src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy
 author wenzelm Wed Dec 30 11:21:54 2015 +0100 (2015-12-30) changeset 61973 0c7e865fa7cb parent 61969 e01015e49041 child 61975 b4b11391c676 permissions -rw-r--r--
more symbols;
```     1 (*  Author: John Harrison, Marco Maggesi, Graziano Gentili, Gianni Ciolli, Valentina Bruno
```
```     2     Ported from "hol_light/Multivariate/canal.ml" by L C Paulson (2014)
```
```     3 *)
```
```     4
```
```     5 section \<open>Complex Analysis Basics\<close>
```
```     6
```
```     7 theory Complex_Analysis_Basics
```
```     8 imports Cartesian_Euclidean_Space
```
```     9 begin
```
```    10
```
```    11
```
```    12 lemma cmod_fact [simp]: "cmod (fact n) = fact n"
```
```    13   by (metis norm_of_nat of_nat_fact)
```
```    14
```
```    15 subsection\<open>General lemmas\<close>
```
```    16
```
```    17 lemma has_derivative_mult_right:
```
```    18   fixes c:: "'a :: real_normed_algebra"
```
```    19   shows "((op * c) has_derivative (op * c)) F"
```
```    20 by (rule has_derivative_mult_right [OF has_derivative_id])
```
```    21
```
```    22 lemma has_derivative_of_real[derivative_intros, simp]:
```
```    23   "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. of_real (f x)) has_derivative (\<lambda>x. of_real (f' x))) F"
```
```    24   using bounded_linear.has_derivative[OF bounded_linear_of_real] .
```
```    25
```
```    26 lemma has_vector_derivative_real_complex:
```
```    27   "DERIV f (of_real a) :> f' \<Longrightarrow> ((\<lambda>x. f (of_real x)) has_vector_derivative f') (at a within s)"
```
```    28   using has_derivative_compose[of of_real of_real a _ f "op * f'"]
```
```    29   by (simp add: scaleR_conv_of_real ac_simps has_vector_derivative_def has_field_derivative_def)
```
```    30
```
```    31 lemma fact_cancel:
```
```    32   fixes c :: "'a::real_field"
```
```    33   shows "of_nat (Suc n) * c / (fact (Suc n)) = c / (fact n)"
```
```    34   by (simp add: of_nat_mult del: of_nat_Suc times_nat.simps)
```
```    35
```
```    36 lemma bilinear_times:
```
```    37   fixes c::"'a::real_algebra" shows "bilinear (\<lambda>x y::'a. x*y)"
```
```    38   by (auto simp: bilinear_def distrib_left distrib_right intro!: linearI)
```
```    39
```
```    40 lemma linear_cnj: "linear cnj"
```
```    41   using bounded_linear.linear[OF bounded_linear_cnj] .
```
```    42
```
```    43 lemma tendsto_Re_upper:
```
```    44   assumes "~ (trivial_limit F)"
```
```    45           "(f \<longlongrightarrow> l) F"
```
```    46           "eventually (\<lambda>x. Re(f x) \<le> b) F"
```
```    47     shows  "Re(l) \<le> b"
```
```    48   by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Re)
```
```    49
```
```    50 lemma tendsto_Re_lower:
```
```    51   assumes "~ (trivial_limit F)"
```
```    52           "(f \<longlongrightarrow> l) F"
```
```    53           "eventually (\<lambda>x. b \<le> Re(f x)) F"
```
```    54     shows  "b \<le> Re(l)"
```
```    55   by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Re)
```
```    56
```
```    57 lemma tendsto_Im_upper:
```
```    58   assumes "~ (trivial_limit F)"
```
```    59           "(f \<longlongrightarrow> l) F"
```
```    60           "eventually (\<lambda>x. Im(f x) \<le> b) F"
```
```    61     shows  "Im(l) \<le> b"
```
```    62   by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Im)
```
```    63
```
```    64 lemma tendsto_Im_lower:
```
```    65   assumes "~ (trivial_limit F)"
```
```    66           "(f \<longlongrightarrow> l) F"
```
```    67           "eventually (\<lambda>x. b \<le> Im(f x)) F"
```
```    68     shows  "b \<le> Im(l)"
```
```    69   by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Im)
```
```    70
```
```    71 lemma lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = op * 0"
```
```    72   by auto
```
```    73
```
```    74 lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = op * 1"
```
```    75   by auto
```
```    76
```
```    77 lemma continuous_mult_left:
```
```    78   fixes c::"'a::real_normed_algebra"
```
```    79   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. c * f x)"
```
```    80 by (rule continuous_mult [OF continuous_const])
```
```    81
```
```    82 lemma continuous_mult_right:
```
```    83   fixes c::"'a::real_normed_algebra"
```
```    84   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x * c)"
```
```    85 by (rule continuous_mult [OF _ continuous_const])
```
```    86
```
```    87 lemma continuous_on_mult_left:
```
```    88   fixes c::"'a::real_normed_algebra"
```
```    89   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c * f x)"
```
```    90 by (rule continuous_on_mult [OF continuous_on_const])
```
```    91
```
```    92 lemma continuous_on_mult_right:
```
```    93   fixes c::"'a::real_normed_algebra"
```
```    94   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x * c)"
```
```    95 by (rule continuous_on_mult [OF _ continuous_on_const])
```
```    96
```
```    97 lemma uniformly_continuous_on_cmul_right [continuous_intros]:
```
```    98   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
```
```    99   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. f x * c)"
```
```   100   using bounded_linear.uniformly_continuous_on[OF bounded_linear_mult_left] .
```
```   101
```
```   102 lemma uniformly_continuous_on_cmul_left[continuous_intros]:
```
```   103   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
```
```   104   assumes "uniformly_continuous_on s f"
```
```   105     shows "uniformly_continuous_on s (\<lambda>x. c * f x)"
```
```   106 by (metis assms bounded_linear.uniformly_continuous_on bounded_linear_mult_right)
```
```   107
```
```   108 lemma continuous_within_norm_id [continuous_intros]: "continuous (at x within S) norm"
```
```   109   by (rule continuous_norm [OF continuous_ident])
```
```   110
```
```   111 lemma continuous_on_norm_id [continuous_intros]: "continuous_on S norm"
```
```   112   by (intro continuous_on_id continuous_on_norm)
```
```   113
```
```   114 subsection\<open>DERIV stuff\<close>
```
```   115
```
```   116 lemma DERIV_zero_connected_constant:
```
```   117   fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
```
```   118   assumes "connected s"
```
```   119       and "open s"
```
```   120       and "finite k"
```
```   121       and "continuous_on s f"
```
```   122       and "\<forall>x\<in>(s - k). DERIV f x :> 0"
```
```   123     obtains c where "\<And>x. x \<in> s \<Longrightarrow> f(x) = c"
```
```   124 using has_derivative_zero_connected_constant [OF assms(1-4)] assms
```
```   125 by (metis DERIV_const has_derivative_const Diff_iff at_within_open frechet_derivative_at has_field_derivative_def)
```
```   126
```
```   127 lemma DERIV_zero_constant:
```
```   128   fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
```
```   129   shows    "\<lbrakk>convex s;
```
```   130              \<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)\<rbrakk>
```
```   131              \<Longrightarrow> \<exists>c. \<forall>x \<in> s. f(x) = c"
```
```   132   by (auto simp: has_field_derivative_def lambda_zero intro: has_derivative_zero_constant)
```
```   133
```
```   134 lemma DERIV_zero_unique:
```
```   135   fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
```
```   136   assumes "convex s"
```
```   137       and d0: "\<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)"
```
```   138       and "a \<in> s"
```
```   139       and "x \<in> s"
```
```   140     shows "f x = f a"
```
```   141   by (rule has_derivative_zero_unique [OF assms(1) _ assms(4,3)])
```
```   142      (metis d0 has_field_derivative_imp_has_derivative lambda_zero)
```
```   143
```
```   144 lemma DERIV_zero_connected_unique:
```
```   145   fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
```
```   146   assumes "connected s"
```
```   147       and "open s"
```
```   148       and d0: "\<And>x. x\<in>s \<Longrightarrow> DERIV f x :> 0"
```
```   149       and "a \<in> s"
```
```   150       and "x \<in> s"
```
```   151     shows "f x = f a"
```
```   152     by (rule has_derivative_zero_unique_connected [OF assms(2,1) _ assms(5,4)])
```
```   153        (metis has_field_derivative_def lambda_zero d0)
```
```   154
```
```   155 lemma DERIV_transform_within:
```
```   156   assumes "(f has_field_derivative f') (at a within s)"
```
```   157       and "0 < d" "a \<in> s"
