author | wenzelm |
Mon, 11 Sep 2023 19:30:48 +0200 | |
changeset 78659 | b5f3d1051b13 |
parent 77277 | c6b50597abbc |
child 78698 | 1b9388e6eb75 |
permissions | -rw-r--r-- |
77277
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1 |
theory Meromorphic |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2 |
imports Laurent_Convergence Riemann_Mapping |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3 |
begin |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
4 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
5 |
lemma analytic_at_cong: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
6 |
assumes "eventually (\<lambda>x. f x = g x) (nhds x)" "x = y" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
7 |
shows "f analytic_on {x} \<longleftrightarrow> g analytic_on {y}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
8 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
9 |
have "g analytic_on {x}" if "f analytic_on {x}" "eventually (\<lambda>x. f x = g x) (nhds x)" for f g |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
10 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
11 |
have "(\<lambda>y. f (x + y)) has_fps_expansion fps_expansion f x" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
12 |
by (rule analytic_at_imp_has_fps_expansion) fact |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
13 |
also have "?this \<longleftrightarrow> (\<lambda>y. g (x + y)) has_fps_expansion fps_expansion f x" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
14 |
using that by (intro has_fps_expansion_cong refl) (auto simp: nhds_to_0' eventually_filtermap) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
15 |
finally show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
16 |
by (rule has_fps_expansion_imp_analytic) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
17 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
18 |
from this[of f g] this[of g f] show ?thesis using assms |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
19 |
by (auto simp: eq_commute) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
20 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
21 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
22 |
definition remove_sings :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex" where |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
23 |
"remove_sings f z = (if \<exists>c. f \<midarrow>z\<rightarrow> c then Lim (at z) f else 0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
24 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
25 |
lemma remove_sings_eqI [intro]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
26 |
assumes "f \<midarrow>z\<rightarrow> c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
27 |
shows "remove_sings f z = c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
28 |
using assms unfolding remove_sings_def by (auto simp: tendsto_Lim) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
29 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
30 |
lemma remove_sings_at_analytic [simp]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
31 |
assumes "f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
32 |
shows "remove_sings f z = f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
33 |
using assms by (intro remove_sings_eqI) (simp add: analytic_at_imp_isCont isContD) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
34 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
35 |
lemma remove_sings_at_pole [simp]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
36 |
assumes "is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
37 |
shows "remove_sings f z = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
38 |
using assms unfolding remove_sings_def is_pole_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
39 |
by (meson at_neq_bot not_tendsto_and_filterlim_at_infinity) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
40 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
41 |
lemma eventually_remove_sings_eq_at: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
42 |
assumes "isolated_singularity_at f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
43 |
shows "eventually (\<lambda>w. remove_sings f w = f w) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
44 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
45 |
from assms obtain r where r: "r > 0" "f analytic_on ball z r - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
46 |
by (auto simp: isolated_singularity_at_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
47 |
hence *: "f analytic_on {w}" if "w \<in> ball z r - {z}" for w |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
48 |
using r that by (auto intro: analytic_on_subset) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
49 |
have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
50 |
using r by (intro eventually_at_in_open) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
51 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
52 |
by eventually_elim (auto simp: remove_sings_at_analytic *) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
53 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
54 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
55 |
lemma eventually_remove_sings_eq_nhds: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
56 |
assumes "f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
57 |
shows "eventually (\<lambda>w. remove_sings f w = f w) (nhds z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
58 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
59 |
from assms obtain A where A: "open A" "z \<in> A" "f holomorphic_on A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
60 |
by (auto simp: analytic_at) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
61 |
have "eventually (\<lambda>z. z \<in> A) (nhds z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
62 |
by (intro eventually_nhds_in_open A) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
63 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
64 |
proof eventually_elim |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
65 |
case (elim w) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
66 |
from elim have "f analytic_on {w}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
67 |
using A analytic_at by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
68 |
thus ?case by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
69 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
70 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
71 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
72 |
lemma remove_sings_compose: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
73 |
assumes "filtermap g (at z) = at z'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
74 |
shows "remove_sings (f \<circ> g) z = remove_sings f z'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
75 |
proof (cases "\<exists>c. f \<midarrow>z'\<rightarrow> c") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
76 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
77 |
then obtain c where c: "f \<midarrow>z'\<rightarrow> c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
78 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
79 |
from c have "remove_sings f z' = c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
80 |
by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
81 |
moreover from c have "remove_sings (f \<circ> g) z = c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
82 |
using c by (intro remove_sings_eqI) (auto simp: filterlim_def filtermap_compose assms) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
83 |
ultimately show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
84 |
by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
85 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
86 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
87 |
hence "\<not>(\<exists>c. (f \<circ> g) \<midarrow>z\<rightarrow> c)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
88 |
by (auto simp: filterlim_def filtermap_compose assms) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
89 |
with False show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
90 |
by (auto simp: remove_sings_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
91 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
92 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
93 |
lemma remove_sings_cong: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
94 |
assumes "eventually (\<lambda>x. f x = g x) (at z)" "z = z'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
95 |
shows "remove_sings f z = remove_sings g z'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
96 |
proof (cases "\<exists>c. f \<midarrow>z\<rightarrow> c") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
97 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
98 |
then obtain c where c: "f \<midarrow>z\<rightarrow> c" by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
99 |
hence "remove_sings f z = c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
100 |
by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
101 |
moreover have "f \<midarrow>z\<rightarrow> c \<longleftrightarrow> g \<midarrow>z'\<rightarrow> c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
102 |
using assms by (intro filterlim_cong refl) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
103 |
with c have "remove_sings g z' = c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
104 |
by (intro remove_sings_eqI) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
105 |
ultimately show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
106 |
by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
107 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
108 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
109 |
have "f \<midarrow>z\<rightarrow> c \<longleftrightarrow> g \<midarrow>z'\<rightarrow> c" for c |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
110 |
using assms by (intro filterlim_cong) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
111 |
with False show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
112 |
by (auto simp: remove_sings_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
113 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
114 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
115 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
116 |
lemma deriv_remove_sings_at_analytic [simp]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
117 |
assumes "f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
118 |
shows "deriv (remove_sings f) z = deriv f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
119 |
apply (rule deriv_cong_ev) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
120 |
apply (rule eventually_remove_sings_eq_nhds) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
121 |
using assms by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
122 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
123 |
lemma isolated_singularity_at_remove_sings [simp, intro]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
124 |
assumes "isolated_singularity_at f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
125 |
shows "isolated_singularity_at (remove_sings f) z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
126 |
using isolated_singularity_at_cong[OF eventually_remove_sings_eq_at[OF assms] refl] assms |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
127 |
by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
128 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
129 |
lemma not_essential_remove_sings_iff [simp]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
130 |
assumes "isolated_singularity_at f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
131 |
shows "not_essential (remove_sings f) z \<longleftrightarrow> not_essential f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
132 |
using not_essential_cong[OF eventually_remove_sings_eq_at[OF assms(1)] refl] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
133 |
by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
134 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
135 |
lemma not_essential_remove_sings [intro]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
136 |
assumes "isolated_singularity_at f z" "not_essential f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
137 |
shows "not_essential (remove_sings f) z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
138 |
by (subst not_essential_remove_sings_iff) (use assms in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
139 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
140 |
lemma |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
141 |
assumes "isolated_singularity_at f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
142 |
shows is_pole_remove_sings_iff [simp]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
143 |
"is_pole (remove_sings f) z \<longleftrightarrow> is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
144 |
and zorder_remove_sings [simp]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
145 |
"zorder (remove_sings f) z = zorder f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
146 |
and zor_poly_remove_sings [simp]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
147 |
"zor_poly (remove_sings f) z = zor_poly f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
148 |
and has_laurent_expansion_remove_sings_iff [simp]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
149 |
"(\<lambda>w. remove_sings f (z + w)) has_laurent_expansion F \<longleftrightarrow> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
150 |
(\<lambda>w. f (z + w)) has_laurent_expansion F" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
151 |
and tendsto_remove_sings_iff [simp]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
152 |
"remove_sings f \<midarrow>z\<rightarrow> c \<longleftrightarrow> f \<midarrow>z\<rightarrow> c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
153 |
by (intro is_pole_cong eventually_remove_sings_eq_at refl zorder_cong |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
154 |
zor_poly_cong has_laurent_expansion_cong' tendsto_cong assms)+ |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
155 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
156 |
lemma get_all_poles_from_remove_sings: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
157 |
fixes f:: "complex \<Rightarrow> complex" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
158 |
defines "ff\<equiv>remove_sings f" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
159 |
assumes f_holo:"f holomorphic_on s - pts" and "finite pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
160 |
"pts\<subseteq>s" "open s" and not_ess:"\<forall>x\<in>pts. not_essential f x" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
161 |
obtains pts' where |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
162 |
"pts' \<subseteq> pts" "finite pts'" "ff holomorphic_on s - pts'" "\<forall>x\<in>pts'. is_pole ff x" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
163 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
164 |
define pts' where "pts' = {x\<in>pts. is_pole f x}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
165 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
166 |
have "pts' \<subseteq> pts" unfolding pts'_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
167 |
then have "finite pts'" using \<open>finite pts\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
168 |
using rev_finite_subset by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
169 |
then have "open (s - pts')" using \<open>open s\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
170 |
by (simp add: finite_imp_closed open_Diff) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
171 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
172 |
have isolated:"isolated_singularity_at f z" if "z\<in>pts" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
173 |
proof (rule isolated_singularity_at_holomorphic) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
174 |
show "f holomorphic_on (s-(pts-{z})) - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
175 |
by (metis Diff_insert f_holo insert_Diff that) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
176 |
show " open (s - (pts - {z}))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
177 |
by (meson assms(3) assms(5) finite_Diff finite_imp_closed open_Diff) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
178 |
show "z \<in> s - (pts - {z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
179 |
using assms(4) that by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
180 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
181 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
182 |
have "ff holomorphic_on s - pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
183 |
proof (rule no_isolated_singularity') |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
184 |
show "(ff \<longlongrightarrow> ff z) (at z within s - pts')" if "z \<in> pts-pts'" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
185 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
186 |
have "at z within s - pts' = at z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
187 |
apply (rule at_within_open) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
188 |
using \<open>open (s - pts')\<close> that \<open>pts\<subseteq>s\<close> by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
189 |
moreover have "ff \<midarrow>z\<rightarrow> ff z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
190 |
unfolding ff_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
191 |
proof (subst tendsto_remove_sings_iff) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
192 |
show "isolated_singularity_at f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
193 |
apply (rule isolated) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
194 |
using that by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
195 |
have "not_essential f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
196 |
using not_ess that by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
197 |
moreover have "\<not>is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
198 |
using that unfolding pts'_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
199 |
ultimately have "\<exists>c. f \<midarrow>z\<rightarrow> c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
200 |
unfolding not_essential_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
201 |
then show "f \<midarrow>z\<rightarrow> remove_sings f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
202 |
using remove_sings_eqI by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
203 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
204 |
ultimately show ?thesis by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
205 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
206 |
have "ff holomorphic_on s - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
207 |
using f_holo |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
208 |
proof (elim holomorphic_transform) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
209 |
fix x assume "x \<in> s - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
210 |
then have "f analytic_on {x}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
211 |
using assms(3) assms(5) f_holo |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
212 |
by (meson finite_imp_closed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
213 |
holomorphic_on_imp_analytic_at open_Diff) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
214 |
from remove_sings_at_analytic[OF this] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
215 |
show "f x = ff x" unfolding ff_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
216 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
217 |
then show "ff holomorphic_on s - pts' - (pts - pts')" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
218 |
apply (elim holomorphic_on_subset) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
219 |
by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
220 |
show "open (s - pts')" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
221 |
by (simp add: \<open>open (s - pts')\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
222 |
show "finite (pts - pts')" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
223 |
by (simp add: assms(3)) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
224 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
225 |
moreover have "\<forall>x\<in>pts'. is_pole ff x" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
226 |
unfolding pts'_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
227 |
using ff_def is_pole_remove_sings_iff isolated by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
228 |
moreover note \<open>pts' \<subseteq> pts\<close> \<open>finite pts'\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
229 |
ultimately show ?thesis using that by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
230 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
231 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
232 |
lemma remove_sings_eq_0_iff: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
233 |
assumes "not_essential f w" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
234 |
shows "remove_sings f w = 0 \<longleftrightarrow> is_pole f w \<or> f \<midarrow>w\<rightarrow> 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
235 |
proof (cases "is_pole f w") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
236 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
237 |
then show ?thesis by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
238 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
239 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
240 |
then obtain c where c:"f \<midarrow>w\<rightarrow> c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
241 |
using \<open>not_essential f w\<close> unfolding not_essential_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
242 |
then show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
243 |
using False remove_sings_eqI by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
244 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
245 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
246 |
definition meromorphic_on:: "[complex \<Rightarrow> complex, complex set, complex set] \<Rightarrow> bool" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
247 |
("_ (meromorphic'_on) _ _" [50,50,50]50) where |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
248 |
"f meromorphic_on D pts \<equiv> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
249 |
open D \<and> pts \<subseteq> D \<and> (\<forall>z\<in>pts. isolated_singularity_at f z \<and> not_essential f z) \<and> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
250 |
(\<forall>z\<in>D. \<not>(z islimpt pts)) \<and> (f holomorphic_on D-pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
251 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
252 |
lemma meromorphic_imp_holomorphic: "f meromorphic_on D pts \<Longrightarrow> f holomorphic_on (D - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
253 |
unfolding meromorphic_on_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
254 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
255 |
lemma meromorphic_imp_closedin_pts: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
256 |
assumes "f meromorphic_on D pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
257 |
shows "closedin (top_of_set D) pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
258 |
by (meson assms closedin_limpt meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
259 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
260 |
lemma meromorphic_imp_open_diff': |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
261 |
assumes "f meromorphic_on D pts" "pts' \<subseteq> pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
262 |
shows "open (D - pts')" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
263 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
264 |
have "D - pts' = D - closure pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
265 |
proof safe |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
266 |
fix x assume x: "x \<in> D" "x \<in> closure pts'" "x \<notin> pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
267 |
hence "x islimpt pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
268 |
by (subst islimpt_in_closure) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
269 |
hence "x islimpt pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
270 |
by (rule islimpt_subset) fact |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
271 |
with assms x show False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
272 |
by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
273 |
qed (use closure_subset in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
274 |
then show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
275 |
using assms meromorphic_on_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
276 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
277 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
278 |
lemma meromorphic_imp_open_diff: "f meromorphic_on D pts \<Longrightarrow> open (D - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
279 |
by (erule meromorphic_imp_open_diff') auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
280 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
281 |
lemma meromorphic_pole_subset: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
282 |
assumes merf: "f meromorphic_on D pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
283 |
shows "{x\<in>D. is_pole f x} \<subseteq> pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
284 |
by (smt (verit) Diff_iff assms mem_Collect_eq meromorphic_imp_open_diff |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
285 |
meromorphic_on_def not_is_pole_holomorphic subsetI) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
286 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
287 |
named_theorems meromorphic_intros |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
288 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
289 |
lemma meromorphic_on_subset: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
290 |
assumes "f meromorphic_on A pts" "open B" "B \<subseteq> A" "pts' = pts \<inter> B" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
291 |
shows "f meromorphic_on B pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
292 |
unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
293 |
proof (intro ballI conjI) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
294 |
fix z assume "z \<in> B" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
295 |
show "\<not>z islimpt pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
296 |
proof |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
297 |
assume "z islimpt pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
298 |
hence "z islimpt pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
299 |
by (rule islimpt_subset) (use \<open>pts' = _\<close> in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
300 |
