author | Manuel Eberl <eberlm@in.tum.de> |
Mon, 02 Dec 2019 17:51:54 +0100 | |
changeset 71201 | 6617fb368a06 |
child 72222 | 01397b6e5eb0 |
permissions | -rw-r--r-- |
71201
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1 |
theory Complex_Singularities |
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2 |
imports Conformal_Mappings |
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3 |
begin |
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4 |
|
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5 |
subsection \<open>Non-essential singular points\<close> |
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6 |
|
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7 |
definition\<^marker>\<open>tag important\<close> is_pole :: |
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8 |
"('a::topological_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> bool" where |
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9 |
"is_pole f a = (LIM x (at a). f x :> at_infinity)" |
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10 |
|
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11 |
lemma is_pole_cong: |
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12 |
assumes "eventually (\<lambda>x. f x = g x) (at a)" "a=b" |
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13 |
shows "is_pole f a \<longleftrightarrow> is_pole g b" |
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14 |
unfolding is_pole_def using assms by (intro filterlim_cong,auto) |
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15 |
|
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16 |
lemma is_pole_transform: |
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17 |
assumes "is_pole f a" "eventually (\<lambda>x. f x = g x) (at a)" "a=b" |
6617fb368a06
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18 |
shows "is_pole g b" |
6617fb368a06
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19 |
using is_pole_cong assms by auto |
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20 |
|
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21 |
lemma is_pole_tendsto: |
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22 |
fixes f::"('a::topological_space \<Rightarrow> 'b::real_normed_div_algebra)" |
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23 |
shows "is_pole f x \<Longrightarrow> ((inverse o f) \<longlongrightarrow> 0) (at x)" |
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24 |
unfolding is_pole_def |
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25 |
by (auto simp add:filterlim_inverse_at_iff[symmetric] comp_def filterlim_at) |
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26 |
|
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27 |
lemma is_pole_inverse_holomorphic: |
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28 |
assumes "open s" |
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29 |
and f_holo:"f holomorphic_on (s-{z})" |
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30 |
and pole:"is_pole f z" |
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31 |
and non_z:"\<forall>x\<in>s-{z}. f x\<noteq>0" |
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32 |
shows "(\<lambda>x. if x=z then 0 else inverse (f x)) holomorphic_on s" |
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33 |
proof - |
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34 |
define g where "g \<equiv> \<lambda>x. if x=z then 0 else inverse (f x)" |
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35 |
have "isCont g z" unfolding isCont_def using is_pole_tendsto[OF pole] |
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36 |
apply (subst Lim_cong_at[where b=z and y=0 and g="inverse \<circ> f"]) |
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37 |
by (simp_all add:g_def) |
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38 |
moreover have "continuous_on (s-{z}) f" using f_holo holomorphic_on_imp_continuous_on by auto |
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39 |
hence "continuous_on (s-{z}) (inverse o f)" unfolding comp_def |
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40 |
by (auto elim!:continuous_on_inverse simp add:non_z) |
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41 |
hence "continuous_on (s-{z}) g" unfolding g_def |
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42 |
apply (subst continuous_on_cong[where t="s-{z}" and g="inverse o f"]) |
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43 |
by auto |
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44 |
ultimately have "continuous_on s g" using open_delete[OF \<open>open s\<close>] \<open>open s\<close> |
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45 |
by (auto simp add:continuous_on_eq_continuous_at) |
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46 |
moreover have "(inverse o f) holomorphic_on (s-{z})" |
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47 |
unfolding comp_def using f_holo |
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48 |
by (auto elim!:holomorphic_on_inverse simp add:non_z) |
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49 |
hence "g holomorphic_on (s-{z})" |
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50 |
apply (subst holomorphic_cong[where t="s-{z}" and g="inverse o f"]) |
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51 |
by (auto simp add:g_def) |
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52 |
ultimately show ?thesis unfolding g_def using \<open>open s\<close> |
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53 |
by (auto elim!: no_isolated_singularity) |
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54 |
qed |
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55 |
|
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56 |
lemma not_is_pole_holomorphic: |
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57 |
assumes "open A" "x \<in> A" "f holomorphic_on A" |
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58 |
shows "\<not>is_pole f x" |
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59 |
proof - |
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60 |
have "continuous_on A f" by (intro holomorphic_on_imp_continuous_on) fact |
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61 |
with assms have "isCont f x" by (simp add: continuous_on_eq_continuous_at) |
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62 |
hence "f \<midarrow>x\<rightarrow> f x" by (simp add: isCont_def) |
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63 |
thus "\<not>is_pole f x" unfolding is_pole_def |
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64 |
using not_tendsto_and_filterlim_at_infinity[of "at x" f "f x"] by auto |
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65 |
qed |
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66 |
|
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67 |
lemma is_pole_inverse_power: "n > 0 \<Longrightarrow> is_pole (\<lambda>z::complex. 1 / (z - a) ^ n) a" |
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68 |
unfolding is_pole_def inverse_eq_divide [symmetric] |
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69 |
by (intro filterlim_compose[OF filterlim_inverse_at_infinity] tendsto_intros) |
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70 |
(auto simp: filterlim_at eventually_at intro!: exI[of _ 1] tendsto_eq_intros) |
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71 |
|
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72 |
lemma is_pole_inverse: "is_pole (\<lambda>z::complex. 1 / (z - a)) a" |
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73 |
using is_pole_inverse_power[of 1 a] by simp |
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74 |
|
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75 |
lemma is_pole_divide: |
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76 |
fixes f :: "'a :: t2_space \<Rightarrow> 'b :: real_normed_field" |
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77 |
assumes "isCont f z" "filterlim g (at 0) (at z)" "f z \<noteq> 0" |
6617fb368a06
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78 |
shows "is_pole (\<lambda>z. f z / g z) z" |
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79 |
proof - |
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80 |
have "filterlim (\<lambda>z. f z * inverse (g z)) at_infinity (at z)" |
6617fb368a06
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81 |
by (intro tendsto_mult_filterlim_at_infinity[of _ "f z"] |
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82 |
filterlim_compose[OF filterlim_inverse_at_infinity])+ |
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83 |
(insert assms, auto simp: isCont_def) |
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84 |
thus ?thesis by (simp add: field_split_simps is_pole_def) |
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85 |
qed |
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86 |
|
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87 |
lemma is_pole_basic: |
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88 |
assumes "f holomorphic_on A" "open A" "z \<in> A" "f z \<noteq> 0" "n > 0" |
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89 |
shows "is_pole (\<lambda>w. f w / (w - z) ^ n) z" |
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90 |
proof (rule is_pole_divide) |
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91 |
have "continuous_on A f" by (rule holomorphic_on_imp_continuous_on) fact |
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92 |
with assms show "isCont f z" by (auto simp: continuous_on_eq_continuous_at) |
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93 |
have "filterlim (\<lambda>w. (w - z) ^ n) (nhds 0) (at z)" |
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94 |
using assms by (auto intro!: tendsto_eq_intros) |
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95 |
thus "filterlim (\<lambda>w. (w - z) ^ n) (at 0) (at z)" |
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96 |
by (intro filterlim_atI tendsto_eq_intros) |
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97 |
(insert assms, auto simp: eventually_at_filter) |
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98 |
qed fact+ |
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99 |
|
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100 |
lemma is_pole_basic': |
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101 |
assumes "f holomorphic_on A" "open A" "0 \<in> A" "f 0 \<noteq> 0" "n > 0" |
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102 |
shows "is_pole (\<lambda>w. f w / w ^ n) 0" |
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103 |
using is_pole_basic[of f A 0] assms by simp |
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104 |
|
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105 |
text \<open>The proposition |
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106 |
\<^term>\<open>\<exists>x. ((f::complex\<Rightarrow>complex) \<longlongrightarrow> x) (at z) \<or> is_pole f z\<close> |
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107 |
can be interpreted as the complex function \<^term>\<open>f\<close> has a non-essential singularity at \<^term>\<open>z\<close> |
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108 |
(i.e. the singularity is either removable or a pole).\<close> |
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109 |
definition not_essential::"[complex \<Rightarrow> complex, complex] \<Rightarrow> bool" where |
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|
110 |
"not_essential f z = (\<exists>x. f\<midarrow>z\<rightarrow>x \<or> is_pole f z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
111 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
112 |
definition isolated_singularity_at::"[complex \<Rightarrow> complex, complex] \<Rightarrow> bool" where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
113 |
"isolated_singularity_at f z = (\<exists>r>0. f analytic_on ball z r-{z})" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
114 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
115 |
named_theorems singularity_intros "introduction rules for singularities" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
116 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
117 |
lemma holomorphic_factor_unique: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
118 |
fixes f::"complex \<Rightarrow> complex" and z::complex and r::real and m n::int |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
119 |
assumes "r>0" "g z\<noteq>0" "h z\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
120 |
and asm:"\<forall>w\<in>ball z r-{z}. f w = g w * (w-z) powr n \<and> g w\<noteq>0 \<and> f w = h w * (w - z) powr m \<and> h w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
121 |
and g_holo:"g holomorphic_on ball z r" and h_holo:"h holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
122 |
shows "n=m" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
123 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
124 |
have [simp]:"at z within ball z r \<noteq> bot" using \<open>r>0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
125 |
by (auto simp add:at_within_ball_bot_iff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
126 |
have False when "n>m" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
127 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
128 |
have "(h \<longlongrightarrow> 0) (at z within ball z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
129 |
proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powr (n - m) * g w"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
130 |
have "\<forall>w\<in>ball z r-{z}. h w = (w-z)powr(n-m) * g w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
131 |
using \<open>n>m\<close> asm \<open>r>0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
132 |
apply (auto simp add:field_simps powr_diff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
133 |
by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
134 |
then show "\<lbrakk>x' \<in> ball z r; 0 < dist x' z;dist x' z < r\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
135 |
\<Longrightarrow> (x' - z) powr (n - m) * g x' = h x'" for x' by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
136 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
137 |
define F where "F \<equiv> at z within ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
138 |
define f' where "f' \<equiv> \<lambda>x. (x - z) powr (n-m)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
139 |
have "f' z=0" using \<open>n>m\<close> unfolding f'_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
140 |
moreover have "continuous F f'" unfolding f'_def F_def continuous_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
141 |
apply (subst Lim_ident_at) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
142 |
using \<open>n>m\<close> by (auto intro!:tendsto_powr_complex_0 tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
143 |
ultimately have "(f' \<longlongrightarrow> 0) F" unfolding F_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
144 |
by (simp add: continuous_within) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
145 |
moreover have "(g \<longlongrightarrow> g z) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
146 |
using holomorphic_on_imp_continuous_on[OF g_holo,unfolded continuous_on_def] \<open>r>0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
147 |
unfolding F_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
148 |
ultimately show " ((\<lambda>w. f' w * g w) \<longlongrightarrow> 0) F" using tendsto_mult by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
149 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
150 |
moreover have "(h \<longlongrightarrow> h z) (at z within ball z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
151 |
using holomorphic_on_imp_continuous_on[OF h_holo] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
152 |
by (auto simp add:continuous_on_def \<open>r>0\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
153 |
ultimately have "h z=0" by (auto intro!: tendsto_unique) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
154 |
thus False using \<open>h z\<noteq>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
155 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
156 |
moreover have False when "m>n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
157 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
158 |
have "(g \<longlongrightarrow> 0) (at z within ball z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
159 |
proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powr (m - n) * h w"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
160 |
have "\<forall>w\<in>ball z r -{z}. g w = (w-z) powr (m-n) * h w" using \<open>m>n\<close> asm |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
161 |
apply (auto simp add:field_simps powr_diff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
162 |
by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
163 |
then show "\<lbrakk>x' \<in> ball z r; 0 < dist x' z;dist x' z < r\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
164 |
\<Longrightarrow> (x' - z) powr (m - n) * h x' = g x'" for x' by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
165 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
166 |
define F where "F \<equiv> at z within ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
167 |
define f' where "f' \<equiv>\<lambda>x. (x - z) powr (m-n)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
168 |
have "f' z=0" using \<open>m>n\<close> unfolding f'_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
169 |
moreover have "continuous F f'" unfolding f'_def F_def continuous_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
170 |
apply (subst Lim_ident_at) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
171 |
using \<open>m>n\<close> by (auto intro!:tendsto_powr_complex_0 tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
172 |
ultimately have "(f' \<longlongrightarrow> 0) F" unfolding F_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
173 |
by (simp add: continuous_within) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
174 |
moreover have "(h \<longlongrightarrow> h z) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
175 |
using holomorphic_on_imp_continuous_on[OF h_holo,unfolded continuous_on_def] \<open>r>0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
176 |
unfolding F_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
177 |
