src/HOL/Analysis/Complex_Analysis_Basics.thy
author paulson <lp15@cam.ac.uk>
Mon May 21 22:52:16 2018 +0100 (12 months ago)
changeset 68255 009f783d1bac
parent 68239 0764ee22a4d1
child 68296 69d680e94961
permissions -rw-r--r--
small clean-up of Complex_Analysis_Basics
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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 continuous_mult_left:
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  fixes c::"'a::real_normed_algebra"
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  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. c * f x)"
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by (rule continuous_mult [OF continuous_const])
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lemma continuous_mult_right:
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  fixes c::"'a::real_normed_algebra"
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  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x * c)"
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by (rule continuous_mult [OF _ continuous_const])
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lemma continuous_on_mult_left:
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  fixes c::"'a::real_normed_algebra"
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  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c * f x)"
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by (rule continuous_on_mult [OF continuous_on_const])
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lemma continuous_on_mult_right:
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  fixes c::"'a::real_normed_algebra"
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  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x * c)"
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by (rule continuous_on_mult [OF _ continuous_on_const])
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lemma uniformly_continuous_on_cmul_right [continuous_intros]:
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  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
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  shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. f x * c)"
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  using bounded_linear.uniformly_continuous_on[OF bounded_linear_mult_left] .
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lemma uniformly_continuous_on_cmul_left[continuous_intros]:
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  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
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  assumes "uniformly_continuous_on s f"
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    shows "uniformly_continuous_on s (\<lambda>x. c * f x)"
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by (metis assms bounded_linear.uniformly_continuous_on bounded_linear_mult_right)
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lemma continuous_within_norm_id [continuous_intros]: "continuous (at x within S) norm"
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  by (rule continuous_norm [OF continuous_ident])
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lemma continuous_on_norm_id [continuous_intros]: "continuous_on S norm"
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  by (intro continuous_on_id continuous_on_norm)
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subsection\<open>DERIV stuff\<close>
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lemma DERIV_zero_connected_constant:
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  fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
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  assumes "connected S"
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      and "open S"
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      and "finite K"
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      and "continuous_on S f"
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      and "\<forall>x\<in>(S - K). DERIV f x :> 0"
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    obtains c where "\<And>x. x \<in> S \<Longrightarrow> f(x) = c"
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using has_derivative_zero_connected_constant [OF assms(1-4)] assms
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by (metis DERIV_const has_derivative_const Diff_iff at_within_open frechet_derivative_at has_field_derivative_def)
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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"
lp15@62534
   289
  unfolding holomorphic_on_def by (metis field_differentiable_linear)
hoelzl@56370
   290
lp15@62217
   291
lemma holomorphic_on_const [simp, holomorphic_intros]: "(\<lambda>z. c) holomorphic_on s"
lp15@62534
   292
  unfolding holomorphic_on_def by (metis field_differentiable_const)
hoelzl@56370
   293
lp15@62217
   294
lemma holomorphic_on_ident [simp, holomorphic_intros]: "(\<lambda>x. x) holomorphic_on s"
lp15@62534
   295
  unfolding holomorphic_on_def by (metis field_differentiable_ident)
hoelzl@56370
   296
lp15@62217
   297
lemma holomorphic_on_id [simp, holomorphic_intros]: "id holomorphic_on s"
hoelzl@56370
   298
  unfolding id_def by (rule holomorphic_on_ident)
hoelzl@56370
   299
hoelzl@56370
   300
lemma holomorphic_on_compose:
hoelzl@56370
   301
  "f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s) \<Longrightarrow> (g o f) holomorphic_on s"
lp15@62534
   302
  using field_differentiable_compose_within[of f _ s g]
hoelzl@56370
   303
  by (auto simp: holomorphic_on_def)
hoelzl@56370
   304
hoelzl@56370
   305
lemma holomorphic_on_compose_gen:
hoelzl@56370
   306
  "f holomorphic_on s \<Longrightarrow> g holomorphic_on t \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> (g o f) holomorphic_on s"
hoelzl@56370
   307
  by (metis holomorphic_on_compose holomorphic_on_subset)
hoelzl@56370
   308
lp15@61520
   309
lemma holomorphic_on_minus [holomorphic_intros]: "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on s"
lp15@62534
   310
  by (metis field_differentiable_minus holomorphic_on_def)
hoelzl@56370
   311
lp15@61520
   312
lemma holomorphic_on_add [holomorphic_intros]:
hoelzl@56370
   313
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on s"
lp15@62534
   314
  unfolding holomorphic_on_def by (metis field_differentiable_add)
hoelzl@56370
   315
lp15@61520
   316
lemma holomorphic_on_diff [holomorphic_intros]:
hoelzl@56370
   317
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) holomorphic_on s"
lp15@62534
   318
  unfolding holomorphic_on_def by (metis field_differentiable_diff)
hoelzl@56370
   319
lp15@61520
   320
lemma holomorphic_on_mult [holomorphic_intros]:
hoelzl@56370
   321
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) holomorphic_on s"
lp15@62534
   322
  unfolding holomorphic_on_def by (metis field_differentiable_mult)
hoelzl@56370
   323
lp15@61520
   324
lemma holomorphic_on_inverse [holomorphic_intros]:
hoelzl@56370
   325
  "\<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
   326
  unfolding holomorphic_on_def by (metis field_differentiable_inverse)
hoelzl@56370
   327
lp15@61520
   328
lemma holomorphic_on_divide [holomorphic_intros]:
hoelzl@56370
   329
  "\<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
   330
  unfolding holomorphic_on_def by (metis field_differentiable_divide)
hoelzl@56370
   331
lp15@61520
   332
lemma holomorphic_on_power [holomorphic_intros]:
hoelzl@56370
   333
  "f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on s"
lp15@62534
   334
  unfolding holomorphic_on_def by (metis field_differentiable_power)
hoelzl@56370
   335
nipkow@64267
   336
lemma holomorphic_on_sum [holomorphic_intros]:
nipkow@64267
   337
  "(\<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
   338
  unfolding holomorphic_on_def by (metis field_differentiable_sum)
hoelzl@56370
   339
eberlm@67135
   340
lemma holomorphic_on_prod [holomorphic_intros]:
eberlm@67135
   341
  "(\<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
   342
  by (induction I rule: infinite_finite_induct) (auto intro: holomorphic_intros)
eberlm@67135
   343
eberlm@66486
   344
lemma holomorphic_pochhammer [holomorphic_intros]:
eberlm@66486
   345
  "f holomorphic_on A \<Longrightarrow> (\<lambda>s. pochhammer (f s) n) holomorphic_on A"
eberlm@66486
   346
  by (induction n) (auto intro!: holomorphic_intros simp: pochhammer_Suc)
eberlm@66486
   347
eberlm@66486
   348
lemma holomorphic_on_scaleR [holomorphic_intros]:
eberlm@66486
   349
  "f holomorphic_on A \<Longrightarrow> (\<lambda>x. c *\<^sub>R f x) holomorphic_on A"
eberlm@66486
   350
  by (auto simp: scaleR_conv_of_real intro!: holomorphic_intros)
eberlm@66486
   351
eberlm@67167
   352
lemma holomorphic_on_Un [holomorphic_intros]:
eberlm@67167
   353
  assumes "f holomorphic_on A" "f holomorphic_on B" "open A" "open B"
eberlm@67167
   354
  shows   "f holomorphic_on (A \<union> B)"
lp15@68239
   355
  using assms by (auto simp: holomorphic_on_def  at_within_open[of _ A]
eberlm@67167
   356
                             at_within_open[of _ B]  at_within_open[of _ "A \<union> B"] open_Un)
eberlm@67167
   357
eberlm@67167
   358
lemma holomorphic_on_If_Un [holomorphic_intros]:
eberlm@67167
   359
  assumes "f holomorphic_on A" "g holomorphic_on B" "open A" "open B"
eberlm@67167
   360
  assumes "\<And>z. z \<in> A \<Longrightarrow> z \<in> B \<Longrightarrow> f z = g z"
eberlm@67167
   361
  shows   "(\<lambda>z. if z \<in> A then f z else g z) holomorphic_on (A \<union> B)" (is "?h holomorphic_on _")
eberlm@67167
   362
proof (intro holomorphic_on_Un)
eberlm@67167
   363
  note \<open>f holomorphic_on A\<close>
eberlm@67167
   364
  also have "f holomorphic_on A \<longleftrightarrow> ?h holomorphic_on A"
eberlm@67167
   365
    by (intro holomorphic_cong) auto
eberlm@67167
   366
  finally show \<dots> .
