src/HOL/Analysis/Complex_Analysis_Basics.thy
author paulson <lp15@cam.ac.uk>
Sat May 26 22:11:55 2018 +0100 (12 months ago)
changeset 68296 69d680e94961
parent 68255 009f783d1bac
child 68721 53ad5c01be3f
permissions -rw-r--r--
tidying and reorganisation around Cauchy Integral Theorem
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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 Equivalence_Lebesgue_Henstock_Integration "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 "((( * ) c) has_derivative (( * ) c)) F"
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by (rule has_derivative_mult_right [OF has_derivative_ident])
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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_field:
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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 "( * ) 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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lemmas has_vector_derivative_real_complex = has_vector_derivative_real_field
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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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  using of_nat_neq_0 by force
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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 lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = ( * ) 0"
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  by auto
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lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = ( * ) 1"
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  by auto
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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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lemmas DERIV_zero_constant = has_field_derivative_zero_constant
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lemma DERIV_zero_unique:
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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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  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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  "(\<And>x. DERIV f x :> 0) \<Longrightarrow> f x = f a"
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  by (metis DERIV_zero_unique UNIV_I 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 continuous_on_Re
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            continuous_on_Im continuous_on_id continuous_on_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 sum_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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definition holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
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           (infixl "(holomorphic'_on)" 50)
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  where "f holomorphic_on s \<equiv> \<forall>x\<in>s. f field_differentiable (at x within s)"
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named_theorems holomorphic_intros "structural introduction rules for holomorphic_on"
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lemma holomorphic_onI [intro?]: "(\<And>x. x \<in> s \<Longrightarrow> f field_differentiable (at x within s)) \<Longrightarrow> f holomorphic_on s"
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  by (simp add: holomorphic_on_def)
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lemma holomorphic_onD [dest?]: "\<lbrakk>f holomorphic_on s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x within s)"
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  by (simp add: holomorphic_on_def)
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lemma holomorphic_on_imp_differentiable_on:
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    "f holomorphic_on s \<Longrightarrow> f differentiable_on s"
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  unfolding holomorphic_on_def differentiable_on_def
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  by (simp add: field_differentiable_imp_differentiable)
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lemma holomorphic_on_imp_differentiable_at:
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   "\<lbrakk>f holomorphic_on s; open s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x)"
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using at_within_open holomorphic_on_def by fastforce
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lemma holomorphic_on_empty [holomorphic_intros]: "f holomorphic_on {}"
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  by (simp add: holomorphic_on_def)
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lemma holomorphic_on_open:
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    "open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> f')"
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  by (auto simp: holomorphic_on_def field_differentiable_def has_field_derivative_def at_within_open [of _ s])
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lemma holomorphic_on_imp_continuous_on:
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    "f holomorphic_on s \<Longrightarrow> continuous_on s f"
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  by (metis field_differentiable_imp_continuous_at continuous_on_eq_continuous_within holomorphic_on_def)
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lemma holomorphic_on_subset [elim]:
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    "f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t"
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  unfolding holomorphic_on_def
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  by (metis field_differentiable_within_subset subsetD)
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lemma holomorphic_transform: "\<lbrakk>f holomorphic_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g holomorphic_on s"
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  by (metis field_differentiable_transform_within linordered_field_no_ub holomorphic_on_def)
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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"
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  by (metis holomorphic_transform)
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lemma holomorphic_on_linear [simp, holomorphic_intros]: "(( * ) c) holomorphic_on s"
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  unfolding holomorphic_on_def by (metis field_differentiable_linear)
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lemma holomorphic_on_const [simp, holomorphic_intros]: "(\<lambda>z. c) holomorphic_on s"
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  unfolding holomorphic_on_def by (metis field_differentiable_const)
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lemma holomorphic_on_ident [simp, holomorphic_intros]: "(\<lambda>x. x) holomorphic_on s"
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  unfolding holomorphic_on_def by (metis field_differentiable_ident)
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lemma holomorphic_on_id [simp, holomorphic_intros]: "id holomorphic_on s"
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  unfolding id_def by (rule holomorphic_on_ident)
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lemma holomorphic_on_compose:
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  "f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s) \<Longrightarrow> (g o f) holomorphic_on s"
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  using field_differentiable_compose_within[of f _ s g]
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  by (auto simp: holomorphic_on_def)
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lemma holomorphic_on_compose_gen:
hoelzl@56370
   286
  "f holomorphic_on s \<Longrightarrow> g holomorphic_on t \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> (g o f) holomorphic_on s"
hoelzl@56370
   287
  by (metis holomorphic_on_compose holomorphic_on_subset)
hoelzl@56370
   288
lp15@61520
   289
lemma holomorphic_on_minus [holomorphic_intros]: "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on s"
lp15@62534
   290
  by (metis field_differentiable_minus holomorphic_on_def)
hoelzl@56370
   291
lp15@61520
   292
lemma holomorphic_on_add [holomorphic_intros]:
hoelzl@56370
   293
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on s"
lp15@62534
   294
  unfolding holomorphic_on_def by (metis field_differentiable_add)
hoelzl@56370
   295
lp15@61520
   296
lemma holomorphic_on_diff [holomorphic_intros]:
hoelzl@56370
   297
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) holomorphic_on s"
lp15@62534
   298
  unfolding holomorphic_on_def by (metis field_differentiable_diff)
hoelzl@56370
   299
lp15@61520
   300
lemma holomorphic_on_mult [holomorphic_intros]:
hoelzl@56370
   301
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) holomorphic_on s"
lp15@62534
   302
  unfolding holomorphic_on_def by (metis field_differentiable_mult)
hoelzl@56370
   303
lp15@61520
   304
lemma holomorphic_on_inverse [holomorphic_intros]:
hoelzl@56370
   305
  "\<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"
lp15@62534
   306
  unfolding holomorphic_on_def by (metis field_differentiable_inverse)
hoelzl@56370
   307
lp15@61520
   308
lemma holomorphic_on_divide [holomorphic_intros]:
hoelzl@56370
   309
  "\<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"
lp15@62534
   310
  unfolding holomorphic_on_def by (metis field_differentiable_divide)
hoelzl@56370
   311
lp15@61520
   312
lemma holomorphic_on_power [holomorphic_intros]:
hoelzl@56370
   313
  "f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on s"
lp15@62534
   314
  unfolding holomorphic_on_def by (metis field_differentiable_power)
hoelzl@56370
   315
nipkow@64267
   316
lemma holomorphic_on_sum [holomorphic_intros]:
nipkow@64267
   317
  "(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) holomorphic_on s"
nipkow@64267
   318
  unfolding holomorphic_on_def by (metis field_differentiable_sum)
hoelzl@56370
   319
eberlm@67135
   320
lemma holomorphic_on_prod [holomorphic_intros]:
eberlm@67135
   321
  "(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. prod (\<lambda>i. f i x) I) holomorphic_on s"
eberlm@67135
   322
  by (induction I rule: infinite_finite_induct) (auto intro: holomorphic_intros)
eberlm@67135
   323
eberlm@66486
   324
lemma holomorphic_pochhammer [holomorphic_intros]:
eberlm@66486
   325
  "f holomorphic_on A \<Longrightarrow> (\<lambda>s. pochhammer (f s) n) holomorphic_on A"
eberlm@66486
   326
  by (induction n) (auto intro!: holomorphic_intros simp: pochhammer_Suc)
eberlm@66486
   327
eberlm@66486
   328
lemma holomorphic_on_scaleR [holomorphic_intros]:
eberlm@66486
   329
  "f holomorphic_on A \<Longrightarrow> (\<lambda>x. c *\<^sub>R f x) holomorphic_on A"
eberlm@66486
   330
  by (auto simp: scaleR_conv_of_real intro!: holomorphic_intros)
eberlm@66486
   331
eberlm@67167
   332
lemma holomorphic_on_Un [holomorphic_intros]:
eberlm@67167
   333
  assumes "f holomorphic_on A" "f holomorphic_on B" "open A" "open B"
eberlm@67167
   334
  shows   "f holomorphic_on (A \<union> B)"
lp15@68239
   335
  using assms by (auto simp: holomorphic_on_def  at_within_open[of _ A]
eberlm@67167
   336
                             at_within_open[of _ B]  at_within_open[of _ "A \<union> B"] open_Un)
eberlm@67167
   337
eberlm@67167
   338
lemma holomorphic_on_If_Un [holomorphic_intros]:
eberlm@67167
   339
  assumes "f holomorphic_on A" "g holomorphic_on B" "open A" "open B"
eberlm@67167
   340
  assumes "\<And>z. z \<in> A \<Longrightarrow> z \<in> B \<Longrightarrow> f z = g z"
eberlm@67167
   341
  shows   "(\<lambda>z. if z \<in> A then f z else g z) holomorphic_on (A \<union> B)" (is "?h holomorphic_on _")
eberlm@67167
   342
proof (intro holomorphic_on_Un)
eberlm@67167
   343
  note \<open>f holomorphic_on A\<close>
eberlm@67167
   344
  also have "f holomorphic_on A \<longleftrightarrow> ?h holomorphic_on A"
eberlm@67167
   345
    by (intro holomorphic_cong) auto
eberlm@67167
   346
  finally show \<dots> .
