-proof -
- have "filterlim (\<lambda>z. f z * inverse (g z)) at_infinity (at z)"
- by (intro tendsto_mult_filterlim_at_infinity[of _ "f z"]
- filterlim_compose[OF filterlim_inverse_at_infinity])+
- (insert assms, auto simp: isCont_def)
- thus ?thesis by (simp add: field_split_simps is_pole_def)
-qed
-
-lemma is_pole_basic:
- assumes "f holomorphic_on A" "open A" "z \<in> A" "f z \<noteq> 0" "n > 0"
- shows "is_pole (\<lambda>w. f w / (w - z) ^ n) z"
-proof (rule is_pole_divide)
- have "continuous_on A f" by (rule holomorphic_on_imp_continuous_on) fact
- with assms show "isCont f z" by (auto simp: continuous_on_eq_continuous_at)
- have "filterlim (\<lambda>w. (w - z) ^ n) (nhds 0) (at z)"
- using assms by (auto intro!: tendsto_eq_intros)
- thus "filterlim (\<lambda>w. (w - z) ^ n) (at 0) (at z)"
- by (intro filterlim_atI tendsto_eq_intros)
- (insert assms, auto simp: eventually_at_filter)
-qed fact+
-
-lemma is_pole_basic':
- assumes "f holomorphic_on A" "open A" "0 \<in> A" "f 0 \<noteq> 0" "n > 0"
- shows "is_pole (\<lambda>w. f w / w ^ n) 0"
- using is_pole_basic[of f A 0] assms by simp
-
-text \<open>The proposition
- \<^term>\<open>\<exists>x. ((f::complex\<Rightarrow>complex) \<longlongrightarrow> x) (at z) \<or> is_pole f z\<close>
-can be interpreted as the complex function \<^term>\<open>f\<close> has a non-essential singularity at \<^term>\<open>z\<close>
-(i.e. the singularity is either removable or a pole).\<close>
-definition not_essential::"[complex \<Rightarrow> complex, complex] \<Rightarrow> bool" where
- "not_essential f z = (\<exists>x. f\<midarrow>z\<rightarrow>x \<or> is_pole f z)"
-
-definition isolated_singularity_at::"[complex \<Rightarrow> complex, complex] \<Rightarrow> bool" where
- "isolated_singularity_at f z = (\<exists>r>0. f analytic_on ball z r-{z})"
-
-named_theorems singularity_intros "introduction rules for singularities"
-
-lemma holomorphic_factor_unique:
- fixes f::"complex \<Rightarrow> complex" and z::complex and r::real and m n::int
- assumes "r>0" "g z\<noteq>0" "h z\<noteq>0"
- and asm:"\<forall>w\<in>ball z r-{z}. f w = g w * (w-z) powr n \<and> g w\<noteq>0 \<and> f w = h w * (w - z) powr m \<and> h w\<noteq>0"
- and g_holo:"g holomorphic_on ball z r" and h_holo:"h holomorphic_on ball z r"
- shows "n=m"
-proof -
- have [simp]:"at z within ball z r \<noteq> bot" using \<open>r>0\<close>
- by (auto simp add:at_within_ball_bot_iff)
- have False when "n>m"
- proof -
- have "(h \<longlongrightarrow> 0) (at z within ball z r)"
- proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powr (n - m) * g w"])
- have "\<forall>w\<in>ball z r-{z}. h w = (w-z)powr(n-m) * g w"
- using \<open>n>m\<close> asm \<open>r>0\<close>
- apply (auto simp add:field_simps powr_diff)
- by force
- then show "\<lbrakk>x' \<in> ball z r; 0 < dist x' z;dist x' z < r\<rbrakk>
- \<Longrightarrow> (x' - z) powr (n - m) * g x' = h x'" for x' by auto
- next
- define F where "F \<equiv> at z within ball z r"
- define f' where "f' \<equiv> \<lambda>x. (x - z) powr (n-m)"
- have "f' z=0" using \<open>n>m\<close> unfolding f'_def by auto
- moreover have "continuous F f'" unfolding f'_def F_def continuous_def
- apply (subst Lim_ident_at)
- using \<open>n>m\<close> by (auto intro!:tendsto_powr_complex_0 tendsto_eq_intros)
- ultimately have "(f' \<longlongrightarrow> 0) F" unfolding F_def
- by (simp add: continuous_within)
- moreover have "(g \<longlongrightarrow> g z) F"
- using holomorphic_on_imp_continuous_on[OF g_holo,unfolded continuous_on_def] \<open>r>0\<close>
- unfolding F_def by auto
- ultimately show " ((\<lambda>w. f' w * g w) \<longlongrightarrow> 0) F" using tendsto_mult by fastforce
- qed
- moreover have "(h \<longlongrightarrow> h z) (at z within ball z r)"
- using holomorphic_on_imp_continuous_on[OF h_holo]
- by (auto simp add:continuous_on_def \<open>r>0\<close>)
- ultimately have "h z=0" by (auto intro!: tendsto_unique)
- thus False using \<open>h z\<noteq>0\<close> by auto
- qed
- moreover have False when "m>n"
- proof -
- have "(g \<longlongrightarrow> 0) (at z within ball z r)"
- proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powr (m - n) * h w"])
- have "\<forall>w\<in>ball z r -{z}. g w = (w-z) powr (m-n) * h w" using \<open>m>n\<close> asm
- apply (auto simp add:field_simps powr_diff)
- by force
- then show "\<lbrakk>x' \<in> ball z r; 0 < dist x' z;dist x' z < r\<rbrakk>
- \<Longrightarrow> (x' - z) powr (m - n) * h x' = g x'" for x' by auto
- next
- define F where "F \<equiv> at z within ball z r"
- define f' where "f' \<equiv>\<lambda>x. (x - z) powr (m-n)"
- have "f' z=0" using \<open>m>n\<close> unfolding f'_def by auto
- moreover have "continuous F f'" unfolding f'_def F_def continuous_def
- apply (subst Lim_ident_at)
- using \<open>m>n\<close> by (auto intro!:tendsto_powr_complex_0 tendsto_eq_intros)
- ultimately have "(f' \<longlongrightarrow> 0) F" unfolding F_def
- by (simp add: continuous_within)
- moreover have "(h \<longlongrightarrow> h z) F"
- using holomorphic_on_imp_continuous_on[OF h_holo,unfolded continuous_on_def] \<open>r>0\<close>
- unfolding F_def by auto
- ultimately show " ((\<lambda>w. f' w * h w) \<longlongrightarrow> 0) F" using tendsto_mult by fastforce
- qed
- moreover have "(g \<longlongrightarrow> g z) (at z within ball z r)"
- using holomorphic_on_imp_continuous_on[OF g_holo]
- by (auto simp add:continuous_on_def \<open>r>0\<close>)
- ultimately have "g z=0" by (auto intro!: tendsto_unique)
- thus False using \<open>g z\<noteq>0\<close> by auto
- qed
- ultimately show "n=m" by fastforce
-qed
-
-lemma holomorphic_factor_puncture:
- assumes f_iso:"isolated_singularity_at f z"
- and "not_essential f z" \<comment> \<open>\<^term>\<open>f\<close> has either a removable singularity or a pole at \<^term>\<open>z\<close>\<close>
- and non_zero:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" \<comment> \<open>\<^term>\<open>f\<close> will not be constantly zero in a neighbour of \<^term>\<open>z\<close>\<close>
- shows "\<exists>!n::int. \<exists>g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
- \<and> (\<forall>w\<in>cball z r-{z}. f w = g w * (w-z) powr n \<and> g w\<noteq>0)"
-proof -
- define P where "P = (\<lambda>f n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
- have imp_unique:"\<exists>!n::int. \<exists>g r. P f n g r" when "\<exists>n g r. P f n g r"
- proof (rule ex_ex1I[OF that])
- fix n1 n2 :: int
- assume g1_asm:"\<exists>g1 r1. P f n1 g1 r1" and g2_asm:"\<exists>g2 r2. P f n2 g2 r2"
- define fac where "fac \<equiv> \<lambda>n g r. \<forall>w\<in>cball z r-{z}. f w = g w * (w - z) powr (of_int n) \<and> g w \<noteq> 0"
- obtain g1 r1 where "0 < r1" and g1_holo: "g1 holomorphic_on cball z r1" and "g1 z\<noteq>0"
- and "fac n1 g1 r1" using g1_asm unfolding P_def fac_def by auto
- obtain g2 r2 where "0 < r2" and g2_holo: "g2 holomorphic_on cball z r2" and "g2 z\<noteq>0"
- and "fac n2 g2 r2" using g2_asm unfolding P_def fac_def by auto
- define r where "r \<equiv> min r1 r2"
- have "r>0" using \<open>r1>0\<close> \<open>r2>0\<close> unfolding r_def by auto
- moreover have "\<forall>w\<in>ball z r-{z}. f w = g1 w * (w-z) powr n1 \<and> g1 w\<noteq>0
- \<and> f w = g2 w * (w - z) powr n2 \<and> g2 w\<noteq>0"
- using \<open>fac n1 g1 r1\<close> \<open>fac n2 g2 r2\<close> unfolding fac_def r_def
- by fastforce
- ultimately show "n1=n2" using g1_holo g2_holo \<open>g1 z\<noteq>0\<close> \<open>g2 z\<noteq>0\<close>
- apply (elim holomorphic_factor_unique)
- by (auto simp add:r_def)
- qed
-
- have P_exist:"\<exists> n g r. P h n g r" when
- "\<exists>z'. (h \<longlongrightarrow> z') (at z)" "isolated_singularity_at h z" "\<exists>\<^sub>Fw in (at z). h w\<noteq>0"
- for h
- proof -
- from that(2) obtain r where "r>0" "h analytic_on ball z r - {z}"
- unfolding isolated_singularity_at_def by auto
- obtain z' where "(h \<longlongrightarrow> z') (at z)" using \<open>\<exists>z'. (h \<longlongrightarrow> z') (at z)\<close> by auto
- define h' where "h'=(\<lambda>x. if x=z then z' else h x)"
- have "h' holomorphic_on ball z r"
- apply (rule no_isolated_singularity'[of "{z}"])
- subgoal by (metis LIM_equal Lim_at_imp_Lim_at_within \<open>h \<midarrow>z\<rightarrow> z'\<close> empty_iff h'_def insert_iff)
- subgoal using \<open>h analytic_on ball z r - {z}\<close> analytic_imp_holomorphic h'_def holomorphic_transform
- by fastforce
- by auto
- have ?thesis when "z'=0"
- proof -
- have "h' z=0" using that unfolding h'_def by auto
- moreover have "\<not> h' constant_on ball z r"
- using \<open>\<exists>\<^sub>Fw in (at z). h w\<noteq>0\<close> unfolding constant_on_def frequently_def eventually_at h'_def
- apply simp
- by (metis \<open>0 < r\<close> centre_in_ball dist_commute mem_ball that)
- moreover note \<open>h' holomorphic_on ball z r\<close>
- ultimately obtain g r1 n where "0 < n" "0 < r1" "ball z r1 \<subseteq> ball z r" and
- g:"g holomorphic_on ball z r1"
- "\<And>w. w \<in> ball z r1 \<Longrightarrow> h' w = (w - z) ^ n * g w"
- "\<And>w. w \<in> ball z r1 \<Longrightarrow> g w \<noteq> 0"
- using holomorphic_factor_zero_nonconstant[of _ "ball z r" z thesis,simplified,
- OF \<open>h' holomorphic_on ball z r\<close> \<open>r>0\<close> \<open>h' z=0\<close> \<open>\<not> h' constant_on ball z r\<close>]
- by (auto simp add:dist_commute)
- define rr where "rr=r1/2"
- have "P h' n g rr"
- unfolding P_def rr_def
- using \<open>n>0\<close> \<open>r1>0\<close> g by (auto simp add:powr_nat)
- then have "P h n g rr"
- unfolding h'_def P_def by auto
- then show ?thesis unfolding P_def by blast
- qed
- moreover have ?thesis when "z'\<noteq>0"
- proof -
- have "h' z\<noteq>0" using that unfolding h'_def by auto
- obtain r1 where "r1>0" "cball z r1 \<subseteq> ball z r" "\<forall>x\<in>cball z r1. h' x\<noteq>0"
- proof -
- have "isCont h' z" "h' z\<noteq>0"
- by (auto simp add: Lim_cong_within \<open>h \<midarrow>z\<rightarrow> z'\<close> \<open>z'\<noteq>0\<close> continuous_at h'_def)
- then obtain r2 where r2:"r2>0" "\<forall>x\<in>ball z r2. h' x\<noteq>0"
- using continuous_at_avoid[of z h' 0 ] unfolding ball_def by auto
- define r1 where "r1=min r2 r / 2"
- have "0 < r1" "cball z r1 \<subseteq> ball z r"
- using \<open>r2>0\<close> \<open>r>0\<close> unfolding r1_def by auto
- moreover have "\<forall>x\<in>cball z r1. h' x \<noteq> 0"
- using r2 unfolding r1_def by simp
- ultimately show ?thesis using that by auto
- qed
- then have "P h' 0 h' r1" using \<open>h' holomorphic_on ball z r\<close> unfolding P_def by auto
- then have "P h 0 h' r1" unfolding P_def h'_def by auto
- then show ?thesis unfolding P_def by blast
- qed
- ultimately show ?thesis by auto
- qed
-
- have ?thesis when "\<exists>x. (f \<longlongrightarrow> x) (at z)"
- apply (rule_tac imp_unique[unfolded P_def])
- using P_exist[OF that(1) f_iso non_zero] unfolding P_def .
- moreover have ?thesis when "is_pole f z"
- proof (rule imp_unique[unfolded P_def])
- obtain e where [simp]:"e>0" and e_holo:"f holomorphic_on ball z e - {z}" and e_nz: "\<forall>x\<in>ball z e-{z}. f x\<noteq>0"
- proof -
- have "\<forall>\<^sub>F z in at z. f z \<noteq> 0"
- using \<open>is_pole f z\<close> filterlim_at_infinity_imp_eventually_ne unfolding is_pole_def
- by auto
- then obtain e1 where e1:"e1>0" "\<forall>x\<in>ball z e1-{z}. f x\<noteq>0"
- using that eventually_at[of "\<lambda>x. f x\<noteq>0" z UNIV,simplified] by (auto simp add:dist_commute)
- obtain e2 where e2:"e2>0" "f holomorphic_on ball z e2 - {z}"
- using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by auto
- define e where "e=min e1 e2"
- show ?thesis
- apply (rule that[of e])
- using e1 e2 unfolding e_def by auto
- qed
-
- define h where "h \<equiv> \<lambda>x. inverse (f x)"
-
- have "\<exists>n g r. P h n g r"
- proof -
- have "h \<midarrow>z\<rightarrow> 0"
- using Lim_transform_within_open assms(2) h_def is_pole_tendsto that by fastforce
- moreover have "\<exists>\<^sub>Fw in (at z). h w\<noteq>0"
- using non_zero
- apply (elim frequently_rev_mp)
- unfolding h_def eventually_at by (auto intro:exI[where x=1])
- moreover have "isolated_singularity_at h z"
- unfolding isolated_singularity_at_def h_def
- apply (rule exI[where x=e])
- using e_holo e_nz \<open>e>0\<close> by (metis open_ball analytic_on_open
- holomorphic_on_inverse open_delete)
- ultimately show ?thesis
- using P_exist[of h] by auto
- qed
- then obtain n g r
- where "0 < r" and
- g_holo:"g holomorphic_on cball z r" and "g z\<noteq>0" and
- g_fac:"(\<forall>w\<in>cball z r-{z}. h w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
- unfolding P_def by auto
- have "P f (-n) (inverse o g) r"
- proof -
- have "f w = inverse (g w) * (w - z) powr of_int (- n)" when "w\<in>cball z r - {z}" for w
- using g_fac[rule_format,of w] that unfolding h_def
- apply (auto simp add:powr_minus )
- by (metis inverse_inverse_eq inverse_mult_distrib)
- then show ?thesis
- unfolding P_def comp_def
- using \<open>r>0\<close> g_holo g_fac \<open>g z\<noteq>0\<close> by (auto intro:holomorphic_intros)
- qed
- then show "\<exists>x g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z \<noteq> 0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int x \<and> g w \<noteq> 0)"
- unfolding P_def by blast
- qed
- ultimately show ?thesis using \<open>not_essential f z\<close> unfolding not_essential_def by presburger
-qed
-
-lemma not_essential_transform:
- assumes "not_essential g z"
- assumes "\<forall>\<^sub>F w in (at z). g w = f w"
- shows "not_essential f z"
- using assms unfolding not_essential_def
- by (simp add: filterlim_cong is_pole_cong)
-
-lemma isolated_singularity_at_transform:
- assumes "isolated_singularity_at g z"
- assumes "\<forall>\<^sub>F w in (at z). g w = f w"
- shows "isolated_singularity_at f z"
-proof -
- obtain r1 where "r1>0" and r1:"g analytic_on ball z r1 - {z}"
- using assms(1) unfolding isolated_singularity_at_def by auto
- obtain r2 where "r2>0" and r2:" \<forall>x. x \<noteq> z \<and> dist x z < r2 \<longrightarrow> g x = f x"
- using assms(2) unfolding eventually_at by auto
- define r3 where "r3=min r1 r2"
- have "r3>0" unfolding r3_def using \<open>r1>0\<close> \<open>r2>0\<close> by auto
- moreover have "f analytic_on ball z r3 - {z}"
- proof -
- have "g holomorphic_on ball z r3 - {z}"
- using r1 unfolding r3_def by (subst (asm) analytic_on_open,auto)
- then have "f holomorphic_on ball z r3 - {z}"
- using r2 unfolding r3_def
- by (auto simp add:dist_commute elim!:holomorphic_transform)
- then show ?thesis by (subst analytic_on_open,auto)
- qed
- ultimately show ?thesis unfolding isolated_singularity_at_def by auto
-qed
-
-lemma not_essential_powr[singularity_intros]:
- assumes "LIM w (at z). f w :> (at x)"
- shows "not_essential (\<lambda>w. (f w) powr (of_int n)) z"
-proof -
- define fp where "fp=(\<lambda>w. (f w) powr (of_int n))"
- have ?thesis when "n>0"
- proof -
- have "(\<lambda>w. (f w) ^ (nat n)) \<midarrow>z\<rightarrow> x ^ nat n"
- using that assms unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
- then have "fp \<midarrow>z\<rightarrow> x ^ nat n" unfolding fp_def
- apply (elim Lim_transform_within[where d=1],simp)
- by (metis less_le powr_0 powr_of_int that zero_less_nat_eq zero_power)
- then show ?thesis unfolding not_essential_def fp_def by auto
- qed
- moreover have ?thesis when "n=0"
- proof -
- have "fp \<midarrow>z\<rightarrow> 1 "
- apply (subst tendsto_cong[where g="\<lambda>_.1"])
- using that filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def by auto
- then show ?thesis unfolding fp_def not_essential_def by auto
- qed
- moreover have ?thesis when "n<0"
- proof (cases "x=0")
- case True
- have "LIM w (at z). inverse ((f w) ^ (nat (-n))) :> at_infinity"
- apply (subst filterlim_inverse_at_iff[symmetric],simp)
- apply (rule filterlim_atI)
- subgoal using assms True that unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
- subgoal using filterlim_at_within_not_equal[OF assms,of 0]
- by (eventually_elim,insert that,auto)
- done
- then have "LIM w (at z). fp w :> at_infinity"
- proof (elim filterlim_mono_eventually)
- show "\<forall>\<^sub>F x in at z. inverse (f x ^ nat (- n)) = fp x"
- using filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def
- apply eventually_elim
- using powr_of_int that by auto
- qed auto
- then show ?thesis unfolding fp_def not_essential_def is_pole_def by auto
- next
- case False
- let ?xx= "inverse (x ^ (nat (-n)))"
- have "(\<lambda>w. inverse ((f w) ^ (nat (-n)))) \<midarrow>z\<rightarrow>?xx"
- using assms False unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
- then have "fp \<midarrow>z\<rightarrow>?xx"
- apply (elim Lim_transform_within[where d=1],simp)
- unfolding fp_def by (metis inverse_zero nat_mono_iff nat_zero_as_int neg_0_less_iff_less
- not_le power_eq_0_iff powr_0 powr_of_int that)
- then show ?thesis unfolding fp_def not_essential_def by auto
- qed
- ultimately show ?thesis by linarith
-qed
-
-lemma isolated_singularity_at_powr[singularity_intros]:
- assumes "isolated_singularity_at f z" "\<forall>\<^sub>F w in (at z). f w\<noteq>0"
- shows "isolated_singularity_at (\<lambda>w. (f w) powr (of_int n)) z"
-proof -
- obtain r1 where "r1>0" "f analytic_on ball z r1 - {z}"
- using assms(1) unfolding isolated_singularity_at_def by auto
- then have r1:"f holomorphic_on ball z r1 - {z}"
- using analytic_on_open[of "ball z r1-{z}" f] by blast
- obtain r2 where "r2>0" and r2:"\<forall>w. w \<noteq> z \<and> dist w z < r2 \<longrightarrow> f w \<noteq> 0"
- using assms(2) unfolding eventually_at by auto
- define r3 where "r3=min r1 r2"
- have "(\<lambda>w. (f w) powr of_int n) holomorphic_on ball z r3 - {z}"
- apply (rule holomorphic_on_powr_of_int)
- subgoal unfolding r3_def using r1 by auto
- subgoal unfolding r3_def using r2 by (auto simp add:dist_commute)
- done
- moreover have "r3>0" unfolding r3_def using \<open>0 < r1\<close> \<open>0 < r2\<close> by linarith
- ultimately show ?thesis unfolding isolated_singularity_at_def
- apply (subst (asm) analytic_on_open[symmetric])
- by auto
-qed
-
-lemma non_zero_neighbour:
- assumes f_iso:"isolated_singularity_at f z"
- and f_ness:"not_essential f z"
- and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
- shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0"
-proof -
- obtain fn fp fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
- and fr: "fp holomorphic_on cball z fr"
- "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
- using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
- have "f w \<noteq> 0" when " w \<noteq> z" "dist w z < fr" for w
- proof -
- have "f w = fp w * (w - z) powr of_int fn" "fp w \<noteq> 0"
- using fr(2)[rule_format, of w] using that by (auto simp add:dist_commute)
- moreover have "(w - z) powr of_int fn \<noteq>0"
- unfolding powr_eq_0_iff using \<open>w\<noteq>z\<close> by auto
- ultimately show ?thesis by auto
- qed
- then show ?thesis using \<open>fr>0\<close> unfolding eventually_at by auto
-qed
-
-lemma non_zero_neighbour_pole:
- assumes "is_pole f z"
- shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0"
- using assms filterlim_at_infinity_imp_eventually_ne[of f "at z" 0]
- unfolding is_pole_def by auto
-
-lemma non_zero_neighbour_alt:
- assumes holo: "f holomorphic_on S"
- and "open S" "connected S" "z \<in> S" "\<beta> \<in> S" "f \<beta> \<noteq> 0"
- shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0 \<and> w\<in>S"
-proof (cases "f z = 0")
- case True
- from isolated_zeros[OF holo \<open>open S\<close> \<open>connected S\<close> \<open>z \<in> S\<close> True \<open>\<beta> \<in> S\<close> \<open>f \<beta> \<noteq> 0\<close>]
- obtain r where "0 < r" "ball z r \<subseteq> S" "\<forall>w \<in> ball z r - {z}.f w \<noteq> 0" by metis
- then show ?thesis unfolding eventually_at
- apply (rule_tac x=r in exI)
- by (auto simp add:dist_commute)
-next
- case False
- obtain r1 where r1:"r1>0" "\<forall>y. dist z y < r1 \<longrightarrow> f y \<noteq> 0"
- using continuous_at_avoid[of z f, OF _ False] assms(2,4) continuous_on_eq_continuous_at
- holo holomorphic_on_imp_continuous_on by blast
- obtain r2 where r2:"r2>0" "ball z r2 \<subseteq> S"
- using assms(2) assms(4) openE by blast
- show ?thesis unfolding eventually_at
- apply (rule_tac x="min r1 r2" in exI)
- using r1 r2 by (auto simp add:dist_commute)
-qed
-
-lemma not_essential_times[singularity_intros]:
- assumes f_ness:"not_essential f z" and g_ness:"not_essential g z"
- assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
- shows "not_essential (\<lambda>w. f w * g w) z"
-proof -
- define fg where "fg = (\<lambda>w. f w * g w)"
- have ?thesis when "\<not> ((\<exists>\<^sub>Fw in (at z). f w\<noteq>0) \<and> (\<exists>\<^sub>Fw in (at z). g w\<noteq>0))"
- proof -
- have "\<forall>\<^sub>Fw in (at z). fg w=0"
- using that[unfolded frequently_def, simplified] unfolding fg_def
- by (auto elim: eventually_rev_mp)
- from tendsto_cong[OF this] have "fg \<midarrow>z\<rightarrow>0" by auto
- then show ?thesis unfolding not_essential_def fg_def by auto
- qed
- moreover have ?thesis when f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" and g_nconst:"\<exists>\<^sub>Fw in (at z). g w\<noteq>0"
- proof -
- obtain fn fp fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
- and fr: "fp holomorphic_on cball z fr"
- "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
- using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
- obtain gn gp gr where [simp]:"gp z \<noteq> 0" and "gr > 0"
- and gr: "gp holomorphic_on cball z gr"
- "\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powr of_int gn \<and> gp w \<noteq> 0"
- using holomorphic_factor_puncture[OF g_iso g_ness g_nconst,THEN ex1_implies_ex] by auto
-
- define r1 where "r1=(min fr gr)"
- have "r1>0" unfolding r1_def using \<open>fr>0\<close> \<open>gr>0\<close> by auto
- have fg_times:"fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" and fgp_nz:"fp w*gp w\<noteq>0"
- when "w\<in>ball z r1 - {z}" for w
- proof -
- have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
- using fr(2)[rule_format,of w] that unfolding r1_def by auto
- moreover have "g w = gp w * (w - z) powr of_int gn" "gp w \<noteq> 0"
- using gr(2)[rule_format, of w] that unfolding r1_def by auto
- ultimately show "fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" "fp w*gp w\<noteq>0"
- unfolding fg_def by (auto simp add:powr_add)
- qed
-
- have [intro]: "fp \<midarrow>z\<rightarrow>fp z" "gp \<midarrow>z\<rightarrow>gp z"
- using fr(1) \<open>fr>0\<close> gr(1) \<open>gr>0\<close>
- by (meson open_ball ball_subset_cball centre_in_ball
- continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on
- holomorphic_on_subset)+
- have ?thesis when "fn+gn>0"
- proof -
- have "(\<lambda>w. (fp w * gp w) * (w - z) ^ (nat (fn+gn))) \<midarrow>z\<rightarrow>0"
- using that by (auto intro!:tendsto_eq_intros)
- then have "fg \<midarrow>z\<rightarrow> 0"
- apply (elim Lim_transform_within[OF _ \<open>r1>0\<close>])
- by (metis (no_types, hide_lams) Diff_iff cball_trivial dist_commute dist_self
- eq_iff_diff_eq_0 fg_times less_le linorder_not_le mem_ball mem_cball powr_of_int
- that)
- then show ?thesis unfolding not_essential_def fg_def by auto
- qed
- moreover have ?thesis when "fn+gn=0"
- proof -
- have "(\<lambda>w. fp w * gp w) \<midarrow>z\<rightarrow>fp z*gp z"
- using that by (auto intro!:tendsto_eq_intros)
- then have "fg \<midarrow>z\<rightarrow> fp z*gp z"
- apply (elim Lim_transform_within[OF _ \<open>r1>0\<close>])
- apply (subst fg_times)
- by (auto simp add:dist_commute that)
- then show ?thesis unfolding not_essential_def fg_def by auto
- qed
- moreover have ?thesis when "fn+gn<0"
- proof -
- have "LIM w (at z). fp w * gp w / (w-z)^nat (-(fn+gn)) :> at_infinity"
- apply (rule filterlim_divide_at_infinity)
- apply (insert that, auto intro!:tendsto_eq_intros filterlim_atI)
- using eventually_at_topological by blast
- then have "is_pole fg z" unfolding is_pole_def
- apply (elim filterlim_transform_within[OF _ _ \<open>r1>0\<close>],simp)
- apply (subst fg_times,simp add:dist_commute)
- apply (subst powr_of_int)
- using that by (auto simp add:field_split_simps)
- then show ?thesis unfolding not_essential_def fg_def by auto
- qed
- ultimately show ?thesis unfolding not_essential_def fg_def by fastforce
- qed
- ultimately show ?thesis by auto
-qed
-
-lemma not_essential_inverse[singularity_intros]:
- assumes f_ness:"not_essential f z"
- assumes f_iso:"isolated_singularity_at f z"
- shows "not_essential (\<lambda>w. inverse (f w)) z"
-proof -
- define vf where "vf = (\<lambda>w. inverse (f w))"
- have ?thesis when "\<not>(\<exists>\<^sub>Fw in (at z). f w\<noteq>0)"
- proof -
- have "\<forall>\<^sub>Fw in (at z). f w=0"
- using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp)
- then have "\<forall>\<^sub>Fw in (at z). vf w=0"
- unfolding vf_def by auto
- from tendsto_cong[OF this] have "vf \<midarrow>z\<rightarrow>0" unfolding vf_def by auto
- then show ?thesis unfolding not_essential_def vf_def by auto
- qed
- moreover have ?thesis when "is_pole f z"
- proof -
- have "vf \<midarrow>z\<rightarrow>0"
- using that filterlim_at filterlim_inverse_at_iff unfolding is_pole_def vf_def by blast
- then show ?thesis unfolding not_essential_def vf_def by auto
- qed
- moreover have ?thesis when "\<exists>x. f\<midarrow>z\<rightarrow>x " and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
- proof -
- from that obtain fz where fz:"f\<midarrow>z\<rightarrow>fz" by auto
- have ?thesis when "fz=0"
- proof -
- have "(\<lambda>w. inverse (vf w)) \<midarrow>z\<rightarrow>0"
- using fz that unfolding vf_def by auto
- moreover have "\<forall>\<^sub>F w in at z. inverse (vf w) \<noteq> 0"
- using non_zero_neighbour[OF f_iso f_ness f_nconst]
- unfolding vf_def by auto
- ultimately have "is_pole vf z"
- using filterlim_inverse_at_iff[of vf "at z"] unfolding filterlim_at is_pole_def by auto
- then show ?thesis unfolding not_essential_def vf_def by auto
- qed
- moreover have ?thesis when "fz\<noteq>0"
- proof -
- have "vf \<midarrow>z\<rightarrow>inverse fz"
- using fz that unfolding vf_def by (auto intro:tendsto_eq_intros)
- then show ?thesis unfolding not_essential_def vf_def by auto
- qed
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis using f_ness unfolding not_essential_def by auto
-qed
-
-lemma isolated_singularity_at_inverse[singularity_intros]:
- assumes f_iso:"isolated_singularity_at f z"
- and f_ness:"not_essential f z"
- shows "isolated_singularity_at (\<lambda>w. inverse (f w)) z"
-proof -
- define vf where "vf = (\<lambda>w. inverse (f w))"
- have ?thesis when "\<not>(\<exists>\<^sub>Fw in (at z). f w\<noteq>0)"
- proof -
- have "\<forall>\<^sub>Fw in (at z). f w=0"
- using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp)
- then have "\<forall>\<^sub>Fw in (at z). vf w=0"
- unfolding vf_def by auto
- then obtain d1 where "d1>0" and d1:"\<forall>x. x \<noteq> z \<and> dist x z < d1 \<longrightarrow> vf x = 0"
- unfolding eventually_at by auto
- then have "vf holomorphic_on ball z d1-{z}"
- apply (rule_tac holomorphic_transform[of "\<lambda>_. 0"])
- by (auto simp add:dist_commute)
- then have "vf analytic_on ball z d1 - {z}"
- by (simp add: analytic_on_open open_delete)
- then show ?thesis using \<open>d1>0\<close> unfolding isolated_singularity_at_def vf_def by auto
- qed
- moreover have ?thesis when f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
- proof -
- have "\<forall>\<^sub>F w in at z. f w \<noteq> 0" using non_zero_neighbour[OF f_iso f_ness f_nconst] .
- then obtain d1 where d1:"d1>0" "\<forall>x. x \<noteq> z \<and> dist x z < d1 \<longrightarrow> f x \<noteq> 0"
- unfolding eventually_at by auto
- obtain d2 where "d2>0" and d2:"f analytic_on ball z d2 - {z}"
- using f_iso unfolding isolated_singularity_at_def by auto
- define d3 where "d3=min d1 d2"
- have "d3>0" unfolding d3_def using \<open>d1>0\<close> \<open>d2>0\<close> by auto
- moreover have "vf analytic_on ball z d3 - {z}"
- unfolding vf_def
- apply (rule analytic_on_inverse)
- subgoal using d2 unfolding d3_def by (elim analytic_on_subset) auto
- subgoal for w using d1 unfolding d3_def by (auto simp add:dist_commute)
- done
- ultimately show ?thesis unfolding isolated_singularity_at_def vf_def by auto
- qed
- ultimately show ?thesis by auto
-qed
-
-lemma not_essential_divide[singularity_intros]:
- assumes f_ness:"not_essential f z" and g_ness:"not_essential g z"
- assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
- shows "not_essential (\<lambda>w. f w / g w) z"
-proof -
- have "not_essential (\<lambda>w. f w * inverse (g w)) z"
- apply (rule not_essential_times[where g="\<lambda>w. inverse (g w)"])
- using assms by (auto intro: isolated_singularity_at_inverse not_essential_inverse)
- then show ?thesis by (simp add:field_simps)
-qed
-
-lemma
- assumes f_iso:"isolated_singularity_at f z"
- and g_iso:"isolated_singularity_at g z"
- shows isolated_singularity_at_times[singularity_intros]:
- "isolated_singularity_at (\<lambda>w. f w * g w) z" and
- isolated_singularity_at_add[singularity_intros]:
- "isolated_singularity_at (\<lambda>w. f w + g w) z"
-proof -
- obtain d1 d2 where "d1>0" "d2>0"
- and d1:"f analytic_on ball z d1 - {z}" and d2:"g analytic_on ball z d2 - {z}"
- using f_iso g_iso unfolding isolated_singularity_at_def by auto
- define d3 where "d3=min d1 d2"
- have "d3>0" unfolding d3_def using \<open>d1>0\<close> \<open>d2>0\<close> by auto
-
- have "(\<lambda>w. f w * g w) analytic_on ball z d3 - {z}"
- apply (rule analytic_on_mult)
- using d1 d2 unfolding d3_def by (auto elim:analytic_on_subset)
- then show "isolated_singularity_at (\<lambda>w. f w * g w) z"
- using \<open>d3>0\<close> unfolding isolated_singularity_at_def by auto
- have "(\<lambda>w. f w + g w) analytic_on ball z d3 - {z}"
- apply (rule analytic_on_add)
- using d1 d2 unfolding d3_def by (auto elim:analytic_on_subset)
- then show "isolated_singularity_at (\<lambda>w. f w + g w) z"
- using \<open>d3>0\<close> unfolding isolated_singularity_at_def by auto
-qed
-
-lemma isolated_singularity_at_uminus[singularity_intros]:
- assumes f_iso:"isolated_singularity_at f z"
- shows "isolated_singularity_at (\<lambda>w. - f w) z"
- using assms unfolding isolated_singularity_at_def using analytic_on_neg by blast
-
-lemma isolated_singularity_at_id[singularity_intros]:
- "isolated_singularity_at (\<lambda>w. w) z"
- unfolding isolated_singularity_at_def by (simp add: gt_ex)
-
-lemma isolated_singularity_at_minus[singularity_intros]:
- assumes f_iso:"isolated_singularity_at f z"
- and g_iso:"isolated_singularity_at g z"
- shows "isolated_singularity_at (\<lambda>w. f w - g w) z"
- using isolated_singularity_at_uminus[THEN isolated_singularity_at_add[OF f_iso,of "\<lambda>w. - g w"]
- ,OF g_iso] by simp
-
-lemma isolated_singularity_at_divide[singularity_intros]:
- assumes f_iso:"isolated_singularity_at f z"
- and g_iso:"isolated_singularity_at g z"
- and g_ness:"not_essential g z"
- shows "isolated_singularity_at (\<lambda>w. f w / g w) z"
- using isolated_singularity_at_inverse[THEN isolated_singularity_at_times[OF f_iso,
- of "\<lambda>w. inverse (g w)"],OF g_iso g_ness] by (simp add:field_simps)
-
-lemma isolated_singularity_at_const[singularity_intros]:
- "isolated_singularity_at (\<lambda>w. c) z"
- unfolding isolated_singularity_at_def by (simp add: gt_ex)
-
-lemma isolated_singularity_at_holomorphic:
- assumes "f holomorphic_on s-{z}" "open s" "z\<in>s"
- shows "isolated_singularity_at f z"
- using assms unfolding isolated_singularity_at_def
- by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)
-
-subsubsection \<open>The order of non-essential singularities (i.e. removable singularities or poles)\<close>
-
-
-definition\<^marker>\<open>tag important\<close> zorder :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> int" where
- "zorder f z = (THE n. (\<exists>h r. r>0 \<and> h holomorphic_on cball z r \<and> h z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w-z) powr (of_int n)
- \<and> h w \<noteq>0)))"
-
-definition\<^marker>\<open>tag important\<close> zor_poly
- ::"[complex \<Rightarrow> complex, complex] \<Rightarrow> complex \<Rightarrow> complex" where
- "zor_poly f z = (SOME h. \<exists>r. r > 0 \<and> h holomorphic_on cball z r \<and> h z \<noteq> 0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w - z) powr (zorder f z)
- \<and> h w \<noteq>0))"
-
-lemma zorder_exist:
- fixes f::"complex \<Rightarrow> complex" and z::complex
- defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z"
- assumes f_iso:"isolated_singularity_at f z"
- and f_ness:"not_essential f z"
- and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
- shows "g z\<noteq>0 \<and> (\<exists>r. r>0 \<and> g holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr n \<and> g w \<noteq>0))"
-proof -
- define P where "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
- have "\<exists>!n. \<exists>g r. P n g r"
- using holomorphic_factor_puncture[OF assms(3-)] unfolding P_def by auto
- then have "\<exists>g r. P n g r"
- unfolding n_def P_def zorder_def
- by (drule_tac theI',argo)
- then have "\<exists>r. P n g r"
- unfolding P_def zor_poly_def g_def n_def
- by (drule_tac someI_ex,argo)
- then obtain r1 where "P n g r1" by auto
- then show ?thesis unfolding P_def by auto
-qed
-
-lemma
- fixes f::"complex \<Rightarrow> complex" and z::complex
- assumes f_iso:"isolated_singularity_at f z"
- and f_ness:"not_essential f z"
- and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
- shows zorder_inverse: "zorder (\<lambda>w. inverse (f w)) z = - zorder f z"
- and zor_poly_inverse: "\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. inverse (f w)) z w
- = inverse (zor_poly f z w)"
-proof -
- define vf where "vf = (\<lambda>w. inverse (f w))"
- define fn vfn where
- "fn = zorder f z" and "vfn = zorder vf z"
- define fp vfp where
- "fp = zor_poly f z" and "vfp = zor_poly vf z"
-
- obtain fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
- and fr: "fp holomorphic_on cball z fr"
- "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
- using zorder_exist[OF f_iso f_ness f_nconst,folded fn_def fp_def]
- by auto
- have fr_inverse: "vf w = (inverse (fp w)) * (w - z) powr (of_int (-fn))"
- and fr_nz: "inverse (fp w)\<noteq>0"
- when "w\<in>ball z fr - {z}" for w
- proof -
- have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
- using fr(2)[rule_format,of w] that by auto
- then show "vf w = (inverse (fp w)) * (w - z) powr (of_int (-fn))" "inverse (fp w)\<noteq>0"
- unfolding vf_def by (auto simp add:powr_minus)
- qed
- obtain vfr where [simp]:"vfp z \<noteq> 0" and "vfr>0" and vfr:"vfp holomorphic_on cball z vfr"
- "(\<forall>w\<in>cball z vfr - {z}. vf w = vfp w * (w - z) powr of_int vfn \<and> vfp w \<noteq> 0)"
- proof -
- have "isolated_singularity_at vf z"
- using isolated_singularity_at_inverse[OF f_iso f_ness] unfolding vf_def .
- moreover have "not_essential vf z"
- using not_essential_inverse[OF f_ness f_iso] unfolding vf_def .
- moreover have "\<exists>\<^sub>F w in at z. vf w \<noteq> 0"
- using f_nconst unfolding vf_def by (auto elim:frequently_elim1)
- ultimately show ?thesis using zorder_exist[of vf z, folded vfn_def vfp_def] that by auto
- qed
-
-
- define r1 where "r1 = min fr vfr"
- have "r1>0" using \<open>fr>0\<close> \<open>vfr>0\<close> unfolding r1_def by simp
- show "vfn = - fn"
- apply (rule holomorphic_factor_unique[of r1 vfp z "\<lambda>w. inverse (fp w)" vf])
- subgoal using \<open>r1>0\<close> by simp
- subgoal by simp
- subgoal by simp
- subgoal
- proof (rule ballI)
- fix w assume "w \<in> ball z r1 - {z}"
- then have "w \<in> ball z fr - {z}" "w \<in> cball z vfr - {z}" unfolding r1_def by auto
- from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)]
- show "vf w = vfp w * (w - z) powr of_int vfn \<and> vfp w \<noteq> 0
- \<and> vf w = inverse (fp w) * (w - z) powr of_int (- fn) \<and> inverse (fp w) \<noteq> 0" by auto
- qed
- subgoal using vfr(1) unfolding r1_def by (auto intro!:holomorphic_intros)
- subgoal using fr unfolding r1_def by (auto intro!:holomorphic_intros)
- done
-
- have "vfp w = inverse (fp w)" when "w\<in>ball z r1-{z}" for w
- proof -
- have "w \<in> ball z fr - {z}" "w \<in> cball z vfr - {z}" "w\<noteq>z" using that unfolding r1_def by auto
- from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)] \<open>vfn = - fn\<close> \<open>w\<noteq>z\<close>
- show ?thesis by auto
- qed
- then show "\<forall>\<^sub>Fw in (at z). vfp w = inverse (fp w)"
- unfolding eventually_at using \<open>r1>0\<close>
- apply (rule_tac x=r1 in exI)
- by (auto simp add:dist_commute)
-qed
-
-lemma
- fixes f g::"complex \<Rightarrow> complex" and z::complex
- assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
- and f_ness:"not_essential f z" and g_ness:"not_essential g z"
- and fg_nconst: "\<exists>\<^sub>Fw in (at z). f w * g w\<noteq> 0"
- shows zorder_times:"zorder (\<lambda>w. f w * g w) z = zorder f z + zorder g z" and
- zor_poly_times:"\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w * g w) z w
- = zor_poly f z w *zor_poly g z w"
-proof -
- define fg where "fg = (\<lambda>w. f w * g w)"
- define fn gn fgn where
- "fn = zorder f z" and "gn = zorder g z" and "fgn = zorder fg z"
- define fp gp fgp where
- "fp = zor_poly f z" and "gp = zor_poly g z" and "fgp = zor_poly fg z"
- have f_nconst:"\<exists>\<^sub>Fw in (at z). f w \<noteq> 0" and g_nconst:"\<exists>\<^sub>Fw in (at z).g w\<noteq> 0"
- using fg_nconst by (auto elim!:frequently_elim1)
- obtain fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
- and fr: "fp holomorphic_on cball z fr"
- "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
- using zorder_exist[OF f_iso f_ness f_nconst,folded fp_def fn_def] by auto
- obtain gr where [simp]:"gp z \<noteq> 0" and "gr > 0"
- and gr: "gp holomorphic_on cball z gr"
- "\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powr of_int gn \<and> gp w \<noteq> 0"
- using zorder_exist[OF g_iso g_ness g_nconst,folded gn_def gp_def] by auto
- define r1 where "r1=min fr gr"
- have "r1>0" unfolding r1_def using \<open>fr>0\<close> \<open>gr>0\<close> by auto
- have fg_times:"fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" and fgp_nz:"fp w*gp w\<noteq>0"
- when "w\<in>ball z r1 - {z}" for w
- proof -
- have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
- using fr(2)[rule_format,of w] that unfolding r1_def by auto
- moreover have "g w = gp w * (w - z) powr of_int gn" "gp w \<noteq> 0"
- using gr(2)[rule_format, of w] that unfolding r1_def by auto
- ultimately show "fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" "fp w*gp w\<noteq>0"
- unfolding fg_def by (auto simp add:powr_add)
- qed
-
- obtain fgr where [simp]:"fgp z \<noteq> 0" and "fgr > 0"
- and fgr: "fgp holomorphic_on cball z fgr"
- "\<forall>w\<in>cball z fgr - {z}. fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0"
- proof -
- have "fgp z \<noteq> 0 \<and> (\<exists>r>0. fgp holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r - {z}. fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0))"
- apply (rule zorder_exist[of fg z, folded fgn_def fgp_def])
- subgoal unfolding fg_def using isolated_singularity_at_times[OF f_iso g_iso] .