ultimately show " ((\<lambda>w. f' w * h w) \<longlongrightarrow> 0) F" using tendsto_mult by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
178 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
179 |
moreover have "(g \<longlongrightarrow> g z) (at z within ball z r)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
180 |
using holomorphic_on_imp_continuous_on[OF g_holo] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
181 |
by (auto simp add:continuous_on_def \<open>r>0\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
182 |
ultimately have "g z=0" by (auto intro!: tendsto_unique) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
183 |
thus False using \<open>g z\<noteq>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
184 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
185 |
ultimately show "n=m" by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
186 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
187 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
188 |
lemma holomorphic_factor_puncture: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
189 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
190 |
and "not_essential f z" \<comment> \<open>\<^term>\<open>f\<close> has either a removable singularity or a pole at \<^term>\<open>z\<close>\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
191 |
and non_zero:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" \<comment> \<open>\<^term>\<open>f\<close> will not be constantly zero in a neighbour of \<^term>\<open>z\<close>\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
192 |
shows "\<exists>!n::int. \<exists>g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
193 |
\<and> (\<forall>w\<in>cball z r-{z}. f w = g w * (w-z) powr n \<and> g w\<noteq>0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
194 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
195 |
define P where "P = (\<lambda>f n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
196 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
197 |
have imp_unique:"\<exists>!n::int. \<exists>g r. P f n g r" when "\<exists>n g r. P f n g r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
198 |
proof (rule ex_ex1I[OF that]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
199 |
fix n1 n2 :: int |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
200 |
assume g1_asm:"\<exists>g1 r1. P f n1 g1 r1" and g2_asm:"\<exists>g2 r2. P f n2 g2 r2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
201 |
define fac where "fac \<equiv> \<lambda>n g r. \<forall>w\<in>cball z r-{z}. f w = g w * (w - z) powr (of_int n) \<and> g w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
202 |
obtain g1 r1 where "0 < r1" and g1_holo: "g1 holomorphic_on cball z r1" and "g1 z\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
203 |
and "fac n1 g1 r1" using g1_asm unfolding P_def fac_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
204 |
obtain g2 r2 where "0 < r2" and g2_holo: "g2 holomorphic_on cball z r2" and "g2 z\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
205 |
and "fac n2 g2 r2" using g2_asm unfolding P_def fac_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
206 |
define r where "r \<equiv> min r1 r2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
207 |
have "r>0" using \<open>r1>0\<close> \<open>r2>0\<close> unfolding r_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
208 |
moreover have "\<forall>w\<in>ball z r-{z}. f w = g1 w * (w-z) powr n1 \<and> g1 w\<noteq>0 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
209 |
\<and> f w = g2 w * (w - z) powr n2 \<and> g2 w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
210 |
using \<open>fac n1 g1 r1\<close> \<open>fac n2 g2 r2\<close> unfolding fac_def r_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
211 |
by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
212 |
ultimately show "n1=n2" using g1_holo g2_holo \<open>g1 z\<noteq>0\<close> \<open>g2 z\<noteq>0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
213 |
apply (elim holomorphic_factor_unique) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
214 |
by (auto simp add:r_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
215 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
216 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
217 |
have P_exist:"\<exists> n g r. P h n g r" when |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
218 |
"\<exists>z'. (h \<longlongrightarrow> z') (at z)" "isolated_singularity_at h z" "\<exists>\<^sub>Fw in (at z). h w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
219 |
for h |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
220 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
221 |
from that(2) obtain r where "r>0" "h analytic_on ball z r - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
222 |
unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
223 |
obtain z' where "(h \<longlongrightarrow> z') (at z)" using \<open>\<exists>z'. (h \<longlongrightarrow> z') (at z)\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
224 |
define h' where "h'=(\<lambda>x. if x=z then z' else h x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
225 |
have "h' holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
226 |
apply (rule no_isolated_singularity'[of "{z}"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
227 |
subgoal by (metis LIM_equal Lim_at_imp_Lim_at_within \<open>h \<midarrow>z\<rightarrow> z'\<close> empty_iff h'_def insert_iff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
228 |
subgoal using \<open>h analytic_on ball z r - {z}\<close> analytic_imp_holomorphic h'_def holomorphic_transform |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
229 |
by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
230 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
231 |
have ?thesis when "z'=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
232 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
233 |
have "h' z=0" using that unfolding h'_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
234 |
moreover have "\<not> h' constant_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
235 |
using \<open>\<exists>\<^sub>Fw in (at z). h w\<noteq>0\<close> unfolding constant_on_def frequently_def eventually_at h'_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
236 |
apply simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
237 |
by (metis \<open>0 < r\<close> centre_in_ball dist_commute mem_ball that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
238 |
moreover note \<open>h' holomorphic_on ball z r\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
239 |
ultimately obtain g r1 n where "0 < n" "0 < r1" "ball z r1 \<subseteq> ball z r" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
240 |
g:"g holomorphic_on ball z r1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
241 |
"\<And>w. w \<in> ball z r1 \<Longrightarrow> h' w = (w - z) ^ n * g w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
242 |
"\<And>w. w \<in> ball z r1 \<Longrightarrow> g w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
243 |
using holomorphic_factor_zero_nonconstant[of _ "ball z r" z thesis,simplified, |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
244 |
OF \<open>h' holomorphic_on ball z r\<close> \<open>r>0\<close> \<open>h' z=0\<close> \<open>\<not> h' constant_on ball z r\<close>] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
245 |
by (auto simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
246 |
define rr where "rr=r1/2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
247 |
have "P h' n g rr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
248 |
unfolding P_def rr_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
249 |
using \<open>n>0\<close> \<open>r1>0\<close> g by (auto simp add:powr_nat) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
250 |
then have "P h n g rr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
251 |
unfolding h'_def P_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
252 |
then show ?thesis unfolding P_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
253 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
254 |
moreover have ?thesis when "z'\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
255 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
256 |
have "h' z\<noteq>0" using that unfolding h'_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
257 |
obtain r1 where "r1>0" "cball z r1 \<subseteq> ball z r" "\<forall>x\<in>cball z r1. h' x\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
258 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
259 |
have "isCont h' z" "h' z\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
260 |
by (auto simp add: Lim_cong_within \<open>h \<midarrow>z\<rightarrow> z'\<close> \<open>z'\<noteq>0\<close> continuous_at h'_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
261 |
then obtain r2 where r2:"r2>0" "\<forall>x\<in>ball z r2. h' x\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
262 |
using continuous_at_avoid[of z h' 0 ] unfolding ball_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
263 |
define r1 where "r1=min r2 r / 2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
264 |
have "0 < r1" "cball z r1 \<subseteq> ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
265 |
using \<open>r2>0\<close> \<open>r>0\<close> unfolding r1_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
266 |
moreover have "\<forall>x\<in>cball z r1. h' x \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
267 |
using r2 unfolding r1_def by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
268 |
ultimately show ?thesis using that by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
269 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
270 |
then have "P h' 0 h' r1" using \<open>h' holomorphic_on ball z r\<close> unfolding P_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
271 |
then have "P h 0 h' r1" unfolding P_def h'_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
272 |
then show ?thesis unfolding P_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
273 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
274 |
ultimately show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
275 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
276 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
277 |
have ?thesis when "\<exists>x. (f \<longlongrightarrow> x) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
278 |
apply (rule_tac imp_unique[unfolded P_def]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
279 |
using P_exist[OF that(1) f_iso non_zero] unfolding P_def . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
280 |
moreover have ?thesis when "is_pole f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
281 |
proof (rule imp_unique[unfolded P_def]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
282 |
obtain e where [simp]:"e>0" and e_holo:"f holomorphic_on ball z e - {z}" and e_nz: "\<forall>x\<in>ball z e-{z}. f x\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
283 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
284 |
have "\<forall>\<^sub>F z in at z. f z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
285 |
using \<open>is_pole f z\<close> filterlim_at_infinity_imp_eventually_ne unfolding is_pole_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
286 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
287 |
then obtain e1 where e1:"e1>0" "\<forall>x\<in>ball z e1-{z}. f x\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
288 |
using that eventually_at[of "\<lambda>x. f x\<noteq>0" z UNIV,simplified] by (auto simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
289 |
obtain e2 where e2:"e2>0" "f holomorphic_on ball z e2 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
290 |
using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
291 |
define e where "e=min e1 e2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
292 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
293 |
apply (rule that[of e]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
294 |
using e1 e2 unfolding e_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
295 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
296 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
297 |
define h where "h \<equiv> \<lambda>x. inverse (f x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
298 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
299 |
have "\<exists>n g r. P h n g r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
300 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
301 |
have "h \<midarrow>z\<rightarrow> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
302 |
using Lim_transform_within_open assms(2) h_def is_pole_tendsto that by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
303 |
moreover have "\<exists>\<^sub>Fw in (at z). h w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
304 |
using non_zero |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
305 |
apply (elim frequently_rev_mp) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
306 |
unfolding h_def eventually_at by (auto intro:exI[where x=1]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
307 |
moreover have "isolated_singularity_at h z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
308 |
unfolding isolated_singularity_at_def h_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
309 |
apply (rule exI[where x=e]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
310 |
using e_holo e_nz \<open>e>0\<close> by (metis open_ball analytic_on_open |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
311 |
holomorphic_on_inverse open_delete) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
312 |
ultimately show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
313 |
using P_exist[of h] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
314 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
315 |
then obtain n g r |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
316 |
where "0 < r" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
317 |
g_holo:"g holomorphic_on cball z r" and "g z\<noteq>0" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
318 |
g_fac:"(\<forall>w\<in>cball z r-{z}. h w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
319 |
unfolding P_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
320 |
have "P f (-n) (inverse o g) r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
321 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
322 |
have "f w = inverse (g w) * (w - z) powr of_int (- n)" when "w\<in>cball z r - {z}" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
323 |
using g_fac[rule_format,of w] that unfolding h_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
324 |
apply (auto simp add:powr_minus ) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
325 |
by (metis inverse_inverse_eq inverse_mult_distrib) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
326 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
327 |
unfolding P_def comp_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
328 |
using \<open>r>0\<close> g_holo g_fac \<open>g z\<noteq>0\<close> by (auto intro:holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
329 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
330 |
then show "\<exists>x g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z \<noteq> 0 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
331 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int x \<and> g w \<noteq> 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
332 |
unfolding P_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
333 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
334 |
ultimately show ?thesis using \<open>not_essential f z\<close> unfolding not_essential_def by presburger |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
335 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
336 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
337 |
lemma not_essential_transform: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
338 |
assumes "not_essential g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
339 |
assumes "\<forall>\<^sub>F w in (at z). g w = f w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
340 |
shows "not_essential f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
341 |
using assms unfolding not_essential_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
342 |
by (simp add: filterlim_cong is_pole_cong) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
343 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
344 |
lemma isolated_singularity_at_transform: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
345 |
assumes "isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
346 |
assumes "\<forall>\<^sub>F w in (at z). g w = f w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
347 |
shows "isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
348 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
349 |
obtain r1 where "r1>0" and r1:"g analytic_on ball z r1 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
350 |
using assms(1) unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
351 |
obtain r2 where "r2>0" and r2:" \<forall>x. x \<noteq> z \<and> dist x z < r2 \<longrightarrow> g x = f x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
352 |
using assms(2) unfolding eventually_at by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
353 |
define r3 where "r3=min r1 r2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
354 |
have "r3>0" unfolding r3_def using \<open>r1>0\<close> \<open>r2>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
355 |
moreover have "f analytic_on ball z r3 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
356 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
357 |
have "g holomorphic_on ball z r3 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
358 |
using r1 unfolding r3_def by (subst (asm) analytic_on_open,auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
359 |
then have "f holomorphic_on ball z r3 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
360 |
using r2 unfolding r3_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
361 |
by (auto simp add:dist_commute elim!:holomorphic_transform) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
362 |
then show ?thesis by (subst analytic_on_open,auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
363 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
364 |
ultimately show ?thesis unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