```
```   158       and "\<And>x. x\<in>s \<Longrightarrow> dist x a < d \<Longrightarrow> f x = g x"
```
```   159     shows "(g has_field_derivative f') (at a within s)"
```
```   160   using assms unfolding has_field_derivative_def
```
```   161   by (blast intro: has_derivative_transform_within)
```
```   162
```
```   163 lemma DERIV_transform_within_open:
```
```   164   assumes "DERIV f a :> f'"
```
```   165       and "open s" "a \<in> s"
```
```   166       and "\<And>x. x\<in>s \<Longrightarrow> f x = g x"
```
```   167     shows "DERIV g a :> f'"
```
```   168   using assms unfolding has_field_derivative_def
```
```   169 by (metis has_derivative_transform_within_open)
```
```   170
```
```   171 lemma DERIV_transform_at:
```
```   172   assumes "DERIV f a :> f'"
```
```   173       and "0 < d"
```
```   174       and "\<And>x. dist x a < d \<Longrightarrow> f x = g x"
```
```   175     shows "DERIV g a :> f'"
```
```   176   by (blast intro: assms DERIV_transform_within)
```
```   177
```
```   178 (*generalising DERIV_isconst_all, which requires type real (using the ordering)*)
```
```   179 lemma DERIV_zero_UNIV_unique:
```
```   180   fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
```
```   181   shows "(\<And>x. DERIV f x :> 0) \<Longrightarrow> f x = f a"
```
```   182 by (metis DERIV_zero_unique UNIV_I assms convex_UNIV)
```
```   183
```
```   184 subsection \<open>Some limit theorems about real part of real series etc.\<close>
```
```   185
```
```   186 (*MOVE? But not to Finite_Cartesian_Product*)
```
```   187 lemma sums_vec_nth :
```
```   188   assumes "f sums a"
```
```   189   shows "(\<lambda>x. f x \$ i) sums a \$ i"
```
```   190 using assms unfolding sums_def
```
```   191 by (auto dest: tendsto_vec_nth [where i=i])
```
```   192
```
```   193 lemma summable_vec_nth :
```
```   194   assumes "summable f"
```
```   195   shows "summable (\<lambda>x. f x \$ i)"
```
```   196 using assms unfolding summable_def
```
```   197 by (blast intro: sums_vec_nth)
```
```   198
```
```   199 subsection \<open>Complex number lemmas\<close>
```
```   200
```
```   201 lemma
```
```   202   shows open_halfspace_Re_lt: "open {z. Re(z) < b}"
```
```   203     and open_halfspace_Re_gt: "open {z. Re(z) > b}"
```
```   204     and closed_halfspace_Re_ge: "closed {z. Re(z) \<ge> b}"
```
```   205     and closed_halfspace_Re_le: "closed {z. Re(z) \<le> b}"
```
```   206     and closed_halfspace_Re_eq: "closed {z. Re(z) = b}"
```
```   207     and open_halfspace_Im_lt: "open {z. Im(z) < b}"
```
```   208     and open_halfspace_Im_gt: "open {z. Im(z) > b}"
```
```   209     and closed_halfspace_Im_ge: "closed {z. Im(z) \<ge> b}"
```
```   210     and closed_halfspace_Im_le: "closed {z. Im(z) \<le> b}"
```
```   211     and closed_halfspace_Im_eq: "closed {z. Im(z) = b}"
```
```   212   by (intro open_Collect_less closed_Collect_le closed_Collect_eq isCont_Re
```
```   213             isCont_Im continuous_ident continuous_const)+
```
```   214
```
```   215 lemma closed_complex_Reals: "closed (\<real> :: complex set)"
```
```   216 proof -
```
```   217   have "(\<real> :: complex set) = {z. Im z = 0}"
```
```   218     by (auto simp: complex_is_Real_iff)
```
```   219   then show ?thesis
```
```   220     by (metis closed_halfspace_Im_eq)
```
```   221 qed
```
```   222
```
```   223 lemma closed_Real_halfspace_Re_le: "closed (\<real> \<inter> {w. Re w \<le> x})"
```
```   224   by (simp add: closed_Int closed_complex_Reals closed_halfspace_Re_le)
```
```   225
```
```   226 lemma closed_Real_halfspace_Re_ge: "closed (\<real> \<inter> {w. x \<le> Re(w)})"
```
```   227   using closed_halfspace_Re_ge
```
```   228   by (simp add: closed_Int closed_complex_Reals)
```
```   229
```
```   230 lemma closed_real_abs_le: "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
```
```   231 proof -
```
```   232   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})"
```
```   233     by auto
```
```   234   then show "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
```
```   235     by (simp add: closed_Int closed_Real_halfspace_Re_ge closed_Real_halfspace_Re_le)
```
```   236 qed
```
```   237
```
```   238 lemma real_lim:
```
```   239   fixes l::complex
```
```   240   assumes "(f \<longlongrightarrow> l) F" and "~(trivial_limit F)" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>"
```
```   241   shows  "l \<in> \<real>"
```
```   242 proof (rule Lim_in_closed_set[OF closed_complex_Reals _ assms(2,1)])
```
```   243   show "eventually (\<lambda>x. f x \<in> \<real>) F"
```
```   244     using assms(3, 4) by (auto intro: eventually_mono)
```
```   245 qed
```
```   246
```
```   247 lemma real_lim_sequentially:
```
```   248   fixes l::complex
```
```   249   shows "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
```
```   250 by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially)
```
```   251
```
```   252 lemma real_series:
```
```   253   fixes l::complex
```
```   254   shows "f sums l \<Longrightarrow> (\<And>n. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
```
```   255 unfolding sums_def
```
```   256 by (metis real_lim_sequentially setsum_in_Reals)
```
```   257
```
```   258 lemma Lim_null_comparison_Re:
```
```   259   assumes "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F" "(g \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F"
```
```   260   by (rule Lim_null_comparison[OF assms(1)] tendsto_eq_intros assms(2))+ simp
```
```   261
```
```   262 subsection\<open>Holomorphic functions\<close>
```
```   263
```
```   264 text{*Could be generalized to real normed fields, but in practice that would only include the reals*}
```
```   265 definition complex_differentiable :: "[complex \<Rightarrow> complex, complex filter] \<Rightarrow> bool"
```
```   266            (infixr "(complex'_differentiable)" 50)
```
```   267   where "f complex_differentiable F \<equiv> \<exists>f'. (f has_field_derivative f') F"
```
```   268
```
```   269 lemma complex_differentiable_imp_continuous_at:
```
```   270     "f complex_differentiable (at x within s) \<Longrightarrow> continuous (at x within s) f"
```
```   271   by (metis DERIV_continuous complex_differentiable_def)
```
```   272
```
```   273 lemma complex_differentiable_within_subset:
```
```   274     "\<lbrakk>f complex_differentiable (at x within s); t \<subseteq> s\<rbrakk>
```
```   275      \<Longrightarrow> f complex_differentiable (at x within t)"
```
```   276   by (metis DERIV_subset complex_differentiable_def)
```
```   277
```
```   278 lemma complex_differentiable_at_within:
```
```   279     "\<lbrakk>f complex_differentiable (at x)\<rbrakk>
```
```   280      \<Longrightarrow> f complex_differentiable (at x within s)"
```
```   281   unfolding complex_differentiable_def
```
```   282   by (metis DERIV_subset top_greatest)
```
```   283
```
```   284 lemma complex_differentiable_linear [derivative_intros]: "(op * c) complex_differentiable F"
```
```   285 proof -
```
```   286   show ?thesis
```
```   287     unfolding complex_differentiable_def has_field_derivative_def mult_commute_abs
```
```   288     by (force intro: has_derivative_mult_right)
```
```   289 qed
```
```   290
```
```   291 lemma complex_differentiable_const [derivative_intros]: "(\<lambda>z. c) complex_differentiable F"
```
```   292   unfolding complex_differentiable_def has_field_derivative_def
```
```   293   by (rule exI [where x=0])
```
```   294      (metis has_derivative_const lambda_zero)
```
```   295
```
```   296 lemma complex_differentiable_ident [derivative_intros]: "(\<lambda>z. z) complex_differentiable F"
```
```   297   unfolding complex_differentiable_def has_field_derivative_def
```
```   298   by (rule exI [where x=1])
```
```   299      (simp add: lambda_one [symmetric])
```
```   300
```
```   301 lemma complex_differentiable_id [derivative_intros]: "id complex_differentiable F"
```
```   302   unfolding id_def by (rule complex_differentiable_ident)
```
```   303
```
```   304 lemma complex_differentiable_minus [derivative_intros]:
```
```   305   "f complex_differentiable F \<Longrightarrow> (\<lambda>z. - (f z)) complex_differentiable F"
```
```   306   using assms unfolding complex_differentiable_def
```
```   307   by (metis field_differentiable_minus)
```
```   308
```
```   309 lemma complex_differentiable_add [derivative_intros]:
```
