section \<open>Conformal Mappings and Consequences of Cauchy's Integral Theorem\<close>
text\<open>By John Harrison et al. Ported from HOL Light by L C Paulson (2016)\<close>
text\<open>Also Cauchy's residue theorem by Wenda Li (2016)\<close>
theory Conformal_Mappings
imports Cauchy_Integral_Formula
begin
subsection \<open>Analytic continuation\<close>
proposition isolated_zeros:
assumes holf: "f holomorphic_on S"
and "open S" "connected S" "\ \ S" "f \ = 0" "\ \ S" "f \ \ 0"
obtains r where "0 < r" and "ball \ r \ S" and
"\z. z \ ball \ r - {\} \ f z \ 0"
proof -
obtain r where "0 < r" and r: "ball \ r \ S"
using \<open>open S\<close> \<open>\<xi> \<in> S\<close> open_contains_ball_eq by blast
have powf: "((\n. (deriv ^^ n) f \ / (fact n) * (z - \)^n) sums f z)" if "z \ ball \ r" for z
by (intro holomorphic_power_series [OF _ that] holomorphic_on_subset [OF holf r])
obtain m where m: "(deriv ^^ m) f \ / (fact m) \ 0"
using holomorphic_fun_eq_0_on_connected [OF holf \<open>open S\<close> \<open>connected S\<close> _ \<open>\<xi> \<in> S\<close> \<open>\<beta> \<in> S\<close>] \<open>f \<beta> \<noteq> 0\<close>
by auto
then have "m \ 0" using assms(5) funpow_0 by fastforce
obtain s where "0 < s" and s: "\z. z \ cball \ s - {\} \ f z \ 0"
using powser_0_nonzero [OF \<open>0 < r\<close> powf \<open>f \<xi> = 0\<close> m]
by (metis \<open>m \<noteq> 0\<close> dist_norm mem_ball norm_minus_commute not_gr_zero)
have "0 < min r s" by (simp add: \<open>0 < r\<close> \<open>0 < s\<close>)
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 "\ \ S"
and "\ 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 \ e \ S"
using \<open>open S\<close> \<open>\<xi> \<in> S\<close> open_contains_cball_eq by blast
define T where "T = cball \ 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' \ \ \ dist x' \ < e; 0 < r\ \ \x'\cball \ e \ U. x' \ \ \ dist x' \ < r"
by (metis \<open>0 < e\<close> IntI dist_commute less_eq_real_def mem_cball min_less_iff_conj)
then have "\ islimpt T" using \\ islimpt U\
by (auto simp: T_def islimpt_approachable)
then have "\ \ closure T"
by (simp add: closure_def)
then have "f \ = 0"
by (auto simp: continuous_constant_on_closure [OF contf])
moreover have "\r. \0 < r; \z. z \ ball \ r - {\} \ f z \ 0\ \ False"
by (metis open_ball \<open>\<xi> islimpt T\<close> centre_in_ball fT0 insertE insert_Diff islimptE)
ultimately show ?thesis
by (metis \<open>open S\<close> \<open>connected S\<close> \<open>\<xi> \<in> S\<close> \<open>w \<in> S\<close> 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 \<open>s \<noteq> {}\<close> obtain \<xi> where "\<xi> \<in> s" by auto
with \<open>open s\<close> have \<xi>: "\<xi> islimpt s"
by (intro interior_limit_point) (auto simp: interior_open)
have "f z - g z = 0"
by (rule analytic_continuation[of "\z. f z - g z" s' s \])
(insert assms \<open>\<xi> \<in> s\<close> \<xi>, auto intro: holomorphic_intros)
thus ?thesis by simp
qed
subsection\<open>Open mapping theorem\<close>
lemma holomorphic_contract_to_zero:
assumes contf: "continuous_on (cball \ r) f"
and holf: "f holomorphic_on ball \ r"
and "0 < r"
and norm_less: "\z. norm(\ - z) = r \ norm(f \) < norm(f z)"
obtains z where "z \ ball \ r" "f z = 0"
proof -
{ assume fnz: "\w. w \ ball \ r \ f w \ 0"
then have "0 < norm (f \)"
by (simp add: \<open>0 < r\<close>)
have fnz': "\w. w \ cball \ 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 \ r) \ {}"
using \<open>0 < r\<close> by simp
define g where [abs_def]: "g z = inverse (f z)" for z
have contg: "continuous_on (cball \ r) g"
unfolding g_def using contf continuous_on_inverse fnz' by blast
have holg: "g holomorphic_on ball \ r"
unfolding g_def using fnz holf holomorphic_on_inverse by blast
have "frontier (cball \ r) \ cball \ r"
by (simp add: subset_iff)
then have contf': "continuous_on (frontier (cball \ r)) f"
and contg': "continuous_on (frontier (cball \ r)) g"
by (blast intro: contf contg continuous_on_subset)+
have froc: "frontier(cball \ r) \ {}"
using \<open>0 < r\<close> by simp
moreover have "continuous_on (frontier (cball \ r)) (norm o f)"
using contf' continuous_on_compose continuous_on_norm_id by blast
ultimately obtain w where w: "w \ frontier(cball \ r)"
and now: "\x. x \ frontier(cball \ 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 \ r)) (norm o g)"
using contg' continuous_on_compose continuous_on_norm_id by blast
then obtain v where v: "v \ frontier(cball \ r)"
and nov: "\x. x \ frontier(cball \ 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 \) \ fact 0 * norm (g v) / r ^ 0"
by (rule Cauchy_inequality [OF holg contg \<open>0 < r\<close>]) (simp add: dist_norm nov)
then have "cmod (g \) \ cmod (g v)"
by simp
moreover have "cmod (\ - w) = r"
