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›)
note ** = has_contour_integral_add [OF has_contour_integral_lmul [OF has_contour_integral_winding_number [OF vpg znotin], of "f z"] *]
show ?thesis
apply (rule has_contour_integral_eq)
using znotin ** 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
by (simp add: winding_number_circlepath assms)
qed
corollary 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 ‹General stepping result for derivative formulas›
lemma Cauchy_next_derivative:
fixes f' :: "complex ⇒ complex"
defines "h ≡ λk w u. f' u / (u-w)^k"
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 ⟹ (h k w 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 "h (Suc k) w contour_integrable_on γ"
and "(f has_field_derivative (k * contour_integral γ (h (Suc k) w))) (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. (h k x z - h k w z) / (x * k - w * k)) contour_integrable_on γ"
if "x ≠ w" "cmod (x-w) < d" for x::complex
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
then have 1: "∀⇩F x in at w. (λz. (h k x z - h k w z) / (x-w) / of_nat k)
contour_integrable_on γ"
unfolding eventually_at
by (force intro: exI [where x=d] simp add: ‹d > 0› dist_norm field_simps)
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))"
by (simp add: algebra_simps)
show ?thesis using ‹i ≤ 1›
apply (simp add: ff_def dist_norm Nat.le_Suc_eq, 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"
by (simp add: field_simps)
} 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 unfolding path_image_def
by (smt (verit, best) dist_norm imageE insert_Diff mem_ball subset_Diff_insert)
then have "d ≤ cmod (x-v) * 2"
by (simp add: field_split_simps)
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)"
by (simp add: field_simps)
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"
by (simp add: norm_mult)
have "x ≠ u"
using uwd ‹0 < d› x d by (force simp: dist_norm ball_def norm_minus_commute)
show ?thesis
using ‹k ≠ 0› ‹x ≠ u› ‹u ≠ w› le_less_trans [OF _ e]
by (simp add: field_simps flip: norm_divide)
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) (λx z. (h k x z - h k w z) / (x-w) / k) (h (Suc k) w) (at w)"
unfolding uniform_limit_iff dist_norm
proof clarify
fix e::real
assume "0 < e"
have *: "cmod ((h k x (γ u) - h k w (γ u)) / ((x-w) * k) - h (Suc k) w (γ u)) < e"
if ec: "cmod ((inverse (γ u - x) ^ k - inverse (γ u - w) ^ k) / ((x-w) * k) -
inverse (γ u - w) * inverse (γ u - w) ^ k) < e / C"
and x: "0 ≤ u" "u ≤ 1"
for x u
proof (cases "(f' (γ u)) = 0")
case True then show ?thesis by (simp add: ‹0 < e› h_def)
next
case False
have "cmod ((h k x (γ u) - h k w (γ u)) / ((x-w) * k) - h (Suc k) w (γ u)) =
cmod (f' (γ u) * ((inverse (γ u - x) ^ k - inverse (γ u - w) ^ k) / ((x-w) * k) -
inverse (γ u - w) * inverse (γ u - w) ^ k))"
by (simp add: h_def field_simps)
also have "… = cmod (f' (γ u)) *
cmod ((inverse (γ u - x) ^ k - inverse (γ u - w) ^ k) / ((x-w) * k) -
inverse (γ u - w) * inverse (γ u - w) ^ k)"
by (simp add: norm_mult)
also have "… < cmod (f' (γ u)) * (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 u in at w.
∀x∈path_image γ.
cmod ((h k u x - h k w x) / (u-w) / of_nat k - h (Suc k) w x) < e"
using twom [OF divide_pos_pos [OF ‹0 < e› ‹C > 0›]] *
unfolding path_image_def h_def
by (force elim: eventually_mono)
qed
show "h (Suc k) w contour_integrable_on γ"
using contour_integral_uniform_limit [OF 1 2 leB γ] by (simp add: h_def)
have *: "(λu. contour_integral γ (λx. (h k u x - h k w x) / (u-w) / k))
─w→ contour_integral γ (h (Suc k) w)"
by (rule contour_integral_uniform_limit [OF 1 2 leB γ]) auto
have **: "contour_integral γ (λx. (h k u x - h k w x) / ((u-w) * k)) =
(f u - f w) / (u-w) / k"
if "dist u w < d" for u
proof -
have "u ∈ S - path_image γ"
by (metis subsetD d dist_commute mem_ball that)
then have "(h k u has_contour_integral f u) γ" "(h k w has_contour_integral f w) γ"
using w by (simp_all add: field_simps int)
then show ?thesis
using contour_integral_unique has_contour_integral_diff
has_contour_integral_div by force
qed
show ?thes2
unfolding has_field_derivative_iff
by (simp add: ‹k ≠ 0› ** Lim_transform_within [OF tendsto_mult_left [OF *] ‹0 < d›])
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
by (simp add: holomorphic_on_imp_continuous_on holomorphic_on_subset)
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)"
by (simp add: algebra_simps)
then have "((λu. f u / (2 * of_real pi * 𝗂 * (u-w)⇧2)) has_contour_integral f' w) (circlepath z r)"
by (simp add: f'_eq)
} 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" "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"
by (simp add: dist_commute ball_subset_ball_iff)
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
qed (simp add: az)
}
then show ?thesis
by (simp add: analytic_on_def)
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
by (simp add: Morera_local_triangle_ball)
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)"
by (simp add: id_def)
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
by (simp add: id_def)
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"
by (auto simp add: Suc)
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
by (simp add: DERIV_imp_deriv)
qed
lemma higher_deriv_add:
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"
by (auto simp add: Suc)
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
by (simp add: DERIV_imp_deriv)
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"
unfolding diff_conv_add_uminus higher_deriv_add
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 (simp add: algebra_simps)
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
by (simp add: DERIV_imp_deriv)
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])
qed (simp add: Suc.IH)
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"
using holomorphic_on_compose[OF _ holo2] ‹(λw. u * w+c) holomorphic_on S›
by (simp add: S analytic_on_open o_def)
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
by (simp add: deriv_compose_linear')
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
lemma higher_deriv_add_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_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-)])
have "continuous_on (S-K) f"
using holf holomorphic_on_imp_continuous_on by auto
then show "continuous_on S f"
by (metis Diff_iff K S at_within_open continuous_on_eq_continuous_at
continuous_within f finite_imp_closed open_Diff)
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_on_subset fcd field_differentiable_def open_interior
has_field_derivative_at_within holomorphic_derivI holomorphic_onI interior_subset)
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))"
by (simp add: field_simps)
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]
by (simp add: power2_eq_square)
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 -
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"
by (metis w dist_norm mem_ball norm_ge_zero not_less_iff_gr_or_eq order_less_le_trans)
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"
by (simp add: holfc holomorphic_on_imp_continuous_on)
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)"
by (metis add_diff_eq diff_add_cancel dist_norm norm_diff_ineq)
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 (simp add: geometric_sum [OF wzu_not1] flip: norm_mult)
apply (simp add: field_simps)
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)"
by (simp add: algebra_simps)
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: "⋀n.
(∑k<n. contour_integral (circlepath z r) (λu. f u / (u-z) ^ Suc k) * (w-z) ^ k) =
contour_integral (circlepath z r) (λu. ∑k<n. (w-z) ^ k * (f u / (u-z) ^ Suc k))"
apply (subst contour_integral_sum)
apply (simp_all only: § finite_lessThan 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 eq
using ‹0 < r› contour_integral_uniform_limit_circlepath [OF eventuallyI ul]
by fastforce
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"
by (simp add: field_simps)
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 "e > 0"
show "eventually (λx. 2/e ≤ cmod (f x)) at_infinity"
by (rule_tac B="2/e" in unbounded)
qed (simp add: dist_norm norm_divide field_split_simps)
have False
using Liouville_weak_0 [OF 1 2] f by simp
}
then show ?thesis
using that by blast
qed
text‹ In particular we get the Fundamental Theorem of Algebra.›
theorem fundamental_theorem_of_algebra:
fixes a :: "nat ⇒ complex"
assumes "a 0 = 0 ∨ (∃i ∈ {1..n}. a i ≠ 0)"
obtains z where "(∑i≤n. a i * z^i) = 0"
using assms
proof (elim disjE bexE)
assume "a 0 = 0" then show ?thesis
by (auto simp: that [of 0])
next
fix i
assume i: "i ∈ {1..n}" and nz: "a i ≠ 0"
have 1: "(λz. ∑i≤n. a i * z^i) holomorphic_on UNIV"
by (rule holomorphic_intros)+
show thesis
proof (rule Liouville_weak_inverse [OF 1])
show "∀⇩F x in at_infinity. B ≤ cmod (∑i≤n. a i * x ^ i)" for B
using i nz by (intro polyfun_extremal exI[of _ i]) auto
qed (use that in auto)
qed
subsection‹Weierstrass convergence theorem›
lemma holomorphic_uniform_limit:
assumes cont: "eventually (λn. continuous_on (cball z r) (f n) ∧ (f n) holomorphic_on ball z r) F"
and ulim: "uniform_limit (cball z r) f g F"
and F: "¬ trivial_limit F"
obtains "continuous_on (cball z r) g" "g holomorphic_on ball z r"
proof (cases r "0::real" rule: linorder_cases)
case less then show ?thesis by (force simp: ball_empty less_imp_le continuous_on_def holomorphic_on_def intro: that)
next
case equal then show ?thesis
by (force simp: holomorphic_on_def intro: that)
next
case greater
have contg: "continuous_on (cball z r) g"
using cont uniform_limit_theorem [OF eventually_mono ulim F] by blast
have "path_image (circlepath z r) ⊆ cball z r"
using ‹0 < r› by auto
then have 1: "continuous_on (path_image (circlepath z r)) (λx. 1 / (2 * complex_of_real pi * 𝗂) * g x)"
by (intro continuous_intros continuous_on_subset [OF contg])
have 2: "((λu. 1 / (2 * of_real pi * 𝗂) * g u / (u-w) ^ 1) has_contour_integral g w) (circlepath z r)"
if w: "w ∈ ball z r" for w
proof -
define d where "d = (r - norm(w-z))"
have "0 < d" "d ≤ r" using w by (auto simp: norm_minus_commute d_def dist_norm)
have dle: "⋀u. cmod (z-u) = r ⟹ d ≤ cmod (u-w)"
unfolding d_def by (metis add_diff_eq diff_add_cancel norm_diff_ineq norm_minus_commute)
have ev_int: "∀⇩F n in F. (λu. f n u / (u-w)) contour_integrable_on circlepath z r"
using w
by (auto intro: eventually_mono [OF cont] Cauchy_higher_derivative_integral_circlepath [where k=0, simplified])
have "⋀e. ⟦0 < r; 0 < d; 0 < e⟧
⟹ ∀⇩F n in F.
∀x∈sphere z r.
