1(* Title: HOL/Complex.thy 2 Author: Jacques D. Fleuriot, 2001 University of Edinburgh 3 Author: Lawrence C Paulson, 2003/4 4*) 5 6section \<open>Complex Numbers: Rectangular and Polar Representations\<close> 7 8theory Complex 9imports Transcendental 10begin 11 12text \<open> 13 We use the \<^theory_text>\<open>codatatype\<close> command to define the type of complex numbers. This 14 allows us to use \<^theory_text>\<open>primcorec\<close> to define complex functions by defining their 15 real and imaginary result separately. 16\<close> 17 18codatatype complex = Complex (Re: real) (Im: real) 19 20lemma complex_surj: "Complex (Re z) (Im z) = z" 21 by (rule complex.collapse) 22 23lemma complex_eqI [intro?]: "Re x = Re y \<Longrightarrow> Im x = Im y \<Longrightarrow> x = y" 24 by (rule complex.expand) simp 25 26lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y" 27 by (auto intro: complex.expand) 28 29 30subsection \<open>Addition and Subtraction\<close> 31 32instantiation complex :: ab_group_add 33begin 34 35primcorec zero_complex 36 where 37 "Re 0 = 0" 38 | "Im 0 = 0" 39 40primcorec plus_complex 41 where 42 "Re (x + y) = Re x + Re y" 43 | "Im (x + y) = Im x + Im y" 44 45primcorec uminus_complex 46 where 47 "Re (- x) = - Re x" 48 | "Im (- x) = - Im x" 49 50primcorec minus_complex 51 where 52 "Re (x - y) = Re x - Re y" 53 | "Im (x - y) = Im x - Im y" 54 55instance 56 by standard (simp_all add: complex_eq_iff) 57 58end 59 60 61subsection \<open>Multiplication and Division\<close> 62 63instantiation complex :: field 64begin 65 66primcorec one_complex 67 where 68 "Re 1 = 1" 69 | "Im 1 = 0" 70 71primcorec times_complex 72 where 73 "Re (x * y) = Re x * Re y - Im x * Im y" 74 | "Im (x * y) = Re x * Im y + Im x * Re y" 75 76primcorec inverse_complex 77 where 78 "Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)" 79 | "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)" 80 81definition "x div y = x * inverse y" for x y :: complex 82 83instance 84 by standard 85 (simp_all add: complex_eq_iff divide_complex_def 86 distrib_left distrib_right right_diff_distrib left_diff_distrib 87 power2_eq_square add_divide_distrib [symmetric]) 88 89end 90 91lemma Re_divide: "Re (x / y) = (Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)" 92 by (simp add: divide_complex_def add_divide_distrib) 93 94lemma Im_divide: "Im (x / y) = (Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)" 95 unfolding divide_complex_def times_complex.sel inverse_complex.sel 96 by (simp add: divide_simps) 97 98lemma Complex_divide: 99 "(x / y) = Complex ((Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)) 100 ((Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2))" 101 by (metis Im_divide Re_divide complex_surj) 102 103lemma Re_power2: "Re (x ^ 2) = (Re x)^2 - (Im x)^2" 104 by (simp add: power2_eq_square) 105 106lemma Im_power2: "Im (x ^ 2) = 2 * Re x * Im x" 107 by (simp add: power2_eq_square) 108 109lemma Re_power_real [simp]: "Im x = 0 \<Longrightarrow> Re (x ^ n) = Re x ^ n " 110 by (induct n) simp_all 111 112lemma Im_power_real [simp]: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0" 113 by (induct n) simp_all 114 115 116subsection \<open>Scalar Multiplication\<close> 117 118instantiation complex :: real_field 119begin 120 121primcorec scaleR_complex 122 where 123 "Re (scaleR r x) = r * Re x" 124 | "Im (scaleR r x) = r * Im x" 125 126instance 127proof 128 fix a b :: real and x y :: complex 129 show "scaleR a (x + y) = scaleR a x + scaleR a y" 130 by (simp add: complex_eq_iff distrib_left) 131 show "scaleR (a + b) x = scaleR a x + scaleR b x" 132 by (simp add: complex_eq_iff distrib_right) 133 show "scaleR a (scaleR b x) = scaleR (a * b) x" 134 by (simp add: complex_eq_iff mult.assoc) 135 show "scaleR 1 x = x" 136 by (simp add: complex_eq_iff) 137 show "scaleR a x * y = scaleR a (x * y)" 138 by (simp add: complex_eq_iff algebra_simps) 139 show "x * scaleR a y = scaleR a (x * y)" 140 by (simp add: complex_eq_iff algebra_simps) 141qed 142 143end 144 145 146subsection \<open>Numerals, Arithmetic, and Embedding from R\<close> 147 148abbreviation complex_of_real :: "real \<Rightarrow> complex" 149 where "complex_of_real \<equiv> of_real" 150 151declare [[coercion "of_real :: real \<Rightarrow> complex"]] 152declare [[coercion "of_rat :: rat \<Rightarrow> complex"]] 153declare [[coercion "of_int :: int \<Rightarrow> complex"]] 154declare [[coercion "of_nat :: nat \<Rightarrow> complex"]] 155 156lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n" 157 by (induct n) simp_all 158 159lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0" 160 by (induct n) simp_all 161 162lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z" 163 by (cases z rule: int_diff_cases) simp 164 165lemma complex_Im_of_int [simp]: "Im (of_int z) = 0" 166 by (cases z rule: int_diff_cases) simp 167 168lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v" 169 using complex_Re_of_int [of "numeral v"] by simp 170 171lemma complex_Im_numeral [simp]: "Im (numeral v) = 0" 172 using complex_Im_of_int [of "numeral v"] by simp 173 174lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" 175 by (simp add: of_real_def) 176 177lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" 178 by (simp add: of_real_def) 179 180lemma Re_divide_numeral [simp]: "Re (z / numeral w) = Re z / numeral w" 181 by (simp add: Re_divide sqr_conv_mult) 182 183lemma Im_divide_numeral [simp]: "Im (z / numeral w) = Im z / numeral w" 184 by (simp add: Im_divide sqr_conv_mult) 185 186lemma Re_divide_of_nat [simp]: "Re (z / of_nat n) = Re z / of_nat n" 187 by (cases n) (simp_all add: Re_divide divide_simps power2_eq_square del: of_nat_Suc) 188 189lemma Im_divide_of_nat [simp]: "Im (z / of_nat n) = Im z / of_nat n" 190 by (cases n) (simp_all add: Im_divide divide_simps power2_eq_square del: of_nat_Suc) 191 192lemma of_real_Re [simp]: "z \<in> \<real> \<Longrightarrow> of_real (Re z) = z" 193 by (auto simp: Reals_def) 194 195lemma complex_Re_fact [simp]: "Re (fact n) = fact n" 196proof - 197 have "(fact n :: complex) = of_real (fact n)" 198 by simp 199 also have "Re \<dots> = fact n" 200 by (subst Re_complex_of_real) simp_all 201 finally show ?thesis . 