eberlm@67167
   347
next
eberlm@67167
   348
  note \<open>g holomorphic_on B\<close>
eberlm@67167
   349
  also have "g holomorphic_on B \<longleftrightarrow> ?h holomorphic_on B"
eberlm@67167
   350
    using assms by (intro holomorphic_cong) auto
eberlm@67167
   351
  finally show \<dots> .
eberlm@67167
   352
qed (insert assms, auto)
eberlm@67167
   353
lp15@67371
   354
lemma leibniz_rule_holomorphic:
lp15@67371
   355
  fixes f::"complex \<Rightarrow> 'b::euclidean_space \<Rightarrow> complex"
lp15@67371
   356
  assumes "\<And>x t. x \<in> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow> ((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"
lp15@67371
   357
  assumes "\<And>x. x \<in> U \<Longrightarrow> (f x) integrable_on cbox a b"
lp15@67371
   358
  assumes "continuous_on (U \<times> (cbox a b)) (\<lambda>(x, t). fx x t)"
lp15@67371
   359
  assumes "convex U"
lp15@67371
   360
  shows "(\<lambda>x. integral (cbox a b) (f x)) holomorphic_on U"
lp15@67371
   361
  using leibniz_rule_field_differentiable[OF assms(1-3) _ assms(4)]
lp15@67371
   362
  by (auto simp: holomorphic_on_def)
lp15@67371
   363
lp15@62534
   364
lemma DERIV_deriv_iff_field_differentiable:
lp15@62534
   365
  "DERIV f x :> deriv f x \<longleftrightarrow> f field_differentiable at x"
lp15@62534
   366
  unfolding field_differentiable_def by (metis DERIV_imp_deriv)
hoelzl@56370
   367
lp15@62533
   368
lemma holomorphic_derivI:
lp15@62533
   369
     "\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk>
lp15@62533
   370
      \<Longrightarrow> (f has_field_derivative deriv f x) (at x within T)"
lp15@62534
   371
by (metis DERIV_deriv_iff_field_differentiable at_within_open  holomorphic_on_def has_field_derivative_at_within)
lp15@62533
   372
hoelzl@56370
   373
lemma complex_derivative_chain:
lp15@62534
   374
  "f field_differentiable at x \<Longrightarrow> g field_differentiable at (f x)
hoelzl@56370
   375
    \<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
lp15@62534
   376
  by (metis DERIV_deriv_iff_field_differentiable DERIV_chain DERIV_imp_deriv)
hoelzl@56370
   377
lp15@62397
   378
lemma deriv_linear [simp]: "deriv (\<lambda>w. c * w) = (\<lambda>z. c)"
hoelzl@56370
   379
  by (metis DERIV_imp_deriv DERIV_cmult_Id)
hoelzl@56370
   380
lp15@62397
   381
lemma deriv_ident [simp]: "deriv (\<lambda>w. w) = (\<lambda>z. 1)"
hoelzl@56370
   382
  by (metis DERIV_imp_deriv DERIV_ident)
hoelzl@56370
   383
lp15@62397
   384
lemma deriv_id [simp]: "deriv id = (\<lambda>z. 1)"
lp15@62397
   385
  by (simp add: id_def)
lp15@62397
   386
lp15@62397
   387
lemma deriv_const [simp]: "deriv (\<lambda>w. c) = (\<lambda>z. 0)"
hoelzl@56370
   388
  by (metis DERIV_imp_deriv DERIV_const)
hoelzl@56370
   389
lp15@62534
   390
lemma deriv_add [simp]:
lp15@62534
   391
  "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
hoelzl@56370
   392
   \<Longrightarrow> deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
lp15@62534
   393
  unfolding DERIV_deriv_iff_field_differentiable[symmetric]
hoelzl@56381
   394
  by (auto intro!: DERIV_imp_deriv derivative_intros)
hoelzl@56370
   395
lp15@62534
   396
lemma deriv_diff [simp]:
lp15@62534
   397
  "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
hoelzl@56370
   398
   \<Longrightarrow> deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
lp15@62534
   399
  unfolding DERIV_deriv_iff_field_differentiable[symmetric]
hoelzl@56381
   400
  by (auto intro!: DERIV_imp_deriv derivative_intros)
hoelzl@56370
   401
lp15@62534
   402
lemma deriv_mult [simp]:
lp15@62534
   403
  "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
hoelzl@56370
   404
   \<Longrightarrow> deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
lp15@62534
   405
  unfolding DERIV_deriv_iff_field_differentiable[symmetric]
hoelzl@56381
   406
  by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
hoelzl@56370
   407
lp15@68255
   408
lemma deriv_cmult:
lp15@62534
   409
  "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z"
lp15@68255
   410
  by simp
hoelzl@56370
   411
lp15@68255
   412
lemma deriv_cmult_right:
lp15@62534
   413
  "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c"
lp15@68255
   414
  by simp
lp15@68255
   415
lp15@68255
   416
lemma deriv_inverse [simp]:
lp15@68255
   417
  "\<lbrakk>f field_differentiable at z; f z \<noteq> 0\<rbrakk>
lp15@68255
   418
   \<Longrightarrow> deriv (\<lambda>w. inverse (f w)) z = - deriv f z / f z ^ 2"
lp15@62534
   419
  unfolding DERIV_deriv_iff_field_differentiable[symmetric]
lp15@68255
   420
  by (safe intro!: DERIV_imp_deriv derivative_eq_intros) (auto simp: divide_simps power2_eq_square)
hoelzl@56370
   421
lp15@68255
   422
lemma deriv_divide [simp]:
lp15@68255
   423
  "\<lbrakk>f field_differentiable at z; g field_differentiable at z; g z \<noteq> 0\<rbrakk>
lp15@68255
   424
   \<Longrightarrow> deriv (\<lambda>w. f w / g w) z = (deriv f z * g z - f z * deriv g z) / g z ^ 2"
lp15@68255
   425
  by (simp add: field_class.field_divide_inverse field_differentiable_inverse)
lp15@68255
   426
     (simp add: divide_simps power2_eq_square)
lp15@68255
   427
lp15@68255
   428
lemma deriv_cdivide_right:
lp15@62534
   429
  "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w / c) z = deriv f z / c"
lp15@68255
   430
  by (simp add: field_class.field_divide_inverse)
lp15@62217
   431
hoelzl@56370
   432
lemma complex_derivative_transform_within_open:
lp15@61609
   433
  "\<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
   434
   \<Longrightarrow> deriv f z = deriv g z"
hoelzl@56370
   435
  unfolding holomorphic_on_def
hoelzl@56370
   436
