src/HOL/Analysis/Complex_Analysis_Basics.thy
 author paulson Tue Oct 25 15:46:07 2016 +0100 (2016-10-25) changeset 64394 141e1ed8d5a0 parent 64267 b9a1486e79be child 65587 16a8991ab398 permissions -rw-r--r--
more new material
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 *)
5 section \<open>Complex Analysis Basics\<close>
7 theory Complex_Analysis_Basics
8 imports Equivalence_Lebesgue_Henstock_Integration "~~/src/HOL/Library/Nonpos_Ints"
9 begin
12 subsection\<open>General lemmas\<close>
14 lemma nonneg_Reals_cmod_eq_Re: "z \<in> \<real>\<^sub>\<ge>\<^sub>0 \<Longrightarrow> norm z = Re z"
15   by (simp add: complex_nonneg_Reals_iff cmod_eq_Re)
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])
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] .
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)
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)
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)
40 lemma linear_cnj: "linear cnj"
41   using bounded_linear.linear[OF bounded_linear_cnj] .
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)
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)
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)
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)
71 lemma lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = op * 0"
72   by auto
74 lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = op * 1"
75   by auto
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])
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])
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])
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])
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] .
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)
108 lemma continuous_within_norm_id [continuous_intros]: "continuous (at x within S) norm"
109   by (rule continuous_norm [OF continuous_ident])
111 lemma continuous_on_norm_id [continuous_intros]: "continuous_on S norm"
112   by (intro continuous_on_id continuous_on_norm)
114 subsection\<open>DERIV stuff\<close>
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)
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)
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)
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)
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)
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)
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)
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 convex_UNIV)
184 subsection \<open>Some limit theorems about real part of real series etc.\<close>
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])
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)
199 subsection \<open>Complex number lemmas\<close>
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 continuous_on_Re
213             continuous_on_Im continuous_on_id continuous_on_const)+
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
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)
226 corollary closed_nonpos_Reals_complex [simp]: "closed (\<real>\<^sub>\<le>\<^sub>0 :: complex set)"
227 proof -
228   have "\<real>\<^sub>\<le>\<^sub>0 = \<real> \<inter> {z. Re(z) \<le> 0}"
229     using complex_nonpos_Reals_iff complex_is_Real_iff by auto
230   then show ?thesis
231     by (metis closed_Real_halfspace_Re_le)
232 qed
234 lemma closed_Real_halfspace_Re_ge: "closed (\<real> \<inter> {w. x \<le> Re(w)})"
235   using closed_halfspace_Re_ge
236   by (simp add: closed_Int closed_complex_Reals)
238 corollary closed_nonneg_Reals_complex [simp]: "closed (\<real>\<^sub>\<ge>\<^sub>0 :: complex set)"
239 proof -
240   have "\<real>\<^sub>\<ge>\<^sub>0 = \<real> \<inter> {z. Re(z) \<ge> 0}"
241     using complex_nonneg_Reals_iff complex_is_Real_iff by auto
242   then show ?thesis
243     by (metis closed_Real_halfspace_Re_ge)
244 qed
246 lemma closed_real_abs_le: "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
247 proof -
248   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})"
249     by auto
250   then show "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
251     by (simp add: closed_Int closed_Real_halfspace_Re_ge closed_Real_halfspace_Re_le)
252 qed
254 lemma real_lim:
255   fixes l::complex
256   assumes "(f \<longlongrightarrow> l) F" and "~(trivial_limit F)" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>"
257   shows  "l \<in> \<real>"
258 proof (rule Lim_in_closed_set[OF closed_complex_Reals _ assms(2,1)])
259   show "eventually (\<lambda>x. f x \<in> \<real>) F"
260     using assms(3, 4) by (auto intro: eventually_mono)
261 qed
263 lemma real_lim_sequentially:
264   fixes l::complex
265   shows "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
266 by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially)
268 lemma real_series:
269   fixes l::complex
270   shows "f sums l \<Longrightarrow> (\<And>n. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
271 unfolding sums_def
272 by (metis real_lim_sequentially sum_in_Reals)
274 lemma Lim_null_comparison_Re:
275   assumes "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F" "(g \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F"
276   by (rule Lim_null_comparison[OF assms(1)] tendsto_eq_intros assms(2))+ simp
278 subsection\<open>Holomorphic functions\<close>
280 subsection\<open>Holomorphic functions\<close>
282 definition holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
283            (infixl "(holomorphic'_on)" 50)
284   where "f holomorphic_on s \<equiv> \<forall>x\<in>s. f field_differentiable (at x within s)"
286 named_theorems holomorphic_intros "structural introduction rules for holomorphic_on"
288 lemma holomorphic_onI [intro?]: "(\<And>x. x \<in> s \<Longrightarrow> f field_differentiable (at x within s)) \<Longrightarrow> f holomorphic_on s"
291 lemma holomorphic_onD [dest?]: "\<lbrakk>f holomorphic_on s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x within s)"
294 lemma holomorphic_on_imp_differentiable_on:
295     "f holomorphic_on s \<Longrightarrow> f differentiable_on s"
296   unfolding holomorphic_on_def differentiable_on_def
299 lemma holomorphic_on_imp_differentiable_at:
300    "\<lbrakk>f holomorphic_on s; open s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x)"
301 using at_within_open holomorphic_on_def by fastforce
303 lemma holomorphic_on_empty [holomorphic_intros]: "f holomorphic_on {}"
306 lemma holomorphic_on_open:
307     "open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> f')"
308   by (auto simp: holomorphic_on_def field_differentiable_def has_field_derivative_def at_within_open [of _ s])
310 lemma holomorphic_on_imp_continuous_on:
311     "f holomorphic_on s \<Longrightarrow> continuous_on s f"
312   by (metis field_differentiable_imp_continuous_at continuous_on_eq_continuous_within holomorphic_on_def)
314 lemma holomorphic_on_subset [elim]:
315     "f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t"
316   unfolding holomorphic_on_def
317   by (metis field_differentiable_within_subset subsetD)
319 lemma holomorphic_transform: "\<lbrakk>f holomorphic_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g holomorphic_on s"
320   by (metis field_differentiable_transform_within linordered_field_no_ub holomorphic_on_def)
322 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"
323   by (metis holomorphic_transform)
325 lemma holomorphic_on_linear [simp, holomorphic_intros]: "(op * c) holomorphic_on s"
326   unfolding holomorphic_on_def by (metis field_differentiable_linear)
328 lemma holomorphic_on_const [simp, holomorphic_intros]: "(\<lambda>z. c) holomorphic_on s"
329   unfolding holomorphic_on_def by (metis field_differentiable_const)
331 lemma holomorphic_on_ident [simp, holomorphic_intros]: "(\<lambda>x. x) holomorphic_on s"
332   unfolding holomorphic_on_def by (metis field_differentiable_ident)
334 lemma holomorphic_on_id [simp, holomorphic_intros]: "id holomorphic_on s"
335   unfolding id_def by (rule holomorphic_on_ident)
337 lemma holomorphic_on_compose:
338   "f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s) \<Longrightarrow> (g o f) holomorphic_on s"
339   using field_differentiable_compose_within[of f _ s g]
340   by (auto simp: holomorphic_on_def)
342 lemma holomorphic_on_compose_gen:
343   "f holomorphic_on s \<Longrightarrow> g holomorphic_on t \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> (g o f) holomorphic_on s"
344   by (metis holomorphic_on_compose holomorphic_on_subset)
346 lemma holomorphic_on_minus [holomorphic_intros]: "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on s"
347   by (metis field_differentiable_minus holomorphic_on_def)
350   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on s"
351   unfolding holomorphic_on_def by (metis field_differentiable_add)
353 lemma holomorphic_on_diff [holomorphic_intros]:
354   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) holomorphic_on s"
355   unfolding holomorphic_on_def by (metis field_differentiable_diff)
357 lemma holomorphic_on_mult [holomorphic_intros]:
358   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) holomorphic_on s"
359   unfolding holomorphic_on_def by (metis field_differentiable_mult)
361 lemma holomorphic_on_inverse [holomorphic_intros]:
362   "\<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"
363   unfolding holomorphic_on_def by (metis field_differentiable_inverse)
365 lemma holomorphic_on_divide [holomorphic_intros]:
366   "\<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"
367   unfolding holomorphic_on_def by (metis field_differentiable_divide)
369 lemma holomorphic_on_power [holomorphic_intros]:
370   "f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on s"
371   unfolding holomorphic_on_def by (metis field_differentiable_power)
373 lemma holomorphic_on_sum [holomorphic_intros]:
374   "(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) holomorphic_on s"
375   unfolding holomorphic_on_def by (metis field_differentiable_sum)
377 lemma DERIV_deriv_iff_field_differentiable:
378   "DERIV f x :> deriv f x \<longleftrightarrow> f field_differentiable at x"
379   unfolding field_differentiable_def by (metis DERIV_imp_deriv)
381 lemma holomorphic_derivI:
382      "\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk>
383       \<Longrightarrow> (f has_field_derivative deriv f x) (at x within T)"
384 by (metis DERIV_deriv_iff_field_differentiable at_within_open  holomorphic_on_def has_field_derivative_at_within)
386 lemma complex_derivative_chain:
387   "f field_differentiable at x \<Longrightarrow> g field_differentiable at (f x)
388     \<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
389   by (metis DERIV_deriv_iff_field_differentiable DERIV_chain DERIV_imp_deriv)
391 lemma deriv_linear [simp]: "deriv (\<lambda>w. c * w) = (\<lambda>z. c)"
392   by (metis DERIV_imp_deriv DERIV_cmult_Id)
394 lemma deriv_ident [simp]: "deriv (\<lambda>w. w) = (\<lambda>z. 1)"
395   by (metis DERIV_imp_deriv DERIV_ident)
397 lemma deriv_id [simp]: "deriv id = (\<lambda>z. 1)"
400 lemma deriv_const [simp]: "deriv (\<lambda>w. c) = (\<lambda>z. 0)"
401   by (metis DERIV_imp_deriv DERIV_const)
404   "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
405    \<Longrightarrow> deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
406   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
407   by (auto intro!: DERIV_imp_deriv derivative_intros)
409 lemma deriv_diff [simp]:
410   "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
411    \<Longrightarrow> deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
412   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
413   by (auto intro!: DERIV_imp_deriv derivative_intros)
415 lemma deriv_mult [simp]:
416   "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
417    \<Longrightarrow> deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
418   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
419   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
421 lemma deriv_cmult [simp]:
422   "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z"
423   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
424   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
426 lemma deriv_cmult_right [simp]:
427   "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c"
428   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
429   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
431 lemma deriv_cdivide_right [simp]:
432   "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w / c) z = deriv f z / c"
433   unfolding Fields.field_class.field_divide_inverse
434   by (blast intro: deriv_cmult_right)
436 lemma complex_derivative_transform_within_open:
437   "\<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>
438    \<Longrightarrow> deriv f z = deriv g z"
439   unfolding holomorphic_on_def
440   by (rule DERIV_imp_deriv)
441      (metis DERIV_deriv_iff_field_differentiable DERIV_transform_within_open at_within_open)
443 lemma deriv_compose_linear:
444   "f field_differentiable at (c * z) \<Longrightarrow> deriv (\<lambda>w. f (c * w)) z = c * deriv f (c * z)"
445 apply (rule DERIV_imp_deriv)
446 apply (simp add: DERIV_deriv_iff_field_differentiable [symmetric])
447 apply (drule DERIV_chain' [of "times c" c z UNIV f "deriv f (c * z)", OF DERIV_cmult_Id])
449 done
451 lemma nonzero_deriv_nonconstant:
452   assumes df: "DERIV f \<xi> :> df" and S: "open S" "\<xi> \<in> S" and "df \<noteq> 0"
453     shows "\<not> f constant_on S"
454 unfolding constant_on_def
455 by (metis \<open>df \<noteq> 0\<close> DERIV_transform_within_open [OF df S] DERIV_const DERIV_unique)
457 lemma holomorphic_nonconstant:
458   assumes holf: "f holomorphic_on S" and "open S" "\<xi> \<in> S" "deriv f \<xi> \<noteq> 0"
459     shows "\<not> f constant_on S"
460     apply (rule nonzero_deriv_nonconstant [of f "deriv f \<xi>" \<xi> S])
461     using assms
462     apply (auto simp: holomorphic_derivI)
463     done
465 subsection\<open>Caratheodory characterization\<close>
467 lemma field_differentiable_caratheodory_at:
468   "f field_differentiable (at z) \<longleftrightarrow>
469          (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)"
470   using CARAT_DERIV [of f]
471   by (simp add: field_differentiable_def has_field_derivative_def)
473 lemma field_differentiable_caratheodory_within:
474   "f field_differentiable (at z within s) \<longleftrightarrow>
475          (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)"
476   using DERIV_caratheodory_within [of f]
477   by (simp add: field_differentiable_def has_field_derivative_def)
479 subsection\<open>Analyticity on a set\<close>
481 definition analytic_on (infixl "(analytic'_on)" 50)
482   where
483    "f analytic_on s \<equiv> \<forall>x \<in> s. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)"
485 lemma analytic_imp_holomorphic: "f analytic_on s \<Longrightarrow> f holomorphic_on s"
486   by (simp add: at_within_open [OF _ open_ball] analytic_on_def holomorphic_on_def)
487      (metis centre_in_ball field_differentiable_at_within)
489 lemma analytic_on_open: "open s \<Longrightarrow> f analytic_on s \<longleftrightarrow> f holomorphic_on s"
490 apply (auto simp: analytic_imp_holomorphic)
491 apply (auto simp: analytic_on_def holomorphic_on_def)
492 by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball)
494 lemma analytic_on_imp_differentiable_at:
495   "f analytic_on s \<Longrightarrow> x \<in> s \<Longrightarrow> f field_differentiable (at x)"
496  apply (auto simp: analytic_on_def holomorphic_on_def)
497 by (metis Topology_Euclidean_Space.open_ball centre_in_ball field_differentiable_within_open)
499 lemma analytic_on_subset: "f analytic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f analytic_on t"
500   by (auto simp: analytic_on_def)
502 lemma analytic_on_Un: "f analytic_on (s \<union> t) \<longleftrightarrow> f analytic_on s \<and> f analytic_on t"
503   by (auto simp: analytic_on_def)
505 lemma analytic_on_Union: "f analytic_on (\<Union>s) \<longleftrightarrow> (\<forall>t \<in> s. f analytic_on t)"
506   by (auto simp: analytic_on_def)
508 lemma analytic_on_UN: "f analytic_on (\<Union>i\<in>I. s i) \<longleftrightarrow> (\<forall>i\<in>I. f analytic_on (s i))"
509   by (auto simp: analytic_on_def)
511 lemma analytic_on_holomorphic:
512   "f analytic_on s \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f holomorphic_on t)"
513   (is "?lhs = ?rhs")
514 proof -
515   have "?lhs \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t)"
516   proof safe
517     assume "f analytic_on s"
518     then show "\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t"
520       apply (rule exI [where x="\<Union>{u. open u \<and> f analytic_on u}"], auto)
521       apply (metis Topology_Euclidean_Space.open_ball analytic_on_open centre_in_ball)
522       by (metis analytic_on_def)
523   next
524     fix t
525     assume "open t" "s \<subseteq> t" "f analytic_on t"
526     then show "f analytic_on s"
527         by (metis analytic_on_subset)
528   qed
529   also have "... \<longleftrightarrow> ?rhs"
530     by (auto simp: analytic_on_open)
531   finally show ?thesis .
532 qed
534 lemma analytic_on_linear: "(op * c) analytic_on s"
535   by (auto simp add: analytic_on_holomorphic holomorphic_on_linear)
537 lemma analytic_on_const: "(\<lambda>z. c) analytic_on s"
538   by (metis analytic_on_def holomorphic_on_const zero_less_one)
540 lemma analytic_on_ident: "(\<lambda>x. x) analytic_on s"
541   by (simp add: analytic_on_def holomorphic_on_ident gt_ex)
543 lemma analytic_on_id: "id analytic_on s"
544   unfolding id_def by (rule analytic_on_ident)
546 lemma analytic_on_compose:
547   assumes f: "f analytic_on s"
548       and g: "g analytic_on (f ` s)"
549     shows "(g o f) analytic_on s"
550 unfolding analytic_on_def
551 proof (intro ballI)
552   fix x
553   assume x: "x \<in> s"
554   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f
555     by (metis analytic_on_def)
556   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g
557     by (metis analytic_on_def g image_eqI x)
558   have "isCont f x"
559     by (metis analytic_on_imp_differentiable_at field_differentiable_imp_continuous_at f x)
560   with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'"
561      by (auto simp: continuous_at_ball)
562   have "g \<circ> f holomorphic_on ball x (min d e)"
563     apply (rule holomorphic_on_compose)
564     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
565     by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball)
566   then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e"
567     by (metis d e min_less_iff_conj)
568 qed
570 lemma analytic_on_compose_gen:
571   "f analytic_on s \<Longrightarrow> g analytic_on t \<Longrightarrow> (\<And>z. z \<in> s \<Longrightarrow> f z \<in> t)
572              \<Longrightarrow> g o f analytic_on s"
573 by (metis analytic_on_compose analytic_on_subset image_subset_iff)
575 lemma analytic_on_neg:
576   "f analytic_on s \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on s"
577 by (metis analytic_on_holomorphic holomorphic_on_minus)
580   assumes f: "f analytic_on s"
581       and g: "g analytic_on s"
582     shows "(\<lambda>z. f z + g z) analytic_on s"
583 unfolding analytic_on_def
584 proof (intro ballI)
585   fix z
586   assume z: "z \<in> s"
587   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
588     by (metis analytic_on_def)
589   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
590     by (metis analytic_on_def g z)
591   have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')"
593     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
594     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
595   then show "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e"
596     by (metis e e' min_less_iff_conj)
597 qed
599 lemma analytic_on_diff:
600   assumes f: "f analytic_on s"
601       and g: "g analytic_on s"
602     shows "(\<lambda>z. f z - g z) analytic_on s"
603 unfolding analytic_on_def
604 proof (intro ballI)
605   fix z
606   assume z: "z \<in> s"
607   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
608     by (metis analytic_on_def)
609   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
610     by (metis analytic_on_def g z)
611   have "(\<lambda>z. f z - g z) holomorphic_on ball z (min e e')"
612     apply (rule holomorphic_on_diff)
613     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
614     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
615   then show "\<exists>e>0. (\<lambda>z. f z - g z) holomorphic_on ball z e"
616     by (metis e e' min_less_iff_conj)
617 qed
619 lemma analytic_on_mult:
620   assumes f: "f analytic_on s"
621       and g: "g analytic_on s"
622     shows "(\<lambda>z. f z * g z) analytic_on s"
623 unfolding analytic_on_def
624 proof (intro ballI)
625   fix z
626   assume z: "z \<in> s"
627   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
628     by (metis analytic_on_def)
629   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
630     by (metis analytic_on_def g z)
631   have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')"
632     apply (rule holomorphic_on_mult)
633     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
634     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
635   then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e"
636     by (metis e e' min_less_iff_conj)
637 qed
639 lemma analytic_on_inverse:
640   assumes f: "f analytic_on s"
641       and nz: "(\<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0)"
642     shows "(\<lambda>z. inverse (f z)) analytic_on s"
643 unfolding analytic_on_def
644 proof (intro ballI)
645   fix z
646   assume z: "z \<in> s"
647   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
648     by (metis analytic_on_def)
649   have "continuous_on (ball z e) f"
650     by (metis fh holomorphic_on_imp_continuous_on)
651   then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0"
652     by (metis Topology_Euclidean_Space.open_ball centre_in_ball continuous_on_open_avoid e z nz)
653   have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')"
654     apply (rule holomorphic_on_inverse)
655     apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball)
656     by (metis nz' mem_ball min_less_iff_conj)
657   then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e"
658     by (metis e e' min_less_iff_conj)
659 qed
661 lemma analytic_on_divide:
662   assumes f: "f analytic_on s"
663       and g: "g analytic_on s"
664       and nz: "(\<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0)"
665     shows "(\<lambda>z. f z / g z) analytic_on s"
666 unfolding divide_inverse
667 by (metis analytic_on_inverse analytic_on_mult f g nz)
669 lemma analytic_on_power:
670   "f analytic_on s \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on s"
671 by (induct n) (auto simp: analytic_on_const analytic_on_mult)
673 lemma analytic_on_sum:
674   "(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on s) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) analytic_on s"
675   by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_const analytic_on_add)
677 lemma deriv_left_inverse:
678   assumes "f holomorphic_on S" and "g holomorphic_on T"
679       and "open S" and "open T"
680       and "f ` S \<subseteq> T"
681       and [simp]: "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z"
682       and "w \<in> S"
683     shows "deriv f w * deriv g (f w) = 1"
684 proof -
685   have "deriv f w * deriv g (f w) = deriv g (f w) * deriv f w"
687   also have "... = deriv (g o f) w"
688     using assms
689     by (metis analytic_on_imp_differentiable_at analytic_on_open complex_derivative_chain image_subset_iff)
690   also have "... = deriv id w"
691     apply (rule complex_derivative_transform_within_open [where s=S])
692     apply (rule assms holomorphic_on_compose_gen holomorphic_intros)+
693     apply simp
694     done
695   also have "... = 1"
696     by simp
697   finally show ?thesis .
698 qed
700 subsection\<open>analyticity at a point\<close>
702 lemma analytic_at_ball:
703   "f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
704 by (metis analytic_on_def singleton_iff)
706 lemma analytic_at:
707     "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
708 by (metis analytic_on_holomorphic empty_subsetI insert_subset)
710 lemma analytic_on_analytic_at:
711     "f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
712 by (metis analytic_at_ball analytic_on_def)
714 lemma analytic_at_two:
715   "f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
716    (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)"
717   (is "?lhs = ?rhs")
718 proof
719   assume ?lhs
720   then obtain s t
721     where st: "open s" "z \<in> s" "f holomorphic_on s"
722               "open t" "z \<in> t" "g holomorphic_on t"
723     by (auto simp: analytic_at)
724   show ?rhs
725     apply (rule_tac x="s \<inter> t" in exI)
726     using st
727     apply (auto simp: Diff_subset holomorphic_on_subset)
728     done
729 next
730   assume ?rhs
731   then show ?lhs
732     by (force simp add: analytic_at)
733 qed
735 subsection\<open>Combining theorems for derivative with ``analytic at'' hypotheses\<close>
737 lemma
738   assumes "f analytic_on {z}" "g analytic_on {z}"
739   shows complex_derivative_add_at: "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
740     and complex_derivative_diff_at: "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
741     and complex_derivative_mult_at: "deriv (\<lambda>w. f w * g w) z =
742            f z * deriv g z + deriv f z * g z"
743 proof -
744   obtain s where s: "open s" "z \<in> s" "f holomorphic_on s" "g holomorphic_on s"
745     using assms by (metis analytic_at_two)
746   show "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
747     apply (rule DERIV_imp_deriv [OF DERIV_add])
748     using s
749     apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
750     done
751   show "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
752     apply (rule DERIV_imp_deriv [OF DERIV_diff])
753     using s
754     apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
755     done
756   show "deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
757     apply (rule DERIV_imp_deriv [OF DERIV_mult'])
758     using s
759     apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
760     done
761 qed
763 lemma deriv_cmult_at:
764   "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. c * f w) z = c * deriv f z"
765 by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
767 lemma deriv_cmult_right_at:
768   "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. f w * c) z = deriv f z * c"
769 by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
771 subsection\<open>Complex differentiation of sequences and series\<close>
773 (* TODO: Could probably be simplified using Uniform_Limit *)
774 lemma has_complex_derivative_sequence:
775   fixes s :: "complex set"
776   assumes cvs: "convex s"
777       and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
778       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"
779       and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
780     shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and>
781                        (g has_field_derivative (g' x)) (at x within s)"
782 proof -
783   from assms obtain x l where x: "x \<in> s" and tf: "((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
784     by blast
785   { fix e::real assume e: "e > 0"
786     then obtain N where N: "\<forall>n\<ge>N. \<forall>x. x \<in> s \<longrightarrow> cmod (f' n x - g' x) \<le> e"
787       by (metis conv)
788     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"
789     proof (rule exI [of _ N], clarify)
790       fix n y h
791       assume "N \<le> n" "y \<in> s"
792       then have "cmod (f' n y - g' y) \<le> e"
793         by (metis N)
794       then have "cmod h * cmod (f' n y - g' y) \<le> cmod h * e"
795         by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
796       then show "cmod (f' n y * h - g' y * h) \<le> e * cmod h"
797         by (simp add: norm_mult [symmetric] field_simps)
798     qed
799   } note ** = this
800   show ?thesis
801   unfolding has_field_derivative_def
802   proof (rule has_derivative_sequence [OF cvs _ _ x])
803     show "\<forall>n. \<forall>x\<in>s. (f n has_derivative (op * (f' n x))) (at x within s)"
804       by (metis has_field_derivative_def df)
805   next show "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
806     by (rule tf)
807   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"
808     by (blast intro: **)
809   qed
810 qed
812 lemma has_complex_derivative_series:
813   fixes s :: "complex set"
814   assumes cvs: "convex s"
815       and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
816       and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s
817                 \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
818       and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) sums l)"
819     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))"
820 proof -
821   from assms obtain x l where x: "x \<in> s" and sf: "((\<lambda>n. f n x) sums l)"
822     by blast
823   { fix e::real assume e: "e > 0"
824     then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> s
825             \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
826       by (metis conv)
827     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"
828     proof (rule exI [of _ N], clarify)
829       fix n y h
830       assume "N \<le> n" "y \<in> s"
831       then have "cmod ((\<Sum>i<n. f' i y) - g' y) \<le> e"
832         by (metis N)
833       then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e"
834         by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
835       then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h"
836         by (simp add: norm_mult [symmetric] field_simps sum_distrib_left)
837     qed
838   } note ** = this
839   show ?thesis
840   unfolding has_field_derivative_def
841   proof (rule has_derivative_series [OF cvs _ _ x])
842     fix n x
843     assume "x \<in> s"
844     then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within s)"
845       by (metis df has_field_derivative_def mult_commute_abs)
846   next show " ((\<lambda>n. f n x) sums l)"
847     by (rule sf)
848   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"
849     by (blast intro: **)
850   qed
851 qed
854 lemma field_differentiable_series:
855   fixes f :: "nat \<Rightarrow> complex \<Rightarrow> complex"
856   assumes "convex s" "open s"
857   assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
858   assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
859   assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)" and x: "x \<in> s"
860   shows   "summable (\<lambda>n. f n x)" and "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)"
861 proof -
862   from assms(4) obtain g' where A: "uniform_limit s (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
863     unfolding uniformly_convergent_on_def by blast
864   from x and \<open>open s\<close> have s: "at x within s = at x" by (rule at_within_open)
865   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)"
866     by (intro has_field_derivative_series[of s f f' g' x0] assms A has_field_derivative_at_within)
867   then obtain g where g: "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>n. f n x) sums g x"
868     "\<And>x. x \<in> s \<Longrightarrow> (g has_field_derivative g' x) (at x within s)" by blast
869   from g[OF x] show "summable (\<lambda>n. f n x)" by (auto simp: summable_def)
870   from g(2)[OF x] have g': "(g has_derivative op * (g' x)) (at x)"
871     by (simp add: has_field_derivative_def s)
872   have "((\<lambda>x. \<Sum>n. f n x) has_derivative op * (g' x)) (at x)"
873     by (rule has_derivative_transform_within_open[OF g' \<open>open s\<close> x])
874        (insert g, auto simp: sums_iff)
875   thus "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)" unfolding differentiable_def
876     by (auto simp: summable_def field_differentiable_def has_field_derivative_def)
877 qed
879 lemma field_differentiable_series':
880   fixes f :: "nat \<Rightarrow> complex \<Rightarrow> complex"
881   assumes "convex s" "open s"
882   assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
883   assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
884   assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)"
885   shows   "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x0)"
886   using field_differentiable_series[OF assms, of x0] \<open>x0 \<in> s\<close> by blast+
888 subsection\<open>Bound theorem\<close>
890 lemma field_differentiable_bound:
891   fixes s :: "complex set"
892   assumes cvs: "convex s"
893       and df:  "\<And>z. z \<in> s \<Longrightarrow> (f has_field_derivative f' z) (at z within s)"
894       and dn:  "\<And>z. z \<in> s \<Longrightarrow> norm (f' z) \<le> B"
895       and "x \<in> s"  "y \<in> s"
896     shows "norm(f x - f y) \<le> B * norm(x - y)"
897   apply (rule differentiable_bound [OF cvs])
898   apply (rule ballI, erule df [unfolded has_field_derivative_def])
899   apply (rule ballI, rule onorm_le, simp add: norm_mult mult_right_mono dn)
900   apply fact
901   apply fact
902   done
904 subsection\<open>Inverse function theorem for complex derivatives\<close>
906 lemma has_complex_derivative_inverse_basic:
907   fixes f :: "complex \<Rightarrow> complex"
908   shows "DERIV f (g y) :> f' \<Longrightarrow>
909         f' \<noteq> 0 \<Longrightarrow>
910         continuous (at y) g \<Longrightarrow>
911         open t \<Longrightarrow>
912         y \<in> t \<Longrightarrow>
913         (\<And>z. z \<in> t \<Longrightarrow> f (g z) = z)
914         \<Longrightarrow> DERIV g y :> inverse (f')"
915   unfolding has_field_derivative_def
916   apply (rule has_derivative_inverse_basic)
917   apply (auto simp:  bounded_linear_mult_right)
918   done
920 (*Used only once, in Multivariate/cauchy.ml. *)
921 lemma has_complex_derivative_inverse_strong:
922   fixes f :: "complex \<Rightarrow> complex"
923   shows "DERIV f x :> f' \<Longrightarrow>
924          f' \<noteq> 0 \<Longrightarrow>
925          open s \<Longrightarrow>
926          x \<in> s \<Longrightarrow>
927          continuous_on s f \<Longrightarrow>
928          (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
929          \<Longrightarrow> DERIV g (f x) :> inverse (f')"
930   unfolding has_field_derivative_def
931   apply (rule has_derivative_inverse_strong [of s x f g ])
932   by auto
934 lemma has_complex_derivative_inverse_strong_x:
935   fixes f :: "complex \<Rightarrow> complex"
936   shows  "DERIV f (g y) :> f' \<Longrightarrow>
937           f' \<noteq> 0 \<Longrightarrow>
938           open s \<Longrightarrow>
939           continuous_on s f \<Longrightarrow>
940           g y \<in> s \<Longrightarrow> f(g y) = y \<Longrightarrow>
941           (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
942           \<Longrightarrow> DERIV g y :> inverse (f')"
943   unfolding has_field_derivative_def
944   apply (rule has_derivative_inverse_strong_x [of s g y f])
945   by auto
947 subsection \<open>Taylor on Complex Numbers\<close>
949 lemma sum_Suc_reindex:
950   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
951     shows  "sum f {0..n} = f 0 - f (Suc n) + sum (\<lambda>i. f (Suc i)) {0..n}"
952 by (induct n) auto
954 lemma complex_taylor:
955   assumes s: "convex s"
956       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)"
957       and B: "\<And>x. x \<in> s \<Longrightarrow> cmod (f (Suc n) x) \<le> B"
958       and w: "w \<in> s"
959       and z: "z \<in> s"
960     shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i)))
961           \<le> B * cmod(z - w)^(Suc n) / fact n"
962 proof -
963   have wzs: "closed_segment w z \<subseteq> s" using assms
964     by (metis convex_contains_segment)
965   { fix u
966     assume "u \<in> closed_segment w z"
967     then have "u \<in> s"
968       by (metis wzs subsetD)
969     have "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / (fact i) +
970                       f (Suc i) u * (z-u)^i / (fact i)) =
971               f (Suc n) u * (z-u) ^ n / (fact n)"
972     proof (induction n)
973       case 0 show ?case by simp
974     next
975       case (Suc n)
976       have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / (fact i) +
977                              f (Suc i) u * (z-u) ^ i / (fact i)) =
978            f (Suc n) u * (z-u) ^ n / (fact n) +
979            f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / (fact (Suc n)) -
980            f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / (fact (Suc n))"
981         using Suc by simp
982       also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n))"
983       proof -
984         have "(fact(Suc n)) *
985              (f(Suc n) u *(z-u) ^ n / (fact n) +
986                f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / (fact(Suc n)) -
987                f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / (fact(Suc n))) =
988             ((fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / (fact n) +
989             ((fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / (fact(Suc n))) -
990             ((fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / (fact(Suc n))"
991           by (simp add: algebra_simps del: fact_Suc)
992         also have "... = ((fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / (fact n) +
993                          (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
994                          (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
995           by (simp del: fact_Suc)
996         also have "... = (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) +
997                          (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
998                          (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
999           by (simp only: fact_Suc of_nat_mult ac_simps) simp
1000         also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
1002         finally show ?thesis
1003         by (simp add: mult_left_cancel [where c = "(fact (Suc n))", THEN iffD1] del: fact_Suc)
1004       qed
1005       finally show ?case .
1006     qed
1007     then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / (fact i)))
1008                 has_field_derivative f (Suc n) u * (z-u) ^ n / (fact n))
1009                (at u within s)"
1010       apply (intro derivative_eq_intros)
1011       apply (blast intro: assms \<open>u \<in> s\<close>)
1012       apply (rule refl)+
1013       apply (auto simp: field_simps)
1014       done
1015   } note sum_deriv = this
1016   { fix u
1017     assume u: "u \<in> closed_segment w z"
1018     then have us: "u \<in> s"
1019       by (metis wzs subsetD)
1020     have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> cmod (f (Suc n) u) * cmod (u - z) ^ n"
1021       by (metis norm_minus_commute order_refl)
1022     also have "... \<le> cmod (f (Suc n) u) * cmod (z - w) ^ n"
1023       by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
1024     also have "... \<le> B * cmod (z - w) ^ n"
1025       by (metis norm_ge_zero zero_le_power mult_right_mono  B [OF us])
1026     finally have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> B * cmod (z - w) ^ n" .
1027   } note cmod_bound = this
1028   have "(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)) = (\<Sum>i\<le>n. (f i z / (fact i)) * 0 ^ i)"
1029     by simp
1030   also have "\<dots> = f 0 z / (fact 0)"
1031     by (subst sum_zero_power) simp
1032   finally have "cmod (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)))
1033                 \<le> cmod ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) -
1034                         (\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))"
1036   also have "... \<le> B * cmod (z - w) ^ n / (fact n) * cmod (w - z)"
1037     apply (rule field_differentiable_bound
1038       [where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / (fact n)"
1039          and s = "closed_segment w z", OF convex_closed_segment])
1040     apply (auto simp: ends_in_segment DERIV_subset [OF sum_deriv wzs]
1041                   norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
1042     done
1043   also have "...  \<le> B * cmod (z - w) ^ Suc n / (fact n)"
1044     by (simp add: algebra_simps norm_minus_commute)
1045   finally show ?thesis .
1046 qed
1048 text\<open>Something more like the traditional MVT for real components\<close>
1050 lemma complex_mvt_line:
1051   assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f'(u)) (at u)"
1052     shows "\<exists>u. u \<in> closed_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
1053 proof -
1054   have twz: "\<And>t. (1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)"
1055     by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
1056   note assms[unfolded has_field_derivative_def, derivative_intros]
1057   show ?thesis
1058     apply (cut_tac mvt_simple
1059                      [of 0 1 "Re o f o (\<lambda>t. (1 - t) *\<^sub>R w +  t *\<^sub>R z)"
1060                       "\<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))"])
1061     apply auto
1062     apply (rule_tac x="(1 - x) *\<^sub>R w + x *\<^sub>R z" in exI)
1063     apply (auto simp: closed_segment_def twz) []
1064     apply (intro derivative_eq_intros has_derivative_at_within, simp_all)
1065     apply (simp add: fun_eq_iff real_vector.scale_right_diff_distrib)
1066     apply (force simp: twz closed_segment_def)
1067     done
1068 qed
1070 lemma complex_taylor_mvt:
1071   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)"
1072     shows "\<exists>u. u \<in> closed_segment w z \<and>
1073             Re (f 0 z) =
1074             Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / (fact i)) +
1075                 (f (Suc n) u * (z-u)^n / (fact n)) * (z - w))"
1076 proof -
1077   { fix u
1078     assume u: "u \<in> closed_segment w z"
1079     have "(\<Sum>i = 0..n.
1080                (f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) /
1081                (fact i)) =
1082           f (Suc 0) u -
1083              (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
1084              (fact (Suc n)) +
1085              (\<Sum>i = 0..n.
1086                  (f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) /
1087                  (fact (Suc i)))"
1088        by (subst sum_Suc_reindex) simp
1089     also have "... = f (Suc 0) u -
1090              (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
1091              (fact (Suc n)) +
1092              (\<Sum>i = 0..n.
1093                  f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i))  -
1094                  f (Suc i) u * (z-u) ^ i / (fact i))"
1095       by (simp only: diff_divide_distrib fact_cancel ac_simps)
1096     also have "... = f (Suc 0) u -
1097              (f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) /
1098              (fact (Suc n)) +
1099              f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n)) - f (Suc 0) u"
1100       by (subst sum_Suc_diff) auto
1101     also have "... = f (Suc n) u * (z-u) ^ n / (fact n)"
1102       by (simp only: algebra_simps diff_divide_distrib fact_cancel)
1103     finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i
1104                              - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / (fact i)) =
1105                   f (Suc n) u * (z - u) ^ n / (fact n)" .
1106     then have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / (fact i)) has_field_derivative
1107                 f (Suc n) u * (z - u) ^ n / (fact n))  (at u)"
1108       apply (intro derivative_eq_intros)+
1109       apply (force intro: u assms)
1110       apply (rule refl)+
1111       apply (auto simp: ac_simps)
1112       done
1113   }
1114   then show ?thesis
1115     apply (cut_tac complex_mvt_line [of w z "\<lambda>u. \<Sum>i = 0..n. f i u * (z-u) ^ i / (fact i)"
1116                "\<lambda>u. (f (Suc n) u * (z-u)^n / (fact n))"])
1117     apply (auto simp add: intro: open_closed_segment)
1118     done
1119 qed
1122 subsection \<open>Polynomal function extremal theorem, from HOL Light\<close>
1124 lemma polyfun_extremal_lemma: (*COMPLEX_POLYFUN_EXTREMAL_LEMMA in HOL Light*)
1125     fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
1126   assumes "0 < e"
1127     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)"
1128 proof (induct n)
1129   case 0 with assms
1130   show ?case
1131     apply (rule_tac x="norm (c 0) / e" in exI)
1132     apply (auto simp: field_simps)
1133     done
1134 next
1135   case (Suc n)
1136   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"
1137     using Suc assms by blast
1138   show ?case
1139   proof (rule exI [where x= "max M (1 + norm(c(Suc n)) / e)"], clarsimp simp del: power_Suc)
1140     fix z::'a
1141     assume z1: "M \<le> norm z" and "1 + norm (c (Suc n)) / e \<le> norm z"
1142     then have z2: "e + norm (c (Suc n)) \<le> e * norm z"
1143       using assms by (simp add: field_simps)
1144     have "norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
1145       using M [OF z1] by simp
1146     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)"
1147       by simp
1148     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)"
1149       by (blast intro: norm_triangle_le elim: )
1150     also have "... \<le> (e + norm (c (Suc n))) * norm z ^ Suc n"
1151       by (simp add: norm_power norm_mult algebra_simps)
1152     also have "... \<le> (e * norm z) * norm z ^ Suc n"
1153       by (metis z2 mult.commute mult_left_mono norm_ge_zero norm_power)
1154     finally show "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc (Suc n)"
1155       by simp
1156   qed
1157 qed
1159 lemma polyfun_extremal: (*COMPLEX_POLYFUN_EXTREMAL in HOL Light*)
1160     fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
1161   assumes k: "c k \<noteq> 0" "1\<le>k" and kn: "k\<le>n"
1162     shows "eventually (\<lambda>z. norm (\<Sum>i\<le>n. c(i) * z^i) \<ge> B) at_infinity"
1163 using kn
1164 proof (induction n)
1165   case 0
1166   then show ?case
1167     using k  by simp
1168 next
1169   case (Suc m)
1170   let ?even = ?case
1171   show ?even
1172   proof (cases "c (Suc m) = 0")
1173     case True
1174     then show ?even using Suc k
1175       by auto (metis antisym_conv less_eq_Suc_le not_le)
1176   next
1177     case False
1178     then obtain M where M:
1179           "\<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"
1180       using polyfun_extremal_lemma [of "norm(c (Suc m)) / 2" c m] Suc
1181       by auto
1182     have "\<exists>b. \<forall>z. b \<le> norm z \<longrightarrow> B \<le> norm (\<Sum>i\<le>Suc m. c i * z^i)"
1183     proof (rule exI [where x="max M (max 1 (\<bar>B\<bar> / (norm(c (Suc m)) / 2)))"], clarsimp simp del: power_Suc)
1184       fix z::'a
1185       assume z1: "M \<le> norm z" "1 \<le> norm z"
1186          and "\<bar>B\<bar> * 2 / norm (c (Suc m)) \<le> norm z"
1187       then have z2: "\<bar>B\<bar> \<le> norm (c (Suc m)) * norm z / 2"
1188         using False by (simp add: field_simps)
1189       have nz: "norm z \<le> norm z ^ Suc m"
1190         by (metis \<open>1 \<le> norm z\<close> One_nat_def less_eq_Suc_le power_increasing power_one_right zero_less_Suc)
1191       have *: "\<And>y x. norm (c (Suc m)) * norm z / 2 \<le> norm y - norm x \<Longrightarrow> B \<le> norm (x + y)"
1192         by (metis abs_le_iff add.commute norm_diff_ineq order_trans z2)
1193       have "norm z * norm (c (Suc m)) + 2 * norm (\<Sum>i\<le>m. c i * z^i)
1194             \<le> norm (c (Suc m)) * norm z + norm (c (Suc m)) * norm z ^ Suc m"
1195         using M [of z] Suc z1  by auto
1196       also have "... \<le> 2 * (norm (c (Suc m)) * norm z ^ Suc m)"
1197         using nz by (simp add: mult_mono del: power_Suc)
1198       finally show "B \<le> norm ((\<Sum>i\<le>m. c i * z^i) + c (Suc m) * z ^ Suc m)"
1199         using Suc.IH
1200         apply (auto simp: eventually_at_infinity)
1201         apply (rule *)
1202         apply (simp add: field_simps norm_mult norm_power)
1203         done
1204     qed
1205     then show ?even