--- a/src/HOL/Analysis/Analysis.thy Mon Mar 11 08:46:20 2024 +0100
+++ b/src/HOL/Analysis/Analysis.thy Mon Mar 11 15:07:02 2024 +0000
@@ -12,6 +12,7 @@
Connected
Abstract_Limits
Isolated
+ Sparse_In
(* Functional Analysis *)
Elementary_Normed_Spaces
Norm_Arith
--- a/src/HOL/Analysis/Complex_Analysis_Basics.thy Mon Mar 11 08:46:20 2024 +0100
+++ b/src/HOL/Analysis/Complex_Analysis_Basics.thy Mon Mar 11 15:07:02 2024 +0000
@@ -451,6 +451,9 @@
lemma analytic_on_id [analytic_intros]: "id analytic_on S"
unfolding id_def by (rule analytic_on_ident)
+lemma analytic_on_scaleR [analytic_intros]: "f analytic_on A \<Longrightarrow> (\<lambda>w. x *\<^sub>R f w) analytic_on A"
+ by (metis analytic_on_holomorphic holomorphic_on_scaleR)
+
lemma analytic_on_compose:
assumes f: "f analytic_on S"
and g: "g analytic_on (f ` S)"
@@ -585,6 +588,10 @@
"(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on S) \<Longrightarrow> (\<lambda>x. prod (\<lambda>i. f i x) I) analytic_on S"
by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_mult)
+lemma analytic_on_gbinomial [analytic_intros]:
+ "f analytic_on A \<Longrightarrow> (\<lambda>w. f w gchoose n) analytic_on A"
+ unfolding gbinomial_prod_rev by (intro analytic_intros) auto
+
lemma deriv_left_inverse:
assumes "f holomorphic_on S" and "g holomorphic_on T"
and "open S" and "open T"
--- a/src/HOL/Analysis/Complex_Transcendental.thy Mon Mar 11 08:46:20 2024 +0100
+++ b/src/HOL/Analysis/Complex_Transcendental.thy Mon Mar 11 15:07:02 2024 +0000
@@ -224,21 +224,20 @@
shows "(\<lambda>x. cos (f x)) holomorphic_on A"
using holomorphic_on_compose[OF assms holomorphic_on_cos] by (simp add: o_def)
-lemma analytic_on_sin [analytic_intros]: "sin analytic_on A"
- using analytic_on_holomorphic holomorphic_on_sin by blast
-
-lemma analytic_on_sin' [analytic_intros]:
- "f analytic_on A \<Longrightarrow> (\<And>z. z \<in> A \<Longrightarrow> f z \<notin> range (\<lambda>n. complex_of_real pi * of_int n)) \<Longrightarrow>
- (\<lambda>z. sin (f z)) analytic_on A"
- using analytic_on_compose_gen[OF _ analytic_on_sin[of UNIV], of f A] by (simp add: o_def)
-
-lemma analytic_on_cos [analytic_intros]: "cos analytic_on A"
- using analytic_on_holomorphic holomorphic_on_cos by blast
-
-lemma analytic_on_cos' [analytic_intros]:
- "f analytic_on A \<Longrightarrow> (\<And>z. z \<in> A \<Longrightarrow> f z \<notin> range (\<lambda>n. complex_of_real pi * of_int n)) \<Longrightarrow>
- (\<lambda>z. cos (f z)) analytic_on A"
- using analytic_on_compose_gen[OF _ analytic_on_cos[of UNIV], of f A] by (simp add: o_def)
+lemma analytic_on_sin [analytic_intros]: "f analytic_on A \<Longrightarrow> (\<lambda>w. sin (f w)) analytic_on A"
+ and analytic_on_cos [analytic_intros]: "f analytic_on A \<Longrightarrow> (\<lambda>w. cos (f w)) analytic_on A"
+ and analytic_on_sinh [analytic_intros]: "f analytic_on A \<Longrightarrow> (\<lambda>w. sinh (f w)) analytic_on A"
+ and analytic_on_cosh [analytic_intros]: "f analytic_on A \<Longrightarrow> (\<lambda>w. cosh (f w)) analytic_on A"
+ unfolding sin_exp_eq cos_exp_eq sinh_def cosh_def
+ by (auto intro!: analytic_intros)
+
+lemma analytic_on_tan [analytic_intros]:
+ "f analytic_on A \<Longrightarrow> (\<And>z. z \<in> A \<Longrightarrow> cos (f z) \<noteq> 0) \<Longrightarrow> (\<lambda>w. tan (f w)) analytic_on A"
+ and analytic_on_cot [analytic_intros]:
+ "f analytic_on A \<Longrightarrow> (\<And>z. z \<in> A \<Longrightarrow> sin (f z) \<noteq> 0) \<Longrightarrow> (\<lambda>w. cot (f w)) analytic_on A"
+ and analytic_on_tanh [analytic_intros]:
+ "f analytic_on A \<Longrightarrow> (\<And>z. z \<in> A \<Longrightarrow> cosh (f z) \<noteq> 0) \<Longrightarrow> (\<lambda>w. tanh (f w)) analytic_on A"
+ unfolding tan_def cot_def tanh_def by (auto intro!: analytic_intros)
subsection\<^marker>\<open>tag unimportant\<close>\<open>More on the Polar Representation of Complex Numbers\<close>
@@ -1252,6 +1251,18 @@
using holomorphic_on_compose_gen[OF _ holomorphic_on_Ln, of f A "- \<real>\<^sub>\<le>\<^sub>0"]
by (auto simp: o_def)
+lemma analytic_on_ln [analytic_intros]:
+ assumes "f analytic_on A" "f ` A \<inter> \<real>\<^sub>\<le>\<^sub>0 = {}"
+ shows "(\<lambda>w. ln (f w)) analytic_on A"
+proof -
+ have *: "ln analytic_on (-\<real>\<^sub>\<le>\<^sub>0)"
+ by (subst analytic_on_open) (auto intro!: holomorphic_intros)
+ have "(ln \<circ> f) analytic_on A"
+ by (rule analytic_on_compose_gen[OF assms(1) *]) (use assms(2) in auto)
+ thus ?thesis
+ by (simp add: o_def)
+qed
+
lemma tendsto_Ln [tendsto_intros]:
assumes "(f \<longlongrightarrow> L) F" "L \<notin> \<real>\<^sub>\<le>\<^sub>0"
shows "((\<lambda>x. Ln (f x)) \<longlongrightarrow> Ln L) F"
--- a/src/HOL/Analysis/Gamma_Function.thy Mon Mar 11 08:46:20 2024 +0100
+++ b/src/HOL/Analysis/Gamma_Function.thy Mon Mar 11 15:07:02 2024 +0000
@@ -1547,6 +1547,54 @@
(auto intro!: holomorphic_on_Polygamma)
+lemma analytic_on_rGamma [analytic_intros]: "f analytic_on A \<Longrightarrow> (\<lambda>w. rGamma (f w)) analytic_on A"
+ using analytic_on_compose[OF _ analytic_rGamma, of f A] by (simp add: o_def)
+
+lemma analytic_on_ln_Gamma [analytic_intros]:
+ "f analytic_on A \<Longrightarrow> (\<And>z. z \<in> A \<Longrightarrow> f z \<notin> \<real>\<^sub>\<le>\<^sub>0) \<Longrightarrow> (\<lambda>w. ln_Gamma (f w)) analytic_on A"
+ by (rule analytic_on_compose[OF _ analytic_ln_Gamma, unfolded o_def]) (auto simp: o_def)
+
+lemma Polygamma_plus_of_nat:
+ assumes "\<forall>k<m. z \<noteq> -of_nat k"
+ shows "Polygamma n (z + of_nat m) =
+ Polygamma n z + (-1) ^ n * fact n * (\<Sum>k<m. 1 / (z + of_nat k) ^ Suc n)"
+ using assms
+proof (induction m)
+ case (Suc m)
+ have "Polygamma n (z + of_nat (Suc m)) = Polygamma n (z + of_nat m + 1)"
+ by (simp add: add_ac)
+ also have "\<dots> = Polygamma n (z + of_nat m) + (-1) ^ n * fact n * (1 / ((z + of_nat m) ^ Suc n))"
+ using Suc.prems by (subst Polygamma_plus1) (auto simp: add_eq_0_iff2)
+ also have "Polygamma n (z + of_nat m) =
+ Polygamma n z + (-1) ^ n * (\<Sum>k<m. 1 / (z + of_nat k) ^ Suc n) * fact n"
+ using Suc.prems by (subst Suc.IH) auto
+ finally show ?case
+ by (simp add: algebra_simps)
+qed auto
+
+lemma tendsto_Gamma [tendsto_intros]:
+ assumes "(f \<longlongrightarrow> c) F" "c \<notin> \<int>\<^sub>\<le>\<^sub>0"
+ shows "((\<lambda>z. Gamma (f z)) \<longlongrightarrow> Gamma c) F"
+ by (intro isCont_tendsto_compose[OF _ assms(1)] continuous_intros assms)
+
+lemma tendsto_Polygamma [tendsto_intros]:
+ fixes f :: "_ \<Rightarrow> 'a :: {real_normed_field,euclidean_space}"
+ assumes "(f \<longlongrightarrow> c) F" "c \<notin> \<int>\<^sub>\<le>\<^sub>0"
+ shows "((\<lambda>z. Polygamma n (f z)) \<longlongrightarrow> Polygamma n c) F"
+ by (intro isCont_tendsto_compose[OF _ assms(1)] continuous_intros assms)
+
+lemma analytic_on_Gamma' [analytic_intros]:
+ assumes "f analytic_on A" "\<forall>x\<in>A. f x \<notin> \<int>\<^sub>\<le>\<^sub>0"
+ shows "(\<lambda>z. Gamma (f z)) analytic_on A"
+ using analytic_on_compose_gen[OF assms(1) analytic_Gamma[of "f ` A"]] assms(2)
+ by (auto simp: o_def)
+
+lemma analytic_on_Polygamma' [analytic_intros]:
+ assumes "f analytic_on A" "\<forall>x\<in>A. f x \<notin> \<int>\<^sub>\<le>\<^sub>0"
+ shows "(\<lambda>z. Polygamma n (f z)) analytic_on A"
+ using analytic_on_compose_gen[OF assms(1) analytic_on_Polygamma[of "f ` A" n]] assms(2)
+ by (auto simp: o_def)
+
subsection\<^marker>\<open>tag unimportant\<close> \<open>The real Gamma function\<close>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Sparse_In.thy Mon Mar 11 15:07:02 2024 +0000
@@ -0,0 +1,242 @@
+theory Sparse_In
+ imports Homotopy
+
+begin
+
+(*TODO: can we remove the definition isolated_points_of from
+ HOL-Complex_Analysis.Complex_Singularities?*)
+(*TODO: more lemmas between sparse_in and discrete?*)
+
+subsection \<open>A set of points sparse in another set\<close>
+
+definition sparse_in:: "'a :: topological_space set \<Rightarrow> 'a set \<Rightarrow> bool"
+ (infixl "(sparse'_in)" 50)
+ where
+ "pts sparse_in A = (\<forall>x\<in>A. \<exists>B. x\<in>B \<and> open B \<and> (\<forall>y\<in>B. \<not> y islimpt pts))"
+
+lemma sparse_in_empty[simp]: "{} sparse_in A"
+ by (meson UNIV_I empty_iff islimpt_def open_UNIV sparse_in_def)
+
+lemma finite_imp_sparse:
+ fixes pts::"'a:: t1_space set"
+ shows "finite pts \<Longrightarrow> pts sparse_in S"
+ by (meson UNIV_I islimpt_finite open_UNIV sparse_in_def)
+
+lemma sparse_in_singleton[simp]: "{x} sparse_in (A::'a:: t1_space set)"
+ by (rule finite_imp_sparse) auto
+
+lemma sparse_in_ball_def:
+ "pts sparse_in D \<longleftrightarrow> (\<forall>x\<in>D. \<exists>e>0. \<forall>y\<in>ball x e. \<not> y islimpt pts)"
+ unfolding sparse_in_def
+ by (meson Elementary_Metric_Spaces.open_ball open_contains_ball_eq subset_eq)
+
+lemma get_sparse_in_cover:
+ assumes "pts sparse_in A"
+ obtains B where "open B" "A \<subseteq> B" "\<forall>y\<in>B. \<not> y islimpt pts"
+proof -
+ obtain getB where getB:"x\<in>getB x" "open (getB x)" "\<forall>y\<in>getB x. \<not> y islimpt pts"
+ if "x\<in>A" for x
+ using assms(1) unfolding sparse_in_def by metis
+ define B where "B = Union (getB ` A)"
+ have "open B" unfolding B_def using getB(2) by blast
+ moreover have "A \<subseteq> B" unfolding B_def using getB(1) by auto
+ moreover have "\<forall>y\<in>B. \<not> y islimpt pts" unfolding B_def by (meson UN_iff getB(3))
+ ultimately show ?thesis using that by blast
+qed
+
+lemma sparse_in_open:
+ assumes "open A"
+ shows "pts sparse_in A \<longleftrightarrow> (\<forall>y\<in>A. \<not>y islimpt pts)"
+ using assms unfolding sparse_in_def by auto
+
+lemma sparse_in_not_in:
+ assumes "pts sparse_in A" "x\<in>A"
+ obtains B where "open B" "x\<in>B" "\<forall>y\<in>B. y\<noteq>x \<longrightarrow> y\<notin>pts"
+ using assms unfolding sparse_in_def
+ by (metis islimptI)
+
+lemma sparse_in_subset:
+ assumes "pts sparse_in A" "B \<subseteq> A"
+ shows "pts sparse_in B"
+ using assms unfolding sparse_in_def by auto
+
+lemma sparse_in_subset2:
+ assumes "pts1 sparse_in D" "pts2 \<subseteq> pts1"
+ shows "pts2 sparse_in D"
+ by (meson assms(1) assms(2) islimpt_subset sparse_in_def)
+
+lemma sparse_in_union:
+ assumes "pts1 sparse_in D1" "pts2 sparse_in D1"
+ shows "(pts1 \<union> pts2) sparse_in (D1 \<inter> D2)"
+ using assms unfolding sparse_in_def islimpt_Un
+ by (metis Int_iff open_Int)
+
+lemma sparse_in_compact_finite:
+ assumes "pts sparse_in A" "compact A"
+ shows "finite (A \<inter> pts)"
+ apply (rule finite_not_islimpt_in_compact[OF \<open>compact A\<close>])
+ using assms unfolding sparse_in_def by blast
+
+lemma sparse_imp_closedin_pts:
+ assumes "pts sparse_in D"
+ shows "closedin (top_of_set D) (D \<inter> pts)"
+ using assms islimpt_subset unfolding closedin_limpt sparse_in_def
+ by fastforce
+
+lemma open_diff_sparse_pts:
+ assumes "open D" "pts sparse_in D"
+ shows "open (D - pts)"
+ using assms sparse_imp_closedin_pts
+ by (metis Diff_Diff_Int Diff_cancel Diff_eq_empty_iff Diff_subset
+ closedin_def double_diff openin_open_eq topspace_euclidean_subtopology)
+
+lemma sparse_imp_countable:
+ fixes D::"'a ::euclidean_space set"
+ assumes "open D" "pts sparse_in D"
+ shows "countable (D \<inter> pts)"
+proof -
+ obtain K :: "nat \<Rightarrow> 'a ::euclidean_space set"
+ where K: "D = (\<Union>n. K n)" "\<And>n. compact (K n)"
+ using assms by (metis open_Union_compact_subsets)
+ then have "D\<inter> pts = (\<Union>n. K n \<inter> pts)"
+ by blast
+ moreover have "\<And>n. finite (K n \<inter> pts)"
+ by (metis K(1) K(2) Union_iff assms(2) rangeI
+ sparse_in_compact_finite sparse_in_subset subsetI)
+ ultimately show ?thesis
+ by (metis countableI_type countable_UN countable_finite)
+qed
+
+lemma sparse_imp_connected:
+ fixes D::"'a ::euclidean_space set"
+ assumes "2 \<le> DIM ('a)" "connected D" "open D" "pts sparse_in D"
+ shows "connected (D - pts)"
+ using assms
+ by (metis Diff_Compl Diff_Diff_Int Diff_eq connected_open_diff_countable
+ sparse_imp_countable)
+
+lemma sparse_in_eventually_iff:
+ assumes "open A"
+ shows "pts sparse_in A \<longleftrightarrow> (\<forall>y\<in>A. (\<forall>\<^sub>F y in at y. y \<notin> pts))"
+ unfolding sparse_in_open[OF \<open>open A\<close>] islimpt_iff_eventually
+ by simp
+
+lemma get_sparse_from_eventually:
+ fixes A::"'a::topological_space set"
+ assumes "\<forall>x\<in>A. \<forall>\<^sub>F z in at x. P z" "open A"
+ obtains pts where "pts sparse_in A" "\<forall>x\<in>A - pts. P x"
+proof -
+ define pts::"'a set" where "pts={x. \<not>P x}"
+ have "pts sparse_in A" "\<forall>x\<in>A - pts. P x"
+ unfolding sparse_in_eventually_iff[OF \<open>open A\<close>] pts_def
+ using assms(1) by simp_all
+ then show ?thesis using that by blast
+qed
+
+lemma sparse_disjoint:
+ assumes "pts \<inter> A = {}" "open A"
+ shows "pts sparse_in A"
+ using assms unfolding sparse_in_eventually_iff[OF \<open>open A\<close>]
+ eventually_at_topological
+ by blast
+
+
+subsection \<open>Co-sparseness filter\<close>
+
+text \<open>
+ The co-sparseness filter allows us to talk about properties that hold on a given set except
+ for an ``insignificant'' number of points that are sparse in that set.
+\<close>
+lemma is_filter_cosparse: "is_filter (\<lambda>P. {x. \<not>P x} sparse_in A)"
+proof (standard, goal_cases)
+ case 1
+ thus ?case by auto
+next
+ case (2 P Q)
+ from sparse_in_union[OF this, of UNIV] show ?case
+ by (auto simp: Un_def)
+next
+ case (3 P Q)
+ from 3(2) show ?case
+ by (rule sparse_in_subset2) (use 3(1) in auto)
+qed
+
+definition cosparse :: "'a set \<Rightarrow> 'a :: topological_space filter" where
+ "cosparse A = Abs_filter (\<lambda>P. {x. \<not>P x} sparse_in A)"
+
+syntax
+ "_eventually_cosparse" :: "pttrn => 'a set => bool => bool" ("(3\<forall>\<^sub>\<approx>_\<in>_./ _)" [0, 0, 10] 10)
+translations
+ "\<forall>\<^sub>\<approx>x\<in>A. P" == "CONST eventually (\<lambda>x. P) (CONST cosparse A)"
+
+syntax
+ "_qeventually_cosparse" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<forall>\<^sub>\<approx>_ | (_)./ _)" [0, 0, 10] 10)
+translations
+ "\<forall>\<^sub>\<approx>x|P. t" => "CONST eventually (\<lambda>x. t) (CONST cosparse {x. P})"
+
+print_translation \<open>
+let
+ fun ev_cosparse_tr' [Abs (x, Tx, t),
+ Const (\<^const_syntax>\<open>cosparse\<close>, _) $ (Const (\<^const_syntax>\<open>Collect\<close>, _) $ Abs (y, Ty, P))] =
+ if x <> y then raise Match
+ else
+ let
+ val x' = Syntax_Trans.mark_bound_body (x, Tx);
+ val t' = subst_bound (x', t);
+ val P' = subst_bound (x', P);
+ in
+ Syntax.const \<^syntax_const>\<open>_qeventually_cosparse\<close> $
+ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'
+ end
+ | ev_cosparse_tr' _ = raise Match;
+in [(\<^const_syntax>\<open>eventually\<close>, K ev_cosparse_tr')] end
+\<close>
+
+lemma eventually_cosparse: "eventually P (cosparse A) \<longleftrightarrow> {x. \<not>P x} sparse_in A"
+ unfolding cosparse_def by (rule eventually_Abs_filter[OF is_filter_cosparse])
+
+lemma eventually_not_in_cosparse:
+ assumes "X sparse_in A"
+ shows "eventually (\<lambda>x. x \<notin> X) (cosparse A)"
+ using assms by (auto simp: eventually_cosparse)
+
+lemma eventually_cosparse_open_eq:
+ "open A \<Longrightarrow> eventually P (cosparse A) \<longleftrightarrow> (\<forall>x\<in>A. eventually P (at x))"
+ unfolding eventually_cosparse
+ by (subst sparse_in_open) (auto simp: islimpt_conv_frequently_at frequently_def)
+
+lemma eventually_cosparse_imp_eventually_at:
+ "eventually P (cosparse A) \<Longrightarrow> x \<in> A \<Longrightarrow> eventually P (at x within B)"
+ unfolding eventually_cosparse sparse_in_def
+ apply (auto simp: islimpt_conv_frequently_at frequently_def)
+ apply (metis UNIV_I eventually_at_topological)
+ done
+
+lemma eventually_in_cosparse:
+ assumes "A \<subseteq> X" "open A"
+ shows "eventually (\<lambda>x. x \<in> X) (cosparse A)"
+proof -
+ have "eventually (\<lambda>x. x \<in> A) (cosparse A)"
+ using assms by (auto simp: eventually_cosparse_open_eq intro: eventually_at_in_open')
+ thus ?thesis
+ by eventually_elim (use assms(1) in blast)
+qed
+
+lemma cosparse_eq_bot_iff: "cosparse A = bot \<longleftrightarrow> (\<forall>x\<in>A. open {x})"
+proof -
+ have "cosparse A = bot \<longleftrightarrow> eventually (\<lambda>_. False) (cosparse A)"
+ by (simp add: trivial_limit_def)
+ also have "\<dots> \<longleftrightarrow> (\<forall>x\<in>A. open {x})"
+ unfolding eventually_cosparse sparse_in_def
+ by (auto simp: islimpt_UNIV_iff)
+ finally show ?thesis .
