author | wenzelm |
Mon, 11 Sep 2023 19:30:48 +0200 | |
changeset 78659 | b5f3d1051b13 |
parent 77322 | 9c295f84d55f |
child 78517 | 28c1f4f5335f |
permissions | -rw-r--r-- |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1 |
theory Complex_Singularities |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2 |
imports Conformal_Mappings |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
3 |
begin |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
4 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
5 |
subsection \<open>Non-essential singular points\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
6 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
7 |
definition\<^marker>\<open>tag important\<close> is_pole :: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
8 |
"('a::topological_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> bool" where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
9 |
"is_pole f a = (LIM x (at a). f x :> at_infinity)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
10 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
11 |
lemma is_pole_cong: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
12 |
assumes "eventually (\<lambda>x. f x = g x) (at a)" "a=b" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
13 |
shows "is_pole f a \<longleftrightarrow> is_pole g b" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
14 |
unfolding is_pole_def using assms by (intro filterlim_cong,auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
15 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
16 |
lemma is_pole_transform: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
17 |
assumes "is_pole f a" "eventually (\<lambda>x. f x = g x) (at a)" "a=b" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
18 |
shows "is_pole g b" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
19 |
using is_pole_cong assms by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
20 |
|
73795 | 21 |
lemma is_pole_shift_iff: |
22 |
fixes f :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector)" |
|
23 |
shows "is_pole (f \<circ> (+) d) a = is_pole f (a + d)" |
|
24 |
by (metis add_diff_cancel_right' filterlim_shift_iff is_pole_def) |
|
25 |
||
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
26 |
lemma is_pole_tendsto: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
27 |
fixes f::"('a::topological_space \<Rightarrow> 'b::real_normed_div_algebra)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
28 |
shows "is_pole f x \<Longrightarrow> ((inverse o f) \<longlongrightarrow> 0) (at x)" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
29 |
unfolding is_pole_def |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
30 |
by (auto simp add:filterlim_inverse_at_iff[symmetric] comp_def filterlim_at) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
31 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
32 |
lemma is_pole_shift_0: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
33 |
fixes f :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
34 |
shows "is_pole f z \<longleftrightarrow> is_pole (\<lambda>x. f (z + x)) 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
35 |
unfolding is_pole_def by (subst at_to_0) (auto simp: filterlim_filtermap add_ac) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
36 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
37 |
lemma is_pole_shift_0': |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
38 |
fixes f :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
39 |
shows "NO_MATCH 0 z \<Longrightarrow> is_pole f z \<longleftrightarrow> is_pole (\<lambda>x. f (z + x)) 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
40 |
by (metis is_pole_shift_0) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
41 |
|
77228
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
42 |
lemma is_pole_compose_iff: |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
43 |
assumes "filtermap g (at x) = (at y)" |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
44 |
shows "is_pole (f \<circ> g) x \<longleftrightarrow> is_pole f y" |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
45 |
unfolding is_pole_def filterlim_def filtermap_compose assms .. |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
46 |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
47 |
lemma is_pole_inverse_holomorphic: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
48 |
assumes "open s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
49 |
and f_holo:"f holomorphic_on (s-{z})" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
50 |
and pole:"is_pole f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
51 |
and non_z:"\<forall>x\<in>s-{z}. f x\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
52 |
shows "(\<lambda>x. if x=z then 0 else inverse (f x)) holomorphic_on s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
53 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
54 |
define g where "g \<equiv> \<lambda>x. if x=z then 0 else inverse (f x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
55 |
have "isCont g z" unfolding isCont_def using is_pole_tendsto[OF pole] |
72222 | 56 |
by (simp add: g_def cong: LIM_cong) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
57 |
moreover have "continuous_on (s-{z}) f" using f_holo holomorphic_on_imp_continuous_on by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
58 |
hence "continuous_on (s-{z}) (inverse o f)" unfolding comp_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
59 |
by (auto elim!:continuous_on_inverse simp add:non_z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
60 |
hence "continuous_on (s-{z}) g" unfolding g_def |
76895 | 61 |
using continuous_on_eq by fastforce |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
62 |
ultimately have "continuous_on s g" using open_delete[OF \<open>open s\<close>] \<open>open s\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
63 |
by (auto simp add:continuous_on_eq_continuous_at) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
64 |
moreover have "(inverse o f) holomorphic_on (s-{z})" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
65 |
unfolding comp_def using f_holo |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
66 |
by (auto elim!:holomorphic_on_inverse simp add:non_z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
67 |
hence "g holomorphic_on (s-{z})" |
76895 | 68 |
using g_def holomorphic_transform by force |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
69 |
ultimately show ?thesis unfolding g_def using \<open>open s\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
70 |
by (auto elim!: no_isolated_singularity) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
71 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
72 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
73 |
lemma not_is_pole_holomorphic: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
74 |
assumes "open A" "x \<in> A" "f holomorphic_on A" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
75 |
shows "\<not>is_pole f x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
76 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
77 |
have "continuous_on A f" by (intro holomorphic_on_imp_continuous_on) fact |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
78 |
with assms have "isCont f x" by (simp add: continuous_on_eq_continuous_at) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
79 |
hence "f \<midarrow>x\<rightarrow> f x" by (simp add: isCont_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
80 |
thus "\<not>is_pole f x" unfolding is_pole_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
81 |
using not_tendsto_and_filterlim_at_infinity[of "at x" f "f x"] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
82 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
83 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
84 |
lemma is_pole_inverse_power: "n > 0 \<Longrightarrow> is_pole (\<lambda>z::complex. 1 / (z - a) ^ n) a" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
85 |
unfolding is_pole_def inverse_eq_divide [symmetric] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
86 |
by (intro filterlim_compose[OF filterlim_inverse_at_infinity] tendsto_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
87 |
(auto simp: filterlim_at eventually_at intro!: exI[of _ 1] tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
88 |
|
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
89 |
lemma is_pole_cmult_iff [simp]: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
90 |
"c \<noteq> 0 \<Longrightarrow> is_pole (\<lambda>z. c * f z :: 'a :: real_normed_field) z \<longleftrightarrow> is_pole f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
91 |
proof |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
92 |
assume *: "c \<noteq> 0" "is_pole (\<lambda>z. c * f z) z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
93 |
have "is_pole (\<lambda>z. inverse c * (c * f z)) z" unfolding is_pole_def |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
94 |
by (rule tendsto_mult_filterlim_at_infinity tendsto_const)+ (use * in \<open>auto simp: is_pole_def\<close>) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
95 |
thus "is_pole f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
96 |
using *(1) by (simp add: field_simps) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
97 |
next |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
98 |
assume *: "c \<noteq> 0" "is_pole f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
99 |
show "is_pole (\<lambda>z. c * f z) z" unfolding is_pole_def |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
100 |
by (rule tendsto_mult_filterlim_at_infinity tendsto_const)+ (use * in \<open>auto simp: is_pole_def\<close>) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
101 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
102 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
103 |
lemma is_pole_uminus_iff [simp]: "is_pole (\<lambda>z. -f z :: 'a :: real_normed_field) z \<longleftrightarrow> is_pole f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
104 |
using is_pole_cmult_iff[of "-1" f] by (simp del: is_pole_cmult_iff) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
105 |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
106 |
lemma is_pole_inverse: "is_pole (\<lambda>z::complex. 1 / (z - a)) a" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
107 |
using is_pole_inverse_power[of 1 a] by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
108 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
109 |
lemma is_pole_divide: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
110 |
fixes f :: "'a :: t2_space \<Rightarrow> 'b :: real_normed_field" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
111 |
assumes "isCont f z" "filterlim g (at 0) (at z)" "f z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
112 |
shows "is_pole (\<lambda>z. f z / g z) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
113 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
114 |
have "filterlim (\<lambda>z. f z * inverse (g z)) at_infinity (at z)" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
115 |
using assms filterlim_compose filterlim_inverse_at_infinity isCont_def |
76895 | 116 |
tendsto_mult_filterlim_at_infinity by blast |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
117 |
thus ?thesis by (simp add: field_split_simps is_pole_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
118 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
119 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
120 |
lemma is_pole_basic: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
121 |
assumes "f holomorphic_on A" "open A" "z \<in> A" "f z \<noteq> 0" "n > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
122 |
shows "is_pole (\<lambda>w. f w / (w - z) ^ n) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
123 |
proof (rule is_pole_divide) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
124 |
have "continuous_on A f" by (rule holomorphic_on_imp_continuous_on) fact |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
125 |
with assms show "isCont f z" by (auto simp: continuous_on_eq_continuous_at) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
126 |
have "filterlim (\<lambda>w. (w - z) ^ n) (nhds 0) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
127 |
using assms by (auto intro!: tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
128 |
thus "filterlim (\<lambda>w. (w - z) ^ n) (at 0) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
129 |
by (intro filterlim_atI tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
130 |
(insert assms, auto simp: eventually_at_filter) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
131 |
qed fact+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
132 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
133 |
lemma is_pole_basic': |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
134 |
assumes "f holomorphic_on A" "open A" "0 \<in> A" "f 0 \<noteq> 0" "n > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
135 |
shows "is_pole (\<lambda>w. f w / w ^ n) 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
136 |
using is_pole_basic[of f A 0] assms by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
137 |
|
77226
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
138 |
lemma is_pole_compose: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
139 |
assumes "is_pole f w" "g \<midarrow>z\<rightarrow> w" "eventually (\<lambda>z. g z \<noteq> w) (at z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
140 |
shows "is_pole (\<lambda>x. f (g x)) z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
141 |
using assms(1) unfolding is_pole_def |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
142 |
by (rule filterlim_compose) (use assms in \<open>auto simp: filterlim_at\<close>) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
143 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
144 |
lemma is_pole_plus_const_iff: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
145 |
"is_pole f z \<longleftrightarrow> is_pole (\<lambda>x. f x + c) z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
146 |
proof |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
147 |
assume "is_pole f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
148 |
then have "filterlim f at_infinity (at z)" unfolding is_pole_def . |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
149 |
moreover have "((\<lambda>_. c) \<longlongrightarrow> c) (at z)" by auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
150 |
ultimately have " LIM x (at z). f x + c :> at_infinity" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
151 |
using tendsto_add_filterlim_at_infinity'[of f "at z"] by auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
152 |
then show "is_pole (\<lambda>x. f x + c) z" unfolding is_pole_def . |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
153 |
next |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
154 |
assume "is_pole (\<lambda>x. f x + c) z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
155 |
then have "filterlim (\<lambda>x. f x + c) at_infinity (at z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
156 |
unfolding is_pole_def . |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
157 |
moreover have "((\<lambda>_. -c) \<longlongrightarrow> -c) (at z)" by auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
158 |
ultimately have " LIM x (at z). f x :> at_infinity" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
159 |
using tendsto_add_filterlim_at_infinity'[of "(\<lambda>x. f x + c)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
160 |
"at z" "(\<lambda>_. - c)" "-c"] |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
161 |
by auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
162 |
then show "is_pole f z" unfolding is_pole_def . |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
163 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
164 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
165 |
lemma is_pole_minus_const_iff: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
166 |
"is_pole (\<lambda>x. f x - c) z \<longleftrightarrow> is_pole f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
167 |
using is_pole_plus_const_iff [of f z "-c"] by simp |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
168 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
169 |
lemma is_pole_alt: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
170 |
"is_pole f x = (\<forall>B>0. \<exists>U. open U \<and> x\<in>U \<and> (\<forall>y\<in>U. y\<noteq>x \<longrightarrow> norm (f y)\<ge>B))" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
171 |
unfolding is_pole_def |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
172 |
unfolding filterlim_at_infinity[of 0,simplified] eventually_at_topological |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
173 |
by auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
174 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
175 |
lemma is_pole_mult_analytic_nonzero1: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
176 |
assumes "is_pole g x" "f analytic_on {x}" "f x \<noteq> 0" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
177 |
shows "is_pole (\<lambda>x. f x * g x) x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
178 |
unfolding is_pole_def |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
179 |
proof (rule tendsto_mult_filterlim_at_infinity) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
180 |
show "f \<midarrow>x\<rightarrow> f x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
181 |
using assms by (simp add: analytic_at_imp_isCont isContD) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
182 |
qed (use assms in \<open>auto simp: is_pole_def\<close>) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
183 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
184 |
lemma is_pole_mult_analytic_nonzero2: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
185 |
assumes "is_pole f x" "g analytic_on {x}" "g x \<noteq> 0" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
186 |
shows "is_pole (\<lambda>x. f x * g x) x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
187 |
by (subst mult.commute, rule is_pole_mult_analytic_nonzero1) (use assms in auto) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
188 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
189 |
lemma is_pole_mult_analytic_nonzero1_iff: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
190 |
assumes "f analytic_on {x}" "f x \<noteq> 0" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
191 |
shows "is_pole (\<lambda>x. f x * g x) x \<longleftrightarrow> is_pole g x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
192 |
proof |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
193 |
assume "is_pole g x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
194 |
thus "is_pole (\<lambda>x. f x * g x) x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
195 |
by (intro is_pole_mult_analytic_nonzero1 assms) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
196 |
next |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
197 |
assume "is_pole (\<lambda>x. f x * g x) x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
198 |
hence "is_pole (\<lambda>x. inverse (f x) * (f x * g x)) x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
199 |
by (rule is_pole_mult_analytic_nonzero1) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
200 |
(use assms in \<open>auto intro!: analytic_intros\<close>) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
201 |
also have "?this \<longleftrightarrow> is_pole g x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
202 |
proof (rule is_pole_cong) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
203 |
have "eventually (\<lambda>x. f x \<noteq> 0) (at x)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
204 |
using assms by (simp add: analytic_at_neq_imp_eventually_neq) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
205 |
thus "eventually (\<lambda>x. inverse (f x) * (f x * g x) = g x) (at x)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
206 |
by eventually_elim auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
207 |
qed auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
208 |
finally show "is_pole g x" . |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
209 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
210 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
211 |
lemma is_pole_mult_analytic_nonzero2_iff: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
212 |
assumes "g analytic_on {x}" "g x \<noteq> 0" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
213 |
shows "is_pole (\<lambda>x. f x * g x) x \<longleftrightarrow> is_pole f x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
214 |
by (subst mult.commute, rule is_pole_mult_analytic_nonzero1_iff) (fact assms)+ |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
215 |
|
77228
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
216 |
lemma frequently_const_imp_not_is_pole: |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
217 |
fixes z :: "'a::first_countable_topology" |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
218 |
assumes "frequently (\<lambda>w. f w = c) (at z)" |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
219 |
shows "\<not> is_pole f z" |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
220 |
proof |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
221 |
assume "is_pole f z" |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
222 |
from assms have "z islimpt {w. f w = c}" |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
223 |
by (simp add: islimpt_conv_frequently_at) |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
224 |
then obtain g where g: "\<And>n. g n \<in> {w. f w = c} - {z}" "g \<longlonglongrightarrow> z" |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
225 |
unfolding islimpt_sequential by blast |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
226 |
then have "(f \<circ> g) \<longlonglongrightarrow> c" |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
227 |
by (simp add: tendsto_eventually) |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
228 |
moreover have *: "filterlim g (at z) sequentially" |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
229 |
using g by (auto simp: filterlim_at) |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
230 |
have "filterlim (f \<circ> g) at_infinity sequentially" |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
231 |
unfolding o_def by (rule filterlim_compose [OF _ *]) |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
232 |
(use \<open>is_pole f z\<close> in \<open>simp add: is_pole_def\<close>) |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
233 |
ultimately show False |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
234 |
using not_tendsto_and_filterlim_at_infinity trivial_limit_sequentially by blast |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
235 |
qed |
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
236 |
|
8c093a4b8ccf
Even more new material from Eberl and Li
paulson <lp15@cam.ac.uk>
parents:
77226
diff
changeset
|
237 |
text \<open>The proposition |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
238 |
\<^term>\<open>\<exists>x. ((f::complex\<Rightarrow>complex) \<longlongrightarrow> x) (at z) \<or> is_pole f z\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
239 |
can be interpreted as the complex function \<^term>\<open>f\<close> has a non-essential singularity at \<^term>\<open>z\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
240 |
(i.e. the singularity is either removable or a pole).\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
241 |
definition not_essential::"[complex \<Rightarrow> complex, complex] \<Rightarrow> bool" where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
242 |
"not_essential f z = (\<exists>x. f\<midarrow>z\<rightarrow>x \<or> is_pole f z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
243 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
244 |
definition isolated_singularity_at::"[complex \<Rightarrow> complex, complex] \<Rightarrow> bool" where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
245 |
"isolated_singularity_at f z = (\<exists>r>0. f analytic_on ball z r-{z})" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
246 |
|
77226
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
247 |
lemma not_essential_cong: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
248 |
assumes "eventually (\<lambda>x. f x = g x) (at z)" "z = z'" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
249 |
shows "not_essential f z \<longleftrightarrow> not_essential g z'" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
250 |
unfolding not_essential_def using assms filterlim_cong is_pole_cong by fastforce |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
251 |
|
77277
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
77228
diff
changeset
|
252 |
lemma not_essential_compose_iff: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
77228
diff
changeset
|
253 |
assumes "filtermap g (at z) = at z'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
77228
diff
changeset
|
254 |
shows "not_essential (f \<circ> g) z = not_essential f z'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
77228
diff
changeset
|
255 |
unfolding not_essential_def filterlim_def filtermap_compose assms is_pole_compose_iff[OF assms] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
77228
diff
changeset
|
256 |
by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
77228
diff
changeset
|
257 |
|
77226
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
258 |
lemma isolated_singularity_at_cong: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
259 |
assumes "eventually (\<lambda>x. f x = g x) (at z)" "z = z'" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
260 |
shows "isolated_singularity_at f z \<longleftrightarrow> isolated_singularity_at g z'" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
261 |
proof - |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
262 |
have "isolated_singularity_at g z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
263 |
if "isolated_singularity_at f z" "eventually (\<lambda>x. f x = g x) (at z)" for f g |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
264 |
proof - |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
265 |
from that(1) obtain r where r: "r > 0" "f analytic_on ball z r - {z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
266 |
by (auto simp: isolated_singularity_at_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
267 |
from that(2) obtain r' where r': "r' > 0" "\<forall>x\<in>ball z r'-{z}. f x = g x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
268 |
unfolding eventually_at_filter eventually_nhds_metric by (auto simp: dist_commute) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
269 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
270 |
have "f holomorphic_on ball z r - {z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
271 |
using r(2) by (subst (asm) analytic_on_open) auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
272 |
hence "f holomorphic_on ball z (min r r') - {z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
273 |
by (rule holomorphic_on_subset) auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
274 |
also have "?this \<longleftrightarrow> g holomorphic_on ball z (min r r') - {z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
275 |
using r' by (intro holomorphic_cong) auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
276 |
also have "\<dots> \<longleftrightarrow> g analytic_on ball z (min r r') - {z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
277 |
by (subst analytic_on_open) auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
278 |
finally show ?thesis |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
279 |
unfolding isolated_singularity_at_def |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
280 |
by (intro exI[of _ "min r r'"]) (use \<open>r > 0\<close> \<open>r' > 0\<close> in auto) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
281 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
282 |
from this[of f g] this[of g f] assms show ?thesis |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
283 |
by (auto simp: eq_commute) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
284 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
285 |
|
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
286 |
lemma removable_singularity: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
287 |
assumes "f holomorphic_on A - {x}" "open A" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
288 |
assumes "f \<midarrow>x\<rightarrow> c" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
289 |
shows "(\<lambda>y. if y = x then c else f y) holomorphic_on A" (is "?g holomorphic_on _") |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
290 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
291 |
have "continuous_on A ?g" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
292 |
unfolding continuous_on_def |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
293 |
proof |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
294 |
fix y assume y: "y \<in> A" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
295 |
show "(?g \<longlongrightarrow> ?g y) (at y within A)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
296 |
proof (cases "y = x") |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
297 |
case False |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
298 |
have "continuous_on (A - {x}) f" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
299 |
using assms(1) by (meson holomorphic_on_imp_continuous_on) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
300 |
hence "(f \<longlongrightarrow> ?g y) (at y within A - {x})" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
301 |
using False y by (auto simp: continuous_on_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
302 |
also have "?this \<longleftrightarrow> (?g \<longlongrightarrow> ?g y) (at y within A - {x})" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
303 |
by (intro filterlim_cong refl) (auto simp: eventually_at_filter) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
304 |
also have "at y within A - {x} = at y within A" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
305 |
using y assms False |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
306 |
by (intro at_within_nhd[of _ "A - {x}"]) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
307 |
finally show ?thesis . |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
308 |
next |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
309 |
case [simp]: True |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
310 |
have "f \<midarrow>x\<rightarrow> c" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
311 |
by fact |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
312 |
also have "?this \<longleftrightarrow> (?g \<longlongrightarrow> ?g y) (at y)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
313 |
by (intro filterlim_cong) (auto simp: eventually_at_filter) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
314 |
finally show ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
315 |
by (meson Lim_at_imp_Lim_at_within) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
316 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
317 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
318 |
moreover { |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
319 |
have "?g holomorphic_on A - {x}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
320 |
using assms(1) holomorphic_transform by fastforce |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
321 |
} |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
322 |
ultimately show ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
323 |
by (rule no_isolated_singularity) (use assms in auto) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
324 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
325 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
326 |
lemma removable_singularity': |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
327 |
assumes "isolated_singularity_at f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
328 |
assumes "f \<midarrow>z\<rightarrow> c" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
329 |
shows "(\<lambda>y. if y = z then c else f y) analytic_on {z}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
330 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
331 |
from assms obtain r where r: "r > 0" "f analytic_on ball z r - {z}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
