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