eberlm@67167
   367
next
eberlm@67167
   368
  note \<open>g holomorphic_on B\<close>
eberlm@67167
   369
  also have "g holomorphic_on B \<longleftrightarrow> ?h holomorphic_on B"
eberlm@67167
   370
    using assms by (intro holomorphic_cong) auto
eberlm@67167
   371
  finally show \<dots> .
eberlm@67167
   372
qed (insert assms, auto)
eberlm@67167
   373
lp15@67371
   374
lemma leibniz_rule_holomorphic:
lp15@67371
   375
  fixes f::"complex \<Rightarrow> 'b::euclidean_space \<Rightarrow> complex"
lp15@67371
   376
  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
   377
  assumes "\<And>x. x \<in> U \<Longrightarrow> (f x) integrable_on cbox a b"
lp15@67371
   378
  assumes "continuous_on (U \<times> (cbox a b)) (\<lambda>(x, t). fx x t)"
lp15@67371
   379
  assumes "convex U"
lp15@67371
   380
  shows "(\<lambda>x. integral (cbox a b) (f x)) holomorphic_on U"
lp15@67371
   381
  using leibniz_rule_field_differentiable[OF assms(1-3) _ assms(4)]
lp15@67371
   382
  by (auto simp: holomorphic_on_def)
lp15@67371
   383
lp15@62534
   384
lemma DERIV_deriv_iff_field_differentiable:
lp15@62534
   385
  "DERIV f x :> deriv f x \<longleftrightarrow> f field_differentiable at x"
lp15@62534
   386
  unfolding field_differentiable_def by (metis DERIV_imp_deriv)
hoelzl@56370
   387
lp15@62533
   388
lemma holomorphic_derivI:
lp15@62533
   389
     "\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk>
lp15@62533
   390
      \<Longrightarrow> (f has_field_derivative deriv f x) (at x within T)"
lp15@62534
   391
by (metis DERIV_deriv_iff_field_differentiable at_within_open  holomorphic_on_def has_field_derivative_at_within)
lp15@62533
   392
hoelzl@56370
   393
lemma complex_derivative_chain:
lp15@62534
   394
  "f field_differentiable at x \<Longrightarrow> g field_differentiable at (f x)
hoelzl@56370
   395
    \<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
lp15@62534
   396
  by (metis DERIV_deriv_iff_field_differentiable DERIV_chain DERIV_imp_deriv)
hoelzl@56370
   397
lp15@62397
   398
lemma deriv_linear [simp]: "deriv (\<lambda>w. c * w) = (\<lambda>z. c)"
hoelzl@56370
   399
  by (metis DERIV_imp_deriv DERIV_cmult_Id)
hoelzl@56370
   400
lp15@62397
   401
lemma deriv_ident [simp]: "deriv (\<lambda>w. w) = (\<lambda>z. 1)"
hoelzl@56370
   402
  by (metis DERIV_imp_deriv DERIV_ident)
hoelzl@56370
   403
lp15@62397
   404
lemma deriv_id [simp]: "deriv id = (\<lambda>z. 1)"
lp15@62397
   405
  by (simp add: id_def)
lp15@62397
   406
lp15@62397
   407
lemma deriv_const [simp]: "deriv (\<lambda>w. c) = (\<lambda>z. 0)"
hoelzl@56370
   408
  by (metis DERIV_imp_deriv DERIV_const)
hoelzl@56370
   409
lp15@62534
   410
lemma deriv_add [simp]:
lp15@62534
   411
  "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
hoelzl@56370
   412
   \<Longrightarrow> deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
lp15@62534
   413
  unfolding DERIV_deriv_iff_field_differentiable[symmetric]
hoelzl@56381
   414
  by (auto intro!: DERIV_imp_deriv derivative_intros)
hoelzl@56370
   415
lp15@62534
   416
lemma deriv_diff [simp]:
lp15@62534
   417
  "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
hoelzl@56370
   418
   \<Longrightarrow> deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
lp15@62534
   419
  unfolding DERIV_deriv_iff_field_differentiable[symmetric]
hoelzl@56381
   420
  by (auto intro!: DERIV_imp_deriv derivative_intros)
hoelzl@56370
   421
lp15@62534
   422
lemma deriv_mult [simp]:
lp15@62534
   423
  "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
hoelzl@56370
   424
   \<Longrightarrow> deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
lp15@62534
   425
  unfolding DERIV_deriv_iff_field_differentiable[symmetric]
hoelzl@56381
   426
  by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
hoelzl@56370
   427
lp15@68255
   428
lemma deriv_cmult:
lp15@62534
   429
  "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z"
lp15@68255
   430
  by simp
hoelzl@56370
   431
lp15@68255
   432
lemma deriv_cmult_right:
lp15@62534
   433
  "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c"
lp15@68255
   434
  by simp
lp15@68255
   435
lp15@68255
   436
lemma deriv_inverse [simp]:
lp15@68255
   437
  "\<lbrakk>f field_differentiable at z; f z \<noteq> 0\<rbrakk>
lp15@68255
   438
   \<Longrightarrow> deriv (\<lambda>w. inverse (f w)) z = - deriv f z / f z ^ 2"
lp15@62534
   439
  unfolding DERIV_deriv_iff_field_differentiable[symmetric]
lp15@68255
   440
  by (safe intro!: DERIV_imp_deriv derivative_eq_intros) (auto simp: divide_simps power2_eq_square)
hoelzl@56370
   441
lp15@68255
   442
lemma deriv_divide [simp]:
lp15@68255
   443
  "\<lbrakk>f field_differentiable at z; g field_differentiable at z; g z \<noteq> 0\<rbrakk>
lp15@68255
   444
   \<Longrightarrow> deriv (\<lambda>w. f w / g w) z = (deriv f z * g z - f z * deriv g z) / g z ^ 2"
lp15@68255
   445
  by (simp add: field_class.field_divide_inverse field_differentiable_inverse)
lp15@68255
   446
     (simp add: divide_simps power2_eq_square)
lp15@68255
   447
lp15@68255
   448
lemma deriv_cdivide_right:
lp15@62534
   449
  "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w / c) z = deriv f z / c"
lp15@68255
   450
  by (simp add: field_class.field_divide_inverse)
lp15@62217
   451
hoelzl@56370
   452
lemma complex_derivative_transform_within_open:
lp15@61609
   453
  "\<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
   454
   \<Longrightarrow> deriv f z = deriv g z"
hoelzl@56370
   455
  unfolding holomorphic_on_def
hoelzl@56370
   456
  by (rule DERIV_imp_deriv)
lp15@62534
   457