332 |
by (auto simp: isolated_singularity_at_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
333 |
have "(\<lambda>y. if y = z then c else f y) holomorphic_on ball z r" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
334 |
proof (rule removable_singularity) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
335 |
show "f holomorphic_on ball z r - {z}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
336 |
using r(2) by (subst (asm) analytic_on_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
337 |
qed (use assms in auto) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
338 |
thus ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
339 |
using r(1) unfolding analytic_at by (intro exI[of _ "ball z r"]) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
340 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
341 |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
342 |
named_theorems singularity_intros "introduction rules for singularities" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
343 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
344 |
lemma holomorphic_factor_unique: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
345 |
fixes f::"complex \<Rightarrow> complex" and z::complex and r::real and m n::int |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
346 |
assumes "r>0" "g z\<noteq>0" "h z\<noteq>0" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
347 |
and asm:"\<forall>w\<in>ball z r-{z}. f w = g w * (w-z) powi n \<and> g w\<noteq>0 \<and> f w = h w * (w - z) powi m \<and> h w\<noteq>0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
348 |
and g_holo:"g holomorphic_on ball z r" and h_holo:"h holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
349 |
shows "n=m" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
350 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
351 |
have [simp]:"at z within ball z r \<noteq> bot" using \<open>r>0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
352 |
by (auto simp add:at_within_ball_bot_iff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
353 |
have False when "n>m" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
354 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
355 |
have "(h \<longlongrightarrow> 0) (at z within ball z r)" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
356 |
proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powi (n - m) * g w"]) |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
357 |
have "\<forall>w\<in>ball z r-{z}. h w = (w-z)powi(n-m) * g w" |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
358 |
using \<open>n>m\<close> asm \<open>r>0\<close> by (simp add: field_simps power_int_diff) force |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
359 |
then show "\<lbrakk>x' \<in> ball z r; 0 < dist x' z;dist x' z < r\<rbrakk> |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
360 |
\<Longrightarrow> (x' - z) powi (n - m) * g x' = h x'" for x' by auto |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
361 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
362 |
define F where "F \<equiv> at z within ball z r" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
363 |
define f' where "f' \<equiv> \<lambda>x. (x - z) powi (n-m)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
364 |
have "f' z=0" using \<open>n>m\<close> unfolding f'_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
365 |
moreover have "continuous F f'" unfolding f'_def F_def continuous_def |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
366 |
using \<open>n>m\<close> |
76895 | 367 |
by (auto simp add: Lim_ident_at intro!:tendsto_powr_complex_0 tendsto_eq_intros) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
368 |
ultimately have "(f' \<longlongrightarrow> 0) F" unfolding F_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
369 |
by (simp add: continuous_within) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
370 |
moreover have "(g \<longlongrightarrow> g z) F" |
77277
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
77228
diff
changeset
|
371 |
unfolding F_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
77228
diff
changeset
|
372 |
using \<open>r>0\<close> centre_in_ball continuous_on_def g_holo holomorphic_on_imp_continuous_on by blast |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
373 |
ultimately show " ((\<lambda>w. f' w * g w) \<longlongrightarrow> 0) F" using tendsto_mult by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
374 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
375 |
moreover have "(h \<longlongrightarrow> h z) (at z within ball z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
376 |
using holomorphic_on_imp_continuous_on[OF h_holo] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
377 |
by (auto simp add:continuous_on_def \<open>r>0\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
378 |
ultimately have "h z=0" by (auto intro!: tendsto_unique) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
379 |
thus False using \<open>h z\<noteq>0\<close> by auto |
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 |
moreover have False when "m>n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
382 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
383 |
have "(g \<longlongrightarrow> 0) (at z within ball z r)" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
384 |
proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powi (m - n) * h w"]) |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
385 |
have "\<forall>w\<in>ball z r -{z}. g w = (w-z) powi (m-n) * h w" using \<open>m>n\<close> asm |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
386 |
by (simp add:field_simps power_int_diff) force |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
387 |
then show "\<lbrakk>x' \<in> ball z r; 0 < dist x' z;dist x' z < r\<rbrakk> |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
388 |
\<Longrightarrow> (x' - z) powi (m - n) * h x' = g x'" for x' by auto |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
389 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
390 |
define F where "F \<equiv> at z within ball z r" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
391 |
define f' where "f' \<equiv>\<lambda>x. (x - z) powi (m-n)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
392 |
have "f' z=0" using \<open>m>n\<close> unfolding f'_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
393 |
moreover have "continuous F f'" unfolding f'_def F_def continuous_def |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
394 |
using \<open>m>n\<close> |
76895 | 395 |
by (auto simp: Lim_ident_at intro!:tendsto_powr_complex_0 tendsto_eq_intros) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
396 |
ultimately have "(f' \<longlongrightarrow> 0) F" unfolding F_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
397 |
by (simp add: continuous_within) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
398 |
moreover have "(h \<longlongrightarrow> h z) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
399 |
using holomorphic_on_imp_continuous_on[OF h_holo,unfolded continuous_on_def] \<open>r>0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
400 |
unfolding F_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
401 |
ultimately show " ((\<lambda>w. f' w * h w) \<longlongrightarrow> 0) F" using tendsto_mult by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
402 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
403 |
moreover have "(g \<longlongrightarrow> g z) (at z within ball z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
404 |
using holomorphic_on_imp_continuous_on[OF g_holo] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
405 |
by (auto simp add:continuous_on_def \<open>r>0\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
406 |
ultimately have "g z=0" by (auto intro!: tendsto_unique) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
407 |
thus False using \<open>g z\<noteq>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
408 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
409 |
ultimately show "n=m" by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
410 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
411 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
412 |
lemma holomorphic_factor_puncture: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
413 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
414 |
and "not_essential f z" \<comment> \<open>\<^term>\<open>f\<close> has either a removable singularity or a pole at \<^term>\<open>z\<close>\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
415 |
and non_zero:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" \<comment> \<open>\<^term>\<open>f\<close> will not be constantly zero in a neighbour of \<^term>\<open>z\<close>\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
416 |
shows "\<exists>!n::int. \<exists>g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0 |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
417 |
\<and> (\<forall>w\<in>cball z r-{z}. f w = g w * (w-z) powi n \<and> g w\<noteq>0)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
418 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
419 |
define P where "P = (\<lambda>f n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0 |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
420 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powi n \<and> g w\<noteq>0))" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
421 |
have imp_unique:"\<exists>!n::int. \<exists>g r. P f n g r" when "\<exists>n g r. P f n g r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
422 |
proof (rule ex_ex1I[OF that]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
423 |
fix n1 n2 :: int |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
424 |
assume g1_asm:"\<exists>g1 r1. P f n1 g1 r1" and g2_asm:"\<exists>g2 r2. P f n2 g2 r2" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
425 |
define fac where "fac \<equiv> \<lambda>n g r. \<forall>w\<in>cball z r-{z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
426 |
obtain g1 r1 where "0 < r1" and g1_holo: "g1 holomorphic_on cball z r1" and "g1 z\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
427 |
and "fac n1 g1 r1" using g1_asm unfolding P_def fac_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
428 |
obtain g2 r2 where "0 < r2" and g2_holo: "g2 holomorphic_on cball z r2" and "g2 z\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
429 |
and "fac n2 g2 r2" using g2_asm unfolding P_def fac_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
430 |
define r where "r \<equiv> min r1 r2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
431 |
have "r>0" using \<open>r1>0\<close> \<open>r2>0\<close> unfolding r_def by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
432 |
moreover have "\<forall>w\<in>ball z r-{z}. f w = g1 w * (w-z) powi n1 \<and> g1 w\<noteq>0 |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
433 |
\<and> f w = g2 w * (w - z) powi n2 \<and> g2 w\<noteq>0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
434 |
using \<open>fac n1 g1 r1\<close> \<open>fac n2 g2 r2\<close> unfolding fac_def r_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
435 |
by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
436 |
ultimately show "n1=n2" using g1_holo g2_holo \<open>g1 z\<noteq>0\<close> \<open>g2 z\<noteq>0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
437 |
apply (elim holomorphic_factor_unique) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
438 |
by (auto simp add:r_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
439 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
440 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
441 |
have P_exist:"\<exists> n g r. P h n g r" when |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
442 |
"\<exists>z'. (h \<longlongrightarrow> z') (at z)" "isolated_singularity_at h z" "\<exists>\<^sub>Fw in (at z). h w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
443 |
for h |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
444 |
proof - |
76895 | 445 |
from that(2) obtain r where "r>0" and r: "h analytic_on ball z r - {z}" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
446 |
unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
447 |
obtain z' where "(h \<longlongrightarrow> z') (at z)" using \<open>\<exists>z'. (h \<longlongrightarrow> z') (at z)\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
448 |
define h' where "h'=(\<lambda>x. if x=z then z' else h x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
449 |
have "h' holomorphic_on ball z r" |
76895 | 450 |
proof (rule no_isolated_singularity'[of "{z}"]) |
451 |
show "\<And>w. w \<in> {z} \<Longrightarrow> (h' \<longlongrightarrow> h' w) (at w within ball z r)" |
|
452 |
by (simp add: LIM_cong Lim_at_imp_Lim_at_within \<open>h \<midarrow>z\<rightarrow> z'\<close> h'_def) |
|
453 |
show "h' holomorphic_on ball z r - {z}" |
|
454 |
using r analytic_imp_holomorphic h'_def holomorphic_transform by fastforce |
|
455 |
qed auto |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
456 |
have ?thesis when "z'=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
457 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
458 |
have "h' z=0" using that unfolding h'_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
459 |
moreover have "\<not> h' constant_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
460 |
using \<open>\<exists>\<^sub>Fw in (at z). h w\<noteq>0\<close> unfolding constant_on_def frequently_def eventually_at h'_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
461 |
by (metis \<open>0 < r\<close> centre_in_ball dist_commute mem_ball that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
462 |
moreover note \<open>h' holomorphic_on ball z r\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
463 |
ultimately obtain g r1 n where "0 < n" "0 < r1" "ball z r1 \<subseteq> ball z r" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
464 |
g:"g holomorphic_on ball z r1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
465 |
"\<And>w. w \<in> ball z r1 \<Longrightarrow> h' w = (w - z) ^ n * g w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
466 |
"\<And>w. w \<in> ball z r1 \<Longrightarrow> g w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
467 |
using holomorphic_factor_zero_nonconstant[of _ "ball z r" z thesis,simplified, |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
468 |
OF \<open>h' holomorphic_on ball z r\<close> \<open>r>0\<close> \<open>h' z=0\<close> \<open>\<not> h' constant_on ball z r\<close>] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
469 |
by (auto simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
470 |
define rr where "rr=r1/2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
471 |
have "P h' n g rr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
472 |
unfolding P_def rr_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
473 |
using \<open>n>0\<close> \<open>r1>0\<close> g by (auto simp add:powr_nat) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
474 |
then have "P h n g rr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
475 |
unfolding h'_def P_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
476 |
then show ?thesis unfolding P_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
477 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
478 |
moreover have ?thesis when "z'\<noteq>0" |
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 |
have "h' z\<noteq>0" using that unfolding h'_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
481 |
obtain r1 where "r1>0" "cball z r1 \<subseteq> ball z r" "\<forall>x\<in>cball z r1. h' x\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
482 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
483 |
have "isCont h' z" "h' z\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
484 |
by (auto simp add: Lim_cong_within \<open>h \<midarrow>z\<rightarrow> z'\<close> \<open>z'\<noteq>0\<close> continuous_at h'_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
485 |
then obtain r2 where r2:"r2>0" "\<forall>x\<in>ball z r2. h' x\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
486 |
using continuous_at_avoid[of z h' 0 ] unfolding ball_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
487 |
define r1 where "r1=min r2 r / 2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
488 |
have "0 < r1" "cball z r1 \<subseteq> ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
489 |
using \<open>r2>0\<close> \<open>r>0\<close> unfolding r1_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
490 |
moreover have "\<forall>x\<in>cball z r1. h' x \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
491 |
using r2 unfolding r1_def by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
492 |
ultimately show ?thesis using that by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
493 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
494 |
then have "P h' 0 h' r1" using \<open>h' holomorphic_on ball z r\<close> unfolding P_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
495 |
then have "P h 0 h' r1" unfolding P_def h'_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
496 |
then show ?thesis unfolding P_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
497 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
498 |
ultimately show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
499 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
500 |
|
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
501 |
have ?thesis when "\<exists>x. (f \<longlongrightarrow> x) (at z)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
502 |
apply (rule_tac imp_unique[unfolded P_def]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
503 |
using P_exist[OF that(1) f_iso non_zero] unfolding P_def . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
504 |
moreover have ?thesis when "is_pole f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
505 |
proof (rule imp_unique[unfolded P_def]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
506 |
obtain e where [simp]:"e>0" and e_holo:"f holomorphic_on ball z e - {z}" and e_nz: "\<forall>x\<in>ball z e-{z}. f x\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
507 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
508 |
have "\<forall>\<^sub>F z in at z. f z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
509 |
using \<open>is_pole f z\<close> filterlim_at_infinity_imp_eventually_ne unfolding is_pole_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
510 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
511 |
then obtain e1 where e1:"e1>0" "\<forall>x\<in>ball z e1-{z}. f x\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
512 |
using that eventually_at[of "\<lambda>x. f x\<noteq>0" z UNIV,simplified] by (auto simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
513 |
obtain e2 where e2:"e2>0" "f holomorphic_on ball z e2 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
514 |
using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
515 |
show ?thesis |
76895 | 516 |
using e1 e2 by (force intro: that[of "min e1 e2"]) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
517 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
518 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
519 |
define h where "h \<equiv> \<lambda>x. inverse (f x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
520 |
have "\<exists>n g r. P h n g r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
521 |
proof - |
76895 | 522 |
have "(\<lambda>x. inverse (f x)) analytic_on ball z e - {z}" |
523 |
by (metis e_holo e_nz open_ball analytic_on_open holomorphic_on_inverse open_delete) |
|
524 |
moreover have "h \<midarrow>z\<rightarrow> 0" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
525 |
using Lim_transform_within_open assms(2) h_def is_pole_tendsto that by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
526 |
moreover have "\<exists>\<^sub>Fw in (at z). h w\<noteq>0" |
76895 | 527 |
using non_zero by (simp add: h_def) |
528 |
ultimately show ?thesis |
|
529 |
using P_exist[of h] \<open>e > 0\<close> |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
530 |
unfolding isolated_singularity_at_def h_def |
76895 | 531 |
by blast |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
532 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
533 |
then obtain n g r |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
534 |
where "0 < r" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
535 |
g_holo:"g holomorphic_on cball z r" and "g z\<noteq>0" and |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
536 |
g_fac:"(\<forall>w\<in>cball z r-{z}. h w = g w * (w - z) powi n \<and> g w \<noteq> 0)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
537 |
unfolding P_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
538 |
have "P f (-n) (inverse o g) r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
539 |
proof - |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
540 |
have "f w = inverse (g w) * (w - z) powi (- n)" when "w\<in>cball z r - {z}" for w |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
541 |
by (metis g_fac h_def inverse_inverse_eq inverse_mult_distrib power_int_minus that) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
542 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
543 |
unfolding P_def comp_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
544 |
using \<open>r>0\<close> g_holo g_fac \<open>g z\<noteq>0\<close> by (auto intro:holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
545 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
546 |
then show "\<exists>x g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z \<noteq> 0 |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
547 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powi x \<and> g w \<noteq> 0)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
548 |
unfolding P_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
549 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
550 |
ultimately show ?thesis using \<open>not_essential f z\<close> unfolding not_essential_def by presburger |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
551 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
552 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
553 |
lemma not_essential_transform: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
554 |
assumes "not_essential g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
555 |
assumes "\<forall>\<^sub>F w in (at z). g w = f w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
556 |
shows "not_essential f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
557 |
using assms unfolding not_essential_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
558 |
by (simp add: filterlim_cong is_pole_cong) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
559 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
560 |
lemma isolated_singularity_at_transform: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
561 |
assumes "isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
562 |
assumes "\<forall>\<^sub>F w in (at z). g w = f w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
563 |
shows "isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
564 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
565 |
obtain r1 where "r1>0" and r1:"g analytic_on ball z r1 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
566 |
using assms(1) unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
567 |
obtain r2 where "r2>0" and r2:" \<forall>x. x \<noteq> z \<and> dist x z < r2 \<longrightarrow> g x = f x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
568 |
using assms(2) unfolding eventually_at by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
569 |
define r3 where "r3=min r1 r2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
570 |
have "r3>0" unfolding r3_def using \<open>r1>0\<close> \<open>r2>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
571 |
moreover have "f analytic_on ball z r3 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
572 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
573 |
have "g holomorphic_on ball z r3 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
574 |
using r1 unfolding r3_def by (subst (asm) analytic_on_open,auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
575 |
then have "f holomorphic_on ball z r3 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
576 |
using r2 unfolding r3_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
577 |
by (auto simp add:dist_commute elim!:holomorphic_transform) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
578 |
then show ?thesis by (subst analytic_on_open,auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
579 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
580 |
ultimately show ?thesis unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
581 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
582 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
583 |
lemma not_essential_powr[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
584 |
assumes "LIM w (at z). f w :> (at x)" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
585 |
shows "not_essential (\<lambda>w. (f w) powi n) z" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
586 |
proof - |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
587 |
define fp where "fp=(\<lambda>w. (f w) powi n)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
588 |
have ?thesis when "n>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
589 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
590 |
have "(\<lambda>w. (f w) ^ (nat n)) \<midarrow>z\<rightarrow> x ^ nat n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
591 |
using that assms unfolding filterlim_at by (auto intro!:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
592 |
then have "fp \<midarrow>z\<rightarrow> x ^ nat n" unfolding fp_def |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
593 |
by (smt (verit) LIM_cong power_int_def that) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
594 |
then show ?thesis unfolding not_essential_def fp_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
595 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
596 |
moreover have ?thesis when "n=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
597 |
proof - |
76895 | 598 |
have "\<forall>\<^sub>F x in at z. fp x = 1" |
599 |
using that filterlim_at_within_not_equal[OF assms] by (auto simp: fp_def) |
|
600 |
then have "fp \<midarrow>z\<rightarrow> 1" |
|
601 |
by (simp add: tendsto_eventually) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
602 |
then show ?thesis unfolding fp_def not_essential_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
603 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
604 |
moreover have ?thesis when "n<0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
605 |
proof (cases "x=0") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
606 |
case True |
76895 | 607 |
have "(\<lambda>x. f x ^ nat (- n)) \<midarrow>z\<rightarrow> 0" |
608 |
using assms True that unfolding filterlim_at by (auto intro!:tendsto_eq_intros) |
|
609 |
moreover have "\<forall>\<^sub>F x in at z. f x ^ nat (- n) \<noteq> 0" |
|
610 |
by (smt (verit) True assms eventually_at_topological filterlim_at power_eq_0_iff) |
|
611 |
ultimately have "LIM w (at z). inverse ((f w) ^ (nat (-n))) :> at_infinity" |
|
612 |
by (metis filterlim_atI filterlim_compose filterlim_inverse_at_infinity) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
613 |
then have "LIM w (at z). fp w :> at_infinity" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
614 |
proof (elim filterlim_mono_eventually) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
615 |
show "\<forall>\<^sub>F x in at z. inverse (f x ^ nat (- n)) = fp x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
616 |
using filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
617 |
by (smt (verit) eventuallyI power_int_def power_inverse that) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
618 |
qed auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
619 |
then show ?thesis unfolding fp_def not_essential_def is_pole_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
620 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
621 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
622 |
let ?xx= "inverse (x ^ (nat (-n)))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
623 |
have "(\<lambda>w. inverse ((f w) ^ (nat (-n)))) \<midarrow>z\<rightarrow>?xx" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
624 |
using assms False unfolding filterlim_at by (auto intro!:tendsto_eq_intros) |
76895 | 625 |
then have "fp \<midarrow>z\<rightarrow> ?xx" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
626 |
by (smt (verit, best) LIM_cong fp_def power_int_def power_inverse that) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
627 |
then show ?thesis unfolding fp_def not_essential_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
628 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
629 |
ultimately show ?thesis by linarith |
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 isolated_singularity_at_powr[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
633 |
assumes "isolated_singularity_at f z" "\<forall>\<^sub>F w in (at z). f w\<noteq>0" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
634 |
shows "isolated_singularity_at (\<lambda>w. (f w) powi n) z" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
635 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
636 |
obtain r1 where "r1>0" "f analytic_on ball z r1 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
637 |
using assms(1) unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
638 |
then have r1:"f holomorphic_on ball z r1 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
639 |
using analytic_on_open[of "ball z r1-{z}" f] by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
640 |
obtain r2 where "r2>0" and r2:"\<forall>w. w \<noteq> z \<and> dist w z < r2 \<longrightarrow> f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
641 |
using assms(2) unfolding eventually_at by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
642 |
define r3 where "r3=min r1 r2" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
643 |
have "(\<lambda>w. (f w) powi n) holomorphic_on ball z r3 - {z}" |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
644 |
by (intro holomorphic_on_power_int) (use r1 r2 in \<open>auto simp: dist_commute r3_def\<close>) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
645 |
moreover have "r3>0" unfolding r3_def using \<open>0 < r1\<close> \<open>0 < r2\<close> by linarith |
76895 | 646 |
ultimately show ?thesis |
647 |
by (meson open_ball analytic_on_open isolated_singularity_at_def open_delete) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
648 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
649 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
650 |
lemma non_zero_neighbour: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
651 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
652 |
and f_ness:"not_essential f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
653 |
and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
654 |
shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
655 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
656 |
obtain fn fp fr where [simp]:"fp z \<noteq> 0" and "fr > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
657 |
and fr: "fp holomorphic_on cball z fr" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
658 |
"\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powi fn \<and> fp w \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
659 |
using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
660 |
have "f w \<noteq> 0" when " w \<noteq> z" "dist w z < fr" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
661 |
proof - |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
662 |
have "f w = fp w * (w - z) powi fn" "fp w \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
663 |
using fr(2)[rule_format, of w] using that by (auto simp add:dist_commute) |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
664 |
moreover have "(w - z) powi fn \<noteq>0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
665 |
unfolding powr_eq_0_iff using \<open>w\<noteq>z\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
666 |
ultimately show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
667 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
668 |
then show ?thesis using \<open>fr>0\<close> unfolding eventually_at by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
669 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
670 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
