# Theory Cauchy_Integral_Formula

```section ‹Cauchy's Integral Formula›
theory Cauchy_Integral_Formula
imports Winding_Numbers
begin

subsection‹Proof›

lemma Cauchy_integral_formula_weak:
assumes S: "convex S" and "finite k" and conf: "continuous_on S f"
and fcd: "(⋀x. x ∈ interior S - k ⟹ f field_differentiable at x)"
and z: "z ∈ interior S - k" and vpg: "valid_path γ"
and pasz: "path_image γ ⊆ S - {z}" and loop: "pathfinish γ = pathstart γ"
shows "((λw. f w / (w - z)) has_contour_integral (2*pi * 𝗂 * winding_number γ z * f z)) γ"
proof -
let ?fz = "λw. (f w - f z)/(w - z)"
obtain f' where f': "(f has_field_derivative f') (at z)"
using fcd [OF z] by (auto simp: field_differentiable_def)
have pas: "path_image γ ⊆ S" and znotin: "z ∉ path_image γ" using pasz by blast+
have c: "continuous (at x within S) (λw. if w = z then f' else (f w - f z) / (w - z))" if "x ∈ S" for x
proof (cases "x = z")
case True then show ?thesis
using LIM_equal [of "z" ?fz "λw. if w = z then f' else ?fz w"] has_field_derivativeD [OF f']
by (force simp add: continuous_within Lim_at_imp_Lim_at_within)
next
case False
then have dxz: "dist x z > 0" by auto
have cf: "continuous (at x within S) f"
using conf continuous_on_eq_continuous_within that by blast
have "continuous (at x within S) (λw. (f w - f z) / (w - z))"
by (rule cf continuous_intros | simp add: False)+
then show ?thesis
using continuous_transform_within [OF _ dxz that] by (force simp: dist_commute)
qed
have fink': "finite (insert z k)" using ‹finite k› by blast
have *: "((λw. if w = z then f' else ?fz w) has_contour_integral 0) γ"
proof (rule Cauchy_theorem_convex [OF _ S fink' _ vpg pas loop])
show "(λw. if w = z then f' else ?fz w) field_differentiable at w"
if "w ∈ interior S - insert z k" for w
proof (rule field_differentiable_transform_within)
show "(λw. ?fz w) field_differentiable at w"
using that by (intro derivative_intros fcd; simp)
qed (use that in ‹auto simp add: dist_pos_lt dist_commute›)
qed (use c in ‹force simp: continuous_on_eq_continuous_within›)
show ?thesis
apply (rule has_contour_integral_eq)
using znotin has_contour_integral_add [OF has_contour_integral_lmul [OF has_contour_integral_winding_number [OF vpg znotin], of "f z"] *]
apply (auto simp: ac_simps divide_simps)
done
qed

theorem Cauchy_integral_formula_convex_simple:
assumes "convex S" and holf: "f holomorphic_on S" and "z ∈ interior S" "valid_path γ" "path_image γ ⊆ S - {z}"
"pathfinish γ = pathstart γ"
shows "((λw. f w / (w - z)) has_contour_integral (2*pi * 𝗂 * winding_number γ z * f z)) γ"
proof -
have "⋀x. x ∈ interior S ⟹ f field_differentiable at x"
using holf at_within_interior holomorphic_onD interior_subset by fastforce
then show ?thesis
using assms
by (intro Cauchy_integral_formula_weak [where k = "{}"]) (auto simp: holomorphic_on_imp_continuous_on)
qed

text‹ Hence the Cauchy formula for points inside a circle.›

theorem Cauchy_integral_circlepath:
assumes contf: "continuous_on (cball z r) f" and holf: "f holomorphic_on (ball z r)" and wz: "norm(w - z) < r"
shows "((λu. f u/(u - w)) has_contour_integral (2 * of_real pi * 𝗂 * f w))
(circlepath z r)"
proof -
have "r > 0"
using assms le_less_trans norm_ge_zero by blast
have "((λu. f u / (u - w)) has_contour_integral (2 * pi) * 𝗂 * winding_number (circlepath z r) w * f w)
(circlepath z r)"
proof (rule Cauchy_integral_formula_weak [where S = "cball z r" and k = "{}"])
show "⋀x. x ∈ interior (cball z r) - {} ⟹
f field_differentiable at x"
using holf holomorphic_on_imp_differentiable_at by auto
have "w ∉ sphere z r"
by simp (metis dist_commute dist_norm not_le order_refl wz)
then show "path_image (circlepath z r) ⊆ cball z r - {w}"
using ‹r > 0› by (auto simp add: cball_def sphere_def)
qed (use wz in ‹simp_all add: dist_norm norm_minus_commute contf›)
then show ?thesis
qed

corollary✐‹tag unimportant› Cauchy_integral_circlepath_simple:
assumes "f holomorphic_on cball z r" "norm(w - z) < r"
shows "((λu. f u/(u - w)) has_contour_integral (2 * of_real pi * 𝗂 * f w))
(circlepath z r)"
using assms by (force simp: holomorphic_on_imp_continuous_on holomorphic_on_subset Cauchy_integral_circlepath)

subsection✐‹tag unimportant› ‹General stepping result for derivative formulas›

lemma Cauchy_next_derivative:
assumes "continuous_on (path_image γ) f'"
and leB: "⋀t. t ∈ {0..1} ⟹ norm (vector_derivative γ (at t)) ≤ B"
and int: "⋀w. w ∈ S - path_image γ ⟹ ((λu. f' u / (u - w)^k) has_contour_integral f w) γ"
and k: "k ≠ 0"
and "open S"
and γ: "valid_path γ"
and w: "w ∈ S - path_image γ"
shows "(λu. f' u / (u - w)^(Suc k)) contour_integrable_on γ"
and "(f has_field_derivative (k * contour_integral γ (λu. f' u/(u - w)^(Suc k))))
(at w)"  (is "?thes2")
proof -
have "open (S - path_image γ)" using ‹open S› closed_valid_path_image γ by blast
then obtain d where "d>0" and d: "ball w d ⊆ S - path_image γ" using w
using open_contains_ball by blast
have [simp]: "⋀n. cmod (1 + of_nat n) = 1 + of_nat n"
by (metis norm_of_nat of_nat_Suc)
have cint: "(λz. (f' z / (z - x) ^ k - f' z / (z - w) ^ k) / (x * k - w * k)) contour_integrable_on γ"
if "x ≠ w" "cmod (x - w) < d" for x
proof -
have "x ∈ S - path_image γ"
by (metis d dist_commute dist_norm mem_ball subsetD that(2))
then show ?thesis
using contour_integrable_diff contour_integrable_div contour_integrable_on_def int w
by meson
qed
have 1: "∀⇩F n in at w. (λx. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k)
contour_integrable_on γ"
unfolding eventually_at
apply (rule_tac x=d in exI)
apply (simp add: ‹d > 0› dist_norm field_simps cint)
done
have bim_g: "bounded (image f' (path_image γ))"
by (simp add: compact_imp_bounded compact_continuous_image compact_valid_path_image assms)
then obtain C where "C > 0" and C: "⋀x. ⟦0 ≤ x; x ≤ 1⟧ ⟹ cmod (f' (γ x)) ≤ C"
by (force simp: bounded_pos path_image_def)
have twom: "∀⇩F n in at w.
