section \<open>Conformal Mappings and Consequences of Cauchy's Integral Theorem\<close>
text\<open>By John Harrison et al. Ported from HOL Light by L C Paulson (2016)\<close>
text\<open>Also Cauchy's residue theorem by Wenda Li (2016)\<close>
theory Conformal_Mappings
imports Cauchy_Integral_Formula
begin
subsection \<open>Analytic continuation\<close>
proposition isolated_zeros:
assumes holf: "f holomorphic_on S"
and "open S" "connected S" "\<xi> \<in> S" "f \<xi> = 0" "\<beta> \<in> S" "f \<beta> \<noteq> 0"
obtains r where "0 < r" and "ball \<xi> r \<subseteq> S" and
"\<And>z. z \<in> ball \<xi> r - {\<xi>} \<Longrightarrow> f z \<noteq> 0"
proof -
obtain r where "0 < r" and r: "ball \<xi> r \<subseteq> S"
using \<open>open S\<close> \<open>\<xi> \<in> S\<close> open_contains_ball_eq by blast
have powf: "((\<lambda>n. (deriv ^^ n) f \<xi> / (fact n) * (z - \<xi>)^n) sums f z)" if "z \<in> ball \<xi> r" for z
apply (rule holomorphic_power_series [OF _ that])
apply (rule holomorphic_on_subset [OF holf r])
done
obtain m where m: "(deriv ^^ m) f \<xi> / (fact m) \<noteq> 0"
using holomorphic_fun_eq_0_on_connected [OF holf \<open>open S\<close> \<open>connected S\<close> _ \<open>\<xi> \<in> S\<close> \<open>\<beta> \<in> S\<close>] \<open>f \<beta> \<noteq> 0\<close>
by auto
then have "m \<noteq> 0" using assms(5) funpow_0 by fastforce
obtain s where "0 < s" and s: "\<And>z. z \<in> cball \<xi> s - {\<xi>} \<Longrightarrow> f z \<noteq> 0"
apply (rule powser_0_nonzero [OF \<open>0 < r\<close> powf \<open>f \<xi> = 0\<close> m])
using \<open>m \<noteq> 0\<close> by (auto simp: dist_commute dist_norm)
have "0 < min r s" by (simp add: \<open>0 < r\<close> \<open>0 < s\<close>)
then show ?thesis
apply (rule that)
using r s by auto
qed
proposition analytic_continuation:
assumes holf: "f holomorphic_on S"
and "open S" and "connected S"
and "U \<subseteq> S" and "\<xi> \<in> S"
and "\<xi> islimpt U"
and fU0 [simp]: "\<And>z. z \<in> U \<Longrightarrow> f z = 0"
and "w \<in> S"
shows "f w = 0"
proof -
obtain e where "0 < e" and e: "cball \<xi> e \<subseteq> S"
using \<open>open S\<close> \<open>\<xi> \<in> S\<close> open_contains_cball_eq by blast
define T where "T = cball \<xi> e \<inter> U"
have contf: "continuous_on (closure T) f"
by (metis T_def closed_cball closure_minimal e holf holomorphic_on_imp_continuous_on
holomorphic_on_subset inf.cobounded1)
have fT0 [simp]: "\<And>x. x \<in> T \<Longrightarrow> f x = 0"
by (simp add: T_def)
have "\<And>r. \<lbrakk>\<forall>e>0. \<exists>x'\<in>U. x' \<noteq> \<xi> \<and> dist x' \<xi> < e; 0 < r\<rbrakk> \<Longrightarrow> \<exists>x'\<in>cball \<xi> e \<inter> U. x' \<noteq> \<xi> \<and> dist x' \<xi> < r"
by (metis \<open>0 < e\<close> IntI dist_commute less_eq_real_def mem_cball min_less_iff_conj)
then have "\<xi> islimpt T" using \<open>\<xi> islimpt U\<close>
by (auto simp: T_def islimpt_approachable)
then have "\<xi> \<in> closure T"
by (simp add: closure_def)
then have "f \<xi> = 0"
by (auto simp: continuous_constant_on_closure [OF contf])
show ?thesis
apply (rule ccontr)
apply (rule isolated_zeros [OF holf \<open>open S\<close> \<open>connected S\<close> \<open>\<xi> \<in> S\<close> \<open>f \<xi> = 0\<close> \<open>w \<in> S\<close>], assumption)
by (metis open_ball \<open>\<xi> islimpt T\<close> centre_in_ball fT0 insertE insert_Diff islimptE)
qed
corollary analytic_continuation_open:
assumes "open s" and "open s'" and "s \<noteq> {}" and "connected s'"
and "s \<subseteq> s'"
assumes "f holomorphic_on s'" and "g holomorphic_on s'"
and "\<And>z. z \<in> s \<Longrightarrow> f z = g z"
assumes "z \<in> s'"
shows "f z = g z"
proof -
from \<open>s \<noteq> {}\<close> obtain \<xi> where "\<xi> \<in> s" by auto
with \<open>open s\<close> have \<xi>: "\<xi> islimpt s"
by (intro interior_limit_point) (auto simp: interior_open)
have "f z - g z = 0"
by (rule analytic_continuation[of "\<lambda>z. f z - g z" s' s \<xi>])
(insert assms \<open>\<xi> \<in> s\<close> \<xi>, auto intro: holomorphic_intros)
thus ?thesis by simp
qed
subsection\<open>Open mapping theorem\<close>
lemma holomorphic_contract_to_zero:
assumes contf: "continuous_on (cball \<xi> r) f"
and holf: "f holomorphic_on ball \<xi> r"
and "0 < r"
and norm_less: "\<And>z. norm(\<xi> - z) = r \<Longrightarrow> norm(f \<xi>) < norm(f z)"
obtains z where "z \<in> ball \<xi> r" "f z = 0"
proof -
{ assume fnz: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w \<noteq> 0"
then have "0 < norm (f \<xi>)"
by (simp add: \<open>0 < r\<close>)
have fnz': "\<And>w. w \<in> cball \<xi> r \<Longrightarrow> f w \<noteq> 0"
by (metis norm_less dist_norm fnz less_eq_real_def mem_ball mem_cball norm_not_less_zero norm_zero)
have "frontier(cball \<xi> r) \<noteq> {}"
using \<open>0 < r\<close> by simp
define g where [abs_def]: "g z = inverse (f z)" for z
have contg: "continuous_on (cball \<xi> r) g"
unfolding g_def using contf continuous_on_inverse fnz' by blast
have holg: "g holomorphic_on ball \<xi> r"
unfolding g_def using fnz holf holomorphic_on_inverse by blast
have "frontier (cball \<xi> r) \<subseteq> cball \<xi> r"
by (simp add: subset_iff)
then have contf': "continuous_on (frontier (cball \<xi> r)) f"
and contg': "continuous_on (frontier (cball \<xi> r)) g"
by (blast intro: contf contg continuous_on_subset)+
have froc: "frontier(cball \<xi> r) \<noteq> {}"
using \<open>0 < r\<close> by simp
moreover have "continuous_on (frontier (cball \<xi> r)) (norm o f)"
using contf' continuous_on_compose continuous_on_norm_id by blast
ultimately obtain w where w: "w \<in> frontier(cball \<xi> r)"
and now: "\<And>x. x \<in> frontier(cball \<xi> r) \<Longrightarrow> norm (f w) \<le> norm (f x)"
apply (rule bexE [OF continuous_attains_inf [OF compact_frontier [OF compact_cball]]])
apply simp
done
then have fw: "0 < norm (f w)"
by (simp add: fnz')
have "continuous_on (frontier (cball \<xi> r)) (norm o g)"
using contg' continuous_on_compose continuous_on_norm_id by blast
then obtain v where v: "v \<in> frontier(cball \<xi> r)"
and nov: "\<And>x. x \<in> frontier(cball \<xi> r) \<Longrightarrow> norm (g v) \<ge> norm (g x)"
apply (rule bexE [OF continuous_attains_sup [OF compact_frontier [OF compact_cball] froc]])
apply simp
done
then have fv: "0 < norm (f v)"
by (simp add: fnz')
have "norm ((deriv ^^ 0) g \<xi>) \<le> fact 0 * norm (g v) / r ^ 0"
by (rule Cauchy_inequality [OF holg contg \<open>0 < r\<close>]) (simp add: dist_norm nov)
then have "cmod (g \<xi>) \<le> norm (g v)"
by simp
with w have wr: "norm (\<xi> - w) = r" and nfw: "norm (f w) \<le> norm (f \<xi>)"
apply (simp_all add: dist_norm)
by (metis \<open>0 < cmod (f \<xi>)\<close> g_def less_imp_inverse_less norm_inverse not_le now order_trans v)
with fw have False
using norm_less by force
}
with that show ?thesis by blast
qed
theorem open_mapping_thm:
assumes holf: "f holomorphic_on S"
and S: "open S" and "connected S"
and "open U" and "U \<subseteq> S"
and fne: "\<not> f constant_on S"
shows "open (f ` U)"
proof -
have *: "open (f ` U)"
if "U \<noteq> {}" and U: "open U" "connected U" and "f holomorphic_on U" and fneU: "\<And>x. \<exists>y \<in> U. f y \<noteq> x"
for U
proof (clarsimp simp: open_contains_ball)
fix \<xi> assume \<xi>: "\<xi> \<in> U"
show "\<exists>e>0. ball (f \<xi>) e \<subseteq> f ` U"
proof -
have hol: "(\<lambda>z. f z - f \<xi>) holomorphic_on U"
by (rule holomorphic_intros that)+
obtain s where "0 < s" and sbU: "ball \<xi> s \<subseteq> U"
and sne: "\<And>z. z \<in> ball \<xi> s - {\<xi>} \<Longrightarrow> (\<lambda>z. f z - f \<xi>) z \<noteq> 0"
using isolated_zeros [OF hol U \<xi>] by (metis fneU right_minus_eq)
obtain r where "0 < r" and r: "cball \<xi> r \<subseteq> ball \<xi> s"
apply (rule_tac r="s/2" in that)
using \<open>0 < s\<close> by auto
have "cball \<xi> r \<subseteq> U"
using sbU r by blast
then have frsbU: "frontier (cball \<xi> r) \<subseteq> U"
using Diff_subset frontier_def order_trans by fastforce
then have cof: "compact (frontier(cball \<xi> r))"
by blast
have frne: "frontier (cball \<xi> r) \<noteq> {}"
using \<open>0 < r\<close> by auto
have contfr: "continuous_on (frontier (cball \<xi> r)) (\<lambda>z. norm (f z - f \<xi>))"
by (metis continuous_on_norm continuous_on_subset frsbU hol holomorphic_on_imp_continuous_on)
obtain w where "norm (\<xi> - w) = r"
and w: "(\<And>z. norm (\<xi> - z) = r \<Longrightarrow> norm (f w - f \<xi>) \<le> norm(f z - f \<xi>))"
apply (rule bexE [OF continuous_attains_inf [OF cof frne contfr]])
apply (simp add: dist_norm)
done
moreover define \<epsilon> where "\<epsilon> \<equiv> norm (f w - f \<xi>) / 3"
ultimately have "0 < \<epsilon>"
using \<open>0 < r\<close> dist_complex_def r sne by auto
have "ball (f \<xi>) \<epsilon> \<subseteq> f ` U"
proof
fix \<gamma>
assume \<gamma>: "\<gamma> \<in> ball (f \<xi>) \<epsilon>"
have *: "cmod (\<gamma> - f \<xi>) < cmod (\<gamma> - f z)" if "cmod (\<xi> - z) = r" for z
proof -
have lt: "cmod (f w - f \<xi>) / 3 < cmod (\<gamma> - f z)"
using w [OF that] \<gamma>
using dist_triangle2 [of "f \<xi>" "\<gamma>" "f z"] dist_triangle2 [of "f \<xi>" "f z" \<gamma>]
by (simp add: \<epsilon>_def dist_norm norm_minus_commute)
show ?thesis
by (metis \<epsilon>_def dist_commute dist_norm less_trans lt mem_ball \<gamma>)
qed
have "continuous_on (cball \<xi> r) (\<lambda>z. \<gamma> - f z)"
apply (rule continuous_intros)+
using \<open>cball \<xi> r \<subseteq> U\<close> \<open>f holomorphic_on U\<close>
apply (blast intro: continuous_on_subset holomorphic_on_imp_continuous_on)
done
moreover have "(\<lambda>z. \<gamma> - f z) holomorphic_on ball \<xi> r"
apply (rule holomorphic_intros)+
apply (metis \<open>cball \<xi> r \<subseteq> U\<close> \<open>f holomorphic_on U\<close> holomorphic_on_subset interior_cball interior_subset)
done
ultimately obtain z where "z \<in> ball \<xi> r" "\<gamma> - f z = 0"
apply (rule holomorphic_contract_to_zero)
apply (blast intro!: \<open>0 < r\<close> *)+
done
then show "\<gamma> \<in> f ` U"
using \<open>cball \<xi> r \<subseteq> U\<close> by fastforce
qed
then show ?thesis using \<open>0 < \<epsilon>\<close> by blast
qed
qed
have "open (f ` X)" if "X \<in> components U" for X
proof -
have holfU: "f holomorphic_on U"
using \<open>U \<subseteq> S\<close> holf holomorphic_on_subset by blast
have "X \<noteq> {}"
using that by (simp add: in_components_nonempty)
moreover have "open X"
using that \<open>open U\<close> open_components by auto
moreover have "connected X"
using that in_components_maximal by blast
moreover have "f holomorphic_on X"
by (meson that holfU holomorphic_on_subset in_components_maximal)
moreover have "\<exists>y\<in>X. f y \<noteq> x" for x
proof (rule ccontr)
