src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy
author paulson <lp15@cam.ac.uk>
Mon Mar 16 15:30:00 2015 +0000 (2015-03-16)
changeset 59730 b7c394c7a619
parent 59615 fdfdf89a83a6
child 60017 b785d6d06430
permissions -rw-r--r--
The factorial function, "fact", now has type "nat => 'a"
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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 {* Complex Analysis Basics *}
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theory Complex_Analysis_Basics
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imports  "~~/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space"
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begin
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lemma cmod_fact [simp]: "cmod (fact n) = fact n"
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  by (metis norm_of_nat of_nat_fact)
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subsection{*General lemmas*}
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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)"
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  using has_derivative_compose[of of_real of_real a UNIV 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 linear_times:
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  fixes c::"'a::real_algebra" shows "linear (\<lambda>x. c * x)"
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  by (auto simp: linearI distrib_left)
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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_mult_left:
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  fixes c::"'a::real_normed_algebra" 
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  shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) ---> c * l) F"
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by (rule tendsto_mult [OF tendsto_const])
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lemma tendsto_mult_right:
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  fixes c::"'a::real_normed_algebra" 
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  shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) ---> l * c) F"
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by (rule tendsto_mult [OF _ tendsto_const])
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lemma tendsto_Re_upper:
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  assumes "~ (trivial_limit F)" 
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          "(f ---> 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 ---> 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 ---> 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 ---> 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 has_real_derivative:
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  fixes f :: "real \<Rightarrow> real" 
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  assumes "(f has_derivative f') F"
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  obtains c where "(f has_real_derivative c) F"
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proof -
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  obtain c where "f' = (\<lambda>x. x * c)"
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    by (metis assms has_derivative_bounded_linear real_bounded_linear)
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  then show ?thesis
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    by (metis assms that has_field_derivative_def mult_commute_abs)
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qed
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lemma has_real_derivative_iff:
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  fixes f :: "real \<Rightarrow> real" 
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  shows "(\<exists>c. (f has_real_derivative c) F) = (\<exists>D. (f has_derivative D) F)"
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  by (metis has_field_derivative_def has_real_derivative)
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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{*DERIV stuff*}
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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 {*Some limit theorems about real part of real series etc.*}
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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 {*Complex number lemmas *}
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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 isCont_ident isCont_const)+
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lemma closed_complex_Reals: "closed (Reals :: complex set)"
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proof -
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  have "(Reals :: 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 real_lim:
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  fixes l::complex
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  assumes "(f ---> 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 ---> 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 ---> 0) F" shows "(f ---> 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{*Holomorphic functions*}
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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:
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    "\<lbrakk>f complex_differentiable (at x within s); t \<subseteq> s\<rbrakk>
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     \<Longrightarrow> f complex_differentiable (at x within t)"
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  by (metis DERIV_subset complex_differentiable_def)
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lemma complex_differentiable_at_within:
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    "\<lbrakk>f complex_differentiable (at x)\<rbrakk>
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     \<Longrightarrow> f complex_differentiable (at x within s)"
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  unfolding complex_differentiable_def
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   296
  by (metis DERIV_subset top_greatest)
lp15@56215
   297
hoelzl@56370
   298
lemma complex_differentiable_linear: "(op * c) complex_differentiable F"
lp15@56215
   299
proof -
hoelzl@56370
   300
  show ?thesis
hoelzl@56370
   301
    unfolding complex_differentiable_def has_field_derivative_def mult_commute_abs
lp15@56215
   302
    by (force intro: has_derivative_mult_right)
lp15@56215
   303
qed
lp15@56215
   304
hoelzl@56370
   305
lemma complex_differentiable_const: "(\<lambda>z. c) complex_differentiable F"
lp15@56215
   306
  unfolding complex_differentiable_def has_field_derivative_def
hoelzl@56369
   307
  by (rule exI [where x=0])
hoelzl@56369
   308
     (metis has_derivative_const lambda_zero) 
lp15@56215
   309
hoelzl@56370
   310
lemma complex_differentiable_ident: "(\<lambda>z. z) complex_differentiable F"
lp15@56215
   311
