--- a/src/HOL/Analysis/Complex_Analysis_Basics.thy Wed Feb 08 15:05:24 2023 +0000
+++ b/src/HOL/Analysis/Complex_Analysis_Basics.thy Thu Feb 09 13:36:25 2023 +0000
@@ -417,14 +417,24 @@
(metis centre_in_ball field_differentiable_at_within)
lemma analytic_on_open: "open S \<Longrightarrow> f analytic_on S \<longleftrightarrow> f holomorphic_on S"
-apply (auto simp: analytic_imp_holomorphic)
-apply (auto simp: analytic_on_def holomorphic_on_def)
-by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball)
+ by (meson analytic_imp_holomorphic analytic_on_def holomorphic_on_subset openE)
lemma analytic_on_imp_differentiable_at:
"f analytic_on S \<Longrightarrow> x \<in> S \<Longrightarrow> f field_differentiable (at x)"
- apply (auto simp: analytic_on_def holomorphic_on_def)
-by (metis open_ball centre_in_ball field_differentiable_within_open)
+ using analytic_on_def holomorphic_on_imp_differentiable_at by auto
+
+lemma analytic_at_imp_isCont:
+ assumes "f analytic_on {z}"
+ shows "isCont f z"
+ using assms by (meson analytic_on_imp_differentiable_at field_differentiable_imp_continuous_at insertI1)
+
+lemma analytic_at_neq_imp_eventually_neq:
+ assumes "f analytic_on {x}" "f x \<noteq> c"
+ shows "eventually (\<lambda>y. f y \<noteq> c) (at x)"
+proof (intro tendsto_imp_eventually_ne)
+ show "f \<midarrow>x\<rightarrow> f x"
+ using assms by (simp add: analytic_at_imp_isCont isContD)
+qed (use assms in auto)
lemma analytic_on_subset: "f analytic_on S \<Longrightarrow> T \<subseteq> S \<Longrightarrow> f analytic_on T"
by (auto simp: analytic_on_def)
@@ -652,15 +662,20 @@
lemma analytic_at_ball:
"f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
-by (metis analytic_on_def singleton_iff)
+ by (metis analytic_on_def singleton_iff)
lemma analytic_at:
- "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
-by (metis analytic_on_holomorphic empty_subsetI insert_subset)
+ "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
+ by (metis analytic_on_holomorphic empty_subsetI insert_subset)
+
+lemma holomorphic_on_imp_analytic_at:
+ assumes "f holomorphic_on A" "open A" "z \<in> A"
+ shows "f analytic_on {z}"
+ using assms by (meson analytic_at)
lemma analytic_on_analytic_at:
- "f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
-by (metis analytic_at_ball analytic_on_def)
+ "f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
+ by (metis analytic_at_ball analytic_on_def)
lemma analytic_at_two:
"f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
--- a/src/HOL/Analysis/Isolated.thy Wed Feb 08 15:05:24 2023 +0000
+++ b/src/HOL/Analysis/Isolated.thy Thu Feb 09 13:36:25 2023 +0000
@@ -14,6 +14,12 @@
definition (in metric_space) uniform_discrete :: "'a set \<Rightarrow> bool" where
"uniform_discrete S \<longleftrightarrow> (\<exists>e>0. \<forall>x\<in>S. \<forall>y\<in>S. dist x y < e \<longrightarrow> x = y)"
+lemma discreteI: "(\<And>x. x \<in> X \<Longrightarrow> x isolated_in X ) \<Longrightarrow> discrete X"
+ unfolding discrete_def by auto
+
+lemma discreteD: "discrete X \<Longrightarrow> x \<in> X \<Longrightarrow> x isolated_in X "
+ unfolding discrete_def by auto
+
lemma uniformI1:
assumes "e>0" "\<And>x y. \<lbrakk>x\<in>S;y\<in>S;dist x y<e\<rbrakk> \<Longrightarrow> x =y "
shows "uniform_discrete S"
@@ -43,6 +49,54 @@
shows "x isolated_in (insert a S) \<longleftrightarrow> x isolated_in S \<or> (x=a \<and> \<not> (x islimpt S))"
by (meson insert_iff islimpt_insert isolated_in_islimpt_iff)
+lemma isolated_inI:
+ assumes "x\<in>S" "open T" "T \<inter> S = {x}"
+ shows "x isolated_in S"
+ using assms unfolding isolated_in_def by auto
+
+lemma isolated_inE:
+ assumes "x isolated_in S"
