author | desharna |
Thu, 08 Jul 2021 08:44:18 +0200 | |
changeset 73933 | fa92bc604c59 |
parent 73932 | fd21b4a93043 |
parent 73928 | 3b76524f5a85 |
child 77228 | 8c093a4b8ccf |
permissions | -rw-r--r-- |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1 |
section \<open>Cauchy's Integral Formula\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2 |
theory Cauchy_Integral_Formula |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
3 |
imports Winding_Numbers |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
4 |
begin |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
5 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
6 |
subsection\<open>Proof\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
7 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
8 |
lemma Cauchy_integral_formula_weak: |
72266 | 9 |
assumes S: "convex S" and "finite k" and conf: "continuous_on S f" |
10 |
and fcd: "(\<And>x. x \<in> interior S - k \<Longrightarrow> f field_differentiable at x)" |
|
11 |
and z: "z \<in> interior S - k" and vpg: "valid_path \<gamma>" |
|
12 |
and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
13 |
shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
14 |
proof - |
72266 | 15 |
let ?fz = "\<lambda>w. (f w - f z)/(w - z)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
16 |
obtain f' where f': "(f has_field_derivative f') (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
17 |
using fcd [OF z] by (auto simp: field_differentiable_def) |
72266 | 18 |
have pas: "path_image \<gamma> \<subseteq> S" and znotin: "z \<notin> path_image \<gamma>" using pasz by blast+ |
19 |
have c: "continuous (at x within S) (\<lambda>w. if w = z then f' else (f w - f z) / (w - z))" if "x \<in> S" for x |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
20 |
proof (cases "x = z") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
21 |
case True then show ?thesis |
72266 | 22 |
using LIM_equal [of "z" ?fz "\<lambda>w. if w = z then f' else ?fz w"] has_field_derivativeD [OF f'] |
23 |
by (force simp add: continuous_within Lim_at_imp_Lim_at_within) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
24 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
25 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
26 |
then have dxz: "dist x z > 0" by auto |
72266 | 27 |
have cf: "continuous (at x within S) f" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
28 |
using conf continuous_on_eq_continuous_within that by blast |
72266 | 29 |
have "continuous (at x within S) (\<lambda>w. (f w - f z) / (w - z))" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
30 |
by (rule cf continuous_intros | simp add: False)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
31 |
then show ?thesis |
72266 | 32 |
apply (rule continuous_transform_within [OF _ dxz that, of ?fz]) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
33 |
apply (force simp: dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
34 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
35 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
36 |
have fink': "finite (insert z k)" using \<open>finite k\<close> by blast |
72266 | 37 |
have *: "((\<lambda>w. if w = z then f' else ?fz w) has_contour_integral 0) \<gamma>" |
38 |
proof (rule Cauchy_theorem_convex [OF _ S fink' _ vpg pas loop]) |
|
39 |
show "(\<lambda>w. if w = z then f' else ?fz w) field_differentiable at w" |
|
40 |
if "w \<in> interior S - insert z k" for w |
|
41 |
proof (rule field_differentiable_transform_within) |
|
42 |
show "(\<lambda>w. ?fz w) field_differentiable at w" |
|
43 |
using that by (intro derivative_intros fcd; simp) |
|
44 |
qed (use that in \<open>auto simp add: dist_pos_lt dist_commute\<close>) |
|
45 |
qed (use c in \<open>force simp: continuous_on_eq_continuous_within\<close>) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
46 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
47 |
apply (rule has_contour_integral_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
48 |
using znotin has_contour_integral_add [OF has_contour_integral_lmul [OF has_contour_integral_winding_number [OF vpg znotin], of "f z"] *] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
49 |
apply (auto simp: ac_simps divide_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
50 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
51 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
52 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
53 |
theorem Cauchy_integral_formula_convex_simple: |
72266 | 54 |
assumes "convex S" and holf: "f holomorphic_on S" and "z \<in> interior S" "valid_path \<gamma>" "path_image \<gamma> \<subseteq> S - {z}" |
55 |
"pathfinish \<gamma> = pathstart \<gamma>" |
|
56 |
shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>" |
|
57 |
proof - |
|
58 |
have "\<And>x. x \<in> interior S \<Longrightarrow> f field_differentiable at x" |
|
59 |
using holf at_within_interior holomorphic_onD interior_subset by fastforce |
|
60 |
then show ?thesis |
|
61 |
using assms |
|
62 |
by (intro Cauchy_integral_formula_weak [where k = "{}"]) (auto simp: holomorphic_on_imp_continuous_on) |
|
63 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
64 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
65 |
text\<open> Hence the Cauchy formula for points inside a circle.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
66 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
67 |
theorem Cauchy_integral_circlepath: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
68 |
assumes contf: "continuous_on (cball z r) f" and holf: "f holomorphic_on (ball z r)" and wz: "norm(w - z) < r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
69 |
shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w)) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
70 |
(circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
71 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
72 |
have "r > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
73 |
using assms le_less_trans norm_ge_zero by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
74 |
have "((\<lambda>u. f u / (u - w)) has_contour_integral (2 * pi) * \<i> * winding_number (circlepath z r) w * f w) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
75 |
(circlepath z r)" |
72266 | 76 |
proof (rule Cauchy_integral_formula_weak [where S = "cball z r" and k = "{}"]) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
77 |
show "\<And>x. x \<in> interior (cball z r) - {} \<Longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
78 |
f field_differentiable at x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
79 |
using holf holomorphic_on_imp_differentiable_at by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
80 |
have "w \<notin> sphere z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
81 |
by simp (metis dist_commute dist_norm not_le order_refl wz) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
82 |
then show "path_image (circlepath z r) \<subseteq> cball z r - {w}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
83 |
using \<open>r > 0\<close> by (auto simp add: cball_def sphere_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
84 |
qed (use wz in \<open>simp_all add: dist_norm norm_minus_commute contf\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
85 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
86 |
by (simp add: winding_number_circlepath assms) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
87 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
88 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
89 |
corollary\<^marker>\<open>tag unimportant\<close> Cauchy_integral_circlepath_simple: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
90 |
assumes "f holomorphic_on cball z r" "norm(w - z) < r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
91 |
shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w)) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
92 |
(circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
93 |
using assms by (force simp: holomorphic_on_imp_continuous_on holomorphic_on_subset Cauchy_integral_circlepath) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
94 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
95 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>General stepping result for derivative formulas\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
96 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
97 |
lemma Cauchy_next_derivative: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
98 |
assumes "continuous_on (path_image \<gamma>) f'" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
99 |
and leB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B" |
72266 | 100 |
and int: "\<And>w. w \<in> S - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f' u / (u - w)^k) has_contour_integral f w) \<gamma>" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
101 |
and k: "k \<noteq> 0" |
72266 | 102 |
and "open S" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
103 |
and \<gamma>: "valid_path \<gamma>" |
72266 | 104 |
and w: "w \<in> S - path_image \<gamma>" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
105 |
shows "(\<lambda>u. f' u / (u - w)^(Suc k)) contour_integrable_on \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
106 |
and "(f has_field_derivative (k * contour_integral \<gamma> (\<lambda>u. f' u/(u - w)^(Suc k)))) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
107 |
(at w)" (is "?thes2") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
108 |
proof - |
72266 | 109 |
have "open (S - path_image \<gamma>)" using \<open>open S\<close> closed_valid_path_image \<gamma> by blast |
110 |
then obtain d where "d>0" and d: "ball w d \<subseteq> S - path_image \<gamma>" using w |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
111 |
using open_contains_ball by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
112 |
have [simp]: "\<And>n. cmod (1 + of_nat n) = 1 + of_nat n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
113 |
by (metis norm_of_nat of_nat_Suc) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
114 |
have cint: "\<And>x. \<lbrakk>x \<noteq> w; cmod (x - w) < d\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
115 |
\<Longrightarrow> (\<lambda>z. (f' z / (z - x) ^ k - f' z / (z - w) ^ k) / (x * k - w * k)) contour_integrable_on \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
116 |
using int w d |
72266 | 117 |
apply (intro contour_integrable_div contour_integrable_diff has_contour_integral_integrable) |
118 |
by (force simp: dist_norm norm_minus_commute) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
119 |
have 1: "\<forall>\<^sub>F n in at w. (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
120 |
contour_integrable_on \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
121 |
unfolding eventually_at |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
122 |
apply (rule_tac x=d in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
123 |
apply (simp add: \<open>d > 0\<close> dist_norm field_simps cint) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
124 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
125 |
have bim_g: "bounded (image f' (path_image \<gamma>))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
126 |
by (simp add: compact_imp_bounded compact_continuous_image compact_valid_path_image assms) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
127 |
then obtain C where "C > 0" and C: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cmod (f' (\<gamma> x)) \<le> C" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
128 |
by (force simp: bounded_pos path_image_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
129 |
have twom: "\<forall>\<^sub>F n in at w. |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
130 |
\<forall>x\<in>path_image \<gamma>. |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
131 |
cmod ((inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k - inverse (x - w) ^ Suc k) < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
132 |
if "0 < e" for e |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
133 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
134 |
have *: "cmod ((inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k) - inverse (x - w) ^ Suc k) < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
135 |
if x: "x \<in> path_image \<gamma>" and "u \<noteq> w" and uwd: "cmod (u - w) < d/2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
136 |
and uw_less: "cmod (u - w) < e * (d/2) ^ (k+2) / (1 + real k)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
137 |
for u x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
138 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
139 |
define ff where [abs_def]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
140 |
"ff n w = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
141 |
(if n = 0 then inverse(x - w)^k |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
142 |
else if n = 1 then k / (x - w)^(Suc k) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
143 |
else (k * of_real(Suc k)) / (x - w)^(k + 2))" for n :: nat and w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
144 |
have km1: "\<And>z::complex. z \<noteq> 0 \<Longrightarrow> z ^ (k - Suc 0) = z ^ k / z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
145 |
by (simp add: field_simps) (metis Suc_pred \<open>k \<noteq> 0\<close> neq0_conv power_Suc) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
146 |
have ff1: "(ff i has_field_derivative ff (Suc i) z) (at z within ball w (d/2))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
147 |
if "z \<in> ball w (d/2)" "i \<le> 1" for i z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
148 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
149 |
have "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
150 |
using \<open>x \<in> path_image \<gamma>\<close> d that ball_divide_subset_numeral by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
151 |
then have xz[simp]: "x \<noteq> z" using \<open>x \<in> path_image \<gamma>\<close> by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
152 |
then have neq: "x * x + z * z \<noteq> x * (z * 2)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
153 |
by (blast intro: dest!: sum_sqs_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
154 |
with xz have "\<And>v. v \<noteq> 0 \<Longrightarrow> (x * x + z * z) * v \<noteq> (x * (z * 2) * v)" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
155 |
then have neqq: "\<And>v. v \<noteq> 0 \<Longrightarrow> x * (x * v) + z * (z * v) \<noteq> x * (z * (2 * v))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
156 |
by (simp add: algebra_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
157 |
show ?thesis using \<open>i \<le> 1\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
158 |
apply (simp add: ff_def dist_norm Nat.le_Suc_eq km1, safe) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
159 |
apply (rule derivative_eq_intros | simp add: km1 | simp add: field_simps neq neqq)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
160 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
161 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
162 |
{ fix a::real and b::real assume ab: "a > 0" "b > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
163 |
then have "k * (1 + real k) * (1 / a) \<le> k * (1 + real k) * (4 / b) \<longleftrightarrow> b \<le> 4 * a" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
164 |
by (subst mult_le_cancel_left_pos) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
165 |
(use \<open>k \<noteq> 0\<close> in \<open>auto simp: divide_simps\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
166 |
with ab have "real k * (1 + real k) / a \<le> (real k * 4 + real k * real k * 4) / b \<longleftrightarrow> b \<le> 4 * a" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
167 |
by (simp add: field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
168 |
} note canc = this |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
169 |
have ff2: "cmod (ff (Suc 1) v) \<le> real (k * (k + 1)) / (d/2) ^ (k + 2)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
170 |
if "v \<in> ball w (d/2)" for v |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
171 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
172 |
have lessd: "\<And>z. cmod (\<gamma> z - v) < d/2 \<Longrightarrow> cmod (w - \<gamma> z) < d" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
173 |
by (metis that norm_minus_commute norm_triangle_half_r dist_norm mem_ball) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
174 |
have "d/2 \<le> cmod (x - v)" using d x that |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
175 |
using lessd d x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
176 |
by (auto simp add: dist_norm path_image_def ball_def not_less [symmetric] del: divide_const_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
177 |
then have "d \<le> cmod (x - v) * 2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
178 |
by (simp add: field_split_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
179 |
then have dpow_le: "d ^ (k+2) \<le> (cmod (x - v) * 2) ^ (k+2)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
180 |
using \<open>0 < d\<close> order_less_imp_le power_mono by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
181 |
have "x \<noteq> v" using that |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
182 |
using \<open>x \<in> path_image \<gamma>\<close> ball_divide_subset_numeral d by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
183 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
184 |
using \<open>d > 0\<close> apply (simp add: ff_def norm_mult norm_divide norm_power dist_norm canc) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
185 |
using dpow_le apply (simp add: field_split_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
186 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
187 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
188 |
have ub: "u \<in> ball w (d/2)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
189 |
using uwd by (simp add: dist_commute dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
190 |
have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k))) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
191 |
\<le> (real k * 4 + real k * real k * 4) * (cmod (u - w) * cmod (u - w)) / (d * (d * (d/2) ^ k))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
192 |
using complex_Taylor [OF _ ff1 ff2 _ ub, of w, simplified] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
193 |
by (simp add: ff_def \<open>0 < d\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
194 |
then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k))) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
195 |
\<le> (cmod (u - w) * real k) * (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
196 |
by (simp add: field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
197 |
then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k))) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
198 |
/ (cmod (u - w) * real k) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
199 |
\<le> (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
200 |
using \<open>k \<noteq> 0\<close> \<open>u \<noteq> w\<close> by (simp add: mult_ac zero_less_mult_iff pos_divide_le_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
201 |
also have "\<dots> < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
202 |
using uw_less \<open>0 < d\<close> by (simp add: mult_ac divide_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
203 |
finally have e: "cmod (inverse (x-u)^k - (inverse (x-w)^k + of_nat k * (u-w) / ((x-w) * (x-w)^k))) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
204 |
/ cmod ((u - w) * real k) < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
205 |
by (simp add: norm_mult) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
206 |
have "x \<noteq> u" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
207 |
using uwd \<open>0 < d\<close> x d by (force simp: dist_norm ball_def norm_minus_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
208 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
209 |
apply (rule le_less_trans [OF _ e]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
210 |
using \<open>k \<noteq> 0\<close> \<open>x \<noteq> u\<close> \<open>u \<noteq> w\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
211 |
apply (simp add: field_simps norm_divide [symmetric]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
212 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
213 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
214 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
215 |
unfolding eventually_at |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
216 |
apply (rule_tac x = "min (d/2) ((e*(d/2)^(k + 2))/(Suc k))" in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
217 |
apply (force simp: \<open>d > 0\<close> dist_norm that simp del: power_Suc intro: *) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
218 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
219 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
220 |
have 2: "uniform_limit (path_image \<gamma>) (\<lambda>n x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) (\<lambda>x. f' x / (x - w) ^ Suc k) (at w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
221 |
unfolding uniform_limit_iff dist_norm |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
222 |
proof clarify |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
223 |
fix e::real |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
224 |
assume "0 < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
225 |
have *: "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
226 |
f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
227 |
if ec: "cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
228 |
inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k) < e / C" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
229 |
and x: "0 \<le> x" "x \<le> 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
230 |
for u x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
231 |
proof (cases "(f' (\<gamma> x)) = 0") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
232 |
case True then show ?thesis by (simp add: \<open>0 < e\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
233 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
234 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
235 |
have "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
236 |
f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
237 |
cmod (f' (\<gamma> x) * ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
238 |
inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
239 |
by (simp add: field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
240 |
also have "\<dots> = cmod (f' (\<gamma> x)) * |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
241 |
cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
242 |
inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
243 |
by (simp add: norm_mult) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
244 |
also have "\<dots> < cmod (f' (\<gamma> x)) * (e/C)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
245 |
using False mult_strict_left_mono [OF ec] by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
246 |
also have "\<dots> \<le> e" using C |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
247 |
by (metis False \<open>0 < e\<close> frac_le less_eq_real_def mult.commute pos_le_divide_eq x zero_less_norm_iff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
248 |
finally show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
249 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
250 |
show "\<forall>\<^sub>F n in at w. |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
251 |
\<forall>x\<in>path_image \<gamma>. |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
252 |
cmod (f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k - f' x / (x - w) ^ Suc k) < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
253 |
using twom [OF divide_pos_pos [OF \<open>0 < e\<close> \<open>C > 0\<close>]] unfolding path_image_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
254 |
by (force intro: * elim: eventually_mono) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
255 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
256 |
show "(\<lambda>u. f' u / (u - w) ^ (Suc k)) contour_integrable_on \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
257 |
by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
258 |
have *: "(\<lambda>n. contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k)) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
259 |
\<midarrow>w\<rightarrow> contour_integral \<gamma> (\<lambda>u. f' u / (u - w) ^ (Suc k))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
260 |
by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
261 |
have **: "contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k)) = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
262 |
(f u - f w) / (u - w) / k" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
263 |
if "dist u w < d" for u |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
264 |
proof - |
72266 | 265 |
have u: "u \<in> S - path_image \<gamma>" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
266 |
by (metis subsetD d dist_commute mem_ball that) |
72266 | 267 |
have \<section>: "((\<lambda>x. f' x * inverse (x - u) ^ k) has_contour_integral f u) \<gamma>" |
268 |
"((\<lambda>x. f' x * inverse (x - w) ^ k) has_contour_integral f w) \<gamma>" |
|
269 |
using u w by (simp_all add: field_simps int) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
270 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
271 |
apply (rule contour_integral_unique) |
72266 | 272 |
apply (simp add: diff_divide_distrib algebra_simps \<section> has_contour_integral_diff has_contour_integral_div) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
273 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
274 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
275 |
show ?thes2 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
276 |
apply (simp add: has_field_derivative_iff del: power_Suc) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
277 |
apply (rule Lim_transform_within [OF tendsto_mult_left [OF *] \<open>0 < d\<close> ]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
278 |
apply (simp add: \<open>k \<noteq> 0\<close> **) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
279 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
280 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
281 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
282 |
lemma Cauchy_next_derivative_circlepath: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
283 |
assumes contf: "continuous_on (path_image (circlepath z r)) f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
284 |
and int: "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>u. f u / (u - w)^k) has_contour_integral g w) (circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
285 |
and k: "k \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
286 |
and w: "w \<in> ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
287 |
shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
288 |
(is "?thes1") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
289 |
and "(g has_field_derivative (k * contour_integral (circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)))) (at w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
290 |
(is "?thes2") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
291 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
292 |
have "r > 0" using w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
293 |
using ball_eq_empty by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
294 |
have wim: "w \<in> ball z r - path_image (circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
295 |
using w by (auto simp: dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
296 |
show ?thes1 ?thes2 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
297 |
by (rule Cauchy_next_derivative [OF contf _ int k open_ball valid_path_circlepath wim, where B = "2 * pi * \<bar>r\<bar>"]; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
298 |
auto simp: vector_derivative_circlepath norm_mult)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
299 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
300 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
301 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
302 |
text\<open> In particular, the first derivative formula.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
303 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
304 |
lemma Cauchy_derivative_integral_circlepath: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
305 |
assumes contf: "continuous_on (cball z r) f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
306 |
and holf: "f holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
307 |
and w: "w \<in> ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
308 |
shows "(\<lambda>u. f u/(u - w)^2) contour_integrable_on (circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
309 |
(is "?thes1") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
310 |
and "(f has_field_derivative (1 / (2 * of_real pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u / (u - w)^2))) (at w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
311 |
(is "?thes2") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
312 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
313 |
have [simp]: "r \<ge> 0" using w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
314 |
using ball_eq_empty by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
315 |
have f: "continuous_on (path_image (circlepath z r)) f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
316 |
by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
317 |
have int: "\<And>w. dist z w < r \<Longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
318 |
((\<lambda>u. f u / (u - w)) has_contour_integral (\<lambda>x. 2 * of_real pi * \<i> * f x) w) (circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
319 |
by (rule Cauchy_integral_circlepath [OF contf holf]) (simp add: dist_norm norm_minus_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
320 |
show ?thes1 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
321 |
apply (simp add: power2_eq_square) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
322 |
apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1, simplified]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
323 |
apply (blast intro: int) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
324 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
325 |
have "((\<lambda>x. 2 * of_real pi * \<i> * f x) has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2)) (at w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
326 |
apply (simp add: power2_eq_square) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
327 |
apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1 and g = "\<lambda>x. 2 * of_real pi * \<i> * f x", simplified]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
328 |
apply (blast intro: int) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
329 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
330 |
then have fder: "(f has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2) / (2 * of_real pi * \<i>)) (at w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
331 |
by (rule DERIV_cdivide [where f = "\<lambda>x. 2 * of_real pi * \<i> * f x" and c = "2 * of_real pi * \<i>", simplified]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
332 |
show ?thes2 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
333 |
by simp (rule fder) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
334 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
335 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
336 |
subsection\<open>Existence of all higher derivatives\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
337 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
338 |
proposition derivative_is_holomorphic: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
339 |
assumes "open S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
340 |
and fder: "\<And>z. z \<in> S \<Longrightarrow> (f has_field_derivative f' z) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
341 |
shows "f' holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
342 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
343 |
have *: "\<exists>h. (f' has_field_derivative h) (at z)" if "z \<in> S" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
344 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
345 |
obtain r where "r > 0" and r: "cball z r \<subseteq> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
346 |
using open_contains_cball \<open>z \<in> S\<close> \<open>open S\<close> by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
347 |
then have holf_cball: "f holomorphic_on cball z r" |
72266 | 348 |
unfolding holomorphic_on_def |
349 |
using field_differentiable_at_within field_differentiable_def fder by fastforce |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
350 |
then have "continuous_on (path_image (circlepath z r)) f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
351 |
using \<open>r > 0\<close> by (force elim: holomorphic_on_subset [THEN holomorphic_on_imp_continuous_on]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
352 |
then have contfpi: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1/(2 * of_real pi*\<i>) * f x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
353 |
by (auto intro: continuous_intros)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
354 |
have contf_cball: "continuous_on (cball z r) f" using holf_cball |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