+qed
+
+lemma cosparse_empty [simp]: "cosparse {} = bot"
+ by (rule filter_eqI) (auto simp: eventually_cosparse sparse_in_def)
+
+lemma cosparse_eq_bot_iff' [simp]: "cosparse (A :: 'a :: perfect_space set) = bot \<longleftrightarrow> A = {}"
+ by (auto simp: cosparse_eq_bot_iff not_open_singleton)
+
+
+end
\ No newline at end of file
--- a/src/HOL/Complex_Analysis/Meromorphic.thy Mon Mar 11 08:46:20 2024 +0100
+++ b/src/HOL/Complex_Analysis/Meromorphic.thy Mon Mar 11 15:07:02 2024 +0000
@@ -1,7 +1,66 @@
-theory Meromorphic
- imports Laurent_Convergence Riemann_Mapping
+theory Meromorphic imports
+ "Laurent_Convergence"
+ "HOL-Analysis.Sparse_In"
begin
+(*TODO: move to topological space? *)
+lemma eventually_nhds_conv_at:
+ "eventually P (nhds x) \<longleftrightarrow> eventually P (at x) \<and> P x"
+ unfolding eventually_at_topological eventually_nhds by fast
+
+(*TODO: to Complex_Singularities? *)
+lemma zorder_uminus [simp]: "zorder (\<lambda>z. -f z) z = zorder f z"
+ using zorder_cmult[of "-1" f] by (simp del: zorder_cmult)
+
+lemma constant_on_imp_analytic_on:
+ assumes "f constant_on A" "open A"
+ shows "f analytic_on A"
+ by (simp add: analytic_on_open assms
+ constant_on_imp_holomorphic_on)
+
+(*TODO: could be moved to Laurent_Convergence*)
+subsection \<open>More Laurent expansions\<close>
+
+lemma has_laurent_expansion_frequently_zero_iff:
+ assumes "(\<lambda>w. f (z + w)) has_laurent_expansion F"
+ shows "frequently (\<lambda>z. f z = 0) (at z) \<longleftrightarrow> F = 0"
+ using assms by (simp add: frequently_def has_laurent_expansion_eventually_nonzero_iff)
+
+lemma has_laurent_expansion_eventually_zero_iff:
+ assumes "(\<lambda>w. f (z + w)) has_laurent_expansion F"
+ shows "eventually (\<lambda>z. f z = 0) (at z) \<longleftrightarrow> F = 0"
+ using assms
+ by (metis has_laurent_expansion_frequently_zero_iff has_laurent_expansion_isolated
+ has_laurent_expansion_not_essential laurent_expansion_def
+ not_essential_frequently_0_imp_eventually_0 not_essential_has_laurent_expansion)
+
+lemma has_laurent_expansion_frequently_nonzero_iff:
+ assumes "(\<lambda>w. f (z + w)) has_laurent_expansion F"
+ shows "frequently (\<lambda>z. f z \<noteq> 0) (at z) \<longleftrightarrow> F \<noteq> 0"
+ using assms by (metis has_laurent_expansion_eventually_zero_iff not_eventually)
+
+lemma has_laurent_expansion_sum_list [laurent_expansion_intros]:
+ assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x has_laurent_expansion F x"
+ shows "(\<lambda>y. \<Sum>x\<leftarrow>xs. f x y) has_laurent_expansion (\<Sum>x\<leftarrow>xs. F x)"
+ using assms by (induction xs) (auto intro!: laurent_expansion_intros)
+
+lemma has_laurent_expansion_prod_list [laurent_expansion_intros]:
+ assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x has_laurent_expansion F x"
+ shows "(\<lambda>y. \<Prod>x\<leftarrow>xs. f x y) has_laurent_expansion (\<Prod>x\<leftarrow>xs. F x)"
+ using assms by (induction xs) (auto intro!: laurent_expansion_intros)
+
+lemma has_laurent_expansion_sum_mset [laurent_expansion_intros]:
+ assumes "\<And>x. x \<in># I \<Longrightarrow> f x has_laurent_expansion F x"
+ shows "(\<lambda>y. \<Sum>x\<in>#I. f x y) has_laurent_expansion (\<Sum>x\<in>#I. F x)"
+ using assms by (induction I) (auto intro!: laurent_expansion_intros)
+
+lemma has_laurent_expansion_prod_mset [laurent_expansion_intros]:
+ assumes "\<And>x. x \<in># I \<Longrightarrow> f x has_laurent_expansion F x"
+ shows "(\<lambda>y. \<Prod>x\<in>#I. f x y) has_laurent_expansion (\<Prod>x\<in>#I. F x)"
+ using assms by (induction I) (auto intro!: laurent_expansion_intros)
+
+subsection \<open>Remove singular points: remove_sings\<close>
+
definition remove_sings :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex" where
"remove_sings f z = (if \<exists>c. f \<midarrow>z\<rightarrow> c then Lim (at z) f else 0)"
@@ -32,7 +91,7 @@
have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
using r by (intro eventually_at_in_open) auto
thus ?thesis
- by eventually_elim (auto simp: remove_sings_at_analytic *)
+ by eventually_elim (auto simp: remove_sings_at_analytic * )
qed
lemma eventually_remove_sings_eq_nhds:
@@ -216,2101 +275,1018 @@
assumes "not_essential f w"
shows "remove_sings f w = 0 \<longleftrightarrow> is_pole f w \<or> f \<midarrow>w\<rightarrow> 0"
proof (cases "is_pole f w")
- case True
- then show ?thesis by simp
-next
case False
then obtain c where c:"f \<midarrow>w\<rightarrow> c"
using \<open>not_essential f w\<close> unfolding not_essential_def by auto
then show ?thesis
using False remove_sings_eqI by auto
-qed
-
-definition meromorphic_on:: "[complex \<Rightarrow> complex, complex set, complex set] \<Rightarrow> bool"
- ("_ (meromorphic'_on) _ _" [50,50,50]50) where
- "f meromorphic_on D pts \<equiv>
- open D \<and> pts \<subseteq> D \<and> (\<forall>z\<in>pts. isolated_singularity_at f z \<and> not_essential f z) \<and>
- (\<forall>z\<in>D. \<not>(z islimpt pts)) \<and> (f holomorphic_on D-pts)"
+qed simp
-lemma meromorphic_imp_holomorphic: "f meromorphic_on D pts \<Longrightarrow> f holomorphic_on (D - pts)"
- unfolding meromorphic_on_def by auto
-
-lemma meromorphic_imp_closedin_pts:
- assumes "f meromorphic_on D pts"
- shows "closedin (top_of_set D) pts"
- by (meson assms closedin_limpt meromorphic_on_def)
+subsection \<open>Meromorphicity\<close>
-lemma meromorphic_imp_open_diff':
- assumes "f meromorphic_on D pts" "pts' \<subseteq> pts"
- shows "open (D - pts')"
-proof -
- have "D - pts' = D - closure pts'"
- proof safe
- fix x assume x: "x \<in> D" "x \<in> closure pts'" "x \<notin> pts'"
- hence "x islimpt pts'"
- by (subst islimpt_in_closure) auto
- hence "x islimpt pts"
- by (rule islimpt_subset) fact
- with assms x show False
- by (auto simp: meromorphic_on_def)
- qed (use closure_subset in auto)
- then show ?thesis
- using assms meromorphic_on_def by auto
-qed
+definition meromorphic_on :: "(complex \<Rightarrow> complex) \<Rightarrow> complex set \<Rightarrow> bool"
+ (infixl "(meromorphic'_on)" 50) where
+ "f meromorphic_on A \<longleftrightarrow> (\<forall>z\<in>A. \<exists>F. (\<lambda>w. f (z + w)) has_laurent_expansion F)"
-lemma meromorphic_imp_open_diff: "f meromorphic_on D pts \<Longrightarrow> open (D - pts)"
- by (erule meromorphic_imp_open_diff') auto
-
-lemma meromorphic_pole_subset:
- assumes merf: "f meromorphic_on D pts"
- shows "{x\<in>D. is_pole f x} \<subseteq> pts"
- by (smt (verit) Diff_iff assms mem_Collect_eq meromorphic_imp_open_diff
- meromorphic_on_def not_is_pole_holomorphic subsetI)
+lemma meromorphic_at_iff: "f meromorphic_on {z} \<longleftrightarrow> isolated_singularity_at f z \<and> not_essential f z"
+ unfolding meromorphic_on_def
+ by (metis has_laurent_expansion_isolated has_laurent_expansion_not_essential
+ insertI1 singletonD not_essential_has_laurent_expansion)
named_theorems meromorphic_intros
-lemma meromorphic_on_subset:
- assumes "f meromorphic_on A pts" "open B" "B \<subseteq> A" "pts' = pts \<inter> B"
- shows "f meromorphic_on B pts'"
- unfolding meromorphic_on_def
-proof (intro ballI conjI)
- fix z assume "z \<in> B"
- show "\<not>z islimpt pts'"
- proof
- assume "z islimpt pts'"
- hence "z islimpt pts"
- by (rule islimpt_subset) (use \<open>pts' = _\<close> in auto)
- thus False using \<open>z \<in> B\<close> \<open>B \<subseteq> A\<close> assms(1)
- by (auto simp: meromorphic_on_def)
- qed
-qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
+lemma meromorphic_on_empty [simp, intro]: "f meromorphic_on {}"
+ by (auto simp: meromorphic_on_def)
+
+lemma meromorphic_on_def':
+ "f meromorphic_on A \<longleftrightarrow> (\<forall>z\<in>A. (\<lambda>w. f (z + w)) has_laurent_expansion laurent_expansion f z)"
+ unfolding meromorphic_on_def using laurent_expansion_eqI by blast
+
+lemma meromorphic_on_meromorphic_at: "f meromorphic_on A \<longleftrightarrow> (\<forall>x\<in>A. f meromorphic_on {x})"
+ by (auto simp: meromorphic_on_def)
-lemma meromorphic_on_superset_pts:
- assumes "f meromorphic_on A pts" "pts \<subseteq> pts'" "pts' \<subseteq> A" "\<forall>x\<in>A. \<not>x islimpt pts'"
- shows "f meromorphic_on A pts'"
- unfolding meromorphic_on_def
-proof (intro conjI ballI impI)
- fix z assume "z \<in> pts'"
- from assms(1) have holo: "f holomorphic_on A - pts" and "open A"
- unfolding meromorphic_on_def by blast+
- have "open (A - pts)"
- by (intro meromorphic_imp_open_diff[OF assms(1)])
+lemma meromorphic_on_altdef:
+ "f meromorphic_on A \<longleftrightarrow> (\<forall>z\<in>A. isolated_singularity_at f z \<and> not_essential f z)"
+ by (subst meromorphic_on_meromorphic_at) (auto simp: meromorphic_at_iff)
- show "isolated_singularity_at f z"
- proof (cases "z \<in> pts")
- case False
- thus ?thesis
- using \<open>open (A - pts)\<close> assms \<open>z \<in> pts'\<close>
- by (intro isolated_singularity_at_holomorphic[of _ "A - pts"] holomorphic_on_subset[OF holo])
- auto
- qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
+lemma meromorphic_on_cong:
+ assumes "\<And>z. z \<in> A \<Longrightarrow> eventually (\<lambda>w. f w = g w) (at z)" "A = B"
+ shows "f meromorphic_on A \<longleftrightarrow> g meromorphic_on B"
+ unfolding meromorphic_on_def using assms
+ by (intro ball_cong refl arg_cong[of _ _ Ex] has_laurent_expansion_cong ext)
+ (simp_all add: at_to_0' eventually_filtermap add_ac)
- show "not_essential f z"
- proof (cases "z \<in> pts")
- case False
- thus ?thesis
- using \<open>open (A - pts)\<close> assms \<open>z \<in> pts'\<close>
- by (intro not_essential_holomorphic[of _ "A - pts"] holomorphic_on_subset[OF holo])
- auto
- qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
-qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
-
-lemma meromorphic_on_no_singularities: "f meromorphic_on A {} \<longleftrightarrow> f holomorphic_on A \<and> open A"
+lemma meromorphic_on_subset: "f meromorphic_on A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> f meromorphic_on B"
by (auto simp: meromorphic_on_def)
-lemma holomorphic_on_imp_meromorphic_on:
- "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"
- by (rule meromorphic_on_superset_pts[where pts = "{}"])
- (auto simp: meromorphic_on_no_singularities)
-
-lemma meromorphic_on_const [meromorphic_intros]:
- assumes "open A" "\<forall>x\<in>A. \<not>x islimpt pts" "pts \<subseteq> A"
- shows "(\<lambda>_. c) meromorphic_on A pts"
- by (rule holomorphic_on_imp_meromorphic_on) (use assms in auto)
-
-lemma meromorphic_on_ident [meromorphic_intros]:
- assumes "open A" "\<forall>x\<in>A. \<not>x islimpt pts" "pts \<subseteq> A"
- shows "(\<lambda>x. x) meromorphic_on A pts"
- by (rule holomorphic_on_imp_meromorphic_on) (use assms in auto)
-
-lemma meromorphic_on_id [meromorphic_intros]:
- assumes "open A" "\<forall>x\<in>A. \<not>x islimpt pts" "pts \<subseteq> A"
- shows "id meromorphic_on A pts"
- using meromorphic_on_ident assms unfolding id_def .
-
-lemma not_essential_add [singularity_intros]:
- assumes f_ness: "not_essential f z" and g_ness: "not_essential g z"
- assumes f_iso: "isolated_singularity_at f z" and g_iso: "isolated_singularity_at g z"
- shows "not_essential (\<lambda>w. f w + g w) z"
-proof -
- have "(\<lambda>w. f (z + w) + g (z + w)) has_laurent_expansion laurent_expansion f z + laurent_expansion g z"
- by (intro not_essential_has_laurent_expansion laurent_expansion_intros assms)
- hence "not_essential (\<lambda>w. f (z + w) + g (z + w)) 0"
- using has_laurent_expansion_not_essential_0 by blast
- thus ?thesis
- by (simp add: not_essential_shift_0)
-qed
-
-lemma meromorphic_on_uminus [meromorphic_intros]:
- assumes "f meromorphic_on A pts"
- shows "(\<lambda>z. -f z) meromorphic_on A pts"
- unfolding meromorphic_on_def
- by (use assms in \<open>auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros\<close>)
-
-lemma meromorphic_on_add [meromorphic_intros]:
- assumes "f meromorphic_on A pts" "g meromorphic_on A pts"
- shows "(\<lambda>z. f z + g z) meromorphic_on A pts"
- unfolding meromorphic_on_def
- by (use assms in \<open>auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros\<close>)
+lemma meromorphic_on_Un:
+ assumes "f meromorphic_on A" "f meromorphic_on B"
+ shows "f meromorphic_on (A \<union> B)"
+ using assms unfolding meromorphic_on_def by blast
-lemma meromorphic_on_add':
- assumes "f meromorphic_on A pts1" "g meromorphic_on A pts2"
- shows "(\<lambda>z. f z + g z) meromorphic_on A (pts1 \<union> pts2)"
-proof (rule meromorphic_intros)
- show "f meromorphic_on A (pts1 \<union> pts2)"
- by (rule meromorphic_on_superset_pts[OF assms(1)])
- (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>)
- show "g meromorphic_on A (pts1 \<union> pts2)"
- by (rule meromorphic_on_superset_pts[OF assms(2)])
- (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>)
-qed
-
-lemma meromorphic_on_add_const [meromorphic_intros]:
- assumes "f meromorphic_on A pts"
- shows "(\<lambda>z. f z + c) meromorphic_on A pts"
- unfolding meromorphic_on_def
- by (use assms in \<open>auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros\<close>)
-
-lemma meromorphic_on_minus_const [meromorphic_intros]:
- assumes "f meromorphic_on A pts"
- shows "(\<lambda>z. f z - c) meromorphic_on A pts"
- using meromorphic_on_add_const[OF assms,of "-c"] by simp
-
-lemma meromorphic_on_diff [meromorphic_intros]:
- assumes "f meromorphic_on A pts" "g meromorphic_on A pts"
- shows "(\<lambda>z. f z - g z) meromorphic_on A pts"
- using meromorphic_on_add[OF assms(1) meromorphic_on_uminus[OF assms(2)]] by simp
-
-lemma meromorphic_on_diff':
- assumes "f meromorphic_on A pts1" "g meromorphic_on A pts2"
- shows "(\<lambda>z. f z - g z) meromorphic_on A (pts1 \<union> pts2)"
-proof (rule meromorphic_intros)
- show "f meromorphic_on A (pts1 \<union> pts2)"
- by (rule meromorphic_on_superset_pts[OF assms(1)])
- (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>)
- show "g meromorphic_on A (pts1 \<union> pts2)"
- by (rule meromorphic_on_superset_pts[OF assms(2)])
- (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>)
-qed
-
-lemma meromorphic_on_mult [meromorphic_intros]:
- assumes "f meromorphic_on A pts" "g meromorphic_on A pts"
- shows "(\<lambda>z. f z * g z) meromorphic_on A pts"
- unfolding meromorphic_on_def
- by (use assms in \<open>auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros\<close>)
-
-lemma meromorphic_on_mult':
- assumes "f meromorphic_on A pts1" "g meromorphic_on A pts2"
- shows "(\<lambda>z. f z * g z) meromorphic_on A (pts1 \<union> pts2)"
-proof (rule meromorphic_intros)
- show "f meromorphic_on A (pts1 \<union> pts2)"
- by (rule meromorphic_on_superset_pts[OF assms(1)])
- (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>)
- show "g meromorphic_on A (pts1 \<union> pts2)"
- by (rule meromorphic_on_superset_pts[OF assms(2)])
- (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>)
-qed
-
+lemma meromorphic_on_Union:
+ assumes "\<And>A. A \<in> B \<Longrightarrow> f meromorphic_on A"
+ shows "f meromorphic_on (\<Union>B)"
+ using assms unfolding meromorphic_on_def by blast
-
-lemma meromorphic_on_imp_not_essential:
- assumes "f meromorphic_on A pts" "z \<in> A"
- shows "not_essential f z"
-proof (cases "z \<in> pts")
- case False
- thus ?thesis
- using not_essential_holomorphic[of f "A - pts" z] meromorphic_imp_open_diff[OF assms(1)] assms
- by (auto simp: meromorphic_on_def)
-qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
-
-lemma meromorphic_imp_analytic: "f meromorphic_on D pts \<Longrightarrow> f analytic_on (D - pts)"
- unfolding meromorphic_on_def
- apply (subst analytic_on_open)
- using meromorphic_imp_open_diff meromorphic_on_id apply blast
- apply auto
- done
-
-lemma not_islimpt_isolated_zeros:
- assumes mero: "f meromorphic_on A pts" and "z \<in> A"
- shows "\<not>z islimpt {w\<in>A. isolated_zero f w}"
-proof
- assume islimpt: "z islimpt {w\<in>A. isolated_zero f w}"
- have holo: "f holomorphic_on A - pts" and "open A"
- using assms by (auto simp: meromorphic_on_def)
- have open': "open (A - (pts - {z}))"
- by (intro meromorphic_imp_open_diff'[OF mero]) auto
- then obtain r where r: "r > 0" "ball z r \<subseteq> A - (pts - {z})"
- using meromorphic_imp_open_diff[OF mero] \<open>z \<in> A\<close> openE by blast
-
- have "not_essential f z"
- using assms by (rule meromorphic_on_imp_not_essential)
- then consider c where "f \<midarrow>z\<rightarrow> c" | "is_pole f z"
- unfolding not_essential_def by blast
- thus False
- proof cases
- assume "is_pole f z"
- hence "eventually (\<lambda>w. f w \<noteq> 0) (at z)"
- by (rule non_zero_neighbour_pole)
- hence "\<not>z islimpt {w. f w = 0}"
- by (simp add: islimpt_conv_frequently_at frequently_def)
- moreover have "z islimpt {w. f w = 0}"
- using islimpt by (rule islimpt_subset) (auto simp: isolated_zero_def)
- ultimately show False by contradiction
- next
- fix c assume c: "f \<midarrow>z\<rightarrow> c"
- define g where "g = (\<lambda>w. if w = z then c else f w)"
- have holo': "g holomorphic_on A - (pts - {z})" unfolding g_def
- by (intro removable_singularity holomorphic_on_subset[OF holo] open' c) auto
+lemma meromorphic_on_UN:
+ assumes "\<And>x. x \<in> X \<Longrightarrow> f meromorphic_on (A x)"
+ shows "f meromorphic_on (\<Union>x\<in>X. A x)"
+ using assms unfolding meromorphic_on_def by blast
- have eq_zero: "g w = 0" if "w \<in> ball z r" for w
- proof (rule analytic_continuation[where f = g])
- show "open (ball z r)" "connected (ball z r)" "{w\<in>ball z r. isolated_zero f w} \<subseteq> ball z r"
- by auto
- have "z islimpt {w\<in>A. isolated_zero f w} \<inter> ball z r"
- using islimpt \<open>r > 0\<close> by (intro islimpt_Int_eventually eventually_at_in_open') auto
- also have "\<dots> = {w\<in>ball z r. isolated_zero f w}"
- using r by auto
- finally show "z islimpt {w\<in>ball z r. isolated_zero f w}"
- by simp
- next
- fix w assume w: "w \<in> {w\<in>ball z r. isolated_zero f w}"
- show "g w = 0"
- proof (cases "w = z")
- case False
- thus ?thesis using w by (auto simp: g_def isolated_zero_def)
- next
- case True
- have "z islimpt {z. f z = 0}"
- using islimpt by (rule islimpt_subset) (auto simp: isolated_zero_def)
- thus ?thesis
- using w by (simp add: isolated_zero_altdef True)
- qed
- qed (use r that in \<open>auto intro!: holomorphic_on_subset[OF holo'] simp: isolated_zero_def\<close>)
-
- have "infinite ({w\<in>A. isolated_zero f w} \<inter> ball z r)"
- using islimpt \<open>r > 0\<close> unfolding islimpt_eq_infinite_ball by blast
- hence "{w\<in>A. isolated_zero f w} \<inter> ball z r \<noteq> {}"
- by force
- then obtain z0 where z0: "z0 \<in> A" "isolated_zero f z0" "z0 \<in> ball z r"
- by blast
- have "\<forall>\<^sub>F y in at z0. y \<in> ball z r - (if z = z0 then {} else {z}) - {z0}"
- using r z0 by (intro eventually_at_in_open) auto
- hence "eventually (\<lambda>w. f w = 0) (at z0)"
- proof eventually_elim
- case (elim w)
- show ?case
- using eq_zero[of w] elim by (auto simp: g_def split: if_splits)
- qed
- hence "eventually (\<lambda>w. f w = 0) (at z0)"
- by (auto simp: g_def eventually_at_filter elim!: eventually_mono split: if_splits)
- moreover from z0 have "eventually (\<lambda>w. f w \<noteq> 0) (at z0)"
- by (simp add: isolated_zero_def)
- ultimately have "eventually (\<lambda>_. False) (at z0)"
- by eventually_elim auto
- thus False
- by simp
- qed
-qed
-
-lemma closedin_isolated_zeros:
- assumes "f meromorphic_on A pts"
- shows "closedin (top_of_set A) {z\<in>A. isolated_zero f z}"
- unfolding closedin_limpt using not_islimpt_isolated_zeros[OF assms] by auto
+lemma meromorphic_on_imp_has_laurent_expansion:
+ assumes "f meromorphic_on A" "z \<in> A"
+ shows "(\<lambda>w. f (z + w)) has_laurent_expansion laurent_expansion f z"
+ using assms laurent_expansion_eqI unfolding meromorphic_on_def by blast
-lemma meromorphic_on_deriv':
- assumes "f meromorphic_on A pts" "open A"
- assumes "\<And>x. x \<in> A - pts \<Longrightarrow> (f has_field_derivative f' x) (at x)"
- shows "f' meromorphic_on A pts"
- unfolding meromorphic_on_def
-proof (intro conjI ballI)
- have "open (A - pts)"
- by (intro meromorphic_imp_open_diff[OF assms(1)])
- thus "f' holomorphic_on A - pts"
- by (rule derivative_is_holomorphic) (use assms in auto)
-next
- fix z assume "z \<in> pts"
- hence "z \<in> A"
- using assms(1) by (auto simp: meromorphic_on_def)
- from \<open>z \<in> pts\<close> obtain r where r: "r > 0" "f analytic_on ball z r - {z}"
- using assms(1) by (auto simp: meromorphic_on_def isolated_singularity_at_def)
-
- have "open (ball z r \<inter> (A - (pts - {z})))"
- by (intro open_Int assms meromorphic_imp_open_diff'[OF assms(1)]) auto
- then obtain r' where r': "r' > 0" "ball z r' \<subseteq> ball z r \<inter> (A - (pts - {z}))"
- using r \<open>z \<in> A\<close> by (subst (asm) open_contains_ball) fastforce
-
- have "open (ball z r' - {z})"
- by auto
- hence "f' holomorphic_on ball z r' - {z}"
- by (rule derivative_is_holomorphic[of _ f]) (use r' in \<open>auto intro!: assms(3)\<close>)
- moreover have "open (ball z r' - {z})"
- by auto
- ultimately show "isolated_singularity_at f' z"
- unfolding isolated_singularity_at_def using \<open>r' > 0\<close>
- by (auto simp: analytic_on_open intro!: exI[of _ r'])
-next
- fix z assume z: "z \<in> pts"
- hence z': "not_essential f z" "z \<in> A"
- using assms by (auto simp: meromorphic_on_def)
- from z'(1) show "not_essential f' z"
- proof (rule not_essential_deriv')
- show "z \<in> A - (pts - {z})"
- using \<open>z \<in> A\<close> by blast
- show "open (A - (pts - {z}))"
- by (intro meromorphic_imp_open_diff'[OF assms(1)]) auto
- qed (use assms in auto)
-qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
-
-lemma meromorphic_on_deriv [meromorphic_intros]:
- assumes "f meromorphic_on A pts" "open A"
- shows "deriv f meromorphic_on A pts"
-proof (intro meromorphic_on_deriv'[OF assms(1)])
- have *: "open (A - pts)"
- by (intro meromorphic_imp_open_diff[OF assms(1)])
- show "(f has_field_derivative deriv f x) (at x)" if "x \<in> A - pts" for x
- using assms(1) by (intro holomorphic_derivI[OF _ * that]) (auto simp: meromorphic_on_def)
-qed fact
-
-lemma meromorphic_on_imp_analytic_at:
- assumes "f meromorphic_on A pts" "z \<in> A - pts"
- shows "f analytic_on {z}"
- using assms by (metis analytic_at meromorphic_imp_open_diff meromorphic_on_def)
-
-lemma meromorphic_compact_finite_pts:
- assumes "f meromorphic_on D pts" "compact S" "S \<subseteq> D"
- shows "finite (S \<inter> pts)"
+lemma meromorphic_on_open_nhd:
+ assumes "f meromorphic_on A"
+ obtains B where "open B" "A \<subseteq> B" "f meromorphic_on B"
proof -
- { assume "infinite (S \<inter> pts)"
- then obtain z where "z \<in> S" and z: "z islimpt (S \<inter> pts)"
- using assms by (metis compact_eq_Bolzano_Weierstrass inf_le1)
- then have False
- using assms by (meson in_mono inf_le2 islimpt_subset meromorphic_on_def) }
- then show ?thesis by metis
-qed
-
-lemma meromorphic_imp_countable:
- assumes "f meromorphic_on D pts"
- shows "countable pts"
-proof -
- obtain K :: "nat \<Rightarrow> complex set" where K: "D = (\<Union>n. K n)" "\<And>n. compact (K n)"
- using assms unfolding meromorphic_on_def by (metis open_Union_compact_subsets)
- then have "pts = (\<Union>n. K n \<inter> pts)"
- using assms meromorphic_on_def by auto
- moreover have "\<And>n. finite (K n \<inter> pts)"
- by (metis K(1) K(2) UN_I assms image_iff meromorphic_compact_finite_pts rangeI subset_eq)
- ultimately show ?thesis
- by (metis countableI_type countable_UN countable_finite)
-qed
-
-lemma meromorphic_imp_connected_diff':
- assumes "f meromorphic_on D pts" "connected D" "pts' \<subseteq> pts"
- shows "connected (D - pts')"
-proof (rule connected_open_diff_countable)
- show "countable pts'"
- by (rule countable_subset [OF assms(3)]) (use assms(1) in \<open>auto simp: meromorphic_imp_countable\<close>)
-qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
-
-lemma meromorphic_imp_connected_diff:
- assumes "f meromorphic_on D pts" "connected D"
- shows "connected (D - pts)"
- using meromorphic_imp_connected_diff'[OF assms order.refl] .