by (metis (no_types) dist_norm frontier_cball mem_sphere w)
ultimately obtain wr: "norm (\ - w) = r" and nfw: "norm (f w) \ norm (f \)"
unfolding g_def
by (metis (no_types) \<open>0 < cmod (f \<xi>)\<close> 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 \<xi> assume \<xi>: "\<xi> \<in> U"
show "\e>0. ball (f \) e \ f ` U"
proof -
have hol: "(\z. f z - f \) holomorphic_on U"
by (rule holomorphic_intros that)+
obtain s where "0 < s" and sbU: "ball \ s \ U"
and sne: "\z. z \ ball \ s - {\} \ (\z. f z - f \) z \ 0"
using isolated_zeros [OF hol U \<xi>] by (metis fneU right_minus_eq)
obtain r where "0 < r" and r: "cball \ r \ ball \ s"
using \<open>0 < s\<close> by (rule_tac r="s/2" in that) auto
have "cball \ r \ U"
using sbU r by blast
then have frsbU: "frontier (cball \ r) \ U"
using Diff_subset frontier_def order_trans by fastforce
then have cof: "compact (frontier(cball \ r))"
by blast
have frne: "frontier (cball \ r) \ {}"
using \<open>0 < r\<close> by auto
have contfr: "continuous_on (frontier (cball \ r)) (\z. norm (f z - f \))"
by (metis continuous_on_norm continuous_on_subset frsbU hol holomorphic_on_imp_continuous_on)
obtain w where "norm (\ - w) = r"
and w: "(\z. norm (\ - z) = r \ norm (f w - f \) \ norm(f z - f \))"
using continuous_attains_inf [OF cof frne contfr] by (auto simp: dist_norm)
moreover define \<epsilon> where "\<epsilon> \<equiv> norm (f w - f \<xi>) / 3"
ultimately have "0 < \"
using \<open>0 < r\<close> dist_complex_def r sne by auto
have "ball (f \) \ \ f ` U"
proof
fix \<gamma>
assume \<gamma>: "\<gamma> \<in> ball (f \<xi>) \<epsilon>"
have *: "cmod (\ - f \) < cmod (\ - f z)" if "cmod (\ - z) = r" for z
proof -
have lt: "cmod (f w - f \) / 3 < cmod (\ - f z)"
using w [OF that] \<gamma>
using dist_triangle2 [of "f \" "\" "f z"] dist_triangle2 [of "f \" "f z" \]
by (simp add: \<epsilon>_def dist_norm norm_minus_commute)
show ?thesis
by (metis \<epsilon>_def dist_commute dist_norm less_trans lt mem_ball \<gamma>)
qed
have "continuous_on (cball \ r) (\z. \ - f z)"
using \<open>cball \<xi> r \<subseteq> U\<close> \<open>f holomorphic_on U\<close>
by (force intro: continuous_intros continuous_on_subset holomorphic_on_imp_continuous_on)
moreover have "(\z. \ - f z) holomorphic_on ball \ r"
using \<open>cball \<xi> r \<subseteq> U\<close> ball_subset_cball holomorphic_on_subset that(4)
by (intro holomorphic_intros) blast
ultimately obtain z where "z \ ball \ r" "\ - f z = 0"
using "*" \<open>0 < r\<close> holomorphic_contract_to_zero by blast
then show "\ \ f ` U"
using \<open>cball \<xi> r \<subseteq> U\<close> by fastforce
qed
then show ?thesis using \<open>0 < \<epsilon>\<close> by blast
qed
qed
have "open (f ` X)" if "X \ components U" for X
proof -
have holfU: "f holomorphic_on U"
using \<open>U \<subseteq> S\<close> holf holomorphic_on_subset by blast
have "X \ {}"
using that by (simp add: in_components_nonempty)
moreover have "open X"
using that \<open>open U\<close> 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 \<open>U \<subseteq> S\<close> in_components_subset that by blast
obtain w where w: "w \ X" using \X \ {}\ by blast
have wis: "w islimpt X"
using w \<open>open X\<close> interior_eq by auto
have hol: "(\z. f z - x) holomorphic_on S"
by (simp add: holf holomorphic_on_diff)
with fne [unfolded constant_on_def]
analytic_continuation[OF hol S \<open>connected S\<close> \<open>X \<subseteq> S\<close> _ wis] not \<open>X \<subseteq> S\<close> 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\<open>No need for \<^term>\<open>S\<close> to be connected. But the nonconstant condition is stronger.\<close>
corollary\<^marker>\<open>tag unimportant\<close> 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 = \(components S)" by simp
with \<open>U \<subseteq> S\<close> have "U = (\<Union>C \<in> components S. C \<inter> U)" by auto
then have "f ` U = (\C \ components S. f ` (C \ U))"
using image_UN by fastforce
moreover
{ fix C assume "C \ components S"
with S \<open>C \<in> components S\<close> open_components in_components_connected
have C: "open C" "connected C" by auto
have "C \ S"
by (metis \<open>C \<in> components S\<close> in_components_maximal)
have nf: "\ f constant_on C"
using \<open>open C\<close> \<open>C \<in> components S\<close> \<open>C \<subseteq> S\<close> fnc in_components_nonempty by blast
have "f holomorphic_on C"
by (metis holf holomorphic_on_subset \<open>C \<subseteq> S\<close>)
then have "open (f ` (C \ U))"
by (meson C \<open>open U\<close> inf_le1 nf open_Int open_mapping_thm)
} ultimately show ?thesis
by force
qed
corollary\<^marker>\<open>tag unimportant\<close> open_mapping_thm3:
assumes holf: "f holomorphic_on S"
and "open S" and injf: "inj_on f S"
shows "open (f ` S)"
proof (rule open_mapping_thm2 [OF holf])
