(* 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>
theory Complex_Analysis_Basics
imports Cartesian_Euclidean_Space
begin
lemma cmod_fact [simp]: "cmod (fact n) = fact n"
by (metis norm_of_nat of_nat_fact)
subsection\<open>General lemmas\<close>
lemma has_derivative_mult_right:
fixes c:: "'a :: real_normed_algebra"
shows "((op * c) has_derivative (op * c)) F"
by (rule has_derivative_mult_right [OF has_derivative_id])
lemma has_derivative_of_real[derivative_intros, simp]:
"(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. of_real (f x)) has_derivative (\<lambda>x. of_real (f' x))) F"
using bounded_linear.has_derivative[OF bounded_linear_of_real] .
lemma has_vector_derivative_real_complex:
"DERIV f (of_real a) :> f' \<Longrightarrow> ((\<lambda>x. f (of_real x)) has_vector_derivative f') (at a within s)"
using has_derivative_compose[of of_real of_real a _ f "op * f'"]
by (simp add: scaleR_conv_of_real ac_simps has_vector_derivative_def has_field_derivative_def)
lemma fact_cancel:
fixes c :: "'a::real_field"
shows "of_nat (Suc n) * c / (fact (Suc n)) = c / (fact n)"
by (simp add: of_nat_mult del: of_nat_Suc times_nat.simps)
lemma bilinear_times:
fixes c::"'a::real_algebra" shows "bilinear (\<lambda>x y::'a. x*y)"
by (auto simp: bilinear_def distrib_left distrib_right intro!: linearI)
lemma linear_cnj: "linear cnj"
using bounded_linear.linear[OF bounded_linear_cnj] .
lemma tendsto_Re_upper:
assumes "~ (trivial_limit F)"
"(f \<longlongrightarrow> l) F"
"eventually (\<lambda>x. Re(f x) \<le> b) F"
shows "Re(l) \<le> b"
by (metis assms tendsto_le [OF _ tendsto_const] tendsto_Re)
lemma tendsto_Re_lower:
assumes "~ (trivial_limit F)"
"(f \<longlongrightarrow> l) F"
"eventually (\<lambda>x. b \<le> Re(f x)) F"
shows "b \<le> Re(l)"
by (metis assms tendsto_le [OF _ _ tendsto_const] tendsto_Re)
lemma tendsto_Im_upper:
assumes "~ (trivial_limit F)"
"(f \<longlongrightarrow> l) F"
"eventually (\<lambda>x. Im(f x) \<le> b) F"
shows "Im(l) \<le> b"
by (metis assms tendsto_le [OF _ tendsto_const] tendsto_Im)
lemma tendsto_Im_lower:
assumes "~ (trivial_limit F)"
"(f \<longlongrightarrow> l) F"
"eventually (\<lambda>x. b \<le> Im(f x)) F"
shows "b \<le> Im(l)"
by (metis assms tendsto_le [OF _ _ tendsto_const] tendsto_Im)
lemma lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = op * 0"
by auto
lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = op * 1"
by auto
lemma continuous_mult_left:
fixes c::"'a::real_normed_algebra"
shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. c * f x)"
by (rule continuous_mult [OF continuous_const])
lemma continuous_mult_right:
fixes c::"'a::real_normed_algebra"
shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x * c)"
by (rule continuous_mult [OF _ continuous_const])
lemma continuous_on_mult_left:
fixes c::"'a::real_normed_algebra"
shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c * f x)"
by (rule continuous_on_mult [OF continuous_on_const])
lemma continuous_on_mult_right:
fixes c::"'a::real_normed_algebra"
shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x * c)"
by (rule continuous_on_mult [OF _ continuous_on_const])
lemma uniformly_continuous_on_cmul_right [continuous_intros]:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. f x * c)"
using bounded_linear.uniformly_continuous_on[OF bounded_linear_mult_left] .
lemma uniformly_continuous_on_cmul_left[continuous_intros]:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
assumes "uniformly_continuous_on s f"
shows "uniformly_continuous_on s (\<lambda>x. c * f x)"
by (metis assms bounded_linear.uniformly_continuous_on bounded_linear_mult_right)
lemma continuous_within_norm_id [continuous_intros]: "continuous (at x within S) norm"
by (rule continuous_norm [OF continuous_ident])
lemma continuous_on_norm_id [continuous_intros]: "continuous_on S norm"
by (intro continuous_on_id continuous_on_norm)
subsection\<open>DERIV stuff\<close>
lemma DERIV_zero_connected_constant:
fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
assumes "connected s"
and "open s"
and "finite k"
and "continuous_on s f"
and "\<forall>x\<in>(s - k). DERIV f x :> 0"
obtains c where "\<And>x. x \<in> s \<Longrightarrow> f(x) = c"
using has_derivative_zero_connected_constant [OF assms(1-4)] assms
by (metis DERIV_const has_derivative_const Diff_iff at_within_open frechet_derivative_at has_field_derivative_def)
lemma DERIV_zero_constant:
fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
shows "\<lbrakk>convex s;
\<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)\<rbrakk>
\<Longrightarrow> \<exists>c. \<forall>x \<in> s. f(x) = c"
by (auto simp: has_field_derivative_def lambda_zero intro: has_derivative_zero_constant)
lemma DERIV_zero_unique:
fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
assumes "convex s"
and d0: "\<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)"
and "a \<in> s"
and "x \<in> s"
shows "f x = f a"
by (rule has_derivative_zero_unique [OF assms(1) _ assms(4,3)])
(metis d0 has_field_derivative_imp_has_derivative lambda_zero)
lemma DERIV_zero_connected_unique:
fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
assumes "connected s"
and "open s"
and d0: "\<And>x. x\<in>s \<Longrightarrow> DERIV f x :> 0"
and "a \<in> s"
and "x \<in> s"
shows "f x = f a"
by (rule has_derivative_zero_unique_connected [OF assms(2,1) _ assms(5,4)])
(metis has_field_derivative_def lambda_zero d0)
lemma DERIV_transform_within:
assumes "(f has_field_derivative f') (at a within s)"
and "0 < d" "a \<in> s"