365 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
366 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
367 |
lemma not_essential_powr[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
368 |
assumes "LIM w (at z). f w :> (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
369 |
shows "not_essential (\<lambda>w. (f w) powr (of_int n)) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
370 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
371 |
define fp where "fp=(\<lambda>w. (f w) powr (of_int n))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
372 |
have ?thesis when "n>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
373 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
374 |
have "(\<lambda>w. (f w) ^ (nat n)) \<midarrow>z\<rightarrow> x ^ nat n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
375 |
using that assms unfolding filterlim_at by (auto intro!:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
376 |
then have "fp \<midarrow>z\<rightarrow> x ^ nat n" unfolding fp_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
377 |
apply (elim Lim_transform_within[where d=1],simp) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
378 |
by (metis less_le powr_0 powr_of_int that zero_less_nat_eq zero_power) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
379 |
then show ?thesis unfolding not_essential_def fp_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
380 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
381 |
moreover have ?thesis when "n=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
382 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
383 |
have "fp \<midarrow>z\<rightarrow> 1 " |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
384 |
apply (subst tendsto_cong[where g="\<lambda>_.1"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
385 |
using that filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
386 |
then show ?thesis unfolding fp_def not_essential_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
387 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
388 |
moreover have ?thesis when "n<0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
389 |
proof (cases "x=0") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
390 |
case True |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
391 |
have "LIM w (at z). inverse ((f w) ^ (nat (-n))) :> at_infinity" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
392 |
apply (subst filterlim_inverse_at_iff[symmetric],simp) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
393 |
apply (rule filterlim_atI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
394 |
subgoal using assms True that unfolding filterlim_at by (auto intro!:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
395 |
subgoal using filterlim_at_within_not_equal[OF assms,of 0] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
396 |
by (eventually_elim,insert that,auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
397 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
398 |
then have "LIM w (at z). fp w :> at_infinity" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
399 |
proof (elim filterlim_mono_eventually) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
400 |
show "\<forall>\<^sub>F x in at z. inverse (f x ^ nat (- n)) = fp x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
401 |
using filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
402 |
apply eventually_elim |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
403 |
using powr_of_int that by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
404 |
qed auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
405 |
then show ?thesis unfolding fp_def not_essential_def is_pole_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
406 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
407 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
408 |
let ?xx= "inverse (x ^ (nat (-n)))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
409 |
have "(\<lambda>w. inverse ((f w) ^ (nat (-n)))) \<midarrow>z\<rightarrow>?xx" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
410 |
using assms False unfolding filterlim_at by (auto intro!:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
411 |
then have "fp \<midarrow>z\<rightarrow>?xx" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
412 |
apply (elim Lim_transform_within[where d=1],simp) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
413 |
unfolding fp_def by (metis inverse_zero nat_mono_iff nat_zero_as_int neg_0_less_iff_less |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
414 |
not_le power_eq_0_iff powr_0 powr_of_int that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
415 |
then show ?thesis unfolding fp_def not_essential_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
416 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
417 |
ultimately show ?thesis by linarith |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
418 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
419 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
420 |
lemma isolated_singularity_at_powr[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
421 |
assumes "isolated_singularity_at f z" "\<forall>\<^sub>F w in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
422 |
shows "isolated_singularity_at (\<lambda>w. (f w) powr (of_int n)) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
423 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
424 |
obtain r1 where "r1>0" "f analytic_on ball z r1 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
425 |
using assms(1) unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
426 |
then have r1:"f holomorphic_on ball z r1 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
427 |
using analytic_on_open[of "ball z r1-{z}" f] by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
428 |
obtain r2 where "r2>0" and r2:"\<forall>w. w \<noteq> z \<and> dist w z < r2 \<longrightarrow> f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
429 |
using assms(2) unfolding eventually_at by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
430 |
define r3 where "r3=min r1 r2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
431 |
have "(\<lambda>w. (f w) powr of_int n) holomorphic_on ball z r3 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
432 |
apply (rule holomorphic_on_powr_of_int) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
433 |
subgoal unfolding r3_def using r1 by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
434 |
subgoal unfolding r3_def using r2 by (auto simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
435 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
436 |
moreover have "r3>0" unfolding r3_def using \<open>0 < r1\<close> \<open>0 < r2\<close> by linarith |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
437 |
ultimately show ?thesis unfolding isolated_singularity_at_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
438 |
apply (subst (asm) analytic_on_open[symmetric]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
439 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
440 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
441 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
442 |
lemma non_zero_neighbour: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
443 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
444 |
and f_ness:"not_essential f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
445 |
and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
446 |
shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
447 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
448 |
obtain fn fp fr where [simp]:"fp z \<noteq> 0" and "fr > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
449 |
and fr: "fp holomorphic_on cball z fr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
450 |
"\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
451 |
using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
452 |
have "f w \<noteq> 0" when " w \<noteq> z" "dist w z < fr" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
453 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
454 |
have "f w = fp w * (w - z) powr of_int fn" "fp w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
455 |
using fr(2)[rule_format, of w] using that by (auto simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
456 |
moreover have "(w - z) powr of_int fn \<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
457 |
unfolding powr_eq_0_iff using \<open>w\<noteq>z\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
458 |
ultimately show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
459 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
460 |
then show ?thesis using \<open>fr>0\<close> unfolding eventually_at by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
461 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
462 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
463 |
lemma non_zero_neighbour_pole: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
464 |
assumes "is_pole f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
465 |
shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
466 |
using assms filterlim_at_infinity_imp_eventually_ne[of f "at z" 0] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
467 |
unfolding is_pole_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
468 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
469 |
lemma non_zero_neighbour_alt: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
470 |
assumes holo: "f holomorphic_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
471 |
and "open S" "connected S" "z \<in> S" "\<beta> \<in> S" "f \<beta> \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
472 |
shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0 \<and> w\<in>S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
473 |
proof (cases "f z = 0") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
474 |
case True |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
475 |
from isolated_zeros[OF holo \<open>open S\<close> \<open>connected S\<close> \<open>z \<in> S\<close> True \<open>\<beta> \<in> S\<close> \<open>f \<beta> \<noteq> 0\<close>] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
476 |
obtain r where "0 < r" "ball z r \<subseteq> S" "\<forall>w \<in> ball z r - {z}.f w \<noteq> 0" by metis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
477 |
then show ?thesis unfolding eventually_at |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
478 |
apply (rule_tac x=r in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
479 |
by (auto simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
480 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
481 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
482 |
obtain r1 where r1:"r1>0" "\<forall>y. dist z y < r1 \<longrightarrow> f y \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
483 |
using continuous_at_avoid[of z f, OF _ False] assms(2,4) continuous_on_eq_continuous_at |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
484 |
holo holomorphic_on_imp_continuous_on by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
485 |
obtain r2 where r2:"r2>0" "ball z r2 \<subseteq> S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
486 |
using assms(2) assms(4) openE by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
487 |
show ?thesis unfolding eventually_at |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
488 |
apply (rule_tac x="min r1 r2" in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
489 |
using r1 r2 by (auto simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
490 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
491 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
492 |
lemma not_essential_times[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
493 |
assumes f_ness:"not_essential f z" and g_ness:"not_essential g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
494 |
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
495 |
shows "not_essential (\<lambda>w. f w * g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
496 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
497 |
define fg where "fg = (\<lambda>w. f w * g w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
498 |
have ?thesis when "\<not> ((\<exists>\<^sub>Fw in (at z). f w\<noteq>0) \<and> (\<exists>\<^sub>Fw in (at z). g w\<noteq>0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
499 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
500 |
have "\<forall>\<^sub>Fw in (at z). fg w=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
501 |
using that[unfolded frequently_def, simplified] unfolding fg_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
502 |
by (auto elim: eventually_rev_mp) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
503 |
from tendsto_cong[OF this] have "fg \<midarrow>z\<rightarrow>0" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
504 |
then show ?thesis unfolding not_essential_def fg_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
505 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
506 |
moreover have ?thesis when f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" and g_nconst:"\<exists>\<^sub>Fw in (at z). g w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
507 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
508 |
obtain fn fp fr where [simp]:"fp z \<noteq> 0" and "fr > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
509 |
and fr: "fp holomorphic_on cball z fr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
510 |
"\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
511 |
using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
512 |
obtain gn gp gr where [simp]:"gp z \<noteq> 0" and "gr > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
513 |
and gr: "gp holomorphic_on cball z gr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
514 |
"\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powr of_int gn \<and> gp w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
515 |
using holomorphic_factor_puncture[OF g_iso g_ness g_nconst,THEN ex1_implies_ex] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
516 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
517 |
define r1 where "r1=(min fr gr)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
518 |
have "r1>0" unfolding r1_def using \<open>fr>0\<close> \<open>gr>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
519 |
have fg_times:"fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" and fgp_nz:"fp w*gp w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
520 |
when "w\<in>ball z r1 - {z}" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
521 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
522 |
have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
523 |
using fr(2)[rule_format,of w] that unfolding r1_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
524 |
moreover have "g w = gp w * (w - z) powr of_int gn" "gp w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
525 |
using gr(2)[rule_format, of w] that unfolding r1_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
526 |
ultimately show "fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" "fp w*gp w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
527 |
unfolding fg_def by (auto simp add:powr_add) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
528 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
529 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
530 |
have [intro]: "fp \<midarrow>z\<rightarrow>fp z" "gp \<midarrow>z\<rightarrow>gp z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
531 |
using fr(1) \<open>fr>0\<close> gr(1) \<open>gr>0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
532 |
by (meson open_ball ball_subset_cball centre_in_ball |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
533 |
continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
534 |
holomorphic_on_subset)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
535 |
have ?thesis when "fn+gn>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
536 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
537 |
have "(\<lambda>w. (fp w * gp w) * (w - z) ^ (nat (fn+gn))) \<midarrow>z\<rightarrow>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
538 |
using that by (auto intro!:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
539 |
then have "fg \<midarrow>z\<rightarrow> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
540 |
apply (elim Lim_transform_within[OF _ \<open>r1>0\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
541 |
by (metis (no_types, hide_lams) Diff_iff cball_trivial dist_commute dist_self |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
542 |
eq_iff_diff_eq_0 fg_times less_le linorder_not_le mem_ball mem_cball powr_of_int |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
543 |
that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
544 |
then show ?thesis unfolding not_essential_def fg_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
545 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
546 |
moreover have ?thesis when "fn+gn=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
547 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
548 |
have "(\<lambda>w. fp w * gp w) \<midarrow>z\<rightarrow>fp z*gp z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
549 |
using that by (auto intro!:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
550 |
then have "fg \<midarrow>z\<rightarrow> fp z*gp z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
551 |
apply (elim Lim_transform_within[OF _ \<open>r1>0\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
552 |
apply (subst fg_times) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
553 |
by (auto simp add:dist_commute that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
554 |
then show ?thesis unfolding not_essential_def fg_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
555 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
556 |
moreover have ?thesis when "fn+gn<0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
557 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
558 |
have "LIM w (at z). fp w * gp w / (w-z)^nat (-(fn+gn)) :> at_infinity" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
559 |
apply (rule filterlim_divide_at_infinity) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
560 |
apply (insert that, auto intro!:tendsto_eq_intros filterlim_atI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
