theory Laurent_Convergence
imports "HOL-Computational_Algebra.Formal_Laurent_Series" "HOL-Library.Landau_Symbols"
Residue_Theorem
begin
definition%important fls_conv_radius :: "complex fls \<Rightarrow> ereal" where
"fls_conv_radius f = fps_conv_radius (fls_regpart f)"
definition%important eval_fls :: "complex fls \<Rightarrow> complex \<Rightarrow> complex" where
"eval_fls F z = eval_fps (fls_base_factor_to_fps F) z * z powi fls_subdegree F"
definition\<^marker>\<open>tag important\<close>
has_laurent_expansion :: "(complex \<Rightarrow> complex) \<Rightarrow> complex fls \<Rightarrow> bool"
(infixl \<open>has'_laurent'_expansion\<close> 60)
where "(f has_laurent_expansion F) \<longleftrightarrow>
fls_conv_radius F > 0 \<and> eventually (\<lambda>z. eval_fls F z = f z) (at 0)"
lemma has_laurent_expansion_schematicI:
"f has_laurent_expansion F \<Longrightarrow> F = G \<Longrightarrow> f has_laurent_expansion G"
by simp
lemma has_laurent_expansion_cong:
assumes "eventually (\<lambda>x. f x = g x) (at 0)" "F = G"
shows "(f has_laurent_expansion F) \<longleftrightarrow> (g has_laurent_expansion G)"
proof -
have "eventually (\<lambda>z. eval_fls F z = g z) (at 0)"
if "eventually (\<lambda>z. eval_fls F z = f z) (at 0)" "eventually (\<lambda>x. f x = g x) (at 0)" for f g
using that by eventually_elim auto
from this[of f g] this[of g f] show ?thesis
using assms by (auto simp: eq_commute has_laurent_expansion_def)
qed
lemma has_laurent_expansion_cong':
assumes "eventually (\<lambda>x. f x = g x) (at z)" "F = G" "z = z'"
shows "((\<lambda>x. f (z + x)) has_laurent_expansion F) \<longleftrightarrow> ((\<lambda>x. g (z' + x)) has_laurent_expansion G)"
by (intro has_laurent_expansion_cong)
(use assms in \<open>auto simp: at_to_0' eventually_filtermap add_ac\<close>)
lemma fls_conv_radius_altdef:
"fls_conv_radius F = fps_conv_radius (fls_base_factor_to_fps F)"
proof -
have "conv_radius (\<lambda>n. fls_nth F (int n)) = conv_radius (\<lambda>n. fls_nth F (int n + fls_subdegree F))"
proof (cases "fls_subdegree F \<ge> 0")
case True
hence "conv_radius (\<lambda>n. fls_nth F (int n + fls_subdegree F)) =
conv_radius (\<lambda>n. fls_nth F (int (n + nat (fls_subdegree F))))"
by auto
thus ?thesis
by (subst (asm) conv_radius_shift) auto
next
case False
hence "conv_radius (\<lambda>n. fls_nth F (int n)) =
conv_radius (\<lambda>n. fls_nth F (fls_subdegree F + int (n + nat (-fls_subdegree F))))"
by auto
thus ?thesis
by (subst (asm) conv_radius_shift) (auto simp: add_ac)
qed
thus ?thesis
by (simp add: fls_conv_radius_def fps_conv_radius_def)
qed
lemma eval_fps_of_nat [simp]: "eval_fps (of_nat n) z = of_nat n"
and eval_fps_of_int [simp]: "eval_fps (of_int m) z = of_int m"
by (simp_all flip: fps_of_nat fps_of_int)
lemma fps_conv_radius_of_nat [simp]: "fps_conv_radius (of_nat n) = \<infinity>"
and fps_conv_radius_of_int [simp]: "fps_conv_radius (of_int m) = \<infinity>"
by (simp_all flip: fps_of_nat fps_of_int)
lemma fps_conv_radius_fls_regpart: "fps_conv_radius (fls_regpart F) = fls_conv_radius F"
by (simp add: fls_conv_radius_def)
lemma fls_conv_radius_0 [simp]: "fls_conv_radius 0 = \<infinity>"
and fls_conv_radius_1 [simp]: "fls_conv_radius 1 = \<infinity>"
and fls_conv_radius_const [simp]: "fls_conv_radius (fls_const c) = \<infinity>"
and fls_conv_radius_numeral [simp]: "fls_conv_radius (numeral num) = \<infinity>"
and fls_conv_radius_of_nat [simp]: "fls_conv_radius (of_nat n) = \<infinity>"
and fls_conv_radius_of_int [simp]: "fls_conv_radius (of_int m) = \<infinity>"
and fls_conv_radius_X [simp]: "fls_conv_radius fls_X = \<infinity>"
and fls_conv_radius_X_inv [simp]: "fls_conv_radius fls_X_inv = \<infinity>"
and fls_conv_radius_X_intpow [simp]: "fls_conv_radius (fls_X_intpow m) = \<infinity>"
by (simp_all add: fls_conv_radius_def fls_X_intpow_regpart)
lemma fls_conv_radius_shift [simp]: "fls_conv_radius (fls_shift n F) = fls_conv_radius F"
unfolding fls_conv_radius_altdef by (subst fls_base_factor_to_fps_shift) (rule refl)
lemma fls_conv_radius_fps_to_fls [simp]: "fls_conv_radius (fps_to_fls F) = fps_conv_radius F"
by (simp add: fls_conv_radius_def)
lemma fls_conv_radius_deriv [simp]: "fls_conv_radius (fls_deriv F) \<ge> fls_conv_radius F"
proof -
have "fls_conv_radius (fls_deriv F) = fps_conv_radius (fls_regpart (fls_deriv F))"
by (simp add: fls_conv_radius_def)
also have "fls_regpart (fls_deriv F) = fps_deriv (fls_regpart F)"
by (intro fps_ext) (auto simp: add_ac)
also have "fps_conv_radius \<dots> \<ge> fls_conv_radius F"
by (simp add: fls_conv_radius_def fps_conv_radius_deriv)
finally show ?thesis .
qed
lemma fls_conv_radius_uminus [simp]: "fls_conv_radius (-F) = fls_conv_radius F"
by (simp add: fls_conv_radius_def)
lemma fls_conv_radius_add: "fls_conv_radius (F + G) \<ge> min (fls_conv_radius F) (fls_conv_radius G)"
by (simp add: fls_conv_radius_def fps_conv_radius_add)
lemma fls_conv_radius_diff: "fls_conv_radius (F - G) \<ge> min (fls_conv_radius F) (fls_conv_radius G)"
by (simp add: fls_conv_radius_def fps_conv_radius_diff)
lemma fls_conv_radius_mult: "fls_conv_radius (F * G) \<ge> min (fls_conv_radius F) (fls_conv_radius G)"
proof (cases "F = 0 \<or> G = 0")
case False
hence [simp]: "F \<noteq> 0" "G \<noteq> 0"
by auto
have "fls_conv_radius (F * G) = fps_conv_radius (fls_regpart (fls_shift (fls_subdegree F + fls_subdegree G) (F * G)))"
by (simp add: fls_conv_radius_altdef)
also have "fls_regpart (fls_shift (fls_subdegree F + fls_subdegree G) (F * G)) =
fls_base_factor_to_fps F * fls_base_factor_to_fps G"
by (simp add: fls_times_def)
also have "fps_conv_radius \<dots> \<ge> min (fls_conv_radius F) (fls_conv_radius G)"
unfolding fls_conv_radius_altdef by (rule fps_conv_radius_mult)
finally show ?thesis .
qed auto
lemma fps_conv_radius_add_ge:
"fps_conv_radius F \<ge> r \<Longrightarrow> fps_conv_radius G \<ge> r \<Longrightarrow> fps_conv_radius (F + G) \<ge> r"
using fps_conv_radius_add[of F G] by (simp add: min_def split: if_splits)
lemma fps_conv_radius_diff_ge:
"fps_conv_radius F \<ge> r \<Longrightarrow> fps_conv_radius G \<ge> r \<Longrightarrow> fps_conv_radius (F - G) \<ge> r"
using fps_conv_radius_diff[of F G] by (simp add: min_def split: if_splits)
lemma fps_conv_radius_mult_ge:
"fps_conv_radius F \<ge> r \<Longrightarrow> fps_conv_radius G \<ge> r \<Longrightarrow> fps_conv_radius (F * G) \<ge> r"
using fps_conv_radius_mult[of F G] by (simp add: min_def split: if_splits)
lemma fls_conv_radius_add_ge:
"fls_conv_radius F \<ge> r \<Longrightarrow> fls_conv_radius G \<ge> r \<Longrightarrow> fls_conv_radius (F + G) \<ge> r"
using fls_conv_radius_add[of F G] by (simp add: min_def split: if_splits)
lemma fls_conv_radius_diff_ge:
"fls_conv_radius F \<ge> r \<Longrightarrow> fls_conv_radius G \<ge> r \<Longrightarrow> fls_conv_radius (F - G) \<ge> r"
using fls_conv_radius_diff[of F G] by (simp add: min_def split: if_splits)
lemma fls_conv_radius_mult_ge:
"fls_conv_radius F \<ge> r \<Longrightarrow> fls_conv_radius G \<ge> r \<Longrightarrow> fls_conv_radius (F * G) \<ge> r"
using fls_conv_radius_mult[of F G] by (simp add: min_def split: if_splits)
lemma fls_conv_radius_power: "fls_conv_radius (F ^ n) \<ge> fls_conv_radius F"
by (induction n) (auto intro!: fls_conv_radius_mult_ge)
lemma eval_fls_0 [simp]: "eval_fls 0 z = 0"
and eval_fls_1 [simp]: "eval_fls 1 z = 1"
and eval_fls_const [simp]: "eval_fls (fls_const c) z = c"
and eval_fls_numeral [simp]: "eval_fls (numeral num) z = numeral num"
and eval_fls_of_nat [simp]: "eval_fls (of_nat n) z = of_nat n"
and eval_fls_of_int [simp]: "eval_fls (of_int m) z = of_int m"
and eval_fls_X [simp]: "eval_fls fls_X z = z"
and eval_fls_X_intpow [simp]: "eval_fls (fls_X_intpow m) z = z powi m"
by (simp_all add: eval_fls_def)
lemma eval_fls_at_0: "eval_fls F 0 = (if fls_subdegree F \<ge> 0 then fls_nth F 0 else 0)"
by (cases "fls_subdegree F = 0")
(simp_all add: eval_fls_def fls_regpart_def eval_fps_at_0)
lemma eval_fps_to_fls:
assumes "norm z < fps_conv_radius F"
shows "eval_fls (fps_to_fls F) z = eval_fps F z"
proof (cases "F = 0")
case [simp]: False
have "eval_fps F z = eval_fps (unit_factor F * normalize F) z"
by (metis unit_factor_mult_normalize)
also have "\<dots> = eval_fps (unit_factor F * fps_X ^ subdegree F) z"
by simp
also have "\<dots> = eval_fps (unit_factor F) z * z ^ subdegree F"
using assms by (subst eval_fps_mult) auto
also have "\<dots> = eval_fls (fps_to_fls F) z"
unfolding eval_fls_def fls_base_factor_to_fps_to_fls fls_subdegree_fls_to_fps
power_int_of_nat ..
finally show ?thesis ..
qed auto
lemma eval_fls_shift:
assumes [simp]: "z \<noteq> 0"
shows "eval_fls (fls_shift n F) z = eval_fls F z * z powi -n"
proof (cases "F = 0")
case [simp]: False
show ?thesis
unfolding eval_fls_def
by (subst fls_base_factor_to_fps_shift, subst fls_shift_subdegree[OF \<open>F \<noteq> 0\<close>], subst power_int_diff)
(auto simp: power_int_minus divide_simps)
qed auto
lemma eval_fls_add:
assumes "ereal (norm z) < fls_conv_radius F" "ereal (norm z) < fls_conv_radius G" "z \<noteq> 0"
shows "eval_fls (F + G) z = eval_fls F z + eval_fls G z"
using assms
proof (induction "fls_subdegree F" "fls_subdegree G" arbitrary: F G rule: linorder_wlog)
case (sym F G)
show ?case
using sym(1)[of G F] sym(2-) by (simp add: add_ac)
next
case (le F G)
show ?case
proof (cases "F = 0 \<or> G = 0")
case False
hence [simp]: "F \<noteq> 0" "G \<noteq> 0"
by auto
note [simp] = \<open>z \<noteq> 0\<close>
define F' G' where "F' = fls_base_factor_to_fps F" "G' = fls_base_factor_to_fps G"
define m n where "m = fls_subdegree F" "n = fls_subdegree G"
have "m \<le> n"
using le by (auto simp: m_n_def)
have conv1: "ereal (cmod z) < fps_conv_radius F'" "ereal (cmod z) < fps_conv_radius G'"
using assms le by (simp_all add: F'_G'_def fls_conv_radius_altdef)
have conv2: "ereal (cmod z) < fps_conv_radius (G' * fps_X ^ nat (n - m))"
using conv1 by (intro less_le_trans[OF _ fps_conv_radius_mult]) auto
have conv3: "ereal (cmod z) < fps_conv_radius (F' + G' * fps_X ^ nat (n - m))"
using conv1 conv2 by (intro less_le_trans[OF _ fps_conv_radius_add]) auto
have "eval_fls F z + eval_fls G z = eval_fps F' z * z powi m + eval_fps G' z * z powi n"
unfolding eval_fls_def m_n_def[symmetric] F'_G'_def[symmetric]
by (simp add: power_int_add algebra_simps)
also have "\<dots> = (eval_fps F' z + eval_fps G' z * z powi (n - m)) * z powi m"
by (simp add: algebra_simps power_int_diff)
also have "eval_fps G' z * z powi (n - m) = eval_fps (G' * fps_X ^ nat (n - m)) z"
using assms \<open>m \<le> n\<close> conv1 by (subst eval_fps_mult) (auto simp: power_int_def)
also have "eval_fps F' z + \<dots> = eval_fps (F' + G' * fps_X ^ nat (n - m)) z"
using conv1 conv2 by (subst eval_fps_add) auto
also have "\<dots> = eval_fls (fps_to_fls (F' + G' * fps_X ^ nat (n - m))) z"
using conv3 by (subst eval_fps_to_fls) auto
also have "\<dots> * z powi m = eval_fls (fls_shift (-m) (fps_to_fls (F' + G' * fps_X ^ nat (n - m)))) z"
by (subst eval_fls_shift) auto
also have "fls_shift (-m) (fps_to_fls (F' + G' * fps_X ^ nat (n - m))) = F + G"
using \<open>m \<le> n\<close>
by (simp add: fls_times_fps_to_fls fps_to_fls_power fls_X_power_conv_shift_1
fls_shifted_times_simps F'_G'_def m_n_def)
finally show ?thesis ..
qed auto
qed
lemma eval_fls_minus:
assumes "ereal (norm z) < fls_conv_radius F"
shows "eval_fls (-F) z = -eval_fls F z"
using assms by (simp add: eval_fls_def eval_fps_minus fls_conv_radius_altdef)
lemma eval_fls_diff:
assumes "ereal (norm z) < fls_conv_radius F" "ereal (norm z) < fls_conv_radius G"
and [simp]: "z \<noteq> 0"
shows "eval_fls (F - G) z = eval_fls F z - eval_fls G z"
proof -
have "eval_fls (F + (-G)) z = eval_fls F z - eval_fls G z"
using assms by (subst eval_fls_add) (auto simp: eval_fls_minus)
thus ?thesis
by simp
qed
lemma eval_fls_mult:
assumes "ereal (norm z) < fls_conv_radius F" "ereal (norm z) < fls_conv_radius G" "z \<noteq> 0"
shows "eval_fls (F * G) z = eval_fls F z * eval_fls G z"
proof (cases "F = 0 \<or> G = 0")
case False
hence [simp]: "F \<noteq> 0" "G \<noteq> 0"
by auto
note [simp] = \<open>z \<noteq> 0\<close>
define F' G' where "F' = fls_base_factor_to_fps F" "G' = fls_base_factor_to_fps G"
define m n where "m = fls_subdegree F" "n = fls_subdegree G"
have "eval_fls F z * eval_fls G z = (eval_fps F' z * eval_fps G' z) * z powi (m + n)"
unfolding eval_fls_def m_n_def[symmetric] F'_G'_def[symmetric]
by (simp add: power_int_add algebra_simps)
also have "\<dots> = eval_fps (F' * G') z * z powi (m + n)"
using assms by (subst eval_fps_mult) (auto simp: F'_G'_def fls_conv_radius_altdef)
also have "\<dots> = eval_fls (F * G) z"
by (simp add: eval_fls_def F'_G'_def m_n_def) (simp add: fls_times_def)
finally show ?thesis ..
