(* Author: John Harrison, Marco Maggesi, Graziano Gentili, Gianni Ciolli, Valentina Bruno
Ported from "hol_light/Multivariate/canal.ml" by L C Paulson (2014)
*)
section \<open>Complex Analysis Basics\<close>
text \<open>Definitions of analytic and holomorphic functions, limit theorems, complex differentiation\<close>
theory Complex_Analysis_Basics
imports Derivative "HOL-Library.Nonpos_Ints" Uncountable_Sets
begin
subsection\<^marker>\<open>tag unimportant\<close>\<open>General lemmas\<close>
lemma nonneg_Reals_cmod_eq_Re: "z \<in> \<real>\<^sub>\<ge>\<^sub>0 \<Longrightarrow> norm z = Re z"
by (simp add: complex_nonneg_Reals_iff cmod_eq_Re)
lemma fact_cancel:
fixes c :: "'a::real_field"
shows "of_nat (Suc n) * c / (fact (Suc n)) = c / (fact n)"
using of_nat_neq_0 by force
lemma vector_derivative_cnj_within:
assumes "at x within A \<noteq> bot" and "f differentiable at x within A"
shows "vector_derivative (\<lambda>z. cnj (f z)) (at x within A) =
cnj (vector_derivative f (at x within A))" (is "_ = cnj ?D")
proof -
let ?D = "vector_derivative f (at x within A)"
from assms have "(f has_vector_derivative ?D) (at x within A)"
by (subst (asm) vector_derivative_works)
hence "((\<lambda>x. cnj (f x)) has_vector_derivative cnj ?D) (at x within A)"
by (rule has_vector_derivative_cnj)
thus ?thesis using assms by (auto dest: vector_derivative_within)
qed
lemma vector_derivative_cnj:
assumes "f differentiable at x"
shows "vector_derivative (\<lambda>z. cnj (f z)) (at x) = cnj (vector_derivative f (at x))"
using assms by (intro vector_derivative_cnj_within) auto
lemma
shows open_halfspace_Re_lt: "open {z. Re(z) < b}"
and open_halfspace_Re_gt: "open {z. Re(z) > b}"
and closed_halfspace_Re_ge: "closed {z. Re(z) \<ge> b}"
and closed_halfspace_Re_le: "closed {z. Re(z) \<le> b}"
and closed_halfspace_Re_eq: "closed {z. Re(z) = b}"
and open_halfspace_Im_lt: "open {z. Im(z) < b}"
and open_halfspace_Im_gt: "open {z. Im(z) > b}"
and closed_halfspace_Im_ge: "closed {z. Im(z) \<ge> b}"
and closed_halfspace_Im_le: "closed {z. Im(z) \<le> b}"
and closed_halfspace_Im_eq: "closed {z. Im(z) = b}"
by (intro open_Collect_less closed_Collect_le closed_Collect_eq continuous_on_Re
continuous_on_Im continuous_on_id continuous_on_const)+
lemma uncountable_halfspace_Im_gt: "uncountable {z. Im z > c}"
proof -
obtain r where r: "r > 0" "ball ((c + 1) *\<^sub>R \<i>) r \<subseteq> {z. Im z > c}"
using open_halfspace_Im_gt[of c] unfolding open_contains_ball by force
then show ?thesis
using countable_subset uncountable_ball by blast
qed
lemma uncountable_halfspace_Im_lt: "uncountable {z. Im z < c}"
proof -
obtain r where r: "r > 0" "ball ((c - 1) *\<^sub>R \<i>) r \<subseteq> {z. Im z < c}"
using open_halfspace_Im_lt[of c] unfolding open_contains_ball by force
then show ?thesis
using countable_subset uncountable_ball by blast
qed
lemma uncountable_halfspace_Re_gt: "uncountable {z. Re z > c}"
proof -
obtain r where r: "r > 0" "ball (of_real(c + 1)) r \<subseteq> {z. Re z > c}"
using open_halfspace_Re_gt[of c] unfolding open_contains_ball by force
then show ?thesis
using countable_subset uncountable_ball by blast
qed
lemma uncountable_halfspace_Re_lt: "uncountable {z. Re z < c}"
proof -
obtain r where r: "r > 0" "ball (of_real(c - 1)) r \<subseteq> {z. Re z < c}"
using open_halfspace_Re_lt[of c] unfolding open_contains_ball by force
then show ?thesis
using countable_subset uncountable_ball by blast
qed
lemma connected_halfspace_Im_gt [intro]: "connected {z. c < Im z}"
by (intro convex_connected convex_halfspace_Im_gt)
lemma connected_halfspace_Im_lt [intro]: "connected {z. c > Im z}"
by (intro convex_connected convex_halfspace_Im_lt)
lemma connected_halfspace_Re_gt [intro]: "connected {z. c < Re z}"
by (intro convex_connected convex_halfspace_Re_gt)
lemma connected_halfspace_Re_lt [intro]: "connected {z. c > Re z}"
by (intro convex_connected convex_halfspace_Re_lt)
lemma closed_complex_Reals: "closed (\<real> :: complex set)"
proof -
have "(\<real> :: complex set) = {z. Im z = 0}"
by (auto simp: complex_is_Real_iff)
then show ?thesis
by (metis closed_halfspace_Im_eq)
qed
lemma closed_Real_halfspace_Re_le: "closed (\<real> \<inter> {w. Re w \<le> x})"
by (simp add: closed_Int closed_complex_Reals closed_halfspace_Re_le)
lemma closed_nonpos_Reals_complex [simp]: "closed (\<real>\<^sub>\<le>\<^sub>0 :: complex set)"
proof -
have "\<real>\<^sub>\<le>\<^sub>0 = \<real> \<inter> {z. Re(z) \<le> 0}"
using complex_nonpos_Reals_iff complex_is_Real_iff by auto
then show ?thesis
by (metis closed_Real_halfspace_Re_le)
qed
lemma closed_Real_halfspace_Re_ge: "closed (\<real> \<inter> {w. x \<le> Re(w)})"
using closed_halfspace_Re_ge
by (simp add: closed_Int closed_complex_Reals)
lemma closed_nonneg_Reals_complex [simp]: "closed (\<real>\<^sub>\<ge>\<^sub>0 :: complex set)"
proof -
have "\<real>\<^sub>\<ge>\<^sub>0 = \<real> \<inter> {z. Re(z) \<ge> 0}"
using complex_nonneg_Reals_iff complex_is_Real_iff by auto
then show ?thesis
by (metis closed_Real_halfspace_Re_ge)
qed
