# Theory Complex_Analysis_Basics

```(*  Author: John Harrison, Marco Maggesi, Graziano Gentili, Gianni Ciolli, Valentina Bruno
Ported from "hol_light/Multivariate/canal.ml" by L C Paulson (2014)
*)

section ‹Complex Analysis Basics›
text ‹Definitions of analytic and holomorphic functions, limit theorems, complex differentiation›

theory Complex_Analysis_Basics
imports Derivative "HOL-Library.Nonpos_Ints" Uncountable_Sets
begin

subsection✐‹tag unimportant›‹General lemmas›

lemma nonneg_Reals_cmod_eq_Re: "z ∈ ℝ⇩≥⇩0 ⟹ norm z = Re z"

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 ≠ bot" and "f differentiable at x within A"
shows   "vector_derivative (λz. 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 "((λx. 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 (λz. 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) ≥ b}"
and closed_halfspace_Re_le: "closed {z. Re(z) ≤ 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) ≥ b}"
and closed_halfspace_Im_le: "closed {z. Im(z) ≤ 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) *⇩R 𝗂) r ⊆ {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) *⇩R 𝗂) r ⊆ {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 ⊆ {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 ⊆ {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 (ℝ :: complex set)"
proof -
have "(ℝ :: 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 (ℝ ∩ {w. Re w ≤ x})"
by (simp add: closed_Int closed_complex_Reals closed_halfspace_Re_le)

lemma closed_nonpos_Reals_complex [simp]: "closed (ℝ⇩≤⇩0 :: complex set)"
proof -
have "ℝ⇩≤⇩0 = ℝ ∩ {z. Re(z) ≤ 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 (ℝ ∩ {w. x ≤ Re(w)})"
using closed_halfspace_Re_ge

lemma closed_nonneg_Reals_complex [simp]: "closed (ℝ⇩≥⇩0 :: complex set)"
proof -
have "ℝ⇩≥⇩0 = ℝ ∩ {z. Re(z) ≥ 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 ∈ ℝ. ¦Re w¦ ≤ r}"
proof -
have "{w ∈ ℝ. ¦Re w¦ ≤ r} = (ℝ ∩ {w. Re w ≤ r}) ∩ (ℝ ∩ {w. Re w ≥ -r})"
by auto
then show "closed {w ∈ ℝ. ¦Re w¦ ≤ 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 ⤏ l) F" and "¬ trivial_limit F" and "eventually P F" and "⋀a. P a ⟹ f a ∈ ℝ"
shows  "l ∈ ℝ"
using Lim_in_closed_set[OF closed_complex_Reals] assms
by (smt (verit) eventually_mono)

lemma real_lim_sequentially:
fixes l::complex
shows "(f ⤏ l) sequentially ⟹ (∃N. ∀n≥N. f n ∈ ℝ) ⟹ l ∈ ℝ"
by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially)

lemma real_series:
fixes l::complex
shows "f sums l ⟹ (⋀n. f n ∈ ℝ) ⟹ l ∈ ℝ"
unfolding sums_def
by (metis real_lim_sequentially sum_in_Reals)

lemma Lim_null_comparison_Re:
assumes "eventually (λx. norm(f x) ≤ Re(g x)) F" "(g ⤏ 0) F" shows "(f ⤏ 0) F"
using Lim_null_comparison assms tendsto_Re by fastforce

subsection‹Holomorphic functions›

definition✐‹tag important› holomorphic_on :: "[complex ⇒ complex, complex set] ⇒ bool"
(infixl "(holomorphic'_on)" 50)
where "f holomorphic_on s ≡ ∀x∈s. f field_differentiable (at x within s)"

named_theorems✐‹tag important› holomorphic_intros "structural introduction rules for holomorphic_on"

lemma holomorphic_onI [intro?]: "(⋀x. x ∈ s ⟹ f field_differentiable (at x within s)) ⟹ f holomorphic_on s"

lemma holomorphic_onD [dest?]: "⟦f holomorphic_on s; x ∈ s⟧ ⟹ f field_differentiable (at x within s)"

lemma holomorphic_on_imp_differentiable_on:
"f holomorphic_on s ⟹ f differentiable_on s"