+ obtains T where "x \<in> S" "open T" "T \<inter> S = {x}"
+ using assms that unfolding isolated_in_def by force
+
+lemma isolated_inE_dist:
+ assumes "x isolated_in S"
+ obtains d where "d > 0" "\<And>y. y \<in> S \<Longrightarrow> dist x y < d \<Longrightarrow> y = x"
+ by (meson assms isolated_in_dist_Ex_iff)
+
+lemma isolated_in_altdef:
+ "x isolated_in S \<longleftrightarrow> (x\<in>S \<and> eventually (\<lambda>y. y \<notin> S) (at x))"
+proof
+ assume "x isolated_in S"
+ from isolated_inE[OF this]
+ obtain T where "x \<in> S" and T:"open T" "T \<inter> S = {x}"
+ by metis
+ have "\<forall>\<^sub>F y in nhds x. y \<in> T"
+ apply (rule eventually_nhds_in_open)
+ using T by auto
+ then have "eventually (\<lambda>y. y \<in> T - {x}) (at x)"
+ unfolding eventually_at_filter by eventually_elim auto
+ then have "eventually (\<lambda>y. y \<notin> S) (at x)"
+ by eventually_elim (use T in auto)
+ then show " x \<in> S \<and> (\<forall>\<^sub>F y in at x. y \<notin> S)" using \<open>x \<in> S\<close> by auto
+next
+ assume "x \<in> S \<and> (\<forall>\<^sub>F y in at x. y \<notin> S)"
+ then have "\<forall>\<^sub>F y in at x. y \<notin> S" "x\<in>S" by auto
+ from this(1) have "eventually (\<lambda>y. y \<notin> S \<or> y = x) (nhds x)"
+ unfolding eventually_at_filter by eventually_elim auto
+ then obtain T where T:"open T" "x \<in> T" "(\<forall>y\<in>T. y \<notin> S \<or> y = x)"
+ unfolding eventually_nhds by auto
+ with \<open>x \<in> S\<close> have "T \<inter> S = {x}"
+ by fastforce
+ with \<open>x\<in>S\<close> \<open>open T\<close>
+ show "x isolated_in S"
+ unfolding isolated_in_def by auto
+qed
+
+lemma discrete_altdef:
+ "discrete S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>\<^sub>F y in at x. y \<notin> S)"
+ unfolding discrete_def isolated_in_altdef by auto
+
(*
TODO.
Other than
@@ -194,4 +248,80 @@
ultimately show ?thesis by fastforce
qed
+definition sparse :: "real \<Rightarrow> 'a :: metric_space set \<Rightarrow> bool"
+ where "sparse \<epsilon> X \<longleftrightarrow> (\<forall>x\<in>X. \<forall>y\<in>X-{x}. dist x y > \<epsilon>)"
+
+lemma sparse_empty [simp, intro]: "sparse \<epsilon> {}"
+ by (auto simp: sparse_def)
+
+lemma sparseI [intro?]:
+ "(\<And>x y. x \<in> X \<Longrightarrow> y \<in> X \<Longrightarrow> x \<noteq> y \<Longrightarrow> dist x y > \<epsilon>) \<Longrightarrow> sparse \<epsilon> X"
+ unfolding sparse_def by auto
+
+lemma sparseD:
+ "sparse \<epsilon> X \<Longrightarrow> x \<in> X \<Longrightarrow> y \<in> X \<Longrightarrow> x \<noteq> y \<Longrightarrow> dist x y > \<epsilon>"
+ unfolding sparse_def by auto
+
+lemma sparseD':
+ "sparse \<epsilon> X \<Longrightarrow> x \<in> X \<Longrightarrow> y \<in> X \<Longrightarrow> dist x y \<le> \<epsilon> \<Longrightarrow> x = y"
+ unfolding sparse_def by force
+
+lemma sparse_singleton [simp, intro]: "sparse \<epsilon> {x}"
+ by (auto simp: sparse_def)
+
+definition setdist_gt where "setdist_gt \<epsilon> X Y \<longleftrightarrow> (\<forall>x\<in>X. \<forall>y\<in>Y. dist x y > \<epsilon>)"
+
+lemma setdist_gt_empty [simp]: "setdist_gt \<epsilon> {} Y" "setdist_gt \<epsilon> X {}"
+ by (auto simp: setdist_gt_def)
+
+lemma setdist_gtI: "(\<And>x y. x \<in> X \<Longrightarrow> y \<in> Y \<Longrightarrow> dist x y > \<epsilon>) \<Longrightarrow> setdist_gt \<epsilon> X Y"
+ unfolding setdist_gt_def by auto
+
+lemma setdist_gtD: "setdist_gt \<epsilon> X Y \<Longrightarrow> x \<in> X \<Longrightarrow> y \<in> Y \<Longrightarrow> dist x y > \<epsilon>"
+ unfolding setdist_gt_def by auto
+
+lemma setdist_gt_setdist: "\<epsilon> < setdist A B \<Longrightarrow> setdist_gt \<epsilon> A B"
+ unfolding setdist_gt_def using setdist_le_dist by fastforce