- subgoal unfolding fg_def using not_essential_times[OF f_ness g_ness f_iso g_iso] .
- subgoal unfolding fg_def using fg_nconst .
- done
- then show ?thesis using that by blast
- qed
- define r2 where "r2 = min fgr r1"
- have "r2>0" using \<open>r1>0\<close> \<open>fgr>0\<close> unfolding r2_def by simp
- show "fgn = fn + gn "
- apply (rule holomorphic_factor_unique[of r2 fgp z "\<lambda>w. fp w * gp w" fg])
- subgoal using \<open>r2>0\<close> by simp
- subgoal by simp
- subgoal by simp
- subgoal
- proof (rule ballI)
- fix w assume "w \<in> ball z r2 - {z}"
- then have "w \<in> ball z r1 - {z}" "w \<in> cball z fgr - {z}" unfolding r2_def by auto
- from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)]
- show "fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0
- \<and> fg w = fp w * gp w * (w - z) powr of_int (fn + gn) \<and> fp w * gp w \<noteq> 0" by auto
- qed
- subgoal using fgr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
- subgoal using fr(1) gr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
- done
-
- have "fgp w = fp w *gp w" when "w\<in>ball z r2-{z}" for w
- proof -
- have "w \<in> ball z r1 - {z}" "w \<in> cball z fgr - {z}" "w\<noteq>z" using that unfolding r2_def by auto
- from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)] \<open>fgn = fn + gn\<close> \<open>w\<noteq>z\<close>
- show ?thesis by auto
- qed
- then show "\<forall>\<^sub>Fw in (at z). fgp w = fp w * gp w"
- using \<open>r2>0\<close> unfolding eventually_at by (auto simp add:dist_commute)
-qed
-
-lemma
- fixes f g::"complex \<Rightarrow> complex" and z::complex
- assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
- and f_ness:"not_essential f z" and g_ness:"not_essential g z"
- and fg_nconst: "\<exists>\<^sub>Fw in (at z). f w * g w\<noteq> 0"
- shows zorder_divide:"zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z" and
- zor_poly_divide:"\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w / g w) z w
- = zor_poly f z w / zor_poly g z w"
-proof -
- have f_nconst:"\<exists>\<^sub>Fw in (at z). f w \<noteq> 0" and g_nconst:"\<exists>\<^sub>Fw in (at z).g w\<noteq> 0"
- using fg_nconst by (auto elim!:frequently_elim1)
- define vg where "vg=(\<lambda>w. inverse (g w))"
- have "zorder (\<lambda>w. f w * vg w) z = zorder f z + zorder vg z"
- apply (rule zorder_times[OF f_iso _ f_ness,of vg])
- subgoal unfolding vg_def using isolated_singularity_at_inverse[OF g_iso g_ness] .
- subgoal unfolding vg_def using not_essential_inverse[OF g_ness g_iso] .
- subgoal unfolding vg_def using fg_nconst by (auto elim!:frequently_elim1)
- done
- then show "zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z"
- using zorder_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def
- by (auto simp add:field_simps)
-
- have "\<forall>\<^sub>F w in at z. zor_poly (\<lambda>w. f w * vg w) z w = zor_poly f z w * zor_poly vg z w"
- apply (rule zor_poly_times[OF f_iso _ f_ness,of vg])
- subgoal unfolding vg_def using isolated_singularity_at_inverse[OF g_iso g_ness] .
- subgoal unfolding vg_def using not_essential_inverse[OF g_ness g_iso] .
- subgoal unfolding vg_def using fg_nconst by (auto elim!:frequently_elim1)
- done
- then show "\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w / g w) z w = zor_poly f z w / zor_poly g z w"
- using zor_poly_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def
- apply eventually_elim
- by (auto simp add:field_simps)
-qed
-
-lemma zorder_exist_zero:
- fixes f::"complex \<Rightarrow> complex" and z::complex
- defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z"
- assumes holo: "f holomorphic_on s" and
- "open s" "connected s" "z\<in>s"
- and non_const: "\<exists>w\<in>s. f w \<noteq> 0"
- shows "(if f z=0 then n > 0 else n=0) \<and> (\<exists>r. r>0 \<and> cball z r \<subseteq> s \<and> g holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r. f w = g w * (w-z) ^ nat n \<and> g w \<noteq>0))"
-proof -
- obtain r where "g z \<noteq> 0" and r: "r>0" "cball z r \<subseteq> s" "g holomorphic_on cball z r"
- "(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
- proof -
- have "g z \<noteq> 0 \<and> (\<exists>r>0. g holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0))"
- proof (rule zorder_exist[of f z,folded g_def n_def])
- show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
- using holo assms(4,6)
- by (meson Diff_subset open_ball analytic_on_holomorphic holomorphic_on_subset openE)
- show "not_essential f z" unfolding not_essential_def
- using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on
- by fastforce
- have "\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w\<in>s"
- proof -
- obtain w where "w\<in>s" "f w\<noteq>0" using non_const by auto
- then show ?thesis
- by (rule non_zero_neighbour_alt[OF holo \<open>open s\<close> \<open>connected s\<close> \<open>z\<in>s\<close>])
- qed
- then show "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
- apply (elim eventually_frequentlyE)
- by auto
- qed
- then obtain r1 where "g z \<noteq> 0" "r1>0" and r1:"g holomorphic_on cball z r1"
- "(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
- by auto
- obtain r2 where r2: "r2>0" "cball z r2 \<subseteq> s"
- using assms(4,6) open_contains_cball_eq by blast
- define r3 where "r3=min r1 r2"
- have "r3>0" "cball z r3 \<subseteq> s" using \<open>r1>0\<close> r2 unfolding r3_def by auto
- moreover have "g holomorphic_on cball z r3"
- using r1(1) unfolding r3_def by auto
- moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
- using r1(2) unfolding r3_def by auto
- ultimately show ?thesis using that[of r3] \<open>g z\<noteq>0\<close> by auto
- qed
-
- have if_0:"if f z=0 then n > 0 else n=0"
- proof -
- have "f\<midarrow> z \<rightarrow> f z"
- by (metis assms(4,6,7) at_within_open continuous_on holo holomorphic_on_imp_continuous_on)
- then have "(\<lambda>w. g w * (w - z) powr of_int n) \<midarrow>z\<rightarrow> f z"
- apply (elim Lim_transform_within_open[where s="ball z r"])
- using r by auto
- moreover have "g \<midarrow>z\<rightarrow>g z"
- by (metis (mono_tags, lifting) open_ball at_within_open_subset
- ball_subset_cball centre_in_ball continuous_on holomorphic_on_imp_continuous_on r(1,3) subsetCE)
- ultimately have "(\<lambda>w. (g w * (w - z) powr of_int n) / g w) \<midarrow>z\<rightarrow> f z/g z"
- apply (rule_tac tendsto_divide)
- using \<open>g z\<noteq>0\<close> by auto
- then have powr_tendsto:"(\<lambda>w. (w - z) powr of_int n) \<midarrow>z\<rightarrow> f z/g z"
- apply (elim Lim_transform_within_open[where s="ball z r"])
- using r by auto
-
- have ?thesis when "n\<ge>0" "f z=0"
- proof -
- have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> f z/g z"
- using powr_tendsto
- apply (elim Lim_transform_within[where d=r])
- by (auto simp add: powr_of_int \<open>n\<ge>0\<close> \<open>r>0\<close>)
- then have *:"(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 0" using \<open>f z=0\<close> by simp
- moreover have False when "n=0"
- proof -
- have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 1"
- using \<open>n=0\<close> by auto
- then show False using * using LIM_unique zero_neq_one by blast
- qed
- ultimately show ?thesis using that by fastforce
- qed
- moreover have ?thesis when "n\<ge>0" "f z\<noteq>0"
- proof -
- have False when "n>0"
- proof -
- have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> f z/g z"
- using powr_tendsto
- apply (elim Lim_transform_within[where d=r])
- by (auto simp add: powr_of_int \<open>n\<ge>0\<close> \<open>r>0\<close>)
- moreover have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 0"
- using \<open>n>0\<close> by (auto intro!:tendsto_eq_intros)
- ultimately show False using \<open>f z\<noteq>0\<close> \<open>g z\<noteq>0\<close> using LIM_unique divide_eq_0_iff by blast
- qed
- then show ?thesis using that by force
- qed
- moreover have False when "n<0"
- proof -
- have "(\<lambda>w. inverse ((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> f z/g z"
- "(\<lambda>w.((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> 0"
- subgoal using powr_tendsto powr_of_int that
- by (elim Lim_transform_within_open[where s=UNIV],auto)
- subgoal using that by (auto intro!:tendsto_eq_intros)
- done
- from tendsto_mult[OF this,simplified]
- have "(\<lambda>x. inverse ((x - z) ^ nat (- n)) * (x - z) ^ nat (- n)) \<midarrow>z\<rightarrow> 0" .
- then have "(\<lambda>x. 1::complex) \<midarrow>z\<rightarrow> 0"
- by (elim Lim_transform_within_open[where s=UNIV],auto)
- then show False using LIM_const_eq by fastforce
- qed
- ultimately show ?thesis by fastforce
- qed
- moreover have "f w = g w * (w-z) ^ nat n \<and> g w \<noteq>0" when "w\<in>cball z r" for w
- proof (cases "w=z")
- case True
- then have "f \<midarrow>z\<rightarrow>f w"
- using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on by fastforce
- then have "(\<lambda>w. g w * (w-z) ^ nat n) \<midarrow>z\<rightarrow>f w"
- proof (elim Lim_transform_within[OF _ \<open>r>0\<close>])
- fix x assume "0 < dist x z" "dist x z < r"
- then have "x \<in> cball z r - {z}" "x\<noteq>z"
- unfolding cball_def by (auto simp add: dist_commute)
- then have "f x = g x * (x - z) powr of_int n"
- using r(4)[rule_format,of x] by simp
- also have "... = g x * (x - z) ^ nat n"
- apply (subst powr_of_int)
- using if_0 \<open>x\<noteq>z\<close> by (auto split:if_splits)
- finally show "f x = g x * (x - z) ^ nat n" .
- qed
- moreover have "(\<lambda>w. g w * (w-z) ^ nat n) \<midarrow>z\<rightarrow> g w * (w-z) ^ nat n"
- using True apply (auto intro!:tendsto_eq_intros)
- by (metis open_ball at_within_open_subset ball_subset_cball centre_in_ball
- continuous_on holomorphic_on_imp_continuous_on r(1) r(3) that)
- ultimately have "f w = g w * (w-z) ^ nat n" using LIM_unique by blast
- then show ?thesis using \<open>g z\<noteq>0\<close> True by auto
- next
- case False
- then have "f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0"
- using r(4) that by auto
- then show ?thesis using False if_0 powr_of_int by (auto split:if_splits)
- qed
- ultimately show ?thesis using r by auto
-qed
-
-lemma zorder_exist_pole:
- fixes f::"complex \<Rightarrow> complex" and z::complex
- defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z"
- assumes holo: "f holomorphic_on s-{z}" and
- "open s" "z\<in>s"
- and "is_pole f z"
- shows "n < 0 \<and> g z\<noteq>0 \<and> (\<exists>r. r>0 \<and> cball z r \<subseteq> s \<and> g holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w / (w-z) ^ nat (- n) \<and> g w \<noteq>0))"
-proof -
- obtain r where "g z \<noteq> 0" and r: "r>0" "cball z r \<subseteq> s" "g holomorphic_on cball z r"
- "(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
- proof -
- have "g z \<noteq> 0 \<and> (\<exists>r>0. g holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0))"
- proof (rule zorder_exist[of f z,folded g_def n_def])
- show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
- using holo assms(4,5)
- by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)
- show "not_essential f z" unfolding not_essential_def
- using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on
- by fastforce
- from non_zero_neighbour_pole[OF \<open>is_pole f z\<close>] show "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
- apply (elim eventually_frequentlyE)
- by auto
- qed
- then obtain r1 where "g z \<noteq> 0" "r1>0" and r1:"g holomorphic_on cball z r1"
- "(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
- by auto
- obtain r2 where r2: "r2>0" "cball z r2 \<subseteq> s"
- using assms(4,5) open_contains_cball_eq by metis
- define r3 where "r3=min r1 r2"
- have "r3>0" "cball z r3 \<subseteq> s" using \<open>r1>0\<close> r2 unfolding r3_def by auto
- moreover have "g holomorphic_on cball z r3"
- using r1(1) unfolding r3_def by auto
- moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
- using r1(2) unfolding r3_def by auto
- ultimately show ?thesis using that[of r3] \<open>g z\<noteq>0\<close> by auto
- qed
-
- have "n<0"
- proof (rule ccontr)
- assume " \<not> n < 0"
- define c where "c=(if n=0 then g z else 0)"
- have [simp]:"g \<midarrow>z\<rightarrow> g z"
- by (metis open_ball at_within_open ball_subset_cball centre_in_ball
- continuous_on holomorphic_on_imp_continuous_on holomorphic_on_subset r(1) r(3) )
- have "\<forall>\<^sub>F x in at z. f x = g x * (x - z) ^ nat n"
- unfolding eventually_at_topological
- apply (rule_tac exI[where x="ball z r"])
- using r powr_of_int \<open>\<not> n < 0\<close> by auto
- moreover have "(\<lambda>x. g x * (x - z) ^ nat n) \<midarrow>z\<rightarrow>c"
- proof (cases "n=0")
- case True
- then show ?thesis unfolding c_def by simp
- next
- case False
- then have "(\<lambda>x. (x - z) ^ nat n) \<midarrow>z\<rightarrow> 0" using \<open>\<not> n < 0\<close>
- by (auto intro!:tendsto_eq_intros)
- from tendsto_mult[OF _ this,of g "g z",simplified]
- show ?thesis unfolding c_def using False by simp
- qed
- ultimately have "f \<midarrow>z\<rightarrow>c" using tendsto_cong by fast
- then show False using \<open>is_pole f z\<close> at_neq_bot not_tendsto_and_filterlim_at_infinity
- unfolding is_pole_def by blast
- qed
- moreover have "\<forall>w\<in>cball z r - {z}. f w = g w / (w-z) ^ nat (- n) \<and> g w \<noteq>0"
- using r(4) \<open>n<0\<close> powr_of_int
- by (metis Diff_iff divide_inverse eq_iff_diff_eq_0 insert_iff linorder_not_le)
- ultimately show ?thesis using r(1-3) \<open>g z\<noteq>0\<close> by auto
-qed
-
-lemma zorder_eqI:
- assumes "open s" "z \<in> s" "g holomorphic_on s" "g z \<noteq> 0"
- assumes fg_eq:"\<And>w. \<lbrakk>w \<in> s;w\<noteq>z\<rbrakk> \<Longrightarrow> f w = g w * (w - z) powr n"
- shows "zorder f z = n"
-proof -
- have "continuous_on s g" by (rule holomorphic_on_imp_continuous_on) fact
- moreover have "open (-{0::complex})" by auto
- ultimately have "open ((g -` (-{0})) \<inter> s)"
- unfolding continuous_on_open_vimage[OF \<open>open s\<close>] by blast
- moreover from assms have "z \<in> (g -` (-{0})) \<inter> s" by auto
- ultimately obtain r where r: "r > 0" "cball z r \<subseteq> s \<inter> (g -` (-{0}))"
- unfolding open_contains_cball by blast
-
- let ?gg= "(\<lambda>w. g w * (w - z) powr n)"
- define P where "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
- have "P n g r"
- unfolding P_def using r assms(3,4,5) by auto
- then have "\<exists>g r. P n g r" by auto
- moreover have unique: "\<exists>!n. \<exists>g r. P n g r" unfolding P_def
- proof (rule holomorphic_factor_puncture)
- have "ball z r-{z} \<subseteq> s" using r using ball_subset_cball by blast
- then have "?gg holomorphic_on ball z r-{z}"
- using \<open>g holomorphic_on s\<close> r by (auto intro!: holomorphic_intros)
- then have "f holomorphic_on ball z r - {z}"
- apply (elim holomorphic_transform)
- using fg_eq \<open>ball z r-{z} \<subseteq> s\<close> by auto
- then show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
- using analytic_on_open open_delete r(1) by blast
- next
- have "not_essential ?gg z"
- proof (intro singularity_intros)
- show "not_essential g z"
- by (meson \<open>continuous_on s g\<close> assms(1) assms(2) continuous_on_eq_continuous_at
- isCont_def not_essential_def)
- show " \<forall>\<^sub>F w in at z. w - z \<noteq> 0" by (simp add: eventually_at_filter)
- then show "LIM w at z. w - z :> at 0"
- unfolding filterlim_at by (auto intro:tendsto_eq_intros)
- show "isolated_singularity_at g z"
- by (meson Diff_subset open_ball analytic_on_holomorphic
- assms(1,2,3) holomorphic_on_subset isolated_singularity_at_def openE)
- qed
- then show "not_essential f z"
- apply (elim not_essential_transform)
- unfolding eventually_at using assms(1,2) assms(5)[symmetric]
- by (metis dist_commute mem_ball openE subsetCE)
- show "\<exists>\<^sub>F w in at z. f w \<noteq> 0" unfolding frequently_at
- proof (rule,rule)
- fix d::real assume "0 < d"
- define z' where "z'=z+min d r / 2"
- have "z' \<noteq> z" " dist z' z < d "
- unfolding z'_def using \<open>d>0\<close> \<open>r>0\<close>
- by (auto simp add:dist_norm)
- moreover have "f z' \<noteq> 0"
- proof (subst fg_eq[OF _ \<open>z'\<noteq>z\<close>])
- have "z' \<in> cball z r" unfolding z'_def using \<open>r>0\<close> \<open>d>0\<close> by (auto simp add:dist_norm)
- then show " z' \<in> s" using r(2) by blast
- show "g z' * (z' - z) powr of_int n \<noteq> 0"
- using P_def \<open>P n g r\<close> \<open>z' \<in> cball z r\<close> calculation(1) by auto
- qed
- ultimately show "\<exists>x\<in>UNIV. x \<noteq> z \<and> dist x z < d \<and> f x \<noteq> 0" by auto
- qed
- qed
- ultimately have "(THE n. \<exists>g r. P n g r) = n"
- by (rule_tac the1_equality)
- then show ?thesis unfolding zorder_def P_def by blast
-qed
-
-lemma residue_pole_order:
- fixes f::"complex \<Rightarrow> complex" and z::complex
- defines "n \<equiv> nat (- zorder f z)" and "h \<equiv> zor_poly f z"
- assumes f_iso:"isolated_singularity_at f z"
- and pole:"is_pole f z"
- shows "residue f z = ((deriv ^^ (n - 1)) h z / fact (n-1))"
-proof -
- define g where "g \<equiv> \<lambda>x. if x=z then 0 else inverse (f x)"
- obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}"
- using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast
- obtain r where "0 < n" "0 < r" and r_cball:"cball z r \<subseteq> ball z e" and h_holo: "h holomorphic_on cball z r"
- and h_divide:"(\<forall>w\<in>cball z r. (w\<noteq>z \<longrightarrow> f w = h w / (w - z) ^ n) \<and> h w \<noteq> 0)"
- proof -
- obtain r where r:"zorder f z < 0" "h z \<noteq> 0" "r>0" "cball z r \<subseteq> ball z e" "h holomorphic_on cball z r"
- "(\<forall>w\<in>cball z r - {z}. f w = h w / (w - z) ^ n \<and> h w \<noteq> 0)"
- using zorder_exist_pole[OF f_holo,simplified,OF \<open>is_pole f z\<close>,folded n_def h_def] by auto
- have "n>0" using \<open>zorder f z < 0\<close> unfolding n_def by simp
- moreover have "(\<forall>w\<in>cball z r. (w\<noteq>z \<longrightarrow> f w = h w / (w - z) ^ n) \<and> h w \<noteq> 0)"
- using \<open>h z\<noteq>0\<close> r(6) by blast
- ultimately show ?thesis using r(3,4,5) that by blast
- qed
- have r_nonzero:"\<And>w. w \<in> ball z r - {z} \<Longrightarrow> f w \<noteq> 0"
- using h_divide by simp
- define c where "c \<equiv> 2 * pi * \<i>"
- define der_f where "der_f \<equiv> ((deriv ^^ (n - 1)) h z / fact (n-1))"
- define h' where "h' \<equiv> \<lambda>u. h u / (u - z) ^ n"
- have "(h' has_contour_integral c / fact (n - 1) * (deriv ^^ (n - 1)) h z) (circlepath z r)"
- unfolding h'_def
- proof (rule Cauchy_has_contour_integral_higher_derivative_circlepath[of z r h z "n-1",
- folded c_def Suc_pred'[OF \<open>n>0\<close>]])
- show "continuous_on (cball z r) h" using holomorphic_on_imp_continuous_on h_holo by simp
- show "h holomorphic_on ball z r" using h_holo by auto
- show " z \<in> ball z r" using \<open>r>0\<close> by auto
- qed
- then have "(h' has_contour_integral c * der_f) (circlepath z r)" unfolding der_f_def by auto
- then have "(f has_contour_integral c * der_f) (circlepath z r)"
- proof (elim has_contour_integral_eq)
- fix x assume "x \<in> path_image (circlepath z r)"
- hence "x\<in>cball z r - {z}" using \<open>r>0\<close> by auto
- then show "h' x = f x" using h_divide unfolding h'_def by auto
- qed
- moreover have "(f has_contour_integral c * residue f z) (circlepath z r)"
- using base_residue[of \<open>ball z e\<close> z,simplified,OF \<open>r>0\<close> f_holo r_cball,folded c_def]
- unfolding c_def by simp
- ultimately have "c * der_f = c * residue f z" using has_contour_integral_unique by blast
- hence "der_f = residue f z" unfolding c_def by auto
- thus ?thesis unfolding der_f_def by auto
-qed
-
-lemma simple_zeroI:
- assumes "open s" "z \<in> s" "g holomorphic_on s" "g z \<noteq> 0"
- assumes "\<And>w. w \<in> s \<Longrightarrow> f w = g w * (w - z)"
- shows "zorder f z = 1"
- using assms(1-4) by (rule zorder_eqI) (use assms(5) in auto)
-
-lemma higher_deriv_power:
- shows "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) w =
- pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)"
-proof (induction j arbitrary: w)
- case 0
- thus ?case by auto
-next
- case (Suc j w)
- have "(deriv ^^ Suc j) (\<lambda>w. (w - z) ^ n) w = deriv ((deriv ^^ j) (\<lambda>w. (w - z) ^ n)) w"
- by simp
- also have "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) =
- (\<lambda>w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))"
- using Suc by (intro Suc.IH ext)
- also {
- have "(\<dots> has_field_derivative of_nat (n - j) *
- pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - Suc j)) (at w)"
- using Suc.prems by (auto intro!: derivative_eq_intros)
- also have "of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j =
- pochhammer (of_nat (Suc n - Suc j)) (Suc j)"
- by (cases "Suc j \<le> n", subst pochhammer_rec)
- (insert Suc.prems, simp_all add: algebra_simps Suc_diff_le pochhammer_0_left)
- finally have "deriv (\<lambda>w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) w =
- \<dots> * (w - z) ^ (n - Suc j)"
- by (rule DERIV_imp_deriv)
- }
- finally show ?case .
-qed
-
-lemma zorder_zero_eqI:
- assumes f_holo:"f holomorphic_on s" and "open s" "z \<in> s"
- assumes zero: "\<And>i. i < nat n \<Longrightarrow> (deriv ^^ i) f z = 0"
- assumes nz: "(deriv ^^ nat n) f z \<noteq> 0" and "n\<ge>0"
- shows "zorder f z = n"
-proof -
- obtain r where [simp]:"r>0" and "ball z r \<subseteq> s"
- using \<open>open s\<close> \<open>z\<in>s\<close> openE by blast
- have nz':"\<exists>w\<in>ball z r. f w \<noteq> 0"
- proof (rule ccontr)
- assume "\<not> (\<exists>w\<in>ball z r. f w \<noteq> 0)"
- then have "eventually (\<lambda>u. f u = 0) (nhds z)"
- using \<open>r>0\<close> unfolding eventually_nhds
- apply (rule_tac x="ball z r" in exI)
- by auto
- then have "(deriv ^^ nat n) f z = (deriv ^^ nat n) (\<lambda>_. 0) z"
- by (intro higher_deriv_cong_ev) auto
- also have "(deriv ^^ nat n) (\<lambda>_. 0) z = 0"
- by (induction n) simp_all
- finally show False using nz by contradiction
- qed
-
- define zn g where "zn = zorder f z" and "g = zor_poly f z"
- obtain e where e_if:"if f z = 0 then 0 < zn else zn = 0" and
- [simp]:"e>0" and "cball z e \<subseteq> ball z r" and
- g_holo:"g holomorphic_on cball z e" and
- e_fac:"(\<forall>w\<in>cball z e. f w = g w * (w - z) ^ nat zn \<and> g w \<noteq> 0)"
- proof -
- have "f holomorphic_on ball z r"
- using f_holo \<open>ball z r \<subseteq> s\<close> by auto
- from that zorder_exist_zero[of f "ball z r" z,simplified,OF this nz',folded zn_def g_def]
- show ?thesis by blast
- qed
- from this(1,2,5) have "zn\<ge>0" "g z\<noteq>0"
- subgoal by (auto split:if_splits)
- subgoal using \<open>0 < e\<close> ball_subset_cball centre_in_ball e_fac by blast
- done
-
- define A where "A = (\<lambda>i. of_nat (i choose (nat zn)) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z)"
- have deriv_A:"(deriv ^^ i) f z = (if zn \<le> int i then A i else 0)" for i
- proof -
- have "eventually (\<lambda>w. w \<in> ball z e) (nhds z)"
- using \<open>cball z e \<subseteq> ball z r\<close> \<open>e>0\<close> by (intro eventually_nhds_in_open) auto
- hence "eventually (\<lambda>w. f w = (w - z) ^ (nat zn) * g w) (nhds z)"
- apply eventually_elim
- by (use e_fac in auto)
- hence "(deriv ^^ i) f z = (deriv ^^ i) (\<lambda>w. (w - z) ^ nat zn * g w) z"
- by (intro higher_deriv_cong_ev) auto
- also have "\<dots> = (\<Sum>j=0..i. of_nat (i choose j) *
- (deriv ^^ j) (\<lambda>w. (w - z) ^ nat zn) z * (deriv ^^ (i - j)) g z)"
- using g_holo \<open>e>0\<close>
- by (intro higher_deriv_mult[of _ "ball z e"]) (auto intro!: holomorphic_intros)
- also have "\<dots> = (\<Sum>j=0..i. if j = nat zn then
- of_nat (i choose nat zn) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z else 0)"
- proof (intro sum.cong refl, goal_cases)
- case (1 j)
- have "(deriv ^^ j) (\<lambda>w. (w - z) ^ nat zn) z =
- pochhammer (of_nat (Suc (nat zn) - j)) j * 0 ^ (nat zn - j)"
- by (subst higher_deriv_power) auto
- also have "\<dots> = (if j = nat zn then fact j else 0)"
- by (auto simp: not_less pochhammer_0_left pochhammer_fact)
- also have "of_nat (i choose j) * \<dots> * (deriv ^^ (i - j)) g z =
- (if j = nat zn then of_nat (i choose (nat zn)) * fact (nat zn)
- * (deriv ^^ (i - nat zn)) g z else 0)"
- by simp
- finally show ?case .
- qed
- also have "\<dots> = (if i \<ge> zn then A i else 0)"
- by (auto simp: A_def)
- finally show "(deriv ^^ i) f z = \<dots>" .
- qed
-
- have False when "n<zn"
- proof -
- have "(deriv ^^ nat n) f z = 0"
- using deriv_A[of "nat n"] that \<open>n\<ge>0\<close> by auto
- with nz show False by auto
- qed
- moreover have "n\<le>zn"
- proof -
- have "g z \<noteq> 0" using e_fac[rule_format,of z] \<open>e>0\<close> by simp
- then have "(deriv ^^ nat zn) f z \<noteq> 0"
- using deriv_A[of "nat zn"] by(auto simp add:A_def)
- then have "nat zn \<ge> nat n" using zero[of "nat zn"] by linarith
- moreover have "zn\<ge>0" using e_if by (auto split:if_splits)
- ultimately show ?thesis using nat_le_eq_zle by blast
- qed
- ultimately show ?thesis unfolding zn_def by fastforce
-qed
-
-lemma
- assumes "eventually (\<lambda>z. f z = g z) (at z)" "z = z'"
- shows zorder_cong:"zorder f z = zorder g z'" and zor_poly_cong:"zor_poly f z = zor_poly g z'"
-proof -
- define P where "P = (\<lambda>ff n h r. 0 < r \<and> h holomorphic_on cball z r \<and> h z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. ff w = h w * (w-z) powr (of_int n) \<and> h w\<noteq>0))"
- have "(\<exists>r. P f n h r) = (\<exists>r. P g n h r)" for n h
- proof -
- have *: "\<exists>r. P g n h r" if "\<exists>r. P f n h r" and "eventually (\<lambda>x. f x = g x) (at z)" for f g
- proof -
- from that(1) obtain r1 where r1_P:"P f n h r1" by auto
- from that(2) obtain r2 where "r2>0" and r2_dist:"\<forall>x. x \<noteq> z \<and> dist x z \<le> r2 \<longrightarrow> f x = g x"
- unfolding eventually_at_le by auto
- define r where "r=min r1 r2"
- have "r>0" "h z\<noteq>0" using r1_P \<open>r2>0\<close> unfolding r_def P_def by auto
- moreover have "h holomorphic_on cball z r"
- using r1_P unfolding P_def r_def by auto
- moreover have "g w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0" when "w\<in>cball z r - {z}" for w
- proof -
- have "f w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0"
- using r1_P that unfolding P_def r_def by auto
- moreover have "f w=g w" using r2_dist[rule_format,of w] that unfolding r_def
- by (simp add: dist_commute)
- ultimately show ?thesis by simp
- qed
- ultimately show ?thesis unfolding P_def by auto
- qed
- from assms have eq': "eventually (\<lambda>z. g z = f z) (at z)"
- by (simp add: eq_commute)
- show ?thesis
- by (rule iffI[OF *[OF _ assms(1)] *[OF _ eq']])
- qed
- then show "zorder f z = zorder g z'" "zor_poly f z = zor_poly g z'"
- using \<open>z=z'\<close> unfolding P_def zorder_def zor_poly_def by auto
-qed
-
-lemma zorder_nonzero_div_power:
- assumes "open s" "z \<in> s" "f holomorphic_on s" "f z \<noteq> 0" "n > 0"
- shows "zorder (\<lambda>w. f w / (w - z) ^ n) z = - n"
- apply (rule zorder_eqI[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>f holomorphic_on s\<close> \<open>f z\<noteq>0\<close>])
- apply (subst powr_of_int)
- using \<open>n>0\<close> by (auto simp add:field_simps)
-
-lemma zor_poly_eq:
- assumes "isolated_singularity_at f z" "not_essential f z" "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
- shows "eventually (\<lambda>w. zor_poly f z w = f w * (w - z) powr - zorder f z) (at z)"
-proof -
- obtain r where r:"r>0"
- "(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w * (w - z) powr of_int (zorder f z))"
- using zorder_exist[OF assms] by blast
- then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w * (w - z) powr - zorder f z"
- by (auto simp: field_simps powr_minus)
- have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
- using r eventually_at_ball'[of r z UNIV] by auto
- thus ?thesis by eventually_elim (insert *, auto)
-qed
-
-lemma zor_poly_zero_eq:
- assumes "f holomorphic_on s" "open s" "connected s" "z \<in> s" "\<exists>w\<in>s. f w \<noteq> 0"
- shows "eventually (\<lambda>w. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)) (at z)"
-proof -
- obtain r where r:"r>0"
- "(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w * (w - z) ^ nat (zorder f z))"
- using zorder_exist_zero[OF assms] by auto
- then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)"
- by (auto simp: field_simps powr_minus)
- have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
- using r eventually_at_ball'[of r z UNIV] by auto
- thus ?thesis by eventually_elim (insert *, auto)
-qed
-
-lemma zor_poly_pole_eq:
- assumes f_iso:"isolated_singularity_at f z" "is_pole f z"
- shows "eventually (\<lambda>w. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)) (at z)"
-proof -
- obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}"
- using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast
- obtain r where r:"r>0"
- "(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w / (w - z) ^ nat (- zorder f z))"
- using zorder_exist_pole[OF f_holo,simplified,OF \<open>is_pole f z\<close>] by auto
- then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)"
- by (auto simp: field_simps)
- have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
- using r eventually_at_ball'[of r z UNIV] by auto
- thus ?thesis by eventually_elim (insert *, auto)
-qed
-
-lemma zor_poly_eqI:
- fixes f :: "complex \<Rightarrow> complex" and z0 :: complex
- defines "n \<equiv> zorder f z0"
- assumes "isolated_singularity_at f z0" "not_essential f z0" "\<exists>\<^sub>F w in at z0. f w \<noteq> 0"
- assumes lim: "((\<lambda>x. f (g x) * (g x - z0) powr - n) \<longlongrightarrow> c) F"
- assumes g: "filterlim g (at z0) F" and "F \<noteq> bot"
- shows "zor_poly f z0 z0 = c"
-proof -
- from zorder_exist[OF assms(2-4)] obtain r where
- r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r"
- "\<And>w. w \<in> cball z0 r - {z0} \<Longrightarrow> f w = zor_poly f z0 w * (w - z0) powr n"
- unfolding n_def by blast
- from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)"
- using eventually_at_ball'[of r z0 UNIV] by auto
- hence "eventually (\<lambda>w. zor_poly f z0 w = f w * (w - z0) powr - n) (at z0)"
- by eventually_elim (insert r, auto simp: field_simps powr_minus)
- moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
- using r by (intro holomorphic_on_imp_continuous_on) auto
- with r(1,2) have "isCont (zor_poly f z0) z0"
- by (auto simp: continuous_on_eq_continuous_at)
- hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
- unfolding isCont_def .
- ultimately have "((\<lambda>w. f w * (w - z0) powr - n) \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
- by (blast intro: Lim_transform_eventually)
- hence "((\<lambda>x. f (g x) * (g x - z0) powr - n) \<longlongrightarrow> zor_poly f z0 z0) F"
- by (rule filterlim_compose[OF _ g])
- from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis .
-qed
-
-lemma zor_poly_zero_eqI:
- fixes f :: "complex \<Rightarrow> complex" and z0 :: complex
- defines "n \<equiv> zorder f z0"
- assumes "f holomorphic_on A" "open A" "connected A" "z0 \<in> A" "\<exists>z\<in>A. f z \<noteq> 0"
- assumes lim: "((\<lambda>x. f (g x) / (g x - z0) ^ nat n) \<longlongrightarrow> c) F"
- assumes g: "filterlim g (at z0) F" and "F \<noteq> bot"
- shows "zor_poly f z0 z0 = c"
-proof -
- from zorder_exist_zero[OF assms(2-6)] obtain r where
- r: "r > 0" "cball z0 r \<subseteq> A" "zor_poly f z0 holomorphic_on cball z0 r"
- "\<And>w. w \<in> cball z0 r \<Longrightarrow> f w = zor_poly f z0 w * (w - z0) ^ nat n"
- unfolding n_def by blast
- from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)"
- using eventually_at_ball'[of r z0 UNIV] by auto
- hence "eventually (\<lambda>w. zor_poly f z0 w = f w / (w - z0) ^ nat n) (at z0)"
- by eventually_elim (insert r, auto simp: field_simps)
- moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
- using r by (intro holomorphic_on_imp_continuous_on) auto
- with r(1,2) have "isCont (zor_poly f z0) z0"
- by (auto simp: continuous_on_eq_continuous_at)
- hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
- unfolding isCont_def .
- ultimately have "((\<lambda>w. f w / (w - z0) ^ nat n) \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
- by (blast intro: Lim_transform_eventually)
- hence "((\<lambda>x. f (g x) / (g x - z0) ^ nat n) \<longlongrightarrow> zor_poly f z0 z0) F"
- by (rule filterlim_compose[OF _ g])
- from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis .
-qed
-
-lemma zor_poly_pole_eqI:
- fixes f :: "complex \<Rightarrow> complex" and z0 :: complex
- defines "n \<equiv> zorder f z0"
- assumes f_iso:"isolated_singularity_at f z0" and "is_pole f z0"
- assumes lim: "((\<lambda>x. f (g x) * (g x - z0) ^ nat (-n)) \<longlongrightarrow> c) F"
- assumes g: "filterlim g (at z0) F" and "F \<noteq> bot"
- shows "zor_poly f z0 z0 = c"
-proof -
- obtain r where r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r"
- proof -
- have "\<exists>\<^sub>F w in at z0. f w \<noteq> 0"
- using non_zero_neighbour_pole[OF \<open>is_pole f z0\<close>] by (auto elim:eventually_frequentlyE)
- moreover have "not_essential f z0" unfolding not_essential_def using \<open>is_pole f z0\<close> by simp
- ultimately show ?thesis using that zorder_exist[OF f_iso,folded n_def] by auto
- qed
- from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)"
- using eventually_at_ball'[of r z0 UNIV] by auto
- have "eventually (\<lambda>w. zor_poly f z0 w = f w * (w - z0) ^ nat (-n)) (at z0)"
- using zor_poly_pole_eq[OF f_iso \<open>is_pole f z0\<close>] unfolding n_def .
- moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
- using r by (intro holomorphic_on_imp_continuous_on) auto
- with r(1,2) have "isCont (zor_poly f z0) z0"
- by (auto simp: continuous_on_eq_continuous_at)
- hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
- unfolding isCont_def .
- ultimately have "((\<lambda>w. f w * (w - z0) ^ nat (-n)) \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
- by (blast intro: Lim_transform_eventually)
- hence "((\<lambda>x. f (g x) * (g x - z0) ^ nat (-n)) \<longlongrightarrow> zor_poly f z0 z0) F"
- by (rule filterlim_compose[OF _ g])
- from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis .
-qed
-
-lemma residue_simple_pole:
- assumes "isolated_singularity_at f z0"
- assumes "is_pole f z0" "zorder f z0 = - 1"
- shows "residue f z0 = zor_poly f z0 z0"
- using assms by (subst residue_pole_order) simp_all
-
-lemma residue_simple_pole_limit:
- assumes "isolated_singularity_at f z0"
- assumes "is_pole f z0" "zorder f z0 = - 1"
- assumes "((\<lambda>x. f (g x) * (g x - z0)) \<longlongrightarrow> c) F"
- assumes "filterlim g (at z0) F" "F \<noteq> bot"
- shows "residue f z0 = c"
-proof -
- have "residue f z0 = zor_poly f z0 z0"
- by (rule residue_simple_pole assms)+
- also have "\<dots> = c"
- apply (rule zor_poly_pole_eqI)
- using assms by auto
- finally show ?thesis .