-
-lemma meromorphic_on_compose [meromorphic_intros]:
- assumes f: "f meromorphic_on A pts" and g: "g holomorphic_on B"
- assumes "open B" and "g ` B \<subseteq> A"
- shows "(\<lambda>x. f (g x)) meromorphic_on B (isolated_points_of (g -` pts \<inter> B))"
- unfolding meromorphic_on_def
-proof (intro ballI conjI)
- fix z assume z: "z \<in> isolated_points_of (g -` pts \<inter> B)"
- hence z': "z \<in> B" "g z \<in> pts"
- using isolated_points_of_subset by blast+
- have g': "g analytic_on {z}"
- using g z' \<open>open B\<close> analytic_at by blast
+ obtain F where F: "\<And>z. z \<in> A \<Longrightarrow> (\<lambda>w. f (z + w)) has_laurent_expansion F z"
+ using assms unfolding meromorphic_on_def by metis
+ have "\<exists>Z. open Z \<and> z \<in> Z \<and> (\<forall>w\<in>Z-{z}. eval_fls (F z) (w - z) = f w)" if z: "z \<in> A" for z
+ proof -
+ obtain Z where Z: "open Z" "0 \<in> Z" "\<And>w. w \<in> Z - {0} \<Longrightarrow> eval_fls (F z) w = f (z + w)"
+ using F[OF z] unfolding has_laurent_expansion_def eventually_at_topological by blast
+ hence "open ((+) z ` Z)" and "z \<in> (+) z ` Z"
+ using open_translation by auto
+ moreover have "eval_fls (F z) (w - z) = f w" if "w \<in> (+) z ` Z - {z}" for w
+ using Z(3)[of "w - z"] that by auto
+ ultimately show ?thesis by blast
+ qed
+ then obtain Z where Z:
+ "\<And>z. z \<in> A \<Longrightarrow> open (Z z) \<and> z \<in> Z z \<and> (\<forall>w\<in>Z z-{z}. eval_fls (F z) (w - z) = f w)"
+ by metis
- show "isolated_singularity_at (\<lambda>x. f (g x)) z"
- by (rule isolated_singularity_at_compose[OF _ g']) (use f z' in \<open>auto simp: meromorphic_on_def\<close>)
- show "not_essential (\<lambda>x. f (g x)) z"
- by (rule not_essential_compose[OF _ g']) (use f z' in \<open>auto simp: meromorphic_on_def\<close>)
-next
- fix z assume z: "z \<in> B"
- hence "g z \<in> A"
- using assms by auto
- hence "\<not>g z islimpt pts"
- using f by (auto simp: meromorphic_on_def)
- hence ev: "eventually (\<lambda>w. w \<notin> pts) (at (g z))"
- by (auto simp: islimpt_conv_frequently_at frequently_def)
- have g': "g analytic_on {z}"
- by (rule holomorphic_on_imp_analytic_at[OF g]) (use assms z in auto)
-
- (* TODO: There's probably a useful lemma somewhere in here to extract... *)
- have "eventually (\<lambda>w. w \<notin> isolated_points_of (g -` pts \<inter> B)) (at z)"
- proof (cases "isolated_zero (\<lambda>w. g w - g z) z")
- case True
- have "eventually (\<lambda>w. w \<notin> pts) (at (g z))"
- using ev by (auto simp: islimpt_conv_frequently_at frequently_def)
- moreover have "g \<midarrow>z\<rightarrow> g z"
- using analytic_at_imp_isCont[OF g'] isContD by blast
- hence lim: "filterlim g (at (g z)) (at z)"
- using True by (auto simp: filterlim_at isolated_zero_def)
- have "eventually (\<lambda>w. g w \<notin> pts) (at z)"
- using ev lim by (rule eventually_compose_filterlim)
- thus ?thesis
- by eventually_elim (auto simp: isolated_points_of_def)
- next
- case False
- have "eventually (\<lambda>w. g w - g z = 0) (nhds z)"
- using False by (rule non_isolated_zero) (auto intro!: analytic_intros g')
- hence "eventually (\<lambda>w. g w = g z \<and> w \<in> B) (nhds z)"
- using eventually_nhds_in_open[OF \<open>open B\<close> \<open>z \<in> B\<close>]
- by eventually_elim auto
- then obtain X where X: "open X" "z \<in> X" "X \<subseteq> B" "\<forall>x\<in>X. g x = g z"
- unfolding eventually_nhds by blast
-
- have "z0 \<notin> isolated_points_of (g -` pts \<inter> B)" if "z0 \<in> X" for z0
- proof (cases "g z \<in> pts")
- case False
- with that have "g z0 \<notin> pts"
- using X by metis
- thus ?thesis
- by (auto simp: isolated_points_of_def)
- next
- case True
- have "eventually (\<lambda>w. w \<in> X) (at z0)"
- by (intro eventually_at_in_open') fact+
- hence "eventually (\<lambda>w. w \<in> g -` pts \<inter> B) (at z0)"
- by eventually_elim (use X True in fastforce)
- hence "frequently (\<lambda>w. w \<in> g -` pts \<inter> B) (at z0)"
- by (meson at_neq_bot eventually_frequently)
- thus "z0 \<notin> isolated_points_of (g -` pts \<inter> B)"
- unfolding isolated_points_of_def by (auto simp: frequently_def)
- qed
- moreover have "eventually (\<lambda>x. x \<in> X) (at z)"
- by (intro eventually_at_in_open') fact+
- ultimately show ?thesis
- by (auto elim!: eventually_mono)
- qed
- thus "\<not>z islimpt isolated_points_of (g -` pts \<inter> B)"
- by (auto simp: islimpt_conv_frequently_at frequently_def)
-next
- have "f \<circ> g analytic_on (\<Union>z\<in>B - isolated_points_of (g -` pts \<inter> B). {z})"
- unfolding analytic_on_UN
- proof
- fix z assume z: "z \<in> B - isolated_points_of (g -` pts \<inter> B)"
- hence "z \<in> B" by blast
- have g': "g analytic_on {z}"
- by (rule holomorphic_on_imp_analytic_at[OF g]) (use assms z in auto)
- show "f \<circ> g analytic_on {z}"
- proof (cases "g z \<in> pts")
- case False
- show ?thesis
- proof (rule analytic_on_compose)
- show "f analytic_on g ` {z}" using False z assms
- by (auto intro!: meromorphic_on_imp_analytic_at[OF f])
- qed fact
- next
- case True
- show ?thesis
- proof (cases "isolated_zero (\<lambda>w. g w - g z) z")
+ define B where "B = (\<Union>z\<in>A. Z z \<inter> eball z (fls_conv_radius (F z)))"
+ show ?thesis
+ proof (rule that[of B])
+ show "open B"
+ using Z unfolding B_def by auto
+ show "A \<subseteq> B"
+ unfolding B_def using F Z by (auto simp: has_laurent_expansion_def zero_ereal_def)
+ show "f meromorphic_on B"
+ unfolding meromorphic_on_def
+ proof
+ fix z assume z: "z \<in> B"
+ show "\<exists>F. (\<lambda>w. f (z + w)) has_laurent_expansion F"
+ proof (cases "z \<in> A")
+ case True
+ thus ?thesis using F by blast
+ next
case False
- hence "eventually (\<lambda>w. g w - g z = 0) (nhds z)"
- by (rule non_isolated_zero) (auto intro!: analytic_intros g')
- hence "f \<circ> g analytic_on {z} \<longleftrightarrow> (\<lambda>_. f (g z)) analytic_on {z}"
- by (intro analytic_at_cong) (auto elim!: eventually_mono)
- thus ?thesis
- by simp
- next
- case True
- hence ev: "eventually (\<lambda>w. g w \<noteq> g z) (at z)"
- by (auto simp: isolated_zero_def)
-
- have "\<not>g z islimpt pts"
- using \<open>g z \<in> pts\<close> f by (auto simp: meromorphic_on_def)
- hence "eventually (\<lambda>w. w \<notin> pts) (at (g z))"
- by (auto simp: islimpt_conv_frequently_at frequently_def)
- moreover have "g \<midarrow>z\<rightarrow> g z"
- using analytic_at_imp_isCont[OF g'] isContD by blast
- with ev have "filterlim g (at (g z)) (at z)"
- by (auto simp: filterlim_at)
- ultimately have "eventually (\<lambda>w. g w \<notin> pts) (at z)"
- using eventually_compose_filterlim by blast
- hence "z \<in> isolated_points_of (g -` pts \<inter> B)"
- using \<open>g z \<in> pts\<close> \<open>z \<in> B\<close>
- by (auto simp: isolated_points_of_def elim!: eventually_mono)
- with z show ?thesis by simp
+ then obtain z0 where z0: "z0 \<in> A" "z \<in> Z z0 - {z0}" "dist z0 z < fls_conv_radius (F z0)"
+ using z False Z unfolding B_def by auto
+ hence "(\<lambda>w. eval_fls (F z0) (w - z0)) analytic_on {z}"
+ by (intro analytic_on_eval_fls' analytic_intros) (auto simp: dist_norm)
+ also have "?this \<longleftrightarrow> f analytic_on {z}"
+ proof (intro analytic_at_cong refl)
+ have "eventually (\<lambda>w. w \<in> Z z0 - {z0}) (nhds z)"
+ using Z[of z0] z0 by (intro eventually_nhds_in_open) auto
+ thus "\<forall>\<^sub>F x in nhds z. eval_fls (F z0) (x - z0) = f x"
+ by eventually_elim (use Z[of z0] z0 in auto)
+ qed
+ finally show ?thesis
+ using analytic_at_imp_has_fps_expansion has_fps_expansion_to_laurent by blast
qed
qed
qed
- also have "\<dots> = B - isolated_points_of (g -` pts \<inter> B)"
- by blast
- finally show "(\<lambda>x. f (g x)) holomorphic_on B - isolated_points_of (g -` pts \<inter> B)"
- unfolding o_def using analytic_imp_holomorphic by blast
-qed (auto simp: isolated_points_of_def \<open>open B\<close>)
-
-lemma meromorphic_on_compose':
- assumes f: "f meromorphic_on A pts" and g: "g holomorphic_on B"
- assumes "open B" and "g ` B \<subseteq> A" and "pts' = (isolated_points_of (g -` pts \<inter> B))"
- shows "(\<lambda>x. f (g x)) meromorphic_on B pts'"
- using meromorphic_on_compose[OF assms(1-4)] assms(5) by simp
-
-lemma meromorphic_on_inverse': "inverse meromorphic_on UNIV 0"
- unfolding meromorphic_on_def
- by (auto intro!: holomorphic_intros singularity_intros not_essential_inverse
- isolated_singularity_at_inverse simp: islimpt_finite)
-
-lemma meromorphic_on_inverse [meromorphic_intros]:
- assumes mero: "f meromorphic_on A pts"
- shows "(\<lambda>z. inverse (f z)) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero f z})"
-proof -
- have "open A"
- using mero by (auto simp: meromorphic_on_def)
- have open': "open (A - pts)"
- by (intro meromorphic_imp_open_diff[OF mero])
- have holo: "f holomorphic_on A - pts"
- using assms by (auto simp: meromorphic_on_def)
- have ana: "f analytic_on A - pts"
- using open' holo by (simp add: analytic_on_open)
-
- show ?thesis
- unfolding meromorphic_on_def
- proof (intro conjI ballI)
- fix z assume z: "z \<in> pts \<union> {z\<in>A. isolated_zero f z}"
- have "isolated_singularity_at f z \<and> not_essential f z"
- proof (cases "z \<in> pts")
- case False
- have "f holomorphic_on A - pts - {z}"
- by (intro holomorphic_on_subset[OF holo]) auto
- hence "isolated_singularity_at f z"
- by (rule isolated_singularity_at_holomorphic)
- (use z False in \<open>auto intro!: meromorphic_imp_open_diff[OF mero]\<close>)
- moreover have "not_essential f z"
- using z False
- by (intro not_essential_holomorphic[OF holo] meromorphic_imp_open_diff[OF mero]) auto
- ultimately show ?thesis by blast
- qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
- thus "isolated_singularity_at (\<lambda>z. inverse (f z)) z" "not_essential (\<lambda>z. inverse (f z)) z"
- by (auto intro!: isolated_singularity_at_inverse not_essential_inverse)
- next
- fix z assume "z \<in> A"
- hence "\<not> z islimpt {z\<in>A. isolated_zero f z}"
- by (rule not_islimpt_isolated_zeros[OF mero])
- thus "\<not> z islimpt pts \<union> {z \<in> A. isolated_zero f z}" using \<open>z \<in> A\<close>
- using mero by (auto simp: islimpt_Un meromorphic_on_def)
- next
- show "pts \<union> {z \<in> A. isolated_zero f z} \<subseteq> A"
- using mero by (auto simp: meromorphic_on_def)
- next
- have "(\<lambda>z. inverse (f z)) analytic_on (\<Union>w\<in>A - (pts \<union> {z \<in> A. isolated_zero f z}) . {w})"
- unfolding analytic_on_UN
- proof (intro ballI)
- fix w assume w: "w \<in> A - (pts \<union> {z \<in> A. isolated_zero f z})"
- show "(\<lambda>z. inverse (f z)) analytic_on {w}"
- proof (cases "f w = 0")
- case False
- thus ?thesis using w
- by (intro analytic_intros analytic_on_subset[OF ana]) auto
- next
- case True
- have "eventually (\<lambda>w. f w = 0) (nhds w)"
- using True w by (intro non_isolated_zero analytic_on_subset[OF ana]) auto
- hence "(\<lambda>z. inverse (f z)) analytic_on {w} \<longleftrightarrow> (\<lambda>_. 0) analytic_on {w}"
- using w by (intro analytic_at_cong refl) auto
- thus ?thesis
- by simp
- qed
- qed
- also have "\<dots> = A - (pts \<union> {z \<in> A. isolated_zero f z})"
- by blast
- finally have "(\<lambda>z. inverse (f z)) analytic_on \<dots>" .
- moreover have "open (A - (pts \<union> {z \<in> A. isolated_zero f z}))"
- using closedin_isolated_zeros[OF mero] open' \<open>open A\<close>
- by (metis (no_types, lifting) Diff_Diff_Int Diff_Un closedin_closed open_Diff open_Int)
- ultimately show "(\<lambda>z. inverse (f z)) holomorphic_on A - (pts \<union> {z \<in> A. isolated_zero f z})"
- by (subst (asm) analytic_on_open) auto
- qed (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un
- intro!: holomorphic_intros singularity_intros\<close>)
-qed
-
-lemma meromorphic_on_inverse'' [meromorphic_intros]:
- assumes "f meromorphic_on A pts" "{z\<in>A. f z = 0} \<subseteq> pts"
- shows "(\<lambda>z. inverse (f z)) meromorphic_on A pts"
-proof -
- have "(\<lambda>z. inverse (f z)) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})"
- by (intro meromorphic_on_inverse assms)
- also have "(pts \<union> {z \<in> A. isolated_zero f z}) = pts"
- using assms(2) by (auto simp: isolated_zero_def)
- finally show ?thesis .
-qed
-
-lemma meromorphic_on_divide [meromorphic_intros]:
- assumes "f meromorphic_on A pts" and "g meromorphic_on A pts"
- shows "(\<lambda>z. f z / g z) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero g z})"
-proof -
- have mero1: "(\<lambda>z. inverse (g z)) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero g z})"
- by (intro meromorphic_intros assms)
- have sparse: "\<forall>x\<in>A. \<not> x islimpt pts \<union> {z\<in>A. isolated_zero g z}" and "pts \<subseteq> A"
- using mero1 by (auto simp: meromorphic_on_def)
- have mero2: "f meromorphic_on A (pts \<union> {z\<in>A. isolated_zero g z})"
- by (rule meromorphic_on_superset_pts[OF assms(1)]) (use sparse \<open>pts \<subseteq> A\<close> in auto)
- have "(\<lambda>z. f z * inverse (g z)) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero g z})"
- by (intro meromorphic_on_mult mero1 mero2)
- thus ?thesis
- by (simp add: field_simps)
qed
-lemma meromorphic_on_divide' [meromorphic_intros]:
- assumes "f meromorphic_on A pts" "g meromorphic_on A pts" "{z\<in>A. g z = 0} \<subseteq> pts"
- shows "(\<lambda>z. f z / g z) meromorphic_on A pts"
-proof -
- have "(\<lambda>z. f z * inverse (g z)) meromorphic_on A pts"
- by (intro meromorphic_intros assms)
- thus ?thesis
- by (simp add: field_simps)
-qed
-
-lemma meromorphic_on_cmult_left [meromorphic_intros]:
- assumes "f meromorphic_on A pts"
- shows "(\<lambda>x. c * f x) meromorphic_on A pts"
- using assms by (intro meromorphic_intros) (auto simp: meromorphic_on_def)
-
-lemma meromorphic_on_cmult_right [meromorphic_intros]:
- assumes "f meromorphic_on A pts"
- shows "(\<lambda>x. f x * c) meromorphic_on A pts"
- using assms by (intro meromorphic_intros) (auto simp: meromorphic_on_def)
-
-lemma meromorphic_on_scaleR [meromorphic_intros]:
- assumes "f meromorphic_on A pts"
- shows "(\<lambda>x. c *\<^sub>R f x) meromorphic_on A pts"
- using assms unfolding scaleR_conv_of_real
- by (intro meromorphic_intros) (auto simp: meromorphic_on_def)
+lemma meromorphic_on_not_essential:
+ assumes "f meromorphic_on {z}"
+ shows "not_essential f z"
+ using assms has_laurent_expansion_not_essential unfolding meromorphic_on_def by blast
-lemma meromorphic_on_sum [meromorphic_intros]:
- assumes "\<And>y. y \<in> I \<Longrightarrow> f y meromorphic_on A pts"
- assumes "I \<noteq> {} \<or> open A \<and> pts \<subseteq> A \<and> (\<forall>x\<in>A. \<not>x islimpt pts)"
- shows "(\<lambda>x. \<Sum>y\<in>I. f y x) meromorphic_on A pts"
-proof -
- have *: "open A \<and> pts \<subseteq> A \<and> (\<forall>x\<in>A. \<not>x islimpt pts)"
- using assms(2)
- proof
- assume "I \<noteq> {}"
- then obtain x where "x \<in> I"
- by blast
- from assms(1)[OF this] show ?thesis
- by (auto simp: meromorphic_on_def)
- qed auto
- show ?thesis
- using assms(1)
- by (induction I rule: infinite_finite_induct) (use * in \<open>auto intro!: meromorphic_intros\<close>)
-qed
-
-lemma meromorphic_on_prod [meromorphic_intros]:
- assumes "\<And>y. y \<in> I \<Longrightarrow> f y meromorphic_on A pts"
- assumes "I \<noteq> {} \<or> open A \<and> pts \<subseteq> A \<and> (\<forall>x\<in>A. \<not>x islimpt pts)"
- shows "(\<lambda>x. \<Prod>y\<in>I. f y x) meromorphic_on A pts"
-proof -
- have *: "open A \<and> pts \<subseteq> A \<and> (\<forall>x\<in>A. \<not>x islimpt pts)"
- using assms(2)
- proof
- assume "I \<noteq> {}"
- then obtain x where "x \<in> I"
- by blast
- from assms(1)[OF this] show ?thesis
- by (auto simp: meromorphic_on_def)
- qed auto
- show ?thesis
- using assms(1)
- by (induction I rule: infinite_finite_induct) (use * in \<open>auto intro!: meromorphic_intros\<close>)
-qed
+lemma meromorphic_on_isolated_singularity:
+ assumes "f meromorphic_on {z}"
+ shows "isolated_singularity_at f z"
+ using assms has_laurent_expansion_isolated unfolding meromorphic_on_def by blast
-lemma meromorphic_on_power [meromorphic_intros]:
- assumes "f meromorphic_on A pts"
- shows "(\<lambda>x. f x ^ n) meromorphic_on A pts"
-proof -
- have "(\<lambda>x. \<Prod>i\<in>{..<n}. f x) meromorphic_on A pts"
- by (intro meromorphic_intros assms(1)) (use assms in \<open>auto simp: meromorphic_on_def\<close>)
- thus ?thesis
- by simp
-qed
-
-lemma meromorphic_on_power_int [meromorphic_intros]:
- assumes "f meromorphic_on A pts"
- shows "(\<lambda>z. f z powi n) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})"
+lemma meromorphic_on_imp_not_islimpt_singularities:
+ assumes "f meromorphic_on A" "z \<in> A"
+ shows "\<not>z islimpt {w. \<not>f analytic_on {w}}"
proof -
- have inv: "(\<lambda>x. inverse (f x)) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})"
- by (intro meromorphic_intros assms)
- have *: "f meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})"
- by (intro meromorphic_on_superset_pts [OF assms(1)])
- (use inv in \<open>auto simp: meromorphic_on_def\<close>)
- show ?thesis
- proof (cases "n \<ge> 0")
- case True
- have "(\<lambda>x. f x ^ nat n) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})"
- by (intro meromorphic_intros *)
- thus ?thesis
- using True by (simp add: power_int_def)
- next
- case False
- have "(\<lambda>x. inverse (f x) ^ nat (-n)) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})"
- by (intro meromorphic_intros assms)
- thus ?thesis
- using False by (simp add: power_int_def)
- qed
-qed
-
-lemma meromorphic_on_power_int' [meromorphic_intros]:
- assumes "f meromorphic_on A pts" "n \<ge> 0 \<or> (\<forall>z\<in>A. isolated_zero f z \<longrightarrow> z \<in> pts)"
- shows "(\<lambda>z. f z powi n) meromorphic_on A pts"
-proof (cases "n \<ge> 0")
- case True
- have "(\<lambda>z. f z ^ nat n) meromorphic_on A pts"
- by (intro meromorphic_intros assms)
- thus ?thesis
- using True by (simp add: power_int_def)
-next
- case False
- have "(\<lambda>z. f z powi n) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero f z})"
- by (rule meromorphic_on_power_int) fact
- also from assms(2) False have "pts \<union> {z\<in>A. isolated_zero f z} = pts"
- by auto
- finally show ?thesis .
