theory Meromorphic imports Laurent_Convergence Riemann_Mapping begin lemma analytic_at_cong: assumes "eventually (λx. f x = g x) (nhds x)" "x = y" shows "f analytic_on {x} ⟷ g analytic_on {y}" proof - have "g analytic_on {x}" if "f analytic_on {x}" "eventually (λx. f x = g x) (nhds x)" for f g proof - have "(λy. f (x + y)) has_fps_expansion fps_expansion f x" by (rule analytic_at_imp_has_fps_expansion) fact also have "?this ⟷ (λy. g (x + y)) has_fps_expansion fps_expansion f x" using that by (intro has_fps_expansion_cong refl) (auto simp: nhds_to_0' eventually_filtermap) finally show ?thesis by (rule has_fps_expansion_imp_analytic) qed from this[of f g] this[of g f] show ?thesis using assms by (auto simp: eq_commute) qed definition remove_sings :: "(complex ⇒ complex) ⇒ complex ⇒ complex" where "remove_sings f z = (if ∃c. f ─z→ c then Lim (at z) f else 0)" lemma remove_sings_eqI [intro]: assumes "f ─z→ c" shows "remove_sings f z = c" using assms unfolding remove_sings_def by (auto simp: tendsto_Lim) lemma remove_sings_at_analytic [simp]: assumes "f analytic_on {z}" shows "remove_sings f z = f z" using assms by (intro remove_sings_eqI) (simp add: analytic_at_imp_isCont isContD) lemma remove_sings_at_pole [simp]: assumes "is_pole f z" shows "remove_sings f z = 0" using assms unfolding remove_sings_def is_pole_def by (meson at_neq_bot not_tendsto_and_filterlim_at_infinity) lemma eventually_remove_sings_eq_at: assumes "isolated_singularity_at f z" shows "eventually (λw. remove_sings f w = f w) (at z)" proof - from assms obtain r where r: "r > 0" "f analytic_on ball z r - {z}" by (auto simp: isolated_singularity_at_def) hence *: "f analytic_on {w}" if "w ∈ ball z r - {z}" for w using r that by (auto intro: analytic_on_subset) have "eventually (λw. w ∈ 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 *) qed lemma eventually_remove_sings_eq_nhds: assumes "f analytic_on {z}" shows "eventually (λw. remove_sings f w = f w) (nhds z)" proof - from assms obtain A where A: "open A" "z ∈ A" "f holomorphic_on A" by (auto simp: analytic_at) have "eventually (λz. z ∈ A) (nhds z)" by (intro eventually_nhds_in_open A) thus ?thesis proof eventually_elim case (elim w) from elim have "f analytic_on {w}" using A analytic_at by blast thus ?case by auto qed qed lemma remove_sings_compose: assumes "filtermap g (at z) = at z'" shows "remove_sings (f ∘ g) z = remove_sings f z'" proof (cases "∃c. f ─z'→ c") case True then obtain c where c: "f ─z'→ c" by auto from c have "remove_sings f z' = c" by blast moreover from c have "remove_sings (f ∘ g) z = c" using c by (intro remove_sings_eqI) (auto simp: filterlim_def filtermap_compose assms) ultimately show ?thesis by simp next case False hence "¬(∃c. (f ∘ g) ─z→ c)" by (auto simp: filterlim_def filtermap_compose assms) with False show ?thesis by (auto simp: remove_sings_def) qed lemma remove_sings_cong: assumes "eventually (λx. f x = g x) (at z)" "z = z'" shows "remove_sings f z = remove_sings g z'" proof (cases "∃c. f ─z→ c") case True then obtain c where c: "f ─z→ c" by blast hence "remove_sings f z = c" by blast moreover have "f ─z→ c ⟷ g ─z'→ c" using assms by (intro filterlim_cong refl) auto with c have "remove_sings g z' = c" by (intro remove_sings_eqI) auto ultimately show ?thesis by simp next case False have "f ─z→ c ⟷ g ─z'→ c" for c using assms by (intro filterlim_cong) auto with False show ?thesis by (auto simp: remove_sings_def) qed lemma deriv_remove_sings_at_analytic [simp]: assumes "f analytic_on {z}" shows "deriv (remove_sings f) z = deriv f z" apply (rule deriv_cong_ev) apply (rule eventually_remove_sings_eq_nhds) using assms by auto lemma isolated_singularity_at_remove_sings [simp, intro]: assumes "isolated_singularity_at f z" shows "isolated_singularity_at (remove_sings f) z" using isolated_singularity_at_cong[OF eventually_remove_sings_eq_at[OF assms] refl] assms by simp lemma not_essential_remove_sings_iff [simp]: assumes "isolated_singularity_at f z" shows "not_essential (remove_sings f) z ⟷ not_essential f z" using not_essential_cong[OF eventually_remove_sings_eq_at[OF assms(1)] refl] by simp lemma not_essential_remove_sings [intro]: assumes "isolated_singularity_at f z" "not_essential f z" shows "not_essential (remove_sings f) z" by (subst not_essential_remove_sings_iff) (use assms in auto) lemma assumes "isolated_singularity_at f z" shows is_pole_remove_sings_iff [simp]: "is_pole (remove_sings f) z ⟷ is_pole f z" and zorder_remove_sings [simp]: "zorder (remove_sings f) z = zorder f z" and zor_poly_remove_sings [simp]: "zor_poly (remove_sings f) z = zor_poly f z" and has_laurent_expansion_remove_sings_iff [simp]: "(λw. remove_sings f (z + w)) has_laurent_expansion F ⟷ (λw. f (z + w)) has_laurent_expansion F" and tendsto_remove_sings_iff [simp]: "remove_sings f ─z→ c ⟷ f ─z→ c" by (intro is_pole_cong