-qed
-
-lemma lhopital_complex_simple:
- assumes "(f has_field_derivative f') (at z)"
- assumes "(g has_field_derivative g') (at z)"
- assumes "f z = 0" "g z = 0" "g' \<noteq> 0" "f' / g' = c"
- shows "((\<lambda>w. f w / g w) \<longlongrightarrow> c) (at z)"
-proof -
- have "eventually (\<lambda>w. w \<noteq> z) (at z)"
- by (auto simp: eventually_at_filter)
- hence "eventually (\<lambda>w. ((f w - f z) / (w - z)) / ((g w - g z) / (w - z)) = f w / g w) (at z)"
- by eventually_elim (simp add: assms field_split_simps)
- moreover have "((\<lambda>w. ((f w - f z) / (w - z)) / ((g w - g z) / (w - z))) \<longlongrightarrow> f' / g') (at z)"
- by (intro tendsto_divide has_field_derivativeD assms)
- ultimately have "((\<lambda>w. f w / g w) \<longlongrightarrow> f' / g') (at z)"
- by (blast intro: Lim_transform_eventually)
- with assms show ?thesis by simp
-qed
-
-lemma
- assumes f_holo:"f holomorphic_on s" and g_holo:"g holomorphic_on s"
- and "open s" "connected s" "z \<in> s"
- assumes g_deriv:"(g has_field_derivative g') (at z)"
- assumes "f z \<noteq> 0" "g z = 0" "g' \<noteq> 0"
- shows porder_simple_pole_deriv: "zorder (\<lambda>w. f w / g w) z = - 1"
- and residue_simple_pole_deriv: "residue (\<lambda>w. f w / g w) z = f z / g'"
-proof -
- have [simp]:"isolated_singularity_at f z" "isolated_singularity_at g z"
- using isolated_singularity_at_holomorphic[OF _ \<open>open s\<close> \<open>z\<in>s\<close>] f_holo g_holo
- by (meson Diff_subset holomorphic_on_subset)+
- have [simp]:"not_essential f z" "not_essential g z"
- unfolding not_essential_def using f_holo g_holo assms(3,5)
- by (meson continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on)+
- have g_nconst:"\<exists>\<^sub>F w in at z. g w \<noteq>0 "
- proof (rule ccontr)
- assume "\<not> (\<exists>\<^sub>F w in at z. g w \<noteq> 0)"
- then have "\<forall>\<^sub>F w in nhds z. g w = 0"
- unfolding eventually_at eventually_nhds frequently_at using \<open>g z = 0\<close>
- by (metis open_ball UNIV_I centre_in_ball dist_commute mem_ball)
- then have "deriv g z = deriv (\<lambda>_. 0) z"
- by (intro deriv_cong_ev) auto
- then have "deriv g z = 0" by auto
- then have "g' = 0" using g_deriv DERIV_imp_deriv by blast
- then show False using \<open>g'\<noteq>0\<close> by auto
- qed
-
- have "zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z"
- proof -
- have "\<forall>\<^sub>F w in at z. f w \<noteq>0 \<and> w\<in>s"
- apply (rule non_zero_neighbour_alt)
- using assms by auto
- with g_nconst have "\<exists>\<^sub>F w in at z. f w * g w \<noteq> 0"
- by (elim frequently_rev_mp eventually_rev_mp,auto)
- then show ?thesis using zorder_divide[of f z g] by auto
- qed
- moreover have "zorder f z=0"
- apply (rule zorder_zero_eqI[OF f_holo \<open>open s\<close> \<open>z\<in>s\<close>])
- using \<open>f z\<noteq>0\<close> by auto
- moreover have "zorder g z=1"
- apply (rule zorder_zero_eqI[OF g_holo \<open>open s\<close> \<open>z\<in>s\<close>])
- subgoal using assms(8) by auto
- subgoal using DERIV_imp_deriv assms(9) g_deriv by auto
- subgoal by simp
- done
- ultimately show "zorder (\<lambda>w. f w / g w) z = - 1" by auto
-
- show "residue (\<lambda>w. f w / g w) z = f z / g'"
- proof (rule residue_simple_pole_limit[where g=id and F="at z",simplified])
- show "zorder (\<lambda>w. f w / g w) z = - 1" by fact
- show "isolated_singularity_at (\<lambda>w. f w / g w) z"
- by (auto intro: singularity_intros)
- show "is_pole (\<lambda>w. f w / g w) z"
- proof (rule is_pole_divide)
- have "\<forall>\<^sub>F x in at z. g x \<noteq> 0"
- apply (rule non_zero_neighbour)
- using g_nconst by auto
- moreover have "g \<midarrow>z\<rightarrow> 0"
- using DERIV_isCont assms(8) continuous_at g_deriv by force
- ultimately show "filterlim g (at 0) (at z)" unfolding filterlim_at by simp
- show "isCont f z"
- using assms(3,5) continuous_on_eq_continuous_at f_holo holomorphic_on_imp_continuous_on
- by auto
- show "f z \<noteq> 0" by fact
- qed
- show "filterlim id (at z) (at z)" by (simp add: filterlim_iff)
- have "((\<lambda>w. (f w * (w - z)) / g w) \<longlongrightarrow> f z / g') (at z)"
- proof (rule lhopital_complex_simple)
- show "((\<lambda>w. f w * (w - z)) has_field_derivative f z) (at z)"
- using assms by (auto intro!: derivative_eq_intros holomorphic_derivI[OF f_holo])
- show "(g has_field_derivative g') (at z)" by fact
- qed (insert assms, auto)
- then show "((\<lambda>w. (f w / g w) * (w - z)) \<longlongrightarrow> f z / g') (at z)"
- by (simp add: field_split_simps)
- qed
-qed
-
-subsection \<open>The argument principle\<close>
-
-theorem argument_principle:
- fixes f::"complex \<Rightarrow> complex" and poles s:: "complex set"
- defines "pz \<equiv> {w. f w = 0 \<or> w \<in> poles}" \<comment> \<open>\<^term>\<open>pz\<close> is the set of poles and zeros\<close>
- assumes "open s" and
- "connected s" and
- f_holo:"f holomorphic_on s-poles" and
- h_holo:"h holomorphic_on s" and
- "valid_path g" and
- loop:"pathfinish g = pathstart g" and
- path_img:"path_image g \<subseteq> s - pz" and
- homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0" and
- finite:"finite pz" and
- poles:"\<forall>p\<in>poles. is_pole f p"
- shows "contour_integral g (\<lambda>x. deriv f x * h x / f x) = 2 * pi * \<i> *
- (\<Sum>p\<in>pz. winding_number g p * h p * zorder f p)"
- (is "?L=?R")
-proof -
- define c where "c \<equiv> 2 * complex_of_real pi * \<i> "
- define ff where "ff \<equiv> (\<lambda>x. deriv f x * h x / f x)"
- define cont where "cont \<equiv> \<lambda>ff p e. (ff has_contour_integral c * zorder f p * h p ) (circlepath p e)"
- define avoid where "avoid \<equiv> \<lambda>p e. \<forall>w\<in>cball p e. w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz)"
-
- have "\<exists>e>0. avoid p e \<and> (p\<in>pz \<longrightarrow> cont ff p e)" when "p\<in>s" for p
- proof -
- obtain e1 where "e1>0" and e1_avoid:"avoid p e1"
- using finite_cball_avoid[OF \<open>open s\<close> finite] \<open>p\<in>s\<close> unfolding avoid_def by auto
- have "\<exists>e2>0. cball p e2 \<subseteq> ball p e1 \<and> cont ff p e2" when "p\<in>pz"
- proof -
- define po where "po \<equiv> zorder f p"
- define pp where "pp \<equiv> zor_poly f p"
- define f' where "f' \<equiv> \<lambda>w. pp w * (w - p) powr po"
- define ff' where "ff' \<equiv> (\<lambda>x. deriv f' x * h x / f' x)"
- obtain r where "pp p\<noteq>0" "r>0" and
- "r<e1" and
- pp_holo:"pp holomorphic_on cball p r" and
- pp_po:"(\<forall>w\<in>cball p r-{p}. f w = pp w * (w - p) powr po \<and> pp w \<noteq> 0)"
- proof -
- have "isolated_singularity_at f p"
- proof -
- have "f holomorphic_on ball p e1 - {p}"
- apply (intro holomorphic_on_subset[OF f_holo])
- using e1_avoid \<open>p\<in>pz\<close> unfolding avoid_def pz_def by force
- then show ?thesis unfolding isolated_singularity_at_def
- using \<open>e1>0\<close> analytic_on_open open_delete by blast
- qed
- moreover have "not_essential f p"
- proof (cases "is_pole f p")
- case True
- then show ?thesis unfolding not_essential_def by auto
- next
- case False
- then have "p\<in>s-poles" using \<open>p\<in>s\<close> poles unfolding pz_def by auto
- moreover have "open (s-poles)"
- using \<open>open s\<close>
- apply (elim open_Diff)
- apply (rule finite_imp_closed)
- using finite unfolding pz_def by simp
- ultimately have "isCont f p"
- using holomorphic_on_imp_continuous_on[OF f_holo] continuous_on_eq_continuous_at
- by auto
- then show ?thesis unfolding isCont_def not_essential_def by auto
- qed
- moreover have "\<exists>\<^sub>F w in at p. f w \<noteq> 0 "
- proof (rule ccontr)
- assume "\<not> (\<exists>\<^sub>F w in at p. f w \<noteq> 0)"
- then have "\<forall>\<^sub>F w in at p. f w= 0" unfolding frequently_def by auto
- then obtain rr where "rr>0" "\<forall>w\<in>ball p rr - {p}. f w =0"
- unfolding eventually_at by (auto simp add:dist_commute)
- then have "ball p rr - {p} \<subseteq> {w\<in>ball p rr-{p}. f w=0}" by blast
- moreover have "infinite (ball p rr - {p})" using \<open>rr>0\<close> using finite_imp_not_open by fastforce
- ultimately have "infinite {w\<in>ball p rr-{p}. f w=0}" using infinite_super by blast
- then have "infinite pz"
- unfolding pz_def infinite_super by auto
- then show False using \<open>finite pz\<close> by auto
- qed
- ultimately obtain r where "pp p \<noteq> 0" and r:"r>0" "pp holomorphic_on cball p r"
- "(\<forall>w\<in>cball p r - {p}. f w = pp w * (w - p) powr of_int po \<and> pp w \<noteq> 0)"
- using zorder_exist[of f p,folded po_def pp_def] by auto
- define r1 where "r1=min r e1 / 2"
- have "r1<e1" unfolding r1_def using \<open>e1>0\<close> \<open>r>0\<close> by auto
- moreover have "r1>0" "pp holomorphic_on cball p r1"
- "(\<forall>w\<in>cball p r1 - {p}. f w = pp w * (w - p) powr of_int po \<and> pp w \<noteq> 0)"
- unfolding r1_def using \<open>e1>0\<close> r by auto
- ultimately show ?thesis using that \<open>pp p\<noteq>0\<close> by auto
- qed
-
- define e2 where "e2 \<equiv> r/2"
- have "e2>0" using \<open>r>0\<close> unfolding e2_def by auto
- define anal where "anal \<equiv> \<lambda>w. deriv pp w * h w / pp w"
- define prin where "prin \<equiv> \<lambda>w. po * h w / (w - p)"
- have "((\<lambda>w. prin w + anal w) has_contour_integral c * po * h p) (circlepath p e2)"
- proof (rule has_contour_integral_add[of _ _ _ _ 0,simplified])
- have "ball p r \<subseteq> s"
- using \<open>r<e1\<close> avoid_def ball_subset_cball e1_avoid by (simp add: subset_eq)
- then have "cball p e2 \<subseteq> s"
- using \<open>r>0\<close> unfolding e2_def by auto
- then have "(\<lambda>w. po * h w) holomorphic_on cball p e2"
- using h_holo by (auto intro!: holomorphic_intros)
- then show "(prin has_contour_integral c * po * h p ) (circlepath p e2)"
- using Cauchy_integral_circlepath_simple[folded c_def, of "\<lambda>w. po * h w"] \<open>e2>0\<close>
- unfolding prin_def by (auto simp add: mult.assoc)
- have "anal holomorphic_on ball p r" unfolding anal_def
- using pp_holo h_holo pp_po \<open>ball p r \<subseteq> s\<close> \<open>pp p\<noteq>0\<close>
- by (auto intro!: holomorphic_intros)
- then show "(anal has_contour_integral 0) (circlepath p e2)"
- using e2_def \<open>r>0\<close>
- by (auto elim!: Cauchy_theorem_disc_simple)
- qed
- then have "cont ff' p e2" unfolding cont_def po_def
- proof (elim has_contour_integral_eq)
- fix w assume "w \<in> path_image (circlepath p e2)"
- then have "w\<in>ball p r" and "w\<noteq>p" unfolding e2_def using \<open>r>0\<close> by auto
- define wp where "wp \<equiv> w-p"
- have "wp\<noteq>0" and "pp w \<noteq>0"
- unfolding wp_def using \<open>w\<noteq>p\<close> \<open>w\<in>ball p r\<close> pp_po by auto
- moreover have der_f':"deriv f' w = po * pp w * (w-p) powr (po - 1) + deriv pp w * (w-p) powr po"
- proof (rule DERIV_imp_deriv)
- have "(pp has_field_derivative (deriv pp w)) (at w)"
- using DERIV_deriv_iff_has_field_derivative pp_holo \<open>w\<noteq>p\<close>
- by (meson open_ball \<open>w \<in> ball p r\<close> ball_subset_cball holomorphic_derivI holomorphic_on_subset)
- then show " (f' has_field_derivative of_int po * pp w * (w - p) powr of_int (po - 1)
- + deriv pp w * (w - p) powr of_int po) (at w)"
- unfolding f'_def using \<open>w\<noteq>p\<close>
- by (auto intro!: derivative_eq_intros DERIV_cong[OF has_field_derivative_powr_of_int])
- qed
- ultimately show "prin w + anal w = ff' w"
- unfolding ff'_def prin_def anal_def
- apply simp
- apply (unfold f'_def)
- apply (fold wp_def)
- apply (auto simp add:field_simps)
- by (metis (no_types, lifting) diff_add_cancel mult.commute powr_add powr_to_1)
- qed
- then have "cont ff p e2" unfolding cont_def
- proof (elim has_contour_integral_eq)
- fix w assume "w \<in> path_image (circlepath p e2)"
- then have "w\<in>ball p r" and "w\<noteq>p" unfolding e2_def using \<open>r>0\<close> by auto
- have "deriv f' w = deriv f w"
- proof (rule complex_derivative_transform_within_open[where s="ball p r - {p}"])
- show "f' holomorphic_on ball p r - {p}" unfolding f'_def using pp_holo
- by (auto intro!: holomorphic_intros)
- next
- have "ball p e1 - {p} \<subseteq> s - poles"
- using ball_subset_cball e1_avoid[unfolded avoid_def] unfolding pz_def
- by auto
- then have "ball p r - {p} \<subseteq> s - poles"
- apply (elim dual_order.trans)
- using \<open>r<e1\<close> by auto
- then show "f holomorphic_on ball p r - {p}" using f_holo
- by auto
- next
- show "open (ball p r - {p})" by auto
- show "w \<in> ball p r - {p}" using \<open>w\<in>ball p r\<close> \<open>w\<noteq>p\<close> by auto
- next
- fix x assume "x \<in> ball p r - {p}"
- then show "f' x = f x"
- using pp_po unfolding f'_def by auto
- qed
- moreover have " f' w = f w "
- using \<open>w \<in> ball p r\<close> ball_subset_cball subset_iff pp_po \<open>w\<noteq>p\<close>
- unfolding f'_def by auto
- ultimately show "ff' w = ff w"
- unfolding ff'_def ff_def by simp
- qed
- moreover have "cball p e2 \<subseteq> ball p e1"
- using \<open>0 < r\<close> \<open>r<e1\<close> e2_def by auto
- ultimately show ?thesis using \<open>e2>0\<close> by auto
- qed
- then obtain e2 where e2:"p\<in>pz \<longrightarrow> e2>0 \<and> cball p e2 \<subseteq> ball p e1 \<and> cont ff p e2"
- by auto
- define e4 where "e4 \<equiv> if p\<in>pz then e2 else e1"
- have "e4>0" using e2 \<open>e1>0\<close> unfolding e4_def by auto
- moreover have "avoid p e4" using e2 \<open>e1>0\<close> e1_avoid unfolding e4_def avoid_def by auto
- moreover have "p\<in>pz \<longrightarrow> cont ff p e4"
- by (auto simp add: e2 e4_def)
- ultimately show ?thesis by auto
- qed
- then obtain get_e where get_e:"\<forall>p\<in>s. get_e p>0 \<and> avoid p (get_e p)
- \<and> (p\<in>pz \<longrightarrow> cont ff p (get_e p))"
- by metis
- define ci where "ci \<equiv> \<lambda>p. contour_integral (circlepath p (get_e p)) ff"
- define w where "w \<equiv> \<lambda>p. winding_number g p"
- have "contour_integral g ff = (\<Sum>p\<in>pz. w p * ci p)" unfolding ci_def w_def
- proof (rule Cauchy_theorem_singularities[OF \<open>open s\<close> \<open>connected s\<close> finite _ \<open>valid_path g\<close> loop
- path_img homo])
- have "open (s - pz)" using open_Diff[OF _ finite_imp_closed[OF finite]] \<open>open s\<close> by auto
- then show "ff holomorphic_on s - pz" unfolding ff_def using f_holo h_holo
- by (auto intro!: holomorphic_intros simp add:pz_def)
- next
- show "\<forall>p\<in>s. 0 < get_e p \<and> (\<forall>w\<in>cball p (get_e p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz))"
- using get_e using avoid_def by blast
- qed
- also have "... = (\<Sum>p\<in>pz. c * w p * h p * zorder f p)"
- proof (rule sum.cong[of pz pz,simplified])
- fix p assume "p \<in> pz"
- show "w p * ci p = c * w p * h p * (zorder f p)"
- proof (cases "p\<in>s")
- assume "p \<in> s"
- have "ci p = c * h p * (zorder f p)" unfolding ci_def
- apply (rule contour_integral_unique)
- using get_e \<open>p\<in>s\<close> \<open>p\<in>pz\<close> unfolding cont_def by (metis mult.assoc mult.commute)
- thus ?thesis by auto
- next
- assume "p\<notin>s"
- then have "w p=0" using homo unfolding w_def by auto
- then show ?thesis by auto
- qed
- qed
- also have "... = c*(\<Sum>p\<in>pz. w p * h p * zorder f p)"
- unfolding sum_distrib_left by (simp add:algebra_simps)
- finally have "contour_integral g ff = c * (\<Sum>p\<in>pz. w p * h p * of_int (zorder f p))" .
- then show ?thesis unfolding ff_def c_def w_def by simp
-qed
-
-subsection \<open>Rouche's theorem \<close>
-
-theorem Rouche_theorem:
- fixes f g::"complex \<Rightarrow> complex" and s:: "complex set"
- defines "fg\<equiv>(\<lambda>p. f p + g p)"
- defines "zeros_fg\<equiv>{p. fg p = 0}" and "zeros_f\<equiv>{p. f p = 0}"
- assumes
- "open s" and "connected s" and
- "finite zeros_fg" and
- "finite zeros_f" and
- f_holo:"f holomorphic_on s" and
- g_holo:"g holomorphic_on s" and
- "valid_path \<gamma>" and
- loop:"pathfinish \<gamma> = pathstart \<gamma>" and
- path_img:"path_image \<gamma> \<subseteq> s " and
- path_less:"\<forall>z\<in>path_image \<gamma>. cmod(f z) > cmod(g z)" and
- homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number \<gamma> z = 0"
- shows "(\<Sum>p\<in>zeros_fg. winding_number \<gamma> p * zorder fg p)
- = (\<Sum>p\<in>zeros_f. winding_number \<gamma> p * zorder f p)"
-proof -
- have path_fg:"path_image \<gamma> \<subseteq> s - zeros_fg"
- proof -
- have False when "z\<in>path_image \<gamma>" and "f z + g z=0" for z
- proof -
- have "cmod (f z) > cmod (g z)" using \<open>z\<in>path_image \<gamma>\<close> path_less by auto
- moreover have "f z = - g z" using \<open>f z + g z =0\<close> by (simp add: eq_neg_iff_add_eq_0)
- then have "cmod (f z) = cmod (g z)" by auto
- ultimately show False by auto
- qed
- then show ?thesis unfolding zeros_fg_def fg_def using path_img by auto
- qed
- have path_f:"path_image \<gamma> \<subseteq> s - zeros_f"
- proof -
- have False when "z\<in>path_image \<gamma>" and "f z =0" for z
- proof -
- have "cmod (g z) < cmod (f z) " using \<open>z\<in>path_image \<gamma>\<close> path_less by auto
- then have "cmod (g z) < 0" using \<open>f z=0\<close> by auto
- then show False by auto
- qed
- then show ?thesis unfolding zeros_f_def using path_img by auto
- qed
- define w where "w \<equiv> \<lambda>p. winding_number \<gamma> p"
- define c where "c \<equiv> 2 * complex_of_real pi * \<i>"
- define h where "h \<equiv> \<lambda>p. g p / f p + 1"
- obtain spikes
- where "finite spikes" and spikes: "\<forall>x\<in>{0..1} - spikes. \<gamma> differentiable at x"
- using \<open>valid_path \<gamma>\<close>
- by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- have h_contour:"((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
- proof -
- have outside_img:"0 \<in> outside (path_image (h o \<gamma>))"
- proof -
- have "h p \<in> ball 1 1" when "p\<in>path_image \<gamma>" for p
- proof -
- have "cmod (g p/f p) <1" using path_less[rule_format,OF that]
- apply (cases "cmod (f p) = 0")
- by (auto simp add: norm_divide)
- then show ?thesis unfolding h_def by (auto simp add:dist_complex_def)
- qed
- then have "path_image (h o \<gamma>) \<subseteq> ball 1 1"
- by (simp add: image_subset_iff path_image_compose)
- moreover have " (0::complex) \<notin> ball 1 1" by (simp add: dist_norm)
- ultimately show "?thesis"
- using convex_in_outside[of "ball 1 1" 0] outside_mono by blast
- qed
- have valid_h:"valid_path (h \<circ> \<gamma>)"
- proof (rule valid_path_compose_holomorphic[OF \<open>valid_path \<gamma>\<close> _ _ path_f])
- show "h holomorphic_on s - zeros_f"
- unfolding h_def using f_holo g_holo
- by (auto intro!: holomorphic_intros simp add:zeros_f_def)
- next
- show "open (s - zeros_f)" using \<open>finite zeros_f\<close> \<open>open s\<close> finite_imp_closed
- by auto
- qed
- have "((\<lambda>z. 1/z) has_contour_integral 0) (h \<circ> \<gamma>)"
- proof -
- have "0 \<notin> path_image (h \<circ> \<gamma>)" using outside_img by (simp add: outside_def)
- then have "((\<lambda>z. 1/z) has_contour_integral c * winding_number (h \<circ> \<gamma>) 0) (h \<circ> \<gamma>)"
- using has_contour_integral_winding_number[of "h o \<gamma>" 0,simplified] valid_h
- unfolding c_def by auto
- moreover have "winding_number (h o \<gamma>) 0 = 0"
- proof -
- have "0 \<in> outside (path_image (h \<circ> \<gamma>))" using outside_img .
- moreover have "path (h o \<gamma>)"
- using valid_h by (simp add: valid_path_imp_path)
- moreover have "pathfinish (h o \<gamma>) = pathstart (h o \<gamma>)"
- by (simp add: loop pathfinish_compose pathstart_compose)
- ultimately show ?thesis using winding_number_zero_in_outside by auto
- qed
- ultimately show ?thesis by auto
- qed
- moreover have "vector_derivative (h \<circ> \<gamma>) (at x) = vector_derivative \<gamma> (at x) * deriv h (\<gamma> x)"
- when "x\<in>{0..1} - spikes" for x
- proof (rule vector_derivative_chain_at_general)
- show "\<gamma> differentiable at x" using that \<open>valid_path \<gamma>\<close> spikes by auto
- next
- define der where "der \<equiv> \<lambda>p. (deriv g p * f p - g p * deriv f p)/(f p * f p)"
- define t where "t \<equiv> \<gamma> x"
- have "f t\<noteq>0" unfolding zeros_f_def t_def
- by (metis DiffD1 image_eqI norm_not_less_zero norm_zero path_defs(4) path_less that)
- moreover have "t\<in>s"
- using contra_subsetD path_image_def path_fg t_def that by fastforce
- ultimately have "(h has_field_derivative der t) (at t)"
- unfolding h_def der_def using g_holo f_holo \<open>open s\<close>
- by (auto intro!: holomorphic_derivI derivative_eq_intros)
- then show "h field_differentiable at (\<gamma> x)"
- unfolding t_def field_differentiable_def by blast
- qed
- then have " ((/) 1 has_contour_integral 0) (h \<circ> \<gamma>)
- = ((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
- unfolding has_contour_integral
- apply (intro has_integral_spike_eq[OF negligible_finite, OF \<open>finite spikes\<close>])
- by auto
- ultimately show ?thesis by auto
- qed
- then have "contour_integral \<gamma> (\<lambda>x. deriv h x / h x) = 0"
- using contour_integral_unique by simp
- moreover have "contour_integral \<gamma> (\<lambda>x. deriv fg x / fg x) = contour_integral \<gamma> (\<lambda>x. deriv f x / f x)
- + contour_integral \<gamma> (\<lambda>p. deriv h p / h p)"
- proof -
- have "(\<lambda>p. deriv f p / f p) contour_integrable_on \<gamma>"
- proof (rule contour_integrable_holomorphic_simple[OF _ _ \<open>valid_path \<gamma>\<close> path_f])
- show "open (s - zeros_f)" using finite_imp_closed[OF \<open>finite zeros_f\<close>] \<open>open s\<close>
- by auto
- then show "(\<lambda>p. deriv f p / f p) holomorphic_on s - zeros_f"
- using f_holo
- by (auto intro!: holomorphic_intros simp add:zeros_f_def)
- qed
- moreover have "(\<lambda>p. deriv h p / h p) contour_integrable_on \<gamma>"
- using h_contour
- by (simp add: has_contour_integral_integrable)
- ultimately have "contour_integral \<gamma> (\<lambda>x. deriv f x / f x + deriv h x / h x) =
- contour_integral \<gamma> (\<lambda>p. deriv f p / f p) + contour_integral \<gamma> (\<lambda>p. deriv h p / h p)"
- using contour_integral_add[of "(\<lambda>p. deriv f p / f p)" \<gamma> "(\<lambda>p. deriv h p / h p)" ]
- by auto
- moreover have "deriv fg p / fg p = deriv f p / f p + deriv h p / h p"
- when "p\<in> path_image \<gamma>" for p
- proof -
- have "fg p\<noteq>0" and "f p\<noteq>0" using path_f path_fg that unfolding zeros_f_def zeros_fg_def
- by auto
- have "h p\<noteq>0"
- proof (rule ccontr)
- assume "\<not> h p \<noteq> 0"
- then have "g p / f p= -1" unfolding h_def by (simp add: add_eq_0_iff2)
- then have "cmod (g p/f p) = 1" by auto
- moreover have "cmod (g p/f p) <1" using path_less[rule_format,OF that]
- apply (cases "cmod (f p) = 0")
- by (auto simp add: norm_divide)
- ultimately show False by auto
- qed
- have der_fg:"deriv fg p = deriv f p + deriv g p" unfolding fg_def
- using f_holo g_holo holomorphic_on_imp_differentiable_at[OF _ \<open>open s\<close>] path_img that
- by auto
- have der_h:"deriv h p = (deriv g p * f p - g p * deriv f p)/(f p * f p)"
- proof -
- define der where "der \<equiv> \<lambda>p. (deriv g p * f p - g p * deriv f p)/(f p * f p)"
- have "p\<in>s" using path_img that by auto
- then have "(h has_field_derivative der p) (at p)"
- unfolding h_def der_def using g_holo f_holo \<open>open s\<close> \<open>f p\<noteq>0\<close>
- by (auto intro!: derivative_eq_intros holomorphic_derivI)
- then show ?thesis unfolding der_def using DERIV_imp_deriv by auto
- qed
- show ?thesis
- apply (simp only:der_fg der_h)
- apply (auto simp add:field_simps \<open>h p\<noteq>0\<close> \<open>f p\<noteq>0\<close> \<open>fg p\<noteq>0\<close>)
- by (auto simp add:field_simps h_def \<open>f p\<noteq>0\<close> fg_def)
- qed
- then have "contour_integral \<gamma> (\<lambda>p. deriv fg p / fg p)
- = contour_integral \<gamma> (\<lambda>p. deriv f p / f p + deriv h p / h p)"
- by (elim contour_integral_eq)
- ultimately show ?thesis by auto
- qed
- moreover have "contour_integral \<gamma> (\<lambda>x. deriv fg x / fg x) = c * (\<Sum>p\<in>zeros_fg. w p * zorder fg p)"
- unfolding c_def zeros_fg_def w_def
- proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close> _ _ \<open>valid_path \<gamma>\<close> loop _ homo
- , of _ "{}" "\<lambda>_. 1",simplified])
- show "fg holomorphic_on s" unfolding fg_def using f_holo g_holo holomorphic_on_add by auto
- show "path_image \<gamma> \<subseteq> s - {p. fg p = 0}" using path_fg unfolding zeros_fg_def .
- show " finite {p. fg p = 0}" using \<open>finite zeros_fg\<close> unfolding zeros_fg_def .
- qed
- moreover have "contour_integral \<gamma> (\<lambda>x. deriv f x / f x) = c * (\<Sum>p\<in>zeros_f. w p * zorder f p)"
- unfolding c_def zeros_f_def w_def
- proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close> _ _ \<open>valid_path \<gamma>\<close> loop _ homo
- , of _ "{}" "\<lambda>_. 1",simplified])
- show "f holomorphic_on s" using f_holo g_holo holomorphic_on_add by auto
- show "path_image \<gamma> \<subseteq> s - {p. f p = 0}" using path_f unfolding zeros_f_def .
- show " finite {p. f p = 0}" using \<open>finite zeros_f\<close> unfolding zeros_f_def .
- qed
- ultimately have " c* (\<Sum>p\<in>zeros_fg. w p * (zorder fg p)) = c* (\<Sum>p\<in>zeros_f. w p * (zorder f p))"
- by auto
- then show ?thesis unfolding c_def using w_def by auto
-qed
-
-
-subsection \<open>Poles and residues of some well-known functions\<close>
-
-(* TODO: add more material here for other functions *)
-lemma is_pole_Gamma: "is_pole Gamma (-of_nat n)"
- unfolding is_pole_def using Gamma_poles .
-
-lemma Gamme_residue:
- "residue Gamma (-of_nat n) = (-1) ^ n / fact n"
-proof (rule residue_simple')
- show "open (- (\<int>\<^sub>\<le>\<^sub>0 - {-of_nat n}) :: complex set)"
- by (intro open_Compl closed_subset_Ints) auto
- show "Gamma holomorphic_on (- (\<int>\<^sub>\<le>\<^sub>0 - {-of_nat n}) - {- of_nat n})"
- by (rule holomorphic_Gamma) auto
- show "(\<lambda>w. Gamma w * (w - (-of_nat n))) \<midarrow>(-of_nat n)\<rightarrow> (- 1) ^ n / fact n"
- using Gamma_residues[of n] by simp
-qed auto
-
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/Contour_Integration.thy Mon Dec 02 17:51:54 2019 +0100
@@ -0,0 +1,1742 @@
+section \<open>Contour integration\<close>
+theory Contour_Integration
+ imports "HOL-Analysis.Analysis"
+begin
+
+lemma lhopital_complex_simple:
+ assumes "(f has_field_derivative f') (at z)"
+ assumes "(g has_field_derivative g') (at z)"
+ assumes "f z = 0" "g z = 0" "g' \<noteq> 0" "f' / g' = c"
+ shows "((\<lambda>w. f w / g w) \<longlongrightarrow> c) (at z)"
+proof -
+ have "eventually (\<lambda>w. w \<noteq> z) (at z)"
+ by (auto simp: eventually_at_filter)
+ hence "eventually (\<lambda>w. ((f w - f z) / (w - z)) / ((g w - g z) / (w - z)) = f w / g w) (at z)"
+ by eventually_elim (simp add: assms field_split_simps)
+ moreover have "((\<lambda>w. ((f w - f z) / (w - z)) / ((g w - g z) / (w - z))) \<longlongrightarrow> f' / g') (at z)"
+ by (intro tendsto_divide has_field_derivativeD assms)
+ ultimately have "((\<lambda>w. f w / g w) \<longlongrightarrow> f' / g') (at z)"
+ by (blast intro: Lim_transform_eventually)
+ with assms show ?thesis by simp
+qed
+
+subsection\<open>Definition\<close>
+
+text\<open>
+ This definition is for complex numbers only, and does not generalise to
+ line integrals in a vector field
+\<close>
+
+definition\<^marker>\<open>tag important\<close> has_contour_integral :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool"
+ (infixr "has'_contour'_integral" 50)
+ where "(f has_contour_integral i) g \<equiv>
+ ((\<lambda>x. f(g x) * vector_derivative g (at x within {0..1}))
+ has_integral i) {0..1}"
+
+definition\<^marker>\<open>tag important\<close> contour_integrable_on
+ (infixr "contour'_integrable'_on" 50)
+ where "f contour_integrable_on g \<equiv> \<exists>i. (f has_contour_integral i) g"
+
+definition\<^marker>\<open>tag important\<close> contour_integral
+ where "contour_integral g f \<equiv> SOME i. (f has_contour_integral i) g \<or> \<not> f contour_integrable_on g \<and> i=0"
+
+lemma not_integrable_contour_integral: "\<not> f contour_integrable_on g \<Longrightarrow> contour_integral g f = 0"
+ unfolding contour_integrable_on_def contour_integral_def by blast
+
+lemma contour_integral_unique: "(f has_contour_integral i) g \<Longrightarrow> contour_integral g f = i"
+ apply (simp add: contour_integral_def has_contour_integral_def contour_integrable_on_def)
+ using has_integral_unique by blast
+
+lemma has_contour_integral_eqpath:
+ "\<lbrakk>(f has_contour_integral y) p; f contour_integrable_on \<gamma>;
+ contour_integral p f = contour_integral \<gamma> f\<rbrakk>
+ \<Longrightarrow> (f has_contour_integral y) \<gamma>"
+using contour_integrable_on_def contour_integral_unique by auto
+
+lemma has_contour_integral_integral:
+ "f contour_integrable_on i \<Longrightarrow> (f has_contour_integral (contour_integral i f)) i"
+ by (metis contour_integral_unique contour_integrable_on_def)
+
+lemma has_contour_integral_unique:
+ "(f has_contour_integral i) g \<Longrightarrow> (f has_contour_integral j) g \<Longrightarrow> i = j"
+ using has_integral_unique
+ by (auto simp: has_contour_integral_def)
+
+lemma has_contour_integral_integrable: "(f has_contour_integral i) g \<Longrightarrow> f contour_integrable_on g"
+ using contour_integrable_on_def by blast
+
+text\<open>Show that we can forget about the localized derivative.\<close>
+
+lemma has_integral_localized_vector_derivative:
+ "((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} \<longleftrightarrow>
+ ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}"
+proof -
+ have *: "{a..b} - {a,b} = interior {a..b}"
+ by (simp add: atLeastAtMost_diff_ends)
+ show ?thesis
+ apply (rule has_integral_spike_eq [of "{a,b}"])
+ apply (auto simp: at_within_interior [of _ "{a..b}"])
+ done
+qed
+
+lemma integrable_on_localized_vector_derivative:
+ "(\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b} \<longleftrightarrow>
+ (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}"
+ by (simp add: integrable_on_def has_integral_localized_vector_derivative)
+
+lemma has_contour_integral:
+ "(f has_contour_integral i) g \<longleftrightarrow>
+ ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
+ by (simp add: has_integral_localized_vector_derivative has_contour_integral_def)
+
+lemma contour_integrable_on:
+ "f contour_integrable_on g \<longleftrightarrow>
+ (\<lambda>t. f(g t) * vector_derivative g (at t)) integrable_on {0..1}"
+ by (simp add: has_contour_integral integrable_on_def contour_integrable_on_def)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Reversing a path\<close>
+
+
+
+lemma has_contour_integral_reversepath:
+ assumes "valid_path g" and f: "(f has_contour_integral i) g"
+ shows "(f has_contour_integral (-i)) (reversepath g)"
+proof -
+ { fix S x
+ assume xs: "g C1_differentiable_on ({0..1} - S)" "x \<notin> (-) 1 ` S" "0 \<le> x" "x \<le> 1"
+ have "vector_derivative (\<lambda>x. g (1 - x)) (at x within {0..1}) =
+ - vector_derivative g (at (1 - x) within {0..1})"
+ proof -
+ obtain f' where f': "(g has_vector_derivative f') (at (1 - x))"
+ using xs
+ by (force simp: has_vector_derivative_def C1_differentiable_on_def)
+ have "(g \<circ> (\<lambda>x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)"
+ by (intro vector_diff_chain_within has_vector_derivative_at_within [OF f'] derivative_eq_intros | simp)+
+ then have mf': "((\<lambda>x. g (1 - x)) has_vector_derivative -f') (at x)"
+ by (simp add: o_def)
+ show ?thesis
+ using xs
+ by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f'])
+ qed
+ } note * = this
+ obtain S where S: "continuous_on {0..1} g" "finite S" "g C1_differentiable_on {0..1} - S"
+ using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
+ have "((\<lambda>x. - (f (g (1 - x)) * vector_derivative g (at (1 - x) within {0..1}))) has_integral -i)
+ {0..1}"
+ using has_integral_affinity01 [where m= "-1" and c=1, OF f [unfolded has_contour_integral_def]]
+ by (simp add: has_integral_neg)
+ then show ?thesis
+ using S
+ apply (clarsimp simp: reversepath_def has_contour_integral_def)
+ apply (rule_tac S = "(\<lambda>x. 1 - x) ` S" in has_integral_spike_finite)
+ apply (auto simp: *)
+ done
+qed
+
+lemma contour_integrable_reversepath:
+ "valid_path g \<Longrightarrow> f contour_integrable_on g \<Longrightarrow> f contour_integrable_on (reversepath g)"
+ using has_contour_integral_reversepath contour_integrable_on_def by blast
+
+lemma contour_integrable_reversepath_eq:
+ "valid_path g \<Longrightarrow> (f contour_integrable_on (reversepath g) \<longleftrightarrow> f contour_integrable_on g)"
+ using contour_integrable_reversepath valid_path_reversepath by fastforce
+
+lemma contour_integral_reversepath:
+ assumes "valid_path g"
+ shows "contour_integral (reversepath g) f = - (contour_integral g f)"
+proof (cases "f contour_integrable_on g")
+ case True then show ?thesis
+ by (simp add: assms contour_integral_unique has_contour_integral_integral has_contour_integral_reversepath)
+next
+ case False then have "\<not> f contour_integrable_on (reversepath g)"
+ by (simp add: assms contour_integrable_reversepath_eq)
+ with False show ?thesis by (simp add: not_integrable_contour_integral)
+qed
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Joining two paths together\<close>
+
+lemma has_contour_integral_join:
+ assumes "(f has_contour_integral i1) g1" "(f has_contour_integral i2) g2"
+ "valid_path g1" "valid_path g2"
+ shows "(f has_contour_integral (i1 + i2)) (g1 +++ g2)"
+proof -
+ obtain s1 s2
+ where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
+ and s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
+ using assms
+ by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ have 1: "((\<lambda>x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}"
+ and 2: "((\<lambda>x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}"
+ using assms
+ by (auto simp: has_contour_integral)
+ have i1: "((\<lambda>x. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}"
+ and i2: "((\<lambda>x. (2*f (g2 (2*x - 1))) * vector_derivative g2 (at (2*x - 1))) has_integral i2) {1/2..1}"
+ using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]]
+ has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]]
+ by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac)
+ have g1: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
+ vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
+ 2 *\<^sub>R vector_derivative g1 (at (z*2))" for z
+ apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>z - 1/2\<bar>"]])
+ apply (simp_all add: dist_real_def abs_if split: if_split_asm)
+ apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def])
+ apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
+ using s1
+ apply (auto simp: algebra_simps vector_derivative_works)
+ done
+ have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
+ vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
+ 2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))" for z
+ apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2 (2*x - 1))" and d = "\<bar>z - 1/2\<bar>"]])
+ apply (simp_all add: dist_real_def abs_if split: if_split_asm)
+ apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def])
+ apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
+ using s2
+ apply (auto simp: algebra_simps vector_derivative_works)
+ done
+ have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}"
+ apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) ((*)2 -` s1)"])
+ using s1
+ apply (force intro: finite_vimageI [where h = "(*)2"] inj_onI)
+ apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
+ done
+ moreover have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}"
+ apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\<lambda>x. 2*x-1) -` s2)"])
+ using s2
+ apply (force intro: finite_vimageI [where h = "\<lambda>x. 2*x-1"] inj_onI)
+ apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2)
+ done
+ ultimately
+ show ?thesis
+ apply (simp add: has_contour_integral)
+ apply (rule has_integral_combine [where c = "1/2"], auto)
+ done
+qed
+
+lemma contour_integrable_joinI:
+ assumes "f contour_integrable_on g1" "f contour_integrable_on g2"
+ "valid_path g1" "valid_path g2"
+ shows "f contour_integrable_on (g1 +++ g2)"
+ using assms
+ by (meson has_contour_integral_join contour_integrable_on_def)
+
+lemma contour_integrable_joinD1:
+ assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g1"
+ shows "f contour_integrable_on g1"
+proof -
+ obtain s1
+ where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
+ using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ have "(\<lambda>x. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
+ using assms
+ apply (auto simp: contour_integrable_on)
+ apply (drule integrable_on_subcbox [where a=0 and b="1/2"])
+ apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified])
+ done
+ then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2))/2) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
+ by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
+ have g1: "\<lbrakk>0 < z; z < 1; z \<notin> s1\<rbrakk> \<Longrightarrow>
+ vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) =
+ 2 *\<^sub>R vector_derivative g1 (at z)" for z
+ apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>(z-1)/2\<bar>"]])
+ apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
+ apply (rule vector_diff_chain_at [of "\<lambda>x. x*2" 2 _ g1, simplified o_def])
+ using s1
+ apply (auto simp: vector_derivative_works has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
+ done
+ show ?thesis
+ using s1
+ apply (auto simp: contour_integrable_on)
+ apply (rule integrable_spike_finite [of "{0,1} \<union> s1", OF _ _ *])
+ apply (auto simp: joinpaths_def scaleR_conv_of_real g1)
+ done
+qed
+
+lemma contour_integrable_joinD2:
+ assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g2"
+ shows "f contour_integrable_on g2"
+proof -
+ obtain s2
+ where s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
+ using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ have "(\<lambda>x. f ((g1 +++ g2) (x/2 + 1/2)) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) integrable_on {0..1}"
+ using assms
+ apply (auto simp: contour_integrable_on)
+ apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto)
+ apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified])
+ apply (simp add: image_affinity_atLeastAtMost_diff)
+ done
+ then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2 + 1/2))/2) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2)))
+ integrable_on {0..1}"
+ by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
+ have g2: "\<lbrakk>0 < z; z < 1; z \<notin> s2\<rbrakk> \<Longrightarrow>
+ vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2+1/2)) =
+ 2 *\<^sub>R vector_derivative g2 (at z)" for z
+ apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2(2*x-1))" and d = "\<bar>z/2\<bar>"]])
+ apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
+ apply (rule vector_diff_chain_at [of "\<lambda>x. x*2-1" 2 _ g2, simplified o_def])
+ using s2
+ apply (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left
+ vector_derivative_works add_divide_distrib)
+ done
+ show ?thesis
+ using s2
+ apply (auto simp: contour_integrable_on)
+ apply (rule integrable_spike_finite [of "{0,1} \<union> s2", OF _ _ *])
+ apply (auto simp: joinpaths_def scaleR_conv_of_real g2)
+ done
+qed
+
+lemma contour_integrable_join [simp]:
+ shows
+ "\<lbrakk>valid_path g1; valid_path g2\<rbrakk>
+ \<Longrightarrow> f contour_integrable_on (g1 +++ g2) \<longleftrightarrow> f contour_integrable_on g1 \<and> f contour_integrable_on g2"
+using contour_integrable_joinD1 contour_integrable_joinD2 contour_integrable_joinI by blast
+
+lemma contour_integral_join [simp]:
+ shows
+ "\<lbrakk>f contour_integrable_on g1; f contour_integrable_on g2; valid_path g1; valid_path g2\<rbrakk>
+ \<Longrightarrow> contour_integral (g1 +++ g2) f = contour_integral g1 f + contour_integral g2 f"
+ by (simp add: has_contour_integral_integral has_contour_integral_join contour_integral_unique)
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Shifting the starting point of a (closed) path\<close>
+
+lemma has_contour_integral_shiftpath:
+ assumes f: "(f has_contour_integral i) g" "valid_path g"
+ and a: "a \<in> {0..1}"
+ shows "(f has_contour_integral i) (shiftpath a g)"
+proof -
+ obtain s
+ where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
+ using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ have *: "((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
+ using assms by (auto simp: has_contour_integral)
+ then have i: "i = integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)) +
+ integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))"
+ apply (rule has_integral_unique)
+ apply (subst add.commute)
+ apply (subst Henstock_Kurzweil_Integration.integral_combine)
+ using assms * integral_unique by auto
+ { fix x
+ have "0 \<le> x \<Longrightarrow> x + a < 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a) ` s \<Longrightarrow>
+ vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))"
+ unfolding shiftpath_def
+ apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x))" and d = "dist(1-a) x"]])
+ apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
+ apply (rule vector_diff_chain_at [of "\<lambda>x. x+a" 1 _ g, simplified o_def scaleR_one])
+ apply (intro derivative_eq_intros | simp)+
+ using g
+ apply (drule_tac x="x+a" in bspec)
+ using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
+ done
+ } note vd1 = this
+ { fix x
+ have "1 < x + a \<Longrightarrow> x \<le> 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a + 1) ` s \<Longrightarrow>
+ vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))"
+ unfolding shiftpath_def
+ apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x-1))" and d = "dist (1-a) x"]])
+ apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
+ apply (rule vector_diff_chain_at [of "\<lambda>x. x+a-1" 1 _ g, simplified o_def scaleR_one])
+ apply (intro derivative_eq_intros | simp)+
+ using g
+ apply (drule_tac x="x+a-1" in bspec)
+ using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
+ done
+ } note vd2 = this
+ have va1: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})"
+ using * a by (fastforce intro: integrable_subinterval_real)
+ have v0a: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})"
+ apply (rule integrable_subinterval_real)
+ using * a by auto
+ have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
+ has_integral integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x))) {0..1 - a}"
+ apply (rule has_integral_spike_finite
+ [where S = "{1-a} \<union> (\<lambda>x. x-a) ` s" and f = "\<lambda>x. f(g(a+x)) * vector_derivative g (at(a+x))"])
+ using s apply blast
+ using a apply (auto simp: algebra_simps vd1)
+ apply (force simp: shiftpath_def add.commute)
+ using has_integral_affinity [where m=1 and c=a, simplified, OF integrable_integral [OF va1]]
+ apply (simp add: image_affinity_atLeastAtMost_diff [where m=1 and c=a, simplified] add.commute)
+ done
+ moreover
+ have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
+ has_integral integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))) {1 - a..1}"
+ apply (rule has_integral_spike_finite
+ [where S = "{1-a} \<union> (\<lambda>x. x-a+1) ` s" and f = "\<lambda>x. f(g(a+x-1)) * vector_derivative g (at(a+x-1))"])
+ using s apply blast
+ using a apply (auto simp: algebra_simps vd2)
+ apply (force simp: shiftpath_def add.commute)