     (metis DERIV_deriv_iff_field_differentiable DERIV_transform_within_open at_within_open)
hoelzl@56370
   458
lp15@62534
   459
lemma deriv_compose_linear:
lp15@62534
   460
  "f field_differentiable at (c * z) \<Longrightarrow> deriv (\<lambda>w. f (c * w)) z = c * deriv f (c * z)"
hoelzl@56370
   461
apply (rule DERIV_imp_deriv)
lp15@68255
   462
  unfolding DERIV_deriv_iff_field_differentiable [symmetric]
lp15@68255
   463
  by (metis (full_types) DERIV_chain2 DERIV_cmult_Id mult.commute)
lp15@68255
   464
hoelzl@56370
   465
lp15@62533
   466
lemma nonzero_deriv_nonconstant:
lp15@62533
   467
  assumes df: "DERIV f \<xi> :> df" and S: "open S" "\<xi> \<in> S" and "df \<noteq> 0"
lp15@62533
   468
    shows "\<not> f constant_on S"
lp15@62533
   469
unfolding constant_on_def
lp15@62533
   470
by (metis \<open>df \<noteq> 0\<close> DERIV_transform_within_open [OF df S] DERIV_const DERIV_unique)
lp15@62533
   471
lp15@62533
   472
lemma holomorphic_nonconstant:
lp15@62533
   473
  assumes holf: "f holomorphic_on S" and "open S" "\<xi> \<in> S" "deriv f \<xi> \<noteq> 0"
lp15@62533
   474
    shows "\<not> f constant_on S"
lp15@68255
   475
  by (rule nonzero_deriv_nonconstant [of f "deriv f \<xi>" \<xi> S])
lp15@68255
   476
    (use assms in \<open>auto simp: holomorphic_derivI\<close>)
lp15@62533
   477
lp15@64394
   478
subsection\<open>Caratheodory characterization\<close>
lp15@64394
   479
lp15@64394
   480
lemma field_differentiable_caratheodory_at:
lp15@64394
   481
  "f field_differentiable (at z) \<longleftrightarrow>
lp15@64394
   482
         (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)"
lp15@64394
   483
  using CARAT_DERIV [of f]
lp15@64394
   484
  by (simp add: field_differentiable_def has_field_derivative_def)
lp15@64394
   485
lp15@64394
   486
lemma field_differentiable_caratheodory_within:
lp15@64394
   487
  "f field_differentiable (at z within s) \<longleftrightarrow>
lp15@64394
   488
         (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)"
lp15@64394
   489
  using DERIV_caratheodory_within [of f]
lp15@64394
   490
  by (simp add: field_differentiable_def has_field_derivative_def)
lp15@64394
   491
wenzelm@60420
   492
subsection\<open>Analyticity on a set\<close>
lp15@56215
   493
lp15@61609
   494
definition analytic_on (infixl "(analytic'_on)" 50)
lp15@68255
   495
  where "f analytic_on S \<equiv> \<forall>x \<in> S. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)"
lp15@56215
   496
lp15@65587
   497
named_theorems analytic_intros "introduction rules for proving analyticity"
lp15@65587
   498
lp15@68255
   499
lemma analytic_imp_holomorphic: "f analytic_on S \<Longrightarrow> f holomorphic_on S"
hoelzl@56370
   500
  by (simp add: at_within_open [OF _ open_ball] analytic_on_def holomorphic_on_def)
lp15@62534
   501
     (metis centre_in_ball field_differentiable_at_within)
lp15@56215
   502
lp15@68255
   503
lemma analytic_on_open: "open S \<Longrightarrow> f analytic_on S \<longleftrightarrow> f holomorphic_on S"
lp15@56215
   504
apply (auto simp: analytic_imp_holomorphic)
lp15@56215
   505
apply (auto simp: analytic_on_def holomorphic_on_def)
lp15@56215
   506
by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball)
lp15@56215
   507
lp15@56215
   508
lemma analytic_on_imp_differentiable_at:
lp15@68255
   509
  "f analytic_on S \<Longrightarrow> x \<in> S \<Longrightarrow> f field_differentiable (at x)"
hoelzl@56370
   510
 apply (auto simp: analytic_on_def holomorphic_on_def)
lp15@66827
   511
by (metis open_ball centre_in_ball field_differentiable_within_open)
lp15@56215
   512
lp15@68255
   513
lemma analytic_on_subset: "f analytic_on S \<Longrightarrow> T \<subseteq> S \<Longrightarrow> f analytic_on T"
lp15@56215
   514
  by (auto simp: analytic_on_def)
lp15@56215
   515
lp15@68255
   516
lemma analytic_on_Un: "f analytic_on (S \<union> T) \<longleftrightarrow> f analytic_on S \<and> f analytic_on T"
lp15@56215
   517
  by (auto simp: analytic_on_def)
lp15@56215
   518
lp15@68255
   519
lemma analytic_on_Union: "f analytic_on (\<Union>\<T>) \<longleftrightarrow> (\<forall>T \<in> \<T>. f analytic_on T)"
hoelzl@56370
   520
  by (auto simp: analytic_on_def)
hoelzl@56370
   521
lp15@68255
   522
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
   523
  by (auto simp: analytic_on_def)
lp15@61609
   524
lp15@56215
   525
lemma analytic_on_holomorphic:
lp15@68255
   526
  "f analytic_on S \<longleftrightarrow> (\<exists>T. open T \<and> S \<subseteq> T \<and> f holomorphic_on T)"
lp15@56215
   527
  (is "?lhs = ?rhs")
lp15@56215
   528
proof -
lp15@68255
   529
  have "?lhs \<longleftrightarrow> (\<exists>T. open T \<and> S \<subseteq> T \<and> f analytic_on T)"
lp15@56215
   530
  proof safe
lp15@68255
   531
    assume "f analytic_on S"
lp15@68255
   532
    then show "\<exists>T. open T \<and> S \<subseteq> T \<and> f analytic_on T"
lp15@56215
   533
      apply (simp add: analytic_on_def)
lp15@68255
   534
      apply (rule exI [where x="\<Union>{U. open U \<and> f analytic_on U}"], auto)
lp15@66827
   535
      apply (metis open_ball analytic_on_open centre_in_ball)
lp15@56215
   536
      by (metis analytic_on_def)
lp15@56215
   537
  next
lp15@68255
   538
    fix T
lp15@68255
   539
    assume "open T" "S \<subseteq> T" "f analytic_on T"
lp15@68255
   540
    then show "f analytic_on S"
lp15@56215
   541
        by (metis analytic_on_subset)
lp15@56215
   542
  qed
lp15@56215
   543
  also have "... \<longleftrightarrow> ?rhs"
lp15@56215
   544
    by (auto simp: analytic_on_open)
lp15@56215
   545
  finally show ?thesis .