∀x∈path_image γ.
cmod ((inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k - inverse (x - w) ^ Suc k) < e"
if "0 < e" for e
proof -
have *: "cmod ((inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k) - inverse (x - w) ^ Suc k)   < e"
if x: "x ∈ path_image γ" and "u ≠ w" and uwd: "cmod (u - w) < d/2"
and uw_less: "cmod (u - w) < e * (d/2) ^ (k+2) / (1 + real k)"
for u x
proof -
define ff where [abs_def]:
"ff n w =
(if n = 0 then inverse(x - w)^k
else if n = 1 then k / (x - w)^(Suc k)
else (k * of_real(Suc k)) / (x - w)^(k + 2))" for n :: nat and w
have km1: "⋀z::complex. z ≠ 0 ⟹ z ^ (k - Suc 0) = z ^ k / z"
by (simp add: field_simps) (metis Suc_pred ‹k ≠ 0› neq0_conv power_Suc)
have ff1: "(ff i has_field_derivative ff (Suc i) z) (at z within ball w (d/2))"
if "z ∈ ball w (d/2)" "i ≤ 1" for i z
proof -
have "z ∉ path_image γ"
using ‹x ∈ path_image γ› d that ball_divide_subset_numeral by blast
then have xz[simp]: "x ≠ z" using ‹x ∈ path_image γ› by blast
then have neq: "x * x + z * z ≠ x * (z * 2)"
by (blast intro: dest!: sum_sqs_eq)
with xz have "⋀v. v ≠ 0 ⟹ (x * x + z * z) * v ≠ (x * (z * 2) * v)" by auto
then have neqq: "⋀v. v ≠ 0 ⟹ x * (x * v) + z * (z * v) ≠ x * (z * (2 * v))"
show ?thesis using ‹i ≤ 1›
apply (simp add: ff_def dist_norm Nat.le_Suc_eq km1, safe)
apply (rule derivative_eq_intros | simp add: km1 | simp add: field_simps neq neqq)+
done
qed
{ fix a::real and b::real assume ab: "a > 0" "b > 0"
then have "k * (1 + real k) * (1 / a) ≤ k * (1 + real k) * (4 / b) ⟷ b ≤ 4 * a"
by (subst mult_le_cancel_left_pos)
(use ‹k ≠ 0› in ‹auto simp: divide_simps›)
with ab have "real k * (1 + real k) / a ≤ (real k * 4 + real k * real k * 4) / b ⟷ b ≤ 4 * a"
} note canc = this
have ff2: "cmod (ff (Suc 1) v) ≤ real (k * (k + 1)) / (d/2) ^ (k + 2)"
if "v ∈ ball w (d/2)" for v
proof -
have lessd: "⋀z. cmod (γ z - v) < d/2 ⟹ cmod (w - γ z) < d"
by (metis that norm_minus_commute norm_triangle_half_r dist_norm mem_ball)
have "d/2 ≤ cmod (x - v)" using d x that
using lessd d x
by (auto simp add: dist_norm path_image_def ball_def not_less [symmetric] del: divide_const_simps)
then have "d ≤ cmod (x - v) * 2"
then have dpow_le: "d ^ (k+2) ≤ (cmod (x - v) * 2) ^ (k+2)"
using ‹0 < d› order_less_imp_le power_mono by blast
have "x ≠ v" using that
using ‹x ∈ path_image γ› ball_divide_subset_numeral d by fastforce
then show ?thesis
using ‹d > 0› apply (simp add: ff_def norm_mult norm_divide norm_power dist_norm canc)
using dpow_le apply (simp add: field_split_simps)
done
qed
have ub: "u ∈ ball w (d/2)"
using uwd by (simp add: dist_commute dist_norm)
have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
≤ (real k * 4 + real k * real k * 4) * (cmod (u - w) * cmod (u - w)) / (d * (d * (d/2) ^ k))"
using complex_Taylor [OF _ ff1 ff2 _ ub, of w, simplified]
by (simp add: ff_def ‹0 < d›)
then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
≤ (cmod (u - w) * real k) * (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)"
then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
/ (cmod (u - w) * real k)
≤ (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)"
using ‹k ≠ 0› ‹u ≠ w› by (simp add: mult_ac zero_less_mult_iff pos_divide_le_eq)
also have "… < e"
using uw_less ‹0 < d› by (simp add: mult_ac divide_simps)
finally have e: "cmod (inverse (x-u)^k - (inverse (x-w)^k + of_nat k * (u-w) / ((x-w) * (x-w)^k)))
/ cmod ((u - w) * real k)   <   e"
have "x ≠ u"
using uwd ‹0 < d› x d by (force simp: dist_norm ball_def norm_minus_commute)
show ?thesis
apply (rule le_less_trans [OF _ e])
using ‹k ≠ 0› ‹x ≠ u› ‹u ≠ w›
apply (simp add: field_simps norm_divide [symmetric])
done
qed
show ?thesis
unfolding eventually_at
apply (rule_tac x = "min (d/2) ((e*(d/2)^(k + 2))/(Suc k))" in exI)
apply (force simp: ‹d > 0› dist_norm that simp del: power_Suc intro: *)
done
qed
have 2: "uniform_limit (path_image γ) (λn x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) (λx. f' x / (x - w) ^ Suc k) (at w)"
unfolding uniform_limit_iff dist_norm
proof clarify
fix e::real
assume "0 < e"
have *: "cmod (f' (γ x) * (inverse (γ x - u) ^ k - inverse (γ x - w) ^ k) / ((u - w) * k) -
f' (γ x) / ((γ x - w) * (γ x - w) ^ k)) < e"
if ec: "cmod ((inverse (γ x - u) ^ k - inverse (γ x - w) ^ k) / ((u - w) * k) -
inverse (γ x - w) * inverse (γ x - w) ^ k) < e / C"
and x: "0 ≤ x" "x ≤ 1"
for u x
proof (cases "(f' (γ x)) = 0")
case True then show ?thesis by (simp add: ‹0 < e›)
next
case False
have "cmod (f' (γ x) * (inverse (γ x - u) ^ k - inverse (γ x - w) ^ k) / ((u - w) * k) -
f' (γ x) / ((γ x - w) * (γ x - w) ^ k)) =
cmod (f' (γ x) * ((inverse (γ x - u) ^ k - inverse (γ x - w) ^ k) / ((u - w) * k) -
inverse (γ x - w) * inverse (γ x - w) ^ k))"
also have "… = cmod (f' (γ x)) *
cmod ((inverse (γ x - u) ^ k - inverse (γ x - w) ^ k) / ((u - w) * k) -
inverse (γ x - w) * inverse (γ x - w) ^ k)"
also have "… < cmod (f' (γ x)) * (e/C)"
using False mult_strict_left_mono [OF ec] by force
also have "… ≤ e" using C
by (metis False ‹0 < e› frac_le less_eq_real_def mult.commute pos_le_divide_eq x zero_less_norm_iff)
finally show ?thesis .
qed
show "∀⇩F n in at w.