and "\<And>x. x\<in>s \<Longrightarrow> dist x a < d \<Longrightarrow> f x = g x"
shows "(g has_field_derivative f') (at a within s)"
using assms unfolding has_field_derivative_def
by (blast intro: has_derivative_transform_within)
lemma DERIV_transform_within_open:
assumes "DERIV f a :> f'"
and "open s" "a \<in> s"
and "\<And>x. x\<in>s \<Longrightarrow> f x = g x"
shows "DERIV g a :> f'"
using assms unfolding has_field_derivative_def
by (metis has_derivative_transform_within_open)
lemma DERIV_transform_at:
assumes "DERIV f a :> f'"
and "0 < d"
and "\<And>x. dist x a < d \<Longrightarrow> f x = g x"
shows "DERIV g a :> f'"
by (blast intro: assms DERIV_transform_within)
(*generalising DERIV_isconst_all, which requires type real (using the ordering)*)
lemma DERIV_zero_UNIV_unique:
fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
shows "(\<And>x. DERIV f x :> 0) \<Longrightarrow> f x = f a"
by (metis DERIV_zero_unique UNIV_I assms convex_UNIV)
subsection \<open>Some limit theorems about real part of real series etc.\<close>
(*MOVE? But not to Finite_Cartesian_Product*)
lemma sums_vec_nth :
assumes "f sums a"
shows "(\<lambda>x. f x $ i) sums a $ i"
using assms unfolding sums_def
by (auto dest: tendsto_vec_nth [where i=i])
lemma summable_vec_nth :
assumes "summable f"
shows "summable (\<lambda>x. f x $ i)"
using assms unfolding summable_def
by (blast intro: sums_vec_nth)
subsection \<open>Complex number lemmas\<close>
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) \<ge> b}"
and closed_halfspace_Re_le: "closed {z. Re(z) \<le> 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) \<ge> b}"
and closed_halfspace_Im_le: "closed {z. Im(z) \<le> b}"
and closed_halfspace_Im_eq: "closed {z. Im(z) = b}"
by (intro open_Collect_less closed_Collect_le closed_Collect_eq isCont_Re
isCont_Im continuous_ident continuous_const)+
lemma closed_complex_Reals: "closed (\<real> :: complex set)"
proof -
have "(\<real> :: 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 (\<real> \<inter> {w. Re w \<le> x})"
by (simp add: closed_Int closed_complex_Reals closed_halfspace_Re_le)
lemma closed_Real_halfspace_Re_ge: "closed (\<real> \<inter> {w. x \<le> Re(w)})"
using closed_halfspace_Re_ge
by (simp add: closed_Int closed_complex_Reals)
lemma closed_real_abs_le: "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
proof -
have "{w \<in> \<real>. \<bar>Re w\<bar> \<le> r} = (\<real> \<inter> {w. Re w \<le> r}) \<inter> (\<real> \<inter> {w. Re w \<ge> -r})"
by auto
then show "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> 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 \<longlongrightarrow> l) F" and "~(trivial_limit F)" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>"
shows "l \<in> \<real>"
proof (rule Lim_in_closed_set[OF closed_complex_Reals _ assms(2,1)])
show "eventually (\<lambda>x. f x \<in> \<real>) F"
using assms(3, 4) by (auto intro: eventually_mono)
qed
lemma real_lim_sequentially:
fixes l::complex
shows "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially)
lemma real_series:
fixes l::complex
shows "f sums l \<Longrightarrow> (\<And>n. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
unfolding sums_def
by (metis real_lim_sequentially setsum_in_Reals)
lemma Lim_null_comparison_Re:
assumes "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F" "(g \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F"
by (rule Lim_null_comparison[OF assms(1)] tendsto_eq_intros assms(2))+ simp
subsection\<open>Holomorphic functions\<close>
text\<open>Could be generalized to real normed fields, but in practice that would only include the reals\<close>
definition complex_differentiable :: "[complex \<Rightarrow> complex, complex filter] \<Rightarrow> bool"
(infixr "(complex'_differentiable)" 50)
where "f complex_differentiable F \<equiv> \<exists>f'. (f has_field_derivative f') F"
lemma complex_differentiable_imp_continuous_at:
"f complex_differentiable (at x within s) \<Longrightarrow> continuous (at x within s) f"
by (metis DERIV_continuous complex_differentiable_def)
lemma complex_differentiable_within_subset:
"\<lbrakk>f complex_differentiable (at x within s); t \<subseteq> s\<rbrakk>
\<Longrightarrow> f complex_differentiable (at x within t)"
by (metis DERIV_subset complex_differentiable_def)
lemma complex_differentiable_at_within:
"\<lbrakk>f complex_differentiable (at x)\<rbrakk>
\<Longrightarrow> f complex_differentiable (at x within s)"
unfolding complex_differentiable_def
by (metis DERIV_subset top_greatest)
lemma complex_differentiable_linear [derivative_intros]: "(op * c) complex_differentiable F"
proof -
show ?thesis
unfolding complex_differentiable_def has_field_derivative_def mult_commute_abs
by (force intro: has_derivative_mult_right)
qed
lemma complex_differentiable_const [derivative_intros]: "(\<lambda>z. c) complex_differentiable F"
unfolding complex_differentiable_def has_field_derivative_def
by (rule exI [where x=0])
(metis has_derivative_const lambda_zero)
lemma complex_differentiable_ident [derivative_intros]: "(\<lambda>z. z) complex_differentiable F"
unfolding complex_differentiable_def has_field_derivative_def
by (rule exI [where x=1])
(simp add: lambda_one [symmetric])
lemma complex_differentiable_id [derivative_intros]: "id complex_differentiable F"
unfolding id_def by (rule complex_differentiable_ident)
lemma complex_differentiable_minus [derivative_intros]:
"f complex_differentiable F \<Longrightarrow> (\<lambda>z. - (f z)) complex_differentiable F"
using assms unfolding complex_differentiable_def