x ≠ w ⟶
cmod (f n x - g x) < e * cmod (x-w)"
apply (rule_tac e1="e * d" in eventually_mono [OF uniform_limitD [OF ulim]])
apply (force simp: dist_norm intro: dle mult_left_mono less_le_trans)+
done
then have ul_less: "uniform_limit (sphere z r) (λn x. f n x / (x-w)) (λx. g x / (x-w)) F"
using greater ‹0 < d›
by (auto simp add: uniform_limit_iff dist_norm norm_divide diff_divide_distrib [symmetric] divide_simps)
have g_cint: "(λu. g u/(u-w)) contour_integrable_on circlepath z r"
by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F ‹0 < r›])
have cif_tends_cig: "((λn. contour_integral(circlepath z r) (λu. f n u / (u-w))) ⤏ contour_integral(circlepath z r) (λu. g u/(u-w))) F"
by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F ‹0 < r›])
have f_tends_cig: "((λn. 2 * of_real pi * 𝗂 * f n w) ⤏ contour_integral (circlepath z r) (λu. g u / (u-w))) F"
proof (rule Lim_transform_eventually)
show "∀⇩F x in F. contour_integral (circlepath z r) (λu. f x u / (u-w))
= 2 * of_real pi * 𝗂 * f x w"
using w‹0 < d› d_def
by (auto intro: eventually_mono [OF cont contour_integral_unique [OF Cauchy_integral_circlepath]])
qed (auto simp: cif_tends_cig)
have "⋀e. 0 < e ⟹ ∀⇩F n in F. dist (f n w) (g w) < e"
by (rule eventually_mono [OF uniform_limitD [OF ulim]]) (use w in auto)
then have "((λn. 2 * of_real pi * 𝗂 * f n w) ⤏ 2 * of_real pi * 𝗂 * g w) F"
by (rule tendsto_mult_left [OF tendstoI])
then have "((λu. g u / (u-w)) has_contour_integral 2 * of_real pi * 𝗂 * g w) (circlepath z r)"
using has_contour_integral_integral [OF g_cint] tendsto_unique [OF F f_tends_cig] w
by fastforce
then have "((λu. g u / (2 * of_real pi * 𝗂 * (u-w))) has_contour_integral g w) (circlepath z r)"
using has_contour_integral_div [where c = "2 * of_real pi * 𝗂"]
by (force simp: field_simps)
then show ?thesis
by (simp add: dist_norm)
qed
show ?thesis
using Cauchy_next_derivative_circlepath(2) [OF 1 2, simplified]
by (fastforce simp add: holomorphic_on_open contg intro: that)
qed
text‹ Version showing that the limit is the limit of the derivatives.›
proposition has_complex_derivative_uniform_limit:
fixes z::complex
assumes cont: "eventually (λn. continuous_on (cball z r) (f n) ∧
(∀w ∈ ball z r. ((f n) has_field_derivative (f' n w)) (at w))) F"
and ulim: "uniform_limit (cball z r) f g F"
and F: "¬ trivial_limit F" and "0 < r"
obtains g' where
"continuous_on (cball z r) g"
"⋀w. w ∈ ball z r ⟹ (g has_field_derivative (g' w)) (at w) ∧ ((λn. f' n w) ⤏ g' w) F"
proof -
let ?conint = "contour_integral (circlepath z r)"
have g: "continuous_on (cball z r) g" "g holomorphic_on ball z r"
by (rule holomorphic_uniform_limit [OF eventually_mono [OF cont] ulim F];
auto simp: holomorphic_on_open field_differentiable_def)+
then obtain g' where g': "⋀x. x ∈ ball z r ⟹ (g has_field_derivative g' x) (at x)"
using DERIV_deriv_iff_has_field_derivative
by (fastforce simp add: holomorphic_on_open)
then have derg: "⋀x. x ∈ ball z r ⟹ deriv g x = g' x"
by (simp add: DERIV_imp_deriv)
have tends_f'n_g': "((λn. f' n w) ⤏ g' w) F" if w: "w ∈ ball z r" for w
proof -
have eq_f': "?conint (λx. f n x / (x-w)⇧2) - ?conint (λx. g x / (x-w)⇧2) = (f' n w - g' w) * (2 * of_real pi * 𝗂)"
if cont_fn: "continuous_on (cball z r) (f n)"
and fnd: "⋀w. w ∈ ball z r ⟹ (f n has_field_derivative f' n w) (at w)" for n
proof -
have hol_fn: "f n holomorphic_on ball z r"
using fnd by (force simp: holomorphic_on_open)
have "(f n has_field_derivative 1 / (2 * of_real pi * 𝗂) * ?conint (λu. f n u / (u-w)⇧2)) (at w)"
by (rule Cauchy_derivative_integral_circlepath [OF cont_fn hol_fn w])
then have f': "f' n w = 1 / (2 * of_real pi * 𝗂) * ?conint (λu. f n u / (u-w)⇧2)"
using DERIV_unique [OF fnd] w by blast
show ?thesis
by (simp add: f' Cauchy_contour_integral_circlepath_2 [OF g w] derg [OF w] field_split_simps)
qed
define d where "d = (r - norm(w-z))^2"
have "d > 0"
using w by (simp add: dist_commute dist_norm d_def)
have dle: "d ≤ cmod ((y-w)⇧2)" if "r = cmod (z-y)" for y
by (smt (verit, best) d_def diff_add_cancel diff_diff_eq2 dist_norm mem_ball
norm_minus_commute norm_power norm_triangle_ineq2 power_mono that w)
have 1: "∀⇩F n in F. (λx. f n x / (x-w)⇧2) contour_integrable_on circlepath z r"
by (force simp: holomorphic_on_open intro: w Cauchy_derivative_integral_circlepath eventually_mono [OF cont])
have 2: "uniform_limit (sphere z r) (λn x. f n x / (x-w)⇧2) (λx. g x / (x-w)⇧2) F"
unfolding uniform_limit_iff
proof clarify
fix e::real
assume "e > 0"
with ‹r > 0›
have "∀⇩F n in F. ∀x. x ≠ w ⟶ cmod (z-x) = r ⟶ cmod (f n x - g x) < e * cmod ((x-w)⇧2)"
by (force simp: ‹0 < d› dist_norm dle intro: less_le_trans eventually_mono [OF uniform_limitD [OF ulim], of "e*d"])
with ‹r > 0› ‹e > 0›
show "∀⇩F n in F. ∀x∈sphere z r. dist (f n x / (x-w)⇧2) (g x / (x-w)⇧2) < e"
by (simp add: norm_divide field_split_simps sphere_def dist_norm)
qed
have "((λn. contour_integral (circlepath z r) (λx. f n x / (x-w)⇧2))
⤏ contour_integral (circlepath z r) ((λx. g x / (x-w)⇧2))) F"
by (rule contour_integral_uniform_limit_circlepath [OF 1 2 F ‹0 < r›])
then have tendsto_0: "((λn. 1 / (2 * of_real pi * 𝗂) * (?conint (λx. f n x / (x-w)⇧2) - ?conint (λx. g x / (x-w)⇧2))) ⤏ 0) F"
using Lim_null by (force intro!: tendsto_mult_right_zero)
have "((λn. f' n w - g' w) ⤏ 0) F"
by (force simp: divide_simps
intro: eq_f' eventually_mono [OF cont] Lim_transform_eventually [OF tendsto_0])
then show ?thesis using Lim_null by blast
qed
obtain g' where "⋀w. w ∈ ball z r ⟹ (g has_field_derivative (g' w)) (at w) ∧ ((λn. f' n w) ⤏ g' w) F"
by (blast intro: tends_f'n_g' g')
then show ?thesis using g
using that by blast
qed
subsection ‹Some more simple/convenient versions for applications›
lemma holomorphic_uniform_sequence:
assumes S: "open S"
and hol_fn: "⋀n. (f n) holomorphic_on S"
and ulim_g: "⋀x. x ∈ S ⟹ ∃d. 0 < d ∧ cball x d ⊆ S ∧ uniform_limit (cball x d) f g sequentially"
shows "g holomorphic_on S"
proof -
have "∃f'. (g has_field_derivative f') (at z)" if "z ∈ S" for z
proof -
obtain r where "0 < r" and r: "cball z r ⊆ S"
and ul: "uniform_limit (cball z r) f g sequentially"
using ulim_g [OF ‹z ∈ S›] by blast
have *: "∀⇩F n in sequentially. continuous_on (cball z r) (f n) ∧ f n holomorphic_on ball z r"
by (smt (verit, best) ball_subset_cball hol_fn holomorphic_on_imp_continuous_on
holomorphic_on_subset not_eventuallyD r)
show ?thesis
using ‹0 < r› centre_in_ball ul
by (auto simp: holomorphic_on_open intro: holomorphic_uniform_limit [OF *])
qed
with S show ?thesis
by (simp add: holomorphic_on_open)
qed
lemma has_complex_derivative_uniform_sequence:
fixes S :: "complex set"
assumes S: "open S"
and hfd: "⋀n x. x ∈ S ⟹ ((f n) has_field_derivative f' n x) (at x)"
and ulim_g: "⋀x. x ∈ S
⟹ ∃d. 0 < d ∧ cball x d ⊆ S ∧ uniform_limit (cball x d) f g sequentially"
shows "∃g'. ∀x ∈ S. (g has_field_derivative g' x) (at x) ∧ ((λn. f' n x) ⤏ g' x) sequentially"
proof -
have y: "∃y. (g has_field_derivative y) (at z) ∧ (λn. f' n z) ⇢ y" if "z ∈ S" for z
proof -
obtain r where "0 < r" and r: "cball z r ⊆ S"
and ul: "uniform_limit (cball z r) f g sequentially"
using ulim_g [OF ‹z ∈ S›] by blast
have *: "∀⇩F n in sequentially. continuous_on (cball z r) (f n) ∧
(∀w ∈ ball z r. ((f n) has_field_derivative (f' n w)) (at w))"
proof (intro eventuallyI conjI ballI)
show "continuous_on (cball z r) (f x)" for x
by (meson S continuous_on_subset hfd holomorphic_on_imp_continuous_on holomorphic_on_open r)
show "w ∈ ball z r ⟹ (f x has_field_derivative f' x w) (at w)" for w x
using ball_subset_cball hfd r by blast
qed
show ?thesis
by (rule has_complex_derivative_uniform_limit [OF *, of g]) (use ‹0 < r› ul in ‹force+›)
qed
show ?thesis
by (rule bchoice) (blast intro: y)
qed
subsection‹On analytic functions defined by a series›
lemma series_and_derivative_comparison:
fixes S :: "complex set"
assumes S: "open S"
and h: "summable h"
and hfd: "⋀n x. x ∈ S ⟹ (f n has_field_derivative f' n x) (at x)"
and to_g: "∀⇩F n in sequentially. ∀x∈S. norm (f n x) ≤ h n"
obtains g g' where "∀x ∈ S. ((λn. f n x) sums g x) ∧ ((λn. f' n x) sums g' x) ∧ (g has_field_derivative g' x) (at x)"
proof -
obtain g where g: "uniform_limit S (λn x. ∑i<n. f i x) g sequentially"
using Weierstrass_m_test_ev [OF to_g h] by force
have *: "∃d>0. cball x d ⊆ S ∧ uniform_limit (cball x d) (λn x. ∑i<n. f i x) g sequentially"
if "x ∈ S" for x
using open_contains_cball [of "S"] ‹x ∈ S› S g uniform_limit_on_subset by blast
have "⋀x. x ∈ S ⟹ (λn. ∑i<n. f i x) ⇢ g x"
by (metis tendsto_uniform_limitI [OF g])
moreover have "∃g'. ∀x∈S. (g has_field_derivative g' x) (at x) ∧ (λn. ∑i<n. f' i x) ⇢ g' x"
by (rule has_complex_derivative_uniform_sequence [OF S]) (auto intro: * hfd DERIV_sum)+
ultimately show ?thesis
by (metis sums_def that)
qed
text‹A version where we only have local uniform/comparative convergence.›
lemma series_and_derivative_comparison_local:
fixes S :: "complex set"
assumes S: "open S"
and hfd: "⋀n x. x ∈ S ⟹ (f n has_field_derivative f' n x) (at x)"
and to_g: "⋀x. x ∈ S ⟹ ∃d h. 0 < d ∧ summable h ∧ (∀⇩F n in sequentially. ∀y∈ball x d ∩ S. norm (f n y) ≤ h n)"
shows "∃g g'. ∀x ∈ S. ((λn. f n x) sums g x) ∧ ((λn. f' n x) sums g' x) ∧ (g has_field_derivative g' x) (at x)"
proof -
have "∃y. (λn. f n z) sums (∑n. f n z) ∧ (λn. f' n z) sums y ∧ ((λx. ∑n. f n x) has_field_derivative y) (at z)"
if "z ∈ S" for z
proof -
obtain d h where "0 < d" "summable h" and le_h: "∀⇩F n in sequentially. ∀y∈ball z d ∩ S. norm (f n y) ≤ h n"
using to_g ‹z ∈ S› by meson
then obtain r where "r>0" and r: "ball z r ⊆ ball z d ∩ S" using ‹z ∈ S› S
by (metis Int_iff open_ball centre_in_ball open_Int open_contains_ball_eq)
have 1: "open (ball z d ∩ S)"
by (simp add: open_Int S)
have 2: "⋀n x. x ∈ ball z d ∩ S ⟹ (f n has_field_derivative f' n x) (at x)"
by (auto simp: hfd)
obtain g g' where gg': "∀x ∈ ball z d ∩ S. ((λn. f n x) sums g x) ∧
((λn. f' n x) sums g' x) ∧ (g has_field_derivative g' x) (at x)"
by (auto intro: le_h series_and_derivative_comparison [OF 1 ‹summable h› hfd])
then have "(λn. f' n z) sums g' z"
by (meson ‹0 < r› centre_in_ball contra_subsetD r)
moreover have "(λn. f n z) sums (∑n. f n z)"
using summable_sums centre_in_ball ‹0 < d› ‹summable h› le_h
by (metis (full_types) Int_iff gg' summable_def that)
moreover have "((λx. ∑n. f n x) has_field_derivative g' z) (at z)"
by (metis (no_types, lifting) "1" r ‹0 < r› gg' has_field_derivative_transform_within_open
open_contains_ball_eq sums_unique)
ultimately show ?thesis by auto
qed
then show ?thesis
by meson
qed
text‹Sometimes convenient to compare with a complex series of positive reals. (?)›
lemma series_and_derivative_comparison_complex:
fixes S :: "complex set"
assumes S: "open S"
and hfd: "⋀n x. x ∈ S ⟹ (f n has_field_derivative f' n x) (at x)"
and to_g: "⋀x. x ∈ S ⟹ ∃d h. 0 < d ∧ summable h ∧ range h ⊆ ℝ⇩≥⇩0 ∧ (∀⇩F n in sequentially. ∀y∈ball x d ∩ S. cmod(f n y) ≤ cmod (h n))"
shows "∃g g'. ∀x ∈ S. ((λn. f n x) sums g x) ∧ ((λn. f' n x) sums g' x) ∧ (g has_field_derivative g' x) (at x)"
apply (rule series_and_derivative_comparison_local [OF S hfd], assumption)
apply (rule ex_forward [OF to_g], assumption)
apply (erule exE)
apply (rule_tac x="Re ∘ h" in exI)
apply (force simp: summable_Re o_def nonneg_Reals_cmod_eq_Re image_subset_iff)
done
text‹Sometimes convenient to compare with a complex series of positive reals. (?)›
lemma series_differentiable_comparison_complex:
fixes S :: "complex set"
assumes S: "open S"
and hfd: "⋀n x. x ∈ S ⟹ f n field_differentiable (at x)"
and to_g: "⋀x. x ∈ S ⟹ ∃d h. 0 < d ∧ summable h ∧ range h ⊆ ℝ⇩≥⇩0 ∧ (∀⇩F n in sequentially. ∀y∈ball x d ∩ S. cmod(f n y) ≤ cmod (h n))"
obtains g where "∀x ∈ S. ((λn. f n x) sums g x) ∧ g field_differentiable (at x)"
proof -
have hfd': "⋀n x. x ∈ S ⟹ (f n has_field_derivative deriv (f n) x) (at x)"
using hfd field_differentiable_derivI by blast
show ?thesis
by (metis field_differentiable_def that series_and_derivative_comparison_complex [OF S hfd' to_g])
qed
text‹In particular, a power series is analytic inside circle of convergence.›
lemma power_series_and_derivative_0:
fixes a :: "nat ⇒ complex" and r::real
assumes "summable (λn. a n * r^n)"
shows "∃g g'. ∀z. cmod z < r ⟶
((λn. a n * z^n) sums g z) ∧ ((λn. of_nat n * a n * z^(n-1)) sums g' z) ∧ (g has_field_derivative g' z) (at z)"
proof (cases "0 < r")
case True
have der: "⋀n z. ((λx. a n * x ^ n) has_field_derivative of_nat n * a n * z ^ (n-1)) (at z)"
by (rule derivative_eq_intros | simp)+
have y_le: "cmod y ≤ cmod (of_real r + of_real (cmod z)) / 2"
if "cmod (z-y) * 2 < r - cmod z" for z y
by (smt (verit, best) field_sum_of_halves norm_minus_commute norm_of_real norm_triangle_ineq2 of_real_add that)
have "summable (λn. a n * complex_of_real r ^ n)"
using assms ‹r > 0› by simp
moreover have "⋀z. cmod z < r ⟹ cmod ((of_real r + of_real (cmod z)) / 2) < cmod (of_real r)"
using ‹r > 0›
by (simp flip: of_real_add)
ultimately have sum: "⋀z. cmod z < r ⟹ summable (λn. of_real (cmod (a n)) * ((of_real r + complex_of_real (cmod z)) / 2) ^ n)"
by (rule power_series_conv_imp_absconv_weak)
have "∃g g'. ∀z ∈ ball 0 r. (λn. (a n) * z ^ n) sums g z ∧
(λn. of_nat n * (a n) * z ^ (n-1)) sums g' z ∧ (g has_field_derivative g' z) (at z)"
apply (rule series_and_derivative_comparison_complex [OF open_ball der])
apply (rule_tac x="(r - norm z)/2" in exI)
apply (rule_tac x="λn. of_real(norm(a n)*((r + norm z)/2)^n)" in exI)
using ‹r > 0›
apply (auto simp: sum eventually_sequentially norm_mult norm_power dist_norm intro!: mult_left_mono power_mono y_le)
done
then show ?thesis
by (simp add: ball_def)
next
case False then show ?thesis
unfolding not_less using less_le_trans norm_not_less_zero by blast
qed
proposition power_series_and_derivative:
fixes a :: "nat ⇒ complex" and r::real
assumes "summable (λn. a n * r^n)"
obtains g g' where "∀z ∈ ball w r.