  by (rule DERIV_imp_deriv)
lp15@62534
   437
     (metis DERIV_deriv_iff_field_differentiable DERIV_transform_within_open at_within_open)
hoelzl@56370
   438
lp15@62534
   439
lemma deriv_compose_linear:
lp15@62534
   440
  "f field_differentiable at (c * z) \<Longrightarrow> deriv (\<lambda>w. f (c * w)) z = c * deriv f (c * z)"
hoelzl@56370
   441
apply (rule DERIV_imp_deriv)
lp15@68255
   442
  unfolding DERIV_deriv_iff_field_differentiable [symmetric]
lp15@68255
   443
  by (metis (full_types) DERIV_chain2 DERIV_cmult_Id mult.commute)
lp15@68255
   444
hoelzl@56370
   445
lp15@62533
   446
lemma nonzero_deriv_nonconstant:
lp15@62533
   447
  assumes df: "DERIV f \<xi> :> df" and S: "open S" "\<xi> \<in> S" and "df \<noteq> 0"
lp15@62533
   448
    shows "\<not> f constant_on S"
lp15@62533
   449
unfolding constant_on_def
lp15@62533
   450
by (metis \<open>df \<noteq> 0\<close> DERIV_transform_within_open [OF df S] DERIV_const DERIV_unique)
lp15@62533
   451
lp15@62533
   452
lemma holomorphic_nonconstant:
lp15@62533
   453
  assumes holf: "f holomorphic_on S" and "open S" "\<xi> \<in> S" "deriv f \<xi> \<noteq> 0"
lp15@62533
   454
    shows "\<not> f constant_on S"
lp15@68255
   455
  by (rule nonzero_deriv_nonconstant [of f "deriv f \<xi>" \<xi> S])
lp15@68255
   456
    (use assms in \<open>auto simp: holomorphic_derivI\<close>)
lp15@62533
   457
lp15@64394
   458
subsection\<open>Caratheodory characterization\<close>
lp15@64394
   459
lp15@64394
   460
lemma field_differentiable_caratheodory_at:
lp15@64394
   461
  "f field_differentiable (at z) \<longleftrightarrow>
lp15@64394
   462
         (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)"
lp15@64394
   463
  using CARAT_DERIV [of f]
lp15@64394
   464
  by (simp add: field_differentiable_def has_field_derivative_def)
lp15@64394
   465
lp15@64394
   466
lemma field_differentiable_caratheodory_within:
lp15@64394
   467
  "f field_differentiable (at z within s) \<longleftrightarrow>
lp15@64394
   468
         (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)"
lp15@64394
   469
  using DERIV_caratheodory_within [of f]
lp15@64394
   470
  by (simp add: field_differentiable_def has_field_derivative_def)
lp15@64394
   471
wenzelm@60420
   472
subsection\<open>Analyticity on a set\<close>
lp15@56215
   473
lp15@61609
   474
definition analytic_on (infixl "(analytic'_on)" 50)
lp15@68255
   475
  where "f analytic_on S \<equiv> \<forall>x \<in> S. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)"
lp15@56215
   476
lp15@65587
   477
named_theorems analytic_intros "introduction rules for proving analyticity"
lp15@65587
   478
lp15@68255
   479
lemma analytic_imp_holomorphic: "f analytic_on S \<Longrightarrow> f holomorphic_on S"
hoelzl@56370
   480
  by (simp add: at_within_open [OF _ open_ball] analytic_on_def holomorphic_on_def)
lp15@62534
   481
     (metis centre_in_ball field_differentiable_at_within)
lp15@56215
   482
lp15@68255
   483
lemma analytic_on_open: "open S \<Longrightarrow> f analytic_on S \<longleftrightarrow> f holomorphic_on S"
lp15@56215
   484
apply (auto simp: analytic_imp_holomorphic)
lp15@56215
   485
apply (auto simp: analytic_on_def holomorphic_on_def)
lp15@56215
   486
by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball)
lp15@56215
   487
lp15@56215
   488
lemma analytic_on_imp_differentiable_at:
lp15@68255
   489
  "f analytic_on S \<Longrightarrow> x \<in> S \<Longrightarrow> f field_differentiable (at x)"
hoelzl@56370
   490
 apply (auto simp: analytic_on_def holomorphic_on_def)
lp15@66827
   491
by (metis open_ball centre_in_ball field_differentiable_within_open)
lp15@56215
   492
lp15@68255
   493
lemma analytic_on_subset: "f analytic_on S \<Longrightarrow> T \<subseteq> S \<Longrightarrow> f analytic_on T"
lp15@56215
   494
  by (auto simp: analytic_on_def)
lp15@56215
   495
lp15@68255
   496
lemma analytic_on_Un: "f analytic_on (S \<union> T) \<longleftrightarrow> f analytic_on S \<and> f analytic_on T"
lp15@56215
   497
  by (auto simp: analytic_on_def)
lp15@56215
   498
lp15@68255
   499
lemma analytic_on_Union: "f analytic_on (\<Union>\<T>) \<longleftrightarrow> (\<forall>T \<in> \<T>. f analytic_on T)"
hoelzl@56370
   500
  by (auto simp: analytic_on_def)
hoelzl@56370
   501
lp15@68255
   502
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
   503
  by (auto simp: analytic_on_def)
lp15@61609
   504
lp15@56215
   505
lemma analytic_on_holomorphic:
lp15@68255
   506
  "f analytic_on S \<longleftrightarrow> (\<exists>T. open T \<and> S \<subseteq> T \<and> f holomorphic_on T)"
lp15@56215
   507
  (is "?lhs = ?rhs")
lp15@56215
   508
proof -
lp15@68255
   509
  have "?lhs \<longleftrightarrow> (\<exists>T. open T \<and> S \<subseteq> T \<and> f analytic_on T)"
lp15@56215
   510
  proof safe
lp15@68255
   511
    assume "f analytic_on S"
lp15@68255
   512
    then show "\<exists>T. open T \<and> S \<subseteq> T \<and> f analytic_on T"
lp15@56215
   513
      apply (simp add: analytic_on_def)
lp15@68255
   514
      apply (rule exI [where x="\<Union>{U. open U \<and> f analytic_on U}"], auto)
lp15@66827
   515
      apply (metis open_ball analytic_on_open centre_in_ball)
lp15@56215
   516
      by (metis analytic_on_def)
lp15@56215
   517
  next
lp15@68255
   518
    fix T
lp15@68255
   519
    assume "open T" "S \<subseteq> T" "f analytic_on T"
lp15@68255
   520
    then show "f analytic_on S"
lp15@56215
   521
        by (metis analytic_on_subset)
lp15@56215
   522
  qed
lp15@56215
   523
  also have "... \<longleftrightarrow> ?rhs"
lp15@56215
   524
    by (auto simp: analytic_on_open)
lp15@56215
   525
  finally show ?thesis .
