author | desharna |
Thu, 24 Nov 2022 10:02:26 +0100 | |
changeset 76574 | 7bc934b99faf |
parent 75168 | ff60b4acd6dd |
child 77166 | 0fb350e7477b |
permissions | -rw-r--r-- |
56215 | 1 |
(* Author: John Harrison, Marco Maggesi, Graziano Gentili, Gianni Ciolli, Valentina Bruno |
2 |
Ported from "hol_light/Multivariate/canal.ml" by L C Paulson (2014) |
|
3 |
*) |
|
4 |
||
60420 | 5 |
section \<open>Complex Analysis Basics\<close> |
71167
b4d409c65a76
Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents:
71030
diff
changeset
|
6 |
text \<open>Definitions of analytic and holomorphic functions, limit theorems, complex differentiation\<close> |
56215 | 7 |
|
8 |
theory Complex_Analysis_Basics |
|
71001
3e374c65f96b
reorganisation to eliminate Brouwer_Fixpoint from complex analysis
paulson <lp15@cam.ac.uk>
parents:
70817
diff
changeset
|
9 |
imports Derivative "HOL-Library.Nonpos_Ints" |
56215 | 10 |
begin |
11 |
||
70136 | 12 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>General lemmas\<close> |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
13 |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
14 |
lemma nonneg_Reals_cmod_eq_Re: "z \<in> \<real>\<^sub>\<ge>\<^sub>0 \<Longrightarrow> norm z = Re z" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
15 |
by (simp add: complex_nonneg_Reals_iff cmod_eq_Re) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
16 |
|
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
17 |
lemma fact_cancel: |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
18 |
fixes c :: "'a::real_field" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
19 |
shows "of_nat (Suc n) * c / (fact (Suc n)) = c / (fact n)" |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
20 |
using of_nat_neq_0 by force |
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56479
diff
changeset
|
21 |
|
68721 | 22 |
lemma vector_derivative_cnj_within: |
23 |
assumes "at x within A \<noteq> bot" and "f differentiable at x within A" |
|
24 |
shows "vector_derivative (\<lambda>z. cnj (f z)) (at x within A) = |
|
25 |
cnj (vector_derivative f (at x within A))" (is "_ = cnj ?D") |
|
26 |
proof - |
|
27 |
let ?D = "vector_derivative f (at x within A)" |
|
28 |
from assms have "(f has_vector_derivative ?D) (at x within A)" |
|
29 |
by (subst (asm) vector_derivative_works) |
|
30 |
hence "((\<lambda>x. cnj (f x)) has_vector_derivative cnj ?D) (at x within A)" |
|
31 |
by (rule has_vector_derivative_cnj) |
|
32 |
thus ?thesis using assms by (auto dest: vector_derivative_within) |
|
33 |
qed |
|
34 |
||
35 |
lemma vector_derivative_cnj: |
|
36 |
assumes "f differentiable at x" |
|
37 |
shows "vector_derivative (\<lambda>z. cnj (f z)) (at x) = cnj (vector_derivative f (at x))" |
|
38 |
using assms by (intro vector_derivative_cnj_within) auto |
|
39 |
||
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
40 |
lemma |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
41 |
shows open_halfspace_Re_lt: "open {z. Re(z) < b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
42 |
and open_halfspace_Re_gt: "open {z. Re(z) > b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
43 |
and closed_halfspace_Re_ge: "closed {z. Re(z) \<ge> b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
44 |
and closed_halfspace_Re_le: "closed {z. Re(z) \<le> b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
45 |
and closed_halfspace_Re_eq: "closed {z. Re(z) = b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
46 |
and open_halfspace_Im_lt: "open {z. Im(z) < b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
47 |
and open_halfspace_Im_gt: "open {z. Im(z) > b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
48 |
and closed_halfspace_Im_ge: "closed {z. Im(z) \<ge> b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
49 |
and closed_halfspace_Im_le: "closed {z. Im(z) \<le> b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
50 |
and closed_halfspace_Im_eq: "closed {z. Im(z) = b}" |
63332 | 51 |
by (intro open_Collect_less closed_Collect_le closed_Collect_eq continuous_on_Re |
52 |
continuous_on_Im continuous_on_id continuous_on_const)+ |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
53 |
|
61070 | 54 |
lemma closed_complex_Reals: "closed (\<real> :: complex set)" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
55 |
proof - |
61070 | 56 |
have "(\<real> :: complex set) = {z. Im z = 0}" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
57 |
by (auto simp: complex_is_Real_iff) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
58 |
then show ?thesis |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
59 |
by (metis closed_halfspace_Im_eq) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
60 |
qed |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
61 |
|
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
62 |
lemma closed_Real_halfspace_Re_le: "closed (\<real> \<inter> {w. Re w \<le> x})" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
63 |
by (simp add: closed_Int closed_complex_Reals closed_halfspace_Re_le) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
64 |
|
69180
922833cc6839
Tagged some theories in HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
69064
diff
changeset
|
65 |
lemma closed_nonpos_Reals_complex [simp]: "closed (\<real>\<^sub>\<le>\<^sub>0 :: complex set)" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
66 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
67 |
have "\<real>\<^sub>\<le>\<^sub>0 = \<real> \<inter> {z. Re(z) \<le> 0}" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
68 |
using complex_nonpos_Reals_iff complex_is_Real_iff by auto |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
69 |
then show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
70 |
by (metis closed_Real_halfspace_Re_le) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
71 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
72 |
|
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
73 |
lemma closed_Real_halfspace_Re_ge: "closed (\<real> \<inter> {w. x \<le> Re(w)})" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
74 |
using closed_halfspace_Re_ge |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
75 |
by (simp add: closed_Int closed_complex_Reals) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
76 |
|
69180
922833cc6839
Tagged some theories in HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
69064
diff
changeset
|
77 |
lemma closed_nonneg_Reals_complex [simp]: "closed (\<real>\<^sub>\<ge>\<^sub>0 :: complex set)" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
78 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
79 |
have "\<real>\<^sub>\<ge>\<^sub>0 = \<real> \<inter> {z. Re(z) \<ge> 0}" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
80 |
using complex_nonneg_Reals_iff complex_is_Real_iff by auto |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
81 |
then show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
82 |
by (metis closed_Real_halfspace_Re_ge) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
83 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
84 |
|
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
85 |
lemma closed_real_abs_le: "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
86 |
proof - |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
87 |
have "{w \<in> \<real>. \<bar>Re w\<bar> \<le> r} = (\<real> \<inter> {w. Re w \<le> r}) \<inter> (\<real> \<inter> {w. Re w \<ge> -r})" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
88 |
by auto |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
89 |
then show "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
90 |
by (simp add: closed_Int closed_Real_halfspace_Re_ge closed_Real_halfspace_Re_le) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
91 |
qed |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
92 |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
93 |
lemma real_lim: |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
94 |
fixes l::complex |
69508 | 95 |
assumes "(f \<longlongrightarrow> l) F" and "\<not> trivial_limit F" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
96 |
shows "l \<in> \<real>" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
97 |
proof (rule Lim_in_closed_set[OF closed_complex_Reals _ assms(2,1)]) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
98 |
show "eventually (\<lambda>x. f x \<in> \<real>) F" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
99 |
using assms(3, 4) by (auto intro: eventually_mono) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
100 |
qed |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
101 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
102 |
lemma real_lim_sequentially: |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
103 |
fixes l::complex |
61973 | 104 |
shows "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
105 |
by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
106 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
107 |
lemma real_series: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
108 |
fixes l::complex |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
109 |
shows "f sums l \<Longrightarrow> (\<And>n. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
110 |
unfolding sums_def |
64267 | 111 |
by (metis real_lim_sequentially sum_in_Reals) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
