section ‹Conformal Mappings and Consequences of Cauchy's Integral Theorem› text‹By John Harrison et al. Ported from HOL Light by L C Paulson (2016)› text‹Also Cauchy's residue theorem by Wenda Li (2016)› theory Conformal_Mappings imports Cauchy_Integral_Formula begin subsection ‹Analytic continuation› proposition isolated_zeros: assumes holf: "f holomorphic_on S" and "open S" "connected S" "ξ ∈ S" "f ξ = 0" "β ∈ S" "f β ≠ 0" obtains r where "0 < r" and "ball ξ r ⊆ S" and "⋀z. z ∈ ball ξ r - {ξ} ⟹ f z ≠ 0" proof - obtain r where "0 < r" and r: "ball ξ r ⊆ S" using ‹open S› ‹ξ ∈ S› open_contains_ball_eq by blast have powf: "((λn. (deriv ^^ n) f ξ / (fact n) * (z - ξ)^n) sums f z)" if "z ∈ ball ξ r" for z by (intro holomorphic_power_series [OF _ that] holomorphic_on_subset [OF holf r]) obtain m where m: "(deriv ^^ m) f ξ / (fact m) ≠ 0" using holomorphic_fun_eq_0_on_connected [OF holf ‹open S› ‹connected S› _ ‹ξ ∈ S› ‹β ∈ S›] ‹f β ≠ 0› by auto then have "m ≠ 0" using assms(5) funpow_0 by fastforce obtain s where "0 < s" and s: "⋀z. z ∈ cball ξ s - {ξ} ⟹ f z ≠ 0" using powser_0_nonzero [OF ‹0 < r› powf ‹f ξ = 0› m] by (metis ‹m ≠ 0› dist_norm mem_ball norm_minus_commute not_gr_zero) have "0 < min r s" by (simp add: ‹0 < r› ‹0 < s›) then show thesis apply (rule that) using r s by auto qed proposition analytic_continuation: assumes holf: "f holomorphic_on S" and "open S" and "connected S" and "U ⊆ S" and "ξ ∈ S" and "ξ islimpt U" and fU0 [simp]: "⋀z. z ∈ U ⟹ f z = 0" and "w ∈ S" shows "f w = 0" proof - obtain e where "0 < e" and e: "cball ξ e ⊆ S" using ‹open S› ‹ξ ∈ S› open_contains_cball_eq by blast define T where "T = cball ξ e ∩ U" have contf: "continuous_on (closure T) f" by (metis T_def closed_cball closure_minimal e holf holomorphic_on_imp_continuous_on holomorphic_on_subset inf.cobounded1) have fT0 [simp]: "⋀x. x ∈ T ⟹ f x = 0" by (simp add: T_def) have "⋀r. ⟦∀e>0. ∃x'∈U. x' ≠ ξ ∧ dist x' ξ < e; 0 < r⟧ ⟹ ∃x'∈cball ξ e ∩ U. x' ≠ ξ ∧ dist x' ξ < r" by (metis ‹0 < e› IntI dist_commute less_eq_real_def mem_cball min_less_iff_conj) then have "ξ islimpt T" using ‹ξ islimpt U› by (auto simp: T_def islimpt_approachable) then have "ξ ∈ closure T" by (simp add: closure_def) then have "f ξ = 0" by (auto simp: continuous_constant_on_closure [OF contf]) moreover have "⋀r. ⟦0 < r; ⋀z. z ∈ ball ξ r - {ξ} ⟹ f z ≠ 0⟧ ⟹ False" by (metis open_ball ‹ξ islimpt T› centre_in_ball fT0 insertE insert_Diff islimptE) ultimately show ?thesis by (metis ‹open S› ‹connected S› ‹ξ ∈ S› ‹w ∈ S› holf isolated_zeros) qed corollary analytic_continuation_open: assumes "open s" and "open s'" and "s ≠ {}" and "connected s'" and "s ⊆ s'" assumes "f holomorphic_on s'" and "g holomorphic_on s'" and "⋀z. z ∈ s ⟹ f z = g z" assumes "z ∈ s'" shows "f z = g z" proof - from ‹s ≠ {}› obtain ξ where "ξ ∈ s" by auto with ‹open s› have ξ: "ξ islimpt s" by (intro interior_limit_point) (auto simp: interior_open) have "f z - g z = 0" by (rule analytic_continuation[of "λz. f z - g z" s' s ξ]) (insert assms ‹ξ ∈ s› ξ, auto intro: holomorphic_intros) thus ?thesis by simp qed corollary analytic_continuation': assumes "f holomorphic_on S" "open S" "connected S" and "U ⊆ S" "ξ ∈ S" "ξ islimpt U" and "f constant_on U" shows "f constant_on S" proof - obtain c where c: "⋀x. x ∈ U ⟹ f x - c = 0" by (metis ‹f constant_on U› constant_on_def diff_self) have "(λz. f z - c) holomorphic_on S" using assms by (intro holomorphic_intros) with c analytic_continuation assms have "⋀x. x ∈ S ⟹ f x - c = 0" by blast then show ?thesis unfolding constant_on_def by force qed lemma holomorphic_compact_finite_zeros: assumes S: "f holomorphic_on S" "open S" "connected S" and "compact K" "K ⊆ S" and "¬ f constant_on S" shows "finite {z∈K. f z = 0}" proof (rule ccontr) assume "infinite {z∈K. f z = 0}" then obtain z where "z ∈ K" and z: "z islimpt {z∈K. f z = 0}" using ‹compact K› by (auto simp: compact_eq_Bolzano_Weierstrass) moreover have "{z∈K. f z = 0} ⊆ S" using ‹K ⊆ S› by blast ultimately show False using assms analytic_continuation [OF S] unfolding constant_on_def by blast qed lemma holomorphic_countable_zeros: assumes S: "f holomorphic_on S" "open S" "connected S" and "fsigma S" and "¬ f constant_on S" shows "countable {z∈S. f z = 0}" proof - obtain F::"nat ⇒ complex set" where F: "range F ⊆ Collect compact" and Seq: "S = (⋃i. F i)" using ‹fsigma S› by (meson fsigma_Union_compact) have fin: "finite {z ∈ F i. f z = 0}" for i using holomorphic_compact_finite_zeros assms F Seq Union_iff by blast have "{z ∈ S. f z = 0} = (⋃i. {z ∈ F i. f z = 0})" using Seq by auto with fin show ?thesis by (simp add: countable_finite) qed lemma holomorphic_countable_equal: assumes "f holomorphic_on S" "g holomorphic_on S" "open S" "connected S" and "fsigma S" and eq: "uncountable {z∈S. f z = g z}" shows "S ⊆ {z∈S. f z = g z}" proof - obtain z where z: "z∈S" "f z = g z" using eq not_finite_existsD uncountable_infinite by blast have "(λx. f x - g x) holomorphic_on S" by (simp add: assms holomorphic_on_diff) then have "(λx. f x - g x) constant_on S" using holomorphic_countable_zeros assms by force with z have "⋀x. x∈S ⟹ f x - g x = 0" unfolding constant_on_def by force then show ?thesis by auto qed lemma holomorphic_countable_equal_UNIV: assumes fg: "f holomorphic_on UNIV" "g holomorphic_on UNIV" and eq: "uncountable {z. f z = g z}" shows "f=g" using holomorphic_countable_equal [OF fg] eq by fastforce subsection‹Open mapping theorem› lemma holomorphic_contract_to_zero: assumes contf: "continuous_on (cball ξ r) f" and holf: "f holomorphic_on ball ξ r" and "0 < r" and norm_less: "⋀z. norm(ξ - z) = r ⟹ norm(f ξ) < norm(f