+
+lemma setdist_gt_mono: "setdist_gt \<epsilon>' A B \<Longrightarrow> \<epsilon> \<le> \<epsilon>' \<Longrightarrow> A' \<subseteq> A \<Longrightarrow> B' \<subseteq> B \<Longrightarrow> setdist_gt \<epsilon> A' B'"
+ by (force simp: setdist_gt_def)
+
+lemma setdist_gt_Un_left: "setdist_gt \<epsilon> (A \<union> B) C \<longleftrightarrow> setdist_gt \<epsilon> A C \<and> setdist_gt \<epsilon> B C"
+ by (auto simp: setdist_gt_def)
+
+lemma setdist_gt_Un_right: "setdist_gt \<epsilon> C (A \<union> B) \<longleftrightarrow> setdist_gt \<epsilon> C A \<and> setdist_gt \<epsilon> C B"
+ by (auto simp: setdist_gt_def)
+
+lemma compact_closed_imp_eventually_setdist_gt_at_right_0:
+ assumes "compact A" "closed B" "A \<inter> B = {}"
+ shows "eventually (\<lambda>\<epsilon>. setdist_gt \<epsilon> A B) (at_right 0)"
+proof (cases "A = {} \<or> B = {}")
+ case False
+ hence "setdist A B > 0"
+ by (metis IntI assms empty_iff in_closed_iff_infdist_zero order_less_le setdist_attains_inf setdist_pos_le setdist_sym)
+ hence "eventually (\<lambda>\<epsilon>. \<epsilon> < setdist A B) (at_right 0)"
+ using eventually_at_right_field by blast
+ thus ?thesis
+ by eventually_elim (auto intro: setdist_gt_setdist)
+qed auto
+
+lemma setdist_gt_symI: "setdist_gt \<epsilon> A B \<Longrightarrow> setdist_gt \<epsilon> B A"
+ by (force simp: setdist_gt_def dist_commute)
+
+lemma setdist_gt_sym: "setdist_gt \<epsilon> A B \<longleftrightarrow> setdist_gt \<epsilon> B A"
+ by (force simp: setdist_gt_def dist_commute)
+
+lemma eventually_setdist_gt_at_right_0_mult_iff:
+ assumes "c > 0"
+ shows "eventually (\<lambda>\<epsilon>. setdist_gt (c * \<epsilon>) A B) (at_right 0) \<longleftrightarrow>
+ eventually (\<lambda>\<epsilon>. setdist_gt \<epsilon> A B) (at_right 0)"
+proof -
+ have "eventually (\<lambda>\<epsilon>. setdist_gt (c * \<epsilon>) A B) (at_right 0) \<longleftrightarrow>
+ eventually (\<lambda>\<epsilon>. setdist_gt \<epsilon> A B) (filtermap ((*) c) (at_right 0))"
+ by (simp add: eventually_filtermap)
+ also have "filtermap ((*) c) (at_right 0) = at_right 0"
+ by (subst filtermap_times_pos_at_right) (use assms in auto)
+ finally show ?thesis .
+qed
+
end
--- a/src/HOL/Complex_Analysis/Complex_Singularities.thy Wed Feb 08 15:05:24 2023 +0000
+++ b/src/HOL/Complex_Analysis/Complex_Singularities.thy Thu Feb 09 13:36:25 2023 +0000
@@ -130,6 +130,84 @@
shows "is_pole (\<lambda>w. f w / w ^ n) 0"
using is_pole_basic[of f A 0] assms by simp
+lemma is_pole_compose:
+ assumes "is_pole f w" "g \<midarrow>z\<rightarrow> w" "eventually (\<lambda>z. g z \<noteq> w) (at z)"
+ shows "is_pole (\<lambda>x. f (g x)) z"
+ using assms(1) unfolding is_pole_def
+ by (rule filterlim_compose) (use assms in \<open>auto simp: filterlim_at\<close>)
+
+lemma is_pole_plus_const_iff:
+ "is_pole f z \<longleftrightarrow> is_pole (\<lambda>x. f x + c) z"
+proof
+ assume "is_pole f z"
+ then have "filterlim f at_infinity (at z)" unfolding is_pole_def .
+ moreover have "((\<lambda>_. c) \<longlongrightarrow> c) (at z)" by auto
+ ultimately have " LIM x (at z). f x + c :> at_infinity"
+ using tendsto_add_filterlim_at_infinity'[of f "at z"] by auto
+ then show "is_pole (\<lambda>x. f x + c) z" unfolding is_pole_def .
+next
+ assume "is_pole (\<lambda>x. f x + c) z"
+ then have "filterlim (\<lambda>x. f x + c) at_infinity (at z)"
+ unfolding is_pole_def .
+ moreover have "((\<lambda>_. -c) \<longlongrightarrow> -c) (at z)" by auto
+ ultimately have " LIM x (at z). f x :> at_infinity"
+ using tendsto_add_filterlim_at_infinity'[of "(\<lambda>x. f x + c)"
+ "at z" "(\<lambda>_. - c)" "-c"]
+ by auto
+ then show "is_pole f z" unfolding is_pole_def .
