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 by (simp add: divide_complex_def diff_divide_distrib) 96 97lemma Complex_divide: 98 "(x / y) = Complex ((Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)) 99 ((Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2))" 100 by (metis Im_divide Re_divide complex_surj) 101 102lemma Re_power2: "Re (x ^ 2) = (Re x)^2 - (Im x)^2" 103 by (simp add: power2_eq_square) 104 105lemma Im_power2: "Im (x ^ 2) = 2 * Re x * Im x" 106 by (simp add: power2_eq_square) 107 108lemma Re_power_real [simp]: "Im x = 0 \<Longrightarrow> Re (x ^ n) = Re x ^ n " 109 by (induct n) simp_all 110 111lemma Im_power_real [simp]: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0" 112 by (induct n) simp_all 113 114 115subsection \<open>Scalar Multiplication\<close> 116 117instantiation complex :: real_field 118begin 119 120primcorec scaleR_complex 121 where 122 "Re (scaleR r x) = r * Re x" 123 | "Im (scaleR r x) = r * Im x" 124 125instance 126proof 127 fix a b :: real and x y :: complex 128 show "scaleR a (x + y) = scaleR a x + scaleR a y" 129 by (simp add: complex_eq_iff distrib_left) 130 show "scaleR (a + b) x = scaleR a x + scaleR b x" 131 by (simp add: complex_eq_iff distrib_right) 132 show "scaleR a (scaleR b x) = scaleR (a * b) x" 133 by (simp add: complex_eq_iff mult.assoc) 134 show "scaleR 1 x = x" 135 by (simp add: complex_eq_iff) 136 show "scaleR a x * y = scaleR a (x * y)" 137 by (simp add: complex_eq_iff algebra_simps) 138 show "x * scaleR a y = scaleR a (x * y)" 139 by (simp add: complex_eq_iff algebra_simps) 140qed 141 142end 143 144 145subsection \<open>Numerals, Arithmetic, and Embedding from R\<close> 146 147abbreviation complex_of_real :: "real \<Rightarrow> complex" 148 where "complex_of_real \<equiv> of_real" 149 150declare [[coercion "of_real :: real \<Rightarrow> complex"]] 151declare [[coercion "of_rat :: rat \<Rightarrow> complex"]] 152declare [[coercion "of_int :: int \<Rightarrow> complex"]] 153declare [[coercion "of_nat :: nat \<Rightarrow> complex"]] 154 155lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n" 156 by (induct n) simp_all 157 158lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0" 159 by (induct n) simp_all 160 161lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z" 162 by (cases z rule: int_diff_cases) simp 163 164lemma complex_Im_of_int [simp]: "Im (of_int z) = 0" 165 by (cases z rule: int_diff_cases) simp 166 167lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v" 168 using complex_Re_of_int [of "numeral v"] by simp 169 170lemma complex_Im_numeral [simp]: "Im (numeral v) = 0" 171 using complex_Im_of_int [of "numeral v"] by simp 172 173lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" 174 by (simp add: of_real_def) 175 176lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" 177 by (simp add: of_real_def) 178 179lemma Re_divide_numeral [simp]: "Re (z / numeral w) = Re z / numeral w" 180 by (simp add: Re_divide sqr_conv_mult) 181 182lemma Im_divide_numeral [simp]: "Im (z / numeral w) = Im z / numeral w" 183 by (simp add: Im_divide sqr_conv_mult) 184 185lemma Re_divide_of_nat [simp]: "Re (z / of_nat n) = Re z / of_nat n" 186 by (cases n) (simp_all add: Re_divide field_split_simps power2_eq_square del: of_nat_Suc) 187 188lemma Im_divide_of_nat [simp]: "Im (z / of_nat n) = Im z / of_nat n" 189 by (cases n) (simp_all add: Im_divide field_split_simps power2_eq_square del: of_nat_Suc) 190 191lemma of_real_Re [simp]: "z \<in> \<real> \<Longrightarrow> of_real (Re z) = z" 192 by (auto simp: Reals_def) 193 194lemma complex_Re_fact [simp]: "Re (fact n) = fact n" 195proof - 196 have "(fact n :: complex) = of_real (fact n)" 197 by simp 198 also have "Re \<dots> = fact n" 199 by (subst Re_complex_of_real) simp_all 200 finally show ?thesis . 