z)" obtains z where "z ∈ ball ξ r" "f z = 0" proof - { assume fnz: "⋀w. w ∈ ball ξ r ⟹ f w ≠ 0" then have "0 < norm (f ξ)" by (simp add: ‹0 < r›) have fnz': "⋀w. w ∈ cball ξ r ⟹ f w ≠ 0" by (metis norm_less dist_norm fnz less_eq_real_def mem_ball mem_cball norm_not_less_zero norm_zero) have "frontier(cball ξ r) ≠ {}" using ‹0 < r› by simp define g where [abs_def]: "g z = inverse (f z)" for z have contg: "continuous_on (cball ξ r) g" unfolding g_def using contf continuous_on_inverse fnz' by blast have holg: "g holomorphic_on ball ξ r" unfolding g_def using fnz holf holomorphic_on_inverse by blast have "frontier (cball ξ r) ⊆ cball ξ r" by (simp add: subset_iff) then have contf': "continuous_on (frontier (cball ξ r)) f" and contg': "continuous_on (frontier (cball ξ r)) g" by (blast intro: contf contg continuous_on_subset)+ have froc: "frontier(cball ξ r) ≠ {}" using ‹0 < r› by simp moreover have "continuous_on (frontier (cball ξ r)) (norm o f)" using contf' continuous_on_compose continuous_on_norm_id by blast ultimately obtain w where w: "w ∈ frontier(cball ξ r)" and now: "⋀x. x ∈ frontier(cball ξ r) ⟹ norm (f w) ≤ norm (f x)" using continuous_attains_inf [OF compact_frontier [OF compact_cball]] by (metis comp_apply) then have fw: "0 < norm (f w)" by (simp add: fnz') have "continuous_on (frontier (cball ξ r)) (norm o g)" using contg' continuous_on_compose continuous_on_norm_id by blast then obtain v where v: "v ∈ frontier(cball ξ r)" and nov: "⋀x. x ∈ frontier(cball ξ r) ⟹ norm (g v) ≥ norm (g x)" using continuous_attains_sup [OF compact_frontier [OF compact_cball] froc] by force then have fv: "0 < norm (f v)" by (simp add: fnz') have "norm ((deriv ^^ 0) g ξ) ≤ fact 0 * norm (g v) / r ^ 0" by (rule Cauchy_inequality [OF holg contg ‹0 < r›]) (simp add: dist_norm nov) then have "cmod (g ξ) ≤ cmod (g v)" by simp moreover have "cmod (ξ - w) = r" by (metis (no_types) dist_norm frontier_cball mem_sphere w) ultimately obtain wr: "norm (ξ - w) = r" and nfw: "norm (f w) ≤ norm (f ξ)" unfolding g_def by (metis (no_types) ‹0 < cmod (f ξ)› less_imp_inverse_less norm_inverse not_le now order_trans v) with fw have False using norm_less by force } with that show ?thesis by blast qed theorem open_mapping_thm: assumes holf: "f holomorphic_on S" and S: "open S" and "connected S" and "open U" and "U ⊆ S" and fne: "¬ f constant_on S" shows "open (f ` U)" proof - have *: "open (f ` U)" if "U ≠ {}" and U: "open U" "connected U" and "f holomorphic_on U" and fneU: "⋀x. ∃y ∈ U. f y ≠ x" for U proof (clarsimp simp: open_contains_ball) fix ξ assume ξ: "ξ ∈ U" show "∃e>0. ball (f ξ) e ⊆ f ` U" proof - have hol: "(λz. f z - f ξ) holomorphic_on U" by (rule holomorphic_intros that)+ obtain s where "0 < s" and sbU: "ball ξ s ⊆ U" and sne: "⋀z. z ∈ ball ξ s - {ξ} ⟹ (λz. f z - f ξ) z ≠ 0" using isolated_zeros [OF hol U ξ] by (metis fneU right_minus_eq) obtain r where "0 < r" and r: "cball ξ r ⊆ ball ξ s" using ‹0 < s› by (rule_tac r="s/2" in that) auto have "cball ξ r ⊆ U" using sbU r by blast then have frsbU: "frontier (cball ξ r) ⊆ U" using Diff_subset frontier_def order_trans by fastforce then have cof: "compact (frontier(cball ξ r))" by blast have frne: "frontier (cball ξ r) ≠ {}" using ‹0 < r› by auto have contfr: "continuous_on (frontier (cball ξ r)) (λz. norm (f z - f ξ))" by (metis continuous_on_norm continuous_on_subset frsbU hol holomorphic_on_imp_continuous_on) obtain w where "norm (ξ - w) = r" and w: "(⋀z. norm (ξ - z) = r ⟹ norm (f w - f ξ) ≤ norm(f z - f ξ))" using continuous_attains_inf [OF cof frne contfr] by (auto simp: dist_norm) moreover define ε where "ε ≡ norm (f w - f ξ) / 3" ultimately have "0 < ε" using ‹0 < r› dist_complex_def r sne by auto have "ball (f ξ) ε ⊆ f ` U" proof fix γ assume γ: "γ ∈ ball (f ξ) ε" have *: "cmod (γ - f ξ) < cmod (γ - f z)" if "cmod (ξ - z) = r" for z proof - have lt: "cmod (f w - f ξ) / 3 < cmod (γ - f z)" using w [OF that] γ using dist_triangle2 [of "f ξ" "γ" "f z"] dist_triangle2 [of "f ξ" "f z" γ] by (simp add: ε_def dist_norm norm_minus_commute) show ?thesis by (metis ε_def dist_commute dist_norm less_trans lt mem_ball γ) qed have "continuous_on (cball ξ r) (λz. γ - f z)" using ‹cball ξ r ⊆ U› ‹f holomorphic_on U› by (force intro: continuous_intros continuous_on_subset holomorphic_on_imp_continuous_on) moreover have "(λz. γ - f z) holomorphic_on ball ξ r" using ‹cball ξ r ⊆ U› ball_subset_cball holomorphic_on_subset that(4) by (intro holomorphic_intros) blast ultimately obtain z where "z ∈ ball ξ r" "γ - f z = 0" using "*" ‹0 < r› holomorphic_contract_to_zero by blast then show "γ ∈ f ` U" using ‹cball ξ r ⊆ U› by fastforce qed then show ?thesis using ‹0 < ε› by blast qed qed have "open (f ` X)" if "X ∈ components U" for X proof - have holfU: "f holomorphic_on U" using ‹U ⊆ S› holf holomorphic_on_subset by blast have "X ≠ {}" using that by (simp add: in_components_nonempty) moreover have "open X" using that ‹open U› open_components by auto moreover have "connected X" using that in_components_maximal by blast moreover have "f holomorphic_on X" by (meson that holfU holomorphic_on_subset in_components_maximal) moreover have "∃y∈X. f y ≠ x" for x proof (rule ccontr) assume not: "¬ (∃y∈X. f y ≠ x)" have "X ⊆ S" using ‹U ⊆ S› in_components_subset that by blast obtain w where w: "w ∈ X" using ‹X ≠ {}› by blast have wis: "w islimpt X" using w ‹open X› interior_eq by auto have hol: "(λz. f z - x) holomorphic_on S" by (simp add: holf holomorphic_on_diff) with fne [unfolded constant_on_def] analytic_continuation[OF hol S ‹connected S› ‹X ⊆ S› _ wis] not ‹X ⊆ S› w show False by auto qed ultimately show ?thesis by (rule *) qed then have "open (f ` ⋃(components U))" by (metis (no_types, lifting) imageE image_Union open_Union) then show ?thesis by force qed text‹No need for \<^term>‹S› to be connected. But the nonconstant condition is stronger.