-qed
-
-lemma has_laurent_expansion_on_imp_meromorphic_on:
- assumes "open A"
- assumes laurent: "\<And>z. z \<in> A \<Longrightarrow> \<exists>F. (\<lambda>w. f (z + w)) has_laurent_expansion F"
- shows "f meromorphic_on A {z\<in>A. \<not>f analytic_on {z}}"
- unfolding meromorphic_on_def
-proof (intro conjI ballI)
- fix z assume "z \<in> {z\<in>A. \<not>f analytic_on {z}}"
- then obtain F where F: "(\<lambda>w. f (z + w)) has_laurent_expansion F"
- using laurent[of z] by blast
- from F show "not_essential f z" "isolated_singularity_at f z"
- using has_laurent_expansion_not_essential has_laurent_expansion_isolated by blast+
-next
- fix z assume z: "z \<in> A"
+ obtain B where B: "open B" "A \<subseteq> B" "f meromorphic_on B"
+ using assms meromorphic_on_open_nhd by blast
obtain F where F: "(\<lambda>w. f (z + w)) has_laurent_expansion F"
- using laurent[of z] \<open>z \<in> A\<close> by blast
+ using B assms(2) unfolding meromorphic_on_def by blast
from F have "isolated_singularity_at f z"
- using has_laurent_expansion_isolated z by blast
+ using has_laurent_expansion_isolated assms(2) by blast
then obtain r where r: "r > 0" "f analytic_on ball z r - {z}"
unfolding isolated_singularity_at_def by blast
have "f analytic_on {w}" if "w \<in> ball z r - {z}" for w
by (rule analytic_on_subset[OF r(2)]) (use that in auto)
hence "eventually (\<lambda>w. f analytic_on {w}) (at z)"
using eventually_at_in_open[of "ball z r" z] \<open>r > 0\<close> by (auto elim!: eventually_mono)
- hence "\<not>z islimpt {w. \<not>f analytic_on {w}}"
+ thus "\<not>z islimpt {w. \<not>f analytic_on {w}}"
by (auto simp: islimpt_conv_frequently_at frequently_def)
- thus "\<not>z islimpt {w\<in>A. \<not>f analytic_on {w}}"
- using islimpt_subset[of z "{w\<in>A. \<not>f analytic_on {w}}" "{w. \<not>f analytic_on {w}}"] by blast
-next
- have "f analytic_on A - {w\<in>A. \<not>f analytic_on {w}}"
+qed
+
+lemma meromorphic_on_imp_sparse_singularities:
+ assumes "f meromorphic_on A"
+ shows "{w. \<not>f analytic_on {w}} sparse_in A"
+ by (metis assms meromorphic_on_imp_not_islimpt_singularities
+ meromorphic_on_open_nhd sparse_in_open sparse_in_subset)
+
+lemma meromorphic_on_imp_sparse_singularities':
+ assumes "f meromorphic_on A"
+ shows "{w\<in>A. \<not>f analytic_on {w}} sparse_in A"
+ using meromorphic_on_imp_sparse_singularities[OF assms]
+ by (rule sparse_in_subset2) auto
+
+lemma meromorphic_onE:
+ assumes "f meromorphic_on A"
+ obtains pts where "pts \<subseteq> A" "pts sparse_in A" "f analytic_on A - pts"
+ "\<And>z. z \<in> A \<Longrightarrow> not_essential f z" "\<And>z. z \<in> A \<Longrightarrow> isolated_singularity_at f z"
+proof (rule that)
+ show "{z \<in> A. \<not> f analytic_on {z}} sparse_in A"
+ using assms by (rule meromorphic_on_imp_sparse_singularities')
+ show "f analytic_on A - {z \<in> A. \<not> f analytic_on {z}}"
by (subst analytic_on_analytic_at) auto
- thus "f holomorphic_on A - {w\<in>A. \<not>f analytic_on {w}}"
- by (meson analytic_imp_holomorphic)
-qed (use assms in auto)
+qed (use assms in \<open>auto intro: meromorphic_on_isolated_singularity meromorphic_on_not_essential meromorphic_on_subset\<close>)
-lemma meromorphic_on_imp_has_laurent_expansion:
- assumes "f meromorphic_on A pts" "z \<in> A"
- shows "(\<lambda>w. f (z + w)) has_laurent_expansion laurent_expansion f z"
-proof (cases "z \<in> pts")
- case True
- thus ?thesis
- using assms by (intro not_essential_has_laurent_expansion) (auto simp: meromorphic_on_def)
-next
- case False
- have "f holomorphic_on (A - pts)"
- using assms by (auto simp: meromorphic_on_def)
- moreover have "z \<in> A - pts" "open (A - pts)"
- using assms(2) False by (auto intro!: meromorphic_imp_open_diff[OF assms(1)])
- ultimately have "f analytic_on {z}"
- unfolding analytic_at by blast
- thus ?thesis
- using isolated_singularity_at_analytic not_essential_analytic
- not_essential_has_laurent_expansion by blast
-qed
-
-lemma
- assumes "isolated_singularity_at f z" "f \<midarrow>z\<rightarrow> c"
- shows eventually_remove_sings_eq_nhds':
- "eventually (\<lambda>w. remove_sings f w = (if w = z then c else f w)) (nhds z)"
- and remove_sings_analytic_at_singularity: "remove_sings f analytic_on {z}"
-proof -
- have "eventually (\<lambda>w. w \<noteq> z) (at z)"
- by (auto simp: eventually_at_filter)
- hence "eventually (\<lambda>w. remove_sings f w = (if w = z then c else f w)) (at z)"
- using eventually_remove_sings_eq_at[OF assms(1)]
- by eventually_elim auto
- moreover have "remove_sings f z = c"
- using assms by auto
- ultimately show ev: "eventually (\<lambda>w. remove_sings f w = (if w = z then c else f w)) (nhds z)"
- by (simp add: eventually_at_filter)
-
- have "(\<lambda>w. if w = z then c else f w) analytic_on {z}"
- by (intro removable_singularity' assms)
- also have "?this \<longleftrightarrow> remove_sings f analytic_on {z}"
- using ev by (intro analytic_at_cong) (auto simp: eq_commute)
- finally show \<dots> .
+lemma meromorphic_onI_weak:
+ assumes "f analytic_on A - pts" "\<And>z. z \<in> pts \<Longrightarrow> not_essential f z" "pts sparse_in A"
+ "pts \<inter> frontier A = {}"
+ shows "f meromorphic_on A"
+ unfolding meromorphic_on_def
+proof
+ fix z assume z: "z \<in> A"
+ show "(\<exists>F. (\<lambda>w. f (z + w)) has_laurent_expansion F)"
+ proof (cases "z \<in> pts")
+ case False
+ have "f analytic_on {z}"
+ using assms(1) by (rule analytic_on_subset) (use False z in auto)
+ thus ?thesis
+ using isolated_singularity_at_analytic not_essential_analytic
+ not_essential_has_laurent_expansion by blast
+ next
+ case True
+ show ?thesis
+ proof (rule exI, rule not_essential_has_laurent_expansion)
+ show "not_essential f z"
+ using assms(2) True by blast
+ next
+ show "isolated_singularity_at f z"
+ proof (rule isolated_singularity_at_holomorphic)
+ show "open (interior A - (pts - {z}))"
+ proof (rule open_diff_sparse_pts)
+ show "pts - {z} sparse_in interior A"
+ using sparse_in_subset sparse_in_subset2 assms interior_subset Diff_subset by metis
+ qed auto
+ next
+ have "f analytic_on interior A - (pts - {z}) - {z}"
+ using assms(1) by (rule analytic_on_subset) (use interior_subset in blast)
+ thus "f holomorphic_on interior A - (pts - {z}) - {z}"
+ by (rule analytic_imp_holomorphic)
+ next
+ from assms(4) and True have "z \<in> interior A"
+ unfolding frontier_def using closure_subset z by blast
+ thus "z \<in> interior A - (pts - {z})"
+ by blast
+ qed
+ qed
+ qed
qed
-lemma remove_sings_meromorphic_on:
- assumes "f meromorphic_on A pts" "\<And>z. z \<in> pts - pts' \<Longrightarrow> \<not>is_pole f z" "pts' \<subseteq> pts"
- shows "remove_sings f meromorphic_on A pts'"
+lemma meromorphic_onI_open:
+ assumes "open A" "f analytic_on A - pts" "\<And>z. z \<in> pts \<Longrightarrow> not_essential f z"
+ assumes "\<And>z. z \<in> A \<Longrightarrow> \<not>z islimpt pts \<inter> A"
+ shows "f meromorphic_on A"
+proof (rule meromorphic_onI_weak)
+ have *: "A - pts \<inter> A = A - pts"
+ by blast
+ show "f analytic_on A - pts \<inter> A"
+ unfolding * by fact
+ show "pts \<inter> A sparse_in A"
+ using assms(1,4) by (subst sparse_in_open) auto
+ show "not_essential f z" if "z \<in> pts \<inter> A" for z
+ using assms(3) that by blast
+ show "pts \<inter> A \<inter> frontier A = {}"
+ using \<open>open A\<close> frontier_disjoint_eq by blast
+qed
+
+lemma meromorphic_at_isCont_imp_analytic:
+ assumes "f meromorphic_on {z}" "isCont f z"
+ shows "f analytic_on {z}"
+proof -
+ have *: "(\<lambda>w. f (z + w)) has_laurent_expansion laurent_expansion f z"
+ using assms by (auto intro: meromorphic_on_imp_has_laurent_expansion)
+ from assms have "\<not>is_pole f z"
+ using is_pole_def not_tendsto_and_filterlim_at_infinity trivial_limit_at by (metis isContD)
+ with * have "fls_subdegree (laurent_expansion f z) \<ge> 0"
+ using has_laurent_expansion_imp_is_pole linorder_not_le by blast
+ hence **: "(\<lambda>w. eval_fls (laurent_expansion f z) (w - z)) analytic_on {z}"
+ by (intro analytic_intros)+ (use * in \<open>auto simp: has_laurent_expansion_def zero_ereal_def\<close>)
+ have "(\<lambda>w. eval_fls (laurent_expansion f z) (w - z)) \<midarrow>z\<rightarrow> eval_fls (laurent_expansion f z) (z - z)"
+ by (intro isContD analytic_at_imp_isCont **)
+ also have "?this \<longleftrightarrow> f \<midarrow>z\<rightarrow> eval_fls (laurent_expansion f z) (z - z)"
+ by (intro filterlim_cong refl)
+ (use * in \<open>auto simp: has_laurent_expansion_def at_to_0' eventually_filtermap add_ac\<close>)
+ finally have "f \<midarrow>z\<rightarrow> eval_fls (laurent_expansion f z) 0"
+ by simp
+ moreover from assms have "f \<midarrow>z\<rightarrow> f z"
+ by (auto intro: isContD)
+ ultimately have ***: "eval_fls (laurent_expansion f z) 0 = f z"
+ by (rule LIM_unique)
+
+ have "eventually (\<lambda>w. f w = eval_fls (laurent_expansion f z) (w - z)) (at z)"
+ using * by (simp add: has_laurent_expansion_def at_to_0' eventually_filtermap add_ac eq_commute)
+ hence "eventually (\<lambda>w. f w = eval_fls (laurent_expansion f z) (w - z)) (nhds z)"
+ unfolding eventually_at_filter by eventually_elim (use *** in auto)
+ hence "f analytic_on {z} \<longleftrightarrow> (\<lambda>w. eval_fls (laurent_expansion f z) (w - z)) analytic_on {z}"
+ by (intro analytic_at_cong refl)
+ with ** show ?thesis
+ by simp
+qed
+
+lemma analytic_on_imp_meromorphic_on:
+ assumes "f analytic_on A"
+ shows "f meromorphic_on A"
+ by (rule meromorphic_onI_weak[of _ _ "{}"]) (use assms in auto)
+
+lemma meromorphic_on_compose:
+ assumes "g meromorphic_on A" "f analytic_on B" "f ` B \<subseteq> A"
+ shows "(\<lambda>w. g (f w)) meromorphic_on B"
unfolding meromorphic_on_def
proof safe
- have "remove_sings f analytic_on {z}" if "z \<in> A - pts'" for z
- proof (cases "z \<in> pts")
+ fix z assume z: "z \<in> B"
+ have "f analytic_on {z}"
+ using assms(2) by (rule analytic_on_subset) (use assms(3) z in auto)
+ hence "(\<lambda>w. f w - f z) analytic_on {z}"
+ by (intro analytic_intros)
+ then obtain F where F: "(\<lambda>w. f (z + w) - f z) has_fps_expansion F"
+ using analytic_at_imp_has_fps_expansion by blast
+
+ from assms(3) and z have "f z \<in> A"
+ by auto
+ with assms(1) obtain G where G: "(\<lambda>w. g (f z + w)) has_laurent_expansion G"
+ using z by (auto simp: meromorphic_on_def)
+
+ have "\<exists>H. ((\<lambda>w. g (f z + w)) \<circ> (\<lambda>w. f (z + w) - f z)) has_laurent_expansion H"
+ proof (cases "F = 0")
case False
- hence *: "f analytic_on {z}"
- using assms meromorphic_imp_open_diff[OF assms(1)] that
- by (force simp: meromorphic_on_def analytic_at)
- have "remove_sings f analytic_on {z} \<longleftrightarrow> f analytic_on {z}"
- by (intro analytic_at_cong eventually_remove_sings_eq_nhds * refl)
- thus ?thesis using * by simp
+ show ?thesis
+ proof (rule exI, rule has_laurent_expansion_compose)
+ show "(\<lambda>w. f (z + w) - f z) has_laurent_expansion fps_to_fls F"
+ using F by (rule has_laurent_expansion_fps)
+ show "fps_nth F 0 = 0"
+ using has_fps_expansion_imp_0_eq_fps_nth_0[OF F] by simp
+ qed fact+
next
case True
- have isol: "isolated_singularity_at f z"
- using True using assms by (auto simp: meromorphic_on_def)
- from assms(1) have "not_essential f z"
- using True by (auto simp: meromorphic_on_def)
- with assms(2) True that obtain c where "f \<midarrow>z\<rightarrow> c"
- by (auto simp: not_essential_def)
- thus "remove_sings f analytic_on {z}"
- by (intro remove_sings_analytic_at_singularity isol)
+ have "(\<lambda>w. g (f z)) has_laurent_expansion fls_const (g (f z))"
+ by auto
+ also have "?this \<longleftrightarrow> (\<lambda>w. ((\<lambda>w. g (f z + w)) \<circ> (\<lambda>w. f (z + w) - f z)) w)
+ has_laurent_expansion fls_const (g (f z))"
+ proof (rule has_laurent_expansion_cong, goal_cases)
+ case 1
+ from F and True have "eventually (\<lambda>w. f (z + w) - f z = 0) (nhds 0)"
+ by (simp add: has_fps_expansion_0_iff)
+ hence "eventually (\<lambda>w. f (z + w) - f z = 0) (at 0)"
+ by (simp add: eventually_nhds_conv_at)
+ thus ?case
+ by eventually_elim auto
+ qed auto
+ finally show ?thesis
+ by blast
qed
- hence "remove_sings f analytic_on A - pts'"
- by (subst analytic_on_analytic_at) auto
- thus "remove_sings f holomorphic_on A - pts'"
- using meromorphic_imp_open_diff'[OF assms(1,3)] by (subst (asm) analytic_on_open)
-qed (use assms islimpt_subset[OF _ assms(3)] in \<open>auto simp: meromorphic_on_def\<close>)
+ thus "\<exists>H. (\<lambda>w. g (f (z + w))) has_laurent_expansion H"
+ by (simp add: o_def)
+qed
+
+lemma constant_on_imp_meromorphic_on:
+ assumes "f constant_on A" "open A"
+ shows "f meromorphic_on A"
+ using assms analytic_on_imp_meromorphic_on
+ constant_on_imp_analytic_on
+ by blast
+
+subsection \<open>Nice meromorphicity\<close>
-lemma remove_sings_holomorphic_on:
- assumes "f meromorphic_on A pts" "\<And>z. z \<in> pts \<Longrightarrow> \<not>is_pole f z"
- shows "remove_sings f holomorphic_on A"
- using remove_sings_meromorphic_on[OF assms(1), of "{}"] assms(2)
- by (auto simp: meromorphic_on_no_singularities)
+text \<open>
+ This is probably very non-standard, but we call a function ``nicely meromorphic'' if it is
+ meromorphic and has no removable singularities. That means that the only singularities are
+ poles.
+\<close>
+definition nicely_meromorphic_on :: "(complex \<Rightarrow> complex) \<Rightarrow> complex set \<Rightarrow> bool"
+ (infixl "(nicely'_meromorphic'_on)" 50)
+ where "f nicely_meromorphic_on A \<longleftrightarrow> f meromorphic_on A
+ \<and> (\<forall>z\<in>A. (is_pole f z \<and> f z=0) \<or> f \<midarrow>z\<rightarrow> f z)"
+
+lemma constant_on_imp_nicely_meromorphic_on:
+ assumes "f constant_on A" "open A"
+ shows "f nicely_meromorphic_on A"
+ by (meson analytic_at_imp_isCont assms
+ constant_on_imp_holomorphic_on
+ constant_on_imp_meromorphic_on
+ holomorphic_on_imp_analytic_at isCont_def
+ nicely_meromorphic_on_def)
-lemma meromorphic_on_Ex_iff:
- "(\<exists>pts. f meromorphic_on A pts) \<longleftrightarrow>
- open A \<and> (\<forall>z\<in>A. \<exists>F. (\<lambda>w. f (z + w)) has_laurent_expansion F)"
+lemma nicely_meromorphic_on_imp_analytic_at:
+ assumes "f nicely_meromorphic_on A" "z \<in> A" "\<not>is_pole f z"
+ shows "f analytic_on {z}"
+proof (rule meromorphic_at_isCont_imp_analytic)
+ show "f meromorphic_on {z}"
+ by (rule meromorphic_on_subset[of _ A]) (use assms in \<open>auto simp: nicely_meromorphic_on_def\<close>)
+next
+ from assms have "f \<midarrow>z\<rightarrow> f z"
+ by (auto simp: nicely_meromorphic_on_def)
+ thus "isCont f z"
+ by (auto simp: isCont_def)
+qed
+
+lemma remove_sings_meromorphic [meromorphic_intros]:
+ assumes "f meromorphic_on A"
+ shows "remove_sings f meromorphic_on A"
+ unfolding meromorphic_on_def
proof safe
- fix pts assume *: "f meromorphic_on A pts"
- from * show "open A"
- by (auto simp: meromorphic_on_def)
- show "\<exists>F. (\<lambda>w. f (z + w)) has_laurent_expansion F" if "z \<in> A" for z
- using that *
- by (intro exI[of _ "laurent_expansion f z"] meromorphic_on_imp_has_laurent_expansion)
-qed (blast intro!: has_laurent_expansion_on_imp_meromorphic_on)
+ fix z assume z: "z \<in> A"
+ show "\<exists>F. (\<lambda>w. remove_sings f (z + w)) has_laurent_expansion F"
+ using assms meromorphic_on_isolated_singularity meromorphic_on_not_essential
+ not_essential_has_laurent_expansion z meromorphic_on_subset by blast
+qed
-lemma is_pole_inverse_holomorphic_pts:
- fixes pts::"complex set" and f::"complex \<Rightarrow> complex"
- defines "g \<equiv> \<lambda>x. (if x\<in>pts then 0 else inverse (f x))"
- assumes mer: "f meromorphic_on D pts"
- and non_z: "\<And>z. z \<in> D - pts \<Longrightarrow> f z \<noteq> 0"
- and all_poles:"\<forall>x. is_pole f x \<longleftrightarrow> x\<in>pts"
- shows "g holomorphic_on D"
+lemma remove_sings_nicely_meromorphic:
+ assumes "f meromorphic_on A"
+ shows "remove_sings f nicely_meromorphic_on A"
proof -
- have "open D" and f_holo: "f holomorphic_on (D-pts)"
- using mer by (auto simp: meromorphic_on_def)
- have "\<exists>r. r>0 \<and> f analytic_on ball z r - {z}
- \<and> (\<forall>x \<in> ball z r - {z}. f x\<noteq>0)" if "z\<in>pts" for z
+ have "remove_sings f meromorphic_on A"
+ by (simp add: assms remove_sings_meromorphic)
+ moreover have "is_pole (remove_sings f) z
+ \<and> remove_sings f z = 0 \<or>
+ remove_sings f \<midarrow>z\<rightarrow> remove_sings f z"
+ if "z\<in>A" for z
+ proof (cases "\<exists>c. f \<midarrow>z\<rightarrow> c")
+ case True
+ then have "remove_sings f \<midarrow>z\<rightarrow> remove_sings f z"
+ by (metis remove_sings_eqI tendsto_remove_sings_iff
+ assms meromorphic_onE that)
+ then show ?thesis by simp
+ next
+ case False
+ then have "is_pole (remove_sings f) z
+ \<and> remove_sings f z = 0"
+ by (meson is_pole_remove_sings_iff remove_sings_def
+ remove_sings_eq_0_iff assms meromorphic_onE that)
+ then show ?thesis by simp
+ qed
+ ultimately show ?thesis
+ unfolding nicely_meromorphic_on_def by simp
+qed
+
+text \<open>
+ A nicely meromorphic function that frequently takes the same value in the neighbourhood of some
+ point is constant.
+\<close>
+lemma frequently_eq_meromorphic_imp_constant:
+ assumes "frequently (\<lambda>z. f z = c) (at z)"
+ assumes "f nicely_meromorphic_on A" "open A" "connected A" "z \<in> A"
+ shows "\<And>w. w \<in> A \<Longrightarrow> f w = c"
+proof -
+ from assms(2) have mero: "f meromorphic_on A"
+ by (auto simp: nicely_meromorphic_on_def)
+ have sparse: "{z. is_pole f z} sparse_in A"
+ using assms(2) mero
+ by (meson assms(3) meromorphic_on_isolated_singularity meromorphic_on_meromorphic_at not_islimpt_poles sparse_in_open)
+
+ have eq: "f w = c" if w: "w \<in> A" "\<not>is_pole f w" for w
proof -
- have "isolated_singularity_at f z" "is_pole f z"
- using mer meromorphic_on_def that all_poles by blast+
- then obtain r1 where "r1>0" and fan: "f analytic_on ball z r1 - {z}"
- by (meson isolated_singularity_at_def)
- obtain r2 where "r2>0" "\<forall>x \<in> ball z r2 - {z}. f x\<noteq>0"
- using non_zero_neighbour_pole[OF \<open>is_pole f z\<close>]
- unfolding eventually_at by (metis Diff_iff UNIV_I dist_commute insertI1 mem_ball)
- define r where "r = min r1 r2"
- have "r>0" by (simp add: \<open>0 < r2\<close> \<open>r1>0\<close> r_def)
- moreover have "f analytic_on ball z r - {z}"
- using r_def by (force intro: analytic_on_subset [OF fan])
- moreover have "\<forall>x \<in> ball z r - {z}. f x\<noteq>0"
- by (simp add: \<open>\<forall>x\<in>ball z r2 - {z}. f x \<noteq> 0\<close> r_def)
- ultimately show ?thesis by auto
+ have "f w - c = 0"
+ proof (rule analytic_continuation[of "\<lambda>w. f w - c"])
+ show "(\<lambda>w. f w - c) holomorphic_on {z\<in>A. \<not>is_pole f z}" using assms(2)
+ by (intro holomorphic_intros)
+ (metis (mono_tags, lifting) analytic_imp_holomorphic analytic_on_analytic_at
+ mem_Collect_eq nicely_meromorphic_on_imp_analytic_at)
+ next
+ from sparse and assms(3) have "open (A - {z. is_pole f z})"
+ by (simp add: open_diff_sparse_pts)
+ also have "A - {z. is_pole f z} = {z\<in>A. \<not>is_pole f z}"
+ by blast
+ finally show "open \<dots>" .