lp15@56215
   526
qed
lp15@56215
   527
lp15@68255
   528
lemma analytic_on_linear [analytic_intros,simp]: "(( * ) c) analytic_on S"
lp15@65587
   529
  by (auto simp add: analytic_on_holomorphic)
lp15@56215
   530
lp15@68255
   531
lemma analytic_on_const [analytic_intros,simp]: "(\<lambda>z. c) analytic_on S"
hoelzl@56370
   532
  by (metis analytic_on_def holomorphic_on_const zero_less_one)
hoelzl@56370
   533
lp15@68255
   534
lemma analytic_on_ident [analytic_intros,simp]: "(\<lambda>x. x) analytic_on S"
lp15@65587
   535
  by (simp add: analytic_on_def gt_ex)
lp15@56215
   536
lp15@68255
   537
lemma analytic_on_id [analytic_intros]: "id analytic_on S"
hoelzl@56370
   538
  unfolding id_def by (rule analytic_on_ident)
lp15@56215
   539
lp15@56215
   540
lemma analytic_on_compose:
lp15@68255
   541
  assumes f: "f analytic_on S"
lp15@68255
   542
      and g: "g analytic_on (f ` S)"
lp15@68255
   543
    shows "(g o f) analytic_on S"
lp15@56215
   544
unfolding analytic_on_def
lp15@56215
   545
proof (intro ballI)
lp15@56215
   546
  fix x
lp15@68255
   547
  assume x: "x \<in> S"
lp15@56215
   548
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f
lp15@56215
   549
    by (metis analytic_on_def)
lp15@56215
   550
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g
lp15@61609
   551
    by (metis analytic_on_def g image_eqI x)
lp15@56215
   552
  have "isCont f x"
lp15@62534
   553
    by (metis analytic_on_imp_differentiable_at field_differentiable_imp_continuous_at f x)
lp15@56215
   554
  with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'"
lp15@56215
   555
     by (auto simp: continuous_at_ball)
lp15@61609
   556
  have "g \<circ> f holomorphic_on ball x (min d e)"
lp15@56215
   557
    apply (rule holomorphic_on_compose)
lp15@56215
   558
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   559
    by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball)
lp15@56215
   560
  then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e"
lp15@61609
   561
    by (metis d e min_less_iff_conj)
lp15@56215
   562
qed
lp15@56215
   563
lp15@56215
   564
lemma analytic_on_compose_gen:
lp15@68255
   565
  "f analytic_on S \<Longrightarrow> g analytic_on T \<Longrightarrow> (\<And>z. z \<in> S \<Longrightarrow> f z \<in> T)
lp15@68255
   566
             \<Longrightarrow> g o f analytic_on S"
lp15@56215
   567
by (metis analytic_on_compose analytic_on_subset image_subset_iff)
lp15@56215
   568
lp15@65587
   569
lemma analytic_on_neg [analytic_intros]:
lp15@68255
   570
  "f analytic_on S \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on S"
lp15@56215
   571
by (metis analytic_on_holomorphic holomorphic_on_minus)
lp15@56215
   572
lp15@65587
   573
lemma analytic_on_add [analytic_intros]:
lp15@68255
   574
  assumes f: "f analytic_on S"
lp15@68255
   575
      and g: "g analytic_on S"
lp15@68255
   576
    shows "(\<lambda>z. f z + g z) analytic_on S"
lp15@56215
   577
unfolding analytic_on_def
lp15@56215
   578
proof (intro ballI)
lp15@56215
   579
  fix z
lp15@68255
   580
  assume z: "z \<in> S"
lp15@56215
   581
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   582
    by (metis analytic_on_def)
lp15@56215
   583
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
lp15@61609
   584
    by (metis analytic_on_def g z)
lp15@61609
   585
  have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')"
lp15@61609
   586
    apply (rule holomorphic_on_add)
lp15@56215
   587
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   588
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   589
  then show "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e"
lp15@56215
   590
    by (metis e e' min_less_iff_conj)
lp15@56215
   591
qed
lp15@56215
   592
lp15@65587
   593
lemma analytic_on_diff [analytic_intros]:
lp15@68255
   594
  assumes f: "f analytic_on S"
lp15@68255
   595
      and g: "g analytic_on S"
lp15@68255
   596
    shows "(\<lambda>z. f z - g z) analytic_on S"
lp15@56215
   597
unfolding analytic_on_def
lp15@56215
   598
proof (intro ballI)
lp15@56215
   599
  fix z
lp15@68255
   600
  assume z: "z \<in> S"
lp15@56215
   601
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   602
    by (metis analytic_on_def)
lp15@56215
   603
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
lp15@61609
   604
    by (metis analytic_on_def g z)
lp15@61609
   605
  have "(\<lambda>z. f z - g z) holomorphic_on ball z (min e e')"
lp15@61609
   606
    apply (rule holomorphic_on_diff)
lp15@56215
   607
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   608
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   609
  then show "\<exists>e>0. (\<lambda>z. f z - g z) holomorphic_on ball z e"
lp15@56215
   610
    by (metis e e' min_less_iff_conj)
lp15@56215
   611
qed
lp15@56215
   612
lp15@65587
   613
lemma analytic_on_mult [analytic_intros]:
lp15@68255
   614
  assumes f: "f analytic_on S"
lp15@68255
   615
      and g: "g analytic_on S"
lp15@68255
   616
    shows "(\<lambda>z. f z * g z) analytic_on S"
lp15@56215
   617
unfolding analytic_on_def
lp15@56215
   618
proof (intro ballI)
lp15@56215
   619
  fix z
lp15@68255
   620
  assume z: "z \<in> S"
lp15@56215
   621
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   622
    by (metis analytic_on_def)
lp15@56215
   623
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
lp15@61609
   624
    by (metis analytic_on_def g z)
lp15@61609
   625
  have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')"
lp15@61609
   626
    apply (rule holomorphic_on_mult)
lp15@56215
   627
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   628
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   629
  then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e"
lp15@56215
   630
    by (metis e e' min_less_iff_conj)
lp15@56215
   631
qed
lp15@56215
   632
lp15@65587
   633
lemma analytic_on_inverse [analytic_intros]:
lp15@68255
   634
  assumes f: "f analytic_on S"
lp15@68255
   635
      and nz: "(\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0)"
lp15@68255
   636
    shows "(\<lambda>z. inverse (f z)) analytic_on S"
lp15@56215
   637
unfolding analytic_on_def
lp15@56215
   638
proof (intro ballI)
lp15@56215
   639
  fix z
lp15@68255
   640
  assume z: "z \<in> S"
lp15@56215
   641
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   642
    by (metis analytic_on_def)
lp15@56215
   643
  have "continuous_on (ball z e) f"
lp15@56215
   644
    by (metis fh holomorphic_on_imp_continuous_on)
lp15@61609
   645
  then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0"
lp15@66827
   646
    by (metis open_ball centre_in_ball continuous_on_open_avoid e z nz)
lp15@61609
   647
  have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')"
lp15@56215
   648
    apply (rule holomorphic_on_inverse)
lp15@56215
   649
    apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball)
lp15@61609
   650
    by (metis nz' mem_ball min_less_iff_conj)
lp15@56215
   651
  then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e"
lp15@56215
   652
    by (metis e e' min_less_iff_conj)
lp15@56215
   653
qed
lp15@56215
   654
lp15@65587
   655
lemma analytic_on_divide [analytic_intros]:
lp15@68255
   656
  assumes f: "f analytic_on S"
lp15@68255
   657
      and g: "g analytic_on S"
lp15@68255
   658
      and nz: "(\<And>z. z \<in> S \<Longrightarrow> g z \<noteq> 0)"
lp15@68255
   659
    shows "(\<lambda>z. f z / g z) analytic_on S"
lp15@56215
   660
unfolding divide_inverse
lp15@56215
   661
by (metis analytic_on_inverse analytic_on_mult f g nz)
lp15@56215
   662
lp15@65587
   663
lemma analytic_on_power [analytic_intros]:
lp15@68255
   664
  "f analytic_on S \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on S"
lp15@65587
   665
by (induct n) (auto simp: analytic_on_mult)
lp15@56215
   666
lp15@65587
   667
lemma analytic_on_sum [analytic_intros]:
lp15@68255
   668
  "(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on S) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) analytic_on S"
hoelzl@56369
   669
  by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_const analytic_on_add)
lp15@56215
   670
lp15@62408
   671
lemma deriv_left_inverse:
lp15@62408
   672
  assumes "f holomorphic_on S" and "g holomorphic_on T"
lp15@62408
   673
      and "open S" and "open T"
lp15@62408
   674
      and "f ` S \<subseteq> T"
lp15@62408
   675
      and [simp]: "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z"
lp15@62408
   676
      and "w \<in> S"
lp15@62408
   677
    shows "deriv f w * deriv g (f w) = 1"
lp15@62408
   678
proof -
lp15@62408
   679
  have "deriv f w * deriv g (f w) = deriv g (f w) * deriv f w"
lp15@62408
   680
    by (simp add: algebra_simps)
lp15@62408
   681
  also have "... = deriv (g o f) w"
lp15@62408
   682
    using assms
lp15@62408
   683
    by (metis analytic_on_imp_differentiable_at analytic_on_open complex_derivative_chain image_subset_iff)
lp15@62408
   684
  also have "... = deriv id w"
lp15@68255
   685
  proof (rule complex_derivative_transform_within_open [where s=S])
lp15@68255
   686
    show "g \<circ> f holomorphic_on S"
lp15@68255
   687
      by (rule assms holomorphic_on_compose_gen holomorphic_intros)+
lp15@68255
   688
  qed (use assms in auto)
lp15@62408
   689
  also have "... = 1"
lp15@62408
   690
    by simp
lp15@62408
   691
  finally show ?thesis .