assume not: "\<not> (\<exists>y\<in>X. f y \<noteq> x)"
have "X \<subseteq> S"
using \<open>U \<subseteq> S\<close> in_components_subset that by blast
obtain w where w: "w \<in> X" using \<open>X \<noteq> {}\<close> by blast
have wis: "w islimpt X"
using w \<open>open X\<close> interior_eq by auto
have hol: "(\<lambda>z. f z - x) holomorphic_on S"
by (simp add: holf holomorphic_on_diff)
with fne [unfolded constant_on_def]
analytic_continuation[OF hol S \<open>connected S\<close> \<open>X \<subseteq> S\<close> _ wis] not \<open>X \<subseteq> S\<close> w
show False by auto
qed
ultimately show ?thesis
by (rule *)
qed
then have "open (f ` \<Union>(components U))"
by (metis (no_types, lifting) imageE image_Union open_Union)
then show ?thesis
by force
qed
text\<open>No need for \<^term>\<open>S\<close> to be connected. But the nonconstant condition is stronger.\<close>
corollary\<^marker>\<open>tag unimportant\<close> open_mapping_thm2:
assumes holf: "f holomorphic_on S"
and S: "open S"
and "open U" "U \<subseteq> S"
and fnc: "\<And>X. \<lbrakk>open X; X \<subseteq> S; X \<noteq> {}\<rbrakk> \<Longrightarrow> \<not> f constant_on X"
shows "open (f ` U)"
proof -
have "S = \<Union>(components S)" by simp
with \<open>U \<subseteq> S\<close> have "U = (\<Union>C \<in> components S. C \<inter> U)" by auto
then have "f ` U = (\<Union>C \<in> components S. f ` (C \<inter> U))"
using image_UN by fastforce
moreover
{ fix C assume "C \<in> components S"
with S \<open>C \<in> components S\<close> open_components in_components_connected
have C: "open C" "connected C" by auto
have "C \<subseteq> S"
by (metis \<open>C \<in> components S\<close> in_components_maximal)
have nf: "\<not> f constant_on C"
apply (rule fnc)
using C \<open>C \<subseteq> S\<close> \<open>C \<in> components S\<close> in_components_nonempty by auto
have "f holomorphic_on C"
by (metis holf holomorphic_on_subset \<open>C \<subseteq> S\<close>)
then have "open (f ` (C \<inter> U))"
apply (rule open_mapping_thm [OF _ C _ _ nf])
apply (simp add: C \<open>open U\<close> open_Int, blast)
done
} ultimately show ?thesis
by force
qed
corollary\<^marker>\<open>tag unimportant\<close> open_mapping_thm3:
assumes holf: "f holomorphic_on S"
and "open S" and injf: "inj_on f S"
shows "open (f ` S)"
apply (rule open_mapping_thm2 [OF holf])
using assms
apply (simp_all add:)
using injective_not_constant subset_inj_on by blast
subsection\<open>Maximum modulus principle\<close>
text\<open>If \<^term>\<open>f\<close> is holomorphic, then its norm (modulus) cannot exhibit a true local maximum that is
properly within the domain of \<^term>\<open>f\<close>.\<close>
proposition maximum_modulus_principle:
assumes holf: "f holomorphic_on S"
and S: "open S" and "connected S"
and "open U" and "U \<subseteq> S" and "\<xi> \<in> U"
and no: "\<And>z. z \<in> U \<Longrightarrow> norm(f z) \<le> norm(f \<xi>)"
shows "f constant_on S"
proof (rule ccontr)
assume "\<not> f constant_on S"
then have "open (f ` U)"
using open_mapping_thm assms by blast
moreover have "\<not> open (f ` U)"
proof -
have "\<exists>t. cmod (f \<xi> - t) < e \<and> t \<notin> f ` U" if "0 < e" for e
apply (rule_tac x="if 0 < Re(f \<xi>) then f \<xi> + (e/2) else f \<xi> - (e/2)" in exI)
using that
apply (simp add: dist_norm)
apply (fastforce simp: cmod_Re_le_iff dest!: no dest: sym)
done
then show ?thesis
unfolding open_contains_ball by (metis \<open>\<xi> \<in> U\<close> contra_subsetD dist_norm imageI mem_ball)
qed
ultimately show False
by blast
qed
proposition maximum_modulus_frontier:
assumes holf: "f holomorphic_on (interior S)"
and contf: "continuous_on (closure S) f"
and bos: "bounded S"
and leB: "\<And>z. z \<in> frontier S \<Longrightarrow> norm(f z) \<le> B"
and "\<xi> \<in> S"
shows "norm(f \<xi>) \<le> B"
proof -
have "compact (closure S)" using bos
by (simp add: bounded_closure compact_eq_bounded_closed)
moreover have "continuous_on (closure S) (cmod \<circ> f)"
using contf continuous_on_compose continuous_on_norm_id by blast
ultimately obtain z where zin: "z \<in> closure S" and z: "\<And>y. y \<in> closure S \<Longrightarrow> (cmod \<circ> f) y \<le> (cmod \<circ> f) z"
using continuous_attains_sup [of "closure S" "norm o f"] \<open>\<xi> \<in> S\<close> by auto
then consider "z \<in> frontier S" | "z \<in> interior S" using frontier_def by auto
then have "norm(f z) \<le> B"
proof cases
case 1 then show ?thesis using leB by blast
next
case 2
have zin: "z \<in> connected_component_set (interior S) z"
by (simp add: 2)
have "f constant_on (connected_component_set (interior S) z)"
apply (rule maximum_modulus_principle [OF _ _ _ _ _ zin])
apply (metis connected_component_subset holf holomorphic_on_subset)
apply (simp_all add: open_connected_component)
by (metis closure_subset comp_eq_dest_lhs interior_subset subsetCE z connected_component_in)
then obtain c where c: "\<And>w. w \<in> connected_component_set (interior S) z \<Longrightarrow> f w = c"
by (auto simp: constant_on_def)
have "f ` closure(connected_component_set (interior S) z) \<subseteq> {c}"
apply (rule image_closure_subset)
apply (meson closure_mono connected_component_subset contf continuous_on_subset interior_subset)
using c
apply auto
done
then have cc: "\<And>w. w \<in> closure(connected_component_set (interior S) z) \<Longrightarrow> f w = c" by blast
have "frontier(connected_component_set (interior S) z) \<noteq> {}"
apply (simp add: frontier_eq_empty)
by (metis "2" bos bounded_interior connected_component_eq_UNIV connected_component_refl not_bounded_UNIV)
then obtain w where w: "w \<in> frontier(connected_component_set (interior S) z)"
by auto
then have "norm (f z) = norm (f w)" by (simp add: "2" c cc frontier_def)
also have "... \<le> B"
apply (rule leB)
using w
using frontier_interior_subset frontier_of_connected_component_subset by blast
finally show ?thesis .
qed
then show ?thesis
using z \<open>\<xi> \<in> S\<close> closure_subset by fastforce
qed
corollary\<^marker>\<open>tag unimportant\<close> maximum_real_frontier:
assumes holf: "f holomorphic_on (interior S)"
and contf: "continuous_on (closure S) f"
and bos: "bounded S"
and leB: "\<And>z. z \<in> frontier S \<Longrightarrow> Re(f z) \<le> B"
and "\<xi> \<in> S"
shows "Re(f \<xi>) \<le> B"
using maximum_modulus_frontier [of "exp o f" S "exp B"]
Transcendental.continuous_on_exp holomorphic_on_compose holomorphic_on_exp assms
by auto
subsection\<^marker>\<open>tag unimportant\<close> \<open>Factoring out a zero according to its order\<close>
lemma holomorphic_factor_order_of_zero:
assumes holf: "f holomorphic_on S"
and os: "open S"
and "\<xi> \<in> S" "0 < n"
and dnz: "(deriv ^^ n) f \<xi> \<noteq> 0"
and dfz: "\<And>i. \<lbrakk>0 < i; i < n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f \<xi> = 0"
obtains g r where "0 < r"
"g holomorphic_on ball \<xi> r"
"\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = (w - \<xi>)^n * g w"
"\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
proof -
obtain r where "r>0" and r: "ball \<xi> r \<subseteq> S" using assms by (blast elim!: openE)
then have holfb: "f holomorphic_on ball \<xi> r"
using holf holomorphic_on_subset by blast
define g where "g w = suminf (\<lambda>i. (deriv ^^ (i + n)) f \<xi> / (fact(i + n)) * (w - \<xi>)^i)" for w
have sumsg: "(\<lambda>i. (deriv ^^ (i + n)) f \<xi> / (fact(i + n)) * (w - \<xi>)^i) sums g w"
and feq: "f w - f \<xi> = (w - \<xi>)^n * g w"
if w: "w \<in> ball \<xi> r" for w
proof -
define powf where "powf = (\<lambda>i. (deriv ^^ i) f \<xi>/(fact i) * (w - \<xi>)^i)"
have sing: "{..<n} - {i. powf i = 0} = (if f \<xi> = 0 then {} else {0})"
unfolding powf_def using \<open>0 < n\<close> dfz by (auto simp: dfz; metis funpow_0 not_gr0)
have "powf sums f w"
unfolding powf_def by (rule holomorphic_power_series [OF holfb w])
moreover have "(\<Sum>i<n. powf i) = f \<xi>"
apply (subst Groups_Big.comm_monoid_add_class.sum.setdiff_irrelevant [symmetric])
apply simp
apply (simp only: dfz sing)
apply (simp add: powf_def)
done
ultimately have fsums: "(\<lambda>i. powf (i+n)) sums (f w - f \<xi>)"
using w sums_iff_shift' by metis
then have *: "summable (\<lambda>i. (w - \<xi>) ^ n * ((deriv ^^ (i + n)) f \<xi> * (w - \<xi>) ^ i / fact (i + n)))"
unfolding powf_def using sums_summable
by (auto simp: power_add mult_ac)
have "summable (\<lambda>i. (deriv ^^ (i + n)) f \<xi> * (w - \<xi>) ^ i / fact (i + n))"
proof (cases "w=\<xi>")
case False then show ?thesis
using summable_mult [OF *, of "1 / (w - \<xi>) ^ n"] by simp
next
case True then show ?thesis
by (auto simp: Power.semiring_1_class.power_0_left intro!: summable_finite [of "{0}"]
split: if_split_asm)
qed
then show sumsg: "(\<lambda>i. (deriv ^^ (i + n)) f \<xi> / (fact(i + n)) * (w - \<xi>)^i) sums g w"
by (simp add: summable_sums_iff g_def)
show "f w - f \<xi> = (w - \<xi>)^n * g w"
apply (rule sums_unique2)
apply (rule fsums [unfolded powf_def])
using sums_mult [OF sumsg, of "(w - \<xi>) ^ n"]
by (auto simp: power_add mult_ac)
qed
then have holg: "g holomorphic_on ball \<xi> r"
by (meson sumsg power_series_holomorphic)
then have contg: "continuous_on (ball \<xi> r) g"
by (blast intro: holomorphic_on_imp_continuous_on)
have "g \<xi> \<noteq> 0"
using dnz unfolding g_def
by (subst suminf_finite [of "{0}"]) auto
obtain d where "0 < d" and d: "\<And>w. w \<in> ball \<xi> d \<Longrightarrow> g w \<noteq> 0"
apply (rule exE [OF continuous_on_avoid [OF contg _ \<open>g \<xi> \<noteq> 0\<close>]])
using \<open>0 < r\<close>
apply force
by (metis \<open>0 < r\<close> less_trans mem_ball not_less_iff_gr_or_eq)
show ?thesis
apply (rule that [where g=g and r ="min r d"])
using \<open>0 < r\<close> \<open>0 < d\<close> holg
apply (auto simp: feq holomorphic_on_subset subset_ball d)
done
qed
lemma holomorphic_factor_order_of_zero_strong:
assumes holf: "f holomorphic_on S" "open S" "\<xi> \<in> S" "0 < n"
and "(deriv ^^ n) f \<xi> \<noteq> 0"
and "\<And>i. \<lbrakk>0 < i; i < n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f \<xi> = 0"
obtains g r where "0 < r"
"g holomorphic_on ball \<xi> r"
"\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = ((w - \<xi>) * g w) ^ n"
"\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
proof -
obtain g r where "0 < r"
and holg: "g holomorphic_on ball \<xi> r"
and feq: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = (w - \<xi>)^n * g w"
and gne: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
by (auto intro: holomorphic_factor_order_of_zero [OF assms])
have con: "continuous_on (ball \<xi> r) (\<lambda>z. deriv g z / g z)"
by (rule continuous_intros) (auto simp: gne holg holomorphic_deriv holomorphic_on_imp_continuous_on)
have cd: "\<And>x. dist \<xi> x < r \<Longrightarrow> (\<lambda>z. deriv g z / g z) field_differentiable at x"