+qed
+
+lemma is_pole_minus_const_iff:
+ "is_pole (\<lambda>x. f x - c) z \<longleftrightarrow> is_pole f z"
+ using is_pole_plus_const_iff [of f z "-c"] by simp
+
+lemma is_pole_alt:
+ "is_pole f x = (\<forall>B>0. \<exists>U. open U \<and> x\<in>U \<and> (\<forall>y\<in>U. y\<noteq>x \<longrightarrow> norm (f y)\<ge>B))"
+ unfolding is_pole_def
+ unfolding filterlim_at_infinity[of 0,simplified] eventually_at_topological
+ by auto
+
+lemma is_pole_mult_analytic_nonzero1:
+ assumes "is_pole g x" "f analytic_on {x}" "f x \<noteq> 0"
+ shows "is_pole (\<lambda>x. f x * g x) x"
+ unfolding is_pole_def
+proof (rule tendsto_mult_filterlim_at_infinity)
+ show "f \<midarrow>x\<rightarrow> f x"
+ using assms by (simp add: analytic_at_imp_isCont isContD)
+qed (use assms in \<open>auto simp: is_pole_def\<close>)
+
+lemma is_pole_mult_analytic_nonzero2:
+ assumes "is_pole f x" "g analytic_on {x}" "g x \<noteq> 0"
+ shows "is_pole (\<lambda>x. f x * g x) x"
+ by (subst mult.commute, rule is_pole_mult_analytic_nonzero1) (use assms in auto)
+
+lemma is_pole_mult_analytic_nonzero1_iff:
+ assumes "f analytic_on {x}" "f x \<noteq> 0"
+ shows "is_pole (\<lambda>x. f x * g x) x \<longleftrightarrow> is_pole g x"
+proof
+ assume "is_pole g x"
+ thus "is_pole (\<lambda>x. f x * g x) x"
+ by (intro is_pole_mult_analytic_nonzero1 assms)
+next
+ assume "is_pole (\<lambda>x. f x * g x) x"
+ hence "is_pole (\<lambda>x. inverse (f x) * (f x * g x)) x"
+ by (rule is_pole_mult_analytic_nonzero1)
+ (use assms in \<open>auto intro!: analytic_intros\<close>)
+ also have "?this \<longleftrightarrow> is_pole g x"
+ proof (rule is_pole_cong)
+ have "eventually (\<lambda>x. f x \<noteq> 0) (at x)"
+ using assms by (simp add: analytic_at_neq_imp_eventually_neq)
+ thus "eventually (\<lambda>x. inverse (f x) * (f x * g x) = g x) (at x)"
+ by eventually_elim auto
+ qed auto
+ finally show "is_pole g x" .
+qed
+
+lemma is_pole_mult_analytic_nonzero2_iff:
+ assumes "g analytic_on {x}" "g x \<noteq> 0"
+ shows "is_pole (\<lambda>x. f x * g x) x \<longleftrightarrow> is_pole f x"
+ by (subst mult.commute, rule is_pole_mult_analytic_nonzero1_iff) (fact assms)+
+
text \<open>The proposition
\<^term>\<open>\<exists>x. ((f::complex\<Rightarrow>complex) \<longlongrightarrow> x) (at z) \<or> is_pole f z\<close>
can be interpreted as the complex function \<^term>\<open>f\<close> has a non-essential singularity at \<^term>\<open>z\<close>
@@ -140,6 +218,39 @@
definition isolated_singularity_at::"[complex \<Rightarrow> complex, complex] \<Rightarrow> bool" where
"isolated_singularity_at f z = (\<exists>r>0. f analytic_on ball z r-{z})"
+lemma not_essential_cong:
+ assumes "eventually (\<lambda>x. f x = g x) (at z)" "z = z'"
+ shows "not_essential f z \<longleftrightarrow> not_essential g z'"
+ unfolding not_essential_def using assms filterlim_cong is_pole_cong by fastforce
+
+lemma isolated_singularity_at_cong:
+ assumes "eventually (\<lambda>x. f x = g x) (at z)" "z = z'"
+ shows "isolated_singularity_at f z \<longleftrightarrow> isolated_singularity_at g z'"
+proof -
+ have "isolated_singularity_at g z"
+ if "isolated_singularity_at f z" "eventually (\<lambda>x. f x = g x) (at z)" for f g
+ proof -
+ from that(1) obtain r where r: "r > 0" "f analytic_on ball z r - {z}"
+ by (auto simp: isolated_singularity_at_def)
+ from that(2) obtain r' where r': "r' > 0" "\<forall>x\<in>ball z r'-{z}. f x = g x"
+ unfolding eventually_at_filter eventually_nhds_metric by (auto simp: dist_commute)
+
+ have "f holomorphic_on ball z r - {z}"
+ using r(2) by (subst (asm) analytic_on_open) auto
+ hence "f holomorphic_on ball z (min r r') - {z}"
+ by (rule holomorphic_on_subset) auto
+ also have "?this \<longleftrightarrow> g holomorphic_on ball z (min r r') - {z}"
+ using r' by (intro holomorphic_cong) auto
+ also have "\<dots> \<longleftrightarrow> g analytic_on ball z (min r r') - {z}"