unfolding holomorphic_on_def differentiable_on_def

lemma holomorphic_on_imp_differentiable_at:
"⟦f holomorphic_on s; open s; x ∈ s⟧ ⟹ f field_differentiable (at x)"
using at_within_open holomorphic_on_def by fastforce

lemma holomorphic_on_empty [holomorphic_intros]: "f holomorphic_on {}"

lemma holomorphic_on_open:
"open s ⟹ f holomorphic_on s ⟷ (∀x ∈ s. ∃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 "⋀n. n ∈ I ⟹ f holomorphic_on A n" "⋀n. n ∈ I ⟹ open (A n)"
shows   "f holomorphic_on (⋃n∈I. A n)"
by (metis UN_E assms holomorphic_on_open open_UN)

lemma holomorphic_on_imp_continuous_on:
"f holomorphic_on s ⟹ 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∈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 ⟹ t ⊆ s ⟹ f holomorphic_on t"
unfolding holomorphic_on_def
by (metis field_differentiable_within_subset subsetD)

lemma holomorphic_transform: "⟦f holomorphic_on s; ⋀x. x ∈ s ⟹ f x = g x⟧ ⟹ g holomorphic_on s"
by (metis field_differentiable_transform_within linordered_field_no_ub holomorphic_on_def)

lemma holomorphic_cong: "s = t ==> (⋀x. x ∈ s ⟹ f x = g x) ⟹ f holomorphic_on s ⟷ 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]: "(λz. c) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_const)

lemma holomorphic_on_ident [simp, holomorphic_intros]: "(λx. 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 ⟹ g holomorphic_on (f ` s) ⟹ (g ∘ 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 ⟹ g holomorphic_on t ⟹ f ` s ⊆ t ⟹ (g ∘ f) holomorphic_on s"
by (metis holomorphic_on_compose holomorphic_on_subset)

lemma holomorphic_on_balls_imp_entire:
assumes "¬bdd_above A" "⋀r. r ∈ A ⟹ 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 ‹¬bdd_above A› obtain r where r: "r ∈ 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 ∈ 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 "⋀r. r > 0 ⟹ f holomorphic_on ball c r"
shows   "f holomorphic_on B"
proof (rule holomorphic_on_balls_imp_entire)
show "¬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 ⟹ (λz. -(f z)) holomorphic_on A"
by (metis field_differentiable_minus holomorphic_on_def)

"⟦f holomorphic_on A; g holomorphic_on A⟧ ⟹ (λz. f z + g z) holomorphic_on A"

lemma holomorphic_on_diff [holomorphic_intros]:
"⟦f holomorphic_on A; g holomorphic_on A⟧ ⟹ (λz. f z - g z) holomorphic_on A"
unfolding holomorphic_on_def by (metis field_differentiable_diff)

lemma holomorphic_on_mult [holomorphic_intros]:
"⟦f holomorphic_on A; g holomorphic_on A⟧ ⟹ (λz. f z * g z) holomorphic_on A"
unfolding holomorphic_on_def by (metis field_differentiable_mult)

lemma holomorphic_on_inverse [holomorphic_intros]:
"⟦f holomorphic_on A; ⋀z. z ∈ A ⟹ f z ≠ 0⟧ ⟹ (λz. inverse (f z)) holomorphic_on A"
unfolding holomorphic_on_def by (metis field_differentiable_inverse)

lemma holomorphic_on_divide [holomorphic_intros]:
"⟦f holomorphic_on A; g holomorphic_on A; ⋀z. z ∈ A ⟹ g z ≠ 0⟧ ⟹ (λz. 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 ⟹ (λz. (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 ≥ 0 ∨ (∀x∈A. f x ≠ 0)" and f: "f holomorphic_on A"
shows   "(λx. f x powi n) holomorphic_on A"
proof (cases "n ≥ 0")
case True
have "(λx. f x ^ nat n) holomorphic_on A"
with True show ?thesis
next
case False
hence "(λx. inverse (f x ^ nat (-n))) holomorphic_on A"
using nz by (auto intro!: holomorphic_intros f)