apply (rule derivative_intros)+
using holg mem_ball apply (blast intro: holomorphic_deriv holomorphic_on_imp_differentiable_at)
apply (metis open_ball at_within_open holg holomorphic_on_def mem_ball)
using gne mem_ball by blast
obtain h where h: "\<And>x. x \<in> ball \<xi> r \<Longrightarrow> (h has_field_derivative deriv g x / g x) (at x)"
apply (rule exE [OF holomorphic_convex_primitive [of "ball \<xi> r" "{}" "\<lambda>z. deriv g z / g z"]])
apply (auto simp: con cd)
apply (metis open_ball at_within_open mem_ball)
done
then have "continuous_on (ball \<xi> r) h"
by (metis open_ball holomorphic_on_imp_continuous_on holomorphic_on_open)
then have con: "continuous_on (ball \<xi> r) (\<lambda>x. exp (h x) / g x)"
by (auto intro!: continuous_intros simp add: holg holomorphic_on_imp_continuous_on gne)
have 0: "dist \<xi> x < r \<Longrightarrow> ((\<lambda>x. exp (h x) / g x) has_field_derivative 0) (at x)" for x
apply (rule h derivative_eq_intros | simp)+
apply (rule DERIV_deriv_iff_field_differentiable [THEN iffD2])
using holg apply (auto simp: holomorphic_on_imp_differentiable_at gne h)
done
obtain c where c: "\<And>x. x \<in> ball \<xi> r \<Longrightarrow> exp (h x) / g x = c"
by (rule DERIV_zero_connected_constant [of "ball \<xi> r" "{}" "\<lambda>x. exp(h x) / g x"]) (auto simp: con 0)
have hol: "(\<lambda>z. exp ((Ln (inverse c) + h z) / of_nat n)) holomorphic_on ball \<xi> r"
apply (rule holomorphic_on_compose [unfolded o_def, where g = exp])
apply (rule holomorphic_intros)+
using h holomorphic_on_open apply blast
apply (rule holomorphic_intros)+
using \<open>0 < n\<close> apply simp
apply (rule holomorphic_intros)+
done
show ?thesis
apply (rule that [where g="\<lambda>z. exp((Ln(inverse c) + h z)/n)" and r =r])
using \<open>0 < r\<close> \<open>0 < n\<close>
apply (auto simp: feq power_mult_distrib exp_divide_power_eq c [symmetric])
apply (rule hol)
apply (simp add: Transcendental.exp_add gne)
done
qed
lemma
fixes k :: "'a::wellorder"
assumes a_def: "a == LEAST x. P x" and P: "P k"
shows def_LeastI: "P a" and def_Least_le: "a \<le> k"
unfolding a_def
by (rule LeastI Least_le; rule P)+
lemma holomorphic_factor_zero_nonconstant:
assumes holf: "f holomorphic_on S" and S: "open S" "connected S"
and "\<xi> \<in> S" "f \<xi> = 0"
and nonconst: "\<not> f constant_on S"
obtains g r n
where "0 < n" "0 < r" "ball \<xi> r \<subseteq> S"
"g holomorphic_on ball \<xi> r"
"\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w = (w - \<xi>)^n * g w"
"\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
proof (cases "\<forall>n>0. (deriv ^^ n) f \<xi> = 0")
case True then show ?thesis
using holomorphic_fun_eq_const_on_connected [OF holf S _ \<open>\<xi> \<in> S\<close>] nonconst by (simp add: constant_on_def)
next
case False
then obtain n0 where "n0 > 0" and n0: "(deriv ^^ n0) f \<xi> \<noteq> 0" by blast
obtain r0 where "r0 > 0" "ball \<xi> r0 \<subseteq> S" using S openE \<open>\<xi> \<in> S\<close> by auto
define n where "n \<equiv> LEAST n. (deriv ^^ n) f \<xi> \<noteq> 0"
have n_ne: "(deriv ^^ n) f \<xi> \<noteq> 0"
by (rule def_LeastI [OF n_def]) (rule n0)
then have "0 < n" using \<open>f \<xi> = 0\<close>
using funpow_0 by fastforce
have n_min: "\<And>k. k < n \<Longrightarrow> (deriv ^^ k) f \<xi> = 0"
using def_Least_le [OF n_def] not_le by blast
then obtain g r1
where "0 < r1" "g holomorphic_on ball \<xi> r1"
"\<And>w. w \<in> ball \<xi> r1 \<Longrightarrow> f w = (w - \<xi>) ^ n * g w"
"\<And>w. w \<in> ball \<xi> r1 \<Longrightarrow> g w \<noteq> 0"
by (auto intro: holomorphic_factor_order_of_zero [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> \<open>n > 0\<close> n_ne] simp: \<open>f \<xi> = 0\<close>)
then show ?thesis
apply (rule_tac g=g and r="min r0 r1" and n=n in that)
using \<open>0 < n\<close> \<open>0 < r0\<close> \<open>0 < r1\<close> \<open>ball \<xi> r0 \<subseteq> S\<close>
apply (auto simp: subset_ball intro: holomorphic_on_subset)
done
qed
lemma holomorphic_lower_bound_difference:
assumes holf: "f holomorphic_on S" and S: "open S" "connected S"
and "\<xi> \<in> S" and "\<phi> \<in> S"
and fne: "f \<phi> \<noteq> f \<xi>"
obtains k n r
where "0 < k" "0 < r"
"ball \<xi> r \<subseteq> S"
"\<And>w. w \<in> ball \<xi> r \<Longrightarrow> k * norm(w - \<xi>)^n \<le> norm(f w - f \<xi>)"
proof -
define n where "n = (LEAST n. 0 < n \<and> (deriv ^^ n) f \<xi> \<noteq> 0)"
obtain n0 where "0 < n0" and n0: "(deriv ^^ n0) f \<xi> \<noteq> 0"
using fne holomorphic_fun_eq_const_on_connected [OF holf S] \<open>\<xi> \<in> S\<close> \<open>\<phi> \<in> S\<close> by blast
then have "0 < n" and n_ne: "(deriv ^^ n) f \<xi> \<noteq> 0"
unfolding n_def by (metis (mono_tags, lifting) LeastI)+
have n_min: "\<And>k. \<lbrakk>0 < k; k < n\<rbrakk> \<Longrightarrow> (deriv ^^ k) f \<xi> = 0"
unfolding n_def by (blast dest: not_less_Least)
then obtain g r
where "0 < r" and holg: "g holomorphic_on ball \<xi> r"
and fne: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = (w - \<xi>) ^ n * g w"
and gnz: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
by (auto intro: holomorphic_factor_order_of_zero [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> \<open>n > 0\<close> n_ne])
obtain e where "e>0" and e: "ball \<xi> e \<subseteq> S" using assms by (blast elim!: openE)
then have holfb: "f holomorphic_on ball \<xi> e"
using holf holomorphic_on_subset by blast
define d where "d = (min e r) / 2"
have "0 < d" using \<open>0 < r\<close> \<open>0 < e\<close> by (simp add: d_def)
have "d < r"
using \<open>0 < r\<close> by (auto simp: d_def)
then have cbb: "cball \<xi> d \<subseteq> ball \<xi> r"
by (auto simp: cball_subset_ball_iff)
then have "g holomorphic_on cball \<xi> d"
by (rule holomorphic_on_subset [OF holg])
then have "closed (g ` cball \<xi> d)"
by (simp add: compact_imp_closed compact_continuous_image holomorphic_on_imp_continuous_on)
moreover have "g ` cball \<xi> d \<noteq> {}"
using \<open>0 < d\<close> by auto
ultimately obtain x where x: "x \<in> g ` cball \<xi> d" and "\<And>y. y \<in> g ` cball \<xi> d \<Longrightarrow> dist 0 x \<le> dist 0 y"
by (rule distance_attains_inf) blast
then have leg: "\<And>w. w \<in> cball \<xi> d \<Longrightarrow> norm x \<le> norm (g w)"
by auto
have "ball \<xi> d \<subseteq> cball \<xi> d" by auto
also have "... \<subseteq> ball \<xi> e" using \<open>0 < d\<close> d_def by auto
also have "... \<subseteq> S" by (rule e)
finally have dS: "ball \<xi> d \<subseteq> S" .
moreover have "x \<noteq> 0" using gnz x \<open>d < r\<close> by auto
ultimately show ?thesis
apply (rule_tac k="norm x" and n=n and r=d in that)
using \<open>d < r\<close> leg
apply (auto simp: \<open>0 < d\<close> fne norm_mult norm_power algebra_simps mult_right_mono)
done
qed
lemma
assumes holf: "f holomorphic_on (S - {\<xi>})" and \<xi>: "\<xi> \<in> interior S"
shows holomorphic_on_extend_lim:
"(\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S - {\<xi>}. g z = f z)) \<longleftrightarrow>
((\<lambda>z. (z - \<xi>) * f z) \<longlongrightarrow> 0) (at \<xi>)"
(is "?P = ?Q")
and holomorphic_on_extend_bounded:
"(\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S - {\<xi>}. g z = f z)) \<longleftrightarrow>
(\<exists>B. eventually (\<lambda>z. norm(f z) \<le> B) (at \<xi>))"
(is "?P = ?R")
proof -
obtain \<delta> where "0 < \<delta>" and \<delta>: "ball \<xi> \<delta> \<subseteq> S"
using \<xi> mem_interior by blast
have "?R" if holg: "g holomorphic_on S" and gf: "\<And>z. z \<in> S - {\<xi>} \<Longrightarrow> g z = f z" for g
proof -
have *: "\<forall>\<^sub>F z in at \<xi>. dist (g z) (g \<xi>) < 1 \<longrightarrow> cmod (f z) \<le> cmod (g \<xi>) + 1"
apply (simp add: eventually_at)
apply (rule_tac x="\<delta>" in exI)
using \<delta> \<open>0 < \<delta>\<close>
apply (clarsimp simp:)
apply (drule_tac c=x in subsetD)
apply (simp add: dist_commute)
by (metis DiffI add.commute diff_le_eq dist_norm gf le_less_trans less_eq_real_def norm_triangle_ineq2 singletonD)
have "continuous_on (interior S) g"
by (meson continuous_on_subset holg holomorphic_on_imp_continuous_on interior_subset)
then have "\<And>x. x \<in> interior S \<Longrightarrow> (g \<longlongrightarrow> g x) (at x)"
using continuous_on_interior continuous_within holg holomorphic_on_imp_continuous_on by blast
then have "(g \<longlongrightarrow> g \<xi>) (at \<xi>)"
by (simp add: \<xi>)
then show ?thesis
apply (rule_tac x="norm(g \<xi>) + 1" in exI)
apply (rule eventually_mp [OF * tendstoD [where e=1]], auto)
done
qed
moreover have "?Q" if "\<forall>\<^sub>F z in at \<xi>. cmod (f z) \<le> B" for B
by (rule lim_null_mult_right_bounded [OF _ that]) (simp add: LIM_zero)
moreover have "?P" if "(\<lambda>z. (z - \<xi>) * f z) \<midarrow>\<xi>\<rightarrow> 0"
proof -
define h where [abs_def]: "h z = (z - \<xi>)^2 * f z" for z
have h0: "(h has_field_derivative 0) (at \<xi>)"
apply (simp add: h_def has_field_derivative_iff)
apply (rule Lim_transform_within [OF that, of 1])
apply (auto simp: field_split_simps power2_eq_square)
done
have holh: "h holomorphic_on S"
proof (simp add: holomorphic_on_def, clarify)
fix z assume "z \<in> S"
show "h field_differentiable at z within S"
proof (cases "z = \<xi>")
case True then show ?thesis
using field_differentiable_at_within field_differentiable_def h0 by blast
next
case False
then have "f field_differentiable at z within S"
using holomorphic_onD [OF holf, of z] \<open>z \<in> S\<close>
unfolding field_differentiable_def has_field_derivative_iff
by (force intro: exI [where x="dist \<xi> z"] elim: Lim_transform_within_set [unfolded eventually_at])
then show ?thesis
by (simp add: h_def power2_eq_square derivative_intros)
qed
qed
define g where [abs_def]: "g z = (if z = \<xi> then deriv h \<xi> else (h z - h \<xi>) / (z - \<xi>))" for z
have holg: "g holomorphic_on S"
unfolding g_def by (rule pole_lemma [OF holh \<xi>])
show ?thesis
apply (rule_tac x="\<lambda>z. if z = \<xi> then deriv g \<xi> else (g z - g \<xi>)/(z - \<xi>)" in exI)
apply (rule conjI)
apply (rule pole_lemma [OF holg \<xi>])
apply (auto simp: g_def power2_eq_square divide_simps)
using h0 apply (simp add: h0 DERIV_imp_deriv h_def power2_eq_square)
done
qed
ultimately show "?P = ?Q" and "?P = ?R"
by meson+
qed
lemma pole_at_infinity:
assumes holf: "f holomorphic_on UNIV" and lim: "((inverse o f) \<longlongrightarrow> l) at_infinity"
obtains a n where "\<And>z. f z = (\<Sum>i\<le>n. a i * z^i)"
proof (cases "l = 0")
case False
with tendsto_inverse [OF lim] show ?thesis
apply (rule_tac a="(\<lambda>n. inverse l)" and n=0 in that)
apply (simp add: Liouville_weak [OF holf, of "inverse l"])
done
next
case True
then have [simp]: "l = 0" .