› corollary✐‹tag unimportant› open_mapping_thm2: assumes holf: "f holomorphic_on S" and S: "open S" and "open U" "U ⊆ S" and fnc: "⋀X. ⟦open X; X ⊆ S; X ≠ {}⟧ ⟹ ¬ f constant_on X" shows "open (f ` U)" proof - have "S = ⋃(components S)" by simp with ‹U ⊆ S› have "U = (⋃C ∈ components S. C ∩ U)" by auto then have "f ` U = (⋃C ∈ components S. f ` (C ∩ U))" using image_UN by fastforce moreover { fix C assume "C ∈ components S" with S ‹C ∈ components S› open_components in_components_connected have C: "open C" "connected C" by auto have "C ⊆ S" by (metis ‹C ∈ components S› in_components_maximal) have nf: "¬ f constant_on C" using ‹open C› ‹C ∈ components S› ‹C ⊆ S› fnc in_components_nonempty by blast have "f holomorphic_on C" by (metis holf holomorphic_on_subset ‹C ⊆ S›) then have "open (f ` (C ∩ U))" by (meson C ‹open U› inf_le1 nf open_Int open_mapping_thm) } ultimately show ?thesis by force qed corollary✐‹tag unimportant› open_mapping_thm3: assumes holf: "f holomorphic_on S" and "open S" and injf: "inj_on f S" shows "open (f ` S)" proof (rule open_mapping_thm2 [OF holf]) show "⋀X. ⟦open X; X ⊆ S; X ≠ {}⟧ ⟹ ¬ f constant_on X" using inj_on_subset injective_not_constant injf by blast qed (use assms in auto) subsection‹Maximum modulus principle› text‹If \<^term>‹f› is holomorphic, then its norm (modulus) cannot exhibit a true local maximum that is properly within the domain of \<^term>‹f›.› proposition maximum_modulus_principle: assumes holf: "f holomorphic_on S" and S: "open S" and "connected S" and "open U" and "U ⊆ S" and "ξ ∈ U" and no: "⋀z. z ∈ U ⟹ norm(f z) ≤ norm(f ξ)" shows "f constant_on S" proof (rule ccontr) assume "¬ f constant_on S" then have "open (f ` U)" using open_mapping_thm assms by blast moreover have "¬ open (f ` U)" proof - have "∃t. cmod (f ξ - t) < e ∧ t ∉ f ` U" if "0 < e" for e using that apply (rule_tac x="if 0 < Re(f ξ) then f ξ + (e/2) else f ξ - (e/2)" in exI) apply (simp add: dist_norm) apply (fastforce simp: cmod_Re_le_iff dest!: no dest: sym) done then show ?thesis unfolding open_contains_ball by (metis ‹ξ ∈ U› contra_subsetD dist_norm imageI mem_ball) qed ultimately show False by blast qed proposition maximum_modulus_frontier: assumes holf: "f holomorphic_on (interior S)" and contf: "continuous_on (closure S) f" and bos: "bounded S" and leB: "⋀z. z ∈ frontier S ⟹ norm(f z) ≤ B" and "ξ ∈ S" shows "norm(f ξ) ≤ B" proof - have "compact (closure S)" using bos by (simp add: bounded_closure compact_eq_bounded_closed) moreover have "continuous_on (closure S) (cmod ∘ f)" using contf continuous_on_compose continuous_on_norm_id by blast ultimately obtain z where "z ∈ closure S" and z: "⋀y. y ∈ closure S ⟹ (cmod ∘ f) y ≤ (cmod ∘ f) z" using continuous_attains_sup [of "closure S" "norm o f"] ‹ξ ∈ S› by auto then consider "z ∈ frontier S" | "z ∈ interior S" using frontier_def by auto then have "norm(f z) ≤ B" proof cases case 1 then show ?thesis using leB by blast next case 2 have "f constant_on (connected_component_set (interior S) z)" proof (rule maximum_modulus_principle) show "f holomorphic_on connected_component_set (interior S) z" by (metis connected_component_subset holf holomorphic_on_subset) show zin: "z ∈ connected_component_set (interior S) z" by (simp add: 2) show "⋀W. W ∈ connected_component_set (interior S) z ⟹ cmod (f W) ≤ cmod (f z)" using closure_def connected_component_subset z by fastforce qed (auto simp: open_connected_component) then obtain c where c: "⋀w. w ∈ connected_component_set (interior S) z ⟹ f w = c" by (auto simp: constant_on_def) have "f ` closure(connected_component_set (interior S) z) ⊆ {c}" proof (rule image_closure_subset) show "continuous_on (closure (connected_component_set (interior S) z)) f" by (meson closure_mono connected_component_subset contf continuous_on_subset interior_subset) qed (use c in auto) then have cc: "⋀w. w ∈ closure(connected_component_set (interior S) z) ⟹ f w = c" by blast have "connected_component (interior S) z z" by (simp add: "2") moreover have "connected_component_set (interior S) z ≠ UNIV" by (metis bos bounded_interior connected_component_eq_UNIV not_bounded_UNIV) ultimately have "frontier(connected_component_set (interior S) z) ≠ {}" by (meson "2" connected_component_eq_empty frontier_not_empty) then obtain w where w: "w ∈ frontier(connected_component_set (interior S) z)" by auto then have "norm (f z) = norm (f w)" by (simp add: "2" c cc frontier_def) also have "... ≤ B" using w frontier_interior_subset frontier_of_connected_component_subset by (blast intro: leB) finally show ?thesis . qed then show ?thesis using z ‹ξ ∈ S› closure_subset by fastforce qed corollary✐‹tag unimportant› maximum_real_frontier: assumes holf: "f holomorphic_on (interior S)" and contf: "continuous_on (closure S) f" and bos: "bounded S" and leB: "⋀z. z ∈ frontier S ⟹ Re(f z) ≤ B" and "ξ ∈ S" shows "Re(f ξ) ≤ B" using maximum_modulus_frontier [of "exp o f" S "exp B"] Transcendental.continuous_on_exp holomorphic_on_compose holomorphic_on_exp assms by auto subsection✐‹tag unimportant› ‹Factoring out a zero according to its order› lemma holomorphic_factor_order_of_zero: assumes holf: "f holomorphic_on S" and os: "open S" and "ξ ∈ S" "0 < n" and dnz: "(deriv ^^ n) f ξ ≠ 0" and dfz: "⋀i. ⟦0 < i; i < n⟧ ⟹ (deriv ^^ i) f ξ = 0" obtains g r where "0 < r" "g holomorphic_on ball ξ r" "⋀w. w ∈ ball ξ r ⟹ f w - f ξ = (w - ξ)^n * g w" "⋀w. w ∈ ball ξ r ⟹ g w ≠ 0" proof - obtain r where "r>0" and r: "ball ξ r ⊆ S" using assms by (blast elim!: openE) then have holfb: "f holomorphic_on ball ξ r" using holf holomorphic_on_subset by blast define g where "g w = suminf (λi. (deriv ^^ (i + n)) f ξ / (fact(i + n)) * (w - ξ)^i)" for w have sumsg: "(λi. (deriv ^^ (i + n)) f ξ / (fact(i + n)) * (w - ξ)^i) sums g w" and feq: "f w - f ξ = (w - ξ)^n * g w" if w: "w ∈ ball ξ r" for w proof - define powf where "powf = (λi. (deriv ^^ i) f ξ/(fact i) * (w - ξ)^i)" have [simp]: "powf 0 = f ξ" by (simp add: powf_def) have sing: "{..