201qed 202 203lemma complex_Im_fact [simp]: "Im (fact n) = 0" 204 by (subst of_nat_fact [symmetric]) (simp only: complex_Im_of_nat) 205 206lemma 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" 207proof (induction A rule: infinite_finite_induct) 208 case (insert x A) 209 hence "Re (prod f (insert x A)) = Re (f x) * Re (prod f A) - Im (f x) * Im (prod f A)" 210 by simp 211 also from insert.prems have "f x \<in> \<real>" by simp 212 hence "Im (f x) = 0" by (auto elim!: Reals_cases) 213 also have "Re (prod f A) = (\<Prod>x\<in>A. Re (f x))" 214 by (intro insert.IH insert.prems) auto 215 finally show ?case using insert.hyps by simp 216qed auto 217 218 219subsection \<open>The Complex Number $i$\<close> 220 221primcorec imaginary_unit :: complex ("\<i>") 222 where 223 "Re \<i> = 0" 224 | "Im \<i> = 1" 225 226lemma Complex_eq: "Complex a b = a + \<i> * b" 227 by (simp add: complex_eq_iff) 228 229lemma complex_eq: "a = Re a + \<i> * Im a" 230 by (simp add: complex_eq_iff) 231 232lemma fun_complex_eq: "f = (\<lambda>x. Re (f x) + \<i> * Im (f x))" 233 by (simp add: fun_eq_iff complex_eq) 234 235lemma i_squared [simp]: "\<i> * \<i> = -1" 236 by (simp add: complex_eq_iff) 237 238lemma power2_i [simp]: "\<i>\<^sup>2 = -1" 239 by (simp add: power2_eq_square) 240 241lemma inverse_i [simp]: "inverse \<i> = - \<i>" 242 by (rule inverse_unique) simp 243 244lemma divide_i [simp]: "x / \<i> = - \<i> * x" 245 by (simp add: divide_complex_def) 246 247lemma complex_i_mult_minus [simp]: "\<i> * (\<i> * x) = - x" 248 by (simp add: mult.assoc [symmetric]) 249 250lemma complex_i_not_zero [simp]: "\<i> \<noteq> 0" 251 by (simp add: complex_eq_iff) 252 253lemma complex_i_not_one [simp]: "\<i> \<noteq> 1" 254 by (simp add: complex_eq_iff) 255 256lemma complex_i_not_numeral [simp]: "\<i> \<noteq> numeral w" 257 by (simp add: complex_eq_iff) 258 259lemma complex_i_not_neg_numeral [simp]: "\<i> \<noteq> - numeral w" 260 by (simp add: complex_eq_iff) 261 262lemma complex_split_polar: "\<exists>r a. z = complex_of_real r * (cos a + \<i> * sin a)" 263 by (simp add: complex_eq_iff polar_Ex) 264 265lemma i_even_power [simp]: "\<i> ^ (n * 2) = (-1) ^ n" 266 by (metis mult.commute power2_i power_mult) 267 268lemma Re_i_times [simp]: "Re (\<i> * z) = - Im z" 269 by simp 270 271lemma Im_i_times [simp]: "Im (\<i> * z) = Re z" 272 by simp 273 274lemma i_times_eq_iff: "\<i> * w = z \<longleftrightarrow> w = - (\<i> * z)" 275 by auto 276 277lemma divide_numeral_i [simp]: "z / (numeral n * \<i>) = - (\<i> * z) / numeral n" 278 by (metis divide_divide_eq_left divide_i mult.commute mult_minus_right) 279 280lemma imaginary_eq_real_iff [simp]: 281 assumes "y \<in> Reals" "x \<in> Reals" 282 shows "\<i> * y = x \<longleftrightarrow> x=0 \<and> y=0" 283 using assms 284 unfolding Reals_def 285 apply clarify 286 apply (rule iffI) 287 apply (metis Im_complex_of_real Im_i_times Re_complex_of_real mult_eq_0_iff of_real_0) 288 by simp 289 290lemma real_eq_imaginary_iff [simp]: 291 assumes "y \<in> Reals" "x \<in> Reals" 292 shows "x = \<i> * y \<longleftrightarrow> x=0 \<and> y=0" 293 using assms imaginary_eq_real_iff by fastforce 294 295subsection \<open>Vector Norm\<close> 296 297instantiation complex :: real_normed_field 298begin 299 300definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)" 301 302abbreviation cmod :: "complex \<Rightarrow> real" 303 where "cmod \<equiv> norm" 304 305definition complex_sgn_def: "sgn x = x /\<^sub>R cmod x" 306 307definition dist_complex_def: "dist x y = cmod (x - y)" 308 309definition uniformity_complex_def [code del]: 310 "(uniformity :: (complex \<times> complex) filter) = (INF e\<in>{0 <..