  unfolding complex_differentiable_def has_field_derivative_def
hoelzl@56369
   312
  by (rule exI [where x=1])
hoelzl@56369
   313
     (simp add: lambda_one [symmetric])
lp15@56215
   314
hoelzl@56370
   315
lemma complex_differentiable_id: "id complex_differentiable F"
hoelzl@56370
   316
  unfolding id_def by (rule complex_differentiable_ident)
hoelzl@56370
   317
lp15@56215
   318
lemma complex_differentiable_minus:
hoelzl@56370
   319
  "f complex_differentiable F \<Longrightarrow> (\<lambda>z. - (f z)) complex_differentiable F"
lp15@56215
   320
  using assms unfolding complex_differentiable_def
lp15@56215
   321
  by (metis field_differentiable_minus)
lp15@56215
   322
lp15@56215
   323
lemma complex_differentiable_add:
lp15@56215
   324
  assumes "f complex_differentiable F" "g complex_differentiable F"
lp15@56215
   325
    shows "(\<lambda>z. f z + g z) complex_differentiable F"
lp15@56215
   326
  using assms unfolding complex_differentiable_def
lp15@56215
   327
  by (metis field_differentiable_add)
lp15@56215
   328
hoelzl@56370
   329
lemma complex_differentiable_setsum:
hoelzl@56370
   330
  "(\<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
   331
  by (induct I rule: infinite_finite_induct)
hoelzl@56370
   332
     (auto intro: complex_differentiable_add complex_differentiable_const)
hoelzl@56370
   333
lp15@56215
   334
lemma complex_differentiable_diff:
lp15@56215
   335
  assumes "f complex_differentiable F" "g complex_differentiable F"
lp15@56215
   336
    shows "(\<lambda>z. f z - g z) complex_differentiable F"
lp15@56215
   337
  using assms unfolding complex_differentiable_def
lp15@56215
   338
  by (metis field_differentiable_diff)
lp15@56215
   339
lp15@56215
   340
lemma complex_differentiable_inverse:
lp15@56215
   341
  assumes "f complex_differentiable (at a within s)" "f a \<noteq> 0"
lp15@56215
   342
  shows "(\<lambda>z. inverse (f z)) complex_differentiable (at a within s)"
lp15@56215
   343
  using assms unfolding complex_differentiable_def
lp15@56215
   344
  by (metis DERIV_inverse_fun)
lp15@56215
   345
lp15@56215
   346
lemma complex_differentiable_mult:
lp15@56215
   347
  assumes "f complex_differentiable (at a within s)" 
lp15@56215
   348
          "g complex_differentiable (at a within s)"
lp15@56215
   349
    shows "(\<lambda>z. f z * g z) complex_differentiable (at a within s)"
lp15@56215
   350
  using assms unfolding complex_differentiable_def
lp15@56215
   351
  by (metis DERIV_mult [of f _ a s g])
lp15@56215
   352
  
lp15@56215
   353
lemma complex_differentiable_divide:
lp15@56215
   354
  assumes "f complex_differentiable (at a within s)" 
lp15@56215
   355
          "g complex_differentiable (at a within s)"
lp15@56215
   356
          "g a \<noteq> 0"
lp15@56215
   357
    shows "(\<lambda>z. f z / g z) complex_differentiable (at a within s)"
lp15@56215
   358
  using assms unfolding complex_differentiable_def
lp15@56215
   359
  by (metis DERIV_divide [of f _ a s g])
lp15@56215
   360
lp15@56215
   361
lemma complex_differentiable_power:
lp15@56215
   362
  assumes "f complex_differentiable (at a within s)" 
lp15@56215
   363
    shows "(\<lambda>z. f z ^ n) complex_differentiable (at a within s)"
lp15@56215
   364
  using assms unfolding complex_differentiable_def
lp15@56215
   365
  by (metis DERIV_power)
lp15@56215
   366
lp15@56215
   367
lemma complex_differentiable_transform_within:
lp15@56215
   368
  "0 < d \<Longrightarrow>
lp15@56215
   369
        x \<in> s \<Longrightarrow>
lp15@56215
   370
        (\<And>x'. x' \<in> s \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x') \<Longrightarrow>
lp15@56215
   371
        f complex_differentiable (at x within s)
lp15@56215
   372
        \<Longrightarrow> g complex_differentiable (at x within s)"
lp15@56215
   373
  unfolding complex_differentiable_def has_field_derivative_def
lp15@56215
   374
  by (blast intro: has_derivative_transform_within)
lp15@56215
   375
lp15@56215
   376
lemma complex_differentiable_compose_within:
lp15@56215
   377
  assumes "f complex_differentiable (at a within s)" 
lp15@56215
   378
          "g complex_differentiable (at (f a) within f`s)"
lp15@56215
   379
    shows "(g o f) complex_differentiable (at a within s)"
lp15@56215
   380
  using assms unfolding complex_differentiable_def
lp15@56215
   381
  by (metis DERIV_image_chain)
lp15@56215
   382
hoelzl@56370
   383
lemma complex_differentiable_compose:
hoelzl@56370
   384
  "f complex_differentiable at z \<Longrightarrow> g complex_differentiable at (f z)
hoelzl@56370
   385
          \<Longrightarrow> (g o f) complex_differentiable at z"
hoelzl@56370
   386
by (metis complex_differentiable_at_within complex_differentiable_compose_within)
hoelzl@56370
   387
lp15@56215
   388
lemma complex_differentiable_within_open:
lp15@56215
   389
     "\<lbrakk>a \<in> s; open s\<rbrakk> \<Longrightarrow> f complex_differentiable at a within s \<longleftrightarrow> 
lp15@56215
   390
                          f complex_differentiable at a"
lp15@56215
   391
  unfolding complex_differentiable_def
lp15@56215
   392
  by (metis at_within_open)
lp15@56215
   393
lp15@56215
   394
subsection{*Caratheodory characterization.*}
lp15@56215
   395
lp15@56215
   396
lemma complex_differentiable_caratheodory_at:
lp15@56215
   397
  "f complex_differentiable (at z) \<longleftrightarrow>
lp15@56215
   398
         (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)"
lp15@56215
   399
  using CARAT_DERIV [of f]
lp15@56215
   400
  by (simp add: complex_differentiable_def has_field_derivative_def)
lp15@56215
   401
lp15@56215
   402
lemma complex_differentiable_caratheodory_within:
lp15@56215
   403
  "f complex_differentiable (at z within s) \<longleftrightarrow>
lp15@56215
   404
         (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)"
lp15@56215
   405
  using DERIV_caratheodory_within [of f]
lp15@56215
   406
  by (simp add: complex_differentiable_def has_field_derivative_def)
lp15@56215
   407
hoelzl@56370
   408
subsection{*Holomorphic*}
hoelzl@56370
   409
hoelzl@56370
   410
definition holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
hoelzl@56370
   411
           (infixl "(holomorphic'_on)" 50)
hoelzl@56370
   412
  where "f holomorphic_on s \<equiv> \<forall>x\<in>s. f complex_differentiable (at x within s)"
hoelzl@56370
   413
  
hoelzl@56370
   414
lemma holomorphic_on_empty: "f holomorphic_on {}"
hoelzl@56370
   415
  by (simp add: holomorphic_on_def)
hoelzl@56370
   416
hoelzl@56370
   417
lemma holomorphic_on_open:
hoelzl@56370
   418
    "open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> f')"
hoelzl@56370
   419
  by (auto simp: holomorphic_on_def complex_differentiable_def has_field_derivative_def at_within_open [of _ s])
hoelzl@56370
   420
hoelzl@56370
   421
lemma holomorphic_on_imp_continuous_on: 
hoelzl@56370
   422
    "f holomorphic_on s \<Longrightarrow> continuous_on s f"
hoelzl@56370
   423
  by (metis complex_differentiable_imp_continuous_at continuous_on_eq_continuous_within holomorphic_on_def) 