112 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
113 |
lemma Lim_null_comparison_Re: |
61973 | 114 |
assumes "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F" "(g \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F" |
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56479
diff
changeset
|
115 |
by (rule Lim_null_comparison[OF assms(1)] tendsto_eq_intros assms(2))+ simp |
56215 | 116 |
|
60420 | 117 |
subsection\<open>Holomorphic functions\<close> |
56215 | 118 |
|
70136 | 119 |
definition\<^marker>\<open>tag important\<close> holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
120 |
(infixl "(holomorphic'_on)" 50) |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
121 |
where "f holomorphic_on s \<equiv> \<forall>x\<in>s. f field_differentiable (at x within s)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
122 |
|
70136 | 123 |
named_theorems\<^marker>\<open>tag important\<close> holomorphic_intros "structural introduction rules for holomorphic_on" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
124 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
125 |
lemma holomorphic_onI [intro?]: "(\<And>x. x \<in> s \<Longrightarrow> f field_differentiable (at x within s)) \<Longrightarrow> f holomorphic_on s" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
126 |
by (simp add: holomorphic_on_def) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
127 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
128 |
lemma holomorphic_onD [dest?]: "\<lbrakk>f holomorphic_on s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x within s)" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
129 |
by (simp add: holomorphic_on_def) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
130 |
|
64394 | 131 |
lemma holomorphic_on_imp_differentiable_on: |
132 |
"f holomorphic_on s \<Longrightarrow> f differentiable_on s" |
|
133 |
unfolding holomorphic_on_def differentiable_on_def |
|
134 |
by (simp add: field_differentiable_imp_differentiable) |
|
135 |
||
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
136 |
lemma holomorphic_on_imp_differentiable_at: |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
137 |
"\<lbrakk>f holomorphic_on s; open s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x)" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
138 |
using at_within_open holomorphic_on_def by fastforce |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
139 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
140 |
lemma holomorphic_on_empty [holomorphic_intros]: "f holomorphic_on {}" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
141 |
by (simp add: holomorphic_on_def) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
142 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
143 |
lemma holomorphic_on_open: |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
144 |
"open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> f')" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
145 |
by (auto simp: holomorphic_on_def field_differentiable_def has_field_derivative_def at_within_open [of _ s]) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
146 |
|
74513
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
147 |
lemma holomorphic_on_UN_open: |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
148 |
assumes "\<And>n. n \<in> I \<Longrightarrow> f holomorphic_on A n" "\<And>n. n \<in> I \<Longrightarrow> open (A n)" |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
149 |
shows "f holomorphic_on (\<Union>n\<in>I. A n)" |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
150 |
proof - |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
151 |
have "f field_differentiable at z within (\<Union>n\<in>I. A n)" if "z \<in> (\<Union>n\<in>I. A n)" for z |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
152 |
proof - |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
153 |
from that obtain n where "n \<in> I" "z \<in> A n" |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
154 |
by blast |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
155 |
hence "f holomorphic_on A n" "open (A n)" |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
156 |
by (simp add: assms)+ |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
157 |
with \<open>z \<in> A n\<close> have "f field_differentiable at z" |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
158 |
by (auto simp: holomorphic_on_open field_differentiable_def) |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
159 |
thus ?thesis |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
160 |
by (meson field_differentiable_at_within) |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
161 |
qed |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
162 |
thus ?thesis |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
163 |
by (auto simp: holomorphic_on_def) |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
164 |
qed |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
165 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
166 |
lemma holomorphic_on_imp_continuous_on: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
167 |
"f holomorphic_on s \<Longrightarrow> continuous_on s f" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
168 |
by (metis field_differentiable_imp_continuous_at continuous_on_eq_continuous_within holomorphic_on_def) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
169 |
|
73885
26171a89466a
A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents:
71633
diff
changeset
|
170 |
lemma holomorphic_closedin_preimage_constant: |
26171a89466a
A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents:
71633
diff
changeset
|
171 |
assumes "f holomorphic_on D" |
26171a89466a
A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents:
71633
diff
changeset
|
172 |
shows "closedin (top_of_set D) {z\<in>D. f z = a}" |
26171a89466a
A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents:
71633
diff
changeset
|
173 |
by (simp add: assms continuous_closedin_preimage_constant holomorphic_on_imp_continuous_on) |
26171a89466a
A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents:
71633
diff
changeset
|
174 |
|
26171a89466a
A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents:
71633
diff
changeset
|
175 |
lemma holomorphic_closed_preimage_constant: |
26171a89466a
A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents:
71633
diff
changeset
|
176 |
assumes "f holomorphic_on UNIV" |
26171a89466a
A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents:
71633
diff
changeset
|
177 |
shows "closed {z. f z = a}" |
26171a89466a
A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents:
71633
diff
changeset
|
178 |
using holomorphic_closedin_preimage_constant [OF assms] by simp |
26171a89466a
A few useful lemmas about derivatives, colinearity and other topics
paulson <lp15@cam.ac.uk>
parents:
71633
diff
changeset
|
179 |
|
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
180 |
lemma holomorphic_on_subset [elim]: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
181 |
"f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
182 |
unfolding holomorphic_on_def |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
183 |
by (metis field_differentiable_within_subset subsetD) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
184 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
185 |
lemma holomorphic_transform: "\<lbrakk>f holomorphic_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g holomorphic_on s" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
186 |
by (metis field_differentiable_transform_within linordered_field_no_ub holomorphic_on_def) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
187 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
188 |
lemma holomorphic_cong: "s = t ==> (\<And>x. x \<in> s \<Longrightarrow> f x = g x) \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> g holomorphic_on t" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
189 |
by (metis holomorphic_transform) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
190 |
|
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
191 |
lemma holomorphic_on_linear [simp, holomorphic_intros]: "((*) c) holomorphic_on s" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
192 |
unfolding holomorphic_on_def by (metis field_differentiable_linear) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
193 |
|
62217 | 194 |
lemma holomorphic_on_const [simp, holomorphic_intros]: "(\<lambda>z. c) holomorphic_on s" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
195 |
unfolding holomorphic_on_def by (metis field_differentiable_const) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
196 |
|
62217 | 197 |
lemma holomorphic_on_ident [simp, holomorphic_intros]: "(\<lambda>x. x) holomorphic_on s" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
198 |
unfolding holomorphic_on_def by (metis field_differentiable_ident) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
199 |
|
62217 | 200 |
lemma holomorphic_on_id [simp, holomorphic_intros]: "id holomorphic_on s" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
201 |
unfolding id_def by (rule holomorphic_on_ident) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
202 |
|
74513
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
203 |