by (metis field_differentiable_minus)
lemma complex_differentiable_add [derivative_intros]:
assumes "f complex_differentiable F" "g complex_differentiable F"
shows "(\<lambda>z. f z + g z) complex_differentiable F"
using assms unfolding complex_differentiable_def
by (metis field_differentiable_add)
lemma complex_differentiable_setsum [derivative_intros]:
"(\<And>i. i \<in> I \<Longrightarrow> (f i) complex_differentiable F) \<Longrightarrow> (\<lambda>z. \<Sum>i\<in>I. f i z) complex_differentiable F"
by (induct I rule: infinite_finite_induct)
(auto intro: complex_differentiable_add complex_differentiable_const)
lemma complex_differentiable_diff [derivative_intros]:
assumes "f complex_differentiable F" "g complex_differentiable F"
shows "(\<lambda>z. f z - g z) complex_differentiable F"
using assms unfolding complex_differentiable_def
by (metis field_differentiable_diff)
lemma complex_differentiable_inverse [derivative_intros]:
assumes "f complex_differentiable (at a within s)" "f a \<noteq> 0"
shows "(\<lambda>z. inverse (f z)) complex_differentiable (at a within s)"
using assms unfolding complex_differentiable_def
by (metis DERIV_inverse_fun)
lemma complex_differentiable_mult [derivative_intros]:
assumes "f complex_differentiable (at a within s)"
"g complex_differentiable (at a within s)"
shows "(\<lambda>z. f z * g z) complex_differentiable (at a within s)"
using assms unfolding complex_differentiable_def
by (metis DERIV_mult [of f _ a s g])
lemma complex_differentiable_divide [derivative_intros]:
assumes "f complex_differentiable (at a within s)"
"g complex_differentiable (at a within s)"
"g a \<noteq> 0"
shows "(\<lambda>z. f z / g z) complex_differentiable (at a within s)"
using assms unfolding complex_differentiable_def
by (metis DERIV_divide [of f _ a s g])
lemma complex_differentiable_power [derivative_intros]:
assumes "f complex_differentiable (at a within s)"
shows "(\<lambda>z. f z ^ n) complex_differentiable (at a within s)"
using assms unfolding complex_differentiable_def
by (metis DERIV_power)
lemma complex_differentiable_transform_within:
"0 < d \<Longrightarrow>
x \<in> s \<Longrightarrow>
(\<And>x'. x' \<in> s \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x') \<Longrightarrow>
f complex_differentiable (at x within s)
\<Longrightarrow> g complex_differentiable (at x within s)"
unfolding complex_differentiable_def has_field_derivative_def
by (blast intro: has_derivative_transform_within)
lemma complex_differentiable_compose_within:
assumes "f complex_differentiable (at a within s)"
"g complex_differentiable (at (f a) within f`s)"
shows "(g o f) complex_differentiable (at a within s)"
using assms unfolding complex_differentiable_def
by (metis DERIV_image_chain)
lemma complex_differentiable_compose:
"f complex_differentiable at z \<Longrightarrow> g complex_differentiable at (f z)
\<Longrightarrow> (g o f) complex_differentiable at z"
by (metis complex_differentiable_at_within complex_differentiable_compose_within)
lemma complex_differentiable_within_open:
"\<lbrakk>a \<in> s; open s\<rbrakk> \<Longrightarrow> f complex_differentiable at a within s \<longleftrightarrow>
f complex_differentiable at a"
unfolding complex_differentiable_def
by (metis at_within_open)
subsection\<open>Caratheodory characterization.\<close>
lemma complex_differentiable_caratheodory_at:
"f complex_differentiable (at z) \<longleftrightarrow>
(\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)"
using CARAT_DERIV [of f]
by (simp add: complex_differentiable_def has_field_derivative_def)
lemma complex_differentiable_caratheodory_within:
"f complex_differentiable (at z within s) \<longleftrightarrow>
(\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)"
using DERIV_caratheodory_within [of f]
by (simp add: complex_differentiable_def has_field_derivative_def)
subsection\<open>Holomorphic\<close>
definition holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
(infixl "(holomorphic'_on)" 50)
where "f holomorphic_on s \<equiv> \<forall>x\<in>s. f complex_differentiable (at x within s)"
named_theorems holomorphic_intros "structural introduction rules for holomorphic_on"
lemma holomorphic_on_empty [holomorphic_intros]: "f holomorphic_on {}"
by (simp add: holomorphic_on_def)
lemma holomorphic_on_open:
"open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> f')"
by (auto simp: holomorphic_on_def complex_differentiable_def has_field_derivative_def at_within_open [of _ s])
lemma holomorphic_on_imp_continuous_on:
"f holomorphic_on s \<Longrightarrow> continuous_on s f"
by (metis complex_differentiable_imp_continuous_at continuous_on_eq_continuous_within holomorphic_on_def)
lemma holomorphic_on_subset:
"f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t"
unfolding holomorphic_on_def
by (metis complex_differentiable_within_subset subsetD)
lemma holomorphic_transform: "\<lbrakk>f holomorphic_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g holomorphic_on s"
by (metis complex_differentiable_transform_within linordered_field_no_ub holomorphic_on_def)
lemma holomorphic_cong: "s = t ==> (\<And>x. x \<in> s \<Longrightarrow> f x = g x) \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> g holomorphic_on t"
by (metis holomorphic_transform)
lemma holomorphic_on_linear [holomorphic_intros]: "(op * c) holomorphic_on s"
unfolding holomorphic_on_def by (metis complex_differentiable_linear)
lemma holomorphic_on_const [holomorphic_intros]: "(\<lambda>z. c) holomorphic_on s"