∀x∈path_image γ.
cmod (f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k - f' x / (x - w) ^ Suc k) < e"
using twom [OF divide_pos_pos [OF ‹0 < e› ‹C > 0›]]   unfolding path_image_def
by (force intro: * elim: eventually_mono)
qed
show "(λu. f' u / (u - w) ^ (Suc k)) contour_integrable_on γ"
by (rule contour_integral_uniform_limit [OF 1 2 leB γ]) auto
have *: "(λn. contour_integral γ (λx. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k))
─w→ contour_integral γ (λu. f' u / (u - w) ^ (Suc k))"
by (rule contour_integral_uniform_limit [OF 1 2 leB γ]) auto
have **: "contour_integral γ (λx. f' x * (inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k)) =
(f u - f w) / (u - w) / k"
if "dist u w < d" for u
proof -
have u: "u ∈ S - path_image γ"
by (metis subsetD d dist_commute mem_ball that)
have §: "((λx. f' x * inverse (x - u) ^ k) has_contour_integral f u) γ"
"((λx. f' x * inverse (x - w) ^ k) has_contour_integral f w) γ"
using u w by (simp_all add: field_simps int)
show ?thesis
apply (rule contour_integral_unique)
apply (simp add: diff_divide_distrib algebra_simps § has_contour_integral_diff has_contour_integral_div)
done
qed
show ?thes2
apply (simp add: has_field_derivative_iff del: power_Suc)
apply (rule Lim_transform_within [OF tendsto_mult_left [OF *] ‹0 < d› ])
apply (simp add: ‹k ≠ 0› **)
done
qed

lemma Cauchy_next_derivative_circlepath:
assumes contf: "continuous_on (path_image (circlepath z r)) f"
and int: "⋀w. w ∈ ball z r ⟹ ((λu. f u / (u - w)^k) has_contour_integral g w) (circlepath z r)"
and k: "k ≠ 0"
and w: "w ∈ ball z r"
shows "(λu. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
(is "?thes1")
and "(g has_field_derivative (k * contour_integral (circlepath z r) (λu. f u/(u - w)^(Suc k)))) (at w)"
(is "?thes2")
proof -
have "r > 0" using w
using ball_eq_empty by fastforce
have wim: "w ∈ ball z r - path_image (circlepath z r)"
using w by (auto simp: dist_norm)
show ?thes1 ?thes2
by (rule Cauchy_next_derivative [OF contf _ int k open_ball valid_path_circlepath wim, where B = "2 * pi * ¦r¦"];
auto simp: vector_derivative_circlepath norm_mult)+
qed

text‹ In particular, the first derivative formula.›

lemma Cauchy_derivative_integral_circlepath:
assumes contf: "continuous_on (cball z r) f"
and holf: "f holomorphic_on ball z r"
and w: "w ∈ ball z r"
shows "(λu. f u/(u - w)^2) contour_integrable_on (circlepath z r)"
(is "?thes1")
and "(f has_field_derivative (1 / (2 * of_real pi * 𝗂) * contour_integral(circlepath z r) (λu. f u / (u - w)^2))) (at w)"
(is "?thes2")
proof -
have [simp]: "r ≥ 0" using w
using ball_eq_empty by fastforce
have f: "continuous_on (path_image (circlepath z r)) f"
by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def)
have int: "⋀w. dist z w < r ⟹
((λu. f u / (u - w)) has_contour_integral (λx. 2 * of_real pi * 𝗂 * f x) w) (circlepath z r)"
by (rule Cauchy_integral_circlepath [OF contf holf]) (simp add: dist_norm norm_minus_commute)
show ?thes1
unfolding power2_eq_square
using int Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1]
by fastforce
have "((λx. 2 * of_real pi * 𝗂 * f x) has_field_derivative contour_integral (circlepath z r) (λu. f u / (u - w)^2)) (at w)"
unfolding power2_eq_square
using int Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1 and g = "λx. 2 * of_real pi * 𝗂 * f x"]
by fastforce
then have fder: "(f has_field_derivative contour_integral (circlepath z r) (λu. f u / (u - w)^2) / (2 * of_real pi * 𝗂)) (at w)"
by (rule DERIV_cdivide [where f = "λx. 2 * of_real pi * 𝗂 * f x" and c = "2 * of_real pi * 𝗂", simplified])
show ?thes2
by simp (rule fder)
qed

subsection‹Existence of all higher derivatives›

proposition derivative_is_holomorphic:
assumes "open S"
and fder: "⋀z. z ∈ S ⟹ (f has_field_derivative f' z) (at z)"
shows "f' holomorphic_on S"
proof -
have *: "∃h. (f' has_field_derivative h) (at z)" if "z ∈ S" for z
proof -
obtain r where "r > 0" and r: "cball z r ⊆ S"
using open_contains_cball ‹z ∈ S› ‹open S› by blast
then have holf_cball: "f holomorphic_on cball z r"
unfolding holomorphic_on_def
using field_differentiable_at_within field_differentiable_def fder by fastforce
then have "continuous_on (path_image (circlepath z r)) f"
using ‹r > 0› by (force elim: holomorphic_on_subset [THEN holomorphic_on_imp_continuous_on])
then have contfpi: "continuous_on (path_image (circlepath z r)) (λx. 1/(2 * of_real pi*𝗂) * f x)"
by (auto intro: continuous_intros)+
have contf_cball: "continuous_on (cball z r) f" using holf_cball
have holf_ball: "f holomorphic_on ball z r" using holf_cball
using ball_subset_cball holomorphic_on_subset by blast
{ fix w  assume w: "w ∈ ball z r"
have intf: "(λu. f u / (u - w)⇧2) contour_integrable_on circlepath z r"
by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
have fder': "(f has_field_derivative 1 / (2 * of_real pi * 𝗂) * contour_integral (circlepath z r) (λu. f u / (u - w)⇧2))
(at w)"
by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
have f'_eq: "f' w = contour_integral (circlepath z r) (λu. f u / (u - w)⇧2) / (2 * of_real pi * 𝗂)"
using fder' ball_subset_cball r w by (force intro: DERIV_unique [OF fder])
have "((λu. f u / (u - w)⇧2 / (2 * of_real pi * 𝗂)) has_contour_integral
contour_integral (circlepath z r) (λu. f u / (u - w)⇧2) / (2 * of_real pi * 𝗂))
(circlepath z r)"
by (rule has_contour_integral_div [OF has_contour_integral_integral [OF intf]])
then have "((λu. f u / (2 * of_real pi * 𝗂 * (u - w)⇧2)) has_contour_integral
contour_integral (circlepath z r) (λu. f u / (u - w)⇧2) / (2 * of_real pi * 𝗂))
(circlepath z r)"
then have "((λu. f u / (2 * of_real pi * 𝗂 * (u - w)⇧2)) has_contour_integral f' w) (circlepath z r)"
} note * = this
show ?thesis
using Cauchy_next_derivative_circlepath [OF contfpi, of 2 f'] ‹0 < r› *
using centre_in_ball mem_ball by force
