(* Author: John Harrison, Marco Maggesi, Graziano Gentili, Gianni Ciolli, Valentina Bruno
Ported from "hol_light/Multivariate/canal.ml" by L C Paulson (2014)
*)
section \<open>Complex Analysis Basics\<close>
text \<open>Definitions of analytic and holomorphic functions, limit theorems, complex differentiation\<close>
theory Complex_Analysis_Basics
imports Derivative "HOL-Library.Nonpos_Ints"
begin
subsection\<^marker>\<open>tag unimportant\<close>\<open>General lemmas\<close>
lemma nonneg_Reals_cmod_eq_Re: "z \ \\<^sub>\\<^sub>0 \ norm z = Re z"
by (simp add: complex_nonneg_Reals_iff cmod_eq_Re)
lemma fact_cancel:
fixes c :: "'a::real_field"
shows "of_nat (Suc n) * c / (fact (Suc n)) = c / (fact n)"
using of_nat_neq_0 by force
lemma vector_derivative_cnj_within:
assumes "at x within A \ bot" and "f differentiable at x within A"
shows "vector_derivative (\z. cnj (f z)) (at x within A) =
cnj (vector_derivative f (at x within A))" (is "_ = cnj ?D")
proof -
let ?D = "vector_derivative f (at x within A)"
from assms have "(f has_vector_derivative ?D) (at x within A)"
by (subst (asm) vector_derivative_works)
hence "((\x. cnj (f x)) has_vector_derivative cnj ?D) (at x within A)"
by (rule has_vector_derivative_cnj)
thus ?thesis using assms by (auto dest: vector_derivative_within)
qed
lemma vector_derivative_cnj:
assumes "f differentiable at x"
shows "vector_derivative (\z. cnj (f z)) (at x) = cnj (vector_derivative f (at x))"
using assms by (intro vector_derivative_cnj_within) auto
lemma
shows open_halfspace_Re_lt: "open {z. Re(z) < b}"
and open_halfspace_Re_gt: "open {z. Re(z) > b}"
and closed_halfspace_Re_ge: "closed {z. Re(z) \ b}"
and closed_halfspace_Re_le: "closed {z. Re(z) \ b}"
and closed_halfspace_Re_eq: "closed {z. Re(z) = b}"
and open_halfspace_Im_lt: "open {z. Im(z) < b}"
and open_halfspace_Im_gt: "open {z. Im(z) > b}"
and closed_halfspace_Im_ge: "closed {z. Im(z) \ b}"
and closed_halfspace_Im_le: "closed {z. Im(z) \ b}"
and closed_halfspace_Im_eq: "closed {z. Im(z) = b}"
by (intro open_Collect_less closed_Collect_le closed_Collect_eq continuous_on_Re
continuous_on_Im continuous_on_id continuous_on_const)+
lemma closed_complex_Reals: "closed (\ :: complex set)"
proof -
have "(\ :: complex set) = {z. Im z = 0}"
by (auto simp: complex_is_Real_iff)
then show ?thesis
by (metis closed_halfspace_Im_eq)
qed
lemma closed_Real_halfspace_Re_le: "closed (\ \ {w. Re w \ x})"
by (simp add: closed_Int closed_complex_Reals closed_halfspace_Re_le)
lemma closed_nonpos_Reals_complex [simp]: "closed (\\<^sub>\\<^sub>0 :: complex set)"
proof -
have "\\<^sub>\\<^sub>0 = \ \ {z. Re(z) \ 0}"
using complex_nonpos_Reals_iff complex_is_Real_iff by auto
then show ?thesis
by (metis closed_Real_halfspace_Re_le)
qed
lemma closed_Real_halfspace_Re_ge: "closed (\ \ {w. x \ Re(w)})"
using closed_halfspace_Re_ge
by (simp add: closed_Int closed_complex_Reals)
lemma closed_nonneg_Reals_complex [simp]: "closed (\\<^sub>\\<^sub>0 :: complex set)"
proof -
have "\\<^sub>\\<^sub>0 = \ \ {z. Re(z) \ 0}"
using complex_nonneg_Reals_iff complex_is_Real_iff by auto
then show ?thesis
by (metis closed_Real_halfspace_Re_ge)
qed
lemma closed_real_abs_le: "closed {w \ \. \Re w\ \ r}"
proof -
have "{w \ \. \Re w\ \ r} = (\ \ {w. Re w \ r}) \ (\ \ {w. Re w \ -r})"
by auto
then show "closed {w \ \. \Re w\ \ r}"
by (simp add: closed_Int closed_Real_halfspace_Re_ge closed_Real_halfspace_Re_le)
qed
lemma real_lim:
fixes l::complex
assumes "(f \ l) F" and "\ trivial_limit F" and "eventually P F" and "\a. P a \ f a \ \"
shows "l \ \"
proof (rule Lim_in_closed_set[OF closed_complex_Reals _ assms(2,1)])