show ?thesis
proof (cases "\<exists>r. 0 < r \<and> (\<forall>z \<in> ball 0 r - {0}. f(inverse z) \<noteq> 0)")
case True
then obtain r where "0 < r" and r: "\<And>z. z \<in> ball 0 r - {0} \<Longrightarrow> f(inverse z) \<noteq> 0"
by auto
have 1: "inverse \<circ> f \<circ> inverse holomorphic_on ball 0 r - {0}"
by (rule holomorphic_on_compose holomorphic_intros holomorphic_on_subset [OF holf] | force simp: r)+
have 2: "0 \<in> interior (ball 0 r)"
using \<open>0 < r\<close> by simp
have "\<exists>B. 0<B \<and> eventually (\<lambda>z. cmod ((inverse \<circ> f \<circ> inverse) z) \<le> B) (at 0)"
apply (rule exI [where x=1])
apply simp
using tendstoD [OF lim [unfolded lim_at_infinity_0] zero_less_one]
apply (rule eventually_mono)
apply (simp add: dist_norm)
done
with holomorphic_on_extend_bounded [OF 1 2]
obtain g where holg: "g holomorphic_on ball 0 r"
and geq: "\<And>z. z \<in> ball 0 r - {0} \<Longrightarrow> g z = (inverse \<circ> f \<circ> inverse) z"
by meson
have ifi0: "(inverse \<circ> f \<circ> inverse) \<midarrow>0\<rightarrow> 0"
using \<open>l = 0\<close> lim lim_at_infinity_0 by blast
have g2g0: "g \<midarrow>0\<rightarrow> g 0"
using \<open>0 < r\<close> centre_in_ball continuous_at continuous_on_eq_continuous_at holg
by (blast intro: holomorphic_on_imp_continuous_on)
have g2g1: "g \<midarrow>0\<rightarrow> 0"
apply (rule Lim_transform_within_open [OF ifi0 open_ball [of 0 r]])
using \<open>0 < r\<close> by (auto simp: geq)
have [simp]: "g 0 = 0"
by (rule tendsto_unique [OF _ g2g0 g2g1]) simp
have "ball 0 r - {0::complex} \<noteq> {}"
using \<open>0 < r\<close>
apply (clarsimp simp: ball_def dist_norm)
apply (drule_tac c="of_real r/2" in subsetD, auto)
done
then obtain w::complex where "w \<noteq> 0" and w: "norm w < r" by force
then have "g w \<noteq> 0" by (simp add: geq r)
obtain B n e where "0 < B" "0 < e" "e \<le> r"
and leg: "\<And>w. norm w < e \<Longrightarrow> B * cmod w ^ n \<le> cmod (g w)"
apply (rule holomorphic_lower_bound_difference [OF holg open_ball connected_ball, of 0 w])
using \<open>0 < r\<close> w \<open>g w \<noteq> 0\<close> by (auto simp: ball_subset_ball_iff)
have "cmod (f z) \<le> cmod z ^ n / B" if "2/e \<le> cmod z" for z
proof -
have ize: "inverse z \<in> ball 0 e - {0}" using that \<open>0 < e\<close>
by (auto simp: norm_divide field_split_simps algebra_simps)
then have [simp]: "z \<noteq> 0" and izr: "inverse z \<in> ball 0 r - {0}" using \<open>e \<le> r\<close>
by auto
then have [simp]: "f z \<noteq> 0"
using r [of "inverse z"] by simp
have [simp]: "f z = inverse (g (inverse z))"
using izr geq [of "inverse z"] by simp
show ?thesis using ize leg [of "inverse z"] \<open>0 < B\<close> \<open>0 < e\<close>
by (simp add: field_split_simps norm_divide algebra_simps)
qed
then show ?thesis
apply (rule_tac a = "\<lambda>k. (deriv ^^ k) f 0 / (fact k)" and n=n in that)
apply (rule_tac A = "2/e" and B = "1/B" in Liouville_polynomial [OF holf], simp)
done
next
case False
then have fi0: "\<And>r. r > 0 \<Longrightarrow> \<exists>z\<in>ball 0 r - {0}. f (inverse z) = 0"
by simp
have fz0: "f z = 0" if "0 < r" and lt1: "\<And>x. x \<noteq> 0 \<Longrightarrow> cmod x < r \<Longrightarrow> inverse (cmod (f (inverse x))) < 1"
for z r
proof -
have f0: "(f \<longlongrightarrow> 0) at_infinity"
proof -
have DIM_complex[intro]: "2 \<le> DIM(complex)" \<comment> \<open>should not be necessary!\<close>
by simp
have "f (inverse x) \<noteq> 0 \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> cmod x < r \<Longrightarrow> 1 < cmod (f (inverse x))" for x
using lt1[of x] by (auto simp: field_simps)
then have **: "cmod (f (inverse x)) \<le> 1 \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> cmod x < r \<Longrightarrow> f (inverse x) = 0" for x
by force
then have *: "(f \<circ> inverse) ` (ball 0 r - {0}) \<subseteq> {0} \<union> - ball 0 1"
by force
have "continuous_on (inverse ` (ball 0 r - {0})) f"
using continuous_on_subset holf holomorphic_on_imp_continuous_on by blast
then have "connected ((f \<circ> inverse) ` (ball 0 r - {0}))"
apply (intro connected_continuous_image continuous_intros)
apply (force intro: connected_punctured_ball)+
done
then have "{0} \<inter> (f \<circ> inverse) ` (ball 0 r - {0}) = {} \<or> - ball 0 1 \<inter> (f \<circ> inverse) ` (ball 0 r - {0}) = {}"
by (rule connected_closedD) (use * in auto)
then have "w \<noteq> 0 \<Longrightarrow> cmod w < r \<Longrightarrow> f (inverse w) = 0" for w
using fi0 **[of w] \<open>0 < r\<close>
apply (auto simp add: inf.commute [of "- ball 0 1"] Diff_eq [symmetric] image_subset_iff dest: less_imp_le)
apply fastforce
apply (drule bspec [of _ _ w])
apply (auto dest: less_imp_le)
done
then show ?thesis
apply (simp add: lim_at_infinity_0)
apply (rule tendsto_eventually)
apply (simp add: eventually_at)
apply (rule_tac x=r in exI)
apply (simp add: \<open>0 < r\<close> dist_norm)
done
qed
obtain w where "w \<in> ball 0 r - {0}" and "f (inverse w) = 0"
using False \<open>0 < r\<close> by blast
then show ?thesis
by (auto simp: f0 Liouville_weak [OF holf, of 0])
qed
show ?thesis
apply (rule that [of "\<lambda>n. 0" 0])
using lim [unfolded lim_at_infinity_0]
apply (simp add: Lim_at dist_norm norm_inverse)
apply (drule_tac x=1 in spec)
using fz0 apply auto
done
qed
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>Entire proper functions are precisely the non-trivial polynomials\<close>
lemma proper_map_polyfun:
fixes c :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,heine_borel}"
assumes "closed S" and "compact K" and c: "c i \<noteq> 0" "1 \<le> i" "i \<le> n"
shows "compact (S \<inter> {z. (\<Sum>i\<le>n. c i * z^i) \<in> K})"
proof -
obtain B where "B > 0" and B: "\<And>x. x \<in> K \<Longrightarrow> norm x \<le> B"
by (metis compact_imp_bounded \<open>compact K\<close> bounded_pos)
have *: "norm x \<le> b"
if "\<And>x. b \<le> norm x \<Longrightarrow> B + 1 \<le> norm (\<Sum>i\<le>n. c i * x ^ i)"
"(\<Sum>i\<le>n. c i * x ^ i) \<in> K" for b x
proof -
have "norm (\<Sum>i\<le>n. c i * x ^ i) \<le> B"
using B that by blast
moreover have "\<not> B + 1 \<le> B"
by simp
ultimately show "norm x \<le> b"
using that by (metis (no_types) less_eq_real_def not_less order_trans)
qed
have "bounded {z. (\<Sum>i\<le>n. c i * z ^ i) \<in> K}"
using Limits.polyfun_extremal [where c=c and B="B+1", OF c]
by (auto simp: bounded_pos eventually_at_infinity_pos *)
moreover have "closed ((\<lambda>z. (\<Sum>i\<le>n. c i * z ^ i)) -` K)"
apply (intro allI continuous_closed_vimage continuous_intros)
using \<open>compact K\<close> compact_eq_bounded_closed by blast
ultimately show ?thesis
using closed_Int_compact [OF \<open>closed S\<close>] compact_eq_bounded_closed
by (auto simp add: vimage_def)
qed
lemma proper_map_polyfun_univ:
fixes c :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,heine_borel}"
assumes "compact K" "c i \<noteq> 0" "1 \<le> i" "i \<le> n"
shows "compact ({z. (\<Sum>i\<le>n. c i * z^i) \<in> K})"
using proper_map_polyfun [of UNIV K c i n] assms by simp
lemma proper_map_polyfun_eq:
assumes "f holomorphic_on UNIV"
shows "(\<forall>k. compact k \<longrightarrow> compact {z. f z \<in> k}) \<longleftrightarrow>
(\<exists>c n. 0 < n \<and> (c n \<noteq> 0) \<and> f = (\<lambda>z. \<Sum>i\<le>n. c i * z^i))"
(is "?lhs = ?rhs")
proof
assume compf [rule_format]: ?lhs
have 2: "\<exists>k. 0 < k \<and> a k \<noteq> 0 \<and> f = (\<lambda>z. \<Sum>i \<le> k. a i * z ^ i)"
if "\<And>z. f z = (\<Sum>i\<le>n. a i * z ^ i)" for a n
proof (cases "\<forall>i\<le>n. 0<i \<longrightarrow> a i = 0")
case True
then have [simp]: "\<And>z. f z = a 0"
by (simp add: that sum.atMost_shift)
have False using compf [of "{a 0}"] by simp
then show ?thesis ..
next
case False
then obtain k where k: "0 < k" "k\<le>n" "a k \<noteq> 0" by force
define m where "m = (GREATEST k. k\<le>n \<and> a k \<noteq> 0)"
have m: "m\<le>n \<and> a m \<noteq> 0"
unfolding m_def
apply (rule GreatestI_nat [where b = n])
using k apply auto
done
have [simp]: "a i = 0" if "m < i" "i \<le> n" for i
using Greatest_le_nat [where b = "n" and P = "\<lambda>k. k\<le>n \<and> a k \<noteq> 0"]
using m_def not_le that by auto
have "k \<le> m"
unfolding m_def
apply (rule Greatest_le_nat [where b = "n"])
using k apply auto
done
with k m show ?thesis
by (rule_tac x=m in exI) (auto simp: that comm_monoid_add_class.sum.mono_neutral_right)
qed
have "((inverse \<circ> f) \<longlongrightarrow> 0) at_infinity"
proof (rule Lim_at_infinityI)
fix e::real assume "0 < e"
with compf [of "cball 0 (inverse e)"]
show "\<exists>B. \<forall>x. B \<le> cmod x \<longrightarrow> dist ((inverse \<circ> f) x) 0 \<le> e"
apply simp
apply (clarsimp simp add: compact_eq_bounded_closed bounded_pos norm_inverse)
apply (rule_tac x="b+1" in exI)
apply (metis inverse_inverse_eq less_add_same_cancel2 less_imp_inverse_less add.commute not_le not_less_iff_gr_or_eq order_trans zero_less_one)
done
qed
then show ?rhs
apply (rule pole_at_infinity [OF assms])
using 2 apply blast
done
next
assume ?rhs
then obtain c n where "0 < n" "c n \<noteq> 0" "f = (\<lambda>z. \<Sum>i\<le>n. c i * z ^ i)" by blast
then have "compact {z. f z \<in> k}" if "compact k" for k
by (auto intro: proper_map_polyfun_univ [OF that])
then show ?lhs by blast
qed
subsection \<open>Relating invertibility and nonvanishing of derivative\<close>
lemma has_complex_derivative_locally_injective:
assumes holf: "f holomorphic_on S"
and S: "\<xi> \<in> S" "open S"
and dnz: "deriv f \<xi> \<noteq> 0"
obtains r where "r > 0" "ball \<xi> r \<subseteq> S" "inj_on f (ball \<xi> r)"
proof -
have *: "\<exists>d>0. \<forall>x. dist \<xi> x < d \<longrightarrow> onorm (\<lambda>v. deriv f x * v - deriv f \<xi> * v) < e" if "e > 0" for e
proof -
have contdf: "continuous_on S (deriv f)"
by (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on \<open>open S\<close>)
obtain \<delta> where "\<delta>>0" and \<delta>: "\<And>x. \<lbrakk>x \<in> S; dist x \<xi> \<le> \<delta>\<rbrakk> \<Longrightarrow> cmod (deriv f x - deriv f \<xi>) \<le> e/2"
using continuous_onE [OF contdf \<open>\<xi> \<in> S\<close>, of "e/2"] \<open>0 < e\<close>
by (metis dist_complex_def half_gt_zero less_imp_le)
obtain \<epsilon> where "\<epsilon>>0" "ball \<xi> \<epsilon> \<subseteq> S"
by (metis openE [OF \<open>open S\<close> \<open>\<xi> \<in> S\<close>])
with \<open>\<delta>>0\<close> have "\<exists>\<delta>>0. \<forall>x. dist \<xi> x < \<delta> \<longrightarrow> onorm (\<lambda>v. deriv f x * v - deriv f \<xi> * v) \<le> e/2"
apply (rule_tac x="min \<delta> \<epsilon>" in exI)
apply (intro conjI allI impI Operator_Norm.onorm_le)
apply simp
apply (simp only: Rings.ring_class.left_diff_distrib [symmetric] norm_mult)
apply (rule mult_right_mono [OF \<delta>])
apply (auto simp: dist_commute Rings.ordered_semiring_class.mult_right_mono \<delta>)
done
with \<open>e>0\<close> show ?thesis by force
qed
have "inj ((*) (deriv f \<xi>))"
using dnz by simp
then obtain g' where g': "linear g'" "g' \<circ> (*) (deriv f \<xi>) = id"
using linear_injective_left_inverse [of "(*) (deriv f \<xi>)"]
by (auto simp: linear_times)
show ?thesis
apply (rule has_derivative_locally_injective [OF S, where f=f and f' = "\<lambda>z h. deriv f z * h" and g' = g'])
using g' *
apply (simp_all add: linear_conv_bounded_linear that)
using DERIV_deriv_iff_field_differentiable has_field_derivative_imp_has_derivative holf
holomorphic_on_imp_differentiable_at \<open>open S\<close> apply blast
done
qed
lemma has_complex_derivative_locally_invertible:
assumes holf: "f holomorphic_on S"
and S: "\<xi> \<in> S" "open S"
and dnz: "deriv f \<xi> \<noteq> 0"
obtains r where "r > 0" "ball \<xi> r \<subseteq> S" "open (f ` (ball \<xi> r))" "inj_on f (ball \<xi> r)"
proof -
obtain r where "r > 0" "ball \<xi> r \<subseteq> S" "inj_on f (ball \<xi> r)"
by (blast intro: that has_complex_derivative_locally_injective [OF assms])
then have \<xi>: "\<xi> \<in> ball \<xi> r" by simp
then have nc: "\<not> f constant_on ball \<xi> r"
using \<open>inj_on f (ball \<xi> r)\<close> injective_not_constant by fastforce
have holf': "f holomorphic_on ball \<xi> r"
using \<open>ball \<xi> r \<subseteq> S\<close> holf holomorphic_on_subset by blast
have "open (f ` ball \<xi> r)"
apply (rule open_mapping_thm [OF holf'])
using nc apply auto
done
then show ?thesis
using \<open>0 < r\<close> \<open>ball \<xi> r \<subseteq> S\<close> \<open>inj_on f (ball \<xi> r)\<close> that by blast
qed
lemma holomorphic_injective_imp_regular:
assumes holf: "f holomorphic_on S"
and "open S" and injf: "inj_on f S"
and "\<xi> \<in> S"
shows "deriv f \<xi> \<noteq> 0"
proof -
obtain r where "r>0" and r: "ball \<xi> r \<subseteq> S" using assms by (blast elim!: openE)
have holf': "f holomorphic_on ball \<xi> r"
using \<open>ball \<xi> r \<subseteq> S\<close> holf holomorphic_on_subset by blast
show ?thesis
proof (cases "\<forall>n>0. (deriv ^^ n) f \<xi> = 0")
case True
have fcon: "f w = f \<xi>" if "w \<in> ball \<xi> r" for w
apply (rule holomorphic_fun_eq_const_on_connected [OF holf'])
using True \<open>0 < r\<close> that by auto
have False
using fcon [of "\<xi> + r/2"] \<open>0 < r\<close> r injf unfolding inj_on_def
by (metis \<open>\<xi> \<in> S\<close> contra_subsetD dist_commute fcon mem_ball perfect_choose_dist)
then show ?thesis ..