lp15@56215
   546
qed
lp15@56215
   547
lp15@68255
   548
lemma analytic_on_linear [analytic_intros,simp]: "(( * ) c) analytic_on S"
lp15@65587
   549
  by (auto simp add: analytic_on_holomorphic)
lp15@56215
   550
lp15@68255
   551
lemma analytic_on_const [analytic_intros,simp]: "(\<lambda>z. c) analytic_on S"
hoelzl@56370
   552
  by (metis analytic_on_def holomorphic_on_const zero_less_one)
hoelzl@56370
   553
lp15@68255
   554
lemma analytic_on_ident [analytic_intros,simp]: "(\<lambda>x. x) analytic_on S"
lp15@65587
   555
  by (simp add: analytic_on_def gt_ex)
lp15@56215
   556
lp15@68255
   557
lemma analytic_on_id [analytic_intros]: "id analytic_on S"
hoelzl@56370
   558
  unfolding id_def by (rule analytic_on_ident)
lp15@56215
   559
lp15@56215
   560
lemma analytic_on_compose:
lp15@68255
   561
  assumes f: "f analytic_on S"
lp15@68255
   562
      and g: "g analytic_on (f ` S)"
lp15@68255
   563
    shows "(g o f) analytic_on S"
lp15@56215
   564
unfolding analytic_on_def
lp15@56215
   565
proof (intro ballI)
lp15@56215
   566
  fix x
lp15@68255
   567
  assume x: "x \<in> S"
lp15@56215
   568
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f
lp15@56215
   569
    by (metis analytic_on_def)
lp15@56215
   570
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g
lp15@61609
   571
    by (metis analytic_on_def g image_eqI x)
lp15@56215
   572
  have "isCont f x"
lp15@62534
   573
    by (metis analytic_on_imp_differentiable_at field_differentiable_imp_continuous_at f x)
lp15@56215
   574
  with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'"
lp15@56215
   575
     by (auto simp: continuous_at_ball)
lp15@61609
   576
  have "g \<circ> f holomorphic_on ball x (min d e)"
lp15@56215
   577
    apply (rule holomorphic_on_compose)
lp15@56215
   578
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   579
    by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball)
lp15@56215
   580
  then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e"
lp15@61609
   581
    by (metis d e min_less_iff_conj)
lp15@56215
   582
qed
lp15@56215
   583
lp15@56215
   584
lemma analytic_on_compose_gen:
lp15@68255
   585
  "f analytic_on S \<Longrightarrow> g analytic_on T \<Longrightarrow> (\<And>z. z \<in> S \<Longrightarrow> f z \<in> T)
lp15@68255
   586
             \<Longrightarrow> g o f analytic_on S"
lp15@56215
   587
by (metis analytic_on_compose analytic_on_subset image_subset_iff)
lp15@56215
   588
lp15@65587
   589
lemma analytic_on_neg [analytic_intros]:
lp15@68255
   590
  "f analytic_on S \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on S"
lp15@56215
   591
by (metis analytic_on_holomorphic holomorphic_on_minus)
lp15@56215
   592
lp15@65587
   593
lemma analytic_on_add [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_add)
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_diff [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_diff)
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_mult [analytic_intros]:
lp15@68255
   634
  assumes f: "f analytic_on S"
lp15@68255
   635
      and g: "g analytic_on S"
lp15@68255
   636
    shows "(\<lambda>z. f z * g 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
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
lp15@61609
   644
    by (metis analytic_on_def g z)
lp15@61609
   645
  have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')"
lp15@61609
   646
    apply (rule holomorphic_on_mult)
lp15@56215
   647
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   648
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   649
  then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e"
lp15@56215
   650
    by (metis e e' min_less_iff_conj)
lp15@56215
   651
qed
lp15@56215
   652
lp15@65587
   653
lemma analytic_on_inverse [analytic_intros]:
lp15@68255
   654
  assumes f: "f analytic_on S"
lp15@68255
   655
      and nz: "(\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0)"
lp15@68255
   656
    shows "(\<lambda>z. inverse (f z)) analytic_on S"
lp15@56215
   657
unfolding analytic_on_def
lp15@56215
   658
proof (intro ballI)
lp15@56215
   659
  fix z
lp15@68255
   660
  assume z: "z \<in> S"
lp15@56215
   661
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   662
    by (metis analytic_on_def)
lp15@56215
   663
  have "continuous_on (ball z e) f"
lp15@56215
   664
    by (metis fh holomorphic_on_imp_continuous_on)
lp15@61609
   665
  then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0"
lp15@66827
   666
    by (metis open_ball centre_in_ball continuous_on_open_avoid e z nz)
lp15@61609
   667
  have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')"
lp15@56215
   668
    apply (rule holomorphic_on_inverse)
lp15@56215
   669
    apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball)
lp15@61609
   670
    by (metis nz' mem_ball min_less_iff_conj)
lp15@56215
   671
  then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e"
lp15@56215
   672
    by (metis e e' min_less_iff_conj)
lp15@56215
   673
qed
lp15@56215
   674
lp15@65587
   675
lemma analytic_on_divide [analytic_intros]:
lp15@68255
   676
  assumes f: "f analytic_on S"
lp15@68255
   677
      and g: "g analytic_on S"
lp15@68255
   678
      and nz: "(\<And>z. z \<in> S \<Longrightarrow> g z \<noteq> 0)"
lp15@68255
   679
    shows "(\<lambda>z. f z / g z) analytic_on S"
lp15@56215
   680
unfolding divide_inverse
lp15@56215
   681
by (metis analytic_on_inverse analytic_on_mult f g nz)
lp15@56215
   682
lp15@65587
   683
lemma analytic_on_power [analytic_intros]:
lp15@68255
   684
  "f analytic_on S \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on S"
lp15@65587
   685
by (induct n) (auto simp: analytic_on_mult)
lp15@56215
   686
lp15@65587
   687
lemma analytic_on_sum [analytic_intros]:
lp15@68255
   688
  "(\<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
   689
  by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_const analytic_on_add)
lp15@56215
   690
lp15@62408
   691
lemma deriv_left_inverse:
lp15@62408
   692
  assumes "f holomorphic_on S" and "g holomorphic_on T"
lp15@62408
   693
      and "open S" and "open T"
lp15@62408
   694
      and "f ` S \<subseteq> T"
lp15@62408
   695
      and [simp]: "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z"
lp15@62408
   696
      and "w \<in> S"
lp15@62408
   697
    shows "deriv f w * deriv g (f w) = 1"
lp15@62408
   698
proof -
lp15@62408
   699
  have "deriv f w * deriv g (f w) = deriv g (f w) * deriv f w"
lp15@62408
   700
    by (simp add: algebra_simps)
lp15@62408
   701
  also have "... = deriv (g o f) w"
lp15@62408
   702
    using assms
lp15@62408
   703
    by (metis analytic_on_imp_differentiable_at analytic_on_open complex_derivative_chain image_subset_iff)
lp15@62408
   704
  also have "... = deriv id w"
lp15@68255
   705
  proof (rule complex_derivative_transform_within_open [where s=S])
lp15@68255
   706
    show "g \<circ> f holomorphic_on S"
lp15@68255
   707
      by (rule assms holomorphic_on_compose_gen holomorphic_intros)+
lp15@68255
   708
  qed (use assms in auto)
lp15@62408
   709
  also have "... = 1"
lp15@62408
   710
    by simp
lp15@62408
   711
  finally show ?thesis .