((λn. a n * (z-w) ^ n) sums g z) ∧ ((λn. of_nat n * a n * (z-w) ^ (n-1)) sums g' z) ∧
(g has_field_derivative g' z) (at z)"
using power_series_and_derivative_0 [OF assms]
apply clarify
apply (rule_tac g="(λz. g(z-w))" in that)
using DERIV_shift [where z="-w"]
apply (auto simp: norm_minus_commute Ball_def dist_norm)
done
proposition power_series_holomorphic:
assumes "⋀w. w ∈ ball z r ⟹ ((λn. a n*(w-z)^n) sums f w)"
shows "f holomorphic_on ball z r"
proof -
have "∃f'. (f has_field_derivative f') (at w)" if w: "dist z w < r" for w
proof -
have wz: "cmod (w-z) < r" using w
by (auto simp: field_split_simps dist_norm norm_minus_commute)
then have "0 ≤ r"
by (meson less_eq_real_def norm_ge_zero order_trans)
have inb: "z + complex_of_real ((dist z w + r) / 2) ∈ ball z r"
using w by (simp add: dist_norm ‹0≤r› flip: of_real_add)
have sum: "summable (λn. a n * of_real (((cmod (z-w) + r) / 2) ^ n))"
using assms [OF inb] by (force simp: summable_def dist_norm)
obtain g g' where gg': "⋀u. u ∈ ball z ((cmod (z-w) + r) / 2) ⟹
(λn. a n * (u-z) ^ n) sums g u ∧
(λn. of_nat n * a n * (u-z) ^ (n-1)) sums g' u ∧ (g has_field_derivative g' u) (at u)"
by (rule power_series_and_derivative [OF sum, of z]) fastforce
have [simp]: "g u = f u" if "cmod (u-w) < (r - cmod (z-w)) / 2" for u
proof -
have less: "cmod (z-u) * 2 < cmod (z-w) + r"
using that dist_triangle2 [of z u w]
by (simp add: dist_norm [symmetric] algebra_simps)
have "(λn. a n * (u-z) ^ n) sums g u" "(λn. a n * (u-z) ^ n) sums f u"
using gg' [of u] less w by (auto simp: assms dist_norm)
then show ?thesis
by (metis sums_unique2)
qed
have "(f has_field_derivative g' w) (at w)"
proof (rule has_field_derivative_transform_within [where d="(r - norm(z-w))/2"])
qed (use w gg' [of w] in ‹(force simp: dist_norm)+›)
then show ?thesis ..
qed
then show ?thesis by (simp add: holomorphic_on_open)
qed
corollary holomorphic_iff_power_series:
"f holomorphic_on ball z r ⟷
(∀w ∈ ball z r. (λn. (deriv ^^ n) f z / (fact n) * (w-z)^n) sums f w)"
using power_series_holomorphic [where a = "λn. (deriv ^^ n) f z / (fact n)"] holomorphic_power_series
by blast
lemma power_series_analytic:
"(⋀w. w ∈ ball z r ⟹ (λn. a n*(w-z)^n) sums f w) ⟹ f analytic_on ball z r"
by (force simp: analytic_on_open intro!: power_series_holomorphic)
lemma analytic_iff_power_series:
"f analytic_on ball z r ⟷
(∀w ∈ ball z r. (λn. (deriv ^^ n) f z / (fact n) * (w-z)^n) sums f w)"
by (simp add: analytic_on_open holomorphic_iff_power_series)
subsection ‹Equality between holomorphic functions, on open ball then connected set›
lemma holomorphic_fun_eq_on_ball:
"⟦f holomorphic_on ball z r; g holomorphic_on ball z r;
w ∈ ball z r;
⋀n. (deriv ^^ n) f z = (deriv ^^ n) g z⟧
⟹ f w = g w"
by (auto simp: holomorphic_iff_power_series sums_unique2 [of "λn. (deriv ^^ n) f z / (fact n) * (w-z)^n"])
lemma holomorphic_fun_eq_0_on_ball:
"⟦f holomorphic_on ball z r; w ∈ ball z r;
⋀n. (deriv ^^ n) f z = 0⟧
⟹ f w = 0"
using holomorphic_fun_eq_on_ball [where g = "λz. 0"] by simp
lemma holomorphic_fun_eq_0_on_connected:
assumes holf: "f holomorphic_on S" and "open S"
and cons: "connected S"
and der: "⋀n. (deriv ^^ n) f z = 0"
and "z ∈ S" "w ∈ S"
shows "f w = 0"
proof -
have *: "ball x e ⊆ (⋂n. {w ∈ S. (deriv ^^ n) f w = 0})"
if "∀u. (deriv ^^ u) f x = 0" "ball x e ⊆ S" for x e
proof -
have "(deriv ^^ m) ((deriv ^^ n) f) x = 0" for m n
by (metis funpow_add o_apply that(1))
then have "⋀x' n. dist x x' < e ⟹ (deriv ^^ n) f x' = 0"
using ‹open S›
by (meson holf holomorphic_fun_eq_0_on_ball holomorphic_higher_deriv holomorphic_on_subset mem_ball that(2))
with that show ?thesis by auto
qed
obtain e where "e>0" and e: "ball w e ⊆ S" using openE [OF ‹open S› ‹w ∈ S›] .
then have holfb: "f holomorphic_on ball w e"
using holf holomorphic_on_subset by blast
have "open (⋂n. {w ∈ S. (deriv ^^ n) f w = 0})"
using ‹open S›
apply (simp add: open_contains_ball Ball_def image_iff)
by (metis (mono_tags) "*" mem_Collect_eq)
then have "openin (top_of_set S) (⋂n. {w ∈ S. (deriv ^^ n) f w = 0})"
by (force intro: open_subset)
moreover have "closedin (top_of_set S) (⋂n. {w ∈ S. (deriv ^^ n) f w = 0})"
using assms
by (auto intro: continuous_closedin_preimage_constant holomorphic_on_imp_continuous_on holomorphic_higher_deriv)
moreover have "(⋂n. {w ∈ S. (deriv ^^ n) f w = 0}) = S ⟹ f w = 0"
using ‹e>0› e by (force intro: holomorphic_fun_eq_0_on_ball [OF holfb])
ultimately show ?thesis
using cons der ‹z ∈ S›
by (auto simp add: connected_clopen)
qed
lemma holomorphic_fun_eq_on_connected:
assumes "f holomorphic_on S" "g holomorphic_on S" and "open S" "connected S"
and "⋀n. (deriv ^^ n) f z = (deriv ^^ n) g z"
and "z ∈ S" "w ∈ S"
shows "f w = g w"
proof (rule holomorphic_fun_eq_0_on_connected [of "λx. f x - g x" S z, simplified])
show "(λx. f x - g x) holomorphic_on S"
by (intro assms holomorphic_intros)
show "⋀n. (deriv ^^ n) (λx. f x - g x) z = 0"
using assms higher_deriv_diff by auto
qed (use assms in auto)
lemma holomorphic_fun_eq_const_on_connected:
assumes holf: "f holomorphic_on S" and "open S"
and cons: "connected S"
and der: "⋀n. 0 < n ⟹ (deriv ^^ n) f z = 0"
and "z ∈ S" "w ∈ S"
shows "f w = f z"
proof (rule holomorphic_fun_eq_0_on_connected [of "λw. f w - f z" S z, simplified])
show "(λw. f w - f z) holomorphic_on S"
by (intro assms holomorphic_intros)
show "⋀n. (deriv ^^ n) (λw. f w - f z) z = 0"
by (subst higher_deriv_diff) (use assms in ‹auto intro: holomorphic_intros›)
qed (use assms in auto)
subsection ‹Some basic lemmas about poles/singularities›
lemma pole_lemma:
assumes holf: "f holomorphic_on S" and a: "a ∈ interior S"
shows "(λz. if z = a then deriv f a
else (f z - f a) / (z-a)) holomorphic_on S" (is "?F holomorphic_on S")
proof -
have *: "?F field_differentiable (at u within S)" if "u ∈ S" "u ≠ a" for u
proof -
have fcd: "f field_differentiable at u within S"
using holf holomorphic_on_def by (simp add: ‹u ∈ S›)
have cd: "(λz. (f z - f a) / (z-a)) field_differentiable at u within S"
by (rule fcd derivative_intros | simp add: that)+
have "0 < dist a u" using that dist_nz by blast
then show ?thesis
by (rule field_differentiable_transform_within [OF _ _ _ cd]) (auto simp: ‹u ∈ S›)
qed
moreover
have "?F field_differentiable at a" if "0 < e" "ball a e ⊆ S" for e
proof -
have holfb: "f holomorphic_on ball a e"
by (rule holomorphic_on_subset [OF holf ‹ball a e ⊆ S›])
have 2: "?F holomorphic_on ball a e - {a}"
using mem_ball that
by (auto simp add: holomorphic_on_def simp flip: field_differentiable_def intro: * field_differentiable_within_subset)
have "isCont (λz. if z = a then deriv f a else (f z - f a) / (z-a)) x"
if "dist a x < e" for x
proof (cases "x=a")
case True
then have "f field_differentiable at a"
using holfb ‹0 < e› holomorphic_on_imp_differentiable_at by auto
with True show ?thesis
by (smt (verit) DERIV_deriv_iff_field_differentiable LIM_equal continuous_at has_field_derivativeD)
next
case False with 2 that show ?thesis
by (simp add: field_differentiable_imp_continuous_at holomorphic_on_imp_differentiable_at open_Diff)
qed
then have 1: "continuous_on (ball a e) ?F"
by (clarsimp simp: continuous_on_eq_continuous_at)
have "?F holomorphic_on ball a e"
by (auto intro: no_isolated_singularity [OF 1 2])
with that show ?thesis
by (simp add: holomorphic_on_imp_differentiable_at)
qed
ultimately show ?thesis
by (metis (lifting) a at_within_interior holomorphic_onI mem_interior)
qed
lemma pole_theorem:
assumes holg: "g holomorphic_on S" and a: "a ∈ interior S"
and eq: "⋀z. z ∈ S - {a} ⟹ g z = (z-a) * f z"
shows "(λz. if z = a then deriv g a
else f z - g a/(z-a)) holomorphic_on S"
using pole_lemma [OF holg a]
by (rule holomorphic_transform) (simp add: eq field_split_simps)
lemma pole_lemma_open:
assumes "f holomorphic_on S" "open S"
shows "(λz. if z = a then deriv f a else (f z - f a)/(z-a)) holomorphic_on S"
proof (cases "a ∈ S")
case True with assms interior_eq pole_lemma
show ?thesis by fastforce
next
case False
then have "(λz. (f z - f a) / (z-a)) field_differentiable at x within S"
if "x ∈ S" for x
using assms that unfolding holomorphic_on_def
by (intro derivative_intros) auto
with False show ?thesis
using holomorphic_on_def holomorphic_transform by presburger
qed
lemma pole_theorem_open:
assumes holg: "g holomorphic_on S" and S: "open S"
and eq: "⋀z. z ∈ S - {a} ⟹ g z = (z-a) * f z"
shows "(λz. if z = a then deriv g a
else f z - g a/(z-a)) holomorphic_on S"
using pole_lemma_open [OF holg S]
by (rule holomorphic_transform) (auto simp: eq divide_simps)
lemma pole_theorem_0:
assumes holg: "g holomorphic_on S" and a: "a ∈ interior S"
and eq: "⋀z. z ∈ S - {a} ⟹ g z = (z-a) * f z"
and [simp]: "f a = deriv g a" "g a = 0"
shows "f holomorphic_on S"
using pole_theorem [OF holg a eq]
by (rule holomorphic_transform) (auto simp: eq field_split_simps)
lemma pole_theorem_open_0:
assumes holg: "g holomorphic_on S" and S: "open S"
and eq: "⋀z. z ∈ S - {a} ⟹ g z = (z-a) * f z"
and [simp]: "f a = deriv g a" "g a = 0"
shows "f holomorphic_on S"
using pole_theorem_open [OF holg S eq]
by (rule holomorphic_transform) (auto simp: eq field_split_simps)
lemma pole_theorem_analytic:
assumes g: "g analytic_on S"
and eq: "⋀z. z ∈ S
⟹ ∃d. 0 < d ∧ (∀w ∈ ball z d - {a}. g w = (w-a) * f w)"
shows "(λz. if z = a then deriv g a else f z - g a/(z-a)) analytic_on S" (is "?F analytic_on S")
unfolding analytic_on_def
proof
fix x
assume "x ∈ S"
with g obtain e where "0 < e" and e: "g holomorphic_on ball x e"
by (auto simp add: analytic_on_def)
obtain d where "0 < d" and d: "⋀w. w ∈ ball x d - {a} ⟹ g w = (w-a) * f w"
using ‹x ∈ S› eq by blast
have "?F holomorphic_on ball x (min d e)"
using d e ‹x ∈ S› by (fastforce simp: holomorphic_on_subset subset_ball intro!: pole_theorem_open)
then show "∃e>0. ?F holomorphic_on ball x e"
using ‹0 < d› ‹0 < e› not_le by fastforce
qed
lemma pole_theorem_analytic_0:
assumes g: "g analytic_on S"
and eq: "⋀z. z ∈ S ⟹ ∃d. 0 < d ∧ (∀w ∈ ball z d - {a}. g w = (w-a) * f w)"
and [simp]: "f a = deriv g a" "g a = 0"
shows "f analytic_on S"
proof -
have [simp]: "(λz. if z = a then deriv g a else f z - g a / (z-a)) = f"
by auto
show ?thesis
using pole_theorem_analytic [OF g eq] by simp
qed
lemma pole_theorem_analytic_open_superset:
assumes g: "g analytic_on S" and "S ⊆ T" "open T"
and eq: "⋀z. z ∈ T - {a} ⟹ g z = (z-a) * f z"
shows "(λz. if z = a then deriv g a else f z - g a/(z-a)) analytic_on S"
proof (rule pole_theorem_analytic [OF g])
fix z
assume "z ∈ S"
then obtain e where "0 < e" and e: "ball z e ⊆ T"
using assms openE by blast
then show "∃d>0. ∀w∈ball z d - {a}. g w = (w-a) * f w"