671 |
lemma non_zero_neighbour_pole: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
672 |
assumes "is_pole f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
673 |
shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
674 |
using assms filterlim_at_infinity_imp_eventually_ne[of f "at z" 0] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
675 |
unfolding is_pole_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
676 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
677 |
lemma non_zero_neighbour_alt: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
678 |
assumes holo: "f holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
679 |
and "open S" "connected S" "z \<in> S" "\<beta> \<in> S" "f \<beta> \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
680 |
shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0 \<and> w\<in>S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
681 |
proof (cases "f z = 0") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
682 |
case True |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
683 |
from isolated_zeros[OF holo \<open>open S\<close> \<open>connected S\<close> \<open>z \<in> S\<close> True \<open>\<beta> \<in> S\<close> \<open>f \<beta> \<noteq> 0\<close>] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
684 |
obtain r where "0 < r" "ball z r \<subseteq> S" "\<forall>w \<in> ball z r - {z}.f w \<noteq> 0" by metis |
76895 | 685 |
then show ?thesis |
686 |
by (smt (verit) open_ball centre_in_ball eventually_at_topological insertE insert_Diff subsetD) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
687 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
688 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
689 |
obtain r1 where r1:"r1>0" "\<forall>y. dist z y < r1 \<longrightarrow> f y \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
690 |
using continuous_at_avoid[of z f, OF _ False] assms(2,4) continuous_on_eq_continuous_at |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
691 |
holo holomorphic_on_imp_continuous_on by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
692 |
obtain r2 where r2:"r2>0" "ball z r2 \<subseteq> S" |
76895 | 693 |
using assms openE by blast |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
694 |
show ?thesis unfolding eventually_at |
76895 | 695 |
by (metis (no_types) dist_commute dual_order.strict_trans linorder_less_linear mem_ball r1 r2 subsetD) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
696 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
697 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
698 |
lemma not_essential_times[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
699 |
assumes f_ness:"not_essential f z" and g_ness:"not_essential g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
700 |
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
701 |
shows "not_essential (\<lambda>w. f w * g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
702 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
703 |
define fg where "fg = (\<lambda>w. f w * g w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
704 |
have ?thesis when "\<not> ((\<exists>\<^sub>Fw in (at z). f w\<noteq>0) \<and> (\<exists>\<^sub>Fw in (at z). g w\<noteq>0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
705 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
706 |
have "\<forall>\<^sub>Fw in (at z). fg w=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
707 |
using that[unfolded frequently_def, simplified] unfolding fg_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
708 |
by (auto elim: eventually_rev_mp) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
709 |
from tendsto_cong[OF this] have "fg \<midarrow>z\<rightarrow>0" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
710 |
then show ?thesis unfolding not_essential_def fg_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
711 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
712 |
moreover have ?thesis when f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" and g_nconst:"\<exists>\<^sub>Fw in (at z). g w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
713 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
714 |
obtain fn fp fr where [simp]:"fp z \<noteq> 0" and "fr > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
715 |
and fr: "fp holomorphic_on cball z fr" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
716 |
"\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powi fn \<and> fp w \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
717 |
using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
718 |
obtain gn gp gr where [simp]:"gp z \<noteq> 0" and "gr > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
719 |
and gr: "gp holomorphic_on cball z gr" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
720 |
"\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powi gn \<and> gp w \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
721 |
using holomorphic_factor_puncture[OF g_iso g_ness g_nconst,THEN ex1_implies_ex] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
722 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
723 |
define r1 where "r1=(min fr gr)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
724 |
have "r1>0" unfolding r1_def using \<open>fr>0\<close> \<open>gr>0\<close> by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
725 |
have fg_times:"fg w = (fp w * gp w) * (w - z) powi (fn+gn)" and fgp_nz:"fp w*gp w\<noteq>0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
726 |
when "w\<in>ball z r1 - {z}" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
727 |
proof - |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
728 |
have "f w = fp w * (w - z) powi fn" "fp w\<noteq>0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
729 |
using fr(2)[rule_format,of w] that unfolding r1_def by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
730 |
moreover have "g w = gp w * (w - z) powi gn" "gp w \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
731 |
using gr(2)[rule_format, of w] that unfolding r1_def by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
732 |
ultimately show "fg w = (fp w * gp w) * (w - z) powi (fn+gn)" "fp w*gp w\<noteq>0" |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
733 |
using that unfolding fg_def by (auto simp add:power_int_add) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
734 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
735 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
736 |
have [intro]: "fp \<midarrow>z\<rightarrow>fp z" "gp \<midarrow>z\<rightarrow>gp z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
737 |
using fr(1) \<open>fr>0\<close> gr(1) \<open>gr>0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
738 |
by (meson open_ball ball_subset_cball centre_in_ball |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
739 |
continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
740 |
holomorphic_on_subset)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
741 |
have ?thesis when "fn+gn>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
742 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
743 |
have "(\<lambda>w. (fp w * gp w) * (w - z) ^ (nat (fn+gn))) \<midarrow>z\<rightarrow>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
744 |
using that by (auto intro!:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
745 |
then have "fg \<midarrow>z\<rightarrow> 0" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
746 |
apply (elim Lim_transform_within[OF _ \<open>r1>0\<close>]) |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
747 |
by (smt (verit, best) Diff_iff dist_commute fg_times mem_ball power_int_def singletonD that zero_less_dist_iff) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
748 |
then show ?thesis unfolding not_essential_def fg_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
749 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
750 |
moreover have ?thesis when "fn+gn=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
751 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
752 |
have "(\<lambda>w. fp w * gp w) \<midarrow>z\<rightarrow>fp z*gp z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
753 |
using that by (auto intro!:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
754 |
then have "fg \<midarrow>z\<rightarrow> fp z*gp z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
755 |
apply (elim Lim_transform_within[OF _ \<open>r1>0\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
756 |
apply (subst fg_times) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
757 |
by (auto simp add:dist_commute that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
758 |
then show ?thesis unfolding not_essential_def fg_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
759 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
760 |
moreover have ?thesis when "fn+gn<0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
761 |
proof - |
76897 | 762 |
have "LIM x at z. (x - z) ^ nat (- (fn + gn)) :> at 0" |
763 |
using eventually_at_topological that |
|
764 |
by (force intro!: tendsto_eq_intros filterlim_atI) |
|
765 |
moreover have "\<exists>c. (\<lambda>c. fp c * gp c) \<midarrow>z\<rightarrow> c \<and> 0 \<noteq> c" |
|
766 |
using \<open>fp \<midarrow>z\<rightarrow> fp z\<close> \<open>gp \<midarrow>z\<rightarrow> gp z\<close> tendsto_mult by fastforce |
|
767 |
ultimately have "LIM w (at z). fp w * gp w / (w-z)^nat (-(fn+gn)) :> at_infinity" |
|
768 |
using filterlim_divide_at_infinity by blast |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
769 |
then have "is_pole fg z" unfolding is_pole_def |
76897 | 770 |
apply (elim filterlim_transform_within[OF _ _ \<open>r1>0\<close>]) |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
771 |
using that |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
772 |
by (simp_all add: dist_commute fg_times of_int_of_nat divide_simps power_int_def del: minus_add_distrib) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
773 |
then show ?thesis unfolding not_essential_def fg_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
774 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
775 |
ultimately show ?thesis unfolding not_essential_def fg_def by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
776 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
777 |
ultimately show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
778 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
779 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
780 |
lemma not_essential_inverse[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
781 |
assumes f_ness:"not_essential f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
782 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
783 |
shows "not_essential (\<lambda>w. inverse (f w)) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
784 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
785 |
define vf where "vf = (\<lambda>w. inverse (f w))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
786 |
have ?thesis when "\<not>(\<exists>\<^sub>Fw in (at z). f w\<noteq>0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
787 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
788 |
have "\<forall>\<^sub>Fw in (at z). f w=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
789 |
using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
790 |
then have "\<forall>\<^sub>Fw in (at z). vf w=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
791 |
unfolding vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
792 |
from tendsto_cong[OF this] have "vf \<midarrow>z\<rightarrow>0" unfolding vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
793 |
then show ?thesis unfolding not_essential_def vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
794 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
795 |
moreover have ?thesis when "is_pole f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
796 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
797 |
have "vf \<midarrow>z\<rightarrow>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
798 |
using that filterlim_at filterlim_inverse_at_iff unfolding is_pole_def vf_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
799 |
then show ?thesis unfolding not_essential_def vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
800 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
801 |
moreover have ?thesis when "\<exists>x. f\<midarrow>z\<rightarrow>x " and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
802 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
803 |
from that obtain fz where fz:"f\<midarrow>z\<rightarrow>fz" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
804 |
have ?thesis when "fz=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
805 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
806 |
have "(\<lambda>w. inverse (vf w)) \<midarrow>z\<rightarrow>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
807 |
using fz that unfolding vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
808 |
moreover have "\<forall>\<^sub>F w in at z. inverse (vf w) \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
809 |
using non_zero_neighbour[OF f_iso f_ness f_nconst] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
810 |
unfolding vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
811 |
ultimately have "is_pole vf z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
812 |
using filterlim_inverse_at_iff[of vf "at z"] unfolding filterlim_at is_pole_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
813 |
then show ?thesis unfolding not_essential_def vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
814 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
815 |
moreover have ?thesis when "fz\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
816 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
817 |
have "vf \<midarrow>z\<rightarrow>inverse fz" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
818 |
using fz that unfolding vf_def by (auto intro:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
819 |
then show ?thesis unfolding not_essential_def vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
820 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
821 |
ultimately show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
822 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
823 |
ultimately show ?thesis using f_ness unfolding not_essential_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
824 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
825 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
826 |
lemma isolated_singularity_at_inverse[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
827 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
828 |
and f_ness:"not_essential f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
829 |
shows "isolated_singularity_at (\<lambda>w. inverse (f w)) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
830 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
831 |
define vf where "vf = (\<lambda>w. inverse (f w))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
832 |
have ?thesis when "\<not>(\<exists>\<^sub>Fw in (at z). f w\<noteq>0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
833 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
834 |
have "\<forall>\<^sub>Fw in (at z). f w=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
835 |
using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
836 |
then have "\<forall>\<^sub>Fw in (at z). vf w=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
837 |
unfolding vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
838 |
then obtain d1 where "d1>0" and d1:"\<forall>x. x \<noteq> z \<and> dist x z < d1 \<longrightarrow> vf x = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
839 |
unfolding eventually_at by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
840 |
then have "vf holomorphic_on ball z d1-{z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
841 |
apply (rule_tac holomorphic_transform[of "\<lambda>_. 0"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
842 |
by (auto simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
843 |
then have "vf analytic_on ball z d1 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
844 |
by (simp add: analytic_on_open open_delete) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
845 |
then show ?thesis using \<open>d1>0\<close> unfolding isolated_singularity_at_def vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
846 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
847 |
moreover have ?thesis when f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
848 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
849 |
have "\<forall>\<^sub>F w in at z. f w \<noteq> 0" using non_zero_neighbour[OF f_iso f_ness f_nconst] . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
850 |
then obtain d1 where d1:"d1>0" "\<forall>x. x \<noteq> z \<and> dist x z < d1 \<longrightarrow> f x \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
851 |
unfolding eventually_at by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
852 |
obtain d2 where "d2>0" and d2:"f analytic_on ball z d2 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
853 |
using f_iso unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
854 |
define d3 where "d3=min d1 d2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
855 |
have "d3>0" unfolding d3_def using \<open>d1>0\<close> \<open>d2>0\<close> by auto |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
856 |
moreover |
76897 | 857 |
have "f analytic_on ball z d3 - {z}" |
858 |
by (smt (verit, best) Diff_iff analytic_on_analytic_at d2 d3_def mem_ball) |
|
859 |
then have "vf analytic_on ball z d3 - {z}" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
860 |
unfolding vf_def |
76897 | 861 |
by (intro analytic_on_inverse; simp add: d1(2) d3_def dist_commute) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
862 |
ultimately show ?thesis unfolding isolated_singularity_at_def vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
863 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
864 |
ultimately show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
865 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
866 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
867 |
lemma not_essential_divide[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
868 |
assumes f_ness:"not_essential f z" and g_ness:"not_essential g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
869 |
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
870 |
shows "not_essential (\<lambda>w. f w / g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
871 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
872 |
have "not_essential (\<lambda>w. f w * inverse (g w)) z" |
76897 | 873 |
by (simp add: f_iso f_ness g_iso g_ness isolated_singularity_at_inverse not_essential_inverse not_essential_times) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
874 |
then show ?thesis by (simp add:field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
875 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
876 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
877 |
lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
878 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
879 |
and g_iso:"isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
880 |
shows isolated_singularity_at_times[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
881 |
"isolated_singularity_at (\<lambda>w. f w * g w) z" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
882 |
isolated_singularity_at_add[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
883 |
"isolated_singularity_at (\<lambda>w. f w + g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
884 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
885 |
obtain d1 d2 where "d1>0" "d2>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
886 |
and d1:"f analytic_on ball z d1 - {z}" and d2:"g analytic_on ball z d2 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
887 |
using f_iso g_iso unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
888 |
define d3 where "d3=min d1 d2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
889 |
have "d3>0" unfolding d3_def using \<open>d1>0\<close> \<open>d2>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
890 |
|
76897 | 891 |
have fan: "f analytic_on ball z d3 - {z}" |
892 |
by (smt (verit, best) Diff_iff analytic_on_analytic_at d1 d3_def mem_ball) |
|
893 |
have gan: "g analytic_on ball z d3 - {z}" |
|
894 |
by (smt (verit, best) Diff_iff analytic_on_analytic_at d2 d3_def mem_ball) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
895 |
have "(\<lambda>w. f w * g w) analytic_on ball z d3 - {z}" |
76897 | 896 |
using analytic_on_mult fan gan by blast |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
897 |
then show "isolated_singularity_at (\<lambda>w. f w * g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
898 |
using \<open>d3>0\<close> unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
899 |
have "(\<lambda>w. f w + g w) analytic_on ball z d3 - {z}" |
76897 | 900 |
using analytic_on_add fan gan by blast |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
901 |
then show "isolated_singularity_at (\<lambda>w. f w + g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
902 |
using \<open>d3>0\<close> unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
903 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
904 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
905 |
lemma isolated_singularity_at_uminus[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
906 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
907 |
shows "isolated_singularity_at (\<lambda>w. - f w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
908 |
using assms unfolding isolated_singularity_at_def using analytic_on_neg by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
909 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
910 |
lemma isolated_singularity_at_id[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
911 |
"isolated_singularity_at (\<lambda>w. w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
912 |
unfolding isolated_singularity_at_def by (simp add: gt_ex) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
913 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
914 |
lemma isolated_singularity_at_minus[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
915 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
916 |
and g_iso:"isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
917 |
shows "isolated_singularity_at (\<lambda>w. f w - g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
918 |
using isolated_singularity_at_uminus[THEN isolated_singularity_at_add[OF f_iso,of "\<lambda>w. - g w"] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
919 |
,OF g_iso] by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
920 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
921 |
lemma isolated_singularity_at_divide[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
922 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
923 |
and g_iso:"isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
924 |
and g_ness:"not_essential g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
925 |
shows "isolated_singularity_at (\<lambda>w. f w / g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
926 |
using isolated_singularity_at_inverse[THEN isolated_singularity_at_times[OF f_iso, |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
927 |
of "\<lambda>w. inverse (g w)"],OF g_iso g_ness] by (simp add:field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
928 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
929 |
lemma isolated_singularity_at_const[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
930 |
"isolated_singularity_at (\<lambda>w. c) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
931 |
unfolding isolated_singularity_at_def by (simp add: gt_ex) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
932 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
933 |
lemma isolated_singularity_at_holomorphic: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
934 |
assumes "f holomorphic_on s-{z}" "open s" "z\<in>s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
935 |
shows "isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
936 |
using assms unfolding isolated_singularity_at_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
937 |
by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
938 |
|
77226
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
939 |
lemma isolated_singularity_at_altdef: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
940 |
"isolated_singularity_at f z \<longleftrightarrow> eventually (\<lambda>z. f analytic_on {z}) (at z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
941 |
proof |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
942 |
assume "isolated_singularity_at f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
943 |
then obtain r where r: "r > 0" "f analytic_on ball z r - {z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
944 |
unfolding isolated_singularity_at_def by blast |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
945 |
have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
946 |
using r(1) by (intro eventually_at_in_open) auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
947 |
thus "eventually (\<lambda>z. f analytic_on {z}) (at z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
948 |
by eventually_elim (use r analytic_on_subset in auto) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
949 |
next |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
950 |
assume "eventually (\<lambda>z. f analytic_on {z}) (at z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
951 |
then obtain A where A: "open A" "z \<in> A" "\<And>w. w \<in> A - {z} \<Longrightarrow> f analytic_on {w}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
952 |
unfolding eventually_at_topological by blast |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
953 |
then show "isolated_singularity_at f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
954 |
by (meson analytic_imp_holomorphic analytic_on_analytic_at isolated_singularity_at_holomorphic) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
955 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
956 |
|
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
957 |
lemma isolated_singularity_at_shift: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
958 |
assumes "isolated_singularity_at (\<lambda>x. f (x + w)) z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
959 |
shows "isolated_singularity_at f (z + w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
960 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
961 |
from assms obtain r where r: "r > 0" and ana: "(\<lambda>x. f (x + w)) analytic_on ball z r - {z}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
962 |
unfolding isolated_singularity_at_def by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
963 |
have "((\<lambda>x. f (x + w)) \<circ> (\<lambda>x. x - w)) analytic_on (ball (z + w) r - {z + w})" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
964 |
by (rule analytic_on_compose_gen[OF _ ana]) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
965 |
(auto simp: dist_norm algebra_simps intro!: analytic_intros) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
966 |
hence "f analytic_on (ball (z + w) r - {z + w})" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
967 |
by (simp add: o_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
968 |
thus ?thesis using r |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
969 |
unfolding isolated_singularity_at_def by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
970 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
971 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
972 |
lemma isolated_singularity_at_shift_iff: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
973 |
"isolated_singularity_at f (z + w) \<longleftrightarrow> isolated_singularity_at (\<lambda>x. f (x + w)) z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
974 |
using isolated_singularity_at_shift[of f w z] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
975 |
isolated_singularity_at_shift[of "\<lambda>x. f (x + w)" "-w" "w + z"] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
976 |
by (auto simp: algebra_simps) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
977 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
978 |
lemma isolated_singularity_at_shift_0: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
979 |
"NO_MATCH 0 z \<Longrightarrow> isolated_singularity_at f z \<longleftrightarrow> isolated_singularity_at (\<lambda>x. f (z + x)) 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
980 |
using isolated_singularity_at_shift_iff[of f 0 z] by (simp add: add_ac) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
981 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
982 |
lemma not_essential_shift: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
983 |
assumes "not_essential (\<lambda>x. f (x + w)) z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
984 |
shows "not_essential f (z + w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
985 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
986 |
from assms consider c where "(\<lambda>x. f (x + w)) \<midarrow>z\<rightarrow> c" | "is_pole (\<lambda>x. f (x + w)) z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
987 |
unfolding not_essential_def by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
988 |
thus ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
989 |
proof cases |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
990 |
case (1 c) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
991 |
hence "f \<midarrow>z + w\<rightarrow> c" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
992 |
by (smt (verit, ccfv_SIG) LIM_cong add.assoc filterlim_at_to_0) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
993 |
thus ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
994 |
by (auto simp: not_essential_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
995 |
next |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
996 |
case 2 |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
997 |
hence "is_pole f (z + w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
998 |
by (subst is_pole_shift_iff [symmetric]) (auto simp: o_def add_ac) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
999 |
thus ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1000 |
by (auto simp: not_essential_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1001 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1002 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1003 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1004 |
lemma not_essential_shift_iff: "not_essential f (z + w) \<longleftrightarrow> not_essential (\<lambda>x. f (x + w)) z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1005 |
using not_essential_shift[of f w z] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1006 |
not_essential_shift[of "\<lambda>x. f (x + w)" "-w" "w + z"] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1007 |
by (auto simp: algebra_simps) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1008 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1009 |
lemma not_essential_shift_0: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1010 |
"NO_MATCH 0 z \<Longrightarrow> not_essential f z \<longleftrightarrow> not_essential (\<lambda>x. f (z + x)) 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1011 |
using not_essential_shift_iff[of f 0 z] by (simp add: add_ac) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1012 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1013 |
lemma not_essential_holomorphic: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1014 |
assumes "f holomorphic_on A" "x \<in> A" "open A" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1015 |
shows "not_essential f x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1016 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1017 |
have "continuous_on A f" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1018 |
using assms holomorphic_on_imp_continuous_on by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1019 |
hence "f \<midarrow>x\<rightarrow> f x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1020 |
using assms continuous_on_eq_continuous_at isContD by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1021 |
thus ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1022 |
by (auto simp: not_essential_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1023 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1024 |