qed
show ?thesis
by (simp add: holomorphic_on_open [OF ‹open S›] *)
qed

lemma holomorphic_deriv [holomorphic_intros]:
"⟦f holomorphic_on S; open S⟧ ⟹ (deriv f) holomorphic_on S"
by (metis DERIV_deriv_iff_field_differentiable at_within_open derivative_is_holomorphic holomorphic_on_def)

lemma holomorphic_deriv_compose:
assumes g: "g holomorphic_on B" and f: "f holomorphic_on A" and "f ` A ⊆ B" "open B"
shows   "(λx. deriv g (f x)) holomorphic_on A"
using holomorphic_on_compose_gen [OF f holomorphic_deriv[OF g]] assms
by (auto simp: o_def)

lemma analytic_deriv [analytic_intros]: "f analytic_on S ⟹ (deriv f) analytic_on S"
using analytic_on_holomorphic holomorphic_deriv by auto

lemma holomorphic_higher_deriv [holomorphic_intros]: "⟦f holomorphic_on S; open S⟧ ⟹ (deriv ^^ n) f holomorphic_on S"
by (induction n) (auto simp: holomorphic_deriv)

lemma analytic_higher_deriv [analytic_intros]: "f analytic_on S ⟹ (deriv ^^ n) f analytic_on S"
unfolding analytic_on_def using holomorphic_higher_deriv by blast

lemma has_field_derivative_higher_deriv:
"⟦f holomorphic_on S; open S; x ∈ S⟧
⟹ ((deriv ^^ n) f has_field_derivative (deriv ^^ (Suc n)) f x) (at x)"
using holomorphic_derivI holomorphic_higher_deriv by fastforce

lemma higher_deriv_cmult:
assumes "f holomorphic_on A" "x ∈ A" "open A"
shows   "(deriv ^^ j) (λx. c * f x) x = c * (deriv ^^ j) f x"
using assms
proof (induction j arbitrary: f x)
case (Suc j f x)
have "deriv ((deriv ^^ j) (λx. c * f x)) x = deriv (λx. c * (deriv ^^ j) f x) x"
using eventually_nhds_in_open[of A x] assms(2,3) Suc.prems
by (intro deriv_cong_ev refl) (auto elim!: eventually_mono simp: Suc.IH)
also have "… = c * deriv ((deriv ^^ j) f) x" using Suc.prems assms(2,3)
by (intro deriv_cmult holomorphic_on_imp_differentiable_at holomorphic_higher_deriv) auto
finally show ?case by simp
qed simp_all

lemma valid_path_compose_holomorphic:
assumes "valid_path g" and holo:"f holomorphic_on S" and "open S" "path_image g ⊆ S"
shows "valid_path (f ∘ g)"
by (meson assms holomorphic_deriv holomorphic_on_imp_continuous_on holomorphic_on_imp_differentiable_at
holomorphic_on_subset subsetD valid_path_compose)

subsection‹Morera's theorem›

lemma Morera_local_triangle_ball:
assumes "⋀z. z ∈ S
⟹ ∃e a. 0 < e ∧ z ∈ ball a e ∧ continuous_on (ball a e) f ∧
(∀b c. closed_segment b c ⊆ ball a e
⟶ contour_integral (linepath a b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c a) f = 0)"
shows "f analytic_on S"
proof -
{ fix z  assume "z ∈ S"
with assms obtain e a where
"0 < e" and z: "z ∈ ball a e" and contf: "continuous_on (ball a e) f"
and 0: "⋀b c. closed_segment b c ⊆ ball a e
⟹ contour_integral (linepath a b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c a) f = 0"
by blast
have az: "dist a z < e" using mem_ball z by blast
have "∃e>0. f holomorphic_on ball z e"
proof (intro exI conjI)
show "f holomorphic_on ball z (e - dist a z)"
proof (rule holomorphic_on_subset)
show "ball z (e - dist a z) ⊆ ball a e"
have sub_ball: "⋀y. dist a y < e ⟹ closed_segment a y ⊆ ball a e"
by (meson ‹0 < e› centre_in_ball convex_ball convex_contains_segment mem_ball)
show "f holomorphic_on ball a e"
using triangle_contour_integrals_starlike_primitive [OF contf _ open_ball, of a]
derivative_is_holomorphic[OF open_ball]
by (force simp add: 0 ‹0 < e› sub_ball)
qed
}
then show ?thesis
qed

lemma Morera_local_triangle:
assumes "⋀z. z ∈ S
⟹ ∃t. open t ∧ z ∈ t ∧ continuous_on t f ∧
(∀a b c. convex hull {a,b,c} ⊆ t
⟶ contour_integral (linepath a b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c a) f = 0)"
shows "f analytic_on S"
proof -
{ fix z  assume "z ∈ S"
with assms obtain t where
"open t" and z: "z ∈ t" and contf: "continuous_on t f"
and 0: "⋀a b c. convex hull {a,b,c} ⊆ t
⟹ contour_integral (linepath a b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c a) f = 0"
by force
then obtain e where "e>0" and e: "ball z e ⊆ t"
using open_contains_ball by blast
have [simp]: "continuous_on (ball z e) f" using contf
using continuous_on_subset e by blast
have eq0: "⋀b c. closed_segment b c ⊆ ball z e ⟹
contour_integral (linepath z b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c z) f = 0"
by (meson 0 z ‹0 < e› centre_in_ball closed_segment_subset convex_ball dual_order.trans e starlike_convex_subset)
have "∃e a. 0 < e ∧ z ∈ ball a e ∧ continuous_on (ball a e) f ∧
(∀b c. closed_segment b c ⊆ ball a e ⟶
contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0)"
using ‹e > 0› eq0 by force
}
then show ?thesis
qed

proposition Morera_triangle:
"⟦continuous_on S f; open S;
⋀a b c. convex hull {a,b,c} ⊆ S
⟶ contour_integral (linepath a b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c a) f = 0⟧
⟹ f analytic_on S"
using Morera_local_triangle by blast

subsection‹Combining theorems for higher derivatives including Leibniz rule›

lemma higher_deriv_linear [simp]:
"(deriv ^^ n) (λw. c*w) = (λz. if n = 0 then c*z else if n = 1 then c else 0)"
by (induction n) auto

lemma higher_deriv_const [simp]: "(deriv ^^ n) (λw. c) = (λw. if n=0 then c else 0)"
by (induction n) auto

lemma higher_deriv_ident [simp]:
"(deriv ^^ n) (λw. w) z = (if n = 0 then z else if n = 1 then 1 else 0)"
proof (induction n)
case (Suc n)
then show ?case by (metis higher_deriv_linear lambda_one)
qed auto

lemma higher_deriv_id [simp]:
"(deriv ^^ n) id z = (if n = 0 then z else if n = 1 then 1 else 0)"

lemma has_complex_derivative_funpow_1:
"⟦(f has_field_derivative 1) (at z); f z = z⟧ ⟹ (f^^n has_field_derivative 1) (at z)"
proof (induction n)
case 0
then show ?case
next
case (Suc n)
then show ?case
by (metis DERIV_chain funpow_Suc_right mult.right_neutral)
qed

lemma higher_deriv_uminus:
assumes "f holomorphic_on S" "open S" and z: "z ∈ S"
shows "(deriv ^^ n) (λw. -(f w)) z = - ((deriv ^^ n) f z)"