<n} - {i. powf i = 0} = (if f ξ = 0 then {} else {0})" unfolding powf_def using ‹0 < n› dfz by (auto simp: dfz; metis funpow_0 not_gr0) have "powf sums f w" unfolding powf_def by (rule holomorphic_power_series [OF holfb w]) moreover have "(∑i<n. powf i) = f ξ" by (subst sum.setdiff_irrelevant [symmetric]; simp add: dfz sing) ultimately have fsums: "(λi. powf (i+n)) sums (f w - f ξ)" using w sums_iff_shift' by metis then have *: "summable (λi. (w - ξ) ^ n * ((deriv ^^ (i + n)) f ξ * (w - ξ) ^ i / fact (i + n)))" unfolding powf_def using sums_summable by (auto simp: power_add mult_ac) have "summable (λi. (deriv ^^ (i + n)) f ξ * (w - ξ) ^ i / fact (i + n))" proof (cases "w=ξ") case False then show ?thesis using summable_mult [OF *, of "1 / (w - ξ) ^ n"] by simp next case True then show ?thesis by (auto simp: Power.semiring_1_class.power_0_left intro!: summable_finite [of "{0}"] split: if_split_asm) qed then show sumsg: "(λi. (deriv ^^ (i + n)) f ξ / (fact(i + n)) * (w - ξ)^i) sums g w" by (simp add: summable_sums_iff g_def) show "f w - f ξ = (w - ξ)^n * g w" using sums_mult [OF sumsg, of "(w - ξ) ^ n"] by (intro sums_unique2 [OF fsums]) (auto simp: power_add mult_ac powf_def) qed then have holg: "g holomorphic_on ball ξ r" by (meson sumsg power_series_holomorphic) then have contg: "continuous_on (ball ξ r) g" by (blast intro: holomorphic_on_imp_continuous_on) have "g ξ ≠ 0" using dnz unfolding g_def by (subst suminf_finite [of "{0}"]) auto obtain d where "0 < d" and d: "⋀w. w ∈ ball ξ d ⟹ g w ≠ 0" using ‹0 < r› continuous_on_avoid [OF contg _ ‹g ξ ≠ 0›] by (metis centre_in_ball le_cases mem_ball mem_ball_leI) show ?thesis proof show "g holomorphic_on ball ξ (min r d)" using holg by (auto simp: feq holomorphic_on_subset subset_ball d) qed (use ‹0 < r› ‹0 < d› in ‹auto simp: feq d›) qed lemma holomorphic_factor_order_of_zero_strong: assumes holf: "f holomorphic_on S" "open S" "ξ ∈ S" "0 < n" and "(deriv ^^ n) f ξ ≠ 0" and "⋀i. ⟦0 < i; i < n⟧ ⟹ (deriv ^^ i) f ξ = 0" obtains g r where "0 < r" "g holomorphic_on ball ξ r" "⋀w. w ∈ ball ξ r ⟹ f w - f ξ = ((w - ξ) * g w) ^ n" "⋀w. w ∈ ball ξ r ⟹ g w ≠ 0" proof - obtain g r where "0 < r" and holg: "g holomorphic_on ball ξ r" and feq: "⋀w. w ∈ ball ξ r ⟹ f w - f ξ = (w - ξ)^n * g w" and gne: "⋀w. w ∈ ball ξ r ⟹ g w ≠ 0" by (auto intro: holomorphic_factor_order_of_zero [OF assms]) have con: "continuous_on (ball ξ r) (λz. deriv g z / g z)" by (rule continuous_intros) (auto simp: gne holg holomorphic_deriv holomorphic_on_imp_continuous_on) have cd: "(λz. deriv g z / g z) field_differentiable at x" if "dist ξ x < r" for x proof (intro derivative_intros) show "deriv g field_differentiable at x" using that holg mem_ball by (blast intro: holomorphic_deriv holomorphic_on_imp_differentiable_at) show "g field_differentiable at x" by (metis that open_ball at_within_open holg holomorphic_on_def mem_ball) qed (simp add: gne that) obtain h where h: "⋀x. x ∈ ball ξ r ⟹ (h has_field_derivative deriv g x / g x) (at x)" using holomorphic_convex_primitive [of "ball ξ r" "{}" "λz. deriv g z / g z"] by (metis (no_types, lifting) Diff_empty at_within_interior cd con convex_ball infinite_imp_nonempty interior_ball mem_ball) then have "continuous_on (ball ξ r) h" by (metis open_ball holomorphic_on_imp_continuous_on holomorphic_on_open) then have con: "continuous_on (ball ξ r) (λx. exp (h x) / g x)" by (auto intro!: continuous_intros simp add: holg holomorphic_on_imp_continuous_on gne) have 0: "dist ξ x < r ⟹ ((λx. exp (h x) / g x) has_field_derivative 0) (at x)" for x apply (rule h derivative_eq_intros DERIV_deriv_iff_field_differentiable [THEN iffD2] | simp)+ using holg by (auto simp: holomorphic_on_imp_differentiable_at gne h) obtain c where c: "⋀x. x ∈ ball ξ r ⟹ exp (h x) / g x = c" by (rule DERIV_zero_connected_constant [of "ball ξ r" "{}" "λx. exp(h x) / g x"]) (auto simp: con 0) have hol: "(λz. exp ((Ln (inverse c) + h z) / of_nat n)) holomorphic_on ball ξ r" proof (intro holomorphic_intros holomorphic_on_compose [unfolded o_def, where g = exp]) show "h holomorphic_on ball ξ r" using h holomorphic_on_open by blast qed (use ‹0 < n› in auto) show ?thesis proof show "⋀w. w ∈ ball ξ r ⟹ f w - f ξ = ((w - ξ) * exp ((Ln (inverse c) + h w) / of_nat n)) ^ n" using ‹0 < n› by (auto simp: feq power_mult_distrib exp_divide_power_eq exp_add gne simp flip: c) qed (use hol ‹0 < r› in auto) qed lemma fixes k :: "'a::wellorder" assumes a_def: "a == LEAST x. P x" and P: "P k" shows def_LeastI: "P a" and def_Least_le: "a ≤ k" unfolding a_def by (rule LeastI Least_le; rule P)+ lemma holomorphic_factor_zero_nonconstant: assumes holf: "f holomorphic_on S" and S: "open S" "connected S" and "ξ ∈ S" "f ξ = 0" and nonconst: "¬ f constant_on S" obtains g r n where "0 < n" "0 < r" "ball ξ r ⊆ S" "g holomorphic_on ball ξ r" "⋀w. w ∈ ball ξ r ⟹ f w = (w - ξ)^n * g w" "⋀w. w ∈ ball ξ r ⟹ g w ≠ 0" proof (cases "∀n>0. (deriv ^^ n) f ξ = 0") case True then show ?thesis using holomorphic_fun_eq_const_on_connected [OF holf S _ ‹ξ ∈ S›] nonconst by (simp add: constant_on_def) next