561 |
using eventually_at_topological by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
562 |
then have "is_pole fg z" unfolding is_pole_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
563 |
apply (elim filterlim_transform_within[OF _ _ \<open>r1>0\<close>],simp) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
564 |
apply (subst fg_times,simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
565 |
apply (subst powr_of_int) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
566 |
using that by (auto simp add:field_split_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
567 |
then show ?thesis unfolding not_essential_def fg_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
568 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
569 |
ultimately show ?thesis unfolding not_essential_def fg_def by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
570 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
571 |
ultimately show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
572 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
573 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
574 |
lemma not_essential_inverse[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
575 |
assumes f_ness:"not_essential f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
576 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
577 |
shows "not_essential (\<lambda>w. inverse (f w)) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
578 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
579 |
define vf where "vf = (\<lambda>w. inverse (f w))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
580 |
have ?thesis when "\<not>(\<exists>\<^sub>Fw in (at z). f w\<noteq>0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
581 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
582 |
have "\<forall>\<^sub>Fw in (at z). f w=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
583 |
using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
584 |
then have "\<forall>\<^sub>Fw in (at z). vf w=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
585 |
unfolding vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
586 |
from tendsto_cong[OF this] have "vf \<midarrow>z\<rightarrow>0" unfolding vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
587 |
then show ?thesis unfolding not_essential_def vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
588 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
589 |
moreover have ?thesis when "is_pole f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
590 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
591 |
have "vf \<midarrow>z\<rightarrow>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
592 |
using that filterlim_at filterlim_inverse_at_iff unfolding is_pole_def vf_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
593 |
then show ?thesis unfolding not_essential_def vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
594 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
595 |
moreover have ?thesis when "\<exists>x. f\<midarrow>z\<rightarrow>x " and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
596 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
597 |
from that obtain fz where fz:"f\<midarrow>z\<rightarrow>fz" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
598 |
have ?thesis when "fz=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
599 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
600 |
have "(\<lambda>w. inverse (vf w)) \<midarrow>z\<rightarrow>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
601 |
using fz that unfolding vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
602 |
moreover have "\<forall>\<^sub>F w in at z. inverse (vf w) \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
603 |
using non_zero_neighbour[OF f_iso f_ness f_nconst] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
604 |
unfolding vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
605 |
ultimately have "is_pole vf z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
606 |
using filterlim_inverse_at_iff[of vf "at z"] unfolding filterlim_at is_pole_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
607 |
then show ?thesis unfolding not_essential_def vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
608 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
609 |
moreover have ?thesis when "fz\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
610 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
611 |
have "vf \<midarrow>z\<rightarrow>inverse fz" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
612 |
using fz that unfolding vf_def by (auto intro:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
613 |
then show ?thesis unfolding not_essential_def vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
614 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
615 |
ultimately show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
616 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
617 |
ultimately show ?thesis using f_ness unfolding not_essential_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
618 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
619 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
620 |
lemma isolated_singularity_at_inverse[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
621 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
622 |
and f_ness:"not_essential f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
623 |
shows "isolated_singularity_at (\<lambda>w. inverse (f w)) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
624 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
625 |
define vf where "vf = (\<lambda>w. inverse (f w))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
626 |
have ?thesis when "\<not>(\<exists>\<^sub>Fw in (at z). f w\<noteq>0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
627 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
628 |
have "\<forall>\<^sub>Fw in (at z). f w=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
629 |
using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
630 |
then have "\<forall>\<^sub>Fw in (at z). vf w=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
631 |
unfolding vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
632 |
then obtain d1 where "d1>0" and d1:"\<forall>x. x \<noteq> z \<and> dist x z < d1 \<longrightarrow> vf x = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
633 |
unfolding eventually_at by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
634 |
then have "vf holomorphic_on ball z d1-{z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
635 |
apply (rule_tac holomorphic_transform[of "\<lambda>_. 0"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
636 |
by (auto simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
637 |
then have "vf analytic_on ball z d1 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
638 |
by (simp add: analytic_on_open open_delete) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
639 |
then show ?thesis using \<open>d1>0\<close> unfolding isolated_singularity_at_def vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
640 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
641 |
moreover have ?thesis when f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
642 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
643 |
have "\<forall>\<^sub>F w in at z. f w \<noteq> 0" using non_zero_neighbour[OF f_iso f_ness f_nconst] . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
644 |
then obtain d1 where d1:"d1>0" "\<forall>x. x \<noteq> z \<and> dist x z < d1 \<longrightarrow> f x \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
645 |
unfolding eventually_at by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
646 |
obtain d2 where "d2>0" and d2:"f analytic_on ball z d2 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
647 |
using f_iso unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
648 |
define d3 where "d3=min d1 d2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
649 |
have "d3>0" unfolding d3_def using \<open>d1>0\<close> \<open>d2>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
650 |
moreover have "vf analytic_on ball z d3 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
651 |
unfolding vf_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
652 |
apply (rule analytic_on_inverse) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
653 |
subgoal using d2 unfolding d3_def by (elim analytic_on_subset) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
654 |
subgoal for w using d1 unfolding d3_def by (auto simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
655 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
656 |
ultimately show ?thesis unfolding isolated_singularity_at_def vf_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
657 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
658 |
ultimately show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
659 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
660 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
661 |
lemma not_essential_divide[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
662 |
assumes f_ness:"not_essential f z" and g_ness:"not_essential g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
663 |
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
664 |
shows "not_essential (\<lambda>w. f w / g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
665 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
666 |
have "not_essential (\<lambda>w. f w * inverse (g w)) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
667 |
apply (rule not_essential_times[where g="\<lambda>w. inverse (g w)"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
668 |
using assms by (auto intro: isolated_singularity_at_inverse not_essential_inverse) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
669 |
then show ?thesis by (simp add:field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
670 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
671 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
672 |
lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
673 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
674 |
and g_iso:"isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
675 |
shows isolated_singularity_at_times[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
676 |
"isolated_singularity_at (\<lambda>w. f w * g w) z" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
677 |
isolated_singularity_at_add[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
678 |
"isolated_singularity_at (\<lambda>w. f w + g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
679 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
680 |
obtain d1 d2 where "d1>0" "d2>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
681 |
and d1:"f analytic_on ball z d1 - {z}" and d2:"g analytic_on ball z d2 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
682 |
using f_iso g_iso unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
683 |
define d3 where "d3=min d1 d2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
684 |
have "d3>0" unfolding d3_def using \<open>d1>0\<close> \<open>d2>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
685 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
686 |
have "(\<lambda>w. f w * g w) analytic_on ball z d3 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
687 |
apply (rule analytic_on_mult) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
688 |
using d1 d2 unfolding d3_def by (auto elim:analytic_on_subset) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
689 |
then show "isolated_singularity_at (\<lambda>w. f w * g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
690 |
using \<open>d3>0\<close> unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
691 |
have "(\<lambda>w. f w + g w) analytic_on ball z d3 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
692 |
apply (rule analytic_on_add) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
693 |
using d1 d2 unfolding d3_def by (auto elim:analytic_on_subset) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
694 |
then show "isolated_singularity_at (\<lambda>w. f w + g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
695 |
using \<open>d3>0\<close> unfolding isolated_singularity_at_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
696 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
697 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
698 |
lemma isolated_singularity_at_uminus[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
699 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
700 |
shows "isolated_singularity_at (\<lambda>w. - f w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
701 |
using assms unfolding isolated_singularity_at_def using analytic_on_neg by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
702 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
703 |
lemma isolated_singularity_at_id[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
704 |
"isolated_singularity_at (\<lambda>w. w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
705 |
unfolding isolated_singularity_at_def by (simp add: gt_ex) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
706 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
707 |
lemma isolated_singularity_at_minus[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
708 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
709 |
and g_iso:"isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
710 |
shows "isolated_singularity_at (\<lambda>w. f w - g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
711 |
using isolated_singularity_at_uminus[THEN isolated_singularity_at_add[OF f_iso,of "\<lambda>w. - g w"] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
712 |
,OF g_iso] by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
713 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
714 |
lemma isolated_singularity_at_divide[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
715 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
716 |
and g_iso:"isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
717 |
and g_ness:"not_essential g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
718 |
shows "isolated_singularity_at (\<lambda>w. f w / g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
719 |
using isolated_singularity_at_inverse[THEN isolated_singularity_at_times[OF f_iso, |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
720 |
of "\<lambda>w. inverse (g w)"],OF g_iso g_ness] by (simp add:field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
721 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
722 |
lemma isolated_singularity_at_const[singularity_intros]: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
723 |
"isolated_singularity_at (\<lambda>w. c) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
724 |
unfolding isolated_singularity_at_def by (simp add: gt_ex) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
725 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
726 |
lemma isolated_singularity_at_holomorphic: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
727 |
assumes "f holomorphic_on s-{z}" "open s" "z\<in>s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
728 |
shows "isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
729 |
using assms unfolding isolated_singularity_at_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
730 |
by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
731 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
732 |
subsubsection \<open>The order of non-essential singularities (i.e. removable singularities or poles)\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
733 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
734 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
735 |
definition\<^marker>\<open>tag important\<close> zorder :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> int" where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
736 |
"zorder f z = (THE n. (\<exists>h r. r>0 \<and> h holomorphic_on cball z r \<and> h z\<noteq>0 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
737 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w-z) powr (of_int n) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
738 |
\<and> h w \<noteq>0)))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
739 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
740 |
definition\<^marker>\<open>tag important\<close> zor_poly |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
741 |
::"[complex \<Rightarrow> complex, complex] \<Rightarrow> complex \<Rightarrow> complex" where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
742 |
"zor_poly f z = (SOME h. \<exists>r. r > 0 \<and> h holomorphic_on cball z r \<and> h z \<noteq> 0 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
743 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w - z) powr (zorder f z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
744 |
\<and> h w \<noteq>0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
745 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
746 |
lemma zorder_exist: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
747 |
fixes f::"complex \<Rightarrow> complex" and z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
748 |
defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
749 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
750 |
and f_ness:"not_essential f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
751 |
and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
752 |
shows "g z\<noteq>0 \<and> (\<exists>r. r>0 \<and> g holomorphic_on cball z r |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
753 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr n \<and> g w \<noteq>0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
754 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
755 |
define P where "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
756 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
757 |
have "\<exists>!n. \<exists>g r. P n g r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
758 |
using holomorphic_factor_puncture[OF assms(3-)] unfolding P_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
759 |
then have "\<exists>g r. P n g r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
760 |
unfolding n_def P_def zorder_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
761 |
by (drule_tac theI',argo) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
762 |