eventually_remove_sings_eq_at refl zorder_cong zor_poly_cong has_laurent_expansion_cong' tendsto_cong assms)+ lemma get_all_poles_from_remove_sings: fixes f:: "complex ⇒ complex" defines "ff≡remove_sings f" assumes f_holo:"f holomorphic_on s - pts" and "finite pts" "pts⊆s" "open s" and not_ess:"∀x∈pts. not_essential f x" obtains pts' where "pts' ⊆ pts" "finite pts'" "ff holomorphic_on s - pts'" "∀x∈pts'. is_pole ff x" proof - define pts' where "pts' = {x∈pts. is_pole f x}" have "pts' ⊆ pts" unfolding pts'_def by auto then have "finite pts'" using ‹finite pts› using rev_finite_subset by blast then have "open (s - pts')" using ‹open s› by (simp add: finite_imp_closed open_Diff) have isolated:"isolated_singularity_at f z" if "z∈pts" for z proof (rule isolated_singularity_at_holomorphic) show "f holomorphic_on (s-(pts-{z})) - {z}" by (metis Diff_insert f_holo insert_Diff that) show " open (s - (pts - {z}))" by (meson assms(3) assms(5) finite_Diff finite_imp_closed open_Diff) show "z ∈ s - (pts - {z})" using assms(4) that by auto qed have "ff holomorphic_on s - pts'" proof (rule no_isolated_singularity') show "(ff ⤏ ff z) (at z within s - pts')" if "z ∈ pts-pts'" for z proof - have "at z within s - pts' = at z" apply (rule at_within_open) using ‹open (s - pts')› that ‹pts⊆s› by auto moreover have "ff ─z→ ff z" unfolding ff_def proof (subst tendsto_remove_sings_iff) show "isolated_singularity_at f z" apply (rule isolated) using that by auto have "not_essential f z" using not_ess that by auto moreover have "¬is_pole f z" using that unfolding pts'_def by auto ultimately have "∃c. f ─z→ c" unfolding not_essential_def by auto then show "f ─z→ remove_sings f z" using remove_sings_eqI by blast qed ultimately show ?thesis by auto qed have "ff holomorphic_on s - pts" using f_holo proof (elim holomorphic_transform) fix x assume "x ∈ s - pts" then have "f analytic_on {x}" using assms(3) assms(5) f_holo by (meson finite_imp_closed holomorphic_on_imp_analytic_at open_Diff) from remove_sings_at_analytic[OF this] show "f x = ff x" unfolding ff_def by auto qed then show "ff holomorphic_on s - pts' - (pts - pts')" apply (elim holomorphic_on_subset) by blast show "open (s - pts')" by (simp add: ‹open (s - pts')›) show "finite (pts - pts')" by (simp add: assms(3)) qed moreover have "∀x∈pts'. is_pole ff x" unfolding pts'_def using ff_def is_pole_remove_sings_iff isolated by blast moreover note ‹pts' ⊆ pts› ‹finite pts'› ultimately show ?thesis using that by auto qed lemma remove_sings_eq_0_iff: assumes "not_essential f w" shows "remove_sings f w = 0 ⟷ is_pole f w ∨ f ─w→ 0" proof (cases "is_pole f w") case True then show ?thesis by simp next case False then obtain c where c:"f ─w→ c" using ‹not_essential f w› unfolding not_essential_def by auto then show ?thesis using False remove_sings_eqI by auto qed definition meromorphic_on:: "[complex ⇒ complex, complex set, complex set] ⇒ bool" ("_ (meromorphic'_on) _ _" [50,50,50]50) where "f meromorphic_on D pts ≡ open D ∧ pts ⊆ D ∧ (∀z∈pts. isolated_singularity_at f z ∧ not_essential f z) ∧ (∀z∈D. ¬(z islimpt pts)) ∧ (f holomorphic_on D-pts)" lemma meromorphic_imp_holomorphic: "f meromorphic_on D pts ⟹ 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) lemma meromorphic_imp_open_diff': assumes "f meromorphic_on D pts" "pts' ⊆ pts" shows "open (D - pts')" proof - have "D - pts' = D - closure pts'" proof safe fix x assume x: "x ∈ D" "x ∈ closure pts'" "x ∉ 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 lemma meromorphic_imp_open_diff: "f meromorphic_on D pts ⟹ open (D - pts)" by (erule meromorphic_imp_open_diff') auto lemma meromorphic_pole_subset: assumes merf: "f meromorphic_on D pts" shows "{x∈D. is_pole f x} ⊆ pts" by (smt (verit) Diff_iff assms mem_Collect_eq meromorphic_imp_open_diff meromorphic_on_def not_is_pole_holomorphic subsetI) named_theorems meromorphic_intros lemma meromorphic_on_subset: assumes "f meromorphic_on A pts" "open B" "B ⊆ A" "pts' = pts ∩ B" shows "f meromorphic_on B pts'" unfolding meromorphic_on_def proof (intro ballI conjI) fix z assume "z ∈ B" show "¬z islimpt pts'" proof assume "z islimpt pts'" hence "z islimpt pts" by (rule islimpt_subset) (use ‹pts' = _› in auto) thus False using ‹z ∈ B› ‹B ⊆ A› assms(1) by (auto simp: meromorphic_on_def) qed qed (use assms in ‹auto simp: meromorphic_on_def›) lemma meromorphic_on_superset_pts: assumes "f meromorphic_on A pts" "pts ⊆ pts'" "pts' ⊆ A" "∀x∈A. ¬x islimpt pts'" shows "f meromorphic_on A pts'" unfolding meromorphic_on_def proof (intro conjI ballI impI) fix z assume "z ∈ 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)]) show "isolated_singularity_at f z" proof (cases "z ∈ pts") case False thus ?thesis using ‹open (A - pts)› assms ‹z ∈ pts'› by (intro isolated_singularity_at_holomorphic[of _ "A - pts"] holomorphic_on_subset[OF holo]) auto qed (use assms in ‹auto simp: meromorphic_on_def›) show "not_essential f z" proof (cases "z ∈ pts") case False thus ?thesis using ‹open (A - pts)› assms ‹z ∈ pts'› by (intro not_essential_holomorphic[of _ "A - pts"] holomorphic_on_subset[OF holo]) auto qed (use assms in ‹auto simp: meromorphic_on_def›) qed (use assms in ‹auto simp: meromorphic_on_def›) lemma meromorphic_on_no_singularities: "f meromorphic_on A {} ⟷ f holomorphic_on A ∧ open A" by (auto simp: meromorphic_on_def) lemma holomorphic_on_imp_meromorphic_on: "f holomorphic_on A ⟹ pts ⊆ A ⟹ open A ⟹ ∀x∈A. ¬x islimpt pts ⟹ 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" "∀x∈A. ¬x islimpt pts" "pts ⊆ A" shows "(λ_. 