202qed 203 204lemma complex_Im_fact [simp]: "Im (fact n) = 0" 205 by (subst of_nat_fact [symmetric]) (simp only: complex_Im_of_nat) 206 207lemma Re_prod_Reals: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<real>) \<Longrightarrow> Re (prod f A) = prod (\<lambda>x. Re (f x)) A" 208proof (induction A rule: infinite_finite_induct) 209 case (insert x A) 210 hence "Re (prod f (insert x A)) = Re (f x) * Re (prod f A) - Im (f x) * Im (prod f A)" 211 by simp 212 also from insert.prems have "f x \<in> \<real>" by simp 213 hence "Im (f x) = 0" by (auto elim!: Reals_cases) 214 also have "Re (prod f A) = (\<Prod>x\<in>A. Re (f x))" 215 by (intro insert.IH insert.prems) auto 216 finally show ?case using insert.hyps by simp 217qed auto 218 219 220subsection \<open>The Complex Number $i$\<close> 221 222primcorec imaginary_unit :: complex ("\<i>") 223 where 224 "Re \<i> = 0" 225 | "Im \<i> = 1" 226 227lemma Complex_eq: "Complex a b = a + \<i> * b" 228 by (simp add: complex_eq_iff) 229 230lemma complex_eq: "a = Re a + \<i> * Im a" 231 by (simp add: complex_eq_iff) 232 233lemma fun_complex_eq: "f = (\<lambda>x. Re (f x) + \<i> * Im (f x))" 234 by (simp add: fun_eq_iff complex_eq) 235 236lemma i_squared [simp]: "\<i> * \<i> = -1" 237 by (simp add: complex_eq_iff) 238 239lemma power2_i [simp]: "\<i>\<^sup>2 = -1" 240 by (simp add: power2_eq_square) 241 242lemma inverse_i [simp]: "inverse \<i> = - \<i>" 243 by (rule inverse_unique) simp 244 245lemma divide_i [simp]: "x / \<i> = - \<i> * x" 246 by (simp add: divide_complex_def) 247 248lemma complex_i_mult_minus [simp]: "\<i> * (\<i> * x) = - x" 249 by (simp add: mult.assoc [symmetric]) 250 251lemma complex_i_not_zero [simp]: "\<i> \<noteq> 0" 252 by (simp add: complex_eq_iff) 253 254lemma complex_i_not_one [simp]: "\<i> \<noteq> 1" 255 by (simp add: complex_eq_iff) 256 257lemma complex_i_not_numeral [simp]: "\<i> \<noteq> numeral w" 258 by (simp add: complex_eq_iff) 259 260lemma complex_i_not_neg_numeral [simp]: "\<i> \<noteq> - numeral w" 261 by (simp add: complex_eq_iff) 262 263lemma complex_split_polar: "\<exists>r a. z = complex_of_real r * (cos a + \<i> * sin a)" 264 by (simp add: complex_eq_iff polar_Ex) 265 266lemma i_even_power [simp]: "\<i> ^ (n * 2) = (-1) ^ n" 267 by (metis mult.commute power2_i power_mult) 268 269lemma Re_i_times [simp]: "Re (\<i> * z) = - Im z" 270 by simp 271 272lemma Im_i_times [simp]: "Im (\<i> * z) = Re z" 273 by simp 274 275lemma i_times_eq_iff: "\<i> * w = z \<longleftrightarrow> w = - (\<i> * z)" 276 by auto 277 278lemma divide_numeral_i [simp]: "z / (numeral n * \<i>) = - (\<i> * z) / numeral n" 279 by (metis divide_divide_eq_left divide_i mult.commute mult_minus_right) 280 281lemma imaginary_eq_real_iff [simp]: 282 assumes "y \<in> Reals" "x \<in> Reals" 283 shows "\<i> * y = x \<longleftrightarrow> x=0 \<and> y=0" 284 using assms 285 unfolding Reals_def 286 apply clarify 287 apply (rule iffI) 288 apply (metis Im_complex_of_real Im_i_times Re_complex_of_real mult_eq_0_iff of_real_0) 289 by simp 290 291lemma real_eq_imaginary_iff [simp]: 292 assumes "y \<in> Reals" "x \<in> Reals" 293 shows "x = \<i> * y \<longleftrightarrow> x=0 \<and> y=0" 294 using assms imaginary_eq_real_iff by fastforce 295 296subsection \<open>Vector Norm\<close> 297 298instantiation complex :: real_normed_field 299begin 300 301definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)" 302 303abbreviation cmod :: "complex \<Rightarrow> real" 304 where "cmod \<equiv> norm" 305 306definition complex_sgn_def: "sgn x = x /\<^sub>R cmod x" 307 308definition dist_complex_def: "dist x y = cmod (x - y)" 309 310definition uniformity_complex_def [code del]: 311 "(uniformity :: (complex \<times> complex) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})" 312 313definition open_complex_def [code del]: 314 "open (U :: complex set) \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)" 315 316instance 317proof 318 fix r :: real and x y :: complex and S :: "complex set" 319 show "(norm x = 0) = (x = 0)" 320 by (simp add: norm_complex_def complex_eq_iff) 321 show "norm (x + y) \<le> norm x + norm y" 322 by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq) 323 show "norm (scaleR r x) = \<bar>r\<bar> * norm x" 324 by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] 325 real_sqrt_mult) 326 show "norm (x * y) = norm x * norm y" 327 by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] 328 power2_eq_square algebra_simps) 329qed (rule complex_sgn_def dist_complex_def open_complex_def uniformity_complex_def)+ 330 331end 332 333declare uniformity_Abort[where 'a = complex, code] 334 335lemma norm_ii [simp]: "norm \<i> = 1" 336 by (simp add: norm_complex_def) 337 338lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1" 339 by (simp add: norm_complex_def) 340 341lemma cmod_complex_polar: "cmod (r * (cos a + \<i> * sin a)) = \<bar>r\<bar>" 342 by (simp add: norm_mult cmod_unit_one) 343 344lemma complex_Re_le_cmod: "Re x \<le> cmod x" 345 unfolding norm_complex_def by (rule real_sqrt_sum_squares_ge1) 346 347lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x" 348 by (rule order_trans [OF _ norm_ge_zero]) simp 349 350lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \<le> cmod a" 351 by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp 352 353lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x" 354 by (simp add: norm_complex_def) 355 356lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x" 357 by (simp add: norm_complex_def) 358 359lemma cmod_le: "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>" 360 apply (subst complex_eq) 361 apply (rule order_trans) 362 apply (rule norm_triangle_ineq) 363 apply (simp add: norm_mult) 364 done 365 366lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>" 