qed auto
lemma eval_fls_power:
assumes "ereal (norm z) < fls_conv_radius F" "z \<noteq> 0"
shows "eval_fls (F ^ n) z = eval_fls F z ^ n"
proof (induction n)
case (Suc n)
have "eval_fls (F ^ Suc n) z = eval_fls (F * F ^ n) z"
by simp
also have "\<dots> = eval_fls F z * eval_fls (F ^ n) z"
using assms by (subst eval_fls_mult) (auto intro!: less_le_trans[OF _ fls_conv_radius_power])
finally show ?case
using Suc by simp
qed auto
lemma eval_fls_eq:
assumes "N \<le> fls_subdegree F" "fls_subdegree F \<ge> 0 \<or> z \<noteq> 0"
assumes "(\<lambda>n. fls_nth F (int n + N) * z powi (int n + N)) sums S"
shows "eval_fls F z = S"
proof (cases "z = 0")
case [simp]: True
have "(\<lambda>n. fls_nth F (int n + N) * z powi (int n + N)) =
(\<lambda>n. if n \<in> (if N \<le> 0 then {nat (-N)} else {}) then fls_nth F (int n + N) else 0)"
by (auto simp: fun_eq_iff split: if_splits)
also have "\<dots> sums (\<Sum>n\<in>(if N \<le> 0 then {nat (-N)} else {}). fls_nth F (int n + N))"
by (rule sums_If_finite_set) auto
also have "\<dots> = fls_nth F 0"
using assms by auto
also have "\<dots> = eval_fls F z"
using assms by (auto simp: eval_fls_def eval_fps_at_0 power_int_0_left_if)
finally show ?thesis
using assms by (simp add: sums_iff)
next
case [simp]: False
define N' where "N' = fls_subdegree F"
define d where "d = nat (N' - N)"
have "(\<lambda>n. fls_nth F (int n + N) * z powi (int n + N)) sums S"
by fact
also have "?this \<longleftrightarrow> (\<lambda>n. fls_nth F (int (n+d) + N) * z powi (int (n+d) + N)) sums S"
by (rule sums_zero_iff_shift [symmetric]) (use assms in \<open>auto simp: d_def N'_def\<close>)
also have "(\<lambda>n. int (n+d) + N) = (\<lambda>n. int n + N')"
using assms by (auto simp: N'_def d_def)
finally have "(\<lambda>n. fls_nth F (int n + N') * z powi (int n + N')) sums S" .
hence "(\<lambda>n. z powi (-N') * (fls_nth F (int n + N') * z powi (int n + N'))) sums (z powi (-N') * S)"
by (intro sums_mult)
hence "(\<lambda>n. fls_nth F (int n + N') * z ^ n) sums (z powi (-N') * S)"
by (simp add: power_int_add power_int_minus field_simps)
thus ?thesis
by (simp add: eval_fls_def eval_fps_def sums_iff power_int_minus N'_def)
qed
lemma norm_summable_fls:
"norm z < fls_conv_radius f \<Longrightarrow> summable (\<lambda>n. norm (fls_nth f n * z ^ n))"
using norm_summable_fps[of z "fls_regpart f"] by (simp add: fls_conv_radius_def)
lemma norm_summable_fls':
"norm z < fls_conv_radius f \<Longrightarrow> summable (\<lambda>n. norm (fls_nth f (n + fls_subdegree f) * z ^ n))"
using norm_summable_fps[of z "fls_base_factor_to_fps f"] by (simp add: fls_conv_radius_altdef)
lemma summable_fls:
"norm z < fls_conv_radius f \<Longrightarrow> summable (\<lambda>n. fls_nth f n * z ^ n)"
by (rule summable_norm_cancel[OF norm_summable_fls])
theorem sums_eval_fls:
fixes f
defines "n \<equiv> fls_subdegree f"
assumes "norm z < fls_conv_radius f" and "z \<noteq> 0 \<or> n \<ge> 0"
shows "(\<lambda>k. fls_nth f (int k + n) * z powi (int k + n)) sums eval_fls f z"
proof (cases "z = 0")
case [simp]: False
have "(\<lambda>k. fps_nth (fls_base_factor_to_fps f) k * z ^ k * z powi n) sums
(eval_fps (fls_base_factor_to_fps f) z * z powi n)"
using assms(2) by (intro sums_eval_fps sums_mult2) (auto simp: fls_conv_radius_altdef)
thus ?thesis
by (simp add: power_int_add n_def eval_fls_def mult_ac)
next
case [simp]: True
with assms have "n \<ge> 0"
by auto
have "(\<lambda>k. fls_nth f (int k + n) * z powi (int k + n)) sums
(\<Sum>k\<in>(if n \<le> 0 then {nat (-n)} else {}). fls_nth f (int k + n) * z powi (int k + n))"
by (intro sums_finite) (auto split: if_splits)
also have "\<dots> = eval_fls f z"
using \<open>n \<ge> 0\<close> by (auto simp: eval_fls_at_0 n_def not_le)
finally show ?thesis .
qed
lemma holomorphic_on_eval_fls:
fixes f
defines "n \<equiv> fls_subdegree f"
assumes "A \<subseteq> eball 0 (fls_conv_radius f) - (if n \<ge> 0 then {} else {0})"
shows "eval_fls f holomorphic_on A"
proof (cases "n \<ge> 0")
case True
have "eval_fls f = (\<lambda>z. eval_fps (fls_base_factor_to_fps f) z * z ^ nat n)"
using True by (simp add: fun_eq_iff eval_fls_def power_int_def n_def)
moreover have "\<dots> holomorphic_on A"
using True assms(2) by (intro holomorphic_intros) (auto simp: fls_conv_radius_altdef)
ultimately show ?thesis
by simp
next
case False
show ?thesis using assms
unfolding eval_fls_def by (intro holomorphic_intros) (auto simp: fls_conv_radius_altdef)
qed
lemma holomorphic_on_eval_fls' [holomorphic_intros]:
assumes "g holomorphic_on A"
assumes "g ` A \<subseteq> eball 0 (fls_conv_radius f) - (if fls_subdegree f \<ge> 0 then {} else {0})"
shows "(\<lambda>x. eval_fls f (g x)) holomorphic_on A"
by (meson assms holomorphic_on_compose holomorphic_on_eval_fls holomorphic_transform o_def)
lemma continuous_on_eval_fls:
fixes f
defines "n \<equiv> fls_subdegree f"
assumes "A \<subseteq> eball 0 (fls_conv_radius f) - (if n \<ge> 0 then {} else {0})"
shows "continuous_on A (eval_fls f)"
using assms holomorphic_on_eval_fls holomorphic_on_imp_continuous_on by blast
lemma continuous_on_eval_fls' [continuous_intros]:
fixes f
defines "n \<equiv> fls_subdegree f"
assumes "g ` A \<subseteq> eball 0 (fls_conv_radius f) - (if n \<ge> 0 then {} else {0})"
assumes "continuous_on A g"
shows "continuous_on A (\<lambda>x. eval_fls f (g x))"
by (metis assms continuous_on_compose2 continuous_on_eval_fls order.refl)
lemmas has_field_derivative_eval_fps' [derivative_intros] =
DERIV_chain2[OF has_field_derivative_eval_fps]
(* TODO: generalise for nonneg subdegree *)
lemma has_field_derivative_eval_fls:
assumes "z \<in> eball 0 (fls_conv_radius f) - {0}"
shows "(eval_fls f has_field_derivative eval_fls (fls_deriv f) z) (at z within A)"
proof -
define g where "g = fls_base_factor_to_fps f"
define n where "n = fls_subdegree f"
have [simp]: "fps_conv_radius g = fls_conv_radius f"
by (simp add: fls_conv_radius_altdef g_def)
have conv1: "fps_conv_radius (fps_deriv g * fps_X) \<ge> fls_conv_radius f"
by (intro fps_conv_radius_mult_ge order.trans[OF _ fps_conv_radius_deriv]) auto
have conv2: "fps_conv_radius (of_int n * g) \<ge> fls_conv_radius f"
by (intro fps_conv_radius_mult_ge) auto
have conv3: "fps_conv_radius (fps_deriv g * fps_X + of_int n * g) \<ge> fls_conv_radius f"
by (intro fps_conv_radius_add_ge conv1 conv2)
have [simp]: "fps_conv_radius g = fls_conv_radius f"
by (simp add: g_def fls_conv_radius_altdef)
have "((\<lambda>z. eval_fps g z * z powi fls_subdegree f) has_field_derivative
(eval_fps (fps_deriv g) z * z powi n + of_int n * z powi (n - 1) * eval_fps g z))
(at z within A)"
using assms by (auto intro!: derivative_eq_intros simp: n_def)
also have "(\<lambda>z. eval_fps g z * z powi fls_subdegree f) = eval_fls f"
by (simp add: eval_fls_def g_def fun_eq_iff)
also have "eval_fps (fps_deriv g) z * z powi n + of_int n * z powi (n - 1) * eval_fps g z =
(z * eval_fps (fps_deriv g) z + of_int n * eval_fps g z) * z powi (n - 1)"
using assms by (auto simp: power_int_diff field_simps)
also have "(z * eval_fps (fps_deriv g) z + of_int n * eval_fps g z) =
eval_fps (fps_deriv g * fps_X + of_int n * g) z"
using conv1 conv2 assms fps_conv_radius_deriv[of g]
by (subst eval_fps_add) (auto simp: eval_fps_mult)
also have "\<dots> = eval_fls (fps_to_fls (fps_deriv g * fps_X + of_int n * g)) z"
using conv3 assms by (subst eval_fps_to_fls) auto
also have "\<dots> * z powi (n - 1) = eval_fls (fls_shift (1 - n) (fps_to_fls (fps_deriv g * fps_X + of_int n * g))) z"
using assms by (subst eval_fls_shift) auto
also have "fls_shift (1 - n) (fps_to_fls (fps_deriv g * fps_X + of_int n * g)) = fls_deriv f"
by (intro fls_eqI) (auto simp: g_def n_def algebra_simps eq_commute[of _ "fls_subdegree f"])
finally show ?thesis .
qed
lemma eval_fls_deriv:
assumes "z \<in> eball 0 (fls_conv_radius F) - {0}"
shows "eval_fls (fls_deriv F) z = deriv (eval_fls F) z"
by (metis DERIV_imp_deriv assms has_field_derivative_eval_fls)
lemma analytic_on_eval_fls:
assumes "A \<subseteq> eball 0 (fls_conv_radius f) - (if fls_subdegree f \<ge> 0 then {} else {0})"
shows "eval_fls f analytic_on A"
proof (rule analytic_on_subset [OF _ assms])
show "eval_fls f analytic_on eball 0 (fls_conv_radius f) - (if fls_subdegree f \<ge> 0 then {} else {0})"
using holomorphic_on_eval_fls[OF order.refl]
by (subst analytic_on_open) auto
qed
lemma analytic_on_eval_fls' [analytic_intros]:
assumes "g analytic_on A"
assumes "g ` A \<subseteq> eball 0 (fls_conv_radius f) - (if fls_subdegree f \<ge> 0 then {} else {0})"
shows "(\<lambda>x. eval_fls f (g x)) analytic_on A"
proof -
have "eval_fls f \<circ> g analytic_on A"
by (intro analytic_on_compose[OF assms(1) analytic_on_eval_fls]) (use assms in auto)
thus ?thesis
by (simp add: o_def)
qed
lemma continuous_eval_fls [continuous_intros]:
assumes "z \<in> eball 0 (fls_conv_radius F) - (if fls_subdegree F \<ge> 0 then {} else {0})"
shows "continuous (at z within A) (eval_fls F)"
proof -
have "isCont (eval_fls F) z"
using continuous_on_eval_fls[OF order.refl] assms
by (subst (asm) continuous_on_eq_continuous_at) auto
thus ?thesis
using continuous_at_imp_continuous_at_within by blast
qed
named_theorems laurent_expansion_intros
lemma has_laurent_expansion_imp_asymp_equiv_0:
assumes F: "f has_laurent_expansion F"
defines "n \<equiv> fls_subdegree F"
shows "f \<sim>[at 0] (\<lambda>z. fls_nth F n * z powi n)"
proof (cases "F = 0")
case True
thus ?thesis using assms
by (auto simp: has_laurent_expansion_def)
next
case [simp]: False
define G where "G = fls_base_factor_to_fps F"
have "fls_conv_radius F > 0"
using F by (auto simp: has_laurent_expansion_def)
hence "isCont (eval_fps G) 0"
by (intro continuous_intros) (auto simp: G_def fps_conv_radius_fls_regpart zero_ereal_def)
hence lim: "eval_fps G \<midarrow>0\<rightarrow> eval_fps G 0"
by (meson isContD)
have [simp]: "fps_nth G 0 \<noteq> 0"
by (auto simp: G_def)
have "f \<sim>[at 0] eval_fls F"
using F by (intro asymp_equiv_refl_ev) (auto simp: has_laurent_expansion_def eq_commute)
also have "\<dots> = (\<lambda>z. eval_fps G z * z powi n)"
by (intro ext) (simp_all add: eval_fls_def G_def n_def)
also have "\<dots> \<sim>[at 0] (\<lambda>z. fps_nth G 0 * z powi n)" using lim
by (intro asymp_equiv_intros tendsto_imp_asymp_equiv_const) (auto simp: eval_fps_at_0)
also have "fps_nth G 0 = fls_nth F n"
by (simp add: G_def n_def)
finally show ?thesis
by simp
qed
lemma has_laurent_expansion_imp_asymp_equiv:
assumes F: "(\<lambda>w. f (z + w)) has_laurent_expansion F"
defines "n \<equiv> fls_subdegree F"
shows "f \<sim>[at z] (\<lambda>w. fls_nth F n * (w - z) powi n)"
using has_laurent_expansion_imp_asymp_equiv_0[OF assms(1)] unfolding n_def
by (simp add: at_to_0[of z] asymp_equiv_filtermap_iff add_ac)
lemmas [tendsto_intros del] = tendsto_power_int
lemma has_laurent_expansion_imp_tendsto_0:
assumes F: "f has_laurent_expansion F" and "fls_subdegree F \<ge> 0"
shows "f \<midarrow>0\<rightarrow> fls_nth F 0"
proof (rule asymp_equiv_tendsto_transfer)
show "(\<lambda>z. fls_nth F (fls_subdegree F) * z powi fls_subdegree F) \<sim>[at 0] f"
by (rule asymp_equiv_symI, rule has_laurent_expansion_imp_asymp_equiv_0) fact
show "(\<lambda>z. fls_nth F (fls_subdegree F) * z powi fls_subdegree F) \<midarrow>0\<rightarrow> fls_nth F 0"
by (rule tendsto_eq_intros refl | use assms(2) in simp)+
(use assms(2) in \<open>auto simp: power_int_0_left_if\<close>)
qed
lemma has_laurent_expansion_imp_tendsto:
assumes F: "(\<lambda>w. f (z + w)) has_laurent_expansion F" and "fls_subdegree F \<ge> 0"
shows "f \<midarrow>z\<rightarrow> fls_nth F 0"
using has_laurent_expansion_imp_tendsto_0[OF assms]
by (simp add: at_to_0[of z] filterlim_filtermap add_ac)
lemma has_laurent_expansion_imp_filterlim_infinity_0:
assumes F: "f has_laurent_expansion F" and "fls_subdegree F < 0"
shows "filterlim f at_infinity (at 0)"
proof (rule asymp_equiv_at_infinity_transfer)
have [simp]: "F \<noteq> 0"
using assms(2) by auto
show "(\<lambda>z. fls_nth F (fls_subdegree F) * z powi fls_subdegree F) \<sim>[at 0] f"
by (rule asymp_equiv_symI, rule has_laurent_expansion_imp_asymp_equiv_0) fact
show "filterlim (\<lambda>z. fls_nth F (fls_subdegree F) * z powi fls_subdegree F) at_infinity (at 0)"
by (rule tendsto_mult_filterlim_at_infinity tendsto_const
filterlim_power_int_neg_at_infinity | use assms(2) in simp)+
(auto simp: eventually_at_filter)
qed
lemma has_laurent_expansion_imp_neg_fls_subdegree:
assumes F: "f has_laurent_expansion F"
and infy:"filterlim f at_infinity (at 0)"
shows "fls_subdegree F < 0"
proof (rule ccontr)
assume asm:"\<not> fls_subdegree F < 0"
define ff where "ff=(\<lambda>z. fls_nth F (fls_subdegree F)
* z powi fls_subdegree F)"
have "(ff \<longlongrightarrow> (if fls_subdegree F =0 then fls_nth F 0 else 0)) (at 0)"
using asm unfolding ff_def
by (auto intro!: tendsto_eq_intros)
moreover have "filterlim ff at_infinity (at 0)"
proof (rule asymp_equiv_at_infinity_transfer)
show "f \<sim>[at 0] ff" unfolding ff_def
using has_laurent_expansion_imp_asymp_equiv_0[OF F] unfolding ff_def .