case False then obtain n0 where "n0 > 0" and n0: "(deriv ^^ n0) f ξ ≠ 0" by blast obtain r0 where "r0 > 0" "ball ξ r0 ⊆ S" using S openE ‹ξ ∈ S› by auto define n where "n ≡ LEAST n. (deriv ^^ n) f ξ ≠ 0" have n_ne: "(deriv ^^ n) f ξ ≠ 0" by (rule def_LeastI [OF n_def]) (rule n0) then have "0 < n" using ‹f ξ = 0› using funpow_0 by fastforce have n_min: "⋀k. k < n ⟹ (deriv ^^ k) f ξ = 0" using def_Least_le [OF n_def] not_le by blast then obtain g r1 where g: "0 < r1" "g holomorphic_on ball ξ r1" and geq: "⋀w. w ∈ ball ξ r1 ⟹ f w = (w - ξ) ^ n * g w" and g0: "⋀w. w ∈ ball ξ r1 ⟹ g w ≠ 0" by (auto intro: holomorphic_factor_order_of_zero [OF holf ‹open S› ‹ξ ∈ S› ‹n > 0› n_ne] simp: ‹f ξ = 0›) show ?thesis proof show "g holomorphic_on ball ξ (min r0 r1)" using g by auto show "⋀w. w ∈ ball ξ (min r0 r1) ⟹ f w = (w - ξ) ^ n * g w" by (simp add: geq) qed (use ‹0 < n› ‹0 < r0› ‹0 < r1› ‹ball ξ r0 ⊆ S› g0 in auto) qed lemma holomorphic_lower_bound_difference: assumes holf: "f holomorphic_on S" and S: "open S" "connected S" and "ξ ∈ S" and "φ ∈ S" and fne: "f φ ≠ f ξ" obtains k n r where "0 < k" "0 < r" "ball ξ r ⊆ S" "⋀w. w ∈ ball ξ r ⟹ k * norm(w - ξ)^n ≤ norm(f w - f ξ)" proof - define n where "n = (LEAST n. 0 < n ∧ (deriv ^^ n) f ξ ≠ 0)" obtain n0 where "0 < n0" and n0: "(deriv ^^ n0) f ξ ≠ 0" using fne holomorphic_fun_eq_const_on_connected [OF holf S] ‹ξ ∈ S› ‹φ ∈ S› by blast then have "0 < n" and n_ne: "(deriv ^^ n) f ξ ≠ 0" unfolding n_def by (metis (mono_tags, lifting) LeastI)+ have n_min: "⋀k. ⟦0 < k; k < n⟧ ⟹ (deriv ^^ k) f ξ = 0" unfolding n_def by (blast dest: not_less_Least) then obtain g r where "0 < r" and holg: "g holomorphic_on ball ξ r" and fne: "⋀w. w ∈ ball ξ r ⟹ f w - f ξ = (w - ξ) ^ n * g w" and gnz: "⋀w. w ∈ ball ξ r ⟹ g w ≠ 0" by (auto intro: holomorphic_factor_order_of_zero [OF holf ‹open S› ‹ξ ∈ S› ‹n > 0› n_ne]) obtain e where "e>0" and e: "ball ξ e ⊆ S" using assms by (blast elim!: openE) then have holfb: "f holomorphic_on ball ξ e" using holf holomorphic_on_subset by blast define d where "d = (min e r) / 2" have "0 < d" using ‹0 < r› ‹0 < e› by (simp add: d_def) have "d < r" using ‹0 < r› by (auto simp: d_def) then have cbb: "cball ξ d ⊆ ball ξ r" by (auto simp: cball_subset_ball_iff) then have "g holomorphic_on cball ξ d" by (rule holomorphic_on_subset [OF holg]) then have "closed (g ` cball ξ d)" by (simp add: compact_imp_closed compact_continuous_image holomorphic_on_imp_continuous_on) moreover have "g ` cball ξ d ≠ {}" using ‹0 < d› by auto ultimately obtain x where x: "x ∈ g ` cball ξ d" and "⋀y. y ∈ g ` cball ξ d ⟹ dist 0 x ≤ dist 0 y" by (rule distance_attains_inf) blast then have leg: "⋀w. w ∈ cball ξ d ⟹ norm x ≤ norm (g w)" by auto have "ball ξ d ⊆ cball ξ d" by auto also have "... ⊆ ball ξ e" using ‹0 < d› d_def by auto also have "... ⊆ S" by (rule e) finally have dS: "ball ξ d ⊆ S" . have "x ≠ 0" using gnz x ‹d < r› by auto show thesis proof show "⋀w. w ∈ ball ξ d ⟹ cmod x * cmod (w - ξ) ^ n ≤ cmod (f w - f ξ)" using ‹d < r› leg by (auto simp: fne norm_mult norm_power algebra_simps mult_right_mono) qed (use dS ‹x ≠ 0› ‹d > 0› in auto) qed lemma assumes holf: "f holomorphic_on (S - {ξ})" and ξ: "ξ ∈ interior S" shows holomorphic_on_extend_lim: "(∃g. g holomorphic_on S ∧ (∀z ∈ S - {ξ}. g z = f z)) ⟷ ((λz. (z - ξ) * f z) ⤏ 0) (at ξ)" (is "?P = ?Q") and holomorphic_on_extend_bounded: "(∃g. g holomorphic_on S ∧ (∀z ∈ S - {ξ}. g z = f z)) ⟷ (∃B. eventually (λz. norm(f z) ≤ B) (at ξ))" (is "?P = ?R") proof - obtain δ where "0 < δ" and δ: "ball ξ δ ⊆ S" using ξ mem_interior by blast have "?R" if holg: "g holomorphic_on S" and gf: "⋀z. z ∈ S - {ξ} ⟹ g z = f z" for g proof - have §: "cmod (f x) ≤ cmod (g ξ) + 1" if "x ≠ ξ" "dist x ξ < δ" "dist (g x) (g ξ) < 1" for x proof - have "x ∈ S" by (metis δ dist_commute mem_ball subsetD that(2)) with that gf [of x] show ?thesis using norm_triangle_ineq2 [of "f x" "g ξ"] dist_complex_def by auto qed then have *: "∀⇩_{F}z in at ξ. dist (g z) (g ξ) < 1 ⟶ cmod (f z) ≤ cmod (g ξ) + 1" using ‹0 < δ› eventually_at by blast have "continuous_on (interior S) g" by (meson continuous_on_subset holg holomorphic_on_imp_continuous_on interior_subset) then have "⋀x. x ∈ interior S ⟹ (g ⤏ g x) (at x)" using continuous_on_interior continuous_within holg holomorphic_on_imp_continuous_on by blast then have "(g ⤏ g ξ) (at ξ)" by (simp add: ξ) then show ?thesis apply (rule_tac x="norm(g ξ) + 1" in exI) apply (rule eventually_mp [OF * tendstoD [where e=1]], auto) done qed moreover have "?Q" if "∀⇩_{F}z in at ξ. cmod (f z) ≤ B" for B by (rule lim_null_mult_right_bounded [OF _ that]) (simp add: LIM_zero) moreover have "?P" if "(λz. (z - ξ) * f z) ─ξ→ 0" proof - define h where [abs_def]: "h z = (z - ξ)^2 * f z" for z have h0: "(h has_field_derivative 0) (at ξ)" apply (simp add: h_def has_field_derivative_iff) apply (auto simp: field_split_simps power2_eq_square Lim_transform_within [OF that, of 1]) done have holh: "h holomorphic_on S" proof (simp add: holomorphic_on_def, clarify) fix z assume "z ∈ S" show "h field_differentiable at z within S" proof (cases "z = ξ") case True then show ?thesis using field_differentiable_at_within field_differentiable_def h0 by blast next case False then have "f field_differentiable at z within S" using holomorphic_onD [OF holf, of z] ‹z ∈ S› unfolding field_differentiable_def has_field_derivative_iff by (force intro: exI [where x="dist ξ z"] elim: Lim_transform_within_set [unfolded eventually_at]) then show ?thesis by (simp add: h_def power2_eq_square derivative_intros) qed qed define g where [abs_def]: "g z = (if z = ξ then deriv h ξ else (h