then have "\<exists>r. P n g r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
763 |
unfolding P_def zor_poly_def g_def n_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
764 |
by (drule_tac someI_ex,argo) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
765 |
then obtain r1 where "P n g r1" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
766 |
then show ?thesis unfolding P_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
767 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
768 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
769 |
lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
770 |
fixes f::"complex \<Rightarrow> complex" and z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
771 |
assumes f_iso:"isolated_singularity_at f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
772 |
and f_ness:"not_essential f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
773 |
and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
774 |
shows zorder_inverse: "zorder (\<lambda>w. inverse (f w)) z = - zorder f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
775 |
and zor_poly_inverse: "\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. inverse (f w)) z w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
776 |
= inverse (zor_poly f z w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
777 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
778 |
define vf where "vf = (\<lambda>w. inverse (f w))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
779 |
define fn vfn where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
780 |
"fn = zorder f z" and "vfn = zorder vf z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
781 |
define fp vfp where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
782 |
"fp = zor_poly f z" and "vfp = zor_poly vf z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
783 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
784 |
obtain fr where [simp]:"fp z \<noteq> 0" and "fr > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
785 |
and fr: "fp holomorphic_on cball z fr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
786 |
"\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
787 |
using zorder_exist[OF f_iso f_ness f_nconst,folded fn_def fp_def] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
788 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
789 |
have fr_inverse: "vf w = (inverse (fp w)) * (w - z) powr (of_int (-fn))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
790 |
and fr_nz: "inverse (fp w)\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
791 |
when "w\<in>ball z fr - {z}" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
792 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
793 |
have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
794 |
using fr(2)[rule_format,of w] that by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
795 |
then show "vf w = (inverse (fp w)) * (w - z) powr (of_int (-fn))" "inverse (fp w)\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
796 |
unfolding vf_def by (auto simp add:powr_minus) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
797 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
798 |
obtain vfr where [simp]:"vfp z \<noteq> 0" and "vfr>0" and vfr:"vfp holomorphic_on cball z vfr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
799 |
"(\<forall>w\<in>cball z vfr - {z}. vf w = vfp w * (w - z) powr of_int vfn \<and> vfp w \<noteq> 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
800 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
801 |
have "isolated_singularity_at vf z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
802 |
using isolated_singularity_at_inverse[OF f_iso f_ness] unfolding vf_def . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
803 |
moreover have "not_essential vf z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
804 |
using not_essential_inverse[OF f_ness f_iso] unfolding vf_def . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
805 |
moreover have "\<exists>\<^sub>F w in at z. vf w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
806 |
using f_nconst unfolding vf_def by (auto elim:frequently_elim1) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
807 |
ultimately show ?thesis using zorder_exist[of vf z, folded vfn_def vfp_def] that by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
808 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
809 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
810 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
811 |
define r1 where "r1 = min fr vfr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
812 |
have "r1>0" using \<open>fr>0\<close> \<open>vfr>0\<close> unfolding r1_def by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
813 |
show "vfn = - fn" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
814 |
apply (rule holomorphic_factor_unique[of r1 vfp z "\<lambda>w. inverse (fp w)" vf]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
815 |
subgoal using \<open>r1>0\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
816 |
subgoal by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
817 |
subgoal by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
818 |
subgoal |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
819 |
proof (rule ballI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
820 |
fix w assume "w \<in> ball z r1 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
821 |
then have "w \<in> ball z fr - {z}" "w \<in> cball z vfr - {z}" unfolding r1_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
822 |
from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
823 |
show "vf w = vfp w * (w - z) powr of_int vfn \<and> vfp w \<noteq> 0 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
824 |
\<and> vf w = inverse (fp w) * (w - z) powr of_int (- fn) \<and> inverse (fp w) \<noteq> 0" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
825 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
826 |
subgoal using vfr(1) unfolding r1_def by (auto intro!:holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
827 |
subgoal using fr unfolding r1_def by (auto intro!:holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
828 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
829 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
830 |
have "vfp w = inverse (fp w)" when "w\<in>ball z r1-{z}" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
831 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
832 |
have "w \<in> ball z fr - {z}" "w \<in> cball z vfr - {z}" "w\<noteq>z" using that unfolding r1_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
833 |
from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)] \<open>vfn = - fn\<close> \<open>w\<noteq>z\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
834 |
show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
835 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
836 |
then show "\<forall>\<^sub>Fw in (at z). vfp w = inverse (fp w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
837 |
unfolding eventually_at using \<open>r1>0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
838 |
apply (rule_tac x=r1 in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
839 |
by (auto simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
840 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
841 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
842 |
lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
843 |
fixes f g::"complex \<Rightarrow> complex" and z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
844 |
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
845 |
and f_ness:"not_essential f z" and g_ness:"not_essential g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
846 |
and fg_nconst: "\<exists>\<^sub>Fw in (at z). f w * g w\<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
847 |
shows zorder_times:"zorder (\<lambda>w. f w * g w) z = zorder f z + zorder g z" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
848 |
zor_poly_times:"\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w * g w) z w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
849 |
= zor_poly f z w *zor_poly g z w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
850 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
851 |
define fg where "fg = (\<lambda>w. f w * g w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
852 |
define fn gn fgn where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
853 |
"fn = zorder f z" and "gn = zorder g z" and "fgn = zorder fg z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
854 |
define fp gp fgp where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
855 |
"fp = zor_poly f z" and "gp = zor_poly g z" and "fgp = zor_poly fg z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
856 |
have f_nconst:"\<exists>\<^sub>Fw in (at z). f w \<noteq> 0" and g_nconst:"\<exists>\<^sub>Fw in (at z).g w\<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
857 |
using fg_nconst by (auto elim!:frequently_elim1) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
858 |
obtain fr where [simp]:"fp z \<noteq> 0" and "fr > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
859 |
and fr: "fp holomorphic_on cball z fr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
860 |
"\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
861 |
using zorder_exist[OF f_iso f_ness f_nconst,folded fp_def fn_def] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
862 |
obtain gr where [simp]:"gp z \<noteq> 0" and "gr > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
863 |
and gr: "gp holomorphic_on cball z gr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
864 |
"\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powr of_int gn \<and> gp w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
865 |
using zorder_exist[OF g_iso g_ness g_nconst,folded gn_def gp_def] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
866 |
define r1 where "r1=min fr gr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
867 |
have "r1>0" unfolding r1_def using \<open>fr>0\<close> \<open>gr>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
868 |
have fg_times:"fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" and fgp_nz:"fp w*gp w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
869 |
when "w\<in>ball z r1 - {z}" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
870 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
871 |
have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
872 |
using fr(2)[rule_format,of w] that unfolding r1_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
873 |
moreover have "g w = gp w * (w - z) powr of_int gn" "gp w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
874 |
using gr(2)[rule_format, of w] that unfolding r1_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
875 |
ultimately show "fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" "fp w*gp w\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
876 |
unfolding fg_def by (auto simp add:powr_add) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
877 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
878 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
879 |
obtain fgr where [simp]:"fgp z \<noteq> 0" and "fgr > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
880 |
and fgr: "fgp holomorphic_on cball z fgr" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
881 |
"\<forall>w\<in>cball z fgr - {z}. fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
882 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
883 |
have "fgp z \<noteq> 0 \<and> (\<exists>r>0. fgp holomorphic_on cball z r |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
884 |
\<and> (\<forall>w\<in>cball z r - {z}. fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
885 |
apply (rule zorder_exist[of fg z, folded fgn_def fgp_def]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
886 |
subgoal unfolding fg_def using isolated_singularity_at_times[OF f_iso g_iso] . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
887 |
subgoal unfolding fg_def using not_essential_times[OF f_ness g_ness f_iso g_iso] . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
888 |
subgoal unfolding fg_def using fg_nconst . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
889 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
890 |
then show ?thesis using that by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
891 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
892 |
define r2 where "r2 = min fgr r1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
893 |
have "r2>0" using \<open>r1>0\<close> \<open>fgr>0\<close> unfolding r2_def by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
894 |
show "fgn = fn + gn " |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
895 |
apply (rule holomorphic_factor_unique[of r2 fgp z "\<lambda>w. fp w * gp w" fg]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
896 |
subgoal using \<open>r2>0\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
897 |
subgoal by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
898 |
subgoal by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
899 |
subgoal |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
900 |
proof (rule ballI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
901 |
fix w assume "w \<in> ball z r2 - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
902 |
then have "w \<in> ball z r1 - {z}" "w \<in> cball z fgr - {z}" unfolding r2_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
903 |
from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
904 |
show "fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
905 |
\<and> fg w = fp w * gp w * (w - z) powr of_int (fn + gn) \<and> fp w * gp w \<noteq> 0" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
906 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
907 |
subgoal using fgr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
908 |
subgoal using fr(1) gr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
909 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
910 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
911 |
have "fgp w = fp w *gp w" when "w\<in>ball z r2-{z}" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
912 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
913 |
have "w \<in> ball z r1 - {z}" "w \<in> cball z fgr - {z}" "w\<noteq>z" using that unfolding r2_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
914 |
from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)] \<open>fgn = fn + gn\<close> \<open>w\<noteq>z\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
915 |
show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
916 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
917 |
then show "\<forall>\<^sub>Fw in (at z). fgp w = fp w * gp w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
918 |
using \<open>r2>0\<close> unfolding eventually_at by (auto simp add:dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
919 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
920 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
921 |
lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
922 |
fixes f g::"complex \<Rightarrow> complex" and z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
923 |
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
924 |
and f_ness:"not_essential f z" and g_ness:"not_essential g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
925 |
and fg_nconst: "\<exists>\<^sub>Fw in (at z). f w * g w\<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
926 |
shows zorder_divide:"zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
927 |
zor_poly_divide:"\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w / g w) z w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
928 |
= zor_poly f z w / zor_poly g z w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
929 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
930 |
have f_nconst:"\<exists>\<^sub>Fw in (at z). f w \<noteq> 0" and g_nconst:"\<exists>\<^sub>Fw in (at z).g w\<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
931 |
using fg_nconst by (auto elim!:frequently_elim1) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
932 |
define vg where "vg=(\<lambda>w. inverse (g w))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
933 |
have "zorder (\<lambda>w. f w * vg w) z = zorder f z + zorder vg z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
934 |
apply (rule zorder_times[OF f_iso _ f_ness,of vg]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
935 |
subgoal unfolding vg_def using isolated_singularity_at_inverse[OF g_iso g_ness] . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
936 |
subgoal unfolding vg_def using not_essential_inverse[OF g_ness g_iso] . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
937 |
subgoal unfolding vg_def using fg_nconst by (auto elim!:frequently_elim1) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
938 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
939 |
then show "zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
940 |
using zorder_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
941 |
by (auto simp add:field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
942 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
943 |
have "\<forall>\<^sub>F w in at z. zor_poly (\<lambda>w. f w * vg w) z w = zor_poly f z w * zor_poly vg z w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
944 |
apply (rule zor_poly_times[OF f_iso _ f_ness,of vg]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
945 |
subgoal unfolding vg_def using isolated_singularity_at_inverse[OF g_iso g_ness] . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
946 |
subgoal unfolding vg_def using not_essential_inverse[OF g_ness g_iso] . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
947 |
subgoal unfolding vg_def using fg_nconst by (auto elim!:frequently_elim1) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
948 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
949 |
then show "\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w / g w) z w = zor_poly f z w / zor_poly g z w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
950 |
using zor_poly_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
951 |
apply eventually_elim |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
952 |
by (auto simp add:field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
953 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
954 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
955 |