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" "∀x∈A. ¬x islimpt pts" "pts ⊆ A" shows "(λx. 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" "∀x∈A. ¬x islimpt pts" "pts ⊆ 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 (λw. f w + g w) z" proof - have "(λw. 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 (λw. 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 "(λz. -f z) meromorphic_on A pts" unfolding meromorphic_on_def by (use assms in ‹auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros›) lemma meromorphic_on_add [meromorphic_intros]: assumes "f meromorphic_on A pts" "g meromorphic_on A pts" shows "(λz. f z + g z) meromorphic_on A pts" unfolding meromorphic_on_def by (use assms in ‹auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros›) lemma meromorphic_on_add': assumes "f meromorphic_on A pts1" "g meromorphic_on A pts2" shows "(λz. f z + g z) meromorphic_on A (pts1 ∪ pts2)" proof (rule meromorphic_intros) show "f meromorphic_on A (pts1 ∪ pts2)" by (rule meromorphic_on_superset_pts[OF assms(1)]) (use assms in ‹auto simp: meromorphic_on_def islimpt_Un›) show "g meromorphic_on A (pts1 ∪ pts2)" by (rule meromorphic_on_superset_pts[OF assms(2)]) (use assms in ‹auto simp: meromorphic_on_def islimpt_Un›) qed lemma meromorphic_on_add_const [meromorphic_intros]: assumes "f meromorphic_on A pts" shows "(λz. f z + c) meromorphic_on A pts" unfolding meromorphic_on_def by (use assms in ‹auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros›) lemma meromorphic_on_minus_const [meromorphic_intros]: assumes "f meromorphic_on A pts" shows "(λz. 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 "(λz. 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 "(λz. f z - g z) meromorphic_on A (pts1 ∪ pts2)" proof (rule meromorphic_intros) show "f meromorphic_on A (pts1 ∪ pts2)" by (rule meromorphic_on_superset_pts[OF assms(1)]) (use assms in ‹auto simp: meromorphic_on_def islimpt_Un›) show "g meromorphic_on A (pts1 ∪ pts2)" by (rule meromorphic_on_superset_pts[OF assms(2)]) (use assms in ‹auto simp: meromorphic_on_def islimpt_Un›) qed lemma meromorphic_on_mult [meromorphic_intros]: assumes "f meromorphic_on A pts" "g meromorphic_on A pts" shows "(λz. f z * g z) meromorphic_on A pts" unfolding meromorphic_on_def by (use assms in ‹auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros›) lemma meromorphic_on_mult': assumes "f meromorphic_on A pts1" "g meromorphic_on A pts2" shows "(λz. f z * g z) meromorphic_on A (pts1 ∪ pts2)" proof (rule meromorphic_intros) show "f meromorphic_on A (pts1 ∪ pts2)" by (rule meromorphic_on_superset_pts[OF assms(1)]) (use assms in ‹auto simp: meromorphic_on_def islimpt_Un›) show "g meromorphic_on A (pts1 ∪ pts2)" by (rule meromorphic_on_superset_pts[OF assms(2)]) (use assms in ‹auto simp: meromorphic_on_def islimpt_Un›) qed lemma meromorphic_on_imp_not_essential: assumes "f meromorphic_on A pts" "z ∈ A" shows "not_essential f z" proof (cases "z ∈ 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 ‹auto simp: meromorphic_on_def›) lemma meromorphic_imp_analytic: "f meromorphic_on D pts ⟹ 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 ∈ A" shows "¬z islimpt {w∈A. isolated_zero f w}" proof assume islimpt: "z islimpt {w∈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 ⊆ A - (pts - {z})" using meromorphic_imp_open_diff[OF mero] ‹z ∈ A› openE by blast have "not_essential f z" using assms by (rule meromorphic_on_imp_not_essential) then consider c where "f ─z→ c" | "is_pole f z" unfolding not_essential_def by blast thus False proof cases assume "is_pole f z" hence "eventually (λw. f w ≠ 0) (at z)" by (rule non_zero_neighbour_pole) hence "¬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 ─z→ c" define g where "g = (λw. 