367 by (simp add: norm_complex_def) 368 369lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>" 370 by (simp add: norm_complex_def) 371 372lemma cmod_power2: "(cmod z)\<^sup>2 = (Re z)\<^sup>2 + (Im z)\<^sup>2" 373 by (simp add: norm_complex_def) 374 375lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z = - cmod z" 376 using abs_Re_le_cmod[of z] by auto 377 378lemma cmod_Re_le_iff: "Im x = Im y \<Longrightarrow> cmod x \<le> cmod y \<longleftrightarrow> \<bar>Re x\<bar> \<le> \<bar>Re y\<bar>" 379 by (metis add.commute add_le_cancel_left norm_complex_def real_sqrt_abs real_sqrt_le_iff) 380 381lemma cmod_Im_le_iff: "Re x = Re y \<Longrightarrow> cmod x \<le> cmod y \<longleftrightarrow> \<bar>Im x\<bar> \<le> \<bar>Im y\<bar>" 382 by (metis add_le_cancel_left norm_complex_def real_sqrt_abs real_sqrt_le_iff) 383 384lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0" 385 by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2]) (auto simp add: norm_complex_def) 386 387lemma abs_sqrt_wlog: "(\<And>x. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)) \<Longrightarrow> P \<bar>x\<bar> (x\<^sup>2)" 388 for x::"'a::linordered_idom" 389 by (metis abs_ge_zero power2_abs) 390 391lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z" 392 unfolding norm_complex_def 393 apply (rule abs_sqrt_wlog [where x="Re z"]) 394 apply (rule abs_sqrt_wlog [where x="Im z"]) 395 apply (rule power2_le_imp_le) 396 apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric]) 397 done 398 399lemma complex_unit_circle: "z \<noteq> 0 \<Longrightarrow> (Re z / cmod z)\<^sup>2 + (Im z / cmod z)\<^sup>2 = 1" 400 by (simp add: norm_complex_def divide_simps complex_eq_iff) 401 402 403text \<open>Properties of complex signum.\<close> 404 405lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" 406 by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute) 407 408lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" 409 by (simp add: complex_sgn_def divide_inverse) 410 411lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" 412 by (simp add: complex_sgn_def divide_inverse) 413 414 415subsection \<open>Absolute value\<close> 416 417instantiation complex :: field_abs_sgn 418begin 419 420definition abs_complex :: "complex \<Rightarrow> complex" 421 where "abs_complex = of_real \<circ> norm" 422 423instance 424 apply standard 425 apply (auto simp add: abs_complex_def complex_sgn_def norm_mult) 426 apply (auto simp add: scaleR_conv_of_real field_simps) 427 done 428 429end 430 431 432subsection \<open>Completeness of the Complexes\<close> 433 434lemma bounded_linear_Re: "bounded_linear Re" 435 by (rule bounded_linear_intro [where K=1]) (simp_all add: norm_complex_def) 436 437lemma bounded_linear_Im: "bounded_linear Im" 438 by (rule bounded_linear_intro [where K=1]) (simp_all add: norm_complex_def) 439 440lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re] 441lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im] 442lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re] 443lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im] 444lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re] 445lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im] 446lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re] 447lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im] 448lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re] 449lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im] 450lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re] 451lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im] 452lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re] 453lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im] 454 455lemma tendsto_Complex [tendsto_intros]: 456 "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) \<longlongrightarrow> Complex a b) F" 457 unfolding Complex_eq by (auto intro!: tendsto_intros) 458 459lemma tendsto_complex_iff: 460 "(f \<longlongrightarrow> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F \<and> ((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F)" 461proof safe 462 assume "((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F" "((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F" 463 from tendsto_Complex[OF this] show "(f \<longlongrightarrow> x) F" 464 unfolding complex.collapse . 465qed (auto intro: tendsto_intros) 466 467lemma continuous_complex_iff: 468 "continuous F f \<longleftrightarrow> continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x))" 469 by (simp only: continuous_def tendsto_complex_iff) 470 471lemma continuous_on_of_real_o_iff [simp]: 472 "continuous_on S (\<lambda>x. complex_of_real (g x)) = continuous_on S g" 473 using continuous_on_Re continuous_on_of_real by fastforce 474 475lemma continuous_on_of_real_id [simp]: 476 "continuous_on S (of_real :: real \<Rightarrow> 'a::real_normed_algebra_1)" 477 by (rule continuous_on_of_real [OF continuous_on_id]) 478 479lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \<longleftrightarrow> 480 ((\<lambda>x. Re (f x)) has_field_derivative (Re x)) F \<and> 481 ((\<lambda>x. Im (f x)) has_field_derivative (Im x)) F" 482 by (simp add: has_vector_derivative_def has_field_derivative_def has_derivative_def 483 tendsto_complex_iff field_simps bounded_linear_scaleR_left bounded_linear_mult_right) 484 485lemma has_field_derivative_Re[derivative_intros]: 486 "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Re (f x)) has_field_derivative (Re D)) F" 487 unfolding has_vector_derivative_complex_iff by safe 488 489lemma has_field_derivative_Im[derivative_intros]: 490 "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Im (f x)) has_field_derivative (Im D)) F" 491 unfolding has_vector_derivative_complex_iff by safe 492 493instance complex :: banach 494proof 495 fix X :: "nat \<Rightarrow> complex" 496 assume X: "Cauchy X" 497 then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) \<longlonglongrightarrow> 498 Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))" 499 by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] 500 Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im) 501 then show "convergent X" 502 unfolding complex.collapse by (rule convergentI) 503qed 504 505declare DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros] 506 507 508subsection \<open>Complex Conjugation\<close> 509 510primcorec cnj :: "complex \<Rightarrow> complex" 511 where 512 "Re (cnj z) = Re z" 513 | "Im (cnj z) = - Im z" 514 515lemma complex_cnj_cancel_iff [simp]: "cnj x = cnj y \<longleftrightarrow> x = y" 516 by (simp add: complex_eq_iff) 517 518lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" 519 by (simp add: complex_eq_iff) 520 521lemma complex_cnj_zero [simp]: "cnj 0 = 0" 522 by (simp add: complex_eq_iff) 523 524lemma complex_cnj_zero_iff [iff]: "cnj z = 0 \<longleftrightarrow> z = 0" 525 by (simp add: complex_eq_iff) 526 527lemma complex_cnj_one_iff [simp]: "cnj z = 1 \<longleftrightarrow> z = 1" 528 by (simp add: complex_eq_iff) 529 530lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y" 531 by (simp add: complex_eq_iff) 532 533lemma cnj_sum [simp]: "cnj (sum f s) = (\<Sum>x\<in>s. cnj (f x))" 534 by (induct s rule: infinite_finite_induct) auto 535 536lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y" 537 by (simp add: complex_eq_iff) 538 539lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x" 540 by (simp add: complex_eq_iff) 541 542lemma complex_cnj_one [simp]: "cnj 1 = 1" 543 by (simp add: complex_eq_iff) 544 545lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y" 546 by (simp add: complex_eq_iff) 547 548lemma cnj_prod [simp]: "cnj (prod f s) = (\<Prod>x\<in>s. cnj (f x))" 549 by (induct s rule: infinite_finite_induct) auto 550 551lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)" 552 by (simp add: complex_eq_iff) 553 554lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y" 555 by (simp add: divide_complex_def) 556 557lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n" 558 by (induct n) simp_all 559 560lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" 561 by (simp add: complex_eq_iff) 562 563lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" 564 by (simp add: complex_eq_iff) 565 566lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w" 567 by (simp add: complex_eq_iff) 568 569lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w" 570 by (simp add: complex_eq_iff) 571 572lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)" 573 by (simp add: complex_eq_iff) 574 575lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" 576 by (simp add: norm_complex_def) 577 578lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" 579 by (simp add: complex_eq_iff) 580 581lemma complex_cnj_i [simp]: "cnj \<i> = - \<i>" 582 by (simp add: complex_eq_iff) 583 584lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" 585 by (simp add: complex_eq_iff) 586 587lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * \<i>" 588 by (simp add: complex_eq_iff) 589 590lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)" 591 by (simp add: complex_eq_iff power2_eq_square) 592 593lemma cnj_add_mult_eq_Re: "z * cnj w + cnj z * w = 2 * Re (z * cnj w)" 594 by (rule complex_eqI) auto 595 596lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2" 597 by (simp add: norm_mult power2_eq_square) 598 599lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" 600 by (simp add: norm_complex_def power2_eq_square) 601 602lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" 603 by simp 604 605lemma complex_cnj_fact [simp]: "cnj (fact n) = fact n" 606 by (subst of_nat_fact [symmetric], subst complex_cnj_of_nat) simp 607 608lemma complex_cnj_pochhammer [simp]: "cnj (pochhammer z n) = pochhammer (cnj z) n" 609 by (induct n arbitrary: z) (simp_all add: pochhammer_rec) 610 611lemma bounded_linear_cnj: "bounded_linear cnj" 612 using complex_cnj_add complex_cnj_scaleR by (rule bounded_linear_intro [where K=1]) simp 613 614lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj] 615 and isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj] 616 and continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj] 617 and continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj] 618 and has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj] 619 620lemma lim_cnj: "((\<lambda>x. cnj(f x)) \<longlongrightarrow> cnj l) F \<longleftrightarrow> (f \<longlongrightarrow> l) F" 621 by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff) 622 623lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)" 624 by (simp add: sums_def lim_cnj cnj_sum [symmetric] del: cnj_sum) 625 626 627subsection \<open>Basic Lemmas\<close> 628 629lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0" 630 by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff) 631 632lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0" 633 by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff) 634 635lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z" 636 by (cases z) 637 (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric] 638 simp del: of_real_power) 639 640lemma complex_div_cnj: "a / b = (a * cnj b) / (norm b)\<^sup>2" 641 using complex_norm_square by auto 642 643lemma Re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0" 644 by (auto simp add: Re_divide) 645 646lemma Im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0" 647 by (auto simp add: Im_divide) 648 649lemma complex_div_gt_0: "(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)" 650proof (cases "b = 0") 651 case True 652 then show ?thesis by auto 653next 654 case False 655 then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2" 656 by (simp add: complex_eq_iff sum_power2_gt_zero_iff) 657 then show ?thesis 658 by (simp add: Re_divide Im_divide zero_less_divide_iff) 659qed 660 661lemma Re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0" 662 and Im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0" 663 