show "filterlim f at_infinity (at 0)" by fact
qed
ultimately show False
using not_tendsto_and_filterlim_at_infinity[of "at (0::complex)"] by auto
qed
lemma has_laurent_expansion_imp_filterlim_infinity:
assumes F: "(\<lambda>w. f (z + w)) has_laurent_expansion F" and "fls_subdegree F < 0"
shows "filterlim f at_infinity (at z)"
using has_laurent_expansion_imp_filterlim_infinity_0[OF assms]
by (simp add: at_to_0[of z] filterlim_filtermap add_ac)
lemma has_laurent_expansion_imp_is_pole_0:
assumes F: "f has_laurent_expansion F" and "fls_subdegree F < 0"
shows "is_pole f 0"
using has_laurent_expansion_imp_filterlim_infinity_0[OF assms]
by (simp add: is_pole_def)
lemma is_pole_0_imp_neg_fls_subdegree:
assumes F: "f has_laurent_expansion F" and "is_pole f 0"
shows "fls_subdegree F < 0"
using F assms(2) has_laurent_expansion_imp_neg_fls_subdegree is_pole_def
by blast
lemma has_laurent_expansion_imp_is_pole:
assumes F: "(\<lambda>x. f (z + x)) has_laurent_expansion F" and "fls_subdegree F < 0"
shows "is_pole f z"
using has_laurent_expansion_imp_is_pole_0[OF assms]
by (simp add: is_pole_shift_0')
lemma is_pole_imp_neg_fls_subdegree:
assumes F: "(\<lambda>x. f (z + x)) has_laurent_expansion F" and "is_pole f z"
shows "fls_subdegree F < 0"
proof -
have "is_pole (\<lambda>x. f (z + x)) 0"
using assms(2) is_pole_shift_0 by blast
then show ?thesis
using F is_pole_0_imp_neg_fls_subdegree by blast
qed
lemma is_pole_fls_subdegree_iff:
assumes "(\<lambda>x. f (z + x)) has_laurent_expansion F"
shows "is_pole f z \<longleftrightarrow> fls_subdegree F < 0"
using assms is_pole_imp_neg_fls_subdegree has_laurent_expansion_imp_is_pole
by auto
lemma
assumes "f has_laurent_expansion F"
shows has_laurent_expansion_isolated_0: "isolated_singularity_at f 0"
and has_laurent_expansion_not_essential_0: "not_essential f 0"
proof -
from assms have "eventually (\<lambda>z. eval_fls F z = f z) (at 0)"
by (auto simp: has_laurent_expansion_def)
then obtain r where r: "r > 0" "\<And>z. z \<in> ball 0 r - {0} \<Longrightarrow> eval_fls F z = f z"
by (auto simp: eventually_at_filter ball_def eventually_nhds_metric)
have "fls_conv_radius F > 0"
using assms by (auto simp: has_laurent_expansion_def)
then obtain R :: real where R: "R > 0" "R \<le> min r (fls_conv_radius F)"
using \<open>r > 0\<close> by (metis dual_order.strict_implies_order ereal_dense2 ereal_less(2) min_def)
have "eval_fls F holomorphic_on ball 0 R - {0}"
using r R by (intro holomorphic_intros ball_eball_mono Diff_mono) (auto simp: ereal_le_less)
also have "?this \<longleftrightarrow> f holomorphic_on ball 0 R - {0}"
using r R by (intro holomorphic_cong) auto
also have "\<dots> \<longleftrightarrow> f analytic_on ball 0 R - {0}"
by (subst analytic_on_open) auto
finally show "isolated_singularity_at f 0"
unfolding isolated_singularity_at_def using \<open>R > 0\<close> by blast
show "not_essential f 0"
proof (cases "fls_subdegree F \<ge> 0")
case True
hence "f \<midarrow>0\<rightarrow> fls_nth F 0"
by (intro has_laurent_expansion_imp_tendsto_0[OF assms])
thus ?thesis
by (auto simp: not_essential_def)
next
case False
hence "is_pole f 0"
by (intro has_laurent_expansion_imp_is_pole_0[OF assms]) auto
thus ?thesis
by (auto simp: not_essential_def)
qed
qed
lemma
assumes "(\<lambda>w. f (z + w)) has_laurent_expansion F"
shows has_laurent_expansion_isolated: "isolated_singularity_at f z"
and has_laurent_expansion_not_essential: "not_essential f z"
using has_laurent_expansion_isolated_0[OF assms] has_laurent_expansion_not_essential_0[OF assms]
by (simp_all add: isolated_singularity_at_shift_0 not_essential_shift_0)
lemma has_laurent_expansion_fps:
assumes "f has_fps_expansion F"
shows "f has_laurent_expansion fps_to_fls F"
proof -
from assms have radius: "0 < fps_conv_radius F" and eval: "\<forall>\<^sub>F z in nhds 0. eval_fps F z = f z"
by (auto simp: has_fps_expansion_def)
from eval have eval': "\<forall>\<^sub>F z in at 0. eval_fps F z = f z"
using eventually_at_filter eventually_mono by fastforce
moreover have "eventually (\<lambda>z. z \<in> eball 0 (fps_conv_radius F) - {0}) (at 0)"
using radius by (intro eventually_at_in_open) (auto simp: zero_ereal_def)
ultimately have "eventually (\<lambda>z. eval_fls (fps_to_fls F) z = f z) (at 0)"
by eventually_elim (auto simp: eval_fps_to_fls)
thus ?thesis using radius
by (auto simp: has_laurent_expansion_def)
qed
lemma has_laurent_expansion_const [simp, intro, laurent_expansion_intros]:
"(\<lambda>_. c) has_laurent_expansion fls_const c"
by (auto simp: has_laurent_expansion_def)
lemma has_laurent_expansion_0 [simp, intro, laurent_expansion_intros]:
"(\<lambda>_. 0) has_laurent_expansion 0"
by (auto simp: has_laurent_expansion_def)
lemma has_fps_expansion_0_iff: "f has_fps_expansion 0 \<longleftrightarrow> eventually (\<lambda>z. f z = 0) (nhds 0)"
by (auto simp: has_fps_expansion_def)
lemma has_laurent_expansion_1 [simp, intro, laurent_expansion_intros]:
"(\<lambda>_. 1) has_laurent_expansion 1"
by (auto simp: has_laurent_expansion_def)
lemma has_laurent_expansion_numeral [simp, intro, laurent_expansion_intros]:
"(\<lambda>_. numeral n) has_laurent_expansion numeral n"
by (auto simp: has_laurent_expansion_def)
lemma has_laurent_expansion_fps_X_power [laurent_expansion_intros]:
"(\<lambda>x. x ^ n) has_laurent_expansion (fls_X_intpow n)"
by (auto simp: has_laurent_expansion_def)
lemma has_laurent_expansion_fps_X_power_int [laurent_expansion_intros]:
"(\<lambda>x. x powi n) has_laurent_expansion (fls_X_intpow n)"
by (auto simp: has_laurent_expansion_def)
lemma has_laurent_expansion_fps_X [laurent_expansion_intros]:
"(\<lambda>x. x) has_laurent_expansion fls_X"
by (auto simp: has_laurent_expansion_def)
lemma has_laurent_expansion_cmult_left [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "(\<lambda>x. c * f x) has_laurent_expansion fls_const c * F"
proof -
from assms have "eventually (\<lambda>z. z \<in> eball 0 (fls_conv_radius F) - {0}) (at 0)"
by (intro eventually_at_in_open) (auto simp: has_laurent_expansion_def zero_ereal_def)
moreover from assms have "eventually (\<lambda>z. eval_fls F z = f z) (at 0)"
by (auto simp: has_laurent_expansion_def)
ultimately have "eventually (\<lambda>z. eval_fls (fls_const c * F) z = c * f z) (at 0)"
by eventually_elim (simp_all add: eval_fls_mult)
with assms show ?thesis
by (auto simp: has_laurent_expansion_def intro!: less_le_trans[OF _ fls_conv_radius_mult])
qed
lemma has_laurent_expansion_cmult_right [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "(\<lambda>x. f x * c) has_laurent_expansion F * fls_const c"
proof -
have "F * fls_const c = fls_const c * F"
by (intro fls_eqI) (auto simp: mult.commute)
with has_laurent_expansion_cmult_left [OF assms] show ?thesis
by (simp add: mult.commute)
qed
lemma has_fps_expansion_scaleR [fps_expansion_intros]:
fixes F :: "'a :: {banach, real_normed_div_algebra, comm_ring_1} fps"
shows "f has_fps_expansion F \<Longrightarrow> (\<lambda>x. c *\<^sub>R f x) has_fps_expansion fps_const (of_real c) * F"
unfolding scaleR_conv_of_real by (intro fps_expansion_intros)
lemma has_laurent_expansion_scaleR [laurent_expansion_intros]:
"f has_laurent_expansion F \<Longrightarrow> (\<lambda>x. c *\<^sub>R f x) has_laurent_expansion fls_const (of_real c) * F"
unfolding scaleR_conv_of_real by (intro laurent_expansion_intros)
lemma has_laurent_expansion_minus [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "(\<lambda>x. - f x) has_laurent_expansion -F"
proof -
from assms have "eventually (\<lambda>x. x \<in> eball 0 (fls_conv_radius F) - {0}) (at 0)"
by (intro eventually_at_in_open) (auto simp: has_laurent_expansion_def zero_ereal_def)
moreover from assms have "eventually (\<lambda>x. eval_fls F x = f x) (at 0)"
by (auto simp: has_laurent_expansion_def)
ultimately have "eventually (\<lambda>x. eval_fls (-F) x = -f x) (at 0)"
by eventually_elim (auto simp: eval_fls_minus)
thus ?thesis using assms by (auto simp: has_laurent_expansion_def)
qed
lemma has_laurent_expansion_add [laurent_expansion_intros]:
assumes "f has_laurent_expansion F" "g has_laurent_expansion G"
shows "(\<lambda>x. f x + g x) has_laurent_expansion F + G"
proof -
from assms have "0 < min (fls_conv_radius F) (fls_conv_radius G)"
by (auto simp: has_laurent_expansion_def)
also have "\<dots> \<le> fls_conv_radius (F + G)"
by (rule fls_conv_radius_add)
finally have radius: "\<dots> > 0" .
from assms have "eventually (\<lambda>x. x \<in> eball 0 (fls_conv_radius F) - {0}) (at 0)"
"eventually (\<lambda>x. x \<in> eball 0 (fls_conv_radius G) - {0}) (at 0)"
by (intro eventually_at_in_open; force simp: has_laurent_expansion_def zero_ereal_def)+
moreover have "eventually (\<lambda>x. eval_fls F x = f x) (at 0)"
and "eventually (\<lambda>x. eval_fls G x = g x) (at 0)"
using assms by (auto simp: has_laurent_expansion_def)
ultimately have "eventually (\<lambda>x. eval_fls (F + G) x = f x + g x) (at 0)"
by eventually_elim (auto simp: eval_fls_add)
with radius show ?thesis by (auto simp: has_laurent_expansion_def)
qed
lemma has_laurent_expansion_diff [laurent_expansion_intros]:
assumes "f has_laurent_expansion F" "g has_laurent_expansion G"
shows "(\<lambda>x. f x - g x) has_laurent_expansion F - G"
using has_laurent_expansion_add[of f F "\<lambda>x. - g x" "-G"] assms
by (simp add: has_laurent_expansion_minus)
lemma has_laurent_expansion_mult [laurent_expansion_intros]:
assumes "f has_laurent_expansion F" "g has_laurent_expansion G"
shows "(\<lambda>x. f x * g x) has_laurent_expansion F * G"
proof -
from assms have "0 < min (fls_conv_radius F) (fls_conv_radius G)"
by (auto simp: has_laurent_expansion_def)
also have "\<dots> \<le> fls_conv_radius (F * G)"
by (rule fls_conv_radius_mult)
finally have radius: "\<dots> > 0" .
from assms have "eventually (\<lambda>x. x \<in> eball 0 (fls_conv_radius F) - {0}) (at 0)"
"eventually (\<lambda>x. x \<in> eball 0 (fls_conv_radius G) - {0}) (at 0)"
by (intro eventually_at_in_open; force simp: has_laurent_expansion_def zero_ereal_def)+
moreover have "eventually (\<lambda>x. eval_fls F x = f x) (at 0)"
and "eventually (\<lambda>x. eval_fls G x = g x) (at 0)"
using assms by (auto simp: has_laurent_expansion_def)
ultimately have "eventually (\<lambda>x. eval_fls (F * G) x = f x * g x) (at 0)"
by eventually_elim (auto simp: eval_fls_mult)
with radius show ?thesis by (auto simp: has_laurent_expansion_def)
qed
lemma has_fps_expansion_power [fps_expansion_intros]:
fixes F :: "'a :: {banach, real_normed_div_algebra, comm_ring_1} fps"
shows "f has_fps_expansion F \<Longrightarrow> (\<lambda>x. f x ^ m) has_fps_expansion F ^ m"
by (induction m) (auto intro!: fps_expansion_intros)
lemma has_laurent_expansion_power [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "(\<lambda>x. f x ^ n) has_laurent_expansion F ^ n"
by (induction n) (auto intro!: laurent_expansion_intros assms)
lemma has_laurent_expansion_sum [laurent_expansion_intros]:
assumes "\<And>x. x \<in> I \<Longrightarrow> f x has_laurent_expansion F x"
shows "(\<lambda>y. \<Sum>x\<in>I. f x y) has_laurent_expansion (\<Sum>x\<in>I. F x)"
using assms by (induction I rule: infinite_finite_induct) (auto intro!: laurent_expansion_intros)
lemma has_laurent_expansion_prod [laurent_expansion_intros]:
assumes "\<And>x. x \<in> I \<Longrightarrow> f x has_laurent_expansion F x"
shows "(\<lambda>y. \<Prod>x\<in>I. f x y) has_laurent_expansion (\<Prod>x\<in>I. F x)"
using assms by (induction I rule: infinite_finite_induct) (auto intro!: laurent_expansion_intros)
lemma has_laurent_expansion_deriv [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "deriv f has_laurent_expansion fls_deriv F"
proof -
have "eventually (\<lambda>z. z \<in> eball 0 (fls_conv_radius F) - {0}) (at 0)"
using assms by (intro eventually_at_in_open)
(auto simp: has_laurent_expansion_def zero_ereal_def)
moreover from assms have "eventually (\<lambda>z. eval_fls F z = f z) (at 0)"
by (auto simp: has_laurent_expansion_def)
then obtain s where "open s" "0 \<in> s" and s: "\<And>w. w \<in> s - {0} \<Longrightarrow> eval_fls F w = f w"
by (auto simp: eventually_nhds eventually_at_filter)
hence "eventually (\<lambda>w. w \<in> s - {0}) (at 0)"
by (intro eventually_at_in_open) auto
ultimately have "eventually (\<lambda>z. eval_fls (fls_deriv F) z = deriv f z) (at 0)"
proof eventually_elim
case (elim z)
hence "eval_fls (fls_deriv F) z = deriv (eval_fls F) z"
by (simp add: eval_fls_deriv)
also have "eventually (\<lambda>w. w \<in> s - {0}) (nhds z)"
using elim and \<open>open s\<close> by (intro eventually_nhds_in_open) auto
hence "eventually (\<lambda>w. eval_fls F w = f w) (nhds z)"
by eventually_elim (use s in auto)
hence "deriv (eval_fls F) z = deriv f z"
by (intro deriv_cong_ev refl)
finally show ?case .
qed
with assms show ?thesis
by (auto simp: has_laurent_expansion_def intro!: less_le_trans[OF _ fls_conv_radius_deriv])
qed
lemma has_laurent_expansion_shift [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "(\<lambda>x. f x * x powi n) has_laurent_expansion (fls_shift (-n) F)"
proof -
have "eventually (\<lambda>x. x \<in> eball 0 (fls_conv_radius F) - {0}) (at 0)"
using assms by (intro eventually_at_in_open) (auto simp: has_laurent_expansion_def zero_ereal_def)
moreover have "eventually (\<lambda>x. eval_fls F x = f x) (at 0)"
using assms by (auto simp: has_laurent_expansion_def)
ultimately have "eventually (\<lambda>x. eval_fls (fls_shift (-n) F) x = f x * x powi n) (at 0)"
by eventually_elim (auto simp: eval_fls_shift assms)
with assms show ?thesis by (auto simp: has_laurent_expansion_def)
qed
lemma has_laurent_expansion_shift' [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "(\<lambda>x. f x * x powi (-n)) has_laurent_expansion (fls_shift n F)"
using has_laurent_expansion_shift[OF assms, of "-n"] by simp
lemma has_laurent_expansion_deriv':
assumes "f has_laurent_expansion F"
assumes "open A" "0 \<in> A" "\<And>x. x \<in> A - {0} \<Longrightarrow> (f has_field_derivative f' x) (at x)"
shows "f' has_laurent_expansion fls_deriv F"
proof -
have "deriv f has_laurent_expansion fls_deriv F"
by (intro laurent_expansion_intros assms)
also have "?this \<longleftrightarrow> ?thesis"
proof (intro has_laurent_expansion_cong refl)
have "eventually (\<lambda>z. z \<in> A - {0}) (at 0)"
by (intro eventually_at_in_open assms)
thus "eventually (\<lambda>z. deriv f z = f' z) (at 0)"
by eventually_elim (auto intro!: DERIV_imp_deriv assms)
qed
finally show ?thesis .