|
77226
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1025 |
lemma not_essential_analytic: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1026 |
assumes "f analytic_on {z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1027 |
shows "not_essential f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1028 |
using analytic_at assms not_essential_holomorphic by blast |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1029 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1030 |
lemma not_essential_id [singularity_intros]: "not_essential (\<lambda>w. w) z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1031 |
by (simp add: not_essential_analytic) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1032 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1033 |
lemma is_pole_imp_not_essential [intro]: "is_pole f z \<Longrightarrow> not_essential f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1034 |
by (auto simp: not_essential_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1035 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1036 |
lemma tendsto_imp_not_essential [intro]: "f \<midarrow>z\<rightarrow> c \<Longrightarrow> not_essential f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1037 |
by (auto simp: not_essential_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1038 |
|
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1039 |
lemma eventually_not_pole: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1040 |
assumes "isolated_singularity_at f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1041 |
shows "eventually (\<lambda>w. \<not>is_pole f w) (at z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1042 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1043 |
from assms obtain r where "r > 0" and r: "f analytic_on ball z r - {z}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1044 |
by (auto simp: isolated_singularity_at_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1045 |
then have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1046 |
by (intro eventually_at_in_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1047 |
thus "eventually (\<lambda>w. \<not>is_pole f w) (at z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1048 |
proof eventually_elim |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1049 |
case (elim w) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1050 |
with r show ?case |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1051 |
using analytic_imp_holomorphic not_is_pole_holomorphic open_delete by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1052 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1053 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1054 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1055 |
lemma not_islimpt_poles: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1056 |
assumes "isolated_singularity_at f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1057 |
shows "\<not>z islimpt {w. is_pole f w}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1058 |
using eventually_not_pole [OF assms] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1059 |
by (auto simp: islimpt_conv_frequently_at frequently_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1060 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1061 |
lemma analytic_at_imp_no_pole: "f analytic_on {z} \<Longrightarrow> \<not>is_pole f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1062 |
using analytic_at not_is_pole_holomorphic by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1063 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1064 |
lemma not_essential_const [singularity_intros]: "not_essential (\<lambda>_. c) z" |
77277
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
77228
diff
changeset
|
1065 |
by blast |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1066 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1067 |
lemma not_essential_uminus [singularity_intros]: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1068 |
assumes f_ness: "not_essential f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1069 |
assumes f_iso:"isolated_singularity_at f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1070 |
shows "not_essential (\<lambda>w. -f w) z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1071 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1072 |
have "not_essential (\<lambda>w. -1 * f w) z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1073 |
by (intro assms singularity_intros) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1074 |
thus ?thesis by simp |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1075 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1076 |
|
77226
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1077 |
lemma isolated_singularity_at_analytic: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1078 |
assumes "f analytic_on {z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1079 |
shows "isolated_singularity_at f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1080 |
proof - |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1081 |
from assms obtain r where r: "r > 0" "f holomorphic_on ball z r" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1082 |
by (auto simp: analytic_on_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1083 |
show ?thesis |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1084 |
by (rule isolated_singularity_at_holomorphic[of f "ball z r"]) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1085 |
(use \<open>r > 0\<close> in \<open>auto intro!: holomorphic_on_subset[OF r(2)]\<close>) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1086 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1087 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1088 |
subsection \<open>The order of non-essential singularities (i.e. removable singularities or poles)\<close> |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1089 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1090 |
definition\<^marker>\<open>tag important\<close> zorder :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> int" where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1091 |
"zorder f z = (THE n. (\<exists>h r. r>0 \<and> h holomorphic_on cball z r \<and> h z\<noteq>0 |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1092 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w-z) powi n |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1093 |
\<and> h w \<noteq>0)))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1094 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1095 |
definition\<^marker>\<open>tag important\<close> zor_poly |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1096 |
::"[complex \<Rightarrow> complex, complex] \<Rightarrow> complex \<Rightarrow> complex" where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1097 |
"zor_poly f z = (SOME h. \<exists>r. r > 0 \<and> h holomorphic_on cball z r \<and> h z \<noteq> 0 |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1098 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w - z) powi (zorder f z) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1099 |
\<and> h w \<noteq>0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1100 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1101 |
lemma zorder_exist: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1102 |
fixes f::"complex \<Rightarrow> complex" and z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1103 |
defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1104 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1105 |
and f_ness:"not_essential f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1106 |
and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1107 |
shows "g z\<noteq>0 \<and> (\<exists>r. r>0 \<and> g holomorphic_on cball z r |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1108 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powi n \<and> g w \<noteq>0))" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1109 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1110 |
define P where "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0 |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1111 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powi n \<and> g w\<noteq>0))" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1112 |
have "\<exists>!n. \<exists>g r. P n g r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1113 |
using holomorphic_factor_puncture[OF assms(3-)] unfolding P_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1114 |
then have "\<exists>g r. P n g r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1115 |
unfolding n_def P_def zorder_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1116 |
by (drule_tac theI',argo) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1117 |
then have "\<exists>r. P n g r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1118 |
unfolding P_def zor_poly_def g_def n_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1119 |
by (drule_tac someI_ex,argo) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1120 |
then obtain r1 where "P n g r1" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1121 |
then show ?thesis unfolding P_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1122 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1123 |
|
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1124 |
lemma zorder_shift: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1125 |
shows "zorder f z = zorder (\<lambda>u. f (u + z)) 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1126 |
unfolding zorder_def |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1127 |
apply (rule arg_cong [of concl: The]) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1128 |
apply (auto simp: fun_eq_iff Ball_def dist_norm) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1129 |
subgoal for x h r |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1130 |
apply (rule_tac x="h o (+)z" in exI) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1131 |
apply (rule_tac x="r" in exI) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1132 |
apply (intro conjI holomorphic_on_compose holomorphic_intros) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1133 |
apply (simp_all flip: cball_translation) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1134 |
apply (simp add: add.commute) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1135 |
done |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1136 |
subgoal for x h r |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1137 |
apply (rule_tac x="h o (\<lambda>u. u-z)" in exI) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1138 |
apply (rule_tac x="r" in exI) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1139 |
apply (intro conjI holomorphic_on_compose holomorphic_intros) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1140 |
apply (simp_all add: flip: cball_translation_subtract) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1141 |
by (metis diff_add_cancel eq_iff_diff_eq_0 norm_minus_commute) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1142 |
done |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1143 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1144 |
lemma zorder_shift': "NO_MATCH 0 z \<Longrightarrow> zorder f z = zorder (\<lambda>u. f (u + z)) 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1145 |
by (rule zorder_shift) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1146 |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1147 |
lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1148 |
fixes f::"complex \<Rightarrow> complex" and z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1149 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1150 |
and f_ness:"not_essential f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1151 |
and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1152 |
shows zorder_inverse: "zorder (\<lambda>w. inverse (f w)) z = - zorder f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1153 |
and zor_poly_inverse: "\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. inverse (f w)) z w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1154 |
= inverse (zor_poly f z w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1155 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1156 |
define vf where "vf = (\<lambda>w. inverse (f w))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1157 |
define fn vfn where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1158 |
"fn = zorder f z" and "vfn = zorder vf z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1159 |
define fp vfp where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1160 |
"fp = zor_poly f z" and "vfp = zor_poly vf z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1161 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1162 |
obtain fr where [simp]:"fp z \<noteq> 0" and "fr > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1163 |
and fr: "fp holomorphic_on cball z fr" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1164 |
"\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powi fn \<and> fp w \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1165 |
using zorder_exist[OF f_iso f_ness f_nconst,folded fn_def fp_def] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1166 |
by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1167 |
have fr_inverse: "vf w = (inverse (fp w)) * (w - z) powi (-fn)" |
76897 | 1168 |
and fr_nz: "inverse (fp w) \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1169 |
when "w\<in>ball z fr - {z}" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1170 |
proof - |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1171 |
have "f w = fp w * (w - z) powi fn" "fp w\<noteq>0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1172 |
using fr(2)[rule_format,of w] that by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1173 |
then show "vf w = (inverse (fp w)) * (w - z) powi (-fn)" "inverse (fp w)\<noteq>0" |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1174 |
by (simp_all add: power_int_minus vf_def) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1175 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1176 |
obtain vfr where [simp]:"vfp z \<noteq> 0" and "vfr>0" and vfr:"vfp holomorphic_on cball z vfr" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1177 |
"(\<forall>w\<in>cball z vfr - {z}. vf w = vfp w * (w - z) powi vfn \<and> vfp w \<noteq> 0)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1178 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1179 |
have "isolated_singularity_at vf z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1180 |
using isolated_singularity_at_inverse[OF f_iso f_ness] unfolding vf_def . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1181 |
moreover have "not_essential vf z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1182 |
using not_essential_inverse[OF f_ness f_iso] unfolding vf_def . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1183 |
moreover have "\<exists>\<^sub>F w in at z. vf w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1184 |
using f_nconst unfolding vf_def by (auto elim:frequently_elim1) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1185 |
ultimately show ?thesis using zorder_exist[of vf z, folded vfn_def vfp_def] that by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1186 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1187 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1188 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1189 |
define r1 where "r1 = min fr vfr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1190 |
have "r1>0" using \<open>fr>0\<close> \<open>vfr>0\<close> unfolding r1_def by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1191 |
show "vfn = - fn" |
76897 | 1192 |
proof (rule holomorphic_factor_unique) |
1193 |
have \<section>: "\<And>w. \<lbrakk>fp w = 0; dist z w < fr\<rbrakk> \<Longrightarrow> False" |
|
1194 |
using fr_nz by force |
|
1195 |
then show "\<forall>w\<in>ball z r1 - {z}. |
|
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1196 |
vf w = vfp w * (w - z) powi vfn \<and> |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1197 |
vfp w \<noteq> 0 \<and> vf w = inverse (fp w) * (w - z) powi (- fn) \<and> |
76897 | 1198 |
inverse (fp w) \<noteq> 0" |
1199 |
using fr_inverse r1_def vfr(2) |
|
1200 |
by (smt (verit) Diff_iff inverse_nonzero_iff_nonzero mem_ball mem_cball) |
|
1201 |
show "vfp holomorphic_on ball z r1" |
|
1202 |
using r1_def vfr(1) by auto |
|
1203 |
show "(\<lambda>w. inverse (fp w)) holomorphic_on ball z r1" |
|
1204 |
by (metis \<section> ball_subset_cball fr(1) holomorphic_on_inverse holomorphic_on_subset mem_ball min.cobounded2 min.commute r1_def subset_ball) |
|
1205 |
qed (use \<open>r1>0\<close> in auto) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1206 |
have "vfp w = inverse (fp w)" when "w\<in>ball z r1-{z}" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1207 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1208 |
have "w \<in> ball z fr - {z}" "w \<in> cball z vfr - {z}" "w\<noteq>z" using that unfolding r1_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1209 |
from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)] \<open>vfn = - fn\<close> \<open>w\<noteq>z\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1210 |
show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1211 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1212 |
then show "\<forall>\<^sub>Fw in (at z). vfp w = inverse (fp w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1213 |
unfolding eventually_at using \<open>r1>0\<close> |
76897 | 1214 |
by (metis DiffI dist_commute mem_ball singletonD) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1215 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1216 |
|
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1217 |
lemma zor_poly_shift: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1218 |
assumes iso1: "isolated_singularity_at f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1219 |
and ness1: "not_essential f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1220 |
and nzero1: "\<exists>\<^sub>F w in at z. f w \<noteq> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1221 |
shows "\<forall>\<^sub>F w in nhds z. zor_poly f z w = zor_poly (\<lambda>u. f (z + u)) 0 (w-z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1222 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1223 |
obtain r1 where "r1>0" "zor_poly f z z \<noteq> 0" and |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1224 |
holo1:"zor_poly f z holomorphic_on cball z r1" and |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1225 |
rball1:"\<forall>w\<in>cball z r1 - {z}. |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1226 |
f w = zor_poly f z w * (w - z) powi (zorder f z) \<and> |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1227 |
zor_poly f z w \<noteq> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1228 |
using zorder_exist[OF iso1 ness1 nzero1] by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1229 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1230 |
define ff where "ff=(\<lambda>u. f (z + u))" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1231 |
have "isolated_singularity_at ff 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1232 |
unfolding ff_def |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1233 |
using iso1 isolated_singularity_at_shift_iff[of f 0 z] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1234 |
by (simp add:algebra_simps) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1235 |
moreover have "not_essential ff 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1236 |
unfolding ff_def |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1237 |
using ness1 not_essential_shift_iff[of f 0 z] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1238 |
by (simp add:algebra_simps) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1239 |
moreover have "\<exists>\<^sub>F w in at 0. ff w \<noteq> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1240 |
unfolding ff_def using nzero1 |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1241 |
by (smt (verit, ccfv_SIG) add.commute eventually_at_to_0 |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1242 |
eventually_mono not_frequently) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1243 |
ultimately obtain r2 where "r2>0" "zor_poly ff 0 0 \<noteq> 0" and |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1244 |
holo2:"zor_poly ff 0 holomorphic_on cball 0 r2" and |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1245 |
rball2:"\<forall>w\<in>cball 0 r2 - {0}. |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1246 |
ff w = zor_poly ff 0 w * w powi (zorder ff 0) \<and> |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1247 |
zor_poly ff 0 w \<noteq> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1248 |
using zorder_exist[of ff 0] by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1249 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1250 |
define r where "r=min r1 r2" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1251 |
have "r>0" using \<open>r1>0\<close> \<open>r2>0\<close> unfolding r_def by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1252 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1253 |
have "zor_poly f z w = zor_poly ff 0 (w - z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1254 |
if "w\<in>ball z r - {z}" for w |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1255 |
proof - |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1256 |
define n where "n \<equiv> zorder f z" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1257 |
|
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1258 |
have "f w = zor_poly f z w * (w - z) powi n" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1259 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1260 |
have "w\<in>cball z r1 - {z}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1261 |
using r_def that by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1262 |
from rball1[rule_format, OF this] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1263 |
show ?thesis unfolding n_def by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1264 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1265 |
|
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1266 |
moreover have "f w = zor_poly ff 0 (w - z) * (w - z) powi n" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1267 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1268 |
have "w-z\<in>cball 0 r2 - {0}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1269 |
using r_def that by (auto simp:dist_complex_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1270 |
from rball2[rule_format, OF this] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1271 |
have "ff (w - z) = zor_poly ff 0 (w - z) * (w - z) |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1272 |
powi (zorder ff 0)" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1273 |
by simp |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1274 |
moreover have "of_int (zorder ff 0) = n" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1275 |
unfolding n_def ff_def by (simp add:zorder_shift' add.commute) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1276 |
ultimately show ?thesis unfolding ff_def by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1277 |
qed |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1278 |
ultimately have "zor_poly f z w * (w - z) powi n |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1279 |
= zor_poly ff 0 (w - z) * (w - z) powi n" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1280 |
by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1281 |
moreover have "(w - z) powi n \<noteq>0" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1282 |
using that by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1283 |
ultimately show ?thesis |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1284 |
using mult_cancel_right by blast |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1285 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1286 |
then have "\<forall>\<^sub>F w in at z. zor_poly f z w |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1287 |
= zor_poly ff 0 (w - z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1288 |
unfolding eventually_at |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1289 |
apply (rule_tac x=r in exI) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1290 |
using \<open>r>0\<close> by (auto simp:dist_commute) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1291 |
moreover have "isCont (zor_poly f z) z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1292 |
using holo1[THEN holomorphic_on_imp_continuous_on] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1293 |
apply (elim continuous_on_interior) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1294 |
using \<open>r1>0\<close> by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1295 |
moreover have "isCont (\<lambda>w. zor_poly ff 0 (w - z)) z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1296 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1297 |
have "isCont (zor_poly ff 0) 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1298 |
using holo2[THEN holomorphic_on_imp_continuous_on] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1299 |
apply (elim continuous_on_interior) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1300 |
using \<open>r2>0\<close> by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1301 |
then show ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1302 |
unfolding isCont_iff by simp |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1303 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1304 |
ultimately show "\<forall>\<^sub>F w in nhds z. zor_poly f z w |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1305 |
= zor_poly ff 0 (w - z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1306 |
by (elim at_within_isCont_imp_nhds;auto) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1307 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1308 |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1309 |
lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1310 |
fixes f g::"complex \<Rightarrow> complex" and z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1311 |
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1312 |
and f_ness:"not_essential f z" and g_ness:"not_essential g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1313 |
and fg_nconst: "\<exists>\<^sub>Fw in (at z). f w * g w\<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1314 |
shows zorder_times:"zorder (\<lambda>w. f w * g w) z = zorder f z + zorder g z" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1315 |
zor_poly_times:"\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w * g w) z w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1316 |
= zor_poly f z w *zor_poly g z w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1317 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1318 |
define fg where "fg = (\<lambda>w. f w * g w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1319 |
define fn gn fgn where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1320 |
"fn = zorder f z" and "gn = zorder g z" and "fgn = zorder fg z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1321 |
define fp gp fgp where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1322 |
"fp = zor_poly f z" and "gp = zor_poly g z" and "fgp = zor_poly fg z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1323 |
have f_nconst:"\<exists>\<^sub>Fw in (at z). f w \<noteq> 0" and g_nconst:"\<exists>\<^sub>Fw in (at z).g w\<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1324 |
using fg_nconst by (auto elim!:frequently_elim1) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1325 |
obtain fr where [simp]:"fp z \<noteq> 0" and "fr > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1326 |
and fr: "fp holomorphic_on cball z fr" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1327 |
"\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powi fn \<and> fp w \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1328 |
using zorder_exist[OF f_iso f_ness f_nconst,folded fp_def fn_def] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1329 |
obtain gr where [simp]:"gp z \<noteq> 0" and "gr > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1330 |
and gr: "gp holomorphic_on cball z gr" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1331 |
"\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powi gn \<and> gp w \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1332 |
using zorder_exist[OF g_iso g_ness g_nconst,folded gn_def gp_def] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1333 |
define r1 where "r1=min fr gr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1334 |
have "r1>0" unfolding r1_def using \<open>fr>0\<close> \<open>gr>0\<close> by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1335 |
have fg_times:"fg w = (fp w * gp w) * (w - z) powi (fn+gn)" and fgp_nz:"fp w*gp w\<noteq>0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1336 |
when "w\<in>ball z r1 - {z}" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1337 |
proof - |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1338 |
have "f w = fp w * (w - z) powi fn" "fp w\<noteq>0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1339 |
using fr(2)[rule_format,of w] that unfolding r1_def by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1340 |
moreover have "g w = gp w * (w - z) powi gn" "gp w \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1341 |
using gr(2)[rule_format, of w] that unfolding r1_def by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1342 |
ultimately show "fg w = (fp w * gp w) * (w - z) powi (fn+gn)" "fp w*gp w\<noteq>0" |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1343 |
using that |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1344 |
unfolding fg_def by (auto simp add:power_int_add) |
71201
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 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1347 |
obtain fgr where [simp]:"fgp z \<noteq> 0" and "fgr > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1348 |
and fgr: "fgp holomorphic_on cball z fgr" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1349 |
"\<forall>w\<in>cball z fgr - {z}. fg w = fgp w * (w - z) powi fgn \<and> fgp w \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1350 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1351 |
have "fgp z \<noteq> 0 \<and> (\<exists>r>0. fgp holomorphic_on cball z r |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1352 |
\<and> (\<forall>w\<in>cball z r - {z}. fg w = fgp w * (w - z) powi fgn \<and> fgp w \<noteq> 0))" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1353 |
apply (rule zorder_exist[of fg z, folded fgn_def fgp_def]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1354 |
subgoal unfolding fg_def using isolated_singularity_at_times[OF f_iso g_iso] . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1355 |
subgoal unfolding fg_def using not_essential_times[OF f_ness g_ness f_iso g_iso] . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1356 |
subgoal unfolding fg_def using fg_nconst . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1357 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1358 |
then show ?thesis using that by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1359 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1360 |
define r2 where "r2 = min fgr r1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1361 |
have "r2>0" using \<open>r1>0\<close> \<open>fgr>0\<close> unfolding r2_def by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1362 |
show "fgn = fn + gn " |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1363 |
apply (rule holomorphic_factor_unique[of r2 fgp z "\<lambda>w. fp w * gp w" fg]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1364 |
subgoal using \<open>r2>0\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1365 |
subgoal by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1366 |
subgoal by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1367 |
subgoal |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1368 |
proof (rule ballI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1369 |
fix w assume "w \<in> ball z r2 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1370 |
then have "w \<in> ball z r1 - {z}" "w \<in> cball z fgr - {z}" unfolding r2_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1371 |
from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)] |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1372 |