show "eventually (\x. f x \ \) F"
using assms(3, 4) by (auto intro: eventually_mono)
qed
lemma real_lim_sequentially:
fixes l::complex
shows "(f \ l) sequentially \ (\N. \n\N. f n \ \) \ l \ \"
by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially)
lemma real_series:
fixes l::complex
shows "f sums l \ (\n. f n \ \) \ l \ \"
unfolding sums_def
by (metis real_lim_sequentially sum_in_Reals)
lemma Lim_null_comparison_Re:
assumes "eventually (\x. norm(f x) \ Re(g x)) F" "(g \ 0) F" shows "(f \ 0) F"
by (rule Lim_null_comparison[OF assms(1)] tendsto_eq_intros assms(2))+ simp
subsection\<open>Holomorphic functions\<close>
definition\<^marker>\<open>tag important\<close> holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
(infixl "(holomorphic'_on)" 50)
where "f holomorphic_on s \ \x\s. f field_differentiable (at x within s)"
named_theorems\<^marker>\<open>tag important\<close> holomorphic_intros "structural introduction rules for holomorphic_on"
lemma holomorphic_onI [intro?]: "(\x. x \ s \ f field_differentiable (at x within s)) \ f holomorphic_on s"
by (simp add: holomorphic_on_def)
lemma holomorphic_onD [dest?]: "\f holomorphic_on s; x \ s\ \ f field_differentiable (at x within s)"
by (simp add: holomorphic_on_def)
lemma holomorphic_on_imp_differentiable_on:
"f holomorphic_on s \ f differentiable_on s"
unfolding holomorphic_on_def differentiable_on_def
by (simp add: field_differentiable_imp_differentiable)
lemma holomorphic_on_imp_differentiable_at:
"\f holomorphic_on s; open s; x \ s\ \ f field_differentiable (at x)"
using at_within_open holomorphic_on_def by fastforce
lemma holomorphic_on_empty [holomorphic_intros]: "f holomorphic_on {}"
by (simp add: holomorphic_on_def)
lemma holomorphic_on_open:
"open s \ f holomorphic_on s \ (\x \ s. \f'. DERIV f x :> f')"
by (auto simp: holomorphic_on_def field_differentiable_def has_field_derivative_def at_within_open [of _ s])
lemma holomorphic_on_imp_continuous_on:
"f holomorphic_on s \ continuous_on s f"
by (metis field_differentiable_imp_continuous_at continuous_on_eq_continuous_within holomorphic_on_def)
lemma holomorphic_on_subset [elim]:
"f holomorphic_on s \ t \ s \ f holomorphic_on t"
unfolding holomorphic_on_def
by (metis field_differentiable_within_subset subsetD)
lemma holomorphic_transform: "\f holomorphic_on s; \x. x \ s \ f x = g x\ \ g holomorphic_on s"
by (metis field_differentiable_transform_within linordered_field_no_ub holomorphic_on_def)
lemma holomorphic_cong: "s = t ==> (\x. x \ s \ f x = g x) \ f holomorphic_on s \ g holomorphic_on t"
by (metis holomorphic_transform)
lemma holomorphic_on_linear [simp, holomorphic_intros]: "((*) c) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_linear)
lemma holomorphic_on_const [simp, holomorphic_intros]: "(\z. c) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_const)
lemma holomorphic_on_ident [simp, holomorphic_intros]: "(\x. x) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_ident)
lemma holomorphic_on_id [simp, holomorphic_intros]: "id holomorphic_on s"
unfolding id_def by (rule holomorphic_on_ident)
lemma holomorphic_on_compose:
"f holomorphic_on s \ g holomorphic_on (f ` s) \ (g o f) holomorphic_on s"
using field_differentiable_compose_within[of f _ s g]
by (auto simp: holomorphic_on_def)
lemma holomorphic_on_compose_gen:
"f holomorphic_on s \ g holomorphic_on t \ f ` s \ t \ (g o f) holomorphic_on s"
by (metis holomorphic_on_compose holomorphic_on_subset)
lemma holomorphic_on_balls_imp_entire:
assumes "\bdd_above A" "\r. r \ A \ f holomorphic_on ball c r"
shows "f holomorphic_on B"
proof (rule holomorphic_on_subset)