qed
definition laurent_expansion :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex fls" where
"laurent_expansion f z =
(if eventually (\<lambda>z. f z = 0) (at z) then 0
else fls_shift (-zorder f z) (fps_to_fls (fps_expansion (zor_poly f z) z)))"
lemma laurent_expansion_cong:
assumes "eventually (\<lambda>w. f w = g w) (at z)" "z = z'"
shows "laurent_expansion f z = laurent_expansion g z'"
unfolding laurent_expansion_def
using zor_poly_cong[OF assms(1,2)] zorder_cong[OF assms] assms
by (intro if_cong refl) (auto elim: eventually_elim2)
theorem not_essential_has_laurent_expansion_0:
assumes "isolated_singularity_at f 0" "not_essential f 0"
shows "f has_laurent_expansion laurent_expansion f 0"
proof (cases "\<exists>\<^sub>F w in at 0. f w \<noteq> 0")
case False
have "(\<lambda>_. 0) has_laurent_expansion 0"
by simp
also have "?this \<longleftrightarrow> f has_laurent_expansion 0"
using False by (intro has_laurent_expansion_cong) (auto simp: frequently_def)
finally show ?thesis
using False by (simp add: laurent_expansion_def frequently_def)
next
case True
define n where "n = zorder f 0"
obtain r where r: "zor_poly f 0 0 \<noteq> 0" "zor_poly f 0 holomorphic_on cball 0 r" "r > 0"
"\<forall>w\<in>cball 0 r - {0}. f w = zor_poly f 0 w * w powi n \<and>
zor_poly f 0 w \<noteq> 0"
using zorder_exist[OF assms True] unfolding n_def by auto
have holo: "zor_poly f 0 holomorphic_on ball 0 r"
by (rule holomorphic_on_subset[OF r(2)]) auto
define F where "F = fps_expansion (zor_poly f 0) 0"
have F: "zor_poly f 0 has_fps_expansion F"
unfolding F_def by (rule has_fps_expansion_fps_expansion[OF _ _ holo]) (use \<open>r > 0\<close> in auto)
have "(\<lambda>z. zor_poly f 0 z * z powi n) has_laurent_expansion fls_shift (-n) (fps_to_fls F)"
by (intro laurent_expansion_intros has_laurent_expansion_fps[OF F])
also have "?this \<longleftrightarrow> f has_laurent_expansion fls_shift (-n) (fps_to_fls F)"
by (intro has_laurent_expansion_cong refl eventually_mono[OF eventually_at_in_open[of "ball 0 r"]])
(use r in \<open>auto simp: complex_powr_of_int\<close>)
finally show ?thesis using True
by (simp add: laurent_expansion_def F_def n_def frequently_def)
qed
lemma not_essential_has_laurent_expansion:
assumes "isolated_singularity_at f z" "not_essential f z"
shows "(\<lambda>x. f (z + x)) has_laurent_expansion laurent_expansion f z"
proof -
from assms(1) have iso:"isolated_singularity_at (\<lambda>x. f (z + x)) 0"
by (simp add: isolated_singularity_at_shift_0)
moreover from assms(2) have ness:"not_essential (\<lambda>x. f (z + x)) 0"
by (simp add: not_essential_shift_0)
ultimately have "(\<lambda>x. f (z + x)) has_laurent_expansion laurent_expansion (\<lambda>x. f (z + x)) 0"
by (rule not_essential_has_laurent_expansion_0)
also have "\<dots> = laurent_expansion f z"
proof (cases "\<exists>\<^sub>F w in at z. f w \<noteq> 0")
case False
then have "\<forall>\<^sub>F w in at z. f w = 0" using not_frequently by force
then have "laurent_expansion (\<lambda>x. f (z + x)) 0 = 0"
by (smt (verit, best) add.commute eventually_at_to_0 eventually_mono
laurent_expansion_def)
moreover have "laurent_expansion f z = 0"
using \<open>\<forall>\<^sub>F w in at z. f w = 0\<close> unfolding laurent_expansion_def by auto
ultimately show ?thesis by auto
next
case True
define df where "df=zor_poly (\<lambda>x. f (z + x)) 0"
define g where "g=(\<lambda>u. u-z)"
have "fps_expansion df 0
= fps_expansion (df o g) z"
proof -
have "\<exists>\<^sub>F w in at 0. f (z + w) \<noteq> 0" using True
by (smt (verit, best) add.commute eventually_at_to_0
eventually_mono not_frequently)
from zorder_exist[OF iso ness this,folded df_def]
obtain r where "r>0" and df_holo:"df holomorphic_on cball 0 r" and "df 0 \<noteq> 0"
"\<forall>w\<in>cball 0 r - {0}.
f (z + w) = df w * w powi (zorder (\<lambda>w. f (z + w)) 0) \<and>
df w \<noteq> 0"
by auto
then have df_nz:"\<forall>w\<in>ball 0 r. df w\<noteq>0" by auto
have "(deriv ^^ n) df 0 = (deriv ^^ n) (df \<circ> g) z" for n
unfolding comp_def g_def
proof (subst higher_deriv_compose_linear'[where u=1 and c="-z",simplified])
show "df holomorphic_on ball 0 r"
using df_holo by auto
show "open (ball z r)" "open (ball 0 r)" "z \<in> ball z r"
using \<open>r>0\<close> by auto
show " \<And>w. w \<in> ball z r \<Longrightarrow> w - z \<in> ball 0 r"
by (simp add: dist_norm)
qed auto
then show ?thesis
unfolding fps_expansion_def by auto
qed
also have "... = fps_expansion (zor_poly f z) z"
proof (rule fps_expansion_cong)
have "\<forall>\<^sub>F w in nhds z. zor_poly f z w
= zor_poly (\<lambda>u. f (z + u)) 0 (w - z)"
apply (rule zor_poly_shift)
using True assms by auto
then show "\<forall>\<^sub>F w in nhds z. (df \<circ> g) w = zor_poly f z w"
unfolding df_def g_def comp_def
by (auto elim:eventually_mono)
qed
finally show ?thesis unfolding df_def
by (auto simp: laurent_expansion_def at_to_0[of z]
eventually_filtermap add_ac zorder_shift')
qed
finally show ?thesis .
qed
lemma has_fps_expansion_to_laurent:
"f has_fps_expansion F \<longleftrightarrow> f has_laurent_expansion fps_to_fls F \<and> f 0 = fps_nth F 0"
proof safe
assume *: "f has_laurent_expansion fps_to_fls F" "f 0 = fps_nth F 0"
have "eventually (\<lambda>z. z \<in> eball 0 (fps_conv_radius F)) (nhds 0)"
using * by (intro eventually_nhds_in_open) (auto simp: has_laurent_expansion_def zero_ereal_def)
moreover have "eventually (\<lambda>z. z \<noteq> 0 \<longrightarrow> eval_fls (fps_to_fls F) z = f z) (nhds 0)"
using * by (auto simp: has_laurent_expansion_def eventually_at_filter)
ultimately have "eventually (\<lambda>z. f z = eval_fps F z) (nhds 0)"
by eventually_elim
(auto simp: has_laurent_expansion_def eventually_at_filter eval_fps_at_0 eval_fps_to_fls *(2))
thus "f has_fps_expansion F"
using * by (auto simp: has_fps_expansion_def has_laurent_expansion_def eq_commute)
next
assume "f has_fps_expansion F"
thus "f 0 = fps_nth F 0"
by (metis eval_fps_at_0 has_fps_expansion_imp_holomorphic)
qed (auto intro: has_laurent_expansion_fps)
lemma eval_fps_fls_base_factor [simp]:
assumes "z \<noteq> 0"
shows "eval_fps (fls_base_factor_to_fps F) z = eval_fls F z * z powi -fls_subdegree F"
using assms unfolding eval_fls_def by (simp add: power_int_minus field_simps)
lemma has_fps_expansion_imp_analytic_0:
assumes "f has_fps_expansion F"
shows "f analytic_on {0}"
by (meson analytic_at_two assms has_fps_expansion_imp_holomorphic)
lemma has_fps_expansion_imp_analytic:
assumes "(\<lambda>x. f (z + x)) has_fps_expansion F"
shows "f analytic_on {z}"
proof -
have "(\<lambda>x. f (z + x)) analytic_on {0}"
by (rule has_fps_expansion_imp_analytic_0) fact
hence "(\<lambda>x. f (z + x)) \<circ> (\<lambda>x. x - z) analytic_on {z}"
by (intro analytic_on_compose_gen analytic_intros) auto
thus ?thesis
by (simp add: o_def)
qed
lemma is_pole_cong_asymp_equiv:
assumes "f \<sim>[at z] g" "z = z'"
shows "is_pole f z = is_pole g z'"
using asymp_equiv_at_infinity_transfer[OF assms(1)]
asymp_equiv_at_infinity_transfer[OF asymp_equiv_symI[OF assms(1)]] assms(2)
unfolding is_pole_def by auto
lemma not_is_pole_const [simp]: "\<not>is_pole (\<lambda>_::'a::perfect_space. c :: complex) z"
using not_tendsto_and_filterlim_at_infinity[of "at z" "\<lambda>_::'a. c" c] by (auto simp: is_pole_def)
lemma has_laurent_expansion_imp_is_pole_iff:
assumes F: "(\<lambda>x. f (z + x)) has_laurent_expansion F"
shows "is_pole f z \<longleftrightarrow> fls_subdegree F < 0"
proof
assume pole: "is_pole f z"
have [simp]: "F \<noteq> 0"
proof
assume "F = 0"
hence "is_pole f z \<longleftrightarrow> is_pole (\<lambda>_. 0 :: complex) z" using assms
by (intro is_pole_cong)
(auto simp: has_laurent_expansion_def at_to_0[of z] eventually_filtermap add_ac)
with pole show False
by simp
qed
note pole
also have "is_pole f z \<longleftrightarrow>
is_pole (\<lambda>w. fls_nth F (fls_subdegree F) * (w - z) powi fls_subdegree F) z"
using has_laurent_expansion_imp_asymp_equiv[OF F] by (intro is_pole_cong_asymp_equiv refl)
also have "\<dots> \<longleftrightarrow> is_pole (\<lambda>w. (w - z) powi fls_subdegree F) z"
by simp
finally have pole': \<dots> .
have False if "fls_subdegree F \<ge> 0"
proof -
have "(\<lambda>w. (w - z) powi fls_subdegree F) holomorphic_on UNIV"
using that by (intro holomorphic_intros) auto
hence "\<not>is_pole (\<lambda>w. (w - z) powi fls_subdegree F) z"
by (meson UNIV_I not_is_pole_holomorphic open_UNIV)
with pole' show False
by simp
qed
thus "fls_subdegree F < 0"
by force
qed (use has_laurent_expansion_imp_is_pole[OF assms] in auto)
lemma analytic_at_imp_has_fps_expansion_0:
assumes "f analytic_on {0}"
shows "f has_fps_expansion fps_expansion f 0"
using assms has_fps_expansion_fps_expansion analytic_at by fast
lemma analytic_at_imp_has_fps_expansion:
assumes "f analytic_on {z}"
shows "(\<lambda>x. f (z + x)) has_fps_expansion fps_expansion f z"
proof -
have "f \<circ> (\<lambda>x. z + x) analytic_on {0}"
by (intro analytic_on_compose_gen[OF _ assms] analytic_intros) auto
hence "(f \<circ> (\<lambda>x. z + x)) has_fps_expansion fps_expansion (f \<circ> (\<lambda>x. z + x)) 0"
unfolding o_def by (intro analytic_at_imp_has_fps_expansion_0) auto
also have "\<dots> = fps_expansion f z"
by (simp add: fps_expansion_def higher_deriv_shift_0')
finally show ?thesis by (simp add: add_ac)
qed
lemma has_laurent_expansion_zorder_0:
assumes "f has_laurent_expansion F" "F \<noteq> 0"
shows "zorder f 0 = fls_subdegree F"
proof -
define G where "G = fls_base_factor_to_fps F"
from assms obtain A where A: "0 \<in> A" "open A" "\<And>x. x \<in> A - {0} \<Longrightarrow> eval_fls F x = f x"
unfolding has_laurent_expansion_def eventually_at_filter eventually_nhds
by blast
show ?thesis
proof (rule zorder_eqI)
show "open (A \<inter> eball 0 (fls_conv_radius F))" "0 \<in> A \<inter> eball 0 (fls_conv_radius F)"
using assms A by (auto simp: has_laurent_expansion_def zero_ereal_def)
show "eval_fps G holomorphic_on A \<inter> eball 0 (fls_conv_radius F)"
by (intro holomorphic_intros) (auto simp: fls_conv_radius_altdef G_def)
show "eval_fps G 0 \<noteq> 0" using \<open>F \<noteq> 0\<close>
by (auto simp: eval_fps_at_0 G_def)
next
fix w :: complex assume "w \<in> A \<inter> eball 0 (fls_conv_radius F)" "w \<noteq> 0"
thus "f w = eval_fps G w * (w - 0) powi (fls_subdegree F)"
using A unfolding G_def
by (subst eval_fps_fls_base_factor)
(auto simp: complex_powr_of_int power_int_minus field_simps)
qed
qed
lemma has_laurent_expansion_zorder:
assumes "(\<lambda>w. f (z + w)) has_laurent_expansion F" "F \<noteq> 0"
shows "zorder f z = fls_subdegree F"
using has_laurent_expansion_zorder_0[OF assms] by (simp add: zorder_shift' add_ac)
lemma has_fps_expansion_zorder_0:
assumes "f has_fps_expansion F" "F \<noteq> 0"
shows "zorder f 0 = int (subdegree F)"
using assms has_laurent_expansion_zorder_0[of f "fps_to_fls F"]
by (auto simp: has_fps_expansion_to_laurent fls_subdegree_fls_to_fps)
lemma has_fps_expansion_zorder:
assumes "(\<lambda>w. f (z + w)) has_fps_expansion F" "F \<noteq> 0"
shows "zorder f z = int (subdegree F)"
using has_fps_expansion_zorder_0[OF assms]
by (simp add: zorder_shift' add_ac)
lemma has_fps_expansion_fls_base_factor_to_fps:
assumes "f has_laurent_expansion F"
defines "n \<equiv> fls_subdegree F"
defines "c \<equiv> fps_nth (fls_base_factor_to_fps F) 0"
shows "(\<lambda>z. if z = 0 then c else f z * z powi -n) has_fps_expansion fls_base_factor_to_fps F"
proof -
have "(\<lambda>z. f z * z powi -n) has_laurent_expansion fls_shift (-(-n)) F"
by (intro laurent_expansion_intros assms)
also have "fls_shift (-(-n)) F = fps_to_fls (fls_base_factor_to_fps F)"
by (simp add: n_def fls_shift_nonneg_subdegree)
also have "(\<lambda>z. f z * z powi - n) has_laurent_expansion fps_to_fls (fls_base_factor_to_fps F) \<longleftrightarrow>
(\<lambda>z. if z = 0 then c else f z * z powi -n) has_laurent_expansion fps_to_fls (fls_base_factor_to_fps F)"
by (intro has_laurent_expansion_cong) (auto simp: eventually_at_filter)
also have "\<dots> \<longleftrightarrow> (\<lambda>z. if z = 0 then c else f z * z powi -n) has_fps_expansion fls_base_factor_to_fps F"
by (subst has_fps_expansion_to_laurent) (auto simp: c_def)
finally show ?thesis .