+ by (subst analytic_on_open) auto
+ finally show ?thesis
+ unfolding isolated_singularity_at_def
+ by (intro exI[of _ "min r r'"]) (use \<open>r > 0\<close> \<open>r' > 0\<close> in auto)
+ qed
+ from this[of f g] this[of g f] assms show ?thesis
+ by (auto simp: eq_commute)
+qed
+
lemma removable_singularity:
assumes "f holomorphic_on A - {x}" "open A"
assumes "f \<midarrow>x\<rightarrow> c"
@@ -795,6 +906,24 @@
using assms unfolding isolated_singularity_at_def
by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)
+lemma isolated_singularity_at_altdef:
+ "isolated_singularity_at f z \<longleftrightarrow> eventually (\<lambda>z. f analytic_on {z}) (at z)"
+proof
+ assume "isolated_singularity_at f z"
+ then obtain r where r: "r > 0" "f analytic_on ball z r - {z}"
+ unfolding isolated_singularity_at_def by blast
+ have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
+ using r(1) by (intro eventually_at_in_open) auto
+ thus "eventually (\<lambda>z. f analytic_on {z}) (at z)"
+ by eventually_elim (use r analytic_on_subset in auto)
+next
+ assume "eventually (\<lambda>z. f analytic_on {z}) (at z)"
+ then obtain A where A: "open A" "z \<in> A" "\<And>w. w \<in> A - {z} \<Longrightarrow> f analytic_on {w}"
+ unfolding eventually_at_topological by blast
+ then show "isolated_singularity_at f z"
+ by (meson analytic_imp_holomorphic analytic_on_analytic_at isolated_singularity_at_holomorphic)
+qed
+
lemma isolated_singularity_at_shift:
assumes "isolated_singularity_at (\<lambda>x. f (x + w)) z"
shows "isolated_singularity_at f (z + w)"
@@ -863,6 +992,20 @@
by (auto simp: not_essential_def)
qed
+lemma not_essential_analytic:
+ assumes "f analytic_on {z}"
+ shows "not_essential f z"
+ using analytic_at assms not_essential_holomorphic by blast
+
+lemma not_essential_id [singularity_intros]: "not_essential (\<lambda>w. w) z"
+ by (simp add: not_essential_analytic)
+
+lemma is_pole_imp_not_essential [intro]: "is_pole f z \<Longrightarrow> not_essential f z"
+ by (auto simp: not_essential_def)
+
+lemma tendsto_imp_not_essential [intro]: "f \<midarrow>z\<rightarrow> c \<Longrightarrow> not_essential f z"
+ by (auto simp: not_essential_def)
+
lemma eventually_not_pole:
assumes "isolated_singularity_at f z"
shows "eventually (\<lambda>w. \<not>is_pole f w) (at z)"
@@ -901,7 +1044,18 @@
thus ?thesis by simp
qed
-subsubsection \<open>The order of non-essential singularities (i.e. removable singularities or poles)\<close>
+lemma isolated_singularity_at_analytic:
+ assumes "f analytic_on {z}"
+ shows "isolated_singularity_at f z"
+proof -
+ from assms obtain r where r: "r > 0" "f holomorphic_on ball z r"
+ by (auto simp: analytic_on_def)
+ show ?thesis
+ by (rule isolated_singularity_at_holomorphic[of f "ball z r"])
+ (use \<open>r > 0\<close> in \<open>auto intro!: holomorphic_on_subset[OF r(2)]\<close>)
+qed
+
+subsection \<open>The order of non-essential singularities (i.e. removable singularities or poles)\<close>
definition\<^marker>\<open>tag important\<close> zorder :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> int" where
"zorder f z = (THE n. (\<exists>h r. r>0 \<and> h holomorphic_on cball z r \<and> h z\<noteq>0
@@ -1658,6 +1812,33 @@
using \<open>z=z'\<close> unfolding P_def zorder_def zor_poly_def by auto
qed
+lemma zorder_times_analytic':
+ assumes "isolated_singularity_at f z" "not_essential f z"
+ assumes "g analytic_on {z}" "frequently (\<lambda>z. f z * g z \<noteq> 0) (at z)"
+ shows "zorder (\<lambda>x. f x * g x) z = zorder f z + zorder g z"
+proof (rule zorder_times)
+ show "isolated_singularity_at g z" "not_essential g z"
+ by (intro isolated_singularity_at_analytic not_essential_analytic assms)+
+qed (use assms in auto)
+
+lemma zorder_cmult:
+ assumes "c \<noteq> 0"
+ shows "zorder (\<lambda>z. c * f z) z = zorder f z"
+proof -
+ define P where
+ "P = (\<lambda>f n h r. 0 < r \<and> h holomorphic_on cball z r \<and>
