src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy
author paulson <lp15@cam.ac.uk>
Fri Mar 21 15:36:00 2014 +0000 (2014-03-21)
changeset 56238 5d147e1e18d1
parent 56223 7696903b9e61
child 56261 918432e3fcfa
permissions -rw-r--r--
a few new lemmas and generalisations of old ones
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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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header {* Complex Analysis Basics *}
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theory Complex_Analysis_Basics
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imports  "~~/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space"
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begin
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subsection {*Complex number lemmas *}
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lemma abs_sqrt_wlog:
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  fixes x::"'a::linordered_idom"
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  assumes "!!x::'a. x\<ge>0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"
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by (metis abs_ge_zero assms power2_abs)
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lemma complex_abs_le_norm: "abs(Re z) + abs(Im z) \<le> sqrt(2) * norm z"
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proof (cases z)
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  case (Complex x y)
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  show ?thesis
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    apply (rule power2_le_imp_le)
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    apply (auto simp: real_sqrt_mult [symmetric] Complex)
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    apply (rule abs_sqrt_wlog [where x=x])
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    apply (rule abs_sqrt_wlog [where x=y])
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    apply (simp add: power2_sum add_commute sum_squares_bound)
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    done
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qed
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lemma continuous_Re: "continuous_on UNIV Re"
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  by (metis (poly_guards_query) bounded_linear.continuous_on bounded_linear_Re 
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            continuous_on_cong continuous_on_id)
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lemma continuous_Im: "continuous_on UNIV Im"
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  by (metis (poly_guards_query) bounded_linear.continuous_on bounded_linear_Im 
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            continuous_on_cong continuous_on_id)
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lemma open_closed_segment: "u \<in> open_segment w z \<Longrightarrow> u \<in> closed_segment w z"
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  by (auto simp add: closed_segment_def open_segment_def)
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lemma has_derivative_Re [has_derivative_intros] : "(Re has_derivative Re) F"
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  by (auto simp add: has_derivative_def bounded_linear_Re)
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lemma has_derivative_Im [has_derivative_intros] : "(Im has_derivative Im) F"
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  by (auto simp add: has_derivative_def bounded_linear_Im)
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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 / of_nat (fact (Suc n)) = c / of_nat (fact n)"
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  apply (subst frac_eq_eq [OF of_nat_fact_not_zero of_nat_fact_not_zero])
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  apply (simp add: algebra_simps of_nat_mult)
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  done
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lemma open_halfspace_Re_lt: "open {z. Re(z) < b}"
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proof -
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  have "{z. Re(z) < b} = Re -`{..<b}"
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    by blast
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  then show ?thesis
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    by (auto simp: continuous_Re continuous_imp_open_vimage [of UNIV])
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qed
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lemma open_halfspace_Re_gt: "open {z. Re(z) > b}"
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proof -
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  have "{z. Re(z) > b} = Re -`{b<..}"
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    by blast
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  then show ?thesis
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    by (auto simp: continuous_Re continuous_imp_open_vimage [of UNIV])
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qed
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lemma closed_halfspace_Re_ge: "closed {z. Re(z) \<ge> b}"
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proof -
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  have "{z. Re(z) \<ge> b} = - {z. Re(z) < b}"
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    by auto
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  then show ?thesis
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    by (simp add: closed_def open_halfspace_Re_lt)
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qed
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lemma closed_halfspace_Re_le: "closed {z. Re(z) \<le> b}"
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proof -
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  have "{z. Re(z) \<le> b} = - {z. Re(z) > b}"
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    by auto
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  then show ?thesis
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    by (simp add: closed_def open_halfspace_Re_gt)
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qed
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lemma closed_halfspace_Re_eq: "closed {z. Re(z) = b}"
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proof -
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  have "{z. Re(z) = b} = {z. Re(z) \<le> b} \<inter> {z. Re(z) \<ge> b}"
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    by auto
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  then show ?thesis
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    by (auto simp: closed_Int closed_halfspace_Re_le closed_halfspace_Re_ge)
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qed
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lemma open_halfspace_Im_lt: "open {z. Im(z) < b}"
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proof -
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  have "{z. Im(z) < b} = Im -`{..<b}"
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    by blast
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  then show ?thesis
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    by (auto simp: continuous_Im continuous_imp_open_vimage [of UNIV])
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qed
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lemma open_halfspace_Im_gt: "open {z. Im(z) > b}"
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proof -
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  have "{z. Im(z) > b} = Im -`{b<..}"
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    by blast
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  then show ?thesis
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    by (auto simp: continuous_Im continuous_imp_open_vimage [of UNIV])
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qed
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lemma closed_halfspace_Im_ge: "closed {z. Im(z) \<ge> b}"
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proof -
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  have "{z. Im(z) \<ge> b} = - {z. Im(z) < b}"
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    by auto
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  then show ?thesis
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    by (simp add: closed_def open_halfspace_Im_lt)
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qed
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lemma closed_halfspace_Im_le: "closed {z. Im(z) \<le> b}"
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proof -
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  have "{z. Im(z) \<le> b} = - {z. Im(z) > b}"
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    by auto
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  then show ?thesis
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    by (simp add: closed_def open_halfspace_Im_gt)
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qed
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lemma closed_halfspace_Im_eq: "closed {z. Im(z) = b}"
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proof -
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  have "{z. Im(z) = b} = {z. Im(z) \<le> b} \<inter> {z. Im(z) \<ge> b}"
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    by auto
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  then show ?thesis
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    by (auto simp: closed_Int closed_halfspace_Im_le closed_halfspace_Im_ge)
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qed
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lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
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  by (metis Complex_in_Reals Im_complex_of_real Reals_cases complex_surj)
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lemma closed_complex_Reals: "closed (Reals :: complex set)"
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proof -
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  have "-(Reals :: complex set) = {z. Im(z) < 0} \<union> {z. 0 < Im(z)}"
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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_def open_Un open_halfspace_Im_gt open_halfspace_Im_lt)
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qed
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lemma linear_times:
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  fixes c::"'a::{real_algebra,real_vector}" shows "linear (\<lambda>x. c * x)"
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  by (auto simp: linearI distrib_left)
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lemma bilinear_times:
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  fixes c::"'a::{real_algebra,real_vector}" shows "bilinear (\<lambda>x y::'a. x*y)"
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  unfolding bilinear_def
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  by (auto simp: distrib_left distrib_right intro!: linearI)
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lemma linear_cnj: "linear cnj"
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  by (auto simp: linearI cnj_def)
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lemma tendsto_mult_left:
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  fixes c::"'a::real_normed_algebra" 
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  shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) ---> c * l) F"
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by (rule tendsto_mult [OF tendsto_const])
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lemma tendsto_mult_right:
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  fixes c::"'a::real_normed_algebra" 
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  shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) ---> l * c) F"
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by (rule tendsto_mult [OF _ tendsto_const])
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lemma tendsto_Re_upper:
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  assumes "~ (trivial_limit F)" 
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          "(f ---> l) F" 
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          "eventually (\<lambda>x. Re(f x) \<le> b) F"
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    shows  "Re(l) \<le> b"