lemma closed_real_abs_le: "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
proof -
have "{w \<in> \<real>. \<bar>Re w\<bar> \<le> r} = (\<real> \<inter> {w. Re w \<le> r}) \<inter> (\<real> \<inter> {w. Re w \<ge> -r})"
by auto
then show "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
by (simp add: closed_Int closed_Real_halfspace_Re_ge closed_Real_halfspace_Re_le)
qed
lemma real_lim:
fixes l::complex
assumes "(f \<longlongrightarrow> l) F" and "\<not> trivial_limit F" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>"
shows "l \<in> \<real>"
using Lim_in_closed_set[OF closed_complex_Reals] assms
by (smt (verit) eventually_mono)
lemma real_lim_sequentially:
fixes l::complex
shows "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially)
lemma real_series:
fixes l::complex
shows "f sums l \<Longrightarrow> (\<And>n. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
unfolding sums_def
by (metis real_lim_sequentially sum_in_Reals)
lemma Lim_null_comparison_Re:
assumes "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F" "(g \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F"
using Lim_null_comparison assms tendsto_Re by fastforce
subsection\<open>Holomorphic functions\<close>
definition\<^marker>\<open>tag important\<close> holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
(infixl "(holomorphic'_on)" 50)
where "f holomorphic_on s \<equiv> \<forall>x\<in>s. f field_differentiable (at x within s)"
named_theorems\<^marker>\<open>tag important\<close> holomorphic_intros "structural introduction rules for holomorphic_on"
lemma holomorphic_onI [intro?]: "(\<And>x. x \<in> s \<Longrightarrow> f field_differentiable (at x within s)) \<Longrightarrow> f holomorphic_on s"
by (simp add: holomorphic_on_def)
lemma holomorphic_onD [dest?]: "\<lbrakk>f holomorphic_on s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x within s)"
by (simp add: holomorphic_on_def)
lemma holomorphic_on_imp_differentiable_on:
"f holomorphic_on s \<Longrightarrow> f differentiable_on s"
unfolding holomorphic_on_def differentiable_on_def
by (simp add: field_differentiable_imp_differentiable)
lemma holomorphic_on_imp_differentiable_at:
"\<lbrakk>f holomorphic_on s; open s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x)"
using at_within_open holomorphic_on_def by fastforce
lemma holomorphic_on_empty [holomorphic_intros]: "f holomorphic_on {}"
by (simp add: holomorphic_on_def)
lemma holomorphic_on_open:
"open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> f')"
by (auto simp: holomorphic_on_def field_differentiable_def has_field_derivative_def at_within_open [of _ s])
lemma holomorphic_on_UN_open:
assumes "\<And>n. n \<in> I \<Longrightarrow> f holomorphic_on A n" "\<And>n. n \<in> I \<Longrightarrow> open (A n)"
shows "f holomorphic_on (\<Union>n\<in>I. A n)"
by (metis UN_E assms holomorphic_on_open open_UN)
lemma holomorphic_on_imp_continuous_on:
"f holomorphic_on s \<Longrightarrow> continuous_on s f"
using differentiable_imp_continuous_on holomorphic_on_imp_differentiable_on by blast
lemma holomorphic_closedin_preimage_constant:
assumes "f holomorphic_on D"
shows "closedin (top_of_set D) {z\<in>D. f z = a}"
by (simp add: assms continuous_closedin_preimage_constant holomorphic_on_imp_continuous_on)
lemma holomorphic_closed_preimage_constant:
assumes "f holomorphic_on UNIV"
shows "closed {z. f z = a}"
using holomorphic_closedin_preimage_constant [OF assms] by simp
lemma holomorphic_on_subset [elim]:
"f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t"
unfolding holomorphic_on_def
by (metis field_differentiable_within_subset subsetD)
lemma holomorphic_transform: "\<lbrakk>f holomorphic_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g holomorphic_on s"
by (metis field_differentiable_transform_within linordered_field_no_ub holomorphic_on_def)
lemma holomorphic_cong: "s = t ==> (\<And>x. x \<in> s \<Longrightarrow> f x = g x) \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> g holomorphic_on t"
by (metis holomorphic_transform)
lemma holomorphic_on_linear [simp, holomorphic_intros]: "((*) c) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_linear)
lemma holomorphic_on_const [simp, holomorphic_intros]: "(\<lambda>z. c) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_const)
lemma holomorphic_on_ident [simp, holomorphic_intros]: "(\<lambda>x. x) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_ident)
lemma holomorphic_on_id [simp, holomorphic_intros]: "id holomorphic_on s"
unfolding id_def by (rule holomorphic_on_ident)
lemma constant_on_imp_holomorphic_on:
assumes "f constant_on A"
shows "f holomorphic_on A"
by (metis assms constant_on_def holomorphic_on_const holomorphic_transform)
lemma holomorphic_on_compose:
"f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s) \<Longrightarrow> (g \<circ> f) holomorphic_on s"
using field_differentiable_compose_within[of f _ s g]
by (auto simp: holomorphic_on_def)
lemma holomorphic_on_compose_gen:
"f holomorphic_on s \<Longrightarrow> g holomorphic_on t \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> (g \<circ> f) holomorphic_on s"
by (metis holomorphic_on_compose holomorphic_on_subset)
lemma holomorphic_on_balls_imp_entire:
assumes "\<not>bdd_above A" "\<And>r. r \<in> A \<Longrightarrow> f holomorphic_on ball c r"
shows "f holomorphic_on B"
proof (rule holomorphic_on_subset)
show "f holomorphic_on UNIV" unfolding holomorphic_on_def
proof
fix z :: complex
from \<open>\<not>bdd_above A\<close> obtain r where r: "r \<in> A" "r > norm (z - c)"
by (meson bdd_aboveI not_le)
with assms(2) have "f holomorphic_on ball c r" by blast
moreover from r have "z \<in> ball c r" by (auto simp: dist_norm norm_minus_commute)
ultimately show "f field_differentiable at z"
by (auto simp: holomorphic_on_def at_within_open[of _ "ball c r"])
qed
qed auto
lemma holomorphic_on_balls_imp_entire':
assumes "\<And>r. r > 0 \<Longrightarrow> f holomorphic_on ball c r"
shows "f holomorphic_on B"
proof (rule holomorphic_on_balls_imp_entire)
show "\<not>bdd_above {(0::real)<..}" unfolding bdd_above_def
by (meson greaterThan_iff gt_ex less_le_not_le order_le_less_trans)
qed (use assms in auto)
lemma holomorphic_on_minus [holomorphic_intros]: "f holomorphic_on A \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on A"
by (metis field_differentiable_minus holomorphic_on_def)
lemma holomorphic_on_add [holomorphic_intros]:
"\<lbrakk>f holomorphic_on A; g holomorphic_on A\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on A"
unfolding holomorphic_on_def by (metis field_differentiable_add)
lemma holomorphic_on_diff [holomorphic_intros]:
"\<lbrakk>f holomorphic_on A; g holomorphic_on A\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) holomorphic_on A"
unfolding holomorphic_on_def by (metis field_differentiable_diff)
lemma holomorphic_on_mult [holomorphic_intros]:
"\<lbrakk>f holomorphic_on A; g holomorphic_on A\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) holomorphic_on A"
unfolding holomorphic_on_def by (metis field_differentiable_mult)
lemma holomorphic_on_inverse [holomorphic_intros]:
"\<lbrakk>f holomorphic_on A; \<And>z. z \<in> A \<Longrightarrow> f z \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>z. inverse (f z)) holomorphic_on A"
unfolding holomorphic_on_def by (metis field_differentiable_inverse)
lemma holomorphic_on_divide [holomorphic_intros]:
"\<lbrakk>f holomorphic_on A; g holomorphic_on A; \<And>z. z \<in> A \<Longrightarrow> g z \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>z. f z / g z) holomorphic_on A"
unfolding holomorphic_on_def by (metis field_differentiable_divide)
lemma holomorphic_on_power [holomorphic_intros]:
"f holomorphic_on A \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on A"
unfolding holomorphic_on_def by (metis field_differentiable_power)
lemma holomorphic_on_power_int [holomorphic_intros]:
assumes nz: "n \<ge> 0 \<or> (\<forall>x\<in>A. f x \<noteq> 0)" and f: "f holomorphic_on A"
shows "(\<lambda>x. f x powi n) holomorphic_on A"
proof (cases "n \<ge> 0")
case True
have "(\<lambda>x. f x ^ nat n) holomorphic_on A"
by (simp add: f holomorphic_on_power)
with True show ?thesis
by (simp add: power_int_def)
next
case False
hence "(\<lambda>x. inverse (f x ^ nat (-n))) holomorphic_on A"
using nz by (auto intro!: holomorphic_intros f)
with False show ?thesis
by (simp add: power_int_def power_inverse)
qed
lemma holomorphic_on_sum [holomorphic_intros]:
"(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on A) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) holomorphic_on A"
unfolding holomorphic_on_def by (metis field_differentiable_sum)
lemma holomorphic_on_prod [holomorphic_intros]:
"(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on A) \<Longrightarrow> (\<lambda>x. prod (\<lambda>i. f i x) I) holomorphic_on A"
by (induction I rule: infinite_finite_induct) (auto intro: holomorphic_intros)
lemma holomorphic_pochhammer [holomorphic_intros]:
"f holomorphic_on A \<Longrightarrow> (\<lambda>s. pochhammer (f s) n) holomorphic_on A"
by (induction n) (auto intro!: holomorphic_intros simp: pochhammer_Suc)
lemma holomorphic_on_scaleR [holomorphic_intros]:
"f holomorphic_on A \<Longrightarrow> (\<lambda>x. c *\<^sub>R f x) holomorphic_on A"
by (auto simp: scaleR_conv_of_real intro!: holomorphic_intros)
lemma holomorphic_on_Un [holomorphic_intros]:
assumes "f holomorphic_on A" "f holomorphic_on B" "open A" "open B"
shows "f holomorphic_on (A \<union> B)"
by (metis Un_iff assms holomorphic_on_open open_Un)
lemma holomorphic_on_If_Un [holomorphic_intros]:
assumes "f holomorphic_on A" "g holomorphic_on B" "open A" "open B"
assumes "\<And>z. z \<in> A \<Longrightarrow> z \<in> B \<Longrightarrow> f z = g z"
shows "(\<lambda>z. if z \<in> A then f z else g z) holomorphic_on (A \<union> B)" (is "?h holomorphic_on _")
proof (intro holomorphic_on_Un)
note \<open>f holomorphic_on A\<close>
also have "f holomorphic_on A \<longleftrightarrow> ?h holomorphic_on A"
by (intro holomorphic_cong) auto
finally show \<dots> .