using eq by auto
qed
lemma pole_theorem_analytic_open_superset_0:
assumes g: "g analytic_on S" "S ⊆ T" "open T" "⋀z. z ∈ T - {a} ⟹ g z = (z-a) * f z"
and [simp]: "f a = deriv g a" "g a = 0"
shows "f analytic_on S"
proof -
have [simp]: "(λz. if z = a then deriv g a else f z - g a / (z-a)) = f"
by auto
have "(λz. if z = a then deriv g a else f z - g a/(z-a)) analytic_on S"
by (rule pole_theorem_analytic_open_superset [OF g])
then show ?thesis by simp
qed
subsection‹General, homology form of Cauchy's theorem›
text‹Proof is based on Dixon's, as presented in Lang's "Complex Analysis" book (page 147).›
lemma contour_integral_continuous_on_linepath_2D:
assumes "open U" and cont_dw: "⋀w. w ∈ U ⟹ F w contour_integrable_on (linepath a b)"
and cond_uu: "continuous_on (U × U) (λ(x,y). F x y)"
and abu: "closed_segment a b ⊆ U"
shows "continuous_on U (λw. contour_integral (linepath a b) (F w))"
proof -
have "∃d>0. ∀x'∈U. dist x' w < d ⟶
dist (contour_integral (linepath a b) (F x'))
(contour_integral (linepath a b) (F w)) ≤ ε"
if "w ∈ U" "0 < ε" "a ≠ b" for w ε
proof -
obtain δ where "δ>0" and δ: "cball w δ ⊆ U" using open_contains_cball ‹open U› ‹w ∈ U› by force
let ?TZ = "cball w δ × closed_segment a b"
have "uniformly_continuous_on ?TZ (λ(x,y). F x y)"
by (metis Sigma_mono δ abu compact_Times compact_cball compact_segment compact_uniformly_continuous
cond_uu continuous_on_subset)
then obtain η where "η>0"
and η: "⋀x x'. ⟦x∈?TZ; x'∈?TZ; dist x' x < η⟧ ⟹
dist ((λ(x,y). F x y) x') ((λ(x,y). F x y) x) < ε/norm(b-a)"
using ‹0 < ε› ‹a ≠ b›
by (auto elim: uniformly_continuous_onE [where e = "ε/norm(b-a)"])
have η: "⟦norm (w - x1) ≤ δ; x2 ∈ closed_segment a b;
norm (w - x1') ≤ δ; x2' ∈ closed_segment a b; norm ((x1', x2') - (x1, x2)) < η⟧
⟹ norm (F x1' x2' - F x1 x2) ≤ ε / cmod (b-a)"
for x1 x2 x1' x2'
using η [of "(x1,x2)" "(x1',x2')"] by (force simp: dist_norm)
have le_ee: "cmod (contour_integral (linepath a b) (λx. F x' x - F w x)) ≤ ε"
if "x' ∈ U" "cmod (x' - w) < δ" "cmod (x' - w) < η" for x'
proof -
have "(λx. F x' x - F w x) contour_integrable_on linepath a b"
by (simp add: ‹w ∈ U› cont_dw contour_integrable_diff that)
then have "cmod (contour_integral (linepath a b) (λx. F x' x - F w x)) ≤ ε/norm(b-a) * norm(b-a)"
using has_contour_integral_bound_linepath [OF has_contour_integral_integral _ η]
using ‹0 < ε› ‹0 < δ› that by (force simp: norm_minus_commute)
also have "… = ε" using ‹a ≠ b› by simp
finally show ?thesis .
qed
show ?thesis
using ‹0 < δ› ‹0 < η› ‹w ∈ U›
apply (intro exI[where x="min δ η"])
by (auto simp: dist_norm contour_integral_diff [OF cont_dw cont_dw, symmetric] intro: le_ee)
qed
then show ?thesis
by (metis (no_types, lifting) continuous_onI continuous_on_iff
contour_integral_trivial dist_self)
qed
text‹This version has \<^term>‹polynomial_function γ› as an additional assumption.›
lemma Cauchy_integral_formula_global_weak:
assumes "open U" and holf: "f holomorphic_on U"
and z: "z ∈ U" and γ: "polynomial_function γ"
and pasz: "path_image γ ⊆ U - {z}" and loop: "pathfinish γ = pathstart γ"
and zero: "⋀w. w ∉ U ⟹ winding_number γ w = 0"
shows "((λw. f w / (w-z)) has_contour_integral (2*pi * 𝗂 * winding_number γ z * f z)) γ"
proof -
obtain γ' where pfγ': "polynomial_function γ'" and γ': "⋀x. (γ has_vector_derivative (γ' x)) (at x)"
using has_vector_derivative_polynomial_function [OF γ] by blast
then have "bounded(path_image γ')"
by (simp add: path_image_def compact_imp_bounded compact_continuous_image continuous_on_polymonial_function)
then obtain B where "B>0" and B: "⋀x. x ∈ path_image γ' ⟹ norm x ≤ B"
using bounded_pos by force
define d where [abs_def]: "d z w = (if w = z then deriv f z else (f w - f z)/(w-z))" for z w
define v where "v = {w. w ∉ path_image γ ∧ winding_number γ w = 0}"
have "path γ" "valid_path γ" using γ
by (auto simp: path_polynomial_function valid_path_polynomial_function)
then have ov: "open v"
by (simp add: v_def open_winding_number_levelsets loop)
have uv_Un: "U ∪ v = UNIV"
using pasz zero by (auto simp: v_def)
have conf: "continuous_on U f"
by (metis holf holomorphic_on_imp_continuous_on)
have hol_d: "(d y) holomorphic_on U" if "y ∈ U" for y
proof -
have *: "(λc. if c = y then deriv f y else (f c - f y) / (c-y)) holomorphic_on U"
by (simp add: holf pole_lemma_open ‹open U›)
then have "isCont (λx. if x = y then deriv f y else (f x - f y) / (x-y)) y"
using at_within_open field_differentiable_imp_continuous_at holomorphic_on_def that ‹open U› by fastforce
then have "continuous_on U (d y)"
using "*" d_def holomorphic_on_imp_continuous_on by auto
moreover have "d y holomorphic_on U - {y}"
proof -
have "(λw. if w = y then deriv f y else (f w - f y) / (w-y)) field_differentiable at w"
if "w ∈ U - {y}" for w
proof (rule field_differentiable_transform_within)
show "(λw. (f w - f y) / (w-y)) field_differentiable at w"
using that ‹open U› holf
by (auto intro!: holomorphic_on_imp_differentiable_at derivative_intros)
show "dist w y > 0"
using that by auto
qed (auto simp: dist_commute)
then show ?thesis
unfolding field_differentiable_def by (simp add: d_def holomorphic_on_open ‹open U› open_delete)
qed
ultimately show ?thesis
by (rule no_isolated_singularity) (auto simp: ‹open U›)
qed
have cint_fxy: "(λx. (f x - f y) / (x-y)) contour_integrable_on γ" if "y ∉ path_image γ" for y
proof (rule contour_integrable_holomorphic_simple [where S = "U-{y}"])
show "(λx. (f x - f y) / (x-y)) holomorphic_on U - {y}"
by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
show "path_image γ ⊆ U - {y}"
using pasz that by blast
qed (auto simp: ‹open U› open_delete ‹valid_path γ›)
define h where
"h z = (if z ∈ U then contour_integral γ (d z) else contour_integral γ (λw. f w/(w-z)))" for z
have U: "((d z) has_contour_integral h z) γ" if "z ∈ U" for z
proof -
have "d z holomorphic_on U"
by (simp add: hol_d that)
with that show ?thesis
by (metis Diff_subset ‹valid_path γ› ‹open U› contour_integrable_holomorphic_simple h_def
has_contour_integral_integral pasz subset_trans)
qed
have V: "((λw. f w / (w-z)) has_contour_integral h z) γ" if z: "z ∈ v" for z
proof -
have 0: "0 = (f z) * 2 * of_real (2 * pi) * 𝗂 * winding_number γ z"
using v_def z by auto
then have "((λx. 1 / (x-z)) has_contour_integral 0) γ"
using z v_def has_contour_integral_winding_number [OF ‹valid_path γ›] by fastforce
then have "((λx. f z * (1 / (x-z))) has_contour_integral 0) γ"
using has_contour_integral_lmul by fastforce
then have "((λx. f z / (x-z)) has_contour_integral 0) γ"
by (simp add: field_split_simps)
moreover have "((λx. (f x - f z) / (x-z)) has_contour_integral contour_integral γ (d z)) γ"
by (metis (no_types, lifting) z cint_fxy contour_integral_eq d_def has_contour_integral_integral mem_Collect_eq v_def)
ultimately have *: "((λx. f z / (x-z) + (f x - f z) / (x-z)) has_contour_integral (0 + contour_integral γ (d z))) γ"
by (rule has_contour_integral_add)
have "((λw. f w / (w-z)) has_contour_integral contour_integral γ (d z)) γ"
if "z ∈ U"
using * by (auto simp: divide_simps has_contour_integral_eq)
moreover have "((λw. f w / (w-z)) has_contour_integral contour_integral γ (λw. f w / (w-z))) γ"
if "z ∉ U"
proof (rule has_contour_integral_integral [OF contour_integrable_holomorphic_simple [where S=U]])
show "(λw. f w / (w-z)) holomorphic_on U"
by (rule holomorphic_intros assms | use that in force)+
qed (use ‹open U› pasz ‹valid_path γ› in auto)
ultimately show ?thesis
using z by (simp add: h_def)
qed
have znot: "z ∉ path_image γ"
using pasz by blast
obtain d0 where "d0>0" and d0: "⋀x y. x ∈ path_image γ ⟹ y ∈ - U ⟹ d0 ≤ dist x y"
using separate_compact_closed [of "path_image γ" "-U"] pasz ‹open U› ‹path γ› compact_path_image
by blast
obtain dd where "0 < dd" and dd: "{y + k | y k. y ∈ path_image γ ∧ k ∈ ball 0 dd} ⊆ U"
proof
show "0 < d0 / 2" using ‹0 < d0› by auto
qed (use ‹0 < d0› d0 in ‹force simp: dist_norm›)
define T where "T ≡ {y + k |y k. y ∈ path_image γ ∧ k ∈ cball 0 (dd / 2)}"
have "⋀x x'. ⟦x ∈ path_image γ; dist x x' * 2 < dd⟧
⟹ ∃y k. x' = y + k ∧ y ∈ path_image γ ∧ dist 0 k * 2 ≤ dd"
by (metis add.commute diff_add_cancel dist_0_norm dist_commute dist_norm less_eq_real_def)
then have subt: "path_image γ ⊆ interior T"
using ‹0 < dd›
apply (clarsimp simp add: mem_interior T_def)
apply (rule_tac x="dd/2" in exI, auto)
done
have "compact T"
unfolding T_def
using ‹valid_path γ› compact_cball compact_sums compact_valid_path_image by blast
have T: "T ⊆ U"
unfolding T_def using ‹0 < dd› dd by fastforce
obtain L where "L>0"
and L: "⋀f B. ⟦f holomorphic_on interior T; ⋀z. z∈interior T ⟹ cmod (f z) ≤ B⟧ ⟹
cmod (contour_integral γ f) ≤ L * B"
using contour_integral_bound_exists [OF open_interior ‹valid_path γ› subt]
by blast
have "bounded(f ` T)"
by (meson ‹compact T› compact_continuous_image compact_imp_bounded conf continuous_on_subset T)
then obtain D where "D>0" and D: "⋀x. x ∈ T ⟹ norm (f x) ≤ D"
by (auto simp: bounded_pos)
obtain C where "C>0" and C: "⋀x. x ∈ T ⟹ norm x ≤ C"
using ‹compact T› bounded_pos compact_imp_bounded by force
have "dist (h y) 0 ≤ e" if "0 < e" and le: "D * L / e + C ≤ cmod y" for e y
proof -
have "D * L / e > 0" using ‹D>0› ‹L>0› ‹e>0› by simp
with le have ybig: "norm y > C" by force
with C have "y ∉ T" by force
then have ynot: "y ∉ path_image γ"
using subt interior_subset by blast
have [simp]: "winding_number γ y = 0"
proof (rule winding_number_zero_outside)
show "path_image γ ⊆ cball 0 C"
by (meson C interior_subset mem_cball_0 subset_eq subt)
qed (use ybig loop ‹path γ› in auto)
have [simp]: "h y = contour_integral γ (λw. f w/(w-y))"
by (rule contour_integral_unique [symmetric]) (simp add: v_def ynot V)
have holint: "(λw. f w / (w-y)) holomorphic_on interior T"
proof (intro holomorphic_intros)
show "f holomorphic_on interior T"
using holf holomorphic_on_subset interior_subset T by blast
qed (use ‹y ∉ T› interior_subset in auto)
have leD: "cmod (f z / (z-y)) ≤ D * (e / L / D)" if z: "z ∈ interior T" for z
proof -
have "D * L / e + cmod z ≤ cmod y"
using le C [of z] z using interior_subset by force
then have DL2: "D * L / e ≤ cmod (z-y)"
using norm_triangle_ineq2 [of y z] by (simp add: norm_minus_commute)
have "cmod (f z / (z-y)) = cmod (f z) * inverse (cmod (z-y))"
by (simp add: norm_mult norm_inverse Fields.field_class.field_divide_inverse)
also have "… ≤ D * (e / L / D)"
proof (rule mult_mono)
show "cmod (f z) ≤ D"
using D interior_subset z by blast
show "inverse (cmod (z-y)) ≤ e / L / D" "D ≥ 0"
using ‹L>0› ‹e>0› ‹D>0› DL2 by (auto simp: norm_divide field_split_simps)
qed auto
finally show ?thesis .