qed
lemma zero_has_laurent_expansion_imp_eq_0:
assumes "(\<lambda>_. 0) has_laurent_expansion F"
shows "F = 0"
proof -
have "at (0 :: complex) \<noteq> bot"
by auto
moreover have "(\<lambda>z. if z = 0 then fls_nth F (fls_subdegree F) else 0) has_fps_expansion
fls_base_factor_to_fps F" (is "?f has_fps_expansion _")
using has_fps_expansion_fls_base_factor_to_fps[OF assms] by (simp cong: if_cong)
hence "isCont ?f 0"
using has_fps_expansion_imp_continuous by blast
hence "?f \<midarrow>0\<rightarrow> fls_nth F (fls_subdegree F)"
by (auto simp: isCont_def)
moreover have "?f \<midarrow>0\<rightarrow> 0 \<longleftrightarrow> (\<lambda>_::complex. 0 :: complex) \<midarrow>0\<rightarrow> 0"
by (intro filterlim_cong) (auto simp: eventually_at_filter)
hence "?f \<midarrow>0\<rightarrow> 0"
by simp
ultimately have "fls_nth F (fls_subdegree F) = 0"
by (rule tendsto_unique)
thus ?thesis
by (meson nth_fls_subdegree_nonzero)
qed
lemma has_laurent_expansion_unique:
assumes "f has_laurent_expansion F" "f has_laurent_expansion G"
shows "F = G"
proof -
from assms have "(\<lambda>x. f x - f x) has_laurent_expansion F - G"
by (intro laurent_expansion_intros)
hence "(\<lambda>_. 0) has_laurent_expansion F - G"
by simp
hence "F - G = 0"
by (rule zero_has_laurent_expansion_imp_eq_0)
thus ?thesis
by simp
qed
lemma laurent_expansion_eqI:
assumes "(\<lambda>x. f (z + x)) has_laurent_expansion F"
shows "laurent_expansion f z = F"
using assms has_laurent_expansion_isolated has_laurent_expansion_not_essential
has_laurent_expansion_unique not_essential_has_laurent_expansion by blast
lemma laurent_expansion_0_eqI:
assumes "f has_laurent_expansion F"
shows "laurent_expansion f 0 = F"
using assms laurent_expansion_eqI[of f 0] by simp
lemma has_laurent_expansion_nonzero_imp_eventually_nonzero:
assumes "f has_laurent_expansion F" "F \<noteq> 0"
shows "eventually (\<lambda>x. f x \<noteq> 0) (at 0)"
proof (rule ccontr)
assume "\<not>eventually (\<lambda>x. f x \<noteq> 0) (at 0)"
with assms have "eventually (\<lambda>x. f x = 0) (at 0)"
by (intro not_essential_frequently_0_imp_eventually_0 has_laurent_expansion_isolated
has_laurent_expansion_not_essential)
(auto simp: frequently_def)
hence "(f has_laurent_expansion 0) \<longleftrightarrow> ((\<lambda>_. 0) has_laurent_expansion 0)"
by (intro has_laurent_expansion_cong) auto
hence "f has_laurent_expansion 0"
by simp
with assms(1) have "F = 0"
using has_laurent_expansion_unique by blast
with \<open>F \<noteq> 0\<close> show False
by contradiction
qed
lemma has_laurent_expansion_eventually_nonzero_iff':
assumes "f has_laurent_expansion F"
shows "eventually (\<lambda>x. f x \<noteq> 0) (at 0) \<longleftrightarrow> F \<noteq> 0 "
proof
assume "\<forall>\<^sub>F x in at 0. f x \<noteq> 0"
moreover have "\<not> (\<forall>\<^sub>F x in at 0. f x \<noteq> 0)" if "F=0"
proof -
have "\<forall>\<^sub>F x in at 0. f x = 0"
using assms that unfolding has_laurent_expansion_def by simp
then show ?thesis unfolding not_eventually
by (auto elim:eventually_frequentlyE)
qed
ultimately show "F \<noteq> 0" by auto
qed (simp add:has_laurent_expansion_nonzero_imp_eventually_nonzero[OF assms])
lemma has_laurent_expansion_eventually_nonzero_iff:
assumes "(\<lambda>w. f (z+w)) has_laurent_expansion F"
shows "eventually (\<lambda>x. f x \<noteq> 0) (at z) \<longleftrightarrow> F \<noteq> 0"
apply (subst eventually_at_to_0)
apply (rule has_laurent_expansion_eventually_nonzero_iff')
using assms by (simp add:add.commute)
lemma has_laurent_expansion_inverse [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "(\<lambda>x. inverse (f x)) has_laurent_expansion inverse F"
proof (cases "F = 0")
case True
thus ?thesis using assms
by (auto simp: has_laurent_expansion_def)
next
case False
define G where "G = laurent_expansion (\<lambda>x. inverse (f x)) 0"
from False have ev: "eventually (\<lambda>z. f z \<noteq> 0) (at 0)"
by (intro has_laurent_expansion_nonzero_imp_eventually_nonzero[OF assms])
have *: "(\<lambda>x. inverse (f x)) has_laurent_expansion G" unfolding G_def
by (intro not_essential_has_laurent_expansion_0 isolated_singularity_at_inverse not_essential_inverse
has_laurent_expansion_isolated_0[OF assms] has_laurent_expansion_not_essential_0[OF assms])
have "(\<lambda>x. f x * inverse (f x)) has_laurent_expansion F * G"
by (intro laurent_expansion_intros assms *)
also have "?this \<longleftrightarrow> (\<lambda>x. 1) has_laurent_expansion F * G"
by (intro has_laurent_expansion_cong refl eventually_mono[OF ev]) auto
finally have "(\<lambda>_. 1) has_laurent_expansion F * G" .
moreover have "(\<lambda>_. 1) has_laurent_expansion 1"
by simp
ultimately have "F * G = 1"
using has_laurent_expansion_unique by blast
hence "G = inverse F"
using inverse_unique by blast
with * show ?thesis
by simp
qed
lemma has_laurent_expansion_power_int [laurent_expansion_intros]:
"f has_laurent_expansion F \<Longrightarrow> (\<lambda>x. f x powi n) has_laurent_expansion (F powi n)"
by (auto simp: power_int_def intro!: laurent_expansion_intros)
lemma has_fps_expansion_0_analytic_continuation:
assumes "f has_fps_expansion 0" "f holomorphic_on A"
assumes "open A" "connected A" "0 \<in> A" "x \<in> A"
shows "f x = 0"
proof -
have "eventually (\<lambda>z. z \<in> A \<and> f z = 0) (nhds 0)" using assms
by (intro eventually_conj eventually_nhds_in_open) (auto simp: has_fps_expansion_def)
then obtain B where B: "open B" "0 \<in> B" "\<forall>z\<in>B. z \<in> A \<and> f z = 0"
unfolding eventually_nhds by blast
show ?thesis
proof (rule analytic_continuation_open[where f = f and g = "\<lambda>_. 0"])
show "B \<noteq> {}"
using \<open>open B\<close> B by auto
show "connected A"
using assms by auto
qed (use assms B in auto)
qed
lemma has_laurent_expansion_0_analytic_continuation:
assumes "f has_laurent_expansion 0" "f holomorphic_on A - {0}"
assumes "open A" "connected A" "0 \<in> A" "x \<in> A - {0}"
shows "f x = 0"
proof -
have "eventually (\<lambda>z. z \<in> A - {0} \<and> f z = 0) (at 0)" using assms
by (intro eventually_conj eventually_at_in_open) (auto simp: has_laurent_expansion_def)
then obtain B where B: "open B" "0 \<in> B" "\<forall>z\<in>B - {0}. z \<in> A - {0} \<and> f z = 0"
unfolding eventually_at_filter eventually_nhds by blast
show ?thesis
proof (rule analytic_continuation_open[where f = f and g = "\<lambda>_. 0"])
show "B - {0} \<noteq> {}"
using \<open>open B\<close> \<open>0 \<in> B\<close> by (metis insert_Diff not_open_singleton)
show "connected (A - {0})"
using assms by (intro connected_open_delete) auto
qed (use assms B in auto)
qed
lemma has_fps_expansion_cong:
assumes "eventually (\<lambda>x. f x = g x) (nhds 0)" "F = G"
shows "f has_fps_expansion F \<longleftrightarrow> g has_fps_expansion G"
using assms(2) by (auto simp: has_fps_expansion_def elim!: eventually_elim2[OF assms(1)])
lemma zor_poly_has_fps_expansion:
assumes "f has_laurent_expansion F" "F \<noteq> 0"
shows "zor_poly f 0 has_fps_expansion fls_base_factor_to_fps F"
proof -
note [simp] = \<open>F \<noteq> 0\<close>
have "eventually (\<lambda>z. f z \<noteq> 0) (at 0)"
by (rule has_laurent_expansion_nonzero_imp_eventually_nonzero[OF assms])
hence freq: "frequently (\<lambda>z. f z \<noteq> 0) (at 0)"
by (rule eventually_frequently[rotated]) auto
have *: "isolated_singularity_at f 0" "not_essential f 0"
using has_laurent_expansion_isolated_0[OF assms(1)] has_laurent_expansion_not_essential_0[OF assms(1)]
by auto
define G where "G = fls_base_factor_to_fps F"
define n where "n = zorder f 0"
have n_altdef: "n = fls_subdegree F"
using has_laurent_expansion_zorder_0 [OF assms(1)] by (simp add: n_def)
obtain r where r: "zor_poly f 0 0 \<noteq> 0" "zor_poly f 0 holomorphic_on cball 0 r" "r > 0"
"\<forall>w\<in>cball 0 r - {0}. f w = zor_poly f 0 w * w powi n \<and>
zor_poly f 0 w \<noteq> 0"
using zorder_exist[OF * freq] unfolding n_def by auto
obtain r' where r': "r' > 0" "\<forall>x\<in>ball 0 r'-{0}. eval_fls F x = f x"
using assms(1) unfolding has_laurent_expansion_def eventually_at_filter eventually_nhds_metric ball_def
by (auto simp: dist_commute)
have holo: "zor_poly f 0 holomorphic_on ball 0 r"
by (rule holomorphic_on_subset[OF r(2)]) auto
have "(\<lambda>z. if z = 0 then fps_nth G 0 else f z * z powi -n) has_fps_expansion G"
unfolding G_def n_altdef by (intro has_fps_expansion_fls_base_factor_to_fps assms)
also have "?this \<longleftrightarrow> zor_poly f 0 has_fps_expansion G"
proof (intro has_fps_expansion_cong)
have "eventually (\<lambda>z. z \<in> ball 0 (min r r')) (nhds 0)"
using \<open>r > 0\<close> \<open>r' > 0\<close> by (intro eventually_nhds_in_open) auto
thus "\<forall>\<^sub>F x in nhds 0. (if x = 0 then G $ 0 else f x * x powi - n) = zor_poly f 0 x"
proof eventually_elim
case (elim w)
have w: "w \<in> ball 0 r" "w \<in> ball 0 r'"
using elim by auto
show ?case
proof (cases "w = 0")
case False
hence "f w = zor_poly f 0 w * w powi n"
using r w by auto
thus ?thesis using False
by (simp add: powr_minus complex_powr_of_int power_int_minus)
next
case [simp]: True
obtain R where R: "R > 0" "R \<le> r" "R \<le> r'" "R \<le> fls_conv_radius F"
using \<open>r > 0\<close> \<open>r' > 0\<close> assms(1) unfolding has_laurent_expansion_def
by (smt (verit, ccfv_SIG) ereal_dense2 ereal_less(2) less_ereal.simps(1) order.strict_implies_order order_trans)
have "eval_fps G 0 = zor_poly f 0 0"
proof (rule analytic_continuation_open[where f = "eval_fps G" and g = "zor_poly f 0"])
show "connected (ball 0 R :: complex set)"
by auto
have "of_real R / 2 \<in> ball 0 R - {0 :: complex}"
using R by auto
thus "ball 0 R - {0 :: complex} \<noteq> {}"
by blast
show "eval_fps G holomorphic_on ball 0 R"
using R less_le_trans[OF _ R(4)] unfolding G_def
by (intro holomorphic_intros) (auto simp: fls_conv_radius_altdef)
show "zor_poly f 0 holomorphic_on ball 0 R"
by (rule holomorphic_on_subset[OF holo]) (use R in auto)
show "eval_fps G z = zor_poly f 0 z" if "z \<in> ball 0 R - {0}" for z
using that r r' R n_altdef unfolding G_def
by (subst eval_fps_fls_base_factor)
(auto simp: complex_powr_of_int field_simps power_int_minus n_def)
qed (use R in auto)
hence "zor_poly f 0 0 = fps_nth G 0"
by (simp add: eval_fps_at_0)
thus ?thesis by simp
qed
qed
qed (use r' in auto)
finally show ?thesis
by (simp add: G_def)
qed
lemma zorder_geI_0:
assumes "f analytic_on {0}" "f holomorphic_on A" "open A" "connected A" "0 \<in> A" "z \<in> A" "f z \<noteq> 0"
assumes "\<And>k. k < n \<Longrightarrow> (deriv ^^ k) f 0 = 0"
shows "zorder f 0 \<ge> n"
proof -
define F where "F = fps_expansion f 0"
from assms have "f has_fps_expansion F"
unfolding F_def using analytic_at_imp_has_fps_expansion_0 by blast
hence laurent: "f has_laurent_expansion fps_to_fls F" and [simp]: "f 0 = fps_nth F 0"
by (simp_all add: has_fps_expansion_to_laurent)
have [simp]: "F \<noteq> 0"
proof
assume [simp]: "F = 0"
hence "f z = 0"
proof (cases "z = 0")
case False
have "f has_laurent_expansion 0"
using laurent by simp
thus ?thesis
proof (rule has_laurent_expansion_0_analytic_continuation)
show "f holomorphic_on A - {0}"
using assms(2) by (rule holomorphic_on_subset) auto
qed (use assms False in auto)
qed auto
with \<open>f z \<noteq> 0\<close> show False by contradiction
qed
have "zorder f 0 = int (subdegree F)"
using has_laurent_expansion_zorder_0[OF laurent] by (simp add: fls_subdegree_fls_to_fps)
also have "subdegree F \<ge> n"
using assms by (intro subdegree_geI \<open>F \<noteq> 0\<close>) (auto simp: F_def fps_expansion_def)
hence "int (subdegree F) \<ge> int n"
by simp
finally show ?thesis .
qed
lemma zorder_geI:
assumes "f analytic_on {x}" "f holomorphic_on A" "open A" "connected A" "x \<in> A" "z \<in> A" "f z \<noteq> 0"
assumes "\<And>k. k < n \<Longrightarrow> (deriv ^^ k) f x = 0"
shows "zorder f x \<ge> n"
proof -
have "zorder f x = zorder (f \<circ> (\<lambda>u. u + x)) 0"
by (subst zorder_shift) (auto simp: o_def)
also have "\<dots> \<ge> n"
proof (rule zorder_geI_0)
show "(f \<circ> (\<lambda>u. u + x)) analytic_on {0}"
by (intro analytic_on_compose_gen[OF _ assms(1)] analytic_intros) auto
show "f \<circ> (\<lambda>u. u + x) holomorphic_on ((+) (-x)) ` A"
by (intro holomorphic_on_compose_gen[OF _ assms(2)] holomorphic_intros) auto
show "connected ((+) (- x) ` A)"
by (intro connected_continuous_image continuous_intros assms)
show "open ((+) (- x) ` A)"
by (intro open_translation assms)
show "z - x \<in> (+) (- x) ` A"
using \<open>z \<in> A\<close> by auto
show "0 \<in> (+) (- x) ` A"
using \<open>x \<in> A\<close> by auto
show "(f \<circ> (\<lambda>u. u + x)) (z - x) \<noteq> 0"
using \<open>f z \<noteq> 0\<close> by auto
next
fix k :: nat assume "k < n"
hence "(deriv ^^ k) f x = 0"
using assms by simp
also have "(deriv ^^ k) f x = (deriv ^^ k) (f \<circ> (+) x) 0"
by (subst higher_deriv_shift_0) auto
finally show "(deriv ^^ k) (f \<circ> (\<lambda>u. u + x)) 0 = 0"
by (subst add.commute) auto
qed
finally show ?thesis .
qed
lemma has_laurent_expansion_divide [laurent_expansion_intros]:
assumes "f has_laurent_expansion F" and "g has_laurent_expansion G"
shows "(\<lambda>x. f x / g x) has_laurent_expansion (F / G)"
proof -
have "(\<lambda>x. f x * inverse (g x)) has_laurent_expansion (F * inverse G)"
by (intro laurent_expansion_intros assms)
thus ?thesis
by (simp add: field_simps)
qed
lemma has_laurent_expansion_residue_0:
assumes "f has_laurent_expansion F"
shows "residue f 0 = fls_residue F"
proof (cases "fls_subdegree F \<ge> 0")
case True
have "residue f 0 = residue (eval_fls F) 0"
using assms by (intro residue_cong) (auto simp: has_laurent_expansion_def eq_commute)
also have "\<dots> = 0"
by (rule residue_holo[OF _ _ holomorphic_on_eval_fls[OF order.refl]])
(use True assms in \<open>auto simp: has_laurent_expansion_def zero_ereal_def\<close>)
also have "\<dots> = fls_residue F"
using True by simp
finally show ?thesis .
next
case False
hence "F \<noteq> 0"
by auto
have *: "zor_poly f 0 has_fps_expansion fls_base_factor_to_fps F"
by (intro zor_poly_has_fps_expansion False assms \<open>F \<noteq> 0\<close>)
have "residue f 0 = (deriv ^^ (nat (-zorder f 0) - 1)) (zor_poly f 0) 0 / fact (nat (- zorder f 0) - 1)"
by (intro residue_pole_order has_laurent_expansion_isolated_0[OF assms]
has_laurent_expansion_imp_is_pole_0[OF assms]) (use False in auto)
also have "\<dots> = fls_residue F"
using has_laurent_expansion_zorder_0[OF assms \<open>F \<noteq> 0\<close>] False
by (subst fps_nth_fps_expansion [OF *, symmetric]) (auto simp: of_nat_diff)
finally show ?thesis .
