src/HOL/Complex.thy
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1 (*  Title:       Complex.thy
2     Author:      Jacques D. Fleuriot
3     Copyright:   2001 University of Edinburgh
4     Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
5 *)
7 header {* Complex Numbers: Rectangular and Polar Representations *}
9 theory Complex
10 imports Transcendental
11 begin
13 datatype complex = Complex real real
15 primrec
16   Re :: "complex \<Rightarrow> real"
17 where
18   Re: "Re (Complex x y) = x"
20 primrec
21   Im :: "complex \<Rightarrow> real"
22 where
23   Im: "Im (Complex x y) = y"
25 lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"
26   by (induct z) simp
28 lemma complex_equality [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"
29   by (induct x, induct y) simp
31 lemma expand_complex_eq: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"
32   by (induct x, induct y) simp
34 lemmas complex_Re_Im_cancel_iff = expand_complex_eq
37 subsection {* Addition and Subtraction *}
40 begin
42 definition
43   complex_zero_def: "0 = Complex 0 0"
45 definition
46   complex_add_def: "x + y = Complex (Re x + Re y) (Im x + Im y)"
48 definition
49   complex_minus_def: "- x = Complex (- Re x) (- Im x)"
51 definition
52   complex_diff_def: "x - (y\<Colon>complex) = x + - y"
54 lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
57 lemma complex_Re_zero [simp]: "Re 0 = 0"
60 lemma complex_Im_zero [simp]: "Im 0 = 0"
64   "Complex a b + Complex c d = Complex (a + c) (b + d)"
67 lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y"
70 lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y"
73 lemma complex_minus [simp]:
74   "- (Complex a b) = Complex (- a) (- b)"
77 lemma complex_Re_minus [simp]: "Re (- x) = - Re x"
80 lemma complex_Im_minus [simp]: "Im (- x) = - Im x"
83 lemma complex_diff [simp]:
84   "Complex a b - Complex c d = Complex (a - c) (b - d)"
87 lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y"
90 lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y"
93 instance
96 end
100 subsection {* Multiplication and Division *}
102 instantiation complex :: "{field, division_by_zero}"
103 begin
105 definition
106   complex_one_def: "1 = Complex 1 0"
108 definition
109   complex_mult_def: "x * y =
110     Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
112 definition
113   complex_inverse_def: "inverse x =
114     Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))"
116 definition
117   complex_divide_def: "x / (y\<Colon>complex) = x * inverse y"
119 lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"
122 lemma complex_Re_one [simp]: "Re 1 = 1"
125 lemma complex_Im_one [simp]: "Im 1 = 0"
128 lemma complex_mult [simp]:
129   "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
132 lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"
135 lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"
138 lemma complex_inverse [simp]:
139   "inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))"
142 lemma complex_Re_inverse:
143   "Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
146 lemma complex_Im_inverse:
147   "Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
150 instance
151   by intro_classes (simp_all add: complex_mult_def
152   right_distrib left_distrib right_diff_distrib left_diff_distrib
153   complex_inverse_def complex_divide_def
155   expand_complex_eq)
157 end
160 subsection {* Numerals and Arithmetic *}
162 instantiation complex :: number_ring
163 begin
165 definition number_of_complex where
166   complex_number_of_def: "number_of w = (of_int w \<Colon> complex)"
168 instance
169   by intro_classes (simp only: complex_number_of_def)
171 end
173 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
174 by (induct n) simp_all
176 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
177 by (induct n) simp_all
179 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
180 by (cases z rule: int_diff_cases) simp
182 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
183 by (cases z rule: int_diff_cases) simp
185 lemma complex_Re_number_of [simp]: "Re (number_of v) = number_of v"
186 unfolding number_of_eq by (rule complex_Re_of_int)
188 lemma complex_Im_number_of [simp]: "Im (number_of v) = 0"
189 unfolding number_of_eq by (rule complex_Im_of_int)
191 lemma Complex_eq_number_of [simp]:
192   "(Complex a b = number_of w) = (a = number_of w \<and> b = 0)"
196 subsection {* Scalar Multiplication *}
198 instantiation complex :: real_field
199 begin
201 definition
202   complex_scaleR_def: "scaleR r x = Complex (r * Re x) (r * Im x)"
204 lemma complex_scaleR [simp]:
205   "scaleR r (Complex a b) = Complex (r * a) (r * b)"
206   unfolding complex_scaleR_def by simp
208 lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"
209   unfolding complex_scaleR_def by simp
211 lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"
212   unfolding complex_scaleR_def by simp
214 instance
215 proof
216   fix a b :: real and x y :: complex
217   show "scaleR a (x + y) = scaleR a x + scaleR a y"
218     by (simp add: expand_complex_eq right_distrib)
219   show "scaleR (a + b) x = scaleR a x + scaleR b x"
220     by (simp add: expand_complex_eq left_distrib)
221   show "scaleR a (scaleR b x) = scaleR (a * b) x"
222     by (simp add: expand_complex_eq mult_assoc)
223   show "scaleR 1 x = x"
225   show "scaleR a x * y = scaleR a (x * y)"
226     by (simp add: expand_complex_eq algebra_simps)
227   show "x * scaleR a y = scaleR a (x * y)"
228     by (simp add: expand_complex_eq algebra_simps)
229 qed
231 end
234 subsection{* Properties of Embedding from Reals *}
236 abbreviation
237   complex_of_real :: "real \<Rightarrow> complex" where
238     "complex_of_real \<equiv> of_real"
240 lemma complex_of_real_def: "complex_of_real r = Complex r 0"
241 by (simp add: of_real_def complex_scaleR_def)
243 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
246 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
250      "Complex x y + complex_of_real r = Complex (x+r) y"
254      "complex_of_real r + Complex x y = Complex (r+x) y"
257 lemma Complex_mult_complex_of_real:
258      "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
261 lemma complex_of_real_mult_Complex:
262      "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
266 subsection {* Vector Norm *}
268 instantiation complex :: real_normed_field
269 begin
271 definition complex_norm_def:
272   "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
274 abbreviation
275   cmod :: "complex \<Rightarrow> real" where
276   "cmod \<equiv> norm"
278 definition complex_sgn_def:
279   "sgn x = x /\<^sub>R cmod x"
281 definition dist_complex_def:
282   "dist x y = cmod (x - y)"
284 definition topo_complex_def [code del]:
285   "topo = {S::complex set. \<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S}"
287 lemmas cmod_def = complex_norm_def
289 lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)"
292 instance proof
293   fix r :: real and x y :: complex
294   show "0 \<le> norm x"
295     by (induct x) simp
296   show "(norm x = 0) = (x = 0)"
297     by (induct x) simp
298   show "norm (x + y) \<le> norm x + norm y"
299     by (induct x, induct y)
301   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
302     by (induct x)
303        (simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult)
304   show "norm (x * y) = norm x * norm y"
305     by (induct x, induct y)
306        (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
307   show "sgn x = x /\<^sub>R cmod x"
308     by (rule complex_sgn_def)
309   show "dist x y = cmod (x - y)"
310     by (rule dist_complex_def)
311   show "topo = {S::complex set. \<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S}"
312     by (rule topo_complex_def)
313 qed
315 end
317 lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1"
318 by simp
320 lemma cmod_complex_polar [simp]:
321      "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
324 lemma complex_Re_le_cmod: "Re x \<le> cmod x"
325 unfolding complex_norm_def
326 by (rule real_sqrt_sum_squares_ge1)
328 lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x"
329 by (rule order_trans [OF _ norm_ge_zero], simp)
331 lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a"
332 by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
334 lemmas real_sum_squared_expand = power2_sum [where 'a=real]
336 lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
337 by (cases x) simp
339 lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
340 by (cases x) simp
342 subsection {* Completeness of the Complexes *}
344 interpretation Re: bounded_linear "Re"
345 apply (unfold_locales, simp, simp)
346 apply (rule_tac x=1 in exI)
348 done
350 interpretation Im: bounded_linear "Im"
351 apply (unfold_locales, simp, simp)
352 apply (rule_tac x=1 in exI)
354 done
356 lemma LIMSEQ_Complex:
357   "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) ----> Complex a b"
358 apply (rule LIMSEQ_I)
359 apply (subgoal_tac "0 < r / sqrt 2")
360 apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe)
361 apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe)
362 apply (rename_tac M N, rule_tac x="max M N" in exI, safe)