hoelzl@56370
   424
hoelzl@56370
   425
lemma holomorphic_on_subset:
hoelzl@56370
   426
    "f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t"
hoelzl@56370
   427
  unfolding holomorphic_on_def
hoelzl@56370
   428
  by (metis complex_differentiable_within_subset subsetD)
hoelzl@56370
   429
hoelzl@56370
   430
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
   431
  by (metis complex_differentiable_transform_within linordered_field_no_ub holomorphic_on_def)
hoelzl@56370
   432
hoelzl@56370
   433
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
   434
  by (metis holomorphic_transform)
hoelzl@56370
   435
hoelzl@56370
   436
lemma holomorphic_on_linear: "(op * c) holomorphic_on s"
hoelzl@56370
   437
  unfolding holomorphic_on_def by (metis complex_differentiable_linear)
hoelzl@56370
   438
hoelzl@56370
   439
lemma holomorphic_on_const: "(\<lambda>z. c) holomorphic_on s"
hoelzl@56370
   440
  unfolding holomorphic_on_def by (metis complex_differentiable_const)
hoelzl@56370
   441
hoelzl@56370
   442
lemma holomorphic_on_ident: "(\<lambda>x. x) holomorphic_on s"
hoelzl@56370
   443
  unfolding holomorphic_on_def by (metis complex_differentiable_ident)
hoelzl@56370
   444
hoelzl@56370
   445
lemma holomorphic_on_id: "id holomorphic_on s"
hoelzl@56370
   446
  unfolding id_def by (rule holomorphic_on_ident)
hoelzl@56370
   447
hoelzl@56370
   448
lemma holomorphic_on_compose:
hoelzl@56370
   449
  "f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s) \<Longrightarrow> (g o f) holomorphic_on s"
hoelzl@56370
   450
  using complex_differentiable_compose_within[of f _ s g]
hoelzl@56370
   451
  by (auto simp: holomorphic_on_def)
hoelzl@56370
   452
hoelzl@56370
   453
lemma holomorphic_on_compose_gen:
hoelzl@56370
   454
  "f holomorphic_on s \<Longrightarrow> g holomorphic_on t \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> (g o f) holomorphic_on s"
hoelzl@56370
   455
  by (metis holomorphic_on_compose holomorphic_on_subset)
hoelzl@56370
   456
hoelzl@56370
   457
lemma holomorphic_on_minus: "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on s"
hoelzl@56370
   458
  by (metis complex_differentiable_minus holomorphic_on_def)
hoelzl@56370
   459
hoelzl@56370
   460
lemma holomorphic_on_add:
hoelzl@56370
   461
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on s"
hoelzl@56370
   462
  unfolding holomorphic_on_def by (metis complex_differentiable_add)
hoelzl@56370
   463
hoelzl@56370
   464
lemma holomorphic_on_diff:
hoelzl@56370
   465
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) holomorphic_on s"
hoelzl@56370
   466
  unfolding holomorphic_on_def by (metis complex_differentiable_diff)
hoelzl@56370
   467
hoelzl@56370
   468
lemma holomorphic_on_mult:
hoelzl@56370
   469
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) holomorphic_on s"
hoelzl@56370
   470
  unfolding holomorphic_on_def by (metis complex_differentiable_mult)
hoelzl@56370
   471
hoelzl@56370
   472
lemma holomorphic_on_inverse:
hoelzl@56370
   473
  "\<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
   474
  unfolding holomorphic_on_def by (metis complex_differentiable_inverse)
hoelzl@56370
   475
hoelzl@56370
   476
lemma holomorphic_on_divide:
hoelzl@56370
   477
  "\<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
   478
  unfolding holomorphic_on_def by (metis complex_differentiable_divide)
hoelzl@56370
   479
hoelzl@56370
   480
lemma holomorphic_on_power:
hoelzl@56370
   481
  "f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on s"
hoelzl@56370
   482
  unfolding holomorphic_on_def by (metis complex_differentiable_power)
hoelzl@56370
   483
hoelzl@56370
   484
lemma holomorphic_on_setsum:
hoelzl@56370
   485
  "(\<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
   486
  unfolding holomorphic_on_def by (metis complex_differentiable_setsum)
hoelzl@56370
   487
hoelzl@56370
   488
definition deriv :: "('a \<Rightarrow> 'a::real_normed_field) \<Rightarrow> 'a \<Rightarrow> 'a" where
hoelzl@56370
   489
  "deriv f x \<equiv> THE D. DERIV f x :> D"
hoelzl@56370
   490
hoelzl@56370
   491
lemma DERIV_imp_deriv: "DERIV f x :> f' \<Longrightarrow> deriv f x = f'"
hoelzl@56370
   492
  unfolding deriv_def by (metis the_equality DERIV_unique)
hoelzl@56370
   493
hoelzl@56370
   494
lemma DERIV_deriv_iff_real_differentiable:
hoelzl@56370
   495
  fixes x :: real
hoelzl@56370
   496
  shows "DERIV f x :> deriv f x \<longleftrightarrow> f differentiable at x"
hoelzl@56370
   497
  unfolding differentiable_def by (metis DERIV_imp_deriv has_real_derivative_iff)
hoelzl@56370
   498
hoelzl@56370
   499
lemma real_derivative_chain:
hoelzl@56370
   500
  fixes x :: real
hoelzl@56370
   501
  shows "f differentiable at x \<Longrightarrow> g differentiable at (f x)
hoelzl@56370
   502
    \<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
hoelzl@56370
   503
  by (metis DERIV_deriv_iff_real_differentiable DERIV_chain DERIV_imp_deriv)
hoelzl@56370
   504
hoelzl@56370
   505
lemma DERIV_deriv_iff_complex_differentiable:
hoelzl@56370
   506
  "DERIV f x :> deriv f x \<longleftrightarrow> f complex_differentiable at x"
hoelzl@56370
   507
  unfolding complex_differentiable_def by (metis DERIV_imp_deriv)
hoelzl@56370
   508
hoelzl@56370
   509
lemma complex_derivative_chain:
hoelzl@56370
   510
  "f complex_differentiable at x \<Longrightarrow> g complex_differentiable at (f x)
hoelzl@56370
   511
    \<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
hoelzl@56370
   512
  by (metis DERIV_deriv_iff_complex_differentiable DERIV_chain DERIV_imp_deriv)
hoelzl@56370
   513
hoelzl@56370
   514
lemma complex_derivative_linear: "deriv (\<lambda>w. c * w) = (\<lambda>z. c)"
hoelzl@56370
   515
  by (metis DERIV_imp_deriv DERIV_cmult_Id)
hoelzl@56370
   516
hoelzl@56370
   517
lemma complex_derivative_ident: "deriv (\<lambda>w. w) = (\<lambda>z. 1)"
hoelzl@56370
   518
  by (metis DERIV_imp_deriv DERIV_ident)
hoelzl@56370
   519
hoelzl@56370
   520
lemma complex_derivative_const: "deriv (\<lambda>w. c) = (\<lambda>z. 0)"
hoelzl@56370
   521
  by (metis DERIV_imp_deriv DERIV_const)
hoelzl@56370
   522
hoelzl@56370
   523
lemma complex_derivative_add:
hoelzl@56370
   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
hoelzl@56370
   529
lemma complex_derivative_diff:
hoelzl@56370
   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
hoelzl@56370
   535
lemma complex_derivative_mult:
hoelzl@56370
   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
hoelzl@56370
   541
lemma complex_derivative_cmult:
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
hoelzl@56370
   546
lemma complex_derivative_cmult_right:
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
hoelzl@56370
   551
lemma complex_derivative_transform_within_open:
hoelzl@56370
   552
  "\<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
   553
   \<Longrightarrow> deriv f z = deriv g z"
hoelzl@56370
   554
  unfolding holomorphic_on_def
hoelzl@56370
   555
  by (rule DERIV_imp_deriv)
hoelzl@56370
   556