show "\X. \open X; X \ S; X \ {}\ \ \ f constant_on X"
using inj_on_subset injective_not_constant injf by blast
qed (use assms in auto)
subsection\<open>Maximum modulus principle\<close>
text\<open>If \<^term>\<open>f\<close> is holomorphic, then its norm (modulus) cannot exhibit a true local maximum that is
properly within the domain of \<^term>\<open>f\<close>.\<close>
proposition maximum_modulus_principle:
assumes holf: "f holomorphic_on S"
and S: "open S" and "connected S"
and "open U" and "U \ S" and "\ \ U"
and no: "\z. z \ U \ norm(f z) \ norm(f \)"
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 \ - t) < e \ t \ f ` U" if "0 < e" for e
using that
apply (rule_tac x="if 0 < Re(f \) then f \ + (e/2) else f \ - (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 \<open>\<xi> \<in> U\<close> 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 "\ \ S"
shows "norm(f \) \ 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"] \<open>\<xi> \<in> S\<close> 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 \<open>\<xi> \<in> S\<close> closure_subset by fastforce
qed
corollary\<^marker>\<open>tag unimportant\<close> 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 "\ \ S"
shows "Re(f \) \ 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\<^marker>\<open>tag unimportant\<close> \<open>Factoring out a zero according to its order\<close>
lemma holomorphic_factor_order_of_zero:
assumes holf: "f holomorphic_on S"
and os: "open S"
and "\ \ S" "0 < n"
and dnz: "(deriv ^^ n) f \ \ 0"
and dfz: "\i. \0 < i; i < n\ \ (deriv ^^ i) f \ = 0"
obtains g r where "0 < r"
"g holomorphic_on ball \ r"
"\w. w \ ball \ r \ f w - f \ = (w - \)^n * g w"
"\w. w \ ball \ r \ g w \ 0"
proof -
obtain r where "r>0" and r: "ball \ r \ S" using assms by (blast elim!: openE)
then have holfb: "f holomorphic_on ball \ r"
using holf holomorphic_on_subset by blast
define g where "g w = suminf (\i. (deriv ^^ (i + n)) f \ / (fact(i + n)) * (w - \)^i)" for w
have sumsg: "(\i. (deriv ^^ (i + n)) f \ / (fact(i + n)) * (w - \)^i) sums g w"
and feq: "f w - f \ = (w - \)^n * g w"
if w: "w \ ball \ r" for w
proof -
define powf where "powf = (\i. (deriv ^^ i) f \/(fact i) * (w - \)^i)"
have [simp]: "powf 0 = f \"
by (simp add: powf_def)
have sing: "{.. = 0 then {} else {0})"
unfolding powf_def using \<open>0 < n\<close> 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"
by (subst sum.setdiff_irrelevant [symmetric]; simp add: dfz sing)
ultimately have fsums: "(\i. powf (i+n)) sums (f w - f \)"
using w sums_iff_shift' by metis
then have *: "summable (\i. (w - \) ^ n * ((deriv ^^ (i + n)) f \ * (w - \) ^ i / fact (i + n)))"
unfolding powf_def using sums_summable
by (auto simp: power_add mult_ac)
have "summable (\i. (deriv ^^ (i + n)) f \ * (w - \) ^ i / fact (i + n))"
proof (cases "w=\")
case False then show ?thesis
using summable_mult [OF *, of "1 / (w - \) ^ 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: "(\i. (deriv ^^ (i + n)) f \ / (fact(i + n)) * (w - \)^i) sums g w"
by (simp add: summable_sums_iff g_def)
show "f w - f \ = (w - \)^n * g w"
using sums_mult [OF sumsg, of "(w - \) ^ n"]
by (intro sums_unique2 [OF fsums]) (auto simp: power_add mult_ac powf_def)
qed
then have holg: "g holomorphic_on ball \ r"
by (meson sumsg power_series_holomorphic)
then have contg: "continuous_on (ball \ r) g"
by (blast intro: holomorphic_on_imp_continuous_on)
have "g \ \ 0"
using dnz unfolding g_def
by (subst suminf_finite [of "{0}"]) auto
obtain d where "0 < d" and d: "\w. w \ ball \ d \ g w \ 0"
using \<open>0 < r\<close> continuous_on_avoid [OF contg _ \<open>g \<xi> \<noteq> 0\<close>]
by (metis centre_in_ball le_cases mem_ball mem_ball_leI)
show ?thesis
proof
show "g holomorphic_on ball \ (min r d)"
using holg by (auto simp: feq holomorphic_on_subset subset_ball d)
qed (use \<open>0 < r\<close> \<open>0 < d\<close> in \<open>auto simp: feq d\<close>)
qed
lemma holomorphic_factor_order_of_zero_strong:
assumes holf: "f holomorphic_on S" "open S" "\ \ S" "0 < n"
and "(deriv ^^ n) f \ \ 0"
and "\i. \0 < i; i < n\ \ (deriv ^^ i) f \ = 0"
obtains g r where "0 < r"
"g holomorphic_on ball \ r"
"\w. w \ ball \ r \ f w - f \ = ((w - \) * g w) ^ n"
"\w. w \ ball \ r \ g w \ 0"
proof -
obtain g r where "0 < r"
and holg: "g holomorphic_on ball \ r"
and feq: "\w. w \ ball \ r \ f w - f \ = (w - \)^n * g w"
and gne: "\w. w \ ball \ r \ g w \ 0"
by (auto intro: holomorphic_factor_order_of_zero [OF assms])
have con: "continuous_on (ball \ r) (\z. deriv g z / g z)"
by (rule continuous_intros) (auto simp: gne holg holomorphic_deriv holomorphic_on_imp_continuous_on)
have cd: "(\z. deriv g z / g z) field_differentiable at x" if "dist \ 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 \ r \ (h has_field_derivative deriv g x / g x) (at x)"