lemma constant_on_imp_holomorphic_on: |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
204 |
assumes "f constant_on A" |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
205 |
shows "f holomorphic_on A" |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
206 |
proof - |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
207 |
from assms obtain c where c: "\<forall>x\<in>A. f x = c" |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
208 |
unfolding constant_on_def by blast |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
209 |
have "f holomorphic_on A \<longleftrightarrow> (\<lambda>_. c) holomorphic_on A" |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
210 |
by (intro holomorphic_cong) (use c in auto) |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
211 |
thus ?thesis |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
212 |
by simp |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
213 |
qed |
67d87d224e00
A few new lemmas plus some refinements
paulson <lp15@cam.ac.uk>
parents:
73885
diff
changeset
|
214 |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
215 |
lemma holomorphic_on_compose: |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
216 |
"f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s) \<Longrightarrow> (g o f) holomorphic_on s" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
217 |
using field_differentiable_compose_within[of f _ s g] |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
218 |
by (auto simp: holomorphic_on_def) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
219 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
220 |
lemma holomorphic_on_compose_gen: |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
221 |
"f holomorphic_on s \<Longrightarrow> g holomorphic_on t \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> (g o f) holomorphic_on s" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
222 |
by (metis holomorphic_on_compose holomorphic_on_subset) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
223 |
|
68721 | 224 |
lemma holomorphic_on_balls_imp_entire: |
225 |
assumes "\<not>bdd_above A" "\<And>r. r \<in> A \<Longrightarrow> f holomorphic_on ball c r" |
|
226 |
shows "f holomorphic_on B" |
|
227 |
proof (rule holomorphic_on_subset) |
|
228 |
show "f holomorphic_on UNIV" unfolding holomorphic_on_def |
|
229 |
proof |
|
230 |
fix z :: complex |
|
231 |
from \<open>\<not>bdd_above A\<close> obtain r where r: "r \<in> A" "r > norm (z - c)" |
|
232 |
by (meson bdd_aboveI not_le) |
|
233 |
with assms(2) have "f holomorphic_on ball c r" by blast |
|
234 |
moreover from r have "z \<in> ball c r" by (auto simp: dist_norm norm_minus_commute) |
|
235 |
ultimately show "f field_differentiable at z" |
|
236 |
by (auto simp: holomorphic_on_def at_within_open[of _ "ball c r"]) |
|
237 |
qed |
|
238 |
qed auto |
|
239 |
||
240 |
lemma holomorphic_on_balls_imp_entire': |
|
241 |
assumes "\<And>r. r > 0 \<Longrightarrow> f holomorphic_on ball c r" |
|
242 |
shows "f holomorphic_on B" |
|
243 |
proof (rule holomorphic_on_balls_imp_entire) |
|
244 |
{ |
|
245 |
fix M :: real |
|
246 |
have "\<exists>x. x > max M 0" by (intro gt_ex) |
|
247 |
hence "\<exists>x>0. x > M" by auto |
|
248 |
} |
|
249 |
thus "\<not>bdd_above {(0::real)<..}" unfolding bdd_above_def |
|
250 |
by (auto simp: not_le) |
|
251 |
qed (insert assms, auto) |
|
252 |
||
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
253 |
lemma holomorphic_on_minus [holomorphic_intros]: "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on s" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
254 |
by (metis field_differentiable_minus holomorphic_on_def) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
255 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
256 |
lemma holomorphic_on_add [holomorphic_intros]: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
257 |
"\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on s" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
258 |
unfolding holomorphic_on_def by (metis field_differentiable_add) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
259 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
260 |
lemma holomorphic_on_diff [holomorphic_intros]: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
261 |
"\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) holomorphic_on s" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
262 |
unfolding holomorphic_on_def by (metis field_differentiable_diff) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
263 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
264 |
lemma holomorphic_on_mult [holomorphic_intros]: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
265 |
"\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) holomorphic_on s" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
266 |
unfolding holomorphic_on_def by (metis field_differentiable_mult) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
267 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
268 |
lemma holomorphic_on_inverse [holomorphic_intros]: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
269 |
"\<lbrakk>f holomorphic_on s; \<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>z. inverse (f z)) holomorphic_on s" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
270 |
unfolding holomorphic_on_def by (metis field_differentiable_inverse) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
271 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
272 |
lemma holomorphic_on_divide [holomorphic_intros]: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
273 |
"\<lbrakk>f holomorphic_on s; g holomorphic_on s; \<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>z. f z / g z) holomorphic_on s" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
274 |
unfolding holomorphic_on_def by (metis field_differentiable_divide) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
275 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
276 |
lemma holomorphic_on_power [holomorphic_intros]: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
277 |
"f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on s" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
278 |
unfolding holomorphic_on_def by (metis field_differentiable_power) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
279 |
|
64267 | 280 |
lemma holomorphic_on_sum [holomorphic_intros]: |
281 |
"(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) holomorphic_on s" |
|
282 |
unfolding holomorphic_on_def by (metis field_differentiable_sum) |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
283 |
|
67135
1a94352812f4
Moved material from AFP to Analysis/Number_Theory
Manuel Eberl <eberlm@in.tum.de>
parents:
66827
diff
changeset
|
284 |
lemma holomorphic_on_prod [holomorphic_intros]: |
1a94352812f4
Moved material from AFP to Analysis/Number_Theory
Manuel Eberl <eberlm@in.tum.de>
parents:
66827
diff
changeset
|
285 |
"(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. prod (\<lambda>i. f i x) I) holomorphic_on s" |
1a94352812f4
Moved material from AFP to Analysis/Number_Theory
Manuel Eberl <eberlm@in.tum.de>
parents:
66827
diff
changeset
|
286 |
by (induction I rule: infinite_finite_induct) (auto intro: holomorphic_intros) |
1a94352812f4
Moved material from AFP to Analysis/Number_Theory
Manuel Eberl <eberlm@in.tum.de>
parents:
66827
diff
changeset
|
287 |
|
66486
ffaaa83543b2
Lemmas about analysis and permutations
Manuel Eberl <eberlm@in.tum.de>
parents:
66453
diff
changeset
|
288 |
lemma holomorphic_pochhammer [holomorphic_intros]: |
ffaaa83543b2
Lemmas about analysis and permutations
Manuel Eberl <eberlm@in.tum.de>
parents:
66453
diff
changeset
|
289 |
"f holomorphic_on A \<Longrightarrow> (\<lambda>s. pochhammer (f s) n) holomorphic_on A" |
ffaaa83543b2
Lemmas about analysis and permutations
Manuel Eberl <eberlm@in.tum.de>
parents:
66453
diff
changeset
|
290 |
by (induction n) (auto intro!: holomorphic_intros simp: pochhammer_Suc) |
ffaaa83543b2
Lemmas about analysis and permutations
Manuel Eberl <eberlm@in.tum.de>
parents:
66453
diff
changeset
|
291 |
|
ffaaa83543b2
Lemmas about analysis and permutations
Manuel Eberl <eberlm@in.tum.de>
parents:
66453
diff
changeset
|
292 |
lemma holomorphic_on_scaleR [holomorphic_intros]: |
ffaaa83543b2
Lemmas about analysis and permutations
Manuel Eberl <eberlm@in.tum.de>
parents:
66453
diff
changeset
|
293 |
"f holomorphic_on A \<Longrightarrow> (\<lambda>x. c *\<^sub>R f x) holomorphic_on A" |
ffaaa83543b2
Lemmas about analysis and permutations
Manuel Eberl <eberlm@in.tum.de>
parents:
66453
diff
changeset
|
294 |
by (auto simp: scaleR_conv_of_real intro!: holomorphic_intros) |
ffaaa83543b2
Lemmas about analysis and permutations
Manuel Eberl <eberlm@in.tum.de>
parents:
66453
diff
changeset
|
295 |
|
67167
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
296 |
lemma holomorphic_on_Un [holomorphic_intros]: |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
297 |
assumes "f holomorphic_on A" "f holomorphic_on B" "open A" "open B" |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
298 |
shows "f holomorphic_on (A \<union> B)" |
68239 | 299 |
using assms by (auto simp: holomorphic_on_def at_within_open[of _ A] |
67167
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
300 |
at_within_open[of _ B] at_within_open[of _ "A \<union> B"] open_Un) |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