next
case False
then obtain n0 where n0: "n0 > 0 \<and> (deriv ^^ n0) f \<xi> \<noteq> 0" by blast
define n where [abs_def]: "n = (LEAST n. n > 0 \<and> (deriv ^^ n) f \<xi> \<noteq> 0)"
have n_ne: "n > 0" "(deriv ^^ n) f \<xi> \<noteq> 0"
using def_LeastI [OF n_def n0] by auto
have n_min: "\<And>k. 0 < k \<Longrightarrow> k < n \<Longrightarrow> (deriv ^^ k) f \<xi> = 0"
using def_Least_le [OF n_def] not_le by auto
obtain g \<delta> where "0 < \<delta>"
and holg: "g holomorphic_on ball \<xi> \<delta>"
and fd: "\<And>w. w \<in> ball \<xi> \<delta> \<Longrightarrow> f w - f \<xi> = ((w - \<xi>) * g w) ^ n"
and gnz: "\<And>w. w \<in> ball \<xi> \<delta> \<Longrightarrow> g w \<noteq> 0"
apply (rule holomorphic_factor_order_of_zero_strong [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> n_ne])
apply (blast intro: n_min)+
done
show ?thesis
proof (cases "n=1")
case True
with n_ne show ?thesis by auto
next
case False
have holgw: "(\<lambda>w. (w - \<xi>) * g w) holomorphic_on ball \<xi> (min r \<delta>)"
apply (rule holomorphic_intros)+
using holg by (simp add: holomorphic_on_subset subset_ball)
have gd: "\<And>w. dist \<xi> w < \<delta> \<Longrightarrow> (g has_field_derivative deriv g w) (at w)"
using holg
by (simp add: DERIV_deriv_iff_field_differentiable holomorphic_on_def at_within_open_NO_MATCH)
have *: "\<And>w. w \<in> ball \<xi> (min r \<delta>)
\<Longrightarrow> ((\<lambda>w. (w - \<xi>) * g w) has_field_derivative ((w - \<xi>) * deriv g w + g w))
(at w)"
by (rule gd derivative_eq_intros | simp)+
have [simp]: "deriv (\<lambda>w. (w - \<xi>) * g w) \<xi> \<noteq> 0"
using * [of \<xi>] \<open>0 < \<delta>\<close> \<open>0 < r\<close> by (simp add: DERIV_imp_deriv gnz)
obtain T where "\<xi> \<in> T" "open T" and Tsb: "T \<subseteq> ball \<xi> (min r \<delta>)" and oimT: "open ((\<lambda>w. (w - \<xi>) * g w) ` T)"
apply (rule has_complex_derivative_locally_invertible [OF holgw, of \<xi>])
using \<open>0 < r\<close> \<open>0 < \<delta>\<close>
apply (simp_all add:)
by (meson open_ball centre_in_ball)
define U where "U = (\<lambda>w. (w - \<xi>) * g w) ` T"
have "open U" by (metis oimT U_def)
have "0 \<in> U"
apply (auto simp: U_def)
apply (rule image_eqI [where x = \<xi>])
apply (auto simp: \<open>\<xi> \<in> T\<close>)
done
then obtain \<epsilon> where "\<epsilon>>0" and \<epsilon>: "cball 0 \<epsilon> \<subseteq> U"
using \<open>open U\<close> open_contains_cball by blast
then have "\<epsilon> * exp(2 * of_real pi * \<i> * (0/n)) \<in> cball 0 \<epsilon>"
"\<epsilon> * exp(2 * of_real pi * \<i> * (1/n)) \<in> cball 0 \<epsilon>"
by (auto simp: norm_mult)
with \<epsilon> have "\<epsilon> * exp(2 * of_real pi * \<i> * (0/n)) \<in> U"
"\<epsilon> * exp(2 * of_real pi * \<i> * (1/n)) \<in> U" by blast+
then obtain y0 y1 where "y0 \<in> T" and y0: "(y0 - \<xi>) * g y0 = \<epsilon> * exp(2 * of_real pi * \<i> * (0/n))"
and "y1 \<in> T" and y1: "(y1 - \<xi>) * g y1 = \<epsilon> * exp(2 * of_real pi * \<i> * (1/n))"
by (auto simp: U_def)
then have "y0 \<in> ball \<xi> \<delta>" "y1 \<in> ball \<xi> \<delta>" using Tsb by auto
moreover have "y0 \<noteq> y1"
using y0 y1 \<open>\<epsilon> > 0\<close> complex_root_unity_eq_1 [of n 1] \<open>n > 0\<close> False by auto
moreover have "T \<subseteq> S"
by (meson Tsb min.cobounded1 order_trans r subset_ball)
ultimately have False
using inj_onD [OF injf, of y0 y1] \<open>y0 \<in> T\<close> \<open>y1 \<in> T\<close>
using fd [of y0] fd [of y1] complex_root_unity [of n 1] n_ne
apply (simp add: y0 y1 power_mult_distrib)
apply (force simp: algebra_simps)
done
then show ?thesis ..
qed
qed
qed
text\<open>Hence a nice clean inverse function theorem\<close>
lemma has_field_derivative_inverse_strong:
fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a"
shows "\<lbrakk>DERIV f x :> f'; f' \<noteq> 0; open S; x \<in> S; continuous_on S f;
\<And>z. z \<in> S \<Longrightarrow> g (f z) = z\<rbrakk>
\<Longrightarrow> DERIV g (f x) :> inverse (f')"
unfolding has_field_derivative_def
by (rule has_derivative_inverse_strong [of S x f g]) auto
lemma has_field_derivative_inverse_strong_x:
fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a"
shows "\<lbrakk>DERIV f (g y) :> f'; f' \<noteq> 0; open S; continuous_on S f; g y \<in> S; f(g y) = y;
\<And>z. z \<in> S \<Longrightarrow> g (f z) = z\<rbrakk>
\<Longrightarrow> DERIV g y :> inverse (f')"
unfolding has_field_derivative_def
by (rule has_derivative_inverse_strong_x [of S g y f]) auto
proposition holomorphic_has_inverse:
assumes holf: "f holomorphic_on S"
and "open S" and injf: "inj_on f S"
obtains g where "g holomorphic_on (f ` S)"
"\<And>z. z \<in> S \<Longrightarrow> deriv f z * deriv g (f z) = 1"
"\<And>z. z \<in> S \<Longrightarrow> g(f z) = z"
proof -
have ofs: "open (f ` S)"
by (rule open_mapping_thm3 [OF assms])
have contf: "continuous_on S f"
by (simp add: holf holomorphic_on_imp_continuous_on)
have *: "(the_inv_into S f has_field_derivative inverse (deriv f z)) (at (f z))" if "z \<in> S" for z
proof -
have 1: "(f has_field_derivative deriv f z) (at z)"
using DERIV_deriv_iff_field_differentiable \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_on_imp_differentiable_at
by blast
have 2: "deriv f z \<noteq> 0"
using \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_injective_imp_regular injf by blast
show ?thesis
apply (rule has_field_derivative_inverse_strong [OF 1 2 \<open>open S\<close> \<open>z \<in> S\<close>])
apply (simp add: holf holomorphic_on_imp_continuous_on)
by (simp add: injf the_inv_into_f_f)
qed
show ?thesis
proof
show "the_inv_into S f holomorphic_on f ` S"
by (simp add: holomorphic_on_open ofs) (blast intro: *)
next
fix z assume "z \<in> S"
have "deriv f z \<noteq> 0"
using \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_injective_imp_regular injf by blast
then show "deriv f z * deriv (the_inv_into S f) (f z) = 1"
using * [OF \<open>z \<in> S\<close>] by (simp add: DERIV_imp_deriv)
next
fix z assume "z \<in> S"
show "the_inv_into S f (f z) = z"
by (simp add: \<open>z \<in> S\<close> injf the_inv_into_f_f)
qed
qed
subsection\<open>The Schwarz Lemma\<close>
lemma Schwarz1:
assumes holf: "f holomorphic_on S"
and contf: "continuous_on (closure S) f"
and S: "open S" "connected S"
and boS: "bounded S"
and "S \<noteq> {}"
obtains w where "w \<in> frontier S"
"\<And>z. z \<in> closure S \<Longrightarrow> norm (f z) \<le> norm (f w)"
proof -
have connf: "continuous_on (closure S) (norm o f)"
using contf continuous_on_compose continuous_on_norm_id by blast
have coc: "compact (closure S)"
by (simp add: \<open>bounded S\<close> bounded_closure compact_eq_bounded_closed)
then obtain x where x: "x \<in> closure S" and xmax: "\<And>z. z \<in> closure S \<Longrightarrow> norm(f z) \<le> norm(f x)"
apply (rule bexE [OF continuous_attains_sup [OF _ _ connf]])
using \<open>S \<noteq> {}\<close> apply auto
done
then show ?thesis
proof (cases "x \<in> frontier S")
case True
then show ?thesis using that xmax by blast
next
case False
then have "x \<in> S"
using \<open>open S\<close> frontier_def interior_eq x by auto
then have "f constant_on S"
apply (rule maximum_modulus_principle [OF holf S \<open>open S\<close> order_refl])
using closure_subset apply (blast intro: xmax)
done
then have "f constant_on (closure S)"
by (rule constant_on_closureI [OF _ contf])
then obtain c where c: "\<And>x. x \<in> closure S \<Longrightarrow> f x = c"
by (meson constant_on_def)
obtain w where "w \<in> frontier S"
by (metis coc all_not_in_conv assms(6) closure_UNIV frontier_eq_empty not_compact_UNIV)
then show ?thesis
by (simp add: c frontier_def that)
qed
qed
lemma Schwarz2:
"\<lbrakk>f holomorphic_on ball 0 r;
0 < s; ball w s \<subseteq> ball 0 r;
\<And>z. norm (w-z) < s \<Longrightarrow> norm(f z) \<le> norm(f w)\<rbrakk>
\<Longrightarrow> f constant_on ball 0 r"
by (rule maximum_modulus_principle [where U = "ball w s" and \<xi> = w]) (simp_all add: dist_norm)
lemma Schwarz3:
assumes holf: "f holomorphic_on (ball 0 r)" and [simp]: "f 0 = 0"
obtains h where "h holomorphic_on (ball 0 r)" and "\<And>z. norm z < r \<Longrightarrow> f z = z * (h z)" and "deriv f 0 = h 0"
proof -
define h where "h z = (if z = 0 then deriv f 0 else f z / z)" for z
have d0: "deriv f 0 = h 0"
by (simp add: h_def)
moreover have "h holomorphic_on (ball 0 r)"
by (rule pole_theorem_open_0 [OF holf, of 0]) (auto simp: h_def)
moreover have "norm z < r \<Longrightarrow> f z = z * h z" for z
by (simp add: h_def)
ultimately show ?thesis
using that by blast
qed
proposition Schwarz_Lemma:
assumes holf: "f holomorphic_on (ball 0 1)" and [simp]: "f 0 = 0"
and no: "\<And>z. norm z < 1 \<Longrightarrow> norm (f z) < 1"
and \<xi>: "norm \<xi> < 1"
shows "norm (f \<xi>) \<le> norm \<xi>" and "norm(deriv f 0) \<le> 1"
and "((\<exists>z. norm z < 1 \<and> z \<noteq> 0 \<and> norm(f z) = norm z)
\<or> norm(deriv f 0) = 1)
\<Longrightarrow> \<exists>\<alpha>. (\<forall>z. norm z < 1 \<longrightarrow> f z = \<alpha> * z) \<and> norm \<alpha> = 1"
(is "?P \<Longrightarrow> ?Q")
proof -
obtain h where holh: "h holomorphic_on (ball 0 1)"
and fz_eq: "\<And>z. norm z < 1 \<Longrightarrow> f z = z * (h z)" and df0: "deriv f 0 = h 0"
by (rule Schwarz3 [OF holf]) auto
have noh_le: "norm (h z) \<le> 1" if z: "norm z < 1" for z
proof -
have "norm (h z) < a" if a: "1 < a" for a
proof -
have "max (inverse a) (norm z) < 1"
using z a by (simp_all add: inverse_less_1_iff)
then obtain r where r: "max (inverse a) (norm z) < r" and "r < 1"
using Rats_dense_in_real by blast
then have nzr: "norm z < r" and ira: "inverse r < a"
using z a less_imp_inverse_less by force+
then have "0 < r"
by (meson norm_not_less_zero not_le order.strict_trans2)
have holh': "h holomorphic_on ball 0 r"
by (meson holh \<open>r < 1\<close> holomorphic_on_subset less_eq_real_def subset_ball)
have conth': "continuous_on (cball 0 r) h"
by (meson \<open>r < 1\<close> dual_order.trans holh holomorphic_on_imp_continuous_on holomorphic_on_subset mem_ball_0 mem_cball_0 not_less subsetI)
obtain w where w: "norm w = r" and lenw: "\<And>z. norm z < r \<Longrightarrow> norm(h z) \<le> norm(h w)"
apply (rule Schwarz1 [OF holh']) using conth' \<open>0 < r\<close> by auto
have "h w = f w / w" using fz_eq \<open>r < 1\<close> nzr w by auto
then have "cmod (h z) < inverse r"
by (metis \<open>0 < r\<close> \<open>r < 1\<close> divide_strict_right_mono inverse_eq_divide
le_less_trans lenw no norm_divide nzr w)
then show ?thesis using ira by linarith
qed
then show "norm (h z) \<le> 1"
using not_le by blast
qed
show "cmod (f \<xi>) \<le> cmod \<xi>"
proof (cases "\<xi> = 0")
case True then show ?thesis by auto
next
case False
then show ?thesis
by (simp add: noh_le fz_eq \<xi> mult_left_le norm_mult)
qed
show no_df0: "norm(deriv f 0) \<le> 1"
by (simp add: \<open>\<And>z. cmod z < 1 \<Longrightarrow> cmod (h z) \<le> 1\<close> df0)
show "?Q" if "?P"
using that
proof
assume "\<exists>z. cmod z < 1 \<and> z \<noteq> 0 \<and> cmod (f z) = cmod z"
then obtain \<gamma> where \<gamma>: "cmod \<gamma> < 1" "\<gamma> \<noteq> 0" "cmod (f \<gamma>) = cmod \<gamma>" by blast
then have [simp]: "norm (h \<gamma>) = 1"
by (simp add: fz_eq norm_mult)