using holomorphic_convex_primitive [of "ball \ r" "{}" "\z. 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 \ r) h"
by (metis open_ball holomorphic_on_imp_continuous_on holomorphic_on_open)
then have con: "continuous_on (ball \ r) (\x. exp (h x) / g x)"
by (auto intro!: continuous_intros simp add: holg holomorphic_on_imp_continuous_on gne)
have 0: "dist \ x < r \ ((\x. 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 \ r \ exp (h x) / g x = c"
by (rule DERIV_zero_connected_constant [of "ball \ r" "{}" "\x. exp(h x) / g x"]) (auto simp: con 0)
have hol: "(\z. exp ((Ln (inverse c) + h z) / of_nat n)) holomorphic_on ball \ r"
proof (intro holomorphic_intros holomorphic_on_compose [unfolded o_def, where g = exp])
show "h holomorphic_on ball \ r"
using h holomorphic_on_open by blast
qed (use \<open>0 < n\<close> in auto)
show ?thesis
proof
show "\w. w \ ball \ r \ f w - f \ = ((w - \) * exp ((Ln (inverse c) + h w) / of_nat n)) ^ n"
using \<open>0 < n\<close>
by (auto simp: feq power_mult_distrib exp_divide_power_eq exp_add gne simp flip: c)
qed (use hol \<open>0 < r\<close> 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 "\ \ S" "f \ = 0"
and nonconst: "\ f constant_on S"
obtains g r n
where "0 < n" "0 < r" "ball \ r \ S"
"g holomorphic_on ball \ r"
"\w. w \ ball \ r \ f w = (w - \)^n * g w"
"\w. w \ ball \ r \ g w \ 0"
proof (cases "\n>0. (deriv ^^ n) f \ = 0")
case True then show ?thesis
using holomorphic_fun_eq_const_on_connected [OF holf S _ \<open>\<xi> \<in> S\<close>] nonconst by (simp add: constant_on_def)
next
case False
then obtain n0 where "n0 > 0" and n0: "(deriv ^^ n0) f \ \ 0" by blast
obtain r0 where "r0 > 0" "ball \ r0 \ S" using S openE \\ \ S\ by auto
define n where "n \ LEAST n. (deriv ^^ n) f \ \ 0"
have n_ne: "(deriv ^^ n) f \ \ 0"
by (rule def_LeastI [OF n_def]) (rule n0)
then have "0 < n" using \<open>f \<xi> = 0\<close>
using funpow_0 by fastforce
have n_min: "\k. k < n \ (deriv ^^ k) f \ = 0"
using def_Least_le [OF n_def] not_le by blast
then obtain g r1
where g: "0 < r1" "g holomorphic_on ball \ r1"
and geq: "\w. w \ ball \ r1 \ f w = (w - \) ^ n * g w"
and g0: "\w. w \ ball \ r1 \ g w \ 0"
by (auto intro: holomorphic_factor_order_of_zero [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> \<open>n > 0\<close> n_ne] simp: \<open>f \<xi> = 0\<close>)
show ?thesis
proof
show "g holomorphic_on ball \ (min r0 r1)"
using g by auto
show "\w. w \ ball \ (min r0 r1) \ f w = (w - \) ^ n * g w"
by (simp add: geq)
qed (use \<open>0 < n\<close> \<open>0 < r0\<close> \<open>0 < r1\<close> \<open>ball \<xi> r0 \<subseteq> S\<close> g0 in auto)
qed
lemma holomorphic_lower_bound_difference:
assumes holf: "f holomorphic_on S" and S: "open S" "connected S"
and "\ \ S" and "\ \ S"
and fne: "f \ \ f \"
obtains k n r
where "0 < k" "0 < r"
"ball \ r \ S"
"\w. w \ ball \ r \ k * norm(w - \)^n \ norm(f w - f \)"
proof -
define n where "n = (LEAST n. 0 < n \ (deriv ^^ n) f \ \ 0)"
obtain n0 where "0 < n0" and n0: "(deriv ^^ n0) f \ \ 0"
using fne holomorphic_fun_eq_const_on_connected [OF holf S] \<open>\<xi> \<in> S\<close> \<open>\<phi> \<in> S\<close> by blast
then have "0 < n" and n_ne: "(deriv ^^ n) f \ \ 0"
unfolding n_def by (metis (mono_tags, lifting) LeastI)+
have n_min: "\k. \0 < k; k < n\ \ (deriv ^^ k) f \ = 0"
unfolding n_def by (blast dest: not_less_Least)
then obtain g r
where "0 < r" and holg: "g holomorphic_on ball \ r"
and fne: "\w. w \ ball \ r \ f w - f \ = (w - \) ^ n * g w"
and gnz: "\w. w \ ball \ r \ g w \ 0"
by (auto intro: holomorphic_factor_order_of_zero [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> \<open>n > 0\<close> n_ne])
obtain e where "e>0" and e: "ball \ e \ S" using assms by (blast elim!: openE)
then have holfb: "f holomorphic_on ball \ e"
using holf holomorphic_on_subset by blast
define d where "d = (min e r) / 2"
have "0 < d" using \<open>0 < r\<close> \<open>0 < e\<close> by (simp add: d_def)
have "d < r"
using \<open>0 < r\<close> by (auto simp: d_def)
then have cbb: "cball \ d \ ball \ r"
by (auto simp: cball_subset_ball_iff)
then have "g holomorphic_on cball \ d"
by (rule holomorphic_on_subset [OF holg])
then have "closed (g ` cball \ d)"
by (simp add: compact_imp_closed compact_continuous_image holomorphic_on_imp_continuous_on)
moreover have "g ` cball \ d \ {}"
using \<open>0 < d\<close> by auto
ultimately obtain x where x: "x \ g ` cball \ d" and "\y. y \ g ` cball \ d \ dist 0 x \ dist 0 y"
by (rule distance_attains_inf) blast
then have leg: "\w. w \ cball \ d \ norm x \ norm (g w)"
by auto
have "ball \ d \ cball \ d" by auto
also have "... \ ball \ e" using \0 < d\ d_def by auto
also have "... \ S" by (rule e)
finally have dS: "ball \ d \ S" .