lp15@62408
   692
qed
lp15@62408
   693
lp15@62408
   694
subsection\<open>analyticity at a point\<close>
lp15@56215
   695
lp15@56215
   696
lemma analytic_at_ball:
lp15@56215
   697
  "f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
lp15@56215
   698
by (metis analytic_on_def singleton_iff)
lp15@56215
   699
lp15@56215
   700
lemma analytic_at:
lp15@56215
   701
    "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
lp15@56215
   702
by (metis analytic_on_holomorphic empty_subsetI insert_subset)
lp15@56215
   703
lp15@56215
   704
lemma analytic_on_analytic_at:
lp15@56215
   705
    "f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
lp15@56215
   706
by (metis analytic_at_ball analytic_on_def)
lp15@56215
   707
lp15@56215
   708
lemma analytic_at_two:
lp15@56215
   709
  "f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
lp15@56215
   710
   (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)"
lp15@56215
   711
  (is "?lhs = ?rhs")
lp15@61609
   712
proof
lp15@56215
   713
  assume ?lhs
lp15@61609
   714
  then obtain s t
lp15@56215
   715
    where st: "open s" "z \<in> s" "f holomorphic_on s"
lp15@56215
   716
              "open t" "z \<in> t" "g holomorphic_on t"
lp15@56215
   717
    by (auto simp: analytic_at)
lp15@56215
   718
  show ?rhs
lp15@56215
   719
    apply (rule_tac x="s \<inter> t" in exI)
lp15@56215
   720
    using st
lp15@56215
   721
    apply (auto simp: Diff_subset holomorphic_on_subset)
lp15@56215
   722
    done
lp15@56215
   723
next
lp15@61609
   724
  assume ?rhs
lp15@56215
   725
  then show ?lhs
lp15@56215
   726
    by (force simp add: analytic_at)
lp15@56215
   727
qed
lp15@56215
   728
wenzelm@60420
   729
subsection\<open>Combining theorems for derivative with ``analytic at'' hypotheses\<close>
lp15@56215
   730
lp15@61609
   731
lemma
lp15@56215
   732
  assumes "f analytic_on {z}" "g analytic_on {z}"
hoelzl@56370
   733
  shows complex_derivative_add_at: "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
hoelzl@56370
   734
    and complex_derivative_diff_at: "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
hoelzl@56370
   735
    and complex_derivative_mult_at: "deriv (\<lambda>w. f w * g w) z =
hoelzl@56370
   736
           f z * deriv g z + deriv f z * g z"
lp15@56215
   737
proof -
lp15@56215
   738
  obtain s where s: "open s" "z \<in> s" "f holomorphic_on s" "g holomorphic_on s"
lp15@56215
   739
    using assms by (metis analytic_at_two)
hoelzl@56370
   740
  show "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
hoelzl@56370
   741
    apply (rule DERIV_imp_deriv [OF DERIV_add])
lp15@56215
   742
    using s
lp15@62534
   743
    apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
lp15@56215
   744
    done
hoelzl@56370
   745
  show "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
hoelzl@56370
   746
    apply (rule DERIV_imp_deriv [OF DERIV_diff])
lp15@56215
   747
    using s
lp15@62534
   748
    apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
lp15@56215
   749
    done
hoelzl@56370
   750
  show "deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
hoelzl@56370
   751
    apply (rule DERIV_imp_deriv [OF DERIV_mult'])
lp15@56215
   752
    using s
lp15@62534
   753
    apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
lp15@56215
   754
    done
lp15@56215
   755
qed
lp15@56215
   756
lp15@62534
   757
lemma deriv_cmult_at:
hoelzl@56370
   758
  "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. c * f w) z = c * deriv f z"
lp15@61848
   759
by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
lp15@56215
   760
lp15@62534
   761
lemma deriv_cmult_right_at:
hoelzl@56370
   762
  "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. f w * c) z = deriv f z * c"
lp15@61848
   763
by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
lp15@56215
   764
wenzelm@60420
   765
subsection\<open>Complex differentiation of sequences and series\<close>
lp15@56215
   766
eberlm@61531
   767
(* TODO: Could probably be simplified using Uniform_Limit *)
lp15@56215
   768
lemma has_complex_derivative_sequence:
lp15@68255
   769
  fixes S :: "complex set"
lp15@68255
   770
  assumes cvs: "convex S"
lp15@68255
   771
      and df:  "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)"
lp15@68255
   772
      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@68255
   773
      and "\<exists>x l. x \<in> S \<and> ((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
lp15@68255
   774
    shows "\<exists>g. \<forall>x \<in> S. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and>
lp15@68255
   775
                       (g has_field_derivative (g' x)) (at x within S)"
lp15@56215
   776
proof -
lp15@68255
   777
  from assms obtain x l where x: "x \<in> S" and tf: "((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
lp15@56215
   778
    by blast
lp15@56215
   779
  { fix e::real assume e: "e > 0"
lp15@68255
   780
    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
   781
      by (metis conv)
lp15@68255
   782
    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
   783
    proof (rule exI [of _ N], clarify)
lp15@56215
   784
      fix n y h
lp15@68255
   785
      assume "N \<le> n" "y \<in> S"
lp15@56215
   786
      then have "cmod (f' n y - g' y) \<le> e"
lp15@56215
   787
        by (metis N)
lp15@56215
   788
      then have "cmod h * cmod (f' n y - g' y) \<le> cmod h * e"
lp15@56215
   789
        by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
lp15@56215
   790
      then show "cmod (f' n y * h - g' y * h) \<le> e * cmod h"
lp15@56215
   791
        by (simp add: norm_mult [symmetric] field_simps)
lp15@56215
   792
    qed
lp15@56215
   793
  } note ** = this
lp15@56215
   794
  show ?thesis
lp15@68055
   795
    unfolding has_field_derivative_def
lp15@56215
   796
  proof (rule has_derivative_sequence [OF cvs _ _ x])
lp15@68239
   797
    show "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
lp15@68239
   798
      by (rule tf)
lp15@68255
   799
  next show "\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
lp15@68239
   800
      unfolding eventually_sequentially by (blast intro: **)
lp15@68055
   801
  qed (metis has_field_derivative_def df)
lp15@56215
   802
qed
lp15@56215
   803
lp15@56215
   804
lemma has_complex_derivative_series:
lp15@68255
   805
  fixes S :: "complex set"
lp15@68255
   806
  assumes cvs: "convex S"
lp15@68255
   807
      and df:  "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)"
lp15@68255
   808
      and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> S
lp15@56215
   809
                \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
lp15@68255
   810
      and "\<exists>x l. x \<in> S \<and> ((\<lambda>n. f n x) sums l)"
lp15@68255
   811
    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
   812
proof -
lp15@68255
   813
  from assms obtain x l where x: "x \<in> S" and sf: "((\<lambda>n. f n x) sums l)"
lp15@56215
   814
    by blast
lp15@56215
   815
  { fix e::real assume e: "e > 0"
lp15@68255
   816
    then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> S
lp15@56215
   817
            \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
lp15@61609
   818
      by (metis conv)
lp15@68255
   819
    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
   820
    proof (rule exI [of _ N], clarify)
lp15@56215
   821
      fix n y h
lp15@68255
   822
      assume "N \<le> n" "y \<in> S"
lp15@56215
   823
      then have "cmod ((\<Sum>i<n. f' i y) - g' y) \<le> e"
lp15@56215
   824
        by (metis N)
lp15@56215
   825
      then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e"
lp15@56215
   826
        by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
lp15@56215
   827
      then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h"
nipkow@64267
   828
        by (simp add: norm_mult [symmetric] field_simps sum_distrib_left)
lp15@56215
   829
    qed
lp15@56215
   830
  } note ** = this
lp15@56215
   831
  show ?thesis
lp15@56215
   832
  unfolding has_field_derivative_def
lp15@56215
   833
  proof (rule has_derivative_series [OF cvs _ _ x])
lp15@56215
   834
    fix n x
lp15@68255
   835
    assume "x \<in> S"
lp15@68255
   836
    then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within S)"
lp15@56215
   837
      by (metis df has_field_derivative_def mult_commute_abs)
lp15@56215
   838
  next show " ((\<lambda>n. f n x) sums l)"
lp15@56215
   839
    by (rule sf)
lp15@68255
   840
  next show "\<And>e. e>0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
lp15@68239
   841
      unfolding eventually_sequentially by (blast intro: **)
lp15@56215
   842
  qed
lp15@56215
   843
qed
lp15@56215
   844
eberlm@61531
   845
lp15@62534
   846
lemma field_differentiable_series:
immler@66252
   847
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach} \<Rightarrow> 'a"
lp15@68255
   848
  assumes "convex S" "open S"
lp15@68255
   849
  assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
lp15@68255
   850
  assumes "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
lp15@68255
   851
  assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)" and x: "x \<in> S"
lp15@68055
   852
  shows  "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)"
eberlm@61531
   853
proof -
lp15@68255
   854
  from assms(4) obtain g' where A: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
eberlm@61531
   855
    unfolding uniformly_convergent_on_def by blast
lp15@68255
   856
  from x and \<open>open S\<close> have S: "at x within S = at x" by (rule at_within_open)
lp15@68255
   857
  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)"
lp15@68255
   858
    by (intro has_field_derivative_series[of S f f' g' x0] assms A has_field_derivative_at_within)
lp15@68255
   859
  then obtain g where g: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. f n x) sums g x"
lp15@68255
   860
    "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative g' x) (at x within S)" by blast
nipkow@67399
   861
  from g(2)[OF x] have g': "(g has_derivative ( * ) (g' x)) (at x)"
lp15@68255
   862
    by (simp add: has_field_derivative_def S)
nipkow@67399
   863
  have "((\<lambda>x. \<Sum>n. f n x) has_derivative ( * ) (g' x)) (at x)"
lp15@68255
   864
    by (rule has_derivative_transform_within_open[OF g' \<open>open S\<close> x])
eberlm@61531
   865
       (insert g, auto simp: sums_iff)
lp15@62534
   866
  thus "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)" unfolding differentiable_def
lp15@62534
   867
    by (auto simp: summable_def field_differentiable_def has_field_derivative_def)
eberlm@61531
   868
qed
eberlm@61531
   869
wenzelm@60420
   870
subsection\<open>Bound theorem\<close>
lp15@56215
   871
lp15@62534
   872
lemma field_differentiable_bound:
lp15@68255
   873
  fixes S :: "'a::real_normed_field set"
lp15@68255
   874
  assumes cvs: "convex S"
lp15@68255
   875
      and df:  "\<And>z. z \<in> S \<Longrightarrow> (f has_field_derivative f' z) (at z within S)"
lp15@68255
   876
      and dn:  "\<And>z. z \<in> S \<Longrightarrow> norm (f' z) \<le> B"
lp15@68255
   877
      and "x \<in> S"  "y \<in> S"
lp15@56215
   878
    shows "norm(f x - f y) \<le> B * norm(x - y)"
lp15@56215
   879
  apply (rule differentiable_bound [OF cvs])
lp15@68239
   880
  apply (erule df [unfolded has_field_derivative_def])
lp15@68239
   881
  apply (rule onorm_le, simp_all add: norm_mult mult_right_mono assms)
lp15@56215
   882
  done
lp15@56215
   883
lp15@62408
   884
subsection\<open>Inverse function theorem for complex derivatives\<close>
lp15@56215
   885
immler@66252
   886
lemma has_field_derivative_inverse_basic:
lp15@56215
   887
  shows "DERIV f (g y) :> f' \<Longrightarrow>
lp15@56215
   888
        f' \<noteq> 0 \<Longrightarrow>
lp15@56215
   889
        continuous (at y) g \<Longrightarrow>
lp15@56215
   890
        open t \<Longrightarrow>
lp15@56215
   891
        y \<in> t \<Longrightarrow>
lp15@56215
   892
        (\<And>z. z \<in> t \<Longrightarrow> f (g z) = z)
lp15@56215
   893
        \<Longrightarrow> DERIV g y :> inverse (f')"
lp15@56215
   894
  unfolding has_field_derivative_def
lp15@56215
   895
  apply (rule has_derivative_inverse_basic)
lp15@56215
   896
  apply (auto simp:  bounded_linear_mult_right)
lp15@56215
   897
  done
lp15@56215
   898
immler@66252
   899
lemma has_field_derivative_inverse_strong:
immler@66252
   900
  fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a"
lp15@56215
   901
  shows "DERIV f x :> f' \<Longrightarrow>
lp15@56215
   902
         f' \<noteq> 0 \<Longrightarrow>
lp15@68255
   903
         open S \<Longrightarrow>
lp15@68255
   904
         x \<in> S \<Longrightarrow>
lp15@68255
   905
         continuous_on S f \<Longrightarrow>
lp15@68255
   906
         (\<And>z. z \<in> S \<Longrightarrow> g (f z) = z)
lp15@56215
   907
         \<Longrightarrow> DERIV g (f x) :> inverse (f')"
lp15@56215
   908
  unfolding has_field_derivative_def
lp15@68255
   909
  apply (rule has_derivative_inverse_strong [of S x f g ])
lp15@56215
   910
  by auto
lp15@56215
   911
immler@66252
   912
lemma has_field_derivative_inverse_strong_x:
immler@66252
   913
  fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a"
lp15@56215
   914
  shows  "DERIV f (g y) :> f' \<Longrightarrow>
lp15@56215
   915
          f' \<noteq> 0 \<Longrightarrow>
lp15@68255
   916
          open S \<Longrightarrow>
lp15@68255
   917
          continuous_on S f \<Longrightarrow>
lp15@68255
   918
          g y \<in> S \<Longrightarrow> f(g y) = y \<Longrightarrow>
lp15@68255
   919
          (\<And>z. z \<in> S \<Longrightarrow> g (f z) = z)
lp15@56215
   920
          \<Longrightarrow> DERIV g y :> inverse (f')"
lp15@56215
   921
  unfolding has_field_derivative_def
lp15@68255
   922
  apply (rule has_derivative_inverse_strong_x [of S g y f])
lp15@56215
   923
  by auto
lp15@56215
   924
wenzelm@60420
   925
subsection \<open>Taylor on Complex Numbers\<close>
lp15@56215
   926
nipkow@64267
   927
lemma sum_Suc_reindex:
lp15@56215
   928
  fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@64267
   929
    shows  "sum f {0..n} = f 0 - f (Suc n) + sum (\<lambda>i. f (Suc i)) {0..n}"
lp15@56215
   930
by (induct n) auto
lp15@56215
   931
immler@66252
   932
lemma field_taylor:
lp15@68255
   933
  assumes S: "convex S"
lp15@68255
   934
      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@68255
   935
      and B: "\<And>x. x \<in> S \<Longrightarrow> norm (f (Suc n) x) \<le> B"
lp15@68255
   936
      and w: "w \<in> S"
lp15@68255
   937
      and z: "z \<in> S"
immler@66252
   938
    shows "norm(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i)))
immler@66252
   939
          \<le> B * norm(z - w)^(Suc n) / fact n"
lp15@56215
   940
proof -
lp15@68255
   941
  have wzs: "closed_segment w z \<subseteq> S" using assms
lp15@56215
   942
    by (metis convex_contains_segment)
lp15@56215
   943
  { fix u
lp15@56215
   944
    assume "u \<in> closed_segment w z"
lp15@68255
   945
    then have "u \<in> S"
lp15@56215
   946
      by (metis wzs subsetD)
lp15@59730
   947
    have "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / (fact i) +
lp15@61609
   948
                      f (Suc i) u * (z-u)^i / (fact i)) =
lp15@59730
   949
              f (Suc n) u * (z-u) ^ n / (fact n)"
lp15@56215
   950
    proof (induction n)
lp15@56215
   951
      case 0 show ?case by simp
lp15@56215
   952
    next
lp15@56215
   953
      case (Suc n)
lp15@59730
   954
      have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / (fact i) +
lp15@61609
   955
                             f (Suc i) u * (z-u) ^ i / (fact i)) =
lp15@59730
   956
           f (Suc n) u * (z-u) ^ n / (fact n) +
lp15@59730
   957
           f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / (fact (Suc n)) -
lp15@59730
   958
           f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / (fact (Suc n))"
hoelzl@56479
   959
        using Suc by simp
lp15@59730
   960
      also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n))"
lp15@56215
   961
      proof -
lp15@59730
   962
        have "(fact(Suc n)) *
lp15@59730
   963
             (f(Suc n) u *(z-u) ^ n / (fact n) +
lp15@59730
   964
               f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / (fact(Suc n)) -
lp15@59730
   965
               f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / (fact(Suc n))) =
lp15@59730
   966
            ((fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / (fact n) +
lp15@59730
   967
            ((fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / (fact(Suc n))) -
lp15@59730
   968
            ((fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / (fact(Suc n))"
haftmann@63367
   969
          by (simp add: algebra_simps del: fact_Suc)
lp15@59730
   970
        also have "... = ((fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / (fact n) +
lp15@59730
   971
                         (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
lp15@59730
   972
                         (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
haftmann@63367
   973
          by (simp del: fact_Suc)
lp15@59730
   974
        also have "... = (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) +
lp15@59730
   975
                         (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
lp15@59730
   976
                         (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
haftmann@63367
   977
          by (simp only: fact_Suc of_nat_mult ac_simps) simp
lp15@56215
   978
        also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
lp15@56215
   979
          by (simp add: algebra_simps)
lp15@56215
   980
        finally show ?thesis
haftmann@63367
   981
        by (simp add: mult_left_cancel [where c = "(fact (Suc n))", THEN iffD1] del: fact_Suc)
lp15@56215
   982
      qed
lp15@56215
   983
      finally show ?case .