lemma zorder_exist_zero: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
956 |
fixes f::"complex \<Rightarrow> complex" and z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
957 |
defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
958 |
assumes holo: "f holomorphic_on s" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
959 |
"open s" "connected s" "z\<in>s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
960 |
and non_const: "\<exists>w\<in>s. f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
961 |
shows "(if f z=0 then n > 0 else n=0) \<and> (\<exists>r. r>0 \<and> cball z r \<subseteq> s \<and> g holomorphic_on cball z r |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
962 |
\<and> (\<forall>w\<in>cball z r. f w = g w * (w-z) ^ nat n \<and> g w \<noteq>0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
963 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
964 |
obtain r where "g z \<noteq> 0" and r: "r>0" "cball z r \<subseteq> s" "g holomorphic_on cball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
965 |
"(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
966 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
967 |
have "g z \<noteq> 0 \<and> (\<exists>r>0. g holomorphic_on cball z r |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
968 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
969 |
proof (rule zorder_exist[of f z,folded g_def n_def]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
970 |
show "isolated_singularity_at f z" unfolding isolated_singularity_at_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
971 |
using holo assms(4,6) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
972 |
by (meson Diff_subset open_ball analytic_on_holomorphic holomorphic_on_subset openE) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
973 |
show "not_essential f z" unfolding not_essential_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
974 |
using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
975 |
by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
976 |
have "\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w\<in>s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
977 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
978 |
obtain w where "w\<in>s" "f w\<noteq>0" using non_const by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
979 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
980 |
by (rule non_zero_neighbour_alt[OF holo \<open>open s\<close> \<open>connected s\<close> \<open>z\<in>s\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
981 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
982 |
then show "\<exists>\<^sub>F w in at z. f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
983 |
apply (elim eventually_frequentlyE) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
984 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
985 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
986 |
then obtain r1 where "g z \<noteq> 0" "r1>0" and r1:"g holomorphic_on cball z r1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
987 |
"(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
988 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
989 |
obtain r2 where r2: "r2>0" "cball z r2 \<subseteq> s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
990 |
using assms(4,6) open_contains_cball_eq by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
991 |
define r3 where "r3=min r1 r2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
992 |
have "r3>0" "cball z r3 \<subseteq> s" using \<open>r1>0\<close> r2 unfolding r3_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
993 |
moreover have "g holomorphic_on cball z r3" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
994 |
using r1(1) unfolding r3_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
995 |
moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
996 |
using r1(2) unfolding r3_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
997 |
ultimately show ?thesis using that[of r3] \<open>g z\<noteq>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
998 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
999 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1000 |
have if_0:"if f z=0 then n > 0 else n=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1001 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1002 |
have "f\<midarrow> z \<rightarrow> f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1003 |
by (metis assms(4,6,7) at_within_open continuous_on holo holomorphic_on_imp_continuous_on) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1004 |
then have "(\<lambda>w. g w * (w - z) powr of_int n) \<midarrow>z\<rightarrow> f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1005 |
apply (elim Lim_transform_within_open[where s="ball z r"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1006 |
using r by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1007 |
moreover have "g \<midarrow>z\<rightarrow>g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1008 |
by (metis (mono_tags, lifting) open_ball at_within_open_subset |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1009 |
ball_subset_cball centre_in_ball continuous_on holomorphic_on_imp_continuous_on r(1,3) subsetCE) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1010 |
ultimately have "(\<lambda>w. (g w * (w - z) powr of_int n) / g w) \<midarrow>z\<rightarrow> f z/g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1011 |
apply (rule_tac tendsto_divide) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1012 |
using \<open>g z\<noteq>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1013 |
then have powr_tendsto:"(\<lambda>w. (w - z) powr of_int n) \<midarrow>z\<rightarrow> f z/g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1014 |
apply (elim Lim_transform_within_open[where s="ball z r"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1015 |
using r by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1016 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1017 |
have ?thesis when "n\<ge>0" "f z=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1018 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1019 |
have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> f z/g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1020 |
using powr_tendsto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1021 |
apply (elim Lim_transform_within[where d=r]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1022 |
by (auto simp add: powr_of_int \<open>n\<ge>0\<close> \<open>r>0\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1023 |
then have *:"(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 0" using \<open>f z=0\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1024 |
moreover have False when "n=0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1025 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1026 |
have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1027 |
using \<open>n=0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1028 |
then show False using * using LIM_unique zero_neq_one by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1029 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1030 |
ultimately show ?thesis using that by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1031 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1032 |
moreover have ?thesis when "n\<ge>0" "f z\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1033 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1034 |
have False when "n>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1035 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1036 |
have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> f z/g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1037 |
using powr_tendsto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1038 |
apply (elim Lim_transform_within[where d=r]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1039 |
by (auto simp add: powr_of_int \<open>n\<ge>0\<close> \<open>r>0\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1040 |
moreover have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1041 |
using \<open>n>0\<close> by (auto intro!:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1042 |
ultimately show False using \<open>f z\<noteq>0\<close> \<open>g z\<noteq>0\<close> using LIM_unique divide_eq_0_iff by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1043 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1044 |
then show ?thesis using that by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1045 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1046 |
moreover have False when "n<0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1047 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1048 |
have "(\<lambda>w. inverse ((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> f z/g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1049 |
"(\<lambda>w.((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1050 |
subgoal using powr_tendsto powr_of_int that |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1051 |
by (elim Lim_transform_within_open[where s=UNIV],auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1052 |
subgoal using that by (auto intro!:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1053 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1054 |
from tendsto_mult[OF this,simplified] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1055 |
have "(\<lambda>x. inverse ((x - z) ^ nat (- n)) * (x - z) ^ nat (- n)) \<midarrow>z\<rightarrow> 0" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1056 |
then have "(\<lambda>x. 1::complex) \<midarrow>z\<rightarrow> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1057 |
by (elim Lim_transform_within_open[where s=UNIV],auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1058 |
then show False using LIM_const_eq by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1059 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1060 |
ultimately show ?thesis by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1061 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1062 |
moreover have "f w = g w * (w-z) ^ nat n \<and> g w \<noteq>0" when "w\<in>cball z r" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1063 |
proof (cases "w=z") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1064 |
case True |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1065 |
then have "f \<midarrow>z\<rightarrow>f w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1066 |
using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1067 |
then have "(\<lambda>w. g w * (w-z) ^ nat n) \<midarrow>z\<rightarrow>f w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1068 |
proof (elim Lim_transform_within[OF _ \<open>r>0\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1069 |
fix x assume "0 < dist x z" "dist x z < r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1070 |
then have "x \<in> cball z r - {z}" "x\<noteq>z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1071 |
unfolding cball_def by (auto simp add: dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1072 |
then have "f x = g x * (x - z) powr of_int n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1073 |
using r(4)[rule_format,of x] by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1074 |
also have "... = g x * (x - z) ^ nat n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1075 |
apply (subst powr_of_int) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1076 |
using if_0 \<open>x\<noteq>z\<close> by (auto split:if_splits) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1077 |
finally show "f x = g x * (x - z) ^ nat n" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1078 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1079 |
moreover have "(\<lambda>w. g w * (w-z) ^ nat n) \<midarrow>z\<rightarrow> g w * (w-z) ^ nat n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1080 |
using True apply (auto intro!:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1081 |
by (metis open_ball at_within_open_subset ball_subset_cball centre_in_ball |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1082 |
continuous_on holomorphic_on_imp_continuous_on r(1) r(3) that) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1083 |
ultimately have "f w = g w * (w-z) ^ nat n" using LIM_unique by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1084 |
then show ?thesis using \<open>g z\<noteq>0\<close> True by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1085 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1086 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1087 |
then have "f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1088 |
using r(4) that by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1089 |
then show ?thesis using False if_0 powr_of_int by (auto split:if_splits) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1090 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1091 |
ultimately show ?thesis using r by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1092 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1093 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1094 |
lemma zorder_exist_pole: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1095 |
fixes f::"complex \<Rightarrow> complex" and z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1096 |
defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1097 |
assumes holo: "f holomorphic_on s-{z}" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1098 |
"open s" "z\<in>s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1099 |
and "is_pole f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1100 |
shows "n < 0 \<and> g z\<noteq>0 \<and> (\<exists>r. r>0 \<and> cball z r \<subseteq> s \<and> g holomorphic_on cball z r |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1101 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w / (w-z) ^ nat (- n) \<and> g w \<noteq>0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1102 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1103 |
obtain r where "g z \<noteq> 0" and r: "r>0" "cball z r \<subseteq> s" "g holomorphic_on cball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1104 |
"(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1105 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1106 |
have "g z \<noteq> 0 \<and> (\<exists>r>0. g holomorphic_on cball z r |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1107 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1108 |
proof (rule zorder_exist[of f z,folded g_def n_def]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1109 |
show "isolated_singularity_at f z" unfolding isolated_singularity_at_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1110 |
using holo assms(4,5) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1111 |
by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1112 |
show "not_essential f z" unfolding not_essential_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1113 |
using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1114 |
by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1115 |
from non_zero_neighbour_pole[OF \<open>is_pole f z\<close>] show "\<exists>\<^sub>F w in at z. f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1116 |
apply (elim eventually_frequentlyE) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1117 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1118 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1119 |
then obtain r1 where "g z \<noteq> 0" "r1>0" and r1:"g holomorphic_on cball z r1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1120 |
"(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1121 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1122 |
obtain r2 where r2: "r2>0" "cball z r2 \<subseteq> s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1123 |
using assms(4,5) open_contains_cball_eq by metis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1124 |
define r3 where "r3=min r1 r2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1125 |
have "r3>0" "cball z r3 \<subseteq> s" using \<open>r1>0\<close> r2 unfolding r3_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1126 |
moreover have "g holomorphic_on cball z r3" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1127 |
using r1(1) unfolding r3_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1128 |
moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1129 |
using r1(2) unfolding r3_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1130 |
ultimately show ?thesis using that[of r3] \<open>g z\<noteq>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1131 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1132 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1133 |
have "n<0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1134 |
proof (rule ccontr) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1135 |
assume " \<not> n < 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1136 |
define c where "c=(if n=0 then g z else 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1137 |
have [simp]:"g \<midarrow>z\<rightarrow> g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1138 |
by (metis open_ball at_within_open ball_subset_cball centre_in_ball |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1139 |
continuous_on holomorphic_on_imp_continuous_on holomorphic_on_subset r(1) r(3) ) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1140 |
have "\<forall>\<^sub>F x in at z. f x = g x * (x - z) ^ nat n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1141 |
unfolding eventually_at_topological |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1142 |
apply (rule_tac exI[where x="ball z r"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1143 |
using r powr_of_int \<open>\<not> n < 0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1144 |
moreover have "(\<lambda>x. g x * (x - z) ^ nat n) \<midarrow>z\<rightarrow>c" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1145 |
proof (cases "n=0") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1146 |
case True |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1147 |
then show ?thesis unfolding c_def by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1148 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1149 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1150 |
then have "(\<lambda>x. (x - z) ^ nat n) \<midarrow>z\<rightarrow> 0" using \<open>\<not> n < 0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1151 |