qed
have "dist (h y) 0 = cmod (contour_integral γ (λw. f w / (w-y)))"
by (simp add: dist_norm)
also have "… ≤ L * (D * (e / L / D))"
by (rule L [OF holint leD])
also have "… = e"
using ‹L>0› ‹0 < D› by auto
finally show ?thesis .
qed
then have "(h ⤏ 0) at_infinity"
by (meson Lim_at_infinityI)
moreover have "h holomorphic_on UNIV"
proof -
have con_ff: "continuous (at (x,z)) (λ(x,y). (f y - f x) / (y-x))"
if "x ∈ U" "z ∈ U" "x ≠ z" for x z
using that conf
apply (simp add: split_def continuous_on_eq_continuous_at ‹open U›)
apply (simp | rule continuous_intros continuous_within_compose2 [where g=f])+
done
have con_fstsnd: "continuous_on UNIV (λx. (fst x - snd x) ::complex)"
by (rule continuous_intros)+
have open_uu_Id: "open (U × U - Id)"
proof (rule open_Diff)
show "open (U × U)"
by (simp add: open_Times ‹open U›)
show "closed (Id :: complex rel)"
using continuous_closed_preimage_constant [OF con_fstsnd closed_UNIV, of 0]
by (auto simp: Id_fstsnd_eq algebra_simps)
qed
have con_derf: "continuous (at z) (deriv f)" if "z ∈ U" for z
by (meson analytic_at analytic_at_imp_isCont assms(1) holf holomorphic_deriv that)
have tendsto_f': "((λ(x,y). if y = x then deriv f (x)
else (f (y) - f (x)) / (y-x)) ⤏ deriv f x)
(at (x, x) within U × U)" if "x ∈ U" for x
proof (rule Lim_withinI)
fix e::real assume "0 < e"
obtain k1 where "k1>0" and k1: "⋀x'. norm (x' - x) ≤ k1 ⟹ norm (deriv f x' - deriv f x) < e"
using ‹0 < e› continuous_within_E [OF con_derf [OF ‹x ∈ U›]]
by (metis UNIV_I dist_norm)
obtain k2 where "k2>0" and k2: "ball x k2 ⊆ U"
by (blast intro: openE [OF ‹open U›] ‹x ∈ U›)
have neq: "norm ((f z' - f x') / (z' - x') - deriv f x) ≤ e"
if "z' ≠ x'" and less_k1: "norm (x'-x, z'-x) < k1" and less_k2: "norm (x'-x, z'-x) < k2"
for x' z'
proof -
have cs_less: "w ∈ closed_segment x' z' ⟹ cmod (w-x) ≤ norm (x'-x, z'-x)" for w
using segment_furthest_le [of w x' z' x]
by (metis (no_types) dist_commute dist_norm norm_fst_le norm_snd_le order_trans)
have derf_le: "w ∈ closed_segment x' z' ⟹ z' ≠ x' ⟹ cmod (deriv f w - deriv f x) ≤ e" for w
by (blast intro: cs_less less_k1 k1 [unfolded divide_const_simps dist_norm] less_imp_le le_less_trans)
have f_has_der: "⋀x. x ∈ U ⟹ (f has_field_derivative deriv f x) (at x within U)"
by (metis DERIV_deriv_iff_field_differentiable at_within_open holf holomorphic_on_def ‹open U›)
have "closed_segment x' z' ⊆ U"
by (rule order_trans [OF _ k2]) (simp add: cs_less le_less_trans [OF _ less_k2] dist_complex_def norm_minus_commute subset_iff)
then have cint_derf: "(deriv f has_contour_integral f z' - f x') (linepath x' z')"
using contour_integral_primitive [OF f_has_der valid_path_linepath] pasz by simp
then have *: "((λx. deriv f x / (z' - x')) has_contour_integral (f z' - f x') / (z' - x')) (linepath x' z')"
by (rule has_contour_integral_div)
have "norm ((f z' - f x') / (z' - x') - deriv f x) ≤ e/norm(z' - x') * norm(z' - x')"
apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_diff [OF *]])
using has_contour_integral_div [where c = "z' - x'", OF has_contour_integral_const_linepath [of "deriv f x" z' x']]
‹e > 0› ‹z' ≠ x'›
apply (auto simp: norm_divide divide_simps derf_le)
done
also have "… ≤ e" using ‹0 < e› by simp
finally show ?thesis .
qed
show "∃d>0. ∀xa∈U × U.
0 < dist xa (x, x) ∧ dist xa (x, x) < d ⟶
dist (case xa of (x, y) ⇒ if y = x then deriv f x else (f y - f x) / (y-x)) (deriv f x) ≤ e"
apply (rule_tac x="min k1 k2" in exI)
using ‹k1>0› ‹k2>0› ‹e>0›
by (force simp: dist_norm neq intro: dual_order.strict_trans2 k1 less_imp_le norm_fst_le)
qed
have con_pa_f: "continuous_on (path_image γ) f"
by (meson holf holomorphic_on_imp_continuous_on holomorphic_on_subset interior_subset subt T)
have le_B: "⋀T. T ∈ {0..1} ⟹ cmod (vector_derivative γ (at T)) ≤ B"
using γ' B by (simp add: path_image_def vector_derivative_at rev_image_eqI)
have f_has_cint: "⋀w. w ∈ v - path_image γ ⟹ ((λu. f u / (u-w) ^ 1) has_contour_integral h w) γ"
by (simp add: V)
have "⋀x y. ⟦x ∈ U; y ∈ U; y ≠ x⟧ ⟹ (λ(x, y). d x y) ─(x, y)→ (f y - f x) / (y - x)"
unfolding d_def
apply (rule Lim_transform_within_open [OF _ open_uu_Id, where f = "(λ(x,y). (f y - f x) / (y-x))"])
using con_ff by (auto simp: continuous_within)
then have cond_uu: "continuous_on (U × U) (λ(x,y). d x y)"
unfolding continuous_on_eq_continuous_within continuous_within d_def
by (fastforce simp add: tendsto_f' intro: Lim_at_imp_Lim_at_within)
have hol_dw: "(λz. d z w) holomorphic_on U" if "w ∈ U" for w
proof -
have "continuous_on U ((λ(x,y). d x y) ∘ (λz. (w,z)))"
by (rule continuous_on_compose continuous_intros continuous_on_subset [OF cond_uu] | force intro: that)+
then have *: "continuous_on U (λz. if w = z then deriv f z else (f w - f z) / (w-z))"
by (rule rev_iffD1 [OF _ continuous_on_cong [OF refl]]) (simp add: d_def field_simps)
have **: "(λz. if w = z then deriv f z else (f w - f z) / (w-z)) field_differentiable at x"
if "x ∈ U" "x ≠ w" for x
proof (rule_tac f = "λx. (f w - f x)/(w-x)" and d = "dist x w" in field_differentiable_transform_within)
show "(λx. (f w - f x) / (w-x)) field_differentiable at x"
using that ‹open U›
by (intro derivative_intros holomorphic_on_imp_differentiable_at [OF holf]; force)
qed (use that ‹open U› in ‹auto simp: dist_commute›)
show ?thesis
unfolding d_def
proof (rule no_isolated_singularity [OF * _ ‹open U›])
show "(λz. if w = z then deriv f z else (f w - f z) / (w-z)) holomorphic_on U - {w}"
by (auto simp: field_differentiable_def [symmetric] holomorphic_on_open open_Diff ‹open U› **)
qed auto
qed
{ fix a b
assume abu: "closed_segment a b ⊆ U"
have cont_cint_d: "continuous_on U (λw. contour_integral (linepath a b) (λz. d z w))"
proof (rule contour_integral_continuous_on_linepath_2D [OF ‹open U› _ _ abu])
show "⋀w. w ∈ U ⟹ (λz. d z w) contour_integrable_on (linepath a b)"
by (metis abu hol_dw continuous_on_subset contour_integrable_continuous_linepath holomorphic_on_imp_continuous_on)
show "continuous_on (U × U) (λ(x, y). d y x)"
by (auto intro: continuous_on_swap_args cond_uu)
qed
have cont_cint_dγ: "continuous_on {0..1} ((λw. contour_integral (linepath a b) (λz. d z w)) ∘ γ)"
by (metis Diff_subset ‹path γ› cont_cint_d continuous_on_compose continuous_on_subset pasz path_def path_image_def)
have "continuous_on {0..1} (λx. vector_derivative γ (at x))"
using pfγ' by (simp add: continuous_on_polymonial_function vector_derivative_at [OF γ'])
then have cint_cint: "(λw. contour_integral (linepath a b) (λz. d z w)) contour_integrable_on γ"
apply (simp add: contour_integrable_on)
apply (rule integrable_continuous_real)
by (rule continuous_on_mult [OF cont_cint_dγ [unfolded o_def]])
have "contour_integral (linepath a b) h = contour_integral (linepath a b) (λz. contour_integral γ (d z))"
using abu by (force simp: h_def intro: contour_integral_eq)
also have "… = contour_integral γ (λw. contour_integral (linepath a b) (λz. d z w))"
proof (rule contour_integral_swap)
show "continuous_on (path_image (linepath a b) × path_image γ) (λ(y1, y2). d y1 y2)"
using abu pasz by (auto intro: continuous_on_subset [OF cond_uu])
show "continuous_on {0..1} (λt. vector_derivative (linepath a b) (at t))"
by (auto intro!: continuous_intros)
show "continuous_on {0..1} (λt. vector_derivative γ (at t))"
by (metis γ' continuous_on_eq path_def path_polynomial_function pfγ' vector_derivative_at)
qed (use ‹valid_path γ› in auto)
finally have cint_h_eq:
"contour_integral (linepath a b) h =
contour_integral γ (λw. contour_integral (linepath a b) (λz. d z w))" .
note cint_cint cint_h_eq
} note cint_h = this
have conthu: "continuous_on U h"
proof (simp add: continuous_on_sequentially, clarify)
fix a x
assume x: "x ∈ U" and au: "∀n. a n ∈ U" and ax: "a ⇢ x"
then have A1: "∀⇩F n in sequentially. d (a n) contour_integrable_on γ"
by (meson U contour_integrable_on_def eventuallyI)
obtain dd where "dd>0" and dd: "cball x dd ⊆ U" using open_contains_cball ‹open U› x by force
have A2: "uniform_limit (path_image γ) (λn. d (a n)) (d x) sequentially"
unfolding uniform_limit_iff dist_norm
proof clarify
fix ee::real
assume "0 < ee"
show "∀⇩F n in sequentially. ∀ξ∈path_image γ. cmod (d (a n) ξ - d x ξ) < ee"
proof -
let ?ddpa = "{(w,z) |w z. w ∈ cball x dd ∧ z ∈ path_image γ}"
have "uniformly_continuous_on ?ddpa (λ(x,y). d x y)"
proof (rule compact_uniformly_continuous [OF continuous_on_subset[OF cond_uu]])
show "compact {(w, z) |w z. w ∈ cball x dd ∧ z ∈ path_image γ}"
using ‹valid_path γ›
by (auto simp: compact_Times compact_valid_path_image simp del: mem_cball)
qed (use dd pasz in auto)
then obtain kk where "kk>0"
and kk: "⋀x x'. ⟦x ∈ ?ddpa; x' ∈ ?ddpa; dist x' x < kk⟧ ⟹
dist ((λ(x,y). d x y) x') ((λ(x,y). d x y) x) < ee"
by (rule uniformly_continuous_onE [where e = ee]) (use ‹0 < ee› in auto)
have kk: "⟦norm (w-x) ≤ dd; z ∈ path_image γ; norm ((w, z) - (x, z)) < kk⟧ ⟹ norm (d w z - d x z) < ee"
for w z
using ‹dd>0› kk [of "(x,z)" "(w,z)"] by (force simp: norm_minus_commute dist_norm)
obtain no where "∀n≥no. dist (a n) x < min dd kk"
using ax unfolding lim_sequentially
by (meson ‹0 < dd› ‹0 < kk› min_less_iff_conj)
then show ?thesis
using ‹dd > 0› ‹kk > 0› by (fastforce simp: eventually_sequentially kk dist_norm)
qed
qed
have "(λn. contour_integral γ (d (a n))) ⇢ contour_integral γ (d x)"
by (rule contour_integral_uniform_limit [OF A1 A2 le_B]) (auto simp: ‹valid_path γ›)
then have tendsto_hx: "(λn. contour_integral γ (d (a n))) ⇢ h x"
by (simp add: h_def x)
then show "(h ∘ a) ⇢ h x"
by (simp add: h_def x au o_def)
qed
show ?thesis
proof (simp add: holomorphic_on_open field_differentiable_def [symmetric], clarify)
fix z0
consider "z0 ∈ v" | "z0 ∈ U" using uv_Un by blast
then show "h field_differentiable at z0"
proof cases
assume "z0 ∈ v" then show ?thesis
using Cauchy_next_derivative [OF con_pa_f le_B f_has_cint _ ov] V f_has_cint ‹valid_path γ›
by (auto simp: field_differentiable_def v_def)
next
assume "z0 ∈ U" then
obtain e where "e>0" and e: "ball z0 e ⊆ U" by (blast intro: openE [OF ‹open U›])
have *: "contour_integral (linepath a b) h + contour_integral (linepath b c) h + contour_integral (linepath c a) h = 0"
if abc_subset: "convex hull {a, b, c} ⊆ ball z0 e" for a b c
proof -
have *: "⋀x1 x2 z. z ∈ U ⟹ closed_segment x1 x2 ⊆ U ⟹ (λw. d w z) contour_integrable_on linepath x1 x2"
using hol_dw holomorphic_on_imp_continuous_on ‹open U›
by (auto intro!: contour_integrable_holomorphic_simple)
have abc: "closed_segment a b ⊆ U" "closed_segment b c ⊆ U" "closed_segment c a ⊆ U"
using that e segments_subset_convex_hull by fastforce+
have eq0: "⋀w. w ∈ U ⟹ contour_integral (linepath a b +++ linepath b c +++ linepath c a) (λz. d z w) = 0"
proof (rule contour_integral_unique [OF Cauchy_theorem_triangle])
show "⋀w. w ∈ U ⟹ (λz. d z w) holomorphic_on convex hull {a, b, c}"
using e abc_subset by (auto intro: holomorphic_on_subset [OF hol_dw])
qed
have "⋀z. z ∈ path_image γ ⟹
contour_integral (linepath a b) (λx. d x z) +
(contour_integral (linepath b c) (λx. d x z) +
contour_integral (linepath c a) (λx. d x z)) = 0"
using abc pasz U "*" eq0 by auto
then show ?thesis
by (simp add: contour_integral_eq_0 cint_h abc contour_integrable_add contour_integral_add [symmetric] add_ac)
qed
show ?thesis
using e ‹e > 0›
by (auto intro!: holomorphic_on_imp_differentiable_at [OF _ open_ball] analytic_imp_holomorphic
Morera_triangle continuous_on_subset [OF conthu] *)
qed
qed
qed
ultimately have [simp]: "h z = 0" for z
by (meson Liouville_weak)
have "((λw. 1 / (w-z)) has_contour_integral complex_of_real (2 * pi) * 𝗂 * winding_number γ z) γ"
by (rule has_contour_integral_winding_number [OF ‹valid_path γ› znot])
then have "((λw. f z * (1 / (w-z))) has_contour_integral complex_of_real (2 * pi) * 𝗂 * winding_number γ z * f z) γ"
by (metis mult.commute has_contour_integral_lmul)
then have 1: "((λw. f z / (w-z)) has_contour_integral complex_of_real (2 * pi) * 𝗂 * winding_number γ z * f z) γ"
by (simp add: field_split_simps)
moreover have 2: "((λw. (f w - f z) / (w-z)) has_contour_integral 0) γ"
using U [OF z] pasz d_def by (force elim: has_contour_integral_eq [where g = "λw. (f w - f z)/(w-z)"])
show ?thesis
using has_contour_integral_add [OF 1 2] by (simp add: diff_divide_distrib)
qed
theorem Cauchy_integral_formula_global:
assumes S: "open S" and holf: "f holomorphic_on S"
and z: "z ∈ S" and vpg: "valid_path γ"
and pasz: "path_image γ ⊆ S - {z}" and loop: "pathfinish γ = pathstart γ"
and zero: "⋀w. w ∉ S ⟹ winding_number γ w = 0"
shows "((λw. f w / (w-z)) has_contour_integral (2*pi * 𝗂 * winding_number γ z * f z)) γ"
proof -
have "path γ" using vpg by (blast intro: valid_path_imp_path)
have hols: "(λw. f w / (w-z)) holomorphic_on S - {z}" "(λw. 1 / (w-z)) holomorphic_on S - {z}"
by (rule holomorphic_intros holomorphic_on_subset [OF holf] | force)+
then have cint_fw: "(λw. f w / (w-z)) contour_integrable_on γ"
by (meson contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on open_delete S vpg pasz)
obtain d where "d>0"
and d: "⋀g h. ⟦valid_path g; valid_path h; ∀t∈{0..1}. cmod (g t - γ t) < d ∧ cmod (h t - γ t) < d;
pathstart h = pathstart g ∧ pathfinish h = pathfinish g⟧
⟹ path_image h ⊆ S - {z} ∧ (∀f. f holomorphic_on S - {z} ⟶ contour_integral h f = contour_integral g f)"
using contour_integral_nearby_ends [OF _ ‹path γ› pasz] S by (simp add: open_Diff) metis
obtain p where polyp: "polynomial_function p"
and ps: "pathstart p = pathstart γ" and pf: "pathfinish p = pathfinish γ" and led: "∀t∈{0..1}. cmod (p t - γ t) < d"
using path_approx_polynomial_function [OF ‹path γ› ‹d > 0›] by metis
then have ploop: "pathfinish p = pathstart p" using loop by auto
have vpp: "valid_path p" using polyp valid_path_polynomial_function by blast
have [simp]: "z ∉ path_image γ" using pasz by blast
have paps: "path_image p ⊆ S - {z}" and cint_eq: "(⋀f. f holomorphic_on S - {z} ⟹ contour_integral p f = contour_integral γ f)"
using pf ps led d [OF vpg vpp] ‹d > 0› by auto
have wn_eq: "winding_number p z = winding_number γ z"
using vpp paps
by (simp add: subset_Diff_insert vpg valid_path_polynomial_function winding_number_valid_path cint_eq hols)
have "winding_number p w = winding_number γ w" if "w ∉ S" for w
proof -
have hol: "(λv. 1 / (v-w)) holomorphic_on S - {z}"
using that by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
have "w ∉ path_image p" "w ∉ path_image γ" using paps pasz that by auto
then show ?thesis
using vpp vpg by (simp add: subset_Diff_insert valid_path_polynomial_function winding_number_valid_path cint_eq [OF hol])
qed
then have wn0: "⋀w. w ∉ S ⟹ winding_number p w = 0"
by (simp add: zero)
show ?thesis
using Cauchy_integral_formula_global_weak [OF S holf z polyp paps ploop wn0] hols
by (metis wn_eq cint_eq has_contour_integral_eqpath cint_fw cint_eq)
qed
theorem Cauchy_theorem_global:
assumes S: "open S" and holf: "f holomorphic_on S"
and vpg: "valid_path γ" and loop: "pathfinish γ = pathstart γ"
and pas: "path_image γ ⊆ S"
and zero: "⋀w. w ∉ S ⟹ winding_number γ w = 0"
shows "(f has_contour_integral 0) γ"
proof -
have "path_image γ ≠ S"
by (metis compact_valid_path_image vpg compact_open path_image_nonempty S)
then obtain z where "z ∈ S" and znot: "z ∉ path_image γ" and pasz: "path_image γ ⊆ S - {z}"
using pas by blast
have hol: "(λw. (w-z) * f w) holomorphic_on S"
by (rule holomorphic_intros holf)+
show ?thesis
using Cauchy_integral_formula_global [OF S hol ‹z ∈ S› vpg pasz loop zero]
by (auto simp: znot elim!: has_contour_integral_eq)
qed
corollary Cauchy_theorem_global_outside:
assumes "open S" "f holomorphic_on S" "valid_path γ" "pathfinish γ = pathstart γ" "path_image γ ⊆ S"
"⋀w. w ∉ S ⟹ w ∈ outside(path_image γ)"
shows "(f has_contour_integral 0) γ"
by (metis Cauchy_theorem_global assms winding_number_zero_in_outside valid_path_imp_path)
lemma simply_connected_imp_winding_number_zero:
assumes "simply_connected S" "path g"
"path_image g ⊆ S" "pathfinish g = pathstart g" "z ∉ S"
shows "winding_number g z = 0"
proof -
have hom: "homotopic_loops S g (linepath (pathstart g) (pathstart g))"
by (meson assms homotopic_paths_imp_homotopic_loops pathfinish_linepath simply_connected_eq_contractible_path)
then have "homotopic_paths (- {z}) g (linepath (pathstart g) (pathstart g))"
by (meson ‹z ∉ S› homotopic_loops_imp_homotopic_paths_null homotopic_paths_subset subset_Compl_singleton)
then have "winding_number g z = winding_number(linepath (pathstart g) (pathstart g)) z"
by (rule winding_number_homotopic_paths)
also have "… = 0"
using assms by (force intro: winding_number_trivial)
finally show ?thesis .
qed
lemma Cauchy_theorem_simply_connected:
assumes "open S" "simply_connected S" "f holomorphic_on S" "valid_path g"
"path_image g ⊆ S" "pathfinish g = pathstart g"
shows "(f has_contour_integral 0) g"
by (meson assms Cauchy_theorem_global simply_connected_imp_winding_number_zero valid_path_imp_path)
proposition holomorphic_logarithm_exists:
assumes A: "convex A" "open A"
and f: "f holomorphic_on A" "⋀x. x ∈ A ⟹ f x ≠ 0"
and z0: "z0 ∈ A"
obtains g where "g holomorphic_on A" and "⋀x. x ∈ A ⟹ exp (g x) = f x"
proof -
note f' = holomorphic_derivI [OF f(1) A(2)]
obtain g where g: "⋀x. x ∈ A ⟹ (g has_field_derivative deriv f x / f x) (at x)"
proof (rule holomorphic_convex_primitive' [OF A])
show "(λx. deriv f x / f x) holomorphic_on A"
by (intro holomorphic_intros f A)
qed (auto simp: A at_within_open[of _ A])
define h where "h = (λx. -g z0 + ln (f z0) + g x)"
from g and A have g_holo: "g holomorphic_on A"
by (auto simp: holomorphic_on_def at_within_open[of _ A] field_differentiable_def)
hence h_holo: "h holomorphic_on A"
by (auto simp: h_def intro!: holomorphic_intros)
note [simp] = at_within_open[OF _ ‹open A›]
have "∃c. ∀x∈A. f x / exp (h x) - 1 = c"
using ‹convex A› z0 f
by (force simp: h_def exp_diff field_simps intro!: has_field_derivative_zero_constant derivative_eq_intros g f')
then obtain c where c: "⋀x. x ∈ A ⟹ f x / exp (h x) - 1 = c"
by blast
from c[OF z0] and z0 and f have "c = 0"
by (simp add: h_def)
with c have "⋀x. x ∈ A ⟹ exp (h x) = f x" by simp
from that[OF h_holo this] show ?thesis .