lp15@56215
   984
    qed
lp15@61609
   985
    then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / (fact i)))
lp15@59730
   986
                has_field_derivative f (Suc n) u * (z-u) ^ n / (fact n))
lp15@68255
   987
               (at u within S)"
hoelzl@56381
   988
      apply (intro derivative_eq_intros)
lp15@68255
   989
      apply (blast intro: assms \<open>u \<in> S\<close>)
lp15@56215
   990
      apply (rule refl)+
lp15@56215
   991
      apply (auto simp: field_simps)
lp15@56215
   992
      done
lp15@56215
   993
  } note sum_deriv = this
lp15@56215
   994
  { fix u
lp15@56215
   995
    assume u: "u \<in> closed_segment w z"
lp15@68255
   996
    then have us: "u \<in> S"
lp15@56215
   997
      by (metis wzs subsetD)
immler@66252
   998
    have "norm (f (Suc n) u) * norm (z - u) ^ n \<le> norm (f (Suc n) u) * norm (u - z) ^ n"
lp15@56215
   999
      by (metis norm_minus_commute order_refl)
immler@66252
  1000
    also have "... \<le> norm (f (Suc n) u) * norm (z - w) ^ n"
lp15@56215
  1001
      by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
immler@66252
  1002
    also have "... \<le> B * norm (z - w) ^ n"
lp15@56215
  1003
      by (metis norm_ge_zero zero_le_power mult_right_mono  B [OF us])
immler@66252
  1004
    finally have "norm (f (Suc n) u) * norm (z - u) ^ n \<le> B * norm (z - w) ^ n" .
lp15@56215
  1005
  } note cmod_bound = this
lp15@59730
  1006
  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
  1007
    by simp
lp15@59730
  1008
  also have "\<dots> = f 0 z / (fact 0)"
nipkow@64267
  1009
    by (subst sum_zero_power) simp
immler@66252
  1010
  finally have "norm (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)))
immler@66252
  1011
                \<le> norm ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) -
lp15@59730
  1012
                        (\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))"
lp15@56215
  1013
    by (simp add: norm_minus_commute)
immler@66252
  1014
  also have "... \<le> B * norm (z - w) ^ n / (fact n) * norm (w - z)"
lp15@62534
  1015
    apply (rule field_differentiable_bound
lp15@59730
  1016
      [where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / (fact n)"
lp15@68255
  1017
         and S = "closed_segment w z", OF convex_closed_segment])
lp15@61609
  1018
    apply (auto simp: ends_in_segment DERIV_subset [OF sum_deriv wzs]
lp15@56215
  1019
                  norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
lp15@56215
  1020
    done
immler@66252
  1021
  also have "...  \<le> B * norm (z - w) ^ Suc n / (fact n)"
lp15@61609
  1022
    by (simp add: algebra_simps norm_minus_commute)
lp15@56215
  1023
  finally show ?thesis .
lp15@56215
  1024
qed
lp15@56215
  1025
immler@66252
  1026
lemma complex_taylor:
lp15@68255
  1027
  assumes S: "convex S"
lp15@68255
  1028
      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@68255
  1029
      and B: "\<And>x. x \<in> S \<Longrightarrow> cmod (f (Suc n) x) \<le> B"
lp15@68255
  1030
      and w: "w \<in> S"
lp15@68255
  1031
      and z: "z \<in> S"
immler@66252
  1032
    shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i)))
immler@66252
  1033
          \<le> B * cmod(z - w)^(Suc n) / fact n"
immler@66252
  1034
  using assms by (rule field_taylor)
immler@66252
  1035
immler@66252
  1036
lp15@62408
  1037
text\<open>Something more like the traditional MVT for real components\<close>
hoelzl@56370
  1038
lp15@56238
  1039
lemma complex_mvt_line:
hoelzl@56369
  1040
  assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f'(u)) (at u)"
paulson@61518
  1041
    shows "\<exists>u. u \<in> closed_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
lp15@56238
  1042
proof -
lp15@56238
  1043
  have twz: "\<And>t. (1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)"
lp15@56238
  1044
    by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
hoelzl@56381
  1045
  note assms[unfolded has_field_derivative_def, derivative_intros]
lp15@56238
  1046
  show ?thesis
lp15@56238
  1047
    apply (cut_tac mvt_simple
lp15@56238
  1048
                     [of 0 1 "Re o f o (\<lambda>t. (1 - t) *\<^sub>R w +  t *\<^sub>R z)"
lp15@56238
  1049
                      "\<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
  1050
    apply auto
lp15@56238
  1051
    apply (rule_tac x="(1 - x) *\<^sub>R w + x *\<^sub>R z" in exI)
paulson@61518
  1052
    apply (auto simp: closed_segment_def twz) []
lp15@67979
  1053
    apply (intro derivative_eq_intros has_derivative_at_withinI, simp_all)
hoelzl@56369
  1054
    apply (simp add: fun_eq_iff real_vector.scale_right_diff_distrib)
paulson@61518
  1055
    apply (force simp: twz closed_segment_def)
lp15@56238
  1056
    done
lp15@56238
  1057
qed
lp15@56238
  1058
lp15@56238
  1059
lemma complex_taylor_mvt:
lp15@56238
  1060
  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
  1061
    shows "\<exists>u. u \<in> closed_segment w z \<and>
lp15@56238
  1062
            Re (f 0 z) =
lp15@59730
  1063
            Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / (fact i)) +
lp15@59730
  1064
                (f (Suc n) u * (z-u)^n / (fact n)) * (z - w))"
lp15@56238
  1065
proof -
lp15@56238
  1066
  { fix u
lp15@56238
  1067
    assume u: "u \<in> closed_segment w z"
lp15@56238
  1068
    have "(\<Sum>i = 0..n.
lp15@56238
  1069
               (f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) /
lp15@59730
  1070
               (fact i)) =
lp15@56238
  1071
          f (Suc 0) u -
lp15@56238
  1072
             (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
lp15@59730
  1073
             (fact (Suc n)) +
lp15@56238
  1074
             (\<Sum>i = 0..n.
lp15@56238
  1075
                 (f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) /
lp15@59730
  1076
                 (fact (Suc i)))"
nipkow@64267
  1077
       by (subst sum_Suc_reindex) simp
lp15@56238
  1078
    also have "... = f (Suc 0) u -
lp15@56238
  1079
             (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
lp15@59730
  1080
             (fact (Suc n)) +
lp15@56238
  1081
             (\<Sum>i = 0..n.