have "x \ 0" using gnz x \d < r\ by auto
show thesis
proof
show "\w. w \ ball \ d \ cmod x * cmod (w - \) ^ n \ cmod (f w - f \)"
using \<open>d < r\<close> leg by (auto simp: fne norm_mult norm_power algebra_simps mult_right_mono)
qed (use dS \<open>x \<noteq> 0\<close> \<open>d > 0\<close> in auto)
qed
lemma
assumes holf: "f holomorphic_on (S - {\})" and \: "\ \ interior S"
shows holomorphic_on_extend_lim:
"(\g. g holomorphic_on S \ (\z \ S - {\}. g z = f z)) \
((\<lambda>z. (z - \<xi>) * f z) \<longlongrightarrow> 0) (at \<xi>)"
(is "?P = ?Q")
and holomorphic_on_extend_bounded:
"(\g. g holomorphic_on S \ (\z \ S - {\}. g z = f z)) \
(\<exists>B. eventually (\<lambda>z. norm(f z) \<le> B) (at \<xi>))"
(is "?P = ?R")
proof -
obtain \<delta> where "0 < \<delta>" and \<delta>: "ball \<xi> \<delta> \<subseteq> S"
using \<xi> mem_interior by blast
have "?R" if holg: "g holomorphic_on S" and gf: "\z. z \ S - {\} \ g z = f z" for g
proof -
have \<section>: "cmod (f x) \<le> cmod (g \<xi>) + 1" if "x \<noteq> \<xi>" "dist x \<xi> < \<delta>" "dist (g x) (g \<xi>) < 1" for x
proof -
have "x \ S"
by (metis \<delta> dist_commute mem_ball subsetD that(2))
with that gf [of x] show ?thesis
using norm_triangle_ineq2 [of "f x" "g \"] dist_complex_def by auto
qed
then have *: "\\<^sub>F z in at \. dist (g z) (g \) < 1 \ cmod (f z) \ cmod (g \) + 1"
using \<open>0 < \<delta>\<close> 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 \) (at \)"
by (simp add: \<xi>)
then show ?thesis
apply (rule_tac x="norm(g \) + 1" in exI)
apply (rule eventually_mp [OF * tendstoD [where e=1]], auto)
done
qed
moreover have "?Q" if "\\<^sub>F z in at \. cmod (f z) \ B" for B
by (rule lim_null_mult_right_bounded [OF _ that]) (simp add: LIM_zero)
moreover have "?P" if "(\z. (z - \) * f z) \\\ 0"
proof -
define h where [abs_def]: "h z = (z - \)^2 * f z" for z
have h0: "(h has_field_derivative 0) (at \)"
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 = \")
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] \<open>z \<in> S\<close>
unfolding field_differentiable_def has_field_derivative_iff
by (force intro: exI [where x="dist \ 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 = \ then deriv h \ else (h z - h \) / (z - \))" for z
have holg: "g holomorphic_on S"
unfolding g_def by (rule pole_lemma [OF holh \<xi>])
have \<section>: "\<forall>z\<in>S - {\<xi>}. (g z - g \<xi>) / (z - \<xi>) = 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 \<xi>])
apply (simp add: \<section>)
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 \<open>0 < r\<close> by simp
have "\B. 0 eventually (\z. 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 \<open>l = 0\<close> lim lim_at_infinity_0 by blast
have g2g0: "g \0\ g 0"
using \<open>0 < r\<close> 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 \<open>0 < r\<close> by (auto simp: geq)
have [simp]: "g 0 = 0"
by (rule tendsto_unique [OF _ g2g0 g2g1]) simp
have "ball 0 r - {0::complex} \ {}"
using \<open>0 < r\<close> 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: \<open>g w \<noteq> 0\<close>)
show "w \ ball 0 r"
using mem_ball_0 w by blast
qed (use \<open>0 < r\<close> in \<open>auto simp: ball_subset_ball_iff\<close>)
have \<section>: "cmod (f z) \<le> cmod z ^ n / B" if "2/e \<le> 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"] \<open>0 < B\<close> \<open>0 < e\<close>
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 \<section> 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 \<open>0 < r\<close> that by force
then show ?thesis
unfolding lim_at_infinity_0
using eventually_at \<open>r > 0\<close> by (force simp add: intro: tendsto_eventually)
qed
obtain w where "w \ ball 0 r - {0}" and "f (inverse w) = 0"
using False \<open>0 < r\<close> 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\<^marker>\<open>tag unimportant\<close> \<open>Entire proper functions are precisely the non-trivial polynomials\<close>
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 \<open>compact K\<close> 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 ((\z. (\i\n. c i * z ^ i)) -` K)"
using \<open>compact K\<close> compact_eq_bounded_closed
by (intro allI continuous_closed_vimage continuous_intros; force)
ultimately show ?thesis
using closed_Int_compact [OF \<open>closed S\<close>] 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}) \
(\<exists>c n. 0 < n \<and> (c n \<noteq> 0) \<and> f = (\<lambda>z. \<Sum>i\<le>n. c i * z^i))"
(is "?lhs = ?rhs")
proof
assume compf [rule_format]: ?lhs
have 2: "\k. 0 < k \ a k \ 0 \ f = (\z. \i \ k. a i * z ^ i)"
if "\z. f z = (\i\n. a i * z ^ i)" for a n
proof (cases "\i\n. 0 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 = "\k. 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 \<section>: "((inverse \<circ> f) \<longlongrightarrow> 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, hide_lams) 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 \<section> 2 show ?rhs by blast