with False show ?thesis
qed

lemma holomorphic_on_sum [holomorphic_intros]:
"(⋀i. i ∈ I ⟹ (f i) holomorphic_on A) ⟹ (λx. sum (λi. f i x) I) holomorphic_on A"
unfolding holomorphic_on_def by (metis field_differentiable_sum)

lemma holomorphic_on_prod [holomorphic_intros]:
"(⋀i. i ∈ I ⟹ (f i) holomorphic_on A) ⟹ (λx. prod (λi. 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 ⟹ (λs. 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 ⟹ (λx. c *⇩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 ∪ 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 "⋀z. z ∈ A ⟹ z ∈ B ⟹ f z = g z"
shows   "(λz. if z ∈ A then f z else g z) holomorphic_on (A ∪ B)" (is "?h holomorphic_on _")
proof (intro holomorphic_on_Un)
note ‹f holomorphic_on A›
also have "f holomorphic_on A ⟷ ?h holomorphic_on A"
by (intro holomorphic_cong) auto
finally show … .
next
note ‹g holomorphic_on B›
also have "g holomorphic_on B ⟷ ?h holomorphic_on B"
using assms by (intro holomorphic_cong) auto
finally show … .
qed (use assms in auto)

lemma holomorphic_derivI:
"⟦f holomorphic_on S; open S; x ∈ S⟧ ⟹ (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:
"⟦f holomorphic_on s; g holomorphic_on s; open s; z ∈ s; ⋀w. w ∈ s ⟹ f w = g w⟧
⟹ 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 ∘ f ∘ 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" "ξ ∈ S" "deriv f ξ ≠ 0"
shows "¬ f constant_on S"
by (rule nonzero_deriv_nonconstant [of f "deriv f ξ" ξ S])
(use assms in ‹auto simp: holomorphic_derivI›)

subsection‹Analyticity on a set›

definition✐‹tag important› analytic_on (infixl "(analytic'_on)" 50)
where "f analytic_on S ≡ ∀x ∈ S. ∃ε. 0 < ε ∧ f holomorphic_on (ball x ε)"

named_theorems✐‹tag important› analytic_intros "introduction rules for proving analyticity"

lemma analytic_imp_holomorphic: "f analytic_on S ⟹ 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 ⟹ f analytic_on S ⟷ 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 ⟹ x ∈ S ⟹ 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 ≠ c"
shows   "eventually (λy. f y ≠ c) (at x)"
using analytic_at_imp_isCont assms isContD tendsto_imp_eventually_ne by blast

lemma analytic_on_subset: "f analytic_on S ⟹ T ⊆ S ⟹ f analytic_on T"
by (auto simp: analytic_on_def)

lemma analytic_on_Un: "f analytic_on (S ∪ T) ⟷ f analytic_on S ∧ f analytic_on T"
by (auto simp: analytic_on_def)

lemma analytic_on_Union: "f analytic_on (⋃𝒯) ⟷ (∀T ∈ 𝒯. f analytic_on T)"
by (auto simp: analytic_on_def)

lemma analytic_on_UN: "f analytic_on (⋃i∈I. S i) ⟷ (∀i∈I. f analytic_on (S i))"
by (auto simp: analytic_on_def)

lemma analytic_on_holomorphic:
"f analytic_on S ⟷ (∃T. open T ∧ S ⊆ T ∧ f holomorphic_on T)"
(is "?lhs = ?rhs")
proof -
have "?lhs ⟷ (∃T. open T ∧ S ⊆ T ∧ f analytic_on T)"
proof safe
assume "f analytic_on S"
then have "∀x ∈ ⋃{U. open U ∧ f analytic_on U}. ∃ε>0. f holomorphic_on ball x ε"
using analytic_on_def by force
moreover have "S ⊆ ⋃{U. open U ∧ f analytic_on U}"
using ‹f analytic_on S›
by (smt (verit, best) open_ball Union_iff analytic_on_def analytic_on_open centre_in_ball mem_Collect_eq subsetI)
ultimately show "∃T. open T ∧ S ⊆ T ∧ 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 ⊆ T" "f analytic_on T"
then show "f analytic_on S"
by (metis analytic_on_subset)
qed
also have "… ⟷ ?rhs"
by (auto simp: analytic_on_open)
finally show ?thesis .