show "f holomorphic_on UNIV" unfolding holomorphic_on_def
proof
fix z :: complex
from \<open>\<not>bdd_above A\<close> obtain r where r: "r \<in> A" "r > norm (z - c)"
by (meson bdd_aboveI not_le)
with assms(2) have "f holomorphic_on ball c r" by blast
moreover from r have "z \ ball c r" by (auto simp: dist_norm norm_minus_commute)
ultimately show "f field_differentiable at z"
by (auto simp: holomorphic_on_def at_within_open[of _ "ball c r"])
qed
qed auto
lemma holomorphic_on_balls_imp_entire':
assumes "\r. r > 0 \ f holomorphic_on ball c r"
shows "f holomorphic_on B"
proof (rule holomorphic_on_balls_imp_entire)
{
fix M :: real
have "\x. x > max M 0" by (intro gt_ex)
hence "\x>0. x > M" by auto
}
thus "\bdd_above {(0::real)<..}" unfolding bdd_above_def
by (auto simp: not_le)
qed (insert assms, auto)
lemma holomorphic_on_minus [holomorphic_intros]: "f holomorphic_on s \ (\z. -(f z)) holomorphic_on s"
by (metis field_differentiable_minus holomorphic_on_def)
lemma holomorphic_on_add [holomorphic_intros]:
"\f holomorphic_on s; g holomorphic_on s\ \ (\z. f z + g z) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_add)
lemma holomorphic_on_diff [holomorphic_intros]:
"\f holomorphic_on s; g holomorphic_on s\ \ (\z. f z - g z) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_diff)
lemma holomorphic_on_mult [holomorphic_intros]:
"\f holomorphic_on s; g holomorphic_on s\ \ (\z. f z * g z) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_mult)
lemma holomorphic_on_inverse [holomorphic_intros]:
"\f holomorphic_on s; \z. z \ s \ f z \ 0\ \ (\z. inverse (f z)) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_inverse)
lemma holomorphic_on_divide [holomorphic_intros]:
"\f holomorphic_on s; g holomorphic_on s; \z. z \ s \ g z \ 0\ \ (\z. f z / g z) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_divide)
lemma holomorphic_on_power [holomorphic_intros]:
"f holomorphic_on s \ (\z. (f z)^n) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_power)
lemma holomorphic_on_sum [holomorphic_intros]:
"(\i. i \ I \ (f i) holomorphic_on s) \ (\x. sum (\i. f i x) I) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_sum)
lemma holomorphic_on_prod [holomorphic_intros]:
"(\i. i \ I \ (f i) holomorphic_on s) \ (\x. prod (\i. f i x) I) holomorphic_on s"
by (induction I rule: infinite_finite_induct) (auto intro: holomorphic_intros)
lemma holomorphic_pochhammer [holomorphic_intros]:
"f holomorphic_on A \ (\s. pochhammer (f s) n) holomorphic_on A"
by (induction n) (auto intro!: holomorphic_intros simp: pochhammer_Suc)
lemma holomorphic_on_scaleR [holomorphic_intros]:
"f holomorphic_on A \ (\x. c *\<^sub>R f x) holomorphic_on A"
by (auto simp: scaleR_conv_of_real intro!: holomorphic_intros)
lemma holomorphic_on_Un [holomorphic_intros]:
assumes "f holomorphic_on A" "f holomorphic_on B" "open A" "open B"
shows "f holomorphic_on (A \ B)"
using assms by (auto simp: holomorphic_on_def at_within_open[of _ A]
at_within_open[of _ B] at_within_open[of _ "A \ B"] open_Un)
lemma holomorphic_on_If_Un [holomorphic_intros]:
assumes "f holomorphic_on A" "g holomorphic_on B" "open A" "open B"
assumes "\z. z \ A \ z \ B \ f z = g z"
shows "(\z. if z \ A then f z else g z) holomorphic_on (A \ B)" (is "?h holomorphic_on _")
proof (intro holomorphic_on_Un)
note \<open>f holomorphic_on A\<close>
also have "f holomorphic_on A \ ?h holomorphic_on A"
by (intro holomorphic_cong) auto
finally show \<dots> .
next
note \<open>g holomorphic_on B\<close>
also have "g holomorphic_on B \ ?h holomorphic_on B"
using assms by (intro holomorphic_cong) auto
finally show \<dots> .