next
assume ?rhs
then obtain c n where "0 < n" "c n \ 0" "f = (\z. \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 \<open>Relating invertibility and nonvanishing of derivative\<close>
lemma has_complex_derivative_locally_injective:
assumes holf: "f holomorphic_on S"
and S: "\ \ S" "open S"
and dnz: "deriv f \ \ 0"
obtains r where "r > 0" "ball \ r \ S" "inj_on f (ball \ r)"
proof -
have *: "\d>0. \x. dist \ x < d \ onorm (\v. deriv f x * v - deriv f \ * 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>open S\<close>)
obtain \<delta> where "\<delta>>0" and \<delta>: "\<And>x. \<lbrakk>x \<in> S; dist x \<xi> \<le> \<delta>\<rbrakk> \<Longrightarrow> cmod (deriv f x - deriv f \<xi>) \<le> e/2"
using continuous_onE [OF contdf \<open>\<xi> \<in> S\<close>, of "e/2"] \<open>0 < e\<close>
by (metis dist_complex_def half_gt_zero less_imp_le)
have \<section>: "\<And>\<zeta> z. \<lbrakk>\<zeta> \<in> S; dist \<xi> \<zeta> < \<delta>\<rbrakk> \<Longrightarrow> cmod (deriv f \<zeta> - deriv f \<xi>) * cmod z \<le> e/2 * cmod z"
by (intro mult_right_mono [OF \<delta>]) (auto simp: dist_commute)
obtain \<epsilon> where "\<epsilon>>0" "ball \<xi> \<epsilon> \<subseteq> S"
by (metis openE [OF \<open>open S\<close> \<open>\<xi> \<in> S\<close>])
with \<open>\<delta>>0\<close> have "\<exists>\<delta>>0. \<forall>x. dist \<xi> x < \<delta> \<longrightarrow> onorm (\<lambda>v. deriv f x * v - deriv f \<xi> * v) \<le> e/2"
using \<section>
apply (rule_tac x="min \ \" in exI)
apply (intro conjI allI impI Operator_Norm.onorm_le)
apply (force simp: mult_right_mono norm_mult [symmetric] left_diff_distrib \<delta>)+
done
with \<open>e>0\<close> show ?thesis by force
qed
have "inj ((*) (deriv f \))"
using dnz by simp
then obtain g' where g': "linear g'" "g' \ (*) (deriv f \) = id"
using linear_injective_left_inverse [of "(*) (deriv f \)"] by auto
show ?thesis
apply (rule has_derivative_locally_injective [OF S, where f=f and f' = "\z h. deriv f z * h" and g' = g'])
using g' *
apply (simp_all add: linear_conv_bounded_linear that)
using \<open>open S\<close> 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: "\ \ S" "open S"
and dnz: "deriv f \ \ 0"
obtains r where "r > 0" "ball \ r \ S" "open (f ` (ball \ r))" "inj_on f (ball \ r)"
proof -
obtain r where "r > 0" "ball \ r \ S" "inj_on f (ball \ r)"
by (blast intro: that has_complex_derivative_locally_injective [OF assms])
then have \<xi>: "\<xi> \<in> ball \<xi> r" by simp
then have nc: "\ f constant_on ball \ r"
using \<open>inj_on f (ball \<xi> r)\<close> injective_not_constant by fastforce
have holf': "f holomorphic_on ball \ r"
using \<open>ball \<xi> r \<subseteq> S\<close> holf holomorphic_on_subset by blast
have "open (f ` ball \ r)"
by (simp add: \<open>inj_on f (ball \<xi> r)\<close> holf' open_mapping_thm3)
then show ?thesis
using \<open>0 < r\<close> \<open>ball \<xi> r \<subseteq> S\<close> \<open>inj_on f (ball \<xi> r)\<close> 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 "\ \ S"
shows "deriv f \ \ 0"
proof -
obtain r where "r>0" and r: "ball \ r \ S" using assms by (blast elim!: openE)
have holf': "f holomorphic_on ball \ r"
using \<open>ball \<xi> r \<subseteq> S\<close> holf holomorphic_on_subset by blast
show ?thesis
proof (cases "\n>0. (deriv ^^ n) f \ = 0")
case True
have fcon: "f w = f \" if "w \ ball \ r" for w
by (meson open_ball True \<open>0 < r\<close> centre_in_ball connected_ball holf'
holomorphic_fun_eq_const_on_connected that)
have False
using fcon [of "\ + r/2"] \0 < r\ r injf unfolding inj_on_def
by (metis \<open>\<xi> \<in> S\<close> 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 \ \ 0" by blast
define n where [abs_def]: "n = (LEAST n. n > 0 \ (deriv ^^ n) f \ \ 0)"
have n_ne: "n > 0" "(deriv ^^ n) f \ \ 0"
using def_LeastI [OF n_def n0] by auto
have n_min: "\k. 0 < k \ k < n \ (deriv ^^ k) f \ = 0"
using def_Least_le [OF n_def] not_le by auto
obtain g \<delta> where "0 < \<delta>"
and holg: "g holomorphic_on ball \ \"
and fd: "\w. w \ ball \ \ \ f w - f \ = ((w - \) * g w) ^ n"
and gnz: "\w. w \ ball \ \ \ g w \ 0"
by (blast intro: n_min holomorphic_factor_order_of_zero_strong [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> 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 \ (min r \)"
using holg by (simp add: holomorphic_on_subset subset_ball)
then have holgw: "(\w. (w - \) * g w) holomorphic_on ball \ (min r \)"
by (intro holomorphic_intros)
have gd: "\w. dist \ w < \ \ (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 \ (min r \)
\<Longrightarrow> ((\<lambda>w. (w - \<xi>) * g w) has_field_derivative ((w - \<xi>) * deriv g w + g w))
(at w)"
by (rule gd derivative_eq_intros | simp)+
have [simp]: "deriv (\w. (w - \) * g w) \ \ 0"
using * [of \<xi>] \<open>0 < \<delta>\<close> \<open>0 < r\<close> by (simp add: DERIV_imp_deriv gnz)