```   310   assumes "f complex_differentiable F" "g complex_differentiable F"
```
```   311     shows "(\<lambda>z. f z + g z) complex_differentiable F"
```
```   312   using assms unfolding complex_differentiable_def
```
```   313   by (metis field_differentiable_add)
```
```   314
```
```   315 lemma complex_differentiable_setsum [derivative_intros]:
```
```   316   "(\<And>i. i \<in> I \<Longrightarrow> (f i) complex_differentiable F) \<Longrightarrow> (\<lambda>z. \<Sum>i\<in>I. f i z) complex_differentiable F"
```
```   317   by (induct I rule: infinite_finite_induct)
```
```   318      (auto intro: complex_differentiable_add complex_differentiable_const)
```
```   319
```
```   320 lemma complex_differentiable_diff [derivative_intros]:
```
```   321   assumes "f complex_differentiable F" "g complex_differentiable F"
```
```   322     shows "(\<lambda>z. f z - g z) complex_differentiable F"
```
```   323   using assms unfolding complex_differentiable_def
```
```   324   by (metis field_differentiable_diff)
```
```   325
```
```   326 lemma complex_differentiable_inverse [derivative_intros]:
```
```   327   assumes "f complex_differentiable (at a within s)" "f a \<noteq> 0"
```
```   328   shows "(\<lambda>z. inverse (f z)) complex_differentiable (at a within s)"
```
```   329   using assms unfolding complex_differentiable_def
```
```   330   by (metis DERIV_inverse_fun)
```
```   331
```
```   332 lemma complex_differentiable_mult [derivative_intros]:
```
```   333   assumes "f complex_differentiable (at a within s)"
```
```   334           "g complex_differentiable (at a within s)"
```
```   335     shows "(\<lambda>z. f z * g z) complex_differentiable (at a within s)"
```
```   336   using assms unfolding complex_differentiable_def
```
```   337   by (metis DERIV_mult [of f _ a s g])
```
```   338
```
```   339 lemma complex_differentiable_divide [derivative_intros]:
```
```   340   assumes "f complex_differentiable (at a within s)"
```
```   341           "g complex_differentiable (at a within s)"
```
```   342           "g a \<noteq> 0"
```
```   343     shows "(\<lambda>z. f z / g z) complex_differentiable (at a within s)"
```
```   344   using assms unfolding complex_differentiable_def
```
```   345   by (metis DERIV_divide [of f _ a s g])
```
```   346
```
```   347 lemma complex_differentiable_power [derivative_intros]:
```
```   348   assumes "f complex_differentiable (at a within s)"
```
```   349     shows "(\<lambda>z. f z ^ n) complex_differentiable (at a within s)"
```
```   350   using assms unfolding complex_differentiable_def
```
```   351   by (metis DERIV_power)
```
```   352
```
```   353 lemma complex_differentiable_transform_within:
```
```   354   "0 < d \<Longrightarrow>
```
```   355         x \<in> s \<Longrightarrow>
```
```   356         (\<And>x'. x' \<in> s \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x') \<Longrightarrow>
```
```   357         f complex_differentiable (at x within s)
```
```   358         \<Longrightarrow> g complex_differentiable (at x within s)"
```
```   359   unfolding complex_differentiable_def has_field_derivative_def
```
```   360   by (blast intro: has_derivative_transform_within)
```
```   361
```
```   362 lemma complex_differentiable_compose_within:
```
```   363   assumes "f complex_differentiable (at a within s)"
```
```   364           "g complex_differentiable (at (f a) within f`s)"
```
```   365     shows "(g o f) complex_differentiable (at a within s)"
```
```   366   using assms unfolding complex_differentiable_def
```
```   367   by (metis DERIV_image_chain)
```
```   368
```
```   369 lemma complex_differentiable_compose:
```
```   370   "f complex_differentiable at z \<Longrightarrow> g complex_differentiable at (f z)
```
```   371           \<Longrightarrow> (g o f) complex_differentiable at z"
```
```   372 by (metis complex_differentiable_at_within complex_differentiable_compose_within)
```
```   373
```
```   374 lemma complex_differentiable_within_open:
```
```   375      "\<lbrakk>a \<in> s; open s\<rbrakk> \<Longrightarrow> f complex_differentiable at a within s \<longleftrightarrow>
```
```   376                           f complex_differentiable at a"
```
```   377   unfolding complex_differentiable_def
```
```   378   by (metis at_within_open)
```
```   379
```
```   380 subsection\<open>Caratheodory characterization.\<close>
```
```   381
```
```   382 lemma complex_differentiable_caratheodory_at:
```
```   383   "f complex_differentiable (at z) \<longleftrightarrow>
```
```   384          (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)"
```
```   385   using CARAT_DERIV [of f]
```
```   386   by (simp add: complex_differentiable_def has_field_derivative_def)
```
```   387
```
```   388 lemma complex_differentiable_caratheodory_within:
```
```   389   "f complex_differentiable (at z within s) \<longleftrightarrow>
```
```   390          (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)"
```
```   391   using DERIV_caratheodory_within [of f]
```
```   392   by (simp add: complex_differentiable_def has_field_derivative_def)
```
```   393
```
```   394 subsection\<open>Holomorphic\<close>
```
```   395
```
```   396 definition holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
```
```   397            (infixl "(holomorphic'_on)" 50)
```
```   398   where "f holomorphic_on s \<equiv> \<forall>x\<in>s. f complex_differentiable (at x within s)"
```
```   399
```
```   400 named_theorems holomorphic_intros "structural introduction rules for holomorphic_on"
```
```   401
```
```   402 lemma holomorphic_on_empty [holomorphic_intros]: "f holomorphic_on {}"
```
```   403   by (simp add: holomorphic_on_def)
```
```   404
```
```   405 lemma holomorphic_on_open:
```
```   406     "open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> f')"
```
```   407   by (auto simp: holomorphic_on_def complex_differentiable_def has_field_derivative_def at_within_open [of _ s])
```
```   408
```
```   409 lemma holomorphic_on_imp_continuous_on:
```
```   410     "f holomorphic_on s \<Longrightarrow> continuous_on s f"
```
```   411   by (metis complex_differentiable_imp_continuous_at continuous_on_eq_continuous_within holomorphic_on_def)
```
```   412
```
```   413 lemma holomorphic_on_subset:
```
```   414     "f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t"
```
```   415   unfolding holomorphic_on_def
```
```   416   by (metis complex_differentiable_within_subset subsetD)
```
```   417
```
```   418 lemma holomorphic_transform: "\<lbrakk>f holomorphic_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g holomorphic_on s"
```
```   419   by (metis complex_differentiable_transform_within linordered_field_no_ub holomorphic_on_def)
```
```   420
```
```   421 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"
```
```   422   by (metis holomorphic_transform)
```
```   423
```
```   424 lemma holomorphic_on_linear [holomorphic_intros]: "(op * c) holomorphic_on s"
```
```   425   unfolding holomorphic_on_def by (metis complex_differentiable_linear)
```
```   426
```
```   427 lemma holomorphic_on_const [holomorphic_intros]: "(\<lambda>z. c) holomorphic_on s"
```
```   428   unfolding holomorphic_on_def by (metis complex_differentiable_const)
```
```   429
```
```   430 lemma holomorphic_on_ident [holomorphic_intros]: "(\<lambda>x. x) holomorphic_on s"
```
```   431   unfolding holomorphic_on_def by (metis complex_differentiable_ident)
```
```   432
```
```   433 lemma holomorphic_on_id [holomorphic_intros]: "id holomorphic_on s"
```
```   434   unfolding id_def by (rule holomorphic_on_ident)
```
```   435
```
```   436 lemma holomorphic_on_compose:
```
```   437   "f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s) \<Longrightarrow> (g o f) holomorphic_on s"
```
```   438   using complex_differentiable_compose_within[of f _ s g]
```
```   439   by (auto simp: holomorphic_on_def)
```
```   440
```
```   441 lemma holomorphic_on_compose_gen:
```
```   442   "f holomorphic_on s \<Longrightarrow> g holomorphic_on t \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> (g o f) holomorphic_on s"
```
```   443   by (metis holomorphic_on_compose holomorphic_on_subset)
```
```   444
```
```   445 lemma holomorphic_on_minus [holomorphic_intros]: "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on s"
```
```   446   by (metis complex_differentiable_minus holomorphic_on_def)
```
```   447
```