     (metis DERIV_deriv_iff_complex_differentiable DERIV_transform_within_open at_within_open)
hoelzl@56370
   557
hoelzl@56370
   558
lemma complex_derivative_compose_linear:
hoelzl@56370
   559
  "f complex_differentiable at (c * z) \<Longrightarrow> deriv (\<lambda>w. f (c * w)) z = c * deriv f (c * z)"
hoelzl@56370
   560
apply (rule DERIV_imp_deriv)
hoelzl@56370
   561
apply (simp add: DERIV_deriv_iff_complex_differentiable [symmetric])
haftmann@59554
   562
apply (drule DERIV_chain' [of "times c" c z UNIV f "deriv f (c * z)", OF DERIV_cmult_Id])
haftmann@59554
   563
apply (simp add: algebra_simps)
hoelzl@56370
   564
done
hoelzl@56370
   565
lp15@59615
   566
subsection{*Analyticity on a set*}
lp15@56215
   567
lp15@56215
   568
definition analytic_on (infixl "(analytic'_on)" 50)  
lp15@56215
   569
  where
lp15@56215
   570
   "f analytic_on s \<equiv> \<forall>x \<in> s. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)"
lp15@56215
   571
hoelzl@56370
   572
lemma analytic_imp_holomorphic: "f analytic_on s \<Longrightarrow> f holomorphic_on s"
hoelzl@56370
   573
  by (simp add: at_within_open [OF _ open_ball] analytic_on_def holomorphic_on_def)
hoelzl@56370
   574
     (metis centre_in_ball complex_differentiable_at_within)
lp15@56215
   575
hoelzl@56370
   576
lemma analytic_on_open: "open s \<Longrightarrow> f analytic_on s \<longleftrightarrow> f holomorphic_on s"
lp15@56215
   577
apply (auto simp: analytic_imp_holomorphic)
lp15@56215
   578
apply (auto simp: analytic_on_def holomorphic_on_def)
lp15@56215
   579
by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball)
lp15@56215
   580
lp15@56215
   581
lemma analytic_on_imp_differentiable_at:
lp15@56215
   582
  "f analytic_on s \<Longrightarrow> x \<in> s \<Longrightarrow> f complex_differentiable (at x)"
hoelzl@56370
   583
 apply (auto simp: analytic_on_def holomorphic_on_def)
lp15@56215
   584
by (metis Topology_Euclidean_Space.open_ball centre_in_ball complex_differentiable_within_open)
lp15@56215
   585
hoelzl@56370
   586
lemma analytic_on_subset: "f analytic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f analytic_on t"
lp15@56215
   587
  by (auto simp: analytic_on_def)
lp15@56215
   588
hoelzl@56370
   589
lemma analytic_on_Un: "f analytic_on (s \<union> t) \<longleftrightarrow> f analytic_on s \<and> f analytic_on t"
lp15@56215
   590
  by (auto simp: analytic_on_def)
lp15@56215
   591
hoelzl@56370
   592
lemma analytic_on_Union: "f analytic_on (\<Union> s) \<longleftrightarrow> (\<forall>t \<in> s. f analytic_on t)"
hoelzl@56370
   593
  by (auto simp: analytic_on_def)
hoelzl@56370
   594
hoelzl@56370
   595
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
   596
  by (auto simp: analytic_on_def)
lp15@56215
   597
  
lp15@56215
   598
lemma analytic_on_holomorphic:
lp15@56215
   599
  "f analytic_on s \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f holomorphic_on t)"
lp15@56215
   600
  (is "?lhs = ?rhs")
lp15@56215
   601
proof -
lp15@56215
   602
  have "?lhs \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t)"
lp15@56215
   603
  proof safe
lp15@56215
   604
    assume "f analytic_on s"
lp15@56215
   605
    then show "\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t"
lp15@56215
   606
      apply (simp add: analytic_on_def)
lp15@56215
   607
      apply (rule exI [where x="\<Union>{u. open u \<and> f analytic_on u}"], auto)
lp15@56215
   608
      apply (metis Topology_Euclidean_Space.open_ball analytic_on_open centre_in_ball)
lp15@56215
   609
      by (metis analytic_on_def)
lp15@56215
   610
  next
lp15@56215
   611
    fix t
lp15@56215
   612
    assume "open t" "s \<subseteq> t" "f analytic_on t" 
lp15@56215
   613
    then show "f analytic_on s"
lp15@56215
   614
        by (metis analytic_on_subset)
lp15@56215
   615
  qed
lp15@56215
   616
  also have "... \<longleftrightarrow> ?rhs"
lp15@56215
   617
    by (auto simp: analytic_on_open)
lp15@56215
   618
  finally show ?thesis .
lp15@56215
   619
qed
lp15@56215
   620
lp15@56215
   621
lemma analytic_on_linear: "(op * c) analytic_on s"
hoelzl@56370
   622
  by (auto simp add: analytic_on_holomorphic holomorphic_on_linear)
lp15@56215
   623
lp15@56215
   624
lemma analytic_on_const: "(\<lambda>z. c) analytic_on s"
hoelzl@56370
   625
  by (metis analytic_on_def holomorphic_on_const zero_less_one)
hoelzl@56370
   626
hoelzl@56370
   627
lemma analytic_on_ident: "(\<lambda>x. x) analytic_on s"
hoelzl@56370
   628
  by (simp add: analytic_on_def holomorphic_on_ident gt_ex)
lp15@56215
   629
lp15@56215
   630
lemma analytic_on_id: "id analytic_on s"
hoelzl@56370
   631
  unfolding id_def by (rule analytic_on_ident)
lp15@56215
   632
lp15@56215
   633
lemma analytic_on_compose:
lp15@56215
   634
  assumes f: "f analytic_on s"
lp15@56215
   635
      and g: "g analytic_on (f ` s)"
lp15@56215
   636
    shows "(g o f) analytic_on s"
lp15@56215
   637
unfolding analytic_on_def
lp15@56215
   638
proof (intro ballI)
lp15@56215
   639
  fix x
lp15@56215
   640
  assume x: "x \<in> s"
lp15@56215
   641
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f
lp15@56215
   642
    by (metis analytic_on_def)
lp15@56215
   643
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g
lp15@56215
   644
    by (metis analytic_on_def g image_eqI x) 
lp15@56215
   645
  have "isCont f x"
lp15@56215
   646
    by (metis analytic_on_imp_differentiable_at complex_differentiable_imp_continuous_at f x)
lp15@56215
   647
  with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'"
lp15@56215
   648
     by (auto simp: continuous_at_ball)
lp15@56215
   649
  have "g \<circ> f holomorphic_on ball x (min d e)" 
lp15@56215
   650
    apply (rule holomorphic_on_compose)
lp15@56215
   651
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   652
    by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball)
lp15@56215
   653
  then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e"
lp15@56215
   654
    by (metis d e min_less_iff_conj) 
lp15@56215
   655
qed
lp15@56215
   656
lp15@56215
   657
lemma analytic_on_compose_gen:
lp15@56215
   658
  "f analytic_on s \<Longrightarrow> g analytic_on t \<Longrightarrow> (\<And>z. z \<in> s \<Longrightarrow> f z \<in> t)
lp15@56215
   659
             \<Longrightarrow> g o f analytic_on s"
lp15@56215
   660
by (metis analytic_on_compose analytic_on_subset image_subset_iff)
lp15@56215
   661
lp15@56215
   662
lemma analytic_on_neg:
lp15@56215
   663
  "f analytic_on s \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on s"
lp15@56215
   664
by (metis analytic_on_holomorphic holomorphic_on_minus)
lp15@56215
   665
lp15@56215
   666
lemma analytic_on_add:
lp15@56215
   667
  assumes f: "f analytic_on s"
lp15@56215
   668
      and g: "g analytic_on s"
lp15@56215
   669
    shows "(\<lambda>z. f z + g z) analytic_on s"
lp15@56215
   670
unfolding analytic_on_def
lp15@56215
   671
proof (intro ballI)
lp15@56215
   672
  fix z
lp15@56215
   673
  assume z: "z \<in> s"
lp15@56215
   674
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   675
    by (metis analytic_on_def)