using complex_div_gt_0 by auto 664 665lemma Re_complex_div_ge_0: "Re (a / b) \<ge> 0 \<longleftrightarrow> Re (a * cnj b) \<ge> 0" 666 by (metis le_less Re_complex_div_eq_0 Re_complex_div_gt_0) 667 668lemma Im_complex_div_ge_0: "Im (a / b) \<ge> 0 \<longleftrightarrow> Im (a * cnj b) \<ge> 0" 669 by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 le_less) 670 671lemma Re_complex_div_lt_0: "Re (a / b) < 0 \<longleftrightarrow> Re (a * cnj b) < 0" 672 by (metis less_asym neq_iff Re_complex_div_eq_0 Re_complex_div_gt_0) 673 674lemma Im_complex_div_lt_0: "Im (a / b) < 0 \<longleftrightarrow> Im (a * cnj b) < 0" 675 by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 less_asym neq_iff) 676 677lemma Re_complex_div_le_0: "Re (a / b) \<le> 0 \<longleftrightarrow> Re (a * cnj b) \<le> 0" 678 by (metis not_le Re_complex_div_gt_0) 679 680lemma Im_complex_div_le_0: "Im (a / b) \<le> 0 \<longleftrightarrow> Im (a * cnj b) \<le> 0" 681 by (metis Im_complex_div_gt_0 not_le) 682 683lemma Re_divide_of_real [simp]: "Re (z / of_real r) = Re z / r" 684 by (simp add: Re_divide power2_eq_square) 685 686lemma Im_divide_of_real [simp]: "Im (z / of_real r) = Im z / r" 687 by (simp add: Im_divide power2_eq_square) 688 689lemma Re_divide_Reals [simp]: "r \<in> \<real> \<Longrightarrow> Re (z / r) = Re z / Re r" 690 by (metis Re_divide_of_real of_real_Re) 691 692lemma Im_divide_Reals [simp]: "r \<in> \<real> \<Longrightarrow> Im (z / r) = Im z / Re r" 693 by (metis Im_divide_of_real of_real_Re) 694 695lemma Re_sum[simp]: "Re (sum f s) = (\<Sum>x\<in>s. Re (f x))" 696 by (induct s rule: infinite_finite_induct) auto 697 698lemma Im_sum[simp]: "Im (sum f s) = (\<Sum>x\<in>s. Im(f x))" 699 by (induct s rule: infinite_finite_induct) auto 700 701lemma sums_complex_iff: "f sums x \<longleftrightarrow> ((\<lambda>x. Re (f x)) sums Re x) \<and> ((\<lambda>x. Im (f x)) sums Im x)" 702 unfolding sums_def tendsto_complex_iff Im_sum Re_sum .. 703 704lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and> summable (\<lambda>x. Im (f x))" 705 unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel) 706 707lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f" 708 unfolding summable_complex_iff by simp 709 710lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))" 711 unfolding summable_complex_iff by blast 712 713lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))" 714 unfolding summable_complex_iff by blast 715 716lemma complex_is_Nat_iff: "z \<in> \<nat> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_nat i)" 717 by (auto simp: Nats_def complex_eq_iff) 718 719lemma complex_is_Int_iff: "z \<in> \<int> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_int i)" 720 by (auto simp: Ints_def complex_eq_iff) 721 722lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0" 723 by (auto simp: Reals_def complex_eq_iff) 724 725lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z" 726 by (auto simp: complex_is_Real_iff complex_eq_iff) 727 728lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm z = \<bar>Re z\<bar>" 729 by (simp add: complex_is_Real_iff norm_complex_def) 730 731lemma Re_Reals_divide: "r \<in> \<real> \<Longrightarrow> Re (r / z) = Re r * Re z / (norm z)\<^sup>2" 732 by (simp add: Re_divide complex_is_Real_iff cmod_power2) 733 734lemma Im_Reals_divide: "r \<in> \<real> \<Longrightarrow> Im (r / z) = -Re r * Im z / (norm z)\<^sup>2" 735 by (simp add: Im_divide complex_is_Real_iff cmod_power2) 736 737lemma series_comparison_complex: 738 fixes f:: "nat \<Rightarrow> 'a::banach" 739 assumes sg: "summable g" 740 and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0" 741 and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)" 742 shows "summable f" 743proof - 744 have g: "\<And>n. cmod (g n) = Re (g n)" 745 using assms by (metis abs_of_nonneg in_Reals_norm) 746 show ?thesis 747 apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N]) 748 using sg 749 apply (auto simp: summable_def) 750 apply (rule_tac x = "Re s" in exI) 751 apply (auto simp: g sums_Re) 752 apply (metis fg g) 753 done 754qed 755 756 757subsection \<open>Polar Form for Complex Numbers\<close> 758 759lemma complex_unimodular_polar: 760 assumes "norm z = 1" 761 obtains t where "0 \<le> t" "t < 2 * pi" "z = Complex (cos t) (sin t)" 762 by (metis cmod_power2 one_power2 complex_surj sincos_total_2pi [of "Re z" "Im z"] assms) 763 764 765subsubsection \<open>$\cos \theta + i \sin \theta$\<close> 766 767primcorec cis :: "real \<Rightarrow> complex" 768 where 769 "Re (cis a) = cos a" 770 | "Im (cis a) = sin a" 771 772lemma cis_zero [simp]: "cis 0 = 1" 773 by (simp add: complex_eq_iff) 774 775lemma norm_cis [simp]: "norm (cis a) = 1" 776 by (simp add: norm_complex_def) 777 778lemma sgn_cis [simp]: "sgn (cis a) = cis a" 779 by (simp add: sgn_div_norm) 780 781lemma cis_neq_zero [simp]: "cis a \<noteq> 0" 782 by (metis norm_cis norm_zero zero_neq_one) 783 784lemma cis_mult: "cis a * cis b = cis (a + b)" 785 by (simp add: complex_eq_iff cos_add sin_add) 786 787lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" 788 by (induct n) (simp_all add: algebra_simps cis_mult) 789 790lemma cis_inverse [simp]: "inverse (cis a) = cis (- a)" 791 by (simp add: complex_eq_iff) 792 793lemma cis_divide: "cis a / cis b = cis (a - b)" 794 by (simp add: divide_complex_def cis_mult) 795 796lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re (cis a ^ n)" 797 by (auto simp add: DeMoivre) 798 799lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im (cis a ^ n)" 800 by (auto simp add: DeMoivre) 801 802lemma cis_pi [simp]: "cis pi = -1" 803 by (simp add: complex_eq_iff) 804 805 806subsubsection \<open>$r(\cos \theta + i \sin \theta)$\<close> 807 808definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex" 809 where "rcis r a = complex_of_real r * cis a" 810 811lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" 812 by (simp add: rcis_def) 813 814lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" 815 by (simp