next
note \<open>g holomorphic_on B\<close>
also have "g holomorphic_on B \<longleftrightarrow> ?h holomorphic_on B"
using assms by (intro holomorphic_cong) auto
finally show \<dots> .
qed (use assms in auto)
lemma holomorphic_derivI:
"\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk> \<Longrightarrow> (f has_field_derivative deriv f x) (at x within T)"
by (metis DERIV_deriv_iff_field_differentiable at_within_open holomorphic_on_def has_field_derivative_at_within)
lemma complex_derivative_transform_within_open:
"\<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>
\<Longrightarrow> deriv f z = deriv g z"
by (smt (verit) DERIV_imp_deriv has_field_derivative_transform_within_open holomorphic_on_open)
lemma holomorphic_on_compose_cnj_cnj:
assumes "f holomorphic_on cnj ` A" "open A"
shows "cnj \<circ> f \<circ> cnj holomorphic_on A"
proof -
have [simp]: "open (cnj ` A)"
unfolding image_cnj_conv_vimage_cnj using assms by (intro open_vimage) auto
show ?thesis
using assms unfolding holomorphic_on_def
by (auto intro!: field_differentiable_cnj_cnj simp: at_within_open_NO_MATCH)
qed
lemma holomorphic_nonconstant:
assumes holf: "f holomorphic_on S" and "open S" "\<xi> \<in> S" "deriv f \<xi> \<noteq> 0"
shows "\<not> f constant_on S"
by (rule nonzero_deriv_nonconstant [of f "deriv f \<xi>" \<xi> S])
(use assms in \<open>auto simp: holomorphic_derivI\<close>)
subsection\<open>Analyticity on a set\<close>
definition\<^marker>\<open>tag important\<close> analytic_on (infixl "(analytic'_on)" 50)
where "f analytic_on S \<equiv> \<forall>x \<in> S. \<exists>\<epsilon>. 0 < \<epsilon> \<and> f holomorphic_on (ball x \<epsilon>)"
named_theorems\<^marker>\<open>tag important\<close> analytic_intros "introduction rules for proving analyticity"
lemma analytic_imp_holomorphic: "f analytic_on S \<Longrightarrow> f holomorphic_on S"
unfolding analytic_on_def holomorphic_on_def
using centre_in_ball field_differentiable_at_within field_differentiable_within_open by blast
lemma analytic_on_open: "open S \<Longrightarrow> f analytic_on S \<longleftrightarrow> f holomorphic_on S"
by (meson analytic_imp_holomorphic analytic_on_def holomorphic_on_subset openE)
lemma constant_on_imp_analytic_on:
assumes "f constant_on A" "open A"
shows "f analytic_on A"
by (simp add: analytic_on_open assms constant_on_imp_holomorphic_on)
lemma analytic_on_imp_differentiable_at:
"f analytic_on S \<Longrightarrow> x \<in> S \<Longrightarrow> f field_differentiable (at x)"
using analytic_on_def holomorphic_on_imp_differentiable_at by auto
lemma analytic_at_imp_isCont:
assumes "f analytic_on {z}"
shows "isCont f z"
by (meson analytic_on_imp_differentiable_at assms field_differentiable_imp_continuous_at insertCI)
lemma analytic_at_neq_imp_eventually_neq:
assumes "f analytic_on {x}" "f x \<noteq> c"
shows "eventually (\<lambda>y. f y \<noteq> c) (at x)"
using analytic_at_imp_isCont assms isContD tendsto_imp_eventually_ne by blast
lemma analytic_on_subset: "f analytic_on S \<Longrightarrow> T \<subseteq> S \<Longrightarrow> f analytic_on T"
by (auto simp: analytic_on_def)
lemma analytic_on_Un: "f analytic_on (S \<union> T) \<longleftrightarrow> f analytic_on S \<and> f analytic_on T"
by (auto simp: analytic_on_def)
lemma analytic_on_Union: "f analytic_on (\<Union>\<T>) \<longleftrightarrow> (\<forall>T \<in> \<T>. f analytic_on T)"
by (auto simp: analytic_on_def)
lemma analytic_on_UN: "f analytic_on (\<Union>i\<in>I. S i) \<longleftrightarrow> (\<forall>i\<in>I. f analytic_on (S i))"
by (auto simp: analytic_on_def)
lemma analytic_on_holomorphic:
"f analytic_on S \<longleftrightarrow> (\<exists>T. open T \<and> S \<subseteq> T \<and> f holomorphic_on T)"
(is "?lhs = ?rhs")
proof -
have "?lhs \<longleftrightarrow> (\<exists>T. open T \<and> S \<subseteq> T \<and> f analytic_on T)"
proof safe
assume "f analytic_on S"
then have "\<forall>x \<in> \<Union>{U. open U \<and> f analytic_on U}. \<exists>\<epsilon>>0. f holomorphic_on ball x \<epsilon>"
using analytic_on_def by force
moreover have "S \<subseteq> \<Union>{U. open U \<and> f analytic_on U}"
using \<open>f analytic_on S\<close>
by (smt (verit, best) open_ball Union_iff analytic_on_def analytic_on_open centre_in_ball mem_Collect_eq subsetI)
ultimately show "\<exists>T. open T \<and> S \<subseteq> T \<and> f analytic_on T"
unfolding analytic_on_def
by (metis (mono_tags, lifting) mem_Collect_eq open_Union)
next
fix T
assume "open T" "S \<subseteq> T" "f analytic_on T"
then show "f analytic_on S"
by (metis analytic_on_subset)
qed
also have "\<dots> \<longleftrightarrow> ?rhs"
by (auto simp: analytic_on_open)
finally show ?thesis .
qed
lemma analytic_on_linear [analytic_intros,simp]: "((*) c) analytic_on S"
by (auto simp add: analytic_on_holomorphic)
lemma analytic_on_const [analytic_intros,simp]: "(\<lambda>z. c) analytic_on S"
by (metis analytic_on_def holomorphic_on_const zero_less_one)
lemma analytic_on_ident [analytic_intros,simp]: "(\<lambda>x. x) analytic_on S"
by (simp add: analytic_on_def gt_ex)