thus False using \<open>z \<in> B\<close> \<open>B \<subseteq> A\<close> assms(1) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
301 |
by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
302 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
303 |
qed (use assms in \<open>auto simp: meromorphic_on_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
304 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
305 |
lemma meromorphic_on_superset_pts: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
306 |
assumes "f meromorphic_on A pts" "pts \<subseteq> pts'" "pts' \<subseteq> A" "\<forall>x\<in>A. \<not>x islimpt pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
307 |
shows "f meromorphic_on A pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
308 |
unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
309 |
proof (intro conjI ballI impI) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
310 |
fix z assume "z \<in> pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
311 |
from assms(1) have holo: "f holomorphic_on A - pts" and "open A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
312 |
unfolding meromorphic_on_def by blast+ |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
313 |
have "open (A - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
314 |
by (intro meromorphic_imp_open_diff[OF assms(1)]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
315 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
316 |
show "isolated_singularity_at f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
317 |
proof (cases "z \<in> pts") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
318 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
319 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
320 |
using \<open>open (A - pts)\<close> assms \<open>z \<in> pts'\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
321 |
by (intro isolated_singularity_at_holomorphic[of _ "A - pts"] holomorphic_on_subset[OF holo]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
322 |
auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
323 |
qed (use assms in \<open>auto simp: meromorphic_on_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
324 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
325 |
show "not_essential f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
326 |
proof (cases "z \<in> pts") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
327 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
328 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
329 |
using \<open>open (A - pts)\<close> assms \<open>z \<in> pts'\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
330 |
by (intro not_essential_holomorphic[of _ "A - pts"] holomorphic_on_subset[OF holo]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
331 |
auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
332 |
qed (use assms in \<open>auto simp: meromorphic_on_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
333 |
qed (use assms in \<open>auto simp: meromorphic_on_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
334 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
335 |
lemma meromorphic_on_no_singularities: "f meromorphic_on A {} \<longleftrightarrow> f holomorphic_on A \<and> open A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
336 |
by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
337 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
338 |
lemma holomorphic_on_imp_meromorphic_on: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
339 |
"f holomorphic_on A \<Longrightarrow> pts \<subseteq> A \<Longrightarrow> open A \<Longrightarrow> \<forall>x\<in>A. \<not>x islimpt pts \<Longrightarrow> f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
340 |
by (rule meromorphic_on_superset_pts[where pts = "{}"]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
341 |
(auto simp: meromorphic_on_no_singularities) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
342 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
343 |
lemma meromorphic_on_const [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
344 |
assumes "open A" "\<forall>x\<in>A. \<not>x islimpt pts" "pts \<subseteq> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
345 |
shows "(\<lambda>_. c) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
346 |
by (rule holomorphic_on_imp_meromorphic_on) (use assms in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
347 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
348 |
lemma meromorphic_on_ident [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
349 |
assumes "open A" "\<forall>x\<in>A. \<not>x islimpt pts" "pts \<subseteq> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
350 |
shows "(\<lambda>x. x) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
351 |
by (rule holomorphic_on_imp_meromorphic_on) (use assms in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
352 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
353 |
lemma meromorphic_on_id [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
354 |
assumes "open A" "\<forall>x\<in>A. \<not>x islimpt pts" "pts \<subseteq> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
355 |
shows "id meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
356 |
using meromorphic_on_ident assms unfolding id_def . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
357 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
358 |
lemma not_essential_add [singularity_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
359 |
assumes f_ness: "not_essential f z" and g_ness: "not_essential g z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
360 |
assumes f_iso: "isolated_singularity_at f z" and g_iso: "isolated_singularity_at g z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
361 |
shows "not_essential (\<lambda>w. f w + g w) z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
362 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
363 |
have "(\<lambda>w. f (z + w) + g (z + w)) has_laurent_expansion laurent_expansion f z + laurent_expansion g z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
364 |
by (intro not_essential_has_laurent_expansion laurent_expansion_intros assms) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
365 |
hence "not_essential (\<lambda>w. f (z + w) + g (z + w)) 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
366 |
using has_laurent_expansion_not_essential_0 by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
367 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
368 |
by (simp add: not_essential_shift_0) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
369 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
370 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
371 |
lemma meromorphic_on_uminus [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
372 |
assumes "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
373 |
shows "(\<lambda>z. -f z) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
374 |
unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
375 |
by (use assms in \<open>auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
376 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
377 |
lemma meromorphic_on_add [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
378 |
assumes "f meromorphic_on A pts" "g meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
379 |
shows "(\<lambda>z. f z + g z) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
380 |
unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
381 |
by (use assms in \<open>auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
382 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
383 |
lemma meromorphic_on_add': |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
384 |
assumes "f meromorphic_on A pts1" "g meromorphic_on A pts2" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
385 |
shows "(\<lambda>z. f z + g z) meromorphic_on A (pts1 \<union> pts2)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
386 |
proof (rule meromorphic_intros) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
387 |
show "f meromorphic_on A (pts1 \<union> pts2)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
388 |
by (rule meromorphic_on_superset_pts[OF assms(1)]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
389 |
(use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
390 |
show "g meromorphic_on A (pts1 \<union> pts2)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
391 |
by (rule meromorphic_on_superset_pts[OF assms(2)]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
392 |
(use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
393 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
394 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
395 |
lemma meromorphic_on_add_const [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
396 |
assumes "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
397 |
shows "(\<lambda>z. f z + c) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
398 |
unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
399 |
by (use assms in \<open>auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
400 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
401 |
lemma meromorphic_on_minus_const [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
402 |
assumes "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
403 |
shows "(\<lambda>z. f z - c) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
404 |
using meromorphic_on_add_const[OF assms,of "-c"] by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
405 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
406 |
lemma meromorphic_on_diff [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
407 |
assumes "f meromorphic_on A pts" "g meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
408 |
shows "(\<lambda>z. f z - g z) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
409 |
using meromorphic_on_add[OF assms(1) meromorphic_on_uminus[OF assms(2)]] by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
410 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
411 |
lemma meromorphic_on_diff': |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
412 |
assumes "f meromorphic_on A pts1" "g meromorphic_on A pts2" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
413 |
shows "(\<lambda>z. f z - g z) meromorphic_on A (pts1 \<union> pts2)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
414 |
proof (rule meromorphic_intros) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
415 |
show "f meromorphic_on A (pts1 \<union> pts2)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
416 |
by (rule meromorphic_on_superset_pts[OF assms(1)]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
417 |
(use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
418 |
show "g meromorphic_on A (pts1 \<union> pts2)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
419 |
by (rule meromorphic_on_superset_pts[OF assms(2)]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
420 |
(use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
421 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
422 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
423 |
lemma meromorphic_on_mult [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
424 |
assumes "f meromorphic_on A pts" "g meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
425 |
shows "(\<lambda>z. f z * g z) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
426 |
unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
427 |
by (use assms in \<open>auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
428 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
429 |
lemma meromorphic_on_mult': |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
430 |
assumes "f meromorphic_on A pts1" "g meromorphic_on A pts2" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
431 |
shows "(\<lambda>z. f z * g z) meromorphic_on A (pts1 \<union> pts2)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
432 |
proof (rule meromorphic_intros) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
433 |
show "f meromorphic_on A (pts1 \<union> pts2)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
434 |
by (rule meromorphic_on_superset_pts[OF assms(1)]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
435 |
(use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
436 |
show "g meromorphic_on A (pts1 \<union> pts2)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
437 |
by (rule meromorphic_on_superset_pts[OF assms(2)]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
438 |
(use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
439 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
440 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
441 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
442 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
443 |
lemma meromorphic_on_imp_not_essential: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
444 |
assumes "f meromorphic_on A pts" "z \<in> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
445 |
shows "not_essential f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
446 |
proof (cases "z \<in> pts") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
447 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
448 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
449 |
using not_essential_holomorphic[of f "A - pts" z] meromorphic_imp_open_diff[OF assms(1)] assms |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
450 |
by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
451 |
qed (use assms in \<open>auto simp: meromorphic_on_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
452 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
453 |
lemma meromorphic_imp_analytic: "f meromorphic_on D pts \<Longrightarrow> f analytic_on (D - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
454 |
unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
455 |
apply (subst analytic_on_open) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
456 |
using meromorphic_imp_open_diff meromorphic_on_id apply blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
457 |
apply auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
458 |
done |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
459 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
460 |
lemma not_islimpt_isolated_zeros: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
461 |
assumes mero: "f meromorphic_on A pts" and "z \<in> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
462 |
shows "\<not>z islimpt {w\<in>A. isolated_zero f w}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
463 |
proof |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
464 |
assume islimpt: "z islimpt {w\<in>A. isolated_zero f w}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
465 |
have holo: "f holomorphic_on A - pts" and "open A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
466 |
using assms by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
467 |
have open': "open (A - (pts - {z}))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
468 |
by (intro meromorphic_imp_open_diff'[OF mero]) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
469 |
then obtain r where r: "r > 0" "ball z r \<subseteq> A - (pts - {z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
470 |
using meromorphic_imp_open_diff[OF mero] \<open>z \<in> A\<close> openE by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
471 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
472 |
have "not_essential f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
473 |
using assms by (rule meromorphic_on_imp_not_essential) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
474 |
then consider c where "f \<midarrow>z\<rightarrow> c" | "is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
475 |
unfolding not_essential_def by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
476 |
thus False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
477 |
proof cases |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
478 |
assume "is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
479 |
hence "eventually (\<lambda>w. f w \<noteq> 0) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
480 |
by (rule non_zero_neighbour_pole) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
481 |
hence "\<not>z islimpt {w. f w = 0}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
482 |
by (simp add: islimpt_conv_frequently_at frequently_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
483 |
moreover have "z islimpt {w. f w = 0}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
484 |
using islimpt by (rule islimpt_subset) (auto simp: isolated_zero_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
485 |
ultimately show False by contradiction |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
486 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
487 |
fix c assume c: "f \<midarrow>z\<rightarrow> c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
488 |
define g where "g = (\<lambda>w. if w = z then c else f w)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
489 |
have holo': "g holomorphic_on A - (pts - {z})" unfolding g_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
490 |
by (intro removable_singularity holomorphic_on_subset[OF holo] open' c) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
491 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
492 |
have eq_zero: "g w = 0" if "w \<in> ball z r" for w |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
493 |
proof (rule analytic_continuation[where f = g]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
494 |
show "open (ball z r)" "connected (ball z r)" "{w\<in>ball z r. isolated_zero f w} \<subseteq> ball z r" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
495 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
496 |
have "z islimpt {w\<in>A. isolated_zero f w} \<inter> ball z r" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
497 |
using islimpt \<open>r > 0\<close> by (intro islimpt_Int_eventually eventually_at_in_open') auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
498 |
also have "\<dots> = {w\<in>ball z r. isolated_zero f w}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
499 |
using r by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
500 |
finally show "z islimpt {w\<in>ball z r. isolated_zero f w}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
501 |
by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
502 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
503 |
fix w assume w: "w \<in> {w\<in>ball z r. isolated_zero f w}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
504 |
show "g w = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
505 |
proof (cases "w = z") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
506 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
507 |
thus ?thesis using w by (auto simp: g_def isolated_zero_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
508 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
509 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
510 |
have "z islimpt {z. f z = 0}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
511 |
using islimpt by (rule islimpt_subset) (auto simp: isolated_zero_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
512 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
513 |
using w by (simp add: isolated_zero_altdef True) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
514 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
515 |
qed (use r that in \<open>auto intro!: holomorphic_on_subset[OF holo'] simp: isolated_zero_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
516 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
517 |
have "infinite ({w\<in>A. isolated_zero f w} \<inter> ball z r)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
518 |
using islimpt \<open>r > 0\<close> unfolding islimpt_eq_infinite_ball by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
519 |
hence "{w\<in>A. isolated_zero f w} \<inter> ball z r \<noteq> {}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
520 |
by force |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
521 |
then obtain z0 where z0: "z0 \<in> A" "isolated_zero f z0" "z0 \<in> ball z r" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
522 |
by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
523 |
have "\<forall>\<^sub>F y in at z0. y \<in> ball z r - (if z = z0 then {} else {z}) - {z0}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
524 |
using r z0 by (intro eventually_at_in_open) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
525 |
hence "eventually (\<lambda>w. f w = 0) (at z0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
526 |
proof eventually_elim |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
527 |
case (elim w) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
528 |
show ?case |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
529 |
using eq_zero[of w] elim by (auto simp: g_def split: if_splits) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
530 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
531 |
hence "eventually (\<lambda>w. f w = 0) (at z0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
532 |
by (auto simp: g_def eventually_at_filter elim!: eventually_mono split: if_splits) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
533 |
moreover from z0 have "eventually (\<lambda>w. f w \<noteq> 0) (at z0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
534 |
by (simp add: isolated_zero_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
535 |
ultimately have "eventually (\<lambda>_. False) (at z0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
536 |
by eventually_elim auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
537 |
thus False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
538 |
by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
539 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
540 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
541 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
542 |
lemma closedin_isolated_zeros: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
543 |
assumes "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
544 |
shows "closedin (top_of_set A) {z\<in>A. isolated_zero f z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
545 |
unfolding closedin_limpt using not_islimpt_isolated_zeros[OF assms] by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
546 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
547 |
lemma meromorphic_on_deriv': |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
548 |
assumes "f meromorphic_on A pts" "open A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
549 |
assumes "\<And>x. x \<in> A - pts \<Longrightarrow> (f has_field_derivative f' x) (at x)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
550 |
shows "f' meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
551 |
unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
552 |
proof (intro conjI ballI) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
553 |
have "open (A - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
554 |
by (intro meromorphic_imp_open_diff[OF assms(1)]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
555 |
thus "f' holomorphic_on A - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
556 |
by (rule derivative_is_holomorphic) (use assms in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
557 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
558 |
fix z assume "z \<in> pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
559 |
hence "z \<in> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
560 |
using assms(1) by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
561 |
from \<open>z \<in> pts\<close> obtain r where r: "r > 0" "f analytic_on ball z r - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
562 |
using assms(1) by (auto simp: meromorphic_on_def isolated_singularity_at_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
563 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
564 |
have "open (ball z r \<inter> (A - (pts - {z})))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
565 |
by (intro open_Int assms meromorphic_imp_open_diff'[OF assms(1)]) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
566 |
then obtain r' where r': "r' > 0" "ball z r' \<subseteq> ball z r \<inter> (A - (pts - {z}))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
567 |
using r \<open>z \<in> A\<close> by (subst (asm) open_contains_ball) fastforce |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
568 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
569 |
have "open (ball z r' - {z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
570 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
571 |
hence "f' holomorphic_on ball z r' - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
572 |
by (rule derivative_is_holomorphic[of _ f]) (use r' in \<open>auto intro!: assms(3)\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
573 |
moreover have "open (ball z r' - {z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
574 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
575 |
ultimately show "isolated_singularity_at f' z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
576 |
unfolding isolated_singularity_at_def using \<open>r' > 0\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
577 |
by (auto simp: analytic_on_open intro!: exI[of _ r']) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
578 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
579 |
fix z assume z: "z \<in> pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
580 |
hence z': "not_essential f z" "z \<in> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
581 |
using assms by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
582 |
from z'(1) show "not_essential f' z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
583 |
proof (rule not_essential_deriv') |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
584 |
show "z \<in> A - (pts - {z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
585 |
using \<open>z \<in> A\<close> by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
586 |
show "open (A - (pts - {z}))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
587 |
by (intro meromorphic_imp_open_diff'[OF assms(1)]) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
588 |
qed (use assms in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
589 |
qed (use assms in \<open>auto simp: meromorphic_on_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
590 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
591 |
lemma meromorphic_on_deriv [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
592 |
assumes "f meromorphic_on A pts" "open A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
593 |
shows "deriv f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
594 |
proof (intro meromorphic_on_deriv'[OF assms(1)]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
595 |
have *: "open (A - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
596 |
by (intro meromorphic_imp_open_diff[OF assms(1)]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
597 |
show "(f has_field_derivative deriv f x) (at x)" if "x \<in> A - pts" for x |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