using z
proof (induction n arbitrary: z)
case 0 then show ?case by simp
next
case (Suc n z)
have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
using Suc.prems assms has_field_derivative_higher_deriv by auto
have "⋀x. x ∈ S ⟹ - (deriv ^^ n) f x = (deriv ^^ n) (λw. - f w) x"
then have "((deriv ^^ n) (λw. - f w) has_field_derivative - deriv ((deriv ^^ n) f) z) (at z)"
using  has_field_derivative_transform_within_open [of "λw. -((deriv ^^ n) f w)"]
using "*" DERIV_minus Suc.prems ‹open S› by blast
then show ?case
qed

fixes z::complex
assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z ∈ S"
shows "(deriv ^^ n) (λw. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
using z
proof (induction n arbitrary: z)
case 0 then show ?case by simp
next
case (Suc n z)
have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
"((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
using Suc.prems assms has_field_derivative_higher_deriv by auto
have "⋀x. x ∈ S ⟹ (deriv ^^ n) f x + (deriv ^^ n) g x = (deriv ^^ n) (λw. f w + g w) x"
then have "((deriv ^^ n) (λw. f w + g w) has_field_derivative
deriv ((deriv ^^ n) f) z + deriv ((deriv ^^ n) g) z) (at z)"
using  has_field_derivative_transform_within_open [of "λw. (deriv ^^ n) f w + (deriv ^^ n) g w"]
using "*" Deriv.field_differentiable_add Suc.prems ‹open S› by blast
then show ?case
qed

lemma higher_deriv_diff:
fixes z::complex
assumes "f holomorphic_on S" "g holomorphic_on S" "open S" "z ∈ S"
shows "(deriv ^^ n) (λw. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
using assms higher_deriv_add higher_deriv_uminus holomorphic_on_minus by presburger

lemma Suc_choose: "Suc n choose k = (n choose k) + (if k = 0 then 0 else (n choose (k - 1)))"
by (cases k) simp_all

lemma higher_deriv_mult:
fixes z::complex
assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z ∈ S"
shows "(deriv ^^ n) (λw. f w * g w) z =
(∑i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
using z
proof (induction n arbitrary: z)
case 0 then show ?case by simp
next
case (Suc n z)
have *: "⋀n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
"⋀n. ((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
using Suc.prems assms has_field_derivative_higher_deriv by auto
have sumeq: "(∑i = 0..n.
of_nat (n choose i) * (deriv ((deriv ^^ i) f) z * (deriv ^^ (n - i)) g z + deriv ((deriv ^^ (n - i)) g) z * (deriv ^^ i) f z)) =
g z * deriv ((deriv ^^ n) f) z + (∑i = 0..n. (deriv ^^ i) f z * (of_nat (Suc n choose i) * (deriv ^^ (Suc n - i)) g z))"
apply (simp add: Suc_choose algebra_simps sum.distrib)
apply (subst (4) sum_Suc_reindex)
apply (auto simp: algebra_simps Suc_diff_le intro: sum.cong)
done
have "((deriv ^^ n) (λw. f w * g w) has_field_derivative
(∑i = 0..Suc n. (Suc n choose i) * (deriv ^^ i) f z * (deriv ^^ (Suc n - i)) g z))
(at z)"
apply (rule has_field_derivative_transform_within_open
[of "λw. (∑i = 0..n. of_nat (n choose i) * (deriv ^^ i) f w * (deriv ^^ (n - i)) g w)" _ _ S])
apply (rule derivative_eq_intros | simp)+
apply (auto intro: DERIV_mult * ‹open S› Suc.prems Suc.IH [symmetric])
by (metis (no_types, lifting) mult.commute sum.cong sumeq)
then show ?case
unfolding funpow.simps o_apply
qed

lemma higher_deriv_transform_within_open:
fixes z::complex
assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z ∈ S"
and fg: "⋀w. w ∈ S ⟹ f w = g w"
shows "(deriv ^^ i) f z = (deriv ^^ i) g z"
using z
by (induction i arbitrary: z)
(auto simp: fg intro: complex_derivative_transform_within_open holomorphic_higher_deriv assms)

lemma higher_deriv_compose_linear':
fixes z::complex
assumes f: "f holomorphic_on T" and S: "open S" and T: "open T" and z: "z ∈ S"
and fg: "⋀w. w ∈ S ⟹ u*w + c ∈ T"
shows "(deriv ^^ n) (λw. f (u*w + c)) z = u^n * (deriv ^^ n) f (u*z + c)"
using z
proof (induction n arbitrary: z)
case 0 then show ?case by simp
next
case (Suc n z)
have holo0: "f holomorphic_on (λw. u * w+c) ` S"
by (meson fg f holomorphic_on_subset image_subset_iff)
have holo2: "(deriv ^^ n) f holomorphic_on (λw. u * w+c) ` S"
by (meson f fg holomorphic_higher_deriv holomorphic_on_subset image_subset_iff T)
have holo3: "(λz. u ^ n * (deriv ^^ n) f (u * z+c)) holomorphic_on S"
by (intro holo2 holomorphic_on_compose [where g="(deriv ^^ n) f", unfolded o_def] holomorphic_intros)
have "(λw. u * w+c) holomorphic_on S" "f holomorphic_on (λw. u * w+c) ` S"
by (rule holo0 holomorphic_intros)+
then have holo1: "(λw. f (u * w+c)) holomorphic_on S"
by (rule holomorphic_on_compose [where g=f, unfolded o_def])
have "deriv ((deriv ^^ n) (λw. f (u * w+c))) z = deriv (λz. u^n * (deriv ^^ n) f (u*z+c)) z"
proof (rule complex_derivative_transform_within_open [OF _ holo3 S Suc.prems])
show "(deriv ^^ n) (λw. f (u * w+c)) holomorphic_on S"
by (rule holomorphic_higher_deriv [OF holo1 S])
also have "… = u^n * deriv (λz. (deriv ^^ n) f (u * z+c)) z"
proof -
have "(deriv ^^ n) f analytic_on T"
by (simp add: analytic_on_open f holomorphic_higher_deriv T)
then have "(λw. (deriv ^^ n) f (u * w+c)) analytic_on S"
proof -
have "(deriv ^^ n) f ∘ (λw. u * w+c) holomorphic_on S"
using holomorphic_on_compose[OF _ holo2] ‹(λw. u * w+c) holomorphic_on S›
by simp
then show ?thesis
by (simp add: S analytic_on_open o_def)
qed
then show ?thesis
by (intro deriv_cmult analytic_on_imp_differentiable_at [OF _ Suc.prems])
qed
also have "… = u * u ^ n * deriv ((deriv ^^ n) f) (u * z+c)"
proof -
have "(deriv ^^ n) f field_differentiable at (u * z+c)"
using Suc.prems T f fg holomorphic_higher_deriv holomorphic_on_imp_differentiable_at by blast
then show ?thesis
qed
finally show ?case
by simp
qed

lemma higher_deriv_compose_linear:
fixes z::complex
assumes f: "f holomorphic_on T" and S: "open S" and T: "open T" and z: "z ∈ S"
and fg: "⋀w. w ∈ S ⟹ u * w ∈ T"
shows "(deriv ^^ n) (λw. f (u * w)) z = u^n * (deriv ^^ n) f (u * z)"