qed (insert assms, auto)
lemma holomorphic_derivI:
"\f holomorphic_on S; open S; x \ S\
\<Longrightarrow> (f has_field_derivative deriv f x) (at x within T)"
by (metis DERIV_deriv_iff_field_differentiable at_within_open holomorphic_on_def has_field_derivative_at_within)
lemma complex_derivative_transform_within_open:
"\f holomorphic_on s; g holomorphic_on s; open s; z \ s; \w. w \ s \ f w = g w\
\<Longrightarrow> deriv f z = deriv g z"
unfolding holomorphic_on_def
by (rule DERIV_imp_deriv)
(metis DERIV_deriv_iff_field_differentiable has_field_derivative_transform_within_open at_within_open)
lemma holomorphic_nonconstant:
assumes holf: "f holomorphic_on S" and "open S" "\ \ S" "deriv f \ \ 0"
shows "\ f constant_on S"
by (rule nonzero_deriv_nonconstant [of f "deriv f \" \ S])
(use assms in \<open>auto simp: holomorphic_derivI\<close>)
subsection\<open>Analyticity on a set\<close>
definition\<^marker>\<open>tag important\<close> analytic_on (infixl "(analytic'_on)" 50)
where "f analytic_on S \ \x \ S. \e. 0 < e \ f holomorphic_on (ball x e)"
named_theorems\<^marker>\<open>tag important\<close> analytic_intros "introduction rules for proving analyticity"
lemma analytic_imp_holomorphic: "f analytic_on S \ f holomorphic_on S"
by (simp add: at_within_open [OF _ open_ball] analytic_on_def holomorphic_on_def)
(metis centre_in_ball field_differentiable_at_within)
lemma analytic_on_open: "open S \ f analytic_on S \ f holomorphic_on S"
apply (auto simp: analytic_imp_holomorphic)
apply (auto simp: analytic_on_def holomorphic_on_def)
by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball)
lemma analytic_on_imp_differentiable_at:
"f analytic_on S \ x \ S \ f field_differentiable (at x)"
apply (auto simp: analytic_on_def holomorphic_on_def)
by (metis open_ball centre_in_ball field_differentiable_within_open)
lemma analytic_on_subset: "f analytic_on S \ T \ S \ f analytic_on T"
by (auto simp: analytic_on_def)
lemma analytic_on_Un: "f analytic_on (S \ T) \ f analytic_on S \ f analytic_on T"
by (auto simp: analytic_on_def)
lemma analytic_on_Union: "f analytic_on (\\) \ (\T \ \. f analytic_on T)"
by (auto simp: analytic_on_def)
lemma analytic_on_UN: "f analytic_on (\i\I. S i) \ (\i\I. f analytic_on (S i))"
by (auto simp: analytic_on_def)
lemma analytic_on_holomorphic:
"f analytic_on S \ (\T. open T \ S \ T \ f holomorphic_on T)"
(is "?lhs = ?rhs")
proof -
have "?lhs \ (\T. open T \ S \ T \ f analytic_on T)"
proof safe
assume "f analytic_on S"
then show "\T. open T \ S \ T \ f analytic_on T"
apply (simp add: analytic_on_def)
apply (rule exI [where x="\{U. open U \ f analytic_on U}"], auto)
apply (metis open_ball analytic_on_open centre_in_ball)
by (metis analytic_on_def)
next
fix T
assume "open T" "S \ T" "f analytic_on T"
then show "f analytic_on S"
by (metis analytic_on_subset)
qed
also have "... \ ?rhs"
by (auto simp: analytic_on_open)
finally show ?thesis .
qed
lemma analytic_on_linear [analytic_intros,simp]: "((*) c) analytic_on S"
by (auto simp add: analytic_on_holomorphic)
lemma analytic_on_const [analytic_intros,simp]: "(\z. c) analytic_on S"
by (metis analytic_on_def holomorphic_on_const zero_less_one)
lemma analytic_on_ident [analytic_intros,simp]: "(\x. x) analytic_on S"
by (simp add: analytic_on_def gt_ex)
lemma analytic_on_id [analytic_intros]: "id analytic_on S"
unfolding id_def by (rule analytic_on_ident)
lemma analytic_on_compose:
assumes f: "f analytic_on S"
and g: "g analytic_on (f ` S)"
shows "(g o f) analytic_on S"
unfolding analytic_on_def
proof (intro ballI)
fix x
assume x: "x \ S"
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f
by (metis analytic_on_def)
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g
by (metis analytic_on_def g image_eqI x)
have "isCont f x"
by (metis analytic_on_imp_differentiable_at field_differentiable_imp_continuous_at f x)
with e' obtain d where d: "0 < d" and fd: "f ` ball x d \ ball (f x) e'"
by (auto simp: continuous_at_ball)
have "g \ f holomorphic_on ball x (min d e)"
apply (rule holomorphic_on_compose)
apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball)
then show "\e>0. g \ f holomorphic_on ball x e"
by (metis d e min_less_iff_conj)
qed
lemma analytic_on_compose_gen:
"f analytic_on S \ g analytic_on T \ (\z. z \ S \ f z \ T)
\<Longrightarrow> g o f analytic_on S"
by (metis analytic_on_compose analytic_on_subset image_subset_iff)
lemma analytic_on_neg [analytic_intros]:
"f analytic_on S \ (\z. -(f z)) analytic_on S"
by (metis analytic_on_holomorphic holomorphic_on_minus)
lemma analytic_on_add [analytic_intros]:
assumes f: "f analytic_on S"
and g: "g analytic_on S"
shows "(\z. f z + g z) analytic_on S"
unfolding analytic_on_def
proof (intro ballI)
fix z
assume z: "z \ S"
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
by (metis analytic_on_def)
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
by (metis analytic_on_def g z)
have "(\z. f z + g z) holomorphic_on ball z (min e e')"
apply (rule holomorphic_on_add)
apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
then show "\e>0. (\z. f z + g z) holomorphic_on ball z e"
by (metis e e' min_less_iff_conj)
qed
lemma analytic_on_diff [analytic_intros]:
assumes f: "f analytic_on S"
and g: "g analytic_on S"
shows "(\z. f z - g z) analytic_on S"
unfolding analytic_on_def
proof (intro ballI)
fix z
assume z: "z \ S"
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
by (metis analytic_on_def)
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
by (metis analytic_on_def g z)
have "(\z. f z - g z) holomorphic_on ball z (min e e')"
apply (rule holomorphic_on_diff)
apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
then show "\e>0. (\z. f z - g z) holomorphic_on ball z e"
by (metis e e' min_less_iff_conj)
qed
lemma analytic_on_mult [analytic_intros]:
assumes f: "f analytic_on S"
and g: "g analytic_on S"
shows "(\z. f z * g z) analytic_on S"
unfolding analytic_on_def
proof (intro ballI)
fix z
assume z: "z \ S"
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
by (metis analytic_on_def)
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
by (metis analytic_on_def g z)
have "(\z. f z * g z) holomorphic_on ball z (min e e')"
apply (rule holomorphic_on_mult)
apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
then show "\e>0. (\z. f z * g z) holomorphic_on ball z e"
by (metis e e' min_less_iff_conj)
qed
lemma analytic_on_inverse [analytic_intros]:
assumes f: "f analytic_on S"
and nz: "(\z. z \ S \ f z \ 0)"
shows "(\z. inverse (f z)) analytic_on S"
unfolding analytic_on_def
proof (intro ballI)
fix z
assume z: "z \ S"
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
by (metis analytic_on_def)
have "continuous_on (ball z e) f"
by (metis fh holomorphic_on_imp_continuous_on)
then obtain e' where e': "0 < e'" and nz': "\y. dist z y < e' \ f y \ 0"
by (metis open_ball centre_in_ball continuous_on_open_avoid e z nz)
have "(\z. inverse (f z)) holomorphic_on ball z (min e e')"
apply (rule holomorphic_on_inverse)
apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball)
by (metis nz' mem_ball min_less_iff_conj)
then show "\e>0. (\z. inverse (f z)) holomorphic_on ball z e"
by (metis e e' min_less_iff_conj)
qed
lemma analytic_on_divide [analytic_intros]:
assumes f: "f analytic_on S"
and g: "g analytic_on S"
and nz: "(\z. z \ S \ g z \ 0)"
shows "(\z. f z / g z) analytic_on S"
unfolding divide_inverse
by (metis analytic_on_inverse analytic_on_mult f g nz)
lemma analytic_on_power [analytic_intros]:
"f analytic_on S \ (\z. (f z) ^ n) analytic_on S"
by (induct n) (auto simp: analytic_on_mult)
lemma analytic_on_sum [analytic_intros]:
"(\i. i \ I \ (f i) analytic_on S) \ (\x. sum (\i. f i x) I) analytic_on S"
by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_add)
lemma deriv_left_inverse:
assumes "f holomorphic_on S" and "g holomorphic_on T"
and "open S" and "open T"
and "f ` S \ T"
and [simp]: "\z. z \ S \ g (f z) = z"
and "w \ S"
shows "deriv f w * deriv g (f w) = 1"
proof -
have "deriv f w * deriv g (f w) = deriv g (f w) * deriv f w"
by (simp add: algebra_simps)
also have "... = deriv (g o f) w"
using assms
by (metis analytic_on_imp_differentiable_at analytic_on_open deriv_chain image_subset_iff)
also have "... = deriv id w"
proof (rule complex_derivative_transform_within_open [where s=S])
show "g \ f holomorphic_on S"
by (rule assms holomorphic_on_compose_gen holomorphic_intros)+
qed (use assms in auto)
also have "... = 1"
by simp
finally show ?thesis .