+ h z \<noteq> 0 \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0))"
+ have *: "P (\<lambda>x. c * f x) n (\<lambda>x. c * h x) r" if "P f n h r" "c \<noteq> 0" for f n h r c
+ using that unfolding P_def by (auto intro!: holomorphic_intros)
+ have "(\<exists>h r. P (\<lambda>x. c * f x) n h r) \<longleftrightarrow> (\<exists>h r. P f n h r)" for n
+ using *[of f n _ _ c] *[of "\<lambda>x. c * f x" n _ _ "inverse c"] \<open>c \<noteq> 0\<close>
+ by (fastforce simp: field_simps)
+ hence "(THE n. \<exists>h r. P (\<lambda>x. c * f x) n h r) = (THE n. \<exists>h r. P f n h r)"
+ by simp
+ thus ?thesis
+ by (simp add: zorder_def P_def)
+qed
+
lemma zorder_nonzero_div_power:
assumes sz: "open s" "z \<in> s" "f holomorphic_on s" "f z \<noteq> 0" and "n > 0"
shows "zorder (\<lambda>w. f w / (w - z) ^ n) z = - n"
@@ -2269,4 +2450,263 @@
qed
qed
+
+lemma deriv_divide_is_pole: \<comment>\<open>Generalises @{thm zorder_deriv}\<close>
+ fixes f g::"complex \<Rightarrow> complex" and z::complex
+ assumes f_iso:"isolated_singularity_at f z"
+ and f_ness:"not_essential f z"
+ and fg_nconst: "\<exists>\<^sub>Fw in (at z). deriv f w * f w \<noteq> 0"
+ and f_ord:"zorder f z \<noteq>0"
+ shows "is_pole (\<lambda>z. deriv f z / f z) z"
+proof (rule neg_zorder_imp_is_pole)
+ define ff where "ff=(\<lambda>w. deriv f w / f w)"
+ show "isolated_singularity_at ff z"
+ using f_iso f_ness unfolding ff_def
+ by (auto intro:singularity_intros)
+ show "not_essential ff z"
+ unfolding ff_def using f_ness f_iso
+ by (auto intro:singularity_intros)
+
+ have "zorder ff z = zorder (deriv f) z - zorder f z"
+ unfolding ff_def using f_iso f_ness fg_nconst
+ apply (rule_tac zorder_divide)
+ by (auto intro:singularity_intros)
+ moreover have "zorder (deriv f) z = zorder f z - 1"
+ proof (rule zorder_deriv_minus_1)
+ show " \<exists>\<^sub>F w in at z. f w \<noteq> 0"
+ using fg_nconst frequently_elim1 by fastforce
+ qed (use f_iso f_ness f_ord in auto)
+ ultimately show "zorder ff z < 0" by auto
+
+ show "\<exists>\<^sub>F w in at z. ff w \<noteq> 0"
+ unfolding ff_def using fg_nconst by auto
+qed
+
+lemma is_pole_deriv_divide_is_pole:
+ fixes f g::"complex \<Rightarrow> complex" and z::complex
+ assumes f_iso:"isolated_singularity_at f z"
+ and "is_pole f z"
+ shows "is_pole (\<lambda>z. deriv f z / f z) z"
+proof (rule deriv_divide_is_pole[OF f_iso])
+ show "not_essential f z"
+ using \<open>is_pole f z\<close> unfolding not_essential_def by auto
+ show "\<exists>\<^sub>F w in at z. deriv f w * f w \<noteq> 0"
+ apply (rule isolated_pole_imp_nzero_times)
+ using assms by auto
+ show "zorder f z \<noteq> 0"
+ using isolated_pole_imp_neg_zorder assms by fastforce
+qed
+
+subsection \<open>Isolated zeroes\<close>
+
+definition isolated_zero :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> bool" where
+ "isolated_zero f z \<longleftrightarrow> f z = 0 \<and> eventually (\<lambda>z. f z \<noteq> 0) (at z)"
+
+lemma isolated_zero_altdef: "isolated_zero f z \<longleftrightarrow> f z = 0 \<and> \<not>z islimpt {z. f z = 0}"
+ unfolding isolated_zero_def eventually_at_filter eventually_nhds islimpt_def by blast
+
+lemma isolated_zero_mult1:
+ assumes "isolated_zero f x" "isolated_zero g x"
+ shows "isolated_zero (\<lambda>x. f x * g x) x"
+proof -
+ have "eventually (\<lambda>x. f x \<noteq> 0) (at x)" "eventually (\<lambda>x. g x \<noteq> 0) (at x)"
+ using assms unfolding isolated_zero_def by auto
+ hence "eventually (\<lambda>x. f x * g x \<noteq> 0) (at x)"
+ by eventually_elim auto
+ with assms show ?thesis
+ by (auto simp: isolated_zero_def)
+qed
+
+lemma isolated_zero_mult2:
+ assumes "isolated_zero f x" "g x \<noteq> 0" "g analytic_on {x}"
+ shows "isolated_zero (\<lambda>x. f x * g x) x"
+proof -
+ have "eventually (\<lambda>x. f x \<noteq> 0) (at x)"
+ using assms unfolding isolated_zero_def by auto