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  by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Re)
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lemma tendsto_Re_lower:
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  assumes "~ (trivial_limit F)" 
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          "(f ---> l) F" 
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          "eventually (\<lambda>x. b \<le> Re(f x)) F"
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    shows  "b \<le> Re(l)"
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  by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Re)
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lemma tendsto_Im_upper:
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  assumes "~ (trivial_limit F)" 
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          "(f ---> l) F" 
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          "eventually (\<lambda>x. Im(f x) \<le> b) F"
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    shows  "Im(l) \<le> b"
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  by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Im)
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lemma tendsto_Im_lower:
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  assumes "~ (trivial_limit F)" 
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          "(f ---> l) F" 
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          "eventually (\<lambda>x. b \<le> Im(f x)) F"
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    shows  "b \<le> Im(l)"
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  by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Im)
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subsection{*General lemmas*}
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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_on_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. f x * c)"
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proof (cases "c=0")
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  case True then show ?thesis
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    by (simp add: uniformly_continuous_on_const)
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next
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  case False show ?thesis
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    apply (rule bounded_linear.uniformly_continuous_on)
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    apply (metis bounded_linear_ident)
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    using assms
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    apply (auto simp: uniformly_continuous_on_def dist_norm)
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    apply (drule_tac x = "e / norm c" in spec, auto)
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    apply (metis divide_pos_pos zero_less_norm_iff False)
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    apply (rule_tac x=d in exI, auto)
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    apply (drule_tac x = x in bspec, assumption)
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    apply (drule_tac x = "x'" in bspec)
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    apply (auto simp: False less_divide_eq)
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    by (metis dual_order.strict_trans2 left_diff_distrib norm_mult_ineq)
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qed
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lemma uniformly_continuous_on_cmul_left[continuous_on_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 (metis continuous_on_eq continuous_on_id continuous_on_norm)
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subsection{*DERIV stuff*}
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(*move some premises to a sensible order. Use more \<And> symbols.*)
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lemma DERIV_continuous_on: "\<lbrakk>\<And>x. x \<in> s \<Longrightarrow> DERIV f x :> D\<rbrakk> \<Longrightarrow> continuous_on s f"
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  by (metis DERIV_continuous continuous_at_imp_continuous_on)
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lemma DERIV_subset: 
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  "(f has_field_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s 
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   \<Longrightarrow> (f has_field_derivative f') (at x within t)"
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  by (simp add: has_field_derivative_def has_derivative_within_subset)
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lemma lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = op * 0"
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  by auto
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lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = op * 1"
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  by auto
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lemma has_derivative_zero_constant:
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  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
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  assumes "convex s"
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      and d0: "\<And>x. x\<in>s \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within s)"
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    shows "\<exists>c. \<forall>x\<in>s. f x = c"
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proof (cases "s={}")
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  case False
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  then obtain x where "x \<in> s"
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    by auto
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  have d0': "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
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    by (metis d0)
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  have "\<And>y. y \<in> s \<Longrightarrow> f x = f y"
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  proof -
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    case goal1
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    then show ?case
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      using differentiable_bound[OF assms(1) d0', of 0 x y] and `x \<in> s`
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      unfolding onorm_zero
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      by auto
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  qed
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  then show ?thesis 
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    by metis
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next
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  case True
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  then show ?thesis by auto
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qed
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lemma has_derivative_zero_unique:
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  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
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  assumes "convex s"
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      and "\<And>x. x\<in>s \<Longrightarrow> (f has_derivative (\<lambda>h. 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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  using assms
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  by (metis has_derivative_zero_constant)
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lemma has_derivative_zero_connected_constant:
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  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
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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). (f has_derivative (\<lambda>h. 0)) (at x within s)"
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    obtains c where "\<And>x. x \<in> s \<Longrightarrow> f(x) = c"
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proof (cases "s = {}")
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  case True
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  then show ?thesis
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by (metis empty_iff that)
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next
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  case False
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  then obtain c where "c \<in> s"
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    by (metis equals0I)
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  then show ?thesis
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    by (metis has_derivative_zero_unique_strong_connected assms that)
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qed
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lemma DERIV_zero_connected_constant:
lp15@56215
   329
  fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
lp15@56215
   330
  assumes "connected s"
lp15@56215
   331
      and "open s"
lp15@56215
   332
      and "finite k"
lp15@56215
   333
      and "continuous_on s f"
lp15@56215
   334
      and "\<forall>x\<in>(s - k). DERIV f x :> 0"
lp15@56215
   335
    obtains c where "\<And>x. x \<in> s \<Longrightarrow> f(x) = c"
lp15@56215
   336
using has_derivative_zero_connected_constant [OF assms(1-4)] assms
lp15@56215
   337
by (metis DERIV_const Derivative.has_derivative_const Diff_iff at_within_open 
lp15@56215
   338
          frechet_derivative_at has_field_derivative_def)
lp15@56215
   339
lp15@56215
   340
lemma DERIV_zero_constant:
lp15@56215
   341
  fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
lp15@56215
   342
  shows    "\<lbrakk>convex s;
lp15@56215
   343
             \<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)\<rbrakk> 
lp15@56215
   344
             \<Longrightarrow> \<exists>c. \<forall>x \<in> s. f(x) = c"
lp15@56215
   345
  unfolding has_field_derivative_def
lp15@56215
   346
  by (auto simp: lambda_zero intro: has_derivative_zero_constant)
lp15@56215
   347
lp15@56215
   348
lemma DERIV_zero_unique:
lp15@56215
   349
  fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
lp15@56215
   350
  assumes "convex s"
lp15@56215
   351
      and d0: "\<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)"
lp15@56215
   352
      and "a \<in> s"
lp15@56215
   353
      and "x \<in> s"
lp15@56215
   354
    shows "f x = f a"
lp15@56215
   355
apply (rule has_derivative_zero_unique [where f=f, OF assms(1) _ assms(3,4)])
lp15@56215
   356
by (metis d0 has_field_derivative_imp_has_derivative lambda_zero)
lp15@56215
   357
lp15@56215
   358
lemma DERIV_zero_connected_unique:
lp15@56215
   359
  fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
lp15@56215
   360
  assumes "connected s"
lp15@56215
   361
      and "open s"
lp15@56215
   362
      and d0: "\<And>x. x\<in>s \<Longrightarrow> DERIV f x :> 0"
lp15@56215
   363
      and "a \<in> s"
lp15@56215
   364
      and "x \<in> s"
lp15@56215
   365
    shows "f x = f a" 
lp15@56215
   366
    apply (rule Integration.has_derivative_zero_unique_strong_connected [of s "{}" f])
lp15@56215
   367
    using assms
lp15@56215
   368
    apply auto
lp15@56215
   369
    apply (metis DERIV_continuous_on)
lp15@56215
   370
  by (metis at_within_open has_field_derivative_def lambda_zero)
lp15@56215
   371
lp15@56215
   372
lemma DERIV_transform_within:
lp15@56215
   373
  assumes "(f has_field_derivative f') (at a within s)"
lp15@56215
   374
      and "0 < d" "a \<in> s"
lp15@56215
   375
      and "\<And>x. x\<in>s \<Longrightarrow> dist x a < d \<Longrightarrow> f x = g x"
lp15@56215
   376
    shows "(g has_field_derivative f') (at a within s)"
lp15@56215
   377
  using assms unfolding has_field_derivative_def
lp15@56215
   378
  by (blast intro: Derivative.has_derivative_transform_within)
lp15@56215
   379
lp15@56215
   380
lemma DERIV_transform_within_open:
lp15@56215
   381
  assumes "DERIV f a :> f'"
lp15@56215
   382
      and "open s" "a \<in> s"
lp15@56215
   383
      and "\<And>x. x\<in>s \<Longrightarrow> f x = g x"
lp15@56215
   384
    shows "DERIV g a :> f'"
lp15@56215
   385
  using assms unfolding has_field_derivative_def
lp15@56215
   386
by (metis has_derivative_transform_within_open)
lp15@56215
   387
lp15@56215
   388
lemma DERIV_transform_at:
lp15@56215
   389
  assumes "DERIV f a :> f'"
lp15@56215
   390
      and "0 < d"
lp15@56215
   391