using higher_deriv_compose_linear' [where c=0] assms by simp

assumes "f analytic_on {z}" "g analytic_on {z}"
shows "(deriv ^^ n) (λw. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
using analytic_at_two assms higher_deriv_add by blast

lemma higher_deriv_diff_at:
assumes "f analytic_on {z}" "g analytic_on {z}"
shows "(deriv ^^ n) (λw. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
using analytic_at_two assms higher_deriv_diff by blast

lemma higher_deriv_uminus_at:
"f analytic_on {z}  ⟹ (deriv ^^ n) (λw. -(f w)) z = - ((deriv ^^ n) f z)"
using higher_deriv_uminus by (auto simp: analytic_at)

lemma higher_deriv_mult_at:
assumes "f analytic_on {z}" "g analytic_on {z}"
shows "(deriv ^^ n) (λw. f w * g w) z =
(∑i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
using analytic_at_two assms higher_deriv_mult by blast

text‹ Nonexistence of isolated singularities and a stronger integral formula.›

proposition no_isolated_singularity:
fixes z::complex
assumes f: "continuous_on S f" and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K"
shows "f holomorphic_on S"
proof -
{ fix z
assume "z ∈ S" and cdf: "⋀x. x ∈ S - K ⟹ f field_differentiable at x"
have "f field_differentiable at z"
proof (cases "z ∈ K")
case False then show ?thesis by (blast intro: cdf ‹z ∈ S›)
next
case True
with finite_set_avoid [OF K, of z]
obtain d where "d>0" and d: "⋀x. ⟦x∈K; x ≠ z⟧ ⟹ d ≤ dist z x"
by blast
obtain e where "e>0" and e: "ball z e ⊆ S"
using  S ‹z ∈ S› by (force simp: open_contains_ball)
have fde: "continuous_on (ball z (min d e)) f"
by (metis Int_iff ball_min_Int continuous_on_subset e f subsetI)
have cont: "{a,b,c} ⊆ ball z (min d e) ⟹ continuous_on (convex hull {a, b, c}) f" for a b c
by (simp add: hull_minimal continuous_on_subset [OF fde])
have fd: "⟦{a,b,c} ⊆ ball z (min d e); x ∈ interior (convex hull {a, b, c}) - K⟧
⟹ f field_differentiable at x" for a b c x
by (metis cdf Diff_iff Int_iff ball_min_Int subsetD convex_ball e interior_mono interior_subset subset_hull)
obtain g where "⋀w. w ∈ ball z (min d e) ⟹ (g has_field_derivative f w) (at w within ball z (min d e))"
apply (rule contour_integral_convex_primitive
[OF convex_ball fde Cauchy_theorem_triangle_cofinite [OF _ K]])
using cont fd by auto
then have "f holomorphic_on ball z (min d e)"
by (metis open_ball at_within_open derivative_is_holomorphic)
then show ?thesis
unfolding holomorphic_on_def
by (metis open_ball ‹0 < d› ‹0 < e› at_within_open centre_in_ball min_less_iff_conj)
qed
}
with holf S K show ?thesis
by (simp add: holomorphic_on_open open_Diff finite_imp_closed field_differentiable_def [symmetric])
qed

lemma no_isolated_singularity':
fixes z::complex
assumes f: "⋀z. z ∈ K ⟹ (f ⤏ f z) (at z within S)"
and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K"
shows "f holomorphic_on S"
proof (rule no_isolated_singularity[OF _ assms(2-)])
show "continuous_on S f" unfolding continuous_on_def
proof
fix z assume z: "z ∈ S"
have "continuous_on (S - K) f"
using holf holomorphic_on_imp_continuous_on by auto
then show "(f ⤏ f z) (at z within S)"
by (metis Diff_iff K S at_within_interior continuous_on_def f finite_imp_closed interior_eq open_Diff z)
qed
qed

proposition Cauchy_integral_formula_convex:
assumes S: "convex S" and K: "finite K" and contf: "continuous_on S f"
and fcd: "(⋀x. x ∈ interior S - K ⟹ f field_differentiable at x)"
and z: "z ∈ interior S" and vpg: "valid_path γ"
and pasz: "path_image γ ⊆ S - {z}" and loop: "pathfinish γ = pathstart γ"
shows "((λw. f w / (w - z)) has_contour_integral (2*pi * 𝗂 * winding_number γ z * f z)) γ"
proof -
have *: "⋀x. x ∈ interior S ⟹ f field_differentiable at x"
unfolding holomorphic_on_open [symmetric] field_differentiable_def
using no_isolated_singularity [where S = "interior S"]
by (meson K contf continuous_at_imp_continuous_on continuous_on_interior fcd
field_differentiable_at_within field_differentiable_def holomorphic_onI
holomorphic_on_imp_differentiable_at open_interior)
show ?thesis
by (rule Cauchy_integral_formula_weak [OF S finite.emptyI contf]) (use * assms in auto)
qed

text‹ Formula for higher derivatives.›

lemma Cauchy_has_contour_integral_higher_derivative_circlepath:
assumes contf: "continuous_on (cball z r) f"
and holf: "f holomorphic_on ball z r"
and w: "w ∈ ball z r"
shows "((λu. f u / (u - w) ^ (Suc k)) has_contour_integral ((2 * pi * 𝗂) / (fact k) * (deriv ^^ k) f w))
(circlepath z r)"
using w
proof (induction k arbitrary: w)
case 0 then show ?case
using assms by (auto simp: Cauchy_integral_circlepath dist_commute dist_norm)
next
case (Suc k)
have [simp]: "r > 0" using w
using ball_eq_empty by fastforce
have f: "continuous_on (path_image (circlepath z r)) f"
by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def less_imp_le)
obtain X where X: "((λu. f u / (u - w) ^ Suc (Suc k)) has_contour_integral X) (circlepath z r)"
using Cauchy_next_derivative_circlepath(1) [OF f Suc.IH _ Suc.prems]
by (auto simp: contour_integrable_on_def)
then have con: "contour_integral (circlepath z r) ((λu. f u / (u - w) ^ Suc (Suc k))) = X"
by (rule contour_integral_unique)
have "⋀n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) w) (at w)"
using Suc.prems assms has_field_derivative_higher_deriv by auto
then have dnf_diff: "⋀n. (deriv ^^ n) f field_differentiable (at w)"
by (force simp: field_differentiable_def)
have "deriv (λw. complex_of_real (2 * pi) * 𝗂 / (fact k) * (deriv ^^ k) f w) w =
of_nat (Suc k) * contour_integral (circlepath z r) (λu. f u / (u - w) ^ Suc (Suc k))"
by (force intro!: DERIV_imp_deriv Cauchy_next_derivative_circlepath [OF f Suc.IH _ Suc.prems])
also have "… = of_nat (Suc k) * X"
by (simp only: con)
finally have "deriv (λw. ((2 * pi) * 𝗂 / (fact k)) * (deriv ^^ k) f w) w = of_nat (Suc k) * X" .