lp15@62408
   712
qed
lp15@62408
   713
lp15@62408
   714
subsection\<open>analyticity at a point\<close>
lp15@56215
   715
lp15@56215
   716
lemma analytic_at_ball:
lp15@56215
   717
  "f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
lp15@56215
   718
by (metis analytic_on_def singleton_iff)
lp15@56215
   719
lp15@56215
   720
lemma analytic_at:
lp15@56215
   721
    "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
lp15@56215
   722
by (metis analytic_on_holomorphic empty_subsetI insert_subset)
lp15@56215
   723
lp15@56215
   724
lemma analytic_on_analytic_at:
lp15@56215
   725
    "f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
lp15@56215
   726
by (metis analytic_at_ball analytic_on_def)
lp15@56215
   727
lp15@56215
   728
lemma analytic_at_two:
lp15@56215
   729
  "f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
lp15@56215
   730
   (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)"
lp15@56215
   731
  (is "?lhs = ?rhs")
lp15@61609
   732
proof
lp15@56215
   733
  assume ?lhs
lp15@61609
   734
  then obtain s t
lp15@56215
   735
    where st: "open s" "z \<in> s" "f holomorphic_on s"
lp15@56215
   736
              "open t" "z \<in> t" "g holomorphic_on t"
lp15@56215
   737
    by (auto simp: analytic_at)
lp15@56215
   738
  show ?rhs
lp15@56215
   739
    apply (rule_tac x="s \<inter> t" in exI)
lp15@56215
   740
    using st
lp15@56215
   741
    apply (auto simp: Diff_subset holomorphic_on_subset)
lp15@56215
   742
    done
lp15@56215
   743
next
lp15@61609
   744
  assume ?rhs
lp15@56215
   745
  then show ?lhs
lp15@56215
   746
    by (force simp add: analytic_at)
lp15@56215
   747
qed
lp15@56215
   748
wenzelm@60420
   749
subsection\<open>Combining theorems for derivative with ``analytic at'' hypotheses\<close>
lp15@56215
   750
lp15@61609
   751
lemma
lp15@56215
   752
  assumes "f analytic_on {z}" "g analytic_on {z}"
hoelzl@56370
   753
  shows complex_derivative_add_at: "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
hoelzl@56370
   754
    and complex_derivative_diff_at: "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
hoelzl@56370
   755
    and complex_derivative_mult_at: "deriv (\<lambda>w. f w * g w) z =
hoelzl@56370
   756
           f z * deriv g z + deriv f z * g z"
lp15@56215
   757
proof -
lp15@56215
   758
  obtain s where s: "open s" "z \<in> s" "f holomorphic_on s" "g holomorphic_on s"
lp15@56215
   759
    using assms by (metis analytic_at_two)
hoelzl@56370
   760
  show "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
hoelzl@56370
   761
    apply (rule DERIV_imp_deriv [OF DERIV_add])
lp15@56215
   762
    using s
lp15@62534
   763
    apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
lp15@56215
   764
    done
hoelzl@56370
   765
  show "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
hoelzl@56370
   766
    apply (rule DERIV_imp_deriv [OF DERIV_diff])
lp15@56215
   767
    using s
lp15@62534
   768
    apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
lp15@56215
   769
    done
hoelzl@56370
   770
  show "deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
hoelzl@56370
   771
    apply (rule DERIV_imp_deriv [OF DERIV_mult'])
lp15@56215
   772
    using s
lp15@62534
   773
    apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
lp15@56215
   774
    done
lp15@56215
   775
qed
lp15@56215
   776
lp15@62534
   777
lemma deriv_cmult_at:
hoelzl@56370
   778
  "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. c * f w) z = c * deriv f z"
lp15@61848
   779
by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
lp15@56215
   780
lp15@62534
   781
lemma deriv_cmult_right_at:
hoelzl@56370
   782
  "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. f w * c) z = deriv f z * c"
lp15@61848
   783
by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
lp15@56215
   784
wenzelm@60420
   785
subsection\<open>Complex differentiation of sequences and series\<close>
lp15@56215
   786
eberlm@61531
   787
(* TODO: Could probably be simplified using Uniform_Limit *)
lp15@56215
   788
lemma has_complex_derivative_sequence:
lp15@68255
   789
  fixes S :: "complex set"
lp15@68255
   790
  assumes cvs: "convex S"
lp15@68255
   791
      and df:  "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)"
lp15@68255
   792
      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
   793
      and "\<exists>x l. x \<in> S \<and> ((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
lp15@68255
   794
    shows "\<exists>g. \<forall>x \<in> S. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and>
lp15@68255
   795
                       (g has_field_derivative (g' x)) (at x within S)"
lp15@56215
   796
proof -
lp15@68255
   797
  from assms obtain x l where x: "x \<in> S" and tf: "((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
lp15@56215
   798
    by blast
lp15@56215
   799
  { fix e::real assume e: "e > 0"
lp15@68255
   800
    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
   801
      by (metis conv)
lp15@68255
   802
    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
   803
    proof (rule exI [of _ N], clarify)
lp15@56215
   804
      fix n y h
lp15@68255
   805
      assume "N \<le> n" "y \<in> S"
lp15@56215
   806
      then have "cmod (f' n y - g' y) \<le> e"
lp15@56215
   807
        by (metis N)
lp15@56215
   808
      then have "cmod h * cmod (f' n y - g' y) \<le> cmod h * e"
lp15@56215
   809
        by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
lp15@56215
   810
      then show "cmod (f' n y * h - g' y * h) \<le> e * cmod h"
lp15@56215
   811
        by (simp add: norm_mult [symmetric] field_simps)
lp15@56215
   812
    qed
lp15@56215
   813
  } note ** = this
lp15@56215
   814
  show ?thesis
lp15@68055
   815
    unfolding has_field_derivative_def
lp15@56215
   816
  proof (rule has_derivative_sequence [OF cvs _ _ x])
lp15@68239
   817
    show "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
lp15@68239
   818
      by (rule tf)
lp15@68255
   819
  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
   820
      unfolding eventually_sequentially by (blast intro: **)
lp15@68055
   821
  qed (metis has_field_derivative_def df)
lp15@56215
   822
qed
lp15@56215
   823
lp15@56215
   824
lemma has_complex_derivative_series:
lp15@68255
   825
  fixes S :: "complex set"
lp15@68255
   826
  assumes cvs: "convex S"
lp15@68255
   827
      and df:  "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)"
lp15@68255
   828
      and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> S
lp15@56215
   829
                \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
lp15@68255
   830
      and "\<exists>x l. x \<in> S \<and> ((\<lambda>n. f n x) sums l)"
lp15@68255
   831
    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
   832
proof -
lp15@68255
   833
  from assms obtain x l where x: "x \<in> S" and sf: "((\<lambda>n. f n x) sums l)"
lp15@56215
   834
    by blast
lp15@56215
   835
  { fix e::real assume e: "e > 0"
lp15@68255
   836
    then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> S
lp15@56215
   837
            \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
lp15@61609
   838
      by (metis conv)
lp15@68255
   839
    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
   840
    proof (rule exI [of _ N], clarify)
lp15@56215
   841
      fix n y h
lp15@68255
   842
      assume "N \<le> n" "y \<in> S"
lp15@56215
   843
      then have "cmod ((\<Sum>i<n. f' i y) - g' y) \<le> e"
lp15@56215
   844
        by (metis N)
lp15@56215
   845
      then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e"
lp15@56215
   846
        by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
lp15@56215
   847
      then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h"
nipkow@64267
   848
        by (simp add: norm_mult [symmetric] field_simps sum_distrib_left)
lp15@56215
   849
    qed
lp15@56215
   850
  } note ** = this
lp15@56215
   851
  show ?thesis
lp15@56215
   852
  unfolding has_field_derivative_def
lp15@56215
   853
  proof (rule has_derivative_series [OF cvs _ _ x])
lp15@56215
   854
    fix n x
lp15@68255
   855
    assume "x \<in> S"
lp15@68255
   856
    then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within S)"
lp15@56215
   857
      by (metis df has_field_derivative_def mult_commute_abs)
lp15@56215
   858
  next show " ((\<lambda>n. f n x) sums l)"
lp15@56215
   859
    by (rule sf)
lp15@68255
   860
  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
   861
      unfolding eventually_sequentially by (blast intro: **)
lp15@56215
   862
  qed
lp15@56215
   863
qed
lp15@56215
   864
eberlm@61531
   865
lp15@62534
   866
lemma field_differentiable_series:
immler@66252
   867
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach} \<Rightarrow> 'a"
lp15@68255
   868
  assumes "convex S" "open S"
lp15@68255
   869
  assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