lp15@56215
   676
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
lp15@56215
   677
    by (metis analytic_on_def g z) 
lp15@56215
   678
  have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')" 
lp15@56215
   679
    apply (rule holomorphic_on_add) 
lp15@56215
   680
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   681
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   682
  then show "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e"
lp15@56215
   683
    by (metis e e' min_less_iff_conj)
lp15@56215
   684
qed
lp15@56215
   685
lp15@56215
   686
lemma analytic_on_diff:
lp15@56215
   687
  assumes f: "f analytic_on s"
lp15@56215
   688
      and g: "g analytic_on s"
lp15@56215
   689
    shows "(\<lambda>z. f z - g z) analytic_on s"
lp15@56215
   690
unfolding analytic_on_def
lp15@56215
   691
proof (intro ballI)
lp15@56215
   692
  fix z
lp15@56215
   693
  assume z: "z \<in> s"
lp15@56215
   694
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   695
    by (metis analytic_on_def)
lp15@56215
   696
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
lp15@56215
   697
    by (metis analytic_on_def g z) 
lp15@56215
   698
  have "(\<lambda>z. f z - g z) holomorphic_on ball z (min e e')" 
lp15@56215
   699
    apply (rule holomorphic_on_diff) 
lp15@56215
   700
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   701
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   702
  then show "\<exists>e>0. (\<lambda>z. f z - g z) holomorphic_on ball z e"
lp15@56215
   703
    by (metis e e' min_less_iff_conj)
lp15@56215
   704
qed
lp15@56215
   705
lp15@56215
   706
lemma analytic_on_mult:
lp15@56215
   707
  assumes f: "f analytic_on s"
lp15@56215
   708
      and g: "g analytic_on s"
lp15@56215
   709
    shows "(\<lambda>z. f z * g z) analytic_on s"
lp15@56215
   710
unfolding analytic_on_def
lp15@56215
   711
proof (intro ballI)
lp15@56215
   712
  fix z
lp15@56215
   713
  assume z: "z \<in> s"
lp15@56215
   714
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   715
    by (metis analytic_on_def)
lp15@56215
   716
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
lp15@56215
   717
    by (metis analytic_on_def g z) 
lp15@56215
   718
  have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')" 
lp15@56215
   719
    apply (rule holomorphic_on_mult) 
lp15@56215
   720
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   721
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   722
  then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e"
lp15@56215
   723
    by (metis e e' min_less_iff_conj)
lp15@56215
   724
qed
lp15@56215
   725
lp15@56215
   726
lemma analytic_on_inverse:
lp15@56215
   727
  assumes f: "f analytic_on s"
lp15@56215
   728
      and nz: "(\<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0)"
lp15@56215
   729
    shows "(\<lambda>z. inverse (f z)) analytic_on s"
lp15@56215
   730
unfolding analytic_on_def
lp15@56215
   731
proof (intro ballI)
lp15@56215
   732
  fix z
lp15@56215
   733
  assume z: "z \<in> s"
lp15@56215
   734
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   735
    by (metis analytic_on_def)
lp15@56215
   736
  have "continuous_on (ball z e) f"
lp15@56215
   737
    by (metis fh holomorphic_on_imp_continuous_on)
lp15@56215
   738
  then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0" 
lp15@56215
   739
    by (metis Topology_Euclidean_Space.open_ball centre_in_ball continuous_on_open_avoid e z nz)  
lp15@56215
   740
  have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')" 
lp15@56215
   741
    apply (rule holomorphic_on_inverse)
lp15@56215
   742
    apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball)
lp15@56215
   743
    by (metis nz' mem_ball min_less_iff_conj) 
lp15@56215
   744
  then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e"
lp15@56215
   745
    by (metis e e' min_less_iff_conj)
lp15@56215
   746
qed
lp15@56215
   747
lp15@56215
   748
lp15@56215
   749
lemma analytic_on_divide:
lp15@56215
   750
  assumes f: "f analytic_on s"
lp15@56215
   751
      and g: "g analytic_on s"
lp15@56215
   752
      and nz: "(\<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0)"
lp15@56215
   753
    shows "(\<lambda>z. f z / g z) analytic_on s"
lp15@56215
   754
unfolding divide_inverse
lp15@56215
   755
by (metis analytic_on_inverse analytic_on_mult f g nz)
lp15@56215
   756
lp15@56215
   757
lemma analytic_on_power:
lp15@56215
   758
  "f analytic_on s \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on s"
lp15@56215
   759
by (induct n) (auto simp: analytic_on_const analytic_on_mult)
lp15@56215
   760
lp15@56215
   761
lemma analytic_on_setsum:
hoelzl@56369
   762
  "(\<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
   763
  by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_const analytic_on_add)
lp15@56215
   764
lp15@56215
   765
subsection{*analyticity at a point.*}
lp15@56215
   766
lp15@56215
   767
lemma analytic_at_ball:
lp15@56215
   768
  "f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
lp15@56215
   769
by (metis analytic_on_def singleton_iff)
lp15@56215
   770
lp15@56215
   771
lemma analytic_at:
lp15@56215
   772
    "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
lp15@56215
   773
by (metis analytic_on_holomorphic empty_subsetI insert_subset)
lp15@56215
   774
lp15@56215
   775
lemma analytic_on_analytic_at:
lp15@56215
   776
    "f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
lp15@56215
   777
by (metis analytic_at_ball analytic_on_def)
lp15@56215
   778
lp15@56215
   779
lemma analytic_at_two:
lp15@56215
   780
  "f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
lp15@56215
   781
   (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)"
lp15@56215
   782
  (is "?lhs = ?rhs")
lp15@56215
   783
proof 
lp15@56215
   784
  assume ?lhs
lp15@56215
   785
  then obtain s t 
lp15@56215
   786
    where st: "open s" "z \<in> s" "f holomorphic_on s"
lp15@56215
   787
              "open t" "z \<in> t" "g holomorphic_on t"
lp15@56215
   788
    by (auto simp: analytic_at)
lp15@56215
   789
  show ?rhs
lp15@56215
   790
    apply (rule_tac x="s \<inter> t" in exI)
lp15@56215
   791
    using st
lp15@56215
   792
    apply (auto simp: Diff_subset holomorphic_on_subset)
lp15@56215
   793
    done
lp15@56215
   794
next
lp15@56215
   795
  assume ?rhs 
lp15@56215
   796
  then show ?lhs
lp15@56215
   797
    by (force simp add: analytic_at)
lp15@56215
   798
qed
lp15@56215
   799
lp15@56215
   800
subsection{*Combining theorems for derivative with ``analytic at'' hypotheses*}
lp15@56215
   801
lp15@56215
   802
lemma 
lp15@56215
   803
  assumes "f analytic_on {z}" "g analytic_on {z}"
hoelzl@56370
   804
  shows complex_derivative_add_at: "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
hoelzl@56370
   805
    and complex_derivative_diff_at: "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
hoelzl@56370
   806
    and complex_derivative_mult_at: "deriv (\<lambda>w. f w * g w) z =
hoelzl@56370
   807