unfolding holomorphic_on_def by (metis complex_differentiable_const)
lemma holomorphic_on_ident [holomorphic_intros]: "(\<lambda>x. x) holomorphic_on s"
unfolding holomorphic_on_def by (metis complex_differentiable_ident)
lemma holomorphic_on_id [holomorphic_intros]: "id holomorphic_on s"
unfolding id_def by (rule holomorphic_on_ident)
lemma holomorphic_on_compose:
"f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s) \<Longrightarrow> (g o f) holomorphic_on s"
using complex_differentiable_compose_within[of f _ s g]
by (auto simp: holomorphic_on_def)
lemma holomorphic_on_compose_gen:
"f holomorphic_on s \<Longrightarrow> g holomorphic_on t \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> (g o f) holomorphic_on s"
by (metis holomorphic_on_compose holomorphic_on_subset)
lemma holomorphic_on_minus [holomorphic_intros]: "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on s"
by (metis complex_differentiable_minus holomorphic_on_def)
lemma holomorphic_on_add [holomorphic_intros]:
"\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on s"
unfolding holomorphic_on_def by (metis complex_differentiable_add)
lemma holomorphic_on_diff [holomorphic_intros]:
"\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) holomorphic_on s"
unfolding holomorphic_on_def by (metis complex_differentiable_diff)
lemma holomorphic_on_mult [holomorphic_intros]:
"\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) holomorphic_on s"
unfolding holomorphic_on_def by (metis complex_differentiable_mult)
lemma holomorphic_on_inverse [holomorphic_intros]:
"\<lbrakk>f holomorphic_on s; \<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>z. inverse (f z)) holomorphic_on s"
unfolding holomorphic_on_def by (metis complex_differentiable_inverse)
lemma holomorphic_on_divide [holomorphic_intros]:
"\<lbrakk>f holomorphic_on s; g holomorphic_on s; \<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>z. f z / g z) holomorphic_on s"
unfolding holomorphic_on_def by (metis complex_differentiable_divide)
lemma holomorphic_on_power [holomorphic_intros]:
"f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on s"
unfolding holomorphic_on_def by (metis complex_differentiable_power)
lemma holomorphic_on_setsum [holomorphic_intros]:
"(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) holomorphic_on s"
unfolding holomorphic_on_def by (metis complex_differentiable_setsum)
lemma DERIV_deriv_iff_complex_differentiable:
"DERIV f x :> deriv f x \<longleftrightarrow> f complex_differentiable at x"
unfolding complex_differentiable_def by (metis DERIV_imp_deriv)
lemma complex_derivative_chain:
"f complex_differentiable at x \<Longrightarrow> g complex_differentiable at (f x)
\<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
by (metis DERIV_deriv_iff_complex_differentiable DERIV_chain DERIV_imp_deriv)
lemma deriv_linear: "deriv (\<lambda>w. c * w) = (\<lambda>z. c)"
by (metis DERIV_imp_deriv DERIV_cmult_Id)
lemma deriv_ident: "deriv (\<lambda>w. w) = (\<lambda>z. 1)"
by (metis DERIV_imp_deriv DERIV_ident)
lemma deriv_const: "deriv (\<lambda>w. c) = (\<lambda>z. 0)"
by (metis DERIV_imp_deriv DERIV_const)
lemma complex_derivative_add:
"\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>
\<Longrightarrow> deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
by (auto intro!: DERIV_imp_deriv derivative_intros)
lemma complex_derivative_diff:
"\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>
\<Longrightarrow> deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
by (auto intro!: DERIV_imp_deriv derivative_intros)
lemma complex_derivative_mult:
"\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>
\<Longrightarrow> deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
lemma complex_derivative_cmult:
"f complex_differentiable at z \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z"
unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
lemma complex_derivative_cmult_right:
"f complex_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c"
unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
lemma complex_derivative_transform_within_open:
"\<lbrakk>f holomorphic_on s; g holomorphic_on s; open s; z \<in> s; \<And>w. w \<in> s \<Longrightarrow> f w = g w\<rbrakk>
\<Longrightarrow> deriv f z = deriv g z"
unfolding holomorphic_on_def
by (rule DERIV_imp_deriv)
(metis DERIV_deriv_iff_complex_differentiable DERIV_transform_within_open at_within_open)
lemma complex_derivative_compose_linear:
"f complex_differentiable at (c * z) \<Longrightarrow> deriv (\<lambda>w. f (c * w)) z = c * deriv f (c * z)"
apply (rule DERIV_imp_deriv)
apply (simp add: DERIV_deriv_iff_complex_differentiable [symmetric])
apply (drule DERIV_chain' [of "times c" c z UNIV f "deriv f (c * z)", OF DERIV_cmult_Id])
apply (simp add: algebra_simps)
done
subsection\<open>Analyticity on a set\<close>
definition analytic_on (infixl "(analytic'_on)" 50)
where
"f analytic_on s \<equiv> \<forall>x \<in> s. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)"
lemma analytic_imp_holomorphic: "f analytic_on s \<Longrightarrow> f holomorphic_on s"
by (simp add: at_within_open [OF _ open_ball] analytic_on_def holomorphic_on_def)
(metis centre_in_ball complex_differentiable_at_within)
lemma analytic_on_open: "open s \<Longrightarrow> f analytic_on s \<longleftrightarrow> f holomorphic_on s"
apply (auto simp: analytic_imp_holomorphic)
apply (auto simp: analytic_on_def holomorphic_on_def)
by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball)
lemma analytic_on_imp_differentiable_at:
"f analytic_on s \<Longrightarrow> x \<in> s \<Longrightarrow> f complex_differentiable (at x)"
apply (auto simp: analytic_on_def holomorphic_on_def)
by (metis Topology_Euclidean_Space.open_ball centre_in_ball complex_differentiable_within_open)
lemma analytic_on_subset: "f analytic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f analytic_on t"
by (auto simp: analytic_on_def)
lemma analytic_on_Un: "f analytic_on (s \<union> t) \<longleftrightarrow> f analytic_on s \<and> f analytic_on t"
by (auto simp: analytic_on_def)
lemma analytic_on_Union: "f analytic_on (\<Union>s) \<longleftrightarrow> (\<forall>t \<in> s. f analytic_on t)"
by (auto simp: analytic_on_def)
lemma analytic_on_UN: "f analytic_on (\<Union>i\<in>I. s i) \<longleftrightarrow> (\<forall>i\<in>I. f analytic_on (s i))"
by (auto simp: analytic_on_def)
lemma analytic_on_holomorphic:
"f analytic_on s \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f holomorphic_on t)"
(is "?lhs = ?rhs")
proof -
have "?lhs \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t)"
proof safe
assume "f analytic_on s"
then show "\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t"
apply (simp add: analytic_on_def)
apply (rule exI [where x="\<Union>{u. open u \<and> f analytic_on u}"], auto)
apply (metis Topology_Euclidean_Space.open_ball analytic_on_open centre_in_ball)
by (metis analytic_on_def)
next
fix t
assume "open t" "s \<subseteq> t" "f analytic_on t"
then show "f analytic_on s"
by (metis analytic_on_subset)
qed
also have "... \<longleftrightarrow> ?rhs"
by (auto simp: analytic_on_open)
finally show ?thesis .
qed
lemma analytic_on_linear: "(op * c) analytic_on s"
by (auto simp add: analytic_on_holomorphic holomorphic_on_linear)
lemma analytic_on_const: "(\<lambda>z. c) analytic_on s"
by (metis analytic_on_def holomorphic_on_const zero_less_one)
lemma analytic_on_ident: "(\<lambda>x. x) analytic_on s"
by (simp add: analytic_on_def holomorphic_on_ident gt_ex)
lemma analytic_on_id: "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 \<in> 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 complex_differentiable_imp_continuous_at f x)
with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'"
by (auto simp: continuous_at_ball)
have "g \<circ> 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 "\<exists>e>0. g \<circ> f holomorphic_on ball x e"
by (metis d e min_less_iff_conj)
qed
lemma analytic_on_compose_gen:
"f analytic_on s \<Longrightarrow> g analytic_on t \<Longrightarrow> (\<And>z. z \<in> s \<Longrightarrow> f z \<in> t)
\<Longrightarrow> g o f analytic_on s"
by (metis analytic_on_compose analytic_on_subset image_subset_iff)
lemma analytic_on_neg:
"f analytic_on s \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on s"
by (metis analytic_on_holomorphic holomorphic_on_minus)
lemma analytic_on_add:
assumes f: "f analytic_on s"
and g: "g analytic_on s"
shows "(\<lambda>z. f z + g z) analytic_on s"
unfolding analytic_on_def
proof (intro ballI)
fix z
assume z: "z \<in> 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 "(\<lambda>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 "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e"
by (metis e e' min_less_iff_conj)
qed
lemma analytic_on_diff:
assumes f: "f analytic_on s"
and g: "g analytic_on s"
shows "(\<lambda>z. f z - g z) analytic_on s"
unfolding analytic_on_def
proof (intro ballI)
fix z
assume z: "z \<in> 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 "(\<lambda>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 "\<exists>e>0. (\<lambda>z. f z - g z) holomorphic_on ball z e"
by (metis e e' min_less_iff_conj)
qed
lemma analytic_on_mult:
assumes f: "f analytic_on s"
and g: "g analytic_on s"
shows "(\<lambda>z. f z * g z) analytic_on s"
unfolding analytic_on_def
proof (intro ballI)
fix z
assume z: "z \<in> 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 "(\<lambda>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 "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e"
by (metis e e' min_less_iff_conj)
qed
lemma analytic_on_inverse:
assumes f: "f analytic_on s"
and nz: "(\<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0)"
shows "(\<lambda>z. inverse (f z)) analytic_on s"
unfolding analytic_on_def
proof (intro ballI)
fix z
assume z: "z \<in> 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': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0"
by (metis Topology_Euclidean_Space.open_ball centre_in_ball continuous_on_open_avoid e z nz)
have "(\<lambda>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 "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e"
by (metis e e' min_less_iff_conj)
qed
lemma analytic_on_divide:
assumes f: "f analytic_on s"
and g: "g analytic_on s"
and nz: "(\<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0)"
shows "(\<lambda>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:
"f analytic_on s \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on s"
by (induct n) (auto simp: analytic_on_const analytic_on_mult)
lemma analytic_on_setsum:
"(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on s) \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) analytic_on s"
by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_const analytic_on_add)
subsection\<open>analyticity at a point.\<close>
lemma analytic_at_ball:
"f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
by (metis analytic_on_def singleton_iff)
lemma analytic_at:
"f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