      and "\<And>x. dist x a < d \<Longrightarrow> f x = g x"
lp15@56215
   392
    shows "DERIV g a :> f'"
lp15@56215
   393
  by (blast intro: assms DERIV_transform_within)
lp15@56215
   394
lp15@56215
   395
lp15@56215
   396
subsection{*Holomorphic functions*}
lp15@56215
   397
lp15@56215
   398
lemma has_derivative_ident[has_derivative_intros, simp]: 
lp15@56215
   399
     "FDERIV complex_of_real x :> complex_of_real"
lp15@56215
   400
  by (simp add: has_derivative_def tendsto_const bounded_linear_of_real)
lp15@56215
   401
lp15@56215
   402
lemma has_real_derivative:
lp15@56215
   403
  fixes f :: "real\<Rightarrow>real" 
lp15@56215
   404
  assumes "(f has_derivative f') F"
lp15@56215
   405
    obtains c where "(f has_derivative (\<lambda>x. x * c)) F"
lp15@56215
   406
proof -
lp15@56215
   407
  obtain c where "f' = (\<lambda>x. x * c)"
lp15@56215
   408
    by (metis assms derivative_linear real_bounded_linear)
lp15@56215
   409
  then show ?thesis
lp15@56215
   410
    by (metis assms that)
lp15@56215
   411
qed
lp15@56215
   412
lp15@56215
   413
lemma has_real_derivative_iff:
lp15@56215
   414
  fixes f :: "real\<Rightarrow>real" 
lp15@56215
   415
  shows "(\<exists>f'. (f has_derivative (\<lambda>x. x * f')) F) = (\<exists>D. (f has_derivative D) F)"
lp15@56215
   416
by (auto elim: has_real_derivative)
lp15@56215
   417
lp15@56215
   418
definition complex_differentiable :: "[complex \<Rightarrow> complex, complex filter] \<Rightarrow> bool"
lp15@56215
   419
           (infixr "(complex'_differentiable)" 50)  
lp15@56215
   420
  where "f complex_differentiable F \<equiv> \<exists>f'. (f has_field_derivative f') F"
lp15@56215
   421
lp15@56215
   422
definition DD :: "['a \<Rightarrow> 'a::real_normed_field, 'a] \<Rightarrow> 'a" --{*for real, complex?*}
lp15@56215
   423
  where "DD f x \<equiv> THE f'. (f has_derivative (\<lambda>x. x * f')) (at x)"
lp15@56215
   424
lp15@56215
   425
definition holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
lp15@56215
   426
           (infixl "(holomorphic'_on)" 50)
lp15@56215
   427
  where "f holomorphic_on s \<equiv> \<forall>x \<in> s. \<exists>f'. (f has_field_derivative f') (at x within s)"
lp15@56215
   428
  
lp15@56215
   429
lemma holomorphic_on_empty: "f holomorphic_on {}"
lp15@56215
   430
  by (simp add: holomorphic_on_def)
lp15@56215
   431
lp15@56215
   432
lemma holomorphic_on_differentiable:
lp15@56215
   433
     "f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. f complex_differentiable (at x within s))"
lp15@56215
   434
unfolding holomorphic_on_def complex_differentiable_def has_field_derivative_def
lp15@56215
   435
by (metis mult_commute_abs)
lp15@56215
   436
lp15@56215
   437
lemma holomorphic_on_open:
lp15@56215
   438
    "open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> f')"
lp15@56215
   439
  by (auto simp: holomorphic_on_def has_field_derivative_def at_within_open [of _ s])
lp15@56215
   440
lp15@56215
   441
lemma complex_differentiable_imp_continuous_at: 
lp15@56215
   442
    "f complex_differentiable (at x) \<Longrightarrow> continuous (at x) f"
lp15@56215
   443
  by (metis DERIV_continuous complex_differentiable_def)
lp15@56215
   444
lp15@56215
   445
lemma holomorphic_on_imp_continuous_on: 
lp15@56215
   446
    "f holomorphic_on s \<Longrightarrow> continuous_on s f"
lp15@56215
   447
  by (metis DERIV_continuous continuous_on_eq_continuous_within holomorphic_on_def) 
lp15@56215
   448
lp15@56215
   449
lemma has_derivative_within_open:
lp15@56215
   450
  "a \<in> s \<Longrightarrow> open s \<Longrightarrow> (f has_field_derivative f') (at a within s) \<longleftrightarrow> DERIV f a :> f'"
lp15@56215
   451
  by (simp add: has_field_derivative_def) (metis has_derivative_within_open)
lp15@56215
   452
lp15@56215
   453
lemma holomorphic_on_subset:
lp15@56215
   454
    "f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t"
lp15@56215
   455
  unfolding holomorphic_on_def
lp15@56215
   456
  by (metis DERIV_subset subsetD)
lp15@56215
   457
lp15@56215
   458
lemma complex_differentiable_within_subset:
lp15@56215
   459
    "\<lbrakk>f complex_differentiable (at x within s); t \<subseteq> s\<rbrakk>
lp15@56215
   460
     \<Longrightarrow> f complex_differentiable (at x within t)"
lp15@56215
   461
  unfolding complex_differentiable_def
lp15@56215
   462
  by (metis DERIV_subset)
lp15@56215
   463
lp15@56215
   464
lemma complex_differentiable_at_within:
lp15@56215
   465
    "\<lbrakk>f complex_differentiable (at x)\<rbrakk>
lp15@56215
   466
     \<Longrightarrow> f complex_differentiable (at x within s)"
lp15@56215
   467
  unfolding complex_differentiable_def
lp15@56215
   468
  by (metis DERIV_subset top_greatest)
lp15@56215
   469
lp15@56215
   470
lp15@56215
   471
lemma has_derivative_mult_right:
lp15@56215
   472
  fixes c:: "'a :: real_normed_algebra"
lp15@56215
   473
  shows "((op * c) has_derivative (op * c)) F"
lp15@56215
   474
by (rule has_derivative_mult_right [OF has_derivative_id])
lp15@56215
   475
lp15@56215
   476
lemma complex_differentiable_linear:
lp15@56215
   477
     "(op * c) complex_differentiable F"
lp15@56215
   478
proof -
lp15@56215
   479
  have "\<And>u::complex. (\<lambda>x. x * u) = op * u"
lp15@56215
   480
    by (rule ext) (simp add: mult_ac)
lp15@56215
   481
  then show ?thesis
lp15@56215
   482
    unfolding complex_differentiable_def has_field_derivative_def
lp15@56215
   483
    by (force intro: has_derivative_mult_right)
lp15@56215
   484
qed
lp15@56215
   485
lp15@56215
   486
lemma complex_differentiable_const:
lp15@56215
   487
  "(\<lambda>z. c) complex_differentiable F"
lp15@56215
   488
  unfolding complex_differentiable_def has_field_derivative_def
lp15@56215
   489
  apply (rule exI [where x=0])
lp15@56215
   490
  by (metis Derivative.has_derivative_const lambda_zero) 
lp15@56215
   491
lp15@56215
   492
lemma complex_differentiable_id:
lp15@56215
   493
  "(\<lambda>z. z) complex_differentiable F"
lp15@56215
   494
  unfolding complex_differentiable_def has_field_derivative_def
lp15@56215
   495
  apply (rule exI [where x=1])
lp15@56215
   496
  apply (simp add: lambda_one [symmetric])
lp15@56215
   497
  done
lp15@56215
   498
lp15@56215
   499
(*DERIV_minus*)
lp15@56215
   500
lemma field_differentiable_minus:
lp15@56215
   501
  assumes "(f has_field_derivative f') F" 
lp15@56215
   502
    shows "((\<lambda>z. - (f z)) has_field_derivative -f') F"
lp15@56215
   503
  apply (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus])
lp15@56215
   504
  using assms 
lp15@56215
   505
  by (auto simp: has_field_derivative_def field_simps mult_commute_abs)
lp15@56215
   506
lp15@56215
   507
(*DERIV_add*)
lp15@56215
   508
lemma field_differentiable_add:
lp15@56215
   509
  assumes "(f has_field_derivative f') F" "(g has_field_derivative g') F"
lp15@56215
   510
    shows "((\<lambda>z. f z + g z) has_field_derivative f' + g') F"
lp15@56215
   511
  apply (rule has_derivative_imp_has_field_derivative[OF has_derivative_add])
lp15@56215
   512
  using assms 
lp15@56215
   513
  by (auto simp: has_field_derivative_def field_simps mult_commute_abs)
lp15@56215
   514
lp15@56215
   515
(*DERIV_diff*)
lp15@56215
   516
lemma field_differentiable_diff:
lp15@56215
   517
  assumes "(f has_field_derivative f') F" "(g has_field_derivative g') F"
lp15@56215
   518
    shows "((\<lambda>z. f z - g z) has_field_derivative f' - g') F"
lp15@56215
   519
by (simp only: assms diff_conv_add_uminus field_differentiable_add field_differentiable_minus)
lp15@56215
   520
lp15@56215
   521
lemma complex_differentiable_minus:
lp15@56215
   522
    "f complex_differentiable F \<Longrightarrow> (\<lambda>z. -(f z)) complex_differentiable F"
lp15@56215
   523
  using assms unfolding complex_differentiable_def
lp15@56215
   524
  by (metis field_differentiable_minus)
lp15@56215
   525
lp15@56215
   526
lemma complex_differentiable_add:
lp15@56215
   527
  assumes "f complex_differentiable F" "g complex_differentiable F"
lp15@56215
   528
    shows "(\<lambda>z. f z + g z) complex_differentiable F"
lp15@56215
   529
  using assms unfolding complex_differentiable_def
lp15@56215
   530
  by (metis field_differentiable_add)
lp15@56215
   531
lp15@56215
   532
lemma complex_differentiable_diff:
lp15@56215
   533
  assumes "f complex_differentiable F" "g complex_differentiable F"
lp15@56215
   534
    shows "(\<lambda>z. f z - g z) complex_differentiable F"
lp15@56215
   535
  using assms unfolding complex_differentiable_def
lp15@56215
   536
  by (metis field_differentiable_diff)
lp15@56215
   537
lp15@56215
   538
lemma complex_differentiable_inverse:
lp15@56215
   539
  assumes "f complex_differentiable (at a within s)" "f a \<noteq> 0"
lp15@56215
   540
  shows "(\<lambda>z. inverse (f z)) complex_differentiable (at a within s)"
lp15@56215
   541
  using assms unfolding complex_differentiable_def
lp15@56215
   542
  by (metis DERIV_inverse_fun)
lp15@56215
   543
lp15@56215
   544
lemma complex_differentiable_mult:
lp15@56215
   545
  assumes "f complex_differentiable (at a within s)" 
lp15@56215
   546
          "g complex_differentiable (at a within s)"
lp15@56215
   547
    shows "(\<lambda>z. f z * g z) complex_differentiable (at a within s)"
lp15@56215
   548
  using assms unfolding complex_differentiable_def
lp15@56215
   549
  by (metis DERIV_mult [of f _ a s g])
lp15@56215
   550
  
lp15@56215
   551
lemma complex_differentiable_divide:
lp15@56215
   552
  assumes "f complex_differentiable (at a within s)" 
lp15@56215
   553
          "g complex_differentiable (at a within s)"
lp15@56215
   554
          "g a \<noteq> 0"
lp15@56215
   555
    shows "(\<lambda>z. f z / g z) complex_differentiable (at a within s)"
lp15@56215
   556
  using assms unfolding complex_differentiable_def
lp15@56215
   557
  by (metis DERIV_divide [of f _ a s g])
lp15@56215
   558
lp15@56215
   559
lemma complex_differentiable_power:
lp15@56215
   560
  assumes "f complex_differentiable (at a within s)" 
lp15@56215
   561
    shows "(\<lambda>z. f z ^ n) complex_differentiable (at a within s)"
lp15@56215
   562
  using assms unfolding complex_differentiable_def
lp15@56215
   563
  by (metis DERIV_power)
lp15@56215
   564
lp15@56215
   565
lemma complex_differentiable_transform_within:
lp15@56215
   566
  "0 < d \<Longrightarrow>
lp15@56215
   567
        x \<in> s \<Longrightarrow>
lp15@56215
   568
        (\<And>x'. x' \<in> s \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x') \<Longrightarrow>
lp15@56215
   569
        f complex_differentiable (at x within s)
lp15@56215
   570
        \<Longrightarrow> g complex_differentiable (at x within s)"
lp15@56215
   571
  unfolding complex_differentiable_def has_field_derivative_def
lp15@56215
   572
  by (blast intro: has_derivative_transform_within)
lp15@56215
   573
lp15@56215
   574
lemma complex_differentiable_compose_within:
lp15@56215
   575
  assumes "f complex_differentiable (at a within s)" 
lp15@56215
   576
          "g complex_differentiable (at (f a) within f`s)"
lp15@56215
   577
    shows "(g o f) complex_differentiable (at a within s)"
lp15@56215
   578
  using assms unfolding complex_differentiable_def
lp15@56215
   579
  by (metis DERIV_image_chain)
lp15@56215
   580
lp15@56215
   581
lemma complex_differentiable_within_open:
lp15@56215
   582
     "\<lbrakk>a \<in> s; open s\<rbrakk> \<Longrightarrow> f complex_differentiable at a within s \<longleftrightarrow> 
lp15@56215
   583
                          f complex_differentiable at a"
lp15@56215
   584
  unfolding complex_differentiable_def
lp15@56215
   585
  by (metis at_within_open)
lp15@56215
   586
lp15@56215
   587
lemma holomorphic_transform:
lp15@56215
   588
     "\<lbrakk>f holomorphic_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g holomorphic_on s"
lp15@56215
   589
  apply (auto simp: holomorphic_on_def has_field_derivative_def)
lp15@56215
   590
  by (metis complex_differentiable_def complex_differentiable_transform_within has_field_derivative_def linordered_field_no_ub)
lp15@56215
   591
lp15@56215
   592
lemma holomorphic_eq:
lp15@56215
   593
     "(\<And>x. x \<in> s \<Longrightarrow> f x = g x) \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> g holomorphic_on s"
lp15@56215
   594
  by (metis holomorphic_transform)
lp15@56215
   595
lp15@56215
   596
subsection{*Holomorphic*}
lp15@56215
   597
lp15@56215
   598
lemma holomorphic_on_linear:
lp15@56215
   599
     "(op * c) holomorphic_on s"
lp15@56215
   600
  unfolding holomorphic_on_def  by (metis DERIV_cmult_Id)
lp15@56215
   601
lp15@56215
   602
lemma holomorphic_on_const:
lp15@56215
   603
     "(\<lambda>z. c) holomorphic_on s"
lp15@56215
   604
  unfolding holomorphic_on_def  
lp15@56215