           f z * deriv g z + deriv f z * g z"
lp15@56215
   808
proof -
lp15@56215
   809
  obtain s where s: "open s" "z \<in> s" "f holomorphic_on s" "g holomorphic_on s"
lp15@56215
   810
    using assms by (metis analytic_at_two)
hoelzl@56370
   811
  show "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
hoelzl@56370
   812
    apply (rule DERIV_imp_deriv [OF DERIV_add])
lp15@56215
   813
    using s
hoelzl@56370
   814
    apply (auto simp: holomorphic_on_open complex_differentiable_def DERIV_deriv_iff_complex_differentiable)
lp15@56215
   815
    done
hoelzl@56370
   816
  show "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
hoelzl@56370
   817
    apply (rule DERIV_imp_deriv [OF DERIV_diff])
lp15@56215
   818
    using s
hoelzl@56370
   819
    apply (auto simp: holomorphic_on_open complex_differentiable_def DERIV_deriv_iff_complex_differentiable)
lp15@56215
   820
    done
hoelzl@56370
   821
  show "deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
hoelzl@56370
   822
    apply (rule DERIV_imp_deriv [OF DERIV_mult'])
lp15@56215
   823
    using s
hoelzl@56370
   824
    apply (auto simp: holomorphic_on_open complex_differentiable_def DERIV_deriv_iff_complex_differentiable)
lp15@56215
   825
    done
lp15@56215
   826
qed
lp15@56215
   827
lp15@56215
   828
lemma complex_derivative_cmult_at:
hoelzl@56370
   829
  "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. c * f w) z = c * deriv f z"
lp15@56215
   830
by (auto simp: complex_derivative_mult_at complex_derivative_const analytic_on_const)
lp15@56215
   831
lp15@56215
   832
lemma complex_derivative_cmult_right_at:
hoelzl@56370
   833
  "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. f w * c) z = deriv f z * c"
lp15@56215
   834
by (auto simp: complex_derivative_mult_at complex_derivative_const analytic_on_const)
lp15@56215
   835
lp15@56215
   836
subsection{*Complex differentiation of sequences and series*}
lp15@56215
   837
lp15@56215
   838
lemma has_complex_derivative_sequence:
lp15@56215
   839
  fixes s :: "complex set"
lp15@56215
   840
  assumes cvs: "convex s"
lp15@56215
   841
      and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
lp15@56215
   842
      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"
lp15@56215
   843
      and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) ---> l) sequentially"
lp15@56215
   844
    shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) ---> g x) sequentially \<and> 
lp15@56215
   845
                       (g has_field_derivative (g' x)) (at x within s)"
lp15@56215
   846
proof -
lp15@56215
   847
  from assms obtain x l where x: "x \<in> s" and tf: "((\<lambda>n. f n x) ---> l) sequentially"
lp15@56215
   848
    by blast
lp15@56215
   849
  { fix e::real assume e: "e > 0"
lp15@56215
   850
    then obtain N where N: "\<forall>n\<ge>N. \<forall>x. x \<in> s \<longrightarrow> cmod (f' n x - g' x) \<le> e"
lp15@56215
   851
      by (metis conv)    
lp15@56215
   852
    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
   853
    proof (rule exI [of _ N], clarify)
lp15@56215
   854
      fix n y h
lp15@56215
   855
      assume "N \<le> n" "y \<in> s"
lp15@56215
   856
      then have "cmod (f' n y - g' y) \<le> e"
lp15@56215
   857
        by (metis N)
lp15@56215
   858
      then have "cmod h * cmod (f' n y - g' y) \<le> cmod h * e"
lp15@56215
   859
        by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
lp15@56215
   860
      then show "cmod (f' n y * h - g' y * h) \<le> e * cmod h"
lp15@56215
   861
        by (simp add: norm_mult [symmetric] field_simps)
lp15@56215
   862
    qed
lp15@56215
   863
  } note ** = this
lp15@56215
   864
  show ?thesis
lp15@56215
   865
  unfolding has_field_derivative_def
lp15@56215
   866
  proof (rule has_derivative_sequence [OF cvs _ _ x])
lp15@56215
   867
    show "\<forall>n. \<forall>x\<in>s. (f n has_derivative (op * (f' n x))) (at x within s)"
lp15@56215
   868
      by (metis has_field_derivative_def df)
lp15@56215
   869
  next show "(\<lambda>n. f n x) ----> l"
lp15@56215
   870
    by (rule tf)
lp15@56215
   871
  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
   872
    by (blast intro: **)
lp15@56215
   873
  qed
lp15@56215
   874
qed
lp15@56215
   875
lp15@56215
   876
lp15@56215
   877
lemma has_complex_derivative_series:
lp15@56215
   878
  fixes s :: "complex set"
lp15@56215
   879
  assumes cvs: "convex s"
lp15@56215
   880
      and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
lp15@56215
   881
      and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s 
lp15@56215
   882
                \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
lp15@56215
   883
      and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) sums l)"
lp15@56215
   884
    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
   885
proof -
lp15@56215
   886
  from assms obtain x l where x: "x \<in> s" and sf: "((\<lambda>n. f n x) sums l)"
lp15@56215
   887
    by blast
lp15@56215
   888
  { fix e::real assume e: "e > 0"
lp15@56215
   889
    then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> s 
lp15@56215
   890
            \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
lp15@56215
   891
      by (metis conv)    
lp15@56215
   892
    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
   893
    proof (rule exI [of _ N], clarify)
lp15@56215
   894
      fix n y h
lp15@56215
   895
      assume "N \<le> n" "y \<in> s"
lp15@56215
   896
      then have "cmod ((\<Sum>i<n. f' i y) - g' y) \<le> e"
lp15@56215
   897
        by (metis N)
lp15@56215
   898
      then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e"
lp15@56215
   899
        by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
lp15@56215
   900
      then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h"
lp15@56215
   901
        by (simp add: norm_mult [symmetric] field_simps setsum_right_distrib)
lp15@56215
   902
    qed
lp15@56215
   903
  } note ** = this
lp15@56215
   904
  show ?thesis
lp15@56215
   905
  unfolding has_field_derivative_def
lp15@56215
   906
  proof (rule has_derivative_series [OF cvs _ _ x])
lp15@56215
   907
    fix n x
lp15@56215
   908
    assume "x \<in> s"
lp15@56215
   909
    then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within s)"
lp15@56215
   910
      by (metis df has_field_derivative_def mult_commute_abs)
lp15@56215
   911
  next show " ((\<lambda>n. f n x) sums l)"
lp15@56215
   912
    by (rule sf)
lp15@56215
   913
  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
   914
    by (blast intro: **)
lp15@56215
   915
  qed
lp15@56215
   916
qed
lp15@56215
   917
lp15@56215
   918
subsection{*Bound theorem*}
lp15@56215
   919
lp15@56215
   920
lemma complex_differentiable_bound:
lp15@56215
   921
  fixes s :: "complex set"
lp15@56215
   922
  assumes cvs: "convex s"
lp15@56215
   923
      and df:  "\<And>z. z \<in> s \<Longrightarrow> (f has_field_derivative f' z) (at z within s)"
lp15@56215
   924
      and dn:  "\<And>z. z \<in> s \<Longrightarrow> norm (f' z) \<le> B"