by (auto intro!:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1152 |
from tendsto_mult[OF _ this,of g "g z",simplified] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1153 |
show ?thesis unfolding c_def using False by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1154 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1155 |
ultimately have "f \<midarrow>z\<rightarrow>c" using tendsto_cong by fast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1156 |
then show False using \<open>is_pole f z\<close> at_neq_bot not_tendsto_and_filterlim_at_infinity |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1157 |
unfolding is_pole_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1158 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1159 |
moreover have "\<forall>w\<in>cball z r - {z}. f w = g w / (w-z) ^ nat (- n) \<and> g w \<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1160 |
using r(4) \<open>n<0\<close> powr_of_int |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1161 |
by (metis Diff_iff divide_inverse eq_iff_diff_eq_0 insert_iff linorder_not_le) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1162 |
ultimately show ?thesis using r(1-3) \<open>g z\<noteq>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1163 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1164 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1165 |
lemma zorder_eqI: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1166 |
assumes "open s" "z \<in> s" "g holomorphic_on s" "g z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1167 |
assumes fg_eq:"\<And>w. \<lbrakk>w \<in> s;w\<noteq>z\<rbrakk> \<Longrightarrow> f w = g w * (w - z) powr n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1168 |
shows "zorder f z = n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1169 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1170 |
have "continuous_on s g" by (rule holomorphic_on_imp_continuous_on) fact |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1171 |
moreover have "open (-{0::complex})" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1172 |
ultimately have "open ((g -` (-{0})) \<inter> s)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1173 |
unfolding continuous_on_open_vimage[OF \<open>open s\<close>] by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1174 |
moreover from assms have "z \<in> (g -` (-{0})) \<inter> s" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1175 |
ultimately obtain r where r: "r > 0" "cball z r \<subseteq> s \<inter> (g -` (-{0}))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1176 |
unfolding open_contains_cball by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1177 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1178 |
let ?gg= "(\<lambda>w. g w * (w - z) powr n)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1179 |
define P where "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1180 |
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1181 |
have "P n g r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1182 |
unfolding P_def using r assms(3,4,5) by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1183 |
then have "\<exists>g r. P n g r" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1184 |
moreover have unique: "\<exists>!n. \<exists>g r. P n g r" unfolding P_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1185 |
proof (rule holomorphic_factor_puncture) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1186 |
have "ball z r-{z} \<subseteq> s" using r using ball_subset_cball by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1187 |
then have "?gg holomorphic_on ball z r-{z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1188 |
using \<open>g holomorphic_on s\<close> r by (auto intro!: holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1189 |
then have "f holomorphic_on ball z r - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1190 |
apply (elim holomorphic_transform) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1191 |
using fg_eq \<open>ball z r-{z} \<subseteq> s\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1192 |
then show "isolated_singularity_at f z" unfolding isolated_singularity_at_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1193 |
using analytic_on_open open_delete r(1) by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1194 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1195 |
have "not_essential ?gg z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1196 |
proof (intro singularity_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1197 |
show "not_essential g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1198 |
by (meson \<open>continuous_on s g\<close> assms(1) assms(2) continuous_on_eq_continuous_at |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1199 |
isCont_def not_essential_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1200 |
show " \<forall>\<^sub>F w in at z. w - z \<noteq> 0" by (simp add: eventually_at_filter) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1201 |
then show "LIM w at z. w - z :> at 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1202 |
unfolding filterlim_at by (auto intro:tendsto_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1203 |
show "isolated_singularity_at g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1204 |
by (meson Diff_subset open_ball analytic_on_holomorphic |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1205 |
assms(1,2,3) holomorphic_on_subset isolated_singularity_at_def openE) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1206 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1207 |
then show "not_essential f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1208 |
apply (elim not_essential_transform) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1209 |
unfolding eventually_at using assms(1,2) assms(5)[symmetric] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1210 |
by (metis dist_commute mem_ball openE subsetCE) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1211 |
show "\<exists>\<^sub>F w in at z. f w \<noteq> 0" unfolding frequently_at |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1212 |
proof (rule,rule) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1213 |
fix d::real assume "0 < d" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1214 |
define z' where "z'=z+min d r / 2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1215 |
have "z' \<noteq> z" " dist z' z < d " |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1216 |
unfolding z'_def using \<open>d>0\<close> \<open>r>0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1217 |
by (auto simp add:dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1218 |
moreover have "f z' \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1219 |
proof (subst fg_eq[OF _ \<open>z'\<noteq>z\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1220 |
have "z' \<in> cball z r" unfolding z'_def using \<open>r>0\<close> \<open>d>0\<close> by (auto simp add:dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1221 |
then show " z' \<in> s" using r(2) by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1222 |
show "g z' * (z' - z) powr of_int n \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1223 |
using P_def \<open>P n g r\<close> \<open>z' \<in> cball z r\<close> calculation(1) by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1224 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1225 |
ultimately show "\<exists>x\<in>UNIV. x \<noteq> z \<and> dist x z < d \<and> f x \<noteq> 0" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1226 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1227 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1228 |
ultimately have "(THE n. \<exists>g r. P n g r) = n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1229 |
by (rule_tac the1_equality) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1230 |
then show ?thesis unfolding zorder_def P_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1231 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1232 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1233 |
lemma simple_zeroI: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1234 |
assumes "open s" "z \<in> s" "g holomorphic_on s" "g z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1235 |
assumes "\<And>w. w \<in> s \<Longrightarrow> f w = g w * (w - z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1236 |
shows "zorder f z = 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1237 |
using assms(1-4) by (rule zorder_eqI) (use assms(5) in auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1238 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1239 |
lemma higher_deriv_power: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1240 |
shows "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) w = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1241 |
pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1242 |
proof (induction j arbitrary: w) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1243 |
case 0 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1244 |
thus ?case by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1245 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1246 |
case (Suc j w) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1247 |
have "(deriv ^^ Suc j) (\<lambda>w. (w - z) ^ n) w = deriv ((deriv ^^ j) (\<lambda>w. (w - z) ^ n)) w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1248 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1249 |
also have "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1250 |
(\<lambda>w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1251 |
using Suc by (intro Suc.IH ext) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1252 |
also { |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1253 |
have "(\<dots> has_field_derivative of_nat (n - j) * |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1254 |
pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - Suc j)) (at w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1255 |
using Suc.prems by (auto intro!: derivative_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1256 |
also have "of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1257 |
pochhammer (of_nat (Suc n - Suc j)) (Suc j)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1258 |
by (cases "Suc j \<le> n", subst pochhammer_rec) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1259 |
(insert Suc.prems, simp_all add: algebra_simps Suc_diff_le pochhammer_0_left) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1260 |
finally have "deriv (\<lambda>w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) w = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1261 |
\<dots> * (w - z) ^ (n - Suc j)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1262 |
by (rule DERIV_imp_deriv) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1263 |
} |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1264 |
finally show ?case . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1265 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1266 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1267 |
lemma zorder_zero_eqI: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1268 |
assumes f_holo:"f holomorphic_on s" and "open s" "z \<in> s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1269 |
assumes zero: "\<And>i. i < nat n \<Longrightarrow> (deriv ^^ i) f z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1270 |
assumes nz: "(deriv ^^ nat n) f z \<noteq> 0" and "n\<ge>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1271 |
shows "zorder f z = n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1272 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1273 |
obtain r where [simp]:"r>0" and "ball z r \<subseteq> s" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1274 |
using \<open>open s\<close> \<open>z\<in>s\<close> openE by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1275 |
have nz':"\<exists>w\<in>ball z r. f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1276 |
proof (rule ccontr) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1277 |
assume "\<not> (\<exists>w\<in>ball z r. f w \<noteq> 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1278 |
then have "eventually (\<lambda>u. f u = 0) (nhds z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1279 |
using \<open>r>0\<close> unfolding eventually_nhds |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1280 |
apply (rule_tac x="ball z r" in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1281 |
by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1282 |
then have "(deriv ^^ nat n) f z = (deriv ^^ nat n) (\<lambda>_. 0) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1283 |
by (intro higher_deriv_cong_ev) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1284 |
also have "(deriv ^^ nat n) (\<lambda>_. 0) z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1285 |
by (induction n) simp_all |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1286 |
finally show False using nz by contradiction |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1287 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1288 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1289 |
define zn g where "zn = zorder f z" and "g = zor_poly f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1290 |
obtain e where e_if:"if f z = 0 then 0 < zn else zn = 0" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1291 |
[simp]:"e>0" and "cball z e \<subseteq> ball z r" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1292 |
g_holo:"g holomorphic_on cball z e" and |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1293 |
e_fac:"(\<forall>w\<in>cball z e. f w = g w * (w - z) ^ nat zn \<and> g w \<noteq> 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1294 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1295 |
have "f holomorphic_on ball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1296 |
using f_holo \<open>ball z r \<subseteq> s\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1297 |
from that zorder_exist_zero[of f "ball z r" z,simplified,OF this nz',folded zn_def g_def] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1298 |
show ?thesis by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1299 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1300 |
from this(1,2,5) have "zn\<ge>0" "g z\<noteq>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1301 |
subgoal by (auto split:if_splits) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1302 |
subgoal using \<open>0 < e\<close> ball_subset_cball centre_in_ball e_fac by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1303 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1304 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1305 |
define A where "A = (\<lambda>i. of_nat (i choose (nat zn)) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1306 |
have deriv_A:"(deriv ^^ i) f z = (if zn \<le> int i then A i else 0)" for i |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1307 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1308 |
have "eventually (\<lambda>w. w \<in> ball z e) (nhds z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1309 |
using \<open>cball z e \<subseteq> ball z r\<close> \<open>e>0\<close> by (intro eventually_nhds_in_open) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1310 |
hence "eventually (\<lambda>w. f w = (w - z) ^ (nat zn) * g w) (nhds z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1311 |
apply eventually_elim |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1312 |
by (use e_fac in auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1313 |
hence "(deriv ^^ i) f z = (deriv ^^ i) (\<lambda>w. (w - z) ^ nat zn * g w) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1314 |
by (intro higher_deriv_cong_ev) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1315 |
also have "\<dots> = (\<Sum>j=0..i. of_nat (i choose j) * |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1316 |
(deriv ^^ j) (\<lambda>w. (w - z) ^ nat zn) z * (deriv ^^ (i - j)) g z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1317 |
using g_holo \<open>e>0\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1318 |
by (intro higher_deriv_mult[of _ "ball z e"]) (auto intro!: holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1319 |
also have "\<dots> = (\<Sum>j=0..i. if j = nat zn then |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1320 |
of_nat (i choose nat zn) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z else 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1321 |
proof (intro sum.cong refl, goal_cases) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1322 |
case (1 j) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1323 |
have "(deriv ^^ j) (\<lambda>w. (w - z) ^ nat zn) z = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1324 |
pochhammer (of_nat (Suc (nat zn) - j)) j * 0 ^ (nat zn - j)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1325 |
by (subst higher_deriv_power) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1326 |
also have "\<dots> = (if j = nat zn then fact j else 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1327 |
by (auto simp: not_less pochhammer_0_left pochhammer_fact) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1328 |
also have "of_nat (i choose j) * \<dots> * (deriv ^^ (i - j)) g z = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1329 |
(if j = nat zn then of_nat (i choose (nat zn)) * fact (nat zn) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1330 |
* (deriv ^^ (i - nat zn)) g z else 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1331 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1332 |
finally show ?case . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1333 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1334 |
also have "\<dots> = (if i \<ge> zn then A i else 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1335 |
by (auto simp: A_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1336 |
finally show "(deriv ^^ i) f z = \<dots>" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1337 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1338 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1339 |
have False when "n<zn" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1340 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1341 |
have "(deriv ^^ nat n) f z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1342 |
using deriv_A[of "nat n"] that \<open>n\<ge>0\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1343 |