qed

lemma analytic_on_linear [analytic_intros,simp]: "((*) c) analytic_on S"

lemma analytic_on_const [analytic_intros,simp]: "(λz. c) analytic_on S"
by (metis analytic_on_def holomorphic_on_const zero_less_one)

lemma analytic_on_ident [analytic_intros,simp]: "(λx. x) analytic_on S"

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 ⟹ (λw. x *⇩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 ∘ f) analytic_on S"
unfolding analytic_on_def
proof (intro ballI)
fix x
assume x: "x ∈ 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 ⊆ ball (f x) e'"
by (auto simp: continuous_at_ball)
have "g ∘ 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 "∃e>0. g ∘ f holomorphic_on ball x e"
by (metis d e min_less_iff_conj)
qed

lemma analytic_on_compose_gen:
"f analytic_on S ⟹ g analytic_on T ⟹ (⋀z. z ∈ S ⟹ f z ∈ T)
⟹ g ∘ 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 ⟹ (λz. -(f z)) analytic_on S"
by (metis analytic_on_holomorphic holomorphic_on_minus)

assumes f: "f analytic_on S"
and g: "g analytic_on S"
shows "(λz. f z + g z) analytic_on S"
unfolding analytic_on_def
proof (intro ballI)
fix z
assume z: "z ∈ 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 "(λz. 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 "∃e>0. (λz. 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 "(λz. f z * g z) analytic_on S"
unfolding analytic_on_def
proof (intro ballI)
fix z
assume z: "z ∈ 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 "(λz. 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 "∃e>0. (λz. 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 "(λz. f z - g z) analytic_on S"
proof -
have "(λz. - g z) analytic_on S"
then have "(λz. f z + - g z) analytic_on S"
then show ?thesis
by fastforce
qed

lemma analytic_on_inverse [analytic_intros]:
assumes f: "f analytic_on S"
and nz: "(⋀z. z ∈ S ⟹ f z ≠ 0)"
shows "(λz. inverse (f z)) analytic_on S"
unfolding analytic_on_def
proof (intro ballI)
fix z
assume z: "z ∈ 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': "⋀y. dist z y < e' ⟹ f y ≠ 0"
by (metis open_ball centre_in_ball continuous_on_open_avoid e z nz)
have "(λz. inverse (f z)) holomorphic_on ball z (min e e')"
using fh holomorphic_on_inverse holomorphic_on_open nz' by fastforce
then show "∃e>0. (λz. 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: "(⋀z. z ∈ S ⟹ g z ≠ 0)"
shows "(λz. 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 ⟹ (λz. (f z) ^ n) analytic_on S"
by (induct n) (auto simp: analytic_on_mult)

lemma analytic_on_power_int [analytic_intros]:
assumes nz: "n ≥ 0 ∨ (∀x∈A. f x ≠ 0)" and f: "f analytic_on A"
shows   "(λx. f x powi n) analytic_on A"
proof (cases "n ≥ 0")
case True
have "(λx. f x ^ nat n) analytic_on A"
using analytic_on_power f by blast
with True show ?thesis
next
case False
hence "(λx. inverse (f x ^ nat (-n))) analytic_on A"
using nz by (auto intro!: analytic_intros f)
with False show ?thesis
qed

lemma analytic_on_sum [analytic_intros]:
"(⋀i. i ∈ I ⟹ (f i) analytic_on S) ⟹ (λx. sum (λi. 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]:
"(⋀i. i ∈ I ⟹ (f i) analytic_on S) ⟹ (λx. prod (λi. 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 ⟹ (λw. 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 ⊆ T"
and [simp]: "⋀z. z ∈ S ⟹ g (f z) = z"
and "w ∈ 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"
also have "… = deriv (g ∘ f) w"
using assms
by (metis analytic_on_imp_differentiable_at analytic_on_open deriv_chain image_subset_iff)
also have "… = deriv id w"
proof (rule complex_derivative_transform_within_open [where s=S])
show "g ∘ f holomorphic_on S"
by (rule assms holomorphic_on_compose_gen holomorphic_intros)+
qed (use assms in auto)
also have "… = 1"
by simp
finally show ?thesis .