```   448 lemma holomorphic_on_add [holomorphic_intros]:
```
```   449   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on s"
```
```   450   unfolding holomorphic_on_def by (metis complex_differentiable_add)
```
```   451
```
```   452 lemma holomorphic_on_diff [holomorphic_intros]:
```
```   453   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) holomorphic_on s"
```
```   454   unfolding holomorphic_on_def by (metis complex_differentiable_diff)
```
```   455
```
```   456 lemma holomorphic_on_mult [holomorphic_intros]:
```
```   457   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) holomorphic_on s"
```
```   458   unfolding holomorphic_on_def by (metis complex_differentiable_mult)
```
```   459
```
```   460 lemma holomorphic_on_inverse [holomorphic_intros]:
```
```   461   "\<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"
```
```   462   unfolding holomorphic_on_def by (metis complex_differentiable_inverse)
```
```   463
```
```   464 lemma holomorphic_on_divide [holomorphic_intros]:
```
```   465   "\<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"
```
```   466   unfolding holomorphic_on_def by (metis complex_differentiable_divide)
```
```   467
```
```   468 lemma holomorphic_on_power [holomorphic_intros]:
```
```   469   "f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on s"
```
```   470   unfolding holomorphic_on_def by (metis complex_differentiable_power)
```
```   471
```
```   472 lemma holomorphic_on_setsum [holomorphic_intros]:
```
```   473   "(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) holomorphic_on s"
```
```   474   unfolding holomorphic_on_def by (metis complex_differentiable_setsum)
```
```   475
```
```   476 lemma DERIV_deriv_iff_complex_differentiable:
```
```   477   "DERIV f x :> deriv f x \<longleftrightarrow> f complex_differentiable at x"
```
```   478   unfolding complex_differentiable_def by (metis DERIV_imp_deriv)
```
```   479
```
```   480 lemma complex_derivative_chain:
```
```   481   "f complex_differentiable at x \<Longrightarrow> g complex_differentiable at (f x)
```
```   482     \<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
```
```   483   by (metis DERIV_deriv_iff_complex_differentiable DERIV_chain DERIV_imp_deriv)
```
```   484
```
```   485 lemma deriv_linear: "deriv (\<lambda>w. c * w) = (\<lambda>z. c)"
```
```   486   by (metis DERIV_imp_deriv DERIV_cmult_Id)
```
```   487
```
```   488 lemma deriv_ident: "deriv (\<lambda>w. w) = (\<lambda>z. 1)"
```
```   489   by (metis DERIV_imp_deriv DERIV_ident)
```
```   490
```
```   491 lemma deriv_const: "deriv (\<lambda>w. c) = (\<lambda>z. 0)"
```
```   492   by (metis DERIV_imp_deriv DERIV_const)
```
```   493
```
```   494 lemma complex_derivative_add:
```
```   495   "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>
```
```   496    \<Longrightarrow> deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
```
```   497   unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
```
```   498   by (auto intro!: DERIV_imp_deriv derivative_intros)
```
```   499
```
```   500 lemma complex_derivative_diff:
```
```   501   "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>
```
```   502    \<Longrightarrow> deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
```
```   503   unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
```
```   504   by (auto intro!: DERIV_imp_deriv derivative_intros)
```
```   505
```
```   506 lemma complex_derivative_mult:
```
```   507   "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>
```
```   508    \<Longrightarrow> deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
```
```   509   unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
```
```   510   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
```
```   511
```
```   512 lemma complex_derivative_cmult:
```
```   513   "f complex_differentiable at z \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z"
```
```   514   unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
```
```   515   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
```
```   516
```
```   517 lemma complex_derivative_cmult_right:
```
```   518   "f complex_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c"
```
```   519   unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
```
```   520   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
```
```   521
```
```   522 lemma complex_derivative_transform_within_open:
```
```   523   "\<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>
```
```   524    \<Longrightarrow> deriv f z = deriv g z"
```
```   525   unfolding holomorphic_on_def
```
```   526   by (rule DERIV_imp_deriv)
```
```   527      (metis DERIV_deriv_iff_complex_differentiable DERIV_transform_within_open at_within_open)
```
```   528
```
```   529 lemma complex_derivative_compose_linear:
```
```   530   "f complex_differentiable at (c * z) \<Longrightarrow> deriv (\<lambda>w. f (c * w)) z = c * deriv f (c * z)"
```
```   531 apply (rule DERIV_imp_deriv)
```
```   532 apply (simp add: DERIV_deriv_iff_complex_differentiable [symmetric])
```
```   533 apply (drule DERIV_chain' [of "times c" c z UNIV f "deriv f (c * z)", OF DERIV_cmult_Id])
```
```   534 apply (simp add: algebra_simps)
```
```   535 done
```
```   536
```
```   537 subsection\<open>Analyticity on a set\<close>
```
```   538
```
```   539 definition analytic_on (infixl "(analytic'_on)" 50)
```
```   540   where
```
```   541    "f analytic_on s \<equiv> \<forall>x \<in> s. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)"
```
```   542
```
```   543 lemma analytic_imp_holomorphic: "f analytic_on s \<Longrightarrow> f holomorphic_on s"
```
```   544   by (simp add: at_within_open [OF _ open_ball] analytic_on_def holomorphic_on_def)
```
```   545      (metis centre_in_ball complex_differentiable_at_within)
```
```   546
```
```   547 lemma analytic_on_open: "open s \<Longrightarrow> f analytic_on s \<longleftrightarrow> f holomorphic_on s"
```
```   548 apply (auto simp: analytic_imp_holomorphic)
```
```   549 apply (auto simp: analytic_on_def holomorphic_on_def)
```
```   550 by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball)
```
```   551
```
```   552 lemma analytic_on_imp_differentiable_at:
```
```   553   "f analytic_on s \<Longrightarrow> x \<in> s \<Longrightarrow> f complex_differentiable (at x)"
```
```   554  apply (auto simp: analytic_on_def holomorphic_on_def)
```
```   555 by (metis Topology_Euclidean_Space.open_ball centre_in_ball complex_differentiable_within_open)
```
```   556
```
```   557 lemma analytic_on_subset: "f analytic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f analytic_on t"
```
```   558   by (auto simp: analytic_on_def)
```
```   559
```
```   560 lemma analytic_on_Un: "f analytic_on (s \<union> t) \<longleftrightarrow> f analytic_on s \<and> f analytic_on t"
```
```   561   by (auto simp: analytic_on_def)
```
```   562
```
```   563 lemma analytic_on_Union: "f analytic_on (\<Union>s) \<longleftrightarrow> (\<forall>t \<in> s. f analytic_on t)"
```
```   564   by (auto simp: analytic_on_def)
```
```   565
```
```   566 lemma analytic_on_UN: "f analytic_on (\<Union>i\<in>I. s i) \<longleftrightarrow> (\<forall>i\<in>I. f analytic_on (s i))"
```
```   567   by (auto simp: analytic_on_def)
```
```   568
```
```   569 lemma analytic_on_holomorphic:
```
```   570   "f analytic_on s \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f holomorphic_on t)"
```
```   571   (is "?lhs = ?rhs")
```
```   572 proof -
```
```   573   have "?lhs \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t)"
```
```   574   proof safe
```
```   575     assume "f analytic_on s"
```
```   576     then show "\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t"
```
```   577       apply (simp add: analytic_on_def)
```
```   578       apply (rule exI [where x="\<Union>{u. open u \<and> f analytic_on u}"], auto)
```
```   579       apply (metis Topology_Euclidean_Space.open_ball analytic_on_open centre_in_ball)
```
```   580       by (metis analytic_on_def)
```
```   581   next
```
```   582     fix t
```
```   583     assume "open t" "s \<subseteq> t" "f analytic_on t"
```
```   584     then show "f analytic_on s"
```
```   585         by (metis analytic_on_subset)
```
```   586   qed
```
```   587   also have "... \<longleftrightarrow> ?rhs"
```
```   588     by (auto simp: analytic_on_open)
```
```   589   finally show ?thesis .