have "ball \<gamma> (1 - cmod \<gamma>) \<subseteq> ball 0 1"
by (simp add: ball_subset_ball_iff)
moreover have "\<And>z. cmod (\<gamma> - z) < 1 - cmod \<gamma> \<Longrightarrow> cmod (h z) \<le> cmod (h \<gamma>)"
apply (simp add: algebra_simps)
by (metis add_diff_cancel_left' diff_diff_eq2 le_less_trans noh_le norm_triangle_ineq4)
ultimately obtain c where c: "\<And>z. norm z < 1 \<Longrightarrow> h z = c"
using Schwarz2 [OF holh, of "1 - norm \<gamma>" \<gamma>, unfolded constant_on_def] \<gamma> by auto
then have "norm c = 1"
using \<gamma> by force
with c show ?thesis
using fz_eq by auto
next
assume [simp]: "cmod (deriv f 0) = 1"
then obtain c where c: "\<And>z. norm z < 1 \<Longrightarrow> h z = c"
using Schwarz2 [OF holh zero_less_one, of 0, unfolded constant_on_def] df0 noh_le
by auto
moreover have "norm c = 1" using df0 c by auto
ultimately show ?thesis
using fz_eq by auto
qed
qed
corollary Schwarz_Lemma':
assumes holf: "f holomorphic_on (ball 0 1)" and [simp]: "f 0 = 0"
and no: "\<And>z. norm z < 1 \<Longrightarrow> norm (f z) < 1"
shows "((\<forall>\<xi>. norm \<xi> < 1 \<longrightarrow> norm (f \<xi>) \<le> norm \<xi>)
\<and> norm(deriv f 0) \<le> 1)
\<and> (((\<exists>z. norm z < 1 \<and> z \<noteq> 0 \<and> norm(f z) = norm z)
\<or> norm(deriv f 0) = 1)
\<longrightarrow> (\<exists>\<alpha>. (\<forall>z. norm z < 1 \<longrightarrow> f z = \<alpha> * z) \<and> norm \<alpha> = 1))"
using Schwarz_Lemma [OF assms]
by (metis (no_types) norm_eq_zero zero_less_one)
subsection\<open>The Schwarz reflection principle\<close>
lemma hol_pal_lem0:
assumes "d \<bullet> a \<le> k" "k \<le> d \<bullet> b"
obtains c where
"c \<in> closed_segment a b" "d \<bullet> c = k"
"\<And>z. z \<in> closed_segment a c \<Longrightarrow> d \<bullet> z \<le> k"
"\<And>z. z \<in> closed_segment c b \<Longrightarrow> k \<le> d \<bullet> z"
proof -
obtain c where cin: "c \<in> closed_segment a b" and keq: "k = d \<bullet> c"
using connected_ivt_hyperplane [of "closed_segment a b" a b d k]
by (auto simp: assms)
have "closed_segment a c \<subseteq> {z. d \<bullet> z \<le> k}" "closed_segment c b \<subseteq> {z. k \<le> d \<bullet> z}"
unfolding segment_convex_hull using assms keq
by (auto simp: convex_halfspace_le convex_halfspace_ge hull_minimal)
then show ?thesis using cin that by fastforce
qed
lemma hol_pal_lem1:
assumes "convex S" "open S"
and abc: "a \<in> S" "b \<in> S" "c \<in> S"
"d \<noteq> 0" and lek: "d \<bullet> a \<le> k" "d \<bullet> b \<le> k" "d \<bullet> c \<le> k"
and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
and contf: "continuous_on S f"
shows "contour_integral (linepath a b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c a) f = 0"
proof -
have "interior (convex hull {a, b, c}) \<subseteq> interior(S \<inter> {x. d \<bullet> x \<le> k})"
apply (rule interior_mono)
apply (rule hull_minimal)
apply (simp add: abc lek)
apply (rule convex_Int [OF \<open>convex S\<close> convex_halfspace_le])
done
also have "... \<subseteq> {z \<in> S. d \<bullet> z < k}"
by (force simp: interior_open [OF \<open>open S\<close>] \<open>d \<noteq> 0\<close>)
finally have *: "interior (convex hull {a, b, c}) \<subseteq> {z \<in> S. d \<bullet> z < k}" .
have "continuous_on (convex hull {a,b,c}) f"
using \<open>convex S\<close> contf abc continuous_on_subset subset_hull
by fastforce
moreover have "f holomorphic_on interior (convex hull {a,b,c})"
by (rule holomorphic_on_subset [OF holf1 *])
ultimately show ?thesis
using Cauchy_theorem_triangle_interior has_chain_integral_chain_integral3
by blast
qed
lemma hol_pal_lem2:
assumes S: "convex S" "open S"
and abc: "a \<in> S" "b \<in> S" "c \<in> S"
and "d \<noteq> 0" and lek: "d \<bullet> a \<le> k" "d \<bullet> b \<le> k"
and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
and holf2: "f holomorphic_on {z. z \<in> S \<and> k < d \<bullet> z}"
and contf: "continuous_on S f"
shows "contour_integral (linepath a b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c a) f = 0"
proof (cases "d \<bullet> c \<le> k")
case True show ?thesis
by (rule hol_pal_lem1 [OF S abc \<open>d \<noteq> 0\<close> lek True holf1 contf])
next
case False
then have "d \<bullet> c > k" by force
obtain a' where a': "a' \<in> closed_segment b c" and "d \<bullet> a' = k"
and ba': "\<And>z. z \<in> closed_segment b a' \<Longrightarrow> d \<bullet> z \<le> k"
and a'c: "\<And>z. z \<in> closed_segment a' c \<Longrightarrow> k \<le> d \<bullet> z"
apply (rule hol_pal_lem0 [of d b k c, OF \<open>d \<bullet> b \<le> k\<close>])
using False by auto
obtain b' where b': "b' \<in> closed_segment a c" and "d \<bullet> b' = k"
and ab': "\<And>z. z \<in> closed_segment a b' \<Longrightarrow> d \<bullet> z \<le> k"
and b'c: "\<And>z. z \<in> closed_segment b' c \<Longrightarrow> k \<le> d \<bullet> z"
apply (rule hol_pal_lem0 [of d a k c, OF \<open>d \<bullet> a \<le> k\<close>])
using False by auto
have a'b': "a' \<in> S \<and> b' \<in> S"
using a' abc b' convex_contains_segment \<open>convex S\<close> by auto
have "continuous_on (closed_segment c a) f"
by (meson abc contf continuous_on_subset convex_contains_segment \<open>convex S\<close>)
then have 1: "contour_integral (linepath c a) f =
contour_integral (linepath c b') f + contour_integral (linepath b' a) f"
apply (rule contour_integral_split_linepath)
using b' by (simp add: closed_segment_commute)
have "continuous_on (closed_segment b c) f"
by (meson abc contf continuous_on_subset convex_contains_segment \<open>convex S\<close>)
then have 2: "contour_integral (linepath b c) f =
contour_integral (linepath b a') f + contour_integral (linepath a' c) f"
by (rule contour_integral_split_linepath [OF _ a'])
have 3: "contour_integral (reversepath (linepath b' a')) f =
- contour_integral (linepath b' a') f"
by (rule contour_integral_reversepath [OF valid_path_linepath])
have fcd_le: "f field_differentiable at x"
if "x \<in> interior S \<and> x \<in> interior {x. d \<bullet> x \<le> k}" for x
proof -
have "f holomorphic_on S \<inter> {c. d \<bullet> c < k}"
by (metis (no_types) Collect_conj_eq Collect_mem_eq holf1)
then have "\<exists>C D. x \<in> interior C \<inter> interior D \<and> f holomorphic_on interior C \<inter> interior D"
using that
by (metis Collect_mem_eq Int_Collect \<open>d \<noteq> 0\<close> interior_halfspace_le interior_open \<open>open S\<close>)
then show "f field_differentiable at x"
by (metis at_within_interior holomorphic_on_def interior_Int interior_interior)
qed
have ab_le: "\<And>x. x \<in> closed_segment a b \<Longrightarrow> d \<bullet> x \<le> k"
proof -
fix x :: complex
assume "x \<in> closed_segment a b"
then have "\<And>C. x \<in> C \<or> b \<notin> C \<or> a \<notin> C \<or> \<not> convex C"
by (meson contra_subsetD convex_contains_segment)
then show "d \<bullet> x \<le> k"
by (metis lek convex_halfspace_le mem_Collect_eq)
qed
have "continuous_on (S \<inter> {x. d \<bullet> x \<le> k}) f" using contf
by (simp add: continuous_on_subset)
then have "(f has_contour_integral 0)
(linepath a b +++ linepath b a' +++ linepath a' b' +++ linepath b' a)"
apply (rule Cauchy_theorem_convex [where K = "{}"])
apply (simp_all add: path_image_join convex_Int convex_halfspace_le \<open>convex S\<close> fcd_le ab_le
closed_segment_subset abc a'b' ba')
by (metis \<open>d \<bullet> a' = k\<close> \<open>d \<bullet> b' = k\<close> convex_contains_segment convex_halfspace_le lek(1) mem_Collect_eq order_refl)
then have 4: "contour_integral (linepath a b) f +
contour_integral (linepath b a') f +
contour_integral (linepath a' b') f +
contour_integral (linepath b' a) f = 0"
by (rule has_chain_integral_chain_integral4)
have fcd_ge: "f field_differentiable at x"
if "x \<in> interior S \<and> x \<in> interior {x. k \<le> d \<bullet> x}" for x
proof -
have f2: "f holomorphic_on S \<inter> {c. k < d \<bullet> c}"
by (metis (full_types) Collect_conj_eq Collect_mem_eq holf2)
have f3: "interior S = S"
by (simp add: interior_open \<open>open S\<close>)
then have "x \<in> S \<inter> interior {c. k \<le> d \<bullet> c}"
using that by simp
then show "f field_differentiable at x"
using f3 f2 unfolding holomorphic_on_def
by (metis (no_types) \<open>d \<noteq> 0\<close> at_within_interior interior_Int interior_halfspace_ge interior_interior)
qed
have "continuous_on (S \<inter> {x. k \<le> d \<bullet> x}) f" using contf
by (simp add: continuous_on_subset)
then have "(f has_contour_integral 0) (linepath a' c +++ linepath c b' +++ linepath b' a')"
apply (rule Cauchy_theorem_convex [where K = "{}"])
apply (simp_all add: path_image_join convex_Int convex_halfspace_ge \<open>convex S\<close>
fcd_ge closed_segment_subset abc a'b' a'c)
by (metis \<open>d \<bullet> a' = k\<close> b'c closed_segment_commute convex_contains_segment
convex_halfspace_ge ends_in_segment(2) mem_Collect_eq order_refl)
then have 5: "contour_integral (linepath a' c) f + contour_integral (linepath c b') f + contour_integral (linepath b' a') f = 0"
by (rule has_chain_integral_chain_integral3)
show ?thesis
using 1 2 3 4 5 by (metis add.assoc eq_neg_iff_add_eq_0 reversepath_linepath)
qed
lemma hol_pal_lem3:
assumes S: "convex S" "open S"
and abc: "a \<in> S" "b \<in> S" "c \<in> S"
and "d \<noteq> 0" and lek: "d \<bullet> a \<le> k"
and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
and holf2: "f holomorphic_on {z. z \<in> S \<and> k < d \<bullet> z}"
and contf: "continuous_on S f"
shows "contour_integral (linepath a b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c a) f = 0"
proof (cases "d \<bullet> b \<le> k")
case True show ?thesis
by (rule hol_pal_lem2 [OF S abc \<open>d \<noteq> 0\<close> lek True holf1 holf2 contf])
next
case False
show ?thesis
proof (cases "d \<bullet> c \<le> k")
case True
have "contour_integral (linepath c a) f +
contour_integral (linepath a b) f +
contour_integral (linepath b c) f = 0"
by (rule hol_pal_lem2 [OF S \<open>c \<in> S\<close> \<open>a \<in> S\<close> \<open>b \<in> S\<close> \<open>d \<noteq> 0\<close> \<open>d \<bullet> c \<le> k\<close> lek holf1 holf2 contf])
then show ?thesis
by (simp add: algebra_simps)
next
case False
have "contour_integral (linepath b c) f +
contour_integral (linepath c a) f +
contour_integral (linepath a b) f = 0"
apply (rule hol_pal_lem2 [OF S \<open>b \<in> S\<close> \<open>c \<in> S\<close> \<open>a \<in> S\<close>, of "-d" "-k"])
using \<open>d \<noteq> 0\<close> \<open>\<not> d \<bullet> b \<le> k\<close> False by (simp_all add: holf1 holf2 contf)
then show ?thesis
by (simp add: algebra_simps)
qed
qed
lemma hol_pal_lem4:
assumes S: "convex S" "open S"
and abc: "a \<in> S" "b \<in> S" "c \<in> S" and "d \<noteq> 0"
and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
and holf2: "f holomorphic_on {z. z \<in> S \<and> k < d \<bullet> z}"
and contf: "continuous_on S f"
shows "contour_integral (linepath a b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c a) f = 0"
proof (cases "d \<bullet> a \<le> k")
case True show ?thesis
by (rule hol_pal_lem3 [OF S abc \<open>d \<noteq> 0\<close> True holf1 holf2 contf])
next
case False
show ?thesis
apply (rule hol_pal_lem3 [OF S abc, of "-d" "-k"])
using \<open>d \<noteq> 0\<close> False by (simp_all add: holf1 holf2 contf)
qed