598 |
using assms(1) by (intro holomorphic_derivI[OF _ * that]) (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
599 |
qed fact |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
600 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
601 |
lemma meromorphic_on_imp_analytic_at: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
602 |
assumes "f meromorphic_on A pts" "z \<in> A - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
603 |
shows "f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
604 |
using assms by (metis analytic_at meromorphic_imp_open_diff meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
605 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
606 |
lemma meromorphic_compact_finite_pts: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
607 |
assumes "f meromorphic_on D pts" "compact S" "S \<subseteq> D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
608 |
shows "finite (S \<inter> pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
609 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
610 |
{ assume "infinite (S \<inter> pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
611 |
then obtain z where "z \<in> S" and z: "z islimpt (S \<inter> pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
612 |
using assms by (metis compact_eq_Bolzano_Weierstrass inf_le1) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
613 |
then have False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
614 |
using assms by (meson in_mono inf_le2 islimpt_subset meromorphic_on_def) } |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
615 |
then show ?thesis by metis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
616 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
617 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
618 |
lemma meromorphic_imp_countable: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
619 |
assumes "f meromorphic_on D pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
620 |
shows "countable pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
621 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
622 |
obtain K :: "nat \<Rightarrow> complex set" where K: "D = (\<Union>n. K n)" "\<And>n. compact (K n)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
623 |
using assms unfolding meromorphic_on_def by (metis open_Union_compact_subsets) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
624 |
then have "pts = (\<Union>n. K n \<inter> pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
625 |
using assms meromorphic_on_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
626 |
moreover have "\<And>n. finite (K n \<inter> pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
627 |
by (metis K(1) K(2) UN_I assms image_iff meromorphic_compact_finite_pts rangeI subset_eq) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
628 |
ultimately show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
629 |
by (metis countableI_type countable_UN countable_finite) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
630 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
631 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
632 |
lemma meromorphic_imp_connected_diff': |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
633 |
assumes "f meromorphic_on D pts" "connected D" "pts' \<subseteq> pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
634 |
shows "connected (D - pts')" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
635 |
proof (rule connected_open_diff_countable) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
636 |
show "countable pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
637 |
by (rule countable_subset [OF assms(3)]) (use assms(1) in \<open>auto simp: meromorphic_imp_countable\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
638 |
qed (use assms in \<open>auto simp: meromorphic_on_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
639 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
640 |
lemma meromorphic_imp_connected_diff: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
641 |
assumes "f meromorphic_on D pts" "connected D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
642 |
shows "connected (D - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
643 |
using meromorphic_imp_connected_diff'[OF assms order.refl] . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
644 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
645 |
lemma meromorphic_on_compose [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
646 |
assumes f: "f meromorphic_on A pts" and g: "g holomorphic_on B" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
647 |
assumes "open B" and "g ` B \<subseteq> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
648 |
shows "(\<lambda>x. f (g x)) meromorphic_on B (isolated_points_of (g -` pts \<inter> B))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
649 |
unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
650 |
proof (intro ballI conjI) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
651 |
fix z assume z: "z \<in> isolated_points_of (g -` pts \<inter> B)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
652 |
hence z': "z \<in> B" "g z \<in> pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
653 |
using isolated_points_of_subset by blast+ |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
654 |
have g': "g analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
655 |
using g z' \<open>open B\<close> analytic_at by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
656 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
657 |
show "isolated_singularity_at (\<lambda>x. f (g x)) z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
658 |
by (rule isolated_singularity_at_compose[OF _ g']) (use f z' in \<open>auto simp: meromorphic_on_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
659 |
show "not_essential (\<lambda>x. f (g x)) z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
660 |
by (rule not_essential_compose[OF _ g']) (use f z' in \<open>auto simp: meromorphic_on_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
661 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
662 |
fix z assume z: "z \<in> B" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
663 |
hence "g z \<in> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
664 |
using assms by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
665 |
hence "\<not>g z islimpt pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
666 |
using f by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
667 |
hence ev: "eventually (\<lambda>w. w \<notin> pts) (at (g z))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
668 |
by (auto simp: islimpt_conv_frequently_at frequently_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
669 |
have g': "g analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
670 |
by (rule holomorphic_on_imp_analytic_at[OF g]) (use assms z in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
671 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
672 |
(* TODO: There's probably a useful lemma somewhere in here to extract... *) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
673 |
have "eventually (\<lambda>w. w \<notin> isolated_points_of (g -` pts \<inter> B)) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
674 |
proof (cases "isolated_zero (\<lambda>w. g w - g z) z") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
675 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
676 |
have "eventually (\<lambda>w. w \<notin> pts) (at (g z))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
677 |
using ev by (auto simp: islimpt_conv_frequently_at frequently_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
678 |
moreover have "g \<midarrow>z\<rightarrow> g z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
679 |
using analytic_at_imp_isCont[OF g'] isContD by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
680 |
hence lim: "filterlim g (at (g z)) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
681 |
using True by (auto simp: filterlim_at isolated_zero_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
682 |
have "eventually (\<lambda>w. g w \<notin> pts) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
683 |
using ev lim by (rule eventually_compose_filterlim) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
684 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
685 |
by eventually_elim (auto simp: isolated_points_of_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
686 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
687 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
688 |
have "eventually (\<lambda>w. g w - g z = 0) (nhds z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
689 |
using False by (rule non_isolated_zero) (auto intro!: analytic_intros g') |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
690 |
hence "eventually (\<lambda>w. g w = g z \<and> w \<in> B) (nhds z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
691 |
using eventually_nhds_in_open[OF \<open>open B\<close> \<open>z \<in> B\<close>] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
692 |
by eventually_elim auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
693 |
then obtain X where X: "open X" "z \<in> X" "X \<subseteq> B" "\<forall>x\<in>X. g x = g z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
694 |
unfolding eventually_nhds by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
695 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
696 |
have "z0 \<notin> isolated_points_of (g -` pts \<inter> B)" if "z0 \<in> X" for z0 |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
697 |
proof (cases "g z \<in> pts") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
698 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
699 |
with that have "g z0 \<notin> pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
700 |
using X by metis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
701 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
702 |
by (auto simp: isolated_points_of_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
703 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
704 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
705 |
have "eventually (\<lambda>w. w \<in> X) (at z0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
706 |
by (intro eventually_at_in_open') fact+ |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
707 |
hence "eventually (\<lambda>w. w \<in> g -` pts \<inter> B) (at z0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
708 |
by eventually_elim (use X True in fastforce) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
709 |
hence "frequently (\<lambda>w. w \<in> g -` pts \<inter> B) (at z0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
710 |
by (meson at_neq_bot eventually_frequently) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
711 |
thus "z0 \<notin> isolated_points_of (g -` pts \<inter> B)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
712 |
unfolding isolated_points_of_def by (auto simp: frequently_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
713 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
714 |
moreover have "eventually (\<lambda>x. x \<in> X) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
715 |
by (intro eventually_at_in_open') fact+ |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
716 |
ultimately show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
717 |
by (auto elim!: eventually_mono) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
718 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
719 |
thus "\<not>z islimpt isolated_points_of (g -` pts \<inter> B)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
720 |
by (auto simp: islimpt_conv_frequently_at frequently_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
721 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
722 |
have "f \<circ> g analytic_on (\<Union>z\<in>B - isolated_points_of (g -` pts \<inter> B). {z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
723 |
unfolding analytic_on_UN |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
724 |
proof |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
725 |
fix z assume z: "z \<in> B - isolated_points_of (g -` pts \<inter> B)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
726 |
hence "z \<in> B" by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
727 |
have g': "g analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
728 |
by (rule holomorphic_on_imp_analytic_at[OF g]) (use assms z in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
729 |
show "f \<circ> g analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
730 |
proof (cases "g z \<in> pts") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
731 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
732 |
show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
733 |
proof (rule analytic_on_compose) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
734 |
show "f analytic_on g ` {z}" using False z assms |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
735 |
by (auto intro!: meromorphic_on_imp_analytic_at[OF f]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
736 |
qed fact |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
737 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
738 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
739 |
show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
740 |
proof (cases "isolated_zero (\<lambda>w. g w - g z) z") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
741 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
742 |
hence "eventually (\<lambda>w. g w - g z = 0) (nhds z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
743 |
by (rule non_isolated_zero) (auto intro!: analytic_intros g') |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
744 |
hence "f \<circ> g analytic_on {z} \<longleftrightarrow> (\<lambda>_. f (g z)) analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
745 |
by (intro analytic_at_cong) (auto elim!: eventually_mono) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
746 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
747 |
by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
748 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
749 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
750 |
hence ev: "eventually (\<lambda>w. g w \<noteq> g z) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
751 |
by (auto simp: isolated_zero_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
752 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
753 |
have "\<not>g z islimpt pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
754 |
using \<open>g z \<in> pts\<close> f by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
755 |
hence "eventually (\<lambda>w. w \<notin> pts) (at (g z))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
756 |
by (auto simp: islimpt_conv_frequently_at frequently_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
757 |
moreover have "g \<midarrow>z\<rightarrow> g z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
758 |
using analytic_at_imp_isCont[OF g'] isContD by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
759 |
with ev have "filterlim g (at (g z)) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
760 |
by (auto simp: filterlim_at) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
761 |
ultimately have "eventually (\<lambda>w. g w \<notin> pts) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
762 |
using eventually_compose_filterlim by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
763 |
hence "z \<in> isolated_points_of (g -` pts \<inter> B)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
764 |
using \<open>g z \<in> pts\<close> \<open>z \<in> B\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
765 |
by (auto simp: isolated_points_of_def elim!: eventually_mono) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
766 |
with z show ?thesis by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
767 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
768 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
769 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
770 |
also have "\<dots> = B - isolated_points_of (g -` pts \<inter> B)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
771 |
by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
772 |
finally show "(\<lambda>x. f (g x)) holomorphic_on B - isolated_points_of (g -` pts \<inter> B)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
773 |
unfolding o_def using analytic_imp_holomorphic by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
774 |
qed (auto simp: isolated_points_of_def \<open>open B\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
775 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
776 |
lemma meromorphic_on_compose': |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
777 |
assumes f: "f meromorphic_on A pts" and g: "g holomorphic_on B" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
778 |
assumes "open B" and "g ` B \<subseteq> A" and "pts' = (isolated_points_of (g -` pts \<inter> B))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
779 |
shows "(\<lambda>x. f (g x)) meromorphic_on B pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
780 |
using meromorphic_on_compose[OF assms(1-4)] assms(5) by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
781 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
782 |
lemma meromorphic_on_inverse': "inverse meromorphic_on UNIV 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
783 |
unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
784 |
by (auto intro!: holomorphic_intros singularity_intros not_essential_inverse |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
785 |
isolated_singularity_at_inverse simp: islimpt_finite) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
786 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
787 |
lemma meromorphic_on_inverse [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
788 |
assumes mero: "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
789 |
shows "(\<lambda>z. inverse (f z)) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero f z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
790 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
791 |
have "open A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
792 |
using mero by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
793 |
have open': "open (A - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
794 |
by (intro meromorphic_imp_open_diff[OF mero]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
795 |
have holo: "f holomorphic_on A - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
796 |
using assms by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
797 |
have ana: "f analytic_on A - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
798 |
using open' holo by (simp add: analytic_on_open) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
799 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
800 |
show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
801 |
unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
802 |
proof (intro conjI ballI) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
803 |
fix z assume z: "z \<in> pts \<union> {z\<in>A. isolated_zero f z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
804 |
have "isolated_singularity_at f z \<and> not_essential f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
805 |
proof (cases "z \<in> pts") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
806 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
807 |
have "f holomorphic_on A - pts - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
808 |
by (intro holomorphic_on_subset[OF holo]) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
809 |
hence "isolated_singularity_at f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
810 |
by (rule isolated_singularity_at_holomorphic) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
811 |
(use z False in \<open>auto intro!: meromorphic_imp_open_diff[OF mero]\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
812 |
moreover have "not_essential f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
813 |
using z False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
814 |
by (intro not_essential_holomorphic[OF holo] meromorphic_imp_open_diff[OF mero]) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
815 |
ultimately show ?thesis by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
816 |
qed (use assms in \<open>auto simp: meromorphic_on_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
817 |
thus "isolated_singularity_at (\<lambda>z. inverse (f z)) z" "not_essential (\<lambda>z. inverse (f z)) z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
818 |
by (auto intro!: isolated_singularity_at_inverse not_essential_inverse) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
819 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
820 |
fix z assume "z \<in> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
821 |
hence "\<not> z islimpt {z\<in>A. isolated_zero f z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
822 |
by (rule not_islimpt_isolated_zeros[OF mero]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
823 |
thus "\<not> z islimpt pts \<union> {z \<in> A. isolated_zero f z}" using \<open>z \<in> A\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
824 |
using mero by (auto simp: islimpt_Un meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
825 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
826 |
show "pts \<union> {z \<in> A. isolated_zero f z} \<subseteq> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
827 |
using mero by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
828 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
829 |
have "(\<lambda>z. inverse (f z)) analytic_on (\<Union>w\<in>A - (pts \<union> {z \<in> A. isolated_zero f z}) . {w})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
830 |
unfolding analytic_on_UN |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
831 |
proof (intro ballI) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
832 |
fix w assume w: "w \<in> A - (pts \<union> {z \<in> A. isolated_zero f z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
833 |
show "(\<lambda>z. inverse (f z)) analytic_on {w}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
834 |
proof (cases "f w = 0") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
835 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
836 |
thus ?thesis using w |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
837 |
by (intro analytic_intros analytic_on_subset[OF ana]) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
838 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
839 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
840 |
have "eventually (\<lambda>w. f w = 0) (nhds w)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
841 |
using True w by (intro non_isolated_zero analytic_on_subset[OF ana]) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
842 |
hence "(\<lambda>z. inverse (f z)) analytic_on {w} \<longleftrightarrow> (\<lambda>_. 0) analytic_on {w}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
843 |
using w by (intro analytic_at_cong refl) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
844 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
845 |
by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
846 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
847 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
848 |
also have "\<dots> = A - (pts \<union> {z \<in> A. isolated_zero f z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
849 |
by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
850 |
finally have "(\<lambda>z. inverse (f z)) analytic_on \<dots>" . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
851 |
moreover have "open (A - (pts \<union> {z \<in> A. isolated_zero f z}))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
852 |
using closedin_isolated_zeros[OF mero] open' \<open>open A\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
853 |
by (metis (no_types, lifting) Diff_Diff_Int Diff_Un closedin_closed open_Diff open_Int) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
854 |
ultimately show "(\<lambda>z. inverse (f z)) holomorphic_on A - (pts \<union> {z \<in> A. isolated_zero f z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
855 |
by (subst (asm) analytic_on_open) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
856 |
qed (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
857 |
intro!: holomorphic_intros singularity_intros\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
858 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
859 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
860 |
lemma meromorphic_on_inverse'' [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
861 |
assumes "f meromorphic_on A pts" "{z\<in>A. f z = 0} \<subseteq> pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
862 |
shows "(\<lambda>z. inverse (f z)) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
863 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
864 |
have "(\<lambda>z. inverse (f z)) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
865 |
by (intro meromorphic_on_inverse assms) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
866 |
also have "(pts \<union> {z \<in> A. isolated_zero f z}) = pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
867 |
using assms(2) by (auto simp: isolated_zero_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
868 |
finally show ?thesis . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
869 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
870 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
871 |
lemma meromorphic_on_divide [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
872 |
assumes "f meromorphic_on A pts" and "g meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
873 |
shows "(\<lambda>z. f z / g z) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero g z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
874 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
875 |
have mero1: "(\<lambda>z. inverse (g z)) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero g z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
876 |
by (intro meromorphic_intros assms) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
877 |
have sparse: "\<forall>x\<in>A. \<not> x islimpt pts \<union> {z\<in>A. isolated_zero g z}" and "pts \<subseteq> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
878 |
using mero1 by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
879 |
have mero2: "f meromorphic_on A (pts \<union> {z\<in>A. isolated_zero g z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
880 |
by (rule meromorphic_on_superset_pts[OF assms(1)]) (use sparse \<open>pts \<subseteq> A\<close> in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
881 |
have "(\<lambda>z. f z * inverse (g z)) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero g z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
882 |