   605
  by (metis DERIV_const)
lp15@56215
   606
lp15@56215
   607
lemma holomorphic_on_id:
lp15@56215
   608
     "id holomorphic_on s"
lp15@56215
   609
  unfolding holomorphic_on_def id_def  
lp15@56215
   610
  by (metis DERIV_ident)
lp15@56215
   611
lp15@56215
   612
lemma holomorphic_on_compose:
lp15@56215
   613
  "f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s)
lp15@56215
   614
           \<Longrightarrow> (g o f) holomorphic_on s"
lp15@56215
   615
  unfolding holomorphic_on_def
lp15@56215
   616
  by (metis DERIV_image_chain imageI)
lp15@56215
   617
lp15@56215
   618
lemma holomorphic_on_compose_gen:
lp15@56215
   619
  "\<lbrakk>f holomorphic_on s; g holomorphic_on t; f ` s \<subseteq> t\<rbrakk> \<Longrightarrow> (g o f) holomorphic_on s"
lp15@56215
   620
  unfolding holomorphic_on_def
lp15@56215
   621
  by (metis DERIV_image_chain DERIV_subset image_subset_iff)
lp15@56215
   622
lp15@56215
   623
lemma holomorphic_on_minus:
lp15@56215
   624
  "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on s"
lp15@56215
   625
  unfolding holomorphic_on_def
lp15@56215
   626
by (metis DERIV_minus)
lp15@56215
   627
lp15@56215
   628
lemma holomorphic_on_add:
lp15@56215
   629
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on s"
lp15@56215
   630
  unfolding holomorphic_on_def
lp15@56215
   631
  by (metis DERIV_add)
lp15@56215
   632
lp15@56215
   633
lemma holomorphic_on_diff:
lp15@56215
   634
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) holomorphic_on s"
lp15@56215
   635
  unfolding holomorphic_on_def
lp15@56215
   636
  by (metis DERIV_diff)
lp15@56215
   637
lp15@56215
   638
lemma holomorphic_on_mult:
lp15@56215
   639
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) holomorphic_on s"
lp15@56215
   640
  unfolding holomorphic_on_def
lp15@56215
   641
  by auto (metis DERIV_mult)
lp15@56215
   642
lp15@56215
   643
lemma holomorphic_on_inverse:
lp15@56215
   644
  "\<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@56215
   645
  unfolding holomorphic_on_def
lp15@56215
   646
  by (metis DERIV_inverse')
lp15@56215
   647
lp15@56215
   648
lemma holomorphic_on_divide:
lp15@56215
   649
  "\<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@56215
   650
  unfolding holomorphic_on_def
lp15@56215
   651
  by (metis (full_types) DERIV_divide)
lp15@56215
   652
lp15@56215
   653
lemma holomorphic_on_power:
lp15@56215
   654
  "f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on s"
lp15@56215
   655
  unfolding holomorphic_on_def
lp15@56215
   656
  by (metis DERIV_power)
lp15@56215
   657
lp15@56215
   658
lemma holomorphic_on_setsum:
lp15@56215
   659
  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s)
lp15@56215
   660
           \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) holomorphic_on s"
lp15@56215
   661
  unfolding holomorphic_on_def
lp15@56215
   662
  apply (induct I rule: finite_induct) 
lp15@56215
   663
  apply (force intro: DERIV_const DERIV_add)+
lp15@56215
   664
  done
lp15@56215
   665
lp15@56215
   666
lemma DERIV_imp_DD: "DERIV f x :> f' \<Longrightarrow> DD f x = f'"
lp15@56215
   667
    apply (simp add: DD_def has_field_derivative_def mult_commute_abs)
lp15@56215
   668
    apply (rule the_equality, assumption)
lp15@56215
   669
    apply (metis DERIV_unique has_field_derivative_def)
lp15@56215
   670
    done
lp15@56215
   671
lp15@56215
   672
lemma DD_iff_derivative_differentiable:
lp15@56215
   673
  fixes f :: "real\<Rightarrow>real" 
lp15@56215
   674
  shows   "DERIV f x :> DD f x \<longleftrightarrow> f differentiable at x"
lp15@56215
   675
unfolding DD_def differentiable_def 
lp15@56215
   676
by (metis (full_types) DD_def DERIV_imp_DD has_field_derivative_def has_real_derivative_iff 
lp15@56215
   677
          mult_commute_abs)
lp15@56215
   678
lp15@56215
   679
lemma DD_iff_derivative_complex_differentiable:
lp15@56215
   680
  fixes f :: "complex\<Rightarrow>complex" 
lp15@56215
   681
  shows "DERIV f x :> DD f x \<longleftrightarrow> f complex_differentiable at x"
lp15@56215
   682
unfolding DD_def complex_differentiable_def
lp15@56215
   683
by (metis DD_def DERIV_imp_DD)
lp15@56215
   684
lp15@56215
   685
lemma complex_differentiable_compose:
lp15@56215
   686
  "f complex_differentiable at z \<Longrightarrow> g complex_differentiable at (f z)
lp15@56215
   687
          \<Longrightarrow> (g o f) complex_differentiable at z"
lp15@56215
   688
by (metis complex_differentiable_at_within complex_differentiable_compose_within)
lp15@56215
   689
lp15@56215
   690
lemma complex_derivative_chain:
lp15@56215
   691
  fixes z::complex
lp15@56215
   692
  shows
lp15@56215
   693
  "f complex_differentiable at z \<Longrightarrow> g complex_differentiable at (f z)
lp15@56215
   694
          \<Longrightarrow> DD (g o f) z = DD g (f z) * DD f z"
lp15@56215
   695
by (metis DD_iff_derivative_complex_differentiable DERIV_chain DERIV_imp_DD)
lp15@56215
   696
lp15@56215
   697
lemma comp_derivative_chain:
lp15@56215
   698
  fixes z::real
lp15@56215
   699
  shows "\<lbrakk>f differentiable at z; g differentiable at (f z)\<rbrakk> 
lp15@56215
   700
         \<Longrightarrow> DD (g o f) z = DD g (f z) * DD f z"
lp15@56215
   701
by (metis DD_iff_derivative_differentiable DERIV_chain DERIV_imp_DD)
lp15@56215
   702
lp15@56215
   703
lemma complex_derivative_linear: "DD (\<lambda>w. c * w) = (\<lambda>z. c)"
lp15@56215
   704
by (metis DERIV_imp_DD DERIV_cmult_Id)
lp15@56215
   705
lp15@56215
   706
lemma complex_derivative_ident: "DD (\<lambda>w. w) = (\<lambda>z. 1)"
lp15@56215
   707
by (metis DERIV_imp_DD DERIV_ident)
lp15@56215
   708
lp15@56215
   709
lemma complex_derivative_const: "DD (\<lambda>w. c) = (\<lambda>z. 0)"
lp15@56215
   710
by (metis DERIV_imp_DD DERIV_const)
lp15@56215
   711
lp15@56215
   712
lemma complex_derivative_add:
lp15@56215
   713
  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>  
lp15@56215
   714
   \<Longrightarrow> DD (\<lambda>w. f w + g w) z = DD f z + DD g z"
lp15@56215
   715
  unfolding complex_differentiable_def
lp15@56215
   716
  by (rule DERIV_imp_DD) (metis (poly_guards_query) DERIV_add DERIV_imp_DD)  
lp15@56215
   717
lp15@56215
   718
lemma complex_derivative_diff:
lp15@56215
   719
  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>  
lp15@56215
   720
   \<Longrightarrow> DD (\<lambda>w. f w - g w) z = DD f z - DD g z"
lp15@56215
   721
  unfolding complex_differentiable_def
lp15@56215
   722
  by (rule DERIV_imp_DD) (metis (poly_guards_query) DERIV_diff DERIV_imp_DD)
lp15@56215
   723
lp15@56215
   724
lemma complex_derivative_mult:
lp15@56215
   725
  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>  
lp15@56215
   726
   \<Longrightarrow> DD (\<lambda>w. f w * g w) z = f z * DD g z + DD f z * g z"
lp15@56215
   727
  unfolding complex_differentiable_def
lp15@56215
   728
  by (rule DERIV_imp_DD) (metis DERIV_imp_DD DERIV_mult')
lp15@56215
   729
lp15@56215
   730
lemma complex_derivative_cmult:
lp15@56215
   731
  "f complex_differentiable at z \<Longrightarrow> DD (\<lambda>w. c * f w) z = c * DD f z"
lp15@56215
   732
  unfolding complex_differentiable_def
lp15@56215
   733
  by (metis DERIV_cmult DERIV_imp_DD)
lp15@56215
   734
lp15@56215
   735
lemma complex_derivative_cmult_right:
lp15@56215
   736
  "f complex_differentiable at z \<Longrightarrow> DD (\<lambda>w. f w * c) z = DD f z * c"
lp15@56215
   737
  unfolding complex_differentiable_def
lp15@56215
   738
  by (metis DERIV_cmult_right DERIV_imp_DD)
lp15@56215
   739
lp15@56215
   740
lemma complex_derivative_transform_within_open:
lp15@56215
   741
  "\<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> 
lp15@56215
   742
   \<Longrightarrow> DD f z = DD g z"
lp15@56215
   743
  unfolding holomorphic_on_def
lp15@56215
   744
  by (rule DERIV_imp_DD) (metis has_derivative_within_open DERIV_imp_DD DERIV_transform_within_open)
lp15@56215
   745
lp15@56215
   746
lemma complex_derivative_compose_linear:
lp15@56215
   747
    "f complex_differentiable at (c * z) \<Longrightarrow> DD (\<lambda>w. f (c * w)) z = c * DD f (c * z)"
lp15@56215
   748
apply (rule DERIV_imp_DD)
lp15@56215
   749
apply (simp add: DD_iff_derivative_complex_differentiable [symmetric])
lp15@56215
   750
apply (metis DERIV_chain' DERIV_cmult_Id comm_semiring_1_class.normalizing_semiring_rules(7))  
lp15@56215
   751
done
lp15@56215
   752
lp15@56215
   753
subsection{*Caratheodory characterization.*}
lp15@56215
   754
lp15@56215
   755
(*REPLACE the original version. BUT IN WHICH FILE??*)
lp15@56215
   756
thm Deriv.CARAT_DERIV
lp15@56215
   757
lp15@56215
   758
lemma complex_differentiable_caratheodory_at:
lp15@56215
   759
  "f complex_differentiable (at z) \<longleftrightarrow>
lp15@56215
   760
         (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)"
lp15@56215
   761
  using CARAT_DERIV [of f]
lp15@56215
   762
  by (simp add: complex_differentiable_def has_field_derivative_def)
lp15@56215
   763
lp15@56215
   764
lemma complex_differentiable_caratheodory_within:
lp15@56215
   765
  "f complex_differentiable (at z within s) \<longleftrightarrow>
lp15@56215
   766
         (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)"
lp15@56215
   767
  using DERIV_caratheodory_within [of f]
lp15@56215
   768
  by (simp add: complex_differentiable_def has_field_derivative_def)
lp15@56215
   769
lp15@56215
   770
subsection{*analyticity on a set*}
lp15@56215
   771
lp15@56215
   772
definition analytic_on (infixl "(analytic'_on)" 50)  
lp15@56215
   773
  where
lp15@56215
   774
   "f analytic_on s \<equiv> \<forall>x \<in> s. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)"
lp15@56215
   775
lp15@56215
   776
lemma analytic_imp_holomorphic:
lp15@56215
   777
     "f analytic_on s \<Longrightarrow> f holomorphic_on s"
lp15@56215
   778
  unfolding analytic_on_def holomorphic_on_def
lp15@56215
   779
  apply (simp add: has_derivative_within_open [OF _ open_ball])
lp15@56215
   780
  apply (metis DERIV_subset dist_self mem_ball top_greatest)
lp15@56215
   781
  done
lp15@56215
   782
lp15@56215
   783
lemma analytic_on_open:
lp15@56215
   784
     "open s \<Longrightarrow> f analytic_on s \<longleftrightarrow> f holomorphic_on s"
lp15@56215
   785
apply (auto simp: analytic_imp_holomorphic)
lp15@56215
   786
apply (auto simp: analytic_on_def holomorphic_on_def)
lp15@56215
   787
by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball)
lp15@56215
   788
lp15@56215
   789
lemma analytic_on_imp_differentiable_at:
lp15@56215
   790
  "f analytic_on s \<Longrightarrow> x \<in> s \<Longrightarrow> f complex_differentiable (at x)"
lp15@56215
   791
 apply (auto simp: analytic_on_def holomorphic_on_differentiable)
lp15@56215
   792
by (metis Topology_Euclidean_Space.open_ball centre_in_ball complex_differentiable_within_open)
lp15@56215
   793
lp15@56215
   794
lemma analytic_on_subset:
lp15@56215
   795
     "f analytic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f analytic_on t"
lp15@56215
   796
  by (auto simp: analytic_on_def)
lp15@56215
   797
lp15@56215
   798
lemma analytic_on_Un:
lp15@56215
   799
    "f analytic_on (s \<union> t) \<longleftrightarrow> f analytic_on s \<and> f analytic_on t"
lp15@56215
   800
  by (auto simp: analytic_on_def)
lp15@56215
   801
lp15@56215
   802
lemma analytic_on_Union:
lp15@56215
   803
  "f analytic_on (\<Union> s) \<longleftrightarrow> (\<forall>t \<in> s. f analytic_on t)"
lp15@56215
   804
  by (auto simp: analytic_on_def)
lp15@56215
   805
  
lp15@56215
   806
lemma analytic_on_holomorphic:
lp15@56215
   807
  "f analytic_on s \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f holomorphic_on t)"
lp15@56215
   808
  (is "?lhs = ?rhs")
lp15@56215
   809
proof -
lp15@56215
   810
  have "?lhs \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t)"
lp15@56215
   811
  proof safe
lp15@56215
   812
    assume "f analytic_on s"
lp15@56215
   813
    then show "\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t"
lp15@56215
   814
      apply (simp add: analytic_on_def)
lp15@56215
   815
      apply (rule exI [where x="\<Union>{u. open u \<and> f analytic_on u}"], auto)
lp15@56215
   816
      apply (metis Topology_Euclidean_Space.open_ball analytic_on_open centre_in_ball)
lp15@56215
   817
      by (metis analytic_on_def)
lp15@56215
   818
  next
lp15@56215
   819
    fix t
lp15@56215
   820
    assume "open t" "s \<subseteq> t" "f analytic_on t" 
lp15@56215
   821
    then show "f analytic_on s"
lp15@56215
   822
        by (metis analytic_on_subset)
lp15@56215
   823
  qed
lp15@56215
   824
  also have "... \<longleftrightarrow> ?rhs"
lp15@56215
   825
    by (auto simp: analytic_on_open)
lp15@56215
   826
  finally show ?thesis .