qed
subsection\<^marker>\<open>tag unimportant\<close>\<open>Analyticity at a point\<close>
lemma analytic_at_ball:
"f analytic_on {z} \ (\e. 0 f holomorphic_on ball z e)"
by (metis analytic_on_def singleton_iff)
lemma analytic_at:
"f analytic_on {z} \ (\s. open s \ z \ s \ f holomorphic_on s)"
by (metis analytic_on_holomorphic empty_subsetI insert_subset)
lemma analytic_on_analytic_at:
"f analytic_on s \ (\z \ s. f analytic_on {z})"
by (metis analytic_at_ball analytic_on_def)
lemma analytic_at_two:
"f analytic_on {z} \ g analytic_on {z} \
(\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then obtain s t
where st: "open s" "z \ s" "f holomorphic_on s"
"open t" "z \ t" "g holomorphic_on t"
by (auto simp: analytic_at)
show ?rhs
apply (rule_tac x="s \ t" in exI)
using st
apply (auto simp: holomorphic_on_subset)
done
next
assume ?rhs
then show ?lhs
by (force simp add: analytic_at)
qed
subsection\<^marker>\<open>tag unimportant\<close>\<open>Combining theorems for derivative with ``analytic at'' hypotheses\<close>
lemma
assumes "f analytic_on {z}" "g analytic_on {z}"
shows complex_derivative_add_at: "deriv (\w. f w + g w) z = deriv f z + deriv g z"
and complex_derivative_diff_at: "deriv (\w. f w - g w) z = deriv f z - deriv g z"
and complex_derivative_mult_at: "deriv (\w. f w * g w) z =
f z * deriv g z + deriv f z * g z"
proof -
obtain s where s: "open s" "z \ s" "f holomorphic_on s" "g holomorphic_on s"
using assms by (metis analytic_at_two)
show "deriv (\w. f w + g w) z = deriv f z + deriv g z"
apply (rule DERIV_imp_deriv [OF DERIV_add])
using s
apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
done
show "deriv (\w. f w - g w) z = deriv f z - deriv g z"
apply (rule DERIV_imp_deriv [OF DERIV_diff])
using s
apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
done
show "deriv (\w. f w * g w) z = f z * deriv g z + deriv f z * g z"
apply (rule DERIV_imp_deriv [OF DERIV_mult'])
using s
apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
done
qed
lemma deriv_cmult_at:
"f analytic_on {z} \ deriv (\w. c * f w) z = c * deriv f z"
by (auto simp: complex_derivative_mult_at)
lemma deriv_cmult_right_at:
"f analytic_on {z} \ deriv (\w. f w * c) z = deriv f z * c"
by (auto simp: complex_derivative_mult_at)
subsection\<^marker>\<open>tag unimportant\<close>\<open>Complex differentiation of sequences and series\<close>
(* TODO: Could probably be simplified using Uniform_Limit *)
lemma has_complex_derivative_sequence:
fixes S :: "complex set"
assumes cvs: "convex S"
and df: "\n x. x \ S \ (f n has_field_derivative f' n x) (at x within S)"
and conv: "\e. 0 < e \ \N. \n x. n \ N \ x \ S \ norm (f' n x - g' x) \ e"
and "\x l. x \ S \ ((\n. f n x) \ l) sequentially"
shows "\g. \x \ S. ((\n. f n x) \ g x) sequentially \
(g has_field_derivative (g' x)) (at x within S)"
proof -
from assms obtain x l where x: "x \ S" and tf: "((\n. f n x) \ l) sequentially"
by blast
{ fix e::real assume e: "e > 0"
then obtain N where N: "\n\N. \x. x \ S \ cmod (f' n x - g' x) \ e"
by (metis conv)
have "\N. \n\N. \x\S. \h. cmod (f' n x * h - g' x * h) \ e * cmod h"
proof (rule exI [of _ N], clarify)
fix n y h
assume "N \ n" "y \ S"
then have "cmod (f' n y - g' y) \ e"
by (metis N)
then have "cmod h * cmod (f' n y - g' y) \ cmod h * e"
by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
then show "cmod (f' n y * h - g' y * h) \ e * cmod h"
by (simp add: norm_mult [symmetric] field_simps)
qed
} note ** = this
show ?thesis
unfolding has_field_derivative_def
proof (rule has_derivative_sequence [OF cvs _ _ x])
show "(\n. f n x) \ l"
by (rule tf)
next show "\e. e > 0 \ \\<^sub>F n in sequentially. \x\S. \h. cmod (f' n x * h - g' x * h) \ e * cmod h"
unfolding eventually_sequentially by (blast intro: **)
qed (metis has_field_derivative_def df)
qed
lemma has_complex_derivative_series:
fixes S :: "complex set"
assumes cvs: "convex S"
and df: "\n x. x \ S \ (f n has_field_derivative f' n x) (at x within S)"
and conv: "\e. 0 < e \ \N. \n x. n \ N \ x \ S
\<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
and "\x l. x \ S \ ((\n. f n x) sums l)"
shows "\g. \x \ S. ((\n. f n x) sums g x) \ ((g has_field_derivative g' x) (at x within S))"
proof -
from assms obtain x l where x: "x \ S" and sf: "((\n. f n x) sums l)"