lp15@56215
   925
      and "x \<in> s"  "y \<in> s"
lp15@56215
   926
    shows "norm(f x - f y) \<le> B * norm(x - y)"
lp15@56215
   927
  apply (rule differentiable_bound [OF cvs])
huffman@56223
   928
  apply (rule ballI, erule df [unfolded has_field_derivative_def])
huffman@56223
   929
  apply (rule ballI, rule onorm_le, simp add: norm_mult mult_right_mono dn)
huffman@56223
   930
  apply fact
huffman@56223
   931
  apply fact
lp15@56215
   932
  done
lp15@56215
   933
lp15@56215
   934
subsection{*Inverse function theorem for complex derivatives.*}
lp15@56215
   935
lp15@56215
   936
lemma has_complex_derivative_inverse_basic:
lp15@56215
   937
  fixes f :: "complex \<Rightarrow> complex"
lp15@56215
   938
  shows "DERIV f (g y) :> f' \<Longrightarrow>
lp15@56215
   939
        f' \<noteq> 0 \<Longrightarrow>
lp15@56215
   940
        continuous (at y) g \<Longrightarrow>
lp15@56215
   941
        open t \<Longrightarrow>
lp15@56215
   942
        y \<in> t \<Longrightarrow>
lp15@56215
   943
        (\<And>z. z \<in> t \<Longrightarrow> f (g z) = z)
lp15@56215
   944
        \<Longrightarrow> DERIV g y :> inverse (f')"
lp15@56215
   945
  unfolding has_field_derivative_def
lp15@56215
   946
  apply (rule has_derivative_inverse_basic)
lp15@56215
   947
  apply (auto simp:  bounded_linear_mult_right)
lp15@56215
   948
  done
lp15@56215
   949
lp15@56215
   950
(*Used only once, in Multivariate/cauchy.ml. *)
lp15@56215
   951
lemma has_complex_derivative_inverse_strong:
lp15@56215
   952
  fixes f :: "complex \<Rightarrow> complex"
lp15@56215
   953
  shows "DERIV f x :> f' \<Longrightarrow>
lp15@56215
   954
         f' \<noteq> 0 \<Longrightarrow>
lp15@56215
   955
         open s \<Longrightarrow>
lp15@56215
   956
         x \<in> s \<Longrightarrow>
lp15@56215
   957
         continuous_on s f \<Longrightarrow>
lp15@56215
   958
         (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
lp15@56215
   959
         \<Longrightarrow> DERIV g (f x) :> inverse (f')"
lp15@56215
   960
  unfolding has_field_derivative_def
lp15@56215
   961
  apply (rule has_derivative_inverse_strong [of s x f g ])
lp15@56215
   962
  using assms 
lp15@56215
   963
  by auto
lp15@56215
   964
lp15@56215
   965
lemma has_complex_derivative_inverse_strong_x:
lp15@56215
   966
  fixes f :: "complex \<Rightarrow> complex"
lp15@56215
   967
  shows  "DERIV f (g y) :> f' \<Longrightarrow>
lp15@56215
   968
          f' \<noteq> 0 \<Longrightarrow>
lp15@56215
   969
          open s \<Longrightarrow>
lp15@56215
   970
          continuous_on s f \<Longrightarrow>
lp15@56215
   971
          g y \<in> s \<Longrightarrow> f(g y) = y \<Longrightarrow>
lp15@56215
   972
          (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
lp15@56215
   973
          \<Longrightarrow> DERIV g y :> inverse (f')"
lp15@56215
   974
  unfolding has_field_derivative_def
lp15@56215
   975
  apply (rule has_derivative_inverse_strong_x [of s g y f])
lp15@56215
   976
  using assms 
lp15@56215
   977
  by auto
lp15@56215
   978
hoelzl@56370
   979
subsection {* Taylor on Complex Numbers *}
lp15@56215
   980
lp15@56215
   981
lemma setsum_Suc_reindex:
lp15@56215
   982
  fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
lp15@56215
   983
    shows  "setsum f {0..n} = f 0 - f (Suc n) + setsum (\<lambda>i. f (Suc i)) {0..n}"
lp15@56215
   984
by (induct n) auto
lp15@56215
   985
lp15@56215
   986
lemma complex_taylor:
lp15@56215
   987
  assumes s: "convex s" 
lp15@56215
   988
      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
   989
      and B: "\<And>x. x \<in> s \<Longrightarrow> cmod (f (Suc n) x) \<le> B"
lp15@56215
   990
      and w: "w \<in> s"
lp15@56215
   991
      and z: "z \<in> s"
lp15@59730
   992
    shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i)))
lp15@56215
   993
          \<le> B * cmod(z - w)^(Suc n) / fact n"
lp15@56215
   994
proof -
lp15@56215
   995
  have wzs: "closed_segment w z \<subseteq> s" using assms
lp15@56215
   996
    by (metis convex_contains_segment)
lp15@56215
   997
  { fix u
lp15@56215
   998
    assume "u \<in> closed_segment w z"
lp15@56215
   999
    then have "u \<in> s"
lp15@56215
  1000
      by (metis wzs subsetD)
lp15@59730
  1001
    have "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / (fact i) +
lp15@59730
  1002
                      f (Suc i) u * (z-u)^i / (fact i)) = 
lp15@59730
  1003
              f (Suc n) u * (z-u) ^ n / (fact n)"
lp15@56215
  1004
    proof (induction n)
lp15@56215
  1005
      case 0 show ?case by simp
lp15@56215
  1006
    next
lp15@56215
  1007
      case (Suc n)
lp15@59730
  1008
      have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / (fact i) +
lp15@59730
  1009
                             f (Suc i) u * (z-u) ^ i / (fact i)) =  
lp15@59730
  1010
           f (Suc n) u * (z-u) ^ n / (fact n) +
lp15@59730
  1011
           f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / (fact (Suc n)) -
lp15@59730
  1012
           f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / (fact (Suc n))"
hoelzl@56479
  1013
        using Suc by simp
lp15@59730
  1014
      also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n))"
lp15@56215
  1015
      proof -
lp15@59730
  1016
        have "(fact(Suc n)) *
lp15@59730
  1017
             (f(Suc n) u *(z-u) ^ n / (fact n) +
lp15@59730
  1018
               f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / (fact(Suc n)) -
lp15@59730
  1019
               f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / (fact(Suc n))) =
lp15@59730
  1020
            ((fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / (fact n) +
lp15@59730
  1021
            ((fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / (fact(Suc n))) -
lp15@59730
  1022
            ((fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / (fact(Suc n))"
lp15@59730
  1023
          by (simp add: algebra_simps del: fact.simps)
lp15@59730
  1024
        also have "... = ((fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / (fact n) +
lp15@59730
  1025
                         (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
lp15@59730
  1026
                         (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
lp15@59730
  1027
          by (simp del: fact.simps)
lp15@59730
  1028
        also have "... = (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) +
lp15@59730
  1029
                         (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
lp15@59730
  1030
                         (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
lp15@59730
  1031
          by (simp only: fact.simps of_nat_mult ac_simps) simp
lp15@56215
  1032
        also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
lp15@56215
  1033
          by (simp add: algebra_simps)
lp15@56215
  1034
        finally show ?thesis
lp15@59730
  1035
        by (simp add: mult_left_cancel [where c = "(fact (Suc n))", THEN iffD1] del: fact.simps)
lp15@56215
  1036
      qed
lp15@56215
  1037
      finally show ?case .