+ next
+ from sparse have "connected (A - {z. is_pole f z})"
+ using assms(3,4) by (intro sparse_imp_connected) auto
+ also have "A - {z. is_pole f z} = {z\<in>A. \<not>is_pole f z}"
+ by blast
+ finally show "connected \<dots>" .
+ next
+ have "eventually (\<lambda>w. w \<in> A) (at z)"
+ using assms by (intro eventually_at_in_open') auto
+ moreover have "eventually (\<lambda>w. \<not>is_pole f w) (at z)" using mero
+ by (metis assms(5) eventually_not_pole meromorphic_onE)
+ ultimately have ev: "eventually (\<lambda>w. w \<in> A \<and> \<not>is_pole f w) (at z)"
+ by eventually_elim auto
+ show "z islimpt {z\<in>A. \<not>is_pole f z \<and> f z = c}"
+ using frequently_eventually_frequently[OF assms(1) ev]
+ unfolding islimpt_conv_frequently_at by (rule frequently_elim1) auto
+ next
+ from assms(1) have "\<not>is_pole f z"
+ by (simp add: frequently_const_imp_not_is_pole)
+ with \<open>z \<in> A\<close> show "z \<in> {z \<in> A. \<not> is_pole f z}"
+ by auto
+ qed (use w in auto)
+ thus "f w = c"
+ by simp
qed
- then obtain get_r where r_pos:"get_r z>0"
- and r_ana:"f analytic_on ball z (get_r z) - {z}"
- and r_nz:"\<forall>x \<in> ball z (get_r z) - {z}. f x\<noteq>0"
- if "z\<in>pts" for z
- by metis
- define p_balls where "p_balls \<equiv> \<Union>z\<in>pts. ball z (get_r z)"
- have g_ball:"g holomorphic_on ball z (get_r z)" if "z\<in>pts" for z
+
+ have not_pole: "\<not>is_pole f w" if w: "w \<in> A" for w
proof -
- have "(\<lambda>x. if x = z then 0 else inverse (f x)) holomorphic_on ball z (get_r z)"
- proof (rule is_pole_inverse_holomorphic)
- show "f holomorphic_on ball z (get_r z) - {z}"
- using analytic_imp_holomorphic r_ana that by blast
- show "is_pole f z"
- using mer meromorphic_on_def that all_poles by force
- show "\<forall>x\<in>ball z (get_r z) - {z}. f x \<noteq> 0"
- using r_nz that by metis
- qed auto
- then show ?thesis unfolding g_def
- by (smt (verit, ccfv_SIG) Diff_iff Elementary_Metric_Spaces.open_ball
- all_poles analytic_imp_holomorphic empty_iff
- holomorphic_transform insert_iff not_is_pole_holomorphic
- open_delete r_ana that)
+ have "eventually (\<lambda>w. \<not>is_pole f w) (at w)"
+ using mero by (metis eventually_not_pole meromorphic_onE that)
+ moreover have "eventually (\<lambda>w. w \<in> A) (at w)"
+ using w \<open>open A\<close> by (intro eventually_at_in_open')
+ ultimately have "eventually (\<lambda>w. f w = c) (at w)"
+ by eventually_elim (auto simp: eq)
+ hence "is_pole f w \<longleftrightarrow> is_pole (\<lambda>_. c) w"
+ by (intro is_pole_cong refl)
+ thus ?thesis
+ by simp
qed
- then have "g holomorphic_on p_balls"
- proof -
- have "g analytic_on p_balls"
- unfolding p_balls_def analytic_on_UN
- using g_ball by (simp add: analytic_on_open)
- moreover have "open p_balls" using p_balls_def by blast
- ultimately show ?thesis
- by (simp add: analytic_imp_holomorphic)
- qed
- moreover have "g holomorphic_on D-pts"
- proof -
- have "(\<lambda>z. inverse (f z)) holomorphic_on D - pts"
- using f_holo holomorphic_on_inverse non_z by blast
- then show ?thesis
- by (metis DiffD2 g_def holomorphic_transform)
- qed
- moreover have "open p_balls"
- using p_balls_def by blast
- ultimately have "g holomorphic_on (p_balls \<union> (D-pts))"
- by (simp add: holomorphic_on_Un meromorphic_imp_open_diff[OF mer])
- moreover have "D \<subseteq> p_balls \<union> (D-pts)"
- unfolding p_balls_def using \<open>\<And>z. z \<in> pts \<Longrightarrow> 0 < get_r z\<close> by force
- ultimately show "g holomorphic_on D" by (meson holomorphic_on_subset)
+
+ show "f w = c" if w: "w \<in> A" for w
+ using eq[OF w not_pole[OF w]] .
qed
-lemma meromorphic_imp_analytic_on:
- assumes "f meromorphic_on D pts"
- shows "f analytic_on (D - pts)"
- by (metis assms analytic_on_open meromorphic_imp_open_diff meromorphic_on_def)
+subsection \<open>Closure properties and proofs for individual functions\<close>
+
+lemma meromorphic_on_const [intro, meromorphic_intros]: "(\<lambda>_. c) meromorphic_on A"
+ by (rule analytic_on_imp_meromorphic_on) auto
+
+lemma meromorphic_on_id [intro, meromorphic_intros]: "(\<lambda>w. w) meromorphic_on A"
+ by (auto simp: meromorphic_on_def intro!: exI laurent_expansion_intros)
+
+lemma meromorphic_on_id' [intro, meromorphic_intros]: "id meromorphic_on A"
+ by (auto simp: meromorphic_on_def intro!: exI laurent_expansion_intros)
+
+lemma meromorphic_on_add [meromorphic_intros]:
+ assumes "f meromorphic_on A" "g meromorphic_on A"
+ shows "(\<lambda>w. f w + g w) meromorphic_on A"
+ unfolding meromorphic_on_def
+ by (rule laurent_expansion_intros exI ballI
+ assms[THEN meromorphic_on_imp_has_laurent_expansion] | assumption)+
+
+lemma meromorphic_on_uminus [meromorphic_intros]:
+ assumes "f meromorphic_on A"
+ shows "(\<lambda>w. -f w) meromorphic_on A"
+ unfolding meromorphic_on_def
+ by (rule laurent_expansion_intros exI ballI
+ assms[THEN meromorphic_on_imp_has_laurent_expansion] | assumption)+
+
+lemma meromorphic_on_diff [meromorphic_intros]:
+ assumes "f meromorphic_on A" "g meromorphic_on A"
+ shows "(\<lambda>w. f w - g w) meromorphic_on A"
+ using meromorphic_on_add[OF assms(1) meromorphic_on_uminus[OF assms(2)]] by simp
+
+lemma meromorphic_on_mult [meromorphic_intros]:
+ assumes "f meromorphic_on A" "g meromorphic_on A"
+ shows "(\<lambda>w. f w * g w) meromorphic_on A"
+ unfolding meromorphic_on_def
+ by (rule laurent_expansion_intros exI ballI
+ assms[THEN meromorphic_on_imp_has_laurent_expansion] | assumption)+
+
+lemma meromorphic_on_power [meromorphic_intros]:
+ assumes "f meromorphic_on A"
+ shows "(\<lambda>w. f w ^ n) meromorphic_on A"
+ unfolding meromorphic_on_def
+ by (rule laurent_expansion_intros exI ballI
+ assms[THEN meromorphic_on_imp_has_laurent_expansion] | assumption)+
+
+lemma meromorphic_on_powi [meromorphic_intros]:
+ assumes "f meromorphic_on A"
+ shows "(\<lambda>w. f w powi n) meromorphic_on A"
+ unfolding meromorphic_on_def
+ by (rule laurent_expansion_intros exI ballI
+ assms[THEN meromorphic_on_imp_has_laurent_expansion] | assumption)+
-lemma meromorphic_imp_constant_on:
- assumes merf: "f meromorphic_on D pts"
- and "f constant_on (D - pts)"
- and "\<forall>x\<in>pts. is_pole f x"
- shows "f constant_on D"
-proof -
- obtain c where c:"\<And>z. z \<in> D-pts \<Longrightarrow> f z = c"
- by (meson assms constant_on_def)
+lemma meromorphic_on_scaleR [meromorphic_intros]:
+ assumes "f meromorphic_on A"
+ shows "(\<lambda>w. scaleR x (f w)) meromorphic_on A"
+ unfolding meromorphic_on_def
+ by (rule laurent_expansion_intros exI ballI
+ assms[THEN meromorphic_on_imp_has_laurent_expansion] | assumption)+
+
+lemma meromorphic_on_inverse [meromorphic_intros]:
+ assumes "f meromorphic_on A"
+ shows "(\<lambda>w. inverse (f w)) meromorphic_on A"
+ unfolding meromorphic_on_def
+ by (rule laurent_expansion_intros exI ballI
+ assms[THEN meromorphic_on_imp_has_laurent_expansion] | assumption)+
+
+lemma meromorphic_on_divide [meromorphic_intros]:
+ assumes "f meromorphic_on A" "g meromorphic_on A"
+ shows "(\<lambda>w. f w / g w) meromorphic_on A"
+ using meromorphic_on_mult[OF assms(1) meromorphic_on_inverse[OF assms(2)]]
+ by (simp add: field_simps)
+
+lemma meromorphic_on_sum [meromorphic_intros]:
+ assumes "\<And>i. i \<in> I \<Longrightarrow> f i meromorphic_on A"
+ shows "(\<lambda>w. \<Sum>i\<in>I. f i w) meromorphic_on A"
+ unfolding meromorphic_on_def
+ by (rule laurent_expansion_intros exI ballI
+ assms[THEN meromorphic_on_imp_has_laurent_expansion] | assumption)+
+
+lemma meromorphic_on_sum_list [meromorphic_intros]:
+ assumes "\<And>i. i \<in> set fs \<Longrightarrow> f i meromorphic_on A"
+ shows "(\<lambda>w. \<Sum>i\<leftarrow>fs. f i w) meromorphic_on A"
+ unfolding meromorphic_on_def
+ by (rule laurent_expansion_intros exI ballI
+ assms[THEN meromorphic_on_imp_has_laurent_expansion] | assumption)+
+
+lemma meromorphic_on_sum_mset [meromorphic_intros]:
+ assumes "\<And>i. i \<in># I \<Longrightarrow> f i meromorphic_on A"
+ shows "(\<lambda>w. \<Sum>i\<in>#I. f i w) meromorphic_on A"
+ unfolding meromorphic_on_def
+ by (rule laurent_expansion_intros exI ballI
+ assms[THEN meromorphic_on_imp_has_laurent_expansion] | assumption)+
+
+lemma meromorphic_on_prod [meromorphic_intros]:
+ assumes "\<And>i. i \<in> I \<Longrightarrow> f i meromorphic_on A"
+ shows "(\<lambda>w. \<Prod>i\<in>I. f i w) meromorphic_on A"
+ unfolding meromorphic_on_def
+ by (rule laurent_expansion_intros exI ballI
+ assms[THEN meromorphic_on_imp_has_laurent_expansion] | assumption)+
+
+lemma meromorphic_on_prod_list [meromorphic_intros]:
+ assumes "\<And>i. i \<in> set fs \<Longrightarrow> f i meromorphic_on A"
+ shows "(\<lambda>w. \<Prod>i\<leftarrow>fs. f i w) meromorphic_on A"
+ unfolding meromorphic_on_def
+ by (rule laurent_expansion_intros exI ballI
+ assms[THEN meromorphic_on_imp_has_laurent_expansion] | assumption)+
- have "f z = c" if "z \<in> D" for z
- proof (cases "is_pole f z")
- case True
- then obtain r0 where "r0 > 0" and r0: "f analytic_on ball z r0 - {z}" and pol: "is_pole f z"
- using merf unfolding meromorphic_on_def isolated_singularity_at_def
- by (metis \<open>z \<in> D\<close> insert_Diff insert_Diff_if insert_iff merf
- meromorphic_imp_open_diff not_is_pole_holomorphic)
- have "open D"
- using merf meromorphic_on_def by auto
- then obtain r where "r > 0" "ball z r \<subseteq> D" "r \<le> r0"
- by (smt (verit, best) \<open>0 < r0\<close> \<open>z \<in> D\<close> openE order_subst2 subset_ball)
- have r: "f analytic_on ball z r - {z}"
- by (meson Diff_mono \<open>r \<le> r0\<close> analytic_on_subset order_refl r0 subset_ball)
- have "ball z r - {z} \<subseteq> -pts"
- using merf r unfolding meromorphic_on_def
- by (meson ComplI Elementary_Metric_Spaces.open_ball
- analytic_imp_holomorphic assms(3) not_is_pole_holomorphic open_delete subsetI)
- with \<open>ball z r \<subseteq> D\<close> have "ball z r - {z} \<subseteq> D-pts"
- by fastforce
- with c have c': "\<And>u. u \<in> ball z r - {z} \<Longrightarrow> f u = c"
- by blast
- have False if "\<forall>\<^sub>F x in at z. cmod c + 1 \<le> cmod (f x)"
- proof -
- have "\<forall>\<^sub>F x in at z within ball z r - {z}. cmod c + 1 \<le> cmod (f x)"
- by (smt (verit, best) Diff_UNIV Diff_eq_empty_iff eventually_at_topological insert_subset that)
- with \<open>r > 0\<close> show ?thesis
- apply (simp add: c' eventually_at_filter topological_space_class.eventually_nhds open_dist)
- by (metis dist_commute min_less_iff_conj perfect_choose_dist)
- qed
- with pol show ?thesis
- by (auto simp: is_pole_def filterlim_at_infinity_conv_norm_at_top filterlim_at_top)
+lemma meromorphic_on_prod_mset [meromorphic_intros]:
+ assumes "\<And>i. i \<in># I \<Longrightarrow> f i meromorphic_on A"
+ shows "(\<lambda>w. \<Prod>i\<in>#I. f i w) meromorphic_on A"
+ unfolding meromorphic_on_def
+ by (rule laurent_expansion_intros exI ballI
+ assms[THEN meromorphic_on_imp_has_laurent_expansion] | assumption)+
+
+lemma meromorphic_on_If [meromorphic_intros]:
+ assumes "f meromorphic_on A" "g meromorphic_on B"
+ assumes "\<And>z. z \<in> A \<Longrightarrow> z \<in> B \<Longrightarrow> f z = g z" "open A" "open B" "C \<subseteq> A \<union> B"
+ shows "(\<lambda>z. if z \<in> A then f z else g z) meromorphic_on C"
+proof (rule meromorphic_on_subset)
+ show "(\<lambda>z. if z \<in> A then f z else g z) meromorphic_on (A \<union> B)"
+ proof (rule meromorphic_on_Un)
+ have "(\<lambda>z. if z \<in> A then f z else g z) meromorphic_on A \<longleftrightarrow> f meromorphic_on A"
+ proof (rule meromorphic_on_cong)
+ fix z assume "z \<in> A"
+ hence "eventually (\<lambda>z. z \<in> A) (at z)"
+ using \<open>open A\<close> by (intro eventually_at_in_open') auto
+ thus "\<forall>\<^sub>F w in at z. (if w \<in> A then f w else g w) = f w"
+ by eventually_elim auto
+ qed auto
+ with assms(1) show "(\<lambda>z. if z \<in> A then f z else g z) meromorphic_on A"
+ by blast
next
- case False
- then show ?thesis by (meson DiffI assms(3) c that)
- qed
- then show ?thesis
- by (simp add: constant_on_def)
+ have "(\<lambda>z. if z \<in> A then f z else g z) meromorphic_on B \<longleftrightarrow> g meromorphic_on B"
+ proof (rule meromorphic_on_cong)
+ fix z assume "z \<in> B"
+ hence "eventually (\<lambda>z. z \<in> B) (at z)"
+ using \<open>open B\<close> by (intro eventually_at_in_open') auto
+ thus "\<forall>\<^sub>F w in at z. (if w \<in> A then f w else g w) = g w"
+ by eventually_elim (use assms(3) in auto)
+ qed auto
+ with assms(2) show "(\<lambda>z. if z \<in> A then f z else g z) meromorphic_on B"
+ by blast
+ qed
+qed fact
+
+lemma meromorphic_on_deriv [meromorphic_intros]:
+ "f meromorphic_on A \<Longrightarrow> deriv f meromorphic_on A"
+ by (metis meromorphic_on_def isolated_singularity_at_deriv meromorphic_on_isolated_singularity
+ meromorphic_on_meromorphic_at meromorphic_on_not_essential not_essential_deriv
+ not_essential_has_laurent_expansion)
+
+lemma meromorphic_on_higher_deriv [meromorphic_intros]:
+ "f meromorphic_on A \<Longrightarrow> (deriv ^^ n) f meromorphic_on A"
+ by (induction n) (auto intro!: meromorphic_intros)
+
+lemma analytic_on_eval_fps [analytic_intros]:
+ assumes "f analytic_on A"
+ assumes "\<And>z. z \<in> A \<Longrightarrow> norm (f z) < fps_conv_radius F"
+ shows "(\<lambda>w. eval_fps F (f w)) analytic_on A"
+ by (rule analytic_on_compose[OF assms(1) analytic_on_eval_fps, unfolded o_def])
+ (use assms(2) in auto)
+
+lemma meromorphic_on_eval_fps [meromorphic_intros]:
+ assumes "f analytic_on A"
+ assumes "\<And>z. z \<in> A \<Longrightarrow> norm (f z) < fps_conv_radius F"
+ shows "(\<lambda>w. eval_fps F (f w)) meromorphic_on A"
+ by (rule analytic_on_imp_meromorphic_on analytic_intros analytic_on_eval_fps assms)+
+
+lemma meromorphic_on_eval_fls [meromorphic_intros]:
+ assumes "f analytic_on A"
+ assumes "\<And>z. z \<in> A \<Longrightarrow> norm (f z) < fls_conv_radius F"
+ shows "(\<lambda>w. eval_fls F (f w)) meromorphic_on A"
+proof (cases "fls_conv_radius F > 0")
+ case False
+ with assms(2) have "A = {}"
+ by (metis all_not_in_conv ereal_less(2) norm_eq_zero order.strict_trans
+ zero_ereal_def zero_less_norm_iff)
+ thus ?thesis
+ by auto
+next
+ case True
+ have F: "eval_fls F has_laurent_expansion F"
+ using True by (rule eval_fls_has_laurent_expansion)
+ show ?thesis
+ proof (rule meromorphic_on_compose[OF _ assms(1)])
+ show "eval_fls F meromorphic_on eball 0 (fls_conv_radius F)"
+ proof (rule meromorphic_onI_open)
+ show "eval_fls F analytic_on eball 0 (fls_conv_radius F) - {0}"
+ by (rule analytic_on_eval_fls) auto
+ show "not_essential (eval_fls F) z" if "z \<in> {0}" for z
+ using that F has_laurent_expansion_not_essential_0 by blast
+ qed (auto simp: islimpt_finite)
+ qed (use assms(2) in auto)
+qed
+
+lemma meromorphic_on_imp_analytic_cosparse:
+ assumes "f meromorphic_on A"
+ shows "eventually (\<lambda>z. f analytic_on {z}) (cosparse A)"
+ unfolding eventually_cosparse using assms meromorphic_on_imp_sparse_singularities by auto
+
+lemma meromorphic_on_imp_not_pole_cosparse:
+ assumes "f meromorphic_on A"
+ shows "eventually (\<lambda>z. \<not>is_pole f z) (cosparse A)"
+proof -
+ have "eventually (\<lambda>z. f analytic_on {z}) (cosparse A)"
+ by (rule meromorphic_on_imp_analytic_cosparse) fact
+ thus ?thesis
+ by eventually_elim (blast dest: analytic_at_imp_no_pole)
+qed
+
+lemma eventually_remove_sings_eq:
+ assumes "f meromorphic_on A"
+ shows "eventually (\<lambda>z. remove_sings f z = f z) (cosparse A)"
+proof -
+ have "eventually (\<lambda>z. f analytic_on {z}) (cosparse A)"
+ using assms by (rule meromorphic_on_imp_analytic_cosparse)
+ thus ?thesis
+ by eventually_elim auto
qed
-lemma meromorphic_isolated:
- assumes merf: "f meromorphic_on D pts" and "p\<in>pts"
- obtains r where "r>0" "ball p r \<subseteq> D" "ball p r \<inter> pts = {p}"
+text \<open>
+ A meromorphic function on a connected domain takes any given value either almost everywhere
+ or almost nowhere.
+\<close>
+lemma meromorphic_imp_constant_or_avoid:
+ assumes mero: "f meromorphic_on A" and A: "open A" "connected A"
+ shows "eventually (\<lambda>z. f z = c) (cosparse A) \<or> eventually (\<lambda>z. f z \<noteq> c) (cosparse A)"
proof -
- have "\<forall>z\<in>D. \<exists>e>0. finite (pts \<inter> ball z e)"
- using merf unfolding meromorphic_on_def islimpt_eq_infinite_ball
- by auto
- then obtain r0 where r0:"r0>0" "finite (pts \<inter> ball p r0)"
- by (metis assms(2) in_mono merf meromorphic_on_def)
- moreover define pts' where "pts' = pts \<inter> ball p r0 - {p}"
- ultimately have "finite pts'"
- by simp
-
- define r1 where "r1=(if pts'={} then r0 else
- min (Min {dist p' p |p'. p'\<in>pts'}/2) r0)"
- have "r1>0 \<and> pts \<inter> ball p r1 - {p} = {}"
- proof (cases "pts'={}")
- case True
- then show ?thesis
- using pts'_def r0(1) r1_def by presburger
- next
- case False
- define S where "S={dist p' p |p'. p'\<in>pts'}"
-
- have nempty:"S \<noteq> {}"
- using False S_def by blast
- have finite:"finite S"
- using \<open>finite pts'\<close> S_def by simp
+ have "eventually (\<lambda>z. f z = c) (cosparse A)" if freq: "frequently (\<lambda>z. f z = c) (cosparse A)"
+ proof -
+ let ?f = "remove_sings f"
+ have ev: "eventually (\<lambda>z. ?f z = f z) (cosparse A)"
+ by (rule eventually_remove_sings_eq) fact
+ have "frequently (\<lambda>z. ?f z = c) (cosparse A)"
+ using frequently_eventually_frequently[OF freq ev] by (rule frequently_elim1) auto
+ then obtain z0 where z0: "z0 \<in> A" "frequently (\<lambda>z. ?f z = c) (at z0)"
+ using A by (auto simp: eventually_cosparse_open_eq frequently_def)
+ have mero': "?f nicely_meromorphic_on A"
+ using mero remove_sings_nicely_meromorphic by blast
+ have eq: "?f w = c" if w: "w \<in> A" for w
+ using frequently_eq_meromorphic_imp_constant[OF z0(2) mero'] A z0(1) w by blast
+ have "eventually (\<lambda>z. z \<in> A) (cosparse A)"
+ by (rule eventually_in_cosparse) (use A in auto)
+ thus "eventually (\<lambda>z. f z = c) (cosparse A)"
+ using ev by eventually_elim (use eq in auto)
+ qed
+ thus ?thesis
+ by (auto simp: frequently_def)
+qed
- have "r1>0"
- proof -
- have "r1=min (Min S/2) r0"
- using False unfolding S_def r1_def by auto
- moreover have "Min S\<in>S"
- using \<open>S\<noteq>{}\<close> \<open>finite S\<close> Min_in by auto
- then have "Min S>0" unfolding S_def
- using pts'_def by force
- ultimately show ?thesis using \<open>r0>0\<close> by auto
- qed
- moreover have "pts \<inter> ball p r1 - {p} = {}"
- proof (rule ccontr)
- assume "pts \<inter> ball p r1 - {p} \<noteq> {}"
- then obtain p' where "p'\<in>pts \<inter> ball p r1 - {p}" by blast
- moreover have "r1\<le>r0" using r1_def by auto
- ultimately have "p'\<in>pts'" unfolding pts'_def
+lemma nicely_meromorphic_imp_constant_or_avoid:
+ assumes "f nicely_meromorphic_on A" "open A" "connected A"
+ shows "(\<forall>x\<in>A. f x = c) \<or> (\<forall>\<^sub>\<approx>x\<in>A. f x \<noteq> c)"
+proof -
+ have "(\<forall>\<^sub>\<approx>x\<in>A. f x = c) \<or> (\<forall>\<^sub>\<approx>x\<in>A. f x \<noteq> c)"
+ by (intro meromorphic_imp_constant_or_avoid)
+ (use assms in \<open>auto simp: nicely_meromorphic_on_def\<close>)
+ thus ?thesis
+ proof
+ assume ev: "\<forall>\<^sub>\<approx>x\<in>A. f x = c"
+ have "\<forall>x\<in>A. f x = c "
+ proof
+ fix x assume x: "x \<in> A"
+ have "not_essential f x"
+ using assms x unfolding nicely_meromorphic_on_def by blast
+ moreover have "is_pole f x \<longleftrightarrow> is_pole (\<lambda>_. c) x"
+ by (intro is_pole_cong) (use ev x in \<open>auto simp: eventually_cosparse_open_eq assms\<close>)
+ hence "\<not>is_pole f x"
by auto
- then have "dist p' p\<ge>Min S"
- using S_def eq_Min_iff local.finite by blast
- moreover have "dist p' p < Min S"
- using \<open>p'\<in>pts \<inter> ball p r1 - {p}\<close> False unfolding r1_def
- apply (fold S_def)
- by (smt (verit, ccfv_threshold) DiffD1 Int_iff dist_commute
- dist_triangle_half_l mem_ball)
- ultimately show False by auto
+ ultimately have "f analytic_on {x}"
+ using assms(1) nicely_meromorphic_on_imp_analytic_at x by blast
+ hence "f \<midarrow>x\<rightarrow> f x"
+ by (intro isContD analytic_at_imp_isCont)
+ also have "?this \<longleftrightarrow> (\<lambda>_. c) \<midarrow>x\<rightarrow> f x"
+ by (intro tendsto_cong) (use ev x in \<open>auto simp: eventually_cosparse_open_eq assms\<close>)
+ finally have "(\<lambda>_. c) \<midarrow>x\<rightarrow> f x" .