by blast
{ fix e::real assume e: "e > 0"
then obtain N where N: "\n x. n \ N \ x \ S
\<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
by (metis conv)
have "\N. \n\N. \x\S. \h. cmod ((\i e * cmod h"
proof (rule exI [of _ N], clarify)
fix n y h
assume "N \ n" "y \ S"
then have "cmod ((\i e"
by (metis N)
then have "cmod h * cmod ((\i cmod h * e"
by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
then show "cmod ((\i e * cmod h"
by (simp add: norm_mult [symmetric] field_simps sum_distrib_left)
qed
} note ** = this
show ?thesis
unfolding has_field_derivative_def
proof (rule has_derivative_series [OF cvs _ _ x])
fix n x
assume "x \ S"
then show "((f n) has_derivative (\z. z * f' n x)) (at x within S)"
by (metis df has_field_derivative_def mult_commute_abs)
next show " ((\n. f n x) sums l)"
by (rule sf)
next show "\e. e>0 \ \\<^sub>F n in sequentially. \x\S. \h. cmod ((\i e * cmod h"
unfolding eventually_sequentially by (blast intro: **)
qed
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>Taylor on Complex Numbers\<close>
lemma sum_Suc_reindex:
fixes f :: "nat \ 'a::ab_group_add"
shows "sum f {0..n} = f 0 - f (Suc n) + sum (\i. f (Suc i)) {0..n}"
by (induct n) auto
lemma field_Taylor:
assumes S: "convex S"
and f: "\i x. x \ S \ i \ n \ (f i has_field_derivative f (Suc i) x) (at x within S)"
and B: "\x. x \ S \ norm (f (Suc n) x) \ B"
and w: "w \ S"
and z: "z \ S"
shows "norm(f 0 z - (\i\n. f i w * (z-w) ^ i / (fact i)))
\<le> B * norm(z - w)^(Suc n) / fact n"
proof -
have wzs: "closed_segment w z \ S" using assms
by (metis convex_contains_segment)
{ fix u
assume "u \ closed_segment w z"
then have "u \ S"
by (metis wzs subsetD)
have "(\i\n. f i u * (- of_nat i * (z-u)^(i - 1)) / (fact i) +
f (Suc i) u * (z-u)^i / (fact i)) =
f (Suc n) u * (z-u) ^ n / (fact n)"
proof (induction n)
case 0 show ?case by simp
next
case (Suc n)
have "(\i\Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / (fact i) +
f (Suc i) u * (z-u) ^ i / (fact i)) =
f (Suc n) u * (z-u) ^ n / (fact n) +
f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / (fact (Suc n)) -
f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / (fact (Suc n))"
using Suc by simp
also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n))"
proof -
have "(fact(Suc n)) *
(f(Suc n) u *(z-u) ^ n / (fact n) +
f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / (fact(Suc n)) -
f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / (fact(Suc n))) =
((fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / (fact n) +
((fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / (fact(Suc n))) -
((fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / (fact(Suc n))"
by (simp add: algebra_simps del: fact_Suc)
also have "... = ((fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / (fact n) +
(f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
(f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
by (simp del: fact_Suc)
also have "... = (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) +
(f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
(f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
by (simp only: fact_Suc of_nat_mult ac_simps) simp
also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
by (simp add: algebra_simps)
finally show ?thesis
by (simp add: mult_left_cancel [where c = "(fact (Suc n))", THEN iffD1] del: fact_Suc)
qed
finally show ?case .
qed
then have "((\v. (\i\n. f i v * (z - v)^i / (fact i)))
has_field_derivative f (Suc n) u * (z-u) ^ n / (fact n))
(at u within S)"
apply (intro derivative_eq_intros)
apply (blast intro: assms \<open>u \<in> S\<close>)
apply (rule refl)+
apply (auto simp: field_simps)
done
} note sum_deriv = this
{ fix u
assume u: "u \ closed_segment w z"
then have us: "u \ S"
by (metis wzs subsetD)
have "norm (f (Suc n) u) * norm (z - u) ^ n \ norm (f (Suc n) u) * norm (u - z) ^ n"
by (metis norm_minus_commute order_refl)
also have "... \ norm (f (Suc n) u) * norm (z - w) ^ n"
by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
also have "... \ B * norm (z - w) ^ n"
by (metis norm_ge_zero zero_le_power mult_right_mono B [OF us])
finally have "norm (f (Suc n) u) * norm (z - u) ^ n \ B * norm (z - w) ^ n" .