355 |
by (simp add: holomorphic_on_imp_continuous_on holomorphic_on_subset) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
356 |
have holf_ball: "f holomorphic_on ball z r" using holf_cball |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
357 |
using ball_subset_cball holomorphic_on_subset by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
358 |
{ fix w assume w: "w \<in> ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
359 |
have intf: "(\<lambda>u. f u / (u - w)\<^sup>2) contour_integrable_on circlepath z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
360 |
by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
361 |
have fder': "(f has_field_derivative 1 / (2 * of_real pi * \<i>) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2)) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
362 |
(at w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
363 |
by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
364 |
have f'_eq: "f' w = contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
365 |
using fder' ball_subset_cball r w by (force intro: DERIV_unique [OF fder]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
366 |
have "((\<lambda>u. f u / (u - w)\<^sup>2 / (2 * of_real pi * \<i>)) has_contour_integral |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
367 |
contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>)) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
368 |
(circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
369 |
by (rule has_contour_integral_div [OF has_contour_integral_integral [OF intf]]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
370 |
then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
371 |
contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>)) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
372 |
(circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
373 |
by (simp add: algebra_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
374 |
then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral f' w) (circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
375 |
by (simp add: f'_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
376 |
} note * = this |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
377 |
show ?thesis |
72266 | 378 |
using Cauchy_next_derivative_circlepath [OF contfpi, of 2 f'] \<open>0 < r\<close> * |
379 |
using centre_in_ball mem_ball by force |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
380 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
381 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
382 |
by (simp add: holomorphic_on_open [OF \<open>open S\<close>] *) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
383 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
384 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
385 |
lemma holomorphic_deriv [holomorphic_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
386 |
"\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv f) holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
387 |
by (metis DERIV_deriv_iff_field_differentiable at_within_open derivative_is_holomorphic holomorphic_on_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
388 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
389 |
lemma analytic_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv f) analytic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
390 |
using analytic_on_holomorphic holomorphic_deriv by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
391 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
392 |
lemma holomorphic_higher_deriv [holomorphic_intros]: "\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv ^^ n) f holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
393 |
by (induction n) (auto simp: holomorphic_deriv) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
394 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
395 |
lemma analytic_higher_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv ^^ n) f analytic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
396 |
unfolding analytic_on_def using holomorphic_higher_deriv by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
397 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
398 |
lemma has_field_derivative_higher_deriv: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
399 |
"\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
400 |
\<Longrightarrow> ((deriv ^^ n) f has_field_derivative (deriv ^^ (Suc n)) f x) (at x)" |
73932
fd21b4a93043
added opaque_combs and renamed hide_lams to opaque_lifting
desharna
parents:
73790
diff
changeset
|
401 |
by (metis (no_types, opaque_lifting) DERIV_deriv_iff_field_differentiable at_within_open comp_apply |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
402 |
funpow.simps(2) holomorphic_higher_deriv holomorphic_on_def) |
73928
3b76524f5a85
Imported lots of material from Stirling_Formula/Gamma_Asymptotics
paulson <lp15@cam.ac.uk>
parents:
73790
diff
changeset
|
403 |
|
3b76524f5a85
Imported lots of material from Stirling_Formula/Gamma_Asymptotics
paulson <lp15@cam.ac.uk>
parents:
73790
diff
changeset
|
404 |
lemma higher_deriv_cmult: |
3b76524f5a85
Imported lots of material from Stirling_Formula/Gamma_Asymptotics
paulson <lp15@cam.ac.uk>
parents:
73790
diff
changeset
|
405 |
assumes "f holomorphic_on A" "x \<in> A" "open A" |
3b76524f5a85
Imported lots of material from Stirling_Formula/Gamma_Asymptotics
paulson <lp15@cam.ac.uk>
parents:
73790
diff
changeset
|
406 |
shows "(deriv ^^ j) (\<lambda>x. c * f x) x = c * (deriv ^^ j) f x" |
3b76524f5a85
Imported lots of material from Stirling_Formula/Gamma_Asymptotics
paulson <lp15@cam.ac.uk>
parents:
73790
diff
changeset
|
407 |
using assms |
3b76524f5a85
Imported lots of material from Stirling_Formula/Gamma_Asymptotics
paulson <lp15@cam.ac.uk>
parents:
73790
diff
changeset
|
408 |
proof (induction j arbitrary: f x) |
3b76524f5a85
Imported lots of material from Stirling_Formula/Gamma_Asymptotics
paulson <lp15@cam.ac.uk>
parents:
73790
diff
changeset
|
409 |
case (Suc j f x) |
3b76524f5a85
Imported lots of material from Stirling_Formula/Gamma_Asymptotics
paulson <lp15@cam.ac.uk>
parents:
73790
diff
changeset
|
410 |
have "deriv ((deriv ^^ j) (\<lambda>x. c * f x)) x = deriv (\<lambda>x. c * (deriv ^^ j) f x) x" |
3b76524f5a85
Imported lots of material from Stirling_Formula/Gamma_Asymptotics
paulson <lp15@cam.ac.uk>
parents:
73790
diff
changeset
|
411 |
using eventually_nhds_in_open[of A x] assms(2,3) Suc.prems |
3b76524f5a85
Imported lots of material from Stirling_Formula/Gamma_Asymptotics
paulson <lp15@cam.ac.uk>
parents:
73790
diff
changeset
|
412 |
by (intro deriv_cong_ev refl) (auto elim!: eventually_mono simp: Suc.IH) |
3b76524f5a85
Imported lots of material from Stirling_Formula/Gamma_Asymptotics
paulson <lp15@cam.ac.uk>
parents:
73790
diff
changeset
|
413 |
also have "\<dots> = c * deriv ((deriv ^^ j) f) x" using Suc.prems assms(2,3) |
3b76524f5a85
Imported lots of material from Stirling_Formula/Gamma_Asymptotics
paulson <lp15@cam.ac.uk>
parents:
73790
diff
changeset
|
414 |
by (intro deriv_cmult holomorphic_on_imp_differentiable_at holomorphic_higher_deriv) auto |
3b76524f5a85
Imported lots of material from Stirling_Formula/Gamma_Asymptotics
paulson <lp15@cam.ac.uk>
parents:
73790
diff
changeset
|
415 |
finally show ?case by simp |
3b76524f5a85
Imported lots of material from Stirling_Formula/Gamma_Asymptotics
paulson <lp15@cam.ac.uk>
parents:
73790
diff
changeset
|
416 |
qed simp_all |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
417 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
418 |
lemma valid_path_compose_holomorphic: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
419 |
assumes "valid_path g" and holo:"f holomorphic_on S" and "open S" "path_image g \<subseteq> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
420 |
shows "valid_path (f \<circ> g)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
421 |
proof (rule valid_path_compose[OF \<open>valid_path g\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
422 |
fix x assume "x \<in> path_image g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
423 |
then show "f field_differentiable at x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
424 |
using analytic_on_imp_differentiable_at analytic_on_open assms holo by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
425 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
426 |
have "deriv f holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
427 |
using holomorphic_deriv holo \<open>open S\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
428 |
then show "continuous_on (path_image g) (deriv f)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
429 |
using assms(4) holomorphic_on_imp_continuous_on holomorphic_on_subset by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
430 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
431 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
432 |
subsection\<open>Morera's theorem\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
433 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
434 |
lemma Morera_local_triangle_ball: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
435 |
assumes "\<And>z. z \<in> S |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
436 |
\<Longrightarrow> \<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
437 |
(\<forall>b c. closed_segment b c \<subseteq> ball a e |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
438 |
\<longrightarrow> contour_integral (linepath a b) f + |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
439 |
contour_integral (linepath b c) f + |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
440 |
contour_integral (linepath c a) f = 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
441 |
shows "f analytic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
442 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
443 |
{ fix z assume "z \<in> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
444 |
with assms obtain e a where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
445 |
"0 < e" and z: "z \<in> ball a e" and contf: "continuous_on (ball a e) f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
446 |
and 0: "\<And>b c. closed_segment b c \<subseteq> ball a e |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
447 |
\<Longrightarrow> contour_integral (linepath a b) f + |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
448 |
contour_integral (linepath b c) f + |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
449 |
contour_integral (linepath c a) f = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
450 |
by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
451 |
have az: "dist a z < e" using mem_ball z by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
452 |
have "\<exists>e>0. f holomorphic_on ball z e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
453 |
proof (intro exI conjI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
454 |
show "f holomorphic_on ball z (e - dist a z)" |
72266 | 455 |
proof (rule holomorphic_on_subset) |
456 |
show "ball z (e - dist a z) \<subseteq> ball a e" |
|
457 |
by (simp add: dist_commute ball_subset_ball_iff) |
|
458 |
have sub_ball: "\<And>y. dist a y < e \<Longrightarrow> closed_segment a y \<subseteq> ball a e" |
|
459 |
by (meson \<open>0 < e\<close> centre_in_ball convex_ball convex_contains_segment mem_ball) |
|
460 |
show "f holomorphic_on ball a e" |
|
461 |
using triangle_contour_integrals_starlike_primitive [OF contf _ open_ball, of a] |
|
462 |
derivative_is_holomorphic[OF open_ball] |
|
463 |
by (force simp add: 0 \<open>0 < e\<close> sub_ball) |
|
464 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
465 |
qed (simp add: az) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
466 |
} |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
467 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
468 |
by (simp add: analytic_on_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
469 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
470 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
471 |
lemma Morera_local_triangle: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
472 |
assumes "\<And>z. z \<in> S |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
473 |
\<Longrightarrow> \<exists>t. open t \<and> z \<in> t \<and> continuous_on t f \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
474 |
(\<forall>a b c. convex hull {a,b,c} \<subseteq> t |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
475 |
\<longrightarrow> contour_integral (linepath a b) f + |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
476 |
contour_integral (linepath b c) f + |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
477 |
contour_integral (linepath c a) f = 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
478 |
shows "f analytic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
479 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
480 |
{ fix z assume "z \<in> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
481 |
with assms obtain t where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
482 |
"open t" and z: "z \<in> t" and contf: "continuous_on t f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
483 |
and 0: "\<And>a b c. convex hull {a,b,c} \<subseteq> t |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
484 |
\<Longrightarrow> contour_integral (linepath a b) f + |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
485 |
contour_integral (linepath b c) f + |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
486 |
contour_integral (linepath c a) f = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
487 |
by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
488 |
then obtain e where "e>0" and e: "ball z e \<subseteq> t" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
489 |
using open_contains_ball by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
490 |
have [simp]: "continuous_on (ball z e) f" using contf |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
491 |
using continuous_on_subset e by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
492 |
have eq0: "\<And>b c. closed_segment b c \<subseteq> ball z e \<Longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
493 |
contour_integral (linepath z b) f + |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
494 |
contour_integral (linepath b c) f + |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
495 |
contour_integral (linepath c z) f = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
496 |
by (meson 0 z \<open>0 < e\<close> centre_in_ball closed_segment_subset convex_ball dual_order.trans e starlike_convex_subset) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
497 |
have "\<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
498 |
(\<forall>b c. closed_segment b c \<subseteq> ball a e \<longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
499 |
contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
500 |
using \<open>e > 0\<close> eq0 by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
501 |
} |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
502 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
503 |
by (simp add: Morera_local_triangle_ball) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
504 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
505 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
506 |
proposition Morera_triangle: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
507 |
"\<lbrakk>continuous_on S f; open S; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
508 |
\<And>a b c. convex hull {a,b,c} \<subseteq> S |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
509 |
\<longrightarrow> contour_integral (linepath a b) f + |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
510 |
contour_integral (linepath b c) f + |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
511 |
contour_integral (linepath c a) f = 0\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
512 |
\<Longrightarrow> f analytic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
513 |
using Morera_local_triangle by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
514 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
515 |
subsection\<open>Combining theorems for higher derivatives including Leibniz rule\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
516 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
517 |
lemma higher_deriv_linear [simp]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
518 |
"(deriv ^^ n) (\<lambda>w. c*w) = (\<lambda>z. if n = 0 then c*z else if n = 1 then c else 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
519 |
by (induction n) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
520 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
521 |
lemma higher_deriv_const [simp]: "(deriv ^^ n) (\<lambda>w. c) = (\<lambda>w. if n=0 then c else 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
522 |
by (induction n) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
523 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
524 |
lemma higher_deriv_ident [simp]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
525 |
"(deriv ^^ n) (\<lambda>w. w) z = (if n = 0 then z else if n = 1 then 1 else 0)" |
72266 | 526 |
proof (induction n) |
527 |
case (Suc n) |
|
528 |
then show ?case by (metis higher_deriv_linear lambda_one) |
|
529 |
qed auto |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
530 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
531 |
lemma higher_deriv_id [simp]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
532 |
"(deriv ^^ n) id z = (if n = 0 then z else if n = 1 then 1 else 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
533 |
by (simp add: id_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
534 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
535 |
lemma has_complex_derivative_funpow_1: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
536 |
"\<lbrakk>(f has_field_derivative 1) (at z); f z = z\<rbrakk> \<Longrightarrow> (f^^n has_field_derivative 1) (at z)" |
72266 | 537 |
proof (induction n) |
538 |
case 0 |
|
539 |
then show ?case |
|
540 |
by (simp add: id_def) |
|
541 |
next |
|
542 |
case (Suc n) |
|
543 |
then show ?case |
|
544 |
by (metis DERIV_chain funpow_Suc_right mult.right_neutral) |
|
545 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
546 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
547 |
lemma higher_deriv_uminus: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
548 |
assumes "f holomorphic_on S" "open S" and z: "z \<in> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
549 |
shows "(deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
550 |
using z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
551 |
proof (induction n arbitrary: z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
552 |
case 0 then show ?case by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
553 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
554 |
case (Suc n z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
555 |
have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
556 |
using Suc.prems assms has_field_derivative_higher_deriv by auto |
72266 | 557 |
have "\<And>x. x \<in> S \<Longrightarrow> - (deriv ^^ n) f x = (deriv ^^ n) (\<lambda>w. - f w) x" |
558 |
by (auto simp add: Suc) |
|
559 |
then have "((deriv ^^ n) (\<lambda>w. - f w) has_field_derivative - deriv ((deriv ^^ n) f) z) (at z)" |
|
560 |
using has_field_derivative_transform_within_open [of "\<lambda>w. -((deriv ^^ n) f w)"] |
|
561 |
using "*" DERIV_minus Suc.prems \<open>open S\<close> by blast |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
562 |
then show ?case |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
563 |
by (simp add: DERIV_imp_deriv) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
564 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
565 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
566 |
lemma higher_deriv_add: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
567 |
fixes z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
568 |
assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
569 |
shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
570 |
using z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
571 |
proof (induction n arbitrary: z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
572 |
case 0 then show ?case by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
573 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
574 |
case (Suc n z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
575 |
have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
576 |
"((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
577 |
using Suc.prems assms has_field_derivative_higher_deriv by auto |
72266 | 578 |
have "\<And>x. x \<in> S \<Longrightarrow> (deriv ^^ n) f x + (deriv ^^ n) g x = (deriv ^^ n) (\<lambda>w. f w + g w) x" |
579 |
by (auto simp add: Suc) |
|
580 |
then have "((deriv ^^ n) (\<lambda>w. f w + g w) has_field_derivative |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
581 |
deriv ((deriv ^^ n) f) z + deriv ((deriv ^^ n) g) z) (at z)" |
72266 | 582 |
using has_field_derivative_transform_within_open [of "\<lambda>w. (deriv ^^ n) f w + (deriv ^^ n) g w"] |
583 |
using "*" Deriv.field_differentiable_add Suc.prems \<open>open S\<close> by blast |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
584 |
then show ?case |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
585 |
by (simp add: DERIV_imp_deriv) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
586 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
587 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
588 |
lemma higher_deriv_diff: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
589 |
fixes z::complex |
72266 | 590 |
assumes "f holomorphic_on S" "g holomorphic_on S" "open S" "z \<in> S" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
591 |
shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z" |
72266 | 592 |
unfolding diff_conv_add_uminus higher_deriv_add |
593 |
using assms higher_deriv_add higher_deriv_uminus holomorphic_on_minus by presburger |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
594 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
595 |
lemma bb: "Suc n choose k = (n choose k) + (if k = 0 then 0 else (n choose (k - 1)))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
596 |
by (cases k) simp_all |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
597 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
598 |
lemma higher_deriv_mult: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
599 |
fixes z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
600 |
assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
601 |
shows "(deriv ^^ n) (\<lambda>w. f w * g w) z = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
602 |
(\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
603 |
using z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
604 |
proof (induction n arbitrary: z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
605 |
case 0 then show ?case by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
606 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
607 |
case (Suc n z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
608 |
have *: "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
609 |
"\<And>n. ((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
610 |
using Suc.prems assms has_field_derivative_higher_deriv by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
611 |
have sumeq: "(\<Sum>i = 0..n. |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
612 |
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)) = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
613 |
g z * deriv ((deriv ^^ n) f) z + (\<Sum>i = 0..n. (deriv ^^ i) f z * (of_nat (Suc n choose i) * (deriv ^^ (Suc n - i)) g z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
614 |
apply (simp add: bb algebra_simps sum.distrib) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
615 |
apply (subst (4) sum_Suc_reindex) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
616 |
apply (auto simp: algebra_simps Suc_diff_le intro: sum.cong) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
617 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
618 |
have "((deriv ^^ n) (\<lambda>w. f w * g w) has_field_derivative |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
619 |
(\<Sum>i = 0..Suc n. (Suc n choose i) * (deriv ^^ i) f z * (deriv ^^ (Suc n - i)) g z)) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
620 |
(at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
621 |
apply (rule has_field_derivative_transform_within_open |
72266 | 622 |
[of "\<lambda>w. (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f w * (deriv ^^ (n - i)) g w)" _ _ S]) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
623 |
apply (simp add: algebra_simps) |
72266 | 624 |
apply (rule derivative_eq_intros | simp)+ |
625 |
apply (auto intro: DERIV_mult * \<open>open S\<close> Suc.prems Suc.IH [symmetric]) |
|
626 |
by (metis (no_types, lifting) mult.commute sum.cong sumeq) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
627 |
then show ?case |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
628 |
unfolding funpow.simps o_apply |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
629 |
by (simp add: DERIV_imp_deriv) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
630 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
631 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
632 |
lemma higher_deriv_transform_within_open: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
633 |
fixes z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
634 |
assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
635 |
and fg: "\<And>w. w \<in> S \<Longrightarrow> f w = g w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
636 |
shows "(deriv ^^ i) f z = (deriv ^^ i) g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
637 |
using z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
638 |
by (induction i arbitrary: z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
639 |
(auto simp: fg intro: complex_derivative_transform_within_open holomorphic_higher_deriv assms) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
640 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
641 |
lemma higher_deriv_compose_linear: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
642 |
fixes z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
643 |
assumes f: "f holomorphic_on T" and S: "open S" and T: "open T" and z: "z \<in> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
644 |
and fg: "\<And>w. w \<in> S \<Longrightarrow> u * w \<in> T" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
645 |
shows "(deriv ^^ n) (\<lambda>w. f (u * w)) z = u^n * (deriv ^^ n) f (u * z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
646 |
using z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
647 |
proof (induction n arbitrary: z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
648 |
case 0 then show ?case by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
649 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
650 |
case (Suc n z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
651 |
have holo0: "f holomorphic_on (*) u ` S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
652 |
by (meson fg f holomorphic_on_subset image_subset_iff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
653 |
have holo2: "(deriv ^^ n) f holomorphic_on (*) u ` S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
654 |
by (meson f fg holomorphic_higher_deriv holomorphic_on_subset image_subset_iff T) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
655 |
have holo3: "(\<lambda>z. u ^ n * (deriv ^^ n) f (u * z)) holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
656 |
by (intro holo2 holomorphic_on_compose [where g="(deriv ^^ n) f", unfolded o_def] holomorphic_intros) |
72266 | 657 |
have "(*) u holomorphic_on S" "f holomorphic_on (*) u ` S" |
658 |
by (rule holo0 holomorphic_intros)+ |
|
659 |
then have holo1: "(\<lambda>w. f (u * w)) holomorphic_on S" |
|
660 |
by (rule holomorphic_on_compose [where g=f, unfolded o_def]) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
661 |
have "deriv ((deriv ^^ n) (\<lambda>w. f (u * w))) z = deriv (\<lambda>z. u^n * (deriv ^^ n) f (u*z)) z" |
72266 | 662 |
proof (rule complex_derivative_transform_within_open [OF _ holo3 S Suc.prems]) |
663 |
show "(deriv ^^ n) (\<lambda>w. f (u * w)) holomorphic_on S" |
|
664 |
by (rule holomorphic_higher_deriv [OF holo1 S]) |
|
665 |
qed (simp add: Suc.IH) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
666 |
also have "\<dots> = u^n * deriv (\<lambda>z. (deriv ^^ n) f (u * z)) z" |
72266 | 667 |
proof - |
668 |
have "(deriv ^^ n) f analytic_on T" |
|
669 |
by (simp add: analytic_on_open f holomorphic_higher_deriv T) |
|
670 |
then have "(\<lambda>w. (deriv ^^ n) f (u * w)) analytic_on S" |
|
671 |
proof - |
|
672 |
have "(deriv ^^ n) f \<circ> (*) u holomorphic_on S" |
|
673 |
by (simp add: holo2 holomorphic_on_compose) |
|
674 |
then show ?thesis |
|
675 |
by (simp add: S analytic_on_open o_def) |
|
676 |
qed |
|
677 |
then show ?thesis |
|
678 |
by (intro deriv_cmult analytic_on_imp_differentiable_at [OF _ Suc.prems]) |
|
679 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
680 |
also have "\<dots> = u * u ^ n * deriv ((deriv ^^ n) f) (u * z)" |
72266 | 681 |
proof - |
682 |
have "(deriv ^^ n) f field_differentiable at (u * z)" |
|
683 |
using Suc.prems T f fg holomorphic_higher_deriv holomorphic_on_imp_differentiable_at by blast |
|
684 |
then show ?thesis |
|
685 |
by (simp add: deriv_compose_linear) |
|
686 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
687 |
finally show ?case |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
688 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
689 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
690 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
691 |
lemma higher_deriv_add_at: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
692 |
assumes "f analytic_on {z}" "g analytic_on {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
693 |
shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
694 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
695 |
have "f analytic_on {z} \<and> g analytic_on {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
696 |
using assms by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
697 |
with higher_deriv_add show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
698 |
by (auto simp: analytic_at_two) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
699 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
700 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
701 |
lemma higher_deriv_diff_at: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
702 |
assumes "f analytic_on {z}" "g analytic_on {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
703 |
shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
704 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
705 |
have "f analytic_on {z} \<and> g analytic_on {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
706 |
using assms by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
707 |
with higher_deriv_diff show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
708 |
by (auto simp: analytic_at_two) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
709 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
710 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
711 |
lemma higher_deriv_uminus_at: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
712 |
"f analytic_on {z} \<Longrightarrow> (deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
713 |
using higher_deriv_uminus |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
714 |
by (auto simp: analytic_at) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
715 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
716 |
lemma higher_deriv_mult_at: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
717 |
assumes "f analytic_on {z}" "g analytic_on {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
718 |
shows "(deriv ^^ n) (\<lambda>w. f w * g w) z = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
719 |
(\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
720 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
721 |
have "f analytic_on {z} \<and> g analytic_on {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
722 |
using assms by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
723 |
with higher_deriv_mult show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
724 |
by (auto simp: analytic_at_two) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
725 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
726 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
727 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
728 |
text\<open> Nonexistence of isolated singularities and a stronger integral formula.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
729 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
730 |
proposition no_isolated_singularity: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
731 |
fixes z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
732 |
assumes f: "continuous_on S f" and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
733 |
shows "f holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
734 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
735 |
{ fix z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
736 |
assume "z \<in> S" and cdf: "\<And>x. x \<in> S - K \<Longrightarrow> f field_differentiable at x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
737 |
have "f field_differentiable at z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
738 |
proof (cases "z \<in> K") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
739 |