add: rcis_def) 816 817lemma rcis_Ex: "\<exists>r a. z = rcis r a" 818 by (simp add: complex_eq_iff polar_Ex) 819 820lemma complex_mod_rcis [simp]: "cmod (rcis r a) = \<bar>r\<bar>" 821 by (simp add: rcis_def norm_mult) 822 823lemma cis_rcis_eq: "cis a = rcis 1 a" 824 by (simp add: rcis_def) 825 826lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1 * r2) (a + b)" 827 by (simp add: rcis_def cis_mult) 828 829lemma rcis_zero_mod [simp]: "rcis 0 a = 0" 830 by (simp add: rcis_def) 831 832lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" 833 by (simp add: rcis_def) 834 835lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0" 836 by (simp add: rcis_def) 837 838lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" 839 by (simp add: rcis_def power_mult_distrib DeMoivre) 840 841lemma rcis_inverse: "inverse(rcis r a) = rcis (1 / r) (- a)" 842 by (simp add: divide_inverse rcis_def) 843 844lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1 / r2) (a - b)" 845 by (simp add: rcis_def cis_divide [symmetric]) 846 847 848subsubsection \<open>Complex exponential\<close> 849 850lemma cis_conv_exp: "cis b = exp (\<i> * b)" 851proof - 852 have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n = 853 of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)" 854 for n :: nat 855 proof - 856 have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)" 857 by (induct n) 858 (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps 859 power2_eq_square add_nonneg_eq_0_iff) 860 then show ?thesis 861 by (simp add: field_simps) 862 qed 863 then show ?thesis 864 using sin_converges [of b] cos_converges [of b] 865 by (auto simp add: Complex_eq cis.ctr exp_def simp del: of_real_mult 866 intro!: sums_unique sums_add sums_mult sums_of_real) 867qed 868 869lemma exp_eq_polar: "exp z = exp (Re z) * cis (Im z)" 870 unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp 871 by (cases z) (simp add: Complex_eq) 872 873lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)" 874 unfolding exp_eq_polar by simp 875 876lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)" 877 unfolding exp_eq_polar by simp 878 879lemma norm_cos_sin [simp]: "norm (Complex (cos t) (sin t)) = 1" 880 by (simp add: norm_complex_def) 881 882lemma norm_exp_eq_Re [simp]: "norm (exp z) = exp (Re z)" 883 by (simp add: cis.code cmod_complex_polar exp_eq_polar Complex_eq) 884 885lemma complex_exp_exists: "\<exists>a r. z = complex_of_real r * exp a" 886 apply (insert rcis_Ex [of z]) 887 apply (auto simp add: exp_eq_polar rcis_def mult.assoc [symmetric]) 888 apply (rule_tac x = "\<i> * complex_of_real a" in exI) 889 apply auto 890 done 891 892lemma exp_pi_i [simp]: "exp (of_real pi * \<i>) = -1" 893 by (metis cis_conv_exp cis_pi mult.commute) 894 895lemma exp_pi_i' [simp]: "exp (\<i> * of_real pi) = -1" 896 using cis_conv_exp cis_pi by auto 897 898lemma exp_two_pi_i [simp]: "exp (2 * of_real pi * \<i>) = 1" 899 by (simp add: exp_eq_polar complex_eq_iff) 900 901lemma exp_two_pi_i' [simp]: "exp (\<i> * (of_real pi * 2)) = 1" 902 by (metis exp_two_pi_i mult.commute) 903 904 905subsubsection \<open>Complex argument\<close> 906 907definition arg :: "complex \<Rightarrow> real" 908 where "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> - pi < a \<and> a \<le> pi))" 909 910lemma arg_zero: "arg 0 = 0" 911 by (simp add: arg_def) 912 913lemma arg_unique: 914 assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi" 915 shows "arg z = x" 916proof - 917 from assms have "z \<noteq> 0" by auto 918 have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x" 919 proof 920 fix a 921 define d where "d = a - x" 922 assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi" 923 from a assms have "- (2*pi) < d \<and> d < 2*pi" 924 unfolding d_def by simp 925 moreover 926 from a assms have "cos a = cos x" and "sin a = sin x" 927 by (simp_all add: complex_eq_iff) 928 then have cos: "cos d = 1" 929 by (simp add: d_def cos_diff) 930 moreover from cos have "sin d = 0" 931 by (rule cos_one_sin_zero) 932 ultimately have "d = 0" 933 by (auto simp: sin_zero_iff elim!: evenE dest!: less_2_cases) 934 then show "a = x" 935 by (simp add: d_def) 936 qed (simp add: assms del: Re_sgn Im_sgn) 937 with \<open>z \<noteq> 0\<close> show "arg z = x" 938 by (simp add: arg_def) 939qed 940 941lemma arg_correct: 942 assumes "z \<noteq> 0" 943 shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi" 944proof (simp add: arg_def assms, rule someI_ex) 945 obtain r a where z: "z = rcis r a" 946 using rcis_Ex by fast 947 with assms have "r \<noteq> 0" by auto 948 define b where "b = (if 0 < r then a else a + pi)" 949 have b: "sgn z = cis b" 950 using \<open>r \<noteq> 0\<close> by (simp add: z b_def rcis_def of_real_def sgn_scaleR sgn_if complex_eq_iff) 951 have cis_2pi_nat: "cis (2 * pi * real_of_nat n) = 1" for n 952 by (induct n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff) 953 have cis_2pi_int: "cis (2 * pi * real_of_int x) = 1" for x 954 by (cases x rule: int_diff_cases) 955 (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat) 956 define c where "c = b - 2 * pi * of_int \<lceil>(b - pi) / (2 * pi)\<rceil>" 957 have "sgn z = cis c" 958 by (simp add: b c_def cis_divide [symmetric] cis_2pi_int) 959 moreover have "- pi < c \<and> c \<le> pi" 960 using ceiling_correct [of "(b - pi) / (2*pi)"] 961 by (simp add: c_def less_divide_eq divide_le_eq algebra_simps del: le_of_int_ceiling) 962 ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" 963 by fast 964qed 965 966lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi" 967 by (cases "z = 0") (simp_all add: arg_zero arg_correct) 968 969lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z" 970 by (simp add: arg_correct) 971 972lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z" 973 by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def) 974 975lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0" 976 using cis_arg [of y] by (simp add: complex_eq_iff) 977 978subsection \<open>Complex n-th roots\<close> 979 980lemma bij_betw_roots_unity: 981 assumes "n > 0" 982 shows "bij_betw (\<lambda>k. cis (2 * pi * real k / real n)) {..<n} {z. z ^ n = 1}" 983 (is "bij_betw ?f _ _") 984 unfolding bij_betw_def 985proof (intro conjI) 986 show inj: "inj_on ?f {..<n}" unfolding inj_on_def 987 proof (safe, goal_cases) 988 case (1 k l) 989 hence kl: "k < n" "l < n" by simp_all 990 from 1 have "1 = ?f k / ?f l" by simp 991 also have "\<dots> = cis (2*pi*(real k - real l)/n)" 992 using assms by (simp add: field_simps cis_divide) 993 finally have "cos (2*pi*(real k - real l) / n) = 1" 994 by (simp add: complex_eq_iff) 995 then obtain m :: int where "2 * pi * (real k - real l) / real n = real_of_int m * 2 * pi" 996 by (subst (asm) cos_one_2pi_int) blast 997 hence "real_of_int (int k - int l) = real_of_int (m * int n)" 998 unfolding of_int_diff of_int_mult using assms by (simp add: divide_simps) 999 also note of_int_eq_iff 1000 finally have *: "abs m * n = abs (int k - int l)" by (simp add: abs_mult) 1001 also have "\<dots> < int n" using kl by linarith 1002 finally have "m = 0" using assms by simp 1003 with * show "k = l" by simp 1004 qed 1005 1006 have subset: "?f ` {..<n} \<subseteq> {z. z ^ n = 1}" 1007 proof safe 1008 fix k :: nat 1009 have "cis (2 * pi * real k / real n) ^ n = cis (2 * pi) ^ k" 1010 using assms by (simp add: DeMoivre mult_ac) 1011 also have "cis (2 * pi) = 1" by (simp add: complex_eq_iff) 1012 finally show "?f k ^ n = 1" by simp 1013 qed 1014 1015 have "n = card {..<n}" by simp 1016 also from assms and subset have "\<dots> \<le> card {z::complex. z ^ n = 1}" 1017 by (intro card_inj_on_le[OF inj]) (auto simp: finite_roots_unity) 1018 finally have card: "card {z::complex. z ^ n = 1} = n" 1019 using assms by (intro antisym card_roots_unity) auto 1020 1021 have "card (?f ` {..<n}) = card {z::complex. z ^ n = 1}" 1022 using card inj by (subst card_image) auto 1023 with subset and assms show "?f ` {..<n} = {z::complex. z ^ n = 1}" 1024 by (intro card_subset_eq finite_roots_unity) auto 1025qed 1026 1027lemma card_roots_unity_eq: 1028 assumes "n > 0" 1029 shows "card {z::complex. z ^ n = 1} = n" 1030 using bij_betw_same_card [OF bij_betw_roots_unity [OF assms]] by simp 1031 1032lemma bij_betw_nth_root_unity: 1033 fixes c :: complex and n :: nat 1034 assumes c: "c \<noteq> 0" and n: "n > 0" 1035 defines "c' \<equiv> root n (norm c) * cis (arg c / n)" 1036 shows "bij_betw (\<lambda>z. c' * z) {z. z ^ n = 1} {z. z ^ n = c}" 1037proof - 1038 have "c' ^ n = of_real (root n (norm c) ^ n) * cis (arg c)" 1039 unfolding of_real_power using n by (simp add: c'_def power_mult_distrib DeMoivre) 1040 also from n have "root n (norm c) ^ n = norm c" by simp 1041 also from c have "of_real \<dots> * cis (arg c) = c" by (simp add: cis_arg Complex.sgn_eq) 1042 finally have [simp]: "c' ^ n = c" . 1043 1044 show ?thesis unfolding bij_betw_def inj_on_def 1045 proof safe 1046 fix z :: complex assume "z ^ n = 1" 1047 hence "(c' * z) ^ n = c' ^ n" by (simp add: power_mult_distrib) 1048 also have "c' ^ n = of_real (root n (norm c) ^ n) * cis (arg c)" 1049 unfolding of_real_power using n by (simp add: c'_def power_mult_distrib DeMoivre) 1050 also from n have "root n (norm c) ^ n = norm c" by simp 1051 also from c have "\<dots> * cis (arg c) = c" by (simp add: cis_arg Complex.sgn_eq) 1052 finally show "(c' * z) ^ n = c" . 1053 next 1054 fix z assume z: "c = z ^ n" 1055 define z' where "z' = z / c'" 1056 from c and n have "c' \<noteq> 0" by (auto simp: c'_def) 1057 with n c have "z = c' * z'" and "z' ^ n = 1" 1058 by (auto simp: z'_def power_divide z) 1059 thus "z \<in> (\<lambda>z. c' * z) ` {z. z ^ n = 1}" by blast 1060 qed (insert c n, auto simp: c'_def) 1061qed 1062 1063lemma finite_nth_roots [intro]: 1064 assumes "n > 0" 1065 shows "finite {z::complex. z ^ n = c}" 1066proof (cases "c = 0") 1067 case True 1068 with assms have "{z::complex. z ^ n = c} = {0}" by auto 1069 thus ?thesis by simp 1070next 1071 case False 1072 from assms have "finite {z::complex. z ^ n = 1}" by (intro finite_roots_unity) simp_all 1073 also have "?this \<longleftrightarrow> ?thesis" 1074 by (rule bij_betw_finite, rule bij_betw_nth_root_unity) fact+ 1075 finally show ?thesis . 1076qed 1077 1078lemma card_nth_roots: 1079 assumes "c \<noteq> 0" "n > 0" 1080 shows "card {z::complex. z ^ n = c} = n" 1081proof - 1082 have "card {z. z ^ n = c} = card {z::complex. z ^ n = 1}" 1083 by (rule sym, rule bij_betw_same_card, rule bij_betw_nth_root_unity) fact+ 1084 also have "\<dots> = n" by (rule card_roots_unity_eq) fact+ 1085 finally show ?thesis . 1086qed 1087 1088lemma sum_roots_unity: 1089 assumes "n > 1" 1090 shows "\<Sum>{z::complex. z ^ n = 1} = 0" 1091proof - 1092 define \<omega> where "\<omega> = cis (2 * pi / real n)" 1093 have [simp]: "\<omega> \<noteq> 1" 1094 proof 1095 assume "\<omega> = 1" 1096 with assms obtain k :: int where "2 * pi / real n = 2 * pi * of_int k" 1097 by (auto simp: \<omega>_def complex_eq_iff cos_one_2pi_int) 1098 with assms have "real n * of_int k = 1" by (simp add: field_simps) 1099 also have "real n * of_int k = of_int (int n * k)" by simp 1100 also have "1 = (of_int 1 :: real)" by simp 1101 also note of_int_eq_iff 1102 finally show False using assms by (auto simp: zmult_eq_1_iff) 1103 qed 1104 1105 have "(\<Sum>z | z ^ n = 1. z :: complex) = (\<Sum>k<n. cis (2 * pi * real k / real n))" 1106 using assms by (intro sum.reindex_bij_betw [symmetric] bij_betw_roots_unity) auto 1107 also have "\<dots> = (\<Sum>k<n. \<omega> ^ k)" 1108 by (intro sum.cong refl) (auto simp: \<omega>_def DeMoivre mult_ac) 1109 also have "\<dots> = (\<omega> ^ n - 1) / (\<omega> - 1)" 1110 by (subst geometric_sum) auto 1111 also have "\<omega> ^ n - 1 = cis (2 * pi) - 1" using assms by (auto simp: \<omega>_def DeMoivre) 1112 also have "\<dots> = 0" by (simp add: complex_eq_iff) 1113 finally show ?thesis by simp 1114qed 1115 1116lemma sum_nth_roots: 1117 assumes "n > 1" 1118 shows "\<Sum>{z::complex. z ^ n = c} = 0" 1119proof (cases "c = 0") 1120 case True 1121 with assms have "{z::complex. z ^ n = c} = {0}" by auto 1122 also have "\<Sum>\<dots> = 0" by simp 1123 finally show ?thesis . 