+ moreover have "eventually (\<lambda>x. g x \<noteq> 0) (at x)"
+ using analytic_at_neq_imp_eventually_neq[of g x 0] assms by auto
+ ultimately have "eventually (\<lambda>x. f x * g x \<noteq> 0) (at x)"
+ by eventually_elim auto
+ thus ?thesis
+ using assms(1) by (auto simp: isolated_zero_def)
+qed
+
+lemma isolated_zero_mult3:
+ assumes "isolated_zero f x" "g x \<noteq> 0" "g analytic_on {x}"
+ shows "isolated_zero (\<lambda>x. g x * f x) x"
+ using isolated_zero_mult2[OF assms] by (simp add: mult_ac)
+
+lemma isolated_zero_prod:
+ assumes "\<And>x. x \<in> I \<Longrightarrow> isolated_zero (f x) z" "I \<noteq> {}" "finite I"
+ shows "isolated_zero (\<lambda>y. \<Prod>x\<in>I. f x y) z"
+ using assms(3,2,1) by (induction rule: finite_ne_induct) (auto intro: isolated_zero_mult1)
+
+lemma non_isolated_zero':
+ assumes "isolated_singularity_at f z" "not_essential f z" "f z = 0" "\<not>isolated_zero f z"
+ shows "eventually (\<lambda>z. f z = 0) (at z)"
+proof (rule not_essential_frequently_0_imp_eventually_0)
+ from assms show "frequently (\<lambda>z. f z = 0) (at z)"
+ by (auto simp: frequently_def isolated_zero_def)
+qed fact+
+
+lemma non_isolated_zero:
+ assumes "\<not>isolated_zero f z" "f analytic_on {z}" "f z = 0"
+ shows "eventually (\<lambda>z. f z = 0) (nhds z)"
+proof -
+ have "eventually (\<lambda>z. f z = 0) (at z)"
+ by (rule non_isolated_zero')
+ (use assms in \<open>auto intro: not_essential_analytic isolated_singularity_at_analytic\<close>)
+ with \<open>f z = 0\<close> show ?thesis
+ unfolding eventually_at_filter by (auto elim!: eventually_mono)
+qed
+
+lemma not_essential_compose:
+ assumes "not_essential f (g z)" "g analytic_on {z}"
+ shows "not_essential (\<lambda>x. f (g x)) z"
+proof (cases "isolated_zero (\<lambda>w. g w - g z) z")
+ case False
+ hence "eventually (\<lambda>w. g w - g z = 0) (nhds z)"
+ by (rule non_isolated_zero) (use assms in \<open>auto intro!: analytic_intros\<close>)
+ hence "not_essential (\<lambda>x. f (g x)) z \<longleftrightarrow> not_essential (\<lambda>_. f (g z)) z"
+ by (intro not_essential_cong refl)
+ (auto elim!: eventually_mono simp: eventually_at_filter)
+ thus ?thesis
+ by (simp add: not_essential_const)
+next
+ case True
+ hence ev: "eventually (\<lambda>w. g w \<noteq> g z) (at z)"
+ by (auto simp: isolated_zero_def)
+ from assms consider c where "f \<midarrow>g z\<rightarrow> c" | "is_pole f (g z)"
+ by (auto simp: not_essential_def)
+ have "isCont g z"
+ by (rule analytic_at_imp_isCont) fact
+ hence lim: "g \<midarrow>z\<rightarrow> g z"
+ using isContD by blast
+
+ from assms(1) consider c where "f \<midarrow>g z\<rightarrow> c" | "is_pole f (g z)"
+ unfolding not_essential_def by blast
+ thus ?thesis
+ proof cases
+ fix c assume "f \<midarrow>g z\<rightarrow> c"
+ hence "(\<lambda>x. f (g x)) \<midarrow>z\<rightarrow> c"
+ by (rule filterlim_compose) (use lim ev in \<open>auto simp: filterlim_at\<close>)
+ thus ?thesis
+ by (auto simp: not_essential_def)
+ next
+ assume "is_pole f (g z)"
+ hence "is_pole (\<lambda>x. f (g x)) z"
+ by (rule is_pole_compose) fact+
+ thus ?thesis
+ by (auto simp: not_essential_def)
+ qed
+qed
+
+subsection \<open>Isolated points\<close>
+
+definition isolated_points_of :: "complex set \<Rightarrow> complex set" where
+ "isolated_points_of A = {z\<in>A. eventually (\<lambda>w. w \<notin> A) (at z)}"
+
+lemma isolated_points_of_altdef: "isolated_points_of A = {z\<in>A. \<not>z islimpt A}"
+ unfolding isolated_points_of_def islimpt_def eventually_at_filter eventually_nhds by blast
+
+lemma isolated_points_of_empty [simp]: "isolated_points_of {} = {}"
+ and isolated_points_of_UNIV [simp]: "isolated_points_of UNIV = {}"
+ by (auto simp: isolated_points_of_def)
+
+lemma isolated_points_of_open_is_empty [simp]: "open A \<Longrightarrow> isolated_points_of A = {}"
+ unfolding isolated_points_of_altdef
+ by (simp add: interior_limit_point interior_open)