qed

subsection✐‹tag unimportant›‹Analyticity at a point›

lemma analytic_at_ball:
"f analytic_on {z} ⟷ (∃e. 0<e ∧ f holomorphic_on ball z e)"
by (metis analytic_on_def singleton_iff)

lemma analytic_at:
"f analytic_on {z} ⟷ (∃s. open s ∧ z ∈ s ∧ 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 ∈ A"
shows   "f analytic_on {z}"
using assms by (meson analytic_at)

lemma analytic_on_analytic_at:
"f analytic_on s ⟷ (∀z ∈ s. f analytic_on {z})"
by (metis analytic_at_ball analytic_on_def)

lemma analytic_at_two:
"f analytic_on {z} ∧ g analytic_on {z} ⟷
(∃S. open S ∧ z ∈ S ∧ f holomorphic_on S ∧ g holomorphic_on S)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then obtain S T
where st: "open S" "z ∈ S" "f holomorphic_on S"
"open T" "z ∈ 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
qed

subsection✐‹tag unimportant›‹Combining theorems for derivative with ``analytic at'' hypotheses›

lemma
assumes "f analytic_on {z}" "g analytic_on {z}"
shows complex_derivative_add_at: "deriv (λw. f w + g w) z = deriv f z + deriv g z"
and complex_derivative_diff_at: "deriv (λw. f w - g w) z = deriv f z - deriv g z"
and complex_derivative_mult_at: "deriv (λw. f w * g w) z =
f z * deriv g z + deriv f z * g z"
proof -
show "deriv (λw. f w + g w) z = deriv f z + deriv g z"
using analytic_on_imp_differentiable_at assms by auto
show "deriv (λw. 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 ∈ S" "f holomorphic_on S" "g holomorphic_on S"
using assms by (metis analytic_at_two)
then show "deriv (λw. 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} ⟹  deriv (λw. c * f w) z = c * deriv f z"
by (auto simp: complex_derivative_mult_at)

lemma deriv_cmult_right_at:
"f analytic_on {z} ⟹  deriv (λw. f w * c) z = deriv f z * c"
by (auto simp: complex_derivative_mult_at)

subsection✐‹tag unimportant›‹Complex differentiation of sequences and series›

(* TODO: Could probably be simplified using Uniform_Limit *)
lemma has_complex_derivative_sequence:
fixes S :: "complex set"
assumes cvs: "convex S"
and df:  "⋀n x. x ∈ S ⟹ (f n has_field_derivative f' n x) (at x within S)"
and conv: "⋀e. 0 < e ⟹ ∃N. ∀n x. n ≥ N ⟶ x ∈ S ⟶ norm (f' n x - g' x) ≤ e"
and "∃x l. x ∈ S ∧ ((λn. f n x) ⤏ l) sequentially"
shows "∃g. ∀x ∈ S. ((λn. f n x) ⤏ g x) sequentially ∧
(g has_field_derivative (g' x)) (at x within S)"
proof -
from assms obtain x l where x: "x ∈ S" and tf: "((λn. f n x) ⤏ l) sequentially"
by blast
show ?thesis
unfolding has_field_derivative_def
proof (rule has_derivative_sequence [OF cvs _ _ x])
show "(λn. f n x) ⇢ l"
by (rule tf)
next
have **: "∃N. ∀n≥N. ∀x∈S. ∀h. cmod (f' n x * h - g' x * h) ≤ ε * cmod h"
if "ε > 0" for ε::real
by (metis that left_diff_distrib mult_right_mono norm_ge_zero norm_mult conv)