qed
subsection‹Cauchy's inequality and more versions of Liouville›
lemma Cauchy_higher_deriv_bound:
assumes holf: "f holomorphic_on (ball z r)"
and contf: "continuous_on (cball z r) f"
and fin : "⋀w. w ∈ ball z r ⟹ f w ∈ ball y B0"
and "0 < r" and "0 < n"
shows "cmod ((deriv ^^ n) f z) ≤ (fact n) * B0 / r^n"
proof -
have "0 < B0"
using ‹0 < r› fin [of z] by (metis ball_eq_empty ex_in_conv fin not_less)
have le_B0: "cmod (f w - y) ≤ B0" if "cmod (w-z) ≤ r" for w
proof (rule continuous_on_closure_norm_le [of "ball z r" "λw. f w - y"], use ‹0 < r› in simp_all)
show "continuous_on (cball z r) (λw. f w - y)"
by (intro continuous_intros contf)
show "dist z w ≤ r"
by (simp add: dist_commute dist_norm that)
qed (use fin in ‹auto simp: dist_norm less_eq_real_def norm_minus_commute›)
have "(deriv ^^ n) f z = (deriv ^^ n) (λw. f w) z - (deriv ^^ n) (λw. y) z"
using ‹0 < n› by simp
also have "... = (deriv ^^ n) (λw. f w - y) z"
using ‹0 < r› higher_deriv_diff holf by auto
finally have "(deriv ^^ n) f z = (deriv ^^ n) (λw. f w - y) z" .
have contf': "continuous_on (cball z r) (λu. f u - y)"
by (rule contf continuous_intros)+
have holf': "(λu. (f u - y)) holomorphic_on (ball z r)"
by (simp add: holf holomorphic_on_diff)
define a where "a = (2 * pi)/(fact n)"
have "0 < a" by (simp add: a_def)
have "B0/r^(Suc n)*2 * pi * r = a*((fact n)*B0/r^n)"
using ‹0 < r› by (simp add: a_def field_split_simps)
have der_dif: "(deriv ^^ n) (λw. f w - y) z = (deriv ^^ n) f z"
using ‹0 < r› ‹0 < n›
by (auto simp: higher_deriv_diff [OF holf holomorphic_on_const])
have "norm ((2 * of_real pi * 𝗂)/(fact n) * (deriv ^^ n) (λw. f w - y) z)
≤ (B0/r^(Suc n)) * (2 * pi * r)"
apply (rule has_contour_integral_bound_circlepath [of "(λu. (f u - y)/(u-z)^(Suc n))" _ z])
using Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf' holf']
using ‹0 < B0› ‹0 < r›
apply (auto simp: norm_divide norm_mult norm_power divide_simps le_B0)
done
then show ?thesis
using ‹0 < r›
by (auto simp: norm_divide norm_mult norm_power field_simps der_dif le_B0)
qed
lemma Cauchy_inequality:
assumes holf: "f holomorphic_on (ball ξ r)"
and contf: "continuous_on (cball ξ r) f"
and "0 < r"
and nof: "⋀x. norm(ξ-x) = r ⟹ norm(f x) ≤ B"
shows "norm ((deriv ^^ n) f ξ) ≤ (fact n) * B / r^n"
proof -
obtain x where "norm (ξ-x) = r"
by (metis ‹0 < r› dist_norm order_less_imp_le vector_choose_dist)
then have "0 ≤ B"
by (metis nof norm_not_less_zero not_le order_trans)
have "ξ ∈ ball ξ r"
using ‹0 < r› by simp
then have "((λu. f u / (u-ξ) ^ Suc n) has_contour_integral (2 * pi) * 𝗂 / fact n * (deriv ^^ n) f ξ)
(circlepath ξ r)"
by (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf])
have "norm ((2 * pi * 𝗂)/(fact n) * (deriv ^^ n) f ξ) ≤ (B / r^(Suc n)) * (2 * pi * r)"
proof (rule has_contour_integral_bound_circlepath)
have "ξ ∈ ball ξ r"
using ‹0 < r› by simp
then show "((λu. f u / (u-ξ) ^ Suc n) has_contour_integral (2 * pi) * 𝗂 / fact n * (deriv ^^ n) f ξ)
(circlepath ξ r)"
by (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf])
show "⋀x. cmod (x-ξ) = r ⟹ cmod (f x / (x-ξ) ^ Suc n) ≤ B / r ^ Suc n"
using ‹0 ≤ B› ‹0 < r›
by (simp add: norm_divide norm_power nof frac_le norm_minus_commute del: power_Suc)
qed (use ‹0 ≤ B› ‹0 < r› in auto)
then show ?thesis using ‹0 < r›
by (simp add: norm_divide norm_mult field_simps)
qed
lemma Liouville_polynomial:
assumes holf: "f holomorphic_on UNIV"
and nof: "⋀z. A ≤ norm z ⟹ norm(f z) ≤ B * norm z ^ n"
shows "f ξ = (∑k≤n. (deriv^^k) f 0 / fact k * ξ ^ k)"
proof (cases rule: le_less_linear [THEN disjE])
assume "B ≤ 0"
then have "⋀z. A ≤ norm z ⟹ norm(f z) = 0"
by (metis nof less_le_trans zero_less_mult_iff neqE norm_not_less_zero norm_power not_le)
then have f0: "(f ⤏ 0) at_infinity"
using Lim_at_infinity by force
then have [simp]: "f = (λw. 0)"
using Liouville_weak [OF holf, of 0]
by (simp add: eventually_at_infinity f0) meson
show ?thesis by simp
next
assume "0 < B"
have "((λk. (deriv ^^ k) f 0 / (fact k) * (ξ - 0)^k) sums f ξ)"
proof (rule holomorphic_power_series [where r = "norm ξ + 1"])
show "f holomorphic_on ball 0 (cmod ξ + 1)" "ξ ∈ ball 0 (cmod ξ + 1)"
using holf holomorphic_on_subset by auto
qed
then have sumsf: "((λk. (deriv ^^ k) f 0 / (fact k) * ξ^k) sums f ξ)" by simp
have "(deriv ^^ k) f 0 / fact k * ξ ^ k = 0" if "k>n" for k
proof (cases "(deriv ^^ k) f 0 = 0")
case True then show ?thesis by simp
next
case False
define w where "w = complex_of_real (fact k * B / cmod ((deriv ^^ k) f 0) + (¦A¦ + 1))"
have "1 ≤ abs (fact k * B / cmod ((deriv ^^ k) f 0) + (¦A¦ + 1))"
using ‹0 < B› by simp
then have wge1: "1 ≤ norm w"
by (metis norm_of_real w_def)
then have "w ≠ 0" by auto
have kB: "0 < fact k * B"
using ‹0 < B› by simp
then have "0 ≤ fact k * B / cmod ((deriv ^^ k) f 0)"
by simp
then have wgeA: "A ≤ cmod w"
by (simp only: w_def norm_of_real)
have "fact k * B / cmod ((deriv ^^ k) f 0) < abs (fact k * B / cmod ((deriv ^^ k) f 0) + (¦A¦ + 1))"
using ‹0 < B› by simp
then have wge: "fact k * B / cmod ((deriv ^^ k) f 0) < norm w"
by (metis norm_of_real w_def)
then have "fact k * B / norm w < cmod ((deriv ^^ k) f 0)"
using False by (simp add: field_split_simps mult.commute split: if_split_asm)
also have "... ≤ fact k * (B * norm w ^ n) / norm w ^ k"
proof (rule Cauchy_inequality)
show "f holomorphic_on ball 0 (cmod w)"
using holf holomorphic_on_subset by force
show "continuous_on (cball 0 (cmod w)) f"
using holf holomorphic_on_imp_continuous_on holomorphic_on_subset by blast
show "⋀x. cmod (0-x) = cmod w ⟹ cmod (f x) ≤ B * cmod w ^ n"
by (metis nof wgeA dist_0_norm dist_norm)
qed (use ‹w ≠ 0› in auto)
also have "... = fact k * B / cmod w ^ (k-n)"
using ‹k>n› by (simp add: divide_simps flip: power_add)
finally have "fact k * B / cmod w < fact k * B / cmod w ^ (k-n)" .
then have "1 / cmod w < 1 / cmod w ^ (k-n)"
by (metis kB divide_inverse inverse_eq_divide mult_less_cancel_left_pos)
then have "cmod w ^ (k-n) < cmod w"
by (smt (verit, best) ‹w ≠ 0› frac_le zero_less_norm_iff)
with self_le_power [OF wge1] show ?thesis
by (meson diff_is_0_eq not_gr0 not_le that)
qed
then have "(deriv ^^ (k + Suc n)) f 0 / fact (k + Suc n) * ξ ^ (k + Suc n) = 0" for k
using not_less_eq by blast
then have "(λi. (deriv ^^ (i + Suc n)) f 0 / fact (i + Suc n) * ξ ^ (i + Suc n)) sums 0"
by (rule sums_0)
with sums_split_initial_segment [OF sumsf, where n = "Suc n"]
show ?thesis
using atLeast0AtMost lessThan_Suc_atMost sums_unique2 by fastforce
qed
text‹Every bounded entire function is a constant function.›
theorem Liouville_theorem:
assumes holf: "f holomorphic_on UNIV"
and bf: "bounded (range f)"
shows "f constant_on UNIV"
using Liouville_polynomial [OF holf, of 0 _ 0, simplified]
by (metis bf bounded_iff constant_on_def rangeI)
text‹A holomorphic function f has only isolated zeros unless f is 0.›
lemma powser_0_nonzero:
fixes a :: "nat ⇒ 'a::{real_normed_field,banach}"
assumes r: "0 < r"
and sm: "⋀x. norm (x-ξ) < r ⟹ (λn. a n * (x-ξ) ^ n) sums (f x)"
and [simp]: "f ξ = 0"
and m0: "a m ≠ 0" and "m>0"
obtains s where "0 < s" and "⋀z. z ∈ cball ξ s - {ξ} ⟹ f z ≠ 0"
proof -
have "r ≤ conv_radius a"
using sm sums_summable by (auto simp: le_conv_radius_iff [where ξ=ξ])
obtain m where am: "a m ≠ 0" and az [simp]: "(⋀n. n<m ⟹ a n = 0)"
proof
show "a (LEAST n. a n ≠ 0) ≠ 0"
by (metis (mono_tags, lifting) m0 LeastI)
qed (fastforce dest!: not_less_Least)
define b where "b i = a (i+m) / a m" for i
define g where "g x = suminf (λi. b i * (x-ξ) ^ i)" for x
have [simp]: "b 0 = 1"
by (simp add: am b_def)
{ fix x::'a
assume "norm (x-ξ) < r"
then have "(λn. (a m * (x-ξ)^m) * (b n * (x-ξ)^n)) sums (f x)"
using am az sm sums_zero_iff_shift [of m "(λn. a n * (x-ξ) ^ n)" "f x"]
by (simp add: b_def monoid_mult_class.power_add algebra_simps)
then have "x ≠ ξ ⟹ (λn. b n * (x-ξ)^n) sums (f x / (a m * (x-ξ)^m))"
using am by (simp add: sums_mult_D)
} note bsums = this
then have "norm (x-ξ) < r ⟹ summable (λn. b n * (x-ξ)^n)" for x
using sums_summable by (cases "x=ξ") auto
then have "r ≤ conv_radius b"
by (simp add: le_conv_radius_iff [where ξ=ξ])
then have "r/2 < conv_radius b"
using not_le order_trans r by fastforce
then have "continuous_on (cball ξ (r/2)) g"
using powser_continuous_suminf [of "r/2" b ξ] by (simp add: g_def)
then obtain s where "s>0" "⋀x. ⟦norm (x-ξ) ≤ s; norm (x-ξ) ≤ r/2⟧ ⟹ dist (g x) (g ξ) < 1/2"
proof (rule continuous_onE)
show "ξ ∈ cball ξ (r / 2)" "1/2 > (0::real)"
using r by auto
qed (auto simp: dist_commute dist_norm)
moreover have "g ξ = 1"
by (simp add: g_def)
ultimately have gnz: "⋀x. ⟦norm (x-ξ) ≤ s; norm (x-ξ) ≤ r/2⟧ ⟹ (g x) ≠ 0"
by fastforce
show ?thesis
proof
have *: "f x ≠ 0" if "x ≠ ξ" "norm (x-ξ) ≤ s" "norm (x-ξ) ≤ r/2" for x
using bsums [of x] that gnz [of x] r sums_iff unfolding g_def by fastforce
show "⋀z. z ∈ cball ξ (min s (r / 2)) - {ξ} ⟹ f z ≠ 0"
by (simp add: "*" dist_norm norm_minus_commute)
qed (use ‹0 < r› ‹0 < s› in auto)
qed
subsection ‹Complex functions and power series›
text ‹
The following defines the power series expansion of a complex function at a given point
(assuming that it is analytic at that point).