by (metis analytic_on_holomorphic empty_subsetI insert_subset)
lemma analytic_on_analytic_at:
"f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
by (metis analytic_at_ball analytic_on_def)
lemma analytic_at_two:
"f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
(\<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 \<in> s" "f holomorphic_on s"
"open t" "z \<in> t" "g holomorphic_on t"
by (auto simp: analytic_at)
show ?rhs
apply (rule_tac x="s \<inter> t" in exI)
using st
apply (auto simp: Diff_subset holomorphic_on_subset)
done
next
assume ?rhs
then show ?lhs
by (force simp add: analytic_at)
qed
subsection\<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 (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
and complex_derivative_diff_at: "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
and complex_derivative_mult_at: "deriv (\<lambda>w. f w * g w) z =
f z * deriv g z + deriv f z * g z"
proof -
obtain s where s: "open s" "z \<in> s" "f holomorphic_on s" "g holomorphic_on s"
using assms by (metis analytic_at_two)
show "deriv (\<lambda>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 complex_differentiable_def DERIV_deriv_iff_complex_differentiable)
done
show "deriv (\<lambda>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 complex_differentiable_def DERIV_deriv_iff_complex_differentiable)
done
show "deriv (\<lambda>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 complex_differentiable_def DERIV_deriv_iff_complex_differentiable)
done
qed
lemma complex_derivative_cmult_at:
"f analytic_on {z} \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z"
by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
lemma complex_derivative_cmult_right_at:
"f analytic_on {z} \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c"
by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
subsection\<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: "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s \<longrightarrow> norm (f' n x - g' x) \<le> e"
and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and>
(g has_field_derivative (g' x)) (at x within s)"
proof -
from assms obtain x l where x: "x \<in> s" and tf: "((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
by blast
{ fix e::real assume e: "e > 0"
then obtain N where N: "\<forall>n\<ge>N. \<forall>x. x \<in> s \<longrightarrow> cmod (f' n x - g' x) \<le> e"
by (metis conv)
have "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
proof (rule exI [of _ N], clarify)
fix n y h
assume "N \<le> n" "y \<in> s"
then have "cmod (f' n y - g' y) \<le> e"
by (metis N)
then have "cmod h * cmod (f' n y - g' y) \<le> 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) \<le> 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 "\<forall>n. \<forall>x\<in>s. (f n has_derivative (op * (f' n x))) (at x within s)"
by (metis has_field_derivative_def df)
next show "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
by (rule tf)
next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
by (blast intro: **)
qed
qed
lemma has_complex_derivative_series:
fixes s :: "complex set"
assumes cvs: "convex s"
and df: "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s
\<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) sums l)"
shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) sums g x) \<and> ((g has_field_derivative g' x) (at x within s))"
proof -
from assms obtain x l where x: "x \<in> s" and sf: "((\<lambda>n. f n x) sums l)"
by blast
{ fix e::real assume e: "e > 0"
then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> s
\<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
by (metis conv)
have "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
proof (rule exI [of _ N], clarify)
fix n y h
assume "N \<le> n" "y \<in> s"
then have "cmod ((\<Sum>i<n. f' i y) - g' y) \<le> e"
by (metis N)
then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e"
by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h"
by (simp add: norm_mult [symmetric] field_simps setsum_right_distrib)
qed
} note ** = this
show ?thesis
unfolding has_field_derivative_def
proof (rule has_derivative_series [OF cvs _ _ x])
fix n x
assume "x \<in> s"
then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within s)"
by (metis df has_field_derivative_def mult_commute_abs)
next show " ((\<lambda>n. f n x) sums l)"
by (rule sf)
next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
by (blast intro: **)
qed
qed
lemma complex_differentiable_series:
fixes f :: "nat \<Rightarrow> complex \<Rightarrow> complex"
assumes "convex s" "open s"
assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)" and x: "x \<in> s"
shows "summable (\<lambda>n. f n x)" and "(\<lambda>x. \<Sum>n. f n x) complex_differentiable (at x)"
proof -
from assms(4) obtain g' where A: "uniform_limit s (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
unfolding uniformly_convergent_on_def by blast
from x and \<open>open s\<close> have s: "at x within s = at x" by (rule at_within_open)
have "\<exists>g. \<forall>x\<in>s. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within s)"
by (intro has_field_derivative_series[of s f f' g' x0] assms A has_field_derivative_at_within)
then obtain g where g: "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>n. f n x) sums g x"
"\<And>x. x \<in> s \<Longrightarrow> (g has_field_derivative g' x) (at x within s)" by blast
from g[OF x] show "summable (\<lambda>n. f n x)" by (auto simp: summable_def)
from g(2)[OF x] have g': "(g has_derivative op * (g' x)) (at x)"