case False then show ?thesis by (blast intro: cdf \<open>z \<in> S\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
740 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
741 |
case True |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
742 |
with finite_set_avoid [OF K, of z] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
743 |
obtain d where "d>0" and d: "\<And>x. \<lbrakk>x\<in>K; x \<noteq> z\<rbrakk> \<Longrightarrow> d \<le> dist z x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
744 |
by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
745 |
obtain e where "e>0" and e: "ball z e \<subseteq> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
746 |
using S \<open>z \<in> S\<close> by (force simp: open_contains_ball) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
747 |
have fde: "continuous_on (ball z (min d e)) f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
748 |
by (metis Int_iff ball_min_Int continuous_on_subset e f subsetI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
749 |
have cont: "{a,b,c} \<subseteq> ball z (min d e) \<Longrightarrow> continuous_on (convex hull {a, b, c}) f" for a b c |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
750 |
by (simp add: hull_minimal continuous_on_subset [OF fde]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
751 |
have fd: "\<lbrakk>{a,b,c} \<subseteq> ball z (min d e); x \<in> interior (convex hull {a, b, c}) - K\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
752 |
\<Longrightarrow> f field_differentiable at x" for a b c x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
753 |
by (metis cdf Diff_iff Int_iff ball_min_Int subsetD convex_ball e interior_mono interior_subset subset_hull) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
754 |
obtain g where "\<And>w. w \<in> ball z (min d e) \<Longrightarrow> (g has_field_derivative f w) (at w within ball z (min d e))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
755 |
apply (rule contour_integral_convex_primitive |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
756 |
[OF convex_ball fde Cauchy_theorem_triangle_cofinite [OF _ K]]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
757 |
using cont fd by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
758 |
then have "f holomorphic_on ball z (min d e)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
759 |
by (metis open_ball at_within_open derivative_is_holomorphic) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
760 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
761 |
unfolding holomorphic_on_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
762 |
by (metis open_ball \<open>0 < d\<close> \<open>0 < e\<close> at_within_open centre_in_ball min_less_iff_conj) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
763 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
764 |
} |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
765 |
with holf S K show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
766 |
by (simp add: holomorphic_on_open open_Diff finite_imp_closed field_differentiable_def [symmetric]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
767 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
768 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
769 |
lemma no_isolated_singularity': |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
770 |
fixes z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
771 |
assumes f: "\<And>z. z \<in> K \<Longrightarrow> (f \<longlongrightarrow> f z) (at z within S)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
772 |
and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
773 |
shows "f holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
774 |
proof (rule no_isolated_singularity[OF _ assms(2-)]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
775 |
show "continuous_on S f" unfolding continuous_on_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
776 |
proof |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
777 |
fix z assume z: "z \<in> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
778 |
show "(f \<longlongrightarrow> f z) (at z within S)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
779 |
proof (cases "z \<in> K") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
780 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
781 |
from holf have "continuous_on (S - K) f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
782 |
by (rule holomorphic_on_imp_continuous_on) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
783 |
with z False have "(f \<longlongrightarrow> f z) (at z within (S - K))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
784 |
by (simp add: continuous_on_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
785 |
also from z K S False have "at z within (S - K) = at z within S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
786 |
by (subst (1 2) at_within_open) (auto intro: finite_imp_closed) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
787 |
finally show "(f \<longlongrightarrow> f z) (at z within S)" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
788 |
qed (insert assms z, simp_all) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
789 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
790 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
791 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
792 |
proposition Cauchy_integral_formula_convex: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
793 |
assumes S: "convex S" and K: "finite K" and contf: "continuous_on S f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
794 |
and fcd: "(\<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
795 |
and z: "z \<in> interior S" and vpg: "valid_path \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
796 |
and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
797 |
shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
798 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
799 |
have *: "\<And>x. x \<in> interior S \<Longrightarrow> f field_differentiable at x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
800 |
unfolding holomorphic_on_open [symmetric] field_differentiable_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
801 |
using no_isolated_singularity [where S = "interior S"] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
802 |
by (meson K contf continuous_at_imp_continuous_on continuous_on_interior fcd |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
803 |
field_differentiable_at_within field_differentiable_def holomorphic_onI |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
804 |
holomorphic_on_imp_differentiable_at open_interior) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
805 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
806 |
by (rule Cauchy_integral_formula_weak [OF S finite.emptyI contf]) (use * assms in auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
807 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
808 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
809 |
text\<open> Formula for higher derivatives.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
810 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
811 |
lemma Cauchy_has_contour_integral_higher_derivative_circlepath: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
812 |
assumes contf: "continuous_on (cball z r) f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
813 |
and holf: "f holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
814 |
and w: "w \<in> ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
815 |
shows "((\<lambda>u. f u / (u - w) ^ (Suc k)) has_contour_integral ((2 * pi * \<i>) / (fact k) * (deriv ^^ k) f w)) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
816 |
(circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
817 |
using w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
818 |
proof (induction k arbitrary: w) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
819 |
case 0 then show ?case |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
820 |
using assms by (auto simp: Cauchy_integral_circlepath dist_commute dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
821 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
822 |
case (Suc k) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
823 |
have [simp]: "r > 0" using w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
824 |
using ball_eq_empty by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
825 |
have f: "continuous_on (path_image (circlepath z r)) f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
826 |
by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def less_imp_le) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
827 |
obtain X where X: "((\<lambda>u. f u / (u - w) ^ Suc (Suc k)) has_contour_integral X) (circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
828 |
using Cauchy_next_derivative_circlepath(1) [OF f Suc.IH _ Suc.prems] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
829 |
by (auto simp: contour_integrable_on_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
830 |
then have con: "contour_integral (circlepath z r) ((\<lambda>u. f u / (u - w) ^ Suc (Suc k))) = X" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
831 |
by (rule contour_integral_unique) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
832 |
have "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) w) (at w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
833 |
using Suc.prems assms has_field_derivative_higher_deriv by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
834 |
then have dnf_diff: "\<And>n. (deriv ^^ n) f field_differentiable (at w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
835 |
by (force simp: field_differentiable_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
836 |
have "deriv (\<lambda>w. complex_of_real (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w) w = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
837 |
of_nat (Suc k) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w) ^ Suc (Suc k))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
838 |
by (force intro!: DERIV_imp_deriv Cauchy_next_derivative_circlepath [OF f Suc.IH _ Suc.prems]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
839 |
also have "\<dots> = of_nat (Suc k) * X" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
840 |
by (simp only: con) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
841 |
finally have "deriv (\<lambda>w. ((2 * pi) * \<i> / (fact k)) * (deriv ^^ k) f w) w = of_nat (Suc k) * X" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
842 |
then have "((2 * pi) * \<i> / (fact k)) * deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
843 |
by (metis deriv_cmult dnf_diff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
844 |
then have "deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X / ((2 * pi) * \<i> / (fact k))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
845 |
by (simp add: field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
846 |
then show ?case |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
847 |
using of_nat_eq_0_iff X by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
848 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
849 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
850 |
lemma Cauchy_higher_derivative_integral_circlepath: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
851 |
assumes contf: "continuous_on (cball z r) f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
852 |
and holf: "f holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
853 |
and w: "w \<in> ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
854 |
shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
855 |
(is "?thes1") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
856 |
and "(deriv ^^ k) f w = (fact k) / (2 * pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
857 |
(is "?thes2") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
858 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
859 |
have *: "((\<lambda>u. f u / (u - w) ^ Suc k) has_contour_integral (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
860 |
(circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
861 |
using Cauchy_has_contour_integral_higher_derivative_circlepath [OF assms] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
862 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
863 |
show ?thes1 using * |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
864 |
using contour_integrable_on_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
865 |
show ?thes2 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
866 |
unfolding contour_integral_unique [OF *] by (simp add: field_split_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
867 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
868 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
869 |
corollary Cauchy_contour_integral_circlepath: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
870 |
assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
871 |
shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)) = (2 * pi * \<i>) * (deriv ^^ k) f w / (fact k)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
872 |
by (simp add: Cauchy_higher_derivative_integral_circlepath [OF assms]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
873 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
874 |
lemma Cauchy_contour_integral_circlepath_2: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
875 |
assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
876 |
shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^2) = (2 * pi * \<i>) * deriv f w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
877 |
using Cauchy_contour_integral_circlepath [OF assms, of 1] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
878 |
by (simp add: power2_eq_square) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
879 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
880 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
881 |
subsection\<open>A holomorphic function is analytic, i.e. has local power series\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
882 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
883 |
theorem holomorphic_power_series: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
884 |
assumes holf: "f holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
885 |
and w: "w \<in> ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
886 |
shows "((\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
887 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
888 |
\<comment> \<open>Replacing \<^term>\<open>r\<close> and the original (weak) premises with stronger ones\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
889 |
obtain r where "r > 0" and holfc: "f holomorphic_on cball z r" and w: "w \<in> ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
890 |
proof |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
891 |
have "cball z ((r + dist w z) / 2) \<subseteq> ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
892 |
using w by (simp add: dist_commute field_sum_of_halves subset_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
893 |
then show "f holomorphic_on cball z ((r + dist w z) / 2)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
894 |
by (rule holomorphic_on_subset [OF holf]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
895 |
have "r > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
896 |
using w by clarsimp (metis dist_norm le_less_trans norm_ge_zero) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
897 |
then show "0 < (r + dist w z) / 2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
898 |
by simp (use zero_le_dist [of w z] in linarith) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
899 |
qed (use w in \<open>auto simp: dist_commute\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
900 |
then have holf: "f holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
901 |
using ball_subset_cball holomorphic_on_subset by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
902 |
have contf: "continuous_on (cball z r) f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
903 |
by (simp add: holfc holomorphic_on_imp_continuous_on) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
904 |
have cint: "\<And>k. (\<lambda>u. f u / (u - z) ^ Suc k) contour_integrable_on circlepath z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
905 |
by (rule Cauchy_higher_derivative_integral_circlepath [OF contf holf]) (simp add: \<open>0 < r\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
906 |
obtain B where "0 < B" and B: "\<And>u. u \<in> cball z r \<Longrightarrow> norm(f u) \<le> B" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
907 |
by (metis (no_types) bounded_pos compact_cball compact_continuous_image compact_imp_bounded contf image_eqI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
908 |
obtain k where k: "0 < k" "k \<le> r" and wz_eq: "norm(w - z) = r - k" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
909 |
and kle: "\<And>u. norm(u - z) = r \<Longrightarrow> k \<le> norm(u - w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
910 |
proof |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
911 |
show "\<And>u. cmod (u - z) = r \<Longrightarrow> r - dist z w \<le> cmod (u - w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
912 |
by (metis add_diff_eq diff_add_cancel dist_norm norm_diff_ineq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
913 |
qed (use w in \<open>auto simp: dist_norm norm_minus_commute\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
914 |
have ul: "uniform_limit (sphere z r) (\<lambda>n x. (\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k))) (\<lambda>x. f x / (x - w)) sequentially" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
915 |
unfolding uniform_limit_iff dist_norm |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
916 |
proof clarify |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
917 |
fix e::real |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
918 |
assume "0 < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
919 |
have rr: "0 \<le> (r - k) / r" "(r - k) / r < 1" using k by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
920 |
obtain n where n: "((r - k) / r) ^ n < e / B * k" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
921 |
using real_arch_pow_inv [of "e/B*k" "(r - k)/r"] \<open>0 < e\<close> \<open>0 < B\<close> k by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
922 |
have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) - f u / (u - w)) < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
923 |
if "n \<le> N" and r: "r = dist z u" for N u |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
924 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
925 |
have N: "((r - k) / r) ^ N < e / B * k" |
72266 | 926 |
using le_less_trans [OF power_decreasing n] |
927 |
using \<open>n \<le> N\<close> k by auto |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
928 |
have u [simp]: "(u \<noteq> z) \<and> (u \<noteq> w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
929 |
using \<open>0 < r\<close> r w by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
930 |
have wzu_not1: "(w - z) / (u - z) \<noteq> 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
931 |
by (metis (no_types) dist_norm divide_eq_1_iff less_irrefl mem_ball norm_minus_commute r w) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
932 |
have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) * (u - w) - f u) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
933 |
= norm ((\<Sum>k<N. (((w - z) / (u - z)) ^ k)) * f u * (u - w) / (u - z) - f u)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
934 |
unfolding sum_distrib_right sum_divide_distrib power_divide by (simp add: algebra_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
935 |
also have "\<dots> = norm ((((w - z) / (u - z)) ^ N - 1) * (u - w) / (((w - z) / (u - z) - 1) * (u - z)) - 1) * norm (f u)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
936 |
using \<open>0 < B\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
937 |
apply (auto simp: geometric_sum [OF wzu_not1]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
938 |
apply (simp add: field_simps norm_mult [symmetric]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
939 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
940 |
also have "\<dots> = norm ((u-z) ^ N * (w - u) - ((w - z) ^ N - (u-z) ^ N) * (u-w)) / (r ^ N * norm (u-w)) * norm (f u)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
941 |
using \<open>0 < r\<close> r by (simp add: divide_simps norm_mult norm_divide norm_power dist_norm norm_minus_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
942 |
also have "\<dots> = norm ((w - z) ^ N * (w - u)) / (r ^ N * norm (u - w)) * norm (f u)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
943 |
by (simp add: algebra_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
944 |
also have "\<dots> = norm (w - z) ^ N * norm (f u) / r ^ N" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
945 |
by (simp add: norm_mult norm_power norm_minus_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
946 |
also have "\<dots> \<le> (((r - k)/r)^N) * B" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
947 |
using \<open>0 < r\<close> w k |
72266 | 948 |
by (simp add: B divide_simps mult_mono r wz_eq) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
949 |
also have "\<dots> < e * k" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
950 |
using \<open>0 < B\<close> N by (simp add: divide_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
951 |
also have "\<dots> \<le> e * norm (u - w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
952 |
using r kle \<open>0 < e\<close> by (simp add: dist_commute dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
953 |
finally show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
954 |
by (simp add: field_split_simps norm_divide del: power_Suc) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
955 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
956 |
with \<open>0 < r\<close> show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>sphere z r. |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
957 |
norm ((\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k)) - f x / (x - w)) < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
958 |
by (auto simp: mult_ac less_imp_le eventually_sequentially Ball_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
959 |
qed |
72266 | 960 |
have \<section>: "\<And>x k. k\<in> {..<x} \<Longrightarrow> |
961 |
(\<lambda>u. (w - z) ^ k * (f u / (u - z) ^ Suc k)) contour_integrable_on circlepath z r" |
|
962 |
using contour_integrable_lmul [OF cint, of "(w - z) ^ a" for a] by (simp add: field_simps) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
963 |
have eq: "\<forall>\<^sub>F x in sequentially. |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
964 |
contour_integral (circlepath z r) (\<lambda>u. \<Sum>k<x. (w - z) ^ k * (f u / (u - z) ^ Suc k)) = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
965 |
(\<Sum>k<x. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z) ^ k)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
966 |
apply (rule eventuallyI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
967 |
apply (subst contour_integral_sum, simp) |
72266 | 968 |
apply (simp_all only: \<section> contour_integral_lmul cint algebra_simps) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
969 |
done |
72266 | 970 |
have "\<And>u k. k \<in> {..<u} \<Longrightarrow> (\<lambda>x. f x / (x - z) ^ Suc k) contour_integrable_on circlepath z r" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
971 |
using \<open>0 < r\<close> by (force intro!: Cauchy_higher_derivative_integral_circlepath [OF contf holf]) |
72266 | 972 |
then have "\<And>u. (\<lambda>y. \<Sum>k<u. (w - z) ^ k * (f y / (y - z) ^ Suc k)) contour_integrable_on circlepath z r" |
973 |
by (intro contour_integrable_sum contour_integrable_lmul, simp) |
|
974 |
then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
975 |
sums contour_integral (circlepath z r) (\<lambda>u. f u/(u - w))" |
72266 | 976 |
unfolding sums_def using \<open>0 < r\<close> |
977 |
by (intro Lim_transform_eventually [OF _ eq] contour_integral_uniform_limit_circlepath [OF eventuallyI ul]) auto |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
978 |
then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
979 |
sums (2 * of_real pi * \<i> * f w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
980 |
using w by (auto simp: dist_commute dist_norm contour_integral_unique [OF Cauchy_integral_circlepath_simple [OF holfc]]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
981 |
then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z)^k / (\<i> * (of_real pi * 2))) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
982 |
sums ((2 * of_real pi * \<i> * f w) / (\<i> * (complex_of_real pi * 2)))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
983 |
by (rule sums_divide) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
984 |
then have "(\<lambda>n. (w - z) ^ n * contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc n) / (\<i> * (of_real pi * 2))) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
985 |
sums f w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
986 |
by (simp add: field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
987 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
988 |
by (simp add: field_simps \<open>0 < r\<close> Cauchy_higher_derivative_integral_circlepath [OF contf holf]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
989 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
990 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
991 |
subsection\<open>The Liouville theorem and the Fundamental Theorem of Algebra\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
992 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
993 |
text\<open> These weak Liouville versions don't even need the derivative formula.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
994 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
995 |
lemma Liouville_weak_0: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
996 |
assumes holf: "f holomorphic_on UNIV" and inf: "(f \<longlongrightarrow> 0) at_infinity" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
997 |
shows "f z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
998 |
proof (rule ccontr) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
999 |
assume fz: "f z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1000 |
with inf [unfolded Lim_at_infinity, rule_format, of "norm(f z)/2"] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1001 |
obtain B where B: "\<And>x. B \<le> cmod x \<Longrightarrow> norm (f x) * 2 < cmod (f z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1002 |
by (auto simp: dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1003 |
define R where "R = 1 + \<bar>B\<bar> + norm z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1004 |
have "R > 0" unfolding R_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1005 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1006 |
have "0 \<le> cmod z + \<bar>B\<bar>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1007 |
by (metis (full_types) add_nonneg_nonneg norm_ge_zero real_norm_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1008 |
then show "0 < 1 + \<bar>B\<bar> + cmod z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1009 |
by linarith |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1010 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1011 |
have *: "((\<lambda>u. f u / (u - z)) has_contour_integral 2 * complex_of_real pi * \<i> * f z) (circlepath z R)" |
72266 | 1012 |
using continuous_on_subset holf holomorphic_on_subset \<open>0 < R\<close> |
1013 |
by (force intro: holomorphic_on_imp_continuous_on Cauchy_integral_circlepath) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1014 |
have "cmod (x - z) = R \<Longrightarrow> cmod (f x) * 2 < cmod (f z)" for x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1015 |
unfolding R_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1016 |
by (rule B) (use norm_triangle_ineq4 [of x z] in auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1017 |
with \<open>R > 0\<close> fz show False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1018 |
using has_contour_integral_bound_circlepath [OF *, of "norm(f z)/2/R"] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1019 |
by (auto simp: less_imp_le norm_mult norm_divide field_split_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1020 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1021 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1022 |
proposition Liouville_weak: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1023 |
assumes "f holomorphic_on UNIV" and "(f \<longlongrightarrow> l) at_infinity" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1024 |
shows "f z = l" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1025 |
using Liouville_weak_0 [of "\<lambda>z. f z - l"] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1026 |
by (simp add: assms holomorphic_on_diff LIM_zero) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1027 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1028 |
proposition Liouville_weak_inverse: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1029 |
assumes "f holomorphic_on UNIV" and unbounded: "\<And>B. eventually (\<lambda>x. norm (f x) \<ge> B) at_infinity" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1030 |
obtains z where "f z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1031 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1032 |
{ assume f: "\<And>z. f z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1033 |
have 1: "(\<lambda>x. 1 / f x) holomorphic_on UNIV" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1034 |
by (simp add: holomorphic_on_divide assms f) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1035 |
have 2: "((\<lambda>x. 1 / f x) \<longlongrightarrow> 0) at_infinity" |
72266 | 1036 |
proof (rule tendstoI [OF eventually_mono]) |
1037 |
fix e::real |
|
1038 |
assume "e > 0" |
|
1039 |
show "eventually (\<lambda>x. 2/e \<le> cmod (f x)) at_infinity" |
|
1040 |
by (rule_tac B="2/e" in unbounded) |
|
1041 |
qed (simp add: dist_norm norm_divide field_split_simps) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1042 |
have False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1043 |
using Liouville_weak_0 [OF 1 2] f by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1044 |
} |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1045 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1046 |
using that by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1047 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1048 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1049 |
text\<open> In particular we get the Fundamental Theorem of Algebra.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1050 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1051 |
theorem fundamental_theorem_of_algebra: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1052 |
fixes a :: "nat \<Rightarrow> complex" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1053 |
assumes "a 0 = 0 \<or> (\<exists>i \<in> {1..n}. a i \<noteq> 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1054 |
obtains z where "(\<Sum>i\<le>n. a i * z^i) = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1055 |
using assms |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1056 |
proof (elim disjE bexE) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1057 |
assume "a 0 = 0" then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1058 |
by (auto simp: that [of 0]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1059 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1060 |
fix i |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1061 |
assume i: "i \<in> {1..n}" and nz: "a i \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1062 |
have 1: "(\<lambda>z. \<Sum>i\<le>n. a i * z^i) holomorphic_on UNIV" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1063 |
by (rule holomorphic_intros)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1064 |
show thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1065 |
proof (rule Liouville_weak_inverse [OF 1]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1066 |
show "\<forall>\<^sub>F x in at_infinity. B \<le> cmod (\<Sum>i\<le>n. a i * x ^ i)" for B |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1067 |
using i nz by (intro polyfun_extremal exI[of _ i]) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1068 |
qed (use that in auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1069 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1070 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1071 |
subsection\<open>Weierstrass convergence theorem\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1072 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1073 |
lemma holomorphic_uniform_limit: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1074 |
assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and> (f n) holomorphic_on ball z r) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1075 |
and ulim: "uniform_limit (cball z r) f g F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1076 |
and F: "\<not> trivial_limit F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1077 |
obtains "continuous_on (cball z r) g" "g holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1078 |
proof (cases r "0::real" rule: linorder_cases) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1079 |
case less then show ?thesis by (force simp: ball_empty less_imp_le continuous_on_def holomorphic_on_def intro: that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1080 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1081 |