with nz show False by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1344 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1345 |
moreover have "n\<le>zn" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1346 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1347 |
have "g z \<noteq> 0" using e_fac[rule_format,of z] \<open>e>0\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1348 |
then have "(deriv ^^ nat zn) f z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1349 |
using deriv_A[of "nat zn"] by(auto simp add:A_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1350 |
then have "nat zn \<ge> nat n" using zero[of "nat zn"] by linarith |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1351 |
moreover have "zn\<ge>0" using e_if by (auto split:if_splits) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1352 |
ultimately show ?thesis using nat_le_eq_zle by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1353 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1354 |
ultimately show ?thesis unfolding zn_def by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1355 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1356 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1357 |
lemma |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1358 |
assumes "eventually (\<lambda>z. f z = g z) (at z)" "z = z'" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1359 |
shows zorder_cong:"zorder f z = zorder g z'" and zor_poly_cong:"zor_poly f z = zor_poly g z'" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1360 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1361 |
define P where "P = (\<lambda>ff n h r. 0 < r \<and> h holomorphic_on cball z r \<and> h z\<noteq>0 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1362 |
\<and> (\<forall>w\<in>cball z r - {z}. ff w = h w * (w-z) powr (of_int n) \<and> h w\<noteq>0))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1363 |
have "(\<exists>r. P f n h r) = (\<exists>r. P g n h r)" for n h |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1364 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1365 |
have *: "\<exists>r. P g n h r" if "\<exists>r. P f n h r" and "eventually (\<lambda>x. f x = g x) (at z)" for f g |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1366 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1367 |
from that(1) obtain r1 where r1_P:"P f n h r1" by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1368 |
from that(2) obtain r2 where "r2>0" and r2_dist:"\<forall>x. x \<noteq> z \<and> dist x z \<le> r2 \<longrightarrow> f x = g x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1369 |
unfolding eventually_at_le by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1370 |
define r where "r=min r1 r2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1371 |
have "r>0" "h z\<noteq>0" using r1_P \<open>r2>0\<close> unfolding r_def P_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1372 |
moreover have "h holomorphic_on cball z r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1373 |
using r1_P unfolding P_def r_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1374 |
moreover have "g w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0" when "w\<in>cball z r - {z}" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1375 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1376 |
have "f w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1377 |
using r1_P that unfolding P_def r_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1378 |
moreover have "f w=g w" using r2_dist[rule_format,of w] that unfolding r_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1379 |
by (simp add: dist_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1380 |
ultimately show ?thesis by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1381 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1382 |
ultimately show ?thesis unfolding P_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1383 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1384 |
from assms have eq': "eventually (\<lambda>z. g z = f z) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1385 |
by (simp add: eq_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1386 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1387 |
by (rule iffI[OF *[OF _ assms(1)] *[OF _ eq']]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1388 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1389 |
then show "zorder f z = zorder g z'" "zor_poly f z = zor_poly g z'" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1390 |
using \<open>z=z'\<close> unfolding P_def zorder_def zor_poly_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1391 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1392 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1393 |
lemma zorder_nonzero_div_power: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1394 |
assumes "open s" "z \<in> s" "f holomorphic_on s" "f z \<noteq> 0" "n > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1395 |
shows "zorder (\<lambda>w. f w / (w - z) ^ n) z = - n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1396 |
apply (rule zorder_eqI[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>f holomorphic_on s\<close> \<open>f z\<noteq>0\<close>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1397 |
apply (subst powr_of_int) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1398 |
using \<open>n>0\<close> by (auto simp add:field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1399 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1400 |
lemma zor_poly_eq: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1401 |
assumes "isolated_singularity_at f z" "not_essential f z" "\<exists>\<^sub>F w in at z. f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1402 |
shows "eventually (\<lambda>w. zor_poly f z w = f w * (w - z) powr - zorder f z) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1403 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1404 |
obtain r where r:"r>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1405 |
"(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w * (w - z) powr of_int (zorder f z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1406 |
using zorder_exist[OF assms] by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1407 |
then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w * (w - z) powr - zorder f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1408 |
by (auto simp: field_simps powr_minus) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1409 |
have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1410 |
using r eventually_at_ball'[of r z UNIV] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1411 |
thus ?thesis by eventually_elim (insert *, auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1412 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1413 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1414 |
lemma zor_poly_zero_eq: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1415 |
assumes "f holomorphic_on s" "open s" "connected s" "z \<in> s" "\<exists>w\<in>s. f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1416 |
shows "eventually (\<lambda>w. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1417 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1418 |
obtain r where r:"r>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1419 |
"(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w * (w - z) ^ nat (zorder f z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1420 |
using zorder_exist_zero[OF assms] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1421 |
then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1422 |
by (auto simp: field_simps powr_minus) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1423 |
have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1424 |
using r eventually_at_ball'[of r z UNIV] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1425 |
thus ?thesis by eventually_elim (insert *, auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1426 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1427 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1428 |
lemma zor_poly_pole_eq: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1429 |
assumes f_iso:"isolated_singularity_at f z" "is_pole f z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1430 |
shows "eventually (\<lambda>w. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1431 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1432 |
obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1433 |
using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1434 |
obtain r where r:"r>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1435 |
"(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w / (w - z) ^ nat (- zorder f z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1436 |
using zorder_exist_pole[OF f_holo,simplified,OF \<open>is_pole f z\<close>] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1437 |
then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1438 |
by (auto simp: field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1439 |
have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1440 |
using r eventually_at_ball'[of r z UNIV] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1441 |
thus ?thesis by eventually_elim (insert *, auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1442 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1443 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1444 |
lemma zor_poly_eqI: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1445 |
fixes f :: "complex \<Rightarrow> complex" and z0 :: complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1446 |
defines "n \<equiv> zorder f z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1447 |
assumes "isolated_singularity_at f z0" "not_essential f z0" "\<exists>\<^sub>F w in at z0. f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1448 |
assumes lim: "((\<lambda>x. f (g x) * (g x - z0) powr - n) \<longlongrightarrow> c) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1449 |
assumes g: "filterlim g (at z0) F" and "F \<noteq> bot" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1450 |
shows "zor_poly f z0 z0 = c" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1451 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1452 |
from zorder_exist[OF assms(2-4)] obtain r where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1453 |
r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1454 |
"\<And>w. w \<in> cball z0 r - {z0} \<Longrightarrow> f w = zor_poly f z0 w * (w - z0) powr n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1455 |
unfolding n_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1456 |
from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1457 |
using eventually_at_ball'[of r z0 UNIV] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1458 |
hence "eventually (\<lambda>w. zor_poly f z0 w = f w * (w - z0) powr - n) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1459 |
by eventually_elim (insert r, auto simp: field_simps powr_minus) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1460 |
moreover have "continuous_on (ball z0 r) (zor_poly f z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1461 |
using r by (intro holomorphic_on_imp_continuous_on) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1462 |
with r(1,2) have "isCont (zor_poly f z0) z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1463 |
by (auto simp: continuous_on_eq_continuous_at) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1464 |
hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1465 |
unfolding isCont_def . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1466 |
ultimately have "((\<lambda>w. f w * (w - z0) powr - n) \<longlongrightarrow> zor_poly f z0 z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1467 |
by (blast intro: Lim_transform_eventually) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1468 |
hence "((\<lambda>x. f (g x) * (g x - z0) powr - n) \<longlongrightarrow> zor_poly f z0 z0) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1469 |
by (rule filterlim_compose[OF _ g]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1470 |
from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1471 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1472 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1473 |
lemma zor_poly_zero_eqI: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1474 |
fixes f :: "complex \<Rightarrow> complex" and z0 :: complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1475 |
defines "n \<equiv> zorder f z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1476 |
assumes "f holomorphic_on A" "open A" "connected A" "z0 \<in> A" "\<exists>z\<in>A. f z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1477 |
assumes lim: "((\<lambda>x. f (g x) / (g x - z0) ^ nat n) \<longlongrightarrow> c) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1478 |
assumes g: "filterlim g (at z0) F" and "F \<noteq> bot" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1479 |
shows "zor_poly f z0 z0 = c" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1480 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1481 |
from zorder_exist_zero[OF assms(2-6)] obtain r where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1482 |
r: "r > 0" "cball z0 r \<subseteq> A" "zor_poly f z0 holomorphic_on cball z0 r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1483 |
"\<And>w. w \<in> cball z0 r \<Longrightarrow> f w = zor_poly f z0 w * (w - z0) ^ nat n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1484 |
unfolding n_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1485 |
from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1486 |
using eventually_at_ball'[of r z0 UNIV] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1487 |
hence "eventually (\<lambda>w. zor_poly f z0 w = f w / (w - z0) ^ nat n) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1488 |
by eventually_elim (insert r, auto simp: field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1489 |
moreover have "continuous_on (ball z0 r) (zor_poly f z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1490 |
using r by (intro holomorphic_on_imp_continuous_on) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1491 |
with r(1,2) have "isCont (zor_poly f z0) z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1492 |
by (auto simp: continuous_on_eq_continuous_at) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1493 |
hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1494 |
unfolding isCont_def . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1495 |
ultimately have "((\<lambda>w. f w / (w - z0) ^ nat n) \<longlongrightarrow> zor_poly f z0 z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1496 |
by (blast intro: Lim_transform_eventually) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1497 |
hence "((\<lambda>x. f (g x) / (g x - z0) ^ nat n) \<longlongrightarrow> zor_poly f z0 z0) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1498 |
by (rule filterlim_compose[OF _ g]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1499 |
from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1500 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1501 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1502 |
lemma zor_poly_pole_eqI: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1503 |
fixes f :: "complex \<Rightarrow> complex" and z0 :: complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1504 |
defines "n \<equiv> zorder f z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1505 |
assumes f_iso:"isolated_singularity_at f z0" and "is_pole f z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1506 |
assumes lim: "((\<lambda>x. f (g x) * (g x - z0) ^ nat (-n)) \<longlongrightarrow> c) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1507 |
assumes g: "filterlim g (at z0) F" and "F \<noteq> bot" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1508 |
shows "zor_poly f z0 z0 = c" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1509 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1510 |
obtain r where r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1511 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1512 |
have "\<exists>\<^sub>F w in at z0. f w \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1513 |
using non_zero_neighbour_pole[OF \<open>is_pole f z0\<close>] by (auto elim:eventually_frequentlyE) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1514 |
moreover have "not_essential f z0" unfolding not_essential_def using \<open>is_pole f z0\<close> by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1515 |
ultimately show ?thesis using that zorder_exist[OF f_iso,folded n_def] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1516 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1517 |
from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1518 |
using eventually_at_ball'[of r z0 UNIV] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1519 |
have "eventually (\<lambda>w. zor_poly f z0 w = f w * (w - z0) ^ nat (-n)) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1520 |
using zor_poly_pole_eq[OF f_iso \<open>is_pole f z0\<close>] unfolding n_def . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1521 |
moreover have "continuous_on (ball z0 r) (zor_poly f z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1522 |
using r by (intro holomorphic_on_imp_continuous_on) auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1523 |
with r(1,2) have "isCont (zor_poly f z0) z0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1524 |
by (auto simp: continuous_on_eq_continuous_at) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1525 |
hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1526 |
unfolding isCont_def . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1527 |
ultimately have "((\<lambda>w. f w * (w - z0) ^ nat (-n)) \<longlongrightarrow> zor_poly f z0 z0) (at z0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1528 |
by (blast intro: Lim_transform_eventually) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1529 |
hence "((\<lambda>x. f (g x) * (g x - z0) ^ nat (-n)) \<longlongrightarrow> zor_poly f z0 z0) F" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1530 |
by (rule filterlim_compose[OF _ g]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1531 |
from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1532 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1533 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1534 |
end |