lp15@68255
   870
  assumes "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
lp15@68255
   871
  assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)" and x: "x \<in> S"
lp15@68055
   872
  shows  "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)"
eberlm@61531
   873
proof -
lp15@68255
   874
  from assms(4) obtain g' where A: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
eberlm@61531
   875
    unfolding uniformly_convergent_on_def by blast
lp15@68255
   876
  from x and \<open>open S\<close> have S: "at x within S = at x" by (rule at_within_open)
lp15@68255
   877
  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
   878
    by (intro has_field_derivative_series[of S f f' g' x0] assms A has_field_derivative_at_within)
lp15@68255
   879
  then obtain g where g: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. f n x) sums g x"
lp15@68255
   880
    "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative g' x) (at x within S)" by blast
nipkow@67399
   881
  from g(2)[OF x] have g': "(g has_derivative ( * ) (g' x)) (at x)"
lp15@68255
   882
    by (simp add: has_field_derivative_def S)
nipkow@67399
   883
  have "((\<lambda>x. \<Sum>n. f n x) has_derivative ( * ) (g' x)) (at x)"
lp15@68255
   884
    by (rule has_derivative_transform_within_open[OF g' \<open>open S\<close> x])
eberlm@61531
   885
       (insert g, auto simp: sums_iff)
lp15@62534
   886
  thus "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)" unfolding differentiable_def
lp15@62534
   887
    by (auto simp: summable_def field_differentiable_def has_field_derivative_def)
eberlm@61531
   888
qed
eberlm@61531
   889
wenzelm@60420
   890
subsection\<open>Bound theorem\<close>
lp15@56215
   891
lp15@62534
   892
lemma field_differentiable_bound:
lp15@68255
   893
  fixes S :: "'a::real_normed_field set"
lp15@68255
   894
  assumes cvs: "convex S"
lp15@68255
   895
      and df:  "\<And>z. z \<in> S \<Longrightarrow> (f has_field_derivative f' z) (at z within S)"
lp15@68255
   896
      and dn:  "\<And>z. z \<in> S \<Longrightarrow> norm (f' z) \<le> B"
lp15@68255
   897
      and "x \<in> S"  "y \<in> S"
lp15@56215
   898
    shows "norm(f x - f y) \<le> B * norm(x - y)"
lp15@56215
   899
  apply (rule differentiable_bound [OF cvs])
lp15@68239
   900
  apply (erule df [unfolded has_field_derivative_def])
lp15@68239
   901
  apply (rule onorm_le, simp_all add: norm_mult mult_right_mono assms)
lp15@56215
   902
  done
lp15@56215
   903
lp15@62408
   904
subsection\<open>Inverse function theorem for complex derivatives\<close>
lp15@56215
   905
immler@66252
   906
lemma has_field_derivative_inverse_basic:
lp15@56215
   907
  shows "DERIV f (g y) :> f' \<Longrightarrow>
lp15@56215
   908
        f' \<noteq> 0 \<Longrightarrow>
lp15@56215
   909
        continuous (at y) g \<Longrightarrow>
lp15@56215
   910
        open t \<Longrightarrow>
lp15@56215
   911
        y \<in> t \<Longrightarrow>
lp15@56215
   912
        (\<And>z. z \<in> t \<Longrightarrow> f (g z) = z)
lp15@56215
   913
        \<Longrightarrow> DERIV g y :> inverse (f')"
lp15@56215
   914
  unfolding has_field_derivative_def
lp15@56215
   915
  apply (rule has_derivative_inverse_basic)
lp15@56215
   916
  apply (auto simp:  bounded_linear_mult_right)
lp15@56215
   917
  done
lp15@56215
   918
immler@66252
   919
lemma has_field_derivative_inverse_strong:
immler@66252
   920
  fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a"
lp15@56215
   921
  shows "DERIV f x :> f' \<Longrightarrow>
lp15@56215
   922
         f' \<noteq> 0 \<Longrightarrow>
lp15@68255
   923
         open S \<Longrightarrow>
lp15@68255
   924
         x \<in> S \<Longrightarrow>
lp15@68255
   925
         continuous_on S f \<Longrightarrow>
lp15@68255
   926
         (\<And>z. z \<in> S \<Longrightarrow> g (f z) = z)
lp15@56215
   927
         \<Longrightarrow> DERIV g (f x) :> inverse (f')"
lp15@56215
   928
  unfolding has_field_derivative_def
lp15@68255
   929
  apply (rule has_derivative_inverse_strong [of S x f g ])
lp15@56215
   930
  by auto
lp15@56215
   931
immler@66252
   932
lemma has_field_derivative_inverse_strong_x:
immler@66252
   933
  fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a"
lp15@56215
   934
  shows  "DERIV f (g y) :> f' \<Longrightarrow>
lp15@56215
   935
          f' \<noteq> 0 \<Longrightarrow>
lp15@68255
   936
          open S \<Longrightarrow>
lp15@68255
   937
          continuous_on S f \<Longrightarrow>
lp15@68255
   938
          g y \<in> S \<Longrightarrow> f(g y) = y \<Longrightarrow>
lp15@68255
   939
          (\<And>z. z \<in> S \<Longrightarrow> g (f z) = z)
lp15@56215
   940
          \<Longrightarrow> DERIV g y :> inverse (f')"
lp15@56215
   941
  unfolding has_field_derivative_def
lp15@68255
   942
  apply (rule has_derivative_inverse_strong_x [of S g y f])
lp15@56215
   943
  by auto
lp15@56215
   944
wenzelm@60420
   945
subsection \<open>Taylor on Complex Numbers\<close>
lp15@56215
   946
nipkow@64267
   947
lemma sum_Suc_reindex:
lp15@56215
   948
  fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@64267
   949
    shows  "sum f {0..n} = f 0 - f (Suc n) + sum (\<lambda>i. f (Suc i)) {0..n}"
lp15@56215
   950
by (induct n) auto
lp15@56215
   951
immler@66252
   952
lemma field_taylor:
lp15@68255
   953
  assumes S: "convex S"
lp15@68255
   954
      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
   955
      and B: "\<And>x. x \<in> S \<Longrightarrow> norm (f (Suc n) x) \<le> B"
lp15@68255
   956
      and w: "w \<in> S"
lp15@68255
   957
      and z: "z \<in> S"
immler@66252
   958
    shows "norm(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i)))
immler@66252
   959
          \<le> B * norm(z - w)^(Suc n) / fact n"
lp15@56215
   960
proof -
lp15@68255
   961
  have wzs: "closed_segment w z \<subseteq> S" using assms
lp15@56215
   962
    by (metis convex_contains_segment)
lp15@56215
   963
  { fix u
lp15@56215
   964
    assume "u \<in> closed_segment w z"
lp15@68255
   965
    then have "u \<in> S"
lp15@56215
   966
      by (metis wzs subsetD)
lp15@59730
   967
    have "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / (fact i) +
lp15@61609
   968
                      f (Suc i) u * (z-u)^i / (fact i)) =
lp15@59730
   969
              f (Suc n) u * (z-u) ^ n / (fact n)"
lp15@56215
   970
    proof (induction n)
lp15@56215
   971
      case 0 show ?case by simp
lp15@56215
   972
    next
lp15@56215
   973
      case (Suc n)
lp15@59730
   974
      have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / (fact i) +
lp15@61609
   975
                             f (Suc i) u * (z-u) ^ i / (fact i)) =
lp15@59730
   976
           f (Suc n) u * (z-u) ^ n / (fact n) +
lp15@59730
   977
           f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / (fact (Suc n)) -
lp15@59730
   978
           f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / (fact (Suc n))"
hoelzl@56479
   979
        using Suc by simp
lp15@59730
   980
      also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n))"
lp15@56215
   981
      proof -
lp15@59730
   982
        have "(fact(Suc n)) *
lp15@59730
   983
             (f(Suc n) u *(z-u) ^ n / (fact n) +
lp15@59730
   984
               f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / (fact(Suc n)) -
lp15@59730
   985
               f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / (fact(Suc n))) =
lp15@59730
   986
            ((fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / (fact n) +
lp15@59730
   987
            ((fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / (fact(Suc n))) -
lp15@59730
   988
            ((fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / (fact(Suc n))"
haftmann@63367
   989
          by (simp add: algebra_simps del: fact_Suc)
lp15@59730
   990
        also have "... = ((fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / (fact n) +
lp15@59730
   991
                         (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
lp15@59730
   992
                         (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
haftmann@63367
   993
          by (simp del: fact_Suc)
lp15@59730
   994
        also have "... = (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) +
lp15@59730
   995
                         (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
lp15@59730
   996
                         (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
haftmann@63367
   997
          by (simp only: fact_Suc of_nat_mult ac_simps) simp
lp15@56215
   998
        also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
lp15@56215
   999
          by (simp add: algebra_simps)
lp15@56215
  1000
        finally show ?thesis
haftmann@63367
  1001
        by (simp add: mult_left_cancel [where c = "(fact (Suc n))", THEN iffD1] del: fact_Suc)
lp15@56215
  1002
      qed
lp15@56215
  1003
      finally show ?case .