then have "((2 * pi) * 𝗂 / (fact k)) * deriv (λw. (deriv ^^ k) f w) w = of_nat (Suc k) * X"
by (metis deriv_cmult dnf_diff)
then have "deriv (λw. (deriv ^^ k) f w) w = of_nat (Suc k) * X / ((2 * pi) * 𝗂 / (fact k))"
then show ?case
using of_nat_eq_0_iff X by fastforce
qed

lemma Cauchy_higher_derivative_integral_circlepath:
assumes contf: "continuous_on (cball z r) f"
and holf: "f holomorphic_on ball z r"
and w: "w ∈ ball z r"
shows "(λu. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
(is "?thes1")
and "(deriv ^^ k) f w = (fact k) / (2 * pi * 𝗂) * contour_integral(circlepath z r) (λu. f u/(u - w)^(Suc k))"
(is "?thes2")
proof -
have *: "((λu. f u / (u - w) ^ Suc k) has_contour_integral (2 * pi) * 𝗂 / (fact k) * (deriv ^^ k) f w)
(circlepath z r)"
using Cauchy_has_contour_integral_higher_derivative_circlepath [OF assms]
by simp
show ?thes1 using *
using contour_integrable_on_def by blast
show ?thes2
unfolding contour_integral_unique [OF *] by (simp add: field_split_simps)
qed

corollary Cauchy_contour_integral_circlepath:
assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w ∈ ball z r"
shows "contour_integral(circlepath z r) (λu. f u/(u - w)^(Suc k)) = (2 * pi * 𝗂) * (deriv ^^ k) f w / (fact k)"
by (simp add: Cauchy_higher_derivative_integral_circlepath [OF assms])

lemma Cauchy_contour_integral_circlepath_2:
assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w ∈ ball z r"
shows "contour_integral(circlepath z r) (λu. f u/(u - w)^2) = (2 * pi * 𝗂) * deriv f w"
using Cauchy_contour_integral_circlepath [OF assms, of 1]

subsection‹A holomorphic function is analytic, i.e. has local power series›

theorem holomorphic_power_series:
assumes holf: "f holomorphic_on ball z r"
and w: "w ∈ ball z r"
shows "((λn. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
proof -
― ‹Replacing \<^term>‹r› and the original (weak) premises with stronger ones›
obtain r where "r > 0" and holfc: "f holomorphic_on cball z r" and w: "w ∈ ball z r"
proof
have "cball z ((r + dist w z) / 2) ⊆ ball z r"
using w by (simp add: dist_commute field_sum_of_halves subset_eq)
then show "f holomorphic_on cball z ((r + dist w z) / 2)"
by (rule holomorphic_on_subset [OF holf])
have "r > 0"
using w by clarsimp (metis dist_norm le_less_trans norm_ge_zero)
then show "0 < (r + dist w z) / 2"
by simp (use zero_le_dist [of w z] in linarith)
qed (use w in ‹auto simp: dist_commute›)
then have holf: "f holomorphic_on ball z r"
using ball_subset_cball holomorphic_on_subset by blast
have contf: "continuous_on (cball z r) f"
have cint: "⋀k. (λu. f u / (u - z) ^ Suc k) contour_integrable_on circlepath z r"
by (rule Cauchy_higher_derivative_integral_circlepath [OF contf holf]) (simp add: ‹0 < r›)
obtain B where "0 < B" and B: "⋀u. u ∈ cball z r ⟹ norm(f u) ≤ B"
by (metis (no_types) bounded_pos compact_cball compact_continuous_image compact_imp_bounded contf image_eqI)
obtain k where k: "0 < k" "k ≤ r" and wz_eq: "norm(w - z) = r - k"
and kle: "⋀u. norm(u - z) = r ⟹ k ≤ norm(u - w)"
proof
show "⋀u. cmod (u - z) = r ⟹ r - dist z w ≤ cmod (u - w)"
qed (use w in ‹auto simp: dist_norm norm_minus_commute›)
have ul: "uniform_limit (sphere z r) (λn x. (∑k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k))) (λx. f x / (x - w)) sequentially"
unfolding uniform_limit_iff dist_norm
proof clarify
fix e::real
assume "0 < e"
have rr: "0 ≤ (r - k) / r" "(r - k) / r < 1" using  k by auto
obtain n where n: "((r - k) / r) ^ n < e / B * k"
using real_arch_pow_inv [of "e/B*k" "(r - k)/r"] ‹0 < e› ‹0 < B› k by force
have "norm ((∑k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) - f u / (u - w)) < e"
if "n ≤ N" and r: "r = dist z u"  for N u
proof -
have N: "((r - k) / r) ^ N < e / B * k"
using le_less_trans [OF power_decreasing n]
using ‹n ≤ N› k by auto
have u [simp]: "(u ≠ z) ∧ (u ≠ w)"
using ‹0 < r› r w by auto
have wzu_not1: "(w - z) / (u - z) ≠ 1"
by (metis (no_types) dist_norm divide_eq_1_iff less_irrefl mem_ball norm_minus_commute r w)
have "norm ((∑k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) * (u - w) - f u)
= norm ((∑k<N. (((w - z) / (u - z)) ^ k)) * f u * (u - w) / (u - z) - f u)"
unfolding sum_distrib_right sum_divide_distrib power_divide by (simp add: algebra_simps)
also have "… = norm ((((w - z) / (u - z)) ^ N - 1) * (u - w) / (((w - z) / (u - z) - 1) * (u - z)) - 1) * norm (f u)"
using ‹0 < B›
apply (auto simp: geometric_sum [OF wzu_not1])
apply (simp add: field_simps norm_mult [symmetric])
done
also have "… = norm ((u-z) ^ N * (w - u) - ((w - z) ^ N - (u-z) ^ N) * (u-w)) / (r ^ N * norm (u-w)) * norm (f u)"
using ‹0 < r› r by (simp add: divide_simps norm_mult norm_divide norm_power dist_norm norm_minus_commute)
also have "… = norm ((w - z) ^ N * (w - u)) / (r ^ N * norm (u - w)) * norm (f u)"
also have "… = norm (w - z) ^ N * norm (f u) / r ^ N"
by (simp add: norm_mult norm_power norm_minus_commute)
also have "… ≤ (((r - k)/r)^N) * B"
using ‹0 < r› w k
by (simp add: B divide_simps mult_mono r wz_eq)
also have "… < e * k"
using ‹0 < B› N by (simp add: divide_simps)
also have "… ≤ e * norm (u - w)"
using r kle ‹0 < e› by (simp add: dist_commute dist_norm)
finally show ?thesis
by (simp add: field_split_simps norm_divide del: power_Suc)
qed
with ‹0 < r› show "∀⇩F n in sequentially. ∀x∈sphere z r.