show "fg w = fgp w * (w - z) powi fgn \<and> fgp w \<noteq> 0 |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1373 |
\<and> fg w = fp w * gp w * (w - z) powi (fn + gn) \<and> fp w * gp w \<noteq> 0" by auto |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1374 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1375 |
subgoal using fgr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1376 |
subgoal using fr(1) gr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1377 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1378 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1379 |
have "fgp w = fp w *gp w" when "w\<in>ball z r2-{z}" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1380 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1381 |
have "w \<in> ball z r1 - {z}" "w \<in> cball z fgr - {z}" "w\<noteq>z" using that unfolding r2_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1382 |
from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)] \<open>fgn = fn + gn\<close> \<open>w\<noteq>z\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1383 |
show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1384 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1385 |
then show "\<forall>\<^sub>Fw in (at z). fgp w = fp w * gp w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1386 |
using \<open>r2>0\<close> unfolding eventually_at by (auto simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1387 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1388 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1389 |
lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1390 |
fixes f g::"complex \<Rightarrow> complex" and z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1391 |
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1392 |
and f_ness:"not_essential f z" and g_ness:"not_essential g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1393 |
and fg_nconst: "\<exists>\<^sub>Fw in (at z). f w * g w\<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1394 |
shows zorder_divide:"zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1395 |
zor_poly_divide:"\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w / g w) z w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1396 |
= zor_poly f z w / zor_poly g z w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1397 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1398 |
have f_nconst:"\<exists>\<^sub>Fw in (at z). f w \<noteq> 0" and g_nconst:"\<exists>\<^sub>Fw in (at z).g w\<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1399 |
using fg_nconst by (auto elim!:frequently_elim1) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1400 |
define vg where "vg=(\<lambda>w. inverse (g w))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1401 |
have "zorder (\<lambda>w. f w * vg w) z = zorder f z + zorder vg z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1402 |
apply (rule zorder_times[OF f_iso _ f_ness,of vg]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1403 |
subgoal unfolding vg_def using isolated_singularity_at_inverse[OF g_iso g_ness] . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1404 |
subgoal unfolding vg_def using not_essential_inverse[OF g_ness g_iso] . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1405 |
subgoal unfolding vg_def using fg_nconst by (auto elim!:frequently_elim1) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1406 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1407 |
then show "zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1408 |
using zorder_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1409 |
by (auto simp add:field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1410 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1411 |
have "\<forall>\<^sub>F w in at z. zor_poly (\<lambda>w. f w * vg w) z w = zor_poly f z w * zor_poly vg z w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1412 |
apply (rule zor_poly_times[OF f_iso _ f_ness,of vg]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1413 |
subgoal unfolding vg_def using isolated_singularity_at_inverse[OF g_iso g_ness] . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1414 |
subgoal unfolding vg_def using not_essential_inverse[OF g_ness g_iso] . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1415 |
subgoal unfolding vg_def using fg_nconst by (auto elim!:frequently_elim1) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1416 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1417 |
then show "\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w / g w) z w = zor_poly f z w / zor_poly g z w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1418 |
using zor_poly_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def |
76897 | 1419 |
by eventually_elim (auto simp add:field_simps) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1420 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1421 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1422 |
lemma zorder_exist_zero: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1423 |
fixes f::"complex \<Rightarrow> complex" and z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1424 |
defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1425 |
assumes holo: "f holomorphic_on s" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1426 |
"open s" "connected s" "z\<in>s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1427 |
and non_const: "\<exists>w\<in>s. f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1428 |
shows "(if f z=0 then n > 0 else n=0) \<and> (\<exists>r. r>0 \<and> cball z r \<subseteq> s \<and> g holomorphic_on cball z r |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1429 |
\<and> (\<forall>w\<in>cball z r. f w = g w * (w-z) ^ nat n \<and> g w \<noteq>0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1430 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1431 |
obtain r where "g z \<noteq> 0" and r: "r>0" "cball z r \<subseteq> s" "g holomorphic_on cball z r" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1432 |
"(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1433 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1434 |
have "g z \<noteq> 0 \<and> (\<exists>r>0. g holomorphic_on cball z r |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1435 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0))" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1436 |
proof (rule zorder_exist[of f z,folded g_def n_def]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1437 |
show "isolated_singularity_at f z" unfolding isolated_singularity_at_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1438 |
using holo assms(4,6) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1439 |
by (meson Diff_subset open_ball analytic_on_holomorphic holomorphic_on_subset openE) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1440 |
show "not_essential f z" unfolding not_essential_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1441 |
using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1442 |
by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1443 |
have "\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w\<in>s" |
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 |
obtain w where "w\<in>s" "f w\<noteq>0" using non_const by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1446 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1447 |
by (rule non_zero_neighbour_alt[OF holo \<open>open s\<close> \<open>connected s\<close> \<open>z\<in>s\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1448 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1449 |
then show "\<exists>\<^sub>F w in at z. f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1450 |
apply (elim eventually_frequentlyE) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1451 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1452 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1453 |
then obtain r1 where "g z \<noteq> 0" "r1>0" and r1:"g holomorphic_on cball z r1" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1454 |
"(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1455 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1456 |
obtain r2 where r2: "r2>0" "cball z r2 \<subseteq> s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1457 |
using assms(4,6) open_contains_cball_eq by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1458 |
define r3 where "r3=min r1 r2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1459 |
have "r3>0" "cball z r3 \<subseteq> s" using \<open>r1>0\<close> r2 unfolding r3_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1460 |
moreover have "g holomorphic_on cball z r3" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1461 |
using r1(1) unfolding r3_def by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1462 |
moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1463 |
using r1(2) unfolding r3_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1464 |
ultimately show ?thesis using that[of r3] \<open>g z\<noteq>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1465 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1466 |
|
76897 | 1467 |
have fz_lim: "f\<midarrow> z \<rightarrow> f z" |
1468 |
by (metis assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on) |
|
1469 |
have gz_lim: "g \<midarrow>z\<rightarrow>g z" |
|
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1470 |
using r |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1471 |
by (meson Elementary_Metric_Spaces.open_ball analytic_at analytic_at_imp_isCont |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1472 |
ball_subset_cball centre_in_ball holomorphic_on_subset isContD) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1473 |
have if_0:"if f z=0 then n > 0 else n=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1474 |
proof - |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1475 |
have "(\<lambda>w. g w * (w - z) powi n) \<midarrow>z\<rightarrow> f z" |
76897 | 1476 |
using fz_lim Lim_transform_within_open[where s="ball z r"] r by fastforce |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1477 |
then have "(\<lambda>w. (g w * (w - z) powi n) / g w) \<midarrow>z\<rightarrow> f z/g z" |
76897 | 1478 |
using gz_lim \<open>g z \<noteq> 0\<close> tendsto_divide by blast |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1479 |
then have powi_tendsto:"(\<lambda>w. (w - z) powi n) \<midarrow>z\<rightarrow> f z/g z" |
76897 | 1480 |
using Lim_transform_within_open[where s="ball z r"] r by fastforce |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1481 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1482 |
have ?thesis when "n\<ge>0" "f z=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1483 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1484 |
have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> f z/g z" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1485 |
using Lim_transform_within[OF powi_tendsto, where d=r] |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1486 |
by (meson power_int_def r(1) that(1)) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1487 |
then have *:"(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 0" using \<open>f z=0\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1488 |
moreover have False when "n=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1489 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1490 |
have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1491 |
using \<open>n=0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1492 |
then show False using * using LIM_unique zero_neq_one by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1493 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1494 |
ultimately show ?thesis using that by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1495 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1496 |
moreover have ?thesis when "n\<ge>0" "f z\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1497 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1498 |
have False when "n>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1499 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1500 |
have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> f z/g z" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1501 |
using Lim_transform_within[OF powi_tendsto, where d=r] |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1502 |
by (meson \<open>0 \<le> n\<close> power_int_def r(1)) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1503 |
moreover have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1504 |
using \<open>n>0\<close> by (auto intro!:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1505 |
ultimately show False using \<open>f z\<noteq>0\<close> \<open>g z\<noteq>0\<close> using LIM_unique divide_eq_0_iff by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1506 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1507 |
then show ?thesis using that by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1508 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1509 |
moreover have False when "n<0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1510 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1511 |
have "(\<lambda>w. inverse ((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> f z/g z" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1512 |
by (smt (verit) LIM_cong power_int_def power_inverse powi_tendsto that) |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1513 |
moreover |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1514 |
have "(\<lambda>w.((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> 0" |
76897 | 1515 |
using that by (auto intro!:tendsto_eq_intros) |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1516 |
ultimately |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1517 |
have "(\<lambda>x. inverse ((x - z) ^ nat (- n)) * (x - z) ^ nat (- n)) \<midarrow>z\<rightarrow> 0" |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1518 |
using tendsto_mult by fastforce |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1519 |
then have "(\<lambda>x. 1::complex) \<midarrow>z\<rightarrow> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1520 |
by (elim Lim_transform_within_open[where s=UNIV],auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1521 |
then show False using LIM_const_eq by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1522 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1523 |
ultimately show ?thesis by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1524 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1525 |
moreover have "f w = g w * (w-z) ^ nat n \<and> g w \<noteq>0" when "w\<in>cball z r" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1526 |
proof (cases "w=z") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1527 |
case True |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1528 |
then have "f \<midarrow>z\<rightarrow>f w" |
76897 | 1529 |
using fz_lim by blast |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1530 |
then have "(\<lambda>w. g w * (w-z) ^ nat n) \<midarrow>z\<rightarrow>f w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1531 |
proof (elim Lim_transform_within[OF _ \<open>r>0\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1532 |
fix x assume "0 < dist x z" "dist x z < r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1533 |
then have "x \<in> cball z r - {z}" "x\<noteq>z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1534 |
unfolding cball_def by (auto simp add: dist_commute) |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1535 |
then have "f x = g x * (x - z) powi n" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1536 |
using r(4)[rule_format,of x] by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1537 |
also have "... = g x * (x - z) ^ nat n" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1538 |
by (smt (verit, best) if_0 int_nat_eq power_int_of_nat) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1539 |
finally show "f x = g x * (x - z) ^ nat n" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1540 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1541 |
moreover have "(\<lambda>w. g w * (w-z) ^ nat n) \<midarrow>z\<rightarrow> g w * (w-z) ^ nat n" |
76897 | 1542 |
using True by (auto intro!:tendsto_eq_intros gz_lim) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1543 |
ultimately have "f w = g w * (w-z) ^ nat n" using LIM_unique by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1544 |
then show ?thesis using \<open>g z\<noteq>0\<close> True by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1545 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1546 |
case False |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1547 |
then have "f w = g w * (w - z) powi n \<and> g w \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1548 |
using r(4) that by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1549 |
then show ?thesis |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1550 |
by (smt (verit, best) False if_0 int_nat_eq power_int_of_nat) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1551 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1552 |
ultimately show ?thesis using r by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1553 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1554 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1555 |
lemma zorder_exist_pole: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1556 |
fixes f::"complex \<Rightarrow> complex" and z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1557 |
defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1558 |
assumes holo: "f holomorphic_on s-{z}" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1559 |
"open s" "z\<in>s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1560 |
and "is_pole f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1561 |
shows "n < 0 \<and> g z\<noteq>0 \<and> (\<exists>r. r>0 \<and> cball z r \<subseteq> s \<and> g holomorphic_on cball z r |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1562 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w / (w-z) ^ nat (- n) \<and> g w \<noteq>0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1563 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1564 |
obtain r where "g z \<noteq> 0" and r: "r>0" "cball z r \<subseteq> s" "g holomorphic_on cball z r" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1565 |
"(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1566 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1567 |
have "g z \<noteq> 0 \<and> (\<exists>r>0. g holomorphic_on cball z r |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1568 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0))" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1569 |
proof (rule zorder_exist[of f z,folded g_def n_def]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1570 |
show "isolated_singularity_at f z" unfolding isolated_singularity_at_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1571 |
using holo assms(4,5) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1572 |
by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1573 |
show "not_essential f z" unfolding not_essential_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1574 |
using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1575 |
by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1576 |
from non_zero_neighbour_pole[OF \<open>is_pole f z\<close>] show "\<exists>\<^sub>F w in at z. f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1577 |
apply (elim eventually_frequentlyE) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1578 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1579 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1580 |
then obtain r1 where "g z \<noteq> 0" "r1>0" and r1:"g holomorphic_on cball z r1" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1581 |
"(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1582 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1583 |
obtain r2 where r2: "r2>0" "cball z r2 \<subseteq> s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1584 |
using assms(4,5) open_contains_cball_eq by metis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1585 |
define r3 where "r3=min r1 r2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1586 |
have "r3>0" "cball z r3 \<subseteq> s" using \<open>r1>0\<close> r2 unfolding r3_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1587 |
moreover have "g holomorphic_on cball z r3" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1588 |
using r1(1) unfolding r3_def by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1589 |
moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1590 |
using r1(2) unfolding r3_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1591 |
ultimately show ?thesis using that[of r3] \<open>g z\<noteq>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1592 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1593 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1594 |
have "n<0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1595 |
proof (rule ccontr) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1596 |
assume " \<not> n < 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1597 |
define c where "c=(if n=0 then g z else 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1598 |
have [simp]:"g \<midarrow>z\<rightarrow> g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1599 |
by (metis open_ball at_within_open ball_subset_cball centre_in_ball |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1600 |
continuous_on holomorphic_on_imp_continuous_on holomorphic_on_subset r(1) r(3) ) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1601 |
have "\<forall>\<^sub>F x in at z. f x = g x * (x - z) ^ nat n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1602 |
unfolding eventually_at_topological |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1603 |
apply (rule_tac exI[where x="ball z r"]) |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1604 |
by (simp add: \<open>\<not> n < 0\<close> linorder_not_le power_int_def r(1) r(4)) |
76897 | 1605 |
moreover have "(\<lambda>x. g x * (x - z) ^ nat n) \<midarrow>z\<rightarrow> c" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1606 |
proof (cases "n=0") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1607 |
case True |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1608 |
then show ?thesis unfolding c_def by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1609 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1610 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1611 |
then have "(\<lambda>x. (x - z) ^ nat n) \<midarrow>z\<rightarrow> 0" using \<open>\<not> n < 0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1612 |
by (auto intro!:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1613 |
from tendsto_mult[OF _ this,of g "g z",simplified] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1614 |
show ?thesis unfolding c_def using False by simp |
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 |
ultimately have "f \<midarrow>z\<rightarrow>c" using tendsto_cong by fast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1617 |
then show False using \<open>is_pole f z\<close> at_neq_bot not_tendsto_and_filterlim_at_infinity |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1618 |
unfolding is_pole_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1619 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1620 |
moreover have "\<forall>w\<in>cball z r - {z}. f w = g w / (w-z) ^ nat (- n) \<and> g w \<noteq>0" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1621 |
using r(4) \<open>n<0\<close> |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1622 |
by (smt (verit) inverse_eq_divide mult.right_neutral power_int_def power_inverse times_divide_eq_right) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1623 |
ultimately show ?thesis using r(1-3) \<open>g z\<noteq>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1624 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1625 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1626 |
lemma zorder_eqI: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1627 |
assumes "open s" "z \<in> s" "g holomorphic_on s" "g z \<noteq> 0" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1628 |
assumes fg_eq:"\<And>w. \<lbrakk>w \<in> s;w\<noteq>z\<rbrakk> \<Longrightarrow> f w = g w * (w - z) powi n" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1629 |
shows "zorder f z = n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1630 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1631 |
have "continuous_on s g" by (rule holomorphic_on_imp_continuous_on) fact |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1632 |
moreover have "open (-{0::complex})" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1633 |
ultimately have "open ((g -` (-{0})) \<inter> s)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1634 |
unfolding continuous_on_open_vimage[OF \<open>open s\<close>] by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1635 |
moreover from assms have "z \<in> (g -` (-{0})) \<inter> s" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1636 |
ultimately obtain r where r: "r > 0" "cball z r \<subseteq> s \<inter> (g -` (-{0}))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1637 |
unfolding open_contains_cball by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1638 |
|
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1639 |
let ?gg= "(\<lambda>w. g w * (w - z) powi n)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1640 |
define P where "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0 |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1641 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powi n \<and> g w\<noteq>0))" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1642 |
have "P n g r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1643 |
unfolding P_def using r assms(3,4,5) by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1644 |
then have "\<exists>g r. P n g r" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1645 |
moreover have unique: "\<exists>!n. \<exists>g r. P n g r" unfolding P_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1646 |
proof (rule holomorphic_factor_puncture) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1647 |
have "ball z r-{z} \<subseteq> s" using r using ball_subset_cball by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1648 |
then have "?gg holomorphic_on ball z r-{z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1649 |
using \<open>g holomorphic_on s\<close> r by (auto intro!: holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1650 |
then have "f holomorphic_on ball z r - {z}" |
76900 | 1651 |
by (smt (verit, best) DiffD2 \<open>ball z r-{z} \<subseteq> s\<close> fg_eq holomorphic_cong singleton_iff subset_iff) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1652 |
then show "isolated_singularity_at f z" unfolding isolated_singularity_at_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1653 |
using analytic_on_open open_delete r(1) by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1654 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1655 |
have "not_essential ?gg z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1656 |
proof (intro singularity_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1657 |
show "not_essential g z" |
76900 | 1658 |
by (meson \<open>continuous_on s g\<close> assms continuous_on_eq_continuous_at |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1659 |
isCont_def not_essential_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1660 |
show " \<forall>\<^sub>F w in at z. w - z \<noteq> 0" by (simp add: eventually_at_filter) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1661 |
then show "LIM w at z. w - z :> at 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1662 |
unfolding filterlim_at by (auto intro:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1663 |
show "isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1664 |
by (meson Diff_subset open_ball analytic_on_holomorphic |
76900 | 1665 |
assms holomorphic_on_subset isolated_singularity_at_def openE) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1666 |
qed |
76900 | 1667 |
moreover |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1668 |
have "\<forall>\<^sub>F w in at z. g w * (w - z) powi n = f w" |
76900 | 1669 |
unfolding eventually_at_topological using assms fg_eq by force |
1670 |
ultimately show "not_essential f z" |
|
1671 |
using not_essential_transform by blast |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1672 |
show "\<exists>\<^sub>F w in at z. f w \<noteq> 0" unfolding frequently_at |
76900 | 1673 |
proof (intro strip) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1674 |
fix d::real assume "0 < d" |
76900 | 1675 |
define z' where "z' \<equiv> z+min d r / 2" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1676 |
have "z' \<noteq> z" " dist z' z < d " |
76900 | 1677 |
unfolding z'_def using \<open>d>0\<close> \<open>r>0\<close> by (auto simp add:dist_norm) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1678 |
moreover have "f z' \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1679 |
proof (subst fg_eq[OF _ \<open>z'\<noteq>z\<close>]) |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
1680 |
have "z' \<in> cball z r" |
76900 | 1681 |
unfolding z'_def using \<open>r>0\<close> \<open>d>0\<close> by (auto simp add:dist_norm) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1682 |
then show " z' \<in> s" using r(2) by blast |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1683 |
show "g z' * (z' - z) powi n \<noteq> 0" |
76900 | 1684 |
using P_def \<open>P n g r\<close> \<open>z' \<in> cball z r\<close> \<open>z' \<noteq> z\<close> by auto |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1685 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1686 |
ultimately show "\<exists>x\<in>UNIV. x \<noteq> z \<and> dist x z < d \<and> f x \<noteq> 0" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1687 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1688 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1689 |
ultimately have "(THE n. \<exists>g r. P n g r) = n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1690 |
by (rule_tac the1_equality) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1691 |
then show ?thesis unfolding zorder_def P_def by blast |
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 simple_zeroI: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1695 |
assumes "open s" "z \<in> s" "g holomorphic_on s" "g z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1696 |
assumes "\<And>w. w \<in> s \<Longrightarrow> f w = g w * (w - z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1697 |
shows "zorder f z = 1" |
76900 | 1698 |
using assms zorder_eqI by force |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1699 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1700 |
lemma higher_deriv_power: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1701 |
shows "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) w = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1702 |
pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1703 |
proof (induction j arbitrary: w) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1704 |
case 0 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1705 |
thus ?case by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1706 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1707 |
case (Suc j w) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1708 |
have "(deriv ^^ Suc j) (\<lambda>w. (w - z) ^ n) w = deriv ((deriv ^^ j) (\<lambda>w. (w - z) ^ n)) w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1709 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1710 |
also have "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1711 |
(\<lambda>w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1712 |
using Suc by (intro Suc.IH ext) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1713 |
also { |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1714 |
have "(\<dots> has_field_derivative of_nat (n - j) * |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1715 |
pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - Suc j)) (at w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1716 |
using Suc.prems by (auto intro!: derivative_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1717 |
also have "of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1718 |
pochhammer (of_nat (Suc n - Suc j)) (Suc j)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1719 |
by (cases "Suc j \<le> n", subst pochhammer_rec) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1720 |