301 |
|
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
302 |
lemma holomorphic_on_If_Un [holomorphic_intros]: |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
303 |
assumes "f holomorphic_on A" "g holomorphic_on B" "open A" "open B" |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
304 |
assumes "\<And>z. z \<in> A \<Longrightarrow> z \<in> B \<Longrightarrow> f z = g z" |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
305 |
shows "(\<lambda>z. if z \<in> A then f z else g z) holomorphic_on (A \<union> B)" (is "?h holomorphic_on _") |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
306 |
proof (intro holomorphic_on_Un) |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
307 |
note \<open>f holomorphic_on A\<close> |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
308 |
also have "f holomorphic_on A \<longleftrightarrow> ?h holomorphic_on A" |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
309 |
by (intro holomorphic_cong) auto |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
310 |
finally show \<dots> . |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
311 |
next |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
312 |
note \<open>g holomorphic_on B\<close> |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
313 |
also have "g holomorphic_on B \<longleftrightarrow> ?h holomorphic_on B" |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
314 |
using assms by (intro holomorphic_cong) auto |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
315 |
finally show \<dots> . |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
316 |
qed (insert assms, auto) |
88d1c9d86f48
Moved analysis material from AFP
Manuel Eberl <eberlm@in.tum.de>
parents:
67135
diff
changeset
|
317 |
|
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
318 |
lemma holomorphic_derivI: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
319 |
"\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
320 |
\<Longrightarrow> (f has_field_derivative deriv f x) (at x within T)" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
321 |
by (metis DERIV_deriv_iff_field_differentiable at_within_open holomorphic_on_def has_field_derivative_at_within) |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
322 |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
323 |
lemma complex_derivative_transform_within_open: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
324 |
"\<lbrakk>f holomorphic_on s; g holomorphic_on s; open s; z \<in> s; \<And>w. w \<in> s \<Longrightarrow> f w = g w\<rbrakk> |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
325 |
\<Longrightarrow> deriv f z = deriv g z" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
326 |
unfolding holomorphic_on_def |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
327 |
by (rule DERIV_imp_deriv) |
71029
934e0044e94b
Moved or deleted some out of place material, also eliminating obsolete naming conventions
paulson <lp15@cam.ac.uk>
parents:
71001
diff
changeset
|
328 |
(metis DERIV_deriv_iff_field_differentiable has_field_derivative_transform_within_open at_within_open) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
329 |
|
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
330 |
lemma holomorphic_nonconstant: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
331 |
assumes holf: "f holomorphic_on S" and "open S" "\<xi> \<in> S" "deriv f \<xi> \<noteq> 0" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
332 |
shows "\<not> f constant_on S" |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
333 |
by (rule nonzero_deriv_nonconstant [of f "deriv f \<xi>" \<xi> S]) |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
334 |
(use assms in \<open>auto simp: holomorphic_derivI\<close>) |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
335 |
|
60420 | 336 |
subsection\<open>Analyticity on a set\<close> |
56215 | 337 |
|
70136 | 338 |
definition\<^marker>\<open>tag important\<close> analytic_on (infixl "(analytic'_on)" 50) |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
339 |
where "f analytic_on S \<equiv> \<forall>x \<in> S. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)" |
56215 | 340 |
|
70136 | 341 |
named_theorems\<^marker>\<open>tag important\<close> analytic_intros "introduction rules for proving analyticity" |
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
342 |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
343 |
lemma analytic_imp_holomorphic: "f analytic_on S \<Longrightarrow> f holomorphic_on S" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
344 |
by (simp add: at_within_open [OF _ open_ball] analytic_on_def holomorphic_on_def) |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
345 |
(metis centre_in_ball field_differentiable_at_within) |
56215 | 346 |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
347 |
lemma analytic_on_open: "open S \<Longrightarrow> f analytic_on S \<longleftrightarrow> f holomorphic_on S" |
56215 | 348 |
apply (auto simp: analytic_imp_holomorphic) |
349 |
apply (auto simp: analytic_on_def holomorphic_on_def) |
|
350 |
by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball) |
|
351 |
||
352 |
lemma analytic_on_imp_differentiable_at: |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
353 |
"f analytic_on S \<Longrightarrow> x \<in> S \<Longrightarrow> f field_differentiable (at x)" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
354 |
apply (auto simp: analytic_on_def holomorphic_on_def) |
66827
c94531b5007d
Divided Topology_Euclidean_Space in two, creating new theory Connected. Also deleted some duplicate / variant theorems
paulson <lp15@cam.ac.uk>
parents:
66486
diff
changeset
|
355 |
by (metis open_ball centre_in_ball field_differentiable_within_open) |
56215 | 356 |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
357 |
lemma analytic_on_subset: "f analytic_on S \<Longrightarrow> T \<subseteq> S \<Longrightarrow> f analytic_on T" |
56215 | 358 |
by (auto simp: analytic_on_def) |
359 |
||
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
360 |
lemma analytic_on_Un: "f analytic_on (S \<union> T) \<longleftrightarrow> f analytic_on S \<and> f analytic_on T" |
56215 | 361 |
by (auto simp: analytic_on_def) |
362 |
||
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
363 |
lemma analytic_on_Union: "f analytic_on (\<Union>\<T>) \<longleftrightarrow> (\<forall>T \<in> \<T>. f analytic_on T)" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
364 |
by (auto simp: analytic_on_def) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
365 |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
366 |
lemma analytic_on_UN: "f analytic_on (\<Union>i\<in>I. S i) \<longleftrightarrow> (\<forall>i\<in>I. f analytic_on (S i))" |
56215 | 367 |
by (auto simp: analytic_on_def) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
368 |
|
56215 | 369 |
lemma analytic_on_holomorphic: |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
370 |
"f analytic_on S \<longleftrightarrow> (\<exists>T. open T \<and> S \<subseteq> T \<and> f holomorphic_on T)" |
56215 | 371 |
(is "?lhs = ?rhs") |
372 |
proof - |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
373 |
have "?lhs \<longleftrightarrow> (\<exists>T. open T \<and> S \<subseteq> T \<and> f analytic_on T)" |
56215 | 374 |
proof safe |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
375 |
assume "f analytic_on S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
376 |
then show "\<exists>T. open T \<and> S \<subseteq> T \<and> f analytic_on T" |
56215 | 377 |
apply (simp add: analytic_on_def) |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
378 |
apply (rule exI [where x="\<Union>{U. open U \<and> f analytic_on U}"], auto) |
66827
c94531b5007d
Divided Topology_Euclidean_Space in two, creating new theory Connected. Also deleted some duplicate / variant theorems
paulson <lp15@cam.ac.uk>
parents:
66486
diff
changeset
|
379 |
apply (metis open_ball analytic_on_open centre_in_ball) |
56215 | 380 |
by (metis analytic_on_def) |
381 |
next |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
382 |
fix T |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
383 |
assume "open T" "S \<subseteq> T" "f analytic_on T" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
384 |
then show "f analytic_on S" |
56215 | 385 |
by (metis analytic_on_subset) |
386 |
qed |
|
387 |
also have "... \<longleftrightarrow> ?rhs" |
|
388 |
by (auto simp: analytic_on_open) |
|
389 |
finally show ?thesis . |
|
390 |
qed |
|
391 |
||
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
392 |
lemma analytic_on_linear [analytic_intros,simp]: "((*) c) analytic_on S" |
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
393 |
by (auto simp add: analytic_on_holomorphic) |
56215 | 394 |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
395 |
lemma analytic_on_const [analytic_intros,simp]: "(\<lambda>z. c) analytic_on S" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
396 |
by (metis analytic_on_def holomorphic_on_const zero_less_one) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
397 |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
398 |
lemma analytic_on_ident [analytic_intros,simp]: "(\<lambda>x. x) analytic_on S" |
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
399 |
by (simp add: analytic_on_def gt_ex) |