}. principal {(x, y). dist x y < e})" 311 312definition open_complex_def [code del]: 313 "open (U :: complex set) \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)" 314 315instance 316proof 317 fix r :: real and x y :: complex and S :: "complex set" 318 show "(norm x = 0) = (x = 0)" 319 by (simp add: norm_complex_def complex_eq_iff) 320 show "norm (x + y) \<le> norm x + norm y" 321 by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq) 322 show "norm (scaleR r x) = \<bar>r\<bar> * norm x" 323 by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] 324 real_sqrt_mult) 325 show "norm (x * y) = norm x * norm y" 326 by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] 327 power2_eq_square algebra_simps) 328qed (rule complex_sgn_def dist_complex_def open_complex_def uniformity_complex_def)+ 329 330end 331 332declare uniformity_Abort[where 'a = complex, code] 333 334lemma norm_ii [simp]: "norm \<i> = 1" 335 by (simp add: norm_complex_def) 336 337lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1" 338 by (simp add: norm_complex_def) 339 340lemma cmod_complex_polar: "cmod (r * (cos a + \<i> * sin a)) = \<bar>r\<bar>" 341 by (simp add: norm_mult cmod_unit_one) 342 343lemma complex_Re_le_cmod: "Re x \<le> cmod x" 344 unfolding norm_complex_def by (rule real_sqrt_sum_squares_ge1) 345 346lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x" 347 by (rule order_trans [OF _ norm_ge_zero]) simp 348 349lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \<le> cmod a" 350 by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp 351 352lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x" 353 by (simp add: norm_complex_def) 354 355lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x" 356 by (simp add: norm_complex_def) 357 358lemma cmod_le: "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>" 359 apply (subst complex_eq) 360 apply (rule order_trans) 361 apply (rule norm_triangle_ineq) 362 apply (simp add: norm_mult) 363 done 364 365lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>" 366 by (simp add: norm_complex_def) 367 368lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>" 369 by (simp add: norm_complex_def) 370 371lemma cmod_power2: "(cmod z)\<^sup>2 = (Re z)\<^sup>2 + (Im z)\<^sup>2" 372 by (simp add: norm_complex_def) 373 374lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z = - cmod z" 375 using abs_Re_le_cmod[of z] by auto 376 377lemma cmod_Re_le_iff: "Im x = Im y \<Longrightarrow> cmod x \<le> cmod y \<longleftrightarrow> \<bar>Re x\<bar> \<le> \<bar>Re y\<bar>" 378 by (metis add.commute add_le_cancel_left norm_complex_def real_sqrt_abs real_sqrt_le_iff) 379 380lemma cmod_Im_le_iff: "Re x = Re y \<Longrightarrow> cmod x \<le> cmod y \<longleftrightarrow> \<bar>Im x\<bar> \<le> \<bar>Im y\<bar>" 381 by (metis add_le_cancel_left norm_complex_def real_sqrt_abs real_sqrt_le_iff) 382 383lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0" 384 by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2]) (auto simp add: norm_complex_def) 385 386lemma abs_sqrt_wlog: "(\<And>x. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)) \<Longrightarrow> P \<bar>x\<bar> (x\<^sup>2)" 387 for x::"'a::linordered_idom" 388 by (metis abs_ge_zero power2_abs) 389 390lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z" 391 unfolding norm_complex_def 392 apply (rule abs_sqrt_wlog [where x="Re z"]) 393 apply (rule abs_sqrt_wlog [where x="Im z"]) 394 apply (rule power2_le_imp_le) 395 apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric]) 396 done 397 398lemma complex_unit_circle: "z \<noteq> 0 \<Longrightarrow> (Re z / cmod z)\<^sup>2 + (Im z / cmod z)\<^sup>2 = 1" 399 by (simp add: norm_complex_def complex_eq_iff power2_eq_square add_divide_distrib [symmetric]) 400 401 402text \<open>Properties of complex signum.