lp15@56215
  1004
    qed
lp15@61609
  1005
    then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / (fact i)))
lp15@59730
  1006
                has_field_derivative f (Suc n) u * (z-u) ^ n / (fact n))
lp15@68255
  1007
               (at u within S)"
hoelzl@56381
  1008
      apply (intro derivative_eq_intros)
lp15@68255
  1009
      apply (blast intro: assms \<open>u \<in> S\<close>)
lp15@56215
  1010
      apply (rule refl)+
lp15@56215
  1011
      apply (auto simp: field_simps)
lp15@56215
  1012
      done
lp15@56215
  1013
  } note sum_deriv = this
lp15@56215
  1014
  { fix u
lp15@56215
  1015
    assume u: "u \<in> closed_segment w z"
lp15@68255
  1016
    then have us: "u \<in> S"
lp15@56215
  1017
      by (metis wzs subsetD)
immler@66252
  1018
    have "norm (f (Suc n) u) * norm (z - u) ^ n \<le> norm (f (Suc n) u) * norm (u - z) ^ n"
lp15@56215
  1019
      by (metis norm_minus_commute order_refl)
immler@66252
  1020
    also have "... \<le> norm (f (Suc n) u) * norm (z - w) ^ n"
lp15@56215
  1021
      by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
immler@66252
  1022
    also have "... \<le> B * norm (z - w) ^ n"
lp15@56215
  1023
      by (metis norm_ge_zero zero_le_power mult_right_mono  B [OF us])
immler@66252
  1024
    finally have "norm (f (Suc n) u) * norm (z - u) ^ n \<le> B * norm (z - w) ^ n" .
lp15@56215
  1025
  } note cmod_bound = this
lp15@59730
  1026
  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
  1027
    by simp
lp15@59730
  1028
  also have "\<dots> = f 0 z / (fact 0)"
nipkow@64267
  1029
    by (subst sum_zero_power) simp
immler@66252
  1030
  finally have "norm (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)))
immler@66252
  1031
                \<le> norm ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) -
lp15@59730
  1032
                        (\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))"
lp15@56215
  1033
    by (simp add: norm_minus_commute)
immler@66252
  1034
  also have "... \<le> B * norm (z - w) ^ n / (fact n) * norm (w - z)"
lp15@62534
  1035
    apply (rule field_differentiable_bound
lp15@59730
  1036
      [where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / (fact n)"
lp15@68255
  1037
         and S = "closed_segment w z", OF convex_closed_segment])
lp15@61609
  1038
    apply (auto simp: ends_in_segment DERIV_subset [OF sum_deriv wzs]
lp15@56215
  1039
                  norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
lp15@56215
  1040
    done
immler@66252
  1041
  also have "...  \<le> B * norm (z - w) ^ Suc n / (fact n)"
lp15@61609
  1042
    by (simp add: algebra_simps norm_minus_commute)
lp15@56215
  1043
  finally show ?thesis .
lp15@56215
  1044
qed
lp15@56215
  1045
immler@66252
  1046
lemma complex_taylor:
lp15@68255
  1047
  assumes S: "convex S"
lp15@68255
  1048
      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
  1049
      and B: "\<And>x. x \<in> S \<Longrightarrow> cmod (f (Suc n) x) \<le> B"
lp15@68255
  1050
      and w: "w \<in> S"
lp15@68255
  1051
      and z: "z \<in> S"
immler@66252
  1052
    shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i)))
immler@66252
  1053
          \<le> B * cmod(z - w)^(Suc n) / fact n"
immler@66252
  1054
  using assms by (rule field_taylor)
immler@66252
  1055
immler@66252
  1056
lp15@62408
  1057
text\<open>Something more like the traditional MVT for real components\<close>
hoelzl@56370
  1058
lp15@56238
  1059
lemma complex_mvt_line:
hoelzl@56369
  1060
  assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f'(u)) (at u)"
paulson@61518
  1061
    shows "\<exists>u. u \<in> closed_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
lp15@56238
  1062
proof -
lp15@56238
  1063
  have twz: "\<And>t. (1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)"
lp15@56238
  1064
    by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
hoelzl@56381
  1065
  note assms[unfolded has_field_derivative_def, derivative_intros]
lp15@56238
  1066
  show ?thesis
lp15@56238
  1067
    apply (cut_tac mvt_simple
lp15@56238
  1068
                     [of 0 1 "Re o f o (\<lambda>t. (1 - t) *\<^sub>R w +  t *\<^sub>R z)"
lp15@56238
  1069
                      "\<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
  1070
    apply auto
lp15@56238
  1071
    apply (rule_tac x="(1 - x) *\<^sub>R w + x *\<^sub>R z" in exI)
paulson@61518
  1072
    apply (auto simp: closed_segment_def twz) []
lp15@67979
  1073
    apply (intro derivative_eq_intros has_derivative_at_withinI, simp_all)
hoelzl@56369
  1074
    apply (simp add: fun_eq_iff real_vector.scale_right_diff_distrib)
paulson@61518
  1075
    apply (force simp: twz closed_segment_def)
lp15@56238
  1076
    done
lp15@56238
  1077
qed
lp15@56238
  1078
lp15@56238
  1079
lemma complex_taylor_mvt:
lp15@56238
  1080
  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
  1081
    shows "\<exists>u. u \<in> closed_segment w z \<and>
lp15@56238
  1082
            Re (f 0 z) =
lp15@59730
  1083
            Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / (fact i)) +
lp15@59730
  1084
                (f (Suc n) u * (z-u)^n / (fact n)) * (z - w))"
lp15@56238
  1085
proof -
lp15@56238
  1086
  { fix u
lp15@56238
  1087
    assume u: "u \<in> closed_segment w z"
lp15@56238
  1088
    have "(\<Sum>i = 0..n.
lp15@56238
  1089
               (f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) /
lp15@59730
  1090
               (fact i)) =
lp15@56238
  1091
          f (Suc 0) u -
lp15@56238
  1092
             (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
lp15@59730
  1093
             (fact (Suc n)) +
lp15@56238
  1094
             (\<Sum>i = 0..n.
lp15@56238
  1095
                 (f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) /
lp15@59730
  1096
                 (fact (Suc i)))"
nipkow@64267
  1097
       by (subst sum_Suc_reindex) simp
lp15@56238
  1098
    also have "... = f (Suc 0) u -
lp15@56238
  1099
             (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
lp15@59730
  1100
             (fact (Suc n)) +
lp15@56238
  1101
             (\<Sum>i = 0..n.