by (intro meromorphic_on_mult mero1 mero2) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
883 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
884 |
by (simp add: field_simps) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
885 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
886 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
887 |
lemma meromorphic_on_divide' [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
888 |
assumes "f meromorphic_on A pts" "g meromorphic_on A pts" "{z\<in>A. g z = 0} \<subseteq> pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
889 |
shows "(\<lambda>z. f z / g z) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
890 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
891 |
have "(\<lambda>z. f z * inverse (g z)) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
892 |
by (intro meromorphic_intros assms) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
893 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
894 |
by (simp add: field_simps) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
895 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
896 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
897 |
lemma meromorphic_on_cmult_left [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
898 |
assumes "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
899 |
shows "(\<lambda>x. c * f x) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
900 |
using assms by (intro meromorphic_intros) (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
901 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
902 |
lemma meromorphic_on_cmult_right [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
903 |
assumes "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
904 |
shows "(\<lambda>x. f x * c) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
905 |
using assms by (intro meromorphic_intros) (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
906 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
907 |
lemma meromorphic_on_scaleR [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
908 |
assumes "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
909 |
shows "(\<lambda>x. c *\<^sub>R f x) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
910 |
using assms unfolding scaleR_conv_of_real |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
911 |
by (intro meromorphic_intros) (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
912 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
913 |
lemma meromorphic_on_sum [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
914 |
assumes "\<And>y. y \<in> I \<Longrightarrow> f y meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
915 |
assumes "I \<noteq> {} \<or> open A \<and> pts \<subseteq> A \<and> (\<forall>x\<in>A. \<not>x islimpt pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
916 |
shows "(\<lambda>x. \<Sum>y\<in>I. f y x) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
917 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
918 |
have *: "open A \<and> pts \<subseteq> A \<and> (\<forall>x\<in>A. \<not>x islimpt pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
919 |
using assms(2) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
920 |
proof |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
921 |
assume "I \<noteq> {}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
922 |
then obtain x where "x \<in> I" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
923 |
by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
924 |
from assms(1)[OF this] show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
925 |
by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
926 |
qed auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
927 |
show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
928 |
using assms(1) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
929 |
by (induction I rule: infinite_finite_induct) (use * in \<open>auto intro!: meromorphic_intros\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
930 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
931 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
932 |
lemma meromorphic_on_prod [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
933 |
assumes "\<And>y. y \<in> I \<Longrightarrow> f y meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
934 |
assumes "I \<noteq> {} \<or> open A \<and> pts \<subseteq> A \<and> (\<forall>x\<in>A. \<not>x islimpt pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
935 |
shows "(\<lambda>x. \<Prod>y\<in>I. f y x) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
936 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
937 |
have *: "open A \<and> pts \<subseteq> A \<and> (\<forall>x\<in>A. \<not>x islimpt pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
938 |
using assms(2) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
939 |
proof |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
940 |
assume "I \<noteq> {}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
941 |
then obtain x where "x \<in> I" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
942 |
by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
943 |
from assms(1)[OF this] show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
944 |
by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
945 |
qed auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
946 |
show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
947 |
using assms(1) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
948 |
by (induction I rule: infinite_finite_induct) (use * in \<open>auto intro!: meromorphic_intros\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
949 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
950 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
951 |
lemma meromorphic_on_power [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
952 |
assumes "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
953 |
shows "(\<lambda>x. f x ^ n) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
954 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
955 |
have "(\<lambda>x. \<Prod>i\<in>{..<n}. f x) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
956 |
by (intro meromorphic_intros assms(1)) (use assms in \<open>auto simp: meromorphic_on_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
957 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
958 |
by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
959 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
960 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
961 |
lemma meromorphic_on_power_int [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
962 |
assumes "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
963 |
shows "(\<lambda>z. f z powi n) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
964 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
965 |
have inv: "(\<lambda>x. inverse (f x)) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
966 |
by (intro meromorphic_intros assms) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
967 |
have *: "f meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
968 |
by (intro meromorphic_on_superset_pts [OF assms(1)]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
969 |
(use inv in \<open>auto simp: meromorphic_on_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
970 |
show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
971 |
proof (cases "n \<ge> 0") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
972 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
973 |
have "(\<lambda>x. f x ^ nat n) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
974 |
by (intro meromorphic_intros *) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
975 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
976 |
using True by (simp add: power_int_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
977 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
978 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
979 |
have "(\<lambda>x. inverse (f x) ^ nat (-n)) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
980 |
by (intro meromorphic_intros assms) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
981 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
982 |
using False by (simp add: power_int_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
983 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
984 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
985 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
986 |
lemma meromorphic_on_power_int' [meromorphic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
987 |
assumes "f meromorphic_on A pts" "n \<ge> 0 \<or> (\<forall>z\<in>A. isolated_zero f z \<longrightarrow> z \<in> pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
988 |
shows "(\<lambda>z. f z powi n) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
989 |
proof (cases "n \<ge> 0") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
990 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
991 |
have "(\<lambda>z. f z ^ nat n) meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
992 |
by (intro meromorphic_intros assms) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
993 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
994 |
using True by (simp add: power_int_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
995 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
996 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
997 |
have "(\<lambda>z. f z powi n) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero f z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
998 |
by (rule meromorphic_on_power_int) fact |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
999 |
also from assms(2) False have "pts \<union> {z\<in>A. isolated_zero f z} = pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1000 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1001 |
finally show ?thesis . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1002 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1003 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1004 |
lemma has_laurent_expansion_on_imp_meromorphic_on: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1005 |
assumes "open A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1006 |
assumes laurent: "\<And>z. z \<in> A \<Longrightarrow> \<exists>F. (\<lambda>w. f (z + w)) has_laurent_expansion F" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1007 |
shows "f meromorphic_on A {z\<in>A. \<not>f analytic_on {z}}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1008 |
unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1009 |
proof (intro conjI ballI) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1010 |
fix z assume "z \<in> {z\<in>A. \<not>f analytic_on {z}}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1011 |
then obtain F where F: "(\<lambda>w. f (z + w)) has_laurent_expansion F" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1012 |
using laurent[of z] by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1013 |
from F show "not_essential f z" "isolated_singularity_at f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1014 |
using has_laurent_expansion_not_essential has_laurent_expansion_isolated by blast+ |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1015 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1016 |
fix z assume z: "z \<in> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1017 |
obtain F where F: "(\<lambda>w. f (z + w)) has_laurent_expansion F" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1018 |
using laurent[of z] \<open>z \<in> A\<close> by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1019 |
from F have "isolated_singularity_at f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1020 |
using has_laurent_expansion_isolated z by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1021 |
then obtain r where r: "r > 0" "f analytic_on ball z r - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1022 |
unfolding isolated_singularity_at_def by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1023 |
have "f analytic_on {w}" if "w \<in> ball z r - {z}" for w |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1024 |
by (rule analytic_on_subset[OF r(2)]) (use that in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1025 |
hence "eventually (\<lambda>w. f analytic_on {w}) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1026 |
using eventually_at_in_open[of "ball z r" z] \<open>r > 0\<close> by (auto elim!: eventually_mono) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1027 |
hence "\<not>z islimpt {w. \<not>f analytic_on {w}}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1028 |
by (auto simp: islimpt_conv_frequently_at frequently_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1029 |
thus "\<not>z islimpt {w\<in>A. \<not>f analytic_on {w}}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1030 |
using islimpt_subset[of z "{w\<in>A. \<not>f analytic_on {w}}" "{w. \<not>f analytic_on {w}}"] by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1031 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1032 |
have "f analytic_on A - {w\<in>A. \<not>f analytic_on {w}}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1033 |
by (subst analytic_on_analytic_at) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1034 |
thus "f holomorphic_on A - {w\<in>A. \<not>f analytic_on {w}}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1035 |
by (meson analytic_imp_holomorphic) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1036 |
qed (use assms in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1037 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1038 |
lemma meromorphic_on_imp_has_laurent_expansion: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1039 |
assumes "f meromorphic_on A pts" "z \<in> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1040 |
shows "(\<lambda>w. f (z + w)) has_laurent_expansion laurent_expansion f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1041 |
proof (cases "z \<in> pts") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1042 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1043 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1044 |
using assms by (intro not_essential_has_laurent_expansion) (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1045 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1046 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1047 |
have "f holomorphic_on (A - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1048 |
using assms by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1049 |
moreover have "z \<in> A - pts" "open (A - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1050 |
using assms(2) False by (auto intro!: meromorphic_imp_open_diff[OF assms(1)]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1051 |
ultimately have "f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1052 |
unfolding analytic_at by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1053 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1054 |
using isolated_singularity_at_analytic not_essential_analytic |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1055 |
not_essential_has_laurent_expansion by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1056 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1057 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1058 |
lemma |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1059 |
assumes "isolated_singularity_at f z" "f \<midarrow>z\<rightarrow> c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1060 |
shows eventually_remove_sings_eq_nhds': |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1061 |
"eventually (\<lambda>w. remove_sings f w = (if w = z then c else f w)) (nhds z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1062 |
and remove_sings_analytic_at_singularity: "remove_sings f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1063 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1064 |
have "eventually (\<lambda>w. w \<noteq> z) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1065 |
by (auto simp: eventually_at_filter) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1066 |
hence "eventually (\<lambda>w. remove_sings f w = (if w = z then c else f w)) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1067 |
using eventually_remove_sings_eq_at[OF assms(1)] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1068 |
by eventually_elim auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1069 |
moreover have "remove_sings f z = c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1070 |
using assms by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1071 |
ultimately show ev: "eventually (\<lambda>w. remove_sings f w = (if w = z then c else f w)) (nhds z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1072 |
by (simp add: eventually_at_filter) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1073 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1074 |
have "(\<lambda>w. if w = z then c else f w) analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1075 |
by (intro removable_singularity' assms) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1076 |
also have "?this \<longleftrightarrow> remove_sings f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1077 |
using ev by (intro analytic_at_cong) (auto simp: eq_commute) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1078 |
finally show \<dots> . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1079 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1080 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1081 |
lemma remove_sings_meromorphic_on: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1082 |
assumes "f meromorphic_on A pts" "\<And>z. z \<in> pts - pts' \<Longrightarrow> \<not>is_pole f z" "pts' \<subseteq> pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1083 |
shows "remove_sings f meromorphic_on A pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1084 |
unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1085 |
proof safe |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1086 |
have "remove_sings f analytic_on {z}" if "z \<in> A - pts'" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1087 |
proof (cases "z \<in> pts") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1088 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1089 |
hence *: "f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1090 |
using assms meromorphic_imp_open_diff[OF assms(1)] that |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1091 |
by (force simp: meromorphic_on_def analytic_at) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1092 |
have "remove_sings f analytic_on {z} \<longleftrightarrow> f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1093 |
by (intro analytic_at_cong eventually_remove_sings_eq_nhds * refl) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1094 |
thus ?thesis using * by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1095 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1096 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1097 |
have isol: "isolated_singularity_at f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1098 |
using True using assms by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1099 |
from assms(1) have "not_essential f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1100 |
using True by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1101 |
with assms(2) True that obtain c where "f \<midarrow>z\<rightarrow> c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1102 |
by (auto simp: not_essential_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1103 |
thus "remove_sings f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1104 |
by (intro remove_sings_analytic_at_singularity isol) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1105 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1106 |
hence "remove_sings f analytic_on A - pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1107 |
by (subst analytic_on_analytic_at) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1108 |
thus "remove_sings f holomorphic_on A - pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1109 |
using meromorphic_imp_open_diff'[OF assms(1,3)] by (subst (asm) analytic_on_open) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1110 |
qed (use assms islimpt_subset[OF _ assms(3)] in \<open>auto simp: meromorphic_on_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1111 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1112 |
lemma remove_sings_holomorphic_on: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1113 |
assumes "f meromorphic_on A pts" "\<And>z. z \<in> pts \<Longrightarrow> \<not>is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1114 |
shows "remove_sings f holomorphic_on A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1115 |
using remove_sings_meromorphic_on[OF assms(1), of "{}"] assms(2) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1116 |
by (auto simp: meromorphic_on_no_singularities) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1117 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1118 |
lemma meromorphic_on_Ex_iff: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1119 |
"(\<exists>pts. f meromorphic_on A pts) \<longleftrightarrow> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1120 |
open A \<and> (\<forall>z\<in>A. \<exists>F. (\<lambda>w. f (z + w)) has_laurent_expansion F)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1121 |
proof safe |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1122 |
fix pts assume *: "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1123 |
from * show "open A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1124 |
by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1125 |
show "\<exists>F. (\<lambda>w. f (z + w)) has_laurent_expansion F" if "z \<in> A" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1126 |
using that * |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1127 |
by (intro exI[of _ "laurent_expansion f z"] meromorphic_on_imp_has_laurent_expansion) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1128 |
qed (blast intro!: has_laurent_expansion_on_imp_meromorphic_on) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1129 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1130 |
lemma is_pole_inverse_holomorphic_pts: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1131 |
fixes pts::"complex set" and f::"complex \<Rightarrow> complex" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1132 |
defines "g \<equiv> \<lambda>x. (if x\<in>pts then 0 else inverse (f x))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1133 |
assumes mer: "f meromorphic_on D pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1134 |
and non_z: "\<And>z. z \<in> D - pts \<Longrightarrow> f z \<noteq> 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1135 |
and all_poles:"\<forall>x. is_pole f x \<longleftrightarrow> x\<in>pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1136 |
shows "g holomorphic_on D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1137 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1138 |
have "open D" and f_holo: "f holomorphic_on (D-pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1139 |
using mer by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1140 |
have "\<exists>r. r>0 \<and> f analytic_on ball z r - {z} |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1141 |
\<and> (\<forall>x \<in> ball z r - {z}. f x\<noteq>0)" if "z\<in>pts" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1142 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1143 |
have "isolated_singularity_at f z" "is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1144 |
using mer meromorphic_on_def that all_poles by blast+ |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1145 |
then obtain r1 where "r1>0" and fan: "f analytic_on ball z r1 - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1146 |
by (meson isolated_singularity_at_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1147 |
obtain r2 where "r2>0" "\<forall>x \<in> ball z r2 - {z}. f x\<noteq>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1148 |
using non_zero_neighbour_pole[OF \<open>is_pole f z\<close>] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1149 |
unfolding eventually_at by (metis Diff_iff UNIV_I dist_commute insertI1 mem_ball) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1150 |
define r where "r = min r1 r2" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1151 |
have "r>0" by (simp add: \<open>0 < r2\<close> \<open>r1>0\<close> r_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1152 |
moreover have "f analytic_on ball z r - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1153 |
using r_def by (force intro: analytic_on_subset [OF fan]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1154 |
moreover have "\<forall>x \<in> ball z r - {z}. f x\<noteq>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1155 |
by (simp add: \<open>\<forall>x\<in>ball z r2 - {z}. f x \<noteq> 0\<close> r_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1156 |
ultimately show ?thesis by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1157 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1158 |
then obtain get_r where r_pos:"get_r z>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1159 |
and r_ana:"f analytic_on ball z (get_r z) - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1160 |
and r_nz:"\<forall>x \<in> ball z (get_r z) - {z}. f x\<noteq>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1161 |
if "z\<in>pts" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1162 |
by metis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1163 |
define p_balls where "p_balls \<equiv> \<Union>z\<in>pts. ball z (get_r z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1164 |
have g_ball:"g holomorphic_on ball z (get_r z)" if "z\<in>pts" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1165 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1166 |
have "(\<lambda>x. if x = z then 0 else inverse (f x)) holomorphic_on ball z (get_r z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1167 |
proof (rule is_pole_inverse_holomorphic) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1168 |
show "f holomorphic_on ball z (get_r z) - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1169 |
using analytic_imp_holomorphic r_ana that by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1170 |
show "is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1171 |
using mer meromorphic_on_def that all_poles by force |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1172 |
show "\<forall>x\<in>ball z (get_r z) - {z}. f x \<noteq> 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1173 |
using r_nz that by metis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1174 |
qed auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1175 |
then show ?thesis unfolding g_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1176 |
by (smt (verit, ccfv_SIG) Diff_iff Elementary_Metric_Spaces.open_ball |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1177 |
all_poles analytic_imp_holomorphic empty_iff |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1178 |
holomorphic_transform insert_iff not_is_pole_holomorphic |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1179 |
open_delete r_ana that) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1180 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1181 |
then have "g holomorphic_on p_balls" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1182 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1183 |
have "g analytic_on p_balls" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1184 |
unfolding p_balls_def analytic_on_UN |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1185 |
using g_ball by (simp add: analytic_on_open) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1186 |
moreover have "open p_balls" using p_balls_def by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1187 |
ultimately show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1188 |
by (simp add: analytic_imp_holomorphic) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1189 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1190 |
moreover have "g holomorphic_on D-pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1191 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1192 |
have "(\<lambda>z. inverse (f z)) holomorphic_on D - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1193 |
using f_holo holomorphic_on_inverse non_z by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1194 |
then show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1195 |
by (metis DiffD2 g_def holomorphic_transform) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1196 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1197 |
moreover have "open p_balls" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1198 |
using p_balls_def by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1199 |
ultimately have "g holomorphic_on (p_balls \<union> (D-pts))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1200 |
by (simp add: holomorphic_on_Un meromorphic_imp_open_diff[OF mer]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1201 |
moreover have "D \<subseteq> p_balls \<union> (D-pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1202 |
unfolding p_balls_def using \<open>\<And>z. z \<in> pts \<Longrightarrow> 0 < get_r z\<close> by force |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1203 |
ultimately show "g holomorphic_on D" by (meson holomorphic_on_subset) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1204 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1205 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1206 |
lemma meromorphic_imp_analytic_on: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1207 |
assumes "f meromorphic_on D pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1208 |
shows "f analytic_on (D - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1209 |
by (metis assms analytic_on_open meromorphic_imp_open_diff meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1210 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1211 |
lemma meromorphic_imp_constant_on: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1212 |
assumes merf: "f meromorphic_on D pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1213 |
and "f constant_on (D - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1214 |
and "\<forall>x\<in>pts. is_pole f x" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1215 |
shows "f constant_on D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1216 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1217 |
obtain c where c:"\<And>z. z \<in> D-pts \<Longrightarrow> f z = c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1218 |
by (meson assms constant_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1219 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1220 |
have "f z = c" if "z \<in> D" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1221 |
proof (cases "is_pole f z") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1222 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1223 |
then obtain r0 where "r0 > 0" and r0: "f analytic_on ball z r0 - {z}" and pol: "is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1224 |
using merf unfolding meromorphic_on_def isolated_singularity_at_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1225 |
by (metis \<open>z \<in> D\<close> insert_Diff insert_Diff_if insert_iff merf |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1226 |
meromorphic_imp_open_diff not_is_pole_holomorphic) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1227 |
have "open D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1228 |
using merf meromorphic_on_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1229 |
then obtain r where "r > 0" "ball z r \<subseteq> D" "r \<le> r0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1230 |
by (smt (verit, best) \<open>0 < r0\<close> \<open>z \<in> D\<close> openE order_subst2 subset_ball) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1231 |
have r: "f analytic_on ball z r - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1232 |
by (meson Diff_mono \<open>r \<le> r0\<close> analytic_on_subset order_refl r0 subset_ball) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1233 |
have "ball z r - {z} \<subseteq> -pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1234 |
using merf r unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1235 |
by (meson ComplI Elementary_Metric_Spaces.open_ball |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1236 |
analytic_imp_holomorphic assms(3) not_is_pole_holomorphic open_delete subsetI) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1237 |
with \<open>ball z r \<subseteq> D\<close> have "ball z r - {z} \<subseteq> D-pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1238 |
by fastforce |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1239 |
with c have c': "\<And>u. u \<in> ball z r - {z} \<Longrightarrow> f u = c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1240 |
by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1241 |
have False if "\<forall>\<^sub>F x in at z. cmod c + 1 \<le> cmod (f x)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1242 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1243 |
have "\<forall>\<^sub>F x in at z within ball z r - {z}. cmod c + 1 \<le> cmod (f x)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1244 |
by (smt (verit, best) Diff_UNIV Diff_eq_empty_iff eventually_at_topological insert_subset that) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1245 |
with \<open>r > 0\<close> show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1246 |
apply (simp add: c' eventually_at_filter topological_space_class.eventually_nhds open_dist) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1247 |
by (metis dist_commute min_less_iff_conj perfect_choose_dist) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1248 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1249 |
with pol show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1250 |
by (auto simp: is_pole_def filterlim_at_infinity_conv_norm_at_top filterlim_at_top) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1251 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1252 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1253 |
then show ?thesis by (meson DiffI assms(3) c that) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1254 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1255 |
then show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1256 |
by (simp add: constant_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1257 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1258 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1259 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1260 |
lemma meromorphic_isolated: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1261 |
assumes merf: "f meromorphic_on D pts" and "p\<in>pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1262 |
obtains r where "r>0" "ball p r \<subseteq> D" "ball p r \<inter> pts = {p}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1263 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1264 |
have "\<forall>z\<in>D. \<exists>e>0. finite (pts \<inter> ball z e)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1265 |
using merf unfolding meromorphic_on_def islimpt_eq_infinite_ball |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1266 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1267 |
then obtain r0 where r0:"r0>0" "finite (pts \<inter> ball p r0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1268 |
by (metis assms(2) in_mono merf meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1269 |
moreover define pts' where "pts' = pts \<inter> ball p r0 - {p}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1270 |
ultimately have "finite pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1271 |
by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1272 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1273 |
define r1 where "r1=(if pts'={} then r0 else |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1274 |
min (Min {dist p' p |p'. p'\<in>pts'}/2) r0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1275 |
have "r1>0 \<and> pts \<inter> ball p r1 - {p} = {}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1276 |
proof (cases "pts'={}") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1277 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1278 |
then show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1279 |
using pts'_def r0(1) r1_def by presburger |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1280 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1281 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1282 |
define S where "S={dist p' p |p'. p'\<in>pts'}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1283 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1284 |
have nempty:"S \<noteq> {}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1285 |
using False S_def by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1286 |
have finite:"finite S" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1287 |
using \<open>finite pts'\<close> S_def by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1288 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1289 |
have "r1>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1290 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1291 |
have "r1=min (Min S/2) r0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1292 |
using False unfolding S_def r1_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1293 |
moreover have "Min S\<in>S" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1294 |
using \<open>S\<noteq>{}\<close> \<open>finite S\<close> Min_in by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1295 |
then have "Min S>0" unfolding S_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1296 |
using pts'_def by force |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1297 |
ultimately show ?thesis using \<open>r0>0\<close> by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1298 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1299 |
moreover have "pts \<inter> ball p r1 - {p} = {}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1300 |
proof (rule ccontr) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1301 |
assume "pts \<inter> ball p r1 - {p} \<noteq> {}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1302 |
then obtain p' where "p'\<in>pts \<inter> ball p r1 - {p}" by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1303 |
moreover have "r1\<le>r0" using r1_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1304 |
ultimately have "p'\<in>pts'" unfolding pts'_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1305 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1306 |
then have "dist p' p\<ge>Min S" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1307 |
using S_def eq_Min_iff local.finite by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1308 |
moreover have "dist p' p < Min S" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1309 |
using \<open>p'\<in>pts \<inter> ball p r1 - {p}\<close> False unfolding r1_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1310 |
apply (fold S_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1311 |
by (smt (verit, ccfv_threshold) DiffD1 Int_iff dist_commute |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1312 |
dist_triangle_half_l mem_ball) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1313 |
ultimately show False by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1314 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1315 |
ultimately show ?thesis by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1316 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1317 |
then have "r1>0" and r1_pts:"pts \<inter> ball p r1 - {p} = {}" by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1318 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1319 |
obtain r2 where "r2>0" "ball p r2 \<subseteq> D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1320 |
by (metis assms(2) merf meromorphic_on_def openE subset_eq) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1321 |
define r where "r=min r1 r2" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1322 |
have "r > 0" unfolding r_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1323 |
by (simp add: \<open>0 < r1\<close> \<open>0 < r2\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1324 |
moreover have "ball p r \<subseteq> D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1325 |
using \<open>ball p r2 \<subseteq> D\<close> r_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1326 |
moreover have "ball p r \<inter> pts = {p}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1327 |
using assms(2) \<open>r>0\<close> r1_pts |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1328 |
unfolding r_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1329 |
ultimately show ?thesis using that by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1330 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1331 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1332 |
lemma meromorphic_pts_closure: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1333 |
assumes merf: "f meromorphic_on D pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1334 |
shows "pts \<subseteq> closure (D - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1335 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1336 |
have "p islimpt (D - pts)" if "p\<in>pts" for p |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1337 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1338 |
obtain r where "r>0" "ball p r \<subseteq> D" "ball p r \<inter> pts = {p}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1339 |
using meromorphic_isolated[OF merf \<open>p\<in>pts\<close>] by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1340 |
from \<open>r>0\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1341 |
have "p islimpt ball p r - {p}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1342 |
by (meson open_ball ball_subset_cball in_mono islimpt_ball |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1343 |
islimpt_punctured le_less open_contains_ball_eq) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1344 |
moreover have " ball p r - {p} \<subseteq> D - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1345 |
using \<open>ball p r \<inter> pts = {p}\<close> \<open>ball p r \<subseteq> D\<close> by fastforce |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1346 |
ultimately show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1347 |
using islimpt_subset by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1348 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1349 |
then show ?thesis by (simp add: islimpt_in_closure subset_eq) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1350 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1351 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1352 |
lemma nconst_imp_nzero_neighbour: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1353 |
assumes merf: "f meromorphic_on D pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1354 |
and f_nconst:"\<not>(\<forall>w\<in>D-pts. f w=0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1355 |
and "z\<in>D" and "connected D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1356 |
shows "(\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w \<in> D - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1357 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1358 |
obtain \<beta> where \<beta>:"\<beta> \<in> D - pts" "f \<beta>\<noteq>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1359 |
using f_nconst by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1360 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1361 |
have ?thesis if "z\<notin>pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1362 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1363 |
have "\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w \<in> D - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1364 |
apply (rule non_zero_neighbour_alt[of f "D-pts" z \<beta>]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1365 |
subgoal using merf meromorphic_on_def by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1366 |
subgoal using merf meromorphic_imp_open_diff by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1367 |
subgoal using assms(4) merf meromorphic_imp_connected_diff by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1368 |
subgoal by (simp add: assms(3) that) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1369 |
using \<beta> by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1370 |
then show ?thesis by (auto elim:eventually_mono) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1371 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1372 |
moreover have ?thesis if "z\<in>pts" "\<not> f \<midarrow>z\<rightarrow> 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1373 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1374 |
have "\<forall>\<^sub>F w in at z. w \<in> D - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1375 |
using merf[unfolded meromorphic_on_def islimpt_iff_eventually] \<open>z\<in>D\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1376 |
using eventually_at_in_open' eventually_elim2 by fastforce |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1377 |
moreover have "\<forall>\<^sub>F w in at z. f w \<noteq> 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1378 |
proof (cases "is_pole f z") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1379 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1380 |
then show ?thesis using non_zero_neighbour_pole by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1381 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1382 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1383 |
moreover have "not_essential f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1384 |
using merf meromorphic_on_def that(1) by fastforce |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1385 |
ultimately obtain c where "c\<noteq>0" "f \<midarrow>z\<rightarrow> c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1386 |
by (metis \<open>\<not> f \<midarrow>z\<rightarrow> 0\<close> not_essential_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1387 |
then show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1388 |
using tendsto_imp_eventually_ne by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1389 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1390 |
ultimately show ?thesis by eventually_elim auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1391 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1392 |
moreover have ?thesis if "z\<in>pts" "f \<midarrow>z\<rightarrow> 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1393 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1394 |
define ff where "ff=(\<lambda>x. if x=z then 0 else f x)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1395 |
define A where "A=D - (pts - {z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1396 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1397 |
have "f holomorphic_on A - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1398 |
by (metis A_def Diff_insert analytic_imp_holomorphic |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1399 |
insert_Diff merf meromorphic_imp_analytic_on that(1)) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1400 |
moreover have "open A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1401 |
using A_def merf meromorphic_imp_open_diff' by force |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1402 |
ultimately have "ff holomorphic_on A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1403 |
using \<open>f \<midarrow>z\<rightarrow> 0\<close> unfolding ff_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1404 |
by (rule removable_singularity) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1405 |
moreover have "connected A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1406 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1407 |
have "connected (D - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1408 |
using assms(4) merf meromorphic_imp_connected_diff by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1409 |
moreover have "D - pts \<subseteq> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1410 |
unfolding A_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1411 |
moreover have "A \<subseteq> closure (D - pts)" unfolding A_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1412 |
by (smt (verit, ccfv_SIG) Diff_empty Diff_insert |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1413 |
closure_subset insert_Diff_single insert_absorb |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1414 |
insert_subset merf meromorphic_pts_closure that(1)) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1415 |
ultimately show ?thesis using connected_intermediate_closure |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1416 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1417 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1418 |
moreover have "z \<in> A" using A_def assms(3) by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1419 |
moreover have "ff z = 0" unfolding ff_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1420 |
moreover have "\<beta> \<in> A " using A_def \<beta>(1) by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1421 |
moreover have "ff \<beta> \<noteq> 0" using \<beta>(1) \<beta>(2) ff_def that(1) by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1422 |
ultimately obtain r where "0 < r" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1423 |
"ball z r \<subseteq> A" "\<And>x. x \<in> ball z r - {z} \<Longrightarrow> ff x \<noteq> 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1424 |
using \<open>open A\<close> isolated_zeros[of ff A z \<beta>] by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1425 |
then show ?thesis unfolding eventually_at ff_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1426 |
by (intro exI[of _ r]) (auto simp: A_def dist_commute ball_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1427 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1428 |
ultimately show ?thesis by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1429 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1430 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1431 |
lemma nconst_imp_nzero_neighbour': |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1432 |
assumes merf: "f meromorphic_on D pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1433 |
and f_nconst:"\<not>(\<forall>w\<in>D-pts. f w=0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1434 |
and "z\<in>D" and "connected D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1435 |
shows "\<forall>\<^sub>F w in at z. f w \<noteq> 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1436 |
using nconst_imp_nzero_neighbour[OF assms] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1437 |
by (auto elim:eventually_mono) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1438 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1439 |
lemma meromorphic_compact_finite_zeros: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1440 |
assumes merf:"f meromorphic_on D pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1441 |
and "compact S" "S \<subseteq> D" "connected D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1442 |
and f_nconst:"\<not>(\<forall>w\<in>D-pts. f w=0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1443 |
shows "finite ({x\<in>S. f x=0})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1444 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1445 |
have "finite ({x\<in>S. f x=0 \<and> x \<notin> pts})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1446 |
proof (rule ccontr) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1447 |
assume "infinite {x \<in> S. f x = 0 \<and> x \<notin> pts}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1448 |
then obtain z where "z\<in>S" and z_lim:"z islimpt {x \<in> S. f x = 0 |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1449 |
\<and> x \<notin> pts}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1450 |
using \<open>compact S\<close> unfolding compact_eq_Bolzano_Weierstrass |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1451 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1452 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1453 |
from z_lim |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1454 |
have "\<exists>\<^sub>F x in at z. f x = 0 \<and> x \<in> S \<and> x \<notin> pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1455 |
unfolding islimpt_iff_eventually not_eventually by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1456 |
moreover have "\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w \<in> D - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1457 |
using nconst_imp_nzero_neighbour[OF merf f_nconst _ \<open>connected D\<close>] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1458 |
\<open>z\<in>S\<close> \<open>S \<subseteq> D\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1459 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1460 |
ultimately have "\<exists>\<^sub>F x in at z. False" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1461 |
by (simp add: eventually_mono frequently_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1462 |
then show False by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1463 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1464 |
moreover have "finite (S \<inter> pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1465 |
using meromorphic_compact_finite_pts[OF merf \<open>compact S\<close> \<open>S \<subseteq> D\<close>] . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1466 |
ultimately have "finite ({x\<in>S. f x=0 \<and> x \<notin> pts} \<union> (S \<inter> pts))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1467 |
unfolding finite_Un by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1468 |
then show ?thesis by (elim rev_finite_subset) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1469 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1470 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1471 |
lemma meromorphic_onI [intro?]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1472 |
assumes "open A" "pts \<subseteq> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1473 |
assumes "f holomorphic_on A - pts" "\<And>z. z \<in> A \<Longrightarrow> \<not>z islimpt pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1474 |
assumes "\<And>z. z \<in> pts \<Longrightarrow> isolated_singularity_at f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1475 |
assumes "\<And>z. z \<in> pts \<Longrightarrow> not_essential f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1476 |
shows "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1477 |
using assms unfolding meromorphic_on_def by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1478 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1479 |
lemma Polygamma_plus_of_nat: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1480 |
assumes "\<forall>k<m. z \<noteq> -of_nat k" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1481 |
shows "Polygamma n (z + of_nat m) = |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1482 |
Polygamma n z + (-1) ^ n * fact n * (\<Sum>k<m. 1 / (z + of_nat k) ^ Suc n)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1483 |
using assms |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1484 |
proof (induction m) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1485 |
case (Suc m) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1486 |
have "Polygamma n (z + of_nat (Suc m)) = Polygamma n (z + of_nat m + 1)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1487 |
by (simp add: add_ac) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1488 |
also have "\<dots> = Polygamma n (z + of_nat m) + (-1) ^ n * fact n * (1 / ((z + of_nat m) ^ Suc n))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1489 |
using Suc.prems by (subst Polygamma_plus1) (auto simp: add_eq_0_iff2) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1490 |
also have "Polygamma n (z + of_nat m) = |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1491 |
Polygamma n z + (-1) ^ n * (\<Sum>k<m. 1 / (z + of_nat k) ^ Suc n) * fact n" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1492 |
using Suc.prems by (subst Suc.IH) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1493 |
finally show ?case |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1494 |
by (simp add: algebra_simps) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1495 |
qed auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1496 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1497 |
lemma tendsto_Gamma [tendsto_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1498 |
assumes "(f \<longlongrightarrow> c) F" "c \<notin> \<int>\<^sub>\<le>\<^sub>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1499 |
shows "((\<lambda>z. Gamma (f z)) \<longlongrightarrow> Gamma c) F" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1500 |
by (intro isCont_tendsto_compose[OF _ assms(1)] continuous_intros assms) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1501 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1502 |
lemma tendsto_Polygamma [tendsto_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1503 |
fixes f :: "_ \<Rightarrow> 'a :: {real_normed_field,euclidean_space}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1504 |
assumes "(f \<longlongrightarrow> c) F" "c \<notin> \<int>\<^sub>\<le>\<^sub>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1505 |
shows "((\<lambda>z. Polygamma n (f z)) \<longlongrightarrow> Polygamma n c) F" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1506 |
by (intro isCont_tendsto_compose[OF _ assms(1)] continuous_intros assms) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1507 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1508 |
lemma analytic_on_Gamma' [analytic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1509 |
assumes "f analytic_on A" "\<forall>x\<in>A. f x \<notin> \<int>\<^sub>\<le>\<^sub>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1510 |
shows "(\<lambda>z. Gamma (f z)) analytic_on A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1511 |
using analytic_on_compose_gen[OF assms(1) analytic_Gamma[of "f ` A"]] assms(2) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1512 |
by (auto simp: o_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1513 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1514 |
lemma analytic_on_Polygamma' [analytic_intros]: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1515 |
assumes "f analytic_on A" "\<forall>x\<in>A. f x \<notin> \<int>\<^sub>\<le>\<^sub>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1516 |
shows "(\<lambda>z. Polygamma n (f z)) analytic_on A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1517 |
using analytic_on_compose_gen[OF assms(1) analytic_on_Polygamma[of "f ` A" n]] assms(2) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1518 |
by (auto simp: o_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1519 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1520 |
lemma |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1521 |
shows is_pole_Polygamma: "is_pole (Polygamma n) (-of_nat m :: complex)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1522 |
and zorder_Polygamma: "zorder (Polygamma n) (-of_nat m) = -int (Suc n)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1523 |
and residue_Polygamma: "residue (Polygamma n) (-of_nat m) = (if n = 0 then -1 else 0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1524 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1525 |
define g1 :: "complex \<Rightarrow> complex" where |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1526 |
"g1 = (\<lambda>z. Polygamma n (z + of_nat (Suc m)) + |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1527 |
(-1) ^ Suc n * fact n * (\<Sum>k<m. 1 / (z + of_nat k) ^ Suc n))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1528 |
define g :: "complex \<Rightarrow> complex" where |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1529 |
"g = (\<lambda>z. g1 z + (-1) ^ Suc n * fact n / (z + of_nat m) ^ Suc n)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1530 |
define F where "F = fps_to_fls (fps_expansion g1 (-of_nat m)) + fls_const ((-1) ^ Suc n * fact n) / fls_X ^ Suc n" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1531 |
have F_altdef: "F = fps_to_fls (fps_expansion g1 (-of_nat m)) + fls_shift (n+1) (fls_const ((-1) ^ Suc n * fact n))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1532 |
by (simp add: F_def del: power_Suc) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1533 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1534 |
have "\<not>(-of_nat m) islimpt (\<int>\<^sub>\<le>\<^sub>0 :: complex set)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1535 |
by (intro discrete_imp_not_islimpt[where e = 1]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1536 |
(auto elim!: nonpos_Ints_cases simp: dist_of_int) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1537 |
hence "eventually (\<lambda>z::complex. z \<notin> \<int>\<^sub>\<le>\<^sub>0) (at (-of_nat m))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1538 |
by (auto simp: islimpt_conv_frequently_at frequently_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1539 |
hence ev: "eventually (\<lambda>z. Polygamma n z = g z) (at (-of_nat m))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1540 |
proof eventually_elim |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1541 |
case (elim z) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1542 |
hence *: "\<forall>k<Suc m. z \<noteq> - of_nat k" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1543 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1544 |
thus ?case |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1545 |
using Polygamma_plus_of_nat[of "Suc m" z n, OF *] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1546 |
by (auto simp: g_def g1_def algebra_simps) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1547 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1548 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1549 |
have "(\<lambda>w. g (-of_nat m + w)) has_laurent_expansion F" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1550 |
unfolding g_def F_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1551 |
by (intro laurent_expansion_intros has_laurent_expansion_fps analytic_at_imp_has_fps_expansion) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1552 |
(auto simp: g1_def intro!: laurent_expansion_intros analytic_intros) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1553 |
also have "?this \<longleftrightarrow> (\<lambda>w. Polygamma n (-of_nat m + w)) has_laurent_expansion F" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1554 |
using ev by (intro has_laurent_expansion_cong refl) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1555 |
(simp_all add: eq_commute at_to_0' eventually_filtermap) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1556 |
finally have *: "(\<lambda>w. Polygamma n (-of_nat m + w)) has_laurent_expansion F" . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1557 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1558 |
have subdegree: "fls_subdegree F = -int (Suc n)" unfolding F_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1559 |
by (subst fls_subdegree_add_eq2) (simp_all add: fls_subdegree_fls_to_fps fls_divide_subdegree) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1560 |
have [simp]: "F \<noteq> 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1561 |
using subdegree by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1562 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1563 |
show "is_pole (Polygamma n) (-of_nat m :: complex)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1564 |
using * by (rule has_laurent_expansion_imp_is_pole) (auto simp: subdegree) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1565 |
show "zorder (Polygamma n) (-of_nat m :: complex) = -int (Suc n)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1566 |
by (subst has_laurent_expansion_zorder[OF *]) (auto simp: subdegree) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1567 |
show "residue (Polygamma n) (-of_nat m :: complex) = (if n = 0 then -1 else 0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1568 |
by (subst has_laurent_expansion_residue[OF *]) (auto simp: F_altdef) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1569 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1570 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1571 |
lemma Gamma_meromorphic_on [meromorphic_intros]: "Gamma meromorphic_on UNIV \<int>\<^sub>\<le>\<^sub>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1572 |
proof |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1573 |
show "\<not>z islimpt \<int>\<^sub>\<le>\<^sub>0" for z :: complex |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1574 |
by (intro discrete_imp_not_islimpt[of 1]) (auto elim!: nonpos_Ints_cases simp: dist_of_int) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1575 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1576 |
fix z :: complex assume z: "z \<in> \<int>\<^sub>\<le>\<^sub>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1577 |
then obtain n where n: "z = -of_nat n" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1578 |
by (elim nonpos_Ints_cases') |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1579 |
show "not_essential Gamma z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1580 |
by (auto simp: n intro!: is_pole_imp_not_essential is_pole_Gamma) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1581 |
have *: "open (-(\<int>\<^sub>\<le>\<^sub>0 - {z}))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1582 |
by (intro open_Compl discrete_imp_closed[of 1]) (auto elim!: nonpos_Ints_cases simp: dist_of_int) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1583 |
have "Gamma holomorphic_on -(\<int>\<^sub>\<le>\<^sub>0 - {z}) - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1584 |
by (intro holomorphic_intros) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1585 |
thus "isolated_singularity_at Gamma z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1586 |
by (rule isolated_singularity_at_holomorphic) (use z * in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1587 |
qed (auto intro!: holomorphic_intros) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1588 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1589 |
lemma Polygamma_meromorphic_on [meromorphic_intros]: "Polygamma n meromorphic_on UNIV \<int>\<^sub>\<le>\<^sub>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1590 |
proof |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1591 |
show "\<not>z islimpt \<int>\<^sub>\<le>\<^sub>0" for z :: complex |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1592 |
by (intro discrete_imp_not_islimpt[of 1]) (auto elim!: nonpos_Ints_cases simp: dist_of_int) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1593 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1594 |
fix z :: complex assume z: "z \<in> \<int>\<^sub>\<le>\<^sub>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1595 |
then obtain m where n: "z = -of_nat m" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1596 |
by (elim nonpos_Ints_cases') |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1597 |
show "not_essential (Polygamma n) z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1598 |
by (auto simp: n intro!: is_pole_imp_not_essential is_pole_Polygamma) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1599 |
have *: "open (-(\<int>\<^sub>\<le>\<^sub>0 - {z}))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1600 |
by (intro open_Compl discrete_imp_closed[of 1]) (auto elim!: nonpos_Ints_cases simp: dist_of_int) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1601 |
have "Polygamma n holomorphic_on -(\<int>\<^sub>\<le>\<^sub>0 - {z}) - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1602 |
by (intro holomorphic_intros) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1603 |
thus "isolated_singularity_at (Polygamma n) z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1604 |
by (rule isolated_singularity_at_holomorphic) (use z * in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1605 |
qed (auto intro!: holomorphic_intros) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1606 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1607 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1608 |
theorem argument_principle': |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1609 |
fixes f::"complex \<Rightarrow> complex" and poles s:: "complex set" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1610 |
\<comment> \<open>\<^term>\<open>pz\<close> is the set of non-essential singularities and zeros\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1611 |
defines "pz \<equiv> {w\<in>s. f w = 0 \<or> w \<in> poles}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1612 |
assumes "open s" and |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1613 |
"connected s" and |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1614 |
f_holo:"f holomorphic_on s-poles" and |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1615 |
h_holo:"h holomorphic_on s" and |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1616 |
"valid_path g" and |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1617 |
loop:"pathfinish g = pathstart g" and |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1618 |
path_img:"path_image g \<subseteq> s - pz" and |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1619 |
homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0" and |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1620 |
finite:"finite pz" and |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1621 |
poles:"\<forall>p\<in>s\<inter>poles. not_essential f p" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1622 |
shows "contour_integral g (\<lambda>x. deriv f x * h x / f x) = 2 * pi * \<i> * |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1623 |
(\<Sum>p\<in>pz. winding_number g p * h p * zorder f p)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1624 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1625 |
define ff where "ff = remove_sings f" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1626 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1627 |
have finite':"finite (s \<inter> poles)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1628 |
using finite unfolding pz_def by (auto elim:rev_finite_subset) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1629 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1630 |
have isolated:"isolated_singularity_at f z" if "z\<in>s" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1631 |
proof (rule isolated_singularity_at_holomorphic) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1632 |
show "f holomorphic_on (s-(poles-{z})) - {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1633 |
by (metis Diff_empty Diff_insert Diff_insert0 Diff_subset |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1634 |
f_holo holomorphic_on_subset insert_Diff) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1635 |
show "open (s - (poles - {z}))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1636 |
by (metis Diff_Diff_Int Int_Diff assms(2) finite' finite_Diff |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1637 |
finite_imp_closed inf.idem open_Diff) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1638 |
show "z \<in> s - (poles - {z})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1639 |
using assms(4) that by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1640 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1641 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1642 |
have not_ess:"not_essential f w" if "w\<in>s" for w |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1643 |
by (metis Diff_Diff_Int Diff_iff Int_Diff Int_absorb assms(2) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1644 |
f_holo finite' finite_imp_closed not_essential_holomorphic |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1645 |
open_Diff poles that) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1646 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1647 |
have nzero:"\<forall>\<^sub>F x in at w. f x \<noteq> 0" if "w\<in>s" for w |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1648 |
proof (rule ccontr) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1649 |
assume "\<not> (\<forall>\<^sub>F x in at w. f x \<noteq> 0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1650 |
then have "\<exists>\<^sub>F x in at w. f x = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1651 |
unfolding not_eventually by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1652 |
moreover have "\<forall>\<^sub>F x in at w. x\<in>s" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1653 |
by (simp add: assms(2) eventually_at_in_open' that) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1654 |
ultimately have "\<exists>\<^sub>F x in at w. x\<in>{w\<in>s. f w = 0}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1655 |
apply (elim frequently_rev_mp) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1656 |
by (auto elim:eventually_mono) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1657 |
from frequently_at_imp_islimpt[OF this] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1658 |
have "w islimpt {w \<in> s. f w = 0}" . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1659 |
then have "infinite({w \<in> s. f w = 0} \<inter> ball w 1)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1660 |
unfolding islimpt_eq_infinite_ball by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1661 |
then have "infinite({w \<in> s. f w = 0})" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1662 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1663 |
then have "infinite pz" unfolding pz_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1664 |
by (smt (verit) Collect_mono_iff rev_finite_subset) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1665 |
then show False using finite by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1666 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1667 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1668 |
obtain pts' where pts':"pts' \<subseteq> s \<inter> poles" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1669 |
"finite pts'" "ff holomorphic_on s - pts'" "\<forall>x\<in>pts'. is_pole ff x" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1670 |
apply (elim get_all_poles_from_remove_sings |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1671 |
[of f,folded ff_def,rotated -1]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1672 |
subgoal using f_holo by fastforce |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1673 |
using \<open>open s\<close> poles finite' by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1674 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1675 |
have pts'_sub_pz:"{w \<in> s. ff w = 0 \<or> w \<in> pts'} \<subseteq> pz" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1676 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1677 |
have "w\<in>poles" if "w\<in>s" "w\<in>pts'" for w |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1678 |
by (meson in_mono le_infE pts'(1) that(2)) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1679 |
moreover have "f w=0" if" w\<in>s" "w\<notin>poles" "ff w=0" for w |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1680 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1681 |
have "\<not> is_pole f w" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1682 |
by (metis DiffI Diff_Diff_Int Diff_subset assms(2) f_holo |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1683 |
finite' finite_imp_closed inf.absorb_iff2 |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1684 |
not_is_pole_holomorphic open_Diff that(1) that(2)) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1685 |
then have "f \<midarrow>w\<rightarrow> 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1686 |
using remove_sings_eq_0_iff[OF not_ess[OF \<open>w\<in>s\<close>]] \<open>ff w=0\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1687 |
unfolding ff_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1688 |
moreover have "f analytic_on {w}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1689 |
using that(1,2) finite' f_holo assms(2) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1690 |
by (metis Diff_Diff_Int Diff_empty Diff_iff Diff_subset |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1691 |
double_diff finite_imp_closed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1692 |
holomorphic_on_imp_analytic_at open_Diff) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1693 |
ultimately show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1694 |
using ff_def remove_sings_at_analytic that(3) by presburger |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1695 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1696 |
ultimately show ?thesis unfolding pz_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1697 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1698 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1699 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1700 |
have "contour_integral g (\<lambda>x. deriv f x * h x / f x) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1701 |
= contour_integral g (\<lambda>x. deriv ff x * h x / ff x)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1702 |
proof (rule contour_integral_eq) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1703 |
fix x assume "x \<in> path_image g" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1704 |
have "f analytic_on {x}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1705 |
proof (rule holomorphic_on_imp_analytic_at[of _ "s-poles"]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1706 |
from finite' |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1707 |
show "open (s - poles)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1708 |
using \<open>open s\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1709 |
by (metis Diff_Compl Diff_Diff_Int Diff_eq finite_imp_closed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1710 |
open_Diff) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1711 |
show "x \<in> s - poles" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1712 |
using path_img \<open>x \<in> path_image g\<close> unfolding pz_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1713 |
qed (use f_holo in simp) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1714 |
then show "deriv f x * h x / f x = deriv ff x * h x / ff x" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1715 |
unfolding ff_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1716 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1717 |
also have "... = complex_of_real (2 * pi) * \<i> * |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1718 |
(\<Sum>p\<in>{w \<in> s. ff w = 0 \<or> w \<in> pts'}. |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1719 |
winding_number g p * h p * of_int (zorder ff p))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1720 |
proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close>, of ff pts' h g]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1721 |
show "path_image g \<subseteq> s - {w \<in> s. ff w = 0 \<or> w \<in> pts'}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1722 |
using path_img pts'_sub_pz by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1723 |
show "finite {w \<in> s. ff w = 0 \<or> w \<in> pts'}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1724 |
using pts'_sub_pz finite |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1725 |
using rev_finite_subset by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1726 |
qed (use pts' assms in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1727 |
also have "... = 2 * pi * \<i> * |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1728 |
(\<Sum>p\<in>pz. winding_number g p * h p * zorder f p)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1729 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1730 |
have "(\<Sum>p\<in>{w \<in> s. ff w = 0 \<or> w \<in> pts'}. |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1731 |
winding_number g p * h p * of_int (zorder ff p)) = |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1732 |
(\<Sum>p\<in>pz. winding_number g p * h p * of_int (zorder f p))" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1733 |
proof (rule sum.mono_neutral_cong_left) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1734 |
have "zorder f w = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1735 |
if "w\<in>s" " f w = 0 \<or> w \<in> poles" "ff w \<noteq> 0" " w \<notin> pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1736 |
for w |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1737 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1738 |
define F where "F=laurent_expansion f w" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1739 |
have has_l:"(\<lambda>x. f (w + x)) has_laurent_expansion F" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1740 |
unfolding F_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1741 |
apply (rule not_essential_has_laurent_expansion) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1742 |
using isolated not_ess \<open>w\<in>s\<close> by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1743 |
from has_laurent_expansion_eventually_nonzero_iff[OF this] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1744 |
have "F \<noteq>0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1745 |
using nzero \<open>w\<in>s\<close> by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1746 |
from tendsto_0_subdegree_iff[OF has_l this] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1747 |
have "f \<midarrow>w\<rightarrow> 0 = (0 < fls_subdegree F)" . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1748 |
moreover have "\<not> (is_pole f w \<or> f \<midarrow>w\<rightarrow> 0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1749 |
using remove_sings_eq_0_iff[OF not_ess[OF \<open>w\<in>s\<close>]] \<open>ff w \<noteq> 0\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1750 |
unfolding ff_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1751 |
moreover have "is_pole f w = (fls_subdegree F < 0)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1752 |
using is_pole_fls_subdegree_iff[OF has_l] . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1753 |
ultimately have "fls_subdegree F = 0" by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1754 |
then show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1755 |
using has_laurent_expansion_zorder[OF has_l \<open>F\<noteq>0\<close>] by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1756 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1757 |
then show "\<forall>i\<in>pz - {w \<in> s. ff w = 0 \<or> w \<in> pts'}. |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1758 |
winding_number g i * h i * of_int (zorder f i) = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1759 |
unfolding pz_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1760 |
show "\<And>x. x \<in> {w \<in> s. ff w = 0 \<or> w \<in> pts'} \<Longrightarrow> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1761 |
winding_number g x * h x * of_int (zorder ff x) = |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1762 |
winding_number g x * h x * of_int (zorder f x)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1763 |
using isolated zorder_remove_sings[of f,folded ff_def] by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1764 |
qed (use pts'_sub_pz finite in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1765 |
then show ?thesis by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1766 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1767 |
finally show ?thesis . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1768 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1769 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1770 |
lemma meromorphic_on_imp_isolated_singularity: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1771 |
assumes "f meromorphic_on D pts" "z \<in> D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1772 |
shows "isolated_singularity_at f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1773 |
by (meson DiffI assms(1) assms(2) holomorphic_on_imp_analytic_at isolated_singularity_at_analytic |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1774 |
meromorphic_imp_open_diff meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1775 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1776 |
lemma meromorphic_imp_not_is_pole: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1777 |
assumes "f meromorphic_on D pts" "z \<in> D - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1778 |
shows "\<not>is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1779 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1780 |
from assms have "f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1781 |
using meromorphic_on_imp_analytic_at by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1782 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1783 |
using analytic_at not_is_pole_holomorphic by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1784 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1785 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1786 |
lemma meromorphic_all_poles_iff_empty [simp]: "f meromorphic_on pts pts \<longleftrightarrow> pts = {}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1787 |
by (auto simp: meromorphic_on_def holomorphic_on_def open_imp_islimpt) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1788 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1789 |
lemma meromorphic_imp_nonsingular_point_exists: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1790 |
assumes "f meromorphic_on A pts" "A \<noteq> {}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1791 |
obtains x where "x \<in> A - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1792 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1793 |
have "A \<noteq> pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1794 |
using assms by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1795 |
moreover have "pts \<subseteq> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1796 |
using assms by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1797 |
ultimately show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1798 |
using that by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1799 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1800 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1801 |
lemma meromorphic_frequently_const_imp_const: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1802 |
assumes "f meromorphic_on A pts" "connected A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1803 |
assumes "frequently (\<lambda>w. f w = c) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1804 |
assumes "z \<in> A - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1805 |
assumes "w \<in> A - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1806 |
shows "f w = c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1807 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1808 |
have "f w - c = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1809 |
proof (rule analytic_continuation[where f = "\<lambda>z. f z - c"]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1810 |
show "(\<lambda>z. f z - c) holomorphic_on (A - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1811 |
by (intro holomorphic_intros meromorphic_imp_holomorphic[OF assms(1)]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1812 |
show [intro]: "open (A - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1813 |
using assms meromorphic_imp_open_diff by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1814 |
show "connected (A - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1815 |
using assms meromorphic_imp_connected_diff by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1816 |
show "{z\<in>A-pts. f z = c} \<subseteq> A - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1817 |
by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1818 |
have "eventually (\<lambda>z. z \<in> A - pts) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1819 |
using assms by (intro eventually_at_in_open') auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1820 |
hence "frequently (\<lambda>z. f z = c \<and> z \<in> A - pts) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1821 |
by (intro frequently_eventually_frequently assms) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1822 |
thus "z islimpt {z\<in>A-pts. f z = c}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1823 |
by (simp add: islimpt_conv_frequently_at conj_commute) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1824 |
qed (use assms in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1825 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1826 |
by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1827 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1828 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1829 |
lemma meromorphic_imp_eventually_neq: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1830 |
assumes "f meromorphic_on A pts" "connected A" "\<not>f constant_on A - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1831 |
assumes "z \<in> A - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1832 |
shows "eventually (\<lambda>z. f z \<noteq> c) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1833 |
proof (rule ccontr) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1834 |
assume "\<not>eventually (\<lambda>z. f z \<noteq> c) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1835 |
hence *: "frequently (\<lambda>z. f z = c) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1836 |
by (auto simp: frequently_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1837 |
have "\<forall>w\<in>A-pts. f w = c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1838 |
using meromorphic_frequently_const_imp_const [OF assms(1,2) * assms(4)] by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1839 |
hence "f constant_on A - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1840 |
by (auto simp: constant_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1841 |
thus False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1842 |
using assms(3) by contradiction |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1843 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1844 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1845 |
lemma meromorphic_frequently_const_imp_const': |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1846 |
assumes "f meromorphic_on A pts" "connected A" "\<forall>w\<in>pts. is_pole f w" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1847 |
assumes "frequently (\<lambda>w. f w = c) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1848 |
assumes "z \<in> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1849 |
assumes "w \<in> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1850 |
shows "f w = c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1851 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1852 |
have "\<not>is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1853 |
using frequently_const_imp_not_is_pole[OF assms(4)] . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1854 |
with assms have z: "z \<in> A - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1855 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1856 |
have *: "f w = c" if "w \<in> A - pts" for w |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1857 |
using that meromorphic_frequently_const_imp_const [OF assms(1,2,4) z] by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1858 |
have "\<not>is_pole f u" if "u \<in> A" for u |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1859 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1860 |
have "is_pole f u \<longleftrightarrow> is_pole (\<lambda>_. c) u" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1861 |
proof (rule is_pole_cong) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1862 |
have "eventually (\<lambda>w. w \<in> A - (pts - {u}) - {u}) (at u)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1863 |
by (intro eventually_at_in_open meromorphic_imp_open_diff' [OF assms(1)]) (use that in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1864 |
thus "eventually (\<lambda>w. f w = c) (at u)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1865 |
by eventually_elim (use * in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1866 |
qed auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1867 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1868 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1869 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1870 |
moreover have "pts \<subseteq> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1871 |
using assms(1) by (simp add: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1872 |
ultimately have "pts = {}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1873 |
using assms(3) by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1874 |
with * and \<open>w \<in> A\<close> show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1875 |
by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1876 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1877 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1878 |
lemma meromorphic_imp_eventually_neq': |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1879 |
assumes "f meromorphic_on A pts" "connected A" "\<forall>w\<in>pts. is_pole f w" "\<not>f constant_on A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1880 |
assumes "z \<in> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1881 |
shows "eventually (\<lambda>z. f z \<noteq> c) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1882 |
proof (rule ccontr) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1883 |
assume "\<not>eventually (\<lambda>z. f z \<noteq> c) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1884 |
hence *: "frequently (\<lambda>z. f z = c) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1885 |
by (auto simp: frequently_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1886 |
have "\<forall>w\<in>A. f w = c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1887 |
using meromorphic_frequently_const_imp_const' [OF assms(1,2,3) * assms(5)] by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1888 |
hence "f constant_on A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1889 |
by (auto simp: constant_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1890 |
thus False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1891 |
using assms(4) by contradiction |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1892 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1893 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1894 |
lemma zorder_eq_0_iff_meromorphic: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1895 |
assumes "f meromorphic_on A pts" "\<forall>z\<in>pts. is_pole f z" "z \<in> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1896 |
assumes "eventually (\<lambda>x. f x \<noteq> 0) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1897 |
shows "zorder f z = 0 \<longleftrightarrow> \<not>is_pole f z \<and> f z \<noteq> 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1898 |
proof (cases "z \<in> pts") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1899 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1900 |
from assms obtain F where F: "(\<lambda>x. f (z + x)) has_laurent_expansion F" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1901 |
by (metis True meromorphic_on_def not_essential_has_laurent_expansion) (* TODO: better lemmas *) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1902 |
from F and assms(4) have [simp]: "F \<noteq> 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1903 |
using has_laurent_expansion_eventually_nonzero_iff by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1904 |
show ?thesis using True assms(2) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1905 |
using is_pole_fls_subdegree_iff [OF F] has_laurent_expansion_zorder [OF F] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1906 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1907 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1908 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1909 |
have ana: "f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1910 |
using meromorphic_on_imp_analytic_at False assms by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1911 |
hence "\<not>is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1912 |
using analytic_at not_is_pole_holomorphic by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1913 |
moreover have "frequently (\<lambda>w. f w \<noteq> 0) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1914 |
using assms(4) by (intro eventually_frequently) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1915 |
ultimately show ?thesis using zorder_eq_0_iff[OF ana] False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1916 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1917 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1918 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1919 |
lemma zorder_pos_iff_meromorphic: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1920 |
assumes "f meromorphic_on A pts" "\<forall>z\<in>pts. is_pole f z" "z \<in> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1921 |
assumes "eventually (\<lambda>x. f x \<noteq> 0) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1922 |
shows "zorder f z > 0 \<longleftrightarrow> \<not>is_pole f z \<and> f z = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1923 |
proof (cases "z \<in> pts") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1924 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1925 |
from assms obtain F where F: "(\<lambda>x. f (z + x)) has_laurent_expansion F" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1926 |
by (metis True meromorphic_on_def not_essential_has_laurent_expansion) (* TODO: better lemmas *) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1927 |
from F and assms(4) have [simp]: "F \<noteq> 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1928 |
using has_laurent_expansion_eventually_nonzero_iff by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1929 |
show ?thesis using True assms(2) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1930 |
using is_pole_fls_subdegree_iff [OF F] has_laurent_expansion_zorder [OF F] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1931 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1932 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1933 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1934 |
have ana: "f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1935 |
using meromorphic_on_imp_analytic_at False assms by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1936 |
hence "\<not>is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1937 |
using analytic_at not_is_pole_holomorphic by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1938 |
moreover have "frequently (\<lambda>w. f w \<noteq> 0) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1939 |
using assms(4) by (intro eventually_frequently) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1940 |
ultimately show ?thesis using zorder_pos_iff'[OF ana] False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1941 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1942 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1943 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1944 |
lemma zorder_neg_iff_meromorphic: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1945 |
assumes "f meromorphic_on A pts" "\<forall>z\<in>pts. is_pole f z" "z \<in> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1946 |
assumes "eventually (\<lambda>x. f x \<noteq> 0) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1947 |
shows "zorder f z < 0 \<longleftrightarrow> is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1948 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1949 |
have "frequently (\<lambda>x. f x \<noteq> 0) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1950 |
using assms by (intro eventually_frequently) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1951 |
moreover from assms have "isolated_singularity_at f z" "not_essential f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1952 |
using meromorphic_on_imp_isolated_singularity meromorphic_on_imp_not_essential by blast+ |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1953 |
ultimately show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1954 |
using isolated_pole_imp_neg_zorder neg_zorder_imp_is_pole by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1955 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1956 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1957 |
lemma meromorphic_on_imp_discrete: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1958 |
assumes mero:"f meromorphic_on S pts" and "connected S" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1959 |
and nconst:"\<not> (\<forall>w\<in>S - pts. f w = c)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1960 |
shows "discrete {x\<in>S. f x=c}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1961 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1962 |
define g where "g=(\<lambda>x. f x - c)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1963 |
have "\<forall>\<^sub>F w in at z. g w \<noteq> 0" if "z \<in> S" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1964 |
proof (rule nconst_imp_nzero_neighbour'[of g S pts z]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1965 |
show "g meromorphic_on S pts" using mero unfolding g_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1966 |
by (auto intro:meromorphic_intros) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1967 |
show "\<not> (\<forall>w\<in>S - pts. g w = 0)" using nconst unfolding g_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1968 |
qed fact+ |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1969 |
then show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1970 |
unfolding discrete_altdef g_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1971 |
using eventually_mono by fastforce |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1972 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1973 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1974 |
lemma meromorphic_isolated_in: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1975 |
assumes merf: "f meromorphic_on D pts" "p\<in>pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1976 |
shows "p isolated_in pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1977 |
by (meson assms isolated_in_islimpt_iff meromorphic_on_def subsetD) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1978 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1979 |
lemma remove_sings_constant_on: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1980 |
assumes merf: "f meromorphic_on D pts" and "connected D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1981 |
and const:"f constant_on (D - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1982 |
shows "(remove_sings f) constant_on D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1983 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1984 |
have remove_sings_const: "remove_sings f constant_on D - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1985 |
using const |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1986 |
by (metis constant_onE merf meromorphic_on_imp_analytic_at remove_sings_at_analytic) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1987 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1988 |
have ?thesis if "D = {}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1989 |
using that unfolding constant_on_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1990 |
moreover have ?thesis if "D\<noteq>{}" "{x\<in>pts. is_pole f x} = {}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1991 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1992 |
obtain \<xi> where "\<xi> \<in> (D - pts)" "\<xi> islimpt (D - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1993 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1994 |
have "open (D - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1995 |
using meromorphic_imp_open_diff[OF merf] . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1996 |
moreover have "(D - pts) \<noteq> {}" using \<open>D\<noteq>{}\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1997 |
by (metis Diff_empty closure_empty merf |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1998 |
meromorphic_pts_closure subset_empty) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1999 |
ultimately show ?thesis using open_imp_islimpt that by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2000 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2001 |
moreover have "remove_sings f holomorphic_on D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2002 |
using remove_sings_holomorphic_on[OF merf] that by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2003 |
moreover note remove_sings_const |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2004 |
moreover have "open D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2005 |
using assms(1) meromorphic_on_def by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2006 |
ultimately show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2007 |
using Conformal_Mappings.analytic_continuation' |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2008 |
[of "remove_sings f" D "D-pts" \<xi>] \<open>connected D\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2009 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2010 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2011 |
moreover have ?thesis if "D\<noteq>{}" "{x\<in>pts. is_pole f x} \<noteq> {}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2012 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2013 |
define PP where "PP={x\<in>D. is_pole f x}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2014 |
have "remove_sings f meromorphic_on D PP" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2015 |
using merf unfolding PP_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2016 |
apply (elim remove_sings_meromorphic_on) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2017 |
subgoal using assms(1) meromorphic_on_def by force |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2018 |
subgoal using meromorphic_pole_subset merf by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2019 |
done |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2020 |
moreover have "remove_sings f constant_on D - PP" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2021 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2022 |
obtain \<xi> where "\<xi> \<in> f ` (D - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2023 |
by (metis Diff_empty Diff_eq_empty_iff \<open>D \<noteq> {}\<close> assms(1) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2024 |
closure_empty ex_in_conv imageI meromorphic_pts_closure) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2025 |
have \<xi>:"\<forall>x\<in>D - pts. f x = \<xi>" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2026 |
by (metis \<open>\<xi> \<in> f ` (D - pts)\<close> assms(3) constant_on_def image_iff) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2027 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2028 |
have "remove_sings f x = \<xi>" if "x\<in>D - PP" for x |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2029 |
proof (cases "x\<in>pts") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2030 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2031 |
then have"x isolated_in pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2032 |
using meromorphic_isolated_in[OF merf] by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2033 |
then obtain T0 where T0:"open T0" "T0 \<inter> pts = {x}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2034 |
unfolding isolated_in_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2035 |
obtain T1 where T1:"open T1" "x\<in>T1" "T1 \<subseteq> D" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2036 |
using merf unfolding meromorphic_on_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2037 |
using True by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2038 |
define T2 where "T2 = T1 \<inter> T0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2039 |
have "open T2" "x\<in>T2" "T2 - {x} \<subseteq> D - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2040 |
using T0 T1 unfolding T2_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2041 |
then have "\<forall>w\<in>T2. w\<noteq>x \<longrightarrow> f w =\<xi>" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2042 |
using \<xi> by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2043 |
then have "\<forall>\<^sub>F x in at x. f x = \<xi>" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2044 |
unfolding eventually_at_topological |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2045 |
using \<open>open T2\<close> \<open>x\<in>T2\<close> by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2046 |
then have "f \<midarrow>x\<rightarrow> \<xi>" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2047 |
using tendsto_eventually by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2048 |
then show ?thesis by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2049 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2050 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2051 |
then show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2052 |
using \<open>\<forall>x\<in>D - pts. f x = \<xi>\<close> assms(1) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2053 |
meromorphic_on_imp_analytic_at that by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2054 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2055 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2056 |
then show ?thesis unfolding constant_on_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2057 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2058 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2059 |
moreover have "is_pole (remove_sings f) x" if "x\<in>PP" for x |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2060 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2061 |
have "isolated_singularity_at f x" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2062 |
by (metis (mono_tags, lifting) DiffI PP_def assms(1) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2063 |
isolated_singularity_at_analytic mem_Collect_eq |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2064 |
meromorphic_on_def meromorphic_on_imp_analytic_at that) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2065 |
then show ?thesis using that unfolding PP_def by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2066 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2067 |
ultimately show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2068 |
using meromorphic_imp_constant_on |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2069 |
[of "remove_sings f" D PP] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2070 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2071 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2072 |
ultimately show ?thesis by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2073 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2074 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2075 |
lemma meromorphic_eq_meromorphic_extend: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2076 |
assumes "f meromorphic_on A pts1" "g meromorphic_on A pts1" "\<not>z islimpt pts2" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2077 |
assumes "\<And>z. z \<in> A - pts2 \<Longrightarrow> f z = g z" "pts1 \<subseteq> pts2" "z \<in> A - pts1" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2078 |
shows "f z = g z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2079 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2080 |
have "g analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2081 |
using assms by (intro meromorphic_on_imp_analytic_at[OF assms(2)]) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2082 |
hence "g \<midarrow>z\<rightarrow> g z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2083 |
using analytic_at_imp_isCont isContD by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2084 |
also have "?this \<longleftrightarrow> f \<midarrow>z\<rightarrow> g z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2085 |
proof (intro filterlim_cong) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2086 |
have "eventually (\<lambda>w. w \<notin> pts2) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2087 |
using assms by (auto simp: islimpt_conv_frequently_at frequently_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2088 |
moreover have "eventually (\<lambda>w. w \<in> A) (at z)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2089 |
using assms by (intro eventually_at_in_open') (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2090 |
ultimately show "\<forall>\<^sub>F x in at z. g x = f x" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2091 |
by eventually_elim (use assms in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2092 |
qed auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2093 |
finally have "f \<midarrow>z\<rightarrow> g z" . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2094 |
moreover have "f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2095 |
using assms by (intro meromorphic_on_imp_analytic_at[OF assms(1)]) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2096 |
hence "f \<midarrow>z\<rightarrow> f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2097 |
using analytic_at_imp_isCont isContD by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2098 |
ultimately show ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2099 |
using tendsto_unique by force |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2100 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2101 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2102 |
lemma meromorphic_constant_on_extend: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2103 |
assumes "f constant_on A - pts1" "f meromorphic_on A pts1" "f meromorphic_on A pts2" "pts2 \<subseteq> pts1" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2104 |
shows "f constant_on A - pts2" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2105 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2106 |
from assms(1) obtain c where c: "\<And>z. z \<in> A - pts1 \<Longrightarrow> f z = c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2107 |
unfolding constant_on_def by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2108 |
have "f z = c" if "z \<in> A - pts2" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2109 |
using assms(3) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2110 |
proof (rule meromorphic_eq_meromorphic_extend[where z = z]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2111 |
show "(\<lambda>a. c) meromorphic_on A pts2" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2112 |
by (intro meromorphic_on_const) (use assms in \<open>auto simp: meromorphic_on_def\<close>) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2113 |
show "\<not>z islimpt pts1" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2114 |
using that assms by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2115 |
qed (use assms c that in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2116 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2117 |
by (auto simp: constant_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2118 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2119 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2120 |
lemma meromorphic_remove_sings_constant_on_imp_constant_on: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2121 |
assumes "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2122 |
assumes "remove_sings f constant_on A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2123 |
shows "f constant_on A - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2124 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2125 |
from assms(2) obtain c where c: "\<And>z. z \<in> A \<Longrightarrow> remove_sings f z = c" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2126 |
by (auto simp: constant_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2127 |
have "f z = c" if "z \<in> A - pts" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2128 |
using meromorphic_on_imp_analytic_at[OF assms(1) that] c[of z] that |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2129 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2130 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2131 |
by (auto simp: constant_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2132 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2133 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2134 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2135 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2136 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2137 |
definition singularities_on :: "complex set \<Rightarrow> (complex \<Rightarrow> complex) \<Rightarrow> complex set" where |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2138 |
"singularities_on A f = |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2139 |
{z\<in>A. isolated_singularity_at f z \<and> not_essential f z \<and> \<not>f analytic_on {z}}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2140 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2141 |
lemma singularities_on_subset: "singularities_on A f \<subseteq> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2142 |
by (auto simp: singularities_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2143 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2144 |
lemma pole_in_singularities_on: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2145 |
assumes "f meromorphic_on A pts" "z \<in> A" "is_pole f z" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2146 |
shows "z \<in> singularities_on A f" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2147 |
unfolding singularities_on_def not_essential_def using assms |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2148 |
using analytic_at_imp_no_pole meromorphic_on_imp_isolated_singularity by force |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2149 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2150 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2151 |
lemma meromorphic_on_subset_pts: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2152 |
assumes "f meromorphic_on A pts" "pts' \<subseteq> pts" "f analytic_on pts - pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2153 |
shows "f meromorphic_on A pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2154 |
proof |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2155 |
show "open A" "pts' \<subseteq> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2156 |
using assms by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2157 |
show "isolated_singularity_at f z" "not_essential f z" if "z \<in> pts'" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2158 |
using assms that by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2159 |
show "\<not>z islimpt pts'" if "z \<in> A" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2160 |
using assms that islimpt_subset unfolding meromorphic_on_def by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2161 |
have "f analytic_on A - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2162 |
using assms(1) meromorphic_imp_analytic by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2163 |
with assms have "f analytic_on (A - pts) \<union> (pts - pts')" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2164 |
by (subst analytic_on_Un) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2165 |
also have "(A - pts) \<union> (pts - pts') = A - pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2166 |
using assms by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2167 |
finally show "f holomorphic_on A - pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2168 |
using analytic_imp_holomorphic by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2169 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2170 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2171 |
lemma meromorphic_on_imp_superset_singularities_on: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2172 |
assumes "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2173 |
shows "singularities_on A f \<subseteq> pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2174 |
proof |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2175 |
fix z assume "z \<in> singularities_on A f" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2176 |
hence "z \<in> A" "\<not>f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2177 |
by (auto simp: singularities_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2178 |
with assms show "z \<in> pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2179 |
by (meson DiffI meromorphic_on_imp_analytic_at) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2180 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2181 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2182 |
lemma meromorphic_on_singularities_on: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2183 |
assumes "f meromorphic_on A pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2184 |
shows "f meromorphic_on A (singularities_on A f)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2185 |
using assms meromorphic_on_imp_superset_singularities_on[OF assms] |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2186 |
proof (rule meromorphic_on_subset_pts) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2187 |
have "f analytic_on {z}" if "z \<in> pts - singularities_on A f" for z |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2188 |
using that assms by (auto simp: singularities_on_def meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2189 |
thus "f analytic_on pts - singularities_on A f" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2190 |
using analytic_on_analytic_at by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2191 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2192 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2193 |
theorem Residue_theorem_inside: |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2194 |
assumes f: "f meromorphic_on s pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2195 |
"simply_connected s" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2196 |
assumes g: "valid_path g" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2197 |
"pathfinish g = pathstart g" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2198 |
"path_image g \<subseteq> s - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2199 |
defines "pts1 \<equiv> pts \<inter> inside (path_image g)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2200 |
shows "finite pts1" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2201 |
and "contour_integral g f = 2 * pi * \<i> * (\<Sum>p\<in>pts1. winding_number g p * residue f p)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2202 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2203 |
note [dest] = valid_path_imp_path |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2204 |
have cl_g [intro]: "closed (path_image g)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2205 |
using g by (auto intro!: closed_path_image) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2206 |
have "open s" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2207 |
using f(1) by (auto simp: meromorphic_on_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2208 |
define pts2 where "pts2 = pts - pts1" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2209 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2210 |
define A where "A = path_image g \<union> inside (path_image g)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2211 |
have "closed A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2212 |
unfolding A_def using g by (intro closed_path_image_Un_inside) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2213 |
moreover have "bounded A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2214 |
unfolding A_def using g by (auto intro!: bounded_path_image bounded_inside) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2215 |
ultimately have 1: "compact A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2216 |
using compact_eq_bounded_closed by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2217 |
have 2: "open (s - pts2)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2218 |
using f by (auto intro!: meromorphic_imp_open_diff' [OF f(1)] simp: pts2_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2219 |
have 3: "A \<subseteq> s - pts2" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2220 |
unfolding A_def pts2_def pts1_def |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2221 |
using f(2) g(3) 2 subset_simply_connected_imp_inside_subset[of s "path_image g"] \<open>open s\<close> |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2222 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2223 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2224 |
obtain \<epsilon> where \<epsilon>: "\<epsilon> > 0" "(\<Union>x\<in>A. ball x \<epsilon>) \<subseteq> s - pts2" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2225 |
using compact_subset_open_imp_ball_epsilon_subset[OF 1 2 3] by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2226 |
define B where "B = (\<Union>x\<in>A. ball x \<epsilon>)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2227 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2228 |
have "finite (A \<inter> pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2229 |
using 1 3 by (intro meromorphic_compact_finite_pts[OF f(1)]) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2230 |
also have "A \<inter> pts = pts1" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2231 |
unfolding pts1_def using g by (auto simp: A_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2232 |
finally show fin: "finite pts1" . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2233 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2234 |
show "contour_integral g f = 2 * pi * \<i> * (\<Sum>p\<in>pts1. winding_number g p * residue f p)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2235 |
proof (rule Residue_theorem) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2236 |
show "open B" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2237 |
by (auto simp: B_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2238 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2239 |
have "connected A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2240 |
unfolding A_def using g |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2241 |
by (intro connected_with_inside closed_path_image connected_path_image) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2242 |
hence "connected (A \<union> B)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2243 |
unfolding B_def using g \<open>\<epsilon> > 0\<close> f(2) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2244 |
by (intro connected_Un_UN connected_path_image valid_path_imp_path) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2245 |
(auto simp: simply_connected_imp_connected) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2246 |
also have "A \<union> B = B" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2247 |
using \<epsilon>(1) by (auto simp: B_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2248 |
finally show "connected B" . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2249 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2250 |
have "f holomorphic_on (s - pts)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2251 |
by (intro meromorphic_imp_holomorphic f) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2252 |
moreover have "B - pts1 \<subseteq> s - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2253 |
using \<epsilon> unfolding B_def by (auto simp: pts1_def pts2_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2254 |
ultimately show "f holomorphic_on (B - pts1)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2255 |
by (rule holomorphic_on_subset) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2256 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2257 |
have "path_image g \<subseteq> A - pts1" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2258 |
using g unfolding pts1_def by (auto simp: A_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2259 |
also have "\<dots> \<subseteq> B - pts1" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2260 |
unfolding B_def using \<epsilon>(1) by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2261 |
finally show "path_image g \<subseteq> B - pts1" . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2262 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2263 |
show "\<forall>z. z \<notin> B \<longrightarrow> winding_number g z = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2264 |
proof safe |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2265 |
fix z assume z: "z \<notin> B" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2266 |
hence "z \<notin> A" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2267 |
using \<epsilon>(1) by (auto simp: B_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2268 |
hence "z \<in> outside (path_image g)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2269 |
unfolding A_def by (simp add: union_with_inside) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2270 |
thus "winding_number g z = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2271 |
using g by (intro winding_number_zero_in_outside) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2272 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2273 |
qed (use g fin in auto) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2274 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2275 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2276 |
theorem Residue_theorem': |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2277 |
assumes f: "f meromorphic_on s pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2278 |
"simply_connected s" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2279 |
assumes g: "valid_path g" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2280 |
"pathfinish g = pathstart g" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2281 |
"path_image g \<subseteq> s - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2282 |
assumes pts': "finite pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2283 |
"pts' \<subseteq> s" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2284 |
"\<And>z. z \<in> pts - pts' \<Longrightarrow> winding_number g z = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2285 |
shows "contour_integral g f = 2 * pi * \<i> * (\<Sum>p\<in>pts'. winding_number g p * residue f p)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2286 |
proof - |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2287 |
note [dest] = valid_path_imp_path |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2288 |
define pts1 where "pts1 = pts \<inter> inside (path_image g)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2289 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2290 |
have "contour_integral g f = 2 * pi * \<i> * (\<Sum>p\<in>pts1. winding_number g p * residue f p)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2291 |
unfolding pts1_def by (intro Residue_theorem_inside[OF f g]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2292 |
also have "(\<Sum>p\<in>pts1. winding_number g p * residue f p) = |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2293 |
(\<Sum>p\<in>pts'. winding_number g p * residue f p)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2294 |
proof (intro sum.mono_neutral_cong refl) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2295 |
show "finite pts1" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2296 |
unfolding pts1_def by (intro Residue_theorem_inside[OF f g]) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2297 |
show "finite pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2298 |
by fact |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2299 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2300 |
fix z assume z: "z \<in> pts' - pts1" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2301 |
show "winding_number g z * residue f z = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2302 |
proof (cases "z \<in> pts") |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2303 |
case True |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2304 |
with z have "z \<notin> path_image g \<union> inside (path_image g)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2305 |
using g(3) by (auto simp: pts1_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2306 |
hence "z \<in> outside (path_image g)" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2307 |
by (simp add: union_with_inside) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2308 |
hence "winding_number g z = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2309 |
using g by (intro winding_number_zero_in_outside) auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2310 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2311 |
by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2312 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2313 |
case False |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2314 |
with z pts' have "z \<in> s - pts" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2315 |
by auto |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2316 |
with f(1) have "f analytic_on {z}" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2317 |
by (intro meromorphic_on_imp_analytic_at) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2318 |
hence "residue f z = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2319 |
using analytic_at residue_holo by blast |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2320 |
thus ?thesis |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2321 |
by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2322 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2323 |
next |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2324 |
fix z assume z: "z \<in> pts1 - pts'" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2325 |
hence "winding_number g z = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2326 |
using pts' by (auto simp: pts1_def) |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2327 |
thus "winding_number g z * residue f z = 0" |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2328 |
by simp |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2329 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2330 |
finally show ?thesis . |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2331 |
qed |
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2332 |
|
c6b50597abbc
More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2333 |
end |