lemma holomorphic_on_paste_across_line:
assumes S: "open S" and "d \<noteq> 0"
and holf1: "f holomorphic_on (S \<inter> {z. d \<bullet> z < k})"
and holf2: "f holomorphic_on (S \<inter> {z. k < d \<bullet> z})"
and contf: "continuous_on S f"
shows "f holomorphic_on S"
proof -
have *: "\<exists>t. open t \<and> p \<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)"
if "p \<in> S" for p
proof -
obtain e where "e>0" and e: "ball p e \<subseteq> S"
using \<open>p \<in> S\<close> openE S by blast
then have "continuous_on (ball p e) f"
using contf continuous_on_subset by blast
moreover have "f holomorphic_on {z. dist p z < e \<and> d \<bullet> z < k}"
apply (rule holomorphic_on_subset [OF holf1])
using e by auto
moreover have "f holomorphic_on {z. dist p z < e \<and> k < d \<bullet> z}"
apply (rule holomorphic_on_subset [OF holf2])
using e by auto
ultimately show ?thesis
apply (rule_tac x="ball p e" in exI)
using \<open>e > 0\<close> e \<open>d \<noteq> 0\<close>
apply (simp add:, clarify)
apply (rule hol_pal_lem4 [of "ball p e" _ _ _ d _ k])
apply (auto simp: subset_hull)
done
qed
show ?thesis
by (blast intro: * Morera_local_triangle analytic_imp_holomorphic)
qed
proposition Schwarz_reflection:
assumes "open S" and cnjs: "cnj ` S \<subseteq> S"
and holf: "f holomorphic_on (S \<inter> {z. 0 < Im z})"
and contf: "continuous_on (S \<inter> {z. 0 \<le> Im z}) f"
and f: "\<And>z. \<lbrakk>z \<in> S; z \<in> \<real>\<rbrakk> \<Longrightarrow> (f z) \<in> \<real>"
shows "(\<lambda>z. if 0 \<le> Im z then f z else cnj(f(cnj z))) holomorphic_on S"
proof -
have 1: "(\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z))) holomorphic_on (S \<inter> {z. 0 < Im z})"
by (force intro: iffD1 [OF holomorphic_cong [OF refl] holf])
have cont_cfc: "continuous_on (S \<inter> {z. Im z \<le> 0}) (cnj o f o cnj)"
apply (intro continuous_intros continuous_on_compose continuous_on_subset [OF contf])
using cnjs apply auto
done
have "cnj \<circ> f \<circ> cnj field_differentiable at x within S \<inter> {z. Im z < 0}"
if "x \<in> S" "Im x < 0" "f field_differentiable at (cnj x) within S \<inter> {z. 0 < Im z}" for x
using that
apply (simp add: field_differentiable_def has_field_derivative_iff Lim_within dist_norm, clarify)
apply (rule_tac x="cnj f'" in exI)
apply (elim all_forward ex_forward conj_forward imp_forward asm_rl, clarify)
apply (drule_tac x="cnj xa" in bspec)
using cnjs apply force
apply (metis complex_cnj_cnj complex_cnj_diff complex_cnj_divide complex_mod_cnj)
done
then have hol_cfc: "(cnj o f o cnj) holomorphic_on (S \<inter> {z. Im z < 0})"
using holf cnjs
by (force simp: holomorphic_on_def)
have 2: "(\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z))) holomorphic_on (S \<inter> {z. Im z < 0})"
apply (rule iffD1 [OF holomorphic_cong [OF refl]])
using hol_cfc by auto
have [simp]: "(S \<inter> {z. 0 \<le> Im z}) \<union> (S \<inter> {z. Im z \<le> 0}) = S"
by force
have "continuous_on ((S \<inter> {z. 0 \<le> Im z}) \<union> (S \<inter> {z. Im z \<le> 0}))
(\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z)))"
apply (rule continuous_on_cases_local)
using cont_cfc contf
apply (simp_all add: closedin_closed_Int closed_halfspace_Im_le closed_halfspace_Im_ge)
using f Reals_cnj_iff complex_is_Real_iff apply auto
done
then have 3: "continuous_on S (\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z)))"
by force
show ?thesis
apply (rule holomorphic_on_paste_across_line [OF \<open>open S\<close>, of "- \<i>" _ 0])
using 1 2 3
apply auto
done
qed
subsection\<open>Bloch's theorem\<close>
lemma Bloch_lemma_0:
assumes holf: "f holomorphic_on cball 0 r" and "0 < r"
and [simp]: "f 0 = 0"
and le: "\<And>z. norm z < r \<Longrightarrow> norm(deriv f z) \<le> 2 * norm(deriv f 0)"
shows "ball 0 ((3 - 2 * sqrt 2) * r * norm(deriv f 0)) \<subseteq> f ` ball 0 r"
proof -
have "sqrt 2 < 3/2"
by (rule real_less_lsqrt) (auto simp: power2_eq_square)
then have sq3: "0 < 3 - 2 * sqrt 2" by simp
show ?thesis
proof (cases "deriv f 0 = 0")
case True then show ?thesis by simp
next
case False
define C where "C = 2 * norm(deriv f 0)"
have "0 < C" using False by (simp add: C_def)
have holf': "f holomorphic_on ball 0 r" using holf
using ball_subset_cball holomorphic_on_subset by blast
then have holdf': "deriv f holomorphic_on ball 0 r"
by (rule holomorphic_deriv [OF _ open_ball])
have "Le1": "norm(deriv f z - deriv f 0) \<le> norm z / (r - norm z) * C"
if "norm z < r" for z
proof -
have T1: "norm(deriv f z - deriv f 0) \<le> norm z / (R - norm z) * C"
if R: "norm z < R" "R < r" for R
proof -
have "0 < R" using R
by (metis less_trans norm_zero zero_less_norm_iff)
have df_le: "\<And>x. norm x < r \<Longrightarrow> norm (deriv f x) \<le> C"
using le by (simp add: C_def)
have hol_df: "deriv f holomorphic_on cball 0 R"
apply (rule holomorphic_on_subset) using R holdf' by auto
have *: "((\<lambda>w. deriv f w / (w - z)) has_contour_integral 2 * pi * \<i> * deriv f z) (circlepath 0 R)"
if "norm z < R" for z
using \<open>0 < R\<close> that Cauchy_integral_formula_convex_simple [OF convex_cball hol_df, of _ "circlepath 0 R"]
by (force simp: winding_number_circlepath)
have **: "((\<lambda>x. deriv f x / (x - z) - deriv f x / x) has_contour_integral
of_real (2 * pi) * \<i> * (deriv f z - deriv f 0))
(circlepath 0 R)"
using has_contour_integral_diff [OF * [of z] * [of 0]] \<open>0 < R\<close> that
by (simp add: algebra_simps)
have [simp]: "\<And>x. norm x = R \<Longrightarrow> x \<noteq> z" using that(1) by blast
have "norm (deriv f x / (x - z) - deriv f x / x)
\<le> C * norm z / (R * (R - norm z))"
if "norm x = R" for x
proof -
have [simp]: "norm (deriv f x * x - deriv f x * (x - z)) =
norm (deriv f x) * norm z"
by (simp add: norm_mult right_diff_distrib')
show ?thesis
using \<open>0 < R\<close> \<open>0 < C\<close> R that
apply (simp add: norm_mult norm_divide divide_simps)
using df_le norm_triangle_ineq2 \<open>0 < C\<close> apply (auto intro!: mult_mono)
done
qed
then show ?thesis
using has_contour_integral_bound_circlepath
[OF **, of "C * norm z/(R*(R - norm z))"]
\<open>0 < R\<close> \<open>0 < C\<close> R
apply (simp add: norm_mult norm_divide)
apply (simp add: divide_simps mult.commute)
done
qed
obtain r' where r': "norm z < r'" "r' < r"
using Rats_dense_in_real [of "norm z" r] \<open>norm z < r\<close> by blast
then have [simp]: "closure {r'<..<r} = {r'..r}" by simp
show ?thesis
apply (rule continuous_ge_on_closure
[where f = "\<lambda>r. norm z / (r - norm z) * C" and s = "{r'<..<r}",
OF _ _ T1])
apply (intro continuous_intros)
using that r'
apply (auto simp: not_le)
done
qed
have "*": "(norm z - norm z^2/(r - norm z)) * norm(deriv f 0) \<le> norm(f z)"
if r: "norm z < r" for z
proof -
have 1: "\<And>x. x \<in> ball 0 r \<Longrightarrow>
((\<lambda>z. f z - deriv f 0 * z) has_field_derivative deriv f x - deriv f 0)
(at x within ball 0 r)"
by (rule derivative_eq_intros holomorphic_derivI holf' | simp)+
have 2: "closed_segment 0 z \<subseteq> ball 0 r"
by (metis \<open>0 < r\<close> convex_ball convex_contains_segment dist_self mem_ball mem_ball_0 that)
have 3: "(\<lambda>t. (norm z)\<^sup>2 * t / (r - norm z) * C) integrable_on {0..1}"
apply (rule integrable_on_cmult_right [where 'b=real, simplified])
apply (rule integrable_on_cdivide [where 'b=real, simplified])
apply (rule integrable_on_cmult_left [where 'b=real, simplified])
apply (rule ident_integrable_on)
done
have 4: "norm (deriv f (x *\<^sub>R z) - deriv f 0) * norm z \<le> norm z * norm z * x * C / (r - norm z)"
if x: "0 \<le> x" "x \<le> 1" for x
proof -
have [simp]: "x * norm z < r"
using r x by (meson le_less_trans mult_le_cancel_right2 norm_not_less_zero)
have "norm (deriv f (x *\<^sub>R z) - deriv f 0) \<le> norm (x *\<^sub>R z) / (r - norm (x *\<^sub>R z)) * C"
apply (rule Le1) using r x \<open>0 < r\<close> by simp
also have "... \<le> norm (x *\<^sub>R z) / (r - norm z) * C"
using r x \<open>0 < r\<close>
apply (simp add: field_split_simps)
by (simp add: \<open>0 < C\<close> mult.assoc mult_left_le_one_le ordered_comm_semiring_class.comm_mult_left_mono)
finally have "norm (deriv f (x *\<^sub>R z) - deriv f 0) * norm z \<le> norm (x *\<^sub>R z) / (r - norm z) * C * norm z"
by (rule mult_right_mono) simp
with x show ?thesis by (simp add: algebra_simps)
qed
have le_norm: "abc \<le> norm d - e \<Longrightarrow> norm(f - d) \<le> e \<Longrightarrow> abc \<le> norm f" for abc d e and f::complex
by (metis add_diff_cancel_left' add_diff_eq diff_left_mono norm_diff_ineq order_trans)
have "norm (integral {0..1} (\<lambda>x. (deriv f (x *\<^sub>R z) - deriv f 0) * z))
\<le> integral {0..1} (\<lambda>t. (norm z)\<^sup>2 * t / (r - norm z) * C)"
apply (rule integral_norm_bound_integral)
using contour_integral_primitive [OF 1, of "linepath 0 z"] 2
apply (simp add: has_contour_integral_linepath has_integral_integrable_integral)
apply (rule 3)
apply (simp add: norm_mult power2_eq_square 4)
done
then have int_le: "norm (f z - deriv f 0 * z) \<le> (norm z)\<^sup>2 * norm(deriv f 0) / ((r - norm z))"
using contour_integral_primitive [OF 1, of "linepath 0 z"] 2
apply (simp add: has_contour_integral_linepath has_integral_integrable_integral C_def)
done
show ?thesis
apply (rule le_norm [OF _ int_le])
using \<open>norm z < r\<close>
apply (simp add: power2_eq_square divide_simps C_def norm_mult)
proof -
have "norm z * (norm (deriv f 0) * (r - norm z - norm z)) \<le> norm z * (norm (deriv f 0) * (r - norm z) - norm (deriv f 0) * norm z)"
by (simp add: algebra_simps)
then show "(norm z * (r - norm z) - norm z * norm z) * norm (deriv f 0) \<le> norm (deriv f 0) * norm z * (r - norm z) - norm z * norm z * norm (deriv f 0)"
by (simp add: algebra_simps)
qed
qed
have sq201 [simp]: "0 < (1 - sqrt 2 / 2)" "(1 - sqrt 2 / 2) < 1"
by (auto simp: sqrt2_less_2)
have 1: "continuous_on (closure (ball 0 ((1 - sqrt 2 / 2) * r))) f"
apply (rule continuous_on_subset [OF holomorphic_on_imp_continuous_on [OF holf]])
apply (subst closure_ball)
using \<open>0 < r\<close> mult_pos_pos sq201
apply (auto simp: cball_subset_cball_iff)
done
have 2: "open (f ` interior (ball 0 ((1 - sqrt 2 / 2) * r)))"
apply (rule open_mapping_thm [OF holf' open_ball connected_ball], force)
using \<open>0 < r\<close> mult_pos_pos sq201 apply (simp add: ball_subset_ball_iff)
using False \<open>0 < r\<close> centre_in_ball holf' holomorphic_nonconstant by blast
have "ball 0 ((3 - 2 * sqrt 2) * r * norm (deriv f 0)) =
ball (f 0) ((3 - 2 * sqrt 2) * r * norm (deriv f 0))"
by simp
also have "... \<subseteq> f ` ball 0 ((1 - sqrt 2 / 2) * r)"
proof -
have 3: "(3 - 2 * sqrt 2) * r * norm (deriv f 0) \<le> norm (f z)"
if "norm z = (1 - sqrt 2 / 2) * r" for z
apply (rule order_trans [OF _ *])
using \<open>0 < r\<close>
apply (simp_all add: field_simps power2_eq_square that)
apply (simp add: mult.assoc [symmetric])
done
show ?thesis
apply (rule ball_subset_open_map_image [OF 1 2 _ bounded_ball])
using \<open>0 < r\<close> sq201 3 apply simp_all
using C_def \<open>0 < C\<close> sq3 apply force
done
qed
also have "... \<subseteq> f ` ball 0 r"
apply (rule image_subsetI [OF imageI], simp)
apply (erule less_le_trans)
using \<open>0 < r\<close> apply (auto simp: field_simps)
done
finally show ?thesis .