qed
lemma has_laurent_expansion_residue:
assumes "(\<lambda>x. f (z + x)) has_laurent_expansion F"
shows "residue f z = fls_residue F"
using has_laurent_expansion_residue_0[OF assms] by (simp add: residue_shift_0')
lemma eval_fls_has_laurent_expansion [laurent_expansion_intros]:
assumes "fls_conv_radius F > 0"
shows "eval_fls F has_laurent_expansion F"
using assms by (auto simp: has_laurent_expansion_def)
lemma fps_expansion_unique_complex:
fixes F G :: "complex fps"
assumes "f has_fps_expansion F" "f has_fps_expansion G"
shows "F = G"
using assms unfolding fps_eq_iff by (auto simp: fps_eq_iff fps_nth_fps_expansion)
lemma fps_expansion_eqI:
assumes "f has_fps_expansion F"
shows "fps_expansion f 0 = F"
using assms unfolding fps_eq_iff
by (auto simp: fps_eq_iff fps_nth_fps_expansion fps_expansion_def)
lemma holomorphic_on_imp_fps_conv_radius_ge:
assumes "f has_fps_expansion F" "f holomorphic_on eball 0 r"
shows "fps_conv_radius F \<ge> r"
proof -
define n where "n = subdegree F"
have "fps_conv_radius (fps_expansion f 0) \<ge> r"
by (intro conv_radius_fps_expansion assms)
also have "fps_expansion f 0 = F"
using assms by (intro fps_expansion_eqI)
finally show ?thesis
by simp
qed
lemma has_fps_expansion_imp_eval_fps_eq:
assumes "f has_fps_expansion F" "norm z < r"
assumes "f holomorphic_on ball 0 r"
shows "eval_fps F z = f z"
proof -
have [simp]: "fps_expansion f 0 = F"
by (rule fps_expansion_eqI) fact
have *: "f holomorphic_on eball 0 (ereal r)"
using assms by simp
from conv_radius_fps_expansion[OF *] have "fps_conv_radius F \<ge> ereal r"
by simp
have "eval_fps (fps_expansion f 0) z = f (0 + z)"
by (rule eval_fps_expansion'[OF *]) (use assms in auto)
thus ?thesis
by simp
qed
lemma has_fps_expansion_imp_sums_complex:
fixes F :: "complex fps"
assumes "f has_fps_expansion F" "f holomorphic_on eball 0 r" "ereal (norm z) < r"
shows "(\<lambda>n. fps_nth F n * z ^ n) sums f z"
proof -
have r: "fps_conv_radius F \<ge> r"
using assms(1,2) by (rule holomorphic_on_imp_fps_conv_radius_ge)
from assms obtain R where R: "norm z < R" "ereal R < r"
using ereal_dense2 less_ereal.simps(1) by blast
have z: "norm z < fps_conv_radius F"
using r R assms(3) by order
have "summable (\<lambda>n. fps_nth F n * z ^ n)"
by (rule summable_fps) (use z in auto)
moreover have "eval_fps F z = f z"
proof (rule has_fps_expansion_imp_eval_fps_eq[where r = R])
have *: "ereal (norm z) < r" if "norm z < R" for z :: complex
using that R ereal_le_less less_imp_le by blast
show "f holomorphic_on ball 0 R"
using assms(2) by (rule holomorphic_on_subset) (use * in auto)
qed (use R assms(1) in auto)
ultimately show ?thesis
unfolding eval_fps_def sums_iff by simp
qed
lemma fls_conv_radius_ge:
assumes "f has_laurent_expansion F"
assumes "f holomorphic_on eball 0 r - {0}"
shows "fls_conv_radius F \<ge> r"
proof -
define n where "n = fls_subdegree F"
define G where "G = fls_base_factor_to_fps F"
define g where "g = (\<lambda>z. if z = 0 then fps_nth G 0 else f z * z powi -n)"
have G: "g has_fps_expansion G"
unfolding G_def g_def n_def
by (intro has_fps_expansion_fls_base_factor_to_fps assms)
have "(\<lambda>z. f z * z powi -n) holomorphic_on eball 0 r - {0}"
by (intro holomorphic_intros assms) auto
also have "?this \<longleftrightarrow> g holomorphic_on eball 0 r - {0}"
by (intro holomorphic_cong) (auto simp: g_def)
finally have "g analytic_on eball 0 r - {0}"
by (subst analytic_on_open) auto
moreover have "g analytic_on {0}"
using G has_fps_expansion_imp_analytic_0 by auto
ultimately have "g analytic_on (eball 0 r - {0} \<union> {0})"
by (subst analytic_on_Un) auto
hence "g analytic_on eball 0 r"
by (rule analytic_on_subset) auto
hence "g holomorphic_on eball 0 r"
by (subst (asm) analytic_on_open) auto
hence "fps_conv_radius (fps_expansion g 0) \<ge> r"
by (intro conv_radius_fps_expansion)
also have "fps_expansion g 0 = G"
using G by (intro fps_expansion_eqI)
finally show ?thesis
by (simp add: fls_conv_radius_altdef G_def)
qed
lemma eval_fls_eqI:
assumes "f has_laurent_expansion F" "f holomorphic_on eball 0 r - {0}"
assumes "z \<in> eball 0 r - {0}"
shows "eval_fls F z = f z"
proof -
have conv: "fls_conv_radius F \<ge> r"
by (intro fls_conv_radius_ge[OF assms(1,2)])
have "(\<lambda>z. eval_fls F z - f z) has_laurent_expansion F - F"
using assms by (intro laurent_expansion_intros assms) (auto simp: has_laurent_expansion_def)
hence "(\<lambda>z. eval_fls F z - f z) has_laurent_expansion 0"
by simp
hence "eval_fls F z - f z = 0"
proof (rule has_laurent_expansion_0_analytic_continuation)
have "ereal 0 \<le> ereal (norm z)"
by simp
also have "norm z < r"
using assms by auto
finally have "r > 0"
by (simp add: zero_ereal_def)
thus "open (eball 0 r :: complex set)" "connected (eball 0 r :: complex set)"
"0 \<in> eball 0 r" "z \<in> eball 0 r - {0}"
using assms by (auto simp: zero_ereal_def)
qed (auto intro!: holomorphic_intros assms less_le_trans[OF _ conv] split: if_splits)
thus ?thesis by simp
qed
lemma fls_nth_as_contour_integral:
assumes F: "f has_laurent_expansion F"
assumes holo: "f holomorphic_on ball 0 r - {0}"
assumes R: "0 < R" "R < r"
shows "((\<lambda>z. f z * z powi (-(n+1))) has_contour_integral
complex_of_real (2 * pi) * \<i> * fls_nth F n) (circlepath 0 R)"
proof -
define I where "I = (\<lambda>z. f z * z powi (-(n+1)))"
have "(I has_contour_integral complex_of_real (2 * pi) * \<i> * residue I 0) (circlepath 0 R)"
proof (rule base_residue)
show "open (ball (0::complex) r)" "0 \<in> ball (0::complex) r"
using R F by (auto simp: has_laurent_expansion_def zero_ereal_def)
qed (use R in \<open>auto intro!: holomorphic_intros holomorphic_on_subset[OF holo]
simp: I_def split: if_splits\<close>)
also have "residue I 0 = fls_residue (fls_shift (n + 1) F)"
unfolding I_def by (intro has_laurent_expansion_residue_0 laurent_expansion_intros F)
also have "\<dots> = fls_nth F n"
by simp
finally show ?thesis
by (simp add: I_def)
qed
lemma tendsto_0_subdegree_iff_0:
assumes F:"f has_laurent_expansion F" and "F\<noteq>0"
shows "(f \<midarrow>0\<rightarrow> 0) \<longleftrightarrow> fls_subdegree F > 0"
proof -
have ?thesis if "is_pole f 0"
proof -
have "fls_subdegree F <0"
using is_pole_0_imp_neg_fls_subdegree[OF F that] .
moreover then have "\<not> f \<midarrow>0\<rightarrow>0"
using \<open>is_pole f 0\<close> F at_neq_bot
has_laurent_expansion_imp_filterlim_infinity_0
not_tendsto_and_filterlim_at_infinity that
by blast
ultimately show ?thesis by auto
qed
moreover have ?thesis if "\<not>is_pole f 0" "\<exists>x. f \<midarrow>0\<rightarrow>x"
proof -
have "fls_subdegree F \<ge>0"
using has_laurent_expansion_imp_is_pole_0[OF F] that(1)
by linarith
have "f \<midarrow>0\<rightarrow>0" if "fls_subdegree F > 0"
using fls_eq0_below_subdegree[OF that]
by (metis F \<open>0 \<le> fls_subdegree F\<close> has_laurent_expansion_imp_tendsto_0)
moreover have "fls_subdegree F > 0" if "f \<midarrow>0\<rightarrow>0"
proof -
have False if "fls_subdegree F = 0"
proof -
have "f \<midarrow>0\<rightarrow> fls_nth F 0"
using has_laurent_expansion_imp_tendsto_0
[OF F \<open>fls_subdegree F \<ge>0\<close>] .
then have "fls_nth F 0 = 0" using \<open>f \<midarrow>0\<rightarrow>0\<close>
using LIM_unique by blast
then have "F = 0"
using nth_fls_subdegree_zero_iff \<open>fls_subdegree F = 0\<close>
by metis
with \<open>F\<noteq>0\<close> show False by auto
qed
with \<open>fls_subdegree F \<ge>0\<close>
show ?thesis by fastforce
qed
ultimately show ?thesis by auto
qed
moreover have "is_pole f 0 \<or> (\<exists>x. f \<midarrow>0\<rightarrow>x)"
proof -
have "not_essential f 0"
using F has_laurent_expansion_not_essential_0 by auto
then show ?thesis unfolding not_essential_def
by auto
qed
ultimately show ?thesis by auto
qed
lemma tendsto_0_subdegree_iff:
assumes F: "(\<lambda>w. f (z+w)) has_laurent_expansion F" and "F \<noteq> 0"
shows "(f \<midarrow>z\<rightarrow> 0) \<longleftrightarrow> fls_subdegree F > 0"
apply (subst Lim_at_zero)
apply (rule tendsto_0_subdegree_iff_0)
using assms by auto
lemma is_pole_0_deriv_divide_iff:
assumes F: "f has_laurent_expansion F" and "F \<noteq> 0"
shows "is_pole (\<lambda>x. deriv f x / f x) 0 \<longleftrightarrow> is_pole f 0 \<or> (f \<midarrow>0\<rightarrow> 0)"
proof -
have "(\<lambda>x. deriv f x / f x) has_laurent_expansion fls_deriv F / F"
using F by (auto intro:laurent_expansion_intros)
have "is_pole (\<lambda>x. deriv f x / f x) 0 \<longleftrightarrow>
fls_subdegree (fls_deriv F / F) < 0"
apply (rule is_pole_fls_subdegree_iff)
using F by (auto intro:laurent_expansion_intros)
also have "... \<longleftrightarrow> is_pole f 0 \<or> (f \<midarrow>0\<rightarrow>0)"
proof (cases "fls_subdegree F = 0")
case True
then have "fls_subdegree (fls_deriv F / F) \<ge> 0"
by (metis diff_zero div_0 \<open>F\<noteq>0\<close> fls_deriv_subdegree0
fls_divide_subdegree)
moreover then have "\<not> is_pole f 0"
by (metis F True is_pole_0_imp_neg_fls_subdegree less_le)
moreover have "\<not> (f \<midarrow>0\<rightarrow>0)"
using tendsto_0_subdegree_iff_0[OF F \<open>F\<noteq>0\<close>] True by auto
ultimately show ?thesis by auto
next
case False
then have "fls_deriv F \<noteq> 0"
by (metis fls_const_subdegree fls_deriv_eq_0_iff)
then have "fls_subdegree (fls_deriv F / F) =
fls_subdegree (fls_deriv F) - fls_subdegree F"
by (rule fls_divide_subdegree[OF _ \<open>F\<noteq>0\<close>])
moreover have "fls_subdegree (fls_deriv F) = fls_subdegree F - 1"
using fls_subdegree_deriv[OF False] .
ultimately have "fls_subdegree (fls_deriv F / F) < 0" by auto
moreover have "f \<midarrow>0\<rightarrow> 0 = (0 < fls_subdegree F)"
using tendsto_0_subdegree_iff_0[OF F \<open>F \<noteq> 0\<close>] .
moreover have "is_pole f 0 = (fls_subdegree F < 0)"
using is_pole_fls_subdegree_iff F by auto
ultimately show ?thesis using False by auto
qed
finally show ?thesis .
qed
lemma is_pole_deriv_divide_iff:
assumes F:"(\<lambda>w. f (z+w)) has_laurent_expansion F" and "F\<noteq>0"
shows "is_pole (\<lambda>x. deriv f x / f x) z \<longleftrightarrow> is_pole f z \<or> (f \<midarrow>z\<rightarrow>0)"
proof -
define ff df where "ff=(\<lambda>w. f (z+w))" and "df=(\<lambda>w. deriv f (z + w))"
have "is_pole (\<lambda>x. deriv f x / f x) z
\<longleftrightarrow> is_pole (\<lambda>x. deriv ff x / ff x) 0"
unfolding ff_def df_def
by (simp add:deriv_shift_0' is_pole_shift_0' comp_def algebra_simps)
moreover have "is_pole f z \<longleftrightarrow> is_pole ff 0"
unfolding ff_def by (auto simp:is_pole_shift_0')
moreover have "(f \<midarrow>z\<rightarrow>0) \<longleftrightarrow> (ff \<midarrow>0\<rightarrow>0)"
unfolding ff_def by (simp add: LIM_offset_zero_iff)
moreover have "is_pole (\<lambda>x. deriv ff x / ff x) 0 = (is_pole ff 0 \<or> ff \<midarrow>0\<rightarrow> 0)"
apply (rule is_pole_0_deriv_divide_iff)
using F ff_def \<open>F\<noteq>0\<close> by blast+
ultimately show ?thesis by auto
qed
lemma subdegree_imp_eventually_deriv_nzero_0:
assumes F:"f has_laurent_expansion F" and "fls_subdegree F\<noteq>0"
shows "eventually (\<lambda>z. deriv f z \<noteq> 0) (at 0)"
proof -
have "deriv f has_laurent_expansion fls_deriv F"
using has_laurent_expansion_deriv[OF F] .
moreover have "fls_deriv F\<noteq>0"
using \<open>fls_subdegree F\<noteq>0\<close>
by (metis fls_const_subdegree fls_deriv_eq_0_iff)
ultimately show ?thesis
using has_laurent_expansion_eventually_nonzero_iff' by blast
qed
lemma subdegree_imp_eventually_deriv_nzero:
assumes F:"(\<lambda>w. f (z+w)) has_laurent_expansion F"
and "fls_subdegree F\<noteq>0"
shows "eventually (\<lambda>w. deriv f w \<noteq> 0) (at z)"
proof -
have "\<forall>\<^sub>F x in at 0. deriv (\<lambda>w. f (z + w)) x \<noteq> 0"
using subdegree_imp_eventually_deriv_nzero_0 assms by auto
then show ?thesis
apply (subst eventually_at_to_0)
by (simp add:deriv_shift_0' comp_def algebra_simps)
qed
lemma has_fps_expansion_imp_asymp_equiv_0:
fixes f :: "complex \<Rightarrow> complex"
assumes F: "f has_fps_expansion F"
defines "n \<equiv> subdegree F"
shows "f \<sim>[nhds 0] (\<lambda>z. fps_nth F n * z ^ n)"
proof -
have F': "f has_laurent_expansion fps_to_fls F"
using F has_laurent_expansion_fps by blast
have "f \<sim>[at 0] (\<lambda>z. fps_nth F n * z ^ n)"
using has_laurent_expansion_imp_asymp_equiv_0[OF F']
by (simp add: fls_subdegree_fls_to_fps n_def)
moreover have "f 0 = fps_nth F n * 0 ^ n"
using F by (auto simp: n_def has_fps_expansion_to_laurent power_0_left)
ultimately show ?thesis
by (auto simp: asymp_equiv_nhds_iff)
qed
lemma has_fps_expansion_imp_tendsto_0:
fixes f :: "complex \<Rightarrow> complex"
assumes "f has_fps_expansion F"
shows "(f \<longlongrightarrow> fps_nth F 0) (nhds 0)"
proof (rule asymp_equiv_tendsto_transfer)
show "(\<lambda>z. fps_nth F (subdegree F) * z ^ subdegree F) \<sim>[nhds 0] f"
by (rule asymp_equiv_symI, rule has_fps_expansion_imp_asymp_equiv_0) fact
have "((\<lambda>z. F $ subdegree F * z ^ subdegree F) \<longlongrightarrow> F $ 0) (at 0)"
by (rule tendsto_eq_intros refl | simp)+ (auto simp: power_0_left)
thus "((\<lambda>z. F $ subdegree F * z ^ subdegree F) \<longlongrightarrow> F $ 0) (nhds 0)"
by (auto simp: tendsto_nhds_iff power_0_left)
qed
lemma has_fps_expansion_imp_0_eq_fps_nth_0:
assumes "f has_fps_expansion F"
shows "f 0 = fps_nth F 0"
proof -
have "eventually (\<lambda>x. f x = eval_fps F x) (nhds 0)"
using assms by (auto simp: has_fps_expansion_def eq_commute)
then obtain A where "open A" "0 \<in> A" "\<forall>x\<in>A. f x = eval_fps F x"
unfolding eventually_nhds by blast
hence "f 0 = eval_fps F 0"
by blast
thus ?thesis
by (simp add: eval_fps_at_0)
qed
lemma fls_nth_compose_aux:
assumes "f has_fps_expansion F"
assumes G: "g has_fps_expansion G" "fps_nth G 0 = 0" "fps_deriv G \<noteq> 0"
assumes "(f \<circ> g) has_laurent_expansion H"
shows "fls_nth H (int n) = fps_nth (fps_compose F G) n"
using assms(1,5)
proof (induction n arbitrary: f F H rule: less_induct)
case (less n f F H)
have [simp]: "g 0 = 0"
using has_fps_expansion_imp_0_eq_fps_nth_0[OF G(1)] G(2) by simp
have ana_f: "f analytic_on {0}"
using less.prems by (meson has_fps_expansion_imp_analytic_0)
have ana_g: "g analytic_on {0}"
using G by (meson has_fps_expansion_imp_analytic_0)
have "(f \<circ> g) has_laurent_expansion fps_to_fls (fps_expansion (f \<circ> g) 0)"
by (rule analytic_at_imp_has_fps_expansion_0 analytic_intros has_laurent_expansion_fps
analytic_on_compose_gen ana_f ana_g)+ auto
with less.prems have "H = fps_to_fls (fps_expansion (f \<circ> g) 0)"
using has_laurent_expansion_unique by blast
also have "fls_subdegree \<dots> \<ge> 0"
by (simp add: fls_subdegree_fls_to_fps)
finally have subdeg: "fls_subdegree H \<ge> 0" .