lemma analytic_on_id [analytic_intros]: "id analytic_on S"
unfolding id_def by (rule analytic_on_ident)
lemma analytic_on_scaleR [analytic_intros]: "f analytic_on A \<Longrightarrow> (\<lambda>w. x *\<^sub>R f w) analytic_on A"
by (metis analytic_on_holomorphic holomorphic_on_scaleR)
lemma analytic_on_compose:
assumes f: "f analytic_on S"
and g: "g analytic_on (f ` S)"
shows "(g \<circ> f) analytic_on S"
unfolding analytic_on_def
proof (intro ballI)
fix x
assume x: "x \<in> S"
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f
by (metis analytic_on_def)
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g
by (metis analytic_on_def g image_eqI x)
have "isCont f x"
by (metis analytic_on_imp_differentiable_at field_differentiable_imp_continuous_at f x)
with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'"
by (auto simp: continuous_at_ball)
have "g \<circ> f holomorphic_on ball x (min d e)"
by (meson fd fh gh holomorphic_on_compose_gen holomorphic_on_subset image_mono min.cobounded1 min.cobounded2 subset_ball)
then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e"
by (metis d e min_less_iff_conj)
qed
lemma analytic_on_compose_gen:
"f analytic_on S \<Longrightarrow> g analytic_on T \<Longrightarrow> (\<And>z. z \<in> S \<Longrightarrow> f z \<in> T)
\<Longrightarrow> g \<circ> f analytic_on S"
by (metis analytic_on_compose analytic_on_subset image_subset_iff)
lemma analytic_on_neg [analytic_intros]:
"f analytic_on S \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on S"
by (metis analytic_on_holomorphic holomorphic_on_minus)
lemma analytic_on_add [analytic_intros]:
assumes f: "f analytic_on S"
and g: "g analytic_on S"
shows "(\<lambda>z. f z + g z) analytic_on S"
unfolding analytic_on_def
proof (intro ballI)
fix z
assume z: "z \<in> S"
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
by (metis analytic_on_def)
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
by (metis analytic_on_def g z)
have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')"
by (metis fh gh holomorphic_on_add holomorphic_on_subset linorder_linear min_def subset_ball)
then show "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e"
by (metis e e' min_less_iff_conj)
qed
lemma analytic_on_mult [analytic_intros]:
assumes f: "f analytic_on S"
and g: "g analytic_on S"
shows "(\<lambda>z. f z * g z) analytic_on S"
unfolding analytic_on_def
proof (intro ballI)
fix z
assume z: "z \<in> S"
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
by (metis analytic_on_def)
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
by (metis analytic_on_def g z)
have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')"
by (metis fh gh holomorphic_on_mult holomorphic_on_subset min.absorb_iff2 min_def subset_ball)
then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e"
by (metis e e' min_less_iff_conj)
qed
lemma analytic_on_diff [analytic_intros]:
assumes f: "f analytic_on S" and g: "g analytic_on S"
shows "(\<lambda>z. f z - g z) analytic_on S"
proof -
have "(\<lambda>z. - g z) analytic_on S"
by (simp add: analytic_on_neg g)
then have "(\<lambda>z. f z + - g z) analytic_on S"
using analytic_on_add f by blast
then show ?thesis
by fastforce
qed
lemma analytic_on_inverse [analytic_intros]:
assumes f: "f analytic_on S"
and nz: "(\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0)"
shows "(\<lambda>z. inverse (f z)) analytic_on S"
unfolding analytic_on_def
proof (intro ballI)
fix z
assume z: "z \<in> S"
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
by (metis analytic_on_def)
have "continuous_on (ball z e) f"
by (metis fh holomorphic_on_imp_continuous_on)
then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0"
by (metis open_ball centre_in_ball continuous_on_open_avoid e z nz)
have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')"
using fh holomorphic_on_inverse holomorphic_on_open nz' by fastforce
then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e"
by (metis e e' min_less_iff_conj)
qed
lemma analytic_on_divide [analytic_intros]:
assumes f: "f analytic_on S" and g: "g analytic_on S"
and nz: "(\<And>z. z \<in> S \<Longrightarrow> g z \<noteq> 0)"
shows "(\<lambda>z. f z / g z) analytic_on S"
unfolding divide_inverse by (metis analytic_on_inverse analytic_on_mult f g nz)
lemma analytic_on_power [analytic_intros]:
"f analytic_on S \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on S"
by (induct n) (auto simp: analytic_on_mult)
lemma analytic_on_power_int [analytic_intros]:
assumes nz: "n \<ge> 0 \<or> (\<forall>x\<in>A. f x \<noteq> 0)" and f: "f analytic_on A"
shows "(\<lambda>x. f x powi n) analytic_on A"
proof (cases "n \<ge> 0")
case True
have "(\<lambda>x. f x ^ nat n) analytic_on A"
using analytic_on_power f by blast
with True show ?thesis
by (simp add: power_int_def)
next
case False
hence "(\<lambda>x. inverse (f x ^ nat (-n))) analytic_on A"
using nz by (auto intro!: analytic_intros f)
with False show ?thesis
by (simp add: power_int_def power_inverse)
qed
lemma analytic_on_sum [analytic_intros]:
"(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on S) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) analytic_on S"
by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_add)
lemma analytic_on_prod [analytic_intros]:
"(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on S) \<Longrightarrow> (\<lambda>x. prod (\<lambda>i. f i x) I) analytic_on S"
by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_mult)
lemma analytic_on_gbinomial [analytic_intros]:
"f analytic_on A \<Longrightarrow> (\<lambda>w. f w gchoose n) analytic_on A"
unfolding gbinomial_prod_rev by (intro analytic_intros) auto
lemma deriv_left_inverse:
assumes "f holomorphic_on S" and "g holomorphic_on T"
and "open S" and "open T"
and "f ` S \<subseteq> T"
and [simp]: "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z"
and "w \<in> S"
shows "deriv f w * deriv g (f w) = 1"
proof -
have "deriv f w * deriv g (f w) = deriv g (f w) * deriv f w"
by (simp add: algebra_simps)
also have "\<dots> = deriv (g \<circ> f) w"
using assms
by (metis analytic_on_imp_differentiable_at analytic_on_open deriv_chain image_subset_iff)
also have "\<dots> = deriv id w"
proof (rule complex_derivative_transform_within_open [where s=S])
show "g \<circ> f holomorphic_on S"
by (rule assms holomorphic_on_compose_gen holomorphic_intros)+
qed (use assms in auto)
also have "\<dots> = 1"
by simp
finally show ?thesis .
qed
subsection\<^marker>\<open>tag unimportant\<close>\<open>Analyticity at a point\<close>
lemma analytic_at_ball:
"f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
by (metis analytic_on_def singleton_iff)
lemma analytic_at:
"f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
by (metis analytic_on_holomorphic empty_subsetI insert_subset)
lemma holomorphic_on_imp_analytic_at:
assumes "f holomorphic_on A" "open A" "z \<in> A"
shows "f analytic_on {z}"
using assms by (meson analytic_at)
lemma analytic_on_analytic_at:
"f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
by (metis analytic_at_ball analytic_on_def)
lemma analytic_at_two:
"f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
(\<exists>S. open S \<and> z \<in> S \<and> f holomorphic_on S \<and> g holomorphic_on S)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then obtain S T
where st: "open S" "z \<in> S" "f holomorphic_on S"
"open T" "z \<in> T" "g holomorphic_on T"
by (auto simp: analytic_at)
then show ?rhs
by (metis Int_iff holomorphic_on_subset inf_le1 inf_le2 open_Int)
next
assume ?rhs
then show ?lhs
by (force simp add: analytic_at)
qed
subsection\<^marker>\<open>tag unimportant\<close>\<open>Combining theorems for derivative with ``analytic at'' hypotheses\<close>
lemma
assumes "f analytic_on {z}" "g analytic_on {z}"
shows complex_derivative_add_at: "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
and complex_derivative_diff_at: "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
and complex_derivative_mult_at: "deriv (\<lambda>w. f w * g w) z =
f z * deriv g z + deriv f z * g z"
proof -
show "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
using analytic_on_imp_differentiable_at assms by auto
show "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
using analytic_on_imp_differentiable_at assms by force
obtain S where "open S" "z \<in> S" "f holomorphic_on S" "g holomorphic_on S"
using assms by (metis analytic_at_two)
then show "deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
by (simp add: DERIV_imp_deriv [OF DERIV_mult'] holomorphic_derivI)
qed
lemma deriv_cmult_at:
"f analytic_on {z} \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z"
by (auto simp: complex_derivative_mult_at)
lemma deriv_cmult_right_at:
"f analytic_on {z} \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c"
by (auto simp: complex_derivative_mult_at)
subsection\<^marker>\<open>tag unimportant\<close>\<open>Complex differentiation of sequences and series\<close>
(* TODO: Could probably be simplified using Uniform_Limit *)
lemma has_complex_derivative_sequence:
fixes S :: "complex set"
assumes cvs: "convex S"
and df: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)"
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"
and "\<exists>x l. x \<in> S \<and> ((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
shows "\<exists>g. \<forall>x \<in> S. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and>
(g has_field_derivative (g' x)) (at x within S)"
proof -
from assms obtain x l where x: "x \<in> S" and tf: "((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
by blast
show ?thesis
unfolding has_field_derivative_def
proof (rule has_derivative_sequence [OF cvs _ _ x])
show "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
by (rule tf)
next
have **: "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. cmod (f' n x * h - g' x * h) \<le> \<epsilon> * cmod h"
if "\<epsilon> > 0" for \<epsilon>::real
by (metis that left_diff_distrib mult_right_mono norm_ge_zero norm_mult conv)
show "\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
unfolding eventually_sequentially by (blast intro: **)