1124next 1125 case False 1126 define c' where "c' = root n (norm c) * cis (arg c / n)" 1127 from False and assms have "(\<Sum>{z. z ^ n = c}) = (\<Sum>z | z ^ n = 1. c' * z)" 1128 by (subst sum.reindex_bij_betw [OF bij_betw_nth_root_unity, symmetric]) 1129 (auto simp: sum_distrib_left finite_roots_unity c'_def) 1130 also from assms have "\<dots> = 0" 1131 by (simp add: sum_distrib_left [symmetric] sum_roots_unity) 1132 finally show ?thesis . 1133qed 1134 1135subsection \<open>Square root of complex numbers\<close> 1136 1137primcorec csqrt :: "complex \<Rightarrow> complex" 1138 where 1139 "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)" 1140 | "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)" 1141 1142lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)" 1143 by (simp add: complex_eq_iff norm_complex_def) 1144 1145lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>" 1146 by (simp add: complex_eq_iff norm_complex_def) 1147 1148lemma of_real_sqrt: "x \<ge> 0 \<Longrightarrow> of_real (sqrt x) = csqrt (of_real x)" 1149 by (simp add: complex_eq_iff norm_complex_def) 1150 1151lemma csqrt_0 [simp]: "csqrt 0 = 0" 1152 by simp 1153 1154lemma csqrt_1 [simp]: "csqrt 1 = 1" 1155 by simp 1156 1157lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2" 1158 by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt) 1159 1160lemma power2_csqrt[simp,algebra]: "(csqrt z)\<^sup>2 = z" 1161proof (cases "Im z = 0") 1162 case True 1163 then show ?thesis 1164 using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"] 1165 by (cases "0::real" "Re z" rule: linorder_cases) 1166 (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re) 1167next 1168 case False 1169 moreover have "cmod z * cmod z - Re z * Re z = Im z * Im z" 1170 by (simp add: norm_complex_def power2_eq_square) 1171 moreover have "\<bar>Re z\<bar> \<le> cmod z" 1172 by (simp add: norm_complex_def) 1173 ultimately show ?thesis 1174 by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq 1175 field_simps real_sqrt_mult[symmetric] real_sqrt_divide) 1176qed 1177 1178lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0" 1179 by auto (metis power2_csqrt power_eq_0_iff) 1180 1181lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1" 1182 by auto (metis power2_csqrt power2_eq_1_iff) 1183 1184lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)" 1185 by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0) 1186 1187lemma Re_csqrt: "0 \<le> Re (csqrt z)" 1188 by (metis csqrt_principal le_less) 1189 1190lemma csqrt_square: 1191 assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)" 1192 shows "csqrt (b^2) = b" 1193proof - 1194 have "csqrt (b^2) = b \<or> csqrt (b^2) = - b" 1195 by (simp add: power2_eq_iff[symmetric]) 1196 moreover have "csqrt (b^2) \<noteq> -b \<or> b = 0" 1197 using csqrt_principal[of "b ^ 2"] assms 1198 by (intro disjCI notI) (auto simp: complex_eq_iff) 1199 ultimately show ?thesis 1200 by auto 1201qed 1202 1203lemma csqrt_unique: "w\<^sup>2 = z \<Longrightarrow> 0 < Re w \<or> Re w = 0 \<and> 0 \<le> Im w \<Longrightarrow> csqrt z = w" 1204 by (auto simp: csqrt_square) 1205 1206lemma csqrt_minus [simp]: 1207 assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)" 1208 shows "csqrt (- x) = \<i> * csqrt x" 1209proof - 1210 have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x" 1211 proof (rule csqrt_square) 1212 have "Im (csqrt x) \<le> 0" 1213 using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod) 1214 then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)" 1215 by (auto simp add: Re_csqrt simp del: csqrt.simps) 1216 qed 1217 also have "(\<i> * csqrt x)^2 = - x" 1218 by (simp add: power_mult_distrib) 1219 finally show ?thesis . 1220qed 1221 1222 1223text \<open>Legacy theorem names\<close> 1224 1225lemmas cmod_def = norm_complex_def 1226 1227lemma legacy_Complex_simps: 1228 shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" 1229 and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)" 1230 and complex_minus: "- (Complex a b) = Complex (- a) (- b)" 1231 and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)" 1232 and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0" 1233 and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0" 1234 and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)" 1235 and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))" 1236 and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0" 1237 and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0" 1238 and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)" 1239 and Complex_eq_i: "Complex x y = \<i> \<longleftrightarrow> x = 0 \<and> y = 1" 1240 and i_mult_Complex: "\<i> * Complex a b = Complex (- b) a" 1241 and Complex_mult_i: "Complex a b * \<i> = Complex (- b) a" 1242 and i_complex_of_real: "\<i> * complex_of_real r = Complex 0 r" 1243 and complex_of_real_i: "complex_of_real r * \<i> = Complex 0 r" 1244 and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y" 1245 and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y" 1246 and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)" 1247 and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)" 1248 and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa \<and> y = 0)" 1249 and complex_cnj: "cnj (Complex a b) = Complex a (- b)" 1250 and Complex_sum': "sum (\<lambda>x. Complex (f x) 0) s = Complex (sum f s) 0" 1251 and Complex_sum: "Complex (sum f s) 0 = sum (\<lambda>x. Complex (f x) 0) s" 1252 and complex_of_real_def: "complex_of_real r = Complex r 0" 1253 and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)" 1254 by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide) 1255 1256lemma Complex_in_Reals: "Complex x 0 \<in> \<real>" 1257 by (metis Reals_of_real complex_of_real_def) 1258 1259end 1260