qed
qed
lemma Bloch_lemma:
assumes holf: "f holomorphic_on cball a r" and "0 < r"
and le: "\<And>z. z \<in> ball a r \<Longrightarrow> norm(deriv f z) \<le> 2 * norm(deriv f a)"
shows "ball (f a) ((3 - 2 * sqrt 2) * r * norm(deriv f a)) \<subseteq> f ` ball a r"
proof -
have fz: "(\<lambda>z. f (a + z)) = f o (\<lambda>z. (a + z))"
by (simp add: o_def)
have hol0: "(\<lambda>z. f (a + z)) holomorphic_on cball 0 r"
unfolding fz by (intro holomorphic_intros holf holomorphic_on_compose | simp)+
then have [simp]: "\<And>x. norm x < r \<Longrightarrow> (\<lambda>z. f (a + z)) field_differentiable at x"
by (metis open_ball at_within_open ball_subset_cball diff_0 dist_norm holomorphic_on_def holomorphic_on_subset mem_ball norm_minus_cancel)
have [simp]: "\<And>z. norm z < r \<Longrightarrow> f field_differentiable at (a + z)"
by (metis holf open_ball add_diff_cancel_left' dist_complex_def holomorphic_on_imp_differentiable_at holomorphic_on_subset interior_cball interior_subset mem_ball norm_minus_commute)
then have [simp]: "f field_differentiable at a"
by (metis add.comm_neutral \<open>0 < r\<close> norm_eq_zero)
have hol1: "(\<lambda>z. f (a + z) - f a) holomorphic_on cball 0 r"
by (intro holomorphic_intros hol0)
then have "ball 0 ((3 - 2 * sqrt 2) * r * norm (deriv (\<lambda>z. f (a + z) - f a) 0))
\<subseteq> (\<lambda>z. f (a + z) - f a) ` ball 0 r"
apply (rule Bloch_lemma_0)
apply (simp_all add: \<open>0 < r\<close>)
apply (simp add: fz deriv_chain)
apply (simp add: dist_norm le)
done
then show ?thesis
apply clarify
apply (drule_tac c="x - f a" in subsetD)
apply (force simp: fz \<open>0 < r\<close> dist_norm deriv_chain field_differentiable_compose)+
done
qed
proposition Bloch_unit:
assumes holf: "f holomorphic_on ball a 1" and [simp]: "deriv f a = 1"
obtains b r where "1/12 < r" and "ball b r \<subseteq> f ` (ball a 1)"
proof -
define r :: real where "r = 249/256"
have "0 < r" "r < 1" by (auto simp: r_def)
define g where "g z = deriv f z * of_real(r - norm(z - a))" for z
have "deriv f holomorphic_on ball a 1"
by (rule holomorphic_deriv [OF holf open_ball])
then have "continuous_on (ball a 1) (deriv f)"
using holomorphic_on_imp_continuous_on by blast
then have "continuous_on (cball a r) (deriv f)"
by (rule continuous_on_subset) (simp add: cball_subset_ball_iff \<open>r < 1\<close>)
then have "continuous_on (cball a r) g"
by (simp add: g_def continuous_intros)
then have 1: "compact (g ` cball a r)"
by (rule compact_continuous_image [OF _ compact_cball])
have 2: "g ` cball a r \<noteq> {}"
using \<open>r > 0\<close> by auto
obtain p where pr: "p \<in> cball a r"
and pge: "\<And>y. y \<in> cball a r \<Longrightarrow> norm (g y) \<le> norm (g p)"
using distance_attains_sup [OF 1 2, of 0] by force
define t where "t = (r - norm(p - a)) / 2"
have "norm (p - a) \<noteq> r"
using pge [of a] \<open>r > 0\<close> by (auto simp: g_def norm_mult)
then have "norm (p - a) < r" using pr
by (simp add: norm_minus_commute dist_norm)
then have "0 < t"
by (simp add: t_def)
have cpt: "cball p t \<subseteq> ball a r"
using \<open>0 < t\<close> by (simp add: cball_subset_ball_iff dist_norm t_def field_simps)
have gen_le_dfp: "norm (deriv f y) * (r - norm (y - a)) / (r - norm (p - a)) \<le> norm (deriv f p)"
if "y \<in> cball a r" for y
proof -
have [simp]: "norm (y - a) \<le> r"
using that by (simp add: dist_norm norm_minus_commute)
have "norm (g y) \<le> norm (g p)"
using pge [OF that] by simp
then have "norm (deriv f y) * abs (r - norm (y - a)) \<le> norm (deriv f p) * abs (r - norm (p - a))"
by (simp only: dist_norm g_def norm_mult norm_of_real)
with that \<open>norm (p - a) < r\<close> show ?thesis
by (simp add: dist_norm field_split_simps)
qed
have le_norm_dfp: "r / (r - norm (p - a)) \<le> norm (deriv f p)"
using gen_le_dfp [of a] \<open>r > 0\<close> by auto
have 1: "f holomorphic_on cball p t"
apply (rule holomorphic_on_subset [OF holf])
using cpt \<open>r < 1\<close> order_subst1 subset_ball by auto
have 2: "norm (deriv f z) \<le> 2 * norm (deriv f p)" if "z \<in> ball p t" for z
proof -
have z: "z \<in> cball a r"
by (meson ball_subset_cball subsetD cpt that)
then have "norm(z - a) < r"
by (metis ball_subset_cball contra_subsetD cpt dist_norm mem_ball norm_minus_commute that)
have "norm (deriv f z) * (r - norm (z - a)) / (r - norm (p - a)) \<le> norm (deriv f p)"
using gen_le_dfp [OF z] by simp
with \<open>norm (z - a) < r\<close> \<open>norm (p - a) < r\<close>
have "norm (deriv f z) \<le> (r - norm (p - a)) / (r - norm (z - a)) * norm (deriv f p)"
by (simp add: field_simps)
also have "... \<le> 2 * norm (deriv f p)"
apply (rule mult_right_mono)
using that \<open>norm (p - a) < r\<close> \<open>norm(z - a) < r\<close>
apply (simp_all add: field_simps t_def dist_norm [symmetric])
using dist_triangle3 [of z a p] by linarith
finally show ?thesis .
qed
have sqrt2: "sqrt 2 < 2113/1494"
by (rule real_less_lsqrt) (auto simp: power2_eq_square)
then have sq3: "0 < 3 - 2 * sqrt 2" by simp
have "1 / 12 / ((3 - 2 * sqrt 2) / 2) < r"
using sq3 sqrt2 by (auto simp: field_simps r_def)
also have "... \<le> cmod (deriv f p) * (r - cmod (p - a))"
using \<open>norm (p - a) < r\<close> le_norm_dfp by (simp add: pos_divide_le_eq)
finally have "1 / 12 < cmod (deriv f p) * (r - cmod (p - a)) * ((3 - 2 * sqrt 2) / 2)"
using pos_divide_less_eq half_gt_zero_iff sq3 by blast
then have **: "1 / 12 < (3 - 2 * sqrt 2) * t * norm (deriv f p)"
using sq3 by (simp add: mult.commute t_def)
have "ball (f p) ((3 - 2 * sqrt 2) * t * norm (deriv f p)) \<subseteq> f ` ball p t"
by (rule Bloch_lemma [OF 1 \<open>0 < t\<close> 2])
also have "... \<subseteq> f ` ball a 1"
apply (rule image_mono)
apply (rule order_trans [OF ball_subset_cball])
apply (rule order_trans [OF cpt])
using \<open>0 < t\<close> \<open>r < 1\<close> apply (simp add: ball_subset_ball_iff dist_norm)
done
finally have "ball (f p) ((3 - 2 * sqrt 2) * t * norm (deriv f p)) \<subseteq> f ` ball a 1" .
with ** show ?thesis
by (rule that)
qed
theorem Bloch:
assumes holf: "f holomorphic_on ball a r" and "0 < r"
and r': "r' \<le> r * norm (deriv f a) / 12"
obtains b where "ball b r' \<subseteq> f ` (ball a r)"
proof (cases "deriv f a = 0")
case True with r' show ?thesis
using ball_eq_empty that by fastforce
next
case False
define C where "C = deriv f a"
have "0 < norm C" using False by (simp add: C_def)
have dfa: "f field_differentiable at a"
apply (rule holomorphic_on_imp_differentiable_at [OF holf])
using \<open>0 < r\<close> by auto
have fo: "(\<lambda>z. f (a + of_real r * z)) = f o (\<lambda>z. (a + of_real r * z))"
by (simp add: o_def)
have holf': "f holomorphic_on (\<lambda>z. a + complex_of_real r * z) ` ball 0 1"
apply (rule holomorphic_on_subset [OF holf])
using \<open>0 < r\<close> apply (force simp: dist_norm norm_mult)
done
have 1: "(\<lambda>z. f (a + r * z) / (C * r)) holomorphic_on ball 0 1"
apply (rule holomorphic_intros holomorphic_on_compose holf' | simp add: fo)+
using \<open>0 < r\<close> by (simp add: C_def False)
have "((\<lambda>z. f (a + of_real r * z) / (C * of_real r)) has_field_derivative
(deriv f (a + of_real r * z) / C)) (at z)"
if "norm z < 1" for z
proof -
have *: "((\<lambda>x. f (a + of_real r * x)) has_field_derivative
(deriv f (a + of_real r * z) * of_real r)) (at z)"
apply (simp add: fo)
apply (rule DERIV_chain [OF field_differentiable_derivI])
apply (rule holomorphic_on_imp_differentiable_at [OF holf], simp)
using \<open>0 < r\<close> apply (simp add: dist_norm norm_mult that)
apply (rule derivative_eq_intros | simp)+
done
show ?thesis
apply (rule derivative_eq_intros * | simp)+
using \<open>0 < r\<close> by (auto simp: C_def False)
qed
have 2: "deriv (\<lambda>z. f (a + of_real r * z) / (C * of_real r)) 0 = 1"
apply (subst deriv_cdivide_right)
apply (simp add: field_differentiable_def fo)
apply (rule exI)
apply (rule DERIV_chain [OF field_differentiable_derivI])
apply (simp add: dfa)
apply (rule derivative_eq_intros | simp add: C_def False fo)+
using \<open>0 < r\<close>
apply (simp add: C_def False fo)
apply (simp add: derivative_intros dfa deriv_chain)
done
have sb1: "(*) (C * r) ` (\<lambda>z. f (a + of_real r * z) / (C * r)) ` ball 0 1
\<subseteq> f ` ball a r"
using \<open>0 < r\<close> by (auto simp: dist_norm norm_mult C_def False)
have sb2: "ball (C * r * b) r' \<subseteq> (*) (C * r) ` ball b t"
if "1 / 12 < t" for b t
proof -
have *: "r * cmod (deriv f a) / 12 \<le> r * (t * cmod (deriv f a))"
using that \<open>0 < r\<close> less_eq_real_def mult.commute mult.right_neutral mult_left_mono norm_ge_zero times_divide_eq_right
by auto
show ?thesis
apply clarify
apply (rule_tac x="x / (C * r)" in image_eqI)
using \<open>0 < r\<close>
apply (simp_all add: dist_norm norm_mult norm_divide C_def False field_simps)
apply (erule less_le_trans)
apply (rule order_trans [OF r' *])
done
qed
show ?thesis
apply (rule Bloch_unit [OF 1 2])
apply (rename_tac t)
apply (rule_tac b="(C * of_real r) * b" in that)
apply (drule image_mono [where f = "\<lambda>z. (C * of_real r) * z"])
using sb1 sb2
apply force
done
qed
corollary Bloch_general:
assumes holf: "f holomorphic_on s" and "a \<in> s"
and tle: "\<And>z. z \<in> frontier s \<Longrightarrow> t \<le> dist a z"
and rle: "r \<le> t * norm(deriv f a) / 12"
obtains b where "ball b r \<subseteq> f ` s"
proof -
consider "r \<le> 0" | "0 < t * norm(deriv f a) / 12" using rle by force
then show ?thesis
proof cases
case 1 then show ?thesis
by (simp add: ball_empty that)
next
case 2
show ?thesis
proof (cases "deriv f a = 0")
case True then show ?thesis
using rle by (simp add: ball_empty that)
next
case False
then have "t > 0"
using 2 by (force simp: zero_less_mult_iff)
have "\<not> ball a t \<subseteq> s \<Longrightarrow> ball a t \<inter> frontier s \<noteq> {}"
apply (rule connected_Int_frontier [of "ball a t" s], simp_all)
using \<open>0 < t\<close> \<open>a \<in> s\<close> centre_in_ball apply blast
done
with tle have *: "ball a t \<subseteq> s" by fastforce
then have 1: "f holomorphic_on ball a t"
using holf using holomorphic_on_subset by blast
show ?thesis
apply (rule Bloch [OF 1 \<open>t > 0\<close> rle])
apply (rule_tac b=b in that)
using * apply force
done
qed
qed
qed
end