56215 | 400 |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
401 |
lemma analytic_on_id [analytic_intros]: "id analytic_on S" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
402 |
unfolding id_def by (rule analytic_on_ident) |
56215 | 403 |
|
404 |
lemma analytic_on_compose: |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
405 |
assumes f: "f analytic_on S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
406 |
and g: "g analytic_on (f ` S)" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
407 |
shows "(g o f) analytic_on S" |
56215 | 408 |
unfolding analytic_on_def |
409 |
proof (intro ballI) |
|
410 |
fix x |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
411 |
assume x: "x \<in> S" |
56215 | 412 |
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f |
413 |
by (metis analytic_on_def) |
|
414 |
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
415 |
by (metis analytic_on_def g image_eqI x) |
56215 | 416 |
have "isCont f x" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
417 |
by (metis analytic_on_imp_differentiable_at field_differentiable_imp_continuous_at f x) |
56215 | 418 |
with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'" |
419 |
by (auto simp: continuous_at_ball) |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
420 |
have "g \<circ> f holomorphic_on ball x (min d e)" |
56215 | 421 |
apply (rule holomorphic_on_compose) |
422 |
apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball) |
|
423 |
by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball) |
|
424 |
then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
425 |
by (metis d e min_less_iff_conj) |
56215 | 426 |
qed |
427 |
||
428 |
lemma analytic_on_compose_gen: |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
429 |
"f analytic_on S \<Longrightarrow> g analytic_on T \<Longrightarrow> (\<And>z. z \<in> S \<Longrightarrow> f z \<in> T) |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
430 |
\<Longrightarrow> g o f analytic_on S" |
56215 | 431 |
by (metis analytic_on_compose analytic_on_subset image_subset_iff) |
432 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
433 |
lemma analytic_on_neg [analytic_intros]: |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
434 |
"f analytic_on S \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on S" |
56215 | 435 |
by (metis analytic_on_holomorphic holomorphic_on_minus) |
436 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
437 |
lemma analytic_on_add [analytic_intros]: |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
438 |
assumes f: "f analytic_on S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
439 |
and g: "g analytic_on S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
440 |
shows "(\<lambda>z. f z + g z) analytic_on S" |
56215 | 441 |
unfolding analytic_on_def |
442 |
proof (intro ballI) |
|
443 |
fix z |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
444 |
assume z: "z \<in> S" |
56215 | 445 |
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f |
446 |
by (metis analytic_on_def) |
|
447 |
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
448 |
by (metis analytic_on_def g z) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
449 |
have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
450 |
apply (rule holomorphic_on_add) |
56215 | 451 |
apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball) |
452 |
by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball) |
|
453 |
then show "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e" |
|
454 |
by (metis e e' min_less_iff_conj) |
|
455 |
qed |
|
456 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
457 |
lemma analytic_on_diff [analytic_intros]: |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
458 |
assumes f: "f analytic_on S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
459 |
and g: "g analytic_on S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
460 |
shows "(\<lambda>z. f z - g z) analytic_on S" |
56215 | 461 |
unfolding analytic_on_def |
462 |
proof (intro ballI) |
|
463 |
fix z |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
464 |
assume z: "z \<in> S" |
56215 | 465 |
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f |
466 |
by (metis analytic_on_def) |
|
467 |
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
468 |
by (metis analytic_on_def g z) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
469 |
have "(\<lambda>z. f z - g z) holomorphic_on ball z (min e e')" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
470 |
apply (rule holomorphic_on_diff) |
56215 | 471 |
apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball) |
472 |
by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball) |
|
473 |
then show "\<exists>e>0. (\<lambda>z. f z - g z) holomorphic_on ball z e" |
|
474 |
by (metis e e' min_less_iff_conj) |
|
475 |
qed |
|
476 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
477 |
lemma analytic_on_mult [analytic_intros]: |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
478 |
assumes f: "f analytic_on S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
479 |
and g: "g analytic_on S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
480 |
shows "(\<lambda>z. f z * g z) analytic_on S" |
56215 | 481 |
unfolding analytic_on_def |
482 |
proof (intro ballI) |
|
483 |
fix z |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
484 |
assume z: "z \<in> S" |
56215 | 485 |
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f |
486 |
by (metis analytic_on_def) |
|
487 |
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
488 |
by (metis analytic_on_def g z) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
489 |
have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
490 |
apply (rule holomorphic_on_mult) |
56215 | 491 |
apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball) |
492 |
by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball) |
|
493 |
then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e" |
|
494 |
by (metis e e' min_less_iff_conj) |
|
495 |
qed |
|
496 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
497 |
lemma analytic_on_inverse [analytic_intros]: |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
498 |
assumes f: "f analytic_on S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
499 |
and nz: "(\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0)" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
500 |
shows "(\<lambda>z. inverse (f z)) analytic_on S" |
56215 | 501 |
unfolding analytic_on_def |
502 |
proof (intro ballI) |
|
503 |
fix z |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
504 |
assume z: "z \<in> S" |
56215 | 505 |
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f |
506 |
by (metis analytic_on_def) |
|
507 |
have "continuous_on (ball z e) f" |
|
508 |
by (metis fh holomorphic_on_imp_continuous_on) |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
509 |
then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0" |
66827
c94531b5007d
Divided Topology_Euclidean_Space in two, creating new theory Connected. Also deleted some duplicate / variant theorems
paulson <lp15@cam.ac.uk>
parents:
66486
diff
changeset
|
510 |
by (metis open_ball centre_in_ball continuous_on_open_avoid e z nz) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
511 |
have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')" |
56215 | 512 |
apply (rule holomorphic_on_inverse) |
513 |
apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball) |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
514 |
by (metis nz' mem_ball min_less_iff_conj) |
56215 | 515 |
then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e" |
516 |
by (metis e e' min_less_iff_conj) |
|
517 |
qed |
|
518 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
519 |
lemma analytic_on_divide [analytic_intros]: |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
520 |
assumes f: "f analytic_on S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
521 |
and g: "g analytic_on S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
522 |
and nz: "(\<And>z. z \<in> S \<Longrightarrow> g z \<noteq> 0)" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
523 |
shows "(\<lambda>z. f z / g z) analytic_on S" |
56215 | 524 |
unfolding divide_inverse |
525 |
by (metis analytic_on_inverse analytic_on_mult f g nz) |
|
526 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
527 |
lemma analytic_on_power [analytic_intros]: |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
528 |
"f analytic_on S \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on S" |
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
529 |
by (induct n) (auto simp: analytic_on_mult) |
56215 | 530 |
|
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
531 |
lemma analytic_on_sum [analytic_intros]: |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
532 |
"(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on S) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) analytic_on S" |
71633 | 533 |
by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_add) |
56215 | 534 |
|
75168
ff60b4acd6dd
Added some theorems (from Wetzel)
paulson <lp15@cam.ac.uk>
parents:
74513
diff
changeset
|
535 |
lemma analytic_on_prod [analytic_intros]: |
ff60b4acd6dd
Added some theorems (from Wetzel)
paulson <lp15@cam.ac.uk>
parents:
74513
diff
changeset
|
536 |