qed (metis has_field_derivative_def df)
qed
lemma has_complex_derivative_series:
fixes S :: "complex set"
assumes cvs: "convex S"
and df: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)"
and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> S
\<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
and "\<exists>x l. x \<in> S \<and> ((\<lambda>n. f n x) sums l)"
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))"
proof -
from assms obtain x l where x: "x \<in> S" and sf: "((\<lambda>n. f n x) sums l)"
by blast
{ fix \<epsilon>::real assume e: "\<epsilon> > 0"
then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> S
\<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> \<epsilon>"
by (metis conv)
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> \<epsilon> * cmod h"
proof (rule exI [of _ N], clarify)
fix n y h
assume "N \<le> n" "y \<in> S"
have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * \<epsilon>"
by (simp add: N \<open>N \<le> n\<close> \<open>y \<in> S\<close> mult_le_cancel_left)
then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> \<epsilon> * cmod h"
by (simp add: norm_mult [symmetric] field_simps sum_distrib_left)
qed
} note ** = this
show ?thesis
unfolding has_field_derivative_def
proof (rule has_derivative_series [OF cvs _ _ x])
fix n x
assume "x \<in> S"
then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within S)"
by (metis df has_field_derivative_def mult_commute_abs)
next show " ((\<lambda>n. f n x) sums l)"
by (rule sf)
next show "\<And>e. e>0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
unfolding eventually_sequentially by (blast intro: **)
qed
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>Taylor on Complex Numbers\<close>
lemma sum_Suc_reindex:
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
shows "sum f {0..n} = f 0 - f (Suc n) + sum (\<lambda>i. f (Suc i)) {0..n}"
by (induct n) auto
lemma field_Taylor:
assumes S: "convex S"
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)"
and B: "\<And>x. x \<in> S \<Longrightarrow> norm (f (Suc n) x) \<le> B"
and w: "w \<in> S"
and z: "z \<in> S"
shows "norm(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i)))
\<le> B * norm(z - w)^(Suc n) / fact n"
proof -
have wzs: "closed_segment w z \<subseteq> S" using assms
by (metis convex_contains_segment)
{ fix u
assume "u \<in> closed_segment w z"
then have "u \<in> S"
by (metis wzs subsetD)
have *: "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / (fact i) +
f (Suc i) u * (z-u)^i / (fact i)) =
f (Suc n) u * (z-u) ^ n / (fact n)"
proof (induction n)
case 0 show ?case by simp
next
case (Suc n)
have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / (fact i) +
f (Suc i) u * (z-u) ^ i / (fact i)) =
f (Suc n) u * (z-u) ^ n / (fact n) +
f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / (fact (Suc n)) -
f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / (fact (Suc n))"
using Suc by simp
also have "\<dots> = f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n))"
proof -
have "(fact(Suc n)) *
(f(Suc n) u *(z-u) ^ n / (fact n) +
f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / (fact(Suc n)) -
f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / (fact(Suc n))) =
((fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / (fact n) +
((fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / (fact(Suc n))) -
((fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / (fact(Suc n))"
by (simp add: algebra_simps del: fact_Suc)
also have "\<dots> = ((fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / (fact n) +
(f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
(f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
by (simp del: fact_Suc)
also have "\<dots> = (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) +
(f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
(f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
by (simp only: fact_Suc of_nat_mult ac_simps) simp
also have "\<dots> = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
by (simp add: algebra_simps)
finally show ?thesis
by (simp add: mult_left_cancel [where c = "(fact (Suc n))", THEN iffD1] del: fact_Suc)
qed
finally show ?case .
qed
have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / (fact i)))
has_field_derivative f (Suc n) u * (z-u) ^ n / (fact n))
(at u within S)"
unfolding * [symmetric]
by (rule derivative_eq_intros assms \<open>u \<in> S\<close> refl | auto simp: field_simps)+
} note sum_deriv = this
{ fix u
assume u: "u \<in> closed_segment w z"
then have us: "u \<in> S"
by (metis wzs subsetD)
have "norm (f (Suc n) u) * norm (z - u) ^ n \<le> norm (f (Suc n) u) * norm (u - z) ^ n"
by (metis norm_minus_commute order_refl)
also have "\<dots> \<le> norm (f (Suc n) u) * norm (z - w) ^ n"
by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
also have "\<dots> \<le> B * norm (z - w) ^ n"
by (metis norm_ge_zero zero_le_power mult_right_mono B [OF us])
finally have "norm (f (Suc n) u) * norm (z - u) ^ n \<le> B * norm (z - w) ^ n" .