} note cmod_bound = this
have "(\i\n. f i z * (z - z) ^ i / (fact i)) = (\i\n. (f i z / (fact i)) * 0 ^ i)"
by simp
also have "\ = f 0 z / (fact 0)"
by (subst sum_zero_power) simp
finally have "norm (f 0 z - (\i\n. f i w * (z - w) ^ i / (fact i)))
\<le> norm ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) -
(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))"
by (simp add: norm_minus_commute)
also have "... \ B * norm (z - w) ^ n / (fact n) * norm (w - z)"
apply (rule field_differentiable_bound
[where f' = "\w. f (Suc n) w * (z - w)^n / (fact n)"
and S = "closed_segment w z", OF convex_closed_segment])
apply (auto simp: DERIV_subset [OF sum_deriv wzs]
norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
done
also have "... \ B * norm (z - w) ^ Suc n / (fact n)"
by (simp add: algebra_simps norm_minus_commute)
finally show ?thesis .
qed
lemma complex_Taylor:
assumes S: "convex S"
and f: "\i x. x \ S \ i \ n \ (f i has_field_derivative f (Suc i) x) (at x within S)"
and B: "\x. x \ S \ cmod (f (Suc n) x) \ B"
and w: "w \ S"
and z: "z \ S"
shows "cmod(f 0 z - (\i\n. f i w * (z-w) ^ i / (fact i)))
\<le> B * cmod(z - w)^(Suc n) / fact n"
using assms by (rule field_Taylor)
text\<open>Something more like the traditional MVT for real components\<close>
lemma complex_mvt_line:
assumes "\u. u \ closed_segment w z \ (f has_field_derivative f'(u)) (at u)"
shows "\u. u \ closed_segment w z \ Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
proof -
have twz: "\t. (1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)"
by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
note assms[unfolded has_field_derivative_def, derivative_intros]
show ?thesis
apply (cut_tac mvt_simple
[of 0 1 "Re o f o (\t. (1 - t) *\<^sub>R w + t *\<^sub>R z)"
"\u. Re o (\h. f'((1 - u) *\<^sub>R w + u *\<^sub>R z) * h) o (\t. t *\<^sub>R (z - w))"])
apply auto
apply (rule_tac x="(1 - x) *\<^sub>R w + x *\<^sub>R z" in exI)
apply (auto simp: closed_segment_def twz) []
apply (intro derivative_eq_intros has_derivative_at_withinI, simp_all)
apply (simp add: fun_eq_iff real_vector.scale_right_diff_distrib)
apply (force simp: twz closed_segment_def)
done
qed
lemma complex_Taylor_mvt:
assumes "\i x. \x \ closed_segment w z; i \ n\ \ ((f i) has_field_derivative f (Suc i) x) (at x)"
shows "\u. u \ closed_segment w z \
Re (f 0 z) =
Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / (fact i)) +
(f (Suc n) u * (z-u)^n / (fact n)) * (z - w))"
proof -
{ fix u
assume u: "u \ closed_segment w z"
have "(\i = 0..n.
(f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) /
(fact i)) =
f (Suc 0) u -
(f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
(fact (Suc n)) +
(\<Sum>i = 0..n.
(f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) /
(fact (Suc i)))"
by (subst sum_Suc_reindex) simp
also have "... = f (Suc 0) u -
(f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
(fact (Suc n)) +
(\<Sum>i = 0..n.
f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i)) -
f (Suc i) u * (z-u) ^ i / (fact i))"
by (simp only: diff_divide_distrib fact_cancel ac_simps)
also have "... = f (Suc 0) u -
(f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) /
(fact (Suc n)) +
f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n)) - f (Suc 0) u"
by (subst sum_Suc_diff) auto
also have "... = f (Suc n) u * (z-u) ^ n / (fact n)"
by (simp only: algebra_simps diff_divide_distrib fact_cancel)
finally have "(\i = 0..n. (f (Suc i) u * (z - u) ^ i
- of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / (fact i)) =
f (Suc n) u * (z - u) ^ n / (fact n)" .
then have "((\u. \i = 0..n. f i u * (z - u) ^ i / (fact i)) has_field_derivative
f (Suc n) u * (z - u) ^ n / (fact n)) (at u)"
apply (intro derivative_eq_intros)+
apply (force intro: u assms)
apply (rule refl)+
apply (auto simp: ac_simps)
done
}
then show ?thesis
apply (cut_tac complex_mvt_line [of w z "\u. \i = 0..n. f i u * (z-u) ^ i / (fact i)"
"\u. (f (Suc n) u * (z-u)^n / (fact n))"])
apply (auto simp add: intro: open_closed_segment)
done
qed
end
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