›
definition fps_expansion :: "(complex ⇒ complex) ⇒ complex ⇒ complex fps" where
"fps_expansion f z0 = Abs_fps (λn. (deriv ^^ n) f z0 / fact n)"
lemma fps_expansion_cong:
assumes "∀⇩F w in nhds x. f w =g w"
shows "fps_expansion f x = fps_expansion g x"
unfolding fps_expansion_def using assms higher_deriv_cong_ev by fastforce
lemma
fixes r :: ereal
assumes "f holomorphic_on eball z0 r"
shows conv_radius_fps_expansion: "fps_conv_radius (fps_expansion f z0) ≥ r"
and eval_fps_expansion: "⋀z. z ∈ eball z0 r ⟹ eval_fps (fps_expansion f z0) (z - z0) = f z"
and eval_fps_expansion': "⋀z. norm z < r ⟹ eval_fps (fps_expansion f z0) z = f (z0 + z)"
proof -
have "(λn. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z"
if "z ∈ ball z0 r'" "ereal r' < r" for z r'
proof -
have "f holomorphic_on ball z0 r'"
using holomorphic_on_subset[OF _ ball_eball_mono] assms that by force
then show ?thesis
using fps_expansion_def holomorphic_power_series that by auto
qed
hence *: "(λn. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z"
if "z ∈ eball z0 r" for z
using that by (subst (asm) eball_conv_UNION_balls) blast
show "fps_conv_radius (fps_expansion f z0) ≥ r" unfolding fps_conv_radius_def
proof (rule conv_radius_geI_ex)
fix r' :: real assume r': "r' > 0" "ereal r' < r"
thus "∃z. norm z = r' ∧ summable (λn. fps_nth (fps_expansion f z0) n * z ^ n)"
using *[of "z0 + of_real r'"]
by (intro exI[of _ "of_real r'"]) (auto simp: summable_def dist_norm)
qed
show "eval_fps (fps_expansion f z0) (z - z0) = f z" if "z ∈ eball z0 r" for z
using *[OF that] by (simp add: eval_fps_def sums_iff)
show "eval_fps (fps_expansion f z0) z = f (z0 + z)" if "ereal (norm z) < r" for z
using *[of "z0 + z"] and that by (simp add: eval_fps_def sums_iff dist_norm)
qed
text ‹
We can now show several more facts about power series expansions (at least in the complex case)
with relative ease that would have been trickier without complex analysis.
›
lemma
fixes f :: "complex fps" and r :: ereal
assumes "⋀z. ereal (norm z) < r ⟹ eval_fps f z ≠ 0"
shows fps_conv_radius_inverse: "fps_conv_radius (inverse f) ≥ min r (fps_conv_radius f)"
and eval_fps_inverse: "⋀z. ereal (norm z) < fps_conv_radius f ⟹ ereal (norm z) < r ⟹
eval_fps (inverse f) z = inverse (eval_fps f z)"
proof -
define R where "R = min (fps_conv_radius f) r"
have *: "fps_conv_radius (inverse f) ≥ min r (fps_conv_radius f) ∧
(∀z∈eball 0 (min (fps_conv_radius f) r). eval_fps (inverse f) z = inverse (eval_fps f z))"
proof (cases "min r (fps_conv_radius f) > 0")
case True
define f' where "f' = fps_expansion (λz. inverse (eval_fps f z)) 0"
have holo: "(λz. inverse (eval_fps f z)) holomorphic_on eball 0 (min r (fps_conv_radius f))"
using assms by (intro holomorphic_intros) auto
from holo have radius: "fps_conv_radius f' ≥ min r (fps_conv_radius f)"
unfolding f'_def by (rule conv_radius_fps_expansion)
have eval_f': "eval_fps f' z = inverse (eval_fps f z)"
if "norm z < fps_conv_radius f" "norm z < r" for z
using that unfolding f'_def by (subst eval_fps_expansion'[OF holo]) auto
have "f * f' = 1"
proof (rule eval_fps_eqD)
from radius and True have "0 < min (fps_conv_radius f) (fps_conv_radius f')"
by (auto simp: min_def split: if_splits)
also have "… ≤ fps_conv_radius (f * f')" by (rule fps_conv_radius_mult)
finally show "… > 0" .
next
from True have "R > 0" by (auto simp: R_def)
hence "eventually (λz. z ∈ eball 0 R) (nhds 0)"
by (intro eventually_nhds_in_open) (auto simp: zero_ereal_def)
thus "eventually (λz. eval_fps (f * f') z = eval_fps 1 z) (nhds 0)"
proof eventually_elim
case (elim z)
hence "eval_fps (f * f') z = eval_fps f z * eval_fps f' z"
using radius by (intro eval_fps_mult)
(auto simp: R_def min_def split: if_splits intro: less_trans)
also have "eval_fps f' z = inverse (eval_fps f z)"
using elim by (intro eval_f') (auto simp: R_def)
also from elim have "eval_fps f z ≠ 0"
by (intro assms) (auto simp: R_def)
hence "eval_fps f z * inverse (eval_fps f z) = eval_fps 1 z"
by simp
finally show "eval_fps (f * f') z = eval_fps 1 z" .
qed
qed simp_all
hence "f' = inverse f"
by (intro fps_inverse_unique [symmetric]) (simp_all add: mult_ac)
with eval_f' and radius show ?thesis by simp
next
case False
hence *: "eball 0 R = {}"
by (intro eball_empty) (auto simp: R_def min_def split: if_splits)
show ?thesis
proof
from False have "min r (fps_conv_radius f) ≤ 0"
by (simp add: min_def)
also have "0 ≤ fps_conv_radius (inverse f)"
by (simp add: fps_conv_radius_def conv_radius_nonneg)
finally show "min r (fps_conv_radius f) ≤ …" .
qed (unfold * [unfolded R_def], auto)
qed
show "fps_conv_radius (inverse f) ≥ min r (fps_conv_radius f)"
using * by blast
show "eval_fps (inverse f) z = inverse (eval_fps f z)"
if "ereal (norm z) < fps_conv_radius f" "ereal (norm z) < r" for z
using that * by auto
qed
lemma
fixes f g :: "complex fps" and r :: ereal
defines "R ≡ Min {r, fps_conv_radius f, fps_conv_radius g}"
assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0"
assumes nz: "⋀z. z ∈ eball 0 r ⟹ eval_fps g z ≠ 0"
shows fps_conv_radius_divide': "fps_conv_radius (f / g) ≥ R"
and eval_fps_divide':
"ereal (norm z) < R ⟹ eval_fps (f / g) z = eval_fps f z / eval_fps g z"
proof -
from nz[of 0] and ‹r > 0› have nz': "fps_nth g 0 ≠ 0"
by (auto simp: eval_fps_at_0 zero_ereal_def)
have "R ≤ min r (fps_conv_radius g)"
by (auto simp: R_def intro: min.coboundedI2)
also have "min r (fps_conv_radius g) ≤ fps_conv_radius (inverse g)"
by (intro fps_conv_radius_inverse assms) (auto simp: zero_ereal_def)
finally have radius: "fps_conv_radius (inverse g) ≥ R" .
have "R ≤ min (fps_conv_radius f) (fps_conv_radius (inverse g))"
by (intro radius min.boundedI) (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
also have "… ≤ fps_conv_radius (f * inverse g)"
by (rule fps_conv_radius_mult)
also have "f * inverse g = f / g"
by (intro fps_divide_unit [symmetric] nz')
finally show "fps_conv_radius (f / g) ≥ R" .
assume z: "ereal (norm z) < R"
have "eval_fps (f * inverse g) z = eval_fps f z * eval_fps (inverse g) z"
using radius by (intro eval_fps_mult less_le_trans[OF z])
(auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
also have "eval_fps (inverse g) z = inverse (eval_fps g z)" using ‹r > 0›
by (intro eval_fps_inverse[where r = r] less_le_trans[OF z] nz)
(auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
also have "f * inverse g = f / g" by fact
finally show "eval_fps (f / g) z = eval_fps f z / eval_fps g z"
by (simp add: field_split_simps)
qed
lemma
fixes f g :: "complex fps" and r :: ereal
defines "R ≡ Min {r, fps_conv_radius f, fps_conv_radius g}"
assumes "subdegree g ≤ subdegree f"
assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0"
assumes "⋀z. z ∈ eball 0 r ⟹ z ≠ 0 ⟹ eval_fps g z ≠ 0"
shows fps_conv_radius_divide: "fps_conv_radius (f / g) ≥ R"
and eval_fps_divide:
"ereal (norm z) < R ⟹ c = fps_nth f (subdegree g) / fps_nth g (subdegree g) ⟹
eval_fps (f / g) z = (if z = 0 then c else eval_fps f z / eval_fps g z)"
proof -
define f' g' where "f' = fps_shift (subdegree g) f" and "g' = fps_shift (subdegree g) g"
have f_eq: "f = f' * fps_X ^ subdegree g" and g_eq: "g = g' * fps_X ^ subdegree g"
unfolding f'_def g'_def by (rule subdegree_decompose' le_refl | fact)+
have subdegree: "subdegree f' = subdegree f - subdegree g" "subdegree g' = 0"
using assms(2) by (simp_all add: f'_def g'_def)
have [simp]: "fps_conv_radius f' = fps_conv_radius f" "fps_conv_radius g' = fps_conv_radius g"
by (simp_all add: f'_def g'_def)
have [simp]: "fps_nth f' 0 = fps_nth f (subdegree g)"
"fps_nth g' 0 = fps_nth g (subdegree g)" by (simp_all add: f'_def g'_def)
have g_nz: "g ≠ 0"
proof -
define z :: complex where "z = (if r = ∞ then 1 else of_real (real_of_ereal r / 2))"
have "z ∈ eball 0 r"
using ‹r > 0› ereal_less_real_iff z_def by fastforce
moreover have "z ≠ 0" using ‹r > 0›
by (cases r) (auto simp: z_def)
ultimately have "eval_fps g z ≠ 0" by (rule assms(6))
thus "g ≠ 0" by auto
qed
have fg: "f / g = f' * inverse g'"
by (subst f_eq, subst (2) g_eq) (insert g_nz, simp add: fps_divide_unit)
have g'_nz: "eval_fps g' z ≠ 0" if z: "norm z < min r (fps_conv_radius g)" for z
proof (cases "z = 0")
case False
with assms and z have "eval_fps g z ≠ 0" by auto
also from z have "eval_fps g z = eval_fps g' z * z ^ subdegree g"
by (subst g_eq) (auto simp: eval_fps_mult)
finally show ?thesis by auto
qed (use ‹g ≠ 0› in ‹auto simp: g'_def eval_fps_at_0›)
have "R ≤ min (min r (fps_conv_radius g)) (fps_conv_radius g')"
by (auto simp: R_def min.coboundedI1 min.coboundedI2)
also have "… ≤ fps_conv_radius (inverse g')"
using g'_nz by (rule fps_conv_radius_inverse)
finally have conv_radius_inv: "R ≤ fps_conv_radius (inverse g')" .
hence "R ≤ fps_conv_radius (f' * inverse g')"
by (intro order.trans[OF _ fps_conv_radius_mult])
(auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
thus "fps_conv_radius (f / g) ≥ R" by (simp add: fg)
fix z c :: complex assume z: "ereal (norm z) < R"
assume c: "c = fps_nth f (subdegree g) / fps_nth g (subdegree g)"
show "eval_fps (f / g) z = (if z = 0 then c else eval_fps f z / eval_fps g z)"
proof (cases "z = 0")
case False
from z and conv_radius_inv have "ereal (norm z) < fps_conv_radius (inverse g')"
by simp
with z have "eval_fps (f / g) z = eval_fps f' z * eval_fps (inverse g') z"
unfolding fg by (subst eval_fps_mult) (auto simp: R_def)
also have "eval_fps (inverse g') z = inverse (eval_fps g' z)"
using z by (intro eval_fps_inverse[of "min r (fps_conv_radius g')"] g'_nz) (auto simp: R_def)
also have "eval_fps f' z * … = eval_fps f z / eval_fps g z"
using z False assms(2) by (simp add: f'_def g'_def eval_fps_shift R_def)
finally show ?thesis using False by simp
qed (simp_all add: eval_fps_at_0 fg field_simps c)
qed
lemma has_fps_expansion_fps_expansion [intro]:
assumes "open A" "0 ∈ A" "f holomorphic_on A"
shows "f has_fps_expansion fps_expansion f 0"
proof -
from assms obtain r where "r > 0 " and r: "ball 0 r ⊆ A"
by (auto simp: open_contains_ball)
with assms have holo: "f holomorphic_on eball 0 (ereal r)"
by auto
have "r ≤ fps_conv_radius (fps_expansion f 0)"
using holo by (intro conv_radius_fps_expansion) auto
then have "… > 0"
by (simp add: ereal_le_less ‹r > 0› zero_ereal_def)
moreover have "eventually (λz. z ∈ ball 0 r) (nhds 0)"
using ‹r > 0› by (intro eventually_nhds_in_open) auto
hence "eventually (λz. eval_fps (fps_expansion f 0) z = f z) (nhds 0)"
by eventually_elim (subst eval_fps_expansion'[OF holo], auto)
ultimately show ?thesis
using ‹r > 0› by (auto simp: has_fps_expansion_def)
qed
lemma fps_conv_radius_tan:
fixes c :: complex
assumes "c ≠ 0"
shows "fps_conv_radius (fps_tan c) ≥ pi / (2 * norm c)"
proof -
have "fps_conv_radius (fps_tan c) ≥
Min {pi / (2 * norm c), fps_conv_radius (fps_sin c), fps_conv_radius (fps_cos c)}"
unfolding fps_tan_def
proof (rule fps_conv_radius_divide)
fix z :: complex assume "z ∈ eball 0 (pi / (2 * norm c))"
with cos_eq_zero_imp_norm_ge[of "c*z"] assms
show "eval_fps (fps_cos c) z ≠ 0" by (auto simp: norm_mult field_simps)
qed (insert assms, auto)
thus ?thesis by (simp add: min_def)
qed
lemma eval_fps_tan:
fixes c :: complex
assumes "norm z < pi / (2 * norm c)"
shows "eval_fps (fps_tan c) z = tan (c * z)"
proof (cases "c = 0")
case False
show ?thesis unfolding fps_tan_def
proof (subst eval_fps_divide'[where r = "pi / (2 * norm c)"])
fix z :: complex assume "z ∈ eball 0 (pi / (2 * norm c))"
with cos_eq_zero_imp_norm_ge[of "c*z"] assms
show "eval_fps (fps_cos c) z ≠ 0" using False by (auto simp: norm_mult field_simps)
qed (use False assms in ‹auto simp: field_simps tan_def›)
qed simp_all
end