lp15@61609
  1102
                 f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i))  -
lp15@59730
  1103
                 f (Suc i) u * (z-u) ^ i / (fact i))"
haftmann@57514
  1104
      by (simp only: diff_divide_distrib fact_cancel ac_simps)
lp15@56238
  1105
    also have "... = f (Suc 0) u -
lp15@56238
  1106
             (f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) /
lp15@59730
  1107
             (fact (Suc n)) +
lp15@59730
  1108
             f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n)) - f (Suc 0) u"
nipkow@64267
  1109
      by (subst sum_Suc_diff) auto
lp15@59730
  1110
    also have "... = f (Suc n) u * (z-u) ^ n / (fact n)"
lp15@56238
  1111
      by (simp only: algebra_simps diff_divide_distrib fact_cancel)
lp15@61609
  1112
    finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i
lp15@59730
  1113
                             - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / (fact i)) =
lp15@59730
  1114
                  f (Suc n) u * (z - u) ^ n / (fact n)" .
lp15@59730
  1115
    then have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / (fact i)) has_field_derivative
lp15@59730
  1116
                f (Suc n) u * (z - u) ^ n / (fact n))  (at u)"
hoelzl@56381
  1117
      apply (intro derivative_eq_intros)+
lp15@56238
  1118
      apply (force intro: u assms)
lp15@56238
  1119
      apply (rule refl)+
haftmann@57514
  1120
      apply (auto simp: ac_simps)
lp15@56238
  1121
      done
lp15@56238
  1122
  }
lp15@56238
  1123
  then show ?thesis
lp15@59730
  1124
    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
  1125
               "\<lambda>u. (f (Suc n) u * (z-u)^n / (fact n))"])
lp15@56238
  1126
    apply (auto simp add: intro: open_closed_segment)
lp15@56238
  1127
    done
lp15@56238
  1128
qed
lp15@56238
  1129
lp15@60017
  1130
wenzelm@60420
  1131
subsection \<open>Polynomal function extremal theorem, from HOL Light\<close>
lp15@60017
  1132
lp15@60017
  1133
lemma polyfun_extremal_lemma: (*COMPLEX_POLYFUN_EXTREMAL_LEMMA in HOL Light*)
lp15@60017
  1134
    fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
lp15@60017
  1135
  assumes "0 < e"
lp15@60017
  1136
    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
  1137
proof (induct n)
lp15@60017
  1138
  case 0 with assms
lp15@60017
  1139
  show ?case
lp15@60017
  1140
    apply (rule_tac x="norm (c 0) / e" in exI)
lp15@60017
  1141
    apply (auto simp: field_simps)
lp15@60017
  1142
    done
lp15@60017
  1143
next
lp15@60017
  1144
  case (Suc n)
lp15@60017
  1145
  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
  1146
    using Suc assms by blast
lp15@60017
  1147
  show ?case
lp15@60017
  1148
  proof (rule exI [where x= "max M (1 + norm(c(Suc n)) / e)"], clarsimp simp del: power_Suc)
lp15@60017
  1149
    fix z::'a
lp15@60017
  1150
    assume z1: "M \<le> norm z" and "1 + norm (c (Suc n)) / e \<le> norm z"
lp15@60017
  1151
    then have z2: "e + norm (c (Suc n)) \<le> e * norm z"
lp15@60017
  1152
      using assms by (simp add: field_simps)
lp15@60017
  1153
    have "norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
lp15@60017
  1154
      using M [OF z1] by simp
lp15@60017
  1155
    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
  1156
      by simp
lp15@60017
  1157
    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
  1158
      by (blast intro: norm_triangle_le elim: )
lp15@60017
  1159
    also have "... \<le> (e + norm (c (Suc n))) * norm z ^ Suc n"
lp15@60017
  1160
      by (simp add: norm_power norm_mult algebra_simps)
lp15@60017
  1161
    also have "... \<le> (e * norm z) * norm z ^ Suc n"
lp15@60017
  1162
      by (metis z2 mult.commute mult_left_mono norm_ge_zero norm_power)
lp15@60017
  1163
    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
  1164
      by simp
lp15@60017
  1165
  qed
lp15@60017
  1166
qed
lp15@60017
  1167
lp15@60017
  1168
lemma polyfun_extremal: (*COMPLEX_POLYFUN_EXTREMAL in HOL Light*)
lp15@60017
  1169
    fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
lp15@60017
  1170
  assumes k: "c k \<noteq> 0" "1\<le>k" and kn: "k\<le>n"
lp15@60017
  1171
    shows "eventually (\<lambda>z. norm (\<Sum>i\<le>n. c(i) * z^i) \<ge> B) at_infinity"
lp15@60017
  1172
using kn
lp15@60017
  1173
proof (induction n)
lp15@60017
  1174
  case 0
lp15@60017
  1175
  then show ?case
lp15@60017
  1176
    using k  by simp
lp15@60017
  1177
next
lp15@60017
  1178
  case (Suc m)
lp15@60017
  1179
  let ?even = ?case
lp15@60017
  1180
  show ?even
lp15@60017
  1181
  proof (cases "c (Suc m) = 0")
lp15@60017
  1182
    case True
lp15@60017
  1183
    then show ?even using Suc k
lp15@60017
  1184
      by auto (metis antisym_conv less_eq_Suc_le not_le)
lp15@60017
  1185
  next
lp15@60017
  1186
    case False
lp15@60017
  1187
    then obtain M where M:
lp15@60017
  1188
          "\<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
  1189
      using polyfun_extremal_lemma [of "norm(c (Suc m)) / 2" c m] Suc
lp15@60017
  1190
      by auto
lp15@60017
  1191
    have "\<exists>b. \<forall>z. b \<le> norm z \<longrightarrow> B \<le> norm (\<Sum>i\<le>Suc m. c i * z^i)"
lp15@60017
  1192
    proof (rule exI [where x="max M (max 1 (\<bar>B\<bar> / (norm(c (Suc m)) / 2)))"], clarsimp simp del: power_Suc)
lp15@60017
  1193
      fix z::'a
lp15@60017
  1194
      assume z1: "M \<le> norm z" "1 \<le> norm z"
lp15@60017
  1195
         and "\<bar>B\<bar> * 2 / norm (c (Suc m)) \<le> norm z"
lp15@60017
  1196
      then have z2: "\<bar>B\<bar> \<le> norm (c (Suc m)) * norm z / 2"
lp15@60017
  1197
        using False by (simp add: field_simps)
lp15@60017
  1198
      have nz: "norm z \<le> norm z ^ Suc m"
wenzelm@60420
  1199
        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
  1200
      have *: "\<And>y x. norm (c (Suc m)) * norm z / 2 \<le> norm y - norm x \<Longrightarrow> B \<le> norm (x + y)"
lp15@60017
  1201
        by (metis abs_le_iff add.commute norm_diff_ineq order_trans z2)
lp15@60017
  1202
      have "norm z * norm (c (Suc m)) + 2 * norm (\<Sum>i\<le>m. c i * z^i)
lp15@60017
  1203
            \<le> norm (c (Suc m)) * norm z + norm (c (Suc m)) * norm z ^ Suc m"
lp15@60017
  1204
        using M [of z] Suc z1  by auto
lp15@60017
  1205
      also have "... \<le> 2 * (norm (c (Suc m)) * norm z ^ Suc m)"
lp15@60017
  1206
        using nz by (simp add: mult_mono del: power_Suc)
lp15@60017
  1207
      finally show "B \<le> norm ((\<Sum>i\<le>m. c i * z^i) + c (Suc m) * z ^ Suc m)"
lp15@60017
  1208
        using Suc.IH
lp15@60017
  1209
        apply (auto simp: eventually_at_infinity)
lp15@60017
  1210
        apply (rule *)
lp15@60017
  1211
        apply (simp add: field_simps norm_mult norm_power)
lp15@60017
  1212
        done
lp15@60017
  1213
    qed
lp15@60017
  1214
    then show ?even
lp15@60017
  1215
      by (simp add: eventually_at_infinity)
lp15@60017
  1216
  qed
lp15@60017
  1217
qed
lp15@60017
  1218
lp15@56215
  1219
end