norm ((∑k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k)) - f x / (x - w)) < e"
by (auto simp: mult_ac less_imp_le eventually_sequentially Ball_def)
qed
have §: "⋀x k. k∈ {..<x} ⟹
(λu. (w - z) ^ k * (f u / (u - z) ^ Suc k)) contour_integrable_on circlepath z r"
using contour_integrable_lmul [OF cint, of "(w - z) ^ a" for a] by (simp add: field_simps)
have eq: "∀⇩F x in sequentially.
contour_integral (circlepath z r) (λu. ∑k<x. (w - z) ^ k * (f u / (u - z) ^ Suc k)) =
(∑k<x. contour_integral (circlepath z r) (λu. f u / (u - z) ^ Suc k) * (w - z) ^ k)"
apply (rule eventuallyI)
apply (subst contour_integral_sum, simp)
apply (simp_all only: § contour_integral_lmul cint algebra_simps)
done
have "⋀u k. k ∈ {..<u} ⟹ (λx. f x / (x - z) ^ Suc k) contour_integrable_on circlepath z r"
using ‹0 < r› by (force intro!: Cauchy_higher_derivative_integral_circlepath [OF contf holf])
then have "⋀u. (λy. ∑k<u. (w - z) ^ k * (f y / (y - z) ^ Suc k)) contour_integrable_on circlepath z r"
by (intro contour_integrable_sum contour_integrable_lmul, simp)
then have "(λk. contour_integral (circlepath z r) (λu. f u/(u - z)^(Suc k)) * (w - z)^k)
sums contour_integral (circlepath z r) (λu. f u/(u - w))"
unfolding sums_def using ‹0 < r›
by (intro Lim_transform_eventually [OF _ eq] contour_integral_uniform_limit_circlepath [OF eventuallyI ul]) auto
then have "(λk. contour_integral (circlepath z r) (λu. f u/(u - z)^(Suc k)) * (w - z)^k)
sums (2 * of_real pi * 𝗂 * f w)"
using w by (auto simp: dist_commute dist_norm contour_integral_unique [OF Cauchy_integral_circlepath_simple [OF holfc]])
then have "(λk. contour_integral (circlepath z r) (λu. f u / (u - z) ^ Suc k) * (w - z)^k / (𝗂 * (of_real pi * 2)))
sums ((2 * of_real pi * 𝗂 * f w) / (𝗂 * (complex_of_real pi * 2)))"
by (rule sums_divide)
then have "(λn. (w - z) ^ n * contour_integral (circlepath z r) (λu. f u / (u - z) ^ Suc n) / (𝗂 * (of_real pi * 2)))
sums f w"
then show ?thesis
by (simp add: field_simps ‹0 < r› Cauchy_higher_derivative_integral_circlepath [OF contf holf])
qed

subsection‹The Liouville theorem and the Fundamental Theorem of Algebra›

text‹ These weak Liouville versions don't even need the derivative formula.›

lemma Liouville_weak_0:
assumes holf: "f holomorphic_on UNIV" and inf: "(f ⤏ 0) at_infinity"
shows "f z = 0"
proof (rule ccontr)
assume fz: "f z ≠ 0"
with inf [unfolded Lim_at_infinity, rule_format, of "norm(f z)/2"]
obtain B where B: "⋀x. B ≤ cmod x ⟹ norm (f x) * 2 < cmod (f z)"
by (auto simp: dist_norm)
define R where "R = 1 + ¦B¦ + norm z"
have "R > 0"
unfolding R_def by (smt (verit) norm_ge_zero)
have *: "((λu. f u / (u - z)) has_contour_integral 2 * complex_of_real pi * 𝗂 * f z) (circlepath z R)"
using continuous_on_subset holf  holomorphic_on_subset ‹0 < R›
by (force intro: holomorphic_on_imp_continuous_on Cauchy_integral_circlepath)
have "cmod (x - z) = R ⟹ cmod (f x) * 2 < cmod (f z)" for x
unfolding R_def by (rule B) (use norm_triangle_ineq4 [of x z] in auto)
with ‹R > 0› fz show False
using has_contour_integral_bound_circlepath [OF *, of "norm(f z)/2/R"]
by (auto simp: less_imp_le norm_mult norm_divide field_split_simps)
qed

proposition Liouville_weak:
assumes "f holomorphic_on UNIV" and "(f ⤏ l) at_infinity"
shows "f z = l"
using Liouville_weak_0 [of "λz. f z - l"]
by (simp add: assms holomorphic_on_diff LIM_zero)

proposition Liouville_weak_inverse:
assumes "f holomorphic_on UNIV" and unbounded: "⋀B. eventually (λx. norm (f x) ≥ B) at_infinity"
obtains z where "f z = 0"
proof -
{ assume f: "⋀z. f z ≠ 0"
have 1: "(λx. 1 / f x) holomorphic_on UNIV"
by (simp add: holomorphic_on_divide assms f)
have 2: "((λx. 1 / f x) ⤏ 0) at_infinity"
proof (rule tendstoI [OF eventually_mono])
fix e::real
assume "```