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 have eq_zero: "g w = 0" if "w ∈ ball z r" for w proof (rule analytic_continuation[where f = g]) show "open (ball z r)" "connected (ball z r)" "{w∈ball z r. isolated_zero f w} ⊆ ball z r" by auto have "z islimpt {w∈A. isolated_zero f w} ∩ ball z r" using islimpt ‹r > 0› by (intro islimpt_Int_eventually eventually_at_in_open') auto also have "… = {w∈ball z r. isolated_zero f w}" using r by auto finally show "z islimpt {w∈ball z r. isolated_zero f w}" by simp next fix w assume w: "w ∈ {w∈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 ‹auto intro!: holomorphic_on_subset[OF holo'] simp: isolated_zero_def›) have "infinite ({w∈A. isolated_zero f w} ∩ ball z r)" using islimpt ‹r > 0› unfolding islimpt_eq_infinite_ball by blast hence "{w∈A. isolated_zero f w} ∩ ball z r ≠ {}" by force then obtain z0 where z0: "z0 ∈ A" "isolated_zero f z0" "z0 ∈ ball z r" by blast have "∀⇩_{F}y in at z0. y ∈ ball z r - (if z = z0 then {} else {z}) - {z0}" using r z0 by (intro eventually_at_in_open) auto hence "eventually (λw. 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 (λw. f w = 0) (at z0)" by (auto simp: g_def eventually_at_filter elim!: eventually_mono split: if_splits) moreover from z0 have "eventually (λw. f w ≠ 0) (at z0)" by (simp add: isolated_zero_def) ultimately have "eventually (λ_. 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∈A. isolated_zero f z}" unfolding closedin_limpt using not_islimpt_isolated_zeros[OF assms] by auto lemma meromorphic_on_deriv': assumes "f meromorphic_on A pts" "open A" assumes "⋀x. x ∈ A - pts ⟹ (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 ∈ pts" hence "z ∈ A" using assms(1) by (auto simp: meromorphic_on_def) from ‹z ∈ pts› 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 ∩ (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' ⊆ ball z r ∩ (A - (pts - {z}))" using r ‹z ∈ A› 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 ‹auto intro!: assms(3)›) moreover have "open (ball z r' - {z})" by auto ultimately show "isolated_singularity_at f' z" unfolding isolated_singularity_at_def using ‹r' > 0› by (auto simp: analytic_on_open intro!: exI[of _ r']) next fix z assume z: "z ∈ pts" hence z': "not_essential f z" "z ∈ A" using assms by (auto simp: meromorphic_on_def) from z'(1) show "not_essential f' z" proof (rule not_essential_deriv') show "z ∈ A - (pts - {z})" using ‹z ∈ A› 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 ‹auto simp: meromorphic_on_def›) 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 ∈ 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 ∈ 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 ⊆ D" shows "finite (S ∩ pts)" proof - { assume "infinite (S ∩ pts)" then obtain z where "z ∈ S" and z: "z islimpt (S ∩ 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 ⇒ complex set" where K: "D = (⋃n. K n)" "⋀n. compact (K n)" using assms unfolding meromorphic_on_def by (metis open_Union_compact_subsets) then have "pts = (⋃n. K n ∩ pts)" using assms meromorphic_on_def by auto moreover have "⋀n. finite (K n ∩ 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' ⊆ 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 ‹auto simp: meromorphic_imp_countable›) qed (use assms in ‹auto simp: meromorphic_on_def›) 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 ⊆ A" shows "(λx. f (g x)) meromorphic_on B (isolated_points_of (g -` pts ∩ B))" unfolding meromorphic_on_def proof (intro ballI conjI) fix z assume z: "z ∈ isolated_points_of (g -` pts ∩ B)" hence z': "z ∈ B" "g z ∈ pts" using isolated_points_of_subset by blast+ have g': "g analytic_on {z}" using g z' ‹open B› analytic_at by blast show "isolated_singularity_at (λx. f (g x)) z" by (rule isolated_singularity_at_compose[OF _ g']) (use f z' in ‹auto simp: meromorphic_on_def›) show "not_essential (λx. f (g x)) z" by (rule not_essential_compose[OF _ g']) (use f z' in ‹auto simp: meromorphic_on_def›) next fix z assume z: "z ∈ B" hence "g z ∈ A" using assms by auto hence "¬g z islimpt pts" using f by (auto simp: meromorphic_on_def) hence ev: "eventually (λw. w ∉ 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 (λw. w ∉ isolated_points_of (g -` pts ∩ B)) (at z)" proof (cases "isolated_zero (λw. g w - g z) z") case True have "eventually (λw. w ∉ pts) (at (g z))" using ev by (auto simp: islimpt_conv_frequently_at frequently_def) moreover have "g ─z→ 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 (λw. g w ∉ 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 (λw. g w - g z = 0) (nhds z)" using False by (rule non_isolated_zero) (auto intro!: analytic_intros g') hence "eventually (λw. g w = g z ∧ w ∈ B) (nhds z)" using eventually_nhds_in_open[OF ‹open B› ‹z ∈ B›] by eventually_elim auto then obtain X where X: "open X" "z ∈ X" "X ⊆ B" "∀x∈X. g x = g z" unfolding eventually_nhds by blast have "z0 ∉ isolated_points_of (g -` pts ∩ B)" if "z0 ∈ X" for z0 proof (cases "g z ∈ pts") case False with that have "g z0 ∉ pts" using X by metis thus ?thesis by (auto simp: isolated_points_of_def) next case True have "eventually (λw. w ∈ X) (at z0)" by (intro eventually_at_in_open') fact+ hence "eventually (λw. w ∈ g -` pts ∩ B) (at z0)" by eventually_elim (use X True in fastforce) hence "frequently (λw. w ∈ g -` pts ∩ B) (at z0)" by (meson at_neq_bot eventually_frequently) thus "z0 ∉ isolated_points_of (g -` pts ∩ B)" unfolding isolated_points_of_def by (auto simp: frequently_def) qed moreover have "eventually (λx. x ∈ X) (at z)" by (intro eventually_at_in_open') fact+ ultimately show ?thesis by (auto elim!: eventually_mono) qed thus "¬z islimpt isolated_points_of (g -` pts ∩ B)" by (auto simp: islimpt_conv_frequently_at frequently_def) next have "f ∘ g analytic_on (⋃z∈B - isolated_points_of (g -` pts ∩ B). {z})" unfolding analytic_on_UN proof fix z assume z: "z ∈ B - isolated_points_of (g -` pts ∩ B)" hence "z ∈ 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 ∘ g analytic_on {z}" proof (cases "g z ∈ 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 (λw. g w - g z) z") case False hence "eventually (λw. g w - g z = 0) (nhds z)" by (rule non_isolated_zero) (auto intro!: analytic_intros g') hence "f ∘ g analytic_on {z} ⟷ (λ_. 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 (λw. g w ≠ g z) (at z)" by (auto simp: isolated_zero_def) have "¬g z islimpt pts" using ‹g z ∈ pts› f by (auto simp: meromorphic_on_def) hence "eventually (λw. w ∉ pts) (at (g z))" by (auto simp: islimpt_conv_frequently_at frequently_def) moreover have "g ─z→ 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 (λw. g w ∉ pts) (at z)" using eventually_compose_filterlim by blast hence "z ∈ isolated_points_of (g -` pts ∩ B)" using ‹g z ∈ pts› ‹z ∈ B› by (auto simp: isolated_points_of_def elim!: eventually_mono) with z show ?thesis by simp qed qed qed also have "… = B - isolated_points_of (g -` pts ∩ B)" by blast finally show "(λx. f (g x)) holomorphic_on B - isolated_points_of (g -` pts ∩ B)" unfolding o_def using analytic_imp_holomorphic by blast qed (auto simp: isolated_points_of_def ‹open B›) lemma meromorphic_on_compose': assumes f: "f meromorphic_on A pts" and g: "g holomorphic_on B" assumes "open B" and "g ` B ⊆ A" and "pts' = (isolated_points_of (g -` pts ∩ B))" shows "(λx. 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 "(λz. inverse (f z)) meromorphic_on A (pts ∪ {z∈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 ∈ pts ∪ {z∈A. isolated_zero f z}" have "isolated_singularity_at f z ∧ not_essential f z" proof (cases "z ∈ 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 ‹auto intro!: meromorphic_imp_open_diff[OF mero]›) 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 ‹auto simp: meromorphic_on_def›) thus "isolated_singularity_at (λz. inverse (f z)) z" "not_essential (λz. inverse (f z)) z" by (auto intro!: isolated_singularity_at_inverse not_essential_inverse) next fix z assume "z ∈ A" hence "¬ z islimpt {z∈A. isolated_zero f z}" by (rule not_islimpt_isolated_zeros[OF mero]) thus "¬ z islimpt pts ∪ {z ∈ A. isolated_zero f z}" using ‹z ∈ A› using mero by (auto simp: islimpt_Un meromorphic_on_def) next show "pts ∪ {z ∈ A. isolated_zero f z} ⊆ A" using mero by (auto simp: meromorphic_on_def) next have "(λz. inverse (f z)) analytic_on (⋃w∈A - (pts ∪ {z ∈ A. isolated_zero f z}) . {w})" unfolding analytic_on_UN proof (intro ballI) fix w assume w: "w ∈ A - (pts ∪ {z ∈ A. isolated_zero f z})" show "(λz. 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 (λw. f w = 0) (nhds w)" using True w by (intro non_isolated_zero analytic_on_subset[OF ana]) auto hence "(λz. inverse (f z)) analytic_on {w} ⟷ (λ_. 0) analytic_on {w}" using w by (intro analytic_at_cong refl) auto thus ?thesis by simp qed qed also have "… = A - (pts ∪ {z ∈ A. isolated_zero f z})" by blast finally have "(λz. inverse (f z)) analytic_on …" . moreover have "open (A - (pts ∪ {z ∈ A. isolated_zero f z}))" using closedin_isolated_zeros[OF mero] open' ‹open A› by (metis (no_types, lifting) Diff_Diff_Int Diff_Un closedin_closed open_Diff open_Int) ultimately show "(λz. inverse (f z)) holomorphic_on A - (pts ∪ {z ∈ A. isolated_zero f z})" by (subst (asm) analytic_on_open) auto qed (use assms in ‹auto simp: meromorphic_on_def islimpt_Un intro!: holomorphic_intros singularity_intros›) qed lemma meromorphic_on_inverse'' [meromorphic_intros]: assumes "f meromorphic_on A pts" "{z∈A. f z = 0} ⊆ pts" shows "(λz. inverse (f z)) meromorphic_on A pts" proof - have "(λz. inverse (f z)) meromorphic_on A (pts ∪ {z ∈ A. isolated_zero f z})" by (intro meromorphic_on_inverse assms) also have "(pts ∪ {z ∈ 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 "(λz. f z / g z) meromorphic_on A (pts ∪ {z∈A. isolated_zero g z})" proof - have mero1: "(λz. inverse (g z)) meromorphic_on A (pts ∪ {z∈A. isolated_zero g z})" by (intro meromorphic_intros assms) have sparse: "∀x∈A. ¬ x islimpt pts ∪ {z∈A. isolated_zero g z}" and "pts ⊆ A" using mero1 by (auto simp: meromorphic_on_def) have mero2: "f meromorphic_on A (pts ∪ {z∈A. isolated_zero g z})" by (rule meromorphic_on_superset_pts[OF assms(1)]) (use sparse ‹pts ⊆ A› in auto) have "(λz. f z * inverse (g z)) meromorphic_on A (pts ∪ {z∈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∈A. g z = 0} ⊆ pts" shows "(λz. f z / g z) meromorphic_on A pts" proof - have "(λz. 