+ moreover have "(\<lambda>_. c) \<midarrow>x\<rightarrow> c"
+ by simp
+ ultimately show "f x = c"
+ using LIM_unique by blast
qed
- ultimately show ?thesis by auto
- qed
- then have "r1>0" and r1_pts:"pts \<inter> ball p r1 - {p} = {}" by auto
+ thus ?thesis
+ by blast
+ qed blast
+qed
- obtain r2 where "r2>0" "ball p r2 \<subseteq> D"
- by (metis assms(2) merf meromorphic_on_def openE subset_eq)
- define r where "r=min r1 r2"
- have "r > 0" unfolding r_def
- by (simp add: \<open>0 < r1\<close> \<open>0 < r2\<close>)
- moreover have "ball p r \<subseteq> D"
- using \<open>ball p r2 \<subseteq> D\<close> r_def by auto
- moreover have "ball p r \<inter> pts = {p}"
- using assms(2) \<open>r>0\<close> r1_pts
- unfolding r_def by auto
+lemma nicely_meromorphic_onE:
+ assumes "f nicely_meromorphic_on A"
+ obtains pts where "pts \<subseteq> A" "pts sparse_in A"
+ "f analytic_on A - pts"
+ "\<And>z. z \<in> pts \<Longrightarrow> is_pole f z \<and> f z=0"
+proof -
+ define pts where "pts = {z \<in> A. \<not> f analytic_on {z}}"
+ have "pts \<subseteq> A" "pts sparse_in A"
+ using assms unfolding pts_def nicely_meromorphic_on_def
+ by (auto intro:meromorphic_on_imp_sparse_singularities')
+ moreover have "f analytic_on A - pts" unfolding pts_def
+ by (subst analytic_on_analytic_at) auto
+ moreover have "\<And>z. z \<in> pts \<Longrightarrow> is_pole f z \<and> f z=0"
+ by (metis (no_types, lifting) remove_sings_eqI
+ remove_sings_eq_0_iff assms is_pole_imp_not_essential
+ mem_Collect_eq nicely_meromorphic_on_def
+ nicely_meromorphic_on_imp_analytic_at pts_def)
ultimately show ?thesis using that by auto
qed
-lemma meromorphic_pts_closure:
- assumes merf: "f meromorphic_on D pts"
- shows "pts \<subseteq> closure (D - pts)"
-proof -
- have "p islimpt (D - pts)" if "p\<in>pts" for p
- proof -
- obtain r where "r>0" "ball p r \<subseteq> D" "ball p r \<inter> pts = {p}"
- using meromorphic_isolated[OF merf \<open>p\<in>pts\<close>] by auto
- from \<open>r>0\<close>
- have "p islimpt ball p r - {p}"
- by (meson open_ball ball_subset_cball in_mono islimpt_ball
- islimpt_punctured le_less open_contains_ball_eq)
- moreover have " ball p r - {p} \<subseteq> D - pts"
- using \<open>ball p r \<inter> pts = {p}\<close> \<open>ball p r \<subseteq> D\<close> by fastforce
- ultimately show ?thesis
- using islimpt_subset by auto
- qed
- then show ?thesis by (simp add: islimpt_in_closure subset_eq)
-qed
-
-lemma nconst_imp_nzero_neighbour:
- assumes merf: "f meromorphic_on D pts"
- and f_nconst:"\<not>(\<forall>w\<in>D-pts. f w=0)"
- and "z\<in>D" and "connected D"
- shows "(\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w \<in> D - pts)"
+lemma nicely_meromorphic_onI_open:
+ assumes "open A" and
+ analytic:"f analytic_on A - pts" and
+ pole:"\<And>x. x\<in>pts \<Longrightarrow> is_pole f x \<and> f x = 0" and
+ isolated:"\<And>x. x\<in>A \<Longrightarrow> isolated_singularity_at f x"
+ shows "f nicely_meromorphic_on A"
proof -
- obtain \<beta> where \<beta>:"\<beta> \<in> D - pts" "f \<beta>\<noteq>0"
- using f_nconst by auto
-
- have ?thesis if "z\<notin>pts"
- proof -
- have "\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w \<in> D - pts"
- apply (rule non_zero_neighbour_alt[of f "D-pts" z \<beta>])
- subgoal using merf meromorphic_on_def by blast
- subgoal using merf meromorphic_imp_open_diff by auto
- subgoal using assms(4) merf meromorphic_imp_connected_diff by blast
- subgoal by (simp add: assms(3) that)
- using \<beta> by auto
- then show ?thesis by (auto elim:eventually_mono)
- qed
- moreover have ?thesis if "z\<in>pts" "\<not> f \<midarrow>z\<rightarrow> 0"
- proof -
- have "\<forall>\<^sub>F w in at z. w \<in> D - pts"
- using merf[unfolded meromorphic_on_def islimpt_iff_eventually] \<open>z\<in>D\<close>
- using eventually_at_in_open' eventually_elim2 by fastforce
- moreover have "\<forall>\<^sub>F w in at z. f w \<noteq> 0"
- proof (cases "is_pole f z")
- case True
- then show ?thesis using non_zero_neighbour_pole by auto
- next
- case False
- moreover have "not_essential f z"
- using merf meromorphic_on_def that(1) by fastforce
- ultimately obtain c where "c\<noteq>0" "f \<midarrow>z\<rightarrow> c"
- by (metis \<open>\<not> f \<midarrow>z\<rightarrow> 0\<close> not_essential_def)
- then show ?thesis
- using tendsto_imp_eventually_ne by auto
- qed
- ultimately show ?thesis by eventually_elim auto
- qed
- moreover have ?thesis if "z\<in>pts" "f \<midarrow>z\<rightarrow> 0"
- proof -
- define ff where "ff=(\<lambda>x. if x=z then 0 else f x)"
- define A where "A=D - (pts - {z})"
-
- have "f holomorphic_on A - {z}"
- by (metis A_def Diff_insert analytic_imp_holomorphic
- insert_Diff merf meromorphic_imp_analytic_on that(1))
- moreover have "open A"
- using A_def merf meromorphic_imp_open_diff' by force
- ultimately have "ff holomorphic_on A"
- using \<open>f \<midarrow>z\<rightarrow> 0\<close> unfolding ff_def
- by (rule removable_singularity)
- moreover have "connected A"
- proof -
- have "connected (D - pts)"
- using assms(4) merf meromorphic_imp_connected_diff by auto
- moreover have "D - pts \<subseteq> A"
- unfolding A_def by auto
- moreover have "A \<subseteq> closure (D - pts)" unfolding A_def
- by (smt (verit, ccfv_SIG) Diff_empty Diff_insert
- closure_subset insert_Diff_single insert_absorb
- insert_subset merf meromorphic_pts_closure that(1))
- ultimately show ?thesis using connected_intermediate_closure
- by auto
- qed
- moreover have "z \<in> A" using A_def assms(3) by blast
- moreover have "ff z = 0" unfolding ff_def by auto
- moreover have "\<beta> \<in> A " using A_def \<beta>(1) by blast
- moreover have "ff \<beta> \<noteq> 0" using \<beta>(1) \<beta>(2) ff_def that(1) by auto
- ultimately obtain r where "0 < r"
- "ball z r \<subseteq> A" "\<And>x. x \<in> ball z r - {z} \<Longrightarrow> ff x \<noteq> 0"
- using \<open>open A\<close> isolated_zeros[of ff A z \<beta>] by auto
- then show ?thesis unfolding eventually_at ff_def
- by (intro exI[of _ r]) (auto simp: A_def dist_commute ball_def)
- qed
- ultimately show ?thesis by auto
+ have "f meromorphic_on A"
+ proof (rule meromorphic_onI_open)
+ show "\<And>z. z \<in> pts \<Longrightarrow> not_essential f z"
+ using pole unfolding not_essential_def by auto
+ show "\<And>z. z \<in> A \<Longrightarrow> \<not> z islimpt pts \<inter> A"
+ by (metis assms(3) assms(4) inf_commute inf_le2
+ islimpt_subset mem_Collect_eq not_islimpt_poles subsetI)
+ qed fact+
+ moreover have "(\<forall>z\<in>A. (is_pole f z \<and> f z=0) \<or> f \<midarrow>z\<rightarrow> f z)"
+ by (meson DiffI analytic analytic_at_imp_isCont
+ analytic_on_analytic_at assms(3) isContD)
+ ultimately show ?thesis unfolding nicely_meromorphic_on_def
+ by auto
qed
-lemma nconst_imp_nzero_neighbour':
- assumes merf: "f meromorphic_on D pts"
- and f_nconst:"\<not>(\<forall>w\<in>D-pts. f w=0)"
- and "z\<in>D" and "connected D"
- shows "\<forall>\<^sub>F w in at z. f w \<noteq> 0"
- using nconst_imp_nzero_neighbour[OF assms]
- by (auto elim:eventually_mono)
+lemma nicely_meromorphic_without_singularities:
+ assumes "f nicely_meromorphic_on A" "\<forall>z\<in>A. \<not> is_pole f z"
+ shows "f analytic_on A"
+ by (meson analytic_on_analytic_at assms
+ nicely_meromorphic_on_imp_analytic_at)
-lemma meromorphic_compact_finite_zeros:
- assumes merf:"f meromorphic_on D pts"
- and "compact S" "S \<subseteq> D" "connected D"
- and f_nconst:"\<not>(\<forall>w\<in>D-pts. f w=0)"
- shows "finite ({x\<in>S. f x=0})"
-proof -
- have "finite ({x\<in>S. f x=0 \<and> x \<notin> pts})"
- proof (rule ccontr)
- assume "infinite {x \<in> S. f x = 0 \<and> x \<notin> pts}"
- then obtain z where "z\<in>S" and z_lim:"z islimpt {x \<in> S. f x = 0
- \<and> x \<notin> pts}"
- using \<open>compact S\<close> unfolding compact_eq_Bolzano_Weierstrass
- by auto
-
- from z_lim
- have "\<exists>\<^sub>F x in at z. f x = 0 \<and> x \<in> S \<and> x \<notin> pts"
- unfolding islimpt_iff_eventually not_eventually by simp
- moreover have "\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w \<in> D - pts"
- using nconst_imp_nzero_neighbour[OF merf f_nconst _ \<open>connected D\<close>]
- \<open>z\<in>S\<close> \<open>S \<subseteq> D\<close>
- by auto
- ultimately have "\<exists>\<^sub>F x in at z. False"
- by (simp add: eventually_mono frequently_def)
- then show False by auto
- qed
- moreover have "finite (S \<inter> pts)"
- using meromorphic_compact_finite_pts[OF merf \<open>compact S\<close> \<open>S \<subseteq> D\<close>] .
- ultimately have "finite ({x\<in>S. f x=0 \<and> x \<notin> pts} \<union> (S \<inter> pts))"
- unfolding finite_Un by auto
- then show ?thesis by (elim rev_finite_subset) auto
-qed
+lemma meromorphic_on_cong':
+ assumes "eventually (\<lambda>z. f z = g z) (cosparse A)" "A = B"
+ shows "f meromorphic_on A \<longleftrightarrow> g meromorphic_on B"
+ unfolding assms(2)[symmetric]
+ by (rule meromorphic_on_cong eventually_cosparse_imp_eventually_at assms)+ auto
-lemma meromorphic_onI [intro?]:
- assumes "open A" "pts \<subseteq> A"
- assumes "f holomorphic_on A - pts" "\<And>z. z \<in> A \<Longrightarrow> \<not>z islimpt pts"
- assumes "\<And>z. z \<in> pts \<Longrightarrow> isolated_singularity_at f z"
- assumes "\<And>z. z \<in> pts \<Longrightarrow> not_essential f z"
- shows "f meromorphic_on A pts"
- using assms unfolding meromorphic_on_def by blast
-lemma Polygamma_plus_of_nat:
- assumes "\<forall>k<m. z \<noteq> -of_nat k"
- shows "Polygamma n (z + of_nat m) =
- Polygamma n z + (-1) ^ n * fact n * (\<Sum>k<m. 1 / (z + of_nat k) ^ Suc n)"
- using assms
-proof (induction m)
- case (Suc m)
- have "Polygamma n (z + of_nat (Suc m)) = Polygamma n (z + of_nat m + 1)"
- by (simp add: add_ac)
- also have "\<dots> = Polygamma n (z + of_nat m) + (-1) ^ n * fact n * (1 / ((z + of_nat m) ^ Suc n))"
- using Suc.prems by (subst Polygamma_plus1) (auto simp: add_eq_0_iff2)
- also have "Polygamma n (z + of_nat m) =
- Polygamma n z + (-1) ^ n * (\<Sum>k<m. 1 / (z + of_nat k) ^ Suc n) * fact n"
- using Suc.prems by (subst Suc.IH) auto
- finally show ?case
- by (simp add: algebra_simps)
-qed auto
+subsection \<open>Meromorphic functions and zorder\<close>
-lemma tendsto_Gamma [tendsto_intros]:
- assumes "(f \<longlongrightarrow> c) F" "c \<notin> \<int>\<^sub>\<le>\<^sub>0"
- shows "((\<lambda>z. Gamma (f z)) \<longlongrightarrow> Gamma c) F"
- by (intro isCont_tendsto_compose[OF _ assms(1)] continuous_intros assms)
-
-lemma tendsto_Polygamma [tendsto_intros]:
- fixes f :: "_ \<Rightarrow> 'a :: {real_normed_field,euclidean_space}"
- assumes "(f \<longlongrightarrow> c) F" "c \<notin> \<int>\<^sub>\<le>\<^sub>0"
- shows "((\<lambda>z. Polygamma n (f z)) \<longlongrightarrow> Polygamma n c) F"
- by (intro isCont_tendsto_compose[OF _ assms(1)] continuous_intros assms)
-
-lemma analytic_on_Gamma' [analytic_intros]:
- assumes "f analytic_on A" "\<forall>x\<in>A. f x \<notin> \<int>\<^sub>\<le>\<^sub>0"
- shows "(\<lambda>z. Gamma (f z)) analytic_on A"
- using analytic_on_compose_gen[OF assms(1) analytic_Gamma[of "f ` A"]] assms(2)
- by (auto simp: o_def)
-
-lemma analytic_on_Polygamma' [analytic_intros]:
- assumes "f analytic_on A" "\<forall>x\<in>A. f x \<notin> \<int>\<^sub>\<le>\<^sub>0"
- shows "(\<lambda>z. Polygamma n (f z)) analytic_on A"
- using analytic_on_compose_gen[OF assms(1) analytic_on_Polygamma[of "f ` A" n]] assms(2)
- by (auto simp: o_def)
-
-lemma
- shows is_pole_Polygamma: "is_pole (Polygamma n) (-of_nat m :: complex)"
- and zorder_Polygamma: "zorder (Polygamma n) (-of_nat m) = -int (Suc n)"
- and residue_Polygamma: "residue (Polygamma n) (-of_nat m) = (if n = 0 then -1 else 0)"
+lemma zorder_power_int:
+ assumes "f meromorphic_on {z}" "frequently (\<lambda>z. f z \<noteq> 0) (at z)"
+ shows "zorder (\<lambda>z. f z powi n) z = n * zorder f z"
proof -
- define g1 :: "complex \<Rightarrow> complex" where
- "g1 = (\<lambda>z. Polygamma n (z + of_nat (Suc m)) +
- (-1) ^ Suc n * fact n * (\<Sum>k<m. 1 / (z + of_nat k) ^ Suc n))"
- define g :: "complex \<Rightarrow> complex" where
- "g = (\<lambda>z. g1 z + (-1) ^ Suc n * fact n / (z + of_nat m) ^ Suc n)"
- define F where "F = fps_to_fls (fps_expansion g1 (-of_nat m)) + fls_const ((-1) ^ Suc n * fact n) / fls_X ^ Suc n"
- have F_altdef: "F = fps_to_fls (fps_expansion g1 (-of_nat m)) + fls_shift (n+1) (fls_const ((-1) ^ Suc n * fact n))"
- by (simp add: F_def del: power_Suc)
-
- have "\<not>(-of_nat m) islimpt (\<int>\<^sub>\<le>\<^sub>0 :: complex set)"
- by (intro discrete_imp_not_islimpt[where e = 1])
- (auto elim!: nonpos_Ints_cases simp: dist_of_int)
- hence "eventually (\<lambda>z::complex. z \<notin> \<int>\<^sub>\<le>\<^sub>0) (at (-of_nat m))"
- by (auto simp: islimpt_conv_frequently_at frequently_def)
- hence ev: "eventually (\<lambda>z. Polygamma n z = g z) (at (-of_nat m))"
- proof eventually_elim
- case (elim z)
- hence *: "\<forall>k<Suc m. z \<noteq> - of_nat k"
- by auto
- thus ?case
- using Polygamma_plus_of_nat[of "Suc m" z n, OF *]
- by (auto simp: g_def g1_def algebra_simps)
- qed
-
- have "(\<lambda>w. g (-of_nat m + w)) has_laurent_expansion F"
- unfolding g_def F_def
- by (intro laurent_expansion_intros has_laurent_expansion_fps analytic_at_imp_has_fps_expansion)
- (auto simp: g1_def intro!: laurent_expansion_intros analytic_intros)
- also have "?this \<longleftrightarrow> (\<lambda>w. Polygamma n (-of_nat m + w)) has_laurent_expansion F"
- using ev by (intro has_laurent_expansion_cong refl)
- (simp_all add: eq_commute at_to_0' eventually_filtermap)
- finally have *: "(\<lambda>w. Polygamma n (-of_nat m + w)) has_laurent_expansion F" .