+ using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]]
+ apply (simp add: image_affinity_atLeastAtMost [where m=1 and c="1-a", simplified])
+ apply (simp add: algebra_simps)
+ done
+ ultimately show ?thesis
+ using a
+ by (auto simp: i has_contour_integral intro: has_integral_combine [where c = "1-a"])
+qed
+
+lemma has_contour_integral_shiftpath_D:
+ assumes "(f has_contour_integral i) (shiftpath a g)"
+ "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
+ shows "(f has_contour_integral i) g"
+proof -
+ obtain s
+ where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
+ using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ { fix x
+ assume x: "0 < x" "x < 1" "x \<notin> s"
+ then have gx: "g differentiable at x"
+ using g by auto
+ have "vector_derivative g (at x within {0..1}) =
+ vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})"
+ apply (rule vector_derivative_at_within_ivl
+ [OF has_vector_derivative_transform_within_open
+ [where f = "(shiftpath (1 - a) (shiftpath a g))" and S = "{0<..<1}-s"]])
+ using s g assms x
+ apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath
+ at_within_interior [of _ "{0..1}"] vector_derivative_works [symmetric])
+ apply (rule differentiable_transform_within [OF gx, of "min x (1-x)"])
+ apply (auto simp: dist_real_def shiftpath_shiftpath abs_if split: if_split_asm)
+ done
+ } note vd = this
+ have fi: "(f has_contour_integral i) (shiftpath (1 - a) (shiftpath a g))"
+ using assms by (auto intro!: has_contour_integral_shiftpath)
+ show ?thesis
+ apply (simp add: has_contour_integral_def)
+ apply (rule has_integral_spike_finite [of "{0,1} \<union> s", OF _ _ fi [unfolded has_contour_integral_def]])
+ using s assms vd
+ apply (auto simp: Path_Connected.shiftpath_shiftpath)
+ done
+qed
+
+lemma has_contour_integral_shiftpath_eq:
+ assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
+ shows "(f has_contour_integral i) (shiftpath a g) \<longleftrightarrow> (f has_contour_integral i) g"
+ using assms has_contour_integral_shiftpath has_contour_integral_shiftpath_D by blast
+
+lemma contour_integrable_on_shiftpath_eq:
+ assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
+ shows "f contour_integrable_on (shiftpath a g) \<longleftrightarrow> f contour_integrable_on g"
+using assms contour_integrable_on_def has_contour_integral_shiftpath_eq by auto
+
+lemma contour_integral_shiftpath:
+ assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
+ shows "contour_integral (shiftpath a g) f = contour_integral g f"
+ using assms
+ by (simp add: contour_integral_def contour_integrable_on_def has_contour_integral_shiftpath_eq)
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>More about straight-line paths\<close>
+
+lemma has_contour_integral_linepath:
+ shows "(f has_contour_integral i) (linepath a b) \<longleftrightarrow>
+ ((\<lambda>x. f(linepath a b x) * (b - a)) has_integral i) {0..1}"
+ by (simp add: has_contour_integral)
+
+lemma has_contour_integral_trivial [iff]: "(f has_contour_integral 0) (linepath a a)"
+ by (simp add: has_contour_integral_linepath)
+
+lemma has_contour_integral_trivial_iff [simp]: "(f has_contour_integral i) (linepath a a) \<longleftrightarrow> i=0"
+ using has_contour_integral_unique by blast
+
+lemma contour_integral_trivial [simp]: "contour_integral (linepath a a) f = 0"
+ using has_contour_integral_trivial contour_integral_unique by blast
+
+
+subsection\<open>Relation to subpath construction\<close>
+
+lemma has_contour_integral_subpath_refl [iff]: "(f has_contour_integral 0) (subpath u u g)"
+ by (simp add: has_contour_integral subpath_def)
+
+lemma contour_integrable_subpath_refl [iff]: "f contour_integrable_on (subpath u u g)"
+ using has_contour_integral_subpath_refl contour_integrable_on_def by blast
+
+lemma contour_integral_subpath_refl [simp]: "contour_integral (subpath u u g) f = 0"
+ by (simp add: contour_integral_unique)
+
+lemma has_contour_integral_subpath:
+ assumes f: "f contour_integrable_on g" and g: "valid_path g"
+ and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
+ shows "(f has_contour_integral integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x)))
+ (subpath u v g)"
+proof (cases "v=u")
+ case True
+ then show ?thesis
+ using f by (simp add: contour_integrable_on_def subpath_def has_contour_integral)
+next
+ case False
+ obtain s where s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x" and fs: "finite s"
+ using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast
+ have *: "((\<lambda>x. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u)))
+ has_integral (1 / (v - u)) * integral {u..v} (\<lambda>t. f (g t) * vector_derivative g (at t)))
+ {0..1}"
+ using f uv
+ apply (simp add: contour_integrable_on subpath_def has_contour_integral)
+ apply (drule integrable_on_subcbox [where a=u and b=v, simplified])
+ apply (simp_all add: has_integral_integral)
+ apply (drule has_integral_affinity [where m="v-u" and c=u, simplified])
+ apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real)
+ apply (simp add: divide_simps False)
+ done
+ { fix x
+ have "x \<in> {0..1} \<Longrightarrow>
+ x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s \<Longrightarrow>
+ vector_derivative (\<lambda>x. g ((v-u) * x + u)) (at x) = (v-u) *\<^sub>R vector_derivative g (at ((v-u) * x + u))"
+ apply (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]])
+ apply (intro derivative_eq_intros | simp)+
+ apply (cut_tac s [of "(v - u) * x + u"])
+ using uv mult_left_le [of x "v-u"]
+ apply (auto simp: vector_derivative_works)
+ done
+ } note vd = this
+ show ?thesis
+ apply (cut_tac has_integral_cmul [OF *, where c = "v-u"])
+ using fs assms
+ apply (simp add: False subpath_def has_contour_integral)
+ apply (rule_tac S = "(\<lambda>t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite)
+ apply (auto simp: inj_on_def False finite_vimageI vd scaleR_conv_of_real)
+ done
+qed
+
+lemma contour_integrable_subpath:
+ assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
+ shows "f contour_integrable_on (subpath u v g)"
+ apply (cases u v rule: linorder_class.le_cases)
+ apply (metis contour_integrable_on_def has_contour_integral_subpath [OF assms])
+ apply (subst reversepath_subpath [symmetric])
+ apply (rule contour_integrable_reversepath)
+ using assms apply (blast intro: valid_path_subpath)
+ apply (simp add: contour_integrable_on_def)
+ using assms apply (blast intro: has_contour_integral_subpath)
+ done
+
+lemma has_integral_contour_integral_subpath:
+ assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
+ shows "(((\<lambda>x. f(g x) * vector_derivative g (at x)))
+ has_integral contour_integral (subpath u v g) f) {u..v}"
+ using assms
+ apply (auto simp: has_integral_integrable_integral)
+ apply (rule integrable_on_subcbox [where a=u and b=v and S = "{0..1}", simplified])
+ apply (auto simp: contour_integral_unique [OF has_contour_integral_subpath] contour_integrable_on)
+ done
+
+lemma contour_integral_subcontour_integral:
+ assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
+ shows "contour_integral (subpath u v g) f =
+ integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x))"
+ using assms has_contour_integral_subpath contour_integral_unique by blast
+
+lemma contour_integral_subpath_combine_less:
+ assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
+ "u<v" "v<w"
+ shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
+ contour_integral (subpath u w g) f"
+ using assms apply (auto simp: contour_integral_subcontour_integral)
+ apply (rule Henstock_Kurzweil_Integration.integral_combine, auto)
+ apply (rule integrable_on_subcbox [where a=u and b=w and S = "{0..1}", simplified])
+ apply (auto simp: contour_integrable_on)
+ done
+
+lemma contour_integral_subpath_combine:
+ assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
+ shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
+ contour_integral (subpath u w g) f"
+proof (cases "u\<noteq>v \<and> v\<noteq>w \<and> u\<noteq>w")
+ case True
+ have *: "subpath v u g = reversepath(subpath u v g) \<and>
+ subpath w u g = reversepath(subpath u w g) \<and>
+ subpath w v g = reversepath(subpath v w g)"
+ by (auto simp: reversepath_subpath)
+ have "u < v \<and> v < w \<or>
+ u < w \<and> w < v \<or>
+ v < u \<and> u < w \<or>
+ v < w \<and> w < u \<or>
+ w < u \<and> u < v \<or>
+ w < v \<and> v < u"
+ using True assms by linarith
+ with assms show ?thesis
+ using contour_integral_subpath_combine_less [of f g u v w]
+ contour_integral_subpath_combine_less [of f g u w v]
+ contour_integral_subpath_combine_less [of f g v u w]
+ contour_integral_subpath_combine_less [of f g v w u]
+ contour_integral_subpath_combine_less [of f g w u v]
+ contour_integral_subpath_combine_less [of f g w v u]
+ apply simp
+ apply (elim disjE)
+ apply (auto simp: * contour_integral_reversepath contour_integrable_subpath
+ valid_path_subpath algebra_simps)
+ done
+next
+ case False
+ then show ?thesis
+ apply (auto)
+ using assms
+ by (metis eq_neg_iff_add_eq_0 contour_integral_reversepath reversepath_subpath valid_path_subpath)
+qed
+
+lemma contour_integral_integral:
+ "contour_integral g f = integral {0..1} (\<lambda>x. f (g x) * vector_derivative g (at x))"
+ by (simp add: contour_integral_def integral_def has_contour_integral contour_integrable_on)
+
+lemma contour_integral_cong:
+ assumes "g = g'" "\<And>x. x \<in> path_image g \<Longrightarrow> f x = f' x"
+ shows "contour_integral g f = contour_integral g' f'"
+ unfolding contour_integral_integral using assms
+ by (intro integral_cong) (auto simp: path_image_def)
+
+
+text \<open>Contour integral along a segment on the real axis\<close>
+
+lemma has_contour_integral_linepath_Reals_iff:
+ fixes a b :: complex and f :: "complex \<Rightarrow> complex"
+ assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
+ shows "(f has_contour_integral I) (linepath a b) \<longleftrightarrow>
+ ((\<lambda>x. f (of_real x)) has_integral I) {Re a..Re b}"
+proof -
+ from assms have [simp]: "of_real (Re a) = a" "of_real (Re b) = b"
+ by (simp_all add: complex_eq_iff)
+ from assms have "a \<noteq> b" by auto
+ have "((\<lambda>x. f (of_real x)) has_integral I) (cbox (Re a) (Re b)) \<longleftrightarrow>
+ ((\<lambda>x. f (a + b * of_real x - a * of_real x)) has_integral I /\<^sub>R (Re b - Re a)) {0..1}"
+ by (subst has_integral_affinity_iff [of "Re b - Re a" _ "Re a", symmetric])
+ (insert assms, simp_all add: field_simps scaleR_conv_of_real)
+ also have "(\<lambda>x. f (a + b * of_real x - a * of_real x)) =
+ (\<lambda>x. (f (a + b * of_real x - a * of_real x) * (b - a)) /\<^sub>R (Re b - Re a))"
+ using \<open>a \<noteq> b\<close> by (auto simp: field_simps fun_eq_iff scaleR_conv_of_real)
+ also have "(\<dots> has_integral I /\<^sub>R (Re b - Re a)) {0..1} \<longleftrightarrow>
+ ((\<lambda>x. f (linepath a b x) * (b - a)) has_integral I) {0..1}" using assms
+ by (subst has_integral_cmul_iff) (auto simp: linepath_def scaleR_conv_of_real algebra_simps)
+ also have "\<dots> \<longleftrightarrow> (f has_contour_integral I) (linepath a b)" unfolding has_contour_integral_def
+ by (intro has_integral_cong) (simp add: vector_derivative_linepath_within)
+ finally show ?thesis by simp
+qed
+
+lemma contour_integrable_linepath_Reals_iff:
+ fixes a b :: complex and f :: "complex \<Rightarrow> complex"
+ assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
+ shows "(f contour_integrable_on linepath a b) \<longleftrightarrow>
+ (\<lambda>x. f (of_real x)) integrable_on {Re a..Re b}"
+ using has_contour_integral_linepath_Reals_iff[OF assms, of f]
+ by (auto simp: contour_integrable_on_def integrable_on_def)
+
+lemma contour_integral_linepath_Reals_eq:
+ fixes a b :: complex and f :: "complex \<Rightarrow> complex"
+ assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
+ shows "contour_integral (linepath a b) f = integral {Re a..Re b} (\<lambda>x. f (of_real x))"
+proof (cases "f contour_integrable_on linepath a b")
+ case True
+ thus ?thesis using has_contour_integral_linepath_Reals_iff[OF assms, of f]
+ using has_contour_integral_integral has_contour_integral_unique by blast
+next
+ case False
+ thus ?thesis using contour_integrable_linepath_Reals_iff[OF assms, of f]
+ by (simp add: not_integrable_contour_integral not_integrable_integral)
+qed
+
+text \<open>Cauchy's theorem where there's a primitive\<close>
+
+lemma contour_integral_primitive_lemma:
+ fixes f :: "complex \<Rightarrow> complex" and g :: "real \<Rightarrow> complex"
+ assumes "a \<le> b"
+ and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
+ and "g piecewise_differentiable_on {a..b}" "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
+ shows "((\<lambda>x. f'(g x) * vector_derivative g (at x within {a..b}))
+ has_integral (f(g b) - f(g a))) {a..b}"
+proof -
+ obtain k where k: "finite k" "\<forall>x\<in>{a..b} - k. g differentiable (at x within {a..b})" and cg: "continuous_on {a..b} g"
+ using assms by (auto simp: piecewise_differentiable_on_def)
+ have cfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
+ apply (rule continuous_on_compose [OF cg, unfolded o_def])
+ using assms
+ apply (metis field_differentiable_def field_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
+ done
+ { fix x::real
+ assume a: "a < x" and b: "x < b" and xk: "x \<notin> k"
+ then have "g differentiable at x within {a..b}"
+ using k by (simp add: differentiable_at_withinI)
+ then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})"
+ by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real)
+ then have gdiff: "(g has_derivative (\<lambda>u. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})"
+ by (simp add: has_vector_derivative_def scaleR_conv_of_real)
+ have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})"
+ using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
+ then have fdiff: "(f has_derivative (*) (f' (g x))) (at (g x) within g ` {a..b})"
+ by (simp add: has_field_derivative_def)
+ have "((\<lambda>x. f (g x)) has_vector_derivative f' (g x) * vector_derivative g (at x within {a..b})) (at x within {a..b})"
+ using diff_chain_within [OF gdiff fdiff]
+ by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac)
+ } note * = this
+ show ?thesis
+ apply (rule fundamental_theorem_of_calculus_interior_strong)
+ using k assms cfg *
+ apply (auto simp: at_within_Icc_at)
+ done
+qed
+
+lemma contour_integral_primitive:
+ assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
+ and "valid_path g" "path_image g \<subseteq> s"
+ shows "(f' has_contour_integral (f(pathfinish g) - f(pathstart g))) g"
+ using assms
+ apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_contour_integral_def)
+ apply (auto intro!: piecewise_C1_imp_differentiable contour_integral_primitive_lemma [of 0 1 s])
+ done
+
+corollary Cauchy_theorem_primitive:
+ assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
+ and "valid_path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g"
+ shows "(f' has_contour_integral 0) g"
+ using assms
+ by (metis diff_self contour_integral_primitive)
+
+text\<open>Existence of path integral for continuous function\<close>
+lemma contour_integrable_continuous_linepath:
+ assumes "continuous_on (closed_segment a b) f"
+ shows "f contour_integrable_on (linepath a b)"
+proof -
+ have "continuous_on {0..1} ((\<lambda>x. f x * (b - a)) \<circ> linepath a b)"
+ apply (rule continuous_on_compose [OF continuous_on_linepath], simp add: linepath_image_01)
+ apply (rule continuous_intros | simp add: assms)+
+ done
+ then show ?thesis
+ apply (simp add: contour_integrable_on_def has_contour_integral_def integrable_on_def [symmetric])
+ apply (rule integrable_continuous [of 0 "1::real", simplified])
+ apply (rule continuous_on_eq [where f = "\<lambda>x. f(linepath a b x)*(b - a)"])
+ apply (auto simp: vector_derivative_linepath_within)
+ done
+qed
+
+lemma has_field_der_id: "((\<lambda>x. x\<^sup>2 / 2) has_field_derivative x) (at x)"
+ by (rule has_derivative_imp_has_field_derivative)
+ (rule derivative_intros | simp)+
+
+lemma contour_integral_id [simp]: "contour_integral (linepath a b) (\<lambda>y. y) = (b^2 - a^2)/2"
+ apply (rule contour_integral_unique)
+ using contour_integral_primitive [of UNIV "\<lambda>x. x^2/2" "\<lambda>x. x" "linepath a b"]
+ apply (auto simp: field_simps has_field_der_id)
+ done
+
+lemma contour_integrable_on_const [iff]: "(\<lambda>x. c) contour_integrable_on (linepath a b)"
+ by (simp add: contour_integrable_continuous_linepath)
+
+lemma contour_integrable_on_id [iff]: "(\<lambda>x. x) contour_integrable_on (linepath a b)"
+ by (simp add: contour_integrable_continuous_linepath)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetical combining theorems\<close>
+
+lemma has_contour_integral_neg:
+ "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. -(f x)) has_contour_integral (-i)) g"
+ by (simp add: has_integral_neg has_contour_integral_def)
+
+lemma has_contour_integral_add:
+ "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
+ \<Longrightarrow> ((\<lambda>x. f1 x + f2 x) has_contour_integral (i1 + i2)) g"
+ by (simp add: has_integral_add has_contour_integral_def algebra_simps)
+
+lemma has_contour_integral_diff:
+ "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
+ \<Longrightarrow> ((\<lambda>x. f1 x - f2 x) has_contour_integral (i1 - i2)) g"
+ by (simp add: has_integral_diff has_contour_integral_def algebra_simps)
+
+lemma has_contour_integral_lmul:
+ "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. c * (f x)) has_contour_integral (c*i)) g"
+apply (simp add: has_contour_integral_def)
+apply (drule has_integral_mult_right)
+apply (simp add: algebra_simps)
+done
+
+lemma has_contour_integral_rmul:
+ "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. (f x) * c) has_contour_integral (i*c)) g"
+apply (drule has_contour_integral_lmul)
+apply (simp add: mult.commute)
+done
+
+lemma has_contour_integral_div:
+ "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. f x/c) has_contour_integral (i/c)) g"
+ by (simp add: field_class.field_divide_inverse) (metis has_contour_integral_rmul)
+
+lemma has_contour_integral_eq:
+ "\<lbrakk>(f has_contour_integral y) p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> (g has_contour_integral y) p"
+apply (simp add: path_image_def has_contour_integral_def)
+by (metis (no_types, lifting) image_eqI has_integral_eq)
+
+lemma has_contour_integral_bound_linepath:
+ assumes "(f has_contour_integral i) (linepath a b)"
+ "0 \<le> B" "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B"
+ shows "norm i \<le> B * norm(b - a)"
+proof -
+ { fix x::real
+ assume x: "0 \<le> x" "x \<le> 1"
+ have "norm (f (linepath a b x)) *
+ norm (vector_derivative (linepath a b) (at x within {0..1})) \<le> B * norm (b - a)"
+ by (auto intro: mult_mono simp: assms linepath_in_path of_real_linepath vector_derivative_linepath_within x)
+ } note * = this
+ have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))"
+ apply (rule has_integral_bound
+ [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
+ using assms * unfolding has_contour_integral_def
+ apply (auto simp: norm_mult)
+ done
+ then show ?thesis
+ by (auto simp: content_real)
+qed
+
+(*UNUSED
+lemma has_contour_integral_bound_linepath_strong:
+ fixes a :: real and f :: "complex \<Rightarrow> real"
+ assumes "(f has_contour_integral i) (linepath a b)"
+ "finite k"
+ "0 \<le> B" "\<And>x::real. x \<in> closed_segment a b - k \<Longrightarrow> norm(f x) \<le> B"
+ shows "norm i \<le> B*norm(b - a)"
+*)
+
+lemma has_contour_integral_const_linepath: "((\<lambda>x. c) has_contour_integral c*(b - a))(linepath a b)"
+ unfolding has_contour_integral_linepath
+ by (metis content_real diff_0_right has_integral_const_real lambda_one of_real_1 scaleR_conv_of_real zero_le_one)
+
+lemma has_contour_integral_0: "((\<lambda>x. 0) has_contour_integral 0) g"
+ by (simp add: has_contour_integral_def)
+
+lemma has_contour_integral_is_0:
+ "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> (f has_contour_integral 0) g"
+ by (rule has_contour_integral_eq [OF has_contour_integral_0]) auto
+
+lemma has_contour_integral_sum:
+ "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a has_contour_integral i a) p\<rbrakk>
+ \<Longrightarrow> ((\<lambda>x. sum (\<lambda>a. f a x) s) has_contour_integral sum i s) p"
+ by (induction s rule: finite_induct) (auto simp: has_contour_integral_0 has_contour_integral_add)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Operations on path integrals\<close>
+
+lemma contour_integral_const_linepath [simp]: "contour_integral (linepath a b) (\<lambda>x. c) = c*(b - a)"
+ by (rule contour_integral_unique [OF has_contour_integral_const_linepath])
+
+lemma contour_integral_neg:
+ "f contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. -(f x)) = -(contour_integral g f)"
+ by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_neg)
+
+lemma contour_integral_add:
+ "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x + f2 x) =
+ contour_integral g f1 + contour_integral g f2"
+ by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_add)
+
+lemma contour_integral_diff:
+ "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x - f2 x) =
+ contour_integral g f1 - contour_integral g f2"
+ by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_diff)
+
+lemma contour_integral_lmul:
+ shows "f contour_integrable_on g
+ \<Longrightarrow> contour_integral g (\<lambda>x. c * f x) = c*contour_integral g f"
+ by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_lmul)
+
+lemma contour_integral_rmul:
+ shows "f contour_integrable_on g
+ \<Longrightarrow> contour_integral g (\<lambda>x. f x * c) = contour_integral g f * c"
+ by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_rmul)
+
+lemma contour_integral_div:
+ shows "f contour_integrable_on g
+ \<Longrightarrow> contour_integral g (\<lambda>x. f x / c) = contour_integral g f / c"
+ by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_div)
+
+lemma contour_integral_eq:
+ "(\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x) \<Longrightarrow> contour_integral p f = contour_integral p g"
+ apply (simp add: contour_integral_def)
+ using has_contour_integral_eq
+ by (metis contour_integral_unique has_contour_integral_integrable has_contour_integral_integral)
+
+lemma contour_integral_eq_0:
+ "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> contour_integral g f = 0"
+ by (simp add: has_contour_integral_is_0 contour_integral_unique)
+
+lemma contour_integral_bound_linepath:
+ shows
+ "\<lbrakk>f contour_integrable_on (linepath a b);
+ 0 \<le> B; \<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
+ \<Longrightarrow> norm(contour_integral (linepath a b) f) \<le> B*norm(b - a)"
+ apply (rule has_contour_integral_bound_linepath [of f])
+ apply (auto simp: has_contour_integral_integral)
+ done
+
+lemma contour_integral_0 [simp]: "contour_integral g (\<lambda>x. 0) = 0"
+ by (simp add: contour_integral_unique has_contour_integral_0)
+
+lemma contour_integral_sum:
+ "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
+ \<Longrightarrow> contour_integral p (\<lambda>x. sum (\<lambda>a. f a x) s) = sum (\<lambda>a. contour_integral p (f a)) s"
+ by (auto simp: contour_integral_unique has_contour_integral_sum has_contour_integral_integral)
+
+lemma contour_integrable_eq:
+ "\<lbrakk>f contour_integrable_on p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g contour_integrable_on p"
+ unfolding contour_integrable_on_def
+ by (metis has_contour_integral_eq)
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetic theorems for path integrability\<close>
+
+lemma contour_integrable_neg:
+ "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. -(f x)) contour_integrable_on g"
+ using has_contour_integral_neg contour_integrable_on_def by blast
+
+lemma contour_integrable_add:
+ "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x + f2 x) contour_integrable_on g"
+ using has_contour_integral_add contour_integrable_on_def
+ by fastforce
+
+lemma contour_integrable_diff:
+ "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x - f2 x) contour_integrable_on g"
+ using has_contour_integral_diff contour_integrable_on_def
+ by fastforce
+
+lemma contour_integrable_lmul:
+ "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. c * f x) contour_integrable_on g"
+ using has_contour_integral_lmul contour_integrable_on_def
+ by fastforce
+
+lemma contour_integrable_rmul:
+ "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x * c) contour_integrable_on g"
+ using has_contour_integral_rmul contour_integrable_on_def
+ by fastforce
+
+lemma contour_integrable_div:
+ "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x / c) contour_integrable_on g"
+ using has_contour_integral_div contour_integrable_on_def
+ by fastforce
+
+lemma contour_integrable_sum:
+ "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
+ \<Longrightarrow> (\<lambda>x. sum (\<lambda>a. f a x) s) contour_integrable_on p"
+ unfolding contour_integrable_on_def
+ by (metis has_contour_integral_sum)
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Reversing a path integral\<close>
+
+lemma has_contour_integral_reverse_linepath:
+ "(f has_contour_integral i) (linepath a b)
+ \<Longrightarrow> (f has_contour_integral (-i)) (linepath b a)"
+ using has_contour_integral_reversepath valid_path_linepath by fastforce
+
+lemma contour_integral_reverse_linepath:
+ "continuous_on (closed_segment a b) f
+ \<Longrightarrow> contour_integral (linepath a b) f = - (contour_integral(linepath b a) f)"
+apply (rule contour_integral_unique)
+apply (rule has_contour_integral_reverse_linepath)
+by (simp add: closed_segment_commute contour_integrable_continuous_linepath has_contour_integral_integral)
+
+
+(* Splitting a path integral in a flat way.*)
+
+lemma has_contour_integral_split:
+ assumes f: "(f has_contour_integral i) (linepath a c)" "(f has_contour_integral j) (linepath c b)"
+ and k: "0 \<le> k" "k \<le> 1"
+ and c: "c - a = k *\<^sub>R (b - a)"
+ shows "(f has_contour_integral (i + j)) (linepath a b)"
+proof (cases "k = 0 \<or> k = 1")
+ case True
+ then show ?thesis
+ using assms by auto
+next
+ case False
+ then have k: "0 < k" "k < 1" "complex_of_real k \<noteq> 1"
+ using assms by auto
+ have c': "c = k *\<^sub>R (b - a) + a"
+ by (metis diff_add_cancel c)
+ have bc: "(b - c) = (1 - k) *\<^sub>R (b - a)"
+ by (simp add: algebra_simps c')
+ { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}"
+ have **: "\<And>x. ((k - x) / k) *\<^sub>R a + (x / k) *\<^sub>R c = (1 - x) *\<^sub>R a + x *\<^sub>R b"
+ using False apply (simp add: c' algebra_simps)
+ apply (simp add: real_vector.scale_left_distrib [symmetric] field_split_simps)
+ done
+ have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}"
+ using k has_integral_affinity01 [OF *, of "inverse k" "0"]
+ apply (simp add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost ** c)
+ apply (auto dest: has_integral_cmul [where c = "inverse k"])
+ done
+ } note fi = this
+ { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}"
+ have **: "\<And>x. (((1 - x) / (1 - k)) *\<^sub>R c + ((x - k) / (1 - k)) *\<^sub>R b) = ((1 - x) *\<^sub>R a + x *\<^sub>R b)"
+ using k
+ apply (simp add: c' field_simps)
+ apply (simp add: scaleR_conv_of_real divide_simps)
+ apply (simp add: field_simps)
+ done
+ have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}"
+ using k has_integral_affinity01 [OF *, of "inverse(1 - k)" "-(k/(1 - k))"]
+ apply (simp add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc)
+ apply (auto dest: has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"])
+ done
+ } note fj = this
+ show ?thesis
+ using f k
+ apply (simp add: has_contour_integral_linepath)
+ apply (simp add: linepath_def)
+ apply (rule has_integral_combine [OF _ _ fi fj], simp_all)
+ done
+qed
+
+lemma continuous_on_closed_segment_transform:
+ assumes f: "continuous_on (closed_segment a b) f"
+ and k: "0 \<le> k" "k \<le> 1"
+ and c: "c - a = k *\<^sub>R (b - a)"
+ shows "continuous_on (closed_segment a c) f"
+proof -
+ have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
+ using c by (simp add: algebra_simps)
+ have "closed_segment a c \<subseteq> closed_segment a b"
+ by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment)
+ then show "continuous_on (closed_segment a c) f"
+ by (rule continuous_on_subset [OF f])
+qed
+
+lemma contour_integral_split:
+ assumes f: "continuous_on (closed_segment a b) f"
+ and k: "0 \<le> k" "k \<le> 1"
+ and c: "c - a = k *\<^sub>R (b - a)"
+ shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
+proof -
+ have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
+ using c by (simp add: algebra_simps)
+ have "closed_segment a c \<subseteq> closed_segment a b"
+ by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment)
+ moreover have "closed_segment c b \<subseteq> closed_segment a b"
+ by (metis c' ends_in_segment(2) in_segment(1) k subset_closed_segment)
+ ultimately
+ have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f"
+ by (auto intro: continuous_on_subset [OF f])
+ show ?thesis
+ by (rule contour_integral_unique) (meson "*" c contour_integrable_continuous_linepath has_contour_integral_integral has_contour_integral_split k)
+qed
+
+lemma contour_integral_split_linepath:
+ assumes f: "continuous_on (closed_segment a b) f"
+ and c: "c \<in> closed_segment a b"
+ shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
+ using c by (auto simp: closed_segment_def algebra_simps intro!: contour_integral_split [OF f])
+
+
+subsection\<open>Reversing the order in a double path integral\<close>
+
+text\<open>The condition is stronger than needed but it's often true in typical situations\<close>
+
+lemma fst_im_cbox [simp]: "cbox c d \<noteq> {} \<Longrightarrow> (fst ` cbox (a,c) (b,d)) = cbox a b"
+ by (auto simp: cbox_Pair_eq)
+
+lemma snd_im_cbox [simp]: "cbox a b \<noteq> {} \<Longrightarrow> (snd ` cbox (a,c) (b,d)) = cbox c d"
+ by (auto simp: cbox_Pair_eq)
+
+proposition contour_integral_swap:
+ assumes fcon: "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1,y2). f y1 y2)"
+ and vp: "valid_path g" "valid_path h"
+ and gvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))"
+ and hvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))"
+ shows "contour_integral g (\<lambda>w. contour_integral h (f w)) =
+ contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
+proof -
+ have gcon: "continuous_on {0..1} g" and hcon: "continuous_on {0..1} h"
+ using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
+ have fgh1: "\<And>x. (\<lambda>t. f (g x) (h t)) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>t. (g x, h t))"
+ by (rule ext) simp
+ have fgh2: "\<And>x. (\<lambda>t. f (g t) (h x)) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>t. (g t, h x))"
+ by (rule ext) simp
+ have fcon_im1: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g x, h t)) ` {0..1}) (\<lambda>(x, y). f x y)"
+ by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
+ have fcon_im2: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g t, h x)) ` {0..1}) (\<lambda>(x, y). f x y)"
+ by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
+ have "\<And>y. y \<in> {0..1} \<Longrightarrow> continuous_on {0..1} (\<lambda>x. f (g x) (h y))"
+ by (subst fgh2) (rule fcon_im2 gcon continuous_intros | simp)+
+ then have vdg: "\<And>y. y \<in> {0..1} \<Longrightarrow> (\<lambda>x. f (g x) (h y) * vector_derivative g (at x)) integrable_on {0..1}"
+ using continuous_on_mult gvcon integrable_continuous_real by blast
+ have "(\<lambda>z. vector_derivative g (at (fst z))) = (\<lambda>x. vector_derivative g (at x)) \<circ> fst"
+ by auto
+ then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\<lambda>x. vector_derivative g (at (fst x)))"
+ apply (rule ssubst)
+ apply (rule continuous_intros | simp add: gvcon)+
+ done
+ have "(\<lambda>z. vector_derivative h (at (snd z))) = (\<lambda>x. vector_derivative h (at x)) \<circ> snd"
+ by auto
+ then have hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\<lambda>x. vector_derivative h (at (snd x)))"
+ apply (rule ssubst)
+ apply (rule continuous_intros | simp add: hvcon)+
+ done
+ have "(\<lambda>x. f (g (fst x)) (h (snd x))) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>w. ((g \<circ> fst) w, (h \<circ> snd) w))"
+ by auto
+ then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))"
+ apply (rule ssubst)
+ apply (rule gcon hcon continuous_intros | simp)+
+ apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon])
+ done
+ have "integral {0..1} (\<lambda>x. contour_integral h (f (g x)) * vector_derivative g (at x)) =
+ integral {0..1} (\<lambda>x. contour_integral h (\<lambda>y. f (g x) y * vector_derivative g (at x)))"
+ proof (rule integral_cong [OF contour_integral_rmul [symmetric]])
+ show "\<And>x. x \<in> {0..1} \<Longrightarrow> f (g x) contour_integrable_on h"
+ unfolding contour_integrable_on
+ apply (rule integrable_continuous_real)
+ apply (rule continuous_on_mult [OF _ hvcon])
+ apply (subst fgh1)
+ apply (rule fcon_im1 hcon continuous_intros | simp)+
+ done
+ qed
+ also have "\<dots> = integral {0..1}
+ (\<lambda>y. contour_integral g (\<lambda>x. f x (h y) * vector_derivative h (at y)))"
+ unfolding contour_integral_integral
+ apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified])
+ apply (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+
+ unfolding integral_mult_left [symmetric]
+ apply (simp only: mult_ac)
+ done
+ also have "\<dots> = contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
+ unfolding contour_integral_integral
+ apply (rule integral_cong)
+ unfolding integral_mult_left [symmetric]
+ apply (simp add: algebra_simps)
+ done
+ finally show ?thesis
+ by (simp add: contour_integral_integral)
+qed
+
+lemma valid_path_negatepath: "valid_path \<gamma> \<Longrightarrow> valid_path (uminus \<circ> \<gamma>)"
+ unfolding o_def using piecewise_C1_differentiable_neg valid_path_def by blast
+
+lemma has_contour_integral_negatepath:
+ assumes \<gamma>: "valid_path \<gamma>" and cint: "((\<lambda>z. f (- z)) has_contour_integral - i) \<gamma>"
+ shows "(f has_contour_integral i) (uminus \<circ> \<gamma>)"
+proof -
+ obtain S where cont: "continuous_on {0..1} \<gamma>" and "finite S" and diff: "\<gamma> C1_differentiable_on {0..1} - S"
+ using \<gamma> by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
+ have "((\<lambda>x. - (f (- \<gamma> x) * vector_derivative \<gamma> (at x within {0..1}))) has_integral i) {0..1}"
+ using cint by (auto simp: has_contour_integral_def dest: has_integral_neg)
+ then
+ have "((\<lambda>x. f (- \<gamma> x) * vector_derivative (uminus \<circ> \<gamma>) (at x within {0..1})) has_integral i) {0..1}"
+ proof (rule rev_iffD1 [OF _ has_integral_spike_eq])
+ show "negligible S"
+ by (simp add: \<open>finite S\<close> negligible_finite)
+ show "f (- \<gamma> x) * vector_derivative (uminus \<circ> \<gamma>) (at x within {0..1}) =
+ - (f (- \<gamma> x) * vector_derivative \<gamma> (at x within {0..1}))"
+ if "x \<in> {0..1} - S" for x
+ proof -
+ have "vector_derivative (uminus \<circ> \<gamma>) (at x within cbox 0 1) = - vector_derivative \<gamma> (at x within cbox 0 1)"
+ proof (rule vector_derivative_within_cbox)
+ show "(uminus \<circ> \<gamma> has_vector_derivative - vector_derivative \<gamma> (at x within cbox 0 1)) (at x within cbox 0 1)"
+ using that unfolding o_def
+ by (metis C1_differentiable_on_eq UNIV_I diff differentiable_subset has_vector_derivative_minus subsetI that vector_derivative_works)
+ qed (use that in auto)
+ then show ?thesis
+ by simp
+ qed
+ qed
+ then show ?thesis by (simp add: has_contour_integral_def)
+qed
+
+lemma contour_integrable_negatepath:
+ assumes \<gamma>: "valid_path \<gamma>" and pi: "(\<lambda>z. f (- z)) contour_integrable_on \<gamma>"
+ shows "f contour_integrable_on (uminus \<circ> \<gamma>)"
+ by (metis \<gamma> add.inverse_inverse contour_integrable_on_def has_contour_integral_negatepath pi)
+
+lemma C1_differentiable_polynomial_function:
+ fixes p :: "real \<Rightarrow> 'a::euclidean_space"
+ shows "polynomial_function p \<Longrightarrow> p C1_differentiable_on S"
+ by (metis continuous_on_polymonial_function C1_differentiable_on_def has_vector_derivative_polynomial_function)
+
+lemma valid_path_polynomial_function:
+ fixes p :: "real \<Rightarrow> 'a::euclidean_space"
+ shows "polynomial_function p \<Longrightarrow> valid_path p"
+by (force simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_polymonial_function C1_differentiable_polynomial_function)
+
+lemma valid_path_subpath_trivial [simp]:
+ fixes g :: "real \<Rightarrow> 'a::euclidean_space"
+ shows "z \<noteq> g x \<Longrightarrow> valid_path (subpath x x g)"
+ by (simp add: subpath_def valid_path_polynomial_function)
+
+subsection\<open>Partial circle path\<close>
+
+definition\<^marker>\<open>tag important\<close> part_circlepath :: "[complex, real, real, real, real] \<Rightarrow> complex"
+ where "part_circlepath z r s t \<equiv> \<lambda>x. z + of_real r * exp (\<i> * of_real (linepath s t x))"
+
+lemma pathstart_part_circlepath [simp]:
+ "pathstart(part_circlepath z r s t) = z + r*exp(\<i> * s)"
+by (metis part_circlepath_def pathstart_def pathstart_linepath)
+
+lemma pathfinish_part_circlepath [simp]:
+ "pathfinish(part_circlepath z r s t) = z + r*exp(\<i>*t)"
+by (metis part_circlepath_def pathfinish_def pathfinish_linepath)
+
+lemma reversepath_part_circlepath[simp]:
+ "reversepath (part_circlepath z r s t) = part_circlepath z r t s"
+ unfolding part_circlepath_def reversepath_def linepath_def
+ by (auto simp:algebra_simps)
+
+lemma has_vector_derivative_part_circlepath [derivative_intros]:
+ "((part_circlepath z r s t) has_vector_derivative
+ (\<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)))
+ (at x within X)"
+ apply (simp add: part_circlepath_def linepath_def scaleR_conv_of_real)
+ apply (rule has_vector_derivative_real_field)
+ apply (rule derivative_eq_intros | simp)+
+ done
+
+lemma differentiable_part_circlepath:
+ "part_circlepath c r a b differentiable at x within A"
+ using has_vector_derivative_part_circlepath[of c r a b x A] differentiableI_vector by blast
+
+lemma vector_derivative_part_circlepath:
+ "vector_derivative (part_circlepath z r s t) (at x) =
+ \<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)"
+ using has_vector_derivative_part_circlepath vector_derivative_at by blast
+
+lemma vector_derivative_part_circlepath01:
+ "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
+ \<Longrightarrow> vector_derivative (part_circlepath z r s t) (at x within {0..1}) =
+ \<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)"
+ using has_vector_derivative_part_circlepath
+ by (auto simp: vector_derivative_at_within_ivl)
+
+lemma valid_path_part_circlepath [simp]: "valid_path (part_circlepath z r s t)"
+ apply (simp add: valid_path_def)
+ apply (rule C1_differentiable_imp_piecewise)
+ apply (auto simp: C1_differentiable_on_eq vector_derivative_works vector_derivative_part_circlepath has_vector_derivative_part_circlepath
+ intro!: continuous_intros)
+ done
+
+lemma path_part_circlepath [simp]: "path (part_circlepath z r s t)"
+ by (simp add: valid_path_imp_path)
+
+proposition path_image_part_circlepath:
+ assumes "s \<le> t"
+ shows "path_image (part_circlepath z r s t) = {z + r * exp(\<i> * of_real x) | x. s \<le> x \<and> x \<le> t}"
+proof -
+ { fix z::real
+ assume "0 \<le> z" "z \<le> 1"
+ with \<open>s \<le> t\<close> have "\<exists>x. (exp (\<i> * linepath s t z) = exp (\<i> * of_real x)) \<and> s \<le> x \<and> x \<le> t"
+ apply (rule_tac x="(1 - z) * s + z * t" in exI)
+ apply (simp add: linepath_def scaleR_conv_of_real algebra_simps)
+ apply (rule conjI)
+ using mult_right_mono apply blast
+ using affine_ineq by (metis "mult.commute")
+ }
+ moreover
+ { fix z
+ assume "s \<le> z" "z \<le> t"
+ then have "z + of_real r * exp (\<i> * of_real z) \<in> (\<lambda>x. z + of_real r * exp (\<i> * linepath s t x)) ` {0..1}"
+ apply (rule_tac x="(z - s)/(t - s)" in image_eqI)
+ apply (simp add: linepath_def scaleR_conv_of_real divide_simps exp_eq)
+ apply (auto simp: field_split_simps)
+ done
+ }
+ ultimately show ?thesis
+ by (fastforce simp add: path_image_def part_circlepath_def)
+qed
+
+lemma path_image_part_circlepath':
+ "path_image (part_circlepath z r s t) = (\<lambda>x. z + r * cis x) ` closed_segment s t"
+proof -
+ have "path_image (part_circlepath z r s t) =
+ (\<lambda>x. z + r * exp(\<i> * of_real x)) ` linepath s t ` {0..1}"
+ by (simp add: image_image path_image_def part_circlepath_def)
+ also have "linepath s t ` {0..1} = closed_segment s t"
+ by (rule linepath_image_01)
+ finally show ?thesis by (simp add: cis_conv_exp)
+qed
+
+lemma path_image_part_circlepath_subset:
+ "\<lbrakk>s \<le> t; 0 \<le> r\<rbrakk> \<Longrightarrow> path_image(part_circlepath z r s t) \<subseteq> sphere z r"
+by (auto simp: path_image_part_circlepath sphere_def dist_norm algebra_simps norm_mult)
+
+lemma in_path_image_part_circlepath:
+ assumes "w \<in> path_image(part_circlepath z r s t)" "s \<le> t" "0 \<le> r"
+ shows "norm(w - z) = r"
+proof -
+ have "w \<in> {c. dist z c = r}"
+ by (metis (no_types) path_image_part_circlepath_subset sphere_def subset_eq assms)
+ thus ?thesis
+ by (simp add: dist_norm norm_minus_commute)
+qed
+
+lemma path_image_part_circlepath_subset':
+ assumes "r \<ge> 0"
+ shows "path_image (part_circlepath z r s t) \<subseteq> sphere z r"
+proof (cases "s \<le> t")
+ case True
+ thus ?thesis using path_image_part_circlepath_subset[of s t r z] assms by simp
+next
+ case False
+ thus ?thesis using path_image_part_circlepath_subset[of t s r z] assms
+ by (subst reversepath_part_circlepath [symmetric], subst path_image_reversepath) simp_all
+qed
+
+lemma part_circlepath_cnj: "cnj (part_circlepath c r a b x) = part_circlepath (cnj c) r (-a) (-b) x"
+ by (simp add: part_circlepath_def exp_cnj linepath_def algebra_simps)
+
+lemma contour_integral_bound_part_circlepath:
+ assumes "f contour_integrable_on part_circlepath c r a b"
+ assumes "B \<ge> 0" "r \<ge> 0" "\<And>x. x \<in> path_image (part_circlepath c r a b) \<Longrightarrow> norm (f x) \<le> B"
+ shows "norm (contour_integral (part_circlepath c r a b) f) \<le> B * r * \<bar>b - a\<bar>"
+proof -
+ let ?I = "integral {0..1} (\<lambda>x. f (part_circlepath c r a b x) * \<i> * of_real (r * (b - a)) *
+ exp (\<i> * linepath a b x))"
+ have "norm ?I \<le> integral {0..1} (\<lambda>x::real. B * 1 * (r * \<bar>b - a\<bar>) * 1)"
+ proof (rule integral_norm_bound_integral, goal_cases)
+ case 1
+ with assms(1) show ?case
+ by (simp add: contour_integrable_on vector_derivative_part_circlepath mult_ac)
+ next
+ case (3 x)
+ with assms(2-) show ?case unfolding norm_mult norm_of_real abs_mult
+ by (intro mult_mono) (auto simp: path_image_def)
+ qed auto
+ also have "?I = contour_integral (part_circlepath c r a b) f"
+ by (simp add: contour_integral_integral vector_derivative_part_circlepath mult_ac)
+ finally show ?thesis by simp
+qed
+
+lemma has_contour_integral_part_circlepath_iff:
+ assumes "a < b"
+ shows "(f has_contour_integral I) (part_circlepath c r a b) \<longleftrightarrow>
+ ((\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) has_integral I) {a..b}"
+proof -
+ have "(f has_contour_integral I) (part_circlepath c r a b) \<longleftrightarrow>
+ ((\<lambda>x. f (part_circlepath c r a b x) * vector_derivative (part_circlepath c r a b)
+ (at x within {0..1})) has_integral I) {0..1}"
+ unfolding has_contour_integral_def ..