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 "(λx. 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 "(λx. 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 "(λx. c *⇩_{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_sum [meromorphic_intros]: assumes "⋀y. y ∈ I ⟹ f y meromorphic_on A pts" assumes "I ≠ {} ∨ open A ∧ pts ⊆ A ∧ (∀x∈A. ¬x islimpt pts)" shows "(λx. ∑y∈I. f y x) meromorphic_on A pts" proof - have *: "open A ∧ pts ⊆ A ∧ (∀x∈A. ¬x islimpt pts)" using assms(2) proof assume "I ≠ {}" then obtain x where "x ∈ 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 ‹auto intro!: meromorphic_intros›) qed lemma meromorphic_on_prod [meromorphic_intros]: assumes "⋀y. y ∈ I ⟹ f y meromorphic_on A pts" assumes "I ≠ {} ∨ open A ∧ pts ⊆ A ∧ (∀x∈A. ¬x islimpt pts)" shows "(λx. ∏y∈I. f y x) meromorphic_on A pts" proof - have *: "open A ∧ pts ⊆ A ∧ (∀x∈A. ¬x islimpt pts)" using assms(2) proof assume "I ≠ {}" then obtain x where "x ∈ 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 ‹auto intro!: meromorphic_intros›) qed lemma meromorphic_on_power [meromorphic_intros]: assumes "f meromorphic_on A pts" shows "(λx. f x ^ n) meromorphic_on A pts" proof - have "(λx. ∏i∈{..<n}. f x) meromorphic_on A pts" by (intro meromorphic_intros assms(1)) (use assms in ‹auto simp: meromorphic_on_def›) thus ?thesis by simp qed lemma meromorphic_on_power_int [meromorphic_intros]: assumes "f meromorphic_on A pts" shows "(λz. f z powi n) meromorphic_on A (pts ∪ {z ∈ A. isolated_zero f z})" proof - have inv: "(λx. inverse (f x)) meromorphic_on A (pts ∪ {z ∈ A. isolated_zero f z})" by (intro meromorphic_intros assms) have *: "f meromorphic_on A (pts ∪ {z ∈ A. isolated_zero f z})" by (intro meromorphic_on_superset_pts [OF assms(1)]) (use inv in ‹auto simp: meromorphic_on_def›) show ?thesis proof (cases "n ≥ 0") case True have "(λx. f x ^ nat n) meromorphic_on A (pts ∪ {z ∈ A. isolated_zero f z})" by (intro meromorphic_intros *) thus ?thesis using True by (simp add: power_int_def) next case False have "(λx. inverse (f x) ^ nat (-n)) meromorphic_on A (pts ∪ {z ∈ 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 ≥ 0 ∨ (∀z∈A. isolated_zero f z ⟶ z ∈ pts)" shows "(λz. f z powi n) meromorphic_on A pts" proof (cases "n ≥ 0") case True have "(λz. 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 "(λz. f z powi n) meromorphic_on A (pts ∪ {z∈A. isolated_zero f z})" by (rule meromorphic_on_power_int) fact also from assms(2) False have "pts ∪ {z∈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: "⋀z. z ∈ A ⟹ ∃F. (λw. f (z + w)) has_laurent_expansion F" shows "f meromorphic_on A {z∈A. ¬f analytic_on {z}}" unfolding meromorphic_on_def proof (intro conjI ballI) fix z assume "z ∈ {z∈A. ¬f analytic_on {z}}" then obtain F where F: "(λw. 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 ∈ A" obtain F where F: "(λw. f (z + w)) has_laurent_expansion F" using laurent[of z] ‹z ∈ A› by blast from F have "isolated_singularity_at f z" using has_laurent_expansion_isolated z 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 ∈ ball z r - {z}" for w by (rule analytic_on_subset[OF r(2)]) (use that in auto) hence "eventually (λw. f analytic_on {w}) (at z)" using eventually_at_in_open[of "ball z r" z] ‹r > 0› by (auto elim!: eventually_mono) hence "¬z islimpt {w. ¬f analytic_on {w}}" by (auto simp: islimpt_conv_frequently_at frequently_def) thus "¬z islimpt {w∈A. ¬f analytic_on {w}}" using islimpt_subset[of z "{w∈A. ¬f analytic_on {w}}" "{w. ¬f analytic_on {w}}"] by blast next have "f analytic_on A - {w∈A. ¬f analytic_on {w}}" by (subst analytic_on_analytic_at) auto thus "f holomorphic_on A - {w∈A. ¬f analytic_on {w}}" by (meson analytic_imp_holomorphic) qed (use assms in auto) lemma meromorphic_on_imp_has_laurent_expansion: assumes "f meromorphic_on A pts" "z ∈ A" shows "(λw. f (z + w)) has_laurent_expansion laurent_expansion f z" proof (cases "z ∈ 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 ∈ 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 ─z→ c" shows eventually_remove_sings_eq_nhds': "eventually (λw. 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 (λw. w ≠ z) (at z)" by (auto simp: eventually_at_filter) hence "eventually (λw. 