"(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on S) \<Longrightarrow> (\<lambda>x. prod (\<lambda>i. f i x) I) analytic_on S" |
ff60b4acd6dd
Added some theorems (from Wetzel)
paulson <lp15@cam.ac.uk>
parents:
74513
diff
changeset
|
537 |
by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_mult) |
ff60b4acd6dd
Added some theorems (from Wetzel)
paulson <lp15@cam.ac.uk>
parents:
74513
diff
changeset
|
538 |
|
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
539 |
lemma deriv_left_inverse: |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
540 |
assumes "f holomorphic_on S" and "g holomorphic_on T" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
541 |
and "open S" and "open T" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
542 |
and "f ` S \<subseteq> T" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
543 |
and [simp]: "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
544 |
and "w \<in> S" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
545 |
shows "deriv f w * deriv g (f w) = 1" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
546 |
proof - |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
547 |
have "deriv f w * deriv g (f w) = deriv g (f w) * deriv f w" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
548 |
by (simp add: algebra_simps) |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
549 |
also have "... = deriv (g o f) w" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
550 |
using assms |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71167
diff
changeset
|
551 |
by (metis analytic_on_imp_differentiable_at analytic_on_open deriv_chain image_subset_iff) |
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
552 |
also have "... = deriv id w" |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
553 |
proof (rule complex_derivative_transform_within_open [where s=S]) |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
554 |
show "g \<circ> f holomorphic_on S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
555 |
by (rule assms holomorphic_on_compose_gen holomorphic_intros)+ |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
556 |
qed (use assms in auto) |
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
557 |
also have "... = 1" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
558 |
by simp |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
559 |
finally show ?thesis . |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
560 |
qed |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
561 |
|
70136 | 562 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>Analyticity at a point\<close> |
56215 | 563 |
|
564 |
lemma analytic_at_ball: |
|
565 |
"f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)" |
|
566 |
by (metis analytic_on_def singleton_iff) |
|
567 |
||
568 |
lemma analytic_at: |
|
569 |
"f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)" |
|
570 |
by (metis analytic_on_holomorphic empty_subsetI insert_subset) |
|
571 |
||
572 |
lemma analytic_on_analytic_at: |
|
573 |
"f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})" |
|
574 |
by (metis analytic_at_ball analytic_on_def) |
|
575 |
||
576 |
lemma analytic_at_two: |
|
577 |
"f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow> |
|
578 |
(\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)" |
|
579 |
(is "?lhs = ?rhs") |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
580 |
proof |
56215 | 581 |
assume ?lhs |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
582 |
then obtain s t |
56215 | 583 |
where st: "open s" "z \<in> s" "f holomorphic_on s" |
584 |
"open t" "z \<in> t" "g holomorphic_on t" |
|
585 |
by (auto simp: analytic_at) |
|
586 |
show ?rhs |
|
587 |
apply (rule_tac x="s \<inter> t" in exI) |
|
588 |
using st |
|
69286 | 589 |
apply (auto simp: holomorphic_on_subset) |
56215 | 590 |
done |
591 |
next |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
592 |
assume ?rhs |
56215 | 593 |
then show ?lhs |
594 |
by (force simp add: analytic_at) |
|
595 |
qed |
|
596 |
||
70136 | 597 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>Combining theorems for derivative with ``analytic at'' hypotheses\<close> |
56215 | 598 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
599 |
lemma |
56215 | 600 |
assumes "f analytic_on {z}" "g analytic_on {z}" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
601 |
shows complex_derivative_add_at: "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
602 |
and complex_derivative_diff_at: "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
603 |
and complex_derivative_mult_at: "deriv (\<lambda>w. f w * g w) z = |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
604 |
f z * deriv g z + deriv f z * g z" |
56215 | 605 |
proof - |
606 |
obtain s where s: "open s" "z \<in> s" "f holomorphic_on s" "g holomorphic_on s" |
|
607 |
using assms by (metis analytic_at_two) |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
608 |
show "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
609 |
apply (rule DERIV_imp_deriv [OF DERIV_add]) |
56215 | 610 |
using s |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
611 |
apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable) |
56215 | 612 |
done |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
613 |
show "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
614 |
apply (rule DERIV_imp_deriv [OF DERIV_diff]) |
56215 | 615 |
using s |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
616 |
apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable) |
56215 | 617 |
done |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
618 |
show "deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
619 |
apply (rule DERIV_imp_deriv [OF DERIV_mult']) |
56215 | 620 |
using s |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
621 |
apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable) |
56215 | 622 |
done |
623 |
qed |
|
624 |
||
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
625 |
lemma deriv_cmult_at: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
626 |
"f analytic_on {z} \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z" |
71633 | 627 |
by (auto simp: complex_derivative_mult_at) |
56215 | 628 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
629 |
lemma deriv_cmult_right_at: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
630 |
"f analytic_on {z} \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c" |
71633 | 631 |
by (auto simp: complex_derivative_mult_at) |
56215 | 632 |
|
70136 | 633 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>Complex differentiation of sequences and series\<close> |
56215 | 634 |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
635 |
(* TODO: Could probably be simplified using Uniform_Limit *) |
56215 | 636 |
lemma has_complex_derivative_sequence: |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
637 |
fixes S :: "complex set" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
638 |
assumes cvs: "convex S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
639 |
and df: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
640 |
and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> S \<longrightarrow> norm (f' n x - g' x) \<le> e" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
641 |
and "\<exists>x l. x \<in> S \<and> ((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
642 |
shows "\<exists>g. \<forall>x \<in> S. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and> |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
643 |
(g has_field_derivative (g' x)) (at x within S)" |
56215 | 644 |
proof - |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
645 |
from assms obtain x l where x: "x \<in> S" and tf: "((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially" |
56215 | 646 |
by blast |
647 |
{ fix e::real assume e: "e > 0" |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
648 |
then obtain N where N: "\<forall>n\<ge>N. \<forall>x. x \<in> S \<longrightarrow> cmod (f' n x - g' x) \<le> e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
649 |
by (metis conv) |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
650 |
have "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h" |
56215 | 651 |
proof (rule exI [of _ N], clarify) |
652 |
fix n y h |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
653 |
assume "N \<le> n" "y \<in> S" |
56215 | 654 |
then have "cmod (f' n y - g' y) \<le> e" |
655 |
by (metis N) |
|
656 |
then have "cmod h * cmod (f' n y - g' y) \<le> cmod h * e" |
|
657 |
by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2) |
|
658 |
then show "cmod (f' n y * h - g' y * h) \<le> e * cmod h" |
|
659 |
by (simp add: norm_mult [symmetric] field_simps) |
|
660 |
qed |
|
661 |
} note ** = this |
|
662 |
show ?thesis |
|
68055 | 663 |
unfolding has_field_derivative_def |
56215 | 664 |
proof (rule has_derivative_sequence [OF cvs _ _ x]) |
68239 | 665 |
show "(\<lambda>n. f n x) \<longlonglongrightarrow> l" |
666 |
by (rule tf) |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
667 |
next show "\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h" |
68239 | 668 |
unfolding eventually_sequentially by (blast intro: **) |
68055 | 669 |
qed (metis has_field_derivative_def df) |
56215 | 670 |
qed |
671 |
||
672 |
lemma has_complex_derivative_series: |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
673 |