```
```   590 qed
```
```   591
```
```   592 lemma analytic_on_linear: "(op * c) analytic_on s"
```
```   593   by (auto simp add: analytic_on_holomorphic holomorphic_on_linear)
```
```   594
```
```   595 lemma analytic_on_const: "(\<lambda>z. c) analytic_on s"
```
```   596   by (metis analytic_on_def holomorphic_on_const zero_less_one)
```
```   597
```
```   598 lemma analytic_on_ident: "(\<lambda>x. x) analytic_on s"
```
```   599   by (simp add: analytic_on_def holomorphic_on_ident gt_ex)
```
```   600
```
```   601 lemma analytic_on_id: "id analytic_on s"
```
```   602   unfolding id_def by (rule analytic_on_ident)
```
```   603
```
```   604 lemma analytic_on_compose:
```
```   605   assumes f: "f analytic_on s"
```
```   606       and g: "g analytic_on (f ` s)"
```
```   607     shows "(g o f) analytic_on s"
```
```   608 unfolding analytic_on_def
```
```   609 proof (intro ballI)
```
```   610   fix x
```
```   611   assume x: "x \<in> s"
```
```   612   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f
```
```   613     by (metis analytic_on_def)
```
```   614   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g
```
```   615     by (metis analytic_on_def g image_eqI x)
```
```   616   have "isCont f x"
```
```   617     by (metis analytic_on_imp_differentiable_at complex_differentiable_imp_continuous_at f x)
```
```   618   with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'"
```
```   619      by (auto simp: continuous_at_ball)
```
```   620   have "g \<circ> f holomorphic_on ball x (min d e)"
```
```   621     apply (rule holomorphic_on_compose)
```
```   622     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
```
```   623     by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball)
```
```   624   then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e"
```
```   625     by (metis d e min_less_iff_conj)
```
```   626 qed
```
```   627
```
```   628 lemma analytic_on_compose_gen:
```
```   629   "f analytic_on s \<Longrightarrow> g analytic_on t \<Longrightarrow> (\<And>z. z \<in> s \<Longrightarrow> f z \<in> t)
```
```   630              \<Longrightarrow> g o f analytic_on s"
```
```   631 by (metis analytic_on_compose analytic_on_subset image_subset_iff)
```
```   632
```
```   633 lemma analytic_on_neg:
```
```   634   "f analytic_on s \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on s"
```
```   635 by (metis analytic_on_holomorphic holomorphic_on_minus)
```
```   636
```
```   637 lemma analytic_on_add:
```
```   638   assumes f: "f analytic_on s"
```
```   639       and g: "g analytic_on s"
```
```   640     shows "(\<lambda>z. f z + g z) analytic_on s"
```
```   641 unfolding analytic_on_def
```
```   642 proof (intro ballI)
```
```   643   fix z
```
```   644   assume z: "z \<in> s"
```
```   645   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
```
```   646     by (metis analytic_on_def)
```
```   647   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
```
```   648     by (metis analytic_on_def g z)
```
```   649   have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')"
```
```   650     apply (rule holomorphic_on_add)
```
```   651     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
```
```   652     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
```
```   653   then show "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e"
```
```   654     by (metis e e' min_less_iff_conj)
```
```   655 qed
```
```   656
```
```   657 lemma analytic_on_diff:
```
```   658   assumes f: "f analytic_on s"
```
```   659       and g: "g analytic_on s"
```
```   660     shows "(\<lambda>z. f z - g z) analytic_on s"
```
```   661 unfolding analytic_on_def
```
```   662 proof (intro ballI)
```
```   663   fix z
```
```   664   assume z: "z \<in> s"
```
```   665   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
```
```   666     by (metis analytic_on_def)
```
```   667   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
```
```   668     by (metis analytic_on_def g z)
```
```   669   have "(\<lambda>z. f z - g z) holomorphic_on ball z (min e e')"
```
```   670     apply (rule holomorphic_on_diff)
```
```   671     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
```
```   672     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
```
```   673   then show "\<exists>e>0. (\<lambda>z. f z - g z) holomorphic_on ball z e"
```
```   674     by (metis e e' min_less_iff_conj)
```
```   675 qed
```
```   676
```
```   677 lemma analytic_on_mult:
```
```   678   assumes f: "f analytic_on s"
```
```   679       and g: "g analytic_on s"
```
```   680     shows "(\<lambda>z. f z * g z) analytic_on s"
```
```   681 unfolding analytic_on_def
```
```   682 proof (intro ballI)
```
```   683   fix z
```
```   684   assume z: "z \<in> s"
```
```   685   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
```
```   686     by (metis analytic_on_def)
```
```   687   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
```
```   688     by (metis analytic_on_def g z)
```
```   689   have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')"
```
```   690     apply (rule holomorphic_on_mult)
```
```   691     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
```
```   692     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
```
```   693   then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e"
```
```   694     by (metis e e' min_less_iff_conj)
```
```   695 qed
```
```   696
```
```   697 lemma analytic_on_inverse:
```
```   698   assumes f: "f analytic_on s"
```
```   699       and nz: "(\<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0)"
```
```   700     shows "(\<lambda>z. inverse (f z)) analytic_on s"
```
```   701 unfolding analytic_on_def
```
```   702 proof (intro ballI)
```
```   703   fix z
```
```   704   assume z: "z \<in> s"
```
```   705   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
```
```   706     by (metis analytic_on_def)
```
```   707   have "continuous_on (ball z e) f"
```
```   708     by (metis fh holomorphic_on_imp_continuous_on)
```
```   709   then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0"
```
```   710     by (metis Topology_Euclidean_Space.open_ball centre_in_ball continuous_on_open_avoid e z nz)
```
```   711   have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')"
```
```   712     apply (rule holomorphic_on_inverse)
```
```   713     apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball)
```
```   714     by (metis nz' mem_ball min_less_iff_conj)
```
```   715   then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e"
```
```   716     by (metis e e' min_less_iff_conj)
```
```   717 qed
```
```   718
```
```   719
```
```   720 lemma analytic_on_divide:
```
```   721   assumes f: "f analytic_on s"
```
```   722       and g: "g analytic_on s"
```
```   723       and nz: "(\<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0)"
```
```   724     shows "(\<lambda>z. f z / g z) analytic_on s"
```
```   725 unfolding divide_inverse
```
```   726 by (metis analytic_on_inverse analytic_on_mult f g nz)
```
```   727
```
```   728 lemma analytic_on_power:
```
```   729   "f analytic_on s \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on s"
```
```   730 by (induct n) (auto simp: analytic_on_const analytic_on_mult)
```
```   731
```
```   732 lemma analytic_on_setsum:
```
```   733   "(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on s) \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) analytic_on s"
```
```   734   by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_const analytic_on_add)
```
```   735
```
```   736 subsection\<open>analyticity at a point.\<close>
```
```   737
```
```   738 lemma analytic_at_ball:
```
```   739   "f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
```
```   740 by (metis analytic_on_def singleton_iff)
```
```   741
```
```   742 lemma analytic_at:
```
```   743     "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
```
```   744 by (metis analytic_on_holomorphic empty_subsetI insert_subset)
```
```   745
```
```   746 lemma analytic_on_analytic_at:
```
```   747     "f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
```
```   748 by (metis analytic_at_ball analytic_on_def)
```
```   749
```
```   750 lemma analytic_at_two:
```
```   751   "f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
```
```   752    (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)"
```
```   753   (is "?lhs = ?rhs")
```
```   754 proof
```
```   755   assume ?lhs
```
```   756   then obtain s t
```
```   757     where st: "open s" "z \<in> s" "f holomorphic_on s"
```
```   758               "open t" "z \<in> t" "g holomorphic_on t"
```
```   759     by (auto simp: analytic_at)
```
```   760   show ?rhs
```
```   761     apply (rule_tac x="s \<inter> t" in exI)
```
```   762     using st
```
```   763     apply (auto simp: Diff_subset holomorphic_on_subset)
```
```   764     done
```
```   765 next
```
```   766   assume ?rhs
```
```   767   then show ?lhs
```
```   768     by (force simp add: analytic_at)
```
```   769 qed
```
```   770
```
```   771 subsection\<open>Combining theorems for derivative with ``analytic at'' hypotheses\<close>
```
```   772
```
```   773 lemma
```
```   774   assumes "f analytic_on {z}" "g analytic_on {z}"
```
```   775   shows complex_derivative_add_at: "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
```
```   776     and complex_derivative_diff_at: "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
```
```   777     and complex_derivative_mult_at: "deriv (\<lambda>w. f w * g w) z =
```
```   778            f z * deriv g z + deriv f z * g z"
```
```   779 proof -
```
```   780   obtain s where s: "open s" "z \<in> s" "f holomorphic_on s" "g holomorphic_on s"
```
```   781     using assms by (metis analytic_at_two)
```
```   782   show "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
```
```   783     apply (rule DERIV_imp_deriv [OF DERIV_add])
```