(insert Suc.prems, simp_all add: algebra_simps Suc_diff_le pochhammer_0_left) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1721 |
finally have "deriv (\<lambda>w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) w = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1722 |
\<dots> * (w - z) ^ (n - Suc j)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1723 |
by (rule DERIV_imp_deriv) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1724 |
} |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1725 |
finally show ?case . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1726 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1727 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1728 |
lemma zorder_zero_eqI: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1729 |
assumes f_holo:"f holomorphic_on s" and "open s" "z \<in> s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1730 |
assumes zero: "\<And>i. i < nat n \<Longrightarrow> (deriv ^^ i) f z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1731 |
assumes nz: "(deriv ^^ nat n) f z \<noteq> 0" and "n\<ge>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1732 |
shows "zorder f z = n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1733 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1734 |
obtain r where [simp]:"r>0" and "ball z r \<subseteq> s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1735 |
using \<open>open s\<close> \<open>z\<in>s\<close> openE by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1736 |
have nz':"\<exists>w\<in>ball z r. f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1737 |
proof (rule ccontr) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1738 |
assume "\<not> (\<exists>w\<in>ball z r. f w \<noteq> 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1739 |
then have "eventually (\<lambda>u. f u = 0) (nhds z)" |
76900 | 1740 |
using open_ball \<open>0 < r\<close> centre_in_ball eventually_nhds by blast |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1741 |
then have "(deriv ^^ nat n) f z = (deriv ^^ nat n) (\<lambda>_. 0) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1742 |
by (intro higher_deriv_cong_ev) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1743 |
also have "(deriv ^^ nat n) (\<lambda>_. 0) z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1744 |
by (induction n) simp_all |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1745 |
finally show False using nz by contradiction |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1746 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1747 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1748 |
define zn g where "zn = zorder f z" and "g = zor_poly f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1749 |
obtain e where e_if:"if f z = 0 then 0 < zn else zn = 0" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1750 |
[simp]:"e>0" and "cball z e \<subseteq> ball z r" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1751 |
g_holo:"g holomorphic_on cball z e" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1752 |
e_fac:"(\<forall>w\<in>cball z e. f w = g w * (w - z) ^ nat zn \<and> g w \<noteq> 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1753 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1754 |
have "f holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1755 |
using f_holo \<open>ball z r \<subseteq> s\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1756 |
from that zorder_exist_zero[of f "ball z r" z,simplified,OF this nz',folded zn_def g_def] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1757 |
show ?thesis by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1758 |
qed |
76900 | 1759 |
then obtain "zn \<ge> 0" "g z\<noteq>0" |
1760 |
by (metis centre_in_cball less_le_not_le order_refl) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1761 |
|
76900 | 1762 |
define A where "A \<equiv> (\<lambda>i. of_nat (i choose (nat zn)) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1763 |
have deriv_A:"(deriv ^^ i) f z = (if zn \<le> int i then A i else 0)" for i |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1764 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1765 |
have "eventually (\<lambda>w. w \<in> ball z e) (nhds z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1766 |
using \<open>cball z e \<subseteq> ball z r\<close> \<open>e>0\<close> by (intro eventually_nhds_in_open) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1767 |
hence "eventually (\<lambda>w. f w = (w - z) ^ (nat zn) * g w) (nhds z)" |
76900 | 1768 |
using e_fac eventually_mono by fastforce |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1769 |
hence "(deriv ^^ i) f z = (deriv ^^ i) (\<lambda>w. (w - z) ^ nat zn * g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1770 |
by (intro higher_deriv_cong_ev) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1771 |
also have "\<dots> = (\<Sum>j=0..i. of_nat (i choose j) * |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1772 |
(deriv ^^ j) (\<lambda>w. (w - z) ^ nat zn) z * (deriv ^^ (i - j)) g z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1773 |
using g_holo \<open>e>0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1774 |
by (intro higher_deriv_mult[of _ "ball z e"]) (auto intro!: holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1775 |
also have "\<dots> = (\<Sum>j=0..i. if j = nat zn then |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1776 |
of_nat (i choose nat zn) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z else 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1777 |
proof (intro sum.cong refl, goal_cases) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1778 |
case (1 j) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1779 |
have "(deriv ^^ j) (\<lambda>w. (w - z) ^ nat zn) z = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1780 |
pochhammer (of_nat (Suc (nat zn) - j)) j * 0 ^ (nat zn - j)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1781 |
by (subst higher_deriv_power) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1782 |
also have "\<dots> = (if j = nat zn then fact j else 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1783 |
by (auto simp: not_less pochhammer_0_left pochhammer_fact) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1784 |
also have "of_nat (i choose j) * \<dots> * (deriv ^^ (i - j)) g z = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1785 |
(if j = nat zn then of_nat (i choose (nat zn)) * fact (nat zn) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1786 |
* (deriv ^^ (i - nat zn)) g z else 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1787 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1788 |
finally show ?case . |
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 |
also have "\<dots> = (if i \<ge> zn then A i else 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1791 |
by (auto simp: A_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1792 |
finally show "(deriv ^^ i) f z = \<dots>" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1793 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1794 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1795 |
have False when "n<zn" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1796 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1797 |
have "(deriv ^^ nat n) f z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1798 |
using deriv_A[of "nat n"] that \<open>n\<ge>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1799 |
with nz show False by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1800 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1801 |
moreover have "n\<le>zn" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1802 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1803 |
have "g z \<noteq> 0" using e_fac[rule_format,of z] \<open>e>0\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1804 |
then have "(deriv ^^ nat zn) f z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1805 |
using deriv_A[of "nat zn"] by(auto simp add:A_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1806 |
then have "nat zn \<ge> nat n" using zero[of "nat zn"] by linarith |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1807 |
moreover have "zn\<ge>0" using e_if by (auto split:if_splits) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1808 |
ultimately show ?thesis using nat_le_eq_zle by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1809 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1810 |
ultimately show ?thesis unfolding zn_def by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1811 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1812 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1813 |
lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1814 |
assumes "eventually (\<lambda>z. f z = g z) (at z)" "z = z'" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1815 |
shows zorder_cong:"zorder f z = zorder g z'" and zor_poly_cong:"zor_poly f z = zor_poly g z'" |
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 |
define P where "P = (\<lambda>ff n h r. 0 < r \<and> h holomorphic_on cball z r \<and> h z\<noteq>0 |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1818 |
\<and> (\<forall>w\<in>cball z r - {z}. ff w = h w * (w-z) powi n \<and> h w\<noteq>0))" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1819 |
have "(\<exists>r. P f n h r) = (\<exists>r. P g n h r)" for n h |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1820 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1821 |
have *: "\<exists>r. P g n h r" if "\<exists>r. P f n h r" and "eventually (\<lambda>x. f x = g x) (at z)" for f g |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1822 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1823 |
from that(1) obtain r1 where r1_P:"P f n h r1" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1824 |
from that(2) obtain r2 where "r2>0" and r2_dist:"\<forall>x. x \<noteq> z \<and> dist x z \<le> r2 \<longrightarrow> f x = g x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1825 |
unfolding eventually_at_le by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1826 |
define r where "r=min r1 r2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1827 |
have "r>0" "h z\<noteq>0" using r1_P \<open>r2>0\<close> unfolding r_def P_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1828 |
moreover have "h holomorphic_on cball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1829 |
using r1_P unfolding P_def r_def by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1830 |
moreover have "g w = h w * (w - z) powi n \<and> h w \<noteq> 0" when "w\<in>cball z r - {z}" for w |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1831 |
proof - |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1832 |
have "f w = h w * (w - z) powi n \<and> h w \<noteq> 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1833 |
using r1_P that unfolding P_def r_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1834 |
moreover have "f w=g w" using r2_dist[rule_format,of w] that unfolding r_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1835 |
by (simp add: dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1836 |
ultimately show ?thesis by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1837 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1838 |
ultimately show ?thesis unfolding P_def by auto |
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 |
from assms have eq': "eventually (\<lambda>z. g z = f z) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1841 |
by (simp add: eq_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1842 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1843 |
by (rule iffI[OF *[OF _ assms(1)] *[OF _ eq']]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1844 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1845 |
then show "zorder f z = zorder g z'" "zor_poly f z = zor_poly g z'" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1846 |
using \<open>z=z'\<close> unfolding P_def zorder_def zor_poly_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1847 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1848 |
|
77226
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1849 |
lemma zorder_times_analytic': |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1850 |
assumes "isolated_singularity_at f z" "not_essential f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1851 |
assumes "g analytic_on {z}" "frequently (\<lambda>z. f z * g z \<noteq> 0) (at z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1852 |
shows "zorder (\<lambda>x. f x * g x) z = zorder f z + zorder g z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1853 |
proof (rule zorder_times) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1854 |
show "isolated_singularity_at g z" "not_essential g z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1855 |
by (intro isolated_singularity_at_analytic not_essential_analytic assms)+ |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1856 |
qed (use assms in auto) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1857 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1858 |
lemma zorder_cmult: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1859 |
assumes "c \<noteq> 0" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1860 |
shows "zorder (\<lambda>z. c * f z) z = zorder f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1861 |
proof - |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1862 |
define P where |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1863 |
"P = (\<lambda>f n h r. 0 < r \<and> h holomorphic_on cball z r \<and> |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1864 |
h z \<noteq> 0 \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w - z) powi n \<and> h w \<noteq> 0))" |
77226
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1865 |
have *: "P (\<lambda>x. c * f x) n (\<lambda>x. c * h x) r" if "P f n h r" "c \<noteq> 0" for f n h r c |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1866 |
using that unfolding P_def by (auto intro!: holomorphic_intros) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1867 |
have "(\<exists>h r. P (\<lambda>x. c * f x) n h r) \<longleftrightarrow> (\<exists>h r. P f n h r)" for n |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1868 |
using *[of f n _ _ c] *[of "\<lambda>x. c * f x" n _ _ "inverse c"] \<open>c \<noteq> 0\<close> |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1869 |
by (fastforce simp: field_simps) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1870 |
hence "(THE n. \<exists>h r. P (\<lambda>x. c * f x) n h r) = (THE n. \<exists>h r. P f n h r)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1871 |
by simp |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1872 |
thus ?thesis |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1873 |
by (simp add: zorder_def P_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1874 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
1875 |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1876 |
lemma zorder_nonzero_div_power: |
76900 | 1877 |
assumes sz: "open s" "z \<in> s" "f holomorphic_on s" "f z \<noteq> 0" and "n > 0" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1878 |
shows "zorder (\<lambda>w. f w / (w - z) ^ n) z = - n" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1879 |
by (intro zorder_eqI [OF sz]) (simp add: inverse_eq_divide power_int_minus) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1880 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1881 |
lemma zor_poly_eq: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1882 |
assumes "isolated_singularity_at f z" "not_essential f z" "\<exists>\<^sub>F w in at z. f w \<noteq> 0" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1883 |
shows "eventually (\<lambda>w. zor_poly f z w = f w * (w - z) powi - zorder f z) (at z)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1884 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1885 |
obtain r where r:"r>0" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1886 |
"(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w * (w - z) powi (zorder f z))" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1887 |
using zorder_exist[OF assms] by blast |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1888 |
then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w * (w - z) powi - zorder f z" |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1889 |
by (auto simp: field_simps power_int_minus) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1890 |
have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1891 |
using r eventually_at_ball'[of r z UNIV] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1892 |
thus ?thesis by eventually_elim (insert *, auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1893 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1894 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1895 |
lemma zor_poly_zero_eq: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1896 |
assumes "f holomorphic_on s" "open s" "connected s" "z \<in> s" "\<exists>w\<in>s. f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1897 |
shows "eventually (\<lambda>w. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1898 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1899 |
obtain r where r:"r>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1900 |
"(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w * (w - z) ^ nat (zorder f z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1901 |
using zorder_exist_zero[OF assms] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1902 |
then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1903 |
by (auto simp: field_simps powr_minus) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1904 |
have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1905 |
using r eventually_at_ball'[of r z UNIV] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1906 |
thus ?thesis by eventually_elim (insert *, auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1907 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1908 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1909 |
lemma zor_poly_pole_eq: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1910 |
assumes f_iso:"isolated_singularity_at f z" "is_pole f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1911 |
shows "eventually (\<lambda>w. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1912 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1913 |
obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1914 |
using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1915 |
obtain r where r:"r>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1916 |
"(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w / (w - z) ^ nat (- zorder f z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1917 |
using zorder_exist_pole[OF f_holo,simplified,OF \<open>is_pole f z\<close>] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1918 |
then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1919 |
by (auto simp: field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1920 |
have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1921 |
using r eventually_at_ball'[of r z UNIV] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1922 |
thus ?thesis by eventually_elim (insert *, auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1923 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1924 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1925 |
lemma zor_poly_eqI: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1926 |
fixes f :: "complex \<Rightarrow> complex" and z0 :: complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1927 |
defines "n \<equiv> zorder f z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1928 |
assumes "isolated_singularity_at f z0" "not_essential f z0" "\<exists>\<^sub>F w in at z0. f w \<noteq> 0" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1929 |
assumes lim: "((\<lambda>x. f (g x) * (g x - z0) powi - n) \<longlongrightarrow> c) F" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1930 |
assumes g: "filterlim g (at z0) F" and "F \<noteq> bot" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1931 |
shows "zor_poly f z0 z0 = c" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1932 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1933 |
from zorder_exist[OF assms(2-4)] obtain r where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1934 |
r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r" |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1935 |
"\<And>w. w \<in> cball z0 r - {z0} \<Longrightarrow> f w = zor_poly f z0 w * (w - z0) powi n" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1936 |
unfolding n_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1937 |
from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1938 |
using eventually_at_ball'[of r z0 UNIV] by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1939 |
hence "eventually (\<lambda>w. zor_poly f z0 w = f w * (w - z0) powi - n) (at z0)" |
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1940 |
by eventually_elim (insert r, auto simp: field_simps power_int_minus) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1941 |
moreover have "continuous_on (ball z0 r) (zor_poly f z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1942 |
using r by (intro holomorphic_on_imp_continuous_on) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1943 |
with r(1,2) have "isCont (zor_poly f z0) z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1944 |
by (auto simp: continuous_on_eq_continuous_at) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1945 |
hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1946 |
unfolding isCont_def . |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1947 |
ultimately have "((\<lambda>w. f w * (w - z0) powi - n) \<longlongrightarrow> zor_poly f z0 z0) (at z0)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1948 |
by (blast intro: Lim_transform_eventually) |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
1949 |
hence "((\<lambda>x. f (g x) * (g x - z0) powi - n) \<longlongrightarrow> zor_poly f z0 z0) F" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1950 |
by (rule filterlim_compose[OF _ g]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1951 |
from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1952 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1953 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1954 |
lemma zor_poly_zero_eqI: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1955 |
fixes f :: "complex \<Rightarrow> complex" and z0 :: complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1956 |
defines "n \<equiv> zorder f z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1957 |
assumes "f holomorphic_on A" "open A" "connected A" "z0 \<in> A" "\<exists>z\<in>A. f z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1958 |
assumes lim: "((\<lambda>x. f (g x) / (g x - z0) ^ nat n) \<longlongrightarrow> c) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1959 |
assumes g: "filterlim g (at z0) F" and "F \<noteq> bot" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1960 |
shows "zor_poly f z0 z0 = c" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1961 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1962 |
from zorder_exist_zero[OF assms(2-6)] obtain r where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1963 |
r: "r > 0" "cball z0 r \<subseteq> A" "zor_poly f z0 holomorphic_on cball z0 r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1964 |
"\<And>w. w \<in> cball z0 r \<Longrightarrow> f w = zor_poly f z0 w * (w - z0) ^ nat n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1965 |
unfolding n_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1966 |
from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1967 |
using eventually_at_ball'[of r z0 UNIV] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1968 |
hence "eventually (\<lambda>w. zor_poly f z0 w = f w / (w - z0) ^ nat n) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1969 |
by eventually_elim (insert r, auto simp: field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1970 |
moreover have "continuous_on (ball z0 r) (zor_poly f z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1971 |
using r by (intro holomorphic_on_imp_continuous_on) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1972 |
with r(1,2) have "isCont (zor_poly f z0) z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1973 |
by (auto simp: continuous_on_eq_continuous_at) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1974 |
hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1975 |
unfolding isCont_def . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1976 |
ultimately have "((\<lambda>w. f w / (w - z0) ^ nat n) \<longlongrightarrow> zor_poly f z0 z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1977 |
by (blast intro: Lim_transform_eventually) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1978 |
hence "((\<lambda>x. f (g x) / (g x - z0) ^ nat n) \<longlongrightarrow> zor_poly f z0 z0) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1979 |
by (rule filterlim_compose[OF _ g]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1980 |
from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1981 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1982 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1983 |
lemma zor_poly_pole_eqI: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1984 |
fixes f :: "complex \<Rightarrow> complex" and z0 :: complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1985 |
defines "n \<equiv> zorder f z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1986 |
assumes f_iso:"isolated_singularity_at f z0" and "is_pole f z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1987 |
assumes lim: "((\<lambda>x. f (g x) * (g x - z0) ^ nat (-n)) \<longlongrightarrow> c) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1988 |
assumes g: "filterlim g (at z0) F" and "F \<noteq> bot" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1989 |
shows "zor_poly f z0 z0 = c" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1990 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1991 |
obtain r where r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1992 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1993 |
have "\<exists>\<^sub>F w in at z0. f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1994 |
using non_zero_neighbour_pole[OF \<open>is_pole f z0\<close>] by (auto elim:eventually_frequentlyE) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1995 |
moreover have "not_essential f z0" unfolding not_essential_def using \<open>is_pole f z0\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1996 |
ultimately show ?thesis using that zorder_exist[OF f_iso,folded n_def] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1997 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1998 |
from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1999 |
using eventually_at_ball'[of r z0 UNIV] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2000 |
have "eventually (\<lambda>w. zor_poly f z0 w = f w * (w - z0) ^ nat (-n)) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2001 |
using zor_poly_pole_eq[OF f_iso \<open>is_pole f z0\<close>] unfolding n_def . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2002 |
moreover have "continuous_on (ball z0 r) (zor_poly f z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2003 |
using r by (intro holomorphic_on_imp_continuous_on) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2004 |
with r(1,2) have "isCont (zor_poly f z0) z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2005 |
by (auto simp: continuous_on_eq_continuous_at) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2006 |
hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2007 |
unfolding isCont_def . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2008 |
ultimately have "((\<lambda>w. f w * (w - z0) ^ nat (-n)) \<longlongrightarrow> zor_poly f z0 z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2009 |
by (blast intro: Lim_transform_eventually) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2010 |
hence "((\<lambda>x. f (g x) * (g x - z0) ^ nat (-n)) \<longlongrightarrow> zor_poly f z0 z0) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2011 |
by (rule filterlim_compose[OF _ g]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2012 |
from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2013 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2014 |
|
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2015 |
lemma |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2016 |
assumes "is_pole f (x :: complex)" "open A" "x \<in> A" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2017 |
assumes "\<And>y. y \<in> A - {x} \<Longrightarrow> (f has_field_derivative f' y) (at y)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2018 |
shows is_pole_deriv': "is_pole f' x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2019 |
and zorder_deriv': "zorder f' x = zorder f x - 1" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2020 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2021 |
have holo: "f holomorphic_on A - {x}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2022 |
using assms by (subst holomorphic_on_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2023 |
obtain r where r: "r > 0" "ball x r \<subseteq> A" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2024 |
using assms(2,3) openE by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2025 |
moreover have "open (ball x r - {x})" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2026 |
by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2027 |
ultimately have "isolated_singularity_at f x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2028 |
by (auto simp: isolated_singularity_at_def analytic_on_open |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2029 |
intro!: exI[of _ r] holomorphic_on_subset[OF holo]) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2030 |
hence ev: "\<forall>\<^sub>F w in at x. zor_poly f x w = f w * (w - x) ^ nat (- zorder f x)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2031 |
using \<open>is_pole f x\<close> zor_poly_pole_eq by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2032 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2033 |
define P where "P = zor_poly f x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2034 |
define n where "n = nat (-zorder f x)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2035 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2036 |
obtain r where r: "r > 0" "cball x r \<subseteq> A" "P holomorphic_on cball x r" "zorder f x < 0" "P x \<noteq> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2037 |
"\<forall>w\<in>cball x r - {x}. f w = P w / (w - x) ^ n \<and> P w \<noteq> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2038 |
unfolding P_def n_def using zorder_exist_pole[OF holo assms(2,3,1)] by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2039 |
have n: "n > 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2040 |
using r(4) by (auto simp: n_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2041 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2042 |
have [derivative_intros]: "(P has_field_derivative deriv P w) (at w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2043 |
if "w \<in> ball x r" for w |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2044 |
using that by (intro holomorphic_derivI[OF holomorphic_on_subset[OF r(3), of "ball x r"]]) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2045 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2046 |
define D where "D = (\<lambda>w. (deriv P w * (w - x) - of_nat n * P w) / (w - x) ^ (n + 1))" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2047 |
define n' where "n' = n - 1" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2048 |
have n': "n = Suc n'" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2049 |
using n by (simp add: n'_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2050 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2051 |
have "eventually (\<lambda>w. w \<in> ball x r) (nhds x)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2052 |
using \<open>r > 0\<close> by (intro eventually_nhds_in_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2053 |
hence ev'': "eventually (\<lambda>w. w \<in> ball x r - {x}) (at x)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2054 |
by (auto simp: eventually_at_filter elim: eventually_mono) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2055 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2056 |