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 (λw. remove_sings f w = (if w = z then c else f w)) (nhds z)" by (simp add: eventually_at_filter) have "(λw. if w = z then c else f w) analytic_on {z}" by (intro removable_singularity' assms) also have "?this ⟷ remove_sings f analytic_on {z}" using ev by (intro analytic_at_cong) (auto simp: eq_commute) finally show … . qed lemma remove_sings_meromorphic_on: assumes "f meromorphic_on A pts" "⋀z. z ∈ pts - pts' ⟹ ¬is_pole f z" "pts' ⊆ pts" shows "remove_sings f meromorphic_on A pts'" unfolding meromorphic_on_def proof safe have "remove_sings f analytic_on {z}" if "z ∈ A - pts'" for z proof (cases "z ∈ pts") 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} ⟷ f analytic_on {z}" by (intro analytic_at_cong eventually_remove_sings_eq_nhds * refl) thus ?thesis using * by simp 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 ─z→ c" by (auto simp: not_essential_def) thus "remove_sings f analytic_on {z}" by (intro remove_sings_analytic_at_singularity isol) 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 ‹auto simp: meromorphic_on_def›) lemma remove_sings_holomorphic_on: assumes "f meromorphic_on A pts" "⋀z. z ∈ pts ⟹ ¬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) lemma meromorphic_on_Ex_iff: "(∃pts. f meromorphic_on A pts) ⟷ open A ∧ (∀z∈A. ∃F. (λw. f (z + w)) has_laurent_expansion F)" proof safe fix pts assume *: "f meromorphic_on A pts" from * show "open A" by (auto simp: meromorphic_on_def) show "∃F. (λw. f (z + w)) has_laurent_expansion F" if "z ∈ 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) lemma is_pole_inverse_holomorphic_pts: fixes pts::"complex set" and f::"complex ⇒ complex" defines "g ≡ λx. (if x∈pts then 0 else inverse (f x))" assumes mer: "f meromorphic_on D pts" and non_z: "⋀z. z ∈ D - pts ⟹ f z ≠ 0" and all_poles:"∀x. is_pole f x ⟷ x∈pts" shows "g holomorphic_on D" proof - have "open D" and f_holo: "f holomorphic_on (D-pts)" using mer by (auto simp: meromorphic_on_def) have "∃r. r>0 ∧ f analytic_on ball z r - {z} ∧ (∀x ∈ ball z r - {z}. f x≠0)" if "z∈pts" for z 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" "∀x ∈ ball z r2 - {z}. f x≠0" using non_zero_neighbour_pole[OF ‹is_pole f z›] 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: ‹0 < r2› ‹r1>0› r_def) moreover have "f analytic_on ball z r - {z}" using r_def by (force intro: analytic_on_subset [OF fan]) moreover have "∀x ∈ ball z r - {z}. f x≠0" by (simp add: ‹∀x∈ball z r2 - {z}. f x ≠ 0› r_def) ultimately show ?thesis by auto 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:"∀x ∈ ball z (get_r z) - {z}. f x≠0" if "z∈pts" for z by metis define p_balls where "p_balls ≡ ⋃z∈pts. ball z (get_r z)" have g_ball:"g holomorphic_on ball z (get_r z)" if "z∈pts" for z proof - have "(λx. 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 "∀x∈ball z (get_r z) - {z}. f x ≠ 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) 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 "(λz. 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 ∪ (D-pts))" by (simp add: holomorphic_on_Un meromorphic_imp_open_diff[OF mer]) moreover have "D ⊆ p_balls ∪ (D-pts)" unfolding p_balls_def using ‹⋀z. z ∈ pts ⟹ 0 < get_r z› by force ultimately show "g holomorphic_on D" by (meson holomorphic_on_subset) 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) lemma meromorphic_imp_constant_on: assumes merf: "f meromorphic_on D pts" and "f constant_on (D - pts)" and "∀x∈pts. is_pole f x" shows "f constant_on D" proof - obtain c where c:"⋀z. z ∈ D-pts ⟹ f z = c" by (meson assms constant_on_def) have "f z = c" if "z ∈ 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 ‹z ∈ D› 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 ⊆ D" "r ≤ r0" by (smt (verit, best) ‹0 < r0› ‹z ∈ D› openE order_subst2 subset_ball) have r: "f analytic_on ball z r - {z}" by (meson Diff_mono ‹r ≤ r0› analytic_on_subset order_refl r0 subset_ball) have "ball z r - {z} ⊆ -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 ‹ball z r ⊆ D› have "ball z r - {z} ⊆ D-pts" by fastforce with c have c': "⋀u. u ∈ ball z r - {z} ⟹ f u = c" by blast have False if "∀⇩_{F}x in at z. cmod c + 1 ≤ cmod (f x)" proof - have "∀⇩_{F}x in at z within ball z r - {z}. cmod c + 1 ≤ cmod (f x)" by (smt (verit, best) Diff_UNIV Diff_eq_empty_iff eventually_at_topological insert_subset that) with ‹r > 0› 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) next case False then show ?thesis by (meson DiffI assms(3) c that) qed then show ?thesis by (simp add: constant_on_def) qed lemma meromorphic_isolated: assumes merf: "f meromorphic_on D pts" and "p∈pts" obtains r where "r>0" "ball p r ⊆ D" "ball p r ∩ pts = {p}" proof - have "∀z∈D. ∃e>0. finite (pts ∩ ball z e)" using merf unfolding meromorphic_on_def islimpt_eq_infinite_ball by auto then obtain r0 where r0:"r0>0" "finite (pts ∩ ball p r0)" by (metis assms(2) in_mono merf meromorphic_on_def) moreover define pts' where "pts' = pts ∩ 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'∈pts'}/2) r0)" have "r1>0 ∧ pts ∩ 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'∈pts'}" have nempty:"S ≠ {}" using False S_def by blast have finite:"finite S" using ‹finite pts'› S_def by simp have "r1>0" proof - have "r1=min (Min