-
- have subdegree: "fls_subdegree F = -int (Suc n)" unfolding F_def
- by (subst fls_subdegree_add_eq2) (simp_all add: fls_subdegree_fls_to_fps fls_divide_subdegree)
- have [simp]: "F \<noteq> 0"
- using subdegree by auto
-
- show "is_pole (Polygamma n) (-of_nat m :: complex)"
- using * by (rule has_laurent_expansion_imp_is_pole) (auto simp: subdegree)
- show "zorder (Polygamma n) (-of_nat m :: complex) = -int (Suc n)"
- by (subst has_laurent_expansion_zorder[OF *]) (auto simp: subdegree)
- show "residue (Polygamma n) (-of_nat m :: complex) = (if n = 0 then -1 else 0)"
- by (subst has_laurent_expansion_residue[OF *]) (auto simp: F_altdef)
+ from assms(1) obtain L where L: "(\<lambda>w. f (z + w)) has_laurent_expansion L"
+ by (auto simp: meromorphic_on_def)
+ from assms(2) and L have [simp]: "L \<noteq> 0"
+ by (metis assms(1) has_laurent_expansion_eventually_nonzero_iff meromorphic_at_iff
+ not_essential_frequently_0_imp_eventually_0 not_eventually not_frequently)
+ from L have L': "(\<lambda>w. f (z + w) powi n) has_laurent_expansion L powi n"
+ by (intro laurent_expansion_intros)
+ have "zorder f z = fls_subdegree L"
+ using L assms(2) \<open>L \<noteq> 0\<close> by (simp add: has_laurent_expansion_zorder)
+ moreover have "zorder (\<lambda>z. f z powi n) z = fls_subdegree (L powi n)"
+ using L' assms(2) \<open>L \<noteq> 0\<close> by (simp add: has_laurent_expansion_zorder)
+ moreover have "fls_subdegree (L powi n) = n * fls_subdegree L"
+ by simp
+ ultimately show ?thesis
+ by simp
qed
-lemma Gamma_meromorphic_on [meromorphic_intros]: "Gamma meromorphic_on UNIV \<int>\<^sub>\<le>\<^sub>0"
-proof
- show "\<not>z islimpt \<int>\<^sub>\<le>\<^sub>0" for z :: complex
- by (intro discrete_imp_not_islimpt[of 1]) (auto elim!: nonpos_Ints_cases simp: dist_of_int)
-next
- fix z :: complex assume z: "z \<in> \<int>\<^sub>\<le>\<^sub>0"
- then obtain n where n: "z = -of_nat n"
- by (elim nonpos_Ints_cases')
- show "not_essential Gamma z"
- by (auto simp: n intro!: is_pole_imp_not_essential is_pole_Gamma)
- have *: "open (-(\<int>\<^sub>\<le>\<^sub>0 - {z}))"
- by (intro open_Compl discrete_imp_closed[of 1]) (auto elim!: nonpos_Ints_cases simp: dist_of_int)
- have "Gamma holomorphic_on -(\<int>\<^sub>\<le>\<^sub>0 - {z}) - {z}"
- by (intro holomorphic_intros) auto
- thus "isolated_singularity_at Gamma z"
- by (rule isolated_singularity_at_holomorphic) (use z * in auto)
-qed (auto intro!: holomorphic_intros)
+lemma zorder_power:
+ assumes "f meromorphic_on {z}" "frequently (\<lambda>z. f z \<noteq> 0) (at z)"
+ shows "zorder (\<lambda>z. f z ^ n) z = n * zorder f z"
+ using zorder_power_int[OF assms, of "int n"] by simp
-lemma Polygamma_meromorphic_on [meromorphic_intros]: "Polygamma n meromorphic_on UNIV \<int>\<^sub>\<le>\<^sub>0"
-proof
- show "\<not>z islimpt \<int>\<^sub>\<le>\<^sub>0" for z :: complex
- by (intro discrete_imp_not_islimpt[of 1]) (auto elim!: nonpos_Ints_cases simp: dist_of_int)
-next
- fix z :: complex assume z: "z \<in> \<int>\<^sub>\<le>\<^sub>0"
- then obtain m where n: "z = -of_nat m"
- by (elim nonpos_Ints_cases')
- show "not_essential (Polygamma n) z"
- by (auto simp: n intro!: is_pole_imp_not_essential is_pole_Polygamma)
- have *: "open (-(\<int>\<^sub>\<le>\<^sub>0 - {z}))"
- by (intro open_Compl discrete_imp_closed[of 1]) (auto elim!: nonpos_Ints_cases simp: dist_of_int)
- have "Polygamma n holomorphic_on -(\<int>\<^sub>\<le>\<^sub>0 - {z}) - {z}"
- by (intro holomorphic_intros) auto
- thus "isolated_singularity_at (Polygamma n) z"
- by (rule isolated_singularity_at_holomorphic) (use z * in auto)
-qed (auto intro!: holomorphic_intros)
+lemma zorder_add1:
+ assumes "f meromorphic_on {z}" "frequently (\<lambda>z. f z \<noteq> 0) (at z)"
+ assumes "g meromorphic_on {z}" "frequently (\<lambda>z. g z \<noteq> 0) (at z)"
+ assumes "zorder f z < zorder g z"
+ shows "zorder (\<lambda>z. f z + g z) z = zorder f z"
+proof -
+ from assms(1) obtain F where F: "(\<lambda>w. f (z + w)) has_laurent_expansion F"
+ by (auto simp: meromorphic_on_def)
+ from assms(3) obtain G where G: "(\<lambda>w. g (z + w)) has_laurent_expansion G"
+ by (auto simp: meromorphic_on_def)
+ have [simp]: "F \<noteq> 0" "G \<noteq> 0"
+ by (metis assms has_laurent_expansion_eventually_nonzero_iff meromorphic_at_iff
+ not_essential_frequently_0_imp_eventually_0 not_eventually not_frequently F G)+
+ have *: "zorder f z = fls_subdegree F" "zorder g z = fls_subdegree G"
+ using F G assms by (simp_all add: has_laurent_expansion_zorder)
+ from assms * have "F \<noteq> -G"
+ by auto
+ hence [simp]: "F + G \<noteq> 0"
+ by (simp add: add_eq_0_iff2)
+ moreover have "zorder (\<lambda>z. f z + g z) z = fls_subdegree (F + G)"
+ using has_laurent_expansion_zorder[OF has_laurent_expansion_add[OF F G]] \<open>F \<noteq> -G\<close> by simp
+ moreover have "fls_subdegree (F + G) = fls_subdegree F"
+ using assms by (simp add: * fls_subdegree_add_eq1)
+ ultimately show ?thesis
+ by (simp add: *)
+qed
+
+lemma zorder_add2:
+ assumes "f meromorphic_on {z}" "frequently (\<lambda>z. f z \<noteq> 0) (at z)"
+ assumes "g meromorphic_on {z}" "frequently (\<lambda>z. g z \<noteq> 0) (at z)"
+ assumes "zorder f z > zorder g z"
+ shows "zorder (\<lambda>z. f z + g z) z = zorder g z"
+ using zorder_add1[OF assms(3,4) assms(1,2)] assms(5-) by (simp add: add.commute)
-theorem argument_principle':
- fixes f::"complex \<Rightarrow> complex" and poles s:: "complex set"
- \<comment> \<open>\<^term>\<open>pz\<close> is the set of non-essential singularities and zeros\<close>
- defines "pz \<equiv> {w\<in>s. f w = 0 \<or> w \<in> poles}"
- assumes "open s" and
- "connected s" and
- f_holo:"f holomorphic_on s-poles" and
- h_holo:"h holomorphic_on s" and
- "valid_path g" and
- loop:"pathfinish g = pathstart g" and
- path_img:"path_image g \<subseteq> s - pz" and
- homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0" and
- finite:"finite pz" and
- poles:"\<forall>p\<in>s\<inter>poles. not_essential f p"
- shows "contour_integral g (\<lambda>x. deriv f x * h x / f x) = 2 * pi * \<i> *
- (\<Sum>p\<in>pz. winding_number g p * h p * zorder f p)"
+lemma zorder_add_ge:
+ fixes f g :: "complex \<Rightarrow> complex"
+ assumes "f meromorphic_on {z}" "frequently (\<lambda>z. f z \<noteq> 0) (at z)"
+ assumes "g meromorphic_on {z}" "frequently (\<lambda>z. g z \<noteq> 0) (at z)"
+ assumes "frequently (\<lambda>z. f z + g z \<noteq> 0) (at z)" "zorder f z \<ge> c" "zorder g z \<ge> c"
+ shows "zorder (\<lambda>z. f z + g z) z \<ge> c"
proof -
- define ff where "ff = remove_sings f"
-
- have finite':"finite (s \<inter> poles)"
- using finite unfolding pz_def by (auto elim:rev_finite_subset)
-
- have isolated:"isolated_singularity_at f z" if "z\<in>s" for z
- proof (rule isolated_singularity_at_holomorphic)
- show "f holomorphic_on (s-(poles-{z})) - {z}"
- by (metis Diff_empty Diff_insert Diff_insert0 Diff_subset
- f_holo holomorphic_on_subset insert_Diff)
- show "open (s - (poles - {z}))"
- by (metis Diff_Diff_Int Int_Diff assms(2) finite' finite_Diff
- finite_imp_closed inf.idem open_Diff)
- show "z \<in> s - (poles - {z})"
- using assms(4) that by auto
- qed
-
- have not_ess:"not_essential f w" if "w\<in>s" for w
- by (metis Diff_Diff_Int Diff_iff Int_Diff Int_absorb assms(2)
- f_holo finite' finite_imp_closed not_essential_holomorphic
- open_Diff poles that)
-
- have nzero:"\<forall>\<^sub>F x in at w. f x \<noteq> 0" if "w\<in>s" for w
- proof (rule ccontr)
- assume "\<not> (\<forall>\<^sub>F x in at w. f x \<noteq> 0)"
- then have "\<exists>\<^sub>F x in at w. f x = 0"
- unfolding not_eventually by simp
- moreover have "\<forall>\<^sub>F x in at w. x\<in>s"
- by (simp add: assms(2) eventually_at_in_open' that)
- ultimately have "\<exists>\<^sub>F x in at w. x\<in>{w\<in>s. f w = 0}"
- apply (elim frequently_rev_mp)
- by (auto elim:eventually_mono)
- from frequently_at_imp_islimpt[OF this]
- have "w islimpt {w \<in> s. f w = 0}" .
- then have "infinite({w \<in> s. f w = 0} \<inter> ball w 1)"
- unfolding islimpt_eq_infinite_ball by auto
- then have "infinite({w \<in> s. f w = 0})"
- by auto
- then have "infinite pz" unfolding pz_def
- by (smt (verit) Collect_mono_iff rev_finite_subset)
- then show False using finite by auto
- qed
-
- obtain pts' where pts':"pts' \<subseteq> s \<inter> poles"
- "finite pts'" "ff holomorphic_on s - pts'" "\<forall>x\<in>pts'. is_pole ff x"
- apply (elim get_all_poles_from_remove_sings
- [of f,folded ff_def,rotated -1])
- subgoal using f_holo by fastforce
- using \<open>open s\<close> poles finite' by auto
+ from assms(1) obtain F where F: "(\<lambda>w. f (z + w)) has_laurent_expansion F"
+ by (auto simp: meromorphic_on_def)
+ from assms(3) obtain G where G: "(\<lambda>w. g (z + w)) has_laurent_expansion G"
+ by (auto simp: meromorphic_on_def)
+ have [simp]: "F \<noteq> 0" "G \<noteq> 0"
+ using assms F G has_laurent_expansion_frequently_nonzero_iff by blast+
+ have FG: "(\<lambda>w. f (z + w) + g (z + w)) has_laurent_expansion F + G"
+ by (intro laurent_expansion_intros F G)
+ have [simp]: "F + G \<noteq> 0"
+ using assms(5) has_laurent_expansion_frequently_nonzero_iff[OF FG] by blast
- have pts'_sub_pz:"{w \<in> s. ff w = 0 \<or> w \<in> pts'} \<subseteq> pz"
- proof -
- have "w\<in>poles" if "w\<in>s" "w\<in>pts'" for w
- by (meson in_mono le_infE pts'(1) that(2))
- moreover have "f w=0" if" w\<in>s" "w\<notin>poles" "ff w=0" for w
- proof -
- have "\<not> is_pole f w"
- by (metis DiffI Diff_Diff_Int Diff_subset assms(2) f_holo
- finite' finite_imp_closed inf.absorb_iff2
- not_is_pole_holomorphic open_Diff that(1) that(2))
- then have "f \<midarrow>w\<rightarrow> 0"
- using remove_sings_eq_0_iff[OF not_ess[OF \<open>w\<in>s\<close>]] \<open>ff w=0\<close>
- unfolding ff_def by auto
- moreover have "f analytic_on {w}"
- using that(1,2) finite' f_holo assms(2)
- by (metis Diff_Diff_Int Diff_empty Diff_iff Diff_subset
- double_diff finite_imp_closed
- holomorphic_on_imp_analytic_at open_Diff)
- ultimately show ?thesis
- using ff_def remove_sings_at_analytic that(3) by presburger
- qed
- ultimately show ?thesis unfolding pz_def by auto
- qed
-
-
- have "contour_integral g (\<lambda>x. deriv f x * h x / f x)
- = contour_integral g (\<lambda>x. deriv ff x * h x / ff x)"
- proof (rule contour_integral_eq)
- fix x assume "x \<in> path_image g"
- have "f analytic_on {x}"
- proof (rule holomorphic_on_imp_analytic_at[of _ "s-poles"])
- from finite'
- show "open (s - poles)"
- using \<open>open s\<close>
- by (metis Diff_Compl Diff_Diff_Int Diff_eq finite_imp_closed
- open_Diff)
- show "x \<in> s - poles"
- using path_img \<open>x \<in> path_image g\<close> unfolding pz_def by auto
- qed (use f_holo in simp)
- then show "deriv f x * h x / f x = deriv ff x * h x / ff x"
- unfolding ff_def by auto
- qed
- also have "... = complex_of_real (2 * pi) * \<i> *
- (\<Sum>p\<in>{w \<in> s. ff w = 0 \<or> w \<in> pts'}.
- winding_number g p * h p * of_int (zorder ff p))"
- proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close>, of ff pts' h g])
- show "path_image g \<subseteq> s - {w \<in> s. ff w = 0 \<or> w \<in> pts'}"
- using path_img pts'_sub_pz by auto
- show "finite {w \<in> s. ff w = 0 \<or> w \<in> pts'}"
- using pts'_sub_pz finite
- using rev_finite_subset by blast
- qed (use pts' assms in auto)
- also have "... = 2 * pi * \<i> *
- (\<Sum>p\<in>pz. winding_number g p * h p * zorder f p)"
- proof -
- have "(\<Sum>p\<in>{w \<in> s. ff w = 0 \<or> w \<in> pts'}.
- winding_number g p * h p * of_int (zorder ff p)) =
- (\<Sum>p\<in>pz. winding_number g p * h p * of_int (zorder f p))"
- proof (rule sum.mono_neutral_cong_left)
- have "zorder f w = 0"
- if "w\<in>s" " f w = 0 \<or> w \<in> poles" "ff w \<noteq> 0" " w \<notin> pts'"
- for w
- proof -
- define F where "F=laurent_expansion f w"
- have has_l:"(\<lambda>x. f (w + x)) has_laurent_expansion F"
- unfolding F_def
- apply (rule not_essential_has_laurent_expansion)
- using isolated not_ess \<open>w\<in>s\<close> by auto
- from has_laurent_expansion_eventually_nonzero_iff[OF this]
- have "F \<noteq>0"
- using nzero \<open>w\<in>s\<close> by auto
- from tendsto_0_subdegree_iff[OF has_l this]
- have "f \<midarrow>w\<rightarrow> 0 = (0 < fls_subdegree F)" .
- moreover have "\<not> (is_pole f w \<or> f \<midarrow>w\<rightarrow> 0)"
- using remove_sings_eq_0_iff[OF not_ess[OF \<open>w\<in>s\<close>]] \<open>ff w \<noteq> 0\<close>
- unfolding ff_def by auto
- moreover have "is_pole f w = (fls_subdegree F < 0)"
- using is_pole_fls_subdegree_iff[OF has_l] .
- ultimately have "fls_subdegree F = 0" by auto
- then show ?thesis
- using has_laurent_expansion_zorder[OF has_l \<open>F\<noteq>0\<close>] by auto
- qed
- then show "\<forall>i\<in>pz - {w \<in> s. ff w = 0 \<or> w \<in> pts'}.
- winding_number g i * h i * of_int (zorder f i) = 0"
- unfolding pz_def by auto
- show "\<And>x. x \<in> {w \<in> s. ff w = 0 \<or> w \<in> pts'} \<Longrightarrow>
- winding_number g x * h x * of_int (zorder ff x) =
- winding_number g x * h x * of_int (zorder f x)"
- using isolated zorder_remove_sings[of f,folded ff_def] by auto
- qed (use pts'_sub_pz finite in auto)
- then show ?thesis by auto
- qed
- finally show ?thesis .
+ have *: "zorder f z = fls_subdegree F" "zorder g z = fls_subdegree G"
+ "zorder (\<lambda>z. f z + g z) z = fls_subdegree (F + G)"
+ using F G FG has_laurent_expansion_zorder by simp_all
+ moreover have "zorder (\<lambda>z. f z + g z) z = fls_subdegree (F + G)"
+ using has_laurent_expansion_zorder[OF has_laurent_expansion_add[OF F G]] by simp
+ moreover have "fls_subdegree (F + G) \<ge> min (fls_subdegree F) (fls_subdegree G)"
+ by (intro fls_plus_subdegree) simp
+ ultimately show ?thesis
+ using assms(6,7) unfolding * by linarith
qed
-lemma meromorphic_on_imp_isolated_singularity:
- assumes "f meromorphic_on D pts" "z \<in> D"
- shows "isolated_singularity_at f z"
- by (meson DiffI assms(1) assms(2) holomorphic_on_imp_analytic_at isolated_singularity_at_analytic
- meromorphic_imp_open_diff meromorphic_on_def)
-
-lemma meromorphic_imp_not_is_pole:
- assumes "f meromorphic_on D pts" "z \<in> D - pts"
- shows "\<not>is_pole f z"
-proof -
- from assms have "f analytic_on {z}"
- using meromorphic_on_imp_analytic_at by blast
+lemma zorder_diff_ge:
+ fixes f g :: "complex \<Rightarrow> complex"
+ assumes "f meromorphic_on {z}" "frequently (\<lambda>z. f z \<noteq> 0) (at z)"
+ assumes "g meromorphic_on {z}" "frequently (\<lambda>z. g z \<noteq> 0) (at z)"
+ assumes "frequently (\<lambda>z. f z \<noteq> g z) (at z)" "zorder f z \<ge> c" "zorder g z \<ge> c"
+ shows "zorder (\<lambda>z. f z - g z) z \<ge> c"
+proof -
+ have "(\<lambda>z. - g z) meromorphic_on {z}"
+ by (auto intro: meromorphic_intros assms)
thus ?thesis
- using analytic_at not_is_pole_holomorphic by blast
+ using zorder_add_ge[of f z "\<lambda>z. -g z" c] assms by simp
qed
-lemma meromorphic_all_poles_iff_empty [simp]: "f meromorphic_on pts pts \<longleftrightarrow> pts = {}"
- by (auto simp: meromorphic_on_def holomorphic_on_def open_imp_islimpt)
-
-lemma meromorphic_imp_nonsingular_point_exists:
- assumes "f meromorphic_on A pts" "A \<noteq> {}"
- obtains x where "x \<in> A - pts"
+lemma zorder_diff1:
+ assumes "f meromorphic_on {z}" "frequently (\<lambda>z. f z \<noteq> 0) (at z)"
+ assumes "g meromorphic_on {z}" "frequently (\<lambda>z. g z \<noteq> 0) (at z)"
+ assumes "zorder f z < zorder g z"
+ shows "zorder (\<lambda>z. f z - g z) z = zorder f z"
proof -
- have "A \<noteq> pts"
- using assms by auto
- moreover have "pts \<subseteq> A"
- using assms by (auto simp: meromorphic_on_def)
- ultimately show ?thesis
- using that by blast
-qed
-
-lemma meromorphic_frequently_const_imp_const:
- assumes "f meromorphic_on A pts" "connected A"
- assumes "frequently (\<lambda>w. f w = c) (at z)"
- assumes "z \<in> A - pts"
- assumes "w \<in> A - pts"
- shows "f w = c"
-proof -
- have "f w - c = 0"
- proof (rule analytic_continuation[where f = "\<lambda>z. f z - c"])
- show "(\<lambda>z. f z - c) holomorphic_on (A - pts)"
- by (intro holomorphic_intros meromorphic_imp_holomorphic[OF assms(1)])
- show [intro]: "open (A - pts)"
- using assms meromorphic_imp_open_diff by blast
- show "connected (A - pts)"
- using assms meromorphic_imp_connected_diff by blast
- show "{z\<in>A-pts. f z = c} \<subseteq> A - pts"
- by blast
- have "eventually (\<lambda>z. z \<in> A - pts) (at z)"
- using assms by (intro eventually_at_in_open') auto
- hence "frequently (\<lambda>z. f z = c \<and> z \<in> A - pts) (at z)"
- by (intro frequently_eventually_frequently assms)
- thus "z islimpt {z\<in>A-pts. f z = c}"
- by (simp add: islimpt_conv_frequently_at conj_commute)
- qed (use assms in auto)
+ have "zorder (\<lambda>z. f z + (-g z)) z = zorder f z"
+ by (intro zorder_add1 meromorphic_intros assms) (use assms in auto)
thus ?thesis
by simp
qed
-lemma meromorphic_imp_eventually_neq:
- assumes "f meromorphic_on A pts" "connected A" "\<not>f constant_on A - pts"
- assumes "z \<in> A - pts"
- shows "eventually (\<lambda>z. f z \<noteq> c) (at z)"
-proof (rule ccontr)
- assume "\<not>eventually (\<lambda>z. f z \<noteq> c) (at z)"
- hence *: "frequently (\<lambda>z. f z = c) (at z)"
- by (auto simp: frequently_def)
- have "\<forall>w\<in>A-pts. f w = c"
- using meromorphic_frequently_const_imp_const [OF assms(1,2) * assms(4)] by blast
- hence "f constant_on A - pts"
- by (auto simp: constant_on_def)
- thus False
- using assms(3) by contradiction
-qed
-
-lemma meromorphic_frequently_const_imp_const':
- assumes "f meromorphic_on A pts" "connected A" "\<forall>w\<in>pts. is_pole f w"
- assumes "frequently (\<lambda>w. f w = c) (at z)"
- assumes "z \<in> A"
- assumes "w \<in> A"
- shows "f w = c"
+lemma zorder_diff2:
+ assumes "f meromorphic_on {z}" "frequently (\<lambda>z. f z \<noteq> 0) (at z)"
+ assumes "g meromorphic_on {z}" "frequently (\<lambda>z. g z \<noteq> 0) (at z)"
+ assumes "zorder f z > zorder g z"
+ shows "zorder (\<lambda>z. f z - g z) z = zorder g z"
proof -
- have "\<not>is_pole f z"
- using frequently_const_imp_not_is_pole[OF assms(4)] .