365 done
367 instance complex :: banach
368 proof
369   fix X :: "nat \<Rightarrow> complex"
370   assume X: "Cauchy X"
371   from Re.Cauchy [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"
372     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
373   from Im.Cauchy [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
374     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
375   have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
376     using LIMSEQ_Complex [OF 1 2] by simp
377   thus "convergent X"
378     by (rule convergentI)
379 qed
382 subsection {* The Complex Number @{term "\<i>"} *}
384 definition
385   "ii" :: complex  ("\<i>") where
386   i_def: "ii \<equiv> Complex 0 1"
388 lemma complex_Re_i [simp]: "Re ii = 0"
391 lemma complex_Im_i [simp]: "Im ii = 1"
394 lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
397 lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
400 lemma complex_i_not_one [simp]: "ii \<noteq> 1"
403 lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w"
406 lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
409 lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
412 lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
413 by (simp add: i_def complex_of_real_def)
415 lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
416 by (simp add: i_def complex_of_real_def)
418 lemma i_squared [simp]: "ii * ii = -1"
421 lemma power2_i [simp]: "ii\<twosuperior> = -1"
424 lemma inverse_i [simp]: "inverse ii = - ii"
425 by (rule inverse_unique, simp)
428 subsection {* Complex Conjugation *}
430 definition
431   cnj :: "complex \<Rightarrow> complex" where
432   "cnj z = Complex (Re z) (- Im z)"
434 lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
437 lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
440 lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
443 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
446 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
449 lemma complex_cnj_zero [simp]: "cnj 0 = 0"
452 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
455 lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
458 lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
461 lemma complex_cnj_minus: "cnj (- x) = - cnj x"
464 lemma complex_cnj_one [simp]: "cnj 1 = 1"
467 lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
470 lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
473 lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
474 by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
476 lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
477 by (induct n, simp_all add: complex_cnj_mult)
479 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
482 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
485 lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w"
488 lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
491 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
494 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
497 lemma complex_cnj_i [simp]: "cnj ii = - ii"
500 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
503 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
506 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
507 by (simp add: expand_complex_eq power2_eq_square)
509 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
510 by (simp add: norm_mult power2_eq_square)
512 interpretation cnj: bounded_linear "cnj"
513 apply (unfold_locales)
515 apply (rule complex_cnj_scaleR)
516 apply (rule_tac x=1 in exI, simp)
517 done
520 subsection{*The Functions @{term sgn} and @{term arg}*}
522 text {*------------ Argand -------------*}
524 definition
525   arg :: "complex => real" where
526   "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)"
528 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
529 by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute)
531 lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
532 by (simp add: i_def complex_of_real_def)
534 lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)"
535 by (simp add: i_def complex_one_def)
537 lemma complex_eq_cancel_iff2 [simp]:
538      "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
541 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
542 by (simp add: complex_sgn_def divide_inverse)
544 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
545 by (simp add: complex_sgn_def divide_inverse)
547 lemma complex_inverse_complex_split:
548      "inverse(complex_of_real x + ii * complex_of_real y) =
549       complex_of_real(x/(x ^ 2 + y ^ 2)) -
550       ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
551 by (simp add: complex_of_real_def i_def diff_minus divide_inverse)
553 (*----------------------------------------------------------------------------*)
554 (* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
555 (* many of the theorems are not used - so should they be kept?                *)
556 (*----------------------------------------------------------------------------*)
558 lemma cos_arg_i_mult_zero_pos:
559    "0 < y ==> cos (arg(Complex 0 y)) = 0"
560 apply (simp add: arg_def abs_if)
561 apply (rule_tac a = "pi/2" in someI2, auto)
562 apply (rule order_less_trans [of _ 0], auto)
563 done
565 lemma cos_arg_i_mult_zero_neg:
566    "y < 0 ==> cos (arg(Complex 0 y)) = 0"
567 apply (simp add: arg_def abs_if)
568 apply (rule_tac a = "- pi/2" in someI2, auto)
569 apply (rule order_trans [of _ 0], auto)
570 done
572 lemma cos_arg_i_mult_zero [simp]:
573      "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
574 by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
577 subsection{*Finally! Polar Form for Complex Numbers*}
579 definition
581   (* abbreviation for (cos a + i sin a) *)
582   cis :: "real => complex" where
583   "cis a = Complex (cos a) (sin a)"
585 definition
586   (* abbreviation for r*(cos a + i sin a) *)
587   rcis :: "[real, real] => complex" where
588   "rcis r a = complex_of_real r * cis a"
590 definition
591   (* e ^ (x + iy) *)
592   expi :: "complex => complex" where
593   "expi z = complex_of_real(exp (Re z)) * cis (Im z)"
595 lemma complex_split_polar:
596      "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
597 apply (induct z)
598 apply (auto simp add: polar_Ex complex_of_real_mult_Complex)
599 done
601 lemma rcis_Ex: "\<exists>r a. z = rcis r a"
602 apply (induct z)
603 apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex)
604 done
606 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
607 by (simp add: rcis_def cis_def)
609 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
610 by (simp add: rcis_def cis_def)
612 lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>"
613 proof -
614   have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)"
615     by (simp only: power_mult_distrib right_distrib)
616   thus ?thesis by simp
617 qed
619 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
622 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
623 by (simp add: cmod_def power2_eq_square)
625 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
626 by simp
629 (*---------------------------------------------------------------------------*)
630 (*  (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b)                      *)
631 (*---------------------------------------------------------------------------*)
633 lemma cis_rcis_eq: "cis a = rcis 1 a"
636 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
638               complex_of_real_def)
640 lemma cis_mult: "cis a * cis b = cis (a + b)"
641 by (simp add: cis_rcis_eq rcis_mult)
643 lemma cis_zero [simp]: "cis 0 = 1"
644 by (simp add: cis_def complex_one_def)
646 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
649 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
652 lemma complex_of_real_minus_one:
653    "complex_of_real (-(1::real)) = -(1::complex)"
654 by (simp add: complex_of_real_def complex_one_def)
656 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
657 by (simp add: mult_assoc [symmetric])
660 lemma cis_real_of_nat_Suc_mult:
661    "cis (real (Suc n) * a) = cis a * cis (real n * a)"
664 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
665 apply (induct_tac "n")
666 apply (auto simp add: cis_real_of_nat_Suc_mult)
667 done
669 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
670 by (simp add: rcis_def power_mult_distrib DeMoivre)
672 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
673 by (simp add: cis_def complex_inverse_complex_split diff_minus)
675 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
676 by (simp add: divide_inverse rcis_def)
678 lemma cis_divide: "cis a / cis b = cis (a - b)"
679 by (simp add: complex_divide_def cis_mult real_diff_def)
681 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
683 apply (case_tac "r2=0", simp)
684 apply (simp add: rcis_inverse rcis_mult real_diff_def)
685 done
687 lemma Re_cis [simp]: "Re(cis a) = cos a"
690 lemma Im_cis [simp]: "Im(cis a) = sin a"
693 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
694 by (auto simp add: DeMoivre)
696 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
697 by (auto simp add: DeMoivre)
699 lemma expi_add: "expi(a + b) = expi(a) * expi(b)"