lp15@56215
   827
qed
lp15@56215
   828
lp15@56215
   829
lemma analytic_on_linear: "(op * c) analytic_on s"
lp15@56215
   830
  apply (simp add: analytic_on_holomorphic holomorphic_on_linear)
lp15@56215
   831
  by (metis open_UNIV top_greatest)
lp15@56215
   832
lp15@56215
   833
lemma analytic_on_const: "(\<lambda>z. c) analytic_on s"
lp15@56215
   834
  unfolding analytic_on_def
lp15@56215
   835
  by (metis holomorphic_on_const zero_less_one)
lp15@56215
   836
lp15@56215
   837
lemma analytic_on_id: "id analytic_on s"
lp15@56215
   838
  unfolding analytic_on_def
lp15@56215
   839
  apply (simp add: holomorphic_on_id)
lp15@56215
   840
  by (metis gt_ex)
lp15@56215
   841
lp15@56215
   842
lemma analytic_on_compose:
lp15@56215
   843
  assumes f: "f analytic_on s"
lp15@56215
   844
      and g: "g analytic_on (f ` s)"
lp15@56215
   845
    shows "(g o f) analytic_on s"
lp15@56215
   846
unfolding analytic_on_def
lp15@56215
   847
proof (intro ballI)
lp15@56215
   848
  fix x
lp15@56215
   849
  assume x: "x \<in> s"
lp15@56215
   850
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f
lp15@56215
   851
    by (metis analytic_on_def)
lp15@56215
   852
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g
lp15@56215
   853
    by (metis analytic_on_def g image_eqI x) 
lp15@56215
   854
  have "isCont f x"
lp15@56215
   855
    by (metis analytic_on_imp_differentiable_at complex_differentiable_imp_continuous_at f x)
lp15@56215
   856
  with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'"
lp15@56215
   857
     by (auto simp: continuous_at_ball)
lp15@56215
   858
  have "g \<circ> f holomorphic_on ball x (min d e)" 
lp15@56215
   859
    apply (rule holomorphic_on_compose)
lp15@56215
   860
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   861
    by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball)
lp15@56215
   862
  then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e"
lp15@56215
   863
    by (metis d e min_less_iff_conj) 
lp15@56215
   864
qed
lp15@56215
   865
lp15@56215
   866
lemma analytic_on_compose_gen:
lp15@56215
   867
  "f analytic_on s \<Longrightarrow> g analytic_on t \<Longrightarrow> (\<And>z. z \<in> s \<Longrightarrow> f z \<in> t)
lp15@56215
   868
             \<Longrightarrow> g o f analytic_on s"
lp15@56215
   869
by (metis analytic_on_compose analytic_on_subset image_subset_iff)
lp15@56215
   870
lp15@56215
   871
lemma analytic_on_neg:
lp15@56215
   872
  "f analytic_on s \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on s"
lp15@56215
   873
by (metis analytic_on_holomorphic holomorphic_on_minus)
lp15@56215
   874
lp15@56215
   875
lemma analytic_on_add:
lp15@56215
   876
  assumes f: "f analytic_on s"
lp15@56215
   877
      and g: "g analytic_on s"
lp15@56215
   878
    shows "(\<lambda>z. f z + g z) analytic_on s"
lp15@56215
   879
unfolding analytic_on_def
lp15@56215
   880
proof (intro ballI)
lp15@56215
   881
  fix z
lp15@56215
   882
  assume z: "z \<in> s"
lp15@56215
   883
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   884
    by (metis analytic_on_def)
lp15@56215
   885
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
lp15@56215
   886
    by (metis analytic_on_def g z) 
lp15@56215
   887
  have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')" 
lp15@56215
   888
    apply (rule holomorphic_on_add) 
lp15@56215
   889
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   890
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   891
  then show "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e"
lp15@56215
   892
    by (metis e e' min_less_iff_conj)
lp15@56215
   893
qed
lp15@56215
   894
lp15@56215
   895
lemma analytic_on_diff:
lp15@56215
   896
  assumes f: "f analytic_on s"
lp15@56215
   897
      and g: "g analytic_on s"
lp15@56215
   898
    shows "(\<lambda>z. f z - g z) analytic_on s"
lp15@56215
   899
unfolding analytic_on_def
lp15@56215
   900
proof (intro ballI)
lp15@56215
   901
  fix z
lp15@56215
   902
  assume z: "z \<in> s"
lp15@56215
   903
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   904
    by (metis analytic_on_def)
lp15@56215
   905
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
lp15@56215
   906
    by (metis analytic_on_def g z) 
lp15@56215
   907
  have "(\<lambda>z. f z - g z) holomorphic_on ball z (min e e')" 
lp15@56215
   908
    apply (rule holomorphic_on_diff) 
lp15@56215
   909
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   910
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   911
  then show "\<exists>e>0. (\<lambda>z. f z - g z) holomorphic_on ball z e"
lp15@56215
   912
    by (metis e e' min_less_iff_conj)
lp15@56215
   913
qed
lp15@56215
   914
lp15@56215
   915
lemma analytic_on_mult:
lp15@56215
   916
  assumes f: "f analytic_on s"
lp15@56215
   917
      and g: "g analytic_on s"
lp15@56215
   918
    shows "(\<lambda>z. f z * g z) analytic_on s"
lp15@56215
   919
unfolding analytic_on_def
lp15@56215
   920
proof (intro ballI)
lp15@56215
   921
  fix z
lp15@56215
   922
  assume z: "z \<in> s"
lp15@56215
   923
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   924
    by (metis analytic_on_def)
lp15@56215
   925
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
lp15@56215
   926
    by (metis analytic_on_def g z) 
lp15@56215
   927
  have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')" 
lp15@56215
   928
    apply (rule holomorphic_on_mult) 
lp15@56215
   929
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   930
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
lp15@56215
   931
  then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e"
lp15@56215
   932
    by (metis e e' min_less_iff_conj)
lp15@56215
   933
qed
lp15@56215
   934
lp15@56215
   935
lemma analytic_on_inverse:
lp15@56215
   936
  assumes f: "f analytic_on s"
lp15@56215
   937
      and nz: "(\<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0)"
lp15@56215
   938
    shows "(\<lambda>z. inverse (f z)) analytic_on s"
lp15@56215
   939
unfolding analytic_on_def
lp15@56215
   940
proof (intro ballI)
lp15@56215
   941
  fix z
lp15@56215
   942
  assume z: "z \<in> s"
lp15@56215
   943
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
lp15@56215
   944
    by (metis analytic_on_def)
lp15@56215
   945
  have "continuous_on (ball z e) f"
lp15@56215
   946
    by (metis fh holomorphic_on_imp_continuous_on)
lp15@56215
   947
  then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0" 
lp15@56215
   948
    by (metis Topology_Euclidean_Space.open_ball centre_in_ball continuous_on_open_avoid e z nz)  
lp15@56215
   949
  have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')" 
lp15@56215
   950
    apply (rule holomorphic_on_inverse)
lp15@56215
   951
    apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball)
lp15@56215
   952
    by (metis nz' mem_ball min_less_iff_conj) 
lp15@56215
   953
  then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e"
lp15@56215
   954
    by (metis e e' min_less_iff_conj)
lp15@56215
   955
qed
lp15@56215
   956
lp15@56215
   957
lp15@56215
   958
lemma analytic_on_divide:
lp15@56215
   959
  assumes f: "f analytic_on s"
lp15@56215
   960
      and g: "g analytic_on s"
lp15@56215
   961
      and nz: "(\<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0)"
lp15@56215
   962
    shows "(\<lambda>z. f z / g z) analytic_on s"
lp15@56215
   963
unfolding divide_inverse
lp15@56215
   964
by (metis analytic_on_inverse analytic_on_mult f g nz)
lp15@56215
   965
lp15@56215
   966
lemma analytic_on_power:
lp15@56215
   967
  "f analytic_on s \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on s"
lp15@56215
   968
by (induct n) (auto simp: analytic_on_const analytic_on_mult)
lp15@56215
   969
lp15@56215
   970
lemma analytic_on_setsum:
lp15@56215
   971
  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on s)
lp15@56215
   972
           \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) analytic_on s"
lp15@56215
   973
  by (induct I rule: finite_induct) (auto simp: analytic_on_const analytic_on_add)
lp15@56215
   974
lp15@56215
   975
subsection{*analyticity at a point.*}
lp15@56215
   976
lp15@56215
   977
lemma analytic_at_ball:
lp15@56215
   978
  "f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
lp15@56215
   979
by (metis analytic_on_def singleton_iff)
lp15@56215
   980
lp15@56215
   981
lemma analytic_at:
lp15@56215
   982
    "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
lp15@56215
   983
by (metis analytic_on_holomorphic empty_subsetI insert_subset)
lp15@56215
   984
lp15@56215
   985
lemma analytic_on_analytic_at:
lp15@56215
   986
    "f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
lp15@56215
   987
by (metis analytic_at_ball analytic_on_def)
lp15@56215
   988
lp15@56215
   989
lemma analytic_at_two:
lp15@56215
   990
  "f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
lp15@56215
   991
   (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)"
lp15@56215
   992
  (is "?lhs = ?rhs")
lp15@56215
   993
proof 
lp15@56215
   994
  assume ?lhs
lp15@56215
   995
  then obtain s t 
lp15@56215
   996
    where st: "open s" "z \<in> s" "f holomorphic_on s"
lp15@56215
   997
              "open t" "z \<in> t" "g holomorphic_on t"
lp15@56215
   998
    by (auto simp: analytic_at)
lp15@56215
   999
  show ?rhs
lp15@56215
  1000
    apply (rule_tac x="s \<inter> t" in exI)
lp15@56215
  1001
    using st
lp15@56215
  1002
    apply (auto simp: Diff_subset holomorphic_on_subset)
lp15@56215
  1003
    done
lp15@56215
  1004
next
lp15@56215
  1005
  assume ?rhs 
lp15@56215
  1006
  then show ?lhs
lp15@56215
  1007
    by (force simp add: analytic_at)
lp15@56215
  1008
qed
lp15@56215
  1009
lp15@56215
  1010
subsection{*Combining theorems for derivative with ``analytic at'' hypotheses*}
lp15@56215
  1011
lp15@56215
  1012
lemma 
lp15@56215
  1013
  assumes "f analytic_on {z}" "g analytic_on {z}"
lp15@56215
  1014
  shows complex_derivative_add_at: "DD (\<lambda>w. f w + g w) z = DD f z + DD g z"
lp15@56215
  1015
    and complex_derivative_diff_at: "DD (\<lambda>w. f w - g w) z = DD f z - DD g z"
lp15@56215
  1016
    and complex_derivative_mult_at: "DD (\<lambda>w. f w * g w) z =
lp15@56215
  1017
           f z * DD g z + DD f z * g z"
lp15@56215
  1018
proof -
lp15@56215
  1019
  obtain s where s: "open s" "z \<in> s" "f holomorphic_on s" "g holomorphic_on s"
lp15@56215
  1020
    using assms by (metis analytic_at_two)