+ also have "\<dots> \<longleftrightarrow> ((\<lambda>x. f (part_circlepath c r a b x) * r * (b - a) * \<i> *
+ cis (linepath a b x)) has_integral I) {0..1}"
+ by (intro has_integral_cong, subst vector_derivative_part_circlepath01)
+ (simp_all add: cis_conv_exp)
+ also have "\<dots> \<longleftrightarrow> ((\<lambda>x. f (c + r * exp (\<i> * linepath (of_real a) (of_real b) x)) *
+ r * \<i> * exp (\<i> * linepath (of_real a) (of_real b) x) *
+ vector_derivative (linepath (of_real a) (of_real b))
+ (at x within {0..1})) has_integral I) {0..1}"
+ by (intro has_integral_cong, subst vector_derivative_linepath_within)
+ (auto simp: part_circlepath_def cis_conv_exp of_real_linepath [symmetric])
+ also have "\<dots> \<longleftrightarrow> ((\<lambda>z. f (c + r * exp (\<i> * z)) * r * \<i> * exp (\<i> * z)) has_contour_integral I)
+ (linepath (of_real a) (of_real b))"
+ by (simp add: has_contour_integral_def)
+ also have "\<dots> \<longleftrightarrow> ((\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) has_integral I) {a..b}" using assms
+ by (subst has_contour_integral_linepath_Reals_iff) (simp_all add: cis_conv_exp)
+ finally show ?thesis .
+qed
+
+lemma contour_integrable_part_circlepath_iff:
+ assumes "a < b"
+ shows "f contour_integrable_on (part_circlepath c r a b) \<longleftrightarrow>
+ (\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}"
+ using assms by (auto simp: contour_integrable_on_def integrable_on_def
+ has_contour_integral_part_circlepath_iff)
+
+lemma contour_integral_part_circlepath_eq:
+ assumes "a < b"
+ shows "contour_integral (part_circlepath c r a b) f =
+ integral {a..b} (\<lambda>t. f (c + r * cis t) * r * \<i> * cis t)"
+proof (cases "f contour_integrable_on part_circlepath c r a b")
+ case True
+ hence "(\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}"
+ using assms by (simp add: contour_integrable_part_circlepath_iff)
+ with True show ?thesis
+ using has_contour_integral_part_circlepath_iff[OF assms]
+ contour_integral_unique has_integral_integrable_integral by blast
+next
+ case False
+ hence "\<not>(\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}"
+ using assms by (simp add: contour_integrable_part_circlepath_iff)
+ with False show ?thesis
+ by (simp add: not_integrable_contour_integral not_integrable_integral)
+qed
+
+lemma contour_integral_part_circlepath_reverse:
+ "contour_integral (part_circlepath c r a b) f = -contour_integral (part_circlepath c r b a) f"
+ by (subst reversepath_part_circlepath [symmetric], subst contour_integral_reversepath) simp_all
+
+lemma contour_integral_part_circlepath_reverse':
+ "b < a \<Longrightarrow> contour_integral (part_circlepath c r a b) f =
+ -contour_integral (part_circlepath c r b a) f"
+ by (rule contour_integral_part_circlepath_reverse)
+
+lemma finite_bounded_log: "finite {z::complex. norm z \<le> b \<and> exp z = w}"
+proof (cases "w = 0")
+ case True then show ?thesis by auto
+next
+ case False
+ have *: "finite {x. cmod (complex_of_real (2 * real_of_int x * pi) * \<i>) \<le> b + cmod (Ln w)}"
+ apply (simp add: norm_mult finite_int_iff_bounded_le)
+ apply (rule_tac x="\<lfloor>(b + cmod (Ln w)) / (2*pi)\<rfloor>" in exI)
+ apply (auto simp: field_split_simps le_floor_iff)
+ done
+ have [simp]: "\<And>P f. {z. P z \<and> (\<exists>n. z = f n)} = f ` {n. P (f n)}"
+ by blast
+ show ?thesis
+ apply (subst exp_Ln [OF False, symmetric])
+ apply (simp add: exp_eq)
+ using norm_add_leD apply (fastforce intro: finite_subset [OF _ *])
+ done
+qed
+
+lemma finite_bounded_log2:
+ fixes a::complex
+ assumes "a \<noteq> 0"
+ shows "finite {z. norm z \<le> b \<and> exp(a*z) = w}"
+proof -
+ have *: "finite ((\<lambda>z. z / a) ` {z. cmod z \<le> b * cmod a \<and> exp z = w})"
+ by (rule finite_imageI [OF finite_bounded_log])
+ show ?thesis
+ by (rule finite_subset [OF _ *]) (force simp: assms norm_mult)
+qed
+
+lemma has_contour_integral_bound_part_circlepath_strong:
+ assumes fi: "(f has_contour_integral i) (part_circlepath z r s t)"
+ and "finite k" and le: "0 \<le> B" "0 < r" "s \<le> t"
+ and B: "\<And>x. x \<in> path_image(part_circlepath z r s t) - k \<Longrightarrow> norm(f x) \<le> B"
+ shows "cmod i \<le> B * r * (t - s)"
+proof -
+ consider "s = t" | "s < t" using \<open>s \<le> t\<close> by linarith
+ then show ?thesis
+ proof cases
+ case 1 with fi [unfolded has_contour_integral]
+ have "i = 0" by (simp add: vector_derivative_part_circlepath)
+ with assms show ?thesis by simp
+ next
+ case 2
+ have [simp]: "\<bar>r\<bar> = r" using \<open>r > 0\<close> by linarith
+ have [simp]: "cmod (complex_of_real t - complex_of_real s) = t-s"
+ by (metis "2" abs_of_pos diff_gt_0_iff_gt norm_of_real of_real_diff)
+ have "finite (part_circlepath z r s t -` {y} \<inter> {0..1})" if "y \<in> k" for y
+ proof -
+ define w where "w = (y - z)/of_real r / exp(\<i> * of_real s)"
+ have fin: "finite (of_real -` {z. cmod z \<le> 1 \<and> exp (\<i> * complex_of_real (t - s) * z) = w})"
+ apply (rule finite_vimageI [OF finite_bounded_log2])
+ using \<open>s < t\<close> apply (auto simp: inj_of_real)
+ done
+ show ?thesis
+ apply (simp add: part_circlepath_def linepath_def vimage_def)
+ apply (rule finite_subset [OF _ fin])
+ using le
+ apply (auto simp: w_def algebra_simps scaleR_conv_of_real exp_add exp_diff)
+ done
+ qed
+ then have fin01: "finite ((part_circlepath z r s t) -` k \<inter> {0..1})"
+ by (rule finite_finite_vimage_IntI [OF \<open>finite k\<close>])
+ have **: "((\<lambda>x. if (part_circlepath z r s t x) \<in> k then 0
+ else f(part_circlepath z r s t x) *
+ vector_derivative (part_circlepath z r s t) (at x)) has_integral i) {0..1}"
+ by (rule has_integral_spike [OF negligible_finite [OF fin01]]) (use fi has_contour_integral in auto)
+ have *: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1; part_circlepath z r s t x \<notin> k\<rbrakk> \<Longrightarrow> cmod (f (part_circlepath z r s t x)) \<le> B"
+ by (auto intro!: B [unfolded path_image_def image_def, simplified])
+ show ?thesis
+ apply (rule has_integral_bound [where 'a=real, simplified, OF _ **, simplified])
+ using assms apply force
+ apply (simp add: norm_mult vector_derivative_part_circlepath)
+ using le * "2" \<open>r > 0\<close> by auto
+ qed
+qed
+
+lemma has_contour_integral_bound_part_circlepath:
+ "\<lbrakk>(f has_contour_integral i) (part_circlepath z r s t);
+ 0 \<le> B; 0 < r; s \<le> t;
+ \<And>x. x \<in> path_image(part_circlepath z r s t) \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
+ \<Longrightarrow> norm i \<le> B*r*(t - s)"
+ by (auto intro: has_contour_integral_bound_part_circlepath_strong)
+
+lemma contour_integrable_continuous_part_circlepath:
+ "continuous_on (path_image (part_circlepath z r s t)) f
+ \<Longrightarrow> f contour_integrable_on (part_circlepath z r s t)"
+ apply (simp add: contour_integrable_on has_contour_integral_def vector_derivative_part_circlepath path_image_def)
+ apply (rule integrable_continuous_real)
+ apply (fast intro: path_part_circlepath [unfolded path_def] continuous_intros continuous_on_compose2 [where g=f, OF _ _ order_refl])
+ done
+
+lemma simple_path_part_circlepath:
+ "simple_path(part_circlepath z r s t) \<longleftrightarrow> (r \<noteq> 0 \<and> s \<noteq> t \<and> \<bar>s - t\<bar> \<le> 2*pi)"
+proof (cases "r = 0 \<or> s = t")
+ case True
+ then show ?thesis
+ unfolding part_circlepath_def simple_path_def
+ by (rule disjE) (force intro: bexI [where x = "1/4"] bexI [where x = "1/3"])+
+next
+ case False then have "r \<noteq> 0" "s \<noteq> t" by auto
+ have *: "\<And>x y z s t. \<i>*((1 - x) * s + x * t) = \<i>*(((1 - y) * s + y * t)) + z \<longleftrightarrow> \<i>*(x - y) * (t - s) = z"
+ by (simp add: algebra_simps)
+ have abs01: "\<And>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1
+ \<Longrightarrow> (x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0 \<longleftrightarrow> \<bar>x - y\<bar> \<in> {0,1})"
+ by auto
+ have **: "\<And>x y. (\<exists>n. (complex_of_real x - of_real y) * (of_real t - of_real s) = 2 * (of_int n * of_real pi)) \<longleftrightarrow>
+ (\<exists>n. \<bar>x - y\<bar> * (t - s) = 2 * (of_int n * pi))"
+ by (force simp: algebra_simps abs_if dest: arg_cong [where f=Re] arg_cong [where f=complex_of_real]
+ intro: exI [where x = "-n" for n])
+ have 1: "\<bar>s - t\<bar> \<le> 2 * pi"
+ if "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow> (\<exists>n. x * (t - s) = 2 * (real_of_int n * pi)) \<longrightarrow> x = 0 \<or> x = 1"
+ proof (rule ccontr)
+ assume "\<not> \<bar>s - t\<bar> \<le> 2 * pi"
+ then have *: "\<And>n. t - s \<noteq> of_int n * \<bar>s - t\<bar>"
+ using False that [of "2*pi / \<bar>t - s\<bar>"]
+ by (simp add: abs_minus_commute divide_simps)
+ show False
+ using * [of 1] * [of "-1"] by auto
+ qed
+ have 2: "\<bar>s - t\<bar> = \<bar>2 * (real_of_int n * pi) / x\<bar>" if "x \<noteq> 0" "x * (t - s) = 2 * (real_of_int n * pi)" for x n
+ proof -
+ have "t-s = 2 * (real_of_int n * pi)/x"
+ using that by (simp add: field_simps)
+ then show ?thesis by (metis abs_minus_commute)
+ qed
+ have abs_away: "\<And>P. (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. P \<bar>x - y\<bar>) \<longleftrightarrow> (\<forall>x::real. 0 \<le> x \<and> x \<le> 1 \<longrightarrow> P x)"
+ by force
+ show ?thesis using False
+ apply (simp add: simple_path_def)
+ apply (simp add: part_circlepath_def linepath_def exp_eq * ** abs01 del: Set.insert_iff)
+ apply (subst abs_away)
+ apply (auto simp: 1)
+ apply (rule ccontr)
+ apply (auto simp: 2 field_split_simps abs_mult dest: of_int_leD)
+ done
+qed
+
+lemma arc_part_circlepath:
+ assumes "r \<noteq> 0" "s \<noteq> t" "\<bar>s - t\<bar> < 2*pi"
+ shows "arc (part_circlepath z r s t)"
+proof -
+ have *: "x = y" if eq: "\<i> * (linepath s t x) = \<i> * (linepath s t y) + 2 * of_int n * complex_of_real pi * \<i>"
+ and x: "x \<in> {0..1}" and y: "y \<in> {0..1}" for x y n
+ proof (rule ccontr)
+ assume "x \<noteq> y"
+ have "(linepath s t x) = (linepath s t y) + 2 * of_int n * complex_of_real pi"
+ by (metis add_divide_eq_iff complex_i_not_zero mult.commute nonzero_mult_div_cancel_left eq)
+ then have "s*y + t*x = s*x + (t*y + of_int n * (pi * 2))"
+ by (force simp: algebra_simps linepath_def dest: arg_cong [where f=Re])
+ with \<open>x \<noteq> y\<close> have st: "s-t = (of_int n * (pi * 2) / (y-x))"
+ by (force simp: field_simps)
+ have "\<bar>real_of_int n\<bar> < \<bar>y - x\<bar>"
+ using assms \<open>x \<noteq> y\<close> by (simp add: st abs_mult field_simps)
+ then show False
+ using assms x y st by (auto dest: of_int_lessD)
+ qed
+ show ?thesis
+ using assms
+ apply (simp add: arc_def)
+ apply (simp add: part_circlepath_def inj_on_def exp_eq)
+ apply (blast intro: *)
+ done
+qed
+
+subsection\<open>Special case of one complete circle\<close>
+
+definition\<^marker>\<open>tag important\<close> circlepath :: "[complex, real, real] \<Rightarrow> complex"
+ where "circlepath z r \<equiv> part_circlepath z r 0 (2*pi)"
+
+lemma circlepath: "circlepath z r = (\<lambda>x. z + r * exp(2 * of_real pi * \<i> * of_real x))"
+ by (simp add: circlepath_def part_circlepath_def linepath_def algebra_simps)
+
+lemma pathstart_circlepath [simp]: "pathstart (circlepath z r) = z + r"
+ by (simp add: circlepath_def)
+
+lemma pathfinish_circlepath [simp]: "pathfinish (circlepath z r) = z + r"
+ by (simp add: circlepath_def) (metis exp_two_pi_i mult.commute)
+
+lemma circlepath_minus: "circlepath z (-r) x = circlepath z r (x + 1/2)"
+proof -
+ have "z + of_real r * exp (2 * pi * \<i> * (x + 1/2)) =
+ z + of_real r * exp (2 * pi * \<i> * x + pi * \<i>)"
+ by (simp add: divide_simps) (simp add: algebra_simps)
+ also have "\<dots> = z - r * exp (2 * pi * \<i> * x)"
+ by (simp add: exp_add)
+ finally show ?thesis
+ by (simp add: circlepath path_image_def sphere_def dist_norm)
+qed
+
+lemma circlepath_add1: "circlepath z r (x+1) = circlepath z r x"
+ using circlepath_minus [of z r "x+1/2"] circlepath_minus [of z "-r" x]
+ by (simp add: add.commute)
+
+lemma circlepath_add_half: "circlepath z r (x + 1/2) = circlepath z r (x - 1/2)"
+ using circlepath_add1 [of z r "x-1/2"]
+ by (simp add: add.commute)
+
+lemma path_image_circlepath_minus_subset:
+ "path_image (circlepath z (-r)) \<subseteq> path_image (circlepath z r)"
+ apply (simp add: path_image_def image_def circlepath_minus, clarify)
+ apply (case_tac "xa \<le> 1/2", force)
+ apply (force simp: circlepath_add_half)+
+ done
+
+lemma path_image_circlepath_minus: "path_image (circlepath z (-r)) = path_image (circlepath z r)"
+ using path_image_circlepath_minus_subset by fastforce
+
+lemma has_vector_derivative_circlepath [derivative_intros]:
+ "((circlepath z r) has_vector_derivative (2 * pi * \<i> * r * exp (2 * of_real pi * \<i> * of_real x)))
+ (at x within X)"
+ apply (simp add: circlepath_def scaleR_conv_of_real)
+ apply (rule derivative_eq_intros)
+ apply (simp add: algebra_simps)
+ done
+
+lemma vector_derivative_circlepath:
+ "vector_derivative (circlepath z r) (at x) =
+ 2 * pi * \<i> * r * exp(2 * of_real pi * \<i> * x)"
+using has_vector_derivative_circlepath vector_derivative_at by blast
+
+lemma vector_derivative_circlepath01:
+ "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
+ \<Longrightarrow> vector_derivative (circlepath z r) (at x within {0..1}) =
+ 2 * pi * \<i> * r * exp(2 * of_real pi * \<i> * x)"
+ using has_vector_derivative_circlepath
+ by (auto simp: vector_derivative_at_within_ivl)
+
+lemma valid_path_circlepath [simp]: "valid_path (circlepath z r)"
+ by (simp add: circlepath_def)
+
+lemma path_circlepath [simp]: "path (circlepath z r)"
+ by (simp add: valid_path_imp_path)
+
+lemma path_image_circlepath_nonneg:
+ assumes "0 \<le> r" shows "path_image (circlepath z r) = sphere z r"
+proof -
+ have *: "x \<in> (\<lambda>u. z + (cmod (x - z)) * exp (\<i> * (of_real u * (of_real pi * 2)))) ` {0..1}" for x
+ proof (cases "x = z")
+ case True then show ?thesis by force
+ next
+ case False
+ define w where "w = x - z"
+ then have "w \<noteq> 0" by (simp add: False)
+ have **: "\<And>t. \<lbrakk>Re w = cos t * cmod w; Im w = sin t * cmod w\<rbrakk> \<Longrightarrow> w = of_real (cmod w) * exp (\<i> * t)"
+ using cis_conv_exp complex_eq_iff by auto
+ show ?thesis
+ apply (rule sincos_total_2pi [of "Re(w/of_real(norm w))" "Im(w/of_real(norm w))"])
+ apply (simp add: divide_simps \<open>w \<noteq> 0\<close> cmod_power2 [symmetric])
+ apply (rule_tac x="t / (2*pi)" in image_eqI)
+ apply (simp add: field_simps \<open>w \<noteq> 0\<close>)
+ using False **
+ apply (auto simp: w_def)
+ done
+ qed
+ show ?thesis
+ unfolding circlepath path_image_def sphere_def dist_norm
+ by (force simp: assms algebra_simps norm_mult norm_minus_commute intro: *)
+qed
+
+lemma path_image_circlepath [simp]:
+ "path_image (circlepath z r) = sphere z \<bar>r\<bar>"
+ using path_image_circlepath_minus
+ by (force simp: path_image_circlepath_nonneg abs_if)
+
+lemma has_contour_integral_bound_circlepath_strong:
+ "\<lbrakk>(f has_contour_integral i) (circlepath z r);
+ finite k; 0 \<le> B; 0 < r;
+ \<And>x. \<lbrakk>norm(x - z) = r; x \<notin> k\<rbrakk> \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
+ \<Longrightarrow> norm i \<le> B*(2*pi*r)"
+ unfolding circlepath_def
+ by (auto simp: algebra_simps in_path_image_part_circlepath dest!: has_contour_integral_bound_part_circlepath_strong)
+
+lemma has_contour_integral_bound_circlepath:
+ "\<lbrakk>(f has_contour_integral i) (circlepath z r);
+ 0 \<le> B; 0 < r; \<And>x. norm(x - z) = r \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
+ \<Longrightarrow> norm i \<le> B*(2*pi*r)"
+ by (auto intro: has_contour_integral_bound_circlepath_strong)
+
+lemma contour_integrable_continuous_circlepath:
+ "continuous_on (path_image (circlepath z r)) f
+ \<Longrightarrow> f contour_integrable_on (circlepath z r)"
+ by (simp add: circlepath_def contour_integrable_continuous_part_circlepath)
+
+lemma simple_path_circlepath: "simple_path(circlepath z r) \<longleftrightarrow> (r \<noteq> 0)"
+ by (simp add: circlepath_def simple_path_part_circlepath)
+
+lemma notin_path_image_circlepath [simp]: "cmod (w - z) < r \<Longrightarrow> w \<notin> path_image (circlepath z r)"
+ by (simp add: sphere_def dist_norm norm_minus_commute)
+
+lemma contour_integral_circlepath:
+ assumes "r > 0"
+ shows "contour_integral (circlepath z r) (\<lambda>w. 1 / (w - z)) = 2 * complex_of_real pi * \<i>"
+proof (rule contour_integral_unique)
+ show "((\<lambda>w. 1 / (w - z)) has_contour_integral 2 * complex_of_real pi * \<i>) (circlepath z r)"
+ unfolding has_contour_integral_def using assms
+ apply (subst has_integral_cong)
+ apply (simp add: vector_derivative_circlepath01)
+ using has_integral_const_real [of _ 0 1] apply (force simp: circlepath)
+ done
+qed
+
+subsection\<open> Uniform convergence of path integral\<close>
+
+text\<open>Uniform convergence when the derivative of the path is bounded, and in particular for the special case of a circle.\<close>
+
+proposition contour_integral_uniform_limit:
+ assumes ev_fint: "eventually (\<lambda>n::'a. (f n) contour_integrable_on \<gamma>) F"
+ and ul_f: "uniform_limit (path_image \<gamma>) f l F"
+ and noleB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
+ and \<gamma>: "valid_path \<gamma>"
+ and [simp]: "\<not> trivial_limit F"
+ shows "l contour_integrable_on \<gamma>" "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
+proof -
+ have "0 \<le> B" by (meson noleB [of 0] atLeastAtMost_iff norm_ge_zero order_refl order_trans zero_le_one)
+ { fix e::real
+ assume "0 < e"
+ then have "0 < e / (\<bar>B\<bar> + 1)" by simp
+ then have "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. cmod (f n x - l x) < e / (\<bar>B\<bar> + 1)"
+ using ul_f [unfolded uniform_limit_iff dist_norm] by auto
+ with ev_fint
+ obtain a where fga: "\<And>x. x \<in> {0..1} \<Longrightarrow> cmod (f a (\<gamma> x) - l (\<gamma> x)) < e / (\<bar>B\<bar> + 1)"
+ and inta: "(\<lambda>t. f a (\<gamma> t) * vector_derivative \<gamma> (at t)) integrable_on {0..1}"
+ using eventually_happens [OF eventually_conj]
+ by (fastforce simp: contour_integrable_on path_image_def)
+ have Ble: "B * e / (\<bar>B\<bar> + 1) \<le> e"
+ using \<open>0 \<le> B\<close> \<open>0 < e\<close> by (simp add: field_split_simps)
+ have "\<exists>h. (\<forall>x\<in>{0..1}. cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - h x) \<le> e) \<and> h integrable_on {0..1}"
+ proof (intro exI conjI ballI)
+ show "cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - f a (\<gamma> x) * vector_derivative \<gamma> (at x)) \<le> e"
+ if "x \<in> {0..1}" for x
+ apply (rule order_trans [OF _ Ble])
+ using noleB [OF that] fga [OF that] \<open>0 \<le> B\<close> \<open>0 < e\<close>
+ apply (simp add: norm_mult left_diff_distrib [symmetric] norm_minus_commute divide_simps)
+ apply (fastforce simp: mult_ac dest: mult_mono [OF less_imp_le])
+ done
+ qed (rule inta)
+ }
+ then show lintg: "l contour_integrable_on \<gamma>"
+ unfolding contour_integrable_on by (metis (mono_tags, lifting)integrable_uniform_limit_real)
+ { fix e::real
+ define B' where "B' = B + 1"
+ have B': "B' > 0" "B' > B" using \<open>0 \<le> B\<close> by (auto simp: B'_def)
+ assume "0 < e"
+ then have ev_no': "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. 2 * cmod (f n x - l x) < e / B'"
+ using ul_f [unfolded uniform_limit_iff dist_norm, rule_format, of "e / B' / 2"] B'
+ by (simp add: field_simps)
+ have ie: "integral {0..1::real} (\<lambda>x. e / 2) < e" using \<open>0 < e\<close> by simp
+ have *: "cmod (f x (\<gamma> t) * vector_derivative \<gamma> (at t) - l (\<gamma> t) * vector_derivative \<gamma> (at t)) \<le> e / 2"
+ if t: "t\<in>{0..1}" and leB': "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) < e / B'" for x t
+ proof -
+ have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) \<le> e * (B/ B')"
+ using mult_mono [OF less_imp_le [OF leB'] noleB] B' \<open>0 < e\<close> t by auto
+ also have "\<dots> < e"
+ by (simp add: B' \<open>0 < e\<close> mult_imp_div_pos_less)
+ finally have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) < e" .
+ then show ?thesis
+ by (simp add: left_diff_distrib [symmetric] norm_mult)
+ qed
+ have le_e: "\<And>x. \<lbrakk>\<forall>xa\<in>{0..1}. 2 * cmod (f x (\<gamma> xa) - l (\<gamma> xa)) < e / B'; f x contour_integrable_on \<gamma>\<rbrakk>
+ \<Longrightarrow> cmod (integral {0..1}
+ (\<lambda>u. f x (\<gamma> u) * vector_derivative \<gamma> (at u) - l (\<gamma> u) * vector_derivative \<gamma> (at u))) < e"
+ apply (rule le_less_trans [OF integral_norm_bound_integral ie])
+ apply (simp add: lintg integrable_diff contour_integrable_on [symmetric])
+ apply (blast intro: *)+
+ done
+ have "\<forall>\<^sub>F x in F. dist (contour_integral \<gamma> (f x)) (contour_integral \<gamma> l) < e"
+ apply (rule eventually_mono [OF eventually_conj [OF ev_no' ev_fint]])
+ apply (simp add: dist_norm contour_integrable_on path_image_def contour_integral_integral)
+ apply (simp add: lintg integral_diff [symmetric] contour_integrable_on [symmetric] le_e)
+ done
+ }
+ then show "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
+ by (rule tendstoI)
+qed
+
+corollary\<^marker>\<open>tag unimportant\<close> contour_integral_uniform_limit_circlepath:
+ assumes "\<forall>\<^sub>F n::'a in F. (f n) contour_integrable_on (circlepath z r)"
+ and "uniform_limit (sphere z r) f l F"
+ and "\<not> trivial_limit F" "0 < r"
+ shows "l contour_integrable_on (circlepath z r)"
+ "((\<lambda>n. contour_integral (circlepath z r) (f n)) \<longlongrightarrow> contour_integral (circlepath z r) l) F"
+ using assms by (auto simp: vector_derivative_circlepath norm_mult intro!: contour_integral_uniform_limit)
+
+end
\ No newline at end of file
--- a/src/HOL/Complex_Analysis/Great_Picard.thy Mon Dec 02 22:40:16 2019 -0500
+++ b/src/HOL/Complex_Analysis/Great_Picard.thy Mon Dec 02 17:51:54 2019 +0100
@@ -4,7 +4,6 @@
theory Great_Picard
imports Conformal_Mappings
-
begin
subsection\<open>Schottky's theorem\<close>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/Residue_Theorem.thy Mon Dec 02 17:51:54 2019 +0100
@@ -0,0 +1,862 @@
+section \<open>The Residue Theorem, the Argument Principle and Rouch\'{e}'s Theorem\<close>
+theory Residue_Theorem
+ imports Complex_Residues
+begin
+
+subsection \<open>Cauchy's residue theorem\<close>
+
+lemma get_integrable_path:
+ assumes "open s" "connected (s-pts)" "finite pts" "f holomorphic_on (s-pts) " "a\<in>s-pts" "b\<in>s-pts"
+ obtains g where "valid_path g" "pathstart g = a" "pathfinish g = b"
+ "path_image g \<subseteq> s-pts" "f contour_integrable_on g" using assms
+proof (induct arbitrary:s thesis a rule:finite_induct[OF \<open>finite pts\<close>])
+ case 1
+ obtain g where "valid_path g" "path_image g \<subseteq> s" "pathstart g = a" "pathfinish g = b"
+ using connected_open_polynomial_connected[OF \<open>open s\<close>,of a b ] \<open>connected (s - {})\<close>
+ valid_path_polynomial_function "1.prems"(6) "1.prems"(7) by auto
+ moreover have "f contour_integrable_on g"
+ using contour_integrable_holomorphic_simple[OF _ \<open>open s\<close> \<open>valid_path g\<close> \<open>path_image g \<subseteq> s\<close>,of f]
+ \<open>f holomorphic_on s - {}\<close>
+ by auto
+ ultimately show ?case using "1"(1)[of g] by auto
+next
+ case idt:(2 p pts)
+ obtain e where "e>0" and e:"\<forall>w\<in>ball a e. w \<in> s \<and> (w \<noteq> a \<longrightarrow> w \<notin> insert p pts)"
+ using finite_ball_avoid[OF \<open>open s\<close> \<open>finite (insert p pts)\<close>, of a]
+ \<open>a \<in> s - insert p pts\<close>
+ by auto
+ define a' where "a' \<equiv> a+e/2"
+ have "a'\<in>s-{p} -pts" using e[rule_format,of "a+e/2"] \<open>e>0\<close>
+ by (auto simp add:dist_complex_def a'_def)
+ then obtain g' where g'[simp]:"valid_path g'" "pathstart g' = a'" "pathfinish g' = b"
+ "path_image g' \<subseteq> s - {p} - pts" "f contour_integrable_on g'"
+ using idt.hyps(3)[of a' "s-{p}"] idt.prems idt.hyps(1)
+ by (metis Diff_insert2 open_delete)
+ define g where "g \<equiv> linepath a a' +++ g'"
+ have "valid_path g" unfolding g_def by (auto intro: valid_path_join)
+ moreover have "pathstart g = a" and "pathfinish g = b" unfolding g_def by auto
+ moreover have "path_image g \<subseteq> s - insert p pts" unfolding g_def
+ proof (rule subset_path_image_join)
+ have "closed_segment a a' \<subseteq> ball a e" using \<open>e>0\<close>
+ by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
+ then show "path_image (linepath a a') \<subseteq> s - insert p pts" using e idt(9)
+ by auto
+ next
+ show "path_image g' \<subseteq> s - insert p pts" using g'(4) by blast
+ qed
+ moreover have "f contour_integrable_on g"
+ proof -
+ have "closed_segment a a' \<subseteq> ball a e" using \<open>e>0\<close>
+ by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
+ then have "continuous_on (closed_segment a a') f"
+ using e idt.prems(6) holomorphic_on_imp_continuous_on[OF idt.prems(5)]
+ apply (elim continuous_on_subset)
+ by auto
+ then have "f contour_integrable_on linepath a a'"
+ using contour_integrable_continuous_linepath by auto
+ then show ?thesis unfolding g_def
+ apply (rule contour_integrable_joinI)
+ by (auto simp add: \<open>e>0\<close>)
+ qed
+ ultimately show ?case using idt.prems(1)[of g] by auto
+qed
+
+lemma Cauchy_theorem_aux:
+ assumes "open s" "connected (s-pts)" "finite pts" "pts \<subseteq> s" "f holomorphic_on s-pts"
+ "valid_path g" "pathfinish g = pathstart g" "path_image g \<subseteq> s-pts"
+ "\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0"
+ "\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
+ shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
+ using assms
+proof (induct arbitrary:s g rule:finite_induct[OF \<open>finite pts\<close>])
+ case 1
+ then show ?case by (simp add: Cauchy_theorem_global contour_integral_unique)
+next
+ case (2 p pts)
+ note fin[simp] = \<open>finite (insert p pts)\<close>
+ and connected = \<open>connected (s - insert p pts)\<close>
+ and valid[simp] = \<open>valid_path g\<close>
+ and g_loop[simp] = \<open>pathfinish g = pathstart g\<close>
+ and holo[simp]= \<open>f holomorphic_on s - insert p pts\<close>
+ and path_img = \<open>path_image g \<subseteq> s - insert p pts\<close>
+ and winding = \<open>\<forall>z. z \<notin> s \<longrightarrow> winding_number g z = 0\<close>
+ and h = \<open>\<forall>pa\<in>s. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s \<and> (w \<noteq> pa \<longrightarrow> w \<notin> insert p pts))\<close>
+ have "h p>0" and "p\<in>s"
+ and h_p: "\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> insert p pts)"
+ using h \<open>insert p pts \<subseteq> s\<close> by auto
+ obtain pg where pg[simp]: "valid_path pg" "pathstart pg = pathstart g" "pathfinish pg=p+h p"
+ "path_image pg \<subseteq> s-insert p pts" "f contour_integrable_on pg"
+ proof -
+ have "p + h p\<in>cball p (h p)" using h[rule_format,of p]
+ by (simp add: \<open>p \<in> s\<close> dist_norm)
+ then have "p + h p \<in> s - insert p pts" using h[rule_format,of p] \<open>insert p pts \<subseteq> s\<close>
+ by fastforce
+ moreover have "pathstart g \<in> s - insert p pts " using path_img by auto
+ ultimately show ?thesis
+ using get_integrable_path[OF \<open>open s\<close> connected fin holo,of "pathstart g" "p+h p"] that
+ by blast
+ qed
+ obtain n::int where "n=winding_number g p"
+ using integer_winding_number[OF _ g_loop,of p] valid path_img
+ by (metis DiffD2 Ints_cases insertI1 subset_eq valid_path_imp_path)
+ define p_circ where "p_circ \<equiv> circlepath p (h p)"
+ define p_circ_pt where "p_circ_pt \<equiv> linepath (p+h p) (p+h p)"
+ define n_circ where "n_circ \<equiv> \<lambda>n. ((+++) p_circ ^^ n) p_circ_pt"
+ define cp where "cp \<equiv> if n\<ge>0 then reversepath (n_circ (nat n)) else n_circ (nat (- n))"
+ have n_circ:"valid_path (n_circ k)"
+ "winding_number (n_circ k) p = k"
+ "pathstart (n_circ k) = p + h p" "pathfinish (n_circ k) = p + h p"
+ "path_image (n_circ k) = (if k=0 then {p + h p} else sphere p (h p))"
+ "p \<notin> path_image (n_circ k)"
+ "\<And>p'. p'\<notin>s - pts \<Longrightarrow> winding_number (n_circ k) p'=0 \<and> p'\<notin>path_image (n_circ k)"
+ "f contour_integrable_on (n_circ k)"
+ "contour_integral (n_circ k) f = k * contour_integral p_circ f"
+ for k
+ proof (induct k)
+ case 0
+ show "valid_path (n_circ 0)"
+ and "path_image (n_circ 0) = (if 0=0 then {p + h p} else sphere p (h p))"
+ and "winding_number (n_circ 0) p = of_nat 0"
+ and "pathstart (n_circ 0) = p + h p"
+ and "pathfinish (n_circ 0) = p + h p"
+ and "p \<notin> path_image (n_circ 0)"
+ unfolding n_circ_def p_circ_pt_def using \<open>h p > 0\<close>
+ by (auto simp add: dist_norm)
+ show "winding_number (n_circ 0) p'=0 \<and> p'\<notin>path_image (n_circ 0)" when "p'\<notin>s- pts" for p'
+ unfolding n_circ_def p_circ_pt_def
+ apply (auto intro!:winding_number_trivial)
+ by (metis Diff_iff pathfinish_in_path_image pg(3) pg(4) subsetCE subset_insertI that)+
+ show "f contour_integrable_on (n_circ 0)"
+ unfolding n_circ_def p_circ_pt_def
+ by (auto intro!:contour_integrable_continuous_linepath simp add:continuous_on_sing)
+ show "contour_integral (n_circ 0) f = of_nat 0 * contour_integral p_circ f"
+ unfolding n_circ_def p_circ_pt_def by auto
+ next
+ case (Suc k)
+ have n_Suc:"n_circ (Suc k) = p_circ +++ n_circ k" unfolding n_circ_def by auto
+ have pcirc:"p \<notin> path_image p_circ" "valid_path p_circ" "pathfinish p_circ = pathstart (n_circ k)"
+ using Suc(3) unfolding p_circ_def using \<open>h p > 0\<close> by (auto simp add: p_circ_def)
+ have pcirc_image:"path_image p_circ \<subseteq> s - insert p pts"
+ proof -
+ have "path_image p_circ \<subseteq> cball p (h p)" using \<open>0 < h p\<close> p_circ_def by auto
+ then show ?thesis using h_p pcirc(1) by auto
+ qed
+ have pcirc_integrable:"f contour_integrable_on p_circ"
+ by (auto simp add:p_circ_def intro!: pcirc_image[unfolded p_circ_def]
+ contour_integrable_continuous_circlepath holomorphic_on_imp_continuous_on
+ holomorphic_on_subset[OF holo])
+ show "valid_path (n_circ (Suc k))"
+ using valid_path_join[OF pcirc(2) Suc(1) pcirc(3)] unfolding n_circ_def by auto
+ show "path_image (n_circ (Suc k))
+ = (if Suc k = 0 then {p + complex_of_real (h p)} else sphere p (h p))"
+ proof -
+ have "path_image p_circ = sphere p (h p)"
+ unfolding p_circ_def using \<open>0 < h p\<close> by auto
+ then show ?thesis unfolding n_Suc using Suc.hyps(5) \<open>h p>0\<close>
+ by (auto simp add: path_image_join[OF pcirc(3)] dist_norm)
+ qed
+ then show "p \<notin> path_image (n_circ (Suc k))" using \<open>h p>0\<close> by auto
+ show "winding_number (n_circ (Suc k)) p = of_nat (Suc k)"
+ proof -
+ have "winding_number p_circ p = 1"
+ by (simp add: \<open>h p > 0\<close> p_circ_def winding_number_circlepath_centre)
+ moreover have "p \<notin> path_image (n_circ k)" using Suc(5) \<open>h p>0\<close> by auto
+ then have "winding_number (p_circ +++ n_circ k) p
+ = winding_number p_circ p + winding_number (n_circ k) p"
+ using valid_path_imp_path Suc.hyps(1) Suc.hyps(2) pcirc
+ apply (intro winding_number_join)
+ by auto
+ ultimately show ?thesis using Suc(2) unfolding n_circ_def
+ by auto
+ qed
+ show "pathstart (n_circ (Suc k)) = p + h p"
+ by (simp add: n_circ_def p_circ_def)
+ show "pathfinish (n_circ (Suc k)) = p + h p"
+ using Suc(4) unfolding n_circ_def by auto
+ show "winding_number (n_circ (Suc k)) p'=0 \<and> p'\<notin>path_image (n_circ (Suc k))" when "p'\<notin>s-pts" for p'
+ proof -
+ have " p' \<notin> path_image p_circ" using \<open>p \<in> s\<close> h p_circ_def that using pcirc_image by blast
+ moreover have "p' \<notin> path_image (n_circ k)"
+ using Suc.hyps(7) that by blast
+ moreover have "winding_number p_circ p' = 0"
+ proof -
+ have "path_image p_circ \<subseteq> cball p (h p)"
+ using h unfolding p_circ_def using \<open>p \<in> s\<close> by fastforce
+ moreover have "p'\<notin>cball p (h p)" using \<open>p \<in> s\<close> h that "2.hyps"(2) by fastforce
+ ultimately show ?thesis unfolding p_circ_def
+ apply (intro winding_number_zero_outside)
+ by auto
+ qed
+ ultimately show ?thesis
+ unfolding n_Suc
+ apply (subst winding_number_join)
+ by (auto simp: valid_path_imp_path pcirc Suc that not_in_path_image_join Suc.hyps(7)[OF that])
+ qed
+ show "f contour_integrable_on (n_circ (Suc k))"
+ unfolding n_Suc
+ by (rule contour_integrable_joinI[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)])
+ show "contour_integral (n_circ (Suc k)) f = (Suc k) * contour_integral p_circ f"
+ unfolding n_Suc
+ by (auto simp add:contour_integral_join[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)]
+ Suc(9) algebra_simps)
+ qed
+ have cp[simp]:"pathstart cp = p + h p" "pathfinish cp = p + h p"
+ "valid_path cp" "path_image cp \<subseteq> s - insert p pts"
+ "winding_number cp p = - n"
+ "\<And>p'. p'\<notin>s - pts \<Longrightarrow> winding_number cp p'=0 \<and> p' \<notin> path_image cp"
+ "f contour_integrable_on cp"
+ "contour_integral cp f = - n * contour_integral p_circ f"
+ proof -
+ show "pathstart cp = p + h p" and "pathfinish cp = p + h p" and "valid_path cp"
+ using n_circ unfolding cp_def by auto
+ next
+ have "sphere p (h p) \<subseteq> s - insert p pts"
+ using h[rule_format,of p] \<open>insert p pts \<subseteq> s\<close> by force
+ moreover have "p + complex_of_real (h p) \<in> s - insert p pts"
+ using pg(3) pg(4) by (metis pathfinish_in_path_image subsetCE)
+ ultimately show "path_image cp \<subseteq> s - insert p pts" unfolding cp_def
+ using n_circ(5) by auto
+ next
+ show "winding_number cp p = - n"
+ unfolding cp_def using winding_number_reversepath n_circ \<open>h p>0\<close>
+ by (auto simp: valid_path_imp_path)
+ next
+ show "winding_number cp p'=0 \<and> p' \<notin> path_image cp" when "p'\<notin>s - pts" for p'
+ unfolding cp_def
+ apply (auto)
+ apply (subst winding_number_reversepath)
+ by (auto simp add: valid_path_imp_path n_circ(7)[OF that] n_circ(1))
+ next
+ show "f contour_integrable_on cp" unfolding cp_def
+ using contour_integrable_reversepath_eq n_circ(1,8) by auto
+ next
+ show "contour_integral cp f = - n * contour_integral p_circ f"
+ unfolding cp_def using contour_integral_reversepath[OF n_circ(1)] n_circ(9)
+ by auto
+ qed
+ define g' where "g' \<equiv> g +++ pg +++ cp +++ (reversepath pg)"
+ have "contour_integral g' f = (\<Sum>p\<in>pts. winding_number g' p * contour_integral (circlepath p (h p)) f)"
+ proof (rule "2.hyps"(3)[of "s-{p}" "g'",OF _ _ \<open>finite pts\<close> ])
+ show "connected (s - {p} - pts)" using connected by (metis Diff_insert2)
+ show "open (s - {p})" using \<open>open s\<close> by auto
+ show " pts \<subseteq> s - {p}" using \<open>insert p pts \<subseteq> s\<close> \<open> p \<notin> pts\<close> by blast
+ show "f holomorphic_on s - {p} - pts" using holo \<open>p \<notin> pts\<close> by (metis Diff_insert2)
+ show "valid_path g'"
+ unfolding g'_def cp_def using n_circ valid pg g_loop
+ by (auto intro!:valid_path_join )
+ show "pathfinish g' = pathstart g'"
+ unfolding g'_def cp_def using pg(2) by simp
+ show "path_image g' \<subseteq> s - {p} - pts"
+ proof -
+ define s' where "s' \<equiv> s - {p} - pts"
+ have s':"s' = s-insert p pts " unfolding s'_def by auto
+ then show ?thesis using path_img pg(4) cp(4)
+ unfolding g'_def
+ apply (fold s'_def s')
+ apply (intro subset_path_image_join)
+ by auto
+ qed
+ note path_join_imp[simp]
+ show "\<forall>z. z \<notin> s - {p} \<longrightarrow> winding_number g' z = 0"
+ proof clarify
+ fix z assume z:"z\<notin>s - {p}"
+ have "winding_number (g +++ pg +++ cp +++ reversepath pg) z = winding_number g z
+ + winding_number (pg +++ cp +++ (reversepath pg)) z"
+ proof (rule winding_number_join)
+ show "path g" using \<open>valid_path g\<close> by (simp add: valid_path_imp_path)
+ show "z \<notin> path_image g" using z path_img by auto
+ show "path (pg +++ cp +++ reversepath pg)" using pg(3) cp
+ by (simp add: valid_path_imp_path)
+ next
+ have "path_image (pg +++ cp +++ reversepath pg) \<subseteq> s - insert p pts"
+ using pg(4) cp(4) by (auto simp:subset_path_image_join)
+ then show "z \<notin> path_image (pg +++ cp +++ reversepath pg)" using z by auto
+ next
+ show "pathfinish g = pathstart (pg +++ cp +++ reversepath pg)" using g_loop by auto
+ qed
+ also have "... = winding_number g z + (winding_number pg z
+ + winding_number (cp +++ (reversepath pg)) z)"
+ proof (subst add_left_cancel,rule winding_number_join)
+ show "path pg" and "path (cp +++ reversepath pg)"
+ and "pathfinish pg = pathstart (cp +++ reversepath pg)"
+ by (auto simp add: valid_path_imp_path)
+ show "z \<notin> path_image pg" using pg(4) z by blast
+ show "z \<notin> path_image (cp +++ reversepath pg)" using z
+ by (metis Diff_iff \<open>z \<notin> path_image pg\<close> contra_subsetD cp(4) insertI1
+ not_in_path_image_join path_image_reversepath singletonD)
+ qed
+ also have "... = winding_number g z + (winding_number pg z
+ + (winding_number cp z + winding_number (reversepath pg) z))"
+ apply (auto intro!:winding_number_join simp: valid_path_imp_path)
+ apply (metis Diff_iff contra_subsetD cp(4) insertI1 singletonD z)
+ by (metis Diff_insert2 Diff_subset contra_subsetD pg(4) z)
+ also have "... = winding_number g z + winding_number cp z"
+ apply (subst winding_number_reversepath)
+ apply (auto simp: valid_path_imp_path)
+ by (metis Diff_iff contra_subsetD insertI1 pg(4) singletonD z)
+ finally have "winding_number g' z = winding_number g z + winding_number cp z"
+ unfolding g'_def .