obtain T where "\ \ T" "open T" and Tsb: "T \ ball \ (min r \)" and oimT: "open ((\w. (w - \) * g w) ` T)"
using \<open>0 < r\<close> \<open>0 < \<delta>\<close> has_complex_derivative_locally_invertible [OF holgw, of \<xi>]
by force
define U where "U = (\w. (w - \) * g w) ` T"
have "open U" by (metis oimT U_def)
moreover have "0 \ U"
using \<open>\<xi> \<in> T\<close> by (auto simp: U_def intro: image_eqI [where x = \<xi>])
ultimately obtain \<epsilon> where "\<epsilon>>0" and \<epsilon>: "cball 0 \<epsilon> \<subseteq> U"
using \<open>open U\<close> open_contains_cball by blast
then have "\ * exp(2 * of_real pi * \ * (0/n)) \ cball 0 \"
"\ * exp(2 * of_real pi * \ * (1/n)) \ cball 0 \"
by (auto simp: norm_mult)
with \<epsilon> have "\<epsilon> * exp(2 * of_real pi * \<i> * (0/n)) \<in> U"
"\ * exp(2 * of_real pi * \ * (1/n)) \ U" by blast+
then obtain y0 y1 where "y0 \ T" and y0: "(y0 - \) * g y0 = \ * exp(2 * of_real pi * \ * (0/n))"
and "y1 \ T" and y1: "(y1 - \) * g y1 = \ * exp(2 * of_real pi * \ * (1/n))"
by (auto simp: U_def)
then have "y0 \ ball \ \" "y1 \ ball \ \" using Tsb by auto
then have "f y0 - f \ = ((y0 - \) * g y0) ^ n" "f y1 - f \ = ((y1 - \) * g y1) ^ n"
using fd by blast+
moreover have "y0 \ y1"
using y0 y1 \<open>\<epsilon> > 0\<close> complex_root_unity_eq_1 [of n 1] \<open>n > 0\<close> 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] \<open>y0 \<in> T\<close> \<open>y1 \<in> T\<close>
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
text\<open>Hence a nice clean inverse function theorem\<close>
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;
\<And>z. z \<in> S \<Longrightarrow> g (f z) = z\<rbrakk>
\<Longrightarrow> 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;
\<And>z. z \<in> S \<Longrightarrow> g (f z) = z\<rbrakk>
\<Longrightarrow> 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 \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_on_imp_differentiable_at
by blast
have 2: "deriv f z \ 0"
using \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_injective_imp_regular injf by blast
show ?thesis
proof (rule has_field_derivative_inverse_strong [OF 1 2 \<open>open S\<close> \<open>z \<in> S\<close>])
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 \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_injective_imp_regular injf by blast
then show "deriv f z * deriv (the_inv_into S f) (f z) = 1"
using * [OF \<open>z \<in> S\<close>] by (simp add: DERIV_imp_deriv)
next
fix z assume "z \ S"
show "the_inv_into S f (f z) = z"
by (simp add: \<open>z \<in> S\<close> injf the_inv_into_f_f)
qed
qed
subsection\<open>The Schwarz Lemma\<close>
lemma Schwarz1:
assumes holf: "f holomorphic_on S"
and contf: "continuous_on (closure S) f"
and S: "open S" "connected S"
and boS: "bounded S"
and "S \ {}"
obtains w where "w \ frontier S"
"\z. z \ closure S \ norm (f z) \ norm (f w)"
proof -
have connf: "continuous_on (closure S) (norm o f)"
using contf continuous_on_compose continuous_on_norm_id by blast
have coc: "compact (closure S)"
by (simp add: \<open>bounded S\<close> bounded_closure compact_eq_bounded_closed)
then obtain x where x: "x \ closure S" and xmax: "\z. z \ closure S \ norm(f z) \ norm(f x)"
using continuous_attains_sup [OF _ _ connf] \<open>S \<noteq> {}\<close> by auto
then show ?thesis
proof (cases "x \ frontier S")
case True
then show ?thesis using that xmax by blast
next
case False
then have "x \ S"
using \<open>open S\<close> frontier_def interior_eq x by auto
then have "f constant_on S"
proof (rule maximum_modulus_principle [OF holf S \<open>open S\<close> order_refl])
show "\z. z \ S \ cmod (f z) \ cmod (f x)"
using closure_subset by (blast intro: xmax)
qed
then have "f constant_on (closure S)"
by (rule constant_on_closureI [OF _ contf])
then obtain c where c: "\x. x \ closure S \ f x = c"
by (meson constant_on_def)
obtain w where "w \ frontier S"
by (metis coc all_not_in_conv assms(6) closure_UNIV frontier_eq_empty not_compact_UNIV)
then show ?thesis
by (simp add: c frontier_def that)
qed
qed
lemma Schwarz2:
"\f holomorphic_on ball 0 r;
0 < s; ball w s \<subseteq> ball 0 r;
\<And>z. norm (w-z) < s \<Longrightarrow> norm(f z) \<le> norm(f w)\<rbrakk>
\<Longrightarrow> f constant_on ball 0 r"
by (rule maximum_modulus_principle [where U = "ball w s" and \<xi> = w]) (simp_all add: dist_norm)
lemma Schwarz3:
assumes holf: "f holomorphic_on (ball 0 r)" and [simp]: "f 0 = 0"
obtains h where "h holomorphic_on (ball 0 r)" and "\z. norm z < r \ f z = z * (h z)" and "deriv f 0 = h 0"
proof -
define h where "h z = (if z = 0 then deriv f 0 else f z / z)" for z
have d0: "deriv f 0 = h 0"
by (simp add: h_def)
moreover have "h holomorphic_on (ball 0 r)"
by (rule pole_theorem_open_0 [OF holf, of 0]) (auto simp: h_def)
moreover have "norm z < r \ f z = z * h z" for z
by (simp add: h_def)