show "⋀e. e > 0 ⟹ ∀⇩F n in sequentially. ∀x∈S. ∀h. cmod (f' n x * h - g' x * h) ≤ 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:  "⋀n x. x ∈ S ⟹ (f n has_field_derivative f' n x) (at x within S)"
and conv: "⋀e. 0 < e ⟹ ∃N. ∀n x. n ≥ N ⟶ x ∈ S
⟶ cmod ((∑i<n. f' i x) - g' x) ≤ e"
and "∃x l. x ∈ S ∧ ((λn. f n x) sums l)"
shows "∃g. ∀x ∈ S. ((λn. f n x) sums g x) ∧ ((g has_field_derivative g' x) (at x within S))"
proof -
from assms obtain x l where x: "x ∈ S" and sf: "((λn. f n x) sums l)"
by blast
{ fix ε::real assume e: "ε > 0"
then obtain N where N: "∀n x. n ≥ N ⟶ x ∈ S
⟶ cmod ((∑i<n. f' i x) - g' x) ≤ ε"
by (metis conv)
have "∃N. ∀n≥N. ∀x∈S. ∀h. cmod ((∑i<n. h * f' i x) - g' x * h) ≤ ε * cmod h"
proof (rule exI [of _ N], clarify)
fix n y h
assume "N ≤ n" "y ∈ S"
have "cmod h * cmod ((∑i<n. f' i y) - g' y) ≤ cmod h * ε"
by (simp add: N ‹N ≤ n› ‹y ∈ S› mult_le_cancel_left)
then show "cmod ((∑i<n. h * f' i y) - g' y * h) ≤ ε * 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 ∈ S"
then show "((f n) has_derivative (λz. z * f' n x)) (at x within S)"
by (metis df has_field_derivative_def mult_commute_abs)
next show " ((λn. f n x) sums l)"
by (rule sf)
next show "⋀e. e>0 ⟹ ∀⇩F n in sequentially. ∀x∈S. ∀h. cmod ((∑i<n. h * f' i x) - g' x * h) ≤ e * cmod h"
unfolding eventually_sequentially by (blast intro: **)
qed
qed

subsection✐‹tag unimportant› ‹Taylor on Complex Numbers›

lemma sum_Suc_reindex:
fixes f :: "nat ⇒ 'a::ab_group_add"
shows "sum f {0..n} = f 0 - f (Suc n) + sum (λi. f (Suc i)) {0..n}"
by (induct n) auto

lemma field_Taylor:
assumes S: "convex S"
and f: "⋀i x. x ∈ S ⟹ i ≤ n ⟹ (f i has_field_derivative f (Suc i) x) (at x within S)"
and B: "⋀x. x ∈ S ⟹ norm (f (Suc n) x) ≤ B"
and w: "w ∈ S"
and z: "z ∈ S"
shows "norm(f 0 z - (∑i≤n. f i w * (z-w) ^ i / (fact i)))
≤ B * norm(z - w)^(Suc n) / fact n"
proof -
have wzs: "closed_segment w z ⊆ S" using assms
by (metis convex_contains_segment)
{ fix u
assume "u ∈ closed_segment w z"
then have "u ∈ S"
by (metis wzs subsetD)
have *: "(∑i≤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 "(∑i≤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 "… = 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 "… = ((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 "… = (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 "… = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
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 "((λv. (∑i≤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 ‹u ∈ S› refl | auto simp: field_simps)+
} note sum_deriv = this
{ fix u
assume u: "u ∈ closed_segment w z"
then have us: "u ∈ S"
by (metis wzs subsetD)
have "norm (f (Suc n) u) * norm (z - u) ^ n ≤ norm (f (Suc n) u) * norm (u - z) ^ n"
by (metis norm_minus_commute order_refl)
also have "… ≤ 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 "… ≤ 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 ≤ B * norm (z - w) ^ n" .