lp15@61609
  1082
                 f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i))  -
lp15@59730
  1083
                 f (Suc i) u * (z-u) ^ i / (fact i))"
haftmann@57514
  1084
      by (simp only: diff_divide_distrib fact_cancel ac_simps)
lp15@56238
  1085
    also have "... = f (Suc 0) u -
lp15@56238
  1086
             (f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) /
lp15@59730
  1087
             (fact (Suc n)) +
lp15@59730
  1088
             f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n)) - f (Suc 0) u"
nipkow@64267
  1089
      by (subst sum_Suc_diff) auto
lp15@59730
  1090
    also have "... = f (Suc n) u * (z-u) ^ n / (fact n)"
lp15@56238
  1091
      by (simp only: algebra_simps diff_divide_distrib fact_cancel)
lp15@61609
  1092
    finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i
lp15@59730
  1093
                             - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / (fact i)) =
lp15@59730
  1094
                  f (Suc n) u * (z - u) ^ n / (fact n)" .
lp15@59730
  1095
    then have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / (fact i)) has_field_derivative
lp15@59730
  1096
                f (Suc n) u * (z - u) ^ n / (fact n))  (at u)"
hoelzl@56381
  1097
      apply (intro derivative_eq_intros)+
lp15@56238
  1098
      apply (force intro: u assms)
lp15@56238
  1099
      apply (rule refl)+
haftmann@57514
  1100
      apply (auto simp: ac_simps)
lp15@56238
  1101
      done
lp15@56238
  1102
  }
lp15@56238
  1103
  then show ?thesis
lp15@59730
  1104
    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
  1105
               "\<lambda>u. (f (Suc n) u * (z-u)^n / (fact n))"])
lp15@56238
  1106
    apply (auto simp add: intro: open_closed_segment)
lp15@56238
  1107
    done
lp15@56238
  1108
qed
lp15@56238
  1109
lp15@60017
  1110
wenzelm@60420
  1111
subsection \<open>Polynomal function extremal theorem, from HOL Light\<close>
lp15@60017
  1112
lp15@60017
  1113
lemma polyfun_extremal_lemma: (*COMPLEX_POLYFUN_EXTREMAL_LEMMA in HOL Light*)
lp15@60017
  1114
    fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
lp15@60017
  1115
  assumes "0 < e"
lp15@60017
  1116
    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
  1117
proof (induct n)
lp15@60017
  1118
  case 0 with assms
lp15@60017
  1119
  show ?case
lp15@60017
  1120
    apply (rule_tac x="norm (c 0) / e" in exI)
lp15@60017
  1121
    apply (auto simp: field_simps)
lp15@60017
  1122
    done
lp15@60017
  1123
next
lp15@60017
  1124
  case (Suc n)
lp15@60017
  1125
  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
  1126
    using Suc assms by blast
lp15@60017
  1127
  show ?case
lp15@60017
  1128
  proof (rule exI [where x= "max M (1 + norm(c(Suc n)) / e)"], clarsimp simp del: power_Suc)
lp15@60017
  1129
    fix z::'a
lp15@60017
  1130
    assume z1: "M \<le> norm z" and "1 + norm (c (Suc n)) / e \<le> norm z"
lp15@60017
  1131
    then have z2: "e + norm (c (Suc n)) \<le> e * norm z"
lp15@60017
  1132
      using assms by (simp add: field_simps)
lp15@60017
  1133
    have "norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
lp15@60017
  1134
      using M [OF z1] by simp
lp15@60017
  1135
    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
  1136
      by simp
lp15@60017
  1137
    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
  1138
      by (blast intro: norm_triangle_le elim: )
lp15@60017
  1139
    also have "... \<le> (e + norm (c (Suc n))) * norm z ^ Suc n"
lp15@60017
  1140
      by (simp add: norm_power norm_mult algebra_simps)
lp15@60017
  1141
    also have "... \<le> (e * norm z) * norm z ^ Suc n"
lp15@60017
  1142
      by (metis z2 mult.commute mult_left_mono norm_ge_zero norm_power)
lp15@60017
  1143
    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
  1144
      by simp
lp15@60017
  1145
  qed
lp15@60017
  1146
qed
lp15@60017
  1147
lp15@60017
  1148
lemma polyfun_extremal: (*COMPLEX_POLYFUN_EXTREMAL in HOL Light*)
lp15@60017
  1149
    fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
lp15@60017
  1150
  assumes k: "c k \<noteq> 0" "1\<le>k" and kn: "k\<le>n"
lp15@60017
  1151
    shows "eventually (\<lambda>z. norm (\<Sum>i\<le>n. c(i) * z^i) \<ge> B) at_infinity"
lp15@60017
  1152
using kn
lp15@60017
  1153
proof (induction n)
lp15@60017
  1154
  case 0
lp15@60017
  1155
  then show ?case
lp15@60017
  1156
    using k  by simp
lp15@60017
  1157
next
lp15@60017
  1158
  case (Suc m)
lp15@60017
  1159
  let ?even = ?case
lp15@60017
  1160
  show ?even
lp15@60017
  1161
  proof (cases "c (Suc m) = 0")
lp15@60017
  1162
    case True
lp15@60017
  1163
    then show ?even using Suc k
lp15@60017
  1164
      by auto (metis antisym_conv less_eq_Suc_le not_le)
lp15@60017
  1165
  next
lp15@60017
  1166
    case False
lp15@60017
  1167
    then obtain M where M:
lp15@60017
  1168
          "\<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
  1169
      using polyfun_extremal_lemma [of "norm(c (Suc m)) / 2" c m] Suc
lp15@60017
  1170
      by auto
lp15@60017
  1171
    have "\<exists>b. \<forall>z. b \<le> norm z \<longrightarrow> B \<le> norm (\<Sum>i\<le>Suc m. c i * z^i)"
lp15@60017
  1172
    proof (rule exI [where x="max M (max 1 (\<bar>B\<bar> / (norm(c (Suc m)) / 2)))"], clarsimp simp del: power_Suc)
lp15@60017
  1173
      fix z::'a
lp15@60017
  1174
      assume z1: "M \<le> norm z" "1 \<le> norm z"
lp15@60017
  1175
         and "\<bar>B\<bar> * 2 / norm (c (Suc m)) \<le> norm z"
lp15@60017
  1176
      then have z2: "\<bar>B\<bar> \<le> norm (c (Suc m)) * norm z / 2"
lp15@60017
  1177
        using False by (simp add: field_simps)
lp15@60017
  1178
      have nz: "norm z \<le> norm z ^ Suc m"
wenzelm@60420
  1179
        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
  1180
      have *: "\<And>y x. norm (c (Suc m)) * norm z / 2 \<le> norm y - norm x \<Longrightarrow> B \<le> norm (x + y)"
lp15@60017
  1181
        by (metis abs_le_iff add.commute norm_diff_ineq order_trans z2)
lp15@60017
  1182
      have "norm z * norm (c (Suc m)) + 2 * norm (\<Sum>i\<le>m. c i * z^i)
lp15@60017
  1183
            \<le> norm (c (Suc m)) * norm z + norm (c (Suc m)) * norm z ^ Suc m"
lp15@60017
  1184
        using M [of z] Suc z1  by auto
lp15@60017
  1185
      also have "... \<le> 2 * (norm (c (Suc m)) * norm z ^ Suc m)"
lp15@60017
  1186
        using nz by (simp add: mult_mono del: power_Suc)
lp15@60017
  1187
      finally show "B \<le> norm ((\<Sum>i\<le>m. c i * z^i) + c (Suc m) * z ^ Suc m)"
lp15@60017
  1188
        using Suc.IH
lp15@60017
  1189
        apply (auto simp: eventually_at_infinity)
lp15@60017
  1190
        apply (rule *)
lp15@60017
  1191
        apply (simp add: field_simps norm_mult norm_power)
lp15@60017
  1192
        done
lp15@60017
  1193
    qed
lp15@60017
  1194
    then show ?even
lp15@60017
  1195
      by (simp add: eventually_at_infinity)
lp15@60017
  1196
  qed
lp15@60017
  1197
qed
lp15@60017
  1198
lp15@56215
  1199
end