ultimately show ?thesis
using that by blast
qed
proposition Schwarz_Lemma:
assumes holf: "f holomorphic_on (ball 0 1)" and [simp]: "f 0 = 0"
and no: "\z. norm z < 1 \ norm (f z) < 1"
and \<xi>: "norm \<xi> < 1"
shows "norm (f \) \ norm \" and "norm(deriv f 0) \ 1"
and "((\z. norm z < 1 \ z \ 0 \ norm(f z) = norm z)
\<or> norm(deriv f 0) = 1)
\<Longrightarrow> \<exists>\<alpha>. (\<forall>z. norm z < 1 \<longrightarrow> f z = \<alpha> * z) \<and> norm \<alpha> = 1"
(is "?P \ ?Q")
proof -
obtain h where holh: "h holomorphic_on (ball 0 1)"
and fz_eq: "\z. norm z < 1 \ f z = z * (h z)" and df0: "deriv f 0 = h 0"
by (rule Schwarz3 [OF holf]) auto
have noh_le: "norm (h z) \ 1" if z: "norm z < 1" for z
proof -
have "norm (h z) < a" if a: "1 < a" for a
proof -
have "max (inverse a) (norm z) < 1"
using z a by (simp_all add: inverse_less_1_iff)
then obtain r where r: "max (inverse a) (norm z) < r" and "r < 1"
using Rats_dense_in_real by blast
then have nzr: "norm z < r" and ira: "inverse r < a"
using z a less_imp_inverse_less by force+
then have "0 < r"
by (meson norm_not_less_zero not_le order.strict_trans2)
have holh': "h holomorphic_on ball 0 r"
by (meson holh \<open>r < 1\<close> holomorphic_on_subset less_eq_real_def subset_ball)
have conth': "continuous_on (cball 0 r) h"
by (meson \<open>r < 1\<close> dual_order.trans holh holomorphic_on_imp_continuous_on holomorphic_on_subset mem_ball_0 mem_cball_0 not_less subsetI)
obtain w where w: "norm w = r" and lenw: "\z. norm z < r \ norm(h z) \ norm(h w)"
apply (rule Schwarz1 [OF holh']) using conth' \<open>0 < r\<close> by auto
have "h w = f w / w" using fz_eq \<open>r < 1\<close> nzr w by auto
then have "cmod (h z) < inverse r"
by (metis \<open>0 < r\<close> \<open>r < 1\<close> divide_strict_right_mono inverse_eq_divide
le_less_trans lenw no norm_divide nzr w)
then show ?thesis using ira by linarith
qed
then show "norm (h z) \ 1"
using not_le by blast
qed
show "cmod (f \) \ cmod \"
proof (cases "\ = 0")
case True then show ?thesis by auto
next
case False
then show ?thesis
by (simp add: noh_le fz_eq \<xi> mult_left_le norm_mult)
qed
show no_df0: "norm(deriv f 0) \ 1"
by (simp add: \<open>\<And>z. cmod z < 1 \<Longrightarrow> cmod (h z) \<le> 1\<close> df0)
show "?Q" if "?P"
using that
proof
assume "\z. cmod z < 1 \ z \ 0 \ cmod (f z) = cmod z"
then obtain \<gamma> where \<gamma>: "cmod \<gamma> < 1" "\<gamma> \<noteq> 0" "cmod (f \<gamma>) = cmod \<gamma>" by blast
then have [simp]: "norm (h \) = 1"
by (simp add: fz_eq norm_mult)
have \<section>: "ball \<gamma> (1 - cmod \<gamma>) \<subseteq> ball 0 1"
by (simp add: ball_subset_ball_iff)
moreover have "\z. cmod (\ - z) < 1 - cmod \ \ cmod (h z) \ cmod (h \)"
by (metis \<open>cmod (h \<gamma>) = 1\<close> \<section> dist_0_norm dist_complex_def in_mono mem_ball noh_le)
ultimately obtain c where c: "\z. norm z < 1 \ h z = c"
using Schwarz2 [OF holh, of "1 - norm \" \, unfolded constant_on_def] \ by auto
then have "norm c = 1"
using \<gamma> by force
with c show ?thesis
using fz_eq by auto
next
assume [simp]: "cmod (deriv f 0) = 1"
then obtain c where c: "\z. norm z < 1 \ h z = c"
using Schwarz2 [OF holh zero_less_one, of 0, unfolded constant_on_def] df0 noh_le
by auto
moreover have "norm c = 1" using df0 c by auto
ultimately show ?thesis
using fz_eq by auto
qed
qed
corollary Schwarz_Lemma':
assumes holf: "f holomorphic_on (ball 0 1)" and [simp]: "f 0 = 0"
and no: "\z. norm z < 1 \ norm (f z) < 1"
shows "((\\. norm \ < 1 \ norm (f \) \ norm \)
\<and> norm(deriv f 0) \<le> 1)
\<and> (((\<exists>z. norm z < 1 \<and> z \<noteq> 0 \<and> norm(f z) = norm z)
\<or> norm(deriv f 0) = 1)
\<longrightarrow> (\<exists>\<alpha>. (\<forall>z. norm z < 1 \<longrightarrow> f z = \<alpha> * z) \<and> norm \<alpha> = 1))"
using Schwarz_Lemma [OF assms]
by (metis (no_types) norm_eq_zero zero_less_one)
subsection\<open>The Schwarz reflection principle\<close>
lemma hol_pal_lem0:
assumes "d \ a \ k" "k \ d \ b"
obtains c where
"c \ closed_segment a b" "d \ c = k"
"\z. z \ closed_segment a c \ d \ z \ k"
"\z. z \ closed_segment c b \ k \ d \ z"
proof -
obtain c where cin: "c \ closed_segment a b" and keq: "k = d \ c"
using connected_ivt_hyperplane [of "closed_segment a b" a b d k]
by (auto simp: assms)
have "closed_segment a c \ {z. d \ z \ k}" "closed_segment c b \ {z. k \ d \ z}"
unfolding segment_convex_hull using assms keq
by (auto simp: convex_halfspace_le convex_halfspace_ge hull_minimal)
then show ?thesis using cin that by fastforce
qed
lemma hol_pal_lem1:
assumes "convex S" "open S"
and abc: "a \ S" "b \ S" "c \ S"
"d \ 0" and lek: "d \ a \ k" "d \ b \ k" "d \ c \ k"
and holf1: "f holomorphic_on {z. z \ S \ d \ z < k}"
and contf: "continuous_on S f"
shows "contour_integral (linepath a b) f +
--> --------------------
--> maximum size reached
--> --------------------
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