z - h ξ) / (z - ξ))" for z have holg: "g holomorphic_on S" unfolding g_def by (rule pole_lemma [OF holh ξ]) have §: "∀z∈S - {ξ}. (g z - g ξ) / (z - ξ) = f z" using h0 by (auto simp: g_def power2_eq_square divide_simps DERIV_imp_deriv h_def) show ?thesis apply (intro exI conjI) apply (rule pole_lemma [OF holg ξ]) apply (simp add: §) done qed ultimately show "?P = ?Q" and "?P = ?R" by meson+ qed lemma pole_at_infinity: assumes holf: "f holomorphic_on UNIV" and lim: "((inverse o f) ⤏ l) at_infinity" obtains a n where "⋀z. f z = (∑i≤n. a i * z^i)" proof (cases "l = 0") case False show thesis proof show "f z = (∑i≤0. inverse l * z ^ i)" for z using False tendsto_inverse [OF lim] by (simp add: Liouville_weak [OF holf]) qed next case True then have [simp]: "l = 0" . show ?thesis proof (cases "∃r. 0 < r ∧ (∀z ∈ ball 0 r - {0}. f(inverse z) ≠ 0)") case True then obtain r where "0 < r" and r: "⋀z. z ∈ ball 0 r - {0} ⟹ f(inverse z) ≠ 0" by auto have 1: "inverse ∘ f ∘ inverse holomorphic_on ball 0 r - {0}" by (rule holomorphic_on_compose holomorphic_intros holomorphic_on_subset [OF holf] | force simp: r)+ have 2: "0 ∈ interior (ball 0 r)" using ‹0 < r› by simp have "∃B. 0<B ∧ eventually (λz. cmod ((inverse ∘ f ∘ inverse) z) ≤ B) (at 0)" apply (rule exI [where x=1]) using tendstoD [OF lim [unfolded lim_at_infinity_0] zero_less_one] by (simp add: eventually_mono) with holomorphic_on_extend_bounded [OF 1 2] obtain g where holg: "g holomorphic_on ball 0 r" and geq: "⋀z. z ∈ ball 0 r - {0} ⟹ g z = (inverse ∘ f ∘ inverse) z" by meson have ifi0: "(inverse ∘ f ∘ inverse) ─0→ 0" using ‹l = 0› lim lim_at_infinity_0 by blast have g2g0: "g ─0→ g 0" using ‹0 < r› centre_in_ball continuous_at continuous_on_eq_continuous_at holg by (blast intro: holomorphic_on_imp_continuous_on) have g2g1: "g ─0→ 0" apply (rule Lim_transform_within_open [OF ifi0 open_ball [of 0 r]]) using ‹0 < r› by (auto simp: geq) have [simp]: "g 0 = 0" by (rule tendsto_unique [OF _ g2g0 g2g1]) simp have "ball 0 r - {0::complex} ≠ {}" using ‹0 < r› by (metis "2" Diff_iff insert_Diff interior_ball interior_singleton) then obtain w::complex where "w ≠ 0" and w: "norm w < r" by force then have "g w ≠ 0" by (simp add: geq r) obtain B n e where "0 < B" "0 < e" "e ≤ r" and leg: "⋀w. norm w < e ⟹ B * cmod w ^ n ≤ cmod (g w)" proof (rule holomorphic_lower_bound_difference [OF holg open_ball connected_ball]) show "g w ≠ g 0" by (simp add: ‹g w ≠ 0›) show "w ∈ ball 0 r" using mem_ball_0 w by blast qed (use ‹0 < r› in ‹auto simp: ball_subset_ball_iff›) have §: "cmod (f z) ≤ cmod z ^ n / B" if "2/e ≤ cmod z" for z proof - have ize: "inverse z ∈ ball 0 e - {0}" using that ‹0 < e› by (auto simp: norm_divide field_split_simps algebra_simps) then have [simp]: "z ≠ 0" and izr: "inverse z ∈ ball 0 r - {0}" using ‹e ≤ r› by auto then have [simp]: "f z ≠ 0" using r [of "inverse z"] by simp have [simp]: "f z = inverse (g (inverse z))" using izr geq [of "inverse z"] by simp show ?thesis using ize leg [of "inverse z"] ‹0 < B› ‹0 < e› by (simp add: field_split_simps norm_divide algebra_simps) qed show thesis proof show "f z = (∑i≤n. (deriv ^^ i) f 0 / fact i * z ^ i)" for z using § by (rule_tac A = "2/e" and B = "1/B" in Liouville_polynomial [OF holf], simp) qed next case False then have fi0: "⋀r. r > 0 ⟹ ∃z∈ball 0 r - {0}. f (inverse z) = 0" by simp have fz0: "f z = 0" if "0 < r" and lt1: "⋀x. x ≠ 0 ⟹ cmod x < r ⟹ inverse (cmod (f (inverse x))) < 1" for z r proof - have f0: "(f ⤏ 0) at_infinity" proof - have DIM_complex[intro]: "2 ≤ DIM(complex)" ― ‹should not be necessary!› by simp have "f (inverse x) ≠ 0 ⟹ x ≠ 0 ⟹ cmod x < r ⟹ 1 < cmod (f (inverse x))" for x using lt1[of x] by (auto simp: field_simps) then have **: "cmod (f (inverse x)) ≤ 1 ⟹ x ≠ 0 ⟹ cmod x < r ⟹ f (inverse x) = 0" for x by force then have *: "(f ∘ inverse) ` (ball 0 r - {0}) ⊆ {0} ∪ - ball 0 1" by force have "continuous_on (inverse ` (ball 0 r - {0})) f" using continuous_on_subset holf holomorphic_on_imp_continuous_on by blast then have "connected ((f ∘ inverse) ` (ball 0 r - {0}))" using connected_punctured_ball by (intro connected_continuous_image continuous_intros; force) then have "{0} ∩ (f ∘ inverse) ` (ball 0 r - {0}) = {} ∨ - ball 0 1 ∩ (f ∘ inverse) ` (ball 0 r - {0}) = {}" by (rule connected_closedD) (use * in auto) then have "f (inverse w) = 0" if "w ≠ 0" "cmod w < r" for w using **[of w] fi0 ‹0 < r› that by force then show ?thesis unfolding lim_at_infinity_0 using eventually_at ‹r > 0› by (force simp add: intro: tendsto_eventually) qed obtain w where "w ∈ ball 0 r - {0}" and "f (inverse w) = 0" using False ‹0 < r› by blast then show ?thesis by (auto simp: f0 Liouville_weak [OF holf, of 0]) qed show thesis proof show "⋀z. f z = (∑i≤0. 0 * z ^ i)" using lim apply (simp add: lim_at_infinity_0 Lim_at dist_norm norm_inverse) using fz0 zero_less_one by blast qed qed qed subsection✐‹tag unimportant› ‹Entire proper functions are precisely the non-trivial polynomials› lemma proper_map_polyfun: fixes c :: "nat ⇒ 'a::{real_normed_div_algebra,heine_borel}" assumes "closed S" and "compact K" and c: "c i ≠ 0" "1 ≤ i" "i ≤ n" shows "compact (S ∩ {z. (∑i≤n. c i * z^i) ∈ K})" proof - obtain B where "B > 0" and B: "⋀x. x ∈ K ⟹ norm x ≤ B" by (metis compact_imp_bounded ‹compact K› bounded_pos) have *: "norm x ≤ b" if "⋀x. b ≤ norm x ⟹ B + 1 ≤ norm (∑i≤n. c i * x ^ i)" "(∑i≤n. c i * x ^ i) ∈ K" for b x proof - have "norm (∑i≤n. c i * x ^ i) ≤ B" using B that by blast moreover have "¬ B + 1 ≤ B" by simp ultimately show "norm x ≤ b" using that by (metis (no_types) less_eq_real_def not_less order_trans) qed have "bounded {z. (∑i≤n. c i * z ^ i) ∈ K}" using Limits.polyfun_extremal [where c=c and B="B+1", OF c] by (auto simp: bounded_pos eventually_at_infinity_pos *) moreover have "closed ((λz. (∑i≤n. c i * z ^ i)) -` K)" using ‹compact K› compact_eq_bounded_closed by (intro allI continuous_closed_vimage continuous_intros; force) ultimately show ?thesis using closed_Int_compact [OF ‹closed S›] compact_eq_bounded_closed by (auto simp add: vimage_def) qed lemma proper_map_polyfun_univ: fixes c :: "nat ⇒ 'a::{real_normed_div_algebra,heine_borel}" assumes "compact K" "c i ≠ 0" "1 ≤ i" "i ≤ n" shows "compact ({z. (∑i≤n. c i * z^i) ∈ K})" using proper_map_polyfun [of UNIV K c i n] assms by simp lemma proper_map_polyfun_eq: assumes "f holomorphic_on UNIV" shows "(∀k. compact k ⟶ compact {z. f z ∈ k}) ⟷ (∃c n. 0 < n ∧ (c n ≠ 0) ∧ f = (λz. ∑i≤n. c i * z^i))" (is "?lhs = ?rhs") proof assume compf [rule_format]: ?lhs have 2: "∃k. 0 < k ∧ a k ≠ 0 ∧ f = (λz. ∑i ≤ k. a i * z ^ i)" if "⋀z. f z = (∑i≤n. a i * z ^ i)" for a n proof (cases "∀i≤n. 0<i ⟶ a i = 0") case True then have [simp]: "⋀z. f z = a 0" by (simp add: that sum.atMost_shift) have False using compf [of "{a 0}"] by simp then show ?thesis .. next case False then obtain k where k: "0 < k" "k≤n" "a k ≠ 0" by force define m where "m = (GREATEST k. k≤n ∧ a k ≠ 0)" have m: "m≤n ∧ a m ≠ 0" unfolding m_def using GreatestI_nat [where b = n] k by (metis (mono_tags, lifting)) have [simp]: "a i = 0" if "m < i" "i ≤ n" for i using Greatest_le_nat [where b = "n" and P = "λk. k≤n ∧ a k ≠ 0"] using m_def not_le that by auto have "k ≤ m" unfolding m_def using Greatest_le_nat [where b = n] k by (metis (mono_tags, lifting)) with k m show ?thesis by (rule_tac x=m in exI) (auto simp: that comm_monoid_add_class.sum.mono_neutral_right) qed have §: "((inverse ∘ f) ⤏ 0) at_infinity" proof (rule Lim_at_infinityI) fix e::real assume "0 < e" with compf [of "cball 0 (inverse e)"] show "∃B. ∀x. B ≤ cmod x ⟶ dist ((inverse ∘ f) x) 0 ≤ e" apply simp apply (clarsimp simp add: compact_eq_bounded_closed bounded_pos norm_inverse) by (metis (no_types, opaque_lifting) inverse_inverse_eq le_less_trans less_eq_real_def less_imp_inverse_less linordered_field_no_ub not_less) qed then obtain a n where "⋀z. f z = (∑i≤n. a i * z^i)" using assms pole_at_infinity by blast with § 2 show ?rhs by blast next assume ?rhs then obtain c n where "0 < n" "c n ≠ 0" "f = (λz. ∑i≤n. c i * z ^ i)" by blast then have "compact {z. f z ∈ k}" if "compact k" for k by (auto intro: proper_map_polyfun_univ [OF that]) then show ?lhs by blast qed subsection ‹Relating invertibility and nonvanishing of derivative› lemma has_complex_derivative_locally_injective: assumes holf: "f holomorphic_on S" and S: "ξ ∈ S" "open S" and dnz: "deriv f ξ ≠ 0" obtains r where "r > 0" "ball ξ r ⊆ S" "inj_on f (ball ξ r)" proof - have *: "∃d>0. ∀x. dist ξ x < d ⟶ onorm (λv. deriv f x * v - deriv f ξ * v) < e" if "e > 0" for e proof - have contdf: "continuous_on S (deriv f)" by (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on ‹open S›) obtain δ where "δ>0" and δ: "⋀x. ⟦x ∈ S; dist x ξ ≤ δ⟧ ⟹ cmod (deriv f x - deriv f ξ) ≤ e/2" using continuous_onE [OF contdf ‹ξ ∈ S›, of "e/2"] ‹0 < e› by (metis dist_complex_def half_gt_zero less_imp_le) have §: "⋀ζ z. ⟦ζ ∈ S; dist ξ ζ < δ⟧ ⟹ cmod (deriv f ζ - deriv f ξ) * cmod z ≤ e/2 * cmod z" by (intro mult_right_mono [OF δ]) (auto simp: dist_commute) obtain ε where "ε>0" "ball ξ ε ⊆ S" by (metis openE [OF ‹open S› ‹ξ ∈ S›]) with ‹δ>0› have "∃δ>0. ∀x. dist ξ x < δ ⟶ onorm (λv. deriv f x * v - deriv f ξ * v) ≤ e/2" using § apply (rule_tac x="min δ ε" in exI) apply (intro conjI allI impI Operator_Norm.onorm_le) apply (force simp: mult_right_mono norm_mult [symmetric] left_diff_distrib δ)+ done with ‹e>0› show ?thesis by force qed have "inj ((*) (deriv f ξ))" using dnz by simp then obtain g' where g': "linear g'" "g' ∘ (*) (deriv f ξ) = id" using linear_injective_left_inverse [of "(*) (deriv f ξ)"] by auto show ?thesis apply (rule has_derivative_locally_injective [OF S, where f=f and f' = "λz h. deriv f z * h" and g' = g']) using g' * apply (simp_all add: linear_conv_bounded_linear that) using ‹open S› has_field_derivative_imp_has_derivative holf holomorphic_derivI by blast qed lemma has_complex_derivative_locally_invertible: assumes holf: "f holomorphic_on S" and S: "ξ ∈ S" "open S" and dnz: "deriv f ξ ≠ 0" obtains r where "r > 0" "ball ξ r ⊆ S" "open (f ` (ball ξ r))" "inj_on f (ball ξ r)" proof - obtain r where "r > 0" "ball ξ r ⊆ S" "inj_on f (ball ξ r)" by (blast intro: that has_complex_derivative_locally_injective [OF assms]) then have ξ: "ξ ∈ ball ξ r" by simp then have nc: "¬ f constant_on ball ξ r" using ‹inj_on f (ball ξ r)› injective_not_constant by fastforce have holf': "f holomorphic_on ball ξ r" using ‹ball ξ r ⊆ S› holf holomorphic_on_subset by blast have "open (f ` ball ξ r)" by (simp add: ‹inj_on f (ball ξ r)› holf' open_mapping_thm3) then show ?thesis using ‹0 < r› ‹ball ξ r ⊆ S› ‹inj_on f (ball ξ r)› that by blast qed lemma holomorphic_injective_imp_regular: assumes holf: "f holomorphic_on S" and "open S" and injf: "inj_on f S" and "ξ ∈ S" shows "deriv f ξ ≠ 0" proof - obtain r where "r>0" and r: "ball ξ r ⊆ S" using assms by (blast elim!: openE) have holf': "f holomorphic_on ball ξ r" using ‹ball ξ r ⊆ S› holf holomorphic_on_subset by blast show ?thesis proof (cases "∀n>0. (deriv ^^ n) f ξ = 0") case True have fcon: "f w = f ξ" if "w ∈ ball ξ r" for w by (meson open_ball True ‹0 < r› centre_in_ball connected_ball holf' holomorphic_fun_eq_const_on_connected that) have False using fcon [of "ξ + r/2"] ‹0 < r› r injf unfolding inj_on_def by (metis ‹ξ ∈ S› contra_subsetD dist_commute fcon mem_ball perfect_choose_dist) then show ?thesis .. next case False then obtain n0 where n0: "n0 > 0 ∧ (deriv ^^ n0) f ξ ≠ 0" by blast define n where [abs_def]: "n = (LEAST n. n > 0 ∧ (deriv ^^ n) f ξ ≠ 0)" have n_ne: "n > 0" "(deriv ^^ n) f ξ ≠ 0" using def_LeastI [OF n_def n0] by auto have n_min: "⋀k. 0 < k ⟹ k < n ⟹ (deriv ^^ k) f ξ = 0" using def_Least_le [OF n_def] not_le by auto obtain g δ where "0 < δ" and holg: "g holomorphic_on ball ξ δ" and fd: "⋀w. w ∈ ball ξ δ ⟹ f w - f ξ = ((w - ξ) * g w) ^ n" and gnz: "⋀w. w ∈ ball ξ δ ⟹ g w ≠ 0" by (blast intro: n_min holomorphic_factor_order_of_zero_strong [OF holf ‹open S› ‹ξ ∈ S› n_ne]) show ?thesis proof (cases "n=1") case True with n_ne show ?thesis by auto next case False have "g holomorphic_on ball ξ (min r δ)" using holg by (simp add: holomorphic_on_subset subset_ball) then have holgw: "(λw. (w - ξ) * g w) holomorphic_on ball ξ (min r δ)" by (intro holomorphic_intros) have gd: "⋀w. dist ξ w < δ ⟹ (g has_field_derivative deriv g w) (at w)" using holg by (simp add: DERIV_deriv_iff_field_differentiable holomorphic_on_def at_within_open_NO_MATCH) have *: "⋀w. w ∈ ball ξ (min r δ) ⟹ ((λw. (w - ξ) * g w) has_field_derivative ((w - ξ) * deriv g w + g w)) (at w)" by (rule gd derivative_eq_intros | simp)+ have [simp]: "deriv (λw. (w - ξ) * g w) ξ ≠ 0" using * [of ξ] ‹0 < δ› ‹0 < r› by (simp add: DERIV_imp_deriv gnz) obtain T where "ξ ∈ T" "open T" and Tsb: "T ⊆ ball ξ (min r δ)" and oimT: "open ((λw. (w - ξ) * g w) ` T)" using ‹0 < r› ‹0 < δ› has_complex_derivative_locally_invertible [OF holgw, of ξ] by force define U where "U = (λw. (w - ξ) * g w) ` T" have "open U" by (metis oimT U_def) moreover have "0 ∈ U" using ‹ξ ∈ T› by (auto simp: U_def intro: image_eqI [where x = ξ]) ultimately obtain ε where "ε>0" and ε: "cball 0 ε ⊆ U" using ‹open U› open_contains_cball by blast then have "ε * exp(2 * of_real pi * 𝗂 * (0/n)) ∈ cball 0 ε" "ε * exp(2 * of_real pi * 𝗂 * (1/n)) ∈ cball 0 ε" by (auto simp: norm_mult) with ε have "ε * exp(2 * of_real pi * 𝗂 * (0/n)) ∈ U" "ε * exp(2 * of_real pi * 𝗂 * (1/n)) ∈ U" by blast+ then obtain y0 y1 where "y0 ∈ T" and y0: "(y0 - ξ) * g y0 = ε * exp(2 * of_real pi * 𝗂 * (0/n))" and "y1 ∈ T" and y1: "(y1 - ξ) * g y1 = ε * exp(2 * of_real pi * 𝗂 * (1/n))" by (auto simp: U_def) then have "y0 ∈ ball ξ δ" "y1 ∈ ball ξ δ" using Tsb by auto then have "f y0 - f ξ = ((y0 - ξ) * g y0) ^ n" "f y1 - f ξ = ((y1 - ξ) * g y1) ^ n" using fd by blast+ moreover have "y0 ≠ y1" using y0 y1 ‹ε > 0› complex_root_unity_eq_1 [of n 1] ‹n > 0› False by auto moreover have "T ⊆ S" by (meson Tsb min.cobounded1 order_trans r subset_ball) ultimately have False using inj_onD [OF injf, of y0 y1] ‹y0 ∈ T› ‹y1 ∈ T› using complex_root_unity [of n 1] apply (simp add: y0 y1 power_mult_distrib) apply (force simp: algebra_simps) done then show ?thesis .. qed qed qed subsubsection ‹Hence a nice clean inverse function theorem› lemma has_field_derivative_inverse_strong: fixes f :: "'a::{euclidean_space,real_normed_field} ⇒ 'a" shows "⟦DERIV f x :> f'; f' ≠ 0; open S; x ∈ S; continuous_on S f; ⋀z. z ∈ S ⟹ g (f z) = z⟧ ⟹ DERIV g (f x) :> inverse (f')" unfolding has_field_derivative_def by (rule has_derivative_inverse_strong [of S x f g]) auto lemma has_field_derivative_inverse_strong_x: fixes f :: "'a::{euclidean_space,real_normed_field} ⇒ 'a" shows "⟦DERIV f (g y) :> f'; f' ≠ 0; open S; continuous_on S f; g y ∈ S; f(g y) = y; ⋀z. z ∈ S ⟹ g (f z) = z⟧ ⟹ DERIV g y :> inverse (f')" unfolding has_field_derivative_def by (rule has_derivative_inverse_strong_x [of S g y f]) auto proposition holomorphic_has_inverse: assumes holf: "f holomorphic_on S" and "open S" and injf: "inj_on f S" obtains g where "g holomorphic_on (f ` S)" "⋀z. z ∈ S ⟹ deriv f z * deriv g (f z) = 1" "⋀z. z ∈ S ⟹ g(f z) = z" proof - have ofs: "open (f ` S)" by (rule open_mapping_thm3 [OF assms]) have contf: "continuous_on S f" by (simp add: holf holomorphic_on_imp_continuous_on) have *: "(the_inv_into S f has_field_derivative inverse (deriv f z)) (at (f z))" if "z ∈ S" for z proof - have 1: "(f has_field_derivative deriv f z) (at z)" using DERIV_deriv_iff_field_differentiable ‹z ∈ S› ‹open S› holf holomorphic_on_imp_differentiable_at by blast have 2: "deriv f z ≠ 0" using ‹z ∈ S› ‹open S› holf holomorphic_injective_imp_regular injf by blast show ?thesis proof (rule has_field_derivative_inverse_strong [OF 1 2 ‹open S› ‹z ∈ S›]) show "continuous_on S f" by (simp add: holf holomorphic_on_imp_continuous_on) show "⋀z. z ∈ S ⟹ the_inv_into S f (f z) = z" by (simp add: injf the_inv_into_f_f) qed qed show ?thesis proof show "the_inv_into S f holomorphic_on f ` S" by (simp add: holomorphic_on_open ofs) (blast intro: *) next fix z assume "z ∈ S" have "deriv f z ≠ 0" using ‹z ∈ S› ‹open S› holf holomorphic_injective_imp_regular injf by blast then show "deriv f z * deriv (the_inv_into S f) (f z) = 1" using * [OF ‹z ∈ S›] by (simp add: DERIV_imp_deriv) next fix z assume "z ∈ S" show "the_inv_into S f (f z) = z" by (simp add: ‹z ∈ S› injf the_inv_into_f_f) qed qed subsubsection ‹ Holomorphism of covering maps and lifts.› lemma covering_space_lift_is_holomorphic: assumes cov: "covering_space C p S" and C: "open C" "p holomorphic_on C" and holf: "f holomorphic_on U" and fim: "f ∈ U → S" and gim: "g ∈ U → C" and contg: "continuous_on U g" and pg_f: "⋀x. x ∈ U ⟹ p(g x) = f x" shows "g holomorphic_on U" unfolding holomorphic_on_def proof (intro strip) fix z assume "z ∈ U" with fim have "f z ∈ S" by blast then obtain T 𝒱 where "f z ∈ T" and opeT: "openin (top_of_set S) T" and UV: "⋃𝒱 = C ∩ p -` T" and <