```   784     using s
```
```   785     apply (auto simp: holomorphic_on_open complex_differentiable_def DERIV_deriv_iff_complex_differentiable)
```
```   786     done
```
```   787   show "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
```
```   788     apply (rule DERIV_imp_deriv [OF DERIV_diff])
```
```   789     using s
```
```   790     apply (auto simp: holomorphic_on_open complex_differentiable_def DERIV_deriv_iff_complex_differentiable)
```
```   791     done
```
```   792   show "deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
```
```   793     apply (rule DERIV_imp_deriv [OF DERIV_mult'])
```
```   794     using s
```
```   795     apply (auto simp: holomorphic_on_open complex_differentiable_def DERIV_deriv_iff_complex_differentiable)
```
```   796     done
```
```   797 qed
```
```   798
```
```   799 lemma complex_derivative_cmult_at:
```
```   800   "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. c * f w) z = c * deriv f z"
```
```   801 by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
```
```   802
```
```   803 lemma complex_derivative_cmult_right_at:
```
```   804   "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. f w * c) z = deriv f z * c"
```
```   805 by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
```
```   806
```
```   807 subsection\<open>Complex differentiation of sequences and series\<close>
```
```   808
```
```   809 (* TODO: Could probably be simplified using Uniform_Limit *)
```
```   810 lemma has_complex_derivative_sequence:
```
```   811   fixes s :: "complex set"
```
```   812   assumes cvs: "convex s"
```
```   813       and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
```
```   814       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"
```
```   815       and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
```
```   816     shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and>
```
```   817                        (g has_field_derivative (g' x)) (at x within s)"
```
```   818 proof -
```
```   819   from assms obtain x l where x: "x \<in> s" and tf: "((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
```
```   820     by blast
```
```   821   { fix e::real assume e: "e > 0"
```
```   822     then obtain N where N: "\<forall>n\<ge>N. \<forall>x. x \<in> s \<longrightarrow> cmod (f' n x - g' x) \<le> e"
```
```   823       by (metis conv)
```
```   824     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"
```
```   825     proof (rule exI [of _ N], clarify)
```
```   826       fix n y h
```
```   827       assume "N \<le> n" "y \<in> s"
```
```   828       then have "cmod (f' n y - g' y) \<le> e"
```
```   829         by (metis N)
```
```   830       then have "cmod h * cmod (f' n y - g' y) \<le> cmod h * e"
```
```   831         by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
```
```   832       then show "cmod (f' n y * h - g' y * h) \<le> e * cmod h"
```
```   833         by (simp add: norm_mult [symmetric] field_simps)
```
```   834     qed
```
```   835   } note ** = this
```
```   836   show ?thesis
```
```   837   unfolding has_field_derivative_def
```
```   838   proof (rule has_derivative_sequence [OF cvs _ _ x])
```
```   839     show "\<forall>n. \<forall>x\<in>s. (f n has_derivative (op * (f' n x))) (at x within s)"
```
```   840       by (metis has_field_derivative_def df)
```
```   841   next show "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
```
```   842     by (rule tf)
```
```   843   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"
```
```   844     by (blast intro: **)
```
```   845   qed
```
```   846 qed
```
```   847
```
```   848
```
```   849 lemma has_complex_derivative_series:
```
```   850   fixes s :: "complex set"
```
```   851   assumes cvs: "convex s"
```
```   852       and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
```
```   853       and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s
```
```   854                 \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
```
```   855       and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) sums l)"
```
```   856     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))"
```
```   857 proof -
```
```   858   from assms obtain x l where x: "x \<in> s" and sf: "((\<lambda>n. f n x) sums l)"
```
```   859     by blast
```
```   860   { fix e::real assume e: "e > 0"
```
```   861     then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> s
```
```   862             \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
```
```   863       by (metis conv)
```
```   864     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"
```
```   865     proof (rule exI [of _ N], clarify)
```
```   866       fix n y h
```
```   867       assume "N \<le> n" "y \<in> s"
```
```   868       then have "cmod ((\<Sum>i<n. f' i y) - g' y) \<le> e"
```
```   869         by (metis N)
```
```   870       then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e"
```
```   871         by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
```
```   872       then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h"
```
```   873         by (simp add: norm_mult [symmetric] field_simps setsum_right_distrib)
```
```   874     qed
```
```   875   } note ** = this
```
```   876   show ?thesis
```
```   877   unfolding has_field_derivative_def
```
```   878   proof (rule has_derivative_series [OF cvs _ _ x])
```
```   879     fix n x
```
```   880     assume "x \<in> s"
```
```   881     then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within s)"
```
```   882       by (metis df has_field_derivative_def mult_commute_abs)
```
```   883   next show " ((\<lambda>n. f n x) sums l)"
```
```   884     by (rule sf)
```
```   885   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"
```
```   886     by (blast intro: **)
```
```   887   qed
```
```   888 qed
```
```   889
```
```   890
```
```   891 lemma complex_differentiable_series:
```
```   892   fixes f :: "nat \<Rightarrow> complex \<Rightarrow> complex"
```
```   893   assumes "convex s" "open s"
```
```   894   assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
```
```   895   assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
```
```   896   assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)" and x: "x \<in> s"
```
```   897   shows   "summable (\<lambda>n. f n x)" and "(\<lambda>x. \<Sum>n. f n x) complex_differentiable (at x)"
```
```   898 proof -
```
```   899   from assms(4) obtain g' where A: "uniform_limit s (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
```
```   900     unfolding uniformly_convergent_on_def by blast
```
```   901   from x and \<open>open s\<close> have s: "at x within s = at x" by (rule at_within_open)
```
```   902   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)"
```
```   903     by (intro has_field_derivative_series[of s f f' g' x0] assms A has_field_derivative_at_within)
```
```   904   then obtain g where g: "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>n. f n x) sums g x"
```
```   905     "\<And>x. x \<in> s \<Longrightarrow> (g has_field_derivative g' x) (at x within s)" by blast
```
```   906   from g[OF x] show "summable (\<lambda>n. f n x)" by (auto simp: summable_def)
```
```   907   from g(2)[OF x] have g': "(g has_derivative op * (g' x)) (at x)"
```
```   908     by (simp add: has_field_derivative_def s)
```
```   909   have "((\<lambda>x. \<Sum>n. f n x) has_derivative op * (g' x)) (at x)"
```
```   910     by (rule has_derivative_transform_within_open[OF \<open>open s\<close> x _ g'])
```
```   911        (insert g, auto simp: sums_iff)
```
```   912   thus "(\<lambda>x. \<Sum>n. f n x) complex_differentiable (at x)" unfolding differentiable_def
```
```   913     by (auto simp: summable_def complex_differentiable_def has_field_derivative_def)
```
```   914 qed
```
```   915
```
```   916 lemma complex_differentiable_series':
```
```   917   fixes f :: "nat \<Rightarrow> complex \<Rightarrow> complex"
```
```   918   assumes "convex s" "open s"
```
```   919   assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
```
```   920   assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
```
```   921   assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)"
```
```   922   shows   "(\<lambda>x. \<Sum>n. f n x) complex_differentiable (at x0)"
```
```   923   using complex_differentiable_series[OF assms, of x0] \<open>x0 \<in> s\<close> by blast+
```
```   924
```
```   925 subsection\<open>Bound theorem\<close>
```
```   926
```
```   927 lemma complex_differentiable_bound:
```
```   928   fixes s :: "complex set"
```
```   929   assumes cvs: "convex s"
```
```   930       and df:  "\<And>z. z \<in> s \<Longrightarrow> (f has_field_derivative f' z) (at z within s)"
```
```   931       and dn:  "\<And>z. z \<in> s \<Longrightarrow> norm (f' z) \<le> B"
```
```   932       and "x \<in> s"  "y \<in> s"
```
```   933     shows "norm(f x - f y) \<le> B * norm(x - y)"
```
```   934   apply (rule differentiable_bound [OF cvs])
```
```   935   apply (rule ballI, erule df [unfolded has_field_derivative_def])
```
```   936   apply (rule ballI, rule onorm_le, simp add: norm_mult mult_right_mono dn)
```
```   937   apply fact
```
```   938   apply fact
```
```   939   done
```
```   940
```
```   941 subsection\<open>Inverse function theorem for complex derivatives.\<close>
```
```   942
```
```   943 lemma has_complex_derivative_inverse_basic:
```
```   944   fixes f :: "complex \<Rightarrow> complex"
```
```   945   shows "DERIV f (g y) :> f' \<Longrightarrow>
```
```   946         f' \<noteq> 0 \<Longrightarrow>
```
```   947         continuous (at y) g \<Longrightarrow>
```
```   948         open t \<Longrightarrow>
```
```   949         y \<in> t \<Longrightarrow>
```
```   950         (\<And>z. z \<in> t \<Longrightarrow> f (g z) = z)
```
```   951         \<Longrightarrow> DERIV g y :> inverse (f')"
```
```   952   unfolding has_field_derivative_def
```