case equal then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1082 |
by (force simp: holomorphic_on_def intro: that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1083 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1084 |
case greater |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1085 |
have contg: "continuous_on (cball z r) g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1086 |
using cont uniform_limit_theorem [OF eventually_mono ulim F] by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1087 |
have "path_image (circlepath z r) \<subseteq> cball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1088 |
using \<open>0 < r\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1089 |
then have 1: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1 / (2 * complex_of_real pi * \<i>) * g x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1090 |
by (intro continuous_intros continuous_on_subset [OF contg]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1091 |
have 2: "((\<lambda>u. 1 / (2 * of_real pi * \<i>) * g u / (u - w) ^ 1) has_contour_integral g w) (circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1092 |
if w: "w \<in> ball z r" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1093 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1094 |
define d where "d = (r - norm(w - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1095 |
have "0 < d" "d \<le> r" using w by (auto simp: norm_minus_commute d_def dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1096 |
have dle: "\<And>u. cmod (z - u) = r \<Longrightarrow> d \<le> cmod (u - w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1097 |
unfolding d_def by (metis add_diff_eq diff_add_cancel norm_diff_ineq norm_minus_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1098 |
have ev_int: "\<forall>\<^sub>F n in F. (\<lambda>u. f n u / (u - w)) contour_integrable_on circlepath z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1099 |
using w |
72266 | 1100 |
by (auto intro: eventually_mono [OF cont] Cauchy_higher_derivative_integral_circlepath [where k=0, simplified]) |
1101 |
have "\<And>e. \<lbrakk>0 < r; 0 < d; 0 < e\<rbrakk> |
|
1102 |
\<Longrightarrow> \<forall>\<^sub>F n in F. |
|
1103 |
\<forall>x\<in>sphere z r. |
|
1104 |
x \<noteq> w \<longrightarrow> |
|
1105 |
cmod (f n x - g x) < e * cmod (x - w)" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1106 |
apply (rule_tac e1="e * d" in eventually_mono [OF uniform_limitD [OF ulim]]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1107 |
apply (force simp: dist_norm intro: dle mult_left_mono less_le_trans)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1108 |
done |
72266 | 1109 |
then have ul_less: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)) (\<lambda>x. g x / (x - w)) F" |
1110 |
using greater \<open>0 < d\<close> |
|
1111 |
by (auto simp add: uniform_limit_iff dist_norm norm_divide diff_divide_distrib [symmetric] divide_simps) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1112 |
have g_cint: "(\<lambda>u. g u/(u - w)) contour_integrable_on circlepath z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1113 |
by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1114 |
have cif_tends_cig: "((\<lambda>n. contour_integral(circlepath z r) (\<lambda>u. f n u / (u - w))) \<longlongrightarrow> contour_integral(circlepath z r) (\<lambda>u. g u/(u - w))) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1115 |
by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1116 |
have f_tends_cig: "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> contour_integral (circlepath z r) (\<lambda>u. g u / (u - w))) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1117 |
proof (rule Lim_transform_eventually) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1118 |
show "\<forall>\<^sub>F x in F. contour_integral (circlepath z r) (\<lambda>u. f x u / (u - w)) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1119 |
= 2 * of_real pi * \<i> * f x w" |
72266 | 1120 |
using w\<open>0 < d\<close> d_def |
1121 |
by (auto intro: eventually_mono [OF cont contour_integral_unique [OF Cauchy_integral_circlepath]]) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1122 |
qed (auto simp: cif_tends_cig) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1123 |
have "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in F. dist (f n w) (g w) < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1124 |
by (rule eventually_mono [OF uniform_limitD [OF ulim]]) (use w in auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1125 |
then have "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> 2 * of_real pi * \<i> * g w) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1126 |
by (rule tendsto_mult_left [OF tendstoI]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1127 |
then have "((\<lambda>u. g u / (u - w)) has_contour_integral 2 * of_real pi * \<i> * g w) (circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1128 |
using has_contour_integral_integral [OF g_cint] tendsto_unique [OF F f_tends_cig] w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1129 |
by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1130 |
then have "((\<lambda>u. g u / (2 * of_real pi * \<i> * (u - w))) has_contour_integral g w) (circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1131 |
using has_contour_integral_div [where c = "2 * of_real pi * \<i>"] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1132 |
by (force simp: field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1133 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1134 |
by (simp add: dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1135 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1136 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1137 |
using Cauchy_next_derivative_circlepath(2) [OF 1 2, simplified] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1138 |
by (fastforce simp add: holomorphic_on_open contg intro: that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1139 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1140 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1141 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1142 |
text\<open> Version showing that the limit is the limit of the derivatives.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1143 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1144 |
proposition has_complex_derivative_uniform_limit: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1145 |
fixes z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1146 |
assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1147 |
(\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1148 |
and ulim: "uniform_limit (cball z r) f g F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1149 |
and F: "\<not> trivial_limit F" and "0 < r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1150 |
obtains g' where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1151 |
"continuous_on (cball z r) g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1152 |
"\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1153 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1154 |
let ?conint = "contour_integral (circlepath z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1155 |
have g: "continuous_on (cball z r) g" "g holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1156 |
by (rule holomorphic_uniform_limit [OF eventually_mono [OF cont] ulim F]; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1157 |
auto simp: holomorphic_on_open field_differentiable_def)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1158 |
then obtain g' where g': "\<And>x. x \<in> ball z r \<Longrightarrow> (g has_field_derivative g' x) (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1159 |
using DERIV_deriv_iff_has_field_derivative |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1160 |
by (fastforce simp add: holomorphic_on_open) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1161 |
then have derg: "\<And>x. x \<in> ball z r \<Longrightarrow> deriv g x = g' x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1162 |
by (simp add: DERIV_imp_deriv) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1163 |
have tends_f'n_g': "((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" if w: "w \<in> ball z r" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1164 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1165 |
have eq_f': "?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2) = (f' n w - g' w) * (2 * of_real pi * \<i>)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1166 |
if cont_fn: "continuous_on (cball z r) (f n)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1167 |
and fnd: "\<And>w. w \<in> ball z r \<Longrightarrow> (f n has_field_derivative f' n w) (at w)" for n |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1168 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1169 |
have hol_fn: "f n holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1170 |
using fnd by (force simp: holomorphic_on_open) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1171 |
have "(f n has_field_derivative 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)) (at w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1172 |
by (rule Cauchy_derivative_integral_circlepath [OF cont_fn hol_fn w]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1173 |
then have f': "f' n w = 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1174 |
using DERIV_unique [OF fnd] w by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1175 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1176 |
by (simp add: f' Cauchy_contour_integral_circlepath_2 [OF g w] derg [OF w] field_split_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1177 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1178 |
define d where "d = (r - norm(w - z))^2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1179 |
have "d > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1180 |
using w by (simp add: dist_commute dist_norm d_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1181 |
have dle: "d \<le> cmod ((y - w)\<^sup>2)" if "r = cmod (z - y)" for y |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1182 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1183 |
have "w \<in> ball z (cmod (z - y))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1184 |
using that w by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1185 |
then have "cmod (w - z) \<le> cmod (z - y)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1186 |
by (simp add: dist_complex_def norm_minus_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1187 |
moreover have "cmod (z - y) - cmod (w - z) \<le> cmod (y - w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1188 |
by (metis diff_add_cancel diff_add_eq_diff_diff_swap norm_minus_commute norm_triangle_ineq2) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1189 |
ultimately show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1190 |
using that by (simp add: d_def norm_power power_mono) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1191 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1192 |
have 1: "\<forall>\<^sub>F n in F. (\<lambda>x. f n x / (x - w)\<^sup>2) contour_integrable_on circlepath z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1193 |
by (force simp: holomorphic_on_open intro: w Cauchy_derivative_integral_circlepath eventually_mono [OF cont]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1194 |
have 2: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)\<^sup>2) (\<lambda>x. g x / (x - w)\<^sup>2) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1195 |
unfolding uniform_limit_iff |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1196 |
proof clarify |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1197 |
fix e::real |
72266 | 1198 |
assume "e > 0" |
1199 |
with \<open>r > 0\<close> |
|
1200 |
have "\<forall>\<^sub>F n in F. \<forall>x. x \<noteq> w \<longrightarrow> cmod (z - x) = r \<longrightarrow> cmod (f n x - g x) < e * cmod ((x - w)\<^sup>2)" |
|
1201 |
by (force simp: \<open>0 < d\<close> dist_norm dle intro: less_le_trans eventually_mono [OF uniform_limitD [OF ulim], of "e*d"]) |
|
1202 |
with \<open>r > 0\<close> \<open>e > 0\<close> |
|
1203 |
show "\<forall>\<^sub>F n in F. \<forall>x\<in>sphere z r. dist (f n x / (x - w)\<^sup>2) (g x / (x - w)\<^sup>2) < e" |
|
1204 |
by (simp add: norm_divide field_split_simps sphere_def dist_norm) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1205 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1206 |
have "((\<lambda>n. contour_integral (circlepath z r) (\<lambda>x. f n x / (x - w)\<^sup>2)) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1207 |
\<longlongrightarrow> contour_integral (circlepath z r) ((\<lambda>x. g x / (x - w)\<^sup>2))) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1208 |
by (rule contour_integral_uniform_limit_circlepath [OF 1 2 F \<open>0 < r\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1209 |
then have tendsto_0: "((\<lambda>n. 1 / (2 * of_real pi * \<i>) * (?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2))) \<longlongrightarrow> 0) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1210 |
using Lim_null by (force intro!: tendsto_mult_right_zero) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1211 |
have "((\<lambda>n. f' n w - g' w) \<longlongrightarrow> 0) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1212 |
apply (rule Lim_transform_eventually [OF tendsto_0]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1213 |
apply (force simp: divide_simps intro: eq_f' eventually_mono [OF cont]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1214 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1215 |
then show ?thesis using Lim_null by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1216 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1217 |
obtain g' where "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1218 |
by (blast intro: tends_f'n_g' g') |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1219 |
then show ?thesis using g |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1220 |
using that by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1221 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1222 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1223 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1224 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Some more simple/convenient versions for applications\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1225 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1226 |
lemma holomorphic_uniform_sequence: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1227 |
assumes S: "open S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1228 |
and hol_fn: "\<And>n. (f n) holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1229 |
and ulim_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1230 |
shows "g holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1231 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1232 |
have "\<exists>f'. (g has_field_derivative f') (at z)" if "z \<in> S" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1233 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1234 |
obtain r where "0 < r" and r: "cball z r \<subseteq> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1235 |
and ul: "uniform_limit (cball z r) f g sequentially" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1236 |
using ulim_g [OF \<open>z \<in> S\<close>] by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1237 |
have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and> f n holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1238 |
proof (intro eventuallyI conjI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1239 |
show "continuous_on (cball z r) (f x)" for x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1240 |
using hol_fn holomorphic_on_imp_continuous_on holomorphic_on_subset r by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1241 |
show "f x holomorphic_on ball z r" for x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1242 |
by (metis hol_fn holomorphic_on_subset interior_cball interior_subset r) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1243 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1244 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1245 |
using \<open>0 < r\<close> centre_in_ball ul |
72266 | 1246 |
by (auto simp: holomorphic_on_open intro: holomorphic_uniform_limit [OF *]) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1247 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1248 |
with S show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1249 |
by (simp add: holomorphic_on_open) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1250 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1251 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1252 |
lemma has_complex_derivative_uniform_sequence: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1253 |
fixes S :: "complex set" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1254 |
assumes S: "open S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1255 |
and hfd: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_field_derivative f' n x) (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1256 |
and ulim_g: "\<And>x. x \<in> S |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1257 |
\<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1258 |
shows "\<exists>g'. \<forall>x \<in> S. (g has_field_derivative g' x) (at x) \<and> ((\<lambda>n. f' n x) \<longlongrightarrow> g' x) sequentially" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1259 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1260 |
have y: "\<exists>y. (g has_field_derivative y) (at z) \<and> (\<lambda>n. f' n z) \<longlonglongrightarrow> y" if "z \<in> S" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1261 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1262 |
obtain r where "0 < r" and r: "cball z r \<subseteq> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1263 |
and ul: "uniform_limit (cball z r) f g sequentially" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1264 |
using ulim_g [OF \<open>z \<in> S\<close>] by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1265 |
have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1266 |
(\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1267 |
proof (intro eventuallyI conjI ballI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1268 |
show "continuous_on (cball z r) (f x)" for x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1269 |
by (meson S continuous_on_subset hfd holomorphic_on_imp_continuous_on holomorphic_on_open r) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1270 |
show "w \<in> ball z r \<Longrightarrow> (f x has_field_derivative f' x w) (at w)" for w x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1271 |
using ball_subset_cball hfd r by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1272 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1273 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1274 |
by (rule has_complex_derivative_uniform_limit [OF *, of g]) (use \<open>0 < r\<close> ul in \<open>force+\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1275 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1276 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1277 |
by (rule bchoice) (blast intro: y) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1278 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1279 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1280 |
subsection\<open>On analytic functions defined by a series\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1281 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1282 |
lemma series_and_derivative_comparison: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1283 |
fixes S :: "complex set" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1284 |
assumes S: "open S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1285 |
and h: "summable h" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1286 |
and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1287 |
and to_g: "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. norm (f n x) \<le> h n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1288 |
obtains g g' where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1289 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1290 |
obtain g where g: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f i x) g sequentially" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1291 |
using Weierstrass_m_test_ev [OF to_g h] by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1292 |
have *: "\<exists>d>0. cball x d \<subseteq> S \<and> uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1293 |
if "x \<in> S" for x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1294 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1295 |
obtain d where "d>0" and d: "cball x d \<subseteq> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1296 |
using open_contains_cball [of "S"] \<open>x \<in> S\<close> S by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1297 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1298 |
proof (intro conjI exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1299 |
show "uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1300 |
using d g uniform_limit_on_subset by (force simp: dist_norm eventually_sequentially) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1301 |
qed (use \<open>d > 0\<close> d in auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1302 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1303 |
have "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i x) \<longlonglongrightarrow> g x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1304 |
by (metis tendsto_uniform_limitI [OF g]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1305 |
moreover have "\<exists>g'. \<forall>x\<in>S. (g has_field_derivative g' x) (at x) \<and> (\<lambda>n. \<Sum>i<n. f' i x) \<longlonglongrightarrow> g' x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1306 |
by (rule has_complex_derivative_uniform_sequence [OF S]) (auto intro: * hfd DERIV_sum)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1307 |
ultimately show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1308 |
by (metis sums_def that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1309 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1310 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1311 |
text\<open>A version where we only have local uniform/comparative convergence.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1312 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1313 |
lemma series_and_derivative_comparison_local: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1314 |
fixes S :: "complex set" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1315 |
assumes S: "open S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1316 |
and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1317 |
and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. norm (f n y) \<le> h n)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1318 |
shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1319 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1320 |
have "\<exists>y. (\<lambda>n. f n z) sums (\<Sum>n. f n z) \<and> (\<lambda>n. f' n z) sums y \<and> ((\<lambda>x. \<Sum>n. f n x) has_field_derivative y) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1321 |
if "z \<in> S" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1322 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1323 |
obtain d h where "0 < d" "summable h" and le_h: "\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball z d \<inter> S. norm (f n y) \<le> h n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1324 |
using to_g \<open>z \<in> S\<close> by meson |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1325 |
then obtain r where "r>0" and r: "ball z r \<subseteq> ball z d \<inter> S" using \<open>z \<in> S\<close> S |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1326 |
by (metis Int_iff open_ball centre_in_ball open_Int open_contains_ball_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1327 |
have 1: "open (ball z d \<inter> S)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1328 |
by (simp add: open_Int S) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1329 |
have 2: "\<And>n x. x \<in> ball z d \<inter> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1330 |
by (auto simp: hfd) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1331 |
obtain g g' where gg': "\<forall>x \<in> ball z d \<inter> S. ((\<lambda>n. f n x) sums g x) \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1332 |
((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1333 |
by (auto intro: le_h series_and_derivative_comparison [OF 1 \<open>summable h\<close> hfd]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1334 |
then have "(\<lambda>n. f' n z) sums g' z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1335 |
by (meson \<open>0 < r\<close> centre_in_ball contra_subsetD r) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1336 |
moreover have "(\<lambda>n. f n z) sums (\<Sum>n. f n z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1337 |
using summable_sums centre_in_ball \<open>0 < d\<close> \<open>summable h\<close> le_h |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1338 |
by (metis (full_types) Int_iff gg' summable_def that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1339 |
moreover have "((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' z) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1340 |
proof (rule has_field_derivative_transform_within) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1341 |
show "\<And>x. dist x z < r \<Longrightarrow> g x = (\<Sum>n. f n x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1342 |
by (metis subsetD dist_commute gg' mem_ball r sums_unique) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1343 |
qed (use \<open>0 < r\<close> gg' \<open>z \<in> S\<close> \<open>0 < d\<close> in auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1344 |
ultimately show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1345 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1346 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1347 |
by (rule_tac x="\<lambda>x. suminf (\<lambda>n. f n x)" in exI) meson |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1348 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1349 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1350 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1351 |
text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1352 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1353 |
lemma series_and_derivative_comparison_complex: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1354 |
fixes S :: "complex set" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1355 |
assumes S: "open S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1356 |
and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1357 |
and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1358 |
shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1359 |
apply (rule series_and_derivative_comparison_local [OF S hfd], assumption) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1360 |
apply (rule ex_forward [OF to_g], assumption) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1361 |
apply (erule exE) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1362 |
apply (rule_tac x="Re \<circ> h" in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1363 |
apply (force simp: summable_Re o_def nonneg_Reals_cmod_eq_Re image_subset_iff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1364 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1365 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1366 |
text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1367 |
lemma series_differentiable_comparison_complex: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1368 |
fixes S :: "complex set" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1369 |
assumes S: "open S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1370 |
and hfd: "\<And>n x. x \<in> S \<Longrightarrow> f n field_differentiable (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1371 |
and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1372 |
obtains g where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> g field_differentiable (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1373 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1374 |
have hfd': "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative deriv (f n) x) (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1375 |
using hfd field_differentiable_derivI by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1376 |
have "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. deriv (f n) x) sums g' x) \<and> (g has_field_derivative g' x) (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1377 |
by (metis series_and_derivative_comparison_complex [OF S hfd' to_g]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1378 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1379 |
using field_differentiable_def that by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1380 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1381 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1382 |
text\<open>In particular, a power series is analytic inside circle of convergence.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1383 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1384 |
lemma power_series_and_derivative_0: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1385 |
fixes a :: "nat \<Rightarrow> complex" and r::real |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1386 |
assumes "summable (\<lambda>n. a n * r^n)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1387 |
shows "\<exists>g g'. \<forall>z. cmod z < r \<longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1388 |
((\<lambda>n. a n * z^n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * z^(n - 1)) sums g' z) \<and> (g has_field_derivative g' z) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1389 |
proof (cases "0 < r") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1390 |
case True |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1391 |
have der: "\<And>n z. ((\<lambda>x. a n * x ^ n) has_field_derivative of_nat n * a n * z ^ (n - 1)) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1392 |
by (rule derivative_eq_intros | simp)+ |
72266 | 1393 |
have y_le: "cmod y \<le> cmod (of_real r + of_real (cmod z)) / 2" |
1394 |
if "cmod (z - y) * 2 < r - cmod z" for z y |
|
1395 |
proof - |
|
1396 |
have "cmod y * 2 \<le> r + cmod z" |
|
1397 |
using norm_triangle_ineq2 [of y z] that |
|
1398 |
by (simp only: diff_le_eq norm_minus_commute mult_2) |
|
1399 |
then show ?thesis |
|
1400 |
using \<open>r > 0\<close> |
|
1401 |
by (auto simp: algebra_simps norm_mult norm_divide norm_power simp flip: of_real_add) |
|
1402 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1403 |
have "summable (\<lambda>n. a n * complex_of_real r ^ n)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1404 |
using assms \<open>r > 0\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1405 |
moreover have "\<And>z. cmod z < r \<Longrightarrow> cmod ((of_real r + of_real (cmod z)) / 2) < cmod (of_real r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1406 |
using \<open>r > 0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1407 |
by (simp flip: of_real_add) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1408 |
ultimately have sum: "\<And>z. cmod z < r \<Longrightarrow> summable (\<lambda>n. of_real (cmod (a n)) * ((of_real r + complex_of_real (cmod z)) / 2) ^ n)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1409 |
by (rule power_series_conv_imp_absconv_weak) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1410 |
have "\<exists>g g'. \<forall>z \<in> ball 0 r. (\<lambda>n. (a n) * z ^ n) sums g z \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1411 |
(\<lambda>n. of_nat n * (a n) * z ^ (n - 1)) sums g' z \<and> (g has_field_derivative g' z) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1412 |
apply (rule series_and_derivative_comparison_complex [OF open_ball der]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1413 |
apply (rule_tac x="(r - norm z)/2" in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1414 |
apply (rule_tac x="\<lambda>n. of_real(norm(a n)*((r + norm z)/2)^n)" in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1415 |
using \<open>r > 0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1416 |
apply (auto simp: sum eventually_sequentially norm_mult norm_power dist_norm intro!: mult_left_mono power_mono y_le) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1417 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1418 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1419 |
by (simp add: ball_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1420 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1421 |
case False then show ?thesis |
72266 | 1422 |
unfolding not_less |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1423 |
using less_le_trans norm_not_less_zero by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1424 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1425 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1426 |
proposition\<^marker>\<open>tag unimportant\<close> power_series_and_derivative: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1427 |
fixes a :: "nat \<Rightarrow> complex" and r::real |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1428 |
assumes "summable (\<lambda>n. a n * r^n)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1429 |
obtains g g' where "\<forall>z \<in> ball w r. |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1430 |
((\<lambda>n. a n * (z - w) ^ n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * (z - w) ^ (n - 1)) sums g' z) \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1431 |
(g has_field_derivative g' z) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1432 |
using power_series_and_derivative_0 [OF assms] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1433 |
apply clarify |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1434 |
apply (rule_tac g="(\<lambda>z. g(z - w))" in that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1435 |
using DERIV_shift [where z="-w"] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1436 |
apply (auto simp: norm_minus_commute Ball_def dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1437 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1438 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1439 |
proposition\<^marker>\<open>tag unimportant\<close> power_series_holomorphic: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1440 |
assumes "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>n. a n*(w - z)^n) sums f w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1441 |
shows "f holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1442 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1443 |
have "\<exists>f'. (f has_field_derivative f') (at w)" if w: "dist z w < r" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1444 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1445 |
have inb: "z + complex_of_real ((dist z w + r) / 2) \<in> ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1446 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1447 |
have wz: "cmod (w - z) < r" using w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1448 |
by (auto simp: field_split_simps dist_norm norm_minus_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1449 |
then have "0 \<le> r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1450 |
by (meson less_eq_real_def norm_ge_zero order_trans) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1451 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1452 |
using w by (simp add: dist_norm \<open>0\<le>r\<close> flip: of_real_add) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1453 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1454 |
have sum: "summable (\<lambda>n. a n * of_real (((cmod (z - w) + r) / 2) ^ n))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1455 |
using assms [OF inb] by (force simp: summable_def dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1456 |
obtain g g' where gg': "\<And>u. u \<in> ball z ((cmod (z - w) + r) / 2) \<Longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1457 |
(\<lambda>n. a n * (u - z) ^ n) sums g u \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1458 |
(\<lambda>n. of_nat n * a n * (u - z) ^ (n - 1)) sums g' u \<and> (g has_field_derivative g' u) (at u)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1459 |
by (rule power_series_and_derivative [OF sum, of z]) fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1460 |
have [simp]: "g u = f u" if "cmod (u - w) < (r - cmod (z - w)) / 2" for u |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1461 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1462 |
have less: "cmod (z - u) * 2 < cmod (z - w) + r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1463 |
using that dist_triangle2 [of z u w] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1464 |
by (simp add: dist_norm [symmetric] algebra_simps) |
72266 | 1465 |
have "(\<lambda>n. a n * (u - z) ^ n) sums g u" "(\<lambda>n. a n * (u - z) ^ n) sums f u" |
1466 |
using gg' [of u] less w by (auto simp: assms dist_norm) |
|
1467 |
then show ?thesis |
|
1468 |
by (metis sums_unique2) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1469 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1470 |
have "(f has_field_derivative g' w) (at w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1471 |
by (rule has_field_derivative_transform_within [where d="(r - norm(z - w))/2"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1472 |
(use w gg' [of w] in \<open>(force simp: dist_norm)+\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1473 |
then show ?thesis .. |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1474 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1475 |
then show ?thesis by (simp add: holomorphic_on_open) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1476 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1477 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1478 |
corollary holomorphic_iff_power_series: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1479 |
"f holomorphic_on ball z r \<longleftrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1480 |
(\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1481 |
apply (intro iffI ballI holomorphic_power_series, assumption+) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1482 |
apply (force intro: power_series_holomorphic [where a = "\<lambda>n. (deriv ^^ n) f z / (fact n)"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1483 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1484 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1485 |
lemma power_series_analytic: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1486 |
"(\<And>w. w \<in> ball z r \<Longrightarrow> (\<lambda>n. a n*(w - z)^n) sums f w) \<Longrightarrow> f analytic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1487 |
by (force simp: analytic_on_open intro!: power_series_holomorphic) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1488 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1489 |
lemma analytic_iff_power_series: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1490 |
"f analytic_on ball z r \<longleftrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1491 |
(\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1492 |
by (simp add: analytic_on_open holomorphic_iff_power_series) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1493 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1494 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Equality between holomorphic functions, on open ball then connected set\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1495 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1496 |
lemma holomorphic_fun_eq_on_ball: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1497 |
"\<lbrakk>f holomorphic_on ball z r; g holomorphic_on ball z r; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1498 |
w \<in> ball z r; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1499 |
\<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1500 |
\<Longrightarrow> f w = g w" |
72266 | 1501 |
by (auto simp: holomorphic_iff_power_series sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"]) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1502 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1503 |
lemma holomorphic_fun_eq_0_on_ball: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1504 |
"\<lbrakk>f holomorphic_on ball z r; w \<in> ball z r; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1505 |
\<And>n. (deriv ^^ n) f z = 0\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1506 |
\<Longrightarrow> f w = 0" |
72266 | 1507 |
by (auto simp: holomorphic_iff_power_series sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"]) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1508 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1509 |
lemma holomorphic_fun_eq_0_on_connected: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1510 |
assumes holf: "f holomorphic_on S" and "open S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1511 |
and cons: "connected S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1512 |
and der: "\<And>n. (deriv ^^ n) f z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1513 |
and "z \<in> S" "w \<in> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1514 |
shows "f w = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1515 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1516 |
have *: "ball x e \<subseteq> (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1517 |
if "\<forall>u. (deriv ^^ u) f x = 0" "ball x e \<subseteq> S" for x e |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1518 |
proof - |
72266 | 1519 |
have "(deriv ^^ m) ((deriv ^^ n) f) x = 0" for m n |
1520 |
by (metis funpow_add o_apply that(1)) |
|
1521 |
then have "\<And>x' n. dist x x' < e \<Longrightarrow> (deriv ^^ n) f x' = 0" |
|
1522 |
using \<open>open S\<close> |
|
1523 |
by (meson holf holomorphic_fun_eq_0_on_ball holomorphic_higher_deriv holomorphic_on_subset mem_ball that(2)) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1524 |
with that show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1525 |
qed |
72266 | 1526 |
obtain e where "e>0" and e: "ball w e \<subseteq> S" using openE [OF \<open>open S\<close> \<open>w \<in> S\<close>] . |
1527 |
then have holfb: "f holomorphic_on ball w e" |
|
1528 |
using holf holomorphic_on_subset by blast |
|
1529 |
have "open (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1530 |
using \<open>open S\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1531 |
apply (simp add: open_contains_ball Ball_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1532 |
apply (erule all_forward) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1533 |
using "*" by auto blast+ |
72266 | 1534 |
then have "openin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})" |
1535 |
by (force intro: open_subset) |
|
1536 |
moreover have "closedin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1537 |
using assms |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1538 |
by (auto intro: continuous_closedin_preimage_constant holomorphic_on_imp_continuous_on holomorphic_higher_deriv) |
72266 | 1539 |
moreover have "(\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}) = S \<Longrightarrow> f w = 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1540 |
using \<open>e>0\<close> e by (force intro: holomorphic_fun_eq_0_on_ball [OF holfb]) |
72266 | 1541 |
ultimately show ?thesis |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1542 |
using cons der \<open>z \<in> S\<close> |
72266 | 1543 |
by (auto simp add: connected_clopen) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1544 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1545 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1546 |
lemma holomorphic_fun_eq_on_connected: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1547 |
assumes "f holomorphic_on S" "g holomorphic_on S" and "open S" "connected S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1548 |
and "\<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1549 |
and "z \<in> S" "w \<in> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1550 |
shows "f w = g w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1551 |
proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>x. f x - g x" S z, simplified]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1552 |
show "(\<lambda>x. f x - g x) holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1553 |
by (intro assms holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1554 |
show "\<And>n. (deriv ^^ n) (\<lambda>x. f x - g x) z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1555 |
using assms higher_deriv_diff by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1556 |
qed (use assms in auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1557 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1558 |
lemma holomorphic_fun_eq_const_on_connected: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1559 |
assumes holf: "f holomorphic_on S" and "open S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1560 |
and cons: "connected S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1561 |
and der: "\<And>n. 0 < n \<Longrightarrow> (deriv ^^ n) f z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1562 |
and "z \<in> S" "w \<in> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1563 |
shows "f w = f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1564 |
proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>w. f w - f z" S z, simplified]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1565 |
show "(\<lambda>w. f w - f z) holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1566 |
by (intro assms holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1567 |
show "\<And>n. (deriv ^^ n) (\<lambda>w. f w - f z) z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1568 |
by (subst higher_deriv_diff) (use assms in \<open>auto intro: holomorphic_intros\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1569 |
qed (use assms in auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1570 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1571 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Some basic lemmas about poles/singularities\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1572 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1573 |
lemma pole_lemma: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1574 |
assumes holf: "f holomorphic_on S" and a: "a \<in> interior S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1575 |
shows "(\<lambda>z. if z = a then deriv f a |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1576 |
else (f z - f a) / (z - a)) holomorphic_on S" (is "?F holomorphic_on S") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1577 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1578 |
have F1: "?F field_differentiable (at u within S)" if "u \<in> S" "u \<noteq> a" for u |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1579 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1580 |
have fcd: "f field_differentiable at u within S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1581 |
using holf holomorphic_on_def by (simp add: \<open>u \<in> S\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1582 |
have cd: "(\<lambda>z. (f z - f a) / (z - a)) field_differentiable at u within S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1583 |
by (rule fcd derivative_intros | simp add: that)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1584 |
have "0 < dist a u" using that dist_nz by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1585 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1586 |
by (rule field_differentiable_transform_within [OF _ _ _ cd]) (auto simp: \<open>u \<in> S\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1587 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1588 |
have F2: "?F field_differentiable at a" if "0 < e" "ball a e \<subseteq> S" for e |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1589 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1590 |
have holfb: "f holomorphic_on ball a e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1591 |
by (rule holomorphic_on_subset [OF holf \<open>ball a e \<subseteq> S\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1592 |
have 2: "?F holomorphic_on ball a e - {a}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1593 |
using mem_ball that |
72266 | 1594 |
by (auto simp add: holomorphic_on_def simp flip: field_differentiable_def intro: F1 field_differentiable_within_subset) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1595 |
have "isCont (\<lambda>z. if z = a then deriv f a else (f z - f a) / (z - a)) x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1596 |
if "dist a x < e" for x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1597 |
proof (cases "x=a") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1598 |
case True |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1599 |
then have "f field_differentiable at a" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1600 |
using holfb \<open>0 < e\<close> holomorphic_on_imp_differentiable_at by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1601 |
with True show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1602 |
by (auto simp: continuous_at has_field_derivative_iff simp flip: DERIV_deriv_iff_field_differentiable |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1603 |
elim: rev_iffD1 [OF _ LIM_equal]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1604 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1605 |
case False with 2 that show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1606 |
by (force simp: holomorphic_on_open open_Diff field_differentiable_def [symmetric] field_differentiable_imp_continuous_at) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1607 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1608 |
then have 1: "continuous_on (ball a e) ?F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1609 |
by (clarsimp simp: continuous_on_eq_continuous_at) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1610 |
have "?F holomorphic_on ball a e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1611 |
by (auto intro: no_isolated_singularity [OF 1 2]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1612 |
with that show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1613 |
by (simp add: holomorphic_on_open field_differentiable_def [symmetric] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1614 |
field_differentiable_at_within) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1615 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1616 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1617 |
proof |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1618 |
fix x assume "x \<in> S" show "?F field_differentiable at x within S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1619 |
proof (cases "x=a") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1620 |
case True then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1621 |
using a by (auto simp: mem_interior intro: field_differentiable_at_within F2) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1622 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1623 |
case False with F1 \<open>x \<in> S\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1624 |
show ?thesis by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1625 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1626 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1627 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1628 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1629 |
lemma pole_theorem: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1630 |
assumes holg: "g holomorphic_on S" and a: "a \<in> interior S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1631 |
and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1632 |
shows "(\<lambda>z. if z = a then deriv g a |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1633 |
else f z - g a/(z - a)) holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1634 |
using pole_lemma [OF holg a] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1635 |
by (rule holomorphic_transform) (simp add: eq field_split_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1636 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1637 |
lemma pole_lemma_open: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1638 |
assumes "f holomorphic_on S" "open S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1639 |
shows "(\<lambda>z. if z = a then deriv f a else (f z - f a)/(z - a)) holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1640 |
proof (cases "a \<in> S") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1641 |
case True with assms interior_eq pole_lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1642 |
show ?thesis by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1643 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1644 |
case False with assms show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1645 |
apply (simp add: holomorphic_on_def field_differentiable_def [symmetric], clarify) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1646 |
apply (rule field_differentiable_transform_within [where f = "\<lambda>z. (f z - f a)/(z - a)" and d = 1]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1647 |
apply (rule derivative_intros | force)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1648 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1649 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1650 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1651 |
lemma pole_theorem_open: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1652 |
assumes holg: "g holomorphic_on S" and S: "open S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1653 |
and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1654 |
shows "(\<lambda>z. if z = a then deriv g a |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1655 |
else f z - g a/(z - a)) holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1656 |
using pole_lemma_open [OF holg S] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1657 |
by (rule holomorphic_transform) (auto simp: eq divide_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1658 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1659 |
lemma pole_theorem_0: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1660 |
assumes holg: "g holomorphic_on S" and a: "a \<in> interior S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1661 |
and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1662 |
and [simp]: "f a = deriv g a" "g a = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1663 |
shows "f holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1664 |
using pole_theorem [OF holg a eq] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1665 |
by (rule holomorphic_transform) (auto simp: eq field_split_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1666 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1667 |
lemma pole_theorem_open_0: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1668 |
assumes holg: "g holomorphic_on S" and S: "open S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1669 |
and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1670 |
and [simp]: "f a = deriv g a" "g a = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1671 |
shows "f holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1672 |
using pole_theorem_open [OF holg S eq] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1673 |
by (rule holomorphic_transform) (auto simp: eq field_split_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1674 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1675 |
lemma pole_theorem_analytic: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1676 |
assumes g: "g analytic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1677 |
and eq: "\<And>z. z \<in> S |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1678 |
\<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1679 |
shows "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S" (is "?F analytic_on S") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1680 |
unfolding analytic_on_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1681 |
proof |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1682 |
fix x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1683 |
assume "x \<in> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1684 |
with g obtain e where "0 < e" and e: "g holomorphic_on ball x e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1685 |
by (auto simp add: analytic_on_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1686 |
obtain d where "0 < d" and d: "\<And>w. w \<in> ball x d - {a} \<Longrightarrow> g w = (w - a) * f w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1687 |
using \<open>x \<in> S\<close> eq by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1688 |
have "?F holomorphic_on ball x (min d e)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1689 |
using d e \<open>x \<in> S\<close> by (fastforce simp: holomorphic_on_subset subset_ball intro!: pole_theorem_open) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1690 |
then show "\<exists>e>0. ?F holomorphic_on ball x e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1691 |
using \<open>0 < d\<close> \<open>0 < e\<close> not_le by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1692 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1693 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1694 |
lemma pole_theorem_analytic_0: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1695 |
assumes g: "g analytic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1696 |
and eq: "\<And>z. z \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1697 |
and [simp]: "f a = deriv g a" "g a = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1698 |
shows "f analytic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1699 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1700 |
have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1701 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1702 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1703 |
using pole_theorem_analytic [OF g eq] by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1704 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1705 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1706 |
lemma pole_theorem_analytic_open_superset: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1707 |
assumes g: "g analytic_on S" and "S \<subseteq> T" "open T" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1708 |
and eq: "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1709 |
shows "(\<lambda>z. if z = a then deriv g a |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1710 |
else f z - g a/(z - a)) analytic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1711 |
proof (rule pole_theorem_analytic [OF g]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1712 |
fix z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1713 |
assume "z \<in> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1714 |
then obtain e where "0 < e" and e: "ball z e \<subseteq> T" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1715 |
using assms openE by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1716 |
then show "\<exists>d>0. \<forall>w\<in>ball z d - {a}. g w = (w - a) * f w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1717 |
using eq by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1718 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1719 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1720 |
lemma pole_theorem_analytic_open_superset_0: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1721 |
assumes g: "g analytic_on S" "S \<subseteq> T" "open T" "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1722 |
and [simp]: "f a = deriv g a" "g a = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1723 |
shows "f analytic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1724 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1725 |
have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1726 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1727 |
have "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1728 |
by (rule pole_theorem_analytic_open_superset [OF g]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1729 |
then show ?thesis by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1730 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1731 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1732 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1733 |
subsection\<open>General, homology form of Cauchy's theorem\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1734 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1735 |
text\<open>Proof is based on Dixon's, as presented in Lang's "Complex Analysis" book (page 147).\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1736 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1737 |
lemma contour_integral_continuous_on_linepath_2D: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1738 |
assumes "open U" and cont_dw: "\<And>w. w \<in> U \<Longrightarrow> F w contour_integrable_on (linepath a b)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1739 |
and cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). F x y)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1740 |
and abu: "closed_segment a b \<subseteq> U" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1741 |
shows "continuous_on U (\<lambda>w. contour_integral (linepath a b) (F w))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1742 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1743 |
have *: "\<exists>d>0. \<forall>x'\<in>U. dist x' w < d \<longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1744 |
dist (contour_integral (linepath a b) (F x')) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1745 |
(contour_integral (linepath a b) (F w)) \<le> \<epsilon>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1746 |
if "w \<in> U" "0 < \<epsilon>" "a \<noteq> b" for w \<epsilon> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1747 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1748 |
obtain \<delta> where "\<delta>>0" and \<delta>: "cball w \<delta> \<subseteq> U" using open_contains_cball \<open>open U\<close> \<open>w \<in> U\<close> by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1749 |
let ?TZ = "cball w \<delta> \<times> closed_segment a b" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1750 |
have "uniformly_continuous_on ?TZ (\<lambda>(x,y). F x y)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1751 |
proof (rule compact_uniformly_continuous) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1752 |
show "continuous_on ?TZ (\<lambda>(x,y). F x y)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1753 |
by (rule continuous_on_subset[OF cond_uu]) (use SigmaE \<delta> abu in blast) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1754 |
show "compact ?TZ" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1755 |
by (simp add: compact_Times) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1756 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1757 |
then obtain \<eta> where "\<eta>>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1758 |
and \<eta>: "\<And>x x'. \<lbrakk>x\<in>?TZ; x'\<in>?TZ; dist x' x < \<eta>\<rbrakk> \<Longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1759 |
dist ((\<lambda>(x,y). F x y) x') ((\<lambda>(x,y). F x y) x) < \<epsilon>/norm(b - a)" |
72266 | 1760 |
using \<open>0 < \<epsilon>\<close> \<open>a \<noteq> b\<close> |
1761 |
by (auto elim: uniformly_continuous_onE [where e = "\<epsilon>/norm(b - a)"]) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1762 |
have \<eta>: "\<lbrakk>norm (w - x1) \<le> \<delta>; x2 \<in> closed_segment a b; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1763 |
norm (w - x1') \<le> \<delta>; x2' \<in> closed_segment a b; norm ((x1', x2') - (x1, x2)) < \<eta>\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1764 |
\<Longrightarrow> norm (F x1' x2' - F x1 x2) \<le> \<epsilon> / cmod (b - a)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1765 |
for x1 x2 x1' x2' |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1766 |
using \<eta> [of "(x1,x2)" "(x1',x2')"] by (force simp: dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1767 |
have le_ee: "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1768 |
if "x' \<in> U" "cmod (x' - w) < \<delta>" "cmod (x' - w) < \<eta>" for x' |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1769 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1770 |
have "(\<lambda>x. F x' x - F w x) contour_integrable_on linepath a b" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1771 |
by (simp add: \<open>w \<in> U\<close> cont_dw contour_integrable_diff that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1772 |
then have "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>/norm(b - a) * norm(b - a)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1773 |
apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_integral _ \<eta>]) |
72266 | 1774 |
using \<open>0 < \<epsilon>\<close> \<open>0 < \<delta>\<close> that by (auto simp: norm_minus_commute) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1775 |
also have "\<dots> = \<epsilon>" using \<open>a \<noteq> b\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1776 |
finally show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1777 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1778 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1779 |
apply (rule_tac x="min \<delta> \<eta>" in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1780 |
using \<open>0 < \<delta>\<close> \<open>0 < \<eta>\<close> |
72266 | 1781 |
by (auto simp: dist_norm contour_integral_diff [OF cont_dw cont_dw, symmetric] \<open>w \<in> U\<close> intro: le_ee) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1782 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1783 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1784 |
proof (cases "a=b") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1785 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1786 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1787 |
by (rule continuous_onI) (use False in \<open>auto intro: *\<close>) |
72266 | 1788 |
qed auto |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1789 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1790 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1791 |
text\<open>This version has \<^term>\<open>polynomial_function \<gamma>\<close> as an additional assumption.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1792 |
lemma Cauchy_integral_formula_global_weak: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1793 |
assumes "open U" and holf: "f holomorphic_on U" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1794 |
and z: "z \<in> U" and \<gamma>: "polynomial_function \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1795 |
and pasz: "path_image \<gamma> \<subseteq> U - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1796 |
and zero: "\<And>w. w \<notin> U \<Longrightarrow> winding_number \<gamma> w = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1797 |
shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1798 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1799 |
obtain \<gamma>' where pf\<gamma>': "polynomial_function \<gamma>'" and \<gamma>': "\<And>x. (\<gamma> has_vector_derivative (\<gamma>' x)) (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1800 |
using has_vector_derivative_polynomial_function [OF \<gamma>] by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1801 |
then have "bounded(path_image \<gamma>')" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1802 |
by (simp add: path_image_def compact_imp_bounded compact_continuous_image continuous_on_polymonial_function) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1803 |
then obtain B where "B>0" and B: "\<And>x. x \<in> path_image \<gamma>' \<Longrightarrow> norm x \<le> B" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1804 |
using bounded_pos by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1805 |
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 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1806 |
define v where "v = {w. w \<notin> path_image \<gamma> \<and> winding_number \<gamma> w = 0}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1807 |
have "path \<gamma>" "valid_path \<gamma>" using \<gamma> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1808 |
by (auto simp: path_polynomial_function valid_path_polynomial_function) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1809 |
then have ov: "open v" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1810 |
by (simp add: v_def open_winding_number_levelsets loop) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1811 |
have uv_Un: "U \<union> v = UNIV" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1812 |
using pasz zero by (auto simp: v_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1813 |
have conf: "continuous_on U f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1814 |
by (metis holf holomorphic_on_imp_continuous_on) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1815 |
have hol_d: "(d y) holomorphic_on U" if "y \<in> U" for y |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1816 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1817 |
have *: "(\<lambda>c. if c = y then deriv f y else (f c - f y) / (c - y)) holomorphic_on U" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1818 |
by (simp add: holf pole_lemma_open \<open>open U\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1819 |
then have "isCont (\<lambda>x. if x = y then deriv f y else (f x - f y) / (x - y)) y" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1820 |
using at_within_open field_differentiable_imp_continuous_at holomorphic_on_def that \<open>open U\<close> by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1821 |
then have "continuous_on U (d y)" |
72266 | 1822 |
using "*" d_def holomorphic_on_imp_continuous_on by auto |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1823 |
moreover have "d y holomorphic_on U - {y}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1824 |
proof - |
72266 | 1825 |
have "(\<lambda>w. if w = y then deriv f y else (f w - f y) / (w - y)) field_differentiable at w" |
1826 |
if "w \<in> U - {y}" for w |
|
1827 |
proof (rule field_differentiable_transform_within) |
|
1828 |
show "(\<lambda>w. (f w - f y) / (w - y)) field_differentiable at w" |
|
1829 |
using that \<open>open U\<close> holf |
|
1830 |
by (auto intro!: holomorphic_on_imp_differentiable_at derivative_intros) |
|
1831 |
show "dist w y > 0" |
|
1832 |
using that by auto |
|
1833 |
qed (auto simp: dist_commute) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1834 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1835 |
unfolding field_differentiable_def by (simp add: d_def holomorphic_on_open \<open>open U\<close> open_delete) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1836 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1837 |
ultimately show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1838 |
by (rule no_isolated_singularity) (auto simp: \<open>open U\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1839 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1840 |
have cint_fxy: "(\<lambda>x. (f x - f y) / (x - y)) contour_integrable_on \<gamma>" if "y \<notin> path_image \<gamma>" for y |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1841 |
proof (rule contour_integrable_holomorphic_simple [where S = "U-{y}"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1842 |
show "(\<lambda>x. (f x - f y) / (x - y)) holomorphic_on U - {y}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1843 |
by (force intro: holomorphic_intros holomorphic_on_subset [OF holf]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1844 |
show "path_image \<gamma> \<subseteq> U - {y}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1845 |
using pasz that by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1846 |
qed (auto simp: \<open>open U\<close> open_delete \<open>valid_path \<gamma>\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1847 |
define h where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1848 |
"h z = (if z \<in> U then contour_integral \<gamma> (d z) else contour_integral \<gamma> (\<lambda>w. f w/(w - z)))" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1849 |
have U: "((d z) has_contour_integral h z) \<gamma>" if "z \<in> U" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1850 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1851 |
have "d z holomorphic_on U" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1852 |
by (simp add: hol_d that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1853 |
with that show ?thesis |
72266 | 1854 |
by (metis Diff_subset \<open>valid_path \<gamma>\<close> \<open>open U\<close> contour_integrable_holomorphic_simple h_def has_contour_integral_integral pasz subset_trans) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1855 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1856 |
have V: "((\<lambda>w. f w / (w - z)) has_contour_integral h z) \<gamma>" if z: "z \<in> v" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1857 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1858 |
have 0: "0 = (f z) * 2 * of_real (2 * pi) * \<i> * winding_number \<gamma> z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1859 |
using v_def z by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1860 |
then have "((\<lambda>x. 1 / (x - z)) has_contour_integral 0) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1861 |
using z v_def has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close>] by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1862 |
then have "((\<lambda>x. f z * (1 / (x - z))) has_contour_integral 0) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1863 |
using has_contour_integral_lmul by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1864 |
then have "((\<lambda>x. f z / (x - z)) has_contour_integral 0) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1865 |
by (simp add: field_split_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1866 |
moreover have "((\<lambda>x. (f x - f z) / (x - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1867 |
using z |
72266 | 1868 |
apply (simp add: v_def) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1869 |
apply (metis (no_types, lifting) contour_integrable_eq d_def has_contour_integral_eq has_contour_integral_integral cint_fxy) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1870 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1871 |
ultimately have *: "((\<lambda>x. f z / (x - z) + (f x - f z) / (x - z)) has_contour_integral (0 + contour_integral \<gamma> (d z))) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1872 |
by (rule has_contour_integral_add) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1873 |
have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>" |
72266 | 1874 |
if "z \<in> U" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1875 |
using * by (auto simp: divide_simps has_contour_integral_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1876 |
moreover have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. f w / (w - z))) \<gamma>" |
72266 | 1877 |
if "z \<notin> U" |
1878 |
proof (rule has_contour_integral_integral [OF contour_integrable_holomorphic_simple [where S=U]]) |
|
1879 |
show "(\<lambda>w. f w / (w - z)) holomorphic_on U" |
|
1880 |
by (rule holomorphic_intros assms | use that in force)+ |
|
1881 |
qed (use \<open>open U\<close> pasz \<open>valid_path \<gamma>\<close> in auto) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1882 |
ultimately show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1883 |
using z by (simp add: h_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1884 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1885 |
have znot: "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1886 |
using pasz by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1887 |
obtain d0 where "d0>0" and d0: "\<And>x y. x \<in> path_image \<gamma> \<Longrightarrow> y \<in> - U \<Longrightarrow> d0 \<le> dist x y" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1888 |
using separate_compact_closed [of "path_image \<gamma>" "-U"] pasz \<open>open U\<close> \<open>path \<gamma>\<close> compact_path_image |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1889 |
by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1890 |
obtain dd where "0 < dd" and dd: "{y + k | y k. y \<in> path_image \<gamma> \<and> k \<in> ball 0 dd} \<subseteq> U" |
72266 | 1891 |
proof |
1892 |
show "0 < d0 / 2" using \<open>0 < d0\<close> by auto |
|
1893 |
qed (use \<open>0 < d0\<close> d0 in \<open>force simp: dist_norm\<close>) |
|
1894 |
define T where "T \<equiv> {y + k |y k. y \<in> path_image \<gamma> \<and> k \<in> cball 0 (dd / 2)}" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1895 |
have "\<And>x x'. \<lbrakk>x \<in> path_image \<gamma>; dist x x' * 2 < dd\<rbrakk> \<Longrightarrow> \<exists>y k. x' = y + k \<and> y \<in> path_image \<gamma> \<and> dist 0 k * 2 \<le> dd" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1896 |
apply (rule_tac x=x in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1897 |
apply (rule_tac x="x'-x" in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1898 |
apply (force simp: dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1899 |
done |
72266 | 1900 |
then have subt: "path_image \<gamma> \<subseteq> interior T" |
1901 |
using \<open>0 < dd\<close> |
|
1902 |
apply (clarsimp simp add: mem_interior T_def) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1903 |
apply (rule_tac x="dd/2" in exI, auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1904 |
done |
72266 | 1905 |
have "compact T" |
1906 |
unfolding T_def |
|
1907 |
by (fastforce simp add: \<open>valid_path \<gamma>\<close> compact_valid_path_image intro!: compact_sums) |
|
1908 |
have T: "T \<subseteq> U" |
|
1909 |
unfolding T_def using \<open>0 < dd\<close> dd by fastforce |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1910 |
obtain L where "L>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1911 |
and L: "\<And>f B. \<lbrakk>f holomorphic_on interior T; \<And>z. z\<in>interior T \<Longrightarrow> cmod (f z) \<le> B\<rbrakk> \<Longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1912 |
cmod (contour_integral \<gamma> f) \<le> L * B" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1913 |
using contour_integral_bound_exists [OF open_interior \<open>valid_path \<gamma>\<close> subt] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1914 |
by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1915 |
have "bounded(f ` T)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1916 |
by (meson \<open>compact T\<close> compact_continuous_image compact_imp_bounded conf continuous_on_subset T) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1917 |
then obtain D where "D>0" and D: "\<And>x. x \<in> T \<Longrightarrow> norm (f x) \<le> D" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1918 |
by (auto simp: bounded_pos) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1919 |
obtain C where "C>0" and C: "\<And>x. x \<in> T \<Longrightarrow> norm x \<le> C" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1920 |
using \<open>compact T\<close> bounded_pos compact_imp_bounded by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1921 |
have "dist (h y) 0 \<le> e" if "0 < e" and le: "D * L / e + C \<le> cmod y" for e y |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1922 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1923 |
have "D * L / e > 0" using \<open>D>0\<close> \<open>L>0\<close> \<open>e>0\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1924 |
with le have ybig: "norm y > C" by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1925 |
with C have "y \<notin> T" by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1926 |
then have ynot: "y \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1927 |
using subt interior_subset by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1928 |
have [simp]: "winding_number \<gamma> y = 0" |
72266 | 1929 |
proof (rule winding_number_zero_outside) |
1930 |
show "path_image \<gamma> \<subseteq> cball 0 C" |
|
1931 |
by (meson C interior_subset mem_cball_0 subset_eq subt) |
|
1932 |
qed (use ybig loop \<open>path \<gamma>\<close> in auto) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1933 |
have [simp]: "h y = contour_integral \<gamma> (\<lambda>w. f w/(w - y))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1934 |
by (rule contour_integral_unique [symmetric]) (simp add: v_def ynot V) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1935 |
have holint: "(\<lambda>w. f w / (w - y)) holomorphic_on interior T" |
72266 | 1936 |
proof (intro holomorphic_intros) |
1937 |
show "f holomorphic_on interior T" |
|
1938 |
using holf holomorphic_on_subset interior_subset T by blast |
|
1939 |
qed (use \<open>y \<notin> T\<close> interior_subset in auto) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1940 |
have leD: "cmod (f z / (z - y)) \<le> D * (e / L / D)" if z: "z \<in> interior T" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1941 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1942 |
have "D * L / e + cmod z \<le> cmod y" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1943 |
using le C [of z] z using interior_subset by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1944 |
then have DL2: "D * L / e \<le> cmod (z - y)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1945 |
using norm_triangle_ineq2 [of y z] by (simp add: norm_minus_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1946 |
have "cmod (f z / (z - y)) = cmod (f z) * inverse (cmod (z - y))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1947 |
by (simp add: norm_mult norm_inverse Fields.field_class.field_divide_inverse) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1948 |
also have "\<dots> \<le> D * (e / L / D)" |
72266 | 1949 |
proof (rule mult_mono) |
1950 |
show "cmod (f z) \<le> D" |
|
1951 |
using D interior_subset z by blast |
|
1952 |
show "inverse (cmod (z - y)) \<le> e / L / D" "D \<ge> 0" |
|
1953 |
using \<open>L>0\<close> \<open>e>0\<close> \<open>D>0\<close> DL2 by (auto simp: norm_divide field_split_simps) |
|
1954 |
qed auto |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1955 |
finally show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1956 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1957 |
have "dist (h y) 0 = cmod (contour_integral \<gamma> (\<lambda>w. f w / (w - y)))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1958 |
by (simp add: dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1959 |
also have "\<dots> \<le> L * (D * (e / L / D))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1960 |
by (rule L [OF holint leD]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1961 |
also have "\<dots> = e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1962 |
using \<open>L>0\<close> \<open>0 < D\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1963 |
finally show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1964 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1965 |
then have "(h \<longlongrightarrow> 0) at_infinity" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1966 |
by (meson Lim_at_infinityI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1967 |
moreover have "h holomorphic_on UNIV" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1968 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1969 |
have con_ff: "continuous (at (x,z)) (\<lambda>(x,y). (f y - f x) / (y - x))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1970 |
if "x \<in> U" "z \<in> U" "x \<noteq> z" for x z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1971 |
using that conf |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1972 |
apply (simp add: split_def continuous_on_eq_continuous_at \<open>open U\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1973 |
apply (simp | rule continuous_intros continuous_within_compose2 [where g=f])+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1974 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1975 |
have con_fstsnd: "continuous_on UNIV (\<lambda>x. (fst x - snd x) ::complex)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1976 |
by (rule continuous_intros)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1977 |
have open_uu_Id: "open (U \<times> U - Id)" |
72266 | 1978 |
proof (rule open_Diff) |
1979 |
show "open (U \<times> U)" |
|
1980 |
by (simp add: open_Times \<open>open U\<close>) |
|
1981 |
show "closed (Id :: complex rel)" |
|
1982 |
using continuous_closed_preimage_constant [OF con_fstsnd closed_UNIV, of 0] |
|
1983 |
by (auto simp: Id_fstsnd_eq algebra_simps) |
|
1984 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1985 |
have con_derf: "continuous (at z) (deriv f)" if "z \<in> U" for z |
72266 | 1986 |
proof (rule continuous_on_interior) |
1987 |
show "continuous_on U (deriv f)" |
|
1988 |
by (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on \<open>open U\<close>) |
|
1989 |
qed (simp add: interior_open that \<open>open U\<close>) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1990 |
have tendsto_f': "((\<lambda>(x,y). if y = x then deriv f (x) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1991 |
else (f (y) - f (x)) / (y - x)) \<longlongrightarrow> deriv f x) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1992 |
(at (x, x) within U \<times> U)" if "x \<in> U" for x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1993 |
proof (rule Lim_withinI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1994 |
fix e::real assume "0 < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1995 |
obtain k1 where "k1>0" and k1: "\<And>x'. norm (x' - x) \<le> k1 \<Longrightarrow> norm (deriv f x' - deriv f x) < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1996 |
using \<open>0 < e\<close> continuous_within_E [OF con_derf [OF \<open>x \<in> U\<close>]] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1997 |
by (metis UNIV_I dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1998 |
obtain k2 where "k2>0" and k2: "ball x k2 \<subseteq> U" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1999 |
by (blast intro: openE [OF \<open>open U\<close>] \<open>x \<in> U\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2000 |
have neq: "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2001 |
if "z' \<noteq> x'" and less_k1: "norm (x'-x, z'-x) < k1" and less_k2: "norm (x'-x, z'-x) < k2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2002 |
for x' z' |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2003 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2004 |
have cs_less: "w \<in> closed_segment x' z' \<Longrightarrow> cmod (w - x) \<le> norm (x'-x, z'-x)" for w |
72266 | 2005 |
using segment_furthest_le [of w x' z' x] |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2006 |
by (metis (no_types) dist_commute dist_norm norm_fst_le norm_snd_le order_trans) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2007 |
have derf_le: "w \<in> closed_segment x' z' \<Longrightarrow> z' \<noteq> x' \<Longrightarrow> cmod (deriv f w - deriv f x) \<le> e" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2008 |
by (blast intro: cs_less less_k1 k1 [unfolded divide_const_simps dist_norm] less_imp_le le_less_trans) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2009 |
have f_has_der: "\<And>x. x \<in> U \<Longrightarrow> (f has_field_derivative deriv f x) (at x within U)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2010 |
by (metis DERIV_deriv_iff_field_differentiable at_within_open holf holomorphic_on_def \<open>open U\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2011 |
have "closed_segment x' z' \<subseteq> U" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2012 |
by (rule order_trans [OF _ k2]) (simp add: cs_less le_less_trans [OF _ less_k2] dist_complex_def norm_minus_commute subset_iff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2013 |
then have cint_derf: "(deriv f has_contour_integral f z' - f x') (linepath x' z')" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2014 |
using contour_integral_primitive [OF f_has_der valid_path_linepath] pasz by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2015 |
then have *: "((\<lambda>x. deriv f x / (z' - x')) has_contour_integral (f z' - f x') / (z' - x')) (linepath x' z')" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2016 |
by (rule has_contour_integral_div) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2017 |
have "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e/norm(z' - x') * norm(z' - x')" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2018 |
apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_diff [OF *]]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2019 |
using has_contour_integral_div [where c = "z' - x'", OF has_contour_integral_const_linepath [of "deriv f x" z' x']] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2020 |
\<open>e > 0\<close> \<open>z' \<noteq> x'\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2021 |
apply (auto simp: norm_divide divide_simps derf_le) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2022 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2023 |
also have "\<dots> \<le> e" using \<open>0 < e\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2024 |
finally show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2025 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2026 |
show "\<exists>d>0. \<forall>xa\<in>U \<times> U. |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2027 |
0 < dist xa (x, x) \<and> dist xa (x, x) < d \<longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2028 |
dist (case xa of (x, y) \<Rightarrow> if y = x then deriv f x else (f y - f x) / (y - x)) (deriv f x) \<le> e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2029 |
apply (rule_tac x="min k1 k2" in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2030 |
using \<open>k1>0\<close> \<open>k2>0\<close> \<open>e>0\<close> |
72266 | 2031 |
by (force simp: dist_norm neq intro: dual_order.strict_trans2 k1 less_imp_le norm_fst_le) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2032 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2033 |
have con_pa_f: "continuous_on (path_image \<gamma>) f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2034 |
by (meson holf holomorphic_on_imp_continuous_on holomorphic_on_subset interior_subset subt T) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2035 |
have le_B: "\<And>T. T \<in> {0..1} \<Longrightarrow> cmod (vector_derivative \<gamma> (at T)) \<le> B" |
72266 | 2036 |
using \<gamma>' B by (simp add: path_image_def vector_derivative_at rev_image_eqI) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2037 |
have f_has_cint: "\<And>w. w \<in> v - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f u / (u - w) ^ 1) has_contour_integral h w) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2038 |
by (simp add: V) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2039 |
have cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). d x y)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2040 |
apply (simp add: continuous_on_eq_continuous_within d_def continuous_within tendsto_f') |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2041 |
apply (simp add: tendsto_within_open_NO_MATCH open_Times \<open>open U\<close>, clarify) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2042 |
apply (rule Lim_transform_within_open [OF _ open_uu_Id, where f = "(\<lambda>(x,y). (f y - f x) / (y - x))"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2043 |
using con_ff |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2044 |
apply (auto simp: continuous_within) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2045 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2046 |
have hol_dw: "(\<lambda>z. d z w) holomorphic_on U" if "w \<in> U" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2047 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2048 |
have "continuous_on U ((\<lambda>(x,y). d x y) \<circ> (\<lambda>z. (w,z)))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2049 |
by (rule continuous_on_compose continuous_intros continuous_on_subset [OF cond_uu] | force intro: that)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2050 |
then have *: "continuous_on U (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2051 |
by (rule rev_iffD1 [OF _ continuous_on_cong [OF refl]]) (simp add: d_def field_simps) |
72266 | 2052 |
have **: "(\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z)) field_differentiable at x" |
2053 |
if "x \<in> U" "x \<noteq> w" for x |
|
2054 |
proof (rule_tac f = "\<lambda>x. (f w - f x)/(w - x)" and d = "dist x w" in field_differentiable_transform_within) |
|
2055 |
show "(\<lambda>x. (f w - f x) / (w - x)) field_differentiable at x" |
|
2056 |
using that \<open>open U\<close> |
|
2057 |
by (intro derivative_intros holomorphic_on_imp_differentiable_at [OF holf]; force) |
|
2058 |
qed (use that \<open>open U\<close> in \<open>auto simp: dist_commute\<close>) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2059 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2060 |
unfolding d_def |
72266 | 2061 |
proof (rule no_isolated_singularity [OF * _ \<open>open U\<close>]) |
2062 |
show "(\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z)) holomorphic_on U - {w}" |
|
2063 |
by (auto simp: field_differentiable_def [symmetric] holomorphic_on_open open_Diff \<open>open U\<close> **) |
|
2064 |
qed auto |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2065 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2066 |
{ fix a b |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2067 |
assume abu: "closed_segment a b \<subseteq> U" |
72266 | 2068 |
have cont_cint_d: "continuous_on U (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))" |
2069 |
proof (rule contour_integral_continuous_on_linepath_2D [OF \<open>open U\<close> _ _ abu]) |
|
2070 |
show "\<And>w. w \<in> U \<Longrightarrow> (\<lambda>z. d z w) contour_integrable_on (linepath a b)" |
|
2071 |
by (metis abu hol_dw continuous_on_subset contour_integrable_continuous_linepath holomorphic_on_imp_continuous_on) |
|
2072 |
show "continuous_on (U \<times> U) (\<lambda>(x, y). d y x)" |
|
2073 |
by (auto intro: continuous_on_swap_args cond_uu) |
|
2074 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2075 |
have cont_cint_d\<gamma>: "continuous_on {0..1} ((\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) \<circ> \<gamma>)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2076 |
proof (rule continuous_on_compose) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2077 |
show "continuous_on {0..1} \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2078 |
using \<open>path \<gamma>\<close> path_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2079 |
show "continuous_on (\<gamma> ` {0..1}) (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2080 |
using pasz unfolding path_image_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2081 |
by (auto intro!: continuous_on_subset [OF cont_cint_d]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2082 |
qed |
72266 | 2083 |
have "continuous_on {0..1} (\<lambda>x. vector_derivative \<gamma> (at x))" |
2084 |
using pf\<gamma>' by (simp add: continuous_on_polymonial_function vector_derivative_at [OF \<gamma>']) |
|
2085 |
then have cint_cint: "(\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) contour_integrable_on \<gamma>" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2086 |
apply (simp add: contour_integrable_on) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2087 |
apply (rule integrable_continuous_real) |
72266 | 2088 |
by (rule continuous_on_mult [OF cont_cint_d\<gamma> [unfolded o_def]]) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2089 |
have "contour_integral (linepath a b) h = contour_integral (linepath a b) (\<lambda>z. contour_integral \<gamma> (d z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2090 |
using abu by (force simp: h_def intro: contour_integral_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2091 |
also have "\<dots> = contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))" |
72266 | 2092 |
proof (rule contour_integral_swap) |
2093 |
show "continuous_on (path_image (linepath a b) \<times> path_image \<gamma>) (\<lambda>(y1, y2). d y1 y2)" |
|
2094 |
using abu pasz by (auto intro: continuous_on_subset [OF cond_uu]) |
|
2095 |
show "continuous_on {0..1} (\<lambda>t. vector_derivative (linepath a b) (at t))" |
|
2096 |
by (auto intro!: continuous_intros) |
|
2097 |
show "continuous_on {0..1} (\<lambda>t. vector_derivative \<gamma> (at t))" |
|
2098 |
by (metis \<gamma>' continuous_on_eq path_def path_polynomial_function pf\<gamma>' vector_derivative_at) |
|
2099 |
qed (use \<open>valid_path \<gamma>\<close> in auto) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2100 |
finally have cint_h_eq: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2101 |
"contour_integral (linepath a b) h = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2102 |
contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2103 |
note cint_cint cint_h_eq |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2104 |
} note cint_h = this |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2105 |
have conthu: "continuous_on U h" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2106 |
proof (simp add: continuous_on_sequentially, clarify) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2107 |
fix a x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2108 |
assume x: "x \<in> U" and au: "\<forall>n. a n \<in> U" and ax: "a \<longlonglongrightarrow> x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2109 |
then have A1: "\<forall>\<^sub>F n in sequentially. d (a n) contour_integrable_on \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2110 |
by (meson U contour_integrable_on_def eventuallyI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2111 |
obtain dd where "dd>0" and dd: "cball x dd \<subseteq> U" using open_contains_cball \<open>open U\<close> x by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2112 |
have A2: "uniform_limit (path_image \<gamma>) (\<lambda>n. d (a n)) (d x) sequentially" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2113 |
unfolding uniform_limit_iff dist_norm |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2114 |
proof clarify |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2115 |
fix ee::real |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2116 |
assume "0 < ee" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2117 |
show "\<forall>\<^sub>F n in sequentially. \<forall>\<xi>\<in>path_image \<gamma>. cmod (d (a n) \<xi> - d x \<xi>) < ee" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2118 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2119 |
let ?ddpa = "{(w,z) |w z. w \<in> cball x dd \<and> z \<in> path_image \<gamma>}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2120 |
have "uniformly_continuous_on ?ddpa (\<lambda>(x,y). d x y)" |
72266 | 2121 |
proof (rule compact_uniformly_continuous [OF continuous_on_subset[OF cond_uu]]) |
2122 |
show "compact {(w, z) |w z. w \<in> cball x dd \<and> z \<in> path_image \<gamma>}" |
|
2123 |
using \<open>valid_path \<gamma>\<close> |
|
2124 |
by (auto simp: compact_Times compact_valid_path_image simp del: mem_cball) |
|
2125 |
qed (use dd pasz in auto) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2126 |
then obtain kk where "kk>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2127 |
and kk: "\<And>x x'. \<lbrakk>x \<in> ?ddpa; x' \<in> ?ddpa; dist x' x < kk\<rbrakk> \<Longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2128 |
dist ((\<lambda>(x,y). d x y) x') ((\<lambda>(x,y). d x y) x) < ee" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2129 |
by (rule uniformly_continuous_onE [where e = ee]) (use \<open>0 < ee\<close> in auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2130 |
have kk: "\<lbrakk>norm (w - x) \<le> dd; z \<in> path_image \<gamma>; norm ((w, z) - (x, z)) < kk\<rbrakk> \<Longrightarrow> norm (d w z - d x z) < ee" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2131 |
for w z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2132 |
using \<open>dd>0\<close> kk [of "(x,z)" "(w,z)"] by (force simp: norm_minus_commute dist_norm) |
72266 | 2133 |
obtain no where "\<forall>n\<ge>no. dist (a n) x < min dd kk" |
2134 |
using ax unfolding lim_sequentially |
|
2135 |
by (meson \<open>0 < dd\<close> \<open>0 < kk\<close> min_less_iff_conj) |
|
2136 |
then show ?thesis |
|
2137 |