\<close> 403 404lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" 405 by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute) 406 407lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" 408 by (simp add: complex_sgn_def divide_inverse) 409 410lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" 411 by (simp add: complex_sgn_def divide_inverse) 412 413 414subsection \<open>Absolute value\<close> 415 416instantiation complex :: field_abs_sgn 417begin 418 419definition abs_complex :: "complex \<Rightarrow> complex" 420 where "abs_complex = of_real \<circ> norm" 421 422instance 423 apply standard 424 apply (auto simp add: abs_complex_def complex_sgn_def norm_mult) 425 apply (auto simp add: scaleR_conv_of_real field_simps) 426 done 427 428end 429 430 431subsection \<open>Completeness of the Complexes\<close> 432 433lemma bounded_linear_Re: "bounded_linear Re" 434 by (rule bounded_linear_intro [where K=1]) (simp_all add: norm_complex_def) 435 436lemma bounded_linear_Im: "bounded_linear Im" 437 by (rule bounded_linear_intro [where K=1]) (simp_all add: norm_complex_def) 438 439lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re] 440lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im] 441lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re] 442lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im] 443lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re] 444lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im] 445lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re] 446lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im] 447lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re] 448lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im] 449lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re] 450lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im] 451lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re] 452lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im] 453 454lemma tendsto_Complex [tendsto_intros]: 455 "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) \<longlongrightarrow> Complex a b) F" 456 unfolding Complex_eq by (auto intro!: tendsto_intros) 457 458lemma tendsto_complex_iff: 459 "(f \<longlongrightarrow> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F \<and> ((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F)" 460proof safe 461 assume "((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F" "((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F" 462 from tendsto_Complex[OF this] show "(f \<longlongrightarrow> x) F" 463 unfolding complex.collapse . 464qed (auto intro: tendsto_intros) 465 466lemma continuous_complex_iff: 467 "continuous F f \<longleftrightarrow> continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x))" 468 by (simp only: continuous_def tendsto_complex_iff) 469 470lemma continuous_on_of_real_o_iff [simp]: 471 "continuous_on S (\<lambda>x. complex_of_real (g x)) = continuous_on S g" 472 using continuous_on_Re continuous_on_of_real by fastforce 473 474lemma continuous_on_of_real_id [simp]: 475 "continuous_on S (of_real :: real \<Rightarrow> 'a::real_normed_algebra_1)" 476 by (rule continuous_on_of_real [OF continuous_on_id]) 477 478lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \<longleftrightarrow> 479 ((\<lambda>x. Re (f x)) has_field_derivative (Re x)) F \<and> 480 ((\<lambda>x. Im (f x)) has_field_derivative (Im x)) F" 481 by (simp add: has_vector_derivative_def has_field_derivative_def has_derivative_def 482 tendsto_complex_iff algebra_simps bounded_linear_scaleR_left bounded_linear_mult_right) 483 484lemma has_field_derivative_Re[derivative_intros]: 485 "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Re (f x)) has_field_derivative (Re D)) F" 486 unfolding has_vector_derivative_complex_iff by safe 487 488lemma has_field_derivative_Im[derivative_intros]: 489 "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Im (f x)) has_field_derivative (Im D)) F" 490 unfolding has_vector_derivative_complex_iff by safe 491 492instance complex :: banach 493proof 494 fix X :: "nat \<Rightarrow> complex" 495 assume X: "Cauchy X" 496 then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) \<longlonglongrightarrow> 497 Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))" 498 by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] 499 Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im) 500 then show "convergent X" 501 unfolding complex.collapse by (rule convergentI) 502qed 503 504declare DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros] 505 506 507subsection \<open>Complex Conjugation\<close> 508 509primcorec cnj :: "complex \<Rightarrow> complex" 510 where 511 "Re (cnj z) = Re z" 512 | "Im (cnj z) = - Im z" 513 514lemma complex_cnj_cancel_iff [simp]: "cnj x = cnj y \<longleftrightarrow> x = y" 515 by (simp add: complex_eq_iff) 516 517lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" 518 by (simp add: complex_eq_iff) 519 520lemma complex_cnj_zero [simp]: "cnj 0 = 0" 521 by (simp add: complex_eq_iff) 522 523lemma complex_cnj_zero_iff [iff]: "cnj z = 0 \<longleftrightarrow> z = 0" 524 by (simp add: complex_eq_iff) 525 526lemma complex_cnj_one_iff [simp]: "cnj z = 1 \<longleftrightarrow> z = 1" 527 by (simp add: complex_eq_iff) 528 529lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y" 530 by (simp add: complex_eq_iff) 531 532lemma cnj_sum [simp]: "cnj (sum f s) = (\<Sum>x\<in>s. cnj (f x))" 533 by (induct s rule: infinite_finite_induct) auto 534 535lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y" 536 by (simp add: complex_eq_iff) 537 538lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x" 539 by (simp add: complex_eq_iff) 540 541lemma complex_cnj_one [simp]: "cnj 1 = 1" 542 by (simp add: complex_eq_iff) 543 544lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y" 545 by (simp add: complex_eq_iff) 546 547lemma cnj_prod [simp]: "cnj (prod f s) = (\<Prod>x\<in>s. cnj (f x))" 548 by (induct s rule: infinite_finite_induct) auto 549 550lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)" 551 by (simp add: complex_eq_iff) 552 553lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y" 554 by (simp add: divide_complex_def) 555 556lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n" 557 by (induct n) simp_all 558 559lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" 560 by (simp add: complex_eq_iff) 561 562lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" 563 by (simp add: complex_eq_iff) 564 565lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w" 566 by (simp add: complex_eq_iff) 567 568lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w" 569 by (simp add: complex_eq_iff) 570 571lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)" 572 by (simp add: complex_eq_iff) 573 574lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" 575 by (simp add: norm_complex_def) 576 577lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" 578 by (simp add: complex_eq_iff) 579 580lemma complex_cnj_i [simp]: "cnj \<i> = - \<i>" 581 by (simp add: complex_eq_iff) 582 583lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" 584 by (simp add: complex_eq_iff) 585 586lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * \<i>" 587 by (simp add: complex_eq_iff) 588 589lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)" 590 by (simp add: complex_eq_iff power2_eq_square) 591 592lemma cnj_add_mult_eq_Re: "z * cnj w + cnