src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy
author paulson <lp15@cam.ac.uk>
Thu Feb 25 13:58:48 2016 +0000 (2016-02-25)
changeset 62408 86f27b264d3d
parent 62397 5ae24f33d343
child 62533 bc25f3916a99
permissions -rw-r--r--
Conformal_mappings: a big development in complex analysis (+ some lemmas)
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(*  Author: John Harrison, Marco Maggesi, Graziano Gentili, Gianni Ciolli, Valentina Bruno
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    Ported from "hol_light/Multivariate/canal.ml" by L C Paulson (2014)
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*)
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section \<open>Complex Analysis Basics\<close>
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theory Complex_Analysis_Basics
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imports Cartesian_Euclidean_Space "~~/src/HOL/Library/Nonpos_Ints"
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begin
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subsection\<open>General lemmas\<close>
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lemma nonneg_Reals_cmod_eq_Re: "z \<in> \<real>\<^sub>\<ge>\<^sub>0 \<Longrightarrow> norm z = Re z"
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  by (simp add: complex_nonneg_Reals_iff cmod_eq_Re)
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lemma has_derivative_mult_right:
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  fixes c:: "'a :: real_normed_algebra"
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  shows "((op * c) has_derivative (op * c)) F"
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by (rule has_derivative_mult_right [OF has_derivative_id])
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lemma has_derivative_of_real[derivative_intros, simp]:
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  "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. of_real (f x)) has_derivative (\<lambda>x. of_real (f' x))) F"
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  using bounded_linear.has_derivative[OF bounded_linear_of_real] .
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lemma has_vector_derivative_real_complex:
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  "DERIV f (of_real a) :> f' \<Longrightarrow> ((\<lambda>x. f (of_real x)) has_vector_derivative f') (at a within s)"
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  using has_derivative_compose[of of_real of_real a _ f "op * f'"]
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  by (simp add: scaleR_conv_of_real ac_simps has_vector_derivative_def has_field_derivative_def)
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lemma fact_cancel:
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  fixes c :: "'a::real_field"
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  shows "of_nat (Suc n) * c / (fact (Suc n)) = c / (fact n)"
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  by (simp add: of_nat_mult del: of_nat_Suc times_nat.simps)
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lemma bilinear_times:
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  fixes c::"'a::real_algebra" shows "bilinear (\<lambda>x y::'a. x*y)"
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  by (auto simp: bilinear_def distrib_left distrib_right intro!: linearI)
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lemma linear_cnj: "linear cnj"
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  using bounded_linear.linear[OF bounded_linear_cnj] .
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lemma tendsto_Re_upper:
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  assumes "~ (trivial_limit F)"
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          "(f \<longlongrightarrow> l) F"
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          "eventually (\<lambda>x. Re(f x) \<le> b) F"
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    shows  "Re(l) \<le> b"
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  by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Re)
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lemma tendsto_Re_lower:
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  assumes "~ (trivial_limit F)"
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          "(f \<longlongrightarrow> l) F"
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          "eventually (\<lambda>x. b \<le> Re(f x)) F"
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    shows  "b \<le> Re(l)"
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  by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Re)
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lemma tendsto_Im_upper:
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  assumes "~ (trivial_limit F)"
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          "(f \<longlongrightarrow> l) F"
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          "eventually (\<lambda>x. Im(f x) \<le> b) F"
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    shows  "Im(l) \<le> b"
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  by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Im)
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lemma tendsto_Im_lower:
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  assumes "~ (trivial_limit F)"
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          "(f \<longlongrightarrow> l) F"
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          "eventually (\<lambda>x. b \<le> Im(f x)) F"
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    shows  "b \<le> Im(l)"
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  by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Im)
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lemma lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = op * 0"
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  by auto
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lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = op * 1"
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  by auto
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lemma continuous_mult_left:
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  fixes c::"'a::real_normed_algebra"
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  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. c * f x)"
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by (rule continuous_mult [OF continuous_const])
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lemma continuous_mult_right:
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  fixes c::"'a::real_normed_algebra"
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  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x * c)"
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by (rule continuous_mult [OF _ continuous_const])
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lemma continuous_on_mult_left:
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  fixes c::"'a::real_normed_algebra"
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  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c * f x)"
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by (rule continuous_on_mult [OF continuous_on_const])
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lemma continuous_on_mult_right:
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  fixes c::"'a::real_normed_algebra"
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  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x * c)"
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by (rule continuous_on_mult [OF _ continuous_on_const])
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lemma uniformly_continuous_on_cmul_right [continuous_intros]:
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  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
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  shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. f x * c)"
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  using bounded_linear.uniformly_continuous_on[OF bounded_linear_mult_left] .
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lemma uniformly_continuous_on_cmul_left[continuous_intros]:
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  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
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  assumes "uniformly_continuous_on s f"
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    shows "uniformly_continuous_on s (\<lambda>x. c * f x)"
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by (metis assms bounded_linear.uniformly_continuous_on bounded_linear_mult_right)
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lemma continuous_within_norm_id [continuous_intros]: "continuous (at x within S) norm"
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  by (rule continuous_norm [OF continuous_ident])
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lemma continuous_on_norm_id [continuous_intros]: "continuous_on S norm"
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  by (intro continuous_on_id continuous_on_norm)
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subsection\<open>DERIV stuff\<close>
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lemma DERIV_zero_connected_constant:
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  fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
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  assumes "connected s"
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      and "open s"
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      and "finite k"
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      and "continuous_on s f"
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      and "\<forall>x\<in>(s - k). DERIV f x :> 0"
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    obtains c where "\<And>x. x \<in> s \<Longrightarrow> f(x) = c"
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using has_derivative_zero_connected_constant [OF assms(1-4)] assms
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by (metis DERIV_const has_derivative_const Diff_iff at_within_open frechet_derivative_at has_field_derivative_def)
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lemma DERIV_zero_constant:
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  fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
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  shows    "\<lbrakk>convex s;
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             \<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)\<rbrakk>
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             \<Longrightarrow> \<exists>c. \<forall>x \<in> s. f(x) = c"
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  by (auto simp: has_field_derivative_def lambda_zero intro: has_derivative_zero_constant)
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lemma DERIV_zero_unique:
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  fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
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  assumes "convex s"
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      and d0: "\<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)"
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      and "a \<in> s"
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      and "x \<in> s"
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    shows "f x = f a"
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  by (rule has_derivative_zero_unique [OF assms(1) _ assms(4,3)])
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     (metis d0 has_field_derivative_imp_has_derivative lambda_zero)
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lemma DERIV_zero_connected_unique:
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  fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
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  assumes "connected s"
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      and "open s"
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      and d0: "\<And>x. x\<in>s \<Longrightarrow> DERIV f x :> 0"
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      and "a \<in> s"
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      and "x \<in> s"
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    shows "f x = f a"
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    by (rule has_derivative_zero_unique_connected [OF assms(2,1) _ assms(5,4)])
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       (metis has_field_derivative_def lambda_zero d0)
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lemma DERIV_transform_within:
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  assumes "(f has_field_derivative f') (at a within s)"
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      and "0 < d" "a \<in> s"
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      and "\<And>x. x\<in>s \<Longrightarrow> dist x a < d \<Longrightarrow> f x = g x"
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    shows "(g has_field_derivative f') (at a within s)"
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  using assms unfolding has_field_derivative_def
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  by (blast intro: has_derivative_transform_within)
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lemma DERIV_transform_within_open:
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  assumes "DERIV f a :> f'"
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      and "open s" "a \<in> s"
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      and "\<And>x. x\<in>s \<Longrightarrow> f x = g x"
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    shows "DERIV g a :> f'"
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  using assms unfolding has_field_derivative_def
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by (metis has_derivative_transform_within_open)
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lemma DERIV_transform_at:
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  assumes "DERIV f a :> f'"
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      and "0 < d"
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      and "\<And>x. dist x a < d \<Longrightarrow> f x = g x"
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    shows "DERIV g a :> f'"
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  by (blast intro: assms DERIV_transform_within)
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(*generalising DERIV_isconst_all, which requires type real (using the ordering)*)
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lemma DERIV_zero_UNIV_unique:
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  fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
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  shows "(\<And>x. DERIV f x :> 0) \<Longrightarrow> f x = f a"
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by (metis DERIV_zero_unique UNIV_I assms convex_UNIV)
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subsection \<open>Some limit theorems about real part of real series etc.\<close>
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(*MOVE? But not to Finite_Cartesian_Product*)
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lemma sums_vec_nth :
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  assumes "f sums a"
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  shows "(\<lambda>x. f x $ i) sums a $ i"
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using assms unfolding sums_def
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by (auto dest: tendsto_vec_nth [where i=i])
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lemma summable_vec_nth :
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  assumes "summable f"
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  shows "summable (\<lambda>x. f x $ i)"
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using assms unfolding summable_def
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by (blast intro: sums_vec_nth)
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subsection \<open>Complex number lemmas\<close>
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lemma
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  shows open_halfspace_Re_lt: "open {z. Re(z) < b}"
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    and open_halfspace_Re_gt: "open {z. Re(z) > b}"
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    and closed_halfspace_Re_ge: "closed {z. Re(z) \<ge> b}"
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    and closed_halfspace_Re_le: "closed {z. Re(z) \<le> b}"
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    and closed_halfspace_Re_eq: "closed {z. Re(z) = b}"
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    and open_halfspace_Im_lt: "open {z. Im(z) < b}"
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    and open_halfspace_Im_gt: "open {z. Im(z) > b}"
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    and closed_halfspace_Im_ge: "closed {z. Im(z) \<ge> b}"
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    and closed_halfspace_Im_le: "closed {z. Im(z) \<le> b}"
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    and closed_halfspace_Im_eq: "closed {z. Im(z) = b}"
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  by (intro open_Collect_less closed_Collect_le closed_Collect_eq isCont_Re
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            isCont_Im continuous_ident continuous_const)+
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lemma closed_complex_Reals: "closed (\<real> :: complex set)"
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proof -
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  have "(\<real> :: complex set) = {z. Im z = 0}"
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    by (auto simp: complex_is_Real_iff)
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  then show ?thesis
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    by (metis closed_halfspace_Im_eq)
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qed
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lemma closed_Real_halfspace_Re_le: "closed (\<real> \<inter> {w. Re w \<le> x})"
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  by (simp add: closed_Int closed_complex_Reals closed_halfspace_Re_le)
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corollary closed_nonpos_Reals_complex [simp]: "closed (\<real>\<^sub>\<le>\<^sub>0 :: complex set)"
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proof -
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  have "\<real>\<^sub>\<le>\<^sub>0 = \<real> \<inter> {z. Re(z) \<le> 0}"
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    using complex_nonpos_Reals_iff complex_is_Real_iff by auto
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  then show ?thesis
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    by (metis closed_Real_halfspace_Re_le)
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qed
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lemma closed_Real_halfspace_Re_ge: "closed (\<real> \<inter> {w. x \<le> Re(w)})"
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  using closed_halfspace_Re_ge
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  by (simp add: closed_Int closed_complex_Reals)
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corollary closed_nonneg_Reals_complex [simp]: "closed (\<real>\<^sub>\<ge>\<^sub>0 :: complex set)"
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proof -
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  have "\<real>\<^sub>\<ge>\<^sub>0 = \<real> \<inter> {z. Re(z) \<ge> 0}"
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    using complex_nonneg_Reals_iff complex_is_Real_iff by auto
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  then show ?thesis
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    by (metis closed_Real_halfspace_Re_ge)
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qed
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lemma closed_real_abs_le: "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
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proof -
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  have "{w \<in> \<real>. \<bar>Re w\<bar> \<le> r} = (\<real> \<inter> {w. Re w \<le> r}) \<inter> (\<real> \<inter> {w. Re w \<ge> -r})"
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    by auto
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  then show "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
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    by (simp add: closed_Int closed_Real_halfspace_Re_ge closed_Real_halfspace_Re_le)
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qed
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lemma real_lim:
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  fixes l::complex
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  assumes "(f \<longlongrightarrow> l) F" and "~(trivial_limit F)" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>"
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  shows  "l \<in> \<real>"
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proof (rule Lim_in_closed_set[OF closed_complex_Reals _ assms(2,1)])
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  show "eventually (\<lambda>x. f x \<in> \<real>) F"
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    using assms(3, 4) by (auto intro: eventually_mono)
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qed
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lemma real_lim_sequentially:
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  fixes l::complex
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  shows "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
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by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially)
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lemma real_series:
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  fixes l::complex
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  shows "f sums l \<Longrightarrow> (\<And>n. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
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unfolding sums_def
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by (metis real_lim_sequentially setsum_in_Reals)
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lemma Lim_null_comparison_Re:
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  assumes "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F" "(g \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F"
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  by (rule Lim_null_comparison[OF assms(1)] tendsto_eq_intros assms(2))+ simp
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subsection\<open>Holomorphic functions\<close>
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text\<open>Could be generalized to real normed fields, but in practice that would only include the reals\<close>
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definition complex_differentiable :: "[complex \<Rightarrow> complex, complex filter] \<Rightarrow> bool"
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           (infixr "(complex'_differentiable)" 50)
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  where "f complex_differentiable F \<equiv> \<exists>f'. (f has_field_derivative f') F"
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lemma complex_differentiable_imp_continuous_at:
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    "f complex_differentiable (at x within s) \<Longrightarrow> continuous (at x within s) f"
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  by (metis DERIV_continuous complex_differentiable_def)
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lemma complex_differentiable_within_subset:
lp15@56215
   290
    "\<lbrakk>f complex_differentiable (at x within s); t \<subseteq> s\<rbrakk>
lp15@56215
   291
     \<Longrightarrow> f complex_differentiable (at x within t)"
hoelzl@56370
   292
  by (metis DERIV_subset complex_differentiable_def)
lp15@56215
   293
lp15@56215
   294
lemma complex_differentiable_at_within:
lp15@56215
   295
    "\<lbrakk>f complex_differentiable (at x)\<rbrakk>
lp15@56215
   296
     \<Longrightarrow> f complex_differentiable (at x within s)"
lp15@56215
   297
  unfolding complex_differentiable_def
lp15@56215
   298
  by (metis DERIV_subset top_greatest)
lp15@56215
   299
lp15@61520
   300
lemma complex_differentiable_linear [derivative_intros]: "(op * c) complex_differentiable F"
lp15@56215
   301
proof -
hoelzl@56370
   302
  show ?thesis
hoelzl@56370
   303
    unfolding complex_differentiable_def has_field_derivative_def mult_commute_abs
lp15@56215
   304
    by (force intro: has_derivative_mult_right)
lp15@56215
   305
qed
lp15@56215
   306
lp15@61520
   307
lemma complex_differentiable_const [derivative_intros]: "(\<lambda>z. c) complex_differentiable F"
lp15@56215
   308
  unfolding complex_differentiable_def has_field_derivative_def
hoelzl@56369
   309
  by (rule exI [where x=0])
lp15@61609
   310
     (metis has_derivative_const lambda_zero)
lp15@56215
   311
lp15@61520
   312
lemma complex_differentiable_ident [derivative_intros]: "(\<lambda>z. z) complex_differentiable F"
lp15@56215
   313
  unfolding complex_differentiable_def has_field_derivative_def
hoelzl@56369
   314
  by (rule exI [where x=1])
hoelzl@56369
   315
     (simp add: lambda_one [symmetric])
lp15@56215
   316
lp15@61520
   317
lemma complex_differentiable_id [derivative_intros]: "id complex_differentiable F"
hoelzl@56370
   318
  unfolding id_def by (rule complex_differentiable_ident)
hoelzl@56370
   319
lp15@61520
   320
lemma complex_differentiable_minus [derivative_intros]:
hoelzl@56370
   321
  "f complex_differentiable F \<Longrightarrow> (\<lambda>z. - (f z)) complex_differentiable F"
lp15@56215
   322
  using assms unfolding complex_differentiable_def
lp15@56215
   323
  by (metis field_differentiable_minus)
lp15@56215
   324
lp15@61520
   325
lemma complex_differentiable_add [derivative_intros]:
lp15@56215
   326
  assumes "f complex_differentiable F" "g complex_differentiable F"
lp15@56215
   327
    shows "(\<lambda>z. f z + g z) complex_differentiable F"
lp15@56215
   328
  using assms unfolding complex_differentiable_def
lp15@56215
   329
  by (metis field_differentiable_add)
lp15@56215
   330
lp15@61520
   331
lemma complex_differentiable_setsum [derivative_intros]:
hoelzl@56370
   332
  "(\<And>i. i \<in> I \<Longrightarrow> (f i) complex_differentiable F) \<Longrightarrow> (\<lambda>z. \<Sum>i\<in>I. f i z) complex_differentiable F"
hoelzl@56370
   333
  by (induct I rule: infinite_finite_induct)
hoelzl@56370
   334
     (auto intro: complex_differentiable_add complex_differentiable_const)
hoelzl@56370
   335
lp15@61520
   336
lemma complex_differentiable_diff [derivative_intros]:
lp15@56215
   337
  assumes "f complex_differentiable F" "g complex_differentiable F"
lp15@56215
   338
    shows "(\<lambda>z. f z - g z) complex_differentiable F"
lp15@56215
   339
  using assms unfolding complex_differentiable_def
lp15@56215
   340
  by (metis field_differentiable_diff)
lp15@56215
   341
lp15@61520
   342
lemma complex_differentiable_inverse [derivative_intros]:
lp15@56215
   343
  assumes "f complex_differentiable (at a within s)" "f a \<noteq> 0"
lp15@56215
   344
  shows "(\<lambda>z. inverse (f z)) complex_differentiable (at a within s)"
lp15@56215
   345
  using assms unfolding complex_differentiable_def
lp15@56215
   346
  by (metis DERIV_inverse_fun)
lp15@56215
   347
lp15@61520
   348
lemma complex_differentiable_mult [derivative_intros]:
lp15@61609
   349
  assumes "f complex_differentiable (at a within s)"
lp15@56215
   350
          "g complex_differentiable (at a within s)"
lp15@56215
   351
    shows "(\<lambda>z. f z * g z) complex_differentiable (at a within s)"
lp15@56215
   352
  using assms unfolding complex_differentiable_def
lp15@56215
   353
  by (metis DERIV_mult [of f _ a s g])
lp15@61609
   354
lp15@61520
   355
lemma complex_differentiable_divide [derivative_intros]:
lp15@61609
   356
  assumes "f complex_differentiable (at a within s)"
lp15@56215
   357
          "g complex_differentiable (at a within s)"
lp15@56215
   358
          "g a \<noteq> 0"
lp15@56215
   359
    shows "(\<lambda>z. f z / g z) complex_differentiable (at a within s)"
lp15@56215
   360
  using assms unfolding complex_differentiable_def
lp15@56215
   361
  by (metis DERIV_divide [of f _ a s g])
lp15@56215
   362
lp15@61520
   363
lemma complex_differentiable_power [derivative_intros]:
lp15@61609
   364
  assumes "f complex_differentiable (at a within s)"
lp15@56215
   365
    shows "(\<lambda>z. f z ^ n) complex_differentiable (at a within s)"
lp15@56215
   366
  using assms unfolding complex_differentiable_def
lp15@56215
   367
  by (metis DERIV_power)
lp15@56215
   368
lp15@56215
   369
lemma complex_differentiable_transform_within:
lp15@56215
   370
  "0 < d \<Longrightarrow>
lp15@56215
   371
        x \<in> s \<Longrightarrow>
lp15@56215
   372
        (\<And>x'. x' \<in> s \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x') \<Longrightarrow>
lp15@56215
   373
        f complex_differentiable (at x within s)
lp15@56215
   374
        \<Longrightarrow> g complex_differentiable (at x within s)"
lp15@56215
   375
  unfolding complex_differentiable_def has_field_derivative_def
lp15@56215
   376
  by (blast intro: has_derivative_transform_within)
lp15@56215
   377
lp15@56215
   378
lemma complex_differentiable_compose_within:
lp15@61609
   379
  assumes "f complex_differentiable (at a within s)"
lp15@56215
   380
          "g complex_differentiable (at (f a) within f`s)"
lp15@56215
   381
    shows "(g o f) complex_differentiable (at a within s)"
lp15@56215
   382
  using assms unfolding complex_differentiable_def
lp15@56215
   383
  by (metis DERIV_image_chain)
lp15@56215
   384
hoelzl@56370
   385
lemma complex_differentiable_compose:
hoelzl@56370
   386
  "f complex_differentiable at z \<Longrightarrow> g complex_differentiable at (f z)
hoelzl@56370
   387
          \<Longrightarrow> (g o f) complex_differentiable at z"
hoelzl@56370
   388
by (metis complex_differentiable_at_within complex_differentiable_compose_within)
hoelzl@56370
   389
lp15@56215
   390
lemma complex_differentiable_within_open:
lp15@61609
   391
     "\<lbrakk>a \<in> s; open s\<rbrakk> \<Longrightarrow> f complex_differentiable at a within s \<longleftrightarrow>
lp15@56215
   392
                          f complex_differentiable at a"
lp15@56215
   393
  unfolding complex_differentiable_def
lp15@56215
   394
  by (metis at_within_open)
lp15@56215
   395
lp15@62408
   396
subsection\<open>Caratheodory characterization\<close>
lp15@56215
   397
lp15@56215
   398
lemma complex_differentiable_caratheodory_at:
lp15@56215
   399
  "f complex_differentiable (at z) \<longleftrightarrow>
lp15@56215
   400
         (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)"
lp15@56215
   401
  using CARAT_DERIV [of f]
lp15@56215
   402
  by (simp add: complex_differentiable_def has_field_derivative_def)
lp15@56215
   403
lp15@56215
   404
lemma complex_differentiable_caratheodory_within:
lp15@56215
   405
  "f complex_differentiable (at z within s) \<longleftrightarrow>
lp15@56215
   406
         (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)"
lp15@56215
   407
  using DERIV_caratheodory_within [of f]
lp15@56215
   408
  by (simp add: complex_differentiable_def has_field_derivative_def)
lp15@56215
   409
wenzelm@60420
   410
subsection\<open>Holomorphic\<close>
hoelzl@56370
   411
hoelzl@56370
   412
definition holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
hoelzl@56370
   413
           (infixl "(holomorphic'_on)" 50)
hoelzl@56370
   414
  where "f holomorphic_on s \<equiv> \<forall>x\<in>s. f complex_differentiable (at x within s)"
lp15@61609
   415
lp15@61520
   416
named_theorems holomorphic_intros "structural introduction rules for holomorphic_on"
lp15@61520
   417
paulson@62131
   418
lemma holomorphic_onI [intro?]: "(\<And>x. x \<in> s \<Longrightarrow> f complex_differentiable (at x within s)) \<Longrightarrow> f holomorphic_on s"
paulson@62131
   419
  by (simp add: holomorphic_on_def)
paulson@62131
   420
paulson@62131
   421
lemma holomorphic_onD [dest?]: "\<lbrakk>f holomorphic_on s; x \<in> s\<rbrakk> \<Longrightarrow> f complex_differentiable (at x within s)"
paulson@62131
   422
  by (simp add: holomorphic_on_def)
paulson@62131
   423
paulson@62131
   424
lemma holomorphic_on_imp_differentiable_at:
paulson@62131
   425
   "\<lbrakk>f holomorphic_on s; open s; x \<in> s\<rbrakk> \<Longrightarrow> f complex_differentiable (at x)"
paulson@62131
   426
using at_within_open holomorphic_on_def by fastforce
paulson@62131
   427
lp15@61520
   428
lemma holomorphic_on_empty [holomorphic_intros]: "f holomorphic_on {}"
hoelzl@56370
   429
  by (simp add: holomorphic_on_def)
hoelzl@56370
   430
hoelzl@56370
   431
lemma holomorphic_on_open:
hoelzl@56370
   432
    "open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> f')"
hoelzl@56370
   433
  by (auto simp: holomorphic_on_def complex_differentiable_def has_field_derivative_def at_within_open [of _ s])
hoelzl@56370
   434
lp15@61609
   435
lemma holomorphic_on_imp_continuous_on:
hoelzl@56370
   436
    "f holomorphic_on s \<Longrightarrow> continuous_on s f"
lp15@61609
   437
  by (metis complex_differentiable_imp_continuous_at continuous_on_eq_continuous_within holomorphic_on_def)
hoelzl@56370
   438
hoelzl@56370
   439
lemma holomorphic_on_subset:
hoelzl@56370
   440
    "f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t"
hoelzl@56370
   441
  unfolding holomorphic_on_def
hoelzl@56370
   442
  by (metis complex_differentiable_within_subset subsetD)
hoelzl@56370
   443
hoelzl@56370
   444
lemma holomorphic_transform: "\<lbrakk>f holomorphic_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g holomorphic_on s"
hoelzl@56370
   445
  by (metis complex_differentiable_transform_within linordered_field_no_ub holomorphic_on_def)
hoelzl@56370
   446
hoelzl@56370
   447
lemma holomorphic_cong: "s = t ==> (\<And>x. x \<in> s \<Longrightarrow> f x = g x) \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> g holomorphic_on t"
hoelzl@56370
   448
  by (metis holomorphic_transform)
hoelzl@56370
   449
lp15@62217
   450
lemma holomorphic_on_linear [simp, holomorphic_intros]: "(op * c) holomorphic_on s"
hoelzl@56370
   451
  unfolding holomorphic_on_def by (metis complex_differentiable_linear)
hoelzl@56370
   452
lp15@62217
   453
lemma holomorphic_on_const [simp, holomorphic_intros]: "(\<lambda>z. c) holomorphic_on s"
hoelzl@56370
   454
  unfolding holomorphic_on_def by (metis complex_differentiable_const)
hoelzl@56370
   455
lp15@62217
   456
lemma holomorphic_on_ident [simp, holomorphic_intros]: "(\<lambda>x. x) holomorphic_on s"
hoelzl@56370
   457
  unfolding holomorphic_on_def by (metis complex_differentiable_ident)
hoelzl@56370
   458
lp15@62217
   459
lemma holomorphic_on_id [simp, holomorphic_intros]: "id holomorphic_on s"
hoelzl@56370
   460
  unfolding id_def by (rule holomorphic_on_ident)
hoelzl@56370
   461
hoelzl@56370
   462
lemma holomorphic_on_compose:
hoelzl@56370
   463
  "f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s) \<Longrightarrow> (g o f) holomorphic_on s"
hoelzl@56370
   464
  using complex_differentiable_compose_within[of f _ s g]
hoelzl@56370
   465
  by (auto simp: holomorphic_on_def)
hoelzl@56370
   466
hoelzl@56370
   467
lemma holomorphic_on_compose_gen:
hoelzl@56370
   468
  "f holomorphic_on s \<Longrightarrow> g holomorphic_on t \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> (g o f) holomorphic_on s"
hoelzl@56370
   469
  by (metis holomorphic_on_compose holomorphic_on_subset)
hoelzl@56370
   470
lp15@61520
   471
lemma holomorphic_on_minus [holomorphic_intros]: "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on s"
hoelzl@56370
   472
  by (metis complex_differentiable_minus holomorphic_on_def)
hoelzl@56370
   473
lp15@61520
   474
lemma holomorphic_on_add [holomorphic_intros]:
hoelzl@56370
   475
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on s"
hoelzl@56370
   476
  unfolding holomorphic_on_def by (metis complex_differentiable_add)
hoelzl@56370
   477
lp15@61520
   478
lemma holomorphic_on_diff [holomorphic_intros]:
hoelzl@56370
   479
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) holomorphic_on s"
hoelzl@56370
   480
  unfolding holomorphic_on_def by (metis complex_differentiable_diff)
hoelzl@56370
   481
lp15@61520
   482
lemma holomorphic_on_mult [holomorphic_intros]:
hoelzl@56370
   483
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) holomorphic_on s"
hoelzl@56370
   484
  unfolding holomorphic_on_def by (metis complex_differentiable_mult)
hoelzl@56370
   485
lp15@61520
   486
lemma holomorphic_on_inverse [holomorphic_intros]:
hoelzl@56370
   487
  "\<lbrakk>f holomorphic_on s; \<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>z. inverse (f z)) holomorphic_on s"
hoelzl@56370
   488
  unfolding holomorphic_on_def by (metis complex_differentiable_inverse)
hoelzl@56370
   489
lp15@61520
   490
lemma holomorphic_on_divide [holomorphic_intros]:
hoelzl@56370
   491
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s; \<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>z. f z / g z) holomorphic_on s"
hoelzl@56370
   492
  unfolding holomorphic_on_def by (metis complex_differentiable_divide)
hoelzl@56370
   493
lp15@61520
   494
lemma holomorphic_on_power [holomorphic_intros]:
hoelzl@56370
   495
  "f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on s"
hoelzl@56370
   496
  unfolding holomorphic_on_def by (metis complex_differentiable_power)
hoelzl@56370
   497
lp15@61520
   498
lemma holomorphic_on_setsum [holomorphic_intros]:
hoelzl@56370
   499
  "(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) holomorphic_on s"
hoelzl@56370
   500
  unfolding holomorphic_on_def by (metis complex_differentiable_setsum)
hoelzl@56370
   501
hoelzl@56370
   502
lemma DERIV_deriv_iff_complex_differentiable:
hoelzl@56370
   503
  "DERIV f x :> deriv f x \<longleftrightarrow> f complex_differentiable at x"
hoelzl@56370
   504
  unfolding complex_differentiable_def by (metis DERIV_imp_deriv)
hoelzl@56370
   505
hoelzl@56370
   506
lemma complex_derivative_chain:
hoelzl@56370
   507
  "f complex_differentiable at x \<Longrightarrow> g complex_differentiable at (f x)
hoelzl@56370
   508
    \<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
hoelzl@56370
   509
  by (metis DERIV_deriv_iff_complex_differentiable DERIV_chain DERIV_imp_deriv)
hoelzl@56370
   510
lp15@62397
   511
lemma deriv_linear [simp]: "deriv (\<lambda>w. c * w) = (\<lambda>z. c)"
hoelzl@56370
   512
  by (metis DERIV_imp_deriv DERIV_cmult_Id)
hoelzl@56370
   513
lp15@62397
   514
lemma deriv_ident [simp]: "deriv (\<lambda>w. w) = (\<lambda>z. 1)"
hoelzl@56370
   515
  by (metis DERIV_imp_deriv DERIV_ident)
hoelzl@56370
   516
lp15@62397
   517
lemma deriv_id [simp]: "deriv id = (\<lambda>z. 1)"
lp15@62397
   518
  by (simp add: id_def)
lp15@62397
   519
lp15@62397
   520
lemma deriv_const [simp]: "deriv (\<lambda>w. c) = (\<lambda>z. 0)"
hoelzl@56370
   521
  by (metis DERIV_imp_deriv DERIV_const)
hoelzl@56370
   522
lp15@62397
   523
lemma complex_derivative_add [simp]:
lp15@61609
   524
  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>
hoelzl@56370
   525
   \<Longrightarrow> deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
hoelzl@56370
   526
  unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
hoelzl@56381
   527
  by (auto intro!: DERIV_imp_deriv derivative_intros)
hoelzl@56370
   528
lp15@62397
   529
lemma complex_derivative_diff [simp]:
lp15@61609
   530
  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>
hoelzl@56370
   531
   \<Longrightarrow> deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
hoelzl@56370
   532
  unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
hoelzl@56381
   533
  by (auto intro!: DERIV_imp_deriv derivative_intros)
hoelzl@56370
   534
lp15@62397
   535
lemma complex_derivative_mult [simp]:
lp15@61609
   536
  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>
hoelzl@56370
   537
   \<Longrightarrow> deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
hoelzl@56370
   538
  unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
hoelzl@56381
   539
  by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
hoelzl@56370
   540
lp15@62397
   541
lemma complex_derivative_cmult [simp]:
hoelzl@56370
   542
  "f complex_differentiable at z \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z"
hoelzl@56370
   543
  unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
hoelzl@56381
   544
  by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
hoelzl@56370
   545
lp15@62397
   546
lemma complex_derivative_cmult_right [simp]:
hoelzl@56370
   547
  "f complex_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c"
hoelzl@56370
   548
  unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
hoelzl@56381
   549
  by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
hoelzl@56370
   550
lp15@62397
   551
lemma complex_derivative_cdivide_right [simp]:
lp15@62217
   552
  "f complex_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w / c) z = deriv f z / c"
lp15@62217
   553
  unfolding Fields.field_class.field_divide_inverse
lp15@62217
   554
  by (blast intro: complex_derivative_cmult_right)
lp15@62217
   555
hoelzl@56370
   556
lemma complex_derivative_transform_within_open:
lp15@61609
   557
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s; open s; z \<in> s; \<And>w. w \<in> s \<Longrightarrow> f w = g w\<rbrakk>
hoelzl@56370
   558
   \<Longrightarrow> deriv f z = deriv g z"
hoelzl@56370
   559
  unfolding holomorphic_on_def
hoelzl@56370
   560
  by (rule DERIV_imp_deriv)
hoelzl@56370
   561
     (metis DERIV_deriv_iff_complex_differentiable DERIV_transform_within_open at_within_open)
hoelzl@56370
   562
hoelzl@56370
   563
lemma complex_derivative_compose_linear:
hoelzl@56370
   564
  "f complex_differentiable at (c * z) \<Longrightarrow> deriv (\<lambda>w. f (c * w)) z = c * deriv f (c * z)"
hoelzl@56370
   565
apply (rule DERIV_imp_deriv)
hoelzl@56370
   566
apply (simp add: DERIV_deriv_iff_complex_differentiable [symmetric])
haftmann@59554
   567
apply (drule DERIV_chain' [of "times c" c z UNIV f "deriv f (c * z)", OF DERIV_cmult_Id])
haftmann@59554
   568
apply (simp add: algebra_simps)
hoelzl@56370
   569
done
hoelzl@56370
   570
wenzelm@60420
   571
subsection\<open>Analyticity on a set\<close>
lp15@56215
   572
lp15@61609
   573
definition analytic_on (infixl "(analytic'_on)" 50)
lp15@56215
   574
  where
lp15@56215
   575
   "f analytic_on s \<equiv> \<forall>x \<in> s. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)"
lp15@56215
   576
hoelzl@56370
   577
lemma analytic_imp_holomorphic: "f analytic_on s \<Longrightarrow> f holomorphic_on s"
hoelzl@56370
   578
  by (simp add: at_within_open [OF _ open_ball] analytic_on_def holomorphic_on_def)
hoelzl@56370
   579
     (metis centre_in_ball complex_differentiable_at_within)
lp15@56215
   580
hoelzl@56370
   581
lemma analytic_on_open: "open s \<Longrightarrow> f analytic_on s \<longleftrightarrow> f holomorphic_on s"
lp15@56215
   582
apply (auto simp: analytic_imp_holomorphic)
lp15@56215
   583
apply (auto simp: analytic_on_def holomorphic_on_def)
lp15@56215
   584
by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball)
lp15@56215
   585
lp15@56215
   586
lemma analytic_on_imp_differentiable_at:
lp15@56215
   587
  "f analytic_on s \<Longrightarrow> x \<in> s \<Longrightarrow> f complex_differentiable (at x)"
hoelzl@56370
   588
 apply (auto simp: analytic_on_def holomorphic_on_def)
lp15@56215
   589
by (metis Topology_Euclidean_Space.open_ball centre_in_ball complex_differentiable_within_open)
lp15@56215
   590
hoelzl@56370
   591
lemma analytic_on_subset: "f analytic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f analytic_on t"
lp15@56215
   592
  by (auto simp: analytic_on_def)
lp15@56215
   593
hoelzl@56370
   594
lemma analytic_on_Un: "f analytic_on (s \<union> t) \<longleftrightarrow> f analytic_on s \<and> f analytic_on t"
lp15@56215
   595
  by (auto simp: analytic_on_def)
lp15@56215
   596
wenzelm@60585
   597
lemma analytic_on_Union: "f analytic_on (\<Union>s) \<longleftrightarrow> (\<forall>t \<in> s. f analytic_on t)"
hoelzl@56370
   598
  by (auto simp: analytic_on_def)
hoelzl@56370
   599
hoelzl@56370
   600
lemma analytic_on_UN: "f analytic_on (\<Union>i\<in>I. s i) \<longleftrightarrow> (\<forall>i\<in>I. f analytic_on (s i))"
lp15@56215
   601
  by (auto simp: analytic_on_def)
lp15@61609
   602
lp15@56215
   603
lemma analytic_on_holomorphic:
lp15@56215
   604
  "f analytic_on s \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f holomorphic_on t)"
lp15@56215
   605
  (is "?lhs = ?rhs")
lp15@56215
   606
proof -
lp15@56215
   607
  have "?lhs \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t)"
lp15@56215
   608
  proof safe
lp15@56215
   609
    assume "f analytic_on s"
lp15@56215
   610
    then show "\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t"
lp15@56215
   611
      apply (simp add: analytic_on_def)
lp15@56215
   612
      apply (rule exI [where x="\<Union>{u. open u \<and> f analytic_on u}"], auto)
lp15@56215
   613
      apply (metis Topology_Euclidean_Space.open_ball analytic_on_open centre_in_ball)
lp15@56215
   614
      by (metis analytic_on_def)
lp15@56215
   615
  next
lp15@56215
   616
    fix t
lp15@61609
   617
    assume "open t" "s \<subseteq> t" "f analytic_on t"
lp15@56215
   618
    then show "f analytic_on s"
lp15@56215
   619
        by (metis analytic_on_subset)
lp15@56215
   620
  qed
lp15@56215
   621
  also have "... \<longleftrightarrow> ?rhs"
lp15@56215
   622
    by (auto simp: analytic_on_open)
lp15@56215
   623
  finally show ?thesis .
lp15@56215
   624
qed
lp15@56215
   625
lp15@56215
   626
lemma analytic_on_linear: "(op * c) analytic_on s"
hoelzl@56370
   627
  by (auto simp add: analytic_on_holomorphic holomorphic_on_linear)
lp15@56215
   628
lp15@56215
   629
lemma analytic_on_const: "(\<lambda>z. c) analytic_on s"
hoelzl@56370
   630
  by (metis analytic_on_def holomorphic_on_const zero_less_one)
hoelzl@56370
   631
hoelzl@56370
   632
lemma analytic_on_ident: "(\<lambda>x. x) analytic_on s"
hoelzl@56370
   633
  by (simp add: analytic_on_def holomorphic_on_ident gt_ex)
lp15@56215
   634
lp15@56215
   635
lemma analytic_on_id: "id analytic_on s"
hoelzl@56370
   636
  unfolding id_def by (rule analytic_on_ident)
lp15@56215
   637
lp15@56215
   638
lemma analytic_on_compose:
lp15@56215
   639
  assumes f: "f analytic_on s"
lp15@56215
   640
      and g: "g analytic_on (f ` s)"
lp15@56215
   641
    shows "(g o f) analytic_on s"
lp15@56215
   642
unfolding analytic_on_def
lp15@56215
   643
proof (intro ballI)
lp15@56215
   644
  fix x
lp15@56215
   645
  assume x: "x \<in> s"
lp15@56215
   646
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f
lp15@56215
   647
    by (metis analytic_on_def)
lp15@56215
   648
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g
lp15@61609
   649
    by (metis analytic_on_def g image_eqI x)
lp15@56215
   650
  have "isCont f x"
lp15@56215
   651
    by (metis analytic_on_imp_differentiable_at complex_differentiable_imp_continuous_at f x)
lp15@56215
   652
  with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'"
lp15@56215
   653
     by (auto simp: continuous_at_ball)
lp15@61609
   654
  have "g \<circ> f holomorphic_on ball x (min d e)"
lp15@56215
   655
    apply (rule holomorphic_on_compose)
lp15@56215
   656
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   657
    by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball)
lp15@56215
   658
  then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e"
lp15@61609
   659
    by (metis d e min_less_iff_conj)
lp15@56215
   660
qed
lp15@56215
   661
lp15@56215
   662
lemma analytic_on_compose_gen:
lp15@56215
   663
  "f analytic_on s \<Longrightarrow> g analytic_on t \<Longrightarrow> (\<And>z. z \<in> s \<Longrightarrow> f z \<in> t)
lp15@56215
   664
             \<Longrightarrow> g o f analytic_on s"
lp15@56215
   665
by (metis analytic_on_compose analytic_on_subset image_subset_iff)
lp15@56215
   666
lp15@56215
   667
lemma analytic_on_neg:
lp15@56215
   668
  "f analytic_on s \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on s"
lp15@56215
   669
by (metis analytic_on_holomorphic holomorphic_on_minus)
lp15@56215
   670
lp15@56215
   671
lemma analytic_on_add:
lp15@56215
   672
  assumes f: "f analytic_on s"
lp15@56215
   673
      and g: "g analytic_on s"
lp15@56215
   674
    shows "(\<lambda>z. f z + g z) analytic_on s"
lp15@56215
   675
unfolding analytic_on_def
lp15@56215
   676
proof (intro ballI)
lp15@56215
   677
  fix z
lp15@56215
   678
  assume z: "z \<in> s"
lp15@56215
   679
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   680
    by (metis analytic_on_def)
lp15@56215
   681
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
lp15@61609
   682
    by (metis analytic_on_def g z)
lp15@61609
   683
  have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')"
lp15@61609
   684
    apply (rule holomorphic_on_add)
lp15@56215
   685
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   686
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   687
  then show "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e"
lp15@56215
   688
    by (metis e e' min_less_iff_conj)
lp15@56215
   689
qed
lp15@56215
   690
lp15@56215
   691
lemma analytic_on_diff:
lp15@56215
   692
  assumes f: "f analytic_on s"
lp15@56215
   693
      and g: "g analytic_on s"
lp15@56215
   694
    shows "(\<lambda>z. f z - g z) analytic_on s"
lp15@56215
   695
unfolding analytic_on_def
lp15@56215
   696
proof (intro ballI)
lp15@56215
   697
  fix z
lp15@56215
   698
  assume z: "z \<in> s"
lp15@56215
   699
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   700
    by (metis analytic_on_def)
lp15@56215
   701
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
lp15@61609
   702
    by (metis analytic_on_def g z)
lp15@61609
   703
  have "(\<lambda>z. f z - g z) holomorphic_on ball z (min e e')"
lp15@61609
   704
    apply (rule holomorphic_on_diff)
lp15@56215
   705
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   706
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   707
  then show "\<exists>e>0. (\<lambda>z. f z - g z) holomorphic_on ball z e"
lp15@56215
   708
    by (metis e e' min_less_iff_conj)
lp15@56215
   709
qed
lp15@56215
   710
lp15@56215
   711
lemma analytic_on_mult:
lp15@56215
   712
  assumes f: "f analytic_on s"
lp15@56215
   713
      and g: "g analytic_on s"
lp15@56215
   714
    shows "(\<lambda>z. f z * g z) analytic_on s"
lp15@56215
   715
unfolding analytic_on_def
lp15@56215
   716
proof (intro ballI)
lp15@56215
   717
  fix z
lp15@56215
   718
  assume z: "z \<in> s"
lp15@56215
   719
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   720
    by (metis analytic_on_def)
lp15@56215
   721
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
lp15@61609
   722
    by (metis analytic_on_def g z)
lp15@61609
   723
  have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')"
lp15@61609
   724
    apply (rule holomorphic_on_mult)
lp15@56215
   725
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   726
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   727
  then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e"
lp15@56215
   728
    by (metis e e' min_less_iff_conj)
lp15@56215
   729
qed
lp15@56215
   730
lp15@56215
   731
lemma analytic_on_inverse:
lp15@56215
   732
  assumes f: "f analytic_on s"
lp15@56215
   733
      and nz: "(\<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0)"
lp15@56215
   734
    shows "(\<lambda>z. inverse (f z)) analytic_on s"
lp15@56215
   735
unfolding analytic_on_def
lp15@56215
   736
proof (intro ballI)
lp15@56215
   737
  fix z
lp15@56215
   738
  assume z: "z \<in> s"
lp15@56215
   739
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   740
    by (metis analytic_on_def)
lp15@56215
   741
  have "continuous_on (ball z e) f"
lp15@56215
   742
    by (metis fh holomorphic_on_imp_continuous_on)
lp15@61609
   743
  then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0"
lp15@61609
   744
    by (metis Topology_Euclidean_Space.open_ball centre_in_ball continuous_on_open_avoid e z nz)
lp15@61609
   745
  have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')"
lp15@56215
   746
    apply (rule holomorphic_on_inverse)
lp15@56215
   747
    apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball)
lp15@61609
   748
    by (metis nz' mem_ball min_less_iff_conj)
lp15@56215
   749
  then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e"
lp15@56215
   750
    by (metis e e' min_less_iff_conj)
lp15@56215
   751
qed
lp15@56215
   752
lp15@56215
   753
lemma analytic_on_divide:
lp15@56215
   754
  assumes f: "f analytic_on s"
lp15@56215
   755
      and g: "g analytic_on s"
lp15@56215
   756
      and nz: "(\<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0)"
lp15@56215
   757
    shows "(\<lambda>z. f z / g z) analytic_on s"
lp15@56215
   758
unfolding divide_inverse
lp15@56215
   759
by (metis analytic_on_inverse analytic_on_mult f g nz)
lp15@56215
   760
lp15@56215
   761
lemma analytic_on_power:
lp15@56215
   762
  "f analytic_on s \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on s"
lp15@56215
   763
by (induct n) (auto simp: analytic_on_const analytic_on_mult)
lp15@56215
   764
lp15@56215
   765
lemma analytic_on_setsum:
hoelzl@56369
   766
  "(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on s) \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) analytic_on s"
hoelzl@56369
   767
  by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_const analytic_on_add)
lp15@56215
   768
lp15@62408
   769
lemma deriv_left_inverse:
lp15@62408
   770
  assumes "f holomorphic_on S" and "g holomorphic_on T"
lp15@62408
   771
      and "open S" and "open T"
lp15@62408
   772
      and "f ` S \<subseteq> T"
lp15@62408
   773
      and [simp]: "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z"
lp15@62408
   774
      and "w \<in> S"
lp15@62408
   775
    shows "deriv f w * deriv g (f w) = 1"
lp15@62408
   776
proof -
lp15@62408
   777
  have "deriv f w * deriv g (f w) = deriv g (f w) * deriv f w"
lp15@62408
   778
    by (simp add: algebra_simps)
lp15@62408
   779
  also have "... = deriv (g o f) w"
lp15@62408
   780
    using assms
lp15@62408
   781
    by (metis analytic_on_imp_differentiable_at analytic_on_open complex_derivative_chain image_subset_iff)
lp15@62408
   782
  also have "... = deriv id w"
lp15@62408
   783
    apply (rule complex_derivative_transform_within_open [where s=S])
lp15@62408
   784
    apply (rule assms holomorphic_on_compose_gen holomorphic_intros)+
lp15@62408
   785
    apply simp
lp15@62408
   786
    done
lp15@62408
   787
  also have "... = 1"
lp15@62408
   788
    by simp
lp15@62408
   789
  finally show ?thesis .
lp15@62408
   790
qed
lp15@62408
   791
lp15@62408
   792
subsection\<open>analyticity at a point\<close>
lp15@56215
   793
lp15@56215
   794
lemma analytic_at_ball:
lp15@56215
   795
  "f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
lp15@56215
   796
by (metis analytic_on_def singleton_iff)
lp15@56215
   797
lp15@56215
   798
lemma analytic_at:
lp15@56215
   799
    "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
lp15@56215
   800
by (metis analytic_on_holomorphic empty_subsetI insert_subset)
lp15@56215
   801
lp15@56215
   802
lemma analytic_on_analytic_at:
lp15@56215
   803
    "f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
lp15@56215
   804
by (metis analytic_at_ball analytic_on_def)
lp15@56215
   805
lp15@56215
   806
lemma analytic_at_two:
lp15@56215
   807
  "f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
lp15@56215
   808
   (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)"
lp15@56215
   809
  (is "?lhs = ?rhs")
lp15@61609
   810
proof
lp15@56215
   811
  assume ?lhs
lp15@61609
   812
  then obtain s t
lp15@56215
   813
    where st: "open s" "z \<in> s" "f holomorphic_on s"
lp15@56215
   814
              "open t" "z \<in> t" "g holomorphic_on t"
lp15@56215
   815
    by (auto simp: analytic_at)
lp15@56215
   816
  show ?rhs
lp15@56215
   817
    apply (rule_tac x="s \<inter> t" in exI)
lp15@56215
   818
    using st
lp15@56215
   819
    apply (auto simp: Diff_subset holomorphic_on_subset)
lp15@56215
   820
    done
lp15@56215
   821
next
lp15@61609
   822
  assume ?rhs
lp15@56215
   823
  then show ?lhs
lp15@56215
   824
    by (force simp add: analytic_at)
lp15@56215
   825
qed
lp15@56215
   826
wenzelm@60420
   827
subsection\<open>Combining theorems for derivative with ``analytic at'' hypotheses\<close>
lp15@56215
   828
lp15@61609
   829
lemma
lp15@56215
   830
  assumes "f analytic_on {z}" "g analytic_on {z}"
hoelzl@56370
   831
  shows complex_derivative_add_at: "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
hoelzl@56370
   832
    and complex_derivative_diff_at: "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
hoelzl@56370
   833
    and complex_derivative_mult_at: "deriv (\<lambda>w. f w * g w) z =
hoelzl@56370
   834
           f z * deriv g z + deriv f z * g z"
lp15@56215
   835
proof -
lp15@56215
   836
  obtain s where s: "open s" "z \<in> s" "f holomorphic_on s" "g holomorphic_on s"
lp15@56215
   837
    using assms by (metis analytic_at_two)
hoelzl@56370
   838
  show "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
hoelzl@56370
   839
    apply (rule DERIV_imp_deriv [OF DERIV_add])
lp15@56215
   840
    using s
hoelzl@56370
   841
    apply (auto simp: holomorphic_on_open complex_differentiable_def DERIV_deriv_iff_complex_differentiable)
lp15@56215
   842
    done
hoelzl@56370
   843
  show "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
hoelzl@56370
   844
    apply (rule DERIV_imp_deriv [OF DERIV_diff])
lp15@56215
   845
    using s
hoelzl@56370
   846
    apply (auto simp: holomorphic_on_open complex_differentiable_def DERIV_deriv_iff_complex_differentiable)
lp15@56215
   847
    done
hoelzl@56370
   848
  show "deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
hoelzl@56370
   849
    apply (rule DERIV_imp_deriv [OF DERIV_mult'])
lp15@56215
   850
    using s
hoelzl@56370
   851
    apply (auto simp: holomorphic_on_open complex_differentiable_def DERIV_deriv_iff_complex_differentiable)
lp15@56215
   852
    done
lp15@56215
   853
qed
lp15@56215
   854
lp15@56215
   855
lemma complex_derivative_cmult_at:
hoelzl@56370
   856
  "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. c * f w) z = c * deriv f z"
lp15@61848
   857
by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
lp15@56215
   858
lp15@56215
   859
lemma complex_derivative_cmult_right_at:
hoelzl@56370
   860
  "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. f w * c) z = deriv f z * c"
lp15@61848
   861
by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
lp15@56215
   862
wenzelm@60420
   863
subsection\<open>Complex differentiation of sequences and series\<close>
lp15@56215
   864
eberlm@61531
   865
(* TODO: Could probably be simplified using Uniform_Limit *)
lp15@56215
   866
lemma has_complex_derivative_sequence:
lp15@56215
   867
  fixes s :: "complex set"
lp15@56215
   868
  assumes cvs: "convex s"
lp15@56215
   869
      and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
lp15@56215
   870
      and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s \<longrightarrow> norm (f' n x - g' x) \<le> e"
wenzelm@61973
   871
      and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
wenzelm@61973
   872
    shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and>
lp15@56215
   873
                       (g has_field_derivative (g' x)) (at x within s)"
lp15@56215
   874
proof -
wenzelm@61973
   875
  from assms obtain x l where x: "x \<in> s" and tf: "((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
lp15@56215
   876
    by blast
lp15@56215
   877
  { fix e::real assume e: "e > 0"
lp15@56215
   878
    then obtain N where N: "\<forall>n\<ge>N. \<forall>x. x \<in> s \<longrightarrow> cmod (f' n x - g' x) \<le> e"
lp15@61609
   879
      by (metis conv)
lp15@56215
   880
    have "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
lp15@56215
   881
    proof (rule exI [of _ N], clarify)
lp15@56215
   882
      fix n y h
lp15@56215
   883
      assume "N \<le> n" "y \<in> s"
lp15@56215
   884
      then have "cmod (f' n y - g' y) \<le> e"
lp15@56215
   885
        by (metis N)
lp15@56215
   886
      then have "cmod h * cmod (f' n y - g' y) \<le> cmod h * e"
lp15@56215
   887
        by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
lp15@56215
   888
      then show "cmod (f' n y * h - g' y * h) \<le> e * cmod h"
lp15@56215
   889
        by (simp add: norm_mult [symmetric] field_simps)
lp15@56215
   890
    qed
lp15@56215
   891
  } note ** = this
lp15@56215
   892
  show ?thesis
lp15@56215
   893
  unfolding has_field_derivative_def
lp15@56215
   894
  proof (rule has_derivative_sequence [OF cvs _ _ x])
lp15@56215
   895
    show "\<forall>n. \<forall>x\<in>s. (f n has_derivative (op * (f' n x))) (at x within s)"
lp15@56215
   896
      by (metis has_field_derivative_def df)
wenzelm@61969
   897
  next show "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
lp15@56215
   898
    by (rule tf)
lp15@56215
   899
  next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
lp15@56215
   900
    by (blast intro: **)
lp15@56215
   901
  qed
lp15@56215
   902
qed
lp15@56215
   903
lp15@56215
   904
lp15@56215
   905
lemma has_complex_derivative_series:
lp15@56215
   906
  fixes s :: "complex set"
lp15@56215
   907
  assumes cvs: "convex s"
lp15@56215
   908
      and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
lp15@61609
   909
      and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s
lp15@56215
   910
                \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
lp15@56215
   911
      and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) sums l)"
lp15@56215
   912
    shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) sums g x) \<and> ((g has_field_derivative g' x) (at x within s))"
lp15@56215
   913
proof -
lp15@56215
   914
  from assms obtain x l where x: "x \<in> s" and sf: "((\<lambda>n. f n x) sums l)"
lp15@56215
   915
    by blast
lp15@56215
   916
  { fix e::real assume e: "e > 0"
lp15@61609
   917
    then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> s
lp15@56215
   918
            \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
lp15@61609
   919
      by (metis conv)
lp15@56215
   920
    have "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
lp15@56215
   921
    proof (rule exI [of _ N], clarify)
lp15@56215
   922
      fix n y h
lp15@56215
   923
      assume "N \<le> n" "y \<in> s"
lp15@56215
   924
      then have "cmod ((\<Sum>i<n. f' i y) - g' y) \<le> e"
lp15@56215
   925
        by (metis N)
lp15@56215
   926
      then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e"
lp15@56215
   927
        by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
lp15@56215
   928
      then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h"
lp15@56215
   929
        by (simp add: norm_mult [symmetric] field_simps setsum_right_distrib)
lp15@56215
   930
    qed
lp15@56215
   931
  } note ** = this
lp15@56215
   932
  show ?thesis
lp15@56215
   933
  unfolding has_field_derivative_def
lp15@56215
   934
  proof (rule has_derivative_series [OF cvs _ _ x])
lp15@56215
   935
    fix n x
lp15@56215
   936
    assume "x \<in> s"
lp15@56215
   937
    then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within s)"
lp15@56215
   938
      by (metis df has_field_derivative_def mult_commute_abs)
lp15@56215
   939
  next show " ((\<lambda>n. f n x) sums l)"
lp15@56215
   940
    by (rule sf)
lp15@56215
   941
  next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
lp15@56215
   942
    by (blast intro: **)
lp15@56215
   943
  qed
lp15@56215
   944
qed
lp15@56215
   945
eberlm@61531
   946
eberlm@61531
   947
lemma complex_differentiable_series:
eberlm@61531
   948
  fixes f :: "nat \<Rightarrow> complex \<Rightarrow> complex"
eberlm@61531
   949
  assumes "convex s" "open s"
eberlm@61531
   950
  assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
eberlm@61531
   951
  assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
eberlm@61531
   952
  assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)" and x: "x \<in> s"
eberlm@61531
   953
  shows   "summable (\<lambda>n. f n x)" and "(\<lambda>x. \<Sum>n. f n x) complex_differentiable (at x)"
eberlm@61531
   954
proof -
eberlm@61531
   955
  from assms(4) obtain g' where A: "uniform_limit s (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
eberlm@61531
   956
    unfolding uniformly_convergent_on_def by blast
wenzelm@61808
   957
  from x and \<open>open s\<close> have s: "at x within s = at x" by (rule at_within_open)
eberlm@61531
   958
  have "\<exists>g. \<forall>x\<in>s. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within s)"
eberlm@61531
   959
    by (intro has_field_derivative_series[of s f f' g' x0] assms A has_field_derivative_at_within)
eberlm@61531
   960
  then obtain g where g: "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>n. f n x) sums g x"
eberlm@61531
   961
    "\<And>x. x \<in> s \<Longrightarrow> (g has_field_derivative g' x) (at x within s)" by blast
eberlm@61531
   962
  from g[OF x] show "summable (\<lambda>n. f n x)" by (auto simp: summable_def)
eberlm@61531
   963
  from g(2)[OF x] have g': "(g has_derivative op * (g' x)) (at x)"
eberlm@61531
   964
    by (simp add: has_field_derivative_def s)
eberlm@61531
   965
  have "((\<lambda>x. \<Sum>n. f n x) has_derivative op * (g' x)) (at x)"
paulson@62087
   966
    by (rule has_derivative_transform_within_open[OF g' \<open>open s\<close> x])
eberlm@61531
   967
       (insert g, auto simp: sums_iff)
eberlm@61531
   968
  thus "(\<lambda>x. \<Sum>n. f n x) complex_differentiable (at x)" unfolding differentiable_def
eberlm@61531
   969
    by (auto simp: summable_def complex_differentiable_def has_field_derivative_def)
eberlm@61531
   970
qed
eberlm@61531
   971
eberlm@61531
   972
lemma complex_differentiable_series':
eberlm@61531
   973
  fixes f :: "nat \<Rightarrow> complex \<Rightarrow> complex"
eberlm@61531
   974
  assumes "convex s" "open s"
eberlm@61531
   975
  assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
eberlm@61531
   976
  assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
eberlm@61531
   977
  assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)"
eberlm@61531
   978
  shows   "(\<lambda>x. \<Sum>n. f n x) complex_differentiable (at x0)"
wenzelm@61808
   979
  using complex_differentiable_series[OF assms, of x0] \<open>x0 \<in> s\<close> by blast+
eberlm@61531
   980
wenzelm@60420
   981
subsection\<open>Bound theorem\<close>
lp15@56215
   982
lp15@56215
   983
lemma complex_differentiable_bound:
lp15@56215
   984
  fixes s :: "complex set"
lp15@56215
   985
  assumes cvs: "convex s"
lp15@56215
   986
      and df:  "\<And>z. z \<in> s \<Longrightarrow> (f has_field_derivative f' z) (at z within s)"
lp15@56215
   987
      and dn:  "\<And>z. z \<in> s \<Longrightarrow> norm (f' z) \<le> B"
lp15@56215
   988
      and "x \<in> s"  "y \<in> s"
lp15@56215
   989
    shows "norm(f x - f y) \<le> B * norm(x - y)"
lp15@56215
   990
  apply (rule differentiable_bound [OF cvs])
huffman@56223
   991
  apply (rule ballI, erule df [unfolded has_field_derivative_def])
huffman@56223
   992
  apply (rule ballI, rule onorm_le, simp add: norm_mult mult_right_mono dn)
huffman@56223
   993
  apply fact
huffman@56223
   994
  apply fact
lp15@56215
   995
  done
lp15@56215
   996
lp15@62408
   997
subsection\<open>Inverse function theorem for complex derivatives\<close>
lp15@56215
   998
lp15@56215
   999
lemma has_complex_derivative_inverse_basic:
lp15@56215
  1000
  fixes f :: "complex \<Rightarrow> complex"
lp15@56215
  1001
  shows "DERIV f (g y) :> f' \<Longrightarrow>
lp15@56215
  1002
        f' \<noteq> 0 \<Longrightarrow>
lp15@56215
  1003
        continuous (at y) g \<Longrightarrow>
lp15@56215
  1004
        open t \<Longrightarrow>
lp15@56215
  1005
        y \<in> t \<Longrightarrow>
lp15@56215
  1006
        (\<And>z. z \<in> t \<Longrightarrow> f (g z) = z)
lp15@56215
  1007
        \<Longrightarrow> DERIV g y :> inverse (f')"
lp15@56215
  1008
  unfolding has_field_derivative_def
lp15@56215
  1009
  apply (rule has_derivative_inverse_basic)
lp15@56215
  1010
  apply (auto simp:  bounded_linear_mult_right)
lp15@56215
  1011
  done
lp15@56215
  1012
lp15@56215
  1013
(*Used only once, in Multivariate/cauchy.ml. *)
lp15@56215
  1014
lemma has_complex_derivative_inverse_strong:
lp15@56215
  1015
  fixes f :: "complex \<Rightarrow> complex"
lp15@56215
  1016
  shows "DERIV f x :> f' \<Longrightarrow>
lp15@56215
  1017
         f' \<noteq> 0 \<Longrightarrow>
lp15@56215
  1018
         open s \<Longrightarrow>
lp15@56215
  1019
         x \<in> s \<Longrightarrow>
lp15@56215
  1020
         continuous_on s f \<Longrightarrow>
lp15@56215
  1021
         (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
lp15@56215
  1022
         \<Longrightarrow> DERIV g (f x) :> inverse (f')"
lp15@56215
  1023
  unfolding has_field_derivative_def
lp15@56215
  1024
  apply (rule has_derivative_inverse_strong [of s x f g ])
lp15@61609
  1025
  using assms
lp15@56215
  1026
  by auto
lp15@56215
  1027
lp15@56215
  1028
lemma has_complex_derivative_inverse_strong_x:
lp15@56215
  1029
  fixes f :: "complex \<Rightarrow> complex"
lp15@56215
  1030
  shows  "DERIV f (g y) :> f' \<Longrightarrow>
lp15@56215
  1031
          f' \<noteq> 0 \<Longrightarrow>
lp15@56215
  1032
          open s \<Longrightarrow>
lp15@56215
  1033
          continuous_on s f \<Longrightarrow>
lp15@56215
  1034
          g y \<in> s \<Longrightarrow> f(g y) = y \<Longrightarrow>
lp15@56215
  1035
          (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
lp15@56215
  1036
          \<Longrightarrow> DERIV g y :> inverse (f')"
lp15@56215
  1037
  unfolding has_field_derivative_def
lp15@56215
  1038
  apply (rule has_derivative_inverse_strong_x [of s g y f])
lp15@61609
  1039
  using assms
lp15@56215
  1040
  by auto
lp15@56215
  1041
wenzelm@60420
  1042
subsection \<open>Taylor on Complex Numbers\<close>
lp15@56215
  1043
lp15@56215
  1044
lemma setsum_Suc_reindex:
lp15@56215
  1045
  fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
lp15@56215
  1046
    shows  "setsum f {0..n} = f 0 - f (Suc n) + setsum (\<lambda>i. f (Suc i)) {0..n}"
lp15@56215
  1047
by (induct n) auto
lp15@56215
  1048
lp15@56215
  1049
lemma complex_taylor:
lp15@61609
  1050
  assumes s: "convex s"
lp15@56215
  1051
      and f: "\<And>i x. x \<in> s \<Longrightarrow> i \<le> n \<Longrightarrow> (f i has_field_derivative f (Suc i) x) (at x within s)"
lp15@56215
  1052
      and B: "\<And>x. x \<in> s \<Longrightarrow> cmod (f (Suc n) x) \<le> B"
lp15@56215
  1053
      and w: "w \<in> s"
lp15@56215
  1054
      and z: "z \<in> s"
lp15@59730
  1055
    shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i)))
lp15@56215
  1056
          \<le> B * cmod(z - w)^(Suc n) / fact n"
lp15@56215
  1057
proof -
lp15@56215
  1058
  have wzs: "closed_segment w z \<subseteq> s" using assms
lp15@56215
  1059
    by (metis convex_contains_segment)
lp15@56215
  1060
  { fix u
lp15@56215
  1061
    assume "u \<in> closed_segment w z"
lp15@56215
  1062
    then have "u \<in> s"
lp15@56215
  1063
      by (metis wzs subsetD)
lp15@59730
  1064
    have "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / (fact i) +
lp15@61609
  1065
                      f (Suc i) u * (z-u)^i / (fact i)) =
lp15@59730
  1066
              f (Suc n) u * (z-u) ^ n / (fact n)"
lp15@56215
  1067
    proof (induction n)
lp15@56215
  1068
      case 0 show ?case by simp
lp15@56215
  1069
    next
lp15@56215
  1070
      case (Suc n)
lp15@59730
  1071
      have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / (fact i) +
lp15@61609
  1072
                             f (Suc i) u * (z-u) ^ i / (fact i)) =
lp15@59730
  1073
           f (Suc n) u * (z-u) ^ n / (fact n) +
lp15@59730
  1074
           f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / (fact (Suc n)) -
lp15@59730
  1075
           f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / (fact (Suc n))"
hoelzl@56479
  1076
        using Suc by simp
lp15@59730
  1077
      also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n))"
lp15@56215
  1078
      proof -
lp15@59730
  1079
        have "(fact(Suc n)) *
lp15@59730
  1080
             (f(Suc n) u *(z-u) ^ n / (fact n) +
lp15@59730
  1081
               f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / (fact(Suc n)) -
lp15@59730
  1082
               f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / (fact(Suc n))) =
lp15@59730
  1083
            ((fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / (fact n) +
lp15@59730
  1084
            ((fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / (fact(Suc n))) -
lp15@59730
  1085
            ((fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / (fact(Suc n))"
lp15@59730
  1086
          by (simp add: algebra_simps del: fact.simps)
lp15@59730
  1087
        also have "... = ((fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / (fact n) +
lp15@59730
  1088
                         (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
lp15@59730
  1089
                         (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
lp15@59730
  1090
          by (simp del: fact.simps)
lp15@59730
  1091
        also have "... = (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) +
lp15@59730
  1092
                         (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
lp15@59730
  1093
                         (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
lp15@59730
  1094
          by (simp only: fact.simps of_nat_mult ac_simps) simp
lp15@56215
  1095
        also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
lp15@56215
  1096
          by (simp add: algebra_simps)
lp15@56215
  1097
        finally show ?thesis
lp15@59730
  1098
        by (simp add: mult_left_cancel [where c = "(fact (Suc n))", THEN iffD1] del: fact.simps)
lp15@56215
  1099
      qed
lp15@56215
  1100
      finally show ?case .
lp15@56215
  1101
    qed
lp15@61609
  1102
    then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / (fact i)))
lp15@59730
  1103
                has_field_derivative f (Suc n) u * (z-u) ^ n / (fact n))
lp15@56215
  1104
               (at u within s)"
hoelzl@56381
  1105
      apply (intro derivative_eq_intros)
wenzelm@60420
  1106
      apply (blast intro: assms \<open>u \<in> s\<close>)
lp15@56215
  1107
      apply (rule refl)+
lp15@56215
  1108
      apply (auto simp: field_simps)
lp15@56215
  1109
      done
lp15@56215
  1110
  } note sum_deriv = this
lp15@56215
  1111
  { fix u
lp15@56215
  1112
    assume u: "u \<in> closed_segment w z"
lp15@56215
  1113
    then have us: "u \<in> s"
lp15@56215
  1114
      by (metis wzs subsetD)
lp15@56215
  1115
    have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> cmod (f (Suc n) u) * cmod (u - z) ^ n"
lp15@56215
  1116
      by (metis norm_minus_commute order_refl)
lp15@56215
  1117
    also have "... \<le> cmod (f (Suc n) u) * cmod (z - w) ^ n"
lp15@56215
  1118
      by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
lp15@56215
  1119
    also have "... \<le> B * cmod (z - w) ^ n"
lp15@56215
  1120
      by (metis norm_ge_zero zero_le_power mult_right_mono  B [OF us])
lp15@56215
  1121
    finally have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> B * cmod (z - w) ^ n" .
lp15@56215
  1122
  } note cmod_bound = this
lp15@59730
  1123
  have "(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)) = (\<Sum>i\<le>n. (f i z / (fact i)) * 0 ^ i)"
lp15@56215
  1124
    by simp
lp15@59730
  1125
  also have "\<dots> = f 0 z / (fact 0)"
lp15@56215
  1126
    by (subst setsum_zero_power) simp
lp15@61609
  1127
  finally have "cmod (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)))
lp15@59730
  1128
                \<le> cmod ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) -
lp15@59730
  1129
                        (\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))"
lp15@56215
  1130
    by (simp add: norm_minus_commute)
lp15@59730
  1131
  also have "... \<le> B * cmod (z - w) ^ n / (fact n) * cmod (w - z)"
lp15@61609
  1132
    apply (rule complex_differentiable_bound
lp15@59730
  1133
      [where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / (fact n)"
paulson@61518
  1134
         and s = "closed_segment w z", OF convex_closed_segment])
lp15@61609
  1135
    apply (auto simp: ends_in_segment DERIV_subset [OF sum_deriv wzs]
lp15@56215
  1136
                  norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
lp15@56215
  1137
    done
lp15@59730
  1138
  also have "...  \<le> B * cmod (z - w) ^ Suc n / (fact n)"
lp15@61609
  1139
    by (simp add: algebra_simps norm_minus_commute)
lp15@56215
  1140
  finally show ?thesis .
lp15@56215
  1141
qed
lp15@56215
  1142
lp15@62408
  1143
text\<open>Something more like the traditional MVT for real components\<close>
hoelzl@56370
  1144
lp15@56238
  1145
lemma complex_mvt_line:
hoelzl@56369
  1146
  assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f'(u)) (at u)"
paulson@61518
  1147
    shows "\<exists>u. u \<in> closed_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
lp15@56238
  1148
proof -
lp15@56238
  1149
  have twz: "\<And>t. (1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)"
lp15@56238
  1150
    by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
hoelzl@56381
  1151
  note assms[unfolded has_field_derivative_def, derivative_intros]
lp15@56238
  1152
  show ?thesis
lp15@56238
  1153
    apply (cut_tac mvt_simple
lp15@56238
  1154
                     [of 0 1 "Re o f o (\<lambda>t. (1 - t) *\<^sub>R w +  t *\<^sub>R z)"
lp15@56238
  1155
                      "\<lambda>u. Re o (\<lambda>h. f'((1 - u) *\<^sub>R w + u *\<^sub>R z) * h) o (\<lambda>t. t *\<^sub>R (z - w))"])
lp15@56238
  1156
    apply auto
lp15@56238
  1157
    apply (rule_tac x="(1 - x) *\<^sub>R w + x *\<^sub>R z" in exI)
paulson@61518
  1158
    apply (auto simp: closed_segment_def twz) []
paulson@61518
  1159
    apply (intro derivative_eq_intros has_derivative_at_within, simp_all)
hoelzl@56369
  1160
    apply (simp add: fun_eq_iff real_vector.scale_right_diff_distrib)
paulson@61518
  1161
    apply (force simp: twz closed_segment_def)
lp15@56238
  1162
    done
lp15@56238
  1163
qed
lp15@56238
  1164
lp15@56238
  1165
lemma complex_taylor_mvt:
lp15@56238
  1166
  assumes "\<And>i x. \<lbrakk>x \<in> closed_segment w z; i \<le> n\<rbrakk> \<Longrightarrow> ((f i) has_field_derivative f (Suc i) x) (at x)"
lp15@56238
  1167
    shows "\<exists>u. u \<in> closed_segment w z \<and>
lp15@56238
  1168
            Re (f 0 z) =
lp15@59730
  1169
            Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / (fact i)) +
lp15@59730
  1170
                (f (Suc n) u * (z-u)^n / (fact n)) * (z - w))"
lp15@56238
  1171
proof -
lp15@56238
  1172
  { fix u
lp15@56238
  1173
    assume u: "u \<in> closed_segment w z"
lp15@56238
  1174
    have "(\<Sum>i = 0..n.
lp15@56238
  1175
               (f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) /
lp15@59730
  1176
               (fact i)) =
lp15@56238
  1177
          f (Suc 0) u -
lp15@56238
  1178
             (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
lp15@59730
  1179
             (fact (Suc n)) +
lp15@56238
  1180
             (\<Sum>i = 0..n.
lp15@56238
  1181
                 (f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) /
lp15@59730
  1182
                 (fact (Suc i)))"
lp15@56238
  1183
       by (subst setsum_Suc_reindex) simp
lp15@56238
  1184
    also have "... = f (Suc 0) u -
lp15@56238
  1185
             (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
lp15@59730
  1186
             (fact (Suc n)) +
lp15@56238
  1187
             (\<Sum>i = 0..n.
lp15@61609
  1188
                 f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i))  -
lp15@59730
  1189
                 f (Suc i) u * (z-u) ^ i / (fact i))"
haftmann@57514
  1190
      by (simp only: diff_divide_distrib fact_cancel ac_simps)
lp15@56238
  1191
    also have "... = f (Suc 0) u -
lp15@56238
  1192
             (f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) /
lp15@59730
  1193
             (fact (Suc n)) +
lp15@59730
  1194
             f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n)) - f (Suc 0) u"
lp15@56238
  1195
      by (subst setsum_Suc_diff) auto
lp15@59730
  1196
    also have "... = f (Suc n) u * (z-u) ^ n / (fact n)"
lp15@56238
  1197
      by (simp only: algebra_simps diff_divide_distrib fact_cancel)
lp15@61609
  1198
    finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i
lp15@59730
  1199
                             - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / (fact i)) =
lp15@59730
  1200
                  f (Suc n) u * (z - u) ^ n / (fact n)" .
lp15@59730
  1201
    then have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / (fact i)) has_field_derivative
lp15@59730
  1202
                f (Suc n) u * (z - u) ^ n / (fact n))  (at u)"
hoelzl@56381
  1203
      apply (intro derivative_eq_intros)+
lp15@56238
  1204
      apply (force intro: u assms)
lp15@56238
  1205
      apply (rule refl)+
haftmann@57514
  1206
      apply (auto simp: ac_simps)
lp15@56238
  1207
      done
lp15@56238
  1208
  }
lp15@56238
  1209
  then show ?thesis
lp15@59730
  1210
    apply (cut_tac complex_mvt_line [of w z "\<lambda>u. \<Sum>i = 0..n. f i u * (z-u) ^ i / (fact i)"
lp15@59730
  1211
               "\<lambda>u. (f (Suc n) u * (z-u)^n / (fact n))"])
lp15@56238
  1212
    apply (auto simp add: intro: open_closed_segment)
lp15@56238
  1213
    done
lp15@56238
  1214
qed
lp15@56238
  1215
lp15@60017
  1216
wenzelm@60420
  1217
subsection \<open>Polynomal function extremal theorem, from HOL Light\<close>
lp15@60017
  1218
lp15@60017
  1219
lemma polyfun_extremal_lemma: (*COMPLEX_POLYFUN_EXTREMAL_LEMMA in HOL Light*)
lp15@60017
  1220
    fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
lp15@60017
  1221
  assumes "0 < e"
lp15@60017
  1222
    shows "\<exists>M. \<forall>z. M \<le> norm(z) \<longrightarrow> norm (\<Sum>i\<le>n. c(i) * z^i) \<le> e * norm(z) ^ (Suc n)"
lp15@60017
  1223
proof (induct n)
lp15@60017
  1224
  case 0 with assms
lp15@60017
  1225
  show ?case
lp15@60017
  1226
    apply (rule_tac x="norm (c 0) / e" in exI)
lp15@60017
  1227
    apply (auto simp: field_simps)
lp15@60017
  1228
    done
lp15@60017
  1229
next
lp15@60017
  1230
  case (Suc n)
lp15@60017
  1231
  obtain M where M: "\<And>z. M \<le> norm z \<Longrightarrow> norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
lp15@60017
  1232
    using Suc assms by blast
lp15@60017
  1233
  show ?case
lp15@60017
  1234
  proof (rule exI [where x= "max M (1 + norm(c(Suc n)) / e)"], clarsimp simp del: power_Suc)
lp15@60017
  1235
    fix z::'a
lp15@60017
  1236
    assume z1: "M \<le> norm z" and "1 + norm (c (Suc n)) / e \<le> norm z"
lp15@60017
  1237
    then have z2: "e + norm (c (Suc n)) \<le> e * norm z"
lp15@60017
  1238
      using assms by (simp add: field_simps)
lp15@60017
  1239
    have "norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
lp15@60017
  1240
      using M [OF z1] by simp
lp15@60017
  1241
    then have "norm (\<Sum>i\<le>n. c i * z^i) + norm (c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc n + norm (c (Suc n) * z ^ Suc n)"
lp15@60017
  1242
      by simp
lp15@60017
  1243
    then have "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc n + norm (c (Suc n) * z ^ Suc n)"
lp15@60017
  1244
      by (blast intro: norm_triangle_le elim: )
lp15@60017
  1245
    also have "... \<le> (e + norm (c (Suc n))) * norm z ^ Suc n"
lp15@60017
  1246
      by (simp add: norm_power norm_mult algebra_simps)
lp15@60017
  1247
    also have "... \<le> (e * norm z) * norm z ^ Suc n"
lp15@60017
  1248
      by (metis z2 mult.commute mult_left_mono norm_ge_zero norm_power)
lp15@60017
  1249
    finally show "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc (Suc n)"
lp15@60162
  1250
      by simp
lp15@60017
  1251
  qed
lp15@60017
  1252
qed
lp15@60017
  1253
lp15@60017
  1254
lemma polyfun_extremal: (*COMPLEX_POLYFUN_EXTREMAL in HOL Light*)
lp15@60017
  1255
    fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
lp15@60017
  1256
  assumes k: "c k \<noteq> 0" "1\<le>k" and kn: "k\<le>n"
lp15@60017
  1257
    shows "eventually (\<lambda>z. norm (\<Sum>i\<le>n. c(i) * z^i) \<ge> B) at_infinity"
lp15@60017
  1258
using kn
lp15@60017
  1259
proof (induction n)
lp15@60017
  1260
  case 0
lp15@60017
  1261
  then show ?case
lp15@60017
  1262
    using k  by simp
lp15@60017
  1263
next
lp15@60017
  1264
  case (Suc m)
lp15@60017
  1265
  let ?even = ?case
lp15@60017
  1266
  show ?even
lp15@60017
  1267
  proof (cases "c (Suc m) = 0")
lp15@60017
  1268
    case True
lp15@60017
  1269
    then show ?even using Suc k
lp15@60017
  1270
      by auto (metis antisym_conv less_eq_Suc_le not_le)
lp15@60017
  1271
  next
lp15@60017
  1272
    case False
lp15@60017
  1273
    then obtain M where M:
lp15@60017
  1274
          "\<And>z. M \<le> norm z \<Longrightarrow> norm (\<Sum>i\<le>m. c i * z^i) \<le> norm (c (Suc m)) / 2 * norm z ^ Suc m"
lp15@60017
  1275
      using polyfun_extremal_lemma [of "norm(c (Suc m)) / 2" c m] Suc
lp15@60017
  1276
      by auto
lp15@60017
  1277
    have "\<exists>b. \<forall>z. b \<le> norm z \<longrightarrow> B \<le> norm (\<Sum>i\<le>Suc m. c i * z^i)"
lp15@60017
  1278
    proof (rule exI [where x="max M (max 1 (\<bar>B\<bar> / (norm(c (Suc m)) / 2)))"], clarsimp simp del: power_Suc)
lp15@60017
  1279
      fix z::'a
lp15@60017
  1280
      assume z1: "M \<le> norm z" "1 \<le> norm z"
lp15@60017
  1281
         and "\<bar>B\<bar> * 2 / norm (c (Suc m)) \<le> norm z"
lp15@60017
  1282
      then have z2: "\<bar>B\<bar> \<le> norm (c (Suc m)) * norm z / 2"
lp15@60017
  1283
        using False by (simp add: field_simps)
lp15@60017
  1284
      have nz: "norm z \<le> norm z ^ Suc m"
wenzelm@60420
  1285
        by (metis \<open>1 \<le> norm z\<close> One_nat_def less_eq_Suc_le power_increasing power_one_right zero_less_Suc)
lp15@60017
  1286
      have *: "\<And>y x. norm (c (Suc m)) * norm z / 2 \<le> norm y - norm x \<Longrightarrow> B \<le> norm (x + y)"
lp15@60017
  1287
        by (metis abs_le_iff add.commute norm_diff_ineq order_trans z2)
lp15@60017
  1288
      have "norm z * norm (c (Suc m)) + 2 * norm (\<Sum>i\<le>m. c i * z^i)
lp15@60017
  1289
            \<le> norm (c (Suc m)) * norm z + norm (c (Suc m)) * norm z ^ Suc m"
lp15@60017
  1290
        using M [of z] Suc z1  by auto
lp15@60017
  1291
      also have "... \<le> 2 * (norm (c (Suc m)) * norm z ^ Suc m)"
lp15@60017
  1292
        using nz by (simp add: mult_mono del: power_Suc)
lp15@60017
  1293
      finally show "B \<le> norm ((\<Sum>i\<le>m. c i * z^i) + c (Suc m) * z ^ Suc m)"
lp15@60017
  1294
        using Suc.IH
lp15@60017
  1295
        apply (auto simp: eventually_at_infinity)
lp15@60017
  1296
        apply (rule *)
lp15@60017
  1297
        apply (simp add: field_simps norm_mult norm_power)
lp15@60017
  1298
        done
lp15@60017
  1299
    qed
lp15@60017
  1300
    then show ?even
lp15@60017
  1301
      by (simp add: eventually_at_infinity)
lp15@60017
  1302
  qed
lp15@60017
  1303
qed
lp15@60017
  1304
lp15@56215
  1305
end