- with assms have z: "z \<in> A - pts"
- by auto
- have *: "f w = c" if "w \<in> A - pts" for w
- using that meromorphic_frequently_const_imp_const [OF assms(1,2,4) z] by auto
- have "\<not>is_pole f u" if "u \<in> A" for u
- proof -
- have "is_pole f u \<longleftrightarrow> is_pole (\<lambda>_. c) u"
- proof (rule is_pole_cong)
- have "eventually (\<lambda>w. w \<in> A - (pts - {u}) - {u}) (at u)"
- by (intro eventually_at_in_open meromorphic_imp_open_diff' [OF assms(1)]) (use that in auto)
- thus "eventually (\<lambda>w. f w = c) (at u)"
- by eventually_elim (use * in auto)
- qed auto
- thus ?thesis
- by auto
- qed
- moreover have "pts \<subseteq> A"
- using assms(1) by (simp add: meromorphic_on_def)
- ultimately have "pts = {}"
- using assms(3) by auto
- with * and \<open>w \<in> A\<close> show ?thesis
- by blast
+ have "zorder (\<lambda>z. f z + (-g z)) z = zorder (\<lambda>z. -g z) z"
+ by (intro zorder_add2 meromorphic_intros assms) (use assms in auto)
+ thus ?thesis
+ by simp
qed
-lemma meromorphic_imp_eventually_neq':
- assumes "f meromorphic_on A pts" "connected A" "\<forall>w\<in>pts. is_pole f w" "\<not>f constant_on A"
- assumes "z \<in> A"
- shows "eventually (\<lambda>z. f z \<noteq> c) (at z)"
-proof (rule ccontr)
- assume "\<not>eventually (\<lambda>z. f z \<noteq> c) (at z)"
- hence *: "frequently (\<lambda>z. f z = c) (at z)"
- by (auto simp: frequently_def)
- have "\<forall>w\<in>A. f w = c"
- using meromorphic_frequently_const_imp_const' [OF assms(1,2,3) * assms(5)] by blast
- hence "f constant_on A"
- by (auto simp: constant_on_def)
- thus False
- using assms(4) by contradiction
-qed
-
-lemma zorder_eq_0_iff_meromorphic:
- assumes "f meromorphic_on A pts" "\<forall>z\<in>pts. is_pole f z" "z \<in> A"
- assumes "eventually (\<lambda>x. f x \<noteq> 0) (at z)"
- shows "zorder f z = 0 \<longleftrightarrow> \<not>is_pole f z \<and> f z \<noteq> 0"
-proof (cases "z \<in> pts")
- case True
- from assms obtain F where F: "(\<lambda>x. f (z + x)) has_laurent_expansion F"
- by (metis True meromorphic_on_def not_essential_has_laurent_expansion) (* TODO: better lemmas *)
- from F and assms(4) have [simp]: "F \<noteq> 0"
- using has_laurent_expansion_eventually_nonzero_iff by blast
- show ?thesis using True assms(2)
- using is_pole_fls_subdegree_iff [OF F] has_laurent_expansion_zorder [OF F]
- by auto
-next
- case False
- have ana: "f analytic_on {z}"
- using meromorphic_on_imp_analytic_at False assms by blast
- hence "\<not>is_pole f z"
- using analytic_at not_is_pole_holomorphic by blast
- moreover have "frequently (\<lambda>w. f w \<noteq> 0) (at z)"
- using assms(4) by (intro eventually_frequently) auto
- ultimately show ?thesis using zorder_eq_0_iff[OF ana] False
- by auto
-qed
-
-lemma zorder_pos_iff_meromorphic:
- assumes "f meromorphic_on A pts" "\<forall>z\<in>pts. is_pole f z" "z \<in> A"
- assumes "eventually (\<lambda>x. f x \<noteq> 0) (at z)"
- shows "zorder f z > 0 \<longleftrightarrow> \<not>is_pole f z \<and> f z = 0"
-proof (cases "z \<in> pts")
- case True
- from assms obtain F where F: "(\<lambda>x. f (z + x)) has_laurent_expansion F"
- by (metis True meromorphic_on_def not_essential_has_laurent_expansion) (* TODO: better lemmas *)
- from F and assms(4) have [simp]: "F \<noteq> 0"
- using has_laurent_expansion_eventually_nonzero_iff by blast
- show ?thesis using True assms(2)
- using is_pole_fls_subdegree_iff [OF F] has_laurent_expansion_zorder [OF F]
- by auto
-next
- case False
- have ana: "f analytic_on {z}"
- using meromorphic_on_imp_analytic_at False assms by blast
- hence "\<not>is_pole f z"
- using analytic_at not_is_pole_holomorphic by blast
- moreover have "frequently (\<lambda>w. f w \<noteq> 0) (at z)"
- using assms(4) by (intro eventually_frequently) auto
- ultimately show ?thesis using zorder_pos_iff'[OF ana] False
- by auto
-qed
-
-lemma zorder_neg_iff_meromorphic:
- assumes "f meromorphic_on A pts" "\<forall>z\<in>pts. is_pole f z" "z \<in> A"
- assumes "eventually (\<lambda>x. f x \<noteq> 0) (at z)"
- shows "zorder f z < 0 \<longleftrightarrow> is_pole f z"
+lemma zorder_mult:
+ assumes "f meromorphic_on {z}" "frequently (\<lambda>z. f z \<noteq> 0) (at z)"
+ assumes "g meromorphic_on {z}" "frequently (\<lambda>z. g z \<noteq> 0) (at z)"
+ shows "zorder (\<lambda>z. f z * g z) z = zorder f z + zorder g z"
proof -
- have "frequently (\<lambda>x. f x \<noteq> 0) (at z)"
- using assms by (intro eventually_frequently) auto
- moreover from assms have "isolated_singularity_at f z" "not_essential f z"
- using meromorphic_on_imp_isolated_singularity meromorphic_on_imp_not_essential by blast+
+ from assms(1) obtain F where F: "(\<lambda>w. f (z + w)) has_laurent_expansion F"
+ by (auto simp: meromorphic_on_def)
+ from assms(3) obtain G where G: "(\<lambda>w. g (z + w)) has_laurent_expansion G"
+ by (auto simp: meromorphic_on_def)
+ have [simp]: "F \<noteq> 0" "G \<noteq> 0"
+ by (metis assms has_laurent_expansion_eventually_nonzero_iff meromorphic_at_iff
+ not_essential_frequently_0_imp_eventually_0 not_eventually not_frequently F G)+
+ have *: "zorder f z = fls_subdegree F" "zorder g z = fls_subdegree G"
+ using F G assms by (simp_all add: has_laurent_expansion_zorder)
+ moreover have "zorder (\<lambda>z. f z * g z) z = fls_subdegree (F * G)"
+ using has_laurent_expansion_zorder[OF has_laurent_expansion_mult[OF F G]] by simp
+ moreover have "fls_subdegree (F * G) = fls_subdegree F + fls_subdegree G"
+ using assms by simp
ultimately show ?thesis
- using isolated_pole_imp_neg_zorder neg_zorder_imp_is_pole by blast
+ by (simp add: *)
qed
-lemma meromorphic_on_imp_discrete:
- assumes mero:"f meromorphic_on S pts" and "connected S"
- and nconst:"\<not> (\<forall>w\<in>S - pts. f w = c)"
- shows "discrete {x\<in>S. f x=c}"
+lemma zorder_divide:
+ assumes "f meromorphic_on {z}" "frequently (\<lambda>z. f z \<noteq> 0) (at z)"
+ assumes "g meromorphic_on {z}" "frequently (\<lambda>z. g z \<noteq> 0) (at z)"
+ shows "zorder (\<lambda>z. f z / g z) z = zorder f z - zorder g z"
proof -
- define g where "g=(\<lambda>x. f x - c)"
- have "\<forall>\<^sub>F w in at z. g w \<noteq> 0" if "z \<in> S" for z
- proof (rule nconst_imp_nzero_neighbour'[of g S pts z])
- show "g meromorphic_on S pts" using mero unfolding g_def
- by (auto intro:meromorphic_intros)
- show "\<not> (\<forall>w\<in>S - pts. g w = 0)" using nconst unfolding g_def by auto
- qed fact+
- then show ?thesis
- unfolding discrete_altdef g_def
- using eventually_mono by fastforce
+ from assms(1) obtain F where F: "(\<lambda>w. f (z + w)) has_laurent_expansion F"
+ by (auto simp: meromorphic_on_def)
+ from assms(3) obtain G where G: "(\<lambda>w. g (z + w)) has_laurent_expansion G"
+ by (auto simp: meromorphic_on_def)
+ have [simp]: "F \<noteq> 0" "G \<noteq> 0"
+ by (metis assms has_laurent_expansion_eventually_nonzero_iff meromorphic_at_iff
+ not_essential_frequently_0_imp_eventually_0 not_eventually not_frequently F G)+
+ have *: "zorder f z = fls_subdegree F" "zorder g z = fls_subdegree G"
+ using F G assms by (simp_all add: has_laurent_expansion_zorder)
+ moreover have "zorder (\<lambda>z. f z / g z) z = fls_subdegree (F / G)"
+ using has_laurent_expansion_zorder[OF has_laurent_expansion_divide[OF F G]] by simp
+ moreover have "fls_subdegree (F / G) = fls_subdegree F - fls_subdegree G"
+ using assms by (simp add: fls_divide_subdegree)
+ ultimately show ?thesis
+ by (simp add: *)
qed
-lemma meromorphic_isolated_in:
- assumes merf: "f meromorphic_on D pts" "p\<in>pts"
- shows "p isolated_in pts"
- by (meson assms isolated_in_islimpt_iff meromorphic_on_def subsetD)
-
-lemma remove_sings_constant_on:
- assumes merf: "f meromorphic_on D pts" and "connected D"
- and const:"f constant_on (D - pts)"
- shows "(remove_sings f) constant_on D"
+lemma constant_on_extend_nicely_meromorphic_on:
+ assumes "f nicely_meromorphic_on B" "f constant_on A"
+ assumes "open A" "open B" "connected B" "A \<noteq> {}" "A \<subseteq> B"
+ shows "f constant_on B"
proof -
- have remove_sings_const: "remove_sings f constant_on D - pts"
- using const
- by (metis constant_onE merf meromorphic_on_imp_analytic_at remove_sings_at_analytic)
-
- have ?thesis if "D = {}"
- using that unfolding constant_on_def by auto
- moreover have ?thesis if "D\<noteq>{}" "{x\<in>pts. is_pole f x} = {}"
- proof -
- obtain \<xi> where "\<xi> \<in> (D - pts)" "\<xi> islimpt (D - pts)"
- proof -
- have "open (D - pts)"
- using meromorphic_imp_open_diff[OF merf] .
- moreover have "(D - pts) \<noteq> {}" using \<open>D\<noteq>{}\<close>
- by (metis Diff_empty closure_empty merf
- meromorphic_pts_closure subset_empty)
- ultimately show ?thesis using open_imp_islimpt that by auto
- qed
- moreover have "remove_sings f holomorphic_on D"
- using remove_sings_holomorphic_on[OF merf] that by auto
- moreover note remove_sings_const
- moreover have "open D"
- using assms(1) meromorphic_on_def by blast
- ultimately show ?thesis
- using Conformal_Mappings.analytic_continuation'
- [of "remove_sings f" D "D-pts" \<xi>] \<open>connected D\<close>
- by auto
- qed
- moreover have ?thesis if "D\<noteq>{}" "{x\<in>pts. is_pole f x} \<noteq> {}"
- proof -
- define PP where "PP={x\<in>D. is_pole f x}"
- have "remove_sings f meromorphic_on D PP"
- using merf unfolding PP_def
- apply (elim remove_sings_meromorphic_on)
- subgoal using assms(1) meromorphic_on_def by force
- subgoal using meromorphic_pole_subset merf by auto
- done
- moreover have "remove_sings f constant_on D - PP"
- proof -
- obtain \<xi> where "\<xi> \<in> f ` (D - pts)"
- by (metis Diff_empty Diff_eq_empty_iff \<open>D \<noteq> {}\<close> assms(1)
- closure_empty ex_in_conv imageI meromorphic_pts_closure)
- have \<xi>:"\<forall>x\<in>D - pts. f x = \<xi>"
- by (metis \<open>\<xi> \<in> f ` (D - pts)\<close> assms(3) constant_on_def image_iff)
+ from assms obtain c where c: "\<And>z. z \<in> A \<Longrightarrow> f z = c"
+ by (auto simp: constant_on_def)
+ have "eventually (\<lambda>z. z \<in> A) (cosparse A)"
+ by (intro eventually_in_cosparse assms order.refl)
+ hence "eventually (\<lambda>z. f z = c) (cosparse A)"
+ by eventually_elim (use c in auto)
+ hence freq: "frequently (\<lambda>z. f z = c) (cosparse A)"
+ by (intro eventually_frequently) (use assms in auto)
+ then obtain z0 where z0: "z0 \<in> A" "frequently (\<lambda>z. f z = c) (at z0)"
+ using assms by (auto simp: frequently_def eventually_cosparse_open_eq)
- have "remove_sings f x = \<xi>" if "x\<in>D - PP" for x
- proof (cases "x\<in>pts")
- case True
- then have"x isolated_in pts"
- using meromorphic_isolated_in[OF merf] by auto
- then obtain T0 where T0:"open T0" "T0 \<inter> pts = {x}"
- unfolding isolated_in_def by auto
- obtain T1 where T1:"open T1" "x\<in>T1" "T1 \<subseteq> D"
- using merf unfolding meromorphic_on_def
- using True by blast
- define T2 where "T2 = T1 \<inter> T0"
- have "open T2" "x\<in>T2" "T2 - {x} \<subseteq> D - pts"
- using T0 T1 unfolding T2_def by auto
- then have "\<forall>w\<in>T2. w\<noteq>x \<longrightarrow> f w =\<xi>"
- using \<xi> by auto
- then have "\<forall>\<^sub>F x in at x. f x = \<xi>"
- unfolding eventually_at_topological
- using \<open>open T2\<close> \<open>x\<in>T2\<close> by auto
- then have "f \<midarrow>x\<rightarrow> \<xi>"
- using tendsto_eventually by auto
- then show ?thesis by blast
- next
- case False
- then show ?thesis
- using \<open>\<forall>x\<in>D - pts. f x = \<xi>\<close> assms(1)
- meromorphic_on_imp_analytic_at that by auto
- qed
-
- then show ?thesis unfolding constant_on_def by auto
- qed
-
- moreover have "is_pole (remove_sings f) x" if "x\<in>PP" for x
- proof -
- have "isolated_singularity_at f x"
- by (metis (mono_tags, lifting) DiffI PP_def assms(1)
- isolated_singularity_at_analytic mem_Collect_eq
- meromorphic_on_def meromorphic_on_imp_analytic_at that)
- then show ?thesis using that unfolding PP_def by simp
- qed
- ultimately show ?thesis
- using meromorphic_imp_constant_on
- [of "remove_sings f" D PP]
- by auto
- qed
- ultimately show ?thesis by auto
-qed
-
-lemma meromorphic_eq_meromorphic_extend:
- assumes "f meromorphic_on A pts1" "g meromorphic_on A pts1" "\<not>z islimpt pts2"
- assumes "\<And>z. z \<in> A - pts2 \<Longrightarrow> f z = g z" "pts1 \<subseteq> pts2" "z \<in> A - pts1"
- shows "f z = g z"
-proof -
- have "g analytic_on {z}"
- using assms by (intro meromorphic_on_imp_analytic_at[OF assms(2)]) auto
- hence "g \<midarrow>z\<rightarrow> g z"
- using analytic_at_imp_isCont isContD by blast
- also have "?this \<longleftrightarrow> f \<midarrow>z\<rightarrow> g z"
- proof (intro filterlim_cong)
- have "eventually (\<lambda>w. w \<notin> pts2) (at z)"
- using assms by (auto simp: islimpt_conv_frequently_at frequently_def)
- moreover have "eventually (\<lambda>w. w \<in> A) (at z)"
- using assms by (intro eventually_at_in_open') (auto simp: meromorphic_on_def)
- ultimately show "\<forall>\<^sub>F x in at z. g x = f x"
- by eventually_elim (use assms in auto)
- qed auto
- finally have "f \<midarrow>z\<rightarrow> g z" .
- moreover have "f analytic_on {z}"
- using assms by (intro meromorphic_on_imp_analytic_at[OF assms(1)]) auto
- hence "f \<midarrow>z\<rightarrow> f z"
- using analytic_at_imp_isCont isContD by blast
- ultimately show ?thesis
- using tendsto_unique by force
-qed
-
-lemma meromorphic_constant_on_extend:
- assumes "f constant_on A - pts1" "f meromorphic_on A pts1" "f meromorphic_on A pts2" "pts2 \<subseteq> pts1"
- shows "f constant_on A - pts2"
-proof -
- from assms(1) obtain c where c: "\<And>z. z \<in> A - pts1 \<Longrightarrow> f z = c"
- unfolding constant_on_def by auto
- have "f z = c" if "z \<in> A - pts2" for z
- using assms(3)
- proof (rule meromorphic_eq_meromorphic_extend[where z = z])
- show "(\<lambda>a. c) meromorphic_on A pts2"
- by (intro meromorphic_on_const) (use assms in \<open>auto simp: meromorphic_on_def\<close>)
- show "\<not>z islimpt pts1"
- using that assms by (auto simp: meromorphic_on_def)
- qed (use assms c that in auto)
- thus ?thesis
+ have "f z = c" if "z \<in> B" for z
+ proof (rule frequently_eq_meromorphic_imp_constant[OF _ assms(1)])
+ show "z0 \<in> B" "frequently (\<lambda>z. f z = c) (at z0)"
+ using z0 assms by auto
+ qed (use assms that in auto)
+ thus "f constant_on B"
by (auto simp: constant_on_def)
qed
-lemma meromorphic_remove_sings_constant_on_imp_constant_on:
- assumes "f meromorphic_on A pts"
- assumes "remove_sings f constant_on A"
- shows "f constant_on A - pts"
-proof -
- from assms(2) obtain c where c: "\<And>z. z \<in> A \<Longrightarrow> remove_sings f z = c"
- by (auto simp: constant_on_def)
- have "f z = c" if "z \<in> A - pts" for z
- using meromorphic_on_imp_analytic_at[OF assms(1) that] c[of z] that
- by auto
- thus ?thesis
- by (auto simp: constant_on_def)
-qed
-
-
-
-
-definition singularities_on :: "complex set \<Rightarrow> (complex \<Rightarrow> complex) \<Rightarrow> complex set" where
- "singularities_on A f =
- {z\<in>A. isolated_singularity_at f z \<and> not_essential f z \<and> \<not>f analytic_on {z}}"
-
-lemma singularities_on_subset: "singularities_on A f \<subseteq> A"
- by (auto simp: singularities_on_def)
-
-lemma pole_in_singularities_on:
- assumes "f meromorphic_on A pts" "z \<in> A" "is_pole f z"
- shows "z \<in> singularities_on A f"
- unfolding singularities_on_def not_essential_def using assms
- using analytic_at_imp_no_pole meromorphic_on_imp_isolated_singularity by force
-
-
-lemma meromorphic_on_subset_pts:
- assumes "f meromorphic_on A pts" "pts' \<subseteq> pts" "f analytic_on pts - pts'"
- shows "f meromorphic_on A pts'"
-proof
- show "open A" "pts' \<subseteq> A"
- using assms by (auto simp: meromorphic_on_def)
- show "isolated_singularity_at f z" "not_essential f z" if "z \<in> pts'" for z
- using assms that by (auto simp: meromorphic_on_def)
- show "\<not>z islimpt pts'" if "z \<in> A" for z
- using assms that islimpt_subset unfolding meromorphic_on_def by blast
- have "f analytic_on A - pts"
- using assms(1) meromorphic_imp_analytic by blast
- with assms have "f analytic_on (A - pts) \<union> (pts - pts')"
- by (subst analytic_on_Un) auto
- also have "(A - pts) \<union> (pts - pts') = A - pts'"
- using assms by (auto simp: meromorphic_on_def)
- finally show "f holomorphic_on A - pts'"
- using analytic_imp_holomorphic by blast
-qed
-
-lemma meromorphic_on_imp_superset_singularities_on:
- assumes "f meromorphic_on A pts"
- shows "singularities_on A f \<subseteq> pts"
-proof
- fix z assume "z \<in> singularities_on A f"
- hence "z \<in> A" "\<not>f analytic_on {z}"
- by (auto simp: singularities_on_def)
- with assms show "z \<in> pts"
- by (meson DiffI meromorphic_on_imp_analytic_at)
-qed
-
-lemma meromorphic_on_singularities_on:
- assumes "f meromorphic_on A pts"
- shows "f meromorphic_on A (singularities_on A f)"
- using assms meromorphic_on_imp_superset_singularities_on[OF assms]
-proof (rule meromorphic_on_subset_pts)
- have "f analytic_on {z}" if "z \<in> pts - singularities_on A f" for z
- using that assms by (auto simp: singularities_on_def meromorphic_on_def)
- thus "f analytic_on pts - singularities_on A f"
- using analytic_on_analytic_at by blast
-qed
-
-theorem Residue_theorem_inside:
- assumes f: "f meromorphic_on s pts"
- "simply_connected s"
- assumes g: "valid_path g"
- "pathfinish g = pathstart g"
- "path_image g \<subseteq> s - pts"
- defines "pts1 \<equiv> pts \<inter> inside (path_image g)"
- shows "finite pts1"
- and "contour_integral g f = 2 * pi * \<i> * (\<Sum>p\<in>pts1. winding_number g p * residue f p)"
-proof -
- note [dest] = valid_path_imp_path
- have cl_g [intro]: "closed (path_image g)"
- using g by (auto intro!: closed_path_image)
- have "open s"
- using f(1) by (auto simp: meromorphic_on_def)
- define pts2 where "pts2 = pts - pts1"
-
- define A where "A = path_image g \<union> inside (path_image g)"
- have "closed A"
- unfolding A_def using g by (intro closed_path_image_Un_inside) auto
- moreover have "bounded A"
- unfolding A_def using g by (auto intro!: bounded_path_image bounded_inside)
- ultimately have 1: "compact A"
- using compact_eq_bounded_closed by blast
- have 2: "open (s - pts2)"
- using f by (auto intro!: meromorphic_imp_open_diff' [OF f(1)] simp: pts2_def)
- have 3: "A \<subseteq> s - pts2"
- unfolding A_def pts2_def pts1_def
- using f(2) g(3) 2 subset_simply_connected_imp_inside_subset[of s "path_image g"] \<open>open s\<close>
- by auto
-
- obtain \<epsilon> where \<epsilon>: "\<epsilon> > 0" "(\<Union>x\<in>A. ball x \<epsilon>) \<subseteq> s - pts2"
- using compact_subset_open_imp_ball_epsilon_subset[OF 1 2 3] by blast
- define B where "B = (\<Union>x\<in>A. ball x \<epsilon>)"
-
- have "finite (A \<inter> pts)"
- using 1 3 by (intro meromorphic_compact_finite_pts[OF f(1)]) auto
- also have "A \<inter> pts = pts1"
- unfolding pts1_def using g by (auto simp: A_def)
- finally show fin: "finite pts1" .
-
- show "contour_integral g f = 2 * pi * \<i> * (\<Sum>p\<in>pts1. winding_number g p * residue f p)"
- proof (rule Residue_theorem)
- show "open B"
- by (auto simp: B_def)
- next
- have "connected A"
- unfolding A_def using g
- by (intro connected_with_inside closed_path_image connected_path_image) auto
- hence "connected (A \<union> B)"
- unfolding B_def using g \<open>\<epsilon> > 0\<close> f(2)
- by (intro connected_Un_UN connected_path_image valid_path_imp_path)
- (auto simp: simply_connected_imp_connected)
- also have "A \<union> B = B"
- using \<epsilon>(1) by (auto simp: B_def)
- finally show "connected B" .
- next
- have "f holomorphic_on (s - pts)"
- by (intro meromorphic_imp_holomorphic f)
- moreover have "B - pts1 \<subseteq> s - pts"
- using \<epsilon> unfolding B_def by (auto simp: pts1_def pts2_def)
- ultimately show "f holomorphic_on (B - pts1)"
- by (rule holomorphic_on_subset)
- next
- have "path_image g \<subseteq> A - pts1"
- using g unfolding pts1_def by (auto simp: A_def)
- also have "\<dots> \<subseteq> B - pts1"
- unfolding B_def using \<epsilon>(1) by auto
- finally show "path_image g \<subseteq> B - pts1" .
- next
- show "\<forall>z. z \<notin> B \<longrightarrow> winding_number g z = 0"
- proof safe
- fix z assume z: "z \<notin> B"
- hence "z \<notin> A"
- using \<epsilon>(1) by (auto simp: B_def)
- hence "z \<in> outside (path_image g)"
- unfolding A_def by (simp add: union_with_inside)
- thus "winding_number g z = 0"
- using g by (intro winding_number_zero_in_outside) auto
- qed
- qed (use g fin in auto)
-qed
-
-theorem Residue_theorem':
- assumes f: "f meromorphic_on s pts"
- "simply_connected s"
- assumes g: "valid_path g"
- "pathfinish g = pathstart g"
- "path_image g \<subseteq> s - pts"
- assumes pts': "finite pts'"
- "pts' \<subseteq> s"
- "\<And>z. z \<in> pts - pts' \<Longrightarrow> winding_number g z = 0"
- shows "contour_integral g f = 2 * pi * \<i> * (\<Sum>p\<in>pts'. winding_number g p * residue f p)"
-proof -
- note [dest] = valid_path_imp_path
- define pts1 where "pts1 = pts \<inter> inside (path_image g)"
-
- have "contour_integral g f = 2 * pi * \<i> * (\<Sum>p\<in>pts1. winding_number g p * residue f p)"
- unfolding pts1_def by (intro Residue_theorem_inside[OF f g])
- also have "(\<Sum>p\<in>pts1. winding_number g p * residue f p) =
- (\<Sum>p\<in>pts'. winding_number g p * residue f p)"
- proof (intro sum.mono_neutral_cong refl)
- show "finite pts1"
- unfolding pts1_def by (intro Residue_theorem_inside[OF f g])
- show "finite pts'"
- by fact
- next
- fix z assume z: "z \<in> pts' - pts1"
- show "winding_number g z * residue f z = 0"
- proof (cases "z \<in> pts")
- case True
- with z have "z \<notin> path_image g \<union> inside (path_image g)"
- using g(3) by (auto simp: pts1_def)
- hence "z \<in> outside (path_image g)"
- by (simp add: union_with_inside)
- hence "winding_number g z = 0"
- using g by (intro winding_number_zero_in_outside) auto
- thus ?thesis
- by simp
- next
- case False
- with z pts' have "z \<in> s - pts"
- by auto
- with f(1) have "f analytic_on {z}"
- by (intro meromorphic_on_imp_analytic_at)
- hence "residue f z = 0"
- using analytic_at residue_holo by blast
- thus ?thesis
- by simp
- qed
- next
- fix z assume z: "z \<in> pts1 - pts'"
- hence "winding_number g z = 0"
- using pts' by (auto simp: pts1_def)
- thus "winding_number g z * residue f z = 0"
- by simp
- qed
- finally show ?thesis .
-qed
-
end
--- a/src/HOL/Deriv.thy Mon Mar 11 08:46:20 2024 +0100
+++ b/src/HOL/Deriv.thy Mon Mar 11 15:07:02 2024 +0000
@@ -98,6 +98,12 @@
unfolding has_derivative_def
by (auto simp add: bounded_linear_compose [OF bounded_linear] scaleR diff dest: tendsto)
+lemma has_derivative_bot [intro]: "bounded_linear f' \<Longrightarrow> (f has_derivative f') bot"
+ by (auto simp: has_derivative_def)
+
+lemma has_field_derivative_bot [simp, intro]: "(f has_field_derivative f') bot"
+ by (auto simp: has_field_derivative_def intro!: has_derivative_bot bounded_linear_mult_right)
+
lemmas has_derivative_scaleR_right [derivative_intros] =
bounded_linear.has_derivative [OF bounded_linear_scaleR_right]
@@ -814,6 +820,13 @@
lemma DERIV_def: "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"
unfolding field_has_derivative_at has_field_derivative_def has_field_derivative_iff ..
+lemma has_field_derivative_unique:
+ assumes "(f has_field_derivative f'1) (at x within A)"
+ assumes "(f has_field_derivative f'2) (at x within A)"
+ assumes "at x within A \<noteq> bot"
+ shows "f'1 = f'2"
+ using assms unfolding has_field_derivative_iff using tendsto_unique by blast
+
text \<open>due to Christian Pardillo Laursen, replacing a proper epsilon-delta horror\<close>
lemma field_derivative_lim_unique:
assumes f: "(f has_field_derivative df) (at z)"
--- a/src/HOL/Nat.thy Mon Mar 11 08:46:20 2024 +0100
+++ b/src/HOL/Nat.thy Mon Mar 11 15:07:02 2024 +0000
@@ -1926,6 +1926,9 @@
lemma Nats_induct [case_names of_nat, induct set: Nats]: "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"
by (rule Nats_cases) auto
+lemma Nats_nonempty [simp]: "\<nat> \<noteq> {}"
+ unfolding Nats_def by auto
+
end
lemma Nats_diff [simp]: