src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy
author paulson <lp15@cam.ac.uk>
Fri Mar 21 15:36:00 2014 +0000 (2014-03-21)
changeset 56238 5d147e1e18d1
parent 56223 7696903b9e61
child 56261 918432e3fcfa
permissions -rw-r--r--
a few new lemmas and generalisations of old ones
     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 header {* Complex Analysis Basics *}
     6 
     7 theory Complex_Analysis_Basics
     8 imports  "~~/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space"
     9 
    10 begin
    11 
    12 subsection {*Complex number lemmas *}
    13 
    14 lemma abs_sqrt_wlog:
    15   fixes x::"'a::linordered_idom"
    16   assumes "!!x::'a. x\<ge>0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"
    17 by (metis abs_ge_zero assms power2_abs)
    18 
    19 lemma complex_abs_le_norm: "abs(Re z) + abs(Im z) \<le> sqrt(2) * norm z"
    20 proof (cases z)
    21   case (Complex x y)
    22   show ?thesis
    23     apply (rule power2_le_imp_le)
    24     apply (auto simp: real_sqrt_mult [symmetric] Complex)
    25     apply (rule abs_sqrt_wlog [where x=x])
    26     apply (rule abs_sqrt_wlog [where x=y])
    27     apply (simp add: power2_sum add_commute sum_squares_bound)
    28     done
    29 qed
    30 
    31 lemma continuous_Re: "continuous_on UNIV Re"
    32   by (metis (poly_guards_query) bounded_linear.continuous_on bounded_linear_Re 
    33             continuous_on_cong continuous_on_id)
    34 
    35 lemma continuous_Im: "continuous_on UNIV Im"
    36   by (metis (poly_guards_query) bounded_linear.continuous_on bounded_linear_Im 
    37             continuous_on_cong continuous_on_id)
    38 
    39 lemma open_closed_segment: "u \<in> open_segment w z \<Longrightarrow> u \<in> closed_segment w z"
    40   by (auto simp add: closed_segment_def open_segment_def)
    41 
    42 lemma has_derivative_Re [has_derivative_intros] : "(Re has_derivative Re) F"
    43   by (auto simp add: has_derivative_def bounded_linear_Re)
    44 
    45 lemma has_derivative_Im [has_derivative_intros] : "(Im has_derivative Im) F"
    46   by (auto simp add: has_derivative_def bounded_linear_Im)
    47 
    48 lemma fact_cancel:
    49   fixes c :: "'a::real_field"
    50   shows "of_nat (Suc n) * c / of_nat (fact (Suc n)) = c / of_nat (fact n)"
    51   apply (subst frac_eq_eq [OF of_nat_fact_not_zero of_nat_fact_not_zero])
    52   apply (simp add: algebra_simps of_nat_mult)
    53   done
    54 
    55 lemma open_halfspace_Re_lt: "open {z. Re(z) < b}"
    56 proof -
    57   have "{z. Re(z) < b} = Re -`{..<b}"
    58     by blast
    59   then show ?thesis
    60     by (auto simp: continuous_Re continuous_imp_open_vimage [of UNIV])
    61 qed
    62 
    63 lemma open_halfspace_Re_gt: "open {z. Re(z) > b}"
    64 proof -
    65   have "{z. Re(z) > b} = Re -`{b<..}"
    66     by blast
    67   then show ?thesis
    68     by (auto simp: continuous_Re continuous_imp_open_vimage [of UNIV])
    69 qed
    70 
    71 lemma closed_halfspace_Re_ge: "closed {z. Re(z) \<ge> b}"
    72 proof -
    73   have "{z. Re(z) \<ge> b} = - {z. Re(z) < b}"
    74     by auto
    75   then show ?thesis
    76     by (simp add: closed_def open_halfspace_Re_lt)
    77 qed
    78 
    79 lemma closed_halfspace_Re_le: "closed {z. Re(z) \<le> b}"
    80 proof -
    81   have "{z. Re(z) \<le> b} = - {z. Re(z) > b}"
    82     by auto
    83   then show ?thesis
    84     by (simp add: closed_def open_halfspace_Re_gt)
    85 qed
    86 
    87 lemma closed_halfspace_Re_eq: "closed {z. Re(z) = b}"
    88 proof -
    89   have "{z. Re(z) = b} = {z. Re(z) \<le> b} \<inter> {z. Re(z) \<ge> b}"
    90     by auto
    91   then show ?thesis
    92     by (auto simp: closed_Int closed_halfspace_Re_le closed_halfspace_Re_ge)
    93 qed
    94 
    95 lemma open_halfspace_Im_lt: "open {z. Im(z) < b}"
    96 proof -
    97   have "{z. Im(z) < b} = Im -`{..<b}"
    98     by blast
    99   then show ?thesis
   100     by (auto simp: continuous_Im continuous_imp_open_vimage [of UNIV])
   101 qed
   102 
   103 lemma open_halfspace_Im_gt: "open {z. Im(z) > b}"
   104 proof -
   105   have "{z. Im(z) > b} = Im -`{b<..}"
   106     by blast
   107   then show ?thesis
   108     by (auto simp: continuous_Im continuous_imp_open_vimage [of UNIV])
   109 qed
   110 
   111 lemma closed_halfspace_Im_ge: "closed {z. Im(z) \<ge> b}"
   112 proof -
   113   have "{z. Im(z) \<ge> b} = - {z. Im(z) < b}"
   114     by auto
   115   then show ?thesis
   116     by (simp add: closed_def open_halfspace_Im_lt)
   117 qed
   118 
   119 lemma closed_halfspace_Im_le: "closed {z. Im(z) \<le> b}"
   120 proof -
   121   have "{z. Im(z) \<le> b} = - {z. Im(z) > b}"
   122     by auto
   123   then show ?thesis
   124     by (simp add: closed_def open_halfspace_Im_gt)
   125 qed
   126 
   127 lemma closed_halfspace_Im_eq: "closed {z. Im(z) = b}"
   128 proof -
   129   have "{z. Im(z) = b} = {z. Im(z) \<le> b} \<inter> {z. Im(z) \<ge> b}"
   130     by auto
   131   then show ?thesis
   132     by (auto simp: closed_Int closed_halfspace_Im_le closed_halfspace_Im_ge)
   133 qed
   134 
   135 lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
   136   by (metis Complex_in_Reals Im_complex_of_real Reals_cases complex_surj)
   137 
   138 lemma closed_complex_Reals: "closed (Reals :: complex set)"
   139 proof -
   140   have "-(Reals :: complex set) = {z. Im(z) < 0} \<union> {z. 0 < Im(z)}"
   141     by (auto simp: complex_is_Real_iff)
   142   then show ?thesis
   143     by (metis closed_def open_Un open_halfspace_Im_gt open_halfspace_Im_lt)
   144 qed
   145 
   146 
   147 lemma linear_times:
   148   fixes c::"'a::{real_algebra,real_vector}" shows "linear (\<lambda>x. c * x)"
   149   by (auto simp: linearI distrib_left)
   150 
   151 lemma bilinear_times:
   152   fixes c::"'a::{real_algebra,real_vector}" shows "bilinear (\<lambda>x y::'a. x*y)"
   153   unfolding bilinear_def
   154   by (auto simp: distrib_left distrib_right intro!: linearI)
   155 
   156 lemma linear_cnj: "linear cnj"
   157   by (auto simp: linearI cnj_def)
   158 
   159 lemma tendsto_mult_left:
   160   fixes c::"'a::real_normed_algebra" 
   161   shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) ---> c * l) F"
   162 by (rule tendsto_mult [OF tendsto_const])
   163 
   164 lemma tendsto_mult_right:
   165   fixes c::"'a::real_normed_algebra" 
   166   shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) ---> l * c) F"
   167 by (rule tendsto_mult [OF _ tendsto_const])
   168 
   169 lemma tendsto_Re_upper:
   170   assumes "~ (trivial_limit F)" 
   171           "(f ---> l) F" 
   172           "eventually (\<lambda>x. Re(f x) \<le> b) F"
   173     shows  "Re(l) \<le> b"
   174   by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Re)
   175 
   176 lemma tendsto_Re_lower:
   177   assumes "~ (trivial_limit F)" 
   178           "(f ---> l) F" 
   179           "eventually (\<lambda>x. b \<le> Re(f x)) F"
   180     shows  "b \<le> Re(l)"
   181   by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Re)
   182 
   183 lemma tendsto_Im_upper:
   184   assumes "~ (trivial_limit F)" 
   185           "(f ---> l) F" 
   186           "eventually (\<lambda>x. Im(f x) \<le> b) F"
   187     shows  "Im(l) \<le> b"
   188   by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Im)
   189 
   190 lemma tendsto_Im_lower:
   191   assumes "~ (trivial_limit F)" 
   192           "(f ---> l) F" 
   193           "eventually (\<lambda>x. b \<le> Im(f x)) F"
   194     shows  "b \<le> Im(l)"
   195   by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Im)
   196 
   197 subsection{*General lemmas*}
   198 
   199 lemma continuous_mult_left:
   200   fixes c::"'a::real_normed_algebra" 
   201   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. c * f x)"
   202 by (rule continuous_mult [OF continuous_const])
   203 
   204 lemma continuous_mult_right:
   205   fixes c::"'a::real_normed_algebra" 
   206   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x * c)"
   207 by (rule continuous_mult [OF _ continuous_const])
   208 
   209 lemma continuous_on_mult_left:
   210   fixes c::"'a::real_normed_algebra" 
   211   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c * f x)"
   212 by (rule continuous_on_mult [OF continuous_on_const])
   213 
   214 lemma continuous_on_mult_right:
   215   fixes c::"'a::real_normed_algebra" 
   216   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x * c)"
   217 by (rule continuous_on_mult [OF _ continuous_on_const])
   218 
   219 lemma uniformly_continuous_on_cmul_right [continuous_on_intros]:
   220   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
   221   assumes "uniformly_continuous_on s f"
   222   shows "uniformly_continuous_on s (\<lambda>x. f x * c)"
   223 proof (cases "c=0")
   224   case True then show ?thesis
   225     by (simp add: uniformly_continuous_on_const)
   226 next
   227   case False show ?thesis
   228     apply (rule bounded_linear.uniformly_continuous_on)
   229     apply (metis bounded_linear_ident)
   230     using assms
   231     apply (auto simp: uniformly_continuous_on_def dist_norm)
   232     apply (drule_tac x = "e / norm c" in spec, auto)
   233     apply (metis divide_pos_pos zero_less_norm_iff False)
   234     apply (rule_tac x=d in exI, auto)
   235     apply (drule_tac x = x in bspec, assumption)
   236     apply (drule_tac x = "x'" in bspec)
   237     apply (auto simp: False less_divide_eq)
   238     by (metis dual_order.strict_trans2 left_diff_distrib norm_mult_ineq)
   239 qed
   240 
   241 lemma uniformly_continuous_on_cmul_left[continuous_on_intros]:
   242   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
   243   assumes "uniformly_continuous_on s f"
   244     shows "uniformly_continuous_on s (\<lambda>x. c * f x)"
   245 by (metis assms bounded_linear.uniformly_continuous_on bounded_linear_mult_right)
   246 
   247 lemma continuous_within_norm_id [continuous_intros]: "continuous (at x within S) norm"
   248   by (rule continuous_norm [OF continuous_ident])
   249 
   250 lemma continuous_on_norm_id [continuous_intros]: "continuous_on S norm"
   251   by (metis continuous_on_eq continuous_on_id continuous_on_norm)
   252 
   253 
   254 subsection{*DERIV stuff*}
   255 
   256 (*move some premises to a sensible order. Use more \<And> symbols.*)
   257 
   258 lemma DERIV_continuous_on: "\<lbrakk>\<And>x. x \<in> s \<Longrightarrow> DERIV f x :> D\<rbrakk> \<Longrightarrow> continuous_on s f"
   259   by (metis DERIV_continuous continuous_at_imp_continuous_on)
   260 
   261 lemma DERIV_subset: 
   262   "(f has_field_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s 
   263    \<Longrightarrow> (f has_field_derivative f') (at x within t)"
   264   by (simp add: has_field_derivative_def has_derivative_within_subset)
   265 
   266 lemma lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = op * 0"
   267   by auto
   268 
   269 lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = op * 1"
   270   by auto
   271 
   272 lemma has_derivative_zero_constant:
   273   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   274   assumes "convex s"
   275       and d0: "\<And>x. x\<in>s \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within s)"
   276     shows "\<exists>c. \<forall>x\<in>s. f x = c"
   277 proof (cases "s={}")
   278   case False
   279   then obtain x where "x \<in> s"
   280     by auto
   281   have d0': "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
   282     by (metis d0)
   283   have "\<And>y. y \<in> s \<Longrightarrow> f x = f y"
   284   proof -
   285     case goal1
   286     then show ?case
   287       using differentiable_bound[OF assms(1) d0', of 0 x y] and `x \<in> s`
   288       unfolding onorm_zero
   289       by auto
   290   qed
   291   then show ?thesis 
   292     by metis
   293 next
   294   case True
   295   then show ?thesis by auto
   296 qed
   297 
   298 lemma has_derivative_zero_unique:
   299   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   300   assumes "convex s"
   301       and "\<And>x. x\<in>s \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within s)"
   302       and "a \<in> s"
   303       and "x \<in> s"
   304     shows "f x = f a"
   305   using assms
   306   by (metis has_derivative_zero_constant)
   307 
   308 lemma has_derivative_zero_connected_constant:
   309   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   310   assumes "connected s"
   311       and "open s"
   312       and "finite k"
   313       and "continuous_on s f"
   314       and "\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)"
   315     obtains c where "\<And>x. x \<in> s \<Longrightarrow> f(x) = c"
   316 proof (cases "s = {}")
   317   case True
   318   then show ?thesis
   319 by (metis empty_iff that)
   320 next
   321   case False
   322   then obtain c where "c \<in> s"
   323     by (metis equals0I)
   324   then show ?thesis
   325     by (metis has_derivative_zero_unique_strong_connected assms that)
   326 qed
   327 
   328 lemma DERIV_zero_connected_constant:
   329   fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
   330   assumes "connected s"
   331       and "open s"
   332       and "finite k"
   333       and "continuous_on s f"
   334       and "\<forall>x\<in>(s - k). DERIV f x :> 0"
   335     obtains c where "\<And>x. x \<in> s \<Longrightarrow> f(x) = c"
   336 using has_derivative_zero_connected_constant [OF assms(1-4)] assms
   337 by (metis DERIV_const Derivative.has_derivative_const Diff_iff at_within_open 
   338           frechet_derivative_at has_field_derivative_def)
   339 
   340 lemma DERIV_zero_constant:
   341   fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
   342   shows    "\<lbrakk>convex s;
   343              \<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)\<rbrakk> 
   344              \<Longrightarrow> \<exists>c. \<forall>x \<in> s. f(x) = c"
   345   unfolding has_field_derivative_def
   346   by (auto simp: lambda_zero intro: has_derivative_zero_constant)
   347 
   348 lemma DERIV_zero_unique:
   349   fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
   350   assumes "convex s"
   351       and d0: "\<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)"
   352       and "a \<in> s"
   353       and "x \<in> s"
   354     shows "f x = f a"
   355 apply (rule has_derivative_zero_unique [where f=f, OF assms(1) _ assms(3,4)])
   356 by (metis d0 has_field_derivative_imp_has_derivative lambda_zero)
   357 
   358 lemma DERIV_zero_connected_unique:
   359   fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
   360   assumes "connected s"
   361       and "open s"
   362       and d0: "\<And>x. x\<in>s \<Longrightarrow> DERIV f x :> 0"
   363       and "a \<in> s"
   364       and "x \<in> s"
   365     shows "f x = f a" 
   366     apply (rule Integration.has_derivative_zero_unique_strong_connected [of s "{}" f])
   367     using assms
   368     apply auto
   369     apply (metis DERIV_continuous_on)
   370   by (metis at_within_open has_field_derivative_def lambda_zero)
   371 
   372 lemma DERIV_transform_within:
   373   assumes "(f has_field_derivative f') (at a within s)"
   374       and "0 < d" "a \<in> s"
   375       and "\<And>x. x\<in>s \<Longrightarrow> dist x a < d \<Longrightarrow> f x = g x"
   376     shows "(g has_field_derivative f') (at a within s)"
   377   using assms unfolding has_field_derivative_def
   378   by (blast intro: Derivative.has_derivative_transform_within)
   379 
   380 lemma DERIV_transform_within_open:
   381   assumes "DERIV f a :> f'"
   382       and "open s" "a \<in> s"
   383       and "\<And>x. x\<in>s \<Longrightarrow> f x = g x"
   384     shows "DERIV g a :> f'"
   385   using assms unfolding has_field_derivative_def
   386 by (metis has_derivative_transform_within_open)
   387 
   388 lemma DERIV_transform_at:
   389   assumes "DERIV f a :> f'"
   390       and "0 < d"
   391       and "\<And>x. dist x a < d \<Longrightarrow> f x = g x"
   392     shows "DERIV g a :> f'"
   393   by (blast intro: assms DERIV_transform_within)
   394 
   395 
   396 subsection{*Holomorphic functions*}
   397 
   398 lemma has_derivative_ident[has_derivative_intros, simp]: 
   399      "FDERIV complex_of_real x :> complex_of_real"
   400   by (simp add: has_derivative_def tendsto_const bounded_linear_of_real)
   401 
   402 lemma has_real_derivative:
   403   fixes f :: "real\<Rightarrow>real" 
   404   assumes "(f has_derivative f') F"
   405     obtains c where "(f has_derivative (\<lambda>x. x * c)) F"
   406 proof -
   407   obtain c where "f' = (\<lambda>x. x * c)"
   408     by (metis assms derivative_linear real_bounded_linear)
   409   then show ?thesis
   410     by (metis assms that)
   411 qed
   412 
   413 lemma has_real_derivative_iff:
   414   fixes f :: "real\<Rightarrow>real" 
   415   shows "(\<exists>f'. (f has_derivative (\<lambda>x. x * f')) F) = (\<exists>D. (f has_derivative D) F)"
   416 by (auto elim: has_real_derivative)
   417 
   418 definition complex_differentiable :: "[complex \<Rightarrow> complex, complex filter] \<Rightarrow> bool"
   419            (infixr "(complex'_differentiable)" 50)  
   420   where "f complex_differentiable F \<equiv> \<exists>f'. (f has_field_derivative f') F"
   421 
   422 definition DD :: "['a \<Rightarrow> 'a::real_normed_field, 'a] \<Rightarrow> 'a" --{*for real, complex?*}
   423   where "DD f x \<equiv> THE f'. (f has_derivative (\<lambda>x. x * f')) (at x)"
   424 
   425 definition holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
   426            (infixl "(holomorphic'_on)" 50)
   427   where "f holomorphic_on s \<equiv> \<forall>x \<in> s. \<exists>f'. (f has_field_derivative f') (at x within s)"
   428   
   429 lemma holomorphic_on_empty: "f holomorphic_on {}"
   430   by (simp add: holomorphic_on_def)
   431 
   432 lemma holomorphic_on_differentiable:
   433      "f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. f complex_differentiable (at x within s))"
   434 unfolding holomorphic_on_def complex_differentiable_def has_field_derivative_def
   435 by (metis mult_commute_abs)
   436 
   437 lemma holomorphic_on_open:
   438     "open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> f')"
   439   by (auto simp: holomorphic_on_def has_field_derivative_def at_within_open [of _ s])
   440 
   441 lemma complex_differentiable_imp_continuous_at: 
   442     "f complex_differentiable (at x) \<Longrightarrow> continuous (at x) f"
   443   by (metis DERIV_continuous complex_differentiable_def)
   444 
   445 lemma holomorphic_on_imp_continuous_on: 
   446     "f holomorphic_on s \<Longrightarrow> continuous_on s f"
   447   by (metis DERIV_continuous continuous_on_eq_continuous_within holomorphic_on_def) 
   448 
   449 lemma has_derivative_within_open:
   450   "a \<in> s \<Longrightarrow> open s \<Longrightarrow> (f has_field_derivative f') (at a within s) \<longleftrightarrow> DERIV f a :> f'"
   451   by (simp add: has_field_derivative_def) (metis has_derivative_within_open)
   452 
   453 lemma holomorphic_on_subset:
   454     "f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t"
   455   unfolding holomorphic_on_def
   456   by (metis DERIV_subset subsetD)
   457 
   458 lemma complex_differentiable_within_subset:
   459     "\<lbrakk>f complex_differentiable (at x within s); t \<subseteq> s\<rbrakk>
   460      \<Longrightarrow> f complex_differentiable (at x within t)"
   461   unfolding complex_differentiable_def
   462   by (metis DERIV_subset)
   463 
   464 lemma complex_differentiable_at_within:
   465     "\<lbrakk>f complex_differentiable (at x)\<rbrakk>
   466      \<Longrightarrow> f complex_differentiable (at x within s)"
   467   unfolding complex_differentiable_def
   468   by (metis DERIV_subset top_greatest)
   469 
   470 
   471 lemma has_derivative_mult_right:
   472   fixes c:: "'a :: real_normed_algebra"
   473   shows "((op * c) has_derivative (op * c)) F"
   474 by (rule has_derivative_mult_right [OF has_derivative_id])
   475 
   476 lemma complex_differentiable_linear:
   477      "(op * c) complex_differentiable F"
   478 proof -
   479   have "\<And>u::complex. (\<lambda>x. x * u) = op * u"
   480     by (rule ext) (simp add: mult_ac)
   481   then show ?thesis
   482     unfolding complex_differentiable_def has_field_derivative_def
   483     by (force intro: has_derivative_mult_right)
   484 qed
   485 
   486 lemma complex_differentiable_const:
   487   "(\<lambda>z. c) complex_differentiable F"
   488   unfolding complex_differentiable_def has_field_derivative_def
   489   apply (rule exI [where x=0])
   490   by (metis Derivative.has_derivative_const lambda_zero) 
   491 
   492 lemma complex_differentiable_id:
   493   "(\<lambda>z. z) complex_differentiable F"
   494   unfolding complex_differentiable_def has_field_derivative_def
   495   apply (rule exI [where x=1])
   496   apply (simp add: lambda_one [symmetric])
   497   done
   498 
   499 (*DERIV_minus*)
   500 lemma field_differentiable_minus:
   501   assumes "(f has_field_derivative f') F" 
   502     shows "((\<lambda>z. - (f z)) has_field_derivative -f') F"
   503   apply (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus])
   504   using assms 
   505   by (auto simp: has_field_derivative_def field_simps mult_commute_abs)
   506 
   507 (*DERIV_add*)
   508 lemma field_differentiable_add:
   509   assumes "(f has_field_derivative f') F" "(g has_field_derivative g') F"
   510     shows "((\<lambda>z. f z + g z) has_field_derivative f' + g') F"
   511   apply (rule has_derivative_imp_has_field_derivative[OF has_derivative_add])
   512   using assms 
   513   by (auto simp: has_field_derivative_def field_simps mult_commute_abs)
   514 
   515 (*DERIV_diff*)
   516 lemma field_differentiable_diff:
   517   assumes "(f has_field_derivative f') F" "(g has_field_derivative g') F"
   518     shows "((\<lambda>z. f z - g z) has_field_derivative f' - g') F"
   519 by (simp only: assms diff_conv_add_uminus field_differentiable_add field_differentiable_minus)
   520 
   521 lemma complex_differentiable_minus:
   522     "f complex_differentiable F \<Longrightarrow> (\<lambda>z. -(f z)) complex_differentiable F"
   523   using assms unfolding complex_differentiable_def
   524   by (metis field_differentiable_minus)
   525 
   526 lemma complex_differentiable_add:
   527   assumes "f complex_differentiable F" "g complex_differentiable F"
   528     shows "(\<lambda>z. f z + g z) complex_differentiable F"
   529   using assms unfolding complex_differentiable_def
   530   by (metis field_differentiable_add)
   531 
   532 lemma complex_differentiable_diff:
   533   assumes "f complex_differentiable F" "g complex_differentiable F"
   534     shows "(\<lambda>z. f z - g z) complex_differentiable F"
   535   using assms unfolding complex_differentiable_def
   536   by (metis field_differentiable_diff)
   537 
   538 lemma complex_differentiable_inverse:
   539   assumes "f complex_differentiable (at a within s)" "f a \<noteq> 0"
   540   shows "(\<lambda>z. inverse (f z)) complex_differentiable (at a within s)"
   541   using assms unfolding complex_differentiable_def
   542   by (metis DERIV_inverse_fun)
   543 
   544 lemma complex_differentiable_mult:
   545   assumes "f complex_differentiable (at a within s)" 
   546           "g complex_differentiable (at a within s)"
   547     shows "(\<lambda>z. f z * g z) complex_differentiable (at a within s)"
   548   using assms unfolding complex_differentiable_def
   549   by (metis DERIV_mult [of f _ a s g])
   550   
   551 lemma complex_differentiable_divide:
   552   assumes "f complex_differentiable (at a within s)" 
   553           "g complex_differentiable (at a within s)"
   554           "g a \<noteq> 0"
   555     shows "(\<lambda>z. f z / g z) complex_differentiable (at a within s)"
   556   using assms unfolding complex_differentiable_def
   557   by (metis DERIV_divide [of f _ a s g])
   558 
   559 lemma complex_differentiable_power:
   560   assumes "f complex_differentiable (at a within s)" 
   561     shows "(\<lambda>z. f z ^ n) complex_differentiable (at a within s)"
   562   using assms unfolding complex_differentiable_def
   563   by (metis DERIV_power)
   564 
   565 lemma complex_differentiable_transform_within:
   566   "0 < d \<Longrightarrow>
   567         x \<in> s \<Longrightarrow>
   568         (\<And>x'. x' \<in> s \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x') \<Longrightarrow>
   569         f complex_differentiable (at x within s)
   570         \<Longrightarrow> g complex_differentiable (at x within s)"
   571   unfolding complex_differentiable_def has_field_derivative_def
   572   by (blast intro: has_derivative_transform_within)
   573 
   574 lemma complex_differentiable_compose_within:
   575   assumes "f complex_differentiable (at a within s)" 
   576           "g complex_differentiable (at (f a) within f`s)"
   577     shows "(g o f) complex_differentiable (at a within s)"
   578   using assms unfolding complex_differentiable_def
   579   by (metis DERIV_image_chain)
   580 
   581 lemma complex_differentiable_within_open:
   582      "\<lbrakk>a \<in> s; open s\<rbrakk> \<Longrightarrow> f complex_differentiable at a within s \<longleftrightarrow> 
   583                           f complex_differentiable at a"
   584   unfolding complex_differentiable_def
   585   by (metis at_within_open)
   586 
   587 lemma holomorphic_transform:
   588      "\<lbrakk>f holomorphic_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g holomorphic_on s"
   589   apply (auto simp: holomorphic_on_def has_field_derivative_def)
   590   by (metis complex_differentiable_def complex_differentiable_transform_within has_field_derivative_def linordered_field_no_ub)
   591 
   592 lemma holomorphic_eq:
   593      "(\<And>x. x \<in> s \<Longrightarrow> f x = g x) \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> g holomorphic_on s"
   594   by (metis holomorphic_transform)
   595 
   596 subsection{*Holomorphic*}
   597 
   598 lemma holomorphic_on_linear:
   599      "(op * c) holomorphic_on s"
   600   unfolding holomorphic_on_def  by (metis DERIV_cmult_Id)
   601 
   602 lemma holomorphic_on_const:
   603      "(\<lambda>z. c) holomorphic_on s"
   604   unfolding holomorphic_on_def  
   605   by (metis DERIV_const)
   606 
   607 lemma holomorphic_on_id:
   608      "id holomorphic_on s"
   609   unfolding holomorphic_on_def id_def  
   610   by (metis DERIV_ident)
   611 
   612 lemma holomorphic_on_compose:
   613   "f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s)
   614            \<Longrightarrow> (g o f) holomorphic_on s"
   615   unfolding holomorphic_on_def
   616   by (metis DERIV_image_chain imageI)
   617 
   618 lemma holomorphic_on_compose_gen:
   619   "\<lbrakk>f holomorphic_on s; g holomorphic_on t; f ` s \<subseteq> t\<rbrakk> \<Longrightarrow> (g o f) holomorphic_on s"
   620   unfolding holomorphic_on_def
   621   by (metis DERIV_image_chain DERIV_subset image_subset_iff)
   622 
   623 lemma holomorphic_on_minus:
   624   "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on s"
   625   unfolding holomorphic_on_def
   626 by (metis DERIV_minus)
   627 
   628 lemma holomorphic_on_add:
   629   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on s"
   630   unfolding holomorphic_on_def
   631   by (metis DERIV_add)
   632 
   633 lemma holomorphic_on_diff:
   634   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) holomorphic_on s"
   635   unfolding holomorphic_on_def
   636   by (metis DERIV_diff)
   637 
   638 lemma holomorphic_on_mult:
   639   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) holomorphic_on s"
   640   unfolding holomorphic_on_def
   641   by auto (metis DERIV_mult)
   642 
   643 lemma holomorphic_on_inverse:
   644   "\<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"
   645   unfolding holomorphic_on_def
   646   by (metis DERIV_inverse')
   647 
   648 lemma holomorphic_on_divide:
   649   "\<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"
   650   unfolding holomorphic_on_def
   651   by (metis (full_types) DERIV_divide)
   652 
   653 lemma holomorphic_on_power:
   654   "f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on s"
   655   unfolding holomorphic_on_def
   656   by (metis DERIV_power)
   657 
   658 lemma holomorphic_on_setsum:
   659   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s)
   660            \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) holomorphic_on s"
   661   unfolding holomorphic_on_def
   662   apply (induct I rule: finite_induct) 
   663   apply (force intro: DERIV_const DERIV_add)+
   664   done
   665 
   666 lemma DERIV_imp_DD: "DERIV f x :> f' \<Longrightarrow> DD f x = f'"
   667     apply (simp add: DD_def has_field_derivative_def mult_commute_abs)
   668     apply (rule the_equality, assumption)
   669     apply (metis DERIV_unique has_field_derivative_def)
   670     done
   671 
   672 lemma DD_iff_derivative_differentiable:
   673   fixes f :: "real\<Rightarrow>real" 
   674   shows   "DERIV f x :> DD f x \<longleftrightarrow> f differentiable at x"
   675 unfolding DD_def differentiable_def 
   676 by (metis (full_types) DD_def DERIV_imp_DD has_field_derivative_def has_real_derivative_iff 
   677           mult_commute_abs)
   678 
   679 lemma DD_iff_derivative_complex_differentiable:
   680   fixes f :: "complex\<Rightarrow>complex" 
   681   shows "DERIV f x :> DD f x \<longleftrightarrow> f complex_differentiable at x"
   682 unfolding DD_def complex_differentiable_def
   683 by (metis DD_def DERIV_imp_DD)
   684 
   685 lemma complex_differentiable_compose:
   686   "f complex_differentiable at z \<Longrightarrow> g complex_differentiable at (f z)
   687           \<Longrightarrow> (g o f) complex_differentiable at z"
   688 by (metis complex_differentiable_at_within complex_differentiable_compose_within)
   689 
   690 lemma complex_derivative_chain:
   691   fixes z::complex
   692   shows
   693   "f complex_differentiable at z \<Longrightarrow> g complex_differentiable at (f z)
   694           \<Longrightarrow> DD (g o f) z = DD g (f z) * DD f z"
   695 by (metis DD_iff_derivative_complex_differentiable DERIV_chain DERIV_imp_DD)
   696 
   697 lemma comp_derivative_chain:
   698   fixes z::real
   699   shows "\<lbrakk>f differentiable at z; g differentiable at (f z)\<rbrakk> 
   700          \<Longrightarrow> DD (g o f) z = DD g (f z) * DD f z"
   701 by (metis DD_iff_derivative_differentiable DERIV_chain DERIV_imp_DD)
   702 
   703 lemma complex_derivative_linear: "DD (\<lambda>w. c * w) = (\<lambda>z. c)"
   704 by (metis DERIV_imp_DD DERIV_cmult_Id)
   705 
   706 lemma complex_derivative_ident: "DD (\<lambda>w. w) = (\<lambda>z. 1)"
   707 by (metis DERIV_imp_DD DERIV_ident)
   708 
   709 lemma complex_derivative_const: "DD (\<lambda>w. c) = (\<lambda>z. 0)"
   710 by (metis DERIV_imp_DD DERIV_const)
   711 
   712 lemma complex_derivative_add:
   713   "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>  
   714    \<Longrightarrow> DD (\<lambda>w. f w + g w) z = DD f z + DD g z"
   715   unfolding complex_differentiable_def
   716   by (rule DERIV_imp_DD) (metis (poly_guards_query) DERIV_add DERIV_imp_DD)  
   717 
   718 lemma complex_derivative_diff:
   719   "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>  
   720    \<Longrightarrow> DD (\<lambda>w. f w - g w) z = DD f z - DD g z"
   721   unfolding complex_differentiable_def
   722   by (rule DERIV_imp_DD) (metis (poly_guards_query) DERIV_diff DERIV_imp_DD)
   723 
   724 lemma complex_derivative_mult:
   725   "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>  
   726    \<Longrightarrow> DD (\<lambda>w. f w * g w) z = f z * DD g z + DD f z * g z"
   727   unfolding complex_differentiable_def
   728   by (rule DERIV_imp_DD) (metis DERIV_imp_DD DERIV_mult')
   729 
   730 lemma complex_derivative_cmult:
   731   "f complex_differentiable at z \<Longrightarrow> DD (\<lambda>w. c * f w) z = c * DD f z"
   732   unfolding complex_differentiable_def
   733   by (metis DERIV_cmult DERIV_imp_DD)
   734 
   735 lemma complex_derivative_cmult_right:
   736   "f complex_differentiable at z \<Longrightarrow> DD (\<lambda>w. f w * c) z = DD f z * c"
   737   unfolding complex_differentiable_def
   738   by (metis DERIV_cmult_right DERIV_imp_DD)
   739 
   740 lemma complex_derivative_transform_within_open:
   741   "\<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> 
   742    \<Longrightarrow> DD f z = DD g z"
   743   unfolding holomorphic_on_def
   744   by (rule DERIV_imp_DD) (metis has_derivative_within_open DERIV_imp_DD DERIV_transform_within_open)
   745 
   746 lemma complex_derivative_compose_linear:
   747     "f complex_differentiable at (c * z) \<Longrightarrow> DD (\<lambda>w. f (c * w)) z = c * DD f (c * z)"
   748 apply (rule DERIV_imp_DD)
   749 apply (simp add: DD_iff_derivative_complex_differentiable [symmetric])
   750 apply (metis DERIV_chain' DERIV_cmult_Id comm_semiring_1_class.normalizing_semiring_rules(7))  
   751 done
   752 
   753 subsection{*Caratheodory characterization.*}
   754 
   755 (*REPLACE the original version. BUT IN WHICH FILE??*)
   756 thm Deriv.CARAT_DERIV
   757 
   758 lemma complex_differentiable_caratheodory_at:
   759   "f complex_differentiable (at z) \<longleftrightarrow>
   760          (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)"
   761   using CARAT_DERIV [of f]
   762   by (simp add: complex_differentiable_def has_field_derivative_def)
   763 
   764 lemma complex_differentiable_caratheodory_within:
   765   "f complex_differentiable (at z within s) \<longleftrightarrow>
   766          (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)"
   767   using DERIV_caratheodory_within [of f]
   768   by (simp add: complex_differentiable_def has_field_derivative_def)
   769 
   770 subsection{*analyticity on a set*}
   771 
   772 definition analytic_on (infixl "(analytic'_on)" 50)  
   773   where
   774    "f analytic_on s \<equiv> \<forall>x \<in> s. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)"
   775 
   776 lemma analytic_imp_holomorphic:
   777      "f analytic_on s \<Longrightarrow> f holomorphic_on s"
   778   unfolding analytic_on_def holomorphic_on_def
   779   apply (simp add: has_derivative_within_open [OF _ open_ball])
   780   apply (metis DERIV_subset dist_self mem_ball top_greatest)
   781   done
   782 
   783 lemma analytic_on_open:
   784      "open s \<Longrightarrow> f analytic_on s \<longleftrightarrow> f holomorphic_on s"
   785 apply (auto simp: analytic_imp_holomorphic)
   786 apply (auto simp: analytic_on_def holomorphic_on_def)
   787 by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball)
   788 
   789 lemma analytic_on_imp_differentiable_at:
   790   "f analytic_on s \<Longrightarrow> x \<in> s \<Longrightarrow> f complex_differentiable (at x)"
   791  apply (auto simp: analytic_on_def holomorphic_on_differentiable)
   792 by (metis Topology_Euclidean_Space.open_ball centre_in_ball complex_differentiable_within_open)
   793 
   794 lemma analytic_on_subset:
   795      "f analytic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f analytic_on t"
   796   by (auto simp: analytic_on_def)
   797 
   798 lemma analytic_on_Un:
   799     "f analytic_on (s \<union> t) \<longleftrightarrow> f analytic_on s \<and> f analytic_on t"
   800   by (auto simp: analytic_on_def)
   801 
   802 lemma analytic_on_Union:
   803   "f analytic_on (\<Union> s) \<longleftrightarrow> (\<forall>t \<in> s. f analytic_on t)"
   804   by (auto simp: analytic_on_def)
   805   
   806 lemma analytic_on_holomorphic:
   807   "f analytic_on s \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f holomorphic_on t)"
   808   (is "?lhs = ?rhs")
   809 proof -
   810   have "?lhs \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t)"
   811   proof safe
   812     assume "f analytic_on s"
   813     then show "\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t"
   814       apply (simp add: analytic_on_def)
   815       apply (rule exI [where x="\<Union>{u. open u \<and> f analytic_on u}"], auto)
   816       apply (metis Topology_Euclidean_Space.open_ball analytic_on_open centre_in_ball)
   817       by (metis analytic_on_def)
   818   next
   819     fix t
   820     assume "open t" "s \<subseteq> t" "f analytic_on t" 
   821     then show "f analytic_on s"
   822         by (metis analytic_on_subset)
   823   qed
   824   also have "... \<longleftrightarrow> ?rhs"
   825     by (auto simp: analytic_on_open)
   826   finally show ?thesis .
   827 qed
   828 
   829 lemma analytic_on_linear: "(op * c) analytic_on s"
   830   apply (simp add: analytic_on_holomorphic holomorphic_on_linear)
   831   by (metis open_UNIV top_greatest)
   832 
   833 lemma analytic_on_const: "(\<lambda>z. c) analytic_on s"
   834   unfolding analytic_on_def
   835   by (metis holomorphic_on_const zero_less_one)
   836 
   837 lemma analytic_on_id: "id analytic_on s"
   838   unfolding analytic_on_def
   839   apply (simp add: holomorphic_on_id)
   840   by (metis gt_ex)
   841 
   842 lemma analytic_on_compose:
   843   assumes f: "f analytic_on s"
   844       and g: "g analytic_on (f ` s)"
   845     shows "(g o f) analytic_on s"
   846 unfolding analytic_on_def
   847 proof (intro ballI)
   848   fix x
   849   assume x: "x \<in> s"
   850   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f
   851     by (metis analytic_on_def)
   852   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g
   853     by (metis analytic_on_def g image_eqI x) 
   854   have "isCont f x"
   855     by (metis analytic_on_imp_differentiable_at complex_differentiable_imp_continuous_at f x)
   856   with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'"
   857      by (auto simp: continuous_at_ball)
   858   have "g \<circ> f holomorphic_on ball x (min d e)" 
   859     apply (rule holomorphic_on_compose)
   860     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
   861     by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball)
   862   then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e"
   863     by (metis d e min_less_iff_conj) 
   864 qed
   865 
   866 lemma analytic_on_compose_gen:
   867   "f analytic_on s \<Longrightarrow> g analytic_on t \<Longrightarrow> (\<And>z. z \<in> s \<Longrightarrow> f z \<in> t)
   868              \<Longrightarrow> g o f analytic_on s"
   869 by (metis analytic_on_compose analytic_on_subset image_subset_iff)
   870 
   871 lemma analytic_on_neg:
   872   "f analytic_on s \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on s"
   873 by (metis analytic_on_holomorphic holomorphic_on_minus)
   874 
   875 lemma analytic_on_add:
   876   assumes f: "f analytic_on s"
   877       and g: "g analytic_on s"
   878     shows "(\<lambda>z. f z + g z) analytic_on s"
   879 unfolding analytic_on_def
   880 proof (intro ballI)
   881   fix z
   882   assume z: "z \<in> s"
   883   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
   884     by (metis analytic_on_def)
   885   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
   886     by (metis analytic_on_def g z) 
   887   have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')" 
   888     apply (rule holomorphic_on_add) 
   889     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
   890     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
   891   then show "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e"
   892     by (metis e e' min_less_iff_conj)
   893 qed
   894 
   895 lemma analytic_on_diff:
   896   assumes f: "f analytic_on s"
   897       and g: "g analytic_on s"
   898     shows "(\<lambda>z. f z - g z) analytic_on s"
   899 unfolding analytic_on_def
   900 proof (intro ballI)
   901   fix z
   902   assume z: "z \<in> s"
   903   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
   904     by (metis analytic_on_def)
   905   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
   906     by (metis analytic_on_def g z) 
   907   have "(\<lambda>z. f z - g z) holomorphic_on ball z (min e e')" 
   908     apply (rule holomorphic_on_diff) 
   909     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
   910     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
   911   then show "\<exists>e>0. (\<lambda>z. f z - g z) holomorphic_on ball z e"
   912     by (metis e e' min_less_iff_conj)
   913 qed
   914 
   915 lemma analytic_on_mult:
   916   assumes f: "f analytic_on s"
   917       and g: "g analytic_on s"
   918     shows "(\<lambda>z. f z * g z) analytic_on s"
   919 unfolding analytic_on_def
   920 proof (intro ballI)
   921   fix z
   922   assume z: "z \<in> s"
   923   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
   924     by (metis analytic_on_def)
   925   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
   926     by (metis analytic_on_def g z) 
   927   have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')" 
   928     apply (rule holomorphic_on_mult) 
   929     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
   930     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
   931   then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e"
   932     by (metis e e' min_less_iff_conj)
   933 qed
   934 
   935 lemma analytic_on_inverse:
   936   assumes f: "f analytic_on s"
   937       and nz: "(\<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0)"
   938     shows "(\<lambda>z. inverse (f z)) analytic_on s"
   939 unfolding analytic_on_def
   940 proof (intro ballI)
   941   fix z
   942   assume z: "z \<in> s"
   943   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
   944     by (metis analytic_on_def)
   945   have "continuous_on (ball z e) f"
   946     by (metis fh holomorphic_on_imp_continuous_on)
   947   then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0" 
   948     by (metis Topology_Euclidean_Space.open_ball centre_in_ball continuous_on_open_avoid e z nz)  
   949   have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')" 
   950     apply (rule holomorphic_on_inverse)
   951     apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball)
   952     by (metis nz' mem_ball min_less_iff_conj) 
   953   then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e"
   954     by (metis e e' min_less_iff_conj)
   955 qed
   956 
   957 
   958 lemma analytic_on_divide:
   959   assumes f: "f analytic_on s"
   960       and g: "g analytic_on s"
   961       and nz: "(\<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0)"
   962     shows "(\<lambda>z. f z / g z) analytic_on s"
   963 unfolding divide_inverse
   964 by (metis analytic_on_inverse analytic_on_mult f g nz)
   965 
   966 lemma analytic_on_power:
   967   "f analytic_on s \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on s"
   968 by (induct n) (auto simp: analytic_on_const analytic_on_mult)
   969 
   970 lemma analytic_on_setsum:
   971   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on s)
   972            \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) analytic_on s"
   973   by (induct I rule: finite_induct) (auto simp: analytic_on_const analytic_on_add)
   974 
   975 subsection{*analyticity at a point.*}
   976 
   977 lemma analytic_at_ball:
   978   "f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
   979 by (metis analytic_on_def singleton_iff)
   980 
   981 lemma analytic_at:
   982     "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
   983 by (metis analytic_on_holomorphic empty_subsetI insert_subset)
   984 
   985 lemma analytic_on_analytic_at:
   986     "f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
   987 by (metis analytic_at_ball analytic_on_def)
   988 
   989 lemma analytic_at_two:
   990   "f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
   991    (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)"
   992   (is "?lhs = ?rhs")
   993 proof 
   994   assume ?lhs
   995   then obtain s t 
   996     where st: "open s" "z \<in> s" "f holomorphic_on s"
   997               "open t" "z \<in> t" "g holomorphic_on t"
   998     by (auto simp: analytic_at)
   999   show ?rhs
  1000     apply (rule_tac x="s \<inter> t" in exI)
  1001     using st
  1002     apply (auto simp: Diff_subset holomorphic_on_subset)
  1003     done
  1004 next
  1005   assume ?rhs 
  1006   then show ?lhs
  1007     by (force simp add: analytic_at)
  1008 qed
  1009 
  1010 subsection{*Combining theorems for derivative with ``analytic at'' hypotheses*}
  1011 
  1012 lemma 
  1013   assumes "f analytic_on {z}" "g analytic_on {z}"
  1014   shows complex_derivative_add_at: "DD (\<lambda>w. f w + g w) z = DD f z + DD g z"
  1015     and complex_derivative_diff_at: "DD (\<lambda>w. f w - g w) z = DD f z - DD g z"
  1016     and complex_derivative_mult_at: "DD (\<lambda>w. f w * g w) z =
  1017            f z * DD g z + DD f z * g z"
  1018 proof -
  1019   obtain s where s: "open s" "z \<in> s" "f holomorphic_on s" "g holomorphic_on s"
  1020     using assms by (metis analytic_at_two)
  1021   show "DD (\<lambda>w. f w + g w) z = DD f z + DD g z"
  1022     apply (rule DERIV_imp_DD [OF DERIV_add])
  1023     using s
  1024     apply (auto simp: holomorphic_on_open complex_differentiable_def DD_iff_derivative_complex_differentiable)
  1025     done
  1026   show "DD (\<lambda>w. f w - g w) z = DD f z - DD g z"
  1027     apply (rule DERIV_imp_DD [OF DERIV_diff])
  1028     using s
  1029     apply (auto simp: holomorphic_on_open complex_differentiable_def DD_iff_derivative_complex_differentiable)
  1030     done
  1031   show "DD (\<lambda>w. f w * g w) z = f z * DD g z + DD f z * g z"
  1032     apply (rule DERIV_imp_DD [OF DERIV_mult'])
  1033     using s
  1034     apply (auto simp: holomorphic_on_open complex_differentiable_def DD_iff_derivative_complex_differentiable)
  1035     done
  1036 qed
  1037 
  1038 lemma complex_derivative_cmult_at:
  1039   "f analytic_on {z} \<Longrightarrow>  DD (\<lambda>w. c * f w) z = c * DD f z"
  1040 by (auto simp: complex_derivative_mult_at complex_derivative_const analytic_on_const)
  1041 
  1042 lemma complex_derivative_cmult_right_at:
  1043   "f analytic_on {z} \<Longrightarrow>  DD (\<lambda>w. f w * c) z = DD f z * c"
  1044 by (auto simp: complex_derivative_mult_at complex_derivative_const analytic_on_const)
  1045 
  1046 text{*A composition lemma for functions of mixed type*}
  1047 lemma has_vector_derivative_real_complex:
  1048   fixes f :: "complex \<Rightarrow> complex"
  1049   assumes "DERIV f (of_real a) :> f'"
  1050   shows "((\<lambda>x. f (of_real x)) has_vector_derivative f') (at a)"
  1051   using diff_chain_at [OF has_derivative_ident, of f "op * f'" a] assms
  1052   unfolding has_field_derivative_def has_vector_derivative_def o_def
  1053   by (auto simp: mult_ac scaleR_conv_of_real)
  1054 
  1055 subsection{*Complex differentiation of sequences and series*}
  1056 
  1057 lemma has_complex_derivative_sequence:
  1058   fixes s :: "complex set"
  1059   assumes cvs: "convex s"
  1060       and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
  1061       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"
  1062       and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) ---> l) sequentially"
  1063     shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) ---> g x) sequentially \<and> 
  1064                        (g has_field_derivative (g' x)) (at x within s)"
  1065 proof -
  1066   from assms obtain x l where x: "x \<in> s" and tf: "((\<lambda>n. f n x) ---> l) sequentially"
  1067     by blast
  1068   { fix e::real assume e: "e > 0"
  1069     then obtain N where N: "\<forall>n\<ge>N. \<forall>x. x \<in> s \<longrightarrow> cmod (f' n x - g' x) \<le> e"
  1070       by (metis conv)    
  1071     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"
  1072     proof (rule exI [of _ N], clarify)
  1073       fix n y h
  1074       assume "N \<le> n" "y \<in> s"
  1075       then have "cmod (f' n y - g' y) \<le> e"
  1076         by (metis N)
  1077       then have "cmod h * cmod (f' n y - g' y) \<le> cmod h * e"
  1078         by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
  1079       then show "cmod (f' n y * h - g' y * h) \<le> e * cmod h"
  1080         by (simp add: norm_mult [symmetric] field_simps)
  1081     qed
  1082   } note ** = this
  1083   show ?thesis
  1084   unfolding has_field_derivative_def
  1085   proof (rule has_derivative_sequence [OF cvs _ _ x])
  1086     show "\<forall>n. \<forall>x\<in>s. (f n has_derivative (op * (f' n x))) (at x within s)"
  1087       by (metis has_field_derivative_def df)
  1088   next show "(\<lambda>n. f n x) ----> l"
  1089     by (rule tf)
  1090   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"
  1091     by (blast intro: **)
  1092   qed
  1093 qed
  1094 
  1095 
  1096 lemma has_complex_derivative_series:
  1097   fixes s :: "complex set"
  1098   assumes cvs: "convex s"
  1099       and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
  1100       and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s 
  1101                 \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
  1102       and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) sums l)"
  1103     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))"
  1104 proof -
  1105   from assms obtain x l where x: "x \<in> s" and sf: "((\<lambda>n. f n x) sums l)"
  1106     by blast
  1107   { fix e::real assume e: "e > 0"
  1108     then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> s 
  1109             \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
  1110       by (metis conv)    
  1111     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"
  1112     proof (rule exI [of _ N], clarify)
  1113       fix n y h
  1114       assume "N \<le> n" "y \<in> s"
  1115       then have "cmod ((\<Sum>i<n. f' i y) - g' y) \<le> e"
  1116         by (metis N)
  1117       then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e"
  1118         by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
  1119       then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h"
  1120         by (simp add: norm_mult [symmetric] field_simps setsum_right_distrib)
  1121     qed
  1122   } note ** = this
  1123   show ?thesis
  1124   unfolding has_field_derivative_def
  1125   proof (rule has_derivative_series [OF cvs _ _ x])
  1126     fix n x
  1127     assume "x \<in> s"
  1128     then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within s)"
  1129       by (metis df has_field_derivative_def mult_commute_abs)
  1130   next show " ((\<lambda>n. f n x) sums l)"
  1131     by (rule sf)
  1132   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"
  1133     by (blast intro: **)
  1134   qed
  1135 qed
  1136 
  1137 subsection{*Bound theorem*}
  1138 
  1139 lemma complex_differentiable_bound:
  1140   fixes s :: "complex set"
  1141   assumes cvs: "convex s"
  1142       and df:  "\<And>z. z \<in> s \<Longrightarrow> (f has_field_derivative f' z) (at z within s)"
  1143       and dn:  "\<And>z. z \<in> s \<Longrightarrow> norm (f' z) \<le> B"
  1144       and "x \<in> s"  "y \<in> s"
  1145     shows "norm(f x - f y) \<le> B * norm(x - y)"
  1146   apply (rule differentiable_bound [OF cvs])
  1147   apply (rule ballI, erule df [unfolded has_field_derivative_def])
  1148   apply (rule ballI, rule onorm_le, simp add: norm_mult mult_right_mono dn)
  1149   apply fact
  1150   apply fact
  1151   done
  1152 
  1153 subsection{*Inverse function theorem for complex derivatives.*}
  1154 
  1155 lemma has_complex_derivative_inverse_basic:
  1156   fixes f :: "complex \<Rightarrow> complex"
  1157   shows "DERIV f (g y) :> f' \<Longrightarrow>
  1158         f' \<noteq> 0 \<Longrightarrow>
  1159         continuous (at y) g \<Longrightarrow>
  1160         open t \<Longrightarrow>
  1161         y \<in> t \<Longrightarrow>
  1162         (\<And>z. z \<in> t \<Longrightarrow> f (g z) = z)
  1163         \<Longrightarrow> DERIV g y :> inverse (f')"
  1164   unfolding has_field_derivative_def
  1165   apply (rule has_derivative_inverse_basic)
  1166   apply (auto simp:  bounded_linear_mult_right)
  1167   done
  1168 
  1169 (*Used only once, in Multivariate/cauchy.ml. *)
  1170 lemma has_complex_derivative_inverse_strong:
  1171   fixes f :: "complex \<Rightarrow> complex"
  1172   shows "DERIV f x :> f' \<Longrightarrow>
  1173          f' \<noteq> 0 \<Longrightarrow>
  1174          open s \<Longrightarrow>
  1175          x \<in> s \<Longrightarrow>
  1176          continuous_on s f \<Longrightarrow>
  1177          (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
  1178          \<Longrightarrow> DERIV g (f x) :> inverse (f')"
  1179   unfolding has_field_derivative_def
  1180   apply (rule has_derivative_inverse_strong [of s x f g ])
  1181   using assms 
  1182   by auto
  1183 
  1184 lemma has_complex_derivative_inverse_strong_x:
  1185   fixes f :: "complex \<Rightarrow> complex"
  1186   shows  "DERIV f (g y) :> f' \<Longrightarrow>
  1187           f' \<noteq> 0 \<Longrightarrow>
  1188           open s \<Longrightarrow>
  1189           continuous_on s f \<Longrightarrow>
  1190           g y \<in> s \<Longrightarrow> f(g y) = y \<Longrightarrow>
  1191           (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
  1192           \<Longrightarrow> DERIV g y :> inverse (f')"
  1193   unfolding has_field_derivative_def
  1194   apply (rule has_derivative_inverse_strong_x [of s g y f])
  1195   using assms 
  1196   by auto
  1197 
  1198 subsection{*Further useful properties of complex conjugation*}
  1199 
  1200 lemma lim_cnj: "((\<lambda>x. cnj(f x)) ---> cnj l) F \<longleftrightarrow> (f ---> l) F"
  1201   by (simp add: tendsto_iff dist_complex_def Complex.complex_cnj_diff [symmetric])
  1202 
  1203 lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
  1204   by (simp add: sums_def lim_cnj cnj_setsum [symmetric])
  1205 
  1206 lemma continuous_within_cnj: "continuous (at z within s) cnj"
  1207 by (metis bounded_linear_cnj linear_continuous_within)
  1208 
  1209 lemma continuous_on_cnj: "continuous_on s cnj"
  1210 by (metis bounded_linear_cnj linear_continuous_on)
  1211 
  1212 subsection{*Some limit theorems about real part of real series etc.*}
  1213 
  1214 lemma real_lim:
  1215   fixes l::complex
  1216   assumes "(f ---> l) F" and " ~(trivial_limit F)" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>"
  1217   shows  "l \<in> \<real>"
  1218 proof -
  1219   have 1: "((\<lambda>i. Im (f i)) ---> Im l) F"
  1220     by (metis assms(1) tendsto_Im) 
  1221   then have  "((\<lambda>i. Im (f i)) ---> 0) F" using assms
  1222     by (metis (mono_tags, lifting) Lim_eventually complex_is_Real_iff eventually_mono)
  1223   then show ?thesis using 1
  1224     by (metis 1 assms(2) complex_is_Real_iff tendsto_unique) 
  1225 qed
  1226 
  1227 lemma real_lim_sequentially:
  1228   fixes l::complex
  1229   shows "(f ---> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
  1230 by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially)
  1231 
  1232 lemma real_series: 
  1233   fixes l::complex
  1234   shows "f sums l \<Longrightarrow> (\<And>n. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
  1235 unfolding sums_def
  1236 by (metis real_lim_sequentially setsum_in_Reals)
  1237 
  1238 
  1239 lemma Lim_null_comparison_Re:
  1240    "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F \<Longrightarrow>  (g ---> 0) F \<Longrightarrow> (f ---> 0) F"
  1241   by (metis Lim_null_comparison complex_Re_zero tendsto_Re)
  1242 
  1243 
  1244 lemma norm_setsum_bound:
  1245   fixes n::nat
  1246   shows" \<lbrakk>\<And>n. N \<le> n \<Longrightarrow> norm (f n) \<le> g n; N \<le> m\<rbrakk>
  1247        \<Longrightarrow> norm (setsum f {m..<n}) \<le> setsum g {m..<n}"
  1248 apply (induct n, auto)
  1249 by (metis dual_order.trans le_cases le_neq_implies_less norm_triangle_mono)
  1250 
  1251 
  1252 (*MOVE? But not to Finite_Cartesian_Product*)
  1253 lemma sums_vec_nth :
  1254   assumes "f sums a"
  1255   shows "(\<lambda>x. f x $ i) sums a $ i"
  1256 using assms unfolding sums_def
  1257 by (auto dest: tendsto_vec_nth [where i=i])
  1258 
  1259 lemma summable_vec_nth :
  1260   assumes "summable f"
  1261   shows "summable (\<lambda>x. f x $ i)"
  1262 using assms unfolding summable_def
  1263 by (blast intro: sums_vec_nth)
  1264 
  1265 lemma sums_Re:
  1266   assumes "f sums a"
  1267   shows "(\<lambda>x. Re (f x)) sums Re a"
  1268 using assms 
  1269 by (auto simp: sums_def Re_setsum [symmetric] isCont_tendsto_compose [of a Re])
  1270 
  1271 lemma sums_Im:
  1272   assumes "f sums a"
  1273   shows "(\<lambda>x. Im (f x)) sums Im a"
  1274 using assms 
  1275 by (auto simp: sums_def Im_setsum [symmetric] isCont_tendsto_compose [of a Im])
  1276 
  1277 lemma summable_Re:
  1278   assumes "summable f"
  1279   shows "summable (\<lambda>x. Re (f x))"
  1280 using assms unfolding summable_def
  1281 by (blast intro: sums_Re)
  1282 
  1283 lemma summable_Im:
  1284   assumes "summable f"
  1285   shows "summable (\<lambda>x. Im (f x))"
  1286 using assms unfolding summable_def
  1287 by (blast intro: sums_Im)
  1288 
  1289 lemma series_comparison_complex:
  1290   fixes f:: "nat \<Rightarrow> 'a::banach"
  1291   assumes sg: "summable g"
  1292      and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"
  1293      and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"
  1294   shows "summable f"
  1295 proof -
  1296   have g: "\<And>n. cmod (g n) = Re (g n)" using assms
  1297     by (metis abs_of_nonneg in_Reals_norm)
  1298   show ?thesis
  1299     apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])
  1300     using sg
  1301     apply (auto simp: summable_def)
  1302     apply (rule_tac x="Re s" in exI)
  1303     apply (auto simp: g sums_Re)
  1304     apply (metis fg g)
  1305     done
  1306 qed
  1307 
  1308 lemma summable_complex_of_real [simp]:
  1309   "summable (\<lambda>n. complex_of_real (f n)) = summable f"
  1310 apply (auto simp: Series.summable_Cauchy)  
  1311 apply (drule_tac x = e in spec, auto)
  1312 apply (rule_tac x=N in exI)
  1313 apply (auto simp: of_real_setsum [symmetric])
  1314 done
  1315 
  1316 
  1317 
  1318 
  1319 lemma setsum_Suc_reindex:
  1320   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
  1321     shows  "setsum f {0..n} = f 0 - f (Suc n) + setsum (\<lambda>i. f (Suc i)) {0..n}"
  1322 by (induct n) auto
  1323 
  1324 
  1325 text{*A kind of complex Taylor theorem.*}
  1326 lemma complex_taylor:
  1327   assumes s: "convex s" 
  1328       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)"
  1329       and B: "\<And>x. x \<in> s \<Longrightarrow> cmod (f (Suc n) x) \<le> B"
  1330       and w: "w \<in> s"
  1331       and z: "z \<in> s"
  1332     shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / of_nat (fact i)))
  1333           \<le> B * cmod(z - w)^(Suc n) / fact n"
  1334 proof -
  1335   have wzs: "closed_segment w z \<subseteq> s" using assms
  1336     by (metis convex_contains_segment)
  1337   { fix u
  1338     assume "u \<in> closed_segment w z"
  1339     then have "u \<in> s"
  1340       by (metis wzs subsetD)
  1341     have "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / of_nat (fact i) +
  1342                       f (Suc i) u * (z-u)^i / of_nat (fact i)) = 
  1343               f (Suc n) u * (z-u) ^ n / of_nat (fact n)"
  1344     proof (induction n)
  1345       case 0 show ?case by simp
  1346     next
  1347       case (Suc n)
  1348       have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / of_nat (fact i) +
  1349                              f (Suc i) u * (z-u) ^ i / of_nat (fact i)) =  
  1350            f (Suc n) u * (z-u) ^ n / of_nat (fact n) +
  1351            f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / of_nat (fact (Suc n)) -
  1352            f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / of_nat (fact (Suc n))"
  1353         using Suc by simp
  1354       also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / of_nat (fact (Suc n))"
  1355       proof -
  1356         have "of_nat(fact(Suc n)) *
  1357              (f(Suc n) u *(z-u) ^ n / of_nat(fact n) +
  1358                f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / of_nat(fact(Suc n)) -
  1359                f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / of_nat(fact(Suc n))) =
  1360             (of_nat(fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / of_nat(fact n) +
  1361             (of_nat(fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / of_nat(fact(Suc n))) -
  1362             (of_nat(fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / of_nat(fact(Suc n))"
  1363           by (simp add: algebra_simps del: fact_Suc)
  1364         also have "... =
  1365                    (of_nat (fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / of_nat (fact n) +
  1366                    (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
  1367                    (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
  1368           by (simp del: fact_Suc)
  1369         also have "... = 
  1370                    (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) +
  1371                    (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
  1372                    (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
  1373           by (simp only: fact_Suc of_nat_mult mult_ac) simp
  1374         also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
  1375           by (simp add: algebra_simps)
  1376         finally show ?thesis
  1377         by (simp add: mult_left_cancel [where c = "of_nat (fact (Suc n))", THEN iffD1] del: fact_Suc)
  1378       qed
  1379       finally show ?case .
  1380     qed
  1381     then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / of_nat (fact i))) 
  1382                 has_field_derivative f (Suc n) u * (z-u) ^ n / of_nat (fact n))
  1383                (at u within s)"
  1384       apply (intro DERIV_intros DERIV_power[THEN DERIV_cong])
  1385       apply (blast intro: assms `u \<in> s`)
  1386       apply (rule refl)+
  1387       apply (auto simp: field_simps)
  1388       done
  1389   } note sum_deriv = this
  1390   { fix u
  1391     assume u: "u \<in> closed_segment w z"
  1392     then have us: "u \<in> s"
  1393       by (metis wzs subsetD)
  1394     have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> cmod (f (Suc n) u) * cmod (u - z) ^ n"
  1395       by (metis norm_minus_commute order_refl)
  1396     also have "... \<le> cmod (f (Suc n) u) * cmod (z - w) ^ n"
  1397       by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
  1398     also have "... \<le> B * cmod (z - w) ^ n"
  1399       by (metis norm_ge_zero zero_le_power mult_right_mono  B [OF us])
  1400     finally have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> B * cmod (z - w) ^ n" .
  1401   } note cmod_bound = this
  1402   have "(\<Sum>i\<le>n. f i z * (z - z) ^ i / of_nat (fact i)) = (\<Sum>i\<le>n. (f i z / of_nat (fact i)) * 0 ^ i)"
  1403     by simp
  1404   also have "\<dots> = f 0 z / of_nat (fact 0)"
  1405     by (subst setsum_zero_power) simp
  1406   finally have "cmod (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / of_nat (fact i))) 
  1407             \<le> cmod ((\<Sum>i\<le>n. f i w * (z - w) ^ i / of_nat (fact i)) -
  1408                     (\<Sum>i\<le>n. f i z * (z - z) ^ i / of_nat (fact i)))"
  1409     by (simp add: norm_minus_commute)
  1410   also have "... \<le> B * cmod (z - w) ^ n / real_of_nat (fact n) * cmod (w - z)"
  1411     apply (rule complex_differentiable_bound 
  1412       [where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / of_nat(fact n)"
  1413          and s = "closed_segment w z", OF convex_segment])
  1414     apply (auto simp: ends_in_segment real_of_nat_def DERIV_subset [OF sum_deriv wzs]
  1415                   norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
  1416     done
  1417   also have "...  \<le> B * cmod (z - w) ^ Suc n / real (fact n)"
  1418     by (simp add: algebra_simps norm_minus_commute real_of_nat_def)
  1419   finally show ?thesis .
  1420 qed
  1421 
  1422 text{* Something more like the traditional MVT for real components.*}
  1423 lemma complex_mvt_line:
  1424   assumes "\<And>u. u \<in> closed_segment w z ==> (f has_field_derivative f'(u)) (at u)"
  1425     shows "\<exists>u. u \<in> open_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
  1426 proof -
  1427   have twz: "\<And>t. (1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)"
  1428     by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
  1429   show ?thesis
  1430     apply (cut_tac mvt_simple
  1431                      [of 0 1 "Re o f o (\<lambda>t. (1 - t) *\<^sub>R w +  t *\<^sub>R z)"
  1432                       "\<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))"])
  1433     apply auto
  1434     apply (rule_tac x="(1 - x) *\<^sub>R w + x *\<^sub>R z" in exI)
  1435     apply (simp add: open_segment_def)
  1436     apply (auto simp add: twz)
  1437     apply (rule has_derivative_at_within)
  1438     apply (intro has_derivative_intros has_derivative_add [OF has_derivative_const, simplified])+
  1439     apply (rule assms [unfolded has_field_derivative_def])
  1440     apply (force simp add: twz closed_segment_def)
  1441     done
  1442 qed
  1443 
  1444 lemma complex_taylor_mvt:
  1445   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)"
  1446     shows "\<exists>u. u \<in> closed_segment w z \<and>
  1447             Re (f 0 z) =
  1448             Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / of_nat (fact i)) +
  1449                 (f (Suc n) u * (z-u)^n / of_nat (fact n)) * (z - w))"
  1450 proof -
  1451   { fix u
  1452     assume u: "u \<in> closed_segment w z"
  1453     have "(\<Sum>i = 0..n.
  1454                (f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) /
  1455                of_nat (fact i)) =
  1456           f (Suc 0) u -
  1457              (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
  1458              of_nat (fact (Suc n)) +
  1459              (\<Sum>i = 0..n.
  1460                  (f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) /
  1461                  of_nat (fact (Suc i)))"
  1462        by (subst setsum_Suc_reindex) simp
  1463     also have "... = f (Suc 0) u -
  1464              (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
  1465              of_nat (fact (Suc n)) +
  1466              (\<Sum>i = 0..n.
  1467                  f (Suc (Suc i)) u * ((z-u) ^ Suc i) / of_nat (fact (Suc i))  - 
  1468                  f (Suc i) u * (z-u) ^ i / of_nat (fact i))"
  1469       by (simp only: diff_divide_distrib fact_cancel mult_ac)
  1470     also have "... = f (Suc 0) u -
  1471              (f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) /
  1472              of_nat (fact (Suc n)) +
  1473              f (Suc (Suc n)) u * (z-u) ^ Suc n / of_nat (fact (Suc n)) - f (Suc 0) u"
  1474       by (subst setsum_Suc_diff) auto
  1475     also have "... = f (Suc n) u * (z-u) ^ n / of_nat (fact n)"
  1476       by (simp only: algebra_simps diff_divide_distrib fact_cancel)
  1477     finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i 
  1478                              - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / of_nat (fact i)) =
  1479                   f (Suc n) u * (z - u) ^ n / of_nat (fact n)" .
  1480     then have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / of_nat (fact i)) has_field_derivative
  1481                 f (Suc n) u * (z - u) ^ n / of_nat (fact n))  (at u)"
  1482       apply (intro DERIV_intros)+
  1483       apply (force intro: u assms)
  1484       apply (rule refl)+
  1485       apply (auto simp: mult_ac)
  1486       done
  1487   }
  1488   then show ?thesis
  1489     apply (cut_tac complex_mvt_line [of w z "\<lambda>u. \<Sum>i = 0..n. f i u * (z-u) ^ i / of_nat (fact i)"
  1490                "\<lambda>u. (f (Suc n) u * (z-u)^n / of_nat (fact n))"])
  1491     apply (auto simp add: intro: open_closed_segment)
  1492     done
  1493 qed
  1494 
  1495 text{*Relations among convergence and absolute convergence for power series.*}
  1496 lemma abel_lemma:
  1497   fixes a :: "nat \<Rightarrow> 'a::real_normed_vector"
  1498   assumes r: "0 \<le> r" and r0: "r < r0" and M: "\<And>n. norm(a n) * r0^n \<le> M"
  1499     shows "summable (\<lambda>n. norm(a(n)) * r^n)"
  1500 proof (rule summable_comparison_test' [of "\<lambda>n. M * (r / r0)^n"])
  1501   show "summable (\<lambda>n. M * (r / r0) ^ n)"
  1502     using assms 
  1503     by (auto simp add: summable_mult summable_geometric)
  1504   next
  1505     fix n
  1506     show "norm (norm (a n) * r ^ n) \<le> M * (r / r0) ^ n"
  1507       using r r0 M [of n]
  1508       apply (auto simp add: abs_mult field_simps power_divide)
  1509       apply (cases "r=0", simp)
  1510       apply (cases n, auto)
  1511       done
  1512 qed
  1513 
  1514 
  1515 end