lp15@56215
  1021
  show "DD (\<lambda>w. f w + g w) z = DD f z + DD g z"
lp15@56215
  1022
    apply (rule DERIV_imp_DD [OF DERIV_add])
lp15@56215
  1023
    using s
lp15@56215
  1024
    apply (auto simp: holomorphic_on_open complex_differentiable_def DD_iff_derivative_complex_differentiable)
lp15@56215
  1025
    done
lp15@56215
  1026
  show "DD (\<lambda>w. f w - g w) z = DD f z - DD g z"
lp15@56215
  1027
    apply (rule DERIV_imp_DD [OF DERIV_diff])
lp15@56215
  1028
    using s
lp15@56215
  1029
    apply (auto simp: holomorphic_on_open complex_differentiable_def DD_iff_derivative_complex_differentiable)
lp15@56215
  1030
    done
lp15@56215
  1031
  show "DD (\<lambda>w. f w * g w) z = f z * DD g z + DD f z * g z"
lp15@56215
  1032
    apply (rule DERIV_imp_DD [OF DERIV_mult'])
lp15@56215
  1033
    using s
lp15@56215
  1034
    apply (auto simp: holomorphic_on_open complex_differentiable_def DD_iff_derivative_complex_differentiable)
lp15@56215
  1035
    done
lp15@56215
  1036
qed
lp15@56215
  1037
lp15@56215
  1038
lemma complex_derivative_cmult_at:
lp15@56215
  1039
  "f analytic_on {z} \<Longrightarrow>  DD (\<lambda>w. c * f w) z = c * DD f z"
lp15@56215
  1040
by (auto simp: complex_derivative_mult_at complex_derivative_const analytic_on_const)
lp15@56215
  1041
lp15@56215
  1042
lemma complex_derivative_cmult_right_at:
lp15@56215
  1043
  "f analytic_on {z} \<Longrightarrow>  DD (\<lambda>w. f w * c) z = DD f z * c"
lp15@56215
  1044
by (auto simp: complex_derivative_mult_at complex_derivative_const analytic_on_const)
lp15@56215
  1045
lp15@56215
  1046
text{*A composition lemma for functions of mixed type*}
lp15@56215
  1047
lemma has_vector_derivative_real_complex:
lp15@56215
  1048
  fixes f :: "complex \<Rightarrow> complex"
lp15@56215
  1049
  assumes "DERIV f (of_real a) :> f'"
lp15@56215
  1050
  shows "((\<lambda>x. f (of_real x)) has_vector_derivative f') (at a)"
lp15@56215
  1051
  using diff_chain_at [OF has_derivative_ident, of f "op * f'" a] assms
lp15@56215
  1052
  unfolding has_field_derivative_def has_vector_derivative_def o_def
lp15@56215
  1053
  by (auto simp: mult_ac scaleR_conv_of_real)
lp15@56215
  1054
lp15@56215
  1055
subsection{*Complex differentiation of sequences and series*}
lp15@56215
  1056
lp15@56215
  1057
lemma has_complex_derivative_sequence:
lp15@56215
  1058
  fixes s :: "complex set"
lp15@56215
  1059
  assumes cvs: "convex s"
lp15@56215
  1060
      and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
lp15@56215
  1061
      and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s \<longrightarrow> norm (f' n x - g' x) \<le> e"
lp15@56215
  1062
      and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) ---> l) sequentially"
lp15@56215
  1063
    shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) ---> g x) sequentially \<and> 
lp15@56215
  1064
                       (g has_field_derivative (g' x)) (at x within s)"
lp15@56215
  1065
proof -
lp15@56215
  1066
  from assms obtain x l where x: "x \<in> s" and tf: "((\<lambda>n. f n x) ---> l) sequentially"
lp15@56215
  1067
    by blast
lp15@56215
  1068
  { fix e::real assume e: "e > 0"
lp15@56215
  1069
    then obtain N where N: "\<forall>n\<ge>N. \<forall>x. x \<in> s \<longrightarrow> cmod (f' n x - g' x) \<le> e"
lp15@56215
  1070
      by (metis conv)    
lp15@56215
  1071
    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
  1072
    proof (rule exI [of _ N], clarify)
lp15@56215
  1073
      fix n y h
lp15@56215
  1074
      assume "N \<le> n" "y \<in> s"
lp15@56215
  1075
      then have "cmod (f' n y - g' y) \<le> e"
lp15@56215
  1076
        by (metis N)
lp15@56215
  1077
      then have "cmod h * cmod (f' n y - g' y) \<le> cmod h * e"
lp15@56215
  1078
        by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
lp15@56215
  1079
      then show "cmod (f' n y * h - g' y * h) \<le> e * cmod h"
lp15@56215
  1080
        by (simp add: norm_mult [symmetric] field_simps)
lp15@56215
  1081
    qed
lp15@56215
  1082
  } note ** = this
lp15@56215
  1083
  show ?thesis
lp15@56215
  1084
  unfolding has_field_derivative_def
lp15@56215
  1085
  proof (rule has_derivative_sequence [OF cvs _ _ x])
lp15@56215
  1086
    show "\<forall>n. \<forall>x\<in>s. (f n has_derivative (op * (f' n x))) (at x within s)"
lp15@56215
  1087
      by (metis has_field_derivative_def df)
lp15@56215
  1088
  next show "(\<lambda>n. f n x) ----> l"
lp15@56215
  1089
    by (rule tf)
lp15@56215
  1090
  next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
lp15@56215
  1091
    by (blast intro: **)
lp15@56215
  1092
  qed
lp15@56215
  1093
qed
lp15@56215
  1094
lp15@56215
  1095
lp15@56215
  1096
lemma has_complex_derivative_series:
lp15@56215
  1097
  fixes s :: "complex set"
lp15@56215
  1098
  assumes cvs: "convex s"
lp15@56215
  1099
      and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
lp15@56215
  1100
      and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s 
lp15@56215
  1101
                \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
lp15@56215
  1102
      and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) sums l)"
lp15@56215
  1103
    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
  1104
proof -
lp15@56215
  1105
  from assms obtain x l where x: "x \<in> s" and sf: "((\<lambda>n. f n x) sums l)"
lp15@56215
  1106
    by blast
lp15@56215
  1107
  { fix e::real assume e: "e > 0"
lp15@56215
  1108
    then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> s 
lp15@56215
  1109
            \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
lp15@56215
  1110
      by (metis conv)    
lp15@56215
  1111
    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
  1112
    proof (rule exI [of _ N], clarify)
lp15@56215
  1113
      fix n y h
lp15@56215
  1114
      assume "N \<le> n" "y \<in> s"
lp15@56215
  1115
      then have "cmod ((\<Sum>i<n. f' i y) - g' y) \<le> e"
lp15@56215
  1116
        by (metis N)
lp15@56215
  1117
      then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e"
lp15@56215
  1118
        by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
lp15@56215
  1119
      then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h"
lp15@56215
  1120
        by (simp add: norm_mult [symmetric] field_simps setsum_right_distrib)
lp15@56215
  1121
    qed
lp15@56215
  1122
  } note ** = this
lp15@56215
  1123
  show ?thesis
lp15@56215
  1124
  unfolding has_field_derivative_def
lp15@56215
  1125
  proof (rule has_derivative_series [OF cvs _ _ x])
lp15@56215
  1126
    fix n x
lp15@56215
  1127
    assume "x \<in> s"
lp15@56215
  1128
    then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within s)"
lp15@56215
  1129
      by (metis df has_field_derivative_def mult_commute_abs)
lp15@56215
  1130
  next show " ((\<lambda>n. f n x) sums l)"
lp15@56215
  1131
    by (rule sf)
lp15@56215
  1132
  next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
lp15@56215
  1133
    by (blast intro: **)
lp15@56215
  1134
  qed
lp15@56215
  1135
qed
lp15@56215
  1136
lp15@56215
  1137
subsection{*Bound theorem*}
lp15@56215
  1138
lp15@56215
  1139
lemma complex_differentiable_bound:
lp15@56215
  1140
  fixes s :: "complex set"
lp15@56215
  1141
  assumes cvs: "convex s"
lp15@56215
  1142
      and df:  "\<And>z. z \<in> s \<Longrightarrow> (f has_field_derivative f' z) (at z within s)"
lp15@56215
  1143
      and dn:  "\<And>z. z \<in> s \<Longrightarrow> norm (f' z) \<le> B"
lp15@56215
  1144
      and "x \<in> s"  "y \<in> s"
lp15@56215
  1145
    shows "norm(f x - f y) \<le> B * norm(x - y)"
lp15@56215
  1146
  apply (rule differentiable_bound [OF cvs])
huffman@56223
  1147
  apply (rule ballI, erule df [unfolded has_field_derivative_def])
huffman@56223
  1148
  apply (rule ballI, rule onorm_le, simp add: norm_mult mult_right_mono dn)
huffman@56223
  1149
  apply fact
huffman@56223
  1150
  apply fact
lp15@56215
  1151
  done
lp15@56215
  1152
lp15@56215
  1153
subsection{*Inverse function theorem for complex derivatives.*}
lp15@56215
  1154
lp15@56215
  1155
lemma has_complex_derivative_inverse_basic:
lp15@56215
  1156
  fixes f :: "complex \<Rightarrow> complex"
lp15@56215
  1157
  shows "DERIV f (g y) :> f' \<Longrightarrow>
lp15@56215
  1158
        f' \<noteq> 0 \<Longrightarrow>
lp15@56215
  1159
        continuous (at y) g \<Longrightarrow>
lp15@56215
  1160
        open t \<Longrightarrow>
lp15@56215
  1161
        y \<in> t \<Longrightarrow>
lp15@56215
  1162
        (\<And>z. z \<in> t \<Longrightarrow> f (g z) = z)
lp15@56215
  1163
        \<Longrightarrow> DERIV g y :> inverse (f')"
lp15@56215
  1164
  unfolding has_field_derivative_def
lp15@56215
  1165
  apply (rule has_derivative_inverse_basic)
lp15@56215
  1166
  apply (auto simp:  bounded_linear_mult_right)
lp15@56215
  1167
  done
lp15@56215
  1168
lp15@56215
  1169
(*Used only once, in Multivariate/cauchy.ml. *)
lp15@56215
  1170
lemma has_complex_derivative_inverse_strong:
lp15@56215
  1171
  fixes f :: "complex \<Rightarrow> complex"
lp15@56215
  1172
  shows "DERIV f x :> f' \<Longrightarrow>
lp15@56215
  1173
         f' \<noteq> 0 \<Longrightarrow>
lp15@56215
  1174
         open s \<Longrightarrow>
lp15@56215
  1175
         x \<in> s \<Longrightarrow>
lp15@56215
  1176
         continuous_on s f \<Longrightarrow>
lp15@56215
  1177
         (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
lp15@56215
  1178
         \<Longrightarrow> DERIV g (f x) :> inverse (f')"
lp15@56215
  1179
  unfolding has_field_derivative_def
lp15@56215
  1180
  apply (rule has_derivative_inverse_strong [of s x f g ])
lp15@56215
  1181
  using assms 
lp15@56215
  1182
  by auto
lp15@56215
  1183
lp15@56215
  1184
lemma has_complex_derivative_inverse_strong_x:
lp15@56215
  1185
  fixes f :: "complex \<Rightarrow> complex"
lp15@56215
  1186
  shows  "DERIV f (g y) :> f' \<Longrightarrow>
lp15@56215
  1187
          f' \<noteq> 0 \<Longrightarrow>
lp15@56215
  1188
          open s \<Longrightarrow>
lp15@56215
  1189
          continuous_on s f \<Longrightarrow>
lp15@56215
  1190
          g y \<in> s \<Longrightarrow> f(g y) = y \<Longrightarrow>
lp15@56215
  1191
          (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
lp15@56215
  1192
          \<Longrightarrow> DERIV g y :> inverse (f')"
lp15@56215
  1193
  unfolding has_field_derivative_def
lp15@56215
  1194
  apply (rule has_derivative_inverse_strong_x [of s g y f])