} note cmod_bound = this
have "(∑i≤n. f i z * (z - z) ^ i / (fact i)) = (∑i≤n. (f i z / (fact i)) * 0 ^ i)"
by simp
also have "… = f 0 z / (fact 0)"
by (subst sum_zero_power) simp
finally have "norm (f 0 z - (∑i≤n. f i w * (z - w) ^ i / (fact i)))
≤ norm ((∑i≤n. f i w * (z - w) ^ i / (fact i)) -
(∑i≤n. f i z * (z - z) ^ i / (fact i)))"
also have "… ≤ B * norm (z - w) ^ n / (fact n) * norm (w - z)"
proof (rule field_differentiable_bound)
show "⋀x. x ∈ closed_segment w z ⟹
((λξ. ∑i≤n. f i ξ * (z - ξ) ^ 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 "…  ≤ B * norm (z - w) ^ Suc n / (fact n)"
finally show ?thesis .
qed

lemma complex_Taylor:
assumes S: "convex S"
and f: "⋀i x. x ∈ S ⟹ i ≤ n ⟹ (f i has_field_derivative f (Suc i) x) (at x within S)"
and B: "⋀x. x ∈ S ⟹ cmod (f (Suc n) x) ≤ B"
and w: "w ∈ S"
and z: "z ∈ S"
shows "cmod(f 0 z - (∑i≤n. f i w * (z-w) ^ i / (fact i))) ≤ B * cmod(z - w)^(Suc n) / fact n"
using assms by (rule field_Taylor)

text‹Something more like the traditional MVT for real components›

lemma complex_mvt_line:
assumes "⋀u. u ∈ closed_segment w z ⟹ (f has_field_derivative f'(u)) (at u)"
shows "∃u. u ∈ closed_segment w z ∧ Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
proof -
define φ where "φ ≡ λt. (1 - t) *⇩R w + t *⇩R z"
have twz: "⋀t. φ t = w + t *⇩R (z - w)"
by (simp add: φ_def real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
note assms[unfolded has_field_derivative_def, derivative_intros]
have *: "⋀x. ⟦0 ≤ x; x ≤ 1⟧
⟹ (Re ∘ f ∘ φ has_derivative Re ∘ (*) (f' (φ x)) ∘ (λt. t *⇩R (z - w)))
(at x within {0..1})"
unfolding φ_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 ∘ f ∘ φ) 1 -
(Re ∘ f ∘ φ) 0 = (Re ∘ (*) (f' (φ x)) ∘ (λt. t *⇩R (z - w))) (1 - 0)"
using mvt_simple [OF zero_less_one *] by force
then show ?thesis
unfolding φ_def
by (smt (verit) comp_apply in_segment(1) scaleR_left_distrib scaleR_one scaleR_zero_left)
qed

lemma complex_Taylor_mvt:
assumes "⋀i x. ⟦x ∈ closed_segment w z; i ≤ n⟧ ⟹ ((f i) has_field_derivative f (Suc i) x) (at x)"
shows "∃u. u ∈ closed_segment w z ∧
Re (f 0 z) =
Re ((∑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 ∈ closed_segment w z"
have "(∑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)) +
(∑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 "… = 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)) +
(∑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 "… = 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 "… = f (Suc n) u * (z-u) ^ n / (fact n)"
by (simp only: algebra_simps diff_divide_distrib fact_cancel)
finally have *: "(∑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 "((λu. ∑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 "λu. ∑i = 0..n. f i u * (z-u) ^ i / (fact i)"
"λu. (f (Suc n) u * (z-u)^n / (fact n))"])
apply (auto simp add: intro: open_closed_segment)
done
qed

end
```