} note cmod_bound = this
have "(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)) = (\<Sum>i\<le>n. (f i z / (fact i)) * 0 ^ i)"
by simp
also have "\<dots> = f 0 z / (fact 0)"
by (subst sum_zero_power) simp
finally have "norm (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)))
\<le> norm ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) -
(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))"
by (simp add: norm_minus_commute)
also have "\<dots> \<le> B * norm (z - w) ^ n / (fact n) * norm (w - z)"
proof (rule field_differentiable_bound)
show "\<And>x. x \<in> closed_segment w z \<Longrightarrow>
((\<lambda>\<xi>. \<Sum>i\<le>n. f i \<xi> * (z - \<xi>) ^ i / fact i) has_field_derivative f (Suc n) x * (z - x) ^ n / fact n)
(at x within closed_segment w z)"
using DERIV_subset sum_deriv wzs by blast
qed (auto simp: norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
also have "\<dots> \<le> B * norm (z - w) ^ Suc n / (fact n)"
by (simp add: algebra_simps norm_minus_commute)
finally show ?thesis .
qed
lemma complex_Taylor:
assumes S: "convex S"
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)"
and B: "\<And>x. x \<in> S \<Longrightarrow> cmod (f (Suc n) x) \<le> B"
and w: "w \<in> S"
and z: "z \<in> S"
shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i))) \<le> B * cmod(z - w)^(Suc n) / fact n"
using assms by (rule field_Taylor)
text\<open>Something more like the traditional MVT for real components\<close>
lemma complex_mvt_line:
assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f'(u)) (at u)"
shows "\<exists>u. u \<in> closed_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
proof -
define \<phi> where "\<phi> \<equiv> \<lambda>t. (1 - t) *\<^sub>R w + t *\<^sub>R z"
have twz: "\<And>t. \<phi> t = w + t *\<^sub>R (z - w)"
by (simp add: \<phi>_def real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
note assms[unfolded has_field_derivative_def, derivative_intros]
have *: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
\<Longrightarrow> (Re \<circ> f \<circ> \<phi> has_derivative Re \<circ> (*) (f' (\<phi> x)) \<circ> (\<lambda>t. t *\<^sub>R (z - w)))
(at x within {0..1})"
unfolding \<phi>_def
by (intro derivative_eq_intros has_derivative_at_withinI)
(auto simp: in_segment scaleR_right_diff_distrib)
obtain x where "0<x" "x<1" "(Re \<circ> f \<circ> \<phi>) 1 -
(Re \<circ> f \<circ> \<phi>) 0 = (Re \<circ> (*) (f' (\<phi> x)) \<circ> (\<lambda>t. t *\<^sub>R (z - w))) (1 - 0)"
using mvt_simple [OF zero_less_one *] by force
then show ?thesis
unfolding \<phi>_def
by (smt (verit) comp_apply in_segment(1) scaleR_left_distrib scaleR_one scaleR_zero_left)
qed
lemma complex_Taylor_mvt:
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)"
shows "\<exists>u. u \<in> closed_segment w z \<and>
Re (f 0 z) =
Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / (fact i)) +
(f (Suc n) u * (z-u)^n / (fact n)) * (z - w))"
proof -
{ fix u
assume u: "u \<in> closed_segment w z"
have "(\<Sum>i = 0..n.
(f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) /
(fact i)) =
f (Suc 0) u -
(f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
(fact (Suc n)) +
(\<Sum>i = 0..n.
(f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) /
(fact (Suc i)))"
by (subst sum_Suc_reindex) simp
also have "\<dots> = f (Suc 0) u -
(f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
(fact (Suc n)) +
(\<Sum>i = 0..n.
f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i)) -
f (Suc i) u * (z-u) ^ i / (fact i))"
by (simp only: diff_divide_distrib fact_cancel ac_simps)
also have "\<dots> = f (Suc 0) u -
(f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) /
(fact (Suc n)) +
f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n)) - f (Suc 0) u"
by (subst sum_Suc_diff) auto
also have "\<dots> = f (Suc n) u * (z-u) ^ n / (fact n)"
by (simp only: algebra_simps diff_divide_distrib fact_cancel)
finally have *: "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i
- of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / (fact i)) =
f (Suc n) u * (z - u) ^ n / (fact n)" .
have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / (fact i)) has_field_derivative
f (Suc n) u * (z - u) ^ n / (fact n)) (at u)"
unfolding * [symmetric]
by (rule derivative_eq_intros assms u refl | auto simp: field_simps)+
}
then show ?thesis
apply (cut_tac complex_mvt_line [of w z "\<lambda>u. \<Sum>i = 0..n. f i u * (z-u) ^ i / (fact i)"
"\<lambda>u. (f (Suc n) u * (z-u)^n / (fact n))"])
apply (auto simp add: intro: open_closed_segment)
done
qed
end