+ moreover have "winding_number g z + winding_number cp z = 0"
+ using winding z \<open>n=winding_number g p\<close> by auto
+ ultimately show "winding_number g' z = 0" unfolding g'_def by auto
+ qed
+ show "\<forall>pa\<in>s - {p}. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s - {p} \<and> (w \<noteq> pa \<longrightarrow> w \<notin> pts))"
+ using h by fastforce
+ qed
+ moreover have "contour_integral g' f = contour_integral g f
+ - winding_number g p * contour_integral p_circ f"
+ proof -
+ have "contour_integral g' f = contour_integral g f
+ + contour_integral (pg +++ cp +++ reversepath pg) f"
+ unfolding g'_def
+ apply (subst contour_integral_join)
+ by (auto simp add:open_Diff[OF \<open>open s\<close>,OF finite_imp_closed[OF fin]]
+ intro!: contour_integrable_holomorphic_simple[OF holo _ _ path_img]
+ contour_integrable_reversepath)
+ also have "... = contour_integral g f + contour_integral pg f
+ + contour_integral (cp +++ reversepath pg) f"
+ apply (subst contour_integral_join)
+ by (auto simp add:contour_integrable_reversepath)
+ also have "... = contour_integral g f + contour_integral pg f
+ + contour_integral cp f + contour_integral (reversepath pg) f"
+ apply (subst contour_integral_join)
+ by (auto simp add:contour_integrable_reversepath)
+ also have "... = contour_integral g f + contour_integral cp f"
+ using contour_integral_reversepath
+ by (auto simp add:contour_integrable_reversepath)
+ also have "... = contour_integral g f - winding_number g p * contour_integral p_circ f"
+ using \<open>n=winding_number g p\<close> by auto
+ finally show ?thesis .
+ qed
+ moreover have "winding_number g' p' = winding_number g p'" when "p'\<in>pts" for p'
+ proof -
+ have [simp]: "p' \<notin> path_image g" "p' \<notin> path_image pg" "p'\<notin>path_image cp"
+ using "2.prems"(8) that
+ apply blast
+ apply (metis Diff_iff Diff_insert2 contra_subsetD pg(4) that)
+ by (meson DiffD2 cp(4) rev_subsetD subset_insertI that)
+ have "winding_number g' p' = winding_number g p'
+ + winding_number (pg +++ cp +++ reversepath pg) p'" unfolding g'_def
+ apply (subst winding_number_join)
+ apply (simp_all add: valid_path_imp_path)
+ apply (intro not_in_path_image_join)
+ by auto
+ also have "... = winding_number g p' + winding_number pg p'
+ + winding_number (cp +++ reversepath pg) p'"
+ apply (subst winding_number_join)
+ apply (simp_all add: valid_path_imp_path)
+ apply (intro not_in_path_image_join)
+ by auto
+ also have "... = winding_number g p' + winding_number pg p'+ winding_number cp p'
+ + winding_number (reversepath pg) p'"
+ apply (subst winding_number_join)
+ by (simp_all add: valid_path_imp_path)
+ also have "... = winding_number g p' + winding_number cp p'"
+ apply (subst winding_number_reversepath)
+ by (simp_all add: valid_path_imp_path)
+ also have "... = winding_number g p'" using that by auto
+ finally show ?thesis .
+ qed
+ ultimately show ?case unfolding p_circ_def
+ apply (subst (asm) sum.cong[OF refl,
+ of pts _ "\<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"])
+ by (auto simp add:sum.insert[OF \<open>finite pts\<close> \<open>p\<notin>pts\<close>] algebra_simps)
+qed
+
+lemma Cauchy_theorem_singularities:
+ assumes "open s" "connected s" "finite pts" and
+ holo:"f holomorphic_on s-pts" and
+ "valid_path g" and
+ loop:"pathfinish g = pathstart g" and
+ "path_image g \<subseteq> s-pts" and
+ homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0" and
+ avoid:"\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
+ shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
+ (is "?L=?R")
+proof -
+ define circ where "circ \<equiv> \<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"
+ define pts1 where "pts1 \<equiv> pts \<inter> s"
+ define pts2 where "pts2 \<equiv> pts - pts1"
+ have "pts=pts1 \<union> pts2" "pts1 \<inter> pts2 = {}" "pts2 \<inter> s={}" "pts1\<subseteq>s"
+ unfolding pts1_def pts2_def by auto
+ have "contour_integral g f = (\<Sum>p\<in>pts1. circ p)" unfolding circ_def
+ proof (rule Cauchy_theorem_aux[OF \<open>open s\<close> _ _ \<open>pts1\<subseteq>s\<close> _ \<open>valid_path g\<close> loop _ homo])
+ have "finite pts1" unfolding pts1_def using \<open>finite pts\<close> by auto
+ then show "connected (s - pts1)"
+ using \<open>open s\<close> \<open>connected s\<close> connected_open_delete_finite[of s] by auto
+ next
+ show "finite pts1" using \<open>pts = pts1 \<union> pts2\<close> assms(3) by auto
+ show "f holomorphic_on s - pts1" by (metis Diff_Int2 Int_absorb holo pts1_def)
+ show "path_image g \<subseteq> s - pts1" using assms(7) pts1_def by auto
+ show "\<forall>p\<in>s. 0 < h p \<and> (\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pts1))"
+ by (simp add: avoid pts1_def)
+ qed
+ moreover have "sum circ pts2=0"
+ proof -
+ have "winding_number g p=0" when "p\<in>pts2" for p
+ using \<open>pts2 \<inter> s={}\<close> that homo[rule_format,of p] by auto
+ thus ?thesis unfolding circ_def
+ apply (intro sum.neutral)
+ by auto
+ qed
+ moreover have "?R=sum circ pts1 + sum circ pts2"
+ unfolding circ_def
+ using sum.union_disjoint[OF _ _ \<open>pts1 \<inter> pts2 = {}\<close>] \<open>finite pts\<close> \<open>pts=pts1 \<union> pts2\<close>
+ by blast
+ ultimately show ?thesis
+ apply (fold circ_def)
+ by auto
+qed
+
+theorem Residue_theorem:
+ fixes s pts::"complex set" and f::"complex \<Rightarrow> complex"
+ and g::"real \<Rightarrow> complex"
+ assumes "open s" "connected s" "finite pts" and
+ holo:"f holomorphic_on s-pts" and
+ "valid_path g" and
+ loop:"pathfinish g = pathstart g" and
+ "path_image g \<subseteq> s-pts" and
+ homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0"
+ shows "contour_integral g f = 2 * pi * \<i> *(\<Sum>p\<in>pts. winding_number g p * residue f p)"
+proof -
+ define c where "c \<equiv> 2 * pi * \<i>"
+ obtain h where avoid:"\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
+ using finite_cball_avoid[OF \<open>open s\<close> \<open>finite pts\<close>] by metis
+ have "contour_integral g f
+ = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
+ using Cauchy_theorem_singularities[OF assms avoid] .
+ also have "... = (\<Sum>p\<in>pts. c * winding_number g p * residue f p)"
+ proof (intro sum.cong)
+ show "pts = pts" by simp
+ next
+ fix x assume "x \<in> pts"
+ show "winding_number g x * contour_integral (circlepath x (h x)) f
+ = c * winding_number g x * residue f x"
+ proof (cases "x\<in>s")
+ case False
+ then have "winding_number g x=0" using homo by auto
+ thus ?thesis by auto
+ next
+ case True
+ have "contour_integral (circlepath x (h x)) f = c* residue f x"
+ using \<open>x\<in>pts\<close> \<open>finite pts\<close> avoid[rule_format,OF True]
+ apply (intro base_residue[of "s-(pts-{x})",THEN contour_integral_unique,folded c_def])
+ by (auto intro:holomorphic_on_subset[OF holo] open_Diff[OF \<open>open s\<close> finite_imp_closed])
+ then show ?thesis by auto
+ qed
+ qed
+ also have "... = c * (\<Sum>p\<in>pts. winding_number g p * residue f p)"
+ by (simp add: sum_distrib_left algebra_simps)
+ finally show ?thesis unfolding c_def .
+qed
+
+subsection \<open>The argument principle\<close>
+
+theorem argument_principle:
+ fixes f::"complex \<Rightarrow> complex" and poles s:: "complex set"
+ defines "pz \<equiv> {w. f w = 0 \<or> w \<in> poles}" \<comment> \<open>\<^term>\<open>pz\<close> is the set of poles and zeros\<close>
+ assumes "open s" and
+ "connected s" and
+ f_holo:"f holomorphic_on s-poles" and
+ h_holo:"h holomorphic_on s" and
+ "valid_path g" and
+ loop:"pathfinish g = pathstart g" and
+ path_img:"path_image g \<subseteq> s - pz" and
+ homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0" and
+ finite:"finite pz" and
+ poles:"\<forall>p\<in>poles. is_pole f p"
+ shows "contour_integral g (\<lambda>x. deriv f x * h x / f x) = 2 * pi * \<i> *
+ (\<Sum>p\<in>pz. winding_number g p * h p * zorder f p)"
+ (is "?L=?R")
+proof -
+ define c where "c \<equiv> 2 * complex_of_real pi * \<i> "
+ define ff where "ff \<equiv> (\<lambda>x. deriv f x * h x / f x)"
+ define cont where "cont \<equiv> \<lambda>ff p e. (ff has_contour_integral c * zorder f p * h p ) (circlepath p e)"
+ define avoid where "avoid \<equiv> \<lambda>p e. \<forall>w\<in>cball p e. w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz)"
+
+ have "\<exists>e>0. avoid p e \<and> (p\<in>pz \<longrightarrow> cont ff p e)" when "p\<in>s" for p
+ proof -
+ obtain e1 where "e1>0" and e1_avoid:"avoid p e1"
+ using finite_cball_avoid[OF \<open>open s\<close> finite] \<open>p\<in>s\<close> unfolding avoid_def by auto
+ have "\<exists>e2>0. cball p e2 \<subseteq> ball p e1 \<and> cont ff p e2" when "p\<in>pz"
+ proof -
+ define po where "po \<equiv> zorder f p"
+ define pp where "pp \<equiv> zor_poly f p"
+ define f' where "f' \<equiv> \<lambda>w. pp w * (w - p) powr po"
+ define ff' where "ff' \<equiv> (\<lambda>x. deriv f' x * h x / f' x)"
+ obtain r where "pp p\<noteq>0" "r>0" and
+ "r<e1" and
+ pp_holo:"pp holomorphic_on cball p r" and
+ pp_po:"(\<forall>w\<in>cball p r-{p}. f w = pp w * (w - p) powr po \<and> pp w \<noteq> 0)"
+ proof -
+ have "isolated_singularity_at f p"
+ proof -
+ have "f holomorphic_on ball p e1 - {p}"
+ apply (intro holomorphic_on_subset[OF f_holo])
+ using e1_avoid \<open>p\<in>pz\<close> unfolding avoid_def pz_def by force
+ then show ?thesis unfolding isolated_singularity_at_def
+ using \<open>e1>0\<close> analytic_on_open open_delete by blast
+ qed
+ moreover have "not_essential f p"
+ proof (cases "is_pole f p")
+ case True
+ then show ?thesis unfolding not_essential_def by auto
+ next
+ case False
+ then have "p\<in>s-poles" using \<open>p\<in>s\<close> poles unfolding pz_def by auto
+ moreover have "open (s-poles)"
+ using \<open>open s\<close>
+ apply (elim open_Diff)
+ apply (rule finite_imp_closed)
+ using finite unfolding pz_def by simp
+ ultimately have "isCont f p"
+ using holomorphic_on_imp_continuous_on[OF f_holo] continuous_on_eq_continuous_at
+ by auto
+ then show ?thesis unfolding isCont_def not_essential_def by auto
+ qed
+ moreover have "\<exists>\<^sub>F w in at p. f w \<noteq> 0 "
+ proof (rule ccontr)
+ assume "\<not> (\<exists>\<^sub>F w in at p. f w \<noteq> 0)"
+ then have "\<forall>\<^sub>F w in at p. f w= 0" unfolding frequently_def by auto
+ then obtain rr where "rr>0" "\<forall>w\<in>ball p rr - {p}. f w =0"
+ unfolding eventually_at by (auto simp add:dist_commute)
+ then have "ball p rr - {p} \<subseteq> {w\<in>ball p rr-{p}. f w=0}" by blast
+ moreover have "infinite (ball p rr - {p})" using \<open>rr>0\<close> using finite_imp_not_open by fastforce
+ ultimately have "infinite {w\<in>ball p rr-{p}. f w=0}" using infinite_super by blast
+ then have "infinite pz"
+ unfolding pz_def infinite_super by auto
+ then show False using \<open>finite pz\<close> by auto
+ qed
+ ultimately obtain r where "pp p \<noteq> 0" and r:"r>0" "pp holomorphic_on cball p r"
+ "(\<forall>w\<in>cball p r - {p}. f w = pp w * (w - p) powr of_int po \<and> pp w \<noteq> 0)"
+ using zorder_exist[of f p,folded po_def pp_def] by auto
+ define r1 where "r1=min r e1 / 2"
+ have "r1<e1" unfolding r1_def using \<open>e1>0\<close> \<open>r>0\<close> by auto
+ moreover have "r1>0" "pp holomorphic_on cball p r1"
+ "(\<forall>w\<in>cball p r1 - {p}. f w = pp w * (w - p) powr of_int po \<and> pp w \<noteq> 0)"
+ unfolding r1_def using \<open>e1>0\<close> r by auto
+ ultimately show ?thesis using that \<open>pp p\<noteq>0\<close> by auto
+ qed
+
+ define e2 where "e2 \<equiv> r/2"
+ have "e2>0" using \<open>r>0\<close> unfolding e2_def by auto
+ define anal where "anal \<equiv> \<lambda>w. deriv pp w * h w / pp w"
+ define prin where "prin \<equiv> \<lambda>w. po * h w / (w - p)"
+ have "((\<lambda>w. prin w + anal w) has_contour_integral c * po * h p) (circlepath p e2)"
+ proof (rule has_contour_integral_add[of _ _ _ _ 0,simplified])
+ have "ball p r \<subseteq> s"
+ using \<open>r<e1\<close> avoid_def ball_subset_cball e1_avoid by (simp add: subset_eq)
+ then have "cball p e2 \<subseteq> s"
+ using \<open>r>0\<close> unfolding e2_def by auto
+ then have "(\<lambda>w. po * h w) holomorphic_on cball p e2"
+ using h_holo by (auto intro!: holomorphic_intros)
+ then show "(prin has_contour_integral c * po * h p ) (circlepath p e2)"
+ using Cauchy_integral_circlepath_simple[folded c_def, of "\<lambda>w. po * h w"] \<open>e2>0\<close>
+ unfolding prin_def by (auto simp add: mult.assoc)
+ have "anal holomorphic_on ball p r" unfolding anal_def
+ using pp_holo h_holo pp_po \<open>ball p r \<subseteq> s\<close> \<open>pp p\<noteq>0\<close>
+ by (auto intro!: holomorphic_intros)
+ then show "(anal has_contour_integral 0) (circlepath p e2)"
+ using e2_def \<open>r>0\<close>
+ by (auto elim!: Cauchy_theorem_disc_simple)
+ qed
+ then have "cont ff' p e2" unfolding cont_def po_def
+ proof (elim has_contour_integral_eq)
+ fix w assume "w \<in> path_image (circlepath p e2)"
+ then have "w\<in>ball p r" and "w\<noteq>p" unfolding e2_def using \<open>r>0\<close> by auto
+ define wp where "wp \<equiv> w-p"
+ have "wp\<noteq>0" and "pp w \<noteq>0"
+ unfolding wp_def using \<open>w\<noteq>p\<close> \<open>w\<in>ball p r\<close> pp_po by auto
+ moreover have der_f':"deriv f' w = po * pp w * (w-p) powr (po - 1) + deriv pp w * (w-p) powr po"
+ proof (rule DERIV_imp_deriv)
+ have "(pp has_field_derivative (deriv pp w)) (at w)"
+ using DERIV_deriv_iff_has_field_derivative pp_holo \<open>w\<noteq>p\<close>
+ by (meson open_ball \<open>w \<in> ball p r\<close> ball_subset_cball holomorphic_derivI holomorphic_on_subset)
+ then show " (f' has_field_derivative of_int po * pp w * (w - p) powr of_int (po - 1)
+ + deriv pp w * (w - p) powr of_int po) (at w)"
+ unfolding f'_def using \<open>w\<noteq>p\<close>
+ by (auto intro!: derivative_eq_intros DERIV_cong[OF has_field_derivative_powr_of_int])
+ qed
+ ultimately show "prin w + anal w = ff' w"
+ unfolding ff'_def prin_def anal_def
+ apply simp
+ apply (unfold f'_def)
+ apply (fold wp_def)
+ apply (auto simp add:field_simps)
+ by (metis (no_types, lifting) diff_add_cancel mult.commute powr_add powr_to_1)
+ qed
+ then have "cont ff p e2" unfolding cont_def
+ proof (elim has_contour_integral_eq)
+ fix w assume "w \<in> path_image (circlepath p e2)"
+ then have "w\<in>ball p r" and "w\<noteq>p" unfolding e2_def using \<open>r>0\<close> by auto
+ have "deriv f' w = deriv f w"
+ proof (rule complex_derivative_transform_within_open[where s="ball p r - {p}"])
+ show "f' holomorphic_on ball p r - {p}" unfolding f'_def using pp_holo
+ by (auto intro!: holomorphic_intros)
+ next
+ have "ball p e1 - {p} \<subseteq> s - poles"
+ using ball_subset_cball e1_avoid[unfolded avoid_def] unfolding pz_def
+ by auto
+ then have "ball p r - {p} \<subseteq> s - poles"
+ apply (elim dual_order.trans)
+ using \<open>r<e1\<close> by auto
+ then show "f holomorphic_on ball p r - {p}" using f_holo
+ by auto
+ next
+ show "open (ball p r - {p})" by auto
+ show "w \<in> ball p r - {p}" using \<open>w\<in>ball p r\<close> \<open>w\<noteq>p\<close> by auto
+ next
+ fix x assume "x \<in> ball p r - {p}"
+ then show "f' x = f x"
+ using pp_po unfolding f'_def by auto
+ qed
+ moreover have " f' w = f w "
+ using \<open>w \<in> ball p r\<close> ball_subset_cball subset_iff pp_po \<open>w\<noteq>p\<close>
+ unfolding f'_def by auto
+ ultimately show "ff' w = ff w"
+ unfolding ff'_def ff_def by simp
+ qed
+ moreover have "cball p e2 \<subseteq> ball p e1"
+ using \<open>0 < r\<close> \<open>r<e1\<close> e2_def by auto
+ ultimately show ?thesis using \<open>e2>0\<close> by auto
+ qed
+ then obtain e2 where e2:"p\<in>pz \<longrightarrow> e2>0 \<and> cball p e2 \<subseteq> ball p e1 \<and> cont ff p e2"
+ by auto
+ define e4 where "e4 \<equiv> if p\<in>pz then e2 else e1"
+ have "e4>0" using e2 \<open>e1>0\<close> unfolding e4_def by auto
+ moreover have "avoid p e4" using e2 \<open>e1>0\<close> e1_avoid unfolding e4_def avoid_def by auto
+ moreover have "p\<in>pz \<longrightarrow> cont ff p e4"
+ by (auto simp add: e2 e4_def)
+ ultimately show ?thesis by auto
+ qed
+ then obtain get_e where get_e:"\<forall>p\<in>s. get_e p>0 \<and> avoid p (get_e p)
+ \<and> (p\<in>pz \<longrightarrow> cont ff p (get_e p))"
+ by metis
+ define ci where "ci \<equiv> \<lambda>p. contour_integral (circlepath p (get_e p)) ff"
+ define w where "w \<equiv> \<lambda>p. winding_number g p"
+ have "contour_integral g ff = (\<Sum>p\<in>pz. w p * ci p)" unfolding ci_def w_def
+ proof (rule Cauchy_theorem_singularities[OF \<open>open s\<close> \<open>connected s\<close> finite _ \<open>valid_path g\<close> loop
+ path_img homo])
+ have "open (s - pz)" using open_Diff[OF _ finite_imp_closed[OF finite]] \<open>open s\<close> by auto
+ then show "ff holomorphic_on s - pz" unfolding ff_def using f_holo h_holo
+ by (auto intro!: holomorphic_intros simp add:pz_def)
+ next
+ show "\<forall>p\<in>s. 0 < get_e p \<and> (\<forall>w\<in>cball p (get_e p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz))"
+ using get_e using avoid_def by blast
+ qed
+ also have "... = (\<Sum>p\<in>pz. c * w p * h p * zorder f p)"
+ proof (rule sum.cong[of pz pz,simplified])
+ fix p assume "p \<in> pz"
+ show "w p * ci p = c * w p * h p * (zorder f p)"
+ proof (cases "p\<in>s")
+ assume "p \<in> s"
+ have "ci p = c * h p * (zorder f p)" unfolding ci_def
+ apply (rule contour_integral_unique)
+ using get_e \<open>p\<in>s\<close> \<open>p\<in>pz\<close> unfolding cont_def by (metis mult.assoc mult.commute)
+ thus ?thesis by auto
+ next
+ assume "p\<notin>s"
+ then have "w p=0" using homo unfolding w_def by auto
+ then show ?thesis by auto
+ qed
+ qed
+ also have "... = c*(\<Sum>p\<in>pz. w p * h p * zorder f p)"
+ unfolding sum_distrib_left by (simp add:algebra_simps)
+ finally have "contour_integral g ff = c * (\<Sum>p\<in>pz. w p * h p * of_int (zorder f p))" .
+ then show ?thesis unfolding ff_def c_def w_def by simp
+qed
+
+subsection \<open>Rouche's theorem \<close>
+
+theorem Rouche_theorem:
+ fixes f g::"complex \<Rightarrow> complex" and s:: "complex set"
+ defines "fg\<equiv>(\<lambda>p. f p + g p)"
+ defines "zeros_fg\<equiv>{p. fg p = 0}" and "zeros_f\<equiv>{p. f p = 0}"
+ assumes
+ "open s" and "connected s" and
+ "finite zeros_fg" and
+ "finite zeros_f" and
+ f_holo:"f holomorphic_on s" and
+ g_holo:"g holomorphic_on s" and
+ "valid_path \<gamma>" and
+ loop:"pathfinish \<gamma> = pathstart \<gamma>" and
+ path_img:"path_image \<gamma> \<subseteq> s " and
+ path_less:"\<forall>z\<in>path_image \<gamma>. cmod(f z) > cmod(g z)" and
+ homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number \<gamma> z = 0"
+ shows "(\<Sum>p\<in>zeros_fg. winding_number \<gamma> p * zorder fg p)
+ = (\<Sum>p\<in>zeros_f. winding_number \<gamma> p * zorder f p)"
+proof -
+ have path_fg:"path_image \<gamma> \<subseteq> s - zeros_fg"
+ proof -
+ have False when "z\<in>path_image \<gamma>" and "f z + g z=0" for z
+ proof -
+ have "cmod (f z) > cmod (g z)" using \<open>z\<in>path_image \<gamma>\<close> path_less by auto
+ moreover have "f z = - g z" using \<open>f z + g z =0\<close> by (simp add: eq_neg_iff_add_eq_0)
+ then have "cmod (f z) = cmod (g z)" by auto
+ ultimately show False by auto
+ qed
+ then show ?thesis unfolding zeros_fg_def fg_def using path_img by auto
+ qed
+ have path_f:"path_image \<gamma> \<subseteq> s - zeros_f"
+ proof -
+ have False when "z\<in>path_image \<gamma>" and "f z =0" for z
+ proof -
+ have "cmod (g z) < cmod (f z) " using \<open>z\<in>path_image \<gamma>\<close> path_less by auto
+ then have "cmod (g z) < 0" using \<open>f z=0\<close> by auto
+ then show False by auto
+ qed
+ then show ?thesis unfolding zeros_f_def using path_img by auto
+ qed
+ define w where "w \<equiv> \<lambda>p. winding_number \<gamma> p"
+ define c where "c \<equiv> 2 * complex_of_real pi * \<i>"
+ define h where "h \<equiv> \<lambda>p. g p / f p + 1"
+ obtain spikes
+ where "finite spikes" and spikes: "\<forall>x\<in>{0..1} - spikes. \<gamma> differentiable at x"
+ using \<open>valid_path \<gamma>\<close>
+ by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ have h_contour:"((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
+ proof -
+ have outside_img:"0 \<in> outside (path_image (h o \<gamma>))"
+ proof -
+ have "h p \<in> ball 1 1" when "p\<in>path_image \<gamma>" for p
+ proof -
+ have "cmod (g p/f p) <1" using path_less[rule_format,OF that]
+ apply (cases "cmod (f p) = 0")
+ by (auto simp add: norm_divide)
+ then show ?thesis unfolding h_def by (auto simp add:dist_complex_def)
+ qed
+ then have "path_image (h o \<gamma>) \<subseteq> ball 1 1"
+ by (simp add: image_subset_iff path_image_compose)
+ moreover have " (0::complex) \<notin> ball 1 1" by (simp add: dist_norm)
+ ultimately show "?thesis"
+ using convex_in_outside[of "ball 1 1" 0] outside_mono by blast
+ qed
+ have valid_h:"valid_path (h \<circ> \<gamma>)"
+ proof (rule valid_path_compose_holomorphic[OF \<open>valid_path \<gamma>\<close> _ _ path_f])
+ show "h holomorphic_on s - zeros_f"
+ unfolding h_def using f_holo g_holo
+ by (auto intro!: holomorphic_intros simp add:zeros_f_def)
+ next
+ show "open (s - zeros_f)" using \<open>finite zeros_f\<close> \<open>open s\<close> finite_imp_closed
+ by auto
+ qed
+ have "((\<lambda>z. 1/z) has_contour_integral 0) (h \<circ> \<gamma>)"
+ proof -
+ have "0 \<notin> path_image (h \<circ> \<gamma>)" using outside_img by (simp add: outside_def)
+ then have "((\<lambda>z. 1/z) has_contour_integral c * winding_number (h \<circ> \<gamma>) 0) (h \<circ> \<gamma>)"
+ using has_contour_integral_winding_number[of "h o \<gamma>" 0,simplified] valid_h
+ unfolding c_def by auto
+ moreover have "winding_number (h o \<gamma>) 0 = 0"
+ proof -
+ have "0 \<in> outside (path_image (h \<circ> \<gamma>))" using outside_img .
+ moreover have "path (h o \<gamma>)"
+ using valid_h by (simp add: valid_path_imp_path)
+ moreover have "pathfinish (h o \<gamma>) = pathstart (h o \<gamma>)"
+ by (simp add: loop pathfinish_compose pathstart_compose)
+ ultimately show ?thesis using winding_number_zero_in_outside by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ moreover have "vector_derivative (h \<circ> \<gamma>) (at x) = vector_derivative \<gamma> (at x) * deriv h (\<gamma> x)"
+ when "x\<in>{0..1} - spikes" for x
+ proof (rule vector_derivative_chain_at_general)
+ show "\<gamma> differentiable at x" using that \<open>valid_path \<gamma>\<close> spikes by auto
+ next
+ define der where "der \<equiv> \<lambda>p. (deriv g p * f p - g p * deriv f p)/(f p * f p)"
+ define t where "t \<equiv> \<gamma> x"
+ have "f t\<noteq>0" unfolding zeros_f_def t_def
+ by (metis DiffD1 image_eqI norm_not_less_zero norm_zero path_defs(4) path_less that)
+ moreover have "t\<in>s"
+ using contra_subsetD path_image_def path_fg t_def that by fastforce
+ ultimately have "(h has_field_derivative der t) (at t)"
+ unfolding h_def der_def using g_holo f_holo \<open>open s\<close>
+ by (auto intro!: holomorphic_derivI derivative_eq_intros)
+ then show "h field_differentiable at (\<gamma> x)"
+ unfolding t_def field_differentiable_def by blast
+ qed
+ then have " ((/) 1 has_contour_integral 0) (h \<circ> \<gamma>)
+ = ((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
+ unfolding has_contour_integral
+ apply (intro has_integral_spike_eq[OF negligible_finite, OF \<open>finite spikes\<close>])
+ by auto
+ ultimately show ?thesis by auto
+ qed
+ then have "contour_integral \<gamma> (\<lambda>x. deriv h x / h x) = 0"
+ using contour_integral_unique by simp
+ moreover have "contour_integral \<gamma> (\<lambda>x. deriv fg x / fg x) = contour_integral \<gamma> (\<lambda>x. deriv f x / f x)
+ + contour_integral \<gamma> (\<lambda>p. deriv h p / h p)"
+ proof -
+ have "(\<lambda>p. deriv f p / f p) contour_integrable_on \<gamma>"
+ proof (rule contour_integrable_holomorphic_simple[OF _ _ \<open>valid_path \<gamma>\<close> path_f])
+ show "open (s - zeros_f)" using finite_imp_closed[OF \<open>finite zeros_f\<close>] \<open>open s\<close>
+ by auto
+ then show "(\<lambda>p. deriv f p / f p) holomorphic_on s - zeros_f"
+ using f_holo
+ by (auto intro!: holomorphic_intros simp add:zeros_f_def)
+ qed
+ moreover have "(\<lambda>p. deriv h p / h p) contour_integrable_on \<gamma>"
+ using h_contour
+ by (simp add: has_contour_integral_integrable)
+ ultimately have "contour_integral \<gamma> (\<lambda>x. deriv f x / f x + deriv h x / h x) =
+ contour_integral \<gamma> (\<lambda>p. deriv f p / f p) + contour_integral \<gamma> (\<lambda>p. deriv h p / h p)"
+ using contour_integral_add[of "(\<lambda>p. deriv f p / f p)" \<gamma> "(\<lambda>p. deriv h p / h p)" ]
+ by auto
+ moreover have "deriv fg p / fg p = deriv f p / f p + deriv h p / h p"
+ when "p\<in> path_image \<gamma>" for p
+ proof -
+ have "fg p\<noteq>0" and "f p\<noteq>0" using path_f path_fg that unfolding zeros_f_def zeros_fg_def
+ by auto
+ have "h p\<noteq>0"
+ proof (rule ccontr)
+ assume "\<not> h p \<noteq> 0"
+ then have "g p / f p= -1" unfolding h_def by (simp add: add_eq_0_iff2)
+ then have "cmod (g p/f p) = 1" by auto
+ moreover have "cmod (g p/f p) <1" using path_less[rule_format,OF that]
+ apply (cases "cmod (f p) = 0")
+ by (auto simp add: norm_divide)
+ ultimately show False by auto
+ qed
+ have der_fg:"deriv fg p = deriv f p + deriv g p" unfolding fg_def
+ using f_holo g_holo holomorphic_on_imp_differentiable_at[OF _ \<open>open s\<close>] path_img that
+ by auto
+ have der_h:"deriv h p = (deriv g p * f p - g p * deriv f p)/(f p * f p)"
+ proof -
+ define der where "der \<equiv> \<lambda>p. (deriv g p * f p - g p * deriv f p)/(f p * f p)"
+ have "p\<in>s" using path_img that by auto
+ then have "(h has_field_derivative der p) (at p)"
+ unfolding h_def der_def using g_holo f_holo \<open>open s\<close> \<open>f p\<noteq>0\<close>
+ by (auto intro!: derivative_eq_intros holomorphic_derivI)
+ then show ?thesis unfolding der_def using DERIV_imp_deriv by auto
+ qed
+ show ?thesis
+ apply (simp only:der_fg der_h)
+ apply (auto simp add:field_simps \<open>h p\<noteq>0\<close> \<open>f p\<noteq>0\<close> \<open>fg p\<noteq>0\<close>)
+ by (auto simp add:field_simps h_def \<open>f p\<noteq>0\<close> fg_def)
+ qed
+ then have "contour_integral \<gamma> (\<lambda>p. deriv fg p / fg p)
+ = contour_integral \<gamma> (\<lambda>p. deriv f p / f p + deriv h p / h p)"
+ by (elim contour_integral_eq)
+ ultimately show ?thesis by auto
+ qed
+ moreover have "contour_integral \<gamma> (\<lambda>x. deriv fg x / fg x) = c * (\<Sum>p\<in>zeros_fg. w p * zorder fg p)"
+ unfolding c_def zeros_fg_def w_def
+ proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close> _ _ \<open>valid_path \<gamma>\<close> loop _ homo
+ , of _ "{}" "\<lambda>_. 1",simplified])
+ show "fg holomorphic_on s" unfolding fg_def using f_holo g_holo holomorphic_on_add by auto
+ show "path_image \<gamma> \<subseteq> s - {p. fg p = 0}" using path_fg unfolding zeros_fg_def .
+ show " finite {p. fg p = 0}" using \<open>finite zeros_fg\<close> unfolding zeros_fg_def .
+ qed
+ moreover have "contour_integral \<gamma> (\<lambda>x. deriv f x / f x) = c * (\<Sum>p\<in>zeros_f. w p * zorder f p)"
+ unfolding c_def zeros_f_def w_def
+ proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close> _ _ \<open>valid_path \<gamma>\<close> loop _ homo
+ , of _ "{}" "\<lambda>_. 1",simplified])
+ show "f holomorphic_on s" using f_holo g_holo holomorphic_on_add by auto
+ show "path_image \<gamma> \<subseteq> s - {p. f p = 0}" using path_f unfolding zeros_f_def .
+ show " finite {p. f p = 0}" using \<open>finite zeros_f\<close> unfolding zeros_f_def .
+ qed
+ ultimately have " c* (\<Sum>p\<in>zeros_fg. w p * (zorder fg p)) = c* (\<Sum>p\<in>zeros_f. w p * (zorder f p))"
+ by auto
+ then show ?thesis unfolding c_def using w_def by auto
+qed
+
+end
\ No newline at end of file
--- a/src/HOL/Complex_Analysis/Riemann_Mapping.thy Mon Dec 02 22:40:16 2019 -0500
+++ b/src/HOL/Complex_Analysis/Riemann_Mapping.thy Mon Dec 02 17:51:54 2019 +0100
@@ -1486,4 +1486,256 @@
ultimately show ?thesis by metis
qed
+
+subsection \<open>Applications to Winding Numbers\<close>
+
+lemma simply_connected_inside_simple_path:
+ fixes p :: "real \<Rightarrow> complex"
+ shows "simple_path p \<Longrightarrow> simply_connected(inside(path_image p))"
+ using Jordan_inside_outside connected_simple_path_image inside_simple_curve_imp_closed simply_connected_eq_frontier_properties
+ by fastforce
+
+lemma simply_connected_Int:
+ fixes S :: "complex set"
+ assumes "open S" "open T" "simply_connected S" "simply_connected T" "connected (S \<inter> T)"
+ shows "simply_connected (S \<inter> T)"
+ using assms by (force simp: simply_connected_eq_winding_number_zero open_Int)
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number defines a continuous logarithm for the path itself\<close>
+
+lemma winding_number_as_continuous_log:
+ assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
+ obtains q where "path q"
+ "pathfinish q - pathstart q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
+ "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = \<zeta> + exp(q t)"
+proof -
+ let ?q = "\<lambda>t. 2 * of_real pi * \<i> * winding_number(subpath 0 t p) \<zeta> + Ln(pathstart p - \<zeta>)"
+ show ?thesis
+ proof
+ have *: "continuous (at t within {0..1}) (\<lambda>x. winding_number (subpath 0 x p) \<zeta>)"
+ if t: "t \<in> {0..1}" for t
+ proof -
+ let ?B = "ball (p t) (norm(p t - \<zeta>))"
+ have "p t \<noteq> \<zeta>"
+ using path_image_def that \<zeta> by blast
+ then have "simply_connected ?B"
+ by (simp add: convex_imp_simply_connected)
+ then have "\<And>f::complex\<Rightarrow>complex. continuous_on ?B f \<and> (\<forall>\<zeta> \<in> ?B. f \<zeta> \<noteq> 0)
+ \<longrightarrow> (\<exists>g. continuous_on ?B g \<and> (\<forall>\<zeta> \<in> ?B. f \<zeta> = exp (g \<zeta>)))"
+ by (simp add: simply_connected_eq_continuous_log)
+ moreover have "continuous_on ?B (\<lambda>w. w - \<zeta>)"
+ by (intro continuous_intros)
+ moreover have "(\<forall>z \<in> ?B. z - \<zeta> \<noteq> 0)"
+ by (auto simp: dist_norm)
+ ultimately obtain g where contg: "continuous_on ?B g"
+ and geq: "\<And>z. z \<in> ?B \<Longrightarrow> z - \<zeta> = exp (g z)" by blast
+ obtain d where "0 < d" and d:
+ "\<And>x. \<lbrakk>x \<in> {0..1}; dist x t < d\<rbrakk> \<Longrightarrow> dist (p x) (p t) < cmod (p t - \<zeta>)"
+ using \<open>path p\<close> t unfolding path_def continuous_on_iff
+ by (metis \<open>p t \<noteq> \<zeta>\<close> right_minus_eq zero_less_norm_iff)
+ have "((\<lambda>x. winding_number (\<lambda>w. subpath 0 x p w - \<zeta>) 0 -
+ winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0) \<longlongrightarrow> 0)
+ (at t within {0..1})"
+ proof (rule Lim_transform_within [OF _ \<open>d > 0\<close>])
+ have "continuous (at t within {0..1}) (g o p)"
+ proof (rule continuous_within_compose)
+ show "continuous (at t within {0..1}) p"
+ using \<open>path p\<close> continuous_on_eq_continuous_within path_def that by blast
+ show "continuous (at (p t) within p ` {0..1}) g"
+ by (metis (no_types, lifting) open_ball UNIV_I \<open>p t \<noteq> \<zeta>\<close> centre_in_ball contg continuous_on_eq_continuous_at continuous_within_topological right_minus_eq zero_less_norm_iff)
+ qed
+ with LIM_zero have "((\<lambda>u. (g (subpath t u p 1) - g (subpath t u p 0))) \<longlongrightarrow> 0) (at t within {0..1})"
+ by (auto simp: subpath_def continuous_within o_def)
+ then show "((\<lambda>u. (g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>)) \<longlongrightarrow> 0)
+ (at t within {0..1})"
+ by (simp add: tendsto_divide_zero)
+ show "(g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>) =
+ winding_number (\<lambda>w. subpath 0 u p w - \<zeta>) 0 - winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0"
+ if "u \<in> {0..1}" "0 < dist u t" "dist u t < d" for u
+ proof -
+ have "closed_segment t u \<subseteq> {0..1}"
+ using closed_segment_eq_real_ivl t that by auto
+ then have piB: "path_image(subpath t u p) \<subseteq> ?B"
+ apply (clarsimp simp add: path_image_subpath_gen)
+ by (metis subsetD le_less_trans \<open>dist u t < d\<close> d dist_commute dist_in_closed_segment)
+ have *: "path (g \<circ> subpath t u p)"
+ apply (rule path_continuous_image)
+ using \<open>path p\<close> t that apply auto[1]
+ using piB contg continuous_on_subset by blast
+ have "(g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>)
+ = winding_number (exp \<circ> g \<circ> subpath t u p) 0"
+ using winding_number_compose_exp [OF *]
+ by (simp add: pathfinish_def pathstart_def o_assoc)
+ also have "... = winding_number (\<lambda>w. subpath t u p w - \<zeta>) 0"
+ proof (rule winding_number_cong)
+ have "exp(g y) = y - \<zeta>" if "y \<in> (subpath t u p) ` {0..1}" for y
+ by (metis that geq path_image_def piB subset_eq)
+ then show "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> (exp \<circ> g \<circ> subpath t u p) x = subpath t u p x - \<zeta>"
+ by auto
+ qed
+ also have "... = winding_number (\<lambda>w. subpath 0 u p w - \<zeta>) 0 -
+ winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0"
+ apply (simp add: winding_number_offset [symmetric])
+ using winding_number_subpath_combine [OF \<open>path p\<close> \<zeta>, of 0 t u] \<open>t \<in> {0..1}\<close> \<open>u \<in> {0..1}\<close>
+ by (simp add: add.commute eq_diff_eq)
+ finally show ?thesis .
+ qed
+ qed
+ then show ?thesis
+ by (subst winding_number_offset) (simp add: continuous_within LIM_zero_iff)
+ qed
+ show "path ?q"
+ unfolding path_def
+ by (intro continuous_intros) (simp add: continuous_on_eq_continuous_within *)
+
+ have "\<zeta> \<noteq> p 0"
+ by (metis \<zeta> pathstart_def pathstart_in_path_image)
+ then show "pathfinish ?q - pathstart ?q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
+ by (simp add: pathfinish_def pathstart_def)
+ show "p t = \<zeta> + exp (?q t)" if "t \<in> {0..1}" for t
+ proof -
+ have "path (subpath 0 t p)"
+ using \<open>path p\<close> that by auto
+ moreover
+ have "\<zeta> \<notin> path_image (subpath 0 t p)"
+ using \<zeta> [unfolded path_image_def] that by (auto simp: path_image_subpath)
+ ultimately show ?thesis
+ using winding_number_exp_2pi [of "subpath 0 t p" \<zeta>] \<open>\<zeta> \<noteq> p 0\<close>
+ by (auto simp: exp_add algebra_simps pathfinish_def pathstart_def subpath_def)
+ qed
+ qed
+qed
+
+subsection \<open>Winding number equality is the same as path/loop homotopy in C - {0}\<close>
+
+lemma winding_number_homotopic_loops_null_eq:
+ assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
+ shows "winding_number p \<zeta> = 0 \<longleftrightarrow> (\<exists>a. homotopic_loops (-{\<zeta>}) p (\<lambda>t. a))"
+ (is "?lhs = ?rhs")
+proof
+ assume [simp]: ?lhs
+ obtain q where "path q"
+ and qeq: "pathfinish q - pathstart q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
+ and peq: "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = \<zeta> + exp(q t)"
+ using winding_number_as_continuous_log [OF assms] by blast
+ have *: "homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r)
+ {0..1} (-{\<zeta>}) ((\<lambda>w. \<zeta> + exp w) \<circ> q) ((\<lambda>w. \<zeta> + exp w) \<circ> (\<lambda>t. 0))"
+ proof (rule homotopic_with_compose_continuous_left)
+ show "homotopic_with_canon (\<lambda>f. pathfinish ((\<lambda>w. \<zeta> + exp w) \<circ> f) = pathstart ((\<lambda>w. \<zeta> + exp w) \<circ> f))
+ {0..1} UNIV q (\<lambda>t. 0)"
+ proof (rule homotopic_with_mono, simp_all add: pathfinish_def pathstart_def)
+ have "homotopic_loops UNIV q (\<lambda>t. 0)"
+ by (rule homotopic_loops_linear) (use qeq \<open>path q\<close> in \<open>auto simp: path_defs\<close>)
+ then have "homotopic_with (\<lambda>r. r 1 = r 0) (top_of_set {0..1}) euclidean q (\<lambda>t. 0)"
+ by (simp add: homotopic_loops_def pathfinish_def pathstart_def)
+ then show "homotopic_with (\<lambda>h. exp (h 1) = exp (h 0)) (top_of_set {0..1}) euclidean q (\<lambda>t. 0)"
+ by (rule homotopic_with_mono) simp
+ qed
+ show "continuous_on UNIV (\<lambda>w. \<zeta> + exp w)"
+ by (rule continuous_intros)+
+ show "range (\<lambda>w. \<zeta> + exp w) \<subseteq> -{\<zeta>}"
+ by auto
+ qed
+ then have "homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r) {0..1} (-{\<zeta>}) p (\<lambda>x. \<zeta> + 1)"
+ by (rule homotopic_with_eq) (auto simp: o_def peq pathfinish_def pathstart_def)
+ then have "homotopic_loops (-{\<zeta>}) p (\<lambda>t. \<zeta> + 1)"
+ by (simp add: homotopic_loops_def)
+ then show ?rhs ..