S/2) r0" using False unfolding S_def r1_def by auto moreover have "Min S∈S" using ‹S≠{}› ‹finite S› Min_in by auto then have "Min S>0" unfolding S_def using pts'_def by force ultimately show ?thesis using ‹r0>0› by auto qed moreover have "pts ∩ ball p r1 - {p} = {}" proof (rule ccontr) assume "pts ∩ ball p r1 - {p} ≠ {}" then obtain p' where "p'∈pts ∩ ball p r1 - {p}" by blast moreover have "r1≤r0" using r1_def by auto ultimately have "p'∈pts'" unfolding pts'_def by auto then have "dist p' p≥Min S" using S_def eq_Min_iff local.finite by blast moreover have "dist p' p < Min S" using ‹p'∈pts ∩ ball p r1 - {p}› 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 qed ultimately show ?thesis by auto qed then have "r1>0" and r1_pts:"pts ∩ ball p r1 - {p} = {}" by auto obtain r2 where "r2>0" "ball p r2 ⊆ 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: ‹0 < r1› ‹0 < r2›) moreover have "ball p r ⊆ D" using ‹ball p r2 ⊆ D› r_def by auto moreover have "ball p r ∩ pts = {p}" using assms(2) ‹r>0› r1_pts unfolding r_def by auto ultimately show ?thesis using that by auto qed lemma meromorphic_pts_closure: assumes merf: "f meromorphic_on D pts" shows "pts ⊆ closure (D - pts)" proof - have "p islimpt (D - pts)" if "p∈pts" for p proof - obtain r where "r>0" "ball p r ⊆ D" "ball p r ∩ pts = {p}" using meromorphic_isolated[OF merf ‹p∈pts›] by auto from ‹r>0› 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} ⊆ D - pts" using ‹ball p r ∩ pts = {p}› ‹ball p r ⊆ D› 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:"¬(∀w∈D-pts. f w=0)" and "z∈D" and "connected D" shows "(∀⇩_{F}w in at z. f w ≠ 0 ∧ w ∈ D - pts)" proof - obtain β where β:"β ∈ D - pts" "f β≠0" using f_nconst by auto have ?thesis if "z∉pts" proof - have "∀⇩_{F}w in at z. f w ≠ 0 ∧ w ∈ D - pts" apply (rule non_zero_neighbour_alt[of f "D-pts" z β]) 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 β by auto then show ?thesis by (auto elim:eventually_mono) qed moreover have ?thesis if "z∈pts" "¬ f ─z→ 0" proof - have "∀⇩_{F}w in at z. w ∈ D - pts" using merf[unfolded meromorphic_on_def islimpt_iff_eventually] ‹z∈D› using eventually_at_in_open' eventually_elim2 by fastforce moreover have "∀⇩_{F}w in at z. f w ≠ 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≠0" "f ─z→ c" by (metis ‹¬ f ─z→ 0› 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∈pts" "f ─z→ 0" proof - define ff where "ff=(λx. 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 ‹f ─z→ 0› 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 ⊆ A" unfolding A_def by auto moreover have "A ⊆ 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 ∈ A" using A_def assms(3) by blast moreover have "ff z = 0" unfolding ff_def by auto moreover have "β ∈ A " using A_def β(1) by blast moreover have "ff β ≠ 0" using β(1) β(2) ff_def that(1) by auto ultimately obtain r where "0 < r" "ball z r ⊆ A" "⋀x. x ∈ ball z r - {z} ⟹ ff x ≠ 0" using ‹open A› isolated_zeros[of ff A z β] 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 qed lemma nconst_imp_nzero_neighbour': assumes merf: "f meromorphic_on D pts" and f_nconst:"¬(∀w∈D-pts. f w=0)" and "z∈D" and "connected D" shows "∀⇩_{F}w in at z. f w ≠ 0" using nconst_imp_nzero_neighbour[OF assms] by (auto elim:eventually_mono) lemma meromorphic_compact_finite_zeros: assumes merf:"f meromorphic_on D pts" and "compact S" "S ⊆ D" "connected D" and f_nconst:"¬(∀w∈D-pts. f w=0)" shows "finite ({x∈S. f x=0})" proof - have "finite ({x∈S. f x=0 ∧ x ∉ pts})" proof (rule ccontr) assume "infinite {x ∈ S. f x = 0 ∧ x ∉ pts}" then obtain z where "z∈S" and z_lim:"z islimpt {x ∈ S. f x = 0 ∧ x ∉ pts}" using ‹compact S› unfolding compact_eq_Bolzano_Weierstrass by auto from z_lim have "∃⇩_{F}x in at z. f x = 0 ∧ x ∈ S ∧ x ∉ pts" unfolding islimpt_iff_eventually not_eventually by simp moreover have "∀⇩_{F}w in at z. f w ≠ 0 ∧ w ∈ D - pts" using nconst_imp_nzero_neighbour[OF merf f_nconst _ ‹connected D›] ‹z∈S› ‹S ⊆ D› by auto ultimately have "∃⇩_{F}x in at z. False" by (simp add: eventually_mono frequently_def) then show False by auto qed moreover have "finite (S ∩ pts)" using meromorphic_compact_finite_pts[OF merf ‹compact S› ‹S ⊆ D›] . ultimately have "finite ({x∈S. f x=0 ∧ x ∉ pts} ∪ (S ∩ pts))" unfolding finite_Un by auto then show ?thesis by (elim rev_finite_subset) auto qed lemma meromorphic_onI [intro?]: assumes "open A" "pts ⊆ A" assumes "f holomorphic_on A - pts" "⋀z. z ∈ A ⟹ ¬z islimpt pts" assumes "⋀z. z ∈ pts ⟹ isolated_singularity_at f z" assumes "⋀z. z ∈ pts ⟹ not_essential f z" shows "f meromorphic_on A pts" using assms unfolding meromorphic_on_def by blast lemma Polygamma_plus_of_nat: assumes "∀k<m. z ≠ -of_nat k" shows "Polygamma n (z + of_nat m) = Polygamma n z + (-1) ^ n * fact n * (∑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 "… = 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 +