{ |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2057 |
fix w assume w: "w \<in> ball x r - {x}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2058 |
have ev': "eventually (\<lambda>w. w \<in> ball x r - {x}) (nhds w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2059 |
using w by (intro eventually_nhds_in_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2060 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2061 |
have "((\<lambda>w. P w / (w - x) ^ n) has_field_derivative D w) (at w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2062 |
apply (rule derivative_eq_intros refl | use w in force)+ |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2063 |
using w |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2064 |
apply (simp add: divide_simps D_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2065 |
apply (simp add: n' algebra_simps) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2066 |
done |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2067 |
also have "?this \<longleftrightarrow> (f has_field_derivative D w) (at w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2068 |
using r by (intro has_field_derivative_cong_ev refl eventually_mono[OF ev']) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2069 |
finally have "(f has_field_derivative D w) (at w)" . |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2070 |
moreover have "(f has_field_derivative f' w) (at w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2071 |
using w r by (intro assms) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2072 |
ultimately have "D w = f' w" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2073 |
using DERIV_unique by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2074 |
} note D_eq = this |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2075 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2076 |
have "is_pole D x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2077 |
unfolding D_def using n \<open>r > 0\<close> \<open>P x \<noteq> 0\<close> |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2078 |
by (intro is_pole_basic[where A = "ball x r"] holomorphic_intros holomorphic_on_subset[OF r(3)]) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2079 |
also have "?this \<longleftrightarrow> is_pole f' x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2080 |
by (intro is_pole_cong eventually_mono[OF ev''] D_eq) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2081 |
finally show "is_pole f' x" . |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2082 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2083 |
have "zorder f' x = -int (Suc n)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2084 |
proof (rule zorder_eqI) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2085 |
show "open (ball x r)" "x \<in> ball x r" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2086 |
using \<open>r > 0\<close> by auto |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
2087 |
show "f' w = (deriv P w * (w - x) - of_nat n * P w) * (w - x) powi (- int (Suc n))" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2088 |
if "w \<in> ball x r" "w \<noteq> x" for w |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
2089 |
using that D_eq[of w] n by (auto simp: D_def power_int_diff power_int_minus powr_nat' divide_simps) |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2090 |
qed (use r n in \<open>auto intro!: holomorphic_intros\<close>) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2091 |
thus "zorder f' x = zorder f x - 1" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2092 |
using n by (simp add: n_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2093 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2094 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2095 |
lemma |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2096 |
assumes "is_pole f (x :: complex)" "isolated_singularity_at f x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2097 |
shows is_pole_deriv: "is_pole (deriv f) x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2098 |
and zorder_deriv: "zorder (deriv f) x = zorder f x - 1" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2099 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2100 |
from assms(2) obtain r where r: "r > 0" "f analytic_on ball x r - {x}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2101 |
by (auto simp: isolated_singularity_at_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2102 |
hence holo: "f holomorphic_on ball x r - {x}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2103 |
by (subst (asm) analytic_on_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2104 |
have *: "x \<in> ball x r" "open (ball x r)" "open (ball x r - {x})" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2105 |
using \<open>r > 0\<close> by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2106 |
show "is_pole (deriv f) x" "zorder (deriv f) x = zorder f x - 1" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2107 |
by (rule is_pole_deriv' zorder_deriv', (rule assms * holomorphic_derivI holo | assumption)+)+ |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2108 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2109 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2110 |
lemma removable_singularity_deriv': |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2111 |
assumes "f \<midarrow>x\<rightarrow> c" "x \<in> A" "open (A :: complex set)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2112 |
assumes "\<And>y. y \<in> A - {x} \<Longrightarrow> (f has_field_derivative f' y) (at y)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2113 |
shows "\<exists>c. f' \<midarrow>x\<rightarrow> c" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2114 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2115 |
have holo: "f holomorphic_on A - {x}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2116 |
using assms by (subst holomorphic_on_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2117 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2118 |
define g where "g = (\<lambda>y. if y = x then c else f y)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2119 |
have deriv_g_eq: "deriv g y = f' y" if "y \<in> A - {x}" for y |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2120 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2121 |
have ev: "eventually (\<lambda>y. y \<in> A - {x}) (nhds y)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2122 |
using that assms by (intro eventually_nhds_in_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2123 |
have "(f has_field_derivative f' y) (at y)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2124 |
using assms that by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2125 |
also have "?this \<longleftrightarrow> (g has_field_derivative f' y) (at y)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2126 |
by (intro has_field_derivative_cong_ev refl eventually_mono[OF ev]) (auto simp: g_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2127 |
finally show ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2128 |
by (intro DERIV_imp_deriv assms) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2129 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2130 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2131 |
have "g holomorphic_on A" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2132 |
unfolding g_def using assms assms(1) holo by (intro removable_singularity) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2133 |
hence "deriv g holomorphic_on A" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2134 |
by (intro holomorphic_deriv assms) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2135 |
hence "continuous_on A (deriv g)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2136 |
by (meson holomorphic_on_imp_continuous_on) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2137 |
hence "(deriv g \<longlongrightarrow> deriv g x) (at x within A)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2138 |
using assms by (auto simp: continuous_on_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2139 |
also have "?this \<longleftrightarrow> (f' \<longlongrightarrow> deriv g x) (at x within A)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2140 |
by (intro filterlim_cong refl) (auto simp: eventually_at_filter deriv_g_eq) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2141 |
finally have "f' \<midarrow>x\<rightarrow> deriv g x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2142 |
using \<open>open A\<close> \<open>x \<in> A\<close> by (meson tendsto_within_open) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2143 |
thus ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2144 |
by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2145 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2146 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2147 |
lemma removable_singularity_deriv: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2148 |
assumes "f \<midarrow>x\<rightarrow> c" "isolated_singularity_at f x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2149 |
shows "\<exists>c. deriv f \<midarrow>x\<rightarrow> c" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2150 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2151 |
from assms(2) obtain r where r: "r > 0" "f analytic_on ball x r - {x}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2152 |
by (auto simp: isolated_singularity_at_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2153 |
hence holo: "f holomorphic_on ball x r - {x}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2154 |
using analytic_imp_holomorphic by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2155 |
show ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2156 |
using assms(1) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2157 |
proof (rule removable_singularity_deriv') |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2158 |
show "x \<in> ball x r" "open (ball x r)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2159 |
using \<open>r > 0\<close> by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2160 |
qed (auto intro!: holomorphic_derivI[OF holo]) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2161 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2162 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2163 |
lemma not_essential_deriv': |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2164 |
assumes "not_essential f x" "x \<in> A" "open A" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2165 |
assumes "\<And>y. y \<in> A - {x} \<Longrightarrow> (f has_field_derivative f' y) (at y)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2166 |
shows "not_essential f' x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2167 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2168 |
have holo: "f holomorphic_on A - {x}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2169 |
using assms by (subst holomorphic_on_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2170 |
from assms consider "is_pole f x" | c where "f \<midarrow>x\<rightarrow> c" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2171 |
by (auto simp: not_essential_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2172 |
thus ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2173 |
proof cases |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2174 |
case 1 |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2175 |
hence "is_pole f' x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2176 |
using is_pole_deriv' assms by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2177 |
thus ?thesis by (auto simp: not_essential_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2178 |
next |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2179 |
case (2 c) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2180 |
from 2 have "\<exists>c. f' \<midarrow>x\<rightarrow> c" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2181 |
by (rule removable_singularity_deriv'[OF _ assms(2-4)]) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2182 |
thus ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2183 |
by (auto simp: not_essential_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2184 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2185 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2186 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2187 |
lemma not_essential_deriv[singularity_intros]: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2188 |
assumes "not_essential f x" "isolated_singularity_at f x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2189 |
shows "not_essential (deriv f) x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2190 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2191 |
from assms(2) obtain r where r: "r > 0" "f analytic_on ball x r - {x}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2192 |
by (auto simp: isolated_singularity_at_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2193 |
hence holo: "f holomorphic_on ball x r - {x}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2194 |
by (subst (asm) analytic_on_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2195 |
show ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2196 |
using assms(1) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2197 |
proof (rule not_essential_deriv') |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2198 |
show "x \<in> ball x r" "open (ball x r)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2199 |
using \<open>r > 0\<close> by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2200 |
qed (auto intro!: holomorphic_derivI[OF holo]) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2201 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2202 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2203 |
lemma not_essential_frequently_0_imp_tendsto_0: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2204 |
fixes f :: "complex \<Rightarrow> complex" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2205 |
assumes sing: "isolated_singularity_at f z" "not_essential f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2206 |
assumes freq: "frequently (\<lambda>z. f z = 0) (at z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2207 |
shows "f \<midarrow>z\<rightarrow> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2208 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2209 |
from freq obtain g :: "nat \<Rightarrow> complex" where g: "filterlim g (at z) at_top" "\<And>n. f (g n) = 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2210 |
using frequently_atE by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2211 |
have "eventually (\<lambda>x. f (g x) = 0) sequentially" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2212 |
using g by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2213 |
hence fg: "(\<lambda>x. f (g x)) \<longlonglongrightarrow> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2214 |
by (simp add: tendsto_eventually) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2215 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2216 |
from assms(2) consider c where "f \<midarrow>z\<rightarrow> c" | "is_pole f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2217 |
unfolding not_essential_def by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2218 |
thus ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2219 |
proof cases |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2220 |
case (1 c) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2221 |
have "(\<lambda>x. f (g x)) \<longlonglongrightarrow> c" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2222 |
by (rule filterlim_compose[OF 1 g(1)]) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2223 |
with fg have "c = 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2224 |
using LIMSEQ_unique by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2225 |
with 1 show ?thesis by simp |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2226 |
next |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2227 |
case 2 |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2228 |
have "filterlim (\<lambda>x. f (g x)) at_infinity sequentially" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2229 |
by (rule filterlim_compose[OF _ g(1)]) (use 2 in \<open>auto simp: is_pole_def\<close>) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2230 |
with fg have False |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2231 |
by (meson not_tendsto_and_filterlim_at_infinity sequentially_bot) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2232 |
thus ?thesis .. |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2233 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2234 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2235 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2236 |
lemma not_essential_frequently_0_imp_eventually_0: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2237 |
fixes f :: "complex \<Rightarrow> complex" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2238 |
assumes sing: "isolated_singularity_at f z" "not_essential f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2239 |
assumes freq: "frequently (\<lambda>z. f z = 0) (at z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2240 |
shows "eventually (\<lambda>z. f z = 0) (at z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2241 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2242 |
from sing obtain r where r: "r > 0" and "f analytic_on ball z r - {z}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2243 |
by (auto simp: isolated_singularity_at_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2244 |
hence holo: "f holomorphic_on ball z r - {z}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2245 |
by (subst (asm) analytic_on_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2246 |
have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2247 |
using r by (intro eventually_at_in_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2248 |
from freq and this have "frequently (\<lambda>w. f w = 0 \<and> w \<in> ball z r - {z}) (at z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2249 |
using frequently_eventually_frequently by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2250 |
hence "frequently (\<lambda>w. w \<in> {w\<in>ball z r - {z}. f w = 0}) (at z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2251 |
by (simp add: conj_commute) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2252 |
hence limpt: "z islimpt {w\<in>ball z r - {z}. f w = 0}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2253 |
using islimpt_conv_frequently_at by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2254 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2255 |
define g where "g = (\<lambda>w. if w = z then 0 else f w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2256 |
have "f \<midarrow>z\<rightarrow> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2257 |
by (intro not_essential_frequently_0_imp_tendsto_0 assms) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2258 |
hence g_holo: "g holomorphic_on ball z r" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2259 |
unfolding g_def by (intro removable_singularity holo) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2260 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2261 |
have g_eq_0: "g w = 0" if "w \<in> ball z r" for w |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2262 |
proof (rule analytic_continuation[where f = g]) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2263 |
show "open (ball z r)" "connected (ball z r)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2264 |
using r by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2265 |
show "z islimpt {w\<in>ball z r - {z}. f w = 0}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2266 |
by fact |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2267 |
show "g w = 0" if "w \<in> {w \<in> ball z r - {z}. f w = 0}" for w |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2268 |
using that by (auto simp: g_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2269 |
qed (use r that g_holo in auto) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2270 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2271 |
have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2272 |
using r by (intro eventually_at_in_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2273 |
thus "eventually (\<lambda>w. f w = 0) (at z)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2274 |
proof eventually_elim |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2275 |
case (elim w) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2276 |
thus ?case using g_eq_0[of w] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2277 |
by (auto simp: g_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2278 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2279 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2280 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2281 |
lemma pole_imp_not_constant: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2282 |
fixes f :: "'a :: {perfect_space} \<Rightarrow> _" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2283 |
assumes "is_pole f x" "open A" "x \<in> A" "A \<subseteq> insert x B" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2284 |
shows "\<not>f constant_on B" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2285 |
proof |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2286 |
assume *: "f constant_on B" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2287 |
then obtain c where c: "\<forall>x\<in>B. f x = c" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2288 |
by (auto simp: constant_on_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2289 |
have "eventually (\<lambda>y. y \<in> A - {x}) (at x)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2290 |
using assms by (intro eventually_at_in_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2291 |
hence "eventually (\<lambda>y. f y = c) (at x)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2292 |
by eventually_elim (use c assms in auto) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2293 |
hence **: "f \<midarrow>x\<rightarrow> c" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2294 |
by (simp add: tendsto_eventually) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2295 |
show False |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2296 |
using not_tendsto_and_filterlim_at_infinity[OF _ ** assms(1)[unfolded is_pole_def]] by simp |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2297 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2298 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2299 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2300 |
lemma neg_zorder_imp_is_pole: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2301 |
assumes iso:"isolated_singularity_at f z" and f_ness:"not_essential f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2302 |
and "zorder f z < 0" and fre_nz:"\<exists>\<^sub>F w in at z. f w \<noteq> 0 " |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2303 |
shows "is_pole f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2304 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2305 |
define P where "P = zor_poly f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2306 |
define n where "n = zorder f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2307 |
have "n<0" unfolding n_def by (simp add: assms(3)) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2308 |
define nn where "nn = nat (-n)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2309 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2310 |
obtain r where "P z \<noteq> 0" "r>0" and r_holo:"P holomorphic_on cball z r" and |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
2311 |
w_Pn:"(\<forall>w\<in>cball z r - {z}. f w = P w * (w - z) powi n \<and> P w \<noteq> 0)" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2312 |
using zorder_exist[OF iso f_ness fre_nz,folded P_def n_def] by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2313 |
|
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
2314 |
have "is_pole (\<lambda>w. P w * (w - z) powi n) z" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2315 |
unfolding is_pole_def |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2316 |
proof (rule tendsto_mult_filterlim_at_infinity) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2317 |
show "P \<midarrow>z\<rightarrow> P z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2318 |
by (meson open_ball \<open>0 < r\<close> ball_subset_cball centre_in_ball |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2319 |
continuous_on_eq_continuous_at continuous_on_subset |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2320 |
holomorphic_on_imp_continuous_on isContD r_holo) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2321 |
show "P z\<noteq>0" by (simp add: \<open>P z \<noteq> 0\<close>) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2322 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2323 |
have "LIM x at z. inverse ((x - z) ^ nat (-n)) :> at_infinity" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2324 |
apply (subst filterlim_inverse_at_iff[symmetric]) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2325 |
using \<open>n<0\<close> |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2326 |
by (auto intro!:tendsto_eq_intros filterlim_atI |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2327 |
simp add:eventually_at_filter) |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
2328 |
then show "LIM x at z. (x - z) powi n :> at_infinity" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2329 |
proof (elim filterlim_mono_eventually) |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
2330 |
have "inverse ((x - z) ^ nat (-n)) = (x - z) powi n" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2331 |
if "x\<noteq>z" for x |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
2332 |
by (metis \<open>n < 0\<close> linorder_not_le power_int_def power_inverse) |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2333 |
then show "\<forall>\<^sub>F x in at z. inverse ((x - z) ^ nat (-n)) |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
2334 |
= (x - z) powi n" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2335 |
by (simp add: eventually_at_filter) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2336 |
qed auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2337 |
qed |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
2338 |
moreover have "\<forall>\<^sub>F w in at z. f w = P w * (w - z) powi n" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2339 |
unfolding eventually_at_le |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2340 |
apply (rule exI[where x=r]) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2341 |
using w_Pn \<open>r>0\<close> by (simp add: dist_commute) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2342 |
ultimately show ?thesis using is_pole_cong by fast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2343 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2344 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2345 |
lemma is_pole_divide_zorder: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2346 |
fixes f g::"complex \<Rightarrow> complex" and z::complex |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2347 |
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2348 |
and f_ness:"not_essential f z" and g_ness:"not_essential g z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2349 |
and fg_nconst: "\<exists>\<^sub>Fw in (at z). f w * g w\<noteq> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2350 |
and z_less:"zorder f z < zorder g z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2351 |
shows "is_pole (\<lambda>z. f z / g z) z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2352 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2353 |
define fn gn fg where "fn=zorder f z" and "gn=zorder g z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2354 |
and "fg=(\<lambda>w. f w / g w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2355 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2356 |
have "isolated_singularity_at fg z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2357 |
unfolding fg_def using f_iso g_iso g_ness |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2358 |
by (auto intro:singularity_intros) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2359 |
moreover have "not_essential fg z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2360 |
unfolding fg_def using f_iso g_iso g_ness f_ness |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2361 |
by (auto intro:singularity_intros) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2362 |
moreover have "zorder fg z < 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2363 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2364 |
have "zorder fg z = fn - gn" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2365 |
using zorder_divide[OF f_iso g_iso f_ness g_ness |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2366 |
fg_nconst,folded fn_def gn_def fg_def] . |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2367 |
then show ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2368 |
using z_less by (simp add: fn_def gn_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2369 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2370 |
moreover have "\<exists>\<^sub>F w in at z. fg w \<noteq> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2371 |
using fg_nconst unfolding fg_def by force |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2372 |
ultimately show "is_pole fg z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2373 |
using neg_zorder_imp_is_pole by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2374 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2375 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2376 |
lemma isolated_pole_imp_nzero_times: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2377 |
assumes f_iso:"isolated_singularity_at f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2378 |
and "is_pole f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2379 |
shows "\<exists>\<^sub>Fw in (at z). deriv f w * f w \<noteq> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2380 |
proof (rule ccontr) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2381 |
assume "\<not> (\<exists>\<^sub>F w in at z. deriv f w * f w \<noteq> 0)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2382 |
then have "\<forall>\<^sub>F x in at z. deriv f x * f x = 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2383 |
unfolding not_frequently by simp |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2384 |
moreover have "\<forall>\<^sub>F w in at z. f w \<noteq> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2385 |
using non_zero_neighbour_pole[OF \<open>is_pole f z\<close>] . |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2386 |
moreover have "\<forall>\<^sub>F w in at z. deriv f w \<noteq> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2387 |
using is_pole_deriv[OF \<open>is_pole f z\<close> f_iso,THEN non_zero_neighbour_pole] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2388 |
. |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2389 |
ultimately have "\<forall>\<^sub>F w in at z. False" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2390 |
apply eventually_elim |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2391 |
by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2392 |
then show False by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2393 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2394 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2395 |
lemma isolated_pole_imp_neg_zorder: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2396 |
assumes f_iso:"isolated_singularity_at f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2397 |
and "is_pole f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2398 |
shows "zorder f z<0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2399 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2400 |
obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2401 |
using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2402 |
show ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2403 |
using zorder_exist_pole[OF f_holo,simplified,OF \<open>is_pole f z\<close>] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2404 |
by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2405 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2406 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2407 |
lemma isolated_singularity_at_deriv[singularity_intros]: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2408 |