lp15@56215
  1195
  using assms 
lp15@56215
  1196
  by auto
lp15@56215
  1197
lp15@56215
  1198
subsection{*Further useful properties of complex conjugation*}
lp15@56215
  1199
lp15@56215
  1200
lemma lim_cnj: "((\<lambda>x. cnj(f x)) ---> cnj l) F \<longleftrightarrow> (f ---> l) F"
lp15@56215
  1201
  by (simp add: tendsto_iff dist_complex_def Complex.complex_cnj_diff [symmetric])
lp15@56215
  1202
lp15@56215
  1203
lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
lp15@56215
  1204
  by (simp add: sums_def lim_cnj cnj_setsum [symmetric])
lp15@56215
  1205
lp15@56215
  1206
lemma continuous_within_cnj: "continuous (at z within s) cnj"
lp15@56215
  1207
by (metis bounded_linear_cnj linear_continuous_within)
lp15@56215
  1208
lp15@56215
  1209
lemma continuous_on_cnj: "continuous_on s cnj"
lp15@56215
  1210
by (metis bounded_linear_cnj linear_continuous_on)
lp15@56215
  1211
lp15@56215
  1212
subsection{*Some limit theorems about real part of real series etc.*}
lp15@56215
  1213
lp15@56215
  1214
lemma real_lim:
lp15@56215
  1215
  fixes l::complex
lp15@56215
  1216
  assumes "(f ---> l) F" and " ~(trivial_limit F)" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>"
lp15@56215
  1217
  shows  "l \<in> \<real>"
lp15@56215
  1218
proof -
lp15@56215
  1219
  have 1: "((\<lambda>i. Im (f i)) ---> Im l) F"
lp15@56215
  1220
    by (metis assms(1) tendsto_Im) 
lp15@56217
  1221
  then have  "((\<lambda>i. Im (f i)) ---> 0) F" using assms
lp15@56217
  1222
    by (metis (mono_tags, lifting) Lim_eventually complex_is_Real_iff eventually_mono)
lp15@56215
  1223
  then show ?thesis using 1
lp15@56215
  1224
    by (metis 1 assms(2) complex_is_Real_iff tendsto_unique) 
lp15@56215
  1225
qed
lp15@56215
  1226
lp15@56215
  1227
lemma real_lim_sequentially:
lp15@56215
  1228
  fixes l::complex
lp15@56215
  1229
  shows "(f ---> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
lp15@56215
  1230
by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially)
lp15@56215
  1231
lp15@56215
  1232
lemma real_series: 
lp15@56215
  1233
  fixes l::complex
lp15@56215
  1234
  shows "f sums l \<Longrightarrow> (\<And>n. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
lp15@56215
  1235
unfolding sums_def
lp15@56215
  1236
by (metis real_lim_sequentially setsum_in_Reals)
lp15@56215
  1237
lp15@56215
  1238
lp15@56215
  1239
lemma Lim_null_comparison_Re:
lp15@56215
  1240
   "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F \<Longrightarrow>  (g ---> 0) F \<Longrightarrow> (f ---> 0) F"
lp15@56215
  1241
  by (metis Lim_null_comparison complex_Re_zero tendsto_Re)
lp15@56215
  1242
lp15@56215
  1243
lp15@56215
  1244
lemma norm_setsum_bound:
lp15@56215
  1245
  fixes n::nat
lp15@56215
  1246
  shows" \<lbrakk>\<And>n. N \<le> n \<Longrightarrow> norm (f n) \<le> g n; N \<le> m\<rbrakk>
lp15@56215
  1247
       \<Longrightarrow> norm (setsum f {m..<n}) \<le> setsum g {m..<n}"
lp15@56215
  1248
apply (induct n, auto)
lp15@56215
  1249
by (metis dual_order.trans le_cases le_neq_implies_less norm_triangle_mono)
lp15@56215
  1250
lp15@56215
  1251
lp15@56217
  1252
(*MOVE? But not to Finite_Cartesian_Product*)
lp15@56215
  1253
lemma sums_vec_nth :
lp15@56215
  1254
  assumes "f sums a"
lp15@56215
  1255
  shows "(\<lambda>x. f x $ i) sums a $ i"
lp15@56215
  1256
using assms unfolding sums_def
lp15@56215
  1257
by (auto dest: tendsto_vec_nth [where i=i])
lp15@56215
  1258
lp15@56215
  1259
lemma summable_vec_nth :
lp15@56215
  1260
  assumes "summable f"
lp15@56215
  1261
  shows "summable (\<lambda>x. f x $ i)"
lp15@56215
  1262
using assms unfolding summable_def
lp15@56215
  1263
by (blast intro: sums_vec_nth)
lp15@56215
  1264
lp15@56215
  1265
lemma sums_Re:
lp15@56215
  1266
  assumes "f sums a"
lp15@56215
  1267
  shows "(\<lambda>x. Re (f x)) sums Re a"
lp15@56215
  1268
using assms 
lp15@56215
  1269
by (auto simp: sums_def Re_setsum [symmetric] isCont_tendsto_compose [of a Re])
lp15@56215
  1270
lp15@56215
  1271
lemma sums_Im:
lp15@56215
  1272
  assumes "f sums a"
lp15@56215
  1273
  shows "(\<lambda>x. Im (f x)) sums Im a"
lp15@56215
  1274
using assms 
lp15@56215
  1275
by (auto simp: sums_def Im_setsum [symmetric] isCont_tendsto_compose [of a Im])
lp15@56215
  1276
lp15@56215
  1277
lemma summable_Re:
lp15@56215
  1278
  assumes "summable f"
lp15@56215
  1279
  shows "summable (\<lambda>x. Re (f x))"
lp15@56215
  1280
using assms unfolding summable_def
lp15@56215
  1281
by (blast intro: sums_Re)
lp15@56215
  1282
lp15@56215
  1283
lemma summable_Im:
lp15@56215
  1284
  assumes "summable f"
lp15@56215
  1285
  shows "summable (\<lambda>x. Im (f x))"
lp15@56215
  1286
using assms unfolding summable_def
lp15@56215
  1287
by (blast intro: sums_Im)
lp15@56215
  1288
lp15@56215
  1289
lemma series_comparison_complex:
lp15@56215
  1290
  fixes f:: "nat \<Rightarrow> 'a::banach"
lp15@56215
  1291
  assumes sg: "summable g"
lp15@56215
  1292
     and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"
lp15@56215
  1293
     and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"
lp15@56215
  1294
  shows "summable f"
lp15@56215
  1295
proof -
lp15@56215
  1296
  have g: "\<And>n. cmod (g n) = Re (g n)" using assms
lp15@56215
  1297
    by (metis abs_of_nonneg in_Reals_norm)
lp15@56215
  1298
  show ?thesis
lp15@56217
  1299
    apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])
lp15@56215
  1300
    using sg
lp15@56215
  1301
    apply (auto simp: summable_def)
lp15@56215
  1302
    apply (rule_tac x="Re s" in exI)
lp15@56215
  1303
    apply (auto simp: g sums_Re)
lp15@56215
  1304
    apply (metis fg g)
lp15@56215
  1305
    done
lp15@56215
  1306
qed
lp15@56215
  1307
lp15@56215
  1308
lemma summable_complex_of_real [simp]:
lp15@56215
  1309
  "summable (\<lambda>n. complex_of_real (f n)) = summable f"
lp15@56215
  1310
apply (auto simp: Series.summable_Cauchy)  
lp15@56215
  1311
apply (drule_tac x = e in spec, auto)
lp15@56215
  1312
apply (rule_tac x=N in exI)
lp15@56215
  1313
apply (auto simp: of_real_setsum [symmetric])
lp15@56215
  1314
done
lp15@56215
  1315
lp15@56215
  1316
lp15@56215
  1317
lp15@56215
  1318
lp15@56215
  1319
lemma setsum_Suc_reindex:
lp15@56215
  1320
  fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
lp15@56215
  1321
    shows  "setsum f {0..n} = f 0 - f (Suc n) + setsum (\<lambda>i. f (Suc i)) {0..n}"
lp15@56215
  1322
by (induct n) auto
lp15@56215
  1323
lp15@56215
  1324
lp15@56217
  1325
text{*A kind of complex Taylor theorem.*}
lp15@56215
  1326
lemma complex_taylor:
lp15@56215
  1327
  assumes s: "convex s" 
lp15@56215
  1328
      and f: "\<And>i x. x \<in> s \<Longrightarrow> i \<le> n \<Longrightarrow> (f i has_field_derivative f (Suc i) x) (at x within s)"
lp15@56215
  1329
      and B: "\<And>x. x \<in> s \<Longrightarrow> cmod (f (Suc n) x) \<le> B"
lp15@56215
  1330
      and w: "w \<in> s"
lp15@56215
  1331
      and z: "z \<in> s"
lp15@56215
  1332
    shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / of_nat (fact i)))
lp15@56215
  1333
          \<le> B * cmod(z - w)^(Suc n) / fact n"
lp15@56215
  1334
proof -
lp15@56215
  1335
  have wzs: "closed_segment w z \<subseteq> s" using assms
lp15@56215
  1336
    by (metis convex_contains_segment)
lp15@56215
  1337
  { fix u
lp15@56215
  1338
    assume "u \<in> closed_segment w z"
lp15@56215
  1339
    then have "u \<in> s"
lp15@56215
  1340
      by (metis wzs subsetD)
lp15@56215
  1341
    have "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / of_nat (fact i) +
lp15@56215
  1342
                      f (Suc i) u * (z-u)^i / of_nat (fact i)) = 
lp15@56215
  1343
              f (Suc n) u * (z-u) ^ n / of_nat (fact n)"
lp15@56215
  1344
    proof (induction n)
lp15@56215
  1345
      case 0 show ?case by simp
lp15@56215
  1346
    next
lp15@56215
  1347
      case (Suc n)
lp15@56215
  1348
      have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / of_nat (fact i) +
lp15@56215
  1349
                             f (Suc i) u * (z-u) ^ i / of_nat (fact i)) =  
lp15@56215
  1350
           f (Suc n) u * (z-u) ^ n / of_nat (fact n) +
lp15@56215
  1351
           f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / of_nat (fact (Suc n)) -
lp15@56215
  1352
           f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / of_nat (fact (Suc n))"
lp15@56215
  1353
        using Suc by simp
lp15@56215
  1354
      also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / of_nat (fact (Suc n))"
lp15@56215
  1355
      proof -
lp15@56215
  1356
        have "of_nat(fact(Suc n)) *
lp15@56215
  1357
             (f(Suc n) u *(z-u) ^ n / of_nat(fact n) +
lp15@56215
  1358
               f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / of_nat(fact(Suc n)) -
lp15@56215
  1359
               f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / of_nat(fact(Suc n))) =
lp15@56215
  1360
            (of_nat(fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / of_nat(fact n) +
lp15@56215
  1361
            (of_nat(fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / of_nat(fact(Suc n))) -
lp15@56215
  1362
            (of_nat(fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / of_nat(fact(Suc n))"
lp15@56215
  1363
          by (simp add: algebra_simps del: fact_Suc)
lp15@56215
  1364
        also have "... =
lp15@56215
  1365
                   (of_nat (fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / of_nat (fact n) +
lp15@56215
  1366
                   (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
lp15@56215
  1367
                   (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
lp15@56215
  1368
          by (simp del: fact_Suc)
lp15@56215
  1369
        also have "... = 
lp15@56215
  1370
                   (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) +
lp15@56215
  1371
                   (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
lp15@56215
  1372
                   (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
lp15@56215
  1373
          by (simp only: fact_Suc of_nat_mult mult_ac) simp
lp15@56215
  1374
        also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
lp15@56215
  1375
          by (simp add: algebra_simps)
lp15@56215
  1376
        finally show ?thesis
lp15@56215
  1377
        by (simp add: mult_left_cancel [where c = "of_nat (fact (Suc n))", THEN iffD1] del: fact_Suc)
lp15@56215
  1378
      qed
lp15@56215
  1379
      finally show ?case .