+next
+ assume ?rhs
+ then obtain a where "homotopic_loops (-{\<zeta>}) p (\<lambda>t. a)" ..
+ then have "winding_number p \<zeta> = winding_number (\<lambda>t. a) \<zeta>" "a \<noteq> \<zeta>"
+ using winding_number_homotopic_loops homotopic_loops_imp_subset by (force simp:)+
+ moreover have "winding_number (\<lambda>t. a) \<zeta> = 0"
+ by (metis winding_number_zero_const \<open>a \<noteq> \<zeta>\<close>)
+ ultimately show ?lhs by metis
+qed
+
+lemma winding_number_homotopic_paths_null_explicit_eq:
+ assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
+ shows "winding_number p \<zeta> = 0 \<longleftrightarrow> homotopic_paths (-{\<zeta>}) p (linepath (pathstart p) (pathstart p))"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ apply (auto simp: winding_number_homotopic_loops_null_eq [OF assms])
+ apply (rule homotopic_loops_imp_homotopic_paths_null)
+ apply (simp add: linepath_refl)
+ done
+next
+ assume ?rhs
+ then show ?lhs
+ by (metis \<zeta> pathstart_in_path_image winding_number_homotopic_paths winding_number_trivial)
+qed
+
+lemma winding_number_homotopic_paths_null_eq:
+ assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
+ shows "winding_number p \<zeta> = 0 \<longleftrightarrow> (\<exists>a. homotopic_paths (-{\<zeta>}) p (\<lambda>t. a))"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ by (auto simp: winding_number_homotopic_paths_null_explicit_eq [OF assms] linepath_refl)
+next
+ assume ?rhs
+ then show ?lhs
+ by (metis \<zeta> homotopic_paths_imp_pathfinish pathfinish_def pathfinish_in_path_image winding_number_homotopic_paths winding_number_zero_const)
+qed
+
+proposition winding_number_homotopic_paths_eq:
+ assumes "path p" and \<zeta>p: "\<zeta> \<notin> path_image p"
+ and "path q" and \<zeta>q: "\<zeta> \<notin> path_image q"
+ and qp: "pathstart q = pathstart p" "pathfinish q = pathfinish p"
+ shows "winding_number p \<zeta> = winding_number q \<zeta> \<longleftrightarrow> homotopic_paths (-{\<zeta>}) p q"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then have "winding_number (p +++ reversepath q) \<zeta> = 0"
+ using assms by (simp add: winding_number_join winding_number_reversepath)
+ moreover
+ have "path (p +++ reversepath q)" "\<zeta> \<notin> path_image (p +++ reversepath q)"
+ using assms by (auto simp: not_in_path_image_join)
+ ultimately obtain a where "homotopic_paths (- {\<zeta>}) (p +++ reversepath q) (linepath a a)"
+ using winding_number_homotopic_paths_null_explicit_eq by blast
+ then show ?rhs
+ using homotopic_paths_imp_pathstart assms
+ by (fastforce simp add: dest: homotopic_paths_imp_homotopic_loops homotopic_paths_loop_parts)
+next
+ assume ?rhs
+ then show ?lhs
+ by (simp add: winding_number_homotopic_paths)
+qed
+
+lemma winding_number_homotopic_loops_eq:
+ assumes "path p" and \<zeta>p: "\<zeta> \<notin> path_image p"
+ and "path q" and \<zeta>q: "\<zeta> \<notin> path_image q"
+ and loops: "pathfinish p = pathstart p" "pathfinish q = pathstart q"
+ shows "winding_number p \<zeta> = winding_number q \<zeta> \<longleftrightarrow> homotopic_loops (-{\<zeta>}) p q"
+ (is "?lhs = ?rhs")
+proof
+ assume L: ?lhs
+ have "pathstart p \<noteq> \<zeta>" "pathstart q \<noteq> \<zeta>"
+ using \<zeta>p \<zeta>q by blast+
+ moreover have "path_connected (-{\<zeta>})"
+ by (simp add: path_connected_punctured_universe)
+ ultimately obtain r where "path r" and rim: "path_image r \<subseteq> -{\<zeta>}"
+ and pas: "pathstart r = pathstart p" and paf: "pathfinish r = pathstart q"
+ by (auto simp: path_connected_def)
+ then have "pathstart r \<noteq> \<zeta>" by blast
+ have "homotopic_loops (- {\<zeta>}) p (r +++ q +++ reversepath r)"
+ proof (rule homotopic_paths_imp_homotopic_loops)
+ show "homotopic_paths (- {\<zeta>}) p (r +++ q +++ reversepath r)"
+ by (metis (mono_tags, hide_lams) \<open>path r\<close> L \<zeta>p \<zeta>q \<open>path p\<close> \<open>path q\<close> homotopic_loops_conjugate loops not_in_path_image_join paf pas path_image_reversepath path_imp_reversepath path_join_eq pathfinish_join pathfinish_reversepath pathstart_join pathstart_reversepath rim subset_Compl_singleton winding_number_homotopic_loops winding_number_homotopic_paths_eq)
+ qed (use loops pas in auto)
+ moreover have "homotopic_loops (- {\<zeta>}) (r +++ q +++ reversepath r) q"
+ using rim \<zeta>q by (auto simp: homotopic_loops_conjugate paf \<open>path q\<close> \<open>path r\<close> loops)
+ ultimately show ?rhs
+ using homotopic_loops_trans by metis
+next
+ assume ?rhs
+ then show ?lhs
+ by (simp add: winding_number_homotopic_loops)
+qed
+
end
--- a/src/HOL/Complex_Analysis/Winding_Numbers.thy Mon Dec 02 22:40:16 2019 -0500
+++ b/src/HOL/Complex_Analysis/Winding_Numbers.thy Mon Dec 02 17:51:54 2019 +0100
@@ -1,23 +1,1274 @@
-section \<open>Winding Numbers\<close>
-
-text\<open>By John Harrison et al. Ported from HOL Light by L C Paulson (2017)\<close>
-
+section \<open>Winding numbers\<close>
theory Winding_Numbers
-imports
- Riemann_Mapping
+ imports Cauchy_Integral_Theorem
begin
-lemma simply_connected_inside_simple_path:
- fixes p :: "real \<Rightarrow> complex"
- shows "simple_path p \<Longrightarrow> simply_connected(inside(path_image p))"
- using Jordan_inside_outside connected_simple_path_image inside_simple_curve_imp_closed simply_connected_eq_frontier_properties
- by fastforce
+subsection \<open>Definition\<close>
+
+definition\<^marker>\<open>tag important\<close> winding_number_prop :: "[real \<Rightarrow> complex, complex, real, real \<Rightarrow> complex, complex] \<Rightarrow> bool" where
+ "winding_number_prop \<gamma> z e p n \<equiv>
+ valid_path p \<and> z \<notin> path_image p \<and>
+ pathstart p = pathstart \<gamma> \<and>
+ pathfinish p = pathfinish \<gamma> \<and>
+ (\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and>
+ contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
+
+definition\<^marker>\<open>tag important\<close> winding_number:: "[real \<Rightarrow> complex, complex] \<Rightarrow> complex" where
+ "winding_number \<gamma> z \<equiv> SOME n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n"
+
+
+lemma winding_number:
+ assumes "path \<gamma>" "z \<notin> path_image \<gamma>" "0 < e"
+ shows "\<exists>p. winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
+proof -
+ have "path_image \<gamma> \<subseteq> UNIV - {z}"
+ using assms by blast
+ then obtain d
+ where d: "d>0"
+ and pi_eq: "\<And>h1 h2. valid_path h1 \<and> valid_path h2 \<and>
+ (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d) \<and>
+ pathstart h2 = pathstart h1 \<and> pathfinish h2 = pathfinish h1 \<longrightarrow>
+ path_image h1 \<subseteq> UNIV - {z} \<and> path_image h2 \<subseteq> UNIV - {z} \<and>
+ (\<forall>f. f holomorphic_on UNIV - {z} \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
+ using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
+ then obtain h where h: "polynomial_function h \<and> pathstart h = pathstart \<gamma> \<and> pathfinish h = pathfinish \<gamma> \<and>
+ (\<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)"
+ using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "d/2"] d by auto
+ define nn where "nn = 1/(2* pi*\<i>) * contour_integral h (\<lambda>w. 1/(w - z))"
+ have "\<exists>n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n"
+ proof (rule_tac x=nn in exI, clarify)
+ fix e::real
+ assume e: "e>0"
+ obtain p where p: "polynomial_function p \<and>
+ pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < min e (d/2))"
+ using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "min e (d/2)"] d \<open>0<e\<close> by auto
+ have "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
+ by (auto simp: intro!: holomorphic_intros)
+ then show "\<exists>p. winding_number_prop \<gamma> z e p nn"
+ apply (rule_tac x=p in exI)
+ using pi_eq [of h p] h p d
+ apply (auto simp: valid_path_polynomial_function norm_minus_commute nn_def winding_number_prop_def)
+ done
+ qed
+ then show ?thesis
+ unfolding winding_number_def by (rule someI2_ex) (blast intro: \<open>0<e\<close>)
+qed
+
+lemma winding_number_unique:
+ assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
+ and pi: "\<And>e. e>0 \<Longrightarrow> \<exists>p. winding_number_prop \<gamma> z e p n"
+ shows "winding_number \<gamma> z = n"
+proof -
+ have "path_image \<gamma> \<subseteq> UNIV - {z}"
+ using assms by blast
+ then obtain e
+ where e: "e>0"
+ and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
+ (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
+ pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
+ contour_integral h2 f = contour_integral h1 f"
+ using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
+ obtain p where p: "winding_number_prop \<gamma> z e p n"
+ using pi [OF e] by blast
+ obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)"
+ using winding_number [OF \<gamma> e] by blast
+ have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
+ using p by (auto simp: winding_number_prop_def)
+ also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))"
+ proof (rule pi_eq)
+ show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
+ by (auto intro!: holomorphic_intros)
+ qed (use p q in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
+ also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
+ using q by (auto simp: winding_number_prop_def)
+ finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
+ then show ?thesis
+ by simp
+qed
+
+(*NB not winding_number_prop here due to the loop in p*)
+lemma winding_number_unique_loop:
+ assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
+ and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ and pi:
+ "\<And>e. e>0 \<Longrightarrow> \<exists>p. valid_path p \<and> z \<notin> path_image p \<and>
+ pathfinish p = pathstart p \<and>
+ (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
+ contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
+ shows "winding_number \<gamma> z = n"
+proof -
+ have "path_image \<gamma> \<subseteq> UNIV - {z}"
+ using assms by blast
+ then obtain e
+ where e: "e>0"
+ and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
+ (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
+ pathfinish h1 = pathstart h1; pathfinish h2 = pathstart h2; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
+ contour_integral h2 f = contour_integral h1 f"
+ using contour_integral_nearby_loops [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
+ obtain p where p:
+ "valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and>
+ (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
+ contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
+ using pi [OF e] by blast
+ obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)"
+ using winding_number [OF \<gamma> e] by blast
+ have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
+ using p by auto
+ also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))"
+ proof (rule pi_eq)
+ show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
+ by (auto intro!: holomorphic_intros)
+ qed (use p q loop in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
+ also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
+ using q by (auto simp: winding_number_prop_def)
+ finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
+ then show ?thesis
+ by simp
+qed
+
+proposition winding_number_valid_path:
+ assumes "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
+ shows "winding_number \<gamma> z = 1/(2*pi*\<i>) * contour_integral \<gamma> (\<lambda>w. 1/(w - z))"
+ by (rule winding_number_unique)
+ (use assms in \<open>auto simp: valid_path_imp_path winding_number_prop_def\<close>)
+
+proposition has_contour_integral_winding_number:
+ assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
+ shows "((\<lambda>w. 1/(w - z)) has_contour_integral (2*pi*\<i>*winding_number \<gamma> z)) \<gamma>"
+by (simp add: winding_number_valid_path has_contour_integral_integral contour_integrable_inversediff assms)
+
+lemma winding_number_trivial [simp]: "z \<noteq> a \<Longrightarrow> winding_number(linepath a a) z = 0"
+ by (simp add: winding_number_valid_path)
+
+lemma winding_number_subpath_trivial [simp]: "z \<noteq> g x \<Longrightarrow> winding_number (subpath x x g) z = 0"
+ by (simp add: path_image_subpath winding_number_valid_path)
+
+lemma winding_number_join:
+ assumes \<gamma>1: "path \<gamma>1" "z \<notin> path_image \<gamma>1"
+ and \<gamma>2: "path \<gamma>2" "z \<notin> path_image \<gamma>2"
+ and "pathfinish \<gamma>1 = pathstart \<gamma>2"
+ shows "winding_number(\<gamma>1 +++ \<gamma>2) z = winding_number \<gamma>1 z + winding_number \<gamma>2 z"
+proof (rule winding_number_unique)
+ show "\<exists>p. winding_number_prop (\<gamma>1 +++ \<gamma>2) z e p
+ (winding_number \<gamma>1 z + winding_number \<gamma>2 z)" if "e > 0" for e
+ proof -
+ obtain p1 where "winding_number_prop \<gamma>1 z e p1 (winding_number \<gamma>1 z)"
+ using \<open>0 < e\<close> \<gamma>1 winding_number by blast
+ moreover
+ obtain p2 where "winding_number_prop \<gamma>2 z e p2 (winding_number \<gamma>2 z)"
+ using \<open>0 < e\<close> \<gamma>2 winding_number by blast
+ ultimately
+ have "winding_number_prop (\<gamma>1+++\<gamma>2) z e (p1+++p2) (winding_number \<gamma>1 z + winding_number \<gamma>2 z)"
+ using assms
+ apply (simp add: winding_number_prop_def not_in_path_image_join contour_integrable_inversediff algebra_simps)
+ apply (auto simp: joinpaths_def)
+ done
+ then show ?thesis
+ by blast
+ qed
+qed (use assms in \<open>auto simp: not_in_path_image_join\<close>)
+
+lemma winding_number_reversepath:
+ assumes "path \<gamma>" "z \<notin> path_image \<gamma>"
+ shows "winding_number(reversepath \<gamma>) z = - (winding_number \<gamma> z)"
+proof (rule winding_number_unique)
+ show "\<exists>p. winding_number_prop (reversepath \<gamma>) z e p (- winding_number \<gamma> z)" if "e > 0" for e
+ proof -
+ obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
+ using \<open>0 < e\<close> assms winding_number by blast
+ then have "winding_number_prop (reversepath \<gamma>) z e (reversepath p) (- winding_number \<gamma> z)"
+ using assms
+ apply (simp add: winding_number_prop_def contour_integral_reversepath contour_integrable_inversediff valid_path_imp_reverse)
+ apply (auto simp: reversepath_def)
+ done
+ then show ?thesis
+ by blast
+ qed
+qed (use assms in auto)
+
+lemma winding_number_shiftpath:
+ assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
+ and "pathfinish \<gamma> = pathstart \<gamma>" "a \<in> {0..1}"
+ shows "winding_number(shiftpath a \<gamma>) z = winding_number \<gamma> z"
+proof (rule winding_number_unique_loop)
+ show "\<exists>p. valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and>
+ (\<forall>t\<in>{0..1}. cmod (shiftpath a \<gamma> t - p t) < e) \<and>
+ contour_integral p (\<lambda>w. 1 / (w - z)) =
+ complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
+ if "e > 0" for e
+ proof -
+ obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
+ using \<open>0 < e\<close> assms winding_number by blast
+ then show ?thesis
+ apply (rule_tac x="shiftpath a p" in exI)
+ using assms that
+ apply (auto simp: winding_number_prop_def path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath contour_integral_shiftpath)
+ apply (simp add: shiftpath_def)
+ done
+ qed
+qed (use assms in \<open>auto simp: path_shiftpath path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath\<close>)
+
+lemma winding_number_split_linepath:
+ assumes "c \<in> closed_segment a b" "z \<notin> closed_segment a b"
+ shows "winding_number(linepath a b) z = winding_number(linepath a c) z + winding_number(linepath c b) z"
+proof -
+ have "z \<notin> closed_segment a c" "z \<notin> closed_segment c b"
+ using assms by (meson convex_contains_segment convex_segment ends_in_segment subsetCE)+
+ then show ?thesis
+ using assms
+ by (simp add: winding_number_valid_path contour_integral_split_linepath [symmetric] continuous_on_inversediff field_simps)
+qed
+
+lemma winding_number_cong:
+ "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> p t = q t) \<Longrightarrow> winding_number p z = winding_number q z"
+ by (simp add: winding_number_def winding_number_prop_def pathstart_def pathfinish_def)
+
+lemma winding_number_constI:
+ assumes "c\<noteq>z" "\<And>t. \<lbrakk>0\<le>t; t\<le>1\<rbrakk> \<Longrightarrow> g t = c"
+ shows "winding_number g z = 0"
+proof -
+ have "winding_number g z = winding_number (linepath c c) z"
+ apply (rule winding_number_cong)
+ using assms unfolding linepath_def by auto
+ moreover have "winding_number (linepath c c) z =0"
+ apply (rule winding_number_trivial)
+ using assms by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma winding_number_offset: "winding_number p z = winding_number (\<lambda>w. p w - z) 0"
+ unfolding winding_number_def
+proof (intro ext arg_cong [where f = Eps] arg_cong [where f = All] imp_cong refl, safe)
+ fix n e g
+ assume "0 < e" and g: "winding_number_prop p z e g n"
+ then show "\<exists>r. winding_number_prop (\<lambda>w. p w - z) 0 e r n"
+ by (rule_tac x="\<lambda>t. g t - z" in exI)
+ (force simp: winding_number_prop_def contour_integral_integral valid_path_def path_defs
+ vector_derivative_def has_vector_derivative_diff_const piecewise_C1_differentiable_diff C1_differentiable_imp_piecewise)
+next
+ fix n e g
+ assume "0 < e" and g: "winding_number_prop (\<lambda>w. p w - z) 0 e g n"
+ then show "\<exists>r. winding_number_prop p z e r n"
+ apply (rule_tac x="\<lambda>t. g t + z" in exI)
+ apply (simp add: winding_number_prop_def contour_integral_integral valid_path_def path_defs
+ piecewise_C1_differentiable_add vector_derivative_def has_vector_derivative_add_const C1_differentiable_imp_piecewise)
+ apply (force simp: algebra_simps)
+ done
+qed
+
+lemma winding_number_negatepath:
+ assumes \<gamma>: "valid_path \<gamma>" and 0: "0 \<notin> path_image \<gamma>"
+ shows "winding_number(uminus \<circ> \<gamma>) 0 = winding_number \<gamma> 0"
+proof -
+ have "(/) 1 contour_integrable_on \<gamma>"
+ using "0" \<gamma> contour_integrable_inversediff by fastforce
+ then have "((\<lambda>z. 1/z) has_contour_integral contour_integral \<gamma> ((/) 1)) \<gamma>"
+ by (rule has_contour_integral_integral)
+ then have "((\<lambda>z. 1 / - z) has_contour_integral - contour_integral \<gamma> ((/) 1)) \<gamma>"
+ using has_contour_integral_neg by auto
+ then show ?thesis
+ using assms
+ apply (simp add: winding_number_valid_path valid_path_negatepath image_def path_defs)
+ apply (simp add: contour_integral_unique has_contour_integral_negatepath)
+ done
+qed
+
+(* A combined theorem deducing several things piecewise.*)
+lemma winding_number_join_pos_combined:
+ "\<lbrakk>valid_path \<gamma>1; z \<notin> path_image \<gamma>1; 0 < Re(winding_number \<gamma>1 z);
+ valid_path \<gamma>2; z \<notin> path_image \<gamma>2; 0 < Re(winding_number \<gamma>2 z); pathfinish \<gamma>1 = pathstart \<gamma>2\<rbrakk>
+ \<Longrightarrow> valid_path(\<gamma>1 +++ \<gamma>2) \<and> z \<notin> path_image(\<gamma>1 +++ \<gamma>2) \<and> 0 < Re(winding_number(\<gamma>1 +++ \<gamma>2) z)"
+ by (simp add: valid_path_join path_image_join winding_number_join valid_path_imp_path)
+
+
+subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Useful sufficient conditions for the winding number to be positive\<close>
+
+lemma Re_winding_number:
+ "\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>\<rbrakk>
+ \<Longrightarrow> Re(winding_number \<gamma> z) = Im(contour_integral \<gamma> (\<lambda>w. 1/(w - z))) / (2*pi)"
+by (simp add: winding_number_valid_path field_simps Re_divide power2_eq_square)
+
+lemma winding_number_pos_le:
+ assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
+ and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> 0 \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
+ shows "0 \<le> Re(winding_number \<gamma> z)"
+proof -
+ have ge0: "0 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" if x: "0 < x" "x < 1" for x
+ using ge by (simp add: Complex.Im_divide algebra_simps x)
+ let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)"
+ let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))"
+ have hi: "(?vd has_integral ?int z) (cbox 0 1)"
+ unfolding box_real
+ apply (subst has_contour_integral [symmetric])
+ using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral)
+ have "0 \<le> Im (?int z)"
+ proof (rule has_integral_component_nonneg [of \<i>, simplified])
+ show "\<And>x. x \<in> cbox 0 1 \<Longrightarrow> 0 \<le> Im (if 0 < x \<and> x < 1 then ?vd x else 0)"
+ by (force simp: ge0)
+ show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else 0) has_integral ?int z) (cbox 0 1)"
+ by (rule has_integral_spike_interior [OF hi]) simp
+ qed
+ then show ?thesis
+ by (simp add: Re_winding_number [OF \<gamma>] field_simps)
+qed
+
+lemma winding_number_pos_lt_lemma:
+ assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
+ and e: "0 < e"
+ and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
+ shows "0 < Re(winding_number \<gamma> z)"
+proof -
+ let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)"
+ let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))"
+ have hi: "(?vd has_integral ?int z) (cbox 0 1)"
+ unfolding box_real
+ apply (subst has_contour_integral [symmetric])
+ using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral)
+ have "e \<le> Im (contour_integral \<gamma> (\<lambda>w. 1 / (w - z)))"
+ proof (rule has_integral_component_le [of \<i> "\<lambda>x. \<i>*e" "\<i>*e" "{0..1}", simplified])
+ show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else \<i> * complex_of_real e) has_integral ?int z) {0..1}"
+ by (rule has_integral_spike_interior [OF hi, simplified box_real]) (use e in simp)
+ show "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow>
+ e \<le> Im (if 0 < x \<and> x < 1 then ?vd x else \<i> * complex_of_real e)"
+ by (simp add: ge)
+ qed (use has_integral_const_real [of _ 0 1] in auto)
+ with e show ?thesis
+ by (simp add: Re_winding_number [OF \<gamma>] field_simps)
+qed
+
+lemma winding_number_pos_lt:
+ assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
+ and e: "0 < e"
+ and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
+ shows "0 < Re (winding_number \<gamma> z)"
+proof -
+ have bm: "bounded ((\<lambda>w. w - z) ` (path_image \<gamma>))"
+ using bounded_translation [of _ "-z"] \<gamma> by (simp add: bounded_valid_path_image)
+ then obtain B where B: "B > 0" and Bno: "\<And>x. x \<in> (\<lambda>w. w - z) ` (path_image \<gamma>) \<Longrightarrow> norm x \<le> B"
+ using bounded_pos [THEN iffD1, OF bm] by blast
+ { fix x::real assume x: "0 < x" "x < 1"
+ then have B2: "cmod (\<gamma> x - z)^2 \<le> B^2" using Bno [of "\<gamma> x - z"]
+ by (simp add: path_image_def power2_eq_square mult_mono')
+ with x have "\<gamma> x \<noteq> z" using \<gamma>
+ using path_image_def by fastforce
+ then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z)) / (cmod (\<gamma> x - z))\<^sup>2"
+ using B ge [OF x] B2 e
+ apply (rule_tac y="e / (cmod (\<gamma> x - z))\<^sup>2" in order_trans)
+ apply (auto simp: divide_left_mono divide_right_mono)
+ done
+ then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
+ by (simp add: complex_div_cnj [of _ "\<gamma> x - z" for x] del: complex_cnj_diff times_complex.sel)
+ } note * = this
+ show ?thesis
+ using e B by (simp add: * winding_number_pos_lt_lemma [OF \<gamma>, of "e/B^2"])
+qed
+
+subsection\<open>The winding number is an integer\<close>
+
+text\<open>Proof from the book Complex Analysis by Lars V. Ahlfors, Chapter 4, section 2.1,
+ Also on page 134 of Serge Lang's book with the name title, etc.\<close>
+
+lemma exp_fg:
+ fixes z::complex
+ assumes g: "(g has_vector_derivative g') (at x within s)"
+ and f: "(f has_vector_derivative (g' / (g x - z))) (at x within s)"
+ and z: "g x \<noteq> z"
+ shows "((\<lambda>x. exp(-f x) * (g x - z)) has_vector_derivative 0) (at x within s)"
+proof -
+ have *: "(exp \<circ> (\<lambda>x. (- f x)) has_vector_derivative - (g' / (g x - z)) * exp (- f x)) (at x within s)"
+ using assms unfolding has_vector_derivative_def scaleR_conv_of_real
+ by (auto intro!: derivative_eq_intros)
+ show ?thesis
+ apply (rule has_vector_derivative_eq_rhs)
+ using z
+ apply (auto intro!: derivative_eq_intros * [unfolded o_def] g)
+ done
+qed
+
+lemma winding_number_exp_integral:
+ fixes z::complex
+ assumes \<gamma>: "\<gamma> piecewise_C1_differentiable_on {a..b}"
+ and ab: "a \<le> b"
+ and z: "z \<notin> \<gamma> ` {a..b}"
+ shows "(\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)) integrable_on {a..b}"
+ (is "?thesis1")
+ "exp (- (integral {a..b} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))) * (\<gamma> b - z) = \<gamma> a - z"
+ (is "?thesis2")
+proof -
+ let ?D\<gamma> = "\<lambda>x. vector_derivative \<gamma> (at x)"
+ have [simp]: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<gamma> x \<noteq> z"
+ using z by force
+ have cong: "continuous_on {a..b} \<gamma>"
+ using \<gamma> by (simp add: piecewise_C1_differentiable_on_def)
+ obtain k where fink: "finite k" and g_C1_diff: "\<gamma> C1_differentiable_on ({a..b} - k)"
+ using \<gamma> by (force simp: piecewise_C1_differentiable_on_def)
+ have \<circ>: "open ({a<..<b} - k)"
+ using \<open>finite k\<close> by (simp add: finite_imp_closed open_Diff)
+ moreover have "{a<..<b} - k \<subseteq> {a..b} - k"
+ by force
+ ultimately have g_diff_at: "\<And>x. \<lbrakk>x \<notin> k; x \<in> {a<..<b}\<rbrakk> \<Longrightarrow> \<gamma> differentiable at x"
+ by (metis Diff_iff differentiable_on_subset C1_diff_imp_diff [OF g_C1_diff] differentiable_on_def at_within_open)
+ { fix w
+ assume "w \<noteq> z"
+ have "continuous_on (ball w (cmod (w - z))) (\<lambda>w. 1 / (w - z))"
+ by (auto simp: dist_norm intro!: continuous_intros)
+ moreover have "\<And>x. cmod (w - x) < cmod (w - z) \<Longrightarrow> \<exists>f'. ((\<lambda>w. 1 / (w - z)) has_field_derivative f') (at x)"
+ by (auto simp: intro!: derivative_eq_intros)
+ ultimately have "\<exists>h. \<forall>y. norm(y - w) < norm(w - z) \<longrightarrow> (h has_field_derivative 1/(y - z)) (at y)"
+ using holomorphic_convex_primitive [of "ball w (norm(w - z))" "{}" "\<lambda>w. 1/(w - z)"]
+ by (force simp: field_differentiable_def Ball_def dist_norm at_within_open_NO_MATCH norm_minus_commute)
+ }
+ then obtain h where h: "\<And>w y. w \<noteq> z \<Longrightarrow> norm(y - w) < norm(w - z) \<Longrightarrow> (h w has_field_derivative 1/(y - z)) (at y)"
+ by meson
+ have exy: "\<exists>y. ((\<lambda>x. inverse (\<gamma> x - z) * ?D\<gamma> x) has_integral y) {a..b}"
+ unfolding integrable_on_def [symmetric]
+ proof (rule contour_integral_local_primitive_any [OF piecewise_C1_imp_differentiable [OF \<gamma>]])
+ show "\<exists>d h. 0 < d \<and>
+ (\<forall>y. cmod (y - w) < d \<longrightarrow> (h has_field_derivative inverse (y - z))(at y within - {z}))"
+ if "w \<in> - {z}" for w
+ apply (rule_tac x="norm(w - z)" in exI)
+ using that inverse_eq_divide has_field_derivative_at_within h
+ by (metis Compl_insert DiffD2 insertCI right_minus_eq zero_less_norm_iff)
+ qed simp
+ have vg_int: "(\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)) integrable_on {a..b}"
+ unfolding box_real [symmetric] divide_inverse_commute
+ by (auto intro!: exy integrable_subinterval simp add: integrable_on_def ab)
+ with ab show ?thesis1
+ by (simp add: divide_inverse_commute integral_def integrable_on_def)
+ { fix t
+ assume t: "t \<in> {a..b}"
+ have cball: "continuous_on (ball (\<gamma> t) (dist (\<gamma> t) z)) (\<lambda>x. inverse (x - z))"
+ using z by (auto intro!: continuous_intros simp: dist_norm)
+ have icd: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow> (\<lambda>w. inverse (w - z)) field_differentiable at x"
+ unfolding field_differentiable_def by (force simp: intro!: derivative_eq_intros)
+ obtain h where h: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow>
+ (h has_field_derivative inverse (x - z)) (at x within {y. cmod (\<gamma> t - y) < cmod (\<gamma> t - z)})"
+ using holomorphic_convex_primitive [where f = "\<lambda>w. inverse(w - z)", OF convex_ball finite.emptyI cball icd]
+ by simp (auto simp: ball_def dist_norm that)
+ { fix x D
+ assume x: "x \<notin> k" "a < x" "x < b"
+ then have "x \<in> interior ({a..b} - k)"
+ using open_subset_interior [OF \<circ>] by fastforce
+ then have con: "isCont ?D\<gamma> x"
+ using g_C1_diff x by (auto simp: C1_differentiable_on_eq intro: continuous_on_interior)
+ then have con_vd: "continuous (at x within {a..b}) (\<lambda>x. ?D\<gamma> x)"
+ by (rule continuous_at_imp_continuous_within)
+ have gdx: "\<gamma> differentiable at x"
+ using x by (simp add: g_diff_at)
+ have "\<And>d. \<lbrakk>x \<notin> k; a < x; x < b;
+ (\<gamma> has_vector_derivative d) (at x); a \<le> t; t \<le> b\<rbrakk>
+ \<Longrightarrow> ((\<lambda>x. integral {a..x}
+ (\<lambda>x. ?D\<gamma> x /
+ (\<gamma> x - z))) has_vector_derivative
+ d / (\<gamma> x - z))
+ (at x within {a..b})"
+ apply (rule has_vector_derivative_eq_rhs)
+ apply (rule integral_has_vector_derivative_continuous_at [where S = "{}", simplified])
+ apply (rule con_vd continuous_intros cong vg_int | simp add: continuous_at_imp_continuous_within has_vector_derivative_continuous vector_derivative_at)+
+ done
+ then have "((\<lambda>c. exp (- integral {a..c} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) * (\<gamma> c - z)) has_derivative (\<lambda>h. 0))
+ (at x within {a..b})"
+ using x gdx t
+ apply (clarsimp simp add: differentiable_iff_scaleR)
+ apply (rule exp_fg [unfolded has_vector_derivative_def, simplified], blast intro: has_derivative_at_withinI)
+ apply (simp_all add: has_vector_derivative_def [symmetric])
+ done
+ } note * = this
+ have "exp (- (integral {a..t} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)))) * (\<gamma> t - z) =\<gamma> a - z"
+ apply (rule has_derivative_zero_unique_strong_interval [of "{a,b} \<union> k" a b])
+ using t
+ apply (auto intro!: * continuous_intros fink cong indefinite_integral_continuous_1 [OF vg_int] simp add: ab)+
+ done
+ }
+ with ab show ?thesis2
+ by (simp add: divide_inverse_commute integral_def)
+qed
+
+lemma winding_number_exp_2pi:
+ "\<lbrakk>path p; z \<notin> path_image p\<rbrakk>
+ \<Longrightarrow> pathfinish p - z = exp (2 * pi * \<i> * winding_number p z) * (pathstart p - z)"
+using winding_number [of p z 1] unfolding valid_path_def path_image_def pathstart_def pathfinish_def winding_number_prop_def
+ by (force dest: winding_number_exp_integral(2) [of _ 0 1 z] simp: field_simps contour_integral_integral exp_minus)
+
+lemma integer_winding_number_eq:
+ assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
+ shows "winding_number \<gamma> z \<in> \<int> \<longleftrightarrow> pathfinish \<gamma> = pathstart \<gamma>"
+proof -
+ obtain p where p: "valid_path p" "z \<notin> path_image p"
+ "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
+ and eq: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
+ using winding_number [OF assms, of 1] unfolding winding_number_prop_def by auto
+ then have wneq: "winding_number \<gamma> z = winding_number p z"
+ using eq winding_number_valid_path by force
+ have iff: "(winding_number \<gamma> z \<in> \<int>) \<longleftrightarrow> (exp (contour_integral p (\<lambda>w. 1 / (w - z))) = 1)"
+ using eq by (simp add: exp_eq_1 complex_is_Int_iff)
+ have "exp (contour_integral p (\<lambda>w. 1 / (w - z))) = (\<gamma> 1 - z) / (\<gamma> 0 - z)"
+ using p winding_number_exp_integral(2) [of p 0 1 z]
+ apply (simp add: valid_path_def path_defs contour_integral_integral exp_minus field_split_simps)
+ by (metis path_image_def pathstart_def pathstart_in_path_image)
+ then have "winding_number p z \<in> \<int> \<longleftrightarrow> pathfinish p = pathstart p"
+ using p wneq iff by (auto simp: path_defs)
+ then show ?thesis using p eq
+ by (auto simp: winding_number_valid_path)
+qed
+
+theorem integer_winding_number:
+ "\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> \<Longrightarrow> winding_number \<gamma> z \<in> \<int>"
+by (metis integer_winding_number_eq)
+
+
+text\<open>If the winding number's magnitude is at least one, then the path must contain points in every direction.*)
+ We can thus bound the winding number of a path that doesn't intersect a given ray. \<close>
+
+lemma winding_number_pos_meets:
+ fixes z::complex
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and 1: "Re (winding_number \<gamma> z) \<ge> 1"
+ and w: "w \<noteq> z"
+ shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
+proof -
+ have [simp]: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> \<gamma> x \<noteq> z"
+ using z by (auto simp: path_image_def)
+ have [simp]: "z \<notin> \<gamma> ` {0..1}"
+ using path_image_def z by auto
+ have gpd: "\<gamma> piecewise_C1_differentiable_on {0..1}"
+ using \<gamma> valid_path_def by blast
+ define r where "r = (w - z) / (\<gamma> 0 - z)"
+ have [simp]: "r \<noteq> 0"
+ using w z by (auto simp: r_def)
+ have cont: "continuous_on {0..1}
+ (\<lambda>x. Im (integral {0..x} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))))"
+ by (intro continuous_intros indefinite_integral_continuous_1 winding_number_exp_integral [OF gpd]; simp)
+ have "Arg2pi r \<le> 2*pi"
+ by (simp add: Arg2pi less_eq_real_def)
+ also have "\<dots> \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))"
+ using 1
+ apply (simp add: winding_number_valid_path [OF \<gamma> z] contour_integral_integral)
+ apply (simp add: Complex.Re_divide field_simps power2_eq_square)
+ done
+ finally have "Arg2pi r \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))" .
+ then have "\<exists>t. t \<in> {0..1} \<and> Im(integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r"
+ by (simp add: Arg2pi_ge_0 cont IVT')
+ then obtain t where t: "t \<in> {0..1}"
+ and eqArg: "Im (integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r"
+ by blast
+ define i where "i = integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
+ have iArg: "Arg2pi r = Im i"
+ using eqArg by (simp add: i_def)
+ have gpdt: "\<gamma> piecewise_C1_differentiable_on {0..t}"
+ by (metis atLeastAtMost_iff atLeastatMost_subset_iff order_refl piecewise_C1_differentiable_on_subset gpd t)
+ have "exp (- i) * (\<gamma> t - z) = \<gamma> 0 - z"
+ unfolding i_def
+ apply (rule winding_number_exp_integral [OF gpdt])
+ using t z unfolding path_image_def by force+
+ then have *: "\<gamma> t - z = exp i * (\<gamma> 0 - z)"
+ by (simp add: exp_minus field_simps)
+ then have "(w - z) = r * (\<gamma> 0 - z)"
+ by (simp add: r_def)
+ then have "z + complex_of_real (exp (Re i)) * (w - z) / complex_of_real (cmod r) = \<gamma> t"
+ apply simp
+ apply (subst Complex_Transcendental.Arg2pi_eq [of r])
+ apply (simp add: iArg)
+ using * apply (simp add: exp_eq_polar field_simps)
+ done
+ with t show ?thesis
+ by (rule_tac x="exp(Re i) / norm r" in exI) (auto simp: path_image_def)
+qed
+
+lemma winding_number_big_meets:
+ fixes z::complex
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "\<bar>Re (winding_number \<gamma> z)\<bar> \<ge> 1"
+ and w: "w \<noteq> z"
+ shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
+proof -
+ { assume "Re (winding_number \<gamma> z) \<le> - 1"
+ then have "Re (winding_number (reversepath \<gamma>) z) \<ge> 1"
+ by (simp add: \<gamma> valid_path_imp_path winding_number_reversepath z)
+ moreover have "valid_path (reversepath \<gamma>)"
+ using \<gamma> valid_path_imp_reverse by auto
+ moreover have "z \<notin> path_image (reversepath \<gamma>)"
+ by (simp add: z)
+ ultimately have "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image (reversepath \<gamma>)"
+ using winding_number_pos_meets w by blast
+ then have ?thesis
+ by simp
+ }
+ then show ?thesis
+ using assms
+ by (simp add: abs_if winding_number_pos_meets split: if_split_asm)
+qed
+
+lemma winding_number_less_1:
+ fixes z::complex
+ shows
+ "\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>; w \<noteq> z;
+ \<And>a::real. 0 < a \<Longrightarrow> z + a*(w - z) \<notin> path_image \<gamma>\<rbrakk>
+ \<Longrightarrow> Re(winding_number \<gamma> z) < 1"
+ by (auto simp: not_less dest: winding_number_big_meets)
+
+text\<open>One way of proving that WN=1 for a loop.\<close>
+lemma winding_number_eq_1:
+ fixes z::complex
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ and 0: "0 < Re(winding_number \<gamma> z)" and 2: "Re(winding_number \<gamma> z) < 2"
+ shows "winding_number \<gamma> z = 1"
+proof -
+ have "winding_number \<gamma> z \<in> Ints"
+ by (simp add: \<gamma> integer_winding_number loop valid_path_imp_path z)
+ then show ?thesis
+ using 0 2 by (auto simp: Ints_def)
+qed
+
+subsection\<open>Continuity of winding number and invariance on connected sets\<close>
-lemma simply_connected_Int:
- fixes S :: "complex set"
- assumes "open S" "open T" "simply_connected S" "simply_connected T" "connected (S \<inter> T)"
- shows "simply_connected (S \<inter> T)"
- using assms by (force simp: simply_connected_eq_winding_number_zero open_Int)
+lemma continuous_at_winding_number:
+ fixes z::complex
+ assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
+ shows "continuous (at z) (winding_number \<gamma>)"
+proof -
+ obtain e where "e>0" and cbg: "cball z e \<subseteq> - path_image \<gamma>"
+ using open_contains_cball [of "- path_image \<gamma>"] z
+ by (force simp: closed_def [symmetric] closed_path_image [OF \<gamma>])
+ then have ppag: "path_image \<gamma> \<subseteq> - cball z (e/2)"
+ by (force simp: cball_def dist_norm)
+ have oc: "open (- cball z (e / 2))"
+ by (simp add: closed_def [symmetric])
+ obtain d where "d>0" and pi_eq:
+ "\<And>h1 h2. \<lbrakk>valid_path h1; valid_path h2;
+ (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d);
+ pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1\<rbrakk>
+ \<Longrightarrow>
+ path_image h1 \<subseteq> - cball z (e / 2) \<and>
+ path_image h2 \<subseteq> - cball z (e / 2) \<and>
+ (\<forall>f. f holomorphic_on - cball z (e / 2) \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
+ using contour_integral_nearby_ends [OF oc \<gamma> ppag] by metis
+ obtain p where p: "valid_path p" "z \<notin> path_image p"
+ "pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma>"
+ and pg: "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (\<gamma> t - p t) < min d e / 2"
+ and pi: "contour_integral p (\<lambda>x. 1 / (x - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
+ using winding_number [OF \<gamma> z, of "min d e / 2"] \<open>d>0\<close> \<open>e>0\<close> by (auto simp: winding_number_prop_def)
+ { fix w
+ assume d2: "cmod (w - z) < d/2" and e2: "cmod (w - z) < e/2"
+ then have wnotp: "w \<notin> path_image p"
+ using cbg \<open>d>0\<close> \<open>e>0\<close>
+ apply (simp add: path_image_def cball_def dist_norm, clarify)
+ apply (frule pg)
+ apply (drule_tac c="\<gamma> x" in subsetD)
+ apply (auto simp: less_eq_real_def norm_minus_commute norm_triangle_half_l)
+ done
+ have wnotg: "w \<notin> path_image \<gamma>"
+ using cbg e2 \<open>e>0\<close> by (force simp: dist_norm norm_minus_commute)
+ { fix k::real
+ assume k: "k>0"
+ then obtain q where q: "valid_path q" "w \<notin> path_image q"
+ "pathstart q = pathstart \<gamma> \<and> pathfinish q = pathfinish \<gamma>"
+ and qg: "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (\<gamma> t - q t) < min k (min d e) / 2"
+ and qi: "contour_integral q (\<lambda>u. 1 / (u - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
+ using winding_number [OF \<gamma> wnotg, of "min k (min d e) / 2"] \<open>d>0\<close> \<open>e>0\<close> k
+ by (force simp: min_divide_distrib_right winding_number_prop_def)
+ have "contour_integral p (\<lambda>u. 1 / (u - w)) = contour_integral q (\<lambda>u. 1 / (u - w))"
+ apply (rule pi_eq [OF \<open>valid_path q\<close> \<open>valid_path p\<close>, THEN conjunct2, THEN conjunct2, rule_format])
+ apply (frule pg)
+ apply (frule qg)
+ using p q \<open>d>0\<close> e2
+ apply (auto simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
+ done
+ then have "contour_integral p (\<lambda>x. 1 / (x - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
+ by (simp add: pi qi)
+ } note pip = this
+ have "path p"
+ using p by (simp add: valid_path_imp_path)
+ then have "winding_number p w = winding_number \<gamma> w"
+ apply (rule winding_number_unique [OF _ wnotp])
+ apply (rule_tac x=p in exI)
+ apply (simp add: p wnotp min_divide_distrib_right pip winding_number_prop_def)
+ done
+ } note wnwn = this
+ obtain pe where "pe>0" and cbp: "cball z (3 / 4 * pe) \<subseteq> - path_image p"
+ using p open_contains_cball [of "- path_image p"]
+ by (force simp: closed_def [symmetric] closed_path_image [OF valid_path_imp_path])
+ obtain L
+ where "L>0"
+ and L: "\<And>f B. \<lbrakk>f holomorphic_on - cball z (3 / 4 * pe);
+ \<forall>z \<in> - cball z (3 / 4 * pe). cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
+ cmod (contour_integral p f) \<le> L * B"
+ using contour_integral_bound_exists [of "- cball z (3/4*pe)" p] cbp \<open>valid_path p\<close> by blast
+ { fix e::real and w::complex
+ assume e: "0 < e" and w: "cmod (w - z) < pe/4" "cmod (w - z) < e * pe\<^sup>2 / (8 * L)"
+ then have [simp]: "w \<notin> path_image p"
+ using cbp p(2) \<open>0 < pe\<close>
+ by (force simp: dist_norm norm_minus_commute path_image_def cball_def)
+ have [simp]: "contour_integral p (\<lambda>x. 1/(x - w)) - contour_integral p (\<lambda>x. 1/(x - z)) =
+ contour_integral p (\<lambda>x. 1/(x - w) - 1/(x - z))"
+ by (simp add: p contour_integrable_inversediff contour_integral_diff)
+ { fix x
+ assume pe: "3/4 * pe < cmod (z - x)"
+ have "cmod (w - x) < pe/4 + cmod (z - x)"
+ by (meson add_less_cancel_right norm_diff_triangle_le order_refl order_trans_rules(21) w(1))
+ then have wx: "cmod (w - x) < 4/3 * cmod (z - x)" using pe by simp
+ have "cmod (z - x) \<le> cmod (z - w) + cmod (w - x)"
+ using norm_diff_triangle_le by blast
+ also have "\<dots> < pe/4 + cmod (w - x)"
+ using w by (simp add: norm_minus_commute)
+ finally have "pe/2 < cmod (w - x)"
+ using pe by auto
+ then have "(pe/2)^2 < cmod (w - x) ^ 2"
+ apply (rule power_strict_mono)
+ using \<open>pe>0\<close> by auto
+ then have pe2: "pe^2 < 4 * cmod (w - x) ^ 2"
+ by (simp add: power_divide)
+ have "8 * L * cmod (w - z) < e * pe\<^sup>2"
+ using w \<open>L>0\<close> by (simp add: field_simps)
+ also have "\<dots> < e * 4 * cmod (w - x) * cmod (w - x)"
+ using pe2 \<open>e>0\<close> by (simp add: power2_eq_square)
+ also have "\<dots> < e * 4 * cmod (w - x) * (4/3 * cmod (z - x))"
+ using wx
+ apply (rule mult_strict_left_mono)
+ using pe2 e not_less_iff_gr_or_eq by fastforce
+ finally have "L * cmod (w - z) < 2/3 * e * cmod (w - x) * cmod (z - x)"
+ by simp
+ also have "\<dots> \<le> e * cmod (w - x) * cmod (z - x)"
+ using e by simp
+ finally have Lwz: "L * cmod (w - z) < e * cmod (w - x) * cmod (z - x)" .