lp15@56215
  1038
    qed
lp15@59730
  1039
    then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / (fact i))) 
lp15@59730
  1040
                has_field_derivative f (Suc n) u * (z-u) ^ n / (fact n))
lp15@56215
  1041
               (at u within s)"
hoelzl@56381
  1042
      apply (intro derivative_eq_intros)
lp15@56215
  1043
      apply (blast intro: assms `u \<in> s`)
lp15@56215
  1044
      apply (rule refl)+
lp15@56215
  1045
      apply (auto simp: field_simps)
lp15@56215
  1046
      done
lp15@56215
  1047
  } note sum_deriv = this
lp15@56215
  1048
  { fix u
lp15@56215
  1049
    assume u: "u \<in> closed_segment w z"
lp15@56215
  1050
    then have us: "u \<in> s"
lp15@56215
  1051
      by (metis wzs subsetD)
lp15@56215
  1052
    have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> cmod (f (Suc n) u) * cmod (u - z) ^ n"
lp15@56215
  1053
      by (metis norm_minus_commute order_refl)
lp15@56215
  1054
    also have "... \<le> cmod (f (Suc n) u) * cmod (z - w) ^ n"
lp15@56215
  1055
      by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
lp15@56215
  1056
    also have "... \<le> B * cmod (z - w) ^ n"
lp15@56215
  1057
      by (metis norm_ge_zero zero_le_power mult_right_mono  B [OF us])
lp15@56215
  1058
    finally have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> B * cmod (z - w) ^ n" .
lp15@56215
  1059
  } note cmod_bound = this
lp15@59730
  1060
  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
  1061
    by simp
lp15@59730
  1062
  also have "\<dots> = f 0 z / (fact 0)"
lp15@56215
  1063
    by (subst setsum_zero_power) simp
lp15@59730
  1064
  finally have "cmod (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i))) 
lp15@59730
  1065
                \<le> cmod ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) -
lp15@59730
  1066
                        (\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))"
lp15@56215
  1067
    by (simp add: norm_minus_commute)
lp15@59730
  1068
  also have "... \<le> B * cmod (z - w) ^ n / (fact n) * cmod (w - z)"
lp15@56215
  1069
    apply (rule complex_differentiable_bound 
lp15@59730
  1070
      [where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / (fact n)"
lp15@56215
  1071
         and s = "closed_segment w z", OF convex_segment])
lp15@56215
  1072
    apply (auto simp: ends_in_segment real_of_nat_def DERIV_subset [OF sum_deriv wzs]
lp15@56215
  1073
                  norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
lp15@56215
  1074
    done
lp15@59730
  1075
  also have "...  \<le> B * cmod (z - w) ^ Suc n / (fact n)"
lp15@56215
  1076
    by (simp add: algebra_simps norm_minus_commute real_of_nat_def)
lp15@56215
  1077
  finally show ?thesis .
lp15@56215
  1078
qed
lp15@56215
  1079
lp15@56238
  1080
text{* Something more like the traditional MVT for real components.*}
hoelzl@56370
  1081
lp15@56238
  1082
lemma complex_mvt_line:
hoelzl@56369
  1083
  assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f'(u)) (at u)"
lp15@56238
  1084
    shows "\<exists>u. u \<in> open_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
lp15@56238
  1085
proof -
lp15@56238
  1086
  have twz: "\<And>t. (1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)"
lp15@56238
  1087
    by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
hoelzl@56381
  1088
  note assms[unfolded has_field_derivative_def, derivative_intros]
lp15@56238
  1089
  show ?thesis
lp15@56238
  1090
    apply (cut_tac mvt_simple
lp15@56238
  1091
                     [of 0 1 "Re o f o (\<lambda>t. (1 - t) *\<^sub>R w +  t *\<^sub>R z)"
lp15@56238
  1092
                      "\<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
  1093
    apply auto
lp15@56238
  1094
    apply (rule_tac x="(1 - x) *\<^sub>R w + x *\<^sub>R z" in exI)
hoelzl@56369
  1095
    apply (auto simp add: open_segment_def twz) []
hoelzl@56381
  1096
    apply (intro derivative_eq_intros has_derivative_at_within)
hoelzl@56369
  1097
    apply simp_all
hoelzl@56369
  1098
    apply (simp add: fun_eq_iff real_vector.scale_right_diff_distrib)
lp15@56238
  1099
    apply (force simp add: twz closed_segment_def)
lp15@56238
  1100
    done
lp15@56238
  1101
qed
lp15@56238
  1102
lp15@56238
  1103
lemma complex_taylor_mvt:
lp15@56238
  1104
  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
  1105
    shows "\<exists>u. u \<in> closed_segment w z \<and>
lp15@56238
  1106
            Re (f 0 z) =
lp15@59730
  1107
            Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / (fact i)) +
lp15@59730
  1108
                (f (Suc n) u * (z-u)^n / (fact n)) * (z - w))"
lp15@56238
  1109
proof -
lp15@56238
  1110
  { fix u
lp15@56238
  1111
    assume u: "u \<in> closed_segment w z"
lp15@56238
  1112
    have "(\<Sum>i = 0..n.
lp15@56238
  1113
               (f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) /
lp15@59730
  1114
               (fact i)) =
lp15@56238
  1115
          f (Suc 0) u -
lp15@56238
  1116
             (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
lp15@59730
  1117
             (fact (Suc n)) +
lp15@56238
  1118
             (\<Sum>i = 0..n.
lp15@56238
  1119
                 (f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) /
lp15@59730
  1120
                 (fact (Suc i)))"
lp15@56238
  1121
       by (subst setsum_Suc_reindex) simp
lp15@56238
  1122
    also have "... = f (Suc 0) u -
lp15@56238
  1123
             (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
lp15@59730
  1124
             (fact (Suc n)) +
lp15@56238
  1125
             (\<Sum>i = 0..n.
lp15@59730
  1126
                 f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i))  - 
lp15@59730
  1127
                 f (Suc i) u * (z-u) ^ i / (fact i))"
haftmann@57514
  1128
      by (simp only: diff_divide_distrib fact_cancel ac_simps)
lp15@56238
  1129
    also have "... = f (Suc 0) u -
lp15@56238
  1130
             (f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) /
lp15@59730
  1131
             (fact (Suc n)) +
lp15@59730
  1132
             f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n)) - f (Suc 0) u"
lp15@56238
  1133
      by (subst setsum_Suc_diff) auto
lp15@59730
  1134
    also have "... = f (Suc n) u * (z-u) ^ n / (fact n)"
lp15@56238
  1135
      by (simp only: algebra_simps diff_divide_distrib fact_cancel)
lp15@56238
  1136
    finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i 
lp15@59730
  1137
                             - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / (fact i)) =
lp15@59730
  1138
                  f (Suc n) u * (z - u) ^ n / (fact n)" .
lp15@59730
  1139
    then have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / (fact i)) has_field_derivative
lp15@59730
  1140
                f (Suc n) u * (z - u) ^ n / (fact n))  (at u)"
hoelzl@56381
  1141
      apply (intro derivative_eq_intros)+
lp15@56238
  1142
      apply (force intro: u assms)
lp15@56238
  1143
      apply (rule refl)+
haftmann@57514
  1144
      apply (auto simp: ac_simps)
lp15@56238
  1145
      done
lp15@56238
  1146
  }
lp15@56238
  1147
  then show ?thesis
lp15@59730
  1148
    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
  1149
               "\<lambda>u. (f (Suc n) u * (z-u)^n / (fact n))"])
lp15@56238
  1150
    apply (auto simp add: intro: open_closed_segment)
lp15@56238
  1151
    done
lp15@56238
  1152
qed
lp15@56238
  1153
lp15@56215
  1154
end