using \<open>dd > 0\<close> \<open>kk > 0\<close> by (fastforce simp: eventually_sequentially kk dist_norm) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2138 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2139 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2140 |
have "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> contour_integral \<gamma> (d x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2141 |
by (rule contour_integral_uniform_limit [OF A1 A2 le_B]) (auto simp: \<open>valid_path \<gamma>\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2142 |
then have tendsto_hx: "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> h x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2143 |
by (simp add: h_def x) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2144 |
then show "(h \<circ> a) \<longlonglongrightarrow> h x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2145 |
by (simp add: h_def x au o_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2146 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2147 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2148 |
proof (simp add: holomorphic_on_open field_differentiable_def [symmetric], clarify) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2149 |
fix z0 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2150 |
consider "z0 \<in> v" | "z0 \<in> U" using uv_Un by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2151 |
then show "h field_differentiable at z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2152 |
proof cases |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2153 |
assume "z0 \<in> v" then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2154 |
using Cauchy_next_derivative [OF con_pa_f le_B f_has_cint _ ov] V f_has_cint \<open>valid_path \<gamma>\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2155 |
by (auto simp: field_differentiable_def v_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2156 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2157 |
assume "z0 \<in> U" then |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2158 |
obtain e where "e>0" and e: "ball z0 e \<subseteq> U" by (blast intro: openE [OF \<open>open U\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2159 |
have *: "contour_integral (linepath a b) h + contour_integral (linepath b c) h + contour_integral (linepath c a) h = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2160 |
if abc_subset: "convex hull {a, b, c} \<subseteq> ball z0 e" for a b c |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2161 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2162 |
have *: "\<And>x1 x2 z. z \<in> U \<Longrightarrow> closed_segment x1 x2 \<subseteq> U \<Longrightarrow> (\<lambda>w. d w z) contour_integrable_on linepath x1 x2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2163 |
using hol_dw holomorphic_on_imp_continuous_on \<open>open U\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2164 |
by (auto intro!: contour_integrable_holomorphic_simple) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2165 |
have abc: "closed_segment a b \<subseteq> U" "closed_segment b c \<subseteq> U" "closed_segment c a \<subseteq> U" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2166 |
using that e segments_subset_convex_hull by fastforce+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2167 |
have eq0: "\<And>w. w \<in> U \<Longrightarrow> contour_integral (linepath a b +++ linepath b c +++ linepath c a) (\<lambda>z. d z w) = 0" |
72266 | 2168 |
proof (rule contour_integral_unique [OF Cauchy_theorem_triangle]) |
2169 |
show "\<And>w. w \<in> U \<Longrightarrow> (\<lambda>z. d z w) holomorphic_on convex hull {a, b, c}" |
|
2170 |
using e abc_subset by (auto intro: holomorphic_on_subset [OF hol_dw]) |
|
2171 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2172 |
have "contour_integral \<gamma> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2173 |
(\<lambda>x. contour_integral (linepath a b) (\<lambda>z. d z x) + |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2174 |
(contour_integral (linepath b c) (\<lambda>z. d z x) + |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2175 |
contour_integral (linepath c a) (\<lambda>z. d z x))) = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2176 |
apply (rule contour_integral_eq_0) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2177 |
using abc pasz U |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2178 |
apply (subst contour_integral_join [symmetric], auto intro: eq0 *)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2179 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2180 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2181 |
by (simp add: cint_h abc contour_integrable_add contour_integral_add [symmetric] add_ac) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2182 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2183 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2184 |
using e \<open>e > 0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2185 |
by (auto intro!: holomorphic_on_imp_differentiable_at [OF _ open_ball] analytic_imp_holomorphic |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2186 |
Morera_triangle continuous_on_subset [OF conthu] *) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2187 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2188 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2189 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2190 |
ultimately have [simp]: "h z = 0" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2191 |
by (meson Liouville_weak) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2192 |
have "((\<lambda>w. 1 / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2193 |
by (rule has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close> znot]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2194 |
then have "((\<lambda>w. f z * (1 / (w - z))) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2195 |
by (metis mult.commute has_contour_integral_lmul) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2196 |
then have 1: "((\<lambda>w. f z / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2197 |
by (simp add: field_split_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2198 |
moreover have 2: "((\<lambda>w. (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2199 |
using U [OF z] pasz d_def by (force elim: has_contour_integral_eq [where g = "\<lambda>w. (f w - f z)/(w - z)"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2200 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2201 |
using has_contour_integral_add [OF 1 2] by (simp add: diff_divide_distrib) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2202 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2203 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2204 |
theorem Cauchy_integral_formula_global: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2205 |
assumes S: "open S" and holf: "f holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2206 |
and z: "z \<in> S" and vpg: "valid_path \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2207 |
and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2208 |
and zero: "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2209 |
shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2210 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2211 |
have "path \<gamma>" using vpg by (blast intro: valid_path_imp_path) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2212 |
have hols: "(\<lambda>w. f w / (w - z)) holomorphic_on S - {z}" "(\<lambda>w. 1 / (w - z)) holomorphic_on S - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2213 |
by (rule holomorphic_intros holomorphic_on_subset [OF holf] | force)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2214 |
then have cint_fw: "(\<lambda>w. f w / (w - z)) contour_integrable_on \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2215 |
by (meson contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on open_delete S vpg pasz) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2216 |
obtain d where "d>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2217 |
and d: "\<And>g h. \<lbrakk>valid_path g; valid_path h; \<forall>t\<in>{0..1}. cmod (g t - \<gamma> t) < d \<and> cmod (h t - \<gamma> t) < d; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2218 |
pathstart h = pathstart g \<and> pathfinish h = pathfinish g\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2219 |
\<Longrightarrow> path_image h \<subseteq> S - {z} \<and> (\<forall>f. f holomorphic_on S - {z} \<longrightarrow> contour_integral h f = contour_integral g f)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2220 |
using contour_integral_nearby_ends [OF _ \<open>path \<gamma>\<close> pasz] S by (simp add: open_Diff) metis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2221 |
obtain p where polyp: "polynomial_function p" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2222 |
and ps: "pathstart p = pathstart \<gamma>" and pf: "pathfinish p = pathfinish \<gamma>" and led: "\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < d" |
72379 | 2223 |
using path_approx_polynomial_function [OF \<open>path \<gamma>\<close> \<open>d > 0\<close>] by metis |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2224 |
then have ploop: "pathfinish p = pathstart p" using loop by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2225 |
have vpp: "valid_path p" using polyp valid_path_polynomial_function by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2226 |
have [simp]: "z \<notin> path_image \<gamma>" using pasz by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2227 |
have paps: "path_image p \<subseteq> S - {z}" and cint_eq: "(\<And>f. f holomorphic_on S - {z} \<Longrightarrow> contour_integral p f = contour_integral \<gamma> f)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2228 |
using pf ps led d [OF vpg vpp] \<open>d > 0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2229 |
have wn_eq: "winding_number p z = winding_number \<gamma> z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2230 |
using vpp paps |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2231 |
by (simp add: subset_Diff_insert vpg valid_path_polynomial_function winding_number_valid_path cint_eq hols) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2232 |
have "winding_number p w = winding_number \<gamma> w" if "w \<notin> S" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2233 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2234 |
have hol: "(\<lambda>v. 1 / (v - w)) holomorphic_on S - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2235 |
using that by (force intro: holomorphic_intros holomorphic_on_subset [OF holf]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2236 |
have "w \<notin> path_image p" "w \<notin> path_image \<gamma>" using paps pasz that by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2237 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2238 |
using vpp vpg by (simp add: subset_Diff_insert valid_path_polynomial_function winding_number_valid_path cint_eq [OF hol]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2239 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2240 |
then have wn0: "\<And>w. w \<notin> S \<Longrightarrow> winding_number p w = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2241 |
by (simp add: zero) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2242 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2243 |
using Cauchy_integral_formula_global_weak [OF S holf z polyp paps ploop wn0] hols |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2244 |
by (metis wn_eq cint_eq has_contour_integral_eqpath cint_fw cint_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2245 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2246 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2247 |
theorem Cauchy_theorem_global: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2248 |
assumes S: "open S" and holf: "f holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2249 |
and vpg: "valid_path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2250 |
and pas: "path_image \<gamma> \<subseteq> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2251 |
and zero: "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2252 |
shows "(f has_contour_integral 0) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2253 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2254 |
obtain z where "z \<in> S" and znot: "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2255 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2256 |
have "compact (path_image \<gamma>)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2257 |
using compact_valid_path_image vpg by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2258 |
then have "path_image \<gamma> \<noteq> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2259 |
by (metis (no_types) compact_open path_image_nonempty S) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2260 |
with pas show ?thesis by (blast intro: that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2261 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2262 |
then have pasz: "path_image \<gamma> \<subseteq> S - {z}" using pas by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2263 |
have hol: "(\<lambda>w. (w - z) * f w) holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2264 |
by (rule holomorphic_intros holf)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2265 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2266 |
using Cauchy_integral_formula_global [OF S hol \<open>z \<in> S\<close> vpg pasz loop zero] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2267 |
by (auto simp: znot elim!: has_contour_integral_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2268 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2269 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2270 |
corollary Cauchy_theorem_global_outside: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2271 |
assumes "open S" "f holomorphic_on S" "valid_path \<gamma>" "pathfinish \<gamma> = pathstart \<gamma>" "path_image \<gamma> \<subseteq> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2272 |
"\<And>w. w \<notin> S \<Longrightarrow> w \<in> outside(path_image \<gamma>)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2273 |
shows "(f has_contour_integral 0) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2274 |
by (metis Cauchy_theorem_global assms winding_number_zero_in_outside valid_path_imp_path) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2275 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2276 |
lemma simply_connected_imp_winding_number_zero: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2277 |
assumes "simply_connected S" "path g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2278 |
"path_image g \<subseteq> S" "pathfinish g = pathstart g" "z \<notin> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2279 |
shows "winding_number g z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2280 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2281 |
have hom: "homotopic_loops S g (linepath (pathstart g) (pathstart g))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2282 |
by (meson assms homotopic_paths_imp_homotopic_loops pathfinish_linepath simply_connected_eq_contractible_path) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2283 |
then have "homotopic_paths (- {z}) g (linepath (pathstart g) (pathstart g))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2284 |
by (meson \<open>z \<notin> S\<close> homotopic_loops_imp_homotopic_paths_null homotopic_paths_subset subset_Compl_singleton) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2285 |
then have "winding_number g z = winding_number(linepath (pathstart g) (pathstart g)) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2286 |
by (rule winding_number_homotopic_paths) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2287 |
also have "\<dots> = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2288 |
using assms by (force intro: winding_number_trivial) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2289 |
finally show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2290 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2291 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2292 |
lemma Cauchy_theorem_simply_connected: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2293 |
assumes "open S" "simply_connected S" "f holomorphic_on S" "valid_path g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2294 |
"path_image g \<subseteq> S" "pathfinish g = pathstart g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2295 |
shows "(f has_contour_integral 0) g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2296 |
using assms |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2297 |
apply (simp add: simply_connected_eq_contractible_path) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2298 |
apply (auto intro!: Cauchy_theorem_null_homotopic [where a = "pathstart g"] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2299 |
homotopic_paths_imp_homotopic_loops) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2300 |
using valid_path_imp_path by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2301 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2302 |
proposition\<^marker>\<open>tag unimportant\<close> holomorphic_logarithm_exists: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2303 |
assumes A: "convex A" "open A" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2304 |
and f: "f holomorphic_on A" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2305 |
and z0: "z0 \<in> A" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2306 |
obtains g where "g holomorphic_on A" and "\<And>x. x \<in> A \<Longrightarrow> exp (g x) = f x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2307 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2308 |
note f' = holomorphic_derivI [OF f(1) A(2)] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2309 |
obtain g where g: "\<And>x. x \<in> A \<Longrightarrow> (g has_field_derivative deriv f x / f x) (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2310 |
proof (rule holomorphic_convex_primitive' [OF A]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2311 |
show "(\<lambda>x. deriv f x / f x) holomorphic_on A" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2312 |
by (intro holomorphic_intros f A) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2313 |
qed (auto simp: A at_within_open[of _ A]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2314 |
define h where "h = (\<lambda>x. -g z0 + ln (f z0) + g x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2315 |
from g and A have g_holo: "g holomorphic_on A" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2316 |
by (auto simp: holomorphic_on_def at_within_open[of _ A] field_differentiable_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2317 |
hence h_holo: "h holomorphic_on A" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2318 |
by (auto simp: h_def intro!: holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2319 |
have "\<exists>c. \<forall>x\<in>A. f x / exp (h x) - 1 = c" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2320 |
proof (rule has_field_derivative_zero_constant, goal_cases) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2321 |
case (2 x) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2322 |
note [simp] = at_within_open[OF _ \<open>open A\<close>] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2323 |
from 2 and z0 and f show ?case |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2324 |
by (auto simp: h_def exp_diff field_simps intro!: derivative_eq_intros g f') |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2325 |
qed fact+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2326 |
then obtain c where c: "\<And>x. x \<in> A \<Longrightarrow> f x / exp (h x) - 1 = c" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2327 |
by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2328 |
from c[OF z0] and z0 and f have "c = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2329 |
by (simp add: h_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2330 |
with c have "\<And>x. x \<in> A \<Longrightarrow> exp (h x) = f x" by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2331 |
from that[OF h_holo this] show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2332 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2333 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2334 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2335 |
(* FIXME mv to Cauchy_Integral_Theorem.thy *) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2336 |
subsection\<open>Cauchy's inequality and more versions of Liouville\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2337 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2338 |
lemma Cauchy_higher_deriv_bound: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2339 |
assumes holf: "f holomorphic_on (ball z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2340 |
and contf: "continuous_on (cball z r) f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2341 |
and fin : "\<And>w. w \<in> ball z r \<Longrightarrow> f w \<in> ball y B0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2342 |
and "0 < r" and "0 < n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2343 |
shows "norm ((deriv ^^ n) f z) \<le> (fact n) * B0 / r^n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2344 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2345 |
have "0 < B0" using \<open>0 < r\<close> fin [of z] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2346 |
by (metis ball_eq_empty ex_in_conv fin not_less) |
72266 | 2347 |
have le_B0: "cmod (f w - y) \<le> B0" if "cmod (w - z) \<le> r" for w |
2348 |
proof (rule continuous_on_closure_norm_le [of "ball z r" "\<lambda>w. f w - y"], use \<open>0 < r\<close> in simp_all) |
|
2349 |
show "continuous_on (cball z r) (\<lambda>w. f w - y)" |
|
2350 |
by (intro continuous_intros contf) |
|
2351 |
show "dist z w \<le> r" |
|
2352 |
by (simp add: dist_commute dist_norm that) |
|
2353 |
qed (use fin in \<open>auto simp: dist_norm less_eq_real_def norm_minus_commute\<close>) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2354 |
have "(deriv ^^ n) f z = (deriv ^^ n) (\<lambda>w. f w) z - (deriv ^^ n) (\<lambda>w. y) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2355 |
using \<open>0 < n\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2356 |
also have "... = (deriv ^^ n) (\<lambda>w. f w - y) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2357 |
by (rule higher_deriv_diff [OF holf, symmetric]) (auto simp: \<open>0 < r\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2358 |
finally have "(deriv ^^ n) f z = (deriv ^^ n) (\<lambda>w. f w - y) z" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2359 |
have contf': "continuous_on (cball z r) (\<lambda>u. f u - y)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2360 |
by (rule contf continuous_intros)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2361 |
have holf': "(\<lambda>u. (f u - y)) holomorphic_on (ball z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2362 |
by (simp add: holf holomorphic_on_diff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2363 |
define a where "a = (2 * pi)/(fact n)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2364 |
have "0 < a" by (simp add: a_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2365 |
have "B0/r^(Suc n)*2 * pi * r = a*((fact n)*B0/r^n)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2366 |
using \<open>0 < r\<close> by (simp add: a_def field_split_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2367 |
have der_dif: "(deriv ^^ n) (\<lambda>w. f w - y) z = (deriv ^^ n) f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2368 |
using \<open>0 < r\<close> \<open>0 < n\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2369 |
by (auto simp: higher_deriv_diff [OF holf holomorphic_on_const]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2370 |
have "norm ((2 * of_real pi * \<i>)/(fact n) * (deriv ^^ n) (\<lambda>w. f w - y) z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2371 |
\<le> (B0/r^(Suc n)) * (2 * pi * r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2372 |
apply (rule has_contour_integral_bound_circlepath [of "(\<lambda>u. (f u - y)/(u - z)^(Suc n))" _ z]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2373 |
using Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf' holf'] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2374 |
using \<open>0 < B0\<close> \<open>0 < r\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2375 |
apply (auto simp: norm_divide norm_mult norm_power divide_simps le_B0) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2376 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2377 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2378 |
using \<open>0 < r\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2379 |
by (auto simp: norm_divide norm_mult norm_power field_simps der_dif le_B0) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2380 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2381 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2382 |
lemma Cauchy_inequality: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2383 |
assumes holf: "f holomorphic_on (ball \<xi> r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2384 |
and contf: "continuous_on (cball \<xi> r) f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2385 |
and "0 < r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2386 |
and nof: "\<And>x. norm(\<xi>-x) = r \<Longrightarrow> norm(f x) \<le> B" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2387 |
shows "norm ((deriv ^^ n) f \<xi>) \<le> (fact n) * B / r^n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2388 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2389 |
obtain x where "norm (\<xi>-x) = r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2390 |
by (metis abs_of_nonneg add_diff_cancel_left' \<open>0 < r\<close> diff_add_cancel |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2391 |
dual_order.strict_implies_order norm_of_real) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2392 |
then have "0 \<le> B" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2393 |
by (metis nof norm_not_less_zero not_le order_trans) |
72266 | 2394 |
have "\<xi> \<in> ball \<xi> r" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2395 |
using \<open>0 < r\<close> by simp |
72266 | 2396 |
then have "((\<lambda>u. f u / (u-\<xi>) ^ Suc n) has_contour_integral (2 * pi) * \<i> / fact n * (deriv ^^ n) f \<xi>) |
2397 |
(circlepath \<xi> r)" |
|
2398 |
by (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf]) |
|
2399 |
have "norm ((2 * pi * \<i>)/(fact n) * (deriv ^^ n) f \<xi>) \<le> (B / r^(Suc n)) * (2 * pi * r)" |
|
2400 |
proof (rule has_contour_integral_bound_circlepath) |
|
2401 |
have "\<xi> \<in> ball \<xi> r" |
|
2402 |
using \<open>0 < r\<close> by simp |
|
2403 |
then show "((\<lambda>u. f u / (u-\<xi>) ^ Suc n) has_contour_integral (2 * pi) * \<i> / fact n * (deriv ^^ n) f \<xi>) |
|
2404 |
(circlepath \<xi> r)" |
|
2405 |
by (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf]) |
|
2406 |
show "\<And>x. cmod (x-\<xi>) = r \<Longrightarrow> cmod (f x / (x-\<xi>) ^ Suc n) \<le> B / r ^ Suc n" |
|
2407 |
using \<open>0 \<le> B\<close> \<open>0 < r\<close> |
|
2408 |
by (simp add: norm_divide norm_power nof frac_le norm_minus_commute del: power_Suc) |
|
2409 |
qed (use \<open>0 \<le> B\<close> \<open>0 < r\<close> in auto) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2410 |
then show ?thesis using \<open>0 < r\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2411 |
by (simp add: norm_divide norm_mult field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2412 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2413 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2414 |
lemma Liouville_polynomial: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2415 |
assumes holf: "f holomorphic_on UNIV" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2416 |
and nof: "\<And>z. A \<le> norm z \<Longrightarrow> norm(f z) \<le> B * norm z ^ n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2417 |
shows "f \<xi> = (\<Sum>k\<le>n. (deriv^^k) f 0 / fact k * \<xi> ^ k)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2418 |
proof (cases rule: le_less_linear [THEN disjE]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2419 |
assume "B \<le> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2420 |
then have "\<And>z. A \<le> norm z \<Longrightarrow> norm(f z) = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2421 |
by (metis nof less_le_trans zero_less_mult_iff neqE norm_not_less_zero norm_power not_le) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2422 |
then have f0: "(f \<longlongrightarrow> 0) at_infinity" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2423 |
using Lim_at_infinity by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2424 |
then have [simp]: "f = (\<lambda>w. 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2425 |
using Liouville_weak [OF holf, of 0] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2426 |
by (simp add: eventually_at_infinity f0) meson |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2427 |
show ?thesis by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2428 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2429 |
assume "0 < B" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2430 |
have "((\<lambda>k. (deriv ^^ k) f 0 / (fact k) * (\<xi> - 0)^k) sums f \<xi>)" |
72266 | 2431 |
proof (rule holomorphic_power_series [where r = "norm \<xi> + 1"]) |
2432 |
show "f holomorphic_on ball 0 (cmod \<xi> + 1)" "\<xi> \<in> ball 0 (cmod \<xi> + 1)" |
|
2433 |
using holf holomorphic_on_subset by auto |
|
2434 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2435 |
then have sumsf: "((\<lambda>k. (deriv ^^ k) f 0 / (fact k) * \<xi>^k) sums f \<xi>)" by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2436 |
have "(deriv ^^ k) f 0 / fact k * \<xi> ^ k = 0" if "k>n" for k |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2437 |
proof (cases "(deriv ^^ k) f 0 = 0") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2438 |
case True then show ?thesis by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2439 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2440 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2441 |
define w where "w = complex_of_real (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2442 |
have "1 \<le> abs (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2443 |
using \<open>0 < B\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2444 |
then have wge1: "1 \<le> norm w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2445 |
by (metis norm_of_real w_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2446 |
then have "w \<noteq> 0" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2447 |
have kB: "0 < fact k * B" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2448 |
using \<open>0 < B\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2449 |
then have "0 \<le> fact k * B / cmod ((deriv ^^ k) f 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2450 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2451 |
then have wgeA: "A \<le> cmod w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2452 |
by (simp only: w_def norm_of_real) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2453 |
have "fact k * B / cmod ((deriv ^^ k) f 0) < abs (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2454 |
using \<open>0 < B\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2455 |
then have wge: "fact k * B / cmod ((deriv ^^ k) f 0) < norm w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2456 |
by (metis norm_of_real w_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2457 |
then have "fact k * B / norm w < cmod ((deriv ^^ k) f 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2458 |
using False by (simp add: field_split_simps mult.commute split: if_split_asm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2459 |
also have "... \<le> fact k * (B * norm w ^ n) / norm w ^ k" |
72266 | 2460 |
proof (rule Cauchy_inequality) |
2461 |
show "f holomorphic_on ball 0 (cmod w)" |
|
2462 |
using holf holomorphic_on_subset by force |
|
2463 |
show "continuous_on (cball 0 (cmod w)) f" |
|
2464 |
using holf holomorphic_on_imp_continuous_on holomorphic_on_subset by blast |
|
2465 |
show "\<And>x. cmod (0 - x) = cmod w \<Longrightarrow> cmod (f x) \<le> B * cmod w ^ n" |
|
2466 |
by (metis nof wgeA dist_0_norm dist_norm) |
|
2467 |
qed (use \<open>w \<noteq> 0\<close> in auto) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2468 |
also have "... = fact k * (B * 1 / cmod w ^ (k-n))" |
72266 | 2469 |
proof - |
2470 |
have "cmod w ^ n / cmod w ^ k = 1 / cmod w ^ (k - n)" |
|
2471 |
using \<open>k>n\<close> \<open>w \<noteq> 0\<close> \<open>0 < B\<close> by (simp add: field_split_simps semiring_normalization_rules) |
|
2472 |
then show ?thesis |
|
2473 |
by (metis times_divide_eq_right) |
|
2474 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2475 |
also have "... = fact k * B / cmod w ^ (k-n)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2476 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2477 |
finally have "fact k * B / cmod w < fact k * B / cmod w ^ (k - n)" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2478 |
then have "1 / cmod w < 1 / cmod w ^ (k - n)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2479 |
by (metis kB divide_inverse inverse_eq_divide mult_less_cancel_left_pos) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2480 |
then have "cmod w ^ (k - n) < cmod w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2481 |
by (metis frac_le le_less_trans norm_ge_zero norm_one not_less order_refl wge1 zero_less_one) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2482 |
with self_le_power [OF wge1] have False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2483 |
by (meson diff_is_0_eq not_gr0 not_le that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2484 |
then show ?thesis by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2485 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2486 |
then have "(deriv ^^ (k + Suc n)) f 0 / fact (k + Suc n) * \<xi> ^ (k + Suc n) = 0" for k |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2487 |
using not_less_eq by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2488 |
then have "(\<lambda>i. (deriv ^^ (i + Suc n)) f 0 / fact (i + Suc n) * \<xi> ^ (i + Suc n)) sums 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2489 |
by (rule sums_0) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2490 |
with sums_split_initial_segment [OF sumsf, where n = "Suc n"] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2491 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2492 |
using atLeast0AtMost lessThan_Suc_atMost sums_unique2 by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2493 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2494 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2495 |
text\<open>Every bounded entire function is a constant function.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2496 |
theorem Liouville_theorem: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2497 |
assumes holf: "f holomorphic_on UNIV" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2498 |
and bf: "bounded (range f)" |
73790
370ce138d1bd
nicer statement of Liouville_theorem
paulson <lp15@cam.ac.uk>
parents:
72379
diff
changeset
|
2499 |
shows "f constant_on UNIV" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2500 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2501 |
obtain B where "\<And>z. cmod (f z) \<le> B" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2502 |
by (meson bf bounded_pos rangeI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2503 |
then show ?thesis |
73790
370ce138d1bd
nicer statement of Liouville_theorem
paulson <lp15@cam.ac.uk>
parents:
72379
diff
changeset
|
2504 |
using Liouville_polynomial [OF holf, of 0 B 0, simplified] |
370ce138d1bd
nicer statement of Liouville_theorem
paulson <lp15@cam.ac.uk>
parents:
72379
diff
changeset
|
2505 |
by (meson constant_on_def) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2506 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2507 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2508 |
text\<open>A holomorphic function f has only isolated zeros unless f is 0.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2509 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2510 |
lemma powser_0_nonzero: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2511 |
fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2512 |
assumes r: "0 < r" |
72266 | 2513 |
and sm: "\<And>x. norm (x-\<xi>) < r \<Longrightarrow> (\<lambda>n. a n * (x-\<xi>) ^ n) sums (f x)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2514 |
and [simp]: "f \<xi> = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2515 |
and m0: "a m \<noteq> 0" and "m>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2516 |
obtains s where "0 < s" and "\<And>z. z \<in> cball \<xi> s - {\<xi>} \<Longrightarrow> f z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2517 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2518 |
have "r \<le> conv_radius a" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2519 |
using sm sums_summable by (auto simp: le_conv_radius_iff [where \<xi>=\<xi>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2520 |
obtain m where am: "a m \<noteq> 0" and az [simp]: "(\<And>n. n<m \<Longrightarrow> a n = 0)" |
72266 | 2521 |
proof |
2522 |
show "a (LEAST n. a n \<noteq> 0) \<noteq> 0" |
|
2523 |
by (metis (mono_tags, lifting) m0 LeastI) |
|
2524 |
qed (fastforce dest!: not_less_Least) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2525 |
define b where "b i = a (i+m) / a m" for i |
72266 | 2526 |
define g where "g x = suminf (\<lambda>i. b i * (x-\<xi>) ^ i)" for x |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2527 |
have [simp]: "b 0 = 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2528 |
by (simp add: am b_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2529 |
{ fix x::'a |
72266 | 2530 |
assume "norm (x-\<xi>) < r" |
2531 |
then have "(\<lambda>n. (a m * (x-\<xi>)^m) * (b n * (x-\<xi>)^n)) sums (f x)" |
|
2532 |
using am az sm sums_zero_iff_shift [of m "(\<lambda>n. a n * (x-\<xi>) ^ n)" "f x"] |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2533 |
by (simp add: b_def monoid_mult_class.power_add algebra_simps) |
72266 | 2534 |
then have "x \<noteq> \<xi> \<Longrightarrow> (\<lambda>n. b n * (x-\<xi>)^n) sums (f x / (a m * (x-\<xi>)^m))" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2535 |
using am by (simp add: sums_mult_D) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2536 |
} note bsums = this |
72266 | 2537 |
then have "norm (x-\<xi>) < r \<Longrightarrow> summable (\<lambda>n. b n * (x-\<xi>)^n)" for x |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2538 |
using sums_summable by (cases "x=\<xi>") auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2539 |
then have "r \<le> conv_radius b" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2540 |
by (simp add: le_conv_radius_iff [where \<xi>=\<xi>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2541 |
then have "r/2 < conv_radius b" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2542 |
using not_le order_trans r by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2543 |
then have "continuous_on (cball \<xi> (r/2)) g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2544 |
using powser_continuous_suminf [of "r/2" b \<xi>] by (simp add: g_def) |
72266 | 2545 |
then obtain s where "s>0" "\<And>x. \<lbrakk>norm (x-\<xi>) \<le> s; norm (x-\<xi>) \<le> r/2\<rbrakk> \<Longrightarrow> dist (g x) (g \<xi>) < 1/2" |
2546 |
proof (rule continuous_onE) |
|
2547 |
show "\<xi> \<in> cball \<xi> (r / 2)" "1/2 > (0::real)" |
|
2548 |
using r by auto |
|
2549 |
qed (auto simp: dist_commute dist_norm) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2550 |
moreover have "g \<xi> = 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2551 |
by (simp add: g_def) |
72266 | 2552 |
ultimately have gnz: "\<And>x. \<lbrakk>norm (x-\<xi>) \<le> s; norm (x-\<xi>) \<le> r/2\<rbrakk> \<Longrightarrow> (g x) \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2553 |
by fastforce |
72266 | 2554 |
have "f x \<noteq> 0" if "x \<noteq> \<xi>" "norm (x-\<xi>) \<le> s" "norm (x-\<xi>) \<le> r/2" for x |
2555 |
using bsums [of x] that gnz [of x] r sums_iff unfolding g_def by fastforce |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2556 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2557 |
apply (rule_tac s="min s (r/2)" in that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2558 |
using \<open>0 < r\<close> \<open>0 < s\<close> by (auto simp: dist_commute dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2559 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2560 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2561 |
subsection \<open>Complex functions and power series\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2562 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2563 |
text \<open> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2564 |
The following defines the power series expansion of a complex function at a given point |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2565 |
(assuming that it is analytic at that point). |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2566 |
\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2567 |
definition\<^marker>\<open>tag important\<close> fps_expansion :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex fps" where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2568 |
"fps_expansion f z0 = Abs_fps (\<lambda>n. (deriv ^^ n) f z0 / fact n)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2569 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2570 |
lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2571 |
fixes r :: ereal |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2572 |
assumes "f holomorphic_on eball z0 r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2573 |
shows conv_radius_fps_expansion: "fps_conv_radius (fps_expansion f z0) \<ge> r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2574 |
and eval_fps_expansion: "\<And>z. z \<in> eball z0 r \<Longrightarrow> eval_fps (fps_expansion f z0) (z - z0) = f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2575 |
and eval_fps_expansion': "\<And>z. norm z < r \<Longrightarrow> eval_fps (fps_expansion f z0) z = f (z0 + z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2576 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2577 |
have "(\<lambda>n. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2578 |
if "z \<in> ball z0 r'" "ereal r' < r" for z r' |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2579 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2580 |
from that(2) have "ereal r' \<le> r" by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2581 |
from assms(1) and this have "f holomorphic_on ball z0 r'" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2582 |
by (rule holomorphic_on_subset[OF _ ball_eball_mono]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2583 |
from holomorphic_power_series [OF this that(1)] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2584 |
show ?thesis by (simp add: fps_expansion_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2585 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2586 |
hence *: "(\<lambda>n. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2587 |
if "z \<in> eball z0 r" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2588 |
using that by (subst (asm) eball_conv_UNION_balls) blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2589 |
show "fps_conv_radius (fps_expansion f z0) \<ge> r" unfolding fps_conv_radius_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2590 |
proof (rule conv_radius_geI_ex) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2591 |
fix r' :: real assume r': "r' > 0" "ereal r' < r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2592 |
thus "\<exists>z. norm z = r' \<and> summable (\<lambda>n. fps_nth (fps_expansion f z0) n * z ^ n)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2593 |
using *[of "z0 + of_real r'"] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2594 |
by (intro exI[of _ "of_real r'"]) (auto simp: summable_def dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2595 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2596 |
show "eval_fps (fps_expansion f z0) (z - z0) = f z" if "z \<in> eball z0 r" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2597 |
using *[OF that] by (simp add: eval_fps_def sums_iff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2598 |
show "eval_fps (fps_expansion f z0) z = f (z0 + z)" if "ereal (norm z) < r" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2599 |
using *[of "z0 + z"] and that by (simp add: eval_fps_def sums_iff dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2600 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2601 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2602 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2603 |
text \<open> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2604 |
We can now show several more facts about power series expansions (at least in the complex case) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2605 |
with relative ease that would have been trickier without complex analysis. |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2606 |
\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2607 |
lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2608 |
fixes f :: "complex fps" and r :: ereal |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2609 |
assumes "\<And>z. ereal (norm z) < r \<Longrightarrow> eval_fps f z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2610 |
shows fps_conv_radius_inverse: "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2611 |
and eval_fps_inverse: "\<And>z. ereal (norm z) < fps_conv_radius f \<Longrightarrow> ereal (norm z) < r \<Longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2612 |
eval_fps (inverse f) z = inverse (eval_fps f z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2613 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2614 |
define R where "R = min (fps_conv_radius f) r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2615 |
have *: "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f) \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2616 |
(\<forall>z\<in>eball 0 (min (fps_conv_radius f) r). eval_fps (inverse f) z = inverse (eval_fps f z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2617 |
proof (cases "min r (fps_conv_radius f) > 0") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2618 |
case True |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2619 |
define f' where "f' = fps_expansion (\<lambda>z. inverse (eval_fps f z)) 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2620 |
have holo: "(\<lambda>z. inverse (eval_fps f z)) holomorphic_on eball 0 (min r (fps_conv_radius f))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2621 |
using assms by (intro holomorphic_intros) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2622 |
from holo have radius: "fps_conv_radius f' \<ge> min r (fps_conv_radius f)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2623 |
unfolding f'_def by (rule conv_radius_fps_expansion) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2624 |
have eval_f': "eval_fps f' z = inverse (eval_fps f z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2625 |
if "norm z < fps_conv_radius f" "norm z < r" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2626 |
using that unfolding f'_def by (subst eval_fps_expansion'[OF holo]) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2627 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2628 |
have "f * f' = 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2629 |
proof (rule eval_fps_eqD) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2630 |
from radius and True have "0 < min (fps_conv_radius f) (fps_conv_radius f')" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2631 |
by (auto simp: min_def split: if_splits) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2632 |
also have "\<dots> \<le> fps_conv_radius (f * f')" by (rule fps_conv_radius_mult) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2633 |
finally show "\<dots> > 0" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2634 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2635 |
from True have "R > 0" by (auto simp: R_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2636 |
hence "eventually (\<lambda>z. z \<in> eball 0 R) (nhds 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2637 |
by (intro eventually_nhds_in_open) (auto simp: zero_ereal_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2638 |
thus "eventually (\<lambda>z. eval_fps (f * f') z = eval_fps 1 z) (nhds 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2639 |
proof eventually_elim |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2640 |
case (elim z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2641 |
hence "eval_fps (f * f') z = eval_fps f z * eval_fps f' z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2642 |
using radius by (intro eval_fps_mult) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2643 |
(auto simp: R_def min_def split: if_splits intro: less_trans) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2644 |
also have "eval_fps f' z = inverse (eval_fps f z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2645 |
using elim by (intro eval_f') (auto simp: R_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2646 |
also from elim have "eval_fps f z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2647 |
by (intro assms) (auto simp: R_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2648 |
hence "eval_fps f z * inverse (eval_fps f z) = eval_fps 1 z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2649 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2650 |
finally show "eval_fps (f * f') z = eval_fps 1 z" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2651 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2652 |
qed simp_all |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2653 |
hence "f' = inverse f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2654 |
by (intro fps_inverse_unique [symmetric]) (simp_all add: mult_ac) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2655 |
with eval_f' and radius show ?thesis by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2656 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2657 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2658 |
hence *: "eball 0 R = {}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2659 |
by (intro eball_empty) (auto simp: R_def min_def split: if_splits) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2660 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2661 |
proof safe |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2662 |
from False have "min r (fps_conv_radius f) \<le> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2663 |
by (simp add: min_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2664 |
also have "0 \<le> fps_conv_radius (inverse f)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2665 |
by (simp add: fps_conv_radius_def conv_radius_nonneg) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2666 |
finally show "min r (fps_conv_radius f) \<le> \<dots>" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2667 |
qed (unfold * [unfolded R_def], auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2668 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2669 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2670 |
from * show "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f)" by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2671 |
from * show "eval_fps (inverse f) z = inverse (eval_fps f z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2672 |
if "ereal (norm z) < fps_conv_radius f" "ereal (norm z) < r" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2673 |
using that by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2674 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2675 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2676 |
lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2677 |
fixes f g :: "complex fps" and r :: ereal |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2678 |
defines "R \<equiv> Min {r, fps_conv_radius f, fps_conv_radius g}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2679 |
assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2680 |
assumes nz: "\<And>z. z \<in> eball 0 r \<Longrightarrow> eval_fps g z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2681 |
shows fps_conv_radius_divide': "fps_conv_radius (f / g) \<ge> R" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2682 |
and eval_fps_divide': |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2683 |
"ereal (norm z) < R \<Longrightarrow> eval_fps (f / g) z = eval_fps f z / eval_fps g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2684 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2685 |
from nz[of 0] and \<open>r > 0\<close> have nz': "fps_nth g 0 \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2686 |
by (auto simp: eval_fps_at_0 zero_ereal_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2687 |
have "R \<le> min r (fps_conv_radius g)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2688 |
by (auto simp: R_def intro: min.coboundedI2) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2689 |
also have "min r (fps_conv_radius g) \<le> fps_conv_radius (inverse g)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2690 |
by (intro fps_conv_radius_inverse assms) (auto simp: zero_ereal_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2691 |
finally have radius: "fps_conv_radius (inverse g) \<ge> R" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2692 |
have "R \<le> min (fps_conv_radius f) (fps_conv_radius (inverse g))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2693 |
by (intro radius min.boundedI) (auto simp: R_def intro: min.coboundedI1 min.coboundedI2) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2694 |
also have "\<dots> \<le> fps_conv_radius (f * inverse g)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2695 |
by (rule fps_conv_radius_mult) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2696 |
also have "f * inverse g = f / g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2697 |
by (intro fps_divide_unit [symmetric] nz') |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2698 |
finally show "fps_conv_radius (f / g) \<ge> R" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2699 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2700 |
assume z: "ereal (norm z) < R" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2701 |
have "eval_fps (f * inverse g) z = eval_fps f z * eval_fps (inverse g) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2702 |
using radius by (intro eval_fps_mult less_le_trans[OF z]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2703 |
(auto simp: R_def intro: min.coboundedI1 min.coboundedI2) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2704 |
also have "eval_fps (inverse g) z = inverse (eval_fps g z)" using \<open>r > 0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2705 |
by (intro eval_fps_inverse[where r = r] less_le_trans[OF z] nz) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2706 |
(auto simp: R_def intro: min.coboundedI1 min.coboundedI2) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2707 |
also have "f * inverse g = f / g" by fact |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2708 |
finally show "eval_fps (f / g) z = eval_fps f z / eval_fps g z" by (simp add: field_split_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2709 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2710 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2711 |
lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2712 |
fixes f g :: "complex fps" and r :: ereal |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2713 |
defines "R \<equiv> Min {r, fps_conv_radius f, fps_conv_radius g}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2714 |
assumes "subdegree g \<le> subdegree f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2715 |
assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2716 |
assumes "\<And>z. z \<in> eball 0 r \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> eval_fps g z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2717 |
shows fps_conv_radius_divide: "fps_conv_radius (f / g) \<ge> R" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2718 |
and eval_fps_divide: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2719 |
"ereal (norm z) < R \<Longrightarrow> c = fps_nth f (subdegree g) / fps_nth g (subdegree g) \<Longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2720 |
eval_fps (f / g) z = (if z = 0 then c else eval_fps f z / eval_fps g z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2721 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2722 |
define f' g' where "f' = fps_shift (subdegree g) f" and "g' = fps_shift (subdegree g) g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2723 |
have f_eq: "f = f' * fps_X ^ subdegree g" and g_eq: "g = g' * fps_X ^ subdegree g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2724 |
unfolding f'_def g'_def by (rule subdegree_decompose' le_refl | fact)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2725 |
have subdegree: "subdegree f' = subdegree f - subdegree g" "subdegree g' = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2726 |
using assms(2) by (simp_all add: f'_def g'_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2727 |
have [simp]: "fps_conv_radius f' = fps_conv_radius f" "fps_conv_radius g' = fps_conv_radius g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2728 |
by (simp_all add: f'_def g'_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2729 |
have [simp]: "fps_nth f' 0 = fps_nth f (subdegree g)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2730 |
"fps_nth g' 0 = fps_nth g (subdegree g)" by (simp_all add: f'_def g'_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2731 |
have g_nz: "g \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2732 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2733 |
define z :: complex where "z = (if r = \<infinity> then 1 else of_real (real_of_ereal r / 2))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2734 |
from \<open>r > 0\<close> have "z \<in> eball 0 r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2735 |
by (cases r) (auto simp: z_def eball_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2736 |
moreover have "z \<noteq> 0" using \<open>r > 0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2737 |
by (cases r) (auto simp: z_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2738 |
ultimately have "eval_fps g z \<noteq> 0" by (rule assms(6)) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2739 |
thus "g \<noteq> 0" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2740 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2741 |
have fg: "f / g = f' * inverse g'" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2742 |
by (subst f_eq, subst (2) g_eq) (insert g_nz, simp add: fps_divide_unit) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2743 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2744 |
have g'_nz: "eval_fps g' z \<noteq> 0" if z: "norm z < min r (fps_conv_radius g)" for z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2745 |
proof (cases "z = 0") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2746 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2747 |
with assms and z have "eval_fps g z \<noteq> 0" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2748 |
also from z have "eval_fps g z = eval_fps g' z * z ^ subdegree g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2749 |
by (subst g_eq) (auto simp: eval_fps_mult) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2750 |
finally show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2751 |
qed (insert \<open>g \<noteq> 0\<close>, auto simp: g'_def eval_fps_at_0) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2752 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2753 |
have "R \<le> min (min r (fps_conv_radius g)) (fps_conv_radius g')" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2754 |
by (auto simp: R_def min.coboundedI1 min.coboundedI2) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2755 |
also have "\<dots> \<le> fps_conv_radius (inverse g')" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2756 |
using g'_nz by (rule fps_conv_radius_inverse) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2757 |
finally have conv_radius_inv: "R \<le> fps_conv_radius (inverse g')" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2758 |
hence "R \<le> fps_conv_radius (f' * inverse g')" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2759 |
by (intro order.trans[OF _ fps_conv_radius_mult]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2760 |
(auto simp: R_def intro: min.coboundedI1 min.coboundedI2) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2761 |
thus "fps_conv_radius (f / g) \<ge> R" by (simp add: fg) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2762 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2763 |
fix z c :: complex assume z: "ereal (norm z) < R" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2764 |
assume c: "c = fps_nth f (subdegree g) / fps_nth g (subdegree g)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2765 |
show "eval_fps (f / g) z = (if z = 0 then c else eval_fps f z / eval_fps g z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2766 |
proof (cases "z = 0") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2767 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2768 |
from z and conv_radius_inv have "ereal (norm z) < fps_conv_radius (inverse g')" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2769 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2770 |
with z have "eval_fps (f / g) z = eval_fps f' z * eval_fps (inverse g') z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2771 |
unfolding fg by (subst eval_fps_mult) (auto simp: R_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2772 |
also have "eval_fps (inverse g') z = inverse (eval_fps g' z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2773 |
using z by (intro eval_fps_inverse[of "min r (fps_conv_radius g')"] g'_nz) (auto simp: R_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2774 |
also have "eval_fps f' z * \<dots> = eval_fps f z / eval_fps g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2775 |
using z False assms(2) by (simp add: f'_def g'_def eval_fps_shift R_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2776 |
finally show ?thesis using False by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2777 |
qed (simp_all add: eval_fps_at_0 fg field_simps c) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2778 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2779 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2780 |
lemma has_fps_expansion_fps_expansion [intro]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2781 |
assumes "open A" "0 \<in> A" "f holomorphic_on A" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2782 |
shows "f has_fps_expansion fps_expansion f 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2783 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2784 |
from assms(1,2) obtain r where r: "r > 0 " "ball 0 r \<subseteq> A" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2785 |
by (auto simp: open_contains_ball) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2786 |
have holo: "f holomorphic_on eball 0 (ereal r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2787 |
using r(2) and assms(3) by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2788 |
from r(1) have "0 < ereal r" by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2789 |
also have "r \<le> fps_conv_radius (fps_expansion f 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2790 |
using holo by (intro conv_radius_fps_expansion) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2791 |
finally have "\<dots> > 0" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2792 |
moreover have "eventually (\<lambda>z. z \<in> ball 0 r) (nhds 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2793 |
using r(1) by (intro eventually_nhds_in_open) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2794 |
hence "eventually (\<lambda>z. eval_fps (fps_expansion f 0) z = f z) (nhds 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2795 |
by eventually_elim (subst eval_fps_expansion'[OF holo], auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2796 |
ultimately show ?thesis using r(1) by (auto simp: has_fps_expansion_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2797 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2798 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2799 |
lemma fps_conv_radius_tan: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2800 |
fixes c :: complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2801 |
assumes "c \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2802 |
shows "fps_conv_radius (fps_tan c) \<ge> pi / (2 * norm c)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2803 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2804 |
have "fps_conv_radius (fps_tan c) \<ge> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2805 |
Min {pi / (2 * norm c), fps_conv_radius (fps_sin c), fps_conv_radius (fps_cos c)}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2806 |
unfolding fps_tan_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2807 |
proof (rule fps_conv_radius_divide) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2808 |
fix z :: complex assume "z \<in> eball 0 (pi / (2 * norm c))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2809 |
with cos_eq_zero_imp_norm_ge[of "c*z"] assms |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2810 |
show "eval_fps (fps_cos c) z \<noteq> 0" by (auto simp: norm_mult field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2811 |
qed (insert assms, auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2812 |
thus ?thesis by (simp add: min_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2813 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2814 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2815 |
lemma eval_fps_tan: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2816 |
fixes c :: complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2817 |
assumes "norm z < pi / (2 * norm c)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2818 |
shows "eval_fps (fps_tan c) z = tan (c * z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2819 |
proof (cases "c = 0") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2820 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2821 |
show ?thesis unfolding fps_tan_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2822 |
proof (subst eval_fps_divide'[where r = "pi / (2 * norm c)"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2823 |
fix z :: complex assume "z \<in> eball 0 (pi / (2 * norm c))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2824 |
with cos_eq_zero_imp_norm_ge[of "c*z"] assms |
72266 | 2825 |
show "eval_fps (fps_cos c) z \<noteq> 0" using False by (auto simp: norm_mult field_simps) |
72379 | 2826 |
qed (use False assms in \<open>auto simp: field_simps tan_def\<close>) |
72266 | 2827 |
qed simp_all |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2828 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2829 |
end |