```   953   apply (rule has_derivative_inverse_basic)
```
```   954   apply (auto simp:  bounded_linear_mult_right)
```
```   955   done
```
```   956
```
```   957 (*Used only once, in Multivariate/cauchy.ml. *)
```
```   958 lemma has_complex_derivative_inverse_strong:
```
```   959   fixes f :: "complex \<Rightarrow> complex"
```
```   960   shows "DERIV f x :> f' \<Longrightarrow>
```
```   961          f' \<noteq> 0 \<Longrightarrow>
```
```   962          open s \<Longrightarrow>
```
```   963          x \<in> s \<Longrightarrow>
```
```   964          continuous_on s f \<Longrightarrow>
```
```   965          (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
```
```   966          \<Longrightarrow> DERIV g (f x) :> inverse (f')"
```
```   967   unfolding has_field_derivative_def
```
```   968   apply (rule has_derivative_inverse_strong [of s x f g ])
```
```   969   using assms
```
```   970   by auto
```
```   971
```
```   972 lemma has_complex_derivative_inverse_strong_x:
```
```   973   fixes f :: "complex \<Rightarrow> complex"
```
```   974   shows  "DERIV f (g y) :> f' \<Longrightarrow>
```
```   975           f' \<noteq> 0 \<Longrightarrow>
```
```   976           open s \<Longrightarrow>
```
```   977           continuous_on s f \<Longrightarrow>
```
```   978           g y \<in> s \<Longrightarrow> f(g y) = y \<Longrightarrow>
```
```   979           (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
```
```   980           \<Longrightarrow> DERIV g y :> inverse (f')"
```
```   981   unfolding has_field_derivative_def
```
```   982   apply (rule has_derivative_inverse_strong_x [of s g y f])
```
```   983   using assms
```
```   984   by auto
```
```   985
```
```   986 subsection \<open>Taylor on Complex Numbers\<close>
```
```   987
```
```   988 lemma setsum_Suc_reindex:
```
```   989   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
```
```   990     shows  "setsum f {0..n} = f 0 - f (Suc n) + setsum (\<lambda>i. f (Suc i)) {0..n}"
```
```   991 by (induct n) auto
```
```   992
```
```   993 lemma complex_taylor:
```
```   994   assumes s: "convex s"
```
```   995       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)"
```
```   996       and B: "\<And>x. x \<in> s \<Longrightarrow> cmod (f (Suc n) x) \<le> B"
```
```   997       and w: "w \<in> s"
```
```   998       and z: "z \<in> s"
```
```   999     shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i)))
```
```  1000           \<le> B * cmod(z - w)^(Suc n) / fact n"
```
```  1001 proof -
```
```  1002   have wzs: "closed_segment w z \<subseteq> s" using assms
```
```  1003     by (metis convex_contains_segment)
```
```  1004   { fix u
```
```  1005     assume "u \<in> closed_segment w z"
```
```  1006     then have "u \<in> s"
```
```  1007       by (metis wzs subsetD)
```
```  1008     have "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / (fact i) +
```
```  1009                       f (Suc i) u * (z-u)^i / (fact i)) =
```
```  1010               f (Suc n) u * (z-u) ^ n / (fact n)"
```
```  1011     proof (induction n)
```
```  1012       case 0 show ?case by simp
```
```  1013     next
```
```  1014       case (Suc n)
```
```  1015       have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / (fact i) +
```
```  1016                              f (Suc i) u * (z-u) ^ i / (fact i)) =
```
```  1017            f (Suc n) u * (z-u) ^ n / (fact n) +
```
```  1018            f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / (fact (Suc n)) -
```
```  1019            f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / (fact (Suc n))"
```
```  1020         using Suc by simp
```
```  1021       also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n))"
```
```  1022       proof -
```
```  1023         have "(fact(Suc n)) *
```
```  1024              (f(Suc n) u *(z-u) ^ n / (fact n) +
```
```  1025                f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / (fact(Suc n)) -
```
```  1026                f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / (fact(Suc n))) =
```
```  1027             ((fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / (fact n) +
```
```  1028             ((fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / (fact(Suc n))) -
```
```  1029             ((fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / (fact(Suc n))"
```
```  1030           by (simp add: algebra_simps del: fact.simps)
```
```  1031         also have "... = ((fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / (fact n) +
```
```  1032                          (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
```
```  1033                          (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
```
```  1034           by (simp del: fact.simps)
```
```  1035         also have "... = (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) +
```
```  1036                          (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
```
```  1037                          (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
```
```  1038           by (simp only: fact.simps of_nat_mult ac_simps) simp
```
```  1039         also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
```
```  1040           by (simp add: algebra_simps)
```
```  1041         finally show ?thesis
```
```  1042         by (simp add: mult_left_cancel [where c = "(fact (Suc n))", THEN iffD1] del: fact.simps)
```
```  1043       qed
```
```  1044       finally show ?case .
```
```  1045     qed
```
```  1046     then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / (fact i)))
```
```  1047                 has_field_derivative f (Suc n) u * (z-u) ^ n / (fact n))
```
```  1048                (at u within s)"
```
```  1049       apply (intro derivative_eq_intros)
```
```  1050       apply (blast intro: assms \<open>u \<in> s\<close>)
```
```  1051       apply (rule refl)+
```
```  1052       apply (auto simp: field_simps)
```
```  1053       done
```
```  1054   } note sum_deriv = this
```
```  1055   { fix u
```
```  1056     assume u: "u \<in> closed_segment w z"
```
```  1057     then have us: "u \<in> s"
```
```  1058       by (metis wzs subsetD)
```
```  1059     have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> cmod (f (Suc n) u) * cmod (u - z) ^ n"
```
```  1060       by (metis norm_minus_commute order_refl)
```
```  1061     also have "... \<le> cmod (f (Suc n) u) * cmod (z - w) ^ n"
```
```  1062       by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
```
```  1063     also have "... \<le> B * cmod (z - w) ^ n"
```
```  1064       by (metis norm_ge_zero zero_le_power mult_right_mono  B [OF us])
```
```  1065     finally have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> B * cmod (z - w) ^ n" .
```
```  1066   } note cmod_bound = this
```
```  1067   have "(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)) = (\<Sum>i\<le>n. (f i z / (fact i)) * 0 ^ i)"
```
```  1068     by simp
```
```  1069   also have "\<dots> = f 0 z / (fact 0)"
```
```  1070     by (subst setsum_zero_power) simp
```
```  1071   finally have "cmod (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)))
```
```  1072                 \<le> cmod ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) -
```
```  1073                         (\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))"
```
```  1074     by (simp add: norm_minus_commute)
```
```  1075   also have "... \<le> B * cmod (z - w) ^ n / (fact n) * cmod (w - z)"
```
```  1076     apply (rule complex_differentiable_bound
```
```  1077       [where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / (fact n)"
```
```  1078          and s = "closed_segment w z", OF convex_closed_segment])
```
```  1079     apply (auto simp: ends_in_segment DERIV_subset [OF sum_deriv wzs]
```
```  1080                   norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
```
```  1081     done
```
```  1082   also have "...  \<le> B * cmod (z - w) ^ Suc n / (fact n)"
```
```  1083     by (simp add: algebra_simps norm_minus_commute)
```
```  1084   finally show ?thesis .
```
```  1085 qed
```
```  1086
```
```  1087 text\<open>Something more like the traditional MVT for real components.\<close>
```
```  1088
```
```  1089 lemma complex_mvt_line:
```
```  1090   assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f'(u)) (at u)"
```
```  1091     shows "\<exists>u. u \<in> closed_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
```
```  1092 proof -
```
```  1093   have twz: "\<And>t. (1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)"
```
```  1094     by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
```
```  1095   note assms[unfolded has_field_derivative_def, derivative_intros]
```
```  1096   show ?thesis
```
```  1097     apply (cut_tac mvt_simple
```
```  1098                      [of 0 1 "Re o f o (\<lambda>t. (1 - t) *\<^sub>R w +  t *\<^sub>R z)"
```
```  1099                       "\<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))"])
```
```  1100     apply auto
```
```  1101     apply (rule_tac x="(1 - x) *\<^sub>R w + x *\<^sub>R z" in exI)
```
```  1102     apply (auto simp: closed_segment_def twz) []
```
```  1103     apply (intro derivative_eq_intros has_derivative_at_within, simp_all)
```
```  1104     apply (simp add: fun_eq_iff real_vector.scale_right_diff_distrib)
```
```  1105     apply (force simp: twz closed_segment_def)
```
```  1106     done
```
```  1107 qed
```
```  1108
```
```  1109 lemma complex_taylor_mvt:
```
```  1110   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)"
```
```  1111     shows "\<exists>u. u \<in> closed_segment w z \<and>
```
```  1112             Re (f 0 z) =
```
```  1113             Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / (fact i)) +
```
```  1114                 (f (Suc n) u * (z-u)^n / (fact n)) * (z - w))"
```
```  1115 proof -
```
```  1116   { fix u
```
```  1117     assume u: "u \<in> closed_segment w z"
```
```  1118     have "(\<Sum>i = 0..n.