+
+lemma isolated_points_of_subset: "isolated_points_of A \<subseteq> A"
+ by (auto simp: isolated_points_of_def)
+
+lemma isolated_points_of_discrete:
+ assumes "discrete A"
+ shows "isolated_points_of A = A"
+ using assms by (auto simp: isolated_points_of_def discrete_altdef)
+
+lemmas uniform_discreteI1 = uniformI1
+lemmas uniform_discreteI2 = uniformI2
+
+lemma isolated_singularity_at_compose:
+ assumes "isolated_singularity_at f (g z)" "g analytic_on {z}"
+ shows "isolated_singularity_at (\<lambda>x. f (g x)) z"
+proof (cases "isolated_zero (\<lambda>w. g w - g z) z")
+ case False
+ hence "eventually (\<lambda>w. g w - g z = 0) (nhds z)"
+ by (rule non_isolated_zero) (use assms in \<open>auto intro!: analytic_intros\<close>)
+ hence "isolated_singularity_at (\<lambda>x. f (g x)) z \<longleftrightarrow> isolated_singularity_at (\<lambda>_. f (g z)) z"
+ by (intro isolated_singularity_at_cong refl)
+ (auto elim!: eventually_mono simp: eventually_at_filter)
+ thus ?thesis
+ by (simp add: isolated_singularity_at_const)
+next
+ case True
+ from assms(1) obtain r where r: "r > 0" "f analytic_on ball (g z) r - {g z}"
+ by (auto simp: isolated_singularity_at_def)
+ hence holo_f: "f holomorphic_on ball (g z) r - {g z}"
+ by (subst (asm) analytic_on_open) auto
+ from assms(2) obtain r' where r': "r' > 0" "g holomorphic_on ball z r'"
+ by (auto simp: analytic_on_def)
+
+ have "continuous_on (ball z r') g"
+ using holomorphic_on_imp_continuous_on r' by blast
+ hence "isCont g z"
+ using r' by (subst (asm) continuous_on_eq_continuous_at) auto
+ hence "g \<midarrow>z\<rightarrow> g z"
+ using isContD by blast
+ hence "eventually (\<lambda>w. g w \<in> ball (g z) r) (at z)"
+ using \<open>r > 0\<close> unfolding tendsto_def by force
+ moreover have "eventually (\<lambda>w. g w \<noteq> g z) (at z)" using True
+ by (auto simp: isolated_zero_def elim!: eventually_mono)
+ ultimately have "eventually (\<lambda>w. g w \<in> ball (g z) r - {g z}) (at z)"
+ by eventually_elim auto
+ then obtain r'' where r'': "r'' > 0" "\<forall>w\<in>ball z r''-{z}. g w \<in> ball (g z) r - {g z}"
+ unfolding eventually_at_filter eventually_nhds_metric ball_def
+ by (auto simp: dist_commute)
+ have "f \<circ> g holomorphic_on ball z (min r' r'') - {z}"
+ proof (rule holomorphic_on_compose_gen)
+ show "g holomorphic_on ball z (min r' r'') - {z}"
+ by (rule holomorphic_on_subset[OF r'(2)]) auto
+ show "f holomorphic_on ball (g z) r - {g z}"
+ by fact
+ show "g ` (ball z (min r' r'') - {z}) \<subseteq> ball (g z) r - {g z}"
+ using r'' by force
+ qed
+ hence "f \<circ> g analytic_on ball z (min r' r'') - {z}"
+ by (subst analytic_on_open) auto
+ thus ?thesis using \<open>r' > 0\<close> \<open>r'' > 0\<close>
+ by (auto simp: isolated_singularity_at_def o_def intro!: exI[of _ "min r' r''"])
+qed
+
+lemma is_pole_power_int_0:
+ assumes "f analytic_on {x}" "isolated_zero f x" "n < 0"
+ shows "is_pole (\<lambda>x. f x powi n) x"
+proof -
+ have "f \<midarrow>x\<rightarrow> f x"
+ using assms(1) by (simp add: analytic_at_imp_isCont isContD)
+ with assms show ?thesis
+ unfolding is_pole_def
+ by (intro filterlim_power_int_neg_at_infinity) (auto simp: isolated_zero_def)
+qed
+
+lemma isolated_zero_imp_not_constant_on:
+ assumes "isolated_zero f x" "x \<in> A" "open A"
+ shows "\<not>f constant_on A"
+proof
+ assume "f constant_on A"
+ then obtain c where c: "\<And>x. x \<in> A \<Longrightarrow> f x = c"
+ by (auto simp: constant_on_def)
+ from assms and c[of x] have [simp]: "c = 0"
+ by (auto simp: isolated_zero_def)
+ have "eventually (\<lambda>x. f x \<noteq> 0) (at x)"
+ using assms by (auto simp: isolated_zero_def)
+ moreover have "eventually (\<lambda>x. x \<in> A) (at x)"
+ using assms by (intro eventually_at_in_open') auto
+ ultimately have "eventually (\<lambda>x. False) (at x)"
+ by eventually_elim (use c in auto)
+ thus False
+ by simp
+qed
+
end