fixes S :: "complex set" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
674 |
assumes cvs: "convex S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
675 |
and df: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
676 |
and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> S |
56215 | 677 |
\<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e" |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
678 |
and "\<exists>x l. x \<in> S \<and> ((\<lambda>n. f n x) sums l)" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
679 |
shows "\<exists>g. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((g has_field_derivative g' x) (at x within S))" |
56215 | 680 |
proof - |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
681 |
from assms obtain x l where x: "x \<in> S" and sf: "((\<lambda>n. f n x) sums l)" |
56215 | 682 |
by blast |
683 |
{ fix e::real assume e: "e > 0" |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
684 |
then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> S |
56215 | 685 |
\<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
686 |
by (metis conv) |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
687 |
have "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h" |
56215 | 688 |
proof (rule exI [of _ N], clarify) |
689 |
fix n y h |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
690 |
assume "N \<le> n" "y \<in> S" |
56215 | 691 |
then have "cmod ((\<Sum>i<n. f' i y) - g' y) \<le> e" |
692 |
by (metis N) |
|
693 |
then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e" |
|
694 |
by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2) |
|
695 |
then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h" |
|
64267 | 696 |
by (simp add: norm_mult [symmetric] field_simps sum_distrib_left) |
56215 | 697 |
qed |
698 |
} note ** = this |
|
699 |
show ?thesis |
|
700 |
unfolding has_field_derivative_def |
|
701 |
proof (rule has_derivative_series [OF cvs _ _ x]) |
|
702 |
fix n x |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
703 |
assume "x \<in> S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
704 |
then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within S)" |
56215 | 705 |
by (metis df has_field_derivative_def mult_commute_abs) |
706 |
next show " ((\<lambda>n. f n x) sums l)" |
|
707 |
by (rule sf) |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
708 |
next show "\<And>e. e>0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h" |
68239 | 709 |
unfolding eventually_sequentially by (blast intro: **) |
56215 | 710 |
qed |
711 |
qed |
|
712 |
||
70136 | 713 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Taylor on Complex Numbers\<close> |
56215 | 714 |
|
64267 | 715 |
lemma sum_Suc_reindex: |
56215 | 716 |
fixes f :: "nat \<Rightarrow> 'a::ab_group_add" |
64267 | 717 |
shows "sum f {0..n} = f 0 - f (Suc n) + sum (\<lambda>i. f (Suc i)) {0..n}" |
56215 | 718 |
by (induct n) auto |
719 |
||
69529 | 720 |
lemma field_Taylor: |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
721 |
assumes S: "convex S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
722 |
and f: "\<And>i x. x \<in> S \<Longrightarrow> i \<le> n \<Longrightarrow> (f i has_field_derivative f (Suc i) x) (at x within S)" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
723 |
and B: "\<And>x. x \<in> S \<Longrightarrow> norm (f (Suc n) x) \<le> B" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
724 |
and w: "w \<in> S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
725 |
and z: "z \<in> S" |
66252 | 726 |
shows "norm(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i))) |
727 |
\<le> B * norm(z - w)^(Suc n) / fact n" |
|
56215 | 728 |
proof - |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
729 |
have wzs: "closed_segment w z \<subseteq> S" using assms |
56215 | 730 |
by (metis convex_contains_segment) |
731 |
{ fix u |
|
732 |
assume "u \<in> closed_segment w z" |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
733 |
then have "u \<in> S" |
56215 | 734 |
by (metis wzs subsetD) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
735 |
have "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / (fact i) + |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
736 |
f (Suc i) u * (z-u)^i / (fact i)) = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
737 |
f (Suc n) u * (z-u) ^ n / (fact n)" |
56215 | 738 |
proof (induction n) |
739 |
case 0 show ?case by simp |
|
740 |
next |
|
741 |
case (Suc n) |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
742 |
have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / (fact i) + |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
743 |
f (Suc i) u * (z-u) ^ i / (fact i)) = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
744 |
f (Suc n) u * (z-u) ^ n / (fact n) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
745 |
f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / (fact (Suc n)) - |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
746 |
f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / (fact (Suc n))" |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56409
diff
changeset
|
747 |
using Suc by simp |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
748 |
also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n))" |
56215 | 749 |
proof - |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
750 |
have "(fact(Suc n)) * |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
751 |
(f(Suc n) u *(z-u) ^ n / (fact n) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
752 |
f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / (fact(Suc n)) - |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
753 |
f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / (fact(Suc n))) = |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
754 |
((fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / (fact n) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
755 |
((fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / (fact(Suc n))) - |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
756 |
((fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / (fact(Suc n))" |
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
63332
diff
changeset
|
757 |
by (simp add: algebra_simps del: fact_Suc) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
758 |
also have "... = ((fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / (fact n) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
759 |
(f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) - |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
760 |
(f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))" |
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
63332
diff
changeset
|
761 |
by (simp del: fact_Suc) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
762 |
also have "... = (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
763 |
(f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) - |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
764 |
(f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))" |
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
63332
diff
changeset
|
765 |
by (simp only: fact_Suc of_nat_mult ac_simps) simp |
56215 | 766 |
also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)" |
767 |
by (simp add: algebra_simps) |
|
768 |
finally show ?thesis |
|
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
63332
diff
changeset
|
769 |
by (simp add: mult_left_cancel [where c = "(fact (Suc n))", THEN iffD1] del: fact_Suc) |
56215 | 770 |
qed |
771 |
finally show ?case . |
|
772 |
qed |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
773 |
then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / (fact i))) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
774 |
has_field_derivative f (Suc n) u * (z-u) ^ n / (fact n)) |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
775 |
(at u within S)" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
776 |
apply (intro derivative_eq_intros) |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
777 |
apply (blast intro: assms \<open>u \<in> S\<close>) |
56215 | 778 |
apply (rule refl)+ |
779 |
apply (auto simp: field_simps) |
|
780 |
done |
|
781 |
} note sum_deriv = this |
|
782 |
{ fix u |
|
783 |
assume u: "u \<in> closed_segment w z" |
|
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
784 |
then have us: "u \<in> S" |
56215 | 785 |
by (metis wzs subsetD) |
66252 | 786 |
have "norm (f (Suc n) u) * norm (z - u) ^ n \<le> norm (f (Suc n) u) * norm (u - z) ^ n" |
56215 | 787 |
by (metis norm_minus_commute order_refl) |
66252 | 788 |
also have "... \<le> norm (f (Suc n) u) * norm (z - w) ^ n" |
56215 | 789 |
by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u]) |
66252 | 790 |
also have "... \<le> B * norm (z - w) ^ n" |
56215 | 791 |
by (metis norm_ge_zero zero_le_power mult_right_mono B [OF us]) |
66252 | 792 |
finally have "norm (f (Suc n) u) * norm (z - u) ^ n \<le> B * norm (z - w) ^ n" . |
56215 | 793 |
} note cmod_bound = this |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
794 |
have "(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)) = (\<Sum>i\<le>n. (f i z / (fact i)) * 0 ^ i)" |
56215 | 795 |
by simp |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
796 |
also have "\<dots> = f 0 z / (fact 0)" |
64267 | 797 |
by (subst sum_zero_power) simp |
66252 | 798 |
finally have "norm (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i))) |
799 |
\<le> norm ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) - |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
800 |
(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))" |
56215 | 801 |
by (simp add: norm_minus_commute) |
66252 | 802 |
also have "... \<le> B * norm (z - w) ^ n / (fact n) * norm (w - z)" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
803 |
apply (rule field_differentiable_bound |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
804 |
[where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / (fact n)" |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
805 |
and S = "closed_segment w z", OF convex_closed_segment]) |
71633 | 806 |
apply (auto simp: DERIV_subset [OF sum_deriv wzs] |
56215 | 807 |
norm_divide norm_mult norm_power divide_le_cancel cmod_bound) |
808 |
done |
|
66252 | 809 |
also have "... \<le> B * norm (z - w) ^ Suc n / (fact n)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
810 |
by (simp add: algebra_simps norm_minus_commute) |
56215 | 811 |
finally show ?thesis . |
812 |
qed |
|
813 |
||
69529 | 814 |
lemma complex_Taylor: |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
815 |
assumes S: "convex S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
816 |
and f: "\<And>i x. x \<in> S \<Longrightarrow> i \<le> n \<Longrightarrow> (f i has_field_derivative f (Suc i) x) (at x within S)" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
817 |
and B: "\<And>x. x \<in> S \<Longrightarrow> cmod (f (Suc n) x) \<le> B" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
818 |
and w: "w \<in> S" |
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
819 |
and z: "z \<in> S" |
66252 | 820 |
shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i))) |
821 |
\<le> B * cmod(z - w)^(Suc n) / fact n" |
|
69529 | 822 |
using assms by (rule field_Taylor) |
66252 | 823 |
|
824 |
||
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
825 |
text\<open>Something more like the traditional MVT for real components\<close> |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
826 |
|
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
827 |
lemma complex_mvt_line: |
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56332
diff
changeset
|
828 |
assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f'(u)) (at u)" |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61235
diff
changeset
|
829 |
shows "\<exists>u. u \<in> closed_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))" |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
830 |
proof - |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
831 |
have twz: "\<And>t. (1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)" |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
832 |
by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib) |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
833 |
note assms[unfolded has_field_derivative_def, derivative_intros] |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
834 |
show ?thesis |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
835 |
apply (cut_tac mvt_simple |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
836 |
[of 0 1 "Re o f o (\<lambda>t. (1 - t) *\<^sub>R w + t *\<^sub>R z)" |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
837 |
"\<lambda>u. Re o (\<lambda>h. f'((1 - u) *\<^sub>R w + u *\<^sub>R z) * h) o (\<lambda>t. t *\<^sub>R (z - w))"]) |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
838 |
apply auto |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
839 |
apply (rule_tac x="(1 - x) *\<^sub>R w + x *\<^sub>R z" in exI) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61235
diff
changeset
|
840 |
apply (auto simp: closed_segment_def twz) [] |
67979
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67968
diff
changeset
|
841 |
apply (intro derivative_eq_intros has_derivative_at_withinI, simp_all) |
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56332
diff
changeset
|
842 |
apply (simp add: fun_eq_iff real_vector.scale_right_diff_distrib) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61235
diff
changeset
|
843 |
apply (force simp: twz closed_segment_def) |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
844 |
done |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
845 |
qed |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
846 |
|
69529 | 847 |
lemma complex_Taylor_mvt: |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
848 |
assumes "\<And>i x. \<lbrakk>x \<in> closed_segment w z; i \<le> n\<rbrakk> \<Longrightarrow> ((f i) has_field_derivative f (Suc i) x) (at x)" |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
849 |
shows "\<exists>u. u \<in> closed_segment w z \<and> |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
850 |
Re (f 0 z) = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
851 |
Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / (fact i)) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
852 |
(f (Suc n) u * (z-u)^n / (fact n)) * (z - w))" |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
853 |
proof - |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
854 |
{ fix u |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
855 |
assume u: "u \<in> closed_segment w z" |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
856 |
have "(\<Sum>i = 0..n. |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
857 |
(f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
858 |
(fact i)) = |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
859 |
f (Suc 0) u - |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
860 |
(f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) / |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
861 |
(fact (Suc n)) + |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
862 |
(\<Sum>i = 0..n. |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
863 |
(f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) / |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
864 |
(fact (Suc i)))" |
64267 | 865 |
by (subst sum_Suc_reindex) simp |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
866 |
also have "... = f (Suc 0) u - |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
867 |
(f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) / |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
868 |
(fact (Suc n)) + |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
869 |
(\<Sum>i = 0..n. |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
870 |
f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i)) - |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
871 |
f (Suc i) u * (z-u) ^ i / (fact i))" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
56889
diff
changeset
|
872 |
by (simp only: diff_divide_distrib fact_cancel ac_simps) |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
873 |
also have "... = f (Suc 0) u - |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
874 |
(f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) / |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
875 |
(fact (Suc n)) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
876 |
f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n)) - f (Suc 0) u" |
64267 | 877 |
by (subst sum_Suc_diff) auto |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
878 |
also have "... = f (Suc n) u * (z-u) ^ n / (fact n)" |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
879 |
by (simp only: algebra_simps diff_divide_distrib fact_cancel) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
880 |
finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
881 |
- of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / (fact i)) = |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
882 |
f (Suc n) u * (z - u) ^ n / (fact n)" . |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
883 |
then have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / (fact i)) has_field_derivative |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
884 |
f (Suc n) u * (z - u) ^ n / (fact n)) (at u)" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
885 |
apply (intro derivative_eq_intros)+ |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
886 |
apply (force intro: u assms) |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
887 |
apply (rule refl)+ |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
56889
diff
changeset
|
888 |
apply (auto simp: ac_simps) |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
889 |
done |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
890 |
} |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
891 |
then show ?thesis |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
892 |
apply (cut_tac complex_mvt_line [of w z "\<lambda>u. \<Sum>i = 0..n. f i u * (z-u) ^ i / (fact i)" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
893 |
"\<lambda>u. (f (Suc n) u * (z-u)^n / (fact n))"]) |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
894 |
apply (auto simp add: intro: open_closed_segment) |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
895 |
done |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
896 |
qed |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
897 |
|
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
898 |
|
56215 | 899 |
end |