+ have "L * cmod (1 / (x - w) - 1 / (x - z)) \<le> e"
+ apply (cases "x=z \<or> x=w")
+ using pe \<open>pe>0\<close> w \<open>L>0\<close>
+ apply (force simp: norm_minus_commute)
+ using wx w(2) \<open>L>0\<close> pe pe2 Lwz
+ apply (auto simp: divide_simps mult_less_0_iff norm_minus_commute norm_divide norm_mult power2_eq_square)
+ done
+ } note L_cmod_le = this
+ have *: "cmod (contour_integral p (\<lambda>x. 1 / (x - w) - 1 / (x - z))) \<le> L * (e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2)"
+ apply (rule L)
+ using \<open>pe>0\<close> w
+ apply (force simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
+ using \<open>pe>0\<close> w \<open>L>0\<close>
+ apply (auto simp: cball_def dist_norm field_simps L_cmod_le simp del: less_divide_eq_numeral1 le_divide_eq_numeral1)
+ done
+ have "cmod (contour_integral p (\<lambda>x. 1 / (x - w)) - contour_integral p (\<lambda>x. 1 / (x - z))) < 2*e"
+ apply simp
+ apply (rule le_less_trans [OF *])
+ using \<open>L>0\<close> e
+ apply (force simp: field_simps)
+ done
+ then have "cmod (winding_number p w - winding_number p z) < e"
+ using pi_ge_two e
+ by (force simp: winding_number_valid_path p field_simps norm_divide norm_mult intro: less_le_trans)
+ } note cmod_wn_diff = this
+ then have "isCont (winding_number p) z"
+ apply (simp add: continuous_at_eps_delta, clarify)
+ apply (rule_tac x="min (pe/4) (e/2*pe^2/L/4)" in exI)
+ using \<open>pe>0\<close> \<open>L>0\<close>
+ apply (simp add: dist_norm cmod_wn_diff)
+ done
+ then show ?thesis
+ apply (rule continuous_transform_within [where d = "min d e / 2"])
+ apply (auto simp: \<open>d>0\<close> \<open>e>0\<close> dist_norm wnwn)
+ done
+qed
+
+corollary continuous_on_winding_number:
+ "path \<gamma> \<Longrightarrow> continuous_on (- path_image \<gamma>) (\<lambda>w. winding_number \<gamma> w)"
+ by (simp add: continuous_at_imp_continuous_on continuous_at_winding_number)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number is constant on a connected region\<close>
+
+lemma winding_number_constant:
+ assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and cs: "connected S" and sg: "S \<inter> path_image \<gamma> = {}"
+ shows "winding_number \<gamma> constant_on S"
+proof -
+ have *: "1 \<le> cmod (winding_number \<gamma> y - winding_number \<gamma> z)"
+ if ne: "winding_number \<gamma> y \<noteq> winding_number \<gamma> z" and "y \<in> S" "z \<in> S" for y z
+ proof -
+ have "winding_number \<gamma> y \<in> \<int>" "winding_number \<gamma> z \<in> \<int>"
+ using that integer_winding_number [OF \<gamma> loop] sg \<open>y \<in> S\<close> by auto
+ with ne show ?thesis
+ by (auto simp: Ints_def simp flip: of_int_diff)
+ qed
+ have cont: "continuous_on S (\<lambda>w. winding_number \<gamma> w)"
+ using continuous_on_winding_number [OF \<gamma>] sg
+ by (meson continuous_on_subset disjoint_eq_subset_Compl)
+ show ?thesis
+ using "*" zero_less_one
+ by (blast intro: continuous_discrete_range_constant [OF cs cont])
+qed
+
+lemma winding_number_eq:
+ "\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; w \<in> S; z \<in> S; connected S; S \<inter> path_image \<gamma> = {}\<rbrakk>
+ \<Longrightarrow> winding_number \<gamma> w = winding_number \<gamma> z"
+ using winding_number_constant by (metis constant_on_def)
+
+lemma open_winding_number_levelsets:
+ assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ shows "open {z. z \<notin> path_image \<gamma> \<and> winding_number \<gamma> z = k}"
+proof -
+ have opn: "open (- path_image \<gamma>)"
+ by (simp add: closed_path_image \<gamma> open_Compl)
+ { fix z assume z: "z \<notin> path_image \<gamma>" and k: "k = winding_number \<gamma> z"
+ obtain e where e: "e>0" "ball z e \<subseteq> - path_image \<gamma>"
+ using open_contains_ball [of "- path_image \<gamma>"] opn z
+ by blast
+ have "\<exists>e>0. \<forall>y. dist y z < e \<longrightarrow> y \<notin> path_image \<gamma> \<and> winding_number \<gamma> y = winding_number \<gamma> z"
+ apply (rule_tac x=e in exI)
+ using e apply (simp add: dist_norm ball_def norm_minus_commute)
+ apply (auto simp: dist_norm norm_minus_commute intro!: winding_number_eq [OF assms, where S = "ball z e"])
+ done
+ } then
+ show ?thesis
+ by (auto simp: open_dist)
+qed
+
+subsection\<open>Winding number is zero "outside" a curve\<close>
+
+proposition winding_number_zero_in_outside:
+ assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and z: "z \<in> outside (path_image \<gamma>)"
+ shows "winding_number \<gamma> z = 0"
+proof -
+ obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
+ using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
+ obtain w::complex where w: "w \<notin> ball 0 (B + 1)"
+ by (metis abs_of_nonneg le_less less_irrefl mem_ball_0 norm_of_real)
+ have "- ball 0 (B + 1) \<subseteq> outside (path_image \<gamma>)"
+ apply (rule outside_subset_convex)
+ using B subset_ball by auto
+ then have wout: "w \<in> outside (path_image \<gamma>)"
+ using w by blast
+ moreover have "winding_number \<gamma> constant_on outside (path_image \<gamma>)"
+ using winding_number_constant [OF \<gamma> loop, of "outside(path_image \<gamma>)"] connected_outside
+ by (metis DIM_complex bounded_path_image dual_order.refl \<gamma> outside_no_overlap)
+ ultimately have "winding_number \<gamma> z = winding_number \<gamma> w"
+ by (metis (no_types, hide_lams) constant_on_def z)
+ also have "\<dots> = 0"
+ proof -
+ have wnot: "w \<notin> path_image \<gamma>" using wout by (simp add: outside_def)
+ { fix e::real assume "0<e"
+ obtain p where p: "polynomial_function p" "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
+ and pg1: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < 1)"
+ and pge: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < e)"
+ using path_approx_polynomial_function [OF \<gamma>, of "min 1 e"] \<open>e>0\<close> by force
+ have pip: "path_image p \<subseteq> ball 0 (B + 1)"
+ using B
+ apply (clarsimp simp add: path_image_def dist_norm ball_def)
+ apply (frule (1) pg1)
+ apply (fastforce dest: norm_add_less)
+ done
+ then have "w \<notin> path_image p" using w by blast
+ then have "\<exists>p. valid_path p \<and> w \<notin> path_image p \<and>
+ pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and>
+ (\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0"
+ apply (rule_tac x=p in exI)
+ apply (simp add: p valid_path_polynomial_function)
+ apply (intro conjI)
+ using pge apply (simp add: norm_minus_commute)
+ apply (rule contour_integral_unique [OF Cauchy_theorem_convex_simple [OF _ convex_ball [of 0 "B+1"]]])
+ apply (rule holomorphic_intros | simp add: dist_norm)+
+ using mem_ball_0 w apply blast
+ using p apply (simp_all add: valid_path_polynomial_function loop pip)
+ done
+ }
+ then show ?thesis
+ by (auto intro: winding_number_unique [OF \<gamma>] simp add: winding_number_prop_def wnot)
+ qed
+ finally show ?thesis .
+qed
+
+corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_const: "a \<noteq> z \<Longrightarrow> winding_number (\<lambda>t. a) z = 0"
+ by (rule winding_number_zero_in_outside)
+ (auto simp: pathfinish_def pathstart_def path_polynomial_function)
+
+corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_outside:
+ "\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> s; path_image \<gamma> \<subseteq> s\<rbrakk> \<Longrightarrow> winding_number \<gamma> z = 0"
+ by (meson convex_in_outside outside_mono subsetCE winding_number_zero_in_outside)
+
+lemma winding_number_zero_at_infinity:
+ assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ shows "\<exists>B. \<forall>z. B \<le> norm z \<longrightarrow> winding_number \<gamma> z = 0"
+proof -
+ obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
+ using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
+ then show ?thesis
+ apply (rule_tac x="B+1" in exI, clarify)
+ apply (rule winding_number_zero_outside [OF \<gamma> convex_cball [of 0 B] loop])
+ apply (meson less_add_one mem_cball_0 not_le order_trans)
+ using ball_subset_cball by blast
+qed
+
+lemma winding_number_zero_point:
+ "\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; open s; path_image \<gamma> \<subseteq> s\<rbrakk>
+ \<Longrightarrow> \<exists>z. z \<in> s \<and> winding_number \<gamma> z = 0"
+ using outside_compact_in_open [of "path_image \<gamma>" s] path_image_nonempty winding_number_zero_in_outside
+ by (fastforce simp add: compact_path_image)
+
+
+text\<open>If a path winds round a set, it winds rounds its inside.\<close>
+lemma winding_number_around_inside:
+ assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ and cls: "closed s" and cos: "connected s" and s_disj: "s \<inter> path_image \<gamma> = {}"
+ and z: "z \<in> s" and wn_nz: "winding_number \<gamma> z \<noteq> 0" and w: "w \<in> s \<union> inside s"
+ shows "winding_number \<gamma> w = winding_number \<gamma> z"
+proof -
+ have ssb: "s \<subseteq> inside(path_image \<gamma>)"
+ proof
+ fix x :: complex
+ assume "x \<in> s"
+ hence "x \<notin> path_image \<gamma>"
+ by (meson disjoint_iff_not_equal s_disj)
+ thus "x \<in> inside (path_image \<gamma>)"
+ using \<open>x \<in> s\<close> by (metis (no_types) ComplI UnE cos \<gamma> loop s_disj union_with_outside winding_number_eq winding_number_zero_in_outside wn_nz z)
+qed
+ show ?thesis
+ apply (rule winding_number_eq [OF \<gamma> loop w])
+ using z apply blast
+ apply (simp add: cls connected_with_inside cos)
+ apply (simp add: Int_Un_distrib2 s_disj, safe)
+ by (meson ssb inside_inside_compact_connected [OF cls, of "path_image \<gamma>"] compact_path_image connected_path_image contra_subsetD disjoint_iff_not_equal \<gamma> inside_no_overlap)
+ qed
+
+
+text\<open>Bounding a WN by 1/2 for a path and point in opposite halfspaces.\<close>
+lemma winding_number_subpath_continuous:
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
+ shows "continuous_on {0..1} (\<lambda>x. winding_number(subpath 0 x \<gamma>) z)"
+proof -
+ have *: "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
+ winding_number (subpath 0 x \<gamma>) z"
+ if x: "0 \<le> x" "x \<le> 1" for x
+ proof -
+ have "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
+ 1 / (2*pi*\<i>) * contour_integral (subpath 0 x \<gamma>) (\<lambda>w. 1/(w - z))"
+ using assms x
+ apply (simp add: contour_integral_subcontour_integral [OF contour_integrable_inversediff])
+ done
+ also have "\<dots> = winding_number (subpath 0 x \<gamma>) z"
+ apply (subst winding_number_valid_path)
+ using assms x
+ apply (simp_all add: path_image_subpath valid_path_subpath)
+ by (force simp: path_image_def)
+ finally show ?thesis .
+ qed
+ show ?thesis
+ apply (rule continuous_on_eq
+ [where f = "\<lambda>x. 1 / (2*pi*\<i>) *
+ integral {0..x} (\<lambda>t. 1/(\<gamma> t - z) * vector_derivative \<gamma> (at t))"])
+ apply (rule continuous_intros)+
+ apply (rule indefinite_integral_continuous_1)
+ apply (rule contour_integrable_inversediff [OF assms, unfolded contour_integrable_on])
+ using assms
+ apply (simp add: *)
+ done
+qed
+
+lemma winding_number_ivt_pos:
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> Re(winding_number \<gamma> z)"
+ shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
+ apply (rule ivt_increasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right], simp)
+ apply (rule winding_number_subpath_continuous [OF \<gamma> z])
+ using assms
+ apply (auto simp: path_image_def image_def)
+ done
+
+lemma winding_number_ivt_neg:
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "Re(winding_number \<gamma> z) \<le> w" "w \<le> 0"
+ shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
+ apply (rule ivt_decreasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right], simp)
+ apply (rule winding_number_subpath_continuous [OF \<gamma> z])
+ using assms
+ apply (auto simp: path_image_def image_def)
+ done
+
+lemma winding_number_ivt_abs:
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> \<bar>Re(winding_number \<gamma> z)\<bar>"
+ shows "\<exists>t \<in> {0..1}. \<bar>Re (winding_number (subpath 0 t \<gamma>) z)\<bar> = w"
+ using assms winding_number_ivt_pos [of \<gamma> z w] winding_number_ivt_neg [of \<gamma> z "-w"]
+ by force
+
+lemma winding_number_lt_half_lemma:
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and az: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
+ shows "Re(winding_number \<gamma> z) < 1/2"
+proof -
+ { assume "Re(winding_number \<gamma> z) \<ge> 1/2"
+ then obtain t::real where t: "0 \<le> t" "t \<le> 1" and sub12: "Re (winding_number (subpath 0 t \<gamma>) z) = 1/2"
+ using winding_number_ivt_pos [OF \<gamma> z, of "1/2"] by auto
+ have gt: "\<gamma> t - z = - (of_real (exp (- (2 * pi * Im (winding_number (subpath 0 t \<gamma>) z)))) * (\<gamma> 0 - z))"
+ using winding_number_exp_2pi [of "subpath 0 t \<gamma>" z]
+ apply (simp add: t \<gamma> valid_path_imp_path)
+ using closed_segment_eq_real_ivl path_image_def t z by (fastforce simp: path_image_subpath Euler sub12)
+ have "b < a \<bullet> \<gamma> 0"
+ proof -
+ have "\<gamma> 0 \<in> {c. b < a \<bullet> c}"
+ by (metis (no_types) pag atLeastAtMost_iff image_subset_iff order_refl path_image_def zero_le_one)
+ thus ?thesis
+ by blast
+ qed
+ moreover have "b < a \<bullet> \<gamma> t"
+ proof -
+ have "\<gamma> t \<in> {c. b < a \<bullet> c}"
+ by (metis (no_types) pag atLeastAtMost_iff image_subset_iff path_image_def t)
+ thus ?thesis
+ by blast
+ qed
+ ultimately have "0 < a \<bullet> (\<gamma> 0 - z)" "0 < a \<bullet> (\<gamma> t - z)" using az
+ by (simp add: inner_diff_right)+
+ then have False
+ by (simp add: gt inner_mult_right mult_less_0_iff)
+ }
+ then show ?thesis by force
+qed
+
+lemma winding_number_lt_half:
+ assumes "valid_path \<gamma>" "a \<bullet> z \<le> b" "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
+ shows "\<bar>Re (winding_number \<gamma> z)\<bar> < 1/2"
+proof -
+ have "z \<notin> path_image \<gamma>" using assms by auto
+ with assms show ?thesis
+ apply (simp add: winding_number_lt_half_lemma abs_if del: less_divide_eq_numeral1)
+ apply (metis complex_inner_1_right winding_number_lt_half_lemma [OF valid_path_imp_reverse, of \<gamma> z a b]
+ winding_number_reversepath valid_path_imp_path inner_minus_left path_image_reversepath)
+ done
+qed
+
+lemma winding_number_le_half:
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
+ and anz: "a \<noteq> 0" and azb: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w \<ge> b}"
+ shows "\<bar>Re (winding_number \<gamma> z)\<bar> \<le> 1/2"
+proof -
+ { assume wnz_12: "\<bar>Re (winding_number \<gamma> z)\<bar> > 1/2"
+ have "isCont (winding_number \<gamma>) z"
+ by (metis continuous_at_winding_number valid_path_imp_path \<gamma> z)
+ then obtain d where "d>0" and d: "\<And>x'. dist x' z < d \<Longrightarrow> dist (winding_number \<gamma> x') (winding_number \<gamma> z) < \<bar>Re(winding_number \<gamma> z)\<bar> - 1/2"
+ using continuous_at_eps_delta wnz_12 diff_gt_0_iff_gt by blast
+ define z' where "z' = z - (d / (2 * cmod a)) *\<^sub>R a"
+ have *: "a \<bullet> z' \<le> b - d / 3 * cmod a"
+ unfolding z'_def inner_mult_right' divide_inverse
+ apply (simp add: field_split_simps algebra_simps dot_square_norm power2_eq_square anz)
+ apply (metis \<open>0 < d\<close> add_increasing azb less_eq_real_def mult_nonneg_nonneg mult_right_mono norm_ge_zero norm_numeral)
+ done
+ have "cmod (winding_number \<gamma> z' - winding_number \<gamma> z) < \<bar>Re (winding_number \<gamma> z)\<bar> - 1/2"
+ using d [of z'] anz \<open>d>0\<close> by (simp add: dist_norm z'_def)
+ then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - cmod (winding_number \<gamma> z' - winding_number \<gamma> z)"
+ by simp
+ then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - \<bar>Re (winding_number \<gamma> z') - Re (winding_number \<gamma> z)\<bar>"
+ using abs_Re_le_cmod [of "winding_number \<gamma> z' - winding_number \<gamma> z"] by simp
+ then have wnz_12': "\<bar>Re (winding_number \<gamma> z')\<bar> > 1/2"
+ by linarith
+ moreover have "\<bar>Re (winding_number \<gamma> z')\<bar> < 1/2"
+ apply (rule winding_number_lt_half [OF \<gamma> *])
+ using azb \<open>d>0\<close> pag
+ apply (auto simp: add_strict_increasing anz field_split_simps dest!: subsetD)
+ done
+ ultimately have False
+ by simp
+ }
+ then show ?thesis by force
+qed
+
+lemma winding_number_lt_half_linepath: "z \<notin> closed_segment a b \<Longrightarrow> \<bar>Re (winding_number (linepath a b) z)\<bar> < 1/2"
+ using separating_hyperplane_closed_point [of "closed_segment a b" z]
+ apply auto
+ apply (simp add: closed_segment_def)
+ apply (drule less_imp_le)
+ apply (frule winding_number_lt_half [OF valid_path_linepath [of a b]])
+ apply (auto simp: segment)
+ done
+
+
+text\<open> Positivity of WN for a linepath.\<close>
+lemma winding_number_linepath_pos_lt:
+ assumes "0 < Im ((b - a) * cnj (b - z))"
+ shows "0 < Re(winding_number(linepath a b) z)"
+proof -
+ have z: "z \<notin> path_image (linepath a b)"
+ using assms
+ by (simp add: closed_segment_def) (force simp: algebra_simps)
+ show ?thesis
+ apply (rule winding_number_pos_lt [OF valid_path_linepath z assms])
+ apply (simp add: linepath_def algebra_simps)
+ done
+qed
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>More winding number properties\<close>
+
+text\<open>including the fact that it's +-1 inside a simple closed curve.\<close>
+
+lemma winding_number_homotopic_paths:
+ assumes "homotopic_paths (-{z}) g h"
+ shows "winding_number g z = winding_number h z"
+proof -
+ have "path g" "path h" using homotopic_paths_imp_path [OF assms] by auto
+ moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
+ using homotopic_paths_imp_subset [OF assms] by auto
+ ultimately obtain d e where "d > 0" "e > 0"
+ and d: "\<And>p. \<lbrakk>path p; pathstart p = pathstart g; pathfinish p = pathfinish g; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
+ \<Longrightarrow> homotopic_paths (-{z}) g p"
+ and e: "\<And>q. \<lbrakk>path q; pathstart q = pathstart h; pathfinish q = pathfinish h; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
+ \<Longrightarrow> homotopic_paths (-{z}) h q"
+ using homotopic_nearby_paths [of g "-{z}"] homotopic_nearby_paths [of h "-{z}"] by force
+ obtain p where p:
+ "valid_path p" "z \<notin> path_image p"
+ "pathstart p = pathstart g" "pathfinish p = pathfinish g"
+ and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
+ and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
+ using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast
+ obtain q where q:
+ "valid_path q" "z \<notin> path_image q"
+ "pathstart q = pathstart h" "pathfinish q = pathfinish h"
+ and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
+ and paq: "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
+ using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast
+ have "homotopic_paths (- {z}) g p"
+ by (simp add: d p valid_path_imp_path norm_minus_commute gp_less)
+ moreover have "homotopic_paths (- {z}) h q"
+ by (simp add: e q valid_path_imp_path norm_minus_commute hq_less)
+ ultimately have "homotopic_paths (- {z}) p q"
+ by (blast intro: homotopic_paths_trans homotopic_paths_sym assms)
+ then have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
+ by (rule Cauchy_theorem_homotopic_paths) (auto intro!: holomorphic_intros simp: p q)
+ then show ?thesis
+ by (simp add: pap paq)
+qed
+
+lemma winding_number_homotopic_loops:
+ assumes "homotopic_loops (-{z}) g h"
+ shows "winding_number g z = winding_number h z"
+proof -
+ have "path g" "path h" using homotopic_loops_imp_path [OF assms] by auto
+ moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
+ using homotopic_loops_imp_subset [OF assms] by auto
+ moreover have gloop: "pathfinish g = pathstart g" and hloop: "pathfinish h = pathstart h"
+ using homotopic_loops_imp_loop [OF assms] by auto
+ ultimately obtain d e where "d > 0" "e > 0"
+ and d: "\<And>p. \<lbrakk>path p; pathfinish p = pathstart p; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
+ \<Longrightarrow> homotopic_loops (-{z}) g p"
+ and e: "\<And>q. \<lbrakk>path q; pathfinish q = pathstart q; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
+ \<Longrightarrow> homotopic_loops (-{z}) h q"
+ using homotopic_nearby_loops [of g "-{z}"] homotopic_nearby_loops [of h "-{z}"] by force
+ obtain p where p:
+ "valid_path p" "z \<notin> path_image p"
+ "pathstart p = pathstart g" "pathfinish p = pathfinish g"
+ and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
+ and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
+ using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast
+ obtain q where q:
+ "valid_path q" "z \<notin> path_image q"
+ "pathstart q = pathstart h" "pathfinish q = pathfinish h"
+ and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
+ and paq: "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
+ using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast
+ have gp: "homotopic_loops (- {z}) g p"
+ by (simp add: gloop d gp_less norm_minus_commute p valid_path_imp_path)
+ have hq: "homotopic_loops (- {z}) h q"
+ by (simp add: e hloop hq_less norm_minus_commute q valid_path_imp_path)
+ have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
+ proof (rule Cauchy_theorem_homotopic_loops)
+ show "homotopic_loops (- {z}) p q"
+ by (blast intro: homotopic_loops_trans homotopic_loops_sym gp hq assms)
+ qed (auto intro!: holomorphic_intros simp: p q)
+ then show ?thesis
+ by (simp add: pap paq)
+qed
+
+lemma winding_number_paths_linear_eq:
+ "\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g;
+ \<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
+ \<Longrightarrow> winding_number h z = winding_number g z"
+ by (blast intro: sym homotopic_paths_linear winding_number_homotopic_paths)
+
+lemma winding_number_loops_linear_eq:
+ "\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h;
+ \<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
+ \<Longrightarrow> winding_number h z = winding_number g z"
+ by (blast intro: sym homotopic_loops_linear winding_number_homotopic_loops)
+
+lemma winding_number_nearby_paths_eq:
+ "\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g;
+ \<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
+ \<Longrightarrow> winding_number h z = winding_number g z"
+ by (metis segment_bound(2) norm_minus_commute not_le winding_number_paths_linear_eq)
+
+lemma winding_number_nearby_loops_eq:
+ "\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h;
+ \<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
+ \<Longrightarrow> winding_number h z = winding_number g z"
+ by (metis segment_bound(2) norm_minus_commute not_le winding_number_loops_linear_eq)
+
+
+lemma winding_number_subpath_combine:
+ "\<lbrakk>path g; z \<notin> path_image g;
+ u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
+ \<Longrightarrow> winding_number (subpath u v g) z + winding_number (subpath v w g) z =
+ winding_number (subpath u w g) z"
+apply (rule trans [OF winding_number_join [THEN sym]
+ winding_number_homotopic_paths [OF homotopic_join_subpaths]])
+ using path_image_subpath_subset by auto
+
+text \<open>Winding numbers of circular contours\<close>
+
+proposition winding_number_part_circlepath_pos_less:
+ assumes "s < t" and no: "norm(w - z) < r"
+ shows "0 < Re (winding_number(part_circlepath z r s t) w)"
+proof -
+ have "0 < r" by (meson no norm_not_less_zero not_le order.strict_trans2)
+ note valid_path_part_circlepath
+ moreover have " w \<notin> path_image (part_circlepath z r s t)"
+ using assms by (auto simp: path_image_def image_def part_circlepath_def norm_mult linepath_def)
+ moreover have "0 < r * (t - s) * (r - cmod (w - z))"
+ using assms by (metis \<open>0 < r\<close> diff_gt_0_iff_gt mult_pos_pos)
+ ultimately show ?thesis
+ apply (rule winding_number_pos_lt [where e = "r*(t - s)*(r - norm(w - z))"])
+ apply (simp add: vector_derivative_part_circlepath right_diff_distrib [symmetric] mult_ac)
+ apply (rule mult_left_mono)+
+ using Re_Im_le_cmod [of "w-z" "linepath s t x" for x]
+ apply (simp add: exp_Euler cos_of_real sin_of_real part_circlepath_def algebra_simps cos_squared_eq [unfolded power2_eq_square])
+ using assms \<open>0 < r\<close> by auto
+qed
+
+lemma winding_number_circlepath_centre: "0 < r \<Longrightarrow> winding_number (circlepath z r) z = 1"
+ apply (rule winding_number_unique_loop)
+ apply (simp_all add: sphere_def valid_path_imp_path)
+ apply (rule_tac x="circlepath z r" in exI)
+ apply (simp add: sphere_def contour_integral_circlepath)
+ done
+
+proposition winding_number_circlepath:
+ assumes "norm(w - z) < r" shows "winding_number(circlepath z r) w = 1"
+proof (cases "w = z")
+ case True then show ?thesis
+ using assms winding_number_circlepath_centre by auto
+next
+ case False
+ have [simp]: "r > 0"
+ using assms le_less_trans norm_ge_zero by blast
+ define r' where "r' = norm(w - z)"
+ have "r' < r"
+ by (simp add: assms r'_def)
+ have disjo: "cball z r' \<inter> sphere z r = {}"
+ using \<open>r' < r\<close> by (force simp: cball_def sphere_def)
+ have "winding_number(circlepath z r) w = winding_number(circlepath z r) z"
+ proof (rule winding_number_around_inside [where s = "cball z r'"])
+ show "winding_number (circlepath z r) z \<noteq> 0"
+ by (simp add: winding_number_circlepath_centre)
+ show "cball z r' \<inter> path_image (circlepath z r) = {}"
+ by (simp add: disjo less_eq_real_def)
+ qed (auto simp: r'_def dist_norm norm_minus_commute)
+ also have "\<dots> = 1"
+ by (simp add: winding_number_circlepath_centre)
+ finally show ?thesis .
+qed
+
+lemma no_bounded_connected_component_imp_winding_number_zero:
+ assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
+ and nb: "\<And>z. bounded (connected_component_set (- s) z) \<longrightarrow> z \<in> s"
+ shows "winding_number g z = 0"
+apply (rule winding_number_zero_in_outside)
+apply (simp_all add: assms)
+by (metis nb [of z] \<open>path_image g \<subseteq> s\<close> \<open>z \<notin> s\<close> contra_subsetD mem_Collect_eq outside outside_mono)
+
+lemma no_bounded_path_component_imp_winding_number_zero:
+ assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
+ and nb: "\<And>z. bounded (path_component_set (- s) z) \<longrightarrow> z \<in> s"
+ shows "winding_number g z = 0"
+apply (rule no_bounded_connected_component_imp_winding_number_zero [OF g])
+by (simp add: bounded_subset nb path_component_subset_connected_component)
subsection\<open>Winding number for a triangle\<close>
@@ -913,241 +2164,4 @@
finally show ?thesis .
qed
-subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number defines a continuous logarithm for the path itself\<close>
-
-lemma winding_number_as_continuous_log:
- assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
- obtains q where "path q"
- "pathfinish q - pathstart q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
- "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = \<zeta> + exp(q t)"
-proof -
- let ?q = "\<lambda>t. 2 * of_real pi * \<i> * winding_number(subpath 0 t p) \<zeta> + Ln(pathstart p - \<zeta>)"
- show ?thesis
- proof
- have *: "continuous (at t within {0..1}) (\<lambda>x. winding_number (subpath 0 x p) \<zeta>)"
- if t: "t \<in> {0..1}" for t
- proof -
- let ?B = "ball (p t) (norm(p t - \<zeta>))"
- have "p t \<noteq> \<zeta>"
- using path_image_def that \<zeta> by blast
- then have "simply_connected ?B"
- by (simp add: convex_imp_simply_connected)
- then have "\<And>f::complex\<Rightarrow>complex. continuous_on ?B f \<and> (\<forall>\<zeta> \<in> ?B. f \<zeta> \<noteq> 0)
- \<longrightarrow> (\<exists>g. continuous_on ?B g \<and> (\<forall>\<zeta> \<in> ?B. f \<zeta> = exp (g \<zeta>)))"
- by (simp add: simply_connected_eq_continuous_log)
- moreover have "continuous_on ?B (\<lambda>w. w - \<zeta>)"
- by (intro continuous_intros)
- moreover have "(\<forall>z \<in> ?B. z - \<zeta> \<noteq> 0)"
- by (auto simp: dist_norm)
- ultimately obtain g where contg: "continuous_on ?B g"
- and geq: "\<And>z. z \<in> ?B \<Longrightarrow> z - \<zeta> = exp (g z)" by blast
- obtain d where "0 < d" and d:
- "\<And>x. \<lbrakk>x \<in> {0..1}; dist x t < d\<rbrakk> \<Longrightarrow> dist (p x) (p t) < cmod (p t - \<zeta>)"
- using \<open>path p\<close> t unfolding path_def continuous_on_iff
- by (metis \<open>p t \<noteq> \<zeta>\<close> right_minus_eq zero_less_norm_iff)
- have "((\<lambda>x. winding_number (\<lambda>w. subpath 0 x p w - \<zeta>) 0 -
- winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0) \<longlongrightarrow> 0)
- (at t within {0..1})"
- proof (rule Lim_transform_within [OF _ \<open>d > 0\<close>])
- have "continuous (at t within {0..1}) (g o p)"
- proof (rule continuous_within_compose)
- show "continuous (at t within {0..1}) p"
- using \<open>path p\<close> continuous_on_eq_continuous_within path_def that by blast
- show "continuous (at (p t) within p ` {0..1}) g"
- by (metis (no_types, lifting) open_ball UNIV_I \<open>p t \<noteq> \<zeta>\<close> centre_in_ball contg continuous_on_eq_continuous_at continuous_within_topological right_minus_eq zero_less_norm_iff)
- qed
- with LIM_zero have "((\<lambda>u. (g (subpath t u p 1) - g (subpath t u p 0))) \<longlongrightarrow> 0) (at t within {0..1})"
- by (auto simp: subpath_def continuous_within o_def)
- then show "((\<lambda>u. (g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>)) \<longlongrightarrow> 0)
- (at t within {0..1})"
- by (simp add: tendsto_divide_zero)
- show "(g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>) =
- winding_number (\<lambda>w. subpath 0 u p w - \<zeta>) 0 - winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0"
- if "u \<in> {0..1}" "0 < dist u t" "dist u t < d" for u
- proof -
- have "closed_segment t u \<subseteq> {0..1}"
- using closed_segment_eq_real_ivl t that by auto
- then have piB: "path_image(subpath t u p) \<subseteq> ?B"
- apply (clarsimp simp add: path_image_subpath_gen)
- by (metis subsetD le_less_trans \<open>dist u t < d\<close> d dist_commute dist_in_closed_segment)
- have *: "path (g \<circ> subpath t u p)"
- apply (rule path_continuous_image)
- using \<open>path p\<close> t that apply auto[1]
- using piB contg continuous_on_subset by blast
- have "(g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>)
- = winding_number (exp \<circ> g \<circ> subpath t u p) 0"
- using winding_number_compose_exp [OF *]
- by (simp add: pathfinish_def pathstart_def o_assoc)
- also have "... = winding_number (\<lambda>w. subpath t u p w - \<zeta>) 0"
- proof (rule winding_number_cong)
- have "exp(g y) = y - \<zeta>" if "y \<in> (subpath t u p) ` {0..1}" for y
- by (metis that geq path_image_def piB subset_eq)
- then show "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> (exp \<circ> g \<circ> subpath t u p) x = subpath t u p x - \<zeta>"
- by auto
- qed
- also have "... = winding_number (\<lambda>w. subpath 0 u p w - \<zeta>) 0 -
- winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0"
- apply (simp add: winding_number_offset [symmetric])
- using winding_number_subpath_combine [OF \<open>path p\<close> \<zeta>, of 0 t u] \<open>t \<in> {0..1}\<close> \<open>u \<in> {0..1}\<close>
- by (simp add: add.commute eq_diff_eq)
- finally show ?thesis .
- qed
- qed
- then show ?thesis
- by (subst winding_number_offset) (simp add: continuous_within LIM_zero_iff)
- qed
- show "path ?q"
- unfolding path_def
- by (intro continuous_intros) (simp add: continuous_on_eq_continuous_within *)
-
- have "\<zeta> \<noteq> p 0"
- by (metis \<zeta> pathstart_def pathstart_in_path_image)
- then show "pathfinish ?q - pathstart ?q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
- by (simp add: pathfinish_def pathstart_def)
- show "p t = \<zeta> + exp (?q t)" if "t \<in> {0..1}" for t
- proof -
- have "path (subpath 0 t p)"
- using \<open>path p\<close> that by auto
- moreover
- have "\<zeta> \<notin> path_image (subpath 0 t p)"
- using \<zeta> [unfolded path_image_def] that by (auto simp: path_image_subpath)
- ultimately show ?thesis
- using winding_number_exp_2pi [of "subpath 0 t p" \<zeta>] \<open>\<zeta> \<noteq> p 0\<close>
- by (auto simp: exp_add algebra_simps pathfinish_def pathstart_def subpath_def)
- qed
- qed
-qed
-
-subsection \<open>Winding number equality is the same as path/loop homotopy in C - {0}\<close>
-
-lemma winding_number_homotopic_loops_null_eq:
- assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
- shows "winding_number p \<zeta> = 0 \<longleftrightarrow> (\<exists>a. homotopic_loops (-{\<zeta>}) p (\<lambda>t. a))"
- (is "?lhs = ?rhs")
-proof
- assume [simp]: ?lhs
- obtain q where "path q"
- and qeq: "pathfinish q - pathstart q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
- and peq: "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = \<zeta> + exp(q t)"
- using winding_number_as_continuous_log [OF assms] by blast
- have *: "homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r)
- {0..1} (-{\<zeta>}) ((\<lambda>w. \<zeta> + exp w) \<circ> q) ((\<lambda>w. \<zeta> + exp w) \<circ> (\<lambda>t. 0))"
- proof (rule homotopic_with_compose_continuous_left)
- show "homotopic_with_canon (\<lambda>f. pathfinish ((\<lambda>w. \<zeta> + exp w) \<circ> f) = pathstart ((\<lambda>w. \<zeta> + exp w) \<circ> f))
- {0..1} UNIV q (\<lambda>t. 0)"
- proof (rule homotopic_with_mono, simp_all add: pathfinish_def pathstart_def)
- have "homotopic_loops UNIV q (\<lambda>t. 0)"
- by (rule homotopic_loops_linear) (use qeq \<open>path q\<close> in \<open>auto simp: path_defs\<close>)
- then have "homotopic_with (\<lambda>r. r 1 = r 0) (top_of_set {0..1}) euclidean q (\<lambda>t. 0)"
- by (simp add: homotopic_loops_def pathfinish_def pathstart_def)
- then show "homotopic_with (\<lambda>h. exp (h 1) = exp (h 0)) (top_of_set {0..1}) euclidean q (\<lambda>t. 0)"
- by (rule homotopic_with_mono) simp
- qed
- show "continuous_on UNIV (\<lambda>w. \<zeta> + exp w)"
- by (rule continuous_intros)+
- show "range (\<lambda>w. \<zeta> + exp w) \<subseteq> -{\<zeta>}"
- by auto
- qed
- then have "homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r) {0..1} (-{\<zeta>}) p (\<lambda>x. \<zeta> + 1)"
- by (rule homotopic_with_eq) (auto simp: o_def peq pathfinish_def pathstart_def)
- then have "homotopic_loops (-{\<zeta>}) p (\<lambda>t. \<zeta> + 1)"
- by (simp add: homotopic_loops_def)
- then show ?rhs ..
-next
- assume ?rhs
- then obtain a where "homotopic_loops (-{\<zeta>}) p (\<lambda>t. a)" ..
- then have "winding_number p \<zeta> = winding_number (\<lambda>t. a) \<zeta>" "a \<noteq> \<zeta>"
- using winding_number_homotopic_loops homotopic_loops_imp_subset by (force simp:)+
- moreover have "winding_number (\<lambda>t. a) \<zeta> = 0"
- by (metis winding_number_zero_const \<open>a \<noteq> \<zeta>\<close>)
- ultimately show ?lhs by metis
-qed
-
-lemma winding_number_homotopic_paths_null_explicit_eq:
- assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
- shows "winding_number p \<zeta> = 0 \<longleftrightarrow> homotopic_paths (-{\<zeta>}) p (linepath (pathstart p) (pathstart p))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- apply (auto simp: winding_number_homotopic_loops_null_eq [OF assms])
- apply (rule homotopic_loops_imp_homotopic_paths_null)
- apply (simp add: linepath_refl)
- done
-next
- assume ?rhs
- then show ?lhs
- by (metis \<zeta> pathstart_in_path_image winding_number_homotopic_paths winding_number_trivial)
-qed
-
-lemma winding_number_homotopic_paths_null_eq:
- assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
- shows "winding_number p \<zeta> = 0 \<longleftrightarrow> (\<exists>a. homotopic_paths (-{\<zeta>}) p (\<lambda>t. a))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- by (auto simp: winding_number_homotopic_paths_null_explicit_eq [OF assms] linepath_refl)
-next
- assume ?rhs
- then show ?lhs
- by (metis \<zeta> homotopic_paths_imp_pathfinish pathfinish_def pathfinish_in_path_image winding_number_homotopic_paths winding_number_zero_const)
-qed
-
-proposition winding_number_homotopic_paths_eq:
- assumes "path p" and \<zeta>p: "\<zeta> \<notin> path_image p"
- and "path q" and \<zeta>q: "\<zeta> \<notin> path_image q"
- and qp: "pathstart q = pathstart p" "pathfinish q = pathfinish p"
- shows "winding_number p \<zeta> = winding_number q \<zeta> \<longleftrightarrow> homotopic_paths (-{\<zeta>}) p q"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then have "winding_number (p +++ reversepath q) \<zeta> = 0"
- using assms by (simp add: winding_number_join winding_number_reversepath)
- moreover
- have "path (p +++ reversepath q)" "\<zeta> \<notin> path_image (p +++ reversepath q)"
- using assms by (auto simp: not_in_path_image_join)
- ultimately obtain a where "homotopic_paths (- {\<zeta>}) (p +++ reversepath q) (linepath a a)"
- using winding_number_homotopic_paths_null_explicit_eq by blast
- then show ?rhs
- using homotopic_paths_imp_pathstart assms
- by (fastforce simp add: dest: homotopic_paths_imp_homotopic_loops homotopic_paths_loop_parts)
-next
- assume ?rhs
- then show ?lhs
- by (simp add: winding_number_homotopic_paths)
-qed
-
-lemma winding_number_homotopic_loops_eq:
- assumes "path p" and \<zeta>p: "\<zeta> \<notin> path_image p"
- and "path q" and \<zeta>q: "\<zeta> \<notin> path_image q"
- and loops: "pathfinish p = pathstart p" "pathfinish q = pathstart q"
- shows "winding_number p \<zeta> = winding_number q \<zeta> \<longleftrightarrow> homotopic_loops (-{\<zeta>}) p q"
- (is "?lhs = ?rhs")
-proof
- assume L: ?lhs
- have "pathstart p \<noteq> \<zeta>" "pathstart q \<noteq> \<zeta>"
- using \<zeta>p \<zeta>q by blast+
- moreover have "path_connected (-{\<zeta>})"
- by (simp add: path_connected_punctured_universe)
- ultimately obtain r where "path r" and rim: "path_image r \<subseteq> -{\<zeta>}"
- and pas: "pathstart r = pathstart p" and paf: "pathfinish r = pathstart q"
- by (auto simp: path_connected_def)
- then have "pathstart r \<noteq> \<zeta>" by blast
- have "homotopic_loops (- {\<zeta>}) p (r +++ q +++ reversepath r)"
- proof (rule homotopic_paths_imp_homotopic_loops)
- show "homotopic_paths (- {\<zeta>}) p (r +++ q +++ reversepath r)"
- by (metis (mono_tags, hide_lams) \<open>path r\<close> L \<zeta>p \<zeta>q \<open>path p\<close> \<open>path q\<close> homotopic_loops_conjugate loops not_in_path_image_join paf pas path_image_reversepath path_imp_reversepath path_join_eq pathfinish_join pathfinish_reversepath pathstart_join pathstart_reversepath rim subset_Compl_singleton winding_number_homotopic_loops winding_number_homotopic_paths_eq)
- qed (use loops pas in auto)
- moreover have "homotopic_loops (- {\<zeta>}) (r +++ q +++ reversepath r) q"
- using rim \<zeta>q by (auto simp: homotopic_loops_conjugate paf \<open>path q\<close> \<open>path r\<close> loops)
- ultimately show ?rhs
- using homotopic_loops_trans by metis
-next
- assume ?rhs
- then show ?lhs
- by (simp add: winding_number_homotopic_loops)
-qed
-
-end
-
+end
\ No newline at end of file