show ?case
proof (cases "n = 0")
case [simp]: True
have lim_g: "g \<midarrow>0\<rightarrow> 0"
using has_laurent_expansion_imp_tendsto_0[of g "fps_to_fls G"] G
by (auto simp: fls_subdegree_fls_to_fps_gt0 has_fps_expansion_to_laurent)
have lim_f: "(f \<longlongrightarrow> fps_nth F 0) (nhds 0)"
by (intro has_fps_expansion_imp_tendsto_0 less.prems)
have "(\<lambda>x. f (g x)) \<midarrow>0\<rightarrow> fps_nth F 0"
by (rule filterlim_compose[OF lim_f lim_g])
moreover have "(f \<circ> g) \<midarrow>0\<rightarrow> fls_nth H 0"
by (intro has_laurent_expansion_imp_tendsto_0 less.prems subdeg)
ultimately have "fps_nth F 0 = fls_nth H 0"
using tendsto_unique by (force simp: o_def)
thus ?thesis
by simp
next
case n: False
define GH where "GH = (fls_deriv H / fls_deriv (fps_to_fls G))"
define GH' where "GH' = fls_regpart GH"
have "(\<lambda>x. deriv (f \<circ> g) x / deriv g x) has_laurent_expansion
fls_deriv H / fls_deriv (fps_to_fls G)"
by (intro laurent_expansion_intros less.prems has_laurent_expansion_fps[of _ G] G)
also have "?this \<longleftrightarrow> (deriv f \<circ> g) has_laurent_expansion fls_deriv H / fls_deriv (fps_to_fls G)"
proof (rule has_laurent_expansion_cong)
from ana_f obtain r1 where r1: "r1 > 0" "f holomorphic_on ball 0 r1"
unfolding analytic_on_def by blast
from ana_g obtain r2 where r2: "r2 > 0" "g holomorphic_on ball 0 r2"
unfolding analytic_on_def by blast
have lim_g: "g \<midarrow>0\<rightarrow> 0"
using has_laurent_expansion_imp_tendsto_0[of g "fps_to_fls G"] G
by (auto simp: fls_subdegree_fls_to_fps_gt0 has_fps_expansion_to_laurent)
moreover have "open (ball 0 r1)" "0 \<in> ball 0 r1"
using r1 by auto
ultimately have "eventually (\<lambda>x. g x \<in> ball 0 r1) (at 0)"
unfolding tendsto_def by blast
moreover have "eventually (\<lambda>x. deriv g x \<noteq> 0) (at 0)"
using G fps_to_fls_eq_0_iff has_fps_expansion_deriv has_fps_expansion_to_laurent
has_laurent_expansion_nonzero_imp_eventually_nonzero by blast
moreover have "eventually (\<lambda>x. x \<in> ball 0 (min r1 r2) - {0}) (at 0)"
by (intro eventually_at_in_open) (use r1 r2 in auto)
ultimately show "eventually (\<lambda>x. deriv (f \<circ> g) x / deriv g x = (deriv f \<circ> g) x) (at 0)"
proof eventually_elim
case (elim x)
thus ?case using r1 r2
by (subst deriv_chain)
(auto simp: field_simps holomorphic_on_def at_within_open[of _ "ball _ _"])
qed
qed auto
finally have GH: "(deriv f \<circ> g) has_laurent_expansion GH"
unfolding GH_def .
have "(deriv f \<circ> g) has_laurent_expansion fps_to_fls (fps_expansion (deriv f \<circ> g) 0)"
by (rule analytic_at_imp_has_fps_expansion_0 analytic_intros has_laurent_expansion_fps
analytic_on_compose_gen ana_f ana_g)+ auto
with GH have "GH = fps_to_fls (fps_expansion (deriv f \<circ> g) 0)"
using has_laurent_expansion_unique by blast
also have "fls_subdegree \<dots> \<ge> 0"
by (simp add: fls_subdegree_fls_to_fps)
finally have subdeg': "fls_subdegree GH \<ge> 0" .
have "deriv f has_fps_expansion fps_deriv F"
by (intro fps_expansion_intros less.prems)
from this and GH have IH: "fls_nth GH (int k) = fps_nth (fps_compose (fps_deriv F) G) k"
if "k < n" for k
by (intro less.IH that)
have "fps_nth (fps_compose (fps_deriv F) G) n = (\<Sum>i=0..n. of_nat (Suc i) * F $ Suc i * G ^ i $ n)"
by (simp add: fps_compose_nth)
have "fps_nth (fps_compose F G) n =
fps_nth (fps_deriv (fps_compose F G)) (n - 1) / of_nat n"
using n by (cases n) (auto simp del: of_nat_Suc)
also have "fps_deriv (fps_compose F G) = fps_compose (fps_deriv F) G * fps_deriv G "
using G by (subst fps_compose_deriv) auto
also have "fps_nth \<dots> (n - 1) = (\<Sum>i=0..n-1. (fps_deriv F oo G) $ i * fps_deriv G $ (n - 1 - i))"
unfolding fps_mult_nth ..
also have "\<dots> = (\<Sum>i=0..n-1. fps_nth GH' i * of_nat (n - i) * G $ (n - i))"
using n by (intro sum.cong) (auto simp: IH Suc_diff_Suc GH'_def)
also have "\<dots> = (\<Sum>i=0..n. fps_nth GH' i * of_nat (n - i) * G $ (n - i))"
by (intro sum.mono_neutral_left) auto
also have "\<dots> = fps_nth (GH' * Abs_fps (\<lambda>i. of_nat i * fps_nth G i)) n"
by (simp add: fps_mult_nth mult_ac)
also have "Abs_fps (\<lambda>i. of_nat i * fps_nth G i) = fps_X * fps_deriv G"
by (simp add: fps_mult_fps_X_deriv_shift)
also have "fps_nth (GH' * (fps_X * fps_deriv G)) n =
fls_nth (fps_to_fls (GH' * (fps_X * fps_deriv G))) (int n)"
by simp
also have "fps_to_fls (GH' * (fps_X * fps_deriv G)) =
GH * fps_to_fls (fps_deriv G) * fls_X"
using subdeg' by (simp add: mult_ac fls_times_fps_to_fls GH'_def)
also have "GH * fps_to_fls (fps_deriv G) = fls_deriv H"
unfolding GH_def using G by (simp add: fls_deriv_fps_to_fls)
also have "fls_deriv H * fls_X = fls_shift (-1) (fls_deriv H)"
using fls_X_times_conv_shift(2) by blast
finally show ?thesis
using n by simp
qed
qed
lemma has_fps_expansion_compose [fps_expansion_intros]:
fixes f g :: "complex \<Rightarrow> complex"
assumes F: "f has_fps_expansion F"
assumes G: "g has_fps_expansion G" "fps_nth G 0 = 0"
shows "(f \<circ> g) has_fps_expansion fps_compose F G"
proof (cases "fps_deriv G = 0")
case False
have [simp]: "g 0 = 0"
using has_fps_expansion_imp_0_eq_fps_nth_0[OF G(1)] G(2) False by simp
have ana_f: "f analytic_on {0}"
using F by (meson has_fps_expansion_imp_analytic_0)
have ana_g: "g analytic_on {0}"
using G by (meson has_fps_expansion_imp_analytic_0)
have fg: "(f \<circ> g) has_fps_expansion fps_expansion (f \<circ> g) 0"
by (rule analytic_at_imp_has_fps_expansion_0 analytic_intros
analytic_on_compose_gen ana_f ana_g)+ auto
have "fls_nth (fps_to_fls (fps_expansion (f \<circ> g) 0)) (int n) = fps_nth (fps_compose F G) n" for n
by (rule fls_nth_compose_aux has_laurent_expansion_fps F G False fg)+
hence "fps_expansion (f \<circ> g) 0 = fps_compose F G"
by (simp add: fps_eq_iff)
thus ?thesis using fg
by simp
next
case True
have [simp]: "f 0 = fps_nth F 0"
using F by (auto dest: has_fps_expansion_imp_0_eq_fps_nth_0)
from True have "fps_nth G n = 0" for n
using G(2) by (cases n) (auto simp del: of_nat_Suc)
hence [simp]: "G = 0"
by (auto simp: fps_eq_iff)
have "(\<lambda>_. f 0) has_fps_expansion fps_const (f 0)"
by (intro fps_expansion_intros)
also have "eventually (\<lambda>x. g x = 0) (nhds 0)"
using G by (auto simp: has_fps_expansion_def)
hence "(\<lambda>_. f 0) has_fps_expansion fps_const (f 0) \<longleftrightarrow> (f \<circ> g) has_fps_expansion fps_const (f 0)"
by (intro has_fps_expansion_cong) (auto elim!: eventually_mono)
thus ?thesis
by simp
qed
lemma has_fps_expansion_fps_to_fls:
assumes "f has_laurent_expansion fps_to_fls F"
shows "(\<lambda>z. if z = 0 then fps_nth F 0 else f z) has_fps_expansion F"
(is "?f' has_fps_expansion _")
proof -
have "f has_laurent_expansion fps_to_fls F \<longleftrightarrow> ?f' has_laurent_expansion fps_to_fls F"
by (intro has_laurent_expansion_cong) (auto simp: eventually_at_filter)
with assms show ?thesis
by (auto simp: has_fps_expansion_to_laurent)
qed
lemma has_laurent_expansion_compose [laurent_expansion_intros]:
fixes f g :: "complex \<Rightarrow> complex"
assumes F: "f has_laurent_expansion F"
assumes G: "g has_laurent_expansion fps_to_fls G" "fps_nth G 0 = 0" "G \<noteq> 0"
shows "(f \<circ> g) has_laurent_expansion fls_compose_fps F G"
proof -
from assms have lim_g: "g \<midarrow>0\<rightarrow> 0"
by (subst tendsto_0_subdegree_iff_0[OF G(1)])
(auto simp: fls_subdegree_fls_to_fps subdegree_pos_iff)
have ev1: "eventually (\<lambda>z. g z \<noteq> 0) (at 0)"
using \<open>G \<noteq> 0\<close> G(1) fps_to_fls_eq_0_iff has_laurent_expansion_fps
has_laurent_expansion_nonzero_imp_eventually_nonzero by blast
moreover have "eventually (\<lambda>z. z \<noteq> 0) (at (0 :: complex))"
by (auto simp: eventually_at_filter)
ultimately have ev: "eventually (\<lambda>z. z \<noteq> 0 \<and> g z \<noteq> 0) (at 0)"
by eventually_elim blast
from ev1 and lim_g have lim_g': "filterlim g (at 0) (at 0)"
by (auto simp: filterlim_at)
define g' where "g' = (\<lambda>z. if z = 0 then fps_nth G 0 else g z)"
show ?thesis
proof (cases "F = 0")
assume [simp]: "F = 0"
have "eventually (\<lambda>z. f z = 0) (at 0)"
using F by (auto simp: has_laurent_expansion_def)
hence "eventually (\<lambda>z. f (g z) = 0) (at 0)"
using lim_g' by (rule eventually_compose_filterlim)
thus ?thesis
by (auto simp: has_laurent_expansion_def)
next
assume [simp]: "F \<noteq> 0"
define n where "n = fls_subdegree F"
define f' where
"f' = (\<lambda>z. if z = 0 then fps_nth (fls_base_factor_to_fps F) 0 else f z * z powi -n)"
have "((\<lambda>z. (f' \<circ> g') z * g z powi n)) has_laurent_expansion fls_compose_fps F G"
unfolding f'_def n_def fls_compose_fps_def g'_def
by (intro fps_expansion_intros laurent_expansion_intros has_fps_expansion_fps_to_fls
has_fps_expansion_fls_base_factor_to_fps assms has_laurent_expansion_fps)
also have "?this \<longleftrightarrow> ?thesis"
by (intro has_laurent_expansion_cong eventually_mono[OF ev])
(auto simp: f'_def power_int_minus g'_def)
finally show ?thesis .
qed
qed
lemma has_laurent_expansion_fls_X_inv [laurent_expansion_intros]:
"inverse has_laurent_expansion fls_X_inv"
using has_laurent_expansion_inverse[OF has_laurent_expansion_fps_X]
by (simp add: fls_inverse_X)
lemma zorder_times_analytic:
assumes "f analytic_on {z}" "g analytic_on {z}"
assumes "eventually (\<lambda>z. f z * g z \<noteq> 0) (at z)"
shows "zorder (\<lambda>z. f z * g z) z = zorder f z + zorder g z"
proof -
have *: "(\<lambda>w. f (z + w)) has_fps_expansion fps_expansion f z"
"(\<lambda>w. g (z + w)) has_fps_expansion fps_expansion g z"
"(\<lambda>w. f (z + w) * g (z + w)) has_fps_expansion fps_expansion f z * fps_expansion g z"
by (intro fps_expansion_intros analytic_at_imp_has_fps_expansion assms)+
have [simp]: "fps_expansion f z \<noteq> 0"
proof
assume "fps_expansion f z = 0"
hence "eventually (\<lambda>z. f z * g z = 0) (at z)" using *(1)
by (auto simp: has_fps_expansion_0_iff nhds_to_0' eventually_filtermap eventually_at_filter
elim: eventually_mono)
with assms(3) have "eventually (\<lambda>z. False) (at z)"
by eventually_elim auto
thus False by simp
qed
have [simp]: "fps_expansion g z \<noteq> 0"
proof
assume "fps_expansion g z = 0"
hence "eventually (\<lambda>z. f z * g z = 0) (at z)" using *(2)
by (auto simp: has_fps_expansion_0_iff nhds_to_0' eventually_filtermap eventually_at_filter
elim: eventually_mono)
with assms(3) have "eventually (\<lambda>z. False) (at z)"
by eventually_elim auto
thus False by simp
qed
from *[THEN has_fps_expansion_zorder] show ?thesis
by auto
qed
lemma zorder_const [simp]: "c \<noteq> 0 \<Longrightarrow> zorder (\<lambda>_. c) z = 0"
by (intro zorder_eqI[where S = UNIV]) auto
lemma zorder_prod_analytic:
assumes "\<And>x. x \<in> A \<Longrightarrow> f x analytic_on {z}"
assumes "eventually (\<lambda>z. (\<Prod>x\<in>A. f x z) \<noteq> 0) (at z)"
shows "zorder (\<lambda>z. \<Prod>x\<in>A. f x z) z = (\<Sum>x\<in>A. zorder (f x) z)"
using assms
proof (induction A rule: infinite_finite_induct)
case (insert x A)
have "zorder (\<lambda>z. f x z * (\<Prod>x\<in>A. f x z)) z = zorder (f x) z + zorder (\<lambda>z. \<Prod>x\<in>A. f x z) z"
using insert.prems insert.hyps by (intro zorder_times_analytic analytic_intros) auto
also have "zorder (\<lambda>z. \<Prod>x\<in>A. f x z) z = (\<Sum>x\<in>A. zorder (f x) z)"
using insert.prems insert.hyps by (intro insert.IH) (auto elim!: eventually_mono)
finally show ?case using insert
by simp
qed auto
lemma zorder_eq_0I:
assumes "g analytic_on {z}" "g z \<noteq> 0"
shows "zorder g z = 0"
using analytic_at assms zorder_eqI by fastforce
lemma zorder_pos_iff:
assumes "f holomorphic_on A" "open A" "z \<in> A" "frequently (\<lambda>z. f z \<noteq> 0) (at z)"
shows "zorder f z > 0 \<longleftrightarrow> f z = 0"
proof -
have "f analytic_on {z}"
using assms analytic_at by blast
hence *: "(\<lambda>w. f (z + w)) has_fps_expansion fps_expansion f z"
using analytic_at_imp_has_fps_expansion by blast
have nz: "fps_expansion f z \<noteq> 0"
proof
assume "fps_expansion f z = 0"
hence "eventually (\<lambda>z. f z = 0) (nhds z)"
using * by (auto simp: has_fps_expansion_def nhds_to_0' eventually_filtermap add_ac)
hence "eventually (\<lambda>z. f z = 0) (at z)"
by (auto simp: eventually_at_filter elim: eventually_mono)
with assms show False
by (auto simp: frequently_def)
qed
from has_fps_expansion_zorder[OF * this] have eq: "zorder f z = int (subdegree (fps_expansion f z))"
by auto
moreover have "subdegree (fps_expansion f z) = 0 \<longleftrightarrow> fps_nth (fps_expansion f z) 0 \<noteq> 0"
using nz by (auto simp: subdegree_eq_0_iff)
moreover have "fps_nth (fps_expansion f z) 0 = f z"
by (auto simp: fps_expansion_def)
ultimately show ?thesis
by auto
qed
lemma zorder_pos_iff':
assumes "f analytic_on {z}" "frequently (\<lambda>z. f z \<noteq> 0) (at z)"
shows "zorder f z > 0 \<longleftrightarrow> f z = 0"
using analytic_at assms zorder_pos_iff by blast
lemma zorder_ge_0:
assumes "f analytic_on {z}" "frequently (\<lambda>z. f z \<noteq> 0) (at z)"
shows "zorder f z \<ge> 0"
proof -
have *: "(\<lambda>w. f (z + w)) has_laurent_expansion fps_to_fls (fps_expansion f z)"
using assms by (simp add: analytic_at_imp_has_fps_expansion has_laurent_expansion_fps)
from * assms(2) have "fps_to_fls (fps_expansion f z) \<noteq> 0"
by (auto simp: has_laurent_expansion_def frequently_def at_to_0' eventually_filtermap add_ac)
with has_laurent_expansion_zorder[OF *] show ?thesis
by (simp add: fls_subdegree_fls_to_fps)
qed
lemma zorder_eq_0_iff:
assumes "f analytic_on {z}" "frequently (\<lambda>w. f w \<noteq> 0) (at z)"
shows "zorder f z = 0 \<longleftrightarrow> f z \<noteq> 0"
using assms zorder_eq_0I zorder_pos_iff' by fastforce
lemma zorder_scale:
assumes "f analytic_on {a * z}" "eventually (\<lambda>w. f w \<noteq> 0) (at (a * z))" "a \<noteq> 0"
shows "zorder (\<lambda>w. f (a * w)) z = zorder f (a * z)"
proof -
from assms(1) obtain r where r: "r > 0" "f holomorphic_on ball (a * z) r"
by (auto simp: analytic_on_def)
have *: "open (ball (a * z) r)" "connected (ball (a * z) r)" "a * z \<in> ball (a * z) r"
using r \<open>a \<noteq> 0\<close> by (auto simp: dist_norm)
from assms(2) have "eventually (\<lambda>w. f w \<noteq> 0 \<and> w \<in> ball (a * z) r - {a * z}) (at (a * z))"
using \<open>r > 0\<close> by (intro eventually_conj eventually_at_in_open) auto
then obtain z0 where "f z0 \<noteq> 0 \<and> z0 \<in> ball (a * z) r - {a * z}"
using eventually_happens[of _ "at (a * z)"] by force
hence **: "\<exists>w\<in>ball (a * z) r. f w \<noteq> 0"
by blast
define n where "n = nat (zorder f (a * z))"
obtain r' where r':
"(if f (a * z) = 0 then 0 < zorder f (a * z) else zorder f (a * z) = 0)"
"r' > 0" "cball (a * z) r' \<subseteq> ball (a * z) r" "zor_poly f (a * z) holomorphic_on cball (a * z) r'"
"\<And>w. w \<in> cball (a * z) r' \<Longrightarrow>
f w = zor_poly f (a * z) w * (w - a * z) ^ n \<and> zor_poly f (a * z) w \<noteq> 0"
unfolding n_def using zorder_exist_zero[OF r(2) * **] by blast
show ?thesis
proof (rule zorder_eqI)
show "open (ball z (r' / norm a))" "z \<in> ball z (r' / norm a)"
using r \<open>r' > 0\<close> \<open>a \<noteq> 0\<close> by auto
have "(*) a ` ball z (r' / cmod a) \<subseteq> cball (a * z) r'"
proof safe
fix w assume "w \<in> ball z (r' / cmod a)"
thus "a * w \<in> cball (a * z) r'"
using dist_mult_left[of a z w] \<open>a \<noteq> 0\<close> by (auto simp: divide_simps mult_ac)
qed
thus "(\<lambda>w. a ^ n * (zor_poly f (a * z) \<circ> (\<lambda>w. a * w)) w) holomorphic_on ball z (r' / norm a)"
using \<open>a \<noteq> 0\<close> by (intro holomorphic_on_compose_gen[OF _ r'(4)] holomorphic_intros) auto
show "a ^ n * (zor_poly f (a * z) \<circ> (\<lambda>w. a * w)) z \<noteq> 0"
using r' \<open>a \<noteq> 0\<close> by auto
show "f (a * w) = a ^ n * (zor_poly f (a * z) \<circ> (*) a) w * (w - z) powi (zorder f (a * z))"
if "w \<in> ball z (r' / norm a)" "w \<noteq> z" for w
proof -
have "f (a * w) = zor_poly f (a * z) (a * w) * (a * (w - z)) ^ n"
using that r'(5)[of "a * w"] dist_mult_left[of a z w] \<open>a \<noteq> 0\<close> unfolding ring_distribs
by (auto simp: divide_simps mult_ac)
also have "\<dots> = a ^ n * zor_poly f (a * z) (a * w) * (w - z) ^ n"
by (subst power_mult_distrib) (auto simp: mult_ac)
also have "(w - z) ^ n = (w - z) powi of_nat n"
by simp
also have "of_nat n = zorder f (a * z)"
using r'(1) by (auto simp: n_def split: if_splits)
finally show ?thesis
unfolding o_def n_def .