```
```  1119                (f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) /
```
```  1120                (fact i)) =
```
```  1121           f (Suc 0) u -
```
```  1122              (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
```
```  1123              (fact (Suc n)) +
```
```  1124              (\<Sum>i = 0..n.
```
```  1125                  (f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) /
```
```  1126                  (fact (Suc i)))"
```
```  1127        by (subst setsum_Suc_reindex) simp
```
```  1128     also have "... = f (Suc 0) u -
```
```  1129              (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
```
```  1130              (fact (Suc n)) +
```
```  1131              (\<Sum>i = 0..n.
```
```  1132                  f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i))  -
```
```  1133                  f (Suc i) u * (z-u) ^ i / (fact i))"
```
```  1134       by (simp only: diff_divide_distrib fact_cancel ac_simps)
```
```  1135     also have "... = f (Suc 0) u -
```
```  1136              (f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) /
```
```  1137              (fact (Suc n)) +
```
```  1138              f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n)) - f (Suc 0) u"
```
```  1139       by (subst setsum_Suc_diff) auto
```
```  1140     also have "... = f (Suc n) u * (z-u) ^ n / (fact n)"
```
```  1141       by (simp only: algebra_simps diff_divide_distrib fact_cancel)
```
```  1142     finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i
```
```  1143                              - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / (fact i)) =
```
```  1144                   f (Suc n) u * (z - u) ^ n / (fact n)" .
```
```  1145     then have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / (fact i)) has_field_derivative
```
```  1146                 f (Suc n) u * (z - u) ^ n / (fact n))  (at u)"
```
```  1147       apply (intro derivative_eq_intros)+
```
```  1148       apply (force intro: u assms)
```
```  1149       apply (rule refl)+
```
```  1150       apply (auto simp: ac_simps)
```
```  1151       done
```
```  1152   }
```
```  1153   then show ?thesis
```
```  1154     apply (cut_tac complex_mvt_line [of w z "\<lambda>u. \<Sum>i = 0..n. f i u * (z-u) ^ i / (fact i)"
```
```  1155                "\<lambda>u. (f (Suc n) u * (z-u)^n / (fact n))"])
```
```  1156     apply (auto simp add: intro: open_closed_segment)
```
```  1157     done
```
```  1158 qed
```
```  1159
```
```  1160
```
```  1161 subsection \<open>Polynomal function extremal theorem, from HOL Light\<close>
```
```  1162
```
```  1163 lemma polyfun_extremal_lemma: (*COMPLEX_POLYFUN_EXTREMAL_LEMMA in HOL Light*)
```
```  1164     fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
```
```  1165   assumes "0 < e"
```
```  1166     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)"
```
```  1167 proof (induct n)
```
```  1168   case 0 with assms
```
```  1169   show ?case
```
```  1170     apply (rule_tac x="norm (c 0) / e" in exI)
```
```  1171     apply (auto simp: field_simps)
```
```  1172     done
```
```  1173 next
```
```  1174   case (Suc n)
```
```  1175   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"
```
```  1176     using Suc assms by blast
```
```  1177   show ?case
```
```  1178   proof (rule exI [where x= "max M (1 + norm(c(Suc n)) / e)"], clarsimp simp del: power_Suc)
```
```  1179     fix z::'a
```
```  1180     assume z1: "M \<le> norm z" and "1 + norm (c (Suc n)) / e \<le> norm z"
```
```  1181     then have z2: "e + norm (c (Suc n)) \<le> e * norm z"
```
```  1182       using assms by (simp add: field_simps)
```
```  1183     have "norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
```
```  1184       using M [OF z1] by simp
```
```  1185     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)"
```
```  1186       by simp
```
```  1187     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)"
```
```  1188       by (blast intro: norm_triangle_le elim: )
```
```  1189     also have "... \<le> (e + norm (c (Suc n))) * norm z ^ Suc n"
```
```  1190       by (simp add: norm_power norm_mult algebra_simps)
```
```  1191     also have "... \<le> (e * norm z) * norm z ^ Suc n"
```
```  1192       by (metis z2 mult.commute mult_left_mono norm_ge_zero norm_power)
```
```  1193     finally show "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc (Suc n)"
```
```  1194       by simp
```
```  1195   qed
```
```  1196 qed
```
```  1197
```
```  1198 lemma polyfun_extremal: (*COMPLEX_POLYFUN_EXTREMAL in HOL Light*)
```
```  1199     fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
```
```  1200   assumes k: "c k \<noteq> 0" "1\<le>k" and kn: "k\<le>n"
```
```  1201     shows "eventually (\<lambda>z. norm (\<Sum>i\<le>n. c(i) * z^i) \<ge> B) at_infinity"
```
```  1202 using kn
```
```  1203 proof (induction n)
```
```  1204   case 0
```
```  1205   then show ?case
```
```  1206     using k  by simp
```
```  1207 next
```
```  1208   case (Suc m)
```
```  1209   let ?even = ?case
```
```  1210   show ?even
```
```  1211   proof (cases "c (Suc m) = 0")
```
```  1212     case True
```
```  1213     then show ?even using Suc k
```
```  1214       by auto (metis antisym_conv less_eq_Suc_le not_le)
```
```  1215   next
```
```  1216     case False
```
```  1217     then obtain M where M:
```
```  1218           "\<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"
```
```  1219       using polyfun_extremal_lemma [of "norm(c (Suc m)) / 2" c m] Suc
```
```  1220       by auto
```
```  1221     have "\<exists>b. \<forall>z. b \<le> norm z \<longrightarrow> B \<le> norm (\<Sum>i\<le>Suc m. c i * z^i)"
```
```  1222     proof (rule exI [where x="max M (max 1 (\<bar>B\<bar> / (norm(c (Suc m)) / 2)))"], clarsimp simp del: power_Suc)
```
```  1223       fix z::'a
```
```  1224       assume z1: "M \<le> norm z" "1 \<le> norm z"
```
```  1225          and "\<bar>B\<bar> * 2 / norm (c (Suc m)) \<le> norm z"
```
```  1226       then have z2: "\<bar>B\<bar> \<le> norm (c (Suc m)) * norm z / 2"
```
```  1227         using False by (simp add: field_simps)
```
```  1228       have nz: "norm z \<le> norm z ^ Suc m"
```
```  1229         by (metis \<open>1 \<le> norm z\<close> One_nat_def less_eq_Suc_le power_increasing power_one_right zero_less_Suc)
```
```  1230       have *: "\<And>y x. norm (c (Suc m)) * norm z / 2 \<le> norm y - norm x \<Longrightarrow> B \<le> norm (x + y)"
```
```  1231         by (metis abs_le_iff add.commute norm_diff_ineq order_trans z2)
```
```  1232       have "norm z * norm (c (Suc m)) + 2 * norm (\<Sum>i\<le>m. c i * z^i)
```
```  1233             \<le> norm (c (Suc m)) * norm z + norm (c (Suc m)) * norm z ^ Suc m"
```
```  1234         using M [of z] Suc z1  by auto
```
```  1235       also have "... \<le> 2 * (norm (c (Suc m)) * norm z ^ Suc m)"
```
```  1236         using nz by (simp add: mult_mono del: power_Suc)
```
```  1237       finally show "B \<le> norm ((\<Sum>i\<le>m. c i * z^i) + c (Suc m) * z ^ Suc m)"
```
```  1238         using Suc.IH
```
```  1239         apply (auto simp: eventually_at_infinity)
```
```  1240         apply (rule *)
```
```  1241         apply (simp add: field_simps norm_mult norm_power)
```
```  1242         done
```
```  1243     qed
```
```  1244     then show ?even
```
```  1245       by (simp add: eventually_at_infinity)
```
```  1246   qed
```
```  1247 qed
```
```  1248
```
```  1249 end
```