lp15@56215
  1380
    qed
lp15@56215
  1381
    then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / of_nat (fact i))) 
lp15@56215
  1382
                has_field_derivative f (Suc n) u * (z-u) ^ n / of_nat (fact n))
lp15@56215
  1383
               (at u within s)"
lp15@56215
  1384
      apply (intro DERIV_intros DERIV_power[THEN DERIV_cong])
lp15@56215
  1385
      apply (blast intro: assms `u \<in> s`)
lp15@56215
  1386
      apply (rule refl)+
lp15@56215
  1387
      apply (auto simp: field_simps)
lp15@56215
  1388
      done
lp15@56215
  1389
  } note sum_deriv = this
lp15@56215
  1390
  { fix u
lp15@56215
  1391
    assume u: "u \<in> closed_segment w z"
lp15@56215
  1392
    then have us: "u \<in> s"
lp15@56215
  1393
      by (metis wzs subsetD)
lp15@56215
  1394
    have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> cmod (f (Suc n) u) * cmod (u - z) ^ n"
lp15@56215
  1395
      by (metis norm_minus_commute order_refl)
lp15@56215
  1396
    also have "... \<le> cmod (f (Suc n) u) * cmod (z - w) ^ n"
lp15@56215
  1397
      by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
lp15@56215
  1398
    also have "... \<le> B * cmod (z - w) ^ n"
lp15@56215
  1399
      by (metis norm_ge_zero zero_le_power mult_right_mono  B [OF us])
lp15@56215
  1400
    finally have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> B * cmod (z - w) ^ n" .
lp15@56215
  1401
  } note cmod_bound = this
lp15@56215
  1402
  have "(\<Sum>i\<le>n. f i z * (z - z) ^ i / of_nat (fact i)) = (\<Sum>i\<le>n. (f i z / of_nat (fact i)) * 0 ^ i)"
lp15@56215
  1403
    by simp
lp15@56215
  1404
  also have "\<dots> = f 0 z / of_nat (fact 0)"
lp15@56215
  1405
    by (subst setsum_zero_power) simp
lp15@56215
  1406
  finally have "cmod (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / of_nat (fact i))) 
lp15@56215
  1407
            \<le> cmod ((\<Sum>i\<le>n. f i w * (z - w) ^ i / of_nat (fact i)) -
lp15@56215
  1408
                    (\<Sum>i\<le>n. f i z * (z - z) ^ i / of_nat (fact i)))"
lp15@56215
  1409
    by (simp add: norm_minus_commute)
lp15@56215
  1410
  also have "... \<le> B * cmod (z - w) ^ n / real_of_nat (fact n) * cmod (w - z)"
lp15@56215
  1411
    apply (rule complex_differentiable_bound 
lp15@56215
  1412
      [where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / of_nat(fact n)"
lp15@56215
  1413
         and s = "closed_segment w z", OF convex_segment])
lp15@56215
  1414
    apply (auto simp: ends_in_segment real_of_nat_def DERIV_subset [OF sum_deriv wzs]
lp15@56215
  1415
                  norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
lp15@56215
  1416
    done
lp15@56215
  1417
  also have "...  \<le> B * cmod (z - w) ^ Suc n / real (fact n)"
lp15@56215
  1418
    by (simp add: algebra_simps norm_minus_commute real_of_nat_def)
lp15@56215
  1419
  finally show ?thesis .
lp15@56215
  1420
qed
lp15@56215
  1421
lp15@56238
  1422
text{* Something more like the traditional MVT for real components.*}
lp15@56238
  1423
lemma complex_mvt_line:
lp15@56238
  1424
  assumes "\<And>u. u \<in> closed_segment w z ==> (f has_field_derivative f'(u)) (at u)"
lp15@56238
  1425
    shows "\<exists>u. u \<in> open_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
lp15@56238
  1426
proof -
lp15@56238
  1427
  have twz: "\<And>t. (1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)"
lp15@56238
  1428
    by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
lp15@56238
  1429
  show ?thesis
lp15@56238
  1430
    apply (cut_tac mvt_simple
lp15@56238
  1431
                     [of 0 1 "Re o f o (\<lambda>t. (1 - t) *\<^sub>R w +  t *\<^sub>R z)"
lp15@56238
  1432
                      "\<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
  1433
    apply auto
lp15@56238
  1434
    apply (rule_tac x="(1 - x) *\<^sub>R w + x *\<^sub>R z" in exI)
lp15@56238
  1435
    apply (simp add: open_segment_def)
lp15@56238
  1436
    apply (auto simp add: twz)
lp15@56238
  1437
    apply (rule has_derivative_at_within)
lp15@56238
  1438
    apply (intro has_derivative_intros has_derivative_add [OF has_derivative_const, simplified])+
lp15@56238
  1439
    apply (rule assms [unfolded has_field_derivative_def])
lp15@56238
  1440
    apply (force simp add: twz closed_segment_def)
lp15@56238
  1441
    done
lp15@56238
  1442
qed
lp15@56238
  1443
lp15@56238
  1444
lemma complex_taylor_mvt:
lp15@56238
  1445
  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
  1446
    shows "\<exists>u. u \<in> closed_segment w z \<and>
lp15@56238
  1447
            Re (f 0 z) =
lp15@56238
  1448
            Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / of_nat (fact i)) +
lp15@56238
  1449
                (f (Suc n) u * (z-u)^n / of_nat (fact n)) * (z - w))"
lp15@56238
  1450
proof -
lp15@56238
  1451
  { fix u
lp15@56238
  1452
    assume u: "u \<in> closed_segment w z"
lp15@56238
  1453
    have "(\<Sum>i = 0..n.
lp15@56238
  1454
               (f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) /
lp15@56238
  1455
               of_nat (fact i)) =
lp15@56238
  1456
          f (Suc 0) u -
lp15@56238
  1457
             (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
lp15@56238
  1458
             of_nat (fact (Suc n)) +
lp15@56238
  1459
             (\<Sum>i = 0..n.
lp15@56238
  1460
                 (f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) /
lp15@56238
  1461
                 of_nat (fact (Suc i)))"
lp15@56238
  1462
       by (subst setsum_Suc_reindex) simp
lp15@56238
  1463
    also have "... = f (Suc 0) u -
lp15@56238
  1464
             (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
lp15@56238
  1465
             of_nat (fact (Suc n)) +
lp15@56238
  1466
             (\<Sum>i = 0..n.
lp15@56238
  1467
                 f (Suc (Suc i)) u * ((z-u) ^ Suc i) / of_nat (fact (Suc i))  - 
lp15@56238
  1468
                 f (Suc i) u * (z-u) ^ i / of_nat (fact i))"
lp15@56238
  1469
      by (simp only: diff_divide_distrib fact_cancel mult_ac)
lp15@56238
  1470
    also have "... = f (Suc 0) u -
lp15@56238
  1471
             (f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) /
lp15@56238
  1472
             of_nat (fact (Suc n)) +
lp15@56238
  1473
             f (Suc (Suc n)) u * (z-u) ^ Suc n / of_nat (fact (Suc n)) - f (Suc 0) u"
lp15@56238
  1474
      by (subst setsum_Suc_diff) auto
lp15@56238
  1475
    also have "... = f (Suc n) u * (z-u) ^ n / of_nat (fact n)"
lp15@56238
  1476
      by (simp only: algebra_simps diff_divide_distrib fact_cancel)
lp15@56238
  1477
    finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i 
lp15@56238
  1478
                             - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / of_nat (fact i)) =
lp15@56238
  1479
                  f (Suc n) u * (z - u) ^ n / of_nat (fact n)" .
lp15@56238
  1480
    then have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / of_nat (fact i)) has_field_derivative
lp15@56238
  1481
                f (Suc n) u * (z - u) ^ n / of_nat (fact n))  (at u)"
lp15@56238
  1482
      apply (intro DERIV_intros)+
lp15@56238
  1483
      apply (force intro: u assms)
lp15@56238
  1484
      apply (rule refl)+
lp15@56238
  1485
      apply (auto simp: mult_ac)
lp15@56238
  1486
      done
lp15@56238
  1487
  }
lp15@56238
  1488
  then show ?thesis
lp15@56238
  1489
    apply (cut_tac complex_mvt_line [of w z "\<lambda>u. \<Sum>i = 0..n. f i u * (z-u) ^ i / of_nat (fact i)"
lp15@56238
  1490
               "\<lambda>u. (f (Suc n) u * (z-u)^n / of_nat (fact n))"])
lp15@56238
  1491
    apply (auto simp add: intro: open_closed_segment)
lp15@56238
  1492
    done
lp15@56238
  1493
qed
lp15@56238
  1494
lp15@56238
  1495
text{*Relations among convergence and absolute convergence for power series.*}
lp15@56238
  1496
lemma abel_lemma:
lp15@56238
  1497
  fixes a :: "nat \<Rightarrow> 'a::real_normed_vector"
lp15@56238
  1498
  assumes r: "0 \<le> r" and r0: "r < r0" and M: "\<And>n. norm(a n) * r0^n \<le> M"
lp15@56238
  1499
    shows "summable (\<lambda>n. norm(a(n)) * r^n)"
lp15@56238
  1500
proof (rule summable_comparison_test' [of "\<lambda>n. M * (r / r0)^n"])
lp15@56238
  1501
  show "summable (\<lambda>n. M * (r / r0) ^ n)"
lp15@56238
  1502
    using assms 
lp15@56238
  1503
    by (auto simp add: summable_mult summable_geometric)
lp15@56238
  1504
  next
lp15@56238
  1505
    fix n
lp15@56238
  1506
    show "norm (norm (a n) * r ^ n) \<le> M * (r / r0) ^ n"
lp15@56238
  1507
      using r r0 M [of n]
lp15@56238
  1508
      apply (auto simp add: abs_mult field_simps power_divide)
lp15@56238
  1509
      apply (cases "r=0", simp)
lp15@56238
  1510
      apply (cases n, auto)
lp15@56238
  1511
      done
lp15@56238
  1512
qed
lp15@56238
  1513
lp15@56238
  1514
lp15@56215
  1515
end