qed
qed
qed
lemma zorder_compose_aux:
assumes "isolated_singularity_at f 0" "not_essential f 0"
assumes G: "g has_fps_expansion G" "G \<noteq> 0" "g 0 = 0"
assumes "eventually (\<lambda>w. f w \<noteq> 0) (at 0)"
shows "zorder (f \<circ> g) 0 = zorder f 0 * subdegree G"
proof -
obtain F where F: "f has_laurent_expansion F"
using not_essential_has_laurent_expansion_0[OF assms(1,2)] by blast
have [simp]: "fps_nth G 0 = 0"
using G \<open>g 0 = 0\<close> by (simp add: has_fps_expansion_imp_0_eq_fps_nth_0)
note [simp] = \<open>G \<noteq> 0\<close> \<open>g 0 = 0\<close>
have [simp]: "F \<noteq> 0"
using has_laurent_expansion_eventually_nonzero_iff[of f 0 F] F assms by simp
have FG: "(f \<circ> g) has_laurent_expansion fls_compose_fps F G"
by (intro has_laurent_expansion_compose has_laurent_expansion_fps F G) auto
have "zorder (f \<circ> g) 0 = fls_subdegree (fls_compose_fps F G)"
using has_laurent_expansion_zorder_0 [OF FG] by (auto simp: fls_compose_fps_eq_0_iff)
also have "\<dots> = fls_subdegree F * int (subdegree G)"
by simp
also have "fls_subdegree F = zorder f 0"
using has_laurent_expansion_zorder_0 [OF F] by auto
finally show ?thesis .
qed
lemma zorder_compose:
assumes "isolated_singularity_at f (g z)" "not_essential f (g z)"
assumes G: "(\<lambda>x. g (z + x) - g z) has_fps_expansion G" "G \<noteq> 0"
assumes "eventually (\<lambda>w. f w \<noteq> 0) (at (g z))"
shows "zorder (f \<circ> g) z = zorder f (g z) * subdegree G"
proof -
define f' where "f' = (\<lambda>w. f (g z + w))"
define g' where "g' = (\<lambda>w. g (z + w) - g z)"
have "zorder f (g z) = zorder f' 0"
by (simp add: f'_def zorder_shift' add_ac)
have "zorder (\<lambda>x. g x - g z) z = zorder g' 0"
by (simp add: g'_def zorder_shift' add_ac)
have "zorder (f \<circ> g) z = zorder (f' \<circ> g') 0"
by (simp add: zorder_shift' f'_def g'_def add_ac o_def)
also have "\<dots> = zorder f' 0 * int (subdegree G)"
proof (rule zorder_compose_aux)
show "isolated_singularity_at f' 0" unfolding f'_def
using assms has_laurent_expansion_isolated_0 not_essential_has_laurent_expansion by blast
show "not_essential f' 0" unfolding f'_def
using assms has_laurent_expansion_not_essential_0 not_essential_has_laurent_expansion by blast
qed (use assms in \<open>auto simp: f'_def g'_def at_to_0' eventually_filtermap add_ac\<close>)
also have "zorder f' 0 = zorder f (g z)"
by (simp add: f'_def zorder_shift' add_ac)
finally show ?thesis .
qed
lemma fps_to_fls_eq_fls_const_iff [simp]: "fps_to_fls F = fls_const c \<longleftrightarrow> F = fps_const c"
using fps_to_fls_eq_iff by fastforce
lemma zorder_compose':
assumes "isolated_singularity_at f (g z)" "not_essential f (g z)"
assumes "g analytic_on {z}"
assumes "eventually (\<lambda>w. f w \<noteq> 0) (at (g z))"
assumes "eventually (\<lambda>w. g w \<noteq> g z) (at z)"
shows "zorder (f \<circ> g) z = zorder f (g z) * zorder (\<lambda>x. g x - g z) z"
proof -
obtain G where G [fps_expansion_intros]: "(\<lambda>x. g (z + x)) has_fps_expansion G"
using assms analytic_at_imp_has_fps_expansion by blast
have G': "(\<lambda>x. g (z + x) - g z) has_fps_expansion G - fps_const (g z)"
by (intro fps_expansion_intros)
hence G'': "(\<lambda>x. g (z + x) - g z) has_laurent_expansion fps_to_fls (G - fps_const (g z))"
using has_laurent_expansion_fps by blast
have nz: "G - fps_const (g z) \<noteq> 0"
using has_laurent_expansion_eventually_nonzero_iff[OF G''] assms by auto
have "zorder (f \<circ> g) z = zorder f (g z) * subdegree (G - fps_const (g z))"
proof (rule zorder_compose)
show "(\<lambda>x. g (z + x) - g z) has_fps_expansion G - fps_const (g z)"
by (intro fps_expansion_intros)
qed (use assms nz in auto)
also have "int (subdegree (G - fps_const (g z))) = fls_subdegree (fps_to_fls G - fls_const (g z))"
by (simp flip: fls_subdegree_fls_to_fps)
also have "\<dots> = zorder (\<lambda>x. g x - g z) z"
using has_laurent_expansion_zorder [OF G''] nz by auto
finally show ?thesis .
qed
lemma analytic_at_cong:
assumes "eventually (\<lambda>x. f x = g x) (nhds x)" "x = y"
shows "f analytic_on {x} \<longleftrightarrow> g analytic_on {y}"
proof -
have "g analytic_on {x}" if "f analytic_on {x}" "eventually (\<lambda>x. f x = g x) (nhds x)" for f g
proof -
have "(\<lambda>y. f (x + y)) has_fps_expansion fps_expansion f x"
by (rule analytic_at_imp_has_fps_expansion) fact
also have "?this \<longleftrightarrow> (\<lambda>y. g (x + y)) has_fps_expansion fps_expansion f x"
using that by (intro has_fps_expansion_cong refl) (auto simp: nhds_to_0' eventually_filtermap)
finally show ?thesis
by (rule has_fps_expansion_imp_analytic)
qed
from this[of f g] this[of g f] show ?thesis using assms
by (auto simp: eq_commute)
qed
lemma has_laurent_expansion_sin' [laurent_expansion_intros]:
"sin has_laurent_expansion fps_to_fls (fps_sin 1)"
using has_fps_expansion_sin' has_fps_expansion_to_laurent by blast
lemma has_laurent_expansion_cos' [laurent_expansion_intros]:
"cos has_laurent_expansion fps_to_fls (fps_cos 1)"
using has_fps_expansion_cos' has_fps_expansion_to_laurent by blast
lemma has_laurent_expansion_sin [laurent_expansion_intros]:
"(\<lambda>z. sin (c * z)) has_laurent_expansion fps_to_fls (fps_sin c)"
by (intro has_laurent_expansion_fps has_fps_expansion_sin)
lemma has_laurent_expansion_cos [laurent_expansion_intros]:
"(\<lambda>z. cos (c * z)) has_laurent_expansion fps_to_fls (fps_cos c)"
by (intro has_laurent_expansion_fps has_fps_expansion_cos)
lemma has_laurent_expansion_tan' [laurent_expansion_intros]:
"tan has_laurent_expansion fps_to_fls (fps_tan 1)"
using has_fps_expansion_tan' has_fps_expansion_to_laurent by blast
lemma has_laurent_expansion_tan [laurent_expansion_intros]:
"(\<lambda>z. tan (c * z)) has_laurent_expansion fps_to_fls (fps_tan c)"
by (intro has_laurent_expansion_fps has_fps_expansion_tan)
subsection \<open>More Laurent expansions\<close>
lemma has_laurent_expansion_frequently_zero_iff:
assumes "(\<lambda>w. f (z + w)) has_laurent_expansion F"
shows "frequently (\<lambda>z. f z = 0) (at z) \<longleftrightarrow> F = 0"
using assms by (simp add: frequently_def has_laurent_expansion_eventually_nonzero_iff)
lemma has_laurent_expansion_eventually_zero_iff:
assumes "(\<lambda>w. f (z + w)) has_laurent_expansion F"
shows "eventually (\<lambda>z. f z = 0) (at z) \<longleftrightarrow> F = 0"
using assms
by (metis has_laurent_expansion_frequently_zero_iff has_laurent_expansion_isolated
has_laurent_expansion_not_essential laurent_expansion_def
not_essential_frequently_0_imp_eventually_0 not_essential_has_laurent_expansion)
lemma has_laurent_expansion_frequently_nonzero_iff:
assumes "(\<lambda>w. f (z + w)) has_laurent_expansion F"
shows "frequently (\<lambda>z. f z \<noteq> 0) (at z) \<longleftrightarrow> F \<noteq> 0"
using assms by (metis has_laurent_expansion_eventually_zero_iff not_eventually)
lemma has_laurent_expansion_sum_list [laurent_expansion_intros]:
assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x has_laurent_expansion F x"
shows "(\<lambda>y. \<Sum>x\<leftarrow>xs. f x y) has_laurent_expansion (\<Sum>x\<leftarrow>xs. F x)"
using assms by (induction xs) (auto intro!: laurent_expansion_intros)
lemma has_laurent_expansion_prod_list [laurent_expansion_intros]:
assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x has_laurent_expansion F x"
shows "(\<lambda>y. \<Prod>x\<leftarrow>xs. f x y) has_laurent_expansion (\<Prod>x\<leftarrow>xs. F x)"
using assms by (induction xs) (auto intro!: laurent_expansion_intros)
lemma has_laurent_expansion_sum_mset [laurent_expansion_intros]:
assumes "\<And>x. x \<in># I \<Longrightarrow> f x has_laurent_expansion F x"
shows "(\<lambda>y. \<Sum>x\<in>#I. f x y) has_laurent_expansion (\<Sum>x\<in>#I. F x)"
using assms by (induction I) (auto intro!: laurent_expansion_intros)
lemma has_laurent_expansion_prod_mset [laurent_expansion_intros]:
assumes "\<And>x. x \<in># I \<Longrightarrow> f x has_laurent_expansion F x"
shows "(\<lambda>y. \<Prod>x\<in>#I. f x y) has_laurent_expansion (\<Prod>x\<in>#I. F x)"
using assms by (induction I) (auto intro!: laurent_expansion_intros)
subsection \<open>Formal convergence versus analytic convergence\<close>
text \<open>
The convergence of a sequence of formal power series and the convergence of the functions
in the complex plane do not imply each other:
\<^item> If we have the sequence of constant power series $(1/n)_{n\geq 0}$, this clearly converges
to the zero function analytically, but as a series of formal power series it is divergent
(since the 0-th coefficient never stabilises).
\<^item> Conversely, the sequence of series $(n! x^n)_{n\geq 0}$ converges formally to $0$,
but the corresponding sequence of functions diverges for every $x \neq 0$.
However, if the sequence of series converges to some limit series $h$ and the corresponding
series of functions converges uniformly to some limit function $g(x)$, then $h$ is also a
series expansion of $g(x)$, i.e.\ in that case, formal and analytic convergence agree.
\<close>
proposition uniform_limit_imp_fps_expansion_eq:
fixes f :: "'a \<Rightarrow> complex fps"
assumes lim1: "(f \<longlongrightarrow> h) F"
assumes lim2: "uniform_limit A (\<lambda>x z. f' x z) g' F"
assumes expansions: "eventually (\<lambda>x. f' x has_fps_expansion f x) F" "g' has_fps_expansion g"
assumes holo: "eventually (\<lambda>x. f' x holomorphic_on A) F"
assumes A: "open A" "0 \<in> A"
assumes nontriv [simp]: "F \<noteq> bot"
shows "g = h"
proof (rule fps_ext)
fix n :: nat
have "eventually (\<lambda>x. fps_nth (f x) n = fps_nth h n) F"
using lim1 unfolding tendsto_fps_iff by blast
hence "eventually (\<lambda>x. (deriv ^^ n) (f' x) 0 / fact n = fps_nth h n) F"
using expansions(1)
proof eventually_elim
case (elim x)
have "fps_nth (f x) n = (deriv ^^ n) (f' x) 0 / fact n"
by (rule fps_nth_fps_expansion) (use elim in auto)
with elim show ?case
by simp
qed
hence "((\<lambda>x. (deriv ^^ n) (f' x) 0 / fact n) \<longlongrightarrow> fps_nth h n) F"
by (simp add: tendsto_eventually)
moreover have "((\<lambda>x. (deriv ^^ n) (f' x) 0) \<longlongrightarrow> (deriv ^^ n) g' 0) F"
using lim2
proof (rule higher_deriv_complex_uniform_limit)
show "eventually (\<lambda>x. f' x holomorphic_on A) F"
using holo by eventually_elim auto
qed (use A in auto)
hence "((\<lambda>x. (deriv ^^ n) (f' x) 0 / fact n) \<longlongrightarrow> (deriv ^^ n) g' 0 / fact n) F"
by (intro tendsto_divide) auto
ultimately have "fps_nth h n = (deriv ^^ n) g' 0 / fact n"
using tendsto_unique[OF nontriv] by blast
also have "\<dots> = fps_nth g n"
by (rule fps_nth_fps_expansion [symmetric]) fact
finally show "fps_nth g n = fps_nth h n" ..
qed
end