by (simp add: has_field_derivative_def s)
have "((\<lambda>x. \<Sum>n. f n x) has_derivative op * (g' x)) (at x)"
by (rule has_derivative_transform_within_open[OF \<open>open s\<close> x _ g'])
(insert g, auto simp: sums_iff)
thus "(\<lambda>x. \<Sum>n. f n x) complex_differentiable (at x)" unfolding differentiable_def
by (auto simp: summable_def complex_differentiable_def has_field_derivative_def)
qed
lemma complex_differentiable_series':
fixes f :: "nat \<Rightarrow> complex \<Rightarrow> complex"
assumes "convex s" "open s"
assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)"
shows "(\<lambda>x. \<Sum>n. f n x) complex_differentiable (at x0)"
using complex_differentiable_series[OF assms, of x0] \<open>x0 \<in> s\<close> by blast+
subsection\<open>Bound theorem\<close>
lemma complex_differentiable_bound:
fixes s :: "complex set"
assumes cvs: "convex s"
and df: "\<And>z. z \<in> s \<Longrightarrow> (f has_field_derivative f' z) (at z within s)"
and dn: "\<And>z. z \<in> s \<Longrightarrow> norm (f' z) \<le> B"
and "x \<in> s" "y \<in> s"
shows "norm(f x - f y) \<le> B * norm(x - y)"
apply (rule differentiable_bound [OF cvs])
apply (rule ballI, erule df [unfolded has_field_derivative_def])
apply (rule ballI, rule onorm_le, simp add: norm_mult mult_right_mono dn)
apply fact
apply fact
done
subsection\<open>Inverse function theorem for complex derivatives.\<close>
lemma has_complex_derivative_inverse_basic:
fixes f :: "complex \<Rightarrow> complex"
shows "DERIV f (g y) :> f' \<Longrightarrow>
f' \<noteq> 0 \<Longrightarrow>
continuous (at y) g \<Longrightarrow>
open t \<Longrightarrow>
y \<in> t \<Longrightarrow>
(\<And>z. z \<in> t \<Longrightarrow> f (g z) = z)
\<Longrightarrow> DERIV g y :> inverse (f')"
unfolding has_field_derivative_def
apply (rule has_derivative_inverse_basic)
apply (auto simp: bounded_linear_mult_right)
done
(*Used only once, in Multivariate/cauchy.ml. *)
lemma has_complex_derivative_inverse_strong:
fixes f :: "complex \<Rightarrow> complex"
shows "DERIV f x :> f' \<Longrightarrow>
f' \<noteq> 0 \<Longrightarrow>
open s \<Longrightarrow>
x \<in> s \<Longrightarrow>
continuous_on s f \<Longrightarrow>
(\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
\<Longrightarrow> DERIV g (f x) :> inverse (f')"
unfolding has_field_derivative_def
apply (rule has_derivative_inverse_strong [of s x f g ])
using assms
by auto
lemma has_complex_derivative_inverse_strong_x:
fixes f :: "complex \<Rightarrow> complex"
shows "DERIV f (g y) :> f' \<Longrightarrow>
f' \<noteq> 0 \<Longrightarrow>
open s \<Longrightarrow>
continuous_on s f \<Longrightarrow>
g y \<in> s \<Longrightarrow> f(g y) = y \<Longrightarrow>
(\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
\<Longrightarrow> DERIV g y :> inverse (f')"
unfolding has_field_derivative_def
apply (rule has_derivative_inverse_strong_x [of s g y f])
using assms
by auto
subsection \<open>Taylor on Complex Numbers\<close>
lemma setsum_Suc_reindex:
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
shows "setsum f {0..n} = f 0 - f (Suc n) + setsum (\<lambda>i. f (Suc i)) {0..n}"
by (induct n) auto
lemma complex_taylor:
assumes s: "convex s"
and f: "\<And>i x. x \<in> s \<Longrightarrow> i \<le> n \<Longrightarrow> (f i has_field_derivative f (Suc i) x) (at x within s)"
and B: "\<And>x. x \<in> s \<Longrightarrow> cmod (f (Suc n) x) \<le> B"
and w: "w \<in> s"
and z: "z \<in> s"
shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i)))
\<le> B * cmod(z - w)^(Suc n) / fact n"
proof -
have wzs: "closed_segment w z \<subseteq> s" using assms
by (metis convex_contains_segment)
{ fix u
assume "u \<in> closed_segment w z"
then have "u \<in> s"
by (metis wzs subsetD)
have "(\<Sum>i\<le>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 "(\<Sum>i\<le>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.simps)
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.simps)
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.simps 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.simps)
qed
finally show ?case .
qed
then have "((\<lambda>v. (\<Sum>i\<le>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 \<in> closed_segment w z"
then have us: "u \<in> s"
by (metis wzs subsetD)
have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> cmod (f (Suc n) u) * cmod (u - z) ^ n"
by (metis norm_minus_commute order_refl)
also have "... \<le> cmod (f (Suc n) u) * cmod (z - w) ^ n"
by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
also have "... \<le> B * cmod (z - w) ^ n"
by (metis norm_ge_zero zero_le_power mult_right_mono B [OF us])
finally have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> B * cmod (z - w) ^ n" .
} note cmod_bound = this
have "(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)) = (\<Sum>i\<le>n. (f i z / (fact i)) * 0 ^ i)"
by simp
also have "\<dots> = f 0 z / (fact 0)"
by (subst setsum_zero_power) simp
finally have "cmod (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)))
\<le> cmod ((\<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 "... \<le> B * cmod (z - w) ^ n / (fact n) * cmod (w - z)"
apply (rule complex_differentiable_bound
[where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / (fact n)"
and s = "closed_segment w z", OF convex_closed_segment])
apply (auto simp: ends_in_segment DERIV_subset [OF sum_deriv wzs]
norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
done
also have "... \<le> B * cmod (z - w) ^ Suc n / (fact n)"
by (simp add: algebra_simps norm_minus_commute)
finally show ?thesis .