assumes "isolated_singularity_at f x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2409 |
shows "isolated_singularity_at (deriv f) x" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2410 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2411 |
obtain r where "r>0" "f analytic_on ball x r - {x}" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2412 |
using assms unfolding isolated_singularity_at_def by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2413 |
from analytic_deriv[OF this(2)] |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2414 |
have "deriv f analytic_on ball x r - {x}" . |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2415 |
with \<open>r>0\<close> show ?thesis |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2416 |
unfolding isolated_singularity_at_def by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2417 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2418 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2419 |
lemma zorder_deriv_minus_1: |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2420 |
fixes f g::"complex \<Rightarrow> complex" and z::complex |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2421 |
assumes f_iso:"isolated_singularity_at f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2422 |
and f_ness:"not_essential f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2423 |
and f_nconst:"\<exists>\<^sub>F w in at z. f w \<noteq> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2424 |
and f_ord:"zorder f z \<noteq>0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2425 |
shows "zorder (deriv f) z = zorder f z - 1" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2426 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2427 |
define P where "P = zor_poly f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2428 |
define n where "n = zorder f z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2429 |
have "n\<noteq>0" unfolding n_def using f_ord by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2430 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2431 |
obtain r where "P z \<noteq> 0" "r>0" and P_holo:"P holomorphic_on cball z r" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2432 |
and "(\<forall>w\<in>cball z r - {z}. f w |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
2433 |
= P w * (w - z) powi n \<and> P w \<noteq> 0)" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2434 |
using zorder_exist[OF f_iso f_ness f_nconst,folded P_def n_def] by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2435 |
from this(4) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2436 |
have f_eq:"(\<forall>w\<in>cball z r - {z}. f w |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2437 |
= P w * (w - z) powi n \<and> P w \<noteq> 0)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2438 |
using complex_powr_of_int f_ord n_def by presburger |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2439 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2440 |
define D where "D = (\<lambda>w. (deriv P w * (w - z) + of_int n * P w) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2441 |
* (w - z) powi (n - 1))" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2442 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2443 |
have deriv_f_eq:"deriv f w = D w" if "w \<in> ball z r - {z}" for w |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2444 |
proof - |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2445 |
have ev': "eventually (\<lambda>w. w \<in> ball z r - {z}) (nhds w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2446 |
using that by (intro eventually_nhds_in_open) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2447 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2448 |
define wz where "wz = w - z" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2449 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2450 |
have "wz \<noteq>0" unfolding wz_def using that by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2451 |
moreover have "(P has_field_derivative deriv P w) (at w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2452 |
by (meson DiffD1 Elementary_Metric_Spaces.open_ball P_holo |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2453 |
ball_subset_cball holomorphic_derivI holomorphic_on_subset that) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2454 |
ultimately have "((\<lambda>w. P w * (w - z) powi n) has_field_derivative D w) (at w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2455 |
unfolding D_def using that |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2456 |
apply (auto intro!: derivative_eq_intros) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2457 |
apply (fold wz_def) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2458 |
by (auto simp:algebra_simps simp flip:power_int_add_1') |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2459 |
also have "?this \<longleftrightarrow> (f has_field_derivative D w) (at w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2460 |
using f_eq |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2461 |
by (intro has_field_derivative_cong_ev refl eventually_mono[OF ev']) auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2462 |
ultimately have "(f has_field_derivative D w) (at w)" by simp |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2463 |
moreover have "(f has_field_derivative deriv f w) (at w)" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2464 |
by (metis DERIV_imp_deriv calculation) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2465 |
ultimately show ?thesis using DERIV_imp_deriv by blast |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2466 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2467 |
|
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2468 |
show "zorder (deriv f) z = n - 1" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2469 |
proof (rule zorder_eqI) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2470 |
show "open (ball z r)" "z \<in> ball z r" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2471 |
using \<open>r > 0\<close> by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2472 |
define g where "g=(\<lambda>w. (deriv P w * (w - z) + of_int n * P w))" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2473 |
show "g holomorphic_on ball z r" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2474 |
unfolding g_def using P_holo |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2475 |
by (auto intro!:holomorphic_intros) |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2476 |
show "g z \<noteq> 0" |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2477 |
unfolding g_def using \<open>P z \<noteq> 0\<close> \<open>n\<noteq>0\<close> by auto |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2478 |
show "deriv f w = |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
2479 |
(deriv P w * (w - z) + of_int n * P w) * (w - z) powi (n - 1)" |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2480 |
if "w \<in> ball z r" "w \<noteq> z" for w |
77322
9c295f84d55f
Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents:
77277
diff
changeset
|
2481 |
using D_def deriv_f_eq that by blast |
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2482 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2483 |
qed |
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2484 |
|
77226
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2485 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2486 |
lemma deriv_divide_is_pole: \<comment>\<open>Generalises @{thm zorder_deriv}\<close> |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2487 |
fixes f g::"complex \<Rightarrow> complex" and z::complex |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2488 |
assumes f_iso:"isolated_singularity_at f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2489 |
and f_ness:"not_essential f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2490 |
and fg_nconst: "\<exists>\<^sub>Fw in (at z). deriv f w * f w \<noteq> 0" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2491 |
and f_ord:"zorder f z \<noteq>0" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2492 |
shows "is_pole (\<lambda>z. deriv f z / f z) z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2493 |
proof (rule neg_zorder_imp_is_pole) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2494 |
define ff where "ff=(\<lambda>w. deriv f w / f w)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2495 |
show "isolated_singularity_at ff z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2496 |
using f_iso f_ness unfolding ff_def |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2497 |
by (auto intro:singularity_intros) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2498 |
show "not_essential ff z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2499 |
unfolding ff_def using f_ness f_iso |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2500 |
by (auto intro:singularity_intros) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2501 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2502 |
have "zorder ff z = zorder (deriv f) z - zorder f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2503 |
unfolding ff_def using f_iso f_ness fg_nconst |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2504 |
apply (rule_tac zorder_divide) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2505 |
by (auto intro:singularity_intros) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2506 |
moreover have "zorder (deriv f) z = zorder f z - 1" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2507 |
proof (rule zorder_deriv_minus_1) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2508 |
show " \<exists>\<^sub>F w in at z. f w \<noteq> 0" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2509 |
using fg_nconst frequently_elim1 by fastforce |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2510 |
qed (use f_iso f_ness f_ord in auto) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2511 |
ultimately show "zorder ff z < 0" by auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2512 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2513 |
show "\<exists>\<^sub>F w in at z. ff w \<noteq> 0" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2514 |
unfolding ff_def using fg_nconst by auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2515 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2516 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2517 |
lemma is_pole_deriv_divide_is_pole: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2518 |
fixes f g::"complex \<Rightarrow> complex" and z::complex |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2519 |
assumes f_iso:"isolated_singularity_at f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2520 |
and "is_pole f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2521 |
shows "is_pole (\<lambda>z. deriv f z / f z) z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2522 |
proof (rule deriv_divide_is_pole[OF f_iso]) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2523 |
show "not_essential f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2524 |
using \<open>is_pole f z\<close> unfolding not_essential_def by auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2525 |
show "\<exists>\<^sub>F w in at z. deriv f w * f w \<noteq> 0" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2526 |
apply (rule isolated_pole_imp_nzero_times) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2527 |
using assms by auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2528 |
show "zorder f z \<noteq> 0" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2529 |
using isolated_pole_imp_neg_zorder assms by fastforce |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2530 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2531 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2532 |
subsection \<open>Isolated zeroes\<close> |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2533 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2534 |
definition isolated_zero :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> bool" where |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2535 |
"isolated_zero f z \<longleftrightarrow> f z = 0 \<and> eventually (\<lambda>z. f z \<noteq> 0) (at z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2536 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2537 |
lemma isolated_zero_altdef: "isolated_zero f z \<longleftrightarrow> f z = 0 \<and> \<not>z islimpt {z. f z = 0}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2538 |
unfolding isolated_zero_def eventually_at_filter eventually_nhds islimpt_def by blast |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2539 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2540 |
lemma isolated_zero_mult1: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2541 |
assumes "isolated_zero f x" "isolated_zero g x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2542 |
shows "isolated_zero (\<lambda>x. f x * g x) x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2543 |
proof - |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2544 |
have "eventually (\<lambda>x. f x \<noteq> 0) (at x)" "eventually (\<lambda>x. g x \<noteq> 0) (at x)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2545 |
using assms unfolding isolated_zero_def by auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2546 |
hence "eventually (\<lambda>x. f x * g x \<noteq> 0) (at x)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2547 |
by eventually_elim auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2548 |
with assms show ?thesis |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2549 |
by (auto simp: isolated_zero_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2550 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2551 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2552 |
lemma isolated_zero_mult2: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2553 |
assumes "isolated_zero f x" "g x \<noteq> 0" "g analytic_on {x}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2554 |
shows "isolated_zero (\<lambda>x. f x * g x) x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2555 |
proof - |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2556 |
have "eventually (\<lambda>x. f x \<noteq> 0) (at x)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2557 |
using assms unfolding isolated_zero_def by auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2558 |
moreover have "eventually (\<lambda>x. g x \<noteq> 0) (at x)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2559 |
using analytic_at_neq_imp_eventually_neq[of g x 0] assms by auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2560 |
ultimately have "eventually (\<lambda>x. f x * g x \<noteq> 0) (at x)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2561 |
by eventually_elim auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2562 |
thus ?thesis |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2563 |
using assms(1) by (auto simp: isolated_zero_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2564 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2565 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2566 |
lemma isolated_zero_mult3: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2567 |
assumes "isolated_zero f x" "g x \<noteq> 0" "g analytic_on {x}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2568 |
shows "isolated_zero (\<lambda>x. g x * f x) x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2569 |
using isolated_zero_mult2[OF assms] by (simp add: mult_ac) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2570 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2571 |
lemma isolated_zero_prod: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2572 |
assumes "\<And>x. x \<in> I \<Longrightarrow> isolated_zero (f x) z" "I \<noteq> {}" "finite I" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2573 |
shows "isolated_zero (\<lambda>y. \<Prod>x\<in>I. f x y) z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2574 |
using assms(3,2,1) by (induction rule: finite_ne_induct) (auto intro: isolated_zero_mult1) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2575 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2576 |
lemma non_isolated_zero': |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2577 |
assumes "isolated_singularity_at f z" "not_essential f z" "f z = 0" "\<not>isolated_zero f z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2578 |
shows "eventually (\<lambda>z. f z = 0) (at z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2579 |
proof (rule not_essential_frequently_0_imp_eventually_0) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2580 |
from assms show "frequently (\<lambda>z. f z = 0) (at z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2581 |
by (auto simp: frequently_def isolated_zero_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2582 |
qed fact+ |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2583 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2584 |
lemma non_isolated_zero: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2585 |
assumes "\<not>isolated_zero f z" "f analytic_on {z}" "f z = 0" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2586 |
shows "eventually (\<lambda>z. f z = 0) (nhds z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2587 |
proof - |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2588 |
have "eventually (\<lambda>z. f z = 0) (at z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2589 |
by (rule non_isolated_zero') |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2590 |
(use assms in \<open>auto intro: not_essential_analytic isolated_singularity_at_analytic\<close>) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2591 |
with \<open>f z = 0\<close> show ?thesis |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2592 |
unfolding eventually_at_filter by (auto elim!: eventually_mono) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2593 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2594 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2595 |
lemma not_essential_compose: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2596 |
assumes "not_essential f (g z)" "g analytic_on {z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2597 |
shows "not_essential (\<lambda>x. f (g x)) z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2598 |
proof (cases "isolated_zero (\<lambda>w. g w - g z) z") |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2599 |
case False |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2600 |
hence "eventually (\<lambda>w. g w - g z = 0) (nhds z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2601 |
by (rule non_isolated_zero) (use assms in \<open>auto intro!: analytic_intros\<close>) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2602 |
hence "not_essential (\<lambda>x. f (g x)) z \<longleftrightarrow> not_essential (\<lambda>_. f (g z)) z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2603 |
by (intro not_essential_cong refl) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2604 |
(auto elim!: eventually_mono simp: eventually_at_filter) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2605 |
thus ?thesis |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2606 |
by (simp add: not_essential_const) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2607 |
next |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2608 |
case True |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2609 |
hence ev: "eventually (\<lambda>w. g w \<noteq> g z) (at z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2610 |
by (auto simp: isolated_zero_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2611 |
from assms consider c where "f \<midarrow>g z\<rightarrow> c" | "is_pole f (g z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2612 |
by (auto simp: not_essential_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2613 |
have "isCont g z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2614 |
by (rule analytic_at_imp_isCont) fact |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2615 |
hence lim: "g \<midarrow>z\<rightarrow> g z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2616 |
using isContD by blast |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2617 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2618 |
from assms(1) consider c where "f \<midarrow>g z\<rightarrow> c" | "is_pole f (g z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2619 |
unfolding not_essential_def by blast |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2620 |
thus ?thesis |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2621 |
proof cases |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2622 |
fix c assume "f \<midarrow>g z\<rightarrow> c" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2623 |
hence "(\<lambda>x. f (g x)) \<midarrow>z\<rightarrow> c" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2624 |
by (rule filterlim_compose) (use lim ev in \<open>auto simp: filterlim_at\<close>) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2625 |
thus ?thesis |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2626 |
by (auto simp: not_essential_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2627 |
next |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2628 |
assume "is_pole f (g z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2629 |
hence "is_pole (\<lambda>x. f (g x)) z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2630 |
by (rule is_pole_compose) fact+ |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2631 |
thus ?thesis |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2632 |
by (auto simp: not_essential_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2633 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2634 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2635 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2636 |
subsection \<open>Isolated points\<close> |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2637 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2638 |
definition isolated_points_of :: "complex set \<Rightarrow> complex set" where |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2639 |
"isolated_points_of A = {z\<in>A. eventually (\<lambda>w. w \<notin> A) (at z)}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2640 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2641 |
lemma isolated_points_of_altdef: "isolated_points_of A = {z\<in>A. \<not>z islimpt A}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2642 |
unfolding isolated_points_of_def islimpt_def eventually_at_filter eventually_nhds by blast |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2643 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2644 |
lemma isolated_points_of_empty [simp]: "isolated_points_of {} = {}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2645 |
and isolated_points_of_UNIV [simp]: "isolated_points_of UNIV = {}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2646 |
by (auto simp: isolated_points_of_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2647 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2648 |
lemma isolated_points_of_open_is_empty [simp]: "open A \<Longrightarrow> isolated_points_of A = {}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2649 |
unfolding isolated_points_of_altdef |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2650 |
by (simp add: interior_limit_point interior_open) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2651 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2652 |
lemma isolated_points_of_subset: "isolated_points_of A \<subseteq> A" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2653 |
by (auto simp: isolated_points_of_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2654 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2655 |
lemma isolated_points_of_discrete: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2656 |
assumes "discrete A" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2657 |
shows "isolated_points_of A = A" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2658 |
using assms by (auto simp: isolated_points_of_def discrete_altdef) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2659 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2660 |
lemmas uniform_discreteI1 = uniformI1 |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2661 |
lemmas uniform_discreteI2 = uniformI2 |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2662 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2663 |
lemma isolated_singularity_at_compose: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2664 |
assumes "isolated_singularity_at f (g z)" "g analytic_on {z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2665 |
shows "isolated_singularity_at (\<lambda>x. f (g x)) z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2666 |
proof (cases "isolated_zero (\<lambda>w. g w - g z) z") |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2667 |
case False |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2668 |
hence "eventually (\<lambda>w. g w - g z = 0) (nhds z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2669 |
by (rule non_isolated_zero) (use assms in \<open>auto intro!: analytic_intros\<close>) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2670 |
hence "isolated_singularity_at (\<lambda>x. f (g x)) z \<longleftrightarrow> isolated_singularity_at (\<lambda>_. f (g z)) z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2671 |
by (intro isolated_singularity_at_cong refl) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2672 |
(auto elim!: eventually_mono simp: eventually_at_filter) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2673 |
thus ?thesis |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2674 |
by (simp add: isolated_singularity_at_const) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2675 |
next |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2676 |
case True |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2677 |
from assms(1) obtain r where r: "r > 0" "f analytic_on ball (g z) r - {g z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2678 |
by (auto simp: isolated_singularity_at_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2679 |
hence holo_f: "f holomorphic_on ball (g z) r - {g z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2680 |
by (subst (asm) analytic_on_open) auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2681 |
from assms(2) obtain r' where r': "r' > 0" "g holomorphic_on ball z r'" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2682 |
by (auto simp: analytic_on_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2683 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2684 |
have "continuous_on (ball z r') g" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2685 |
using holomorphic_on_imp_continuous_on r' by blast |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2686 |
hence "isCont g z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2687 |
using r' by (subst (asm) continuous_on_eq_continuous_at) auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2688 |
hence "g \<midarrow>z\<rightarrow> g z" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2689 |
using isContD by blast |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2690 |
hence "eventually (\<lambda>w. g w \<in> ball (g z) r) (at z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2691 |
using \<open>r > 0\<close> unfolding tendsto_def by force |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2692 |
moreover have "eventually (\<lambda>w. g w \<noteq> g z) (at z)" using True |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2693 |
by (auto simp: isolated_zero_def elim!: eventually_mono) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2694 |
ultimately have "eventually (\<lambda>w. g w \<in> ball (g z) r - {g z}) (at z)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2695 |
by eventually_elim auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2696 |
then obtain r'' where r'': "r'' > 0" "\<forall>w\<in>ball z r''-{z}. g w \<in> ball (g z) r - {g z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2697 |
unfolding eventually_at_filter eventually_nhds_metric ball_def |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2698 |
by (auto simp: dist_commute) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2699 |
have "f \<circ> g holomorphic_on ball z (min r' r'') - {z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2700 |
proof (rule holomorphic_on_compose_gen) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2701 |
show "g holomorphic_on ball z (min r' r'') - {z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2702 |
by (rule holomorphic_on_subset[OF r'(2)]) auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2703 |
show "f holomorphic_on ball (g z) r - {g z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2704 |
by fact |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2705 |
show "g ` (ball z (min r' r'') - {z}) \<subseteq> ball (g z) r - {g z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2706 |
using r'' by force |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2707 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2708 |
hence "f \<circ> g analytic_on ball z (min r' r'') - {z}" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2709 |
by (subst analytic_on_open) auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2710 |
thus ?thesis using \<open>r' > 0\<close> \<open>r'' > 0\<close> |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2711 |
by (auto simp: isolated_singularity_at_def o_def intro!: exI[of _ "min r' r''"]) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2712 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2713 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2714 |
lemma is_pole_power_int_0: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2715 |
assumes "f analytic_on {x}" "isolated_zero f x" "n < 0" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2716 |
shows "is_pole (\<lambda>x. f x powi n) x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2717 |
proof - |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2718 |
have "f \<midarrow>x\<rightarrow> f x" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2719 |
using assms(1) by (simp add: analytic_at_imp_isCont isContD) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2720 |
with assms show ?thesis |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2721 |
unfolding is_pole_def |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2722 |
by (intro filterlim_power_int_neg_at_infinity) (auto simp: isolated_zero_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2723 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2724 |
|
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2725 |
lemma isolated_zero_imp_not_constant_on: |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2726 |
assumes "isolated_zero f x" "x \<in> A" "open A" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2727 |
shows "\<not>f constant_on A" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2728 |
proof |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2729 |
assume "f constant_on A" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2730 |
then obtain c where c: "\<And>x. x \<in> A \<Longrightarrow> f x = c" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2731 |
by (auto simp: constant_on_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2732 |
from assms and c[of x] have [simp]: "c = 0" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2733 |
by (auto simp: isolated_zero_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2734 |
have "eventually (\<lambda>x. f x \<noteq> 0) (at x)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2735 |
using assms by (auto simp: isolated_zero_def) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2736 |
moreover have "eventually (\<lambda>x. x \<in> A) (at x)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2737 |
using assms by (intro eventually_at_in_open') auto |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2738 |
ultimately have "eventually (\<lambda>x. False) (at x)" |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2739 |
by eventually_elim (use c in auto) |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2740 |
thus False |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2741 |
by simp |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2742 |
qed |
69956724ad4f
More material for Analysis and Complex_Analysis
paulson <lp15@cam.ac.uk>
parents:
77223
diff
changeset
|
2743 |
|
77223
607e1e345e8f
Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents:
76900
diff
changeset
|
2744 |
end |