qed
text\<open>Something more like the traditional MVT for real components.\<close>
lemma complex_mvt_line:
assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f'(u)) (at u)"
shows "\<exists>u. u \<in> closed_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
proof -
have twz: "\<And>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 (\<lambda>t. (1 - t) *\<^sub>R w + t *\<^sub>R z)"
"\<lambda>u. Re o (\<lambda>h. f'((1 - u) *\<^sub>R w + u *\<^sub>R z) * h) o (\<lambda>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_within, 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 "\<And>i x. \<lbrakk>x \<in> closed_segment w z; i \<le> n\<rbrakk> \<Longrightarrow> ((f i) has_field_derivative f (Suc i) x) (at x)"
shows "\<exists>u. u \<in> closed_segment w z \<and>
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 \<in> closed_segment w z"
have "(\<Sum>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 setsum_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 setsum_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 "(\<Sum>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 "((\<lambda>u. \<Sum>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 "\<lambda>u. \<Sum>i = 0..n. f i u * (z-u) ^ i / (fact i)"
"\<lambda>u. (f (Suc n) u * (z-u)^n / (fact n))"])
apply (auto simp add: intro: open_closed_segment)
done
qed
subsection \<open>Polynomal function extremal theorem, from HOL Light\<close>
lemma polyfun_extremal_lemma: (*COMPLEX_POLYFUN_EXTREMAL_LEMMA in HOL Light*)
fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
assumes "0 < e"
shows "\<exists>M. \<forall>z. M \<le> norm(z) \<longrightarrow> norm (\<Sum>i\<le>n. c(i) * z^i) \<le> e * norm(z) ^ (Suc n)"
proof (induct n)
case 0 with assms
show ?case
apply (rule_tac x="norm (c 0) / e" in exI)
apply (auto simp: field_simps)
done
next
case (Suc n)
obtain M where M: "\<And>z. M \<le> norm z \<Longrightarrow> norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
using Suc assms by blast
show ?case
proof (rule exI [where x= "max M (1 + norm(c(Suc n)) / e)"], clarsimp simp del: power_Suc)
fix z::'a
assume z1: "M \<le> norm z" and "1 + norm (c (Suc n)) / e \<le> norm z"
then have z2: "e + norm (c (Suc n)) \<le> e * norm z"
using assms by (simp add: field_simps)
have "norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
using M [OF z1] by simp
then have "norm (\<Sum>i\<le>n. c i * z^i) + norm (c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc n + norm (c (Suc n) * z ^ Suc n)"
by simp
then have "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc n + norm (c (Suc n) * z ^ Suc n)"
by (blast intro: norm_triangle_le elim: )
also have "... \<le> (e + norm (c (Suc n))) * norm z ^ Suc n"
by (simp add: norm_power norm_mult algebra_simps)
also have "... \<le> (e * norm z) * norm z ^ Suc n"
by (metis z2 mult.commute mult_left_mono norm_ge_zero norm_power)
finally show "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc (Suc n)"
by simp
qed
qed
lemma polyfun_extremal: (*COMPLEX_POLYFUN_EXTREMAL in HOL Light*)
fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
assumes k: "c k \<noteq> 0" "1\<le>k" and kn: "k\<le>n"
shows "eventually (\<lambda>z. norm (\<Sum>i\<le>n. c(i) * z^i) \<ge> B) at_infinity"
using kn
proof (induction n)
case 0
then show ?case
using k by simp
next
case (Suc m)
let ?even = ?case
show ?even
proof (cases "c (Suc m) = 0")
case True
then show ?even using Suc k
by auto (metis antisym_conv less_eq_Suc_le not_le)
next
case False
then obtain M where M:
"\<And>z. M \<le> norm z \<Longrightarrow> norm (\<Sum>i\<le>m. c i * z^i) \<le> norm (c (Suc m)) / 2 * norm z ^ Suc m"
using polyfun_extremal_lemma [of "norm(c (Suc m)) / 2" c m] Suc
by auto
have "\<exists>b. \<forall>z. b \<le> norm z \<longrightarrow> B \<le> norm (\<Sum>i\<le>Suc m. c i * z^i)"
proof (rule exI [where x="max M (max 1 (\<bar>B\<bar> / (norm(c (Suc m)) / 2)))"], clarsimp simp del: power_Suc)
fix z::'a
assume z1: "M \<le> norm z" "1 \<le> norm z"
and "\<bar>B\<bar> * 2 / norm (c (Suc m)) \<le> norm z"
then have z2: "\<bar>B\<bar> \<le> norm (c (Suc m)) * norm z / 2"
using False by (simp add: field_simps)
have nz: "norm z \<le> norm z ^ Suc m"
by (metis \<open>1 \<le> norm z\<close> One_nat_def less_eq_Suc_le power_increasing power_one_right zero_less_Suc)
have *: "\<And>y x. norm (c (Suc m)) * norm z / 2 \<le> norm y - norm x \<Longrightarrow> B \<le> norm (x + y)"
by (metis abs_le_iff add.commute norm_diff_ineq order_trans z2)
have "norm z * norm (c (Suc m)) + 2 * norm (\<Sum>i\<le>m. c i * z^i)
\<le> norm (c (Suc m)) * norm z + norm (c (Suc m)) * norm z ^ Suc m"
using M [of z] Suc z1 by auto
also have "... \<le> 2 * (norm (c (Suc m)) * norm z ^ Suc m)"
using nz by (simp add: mult_mono del: power_Suc)
finally show "B \<le> norm ((\<Sum>i\<le>m. c i * z^i) + c (Suc m) * z ^ Suc m)"
using Suc.IH
apply (auto simp: eventually_at_infinity)
apply (rule *)
apply (simp add: field_simps norm_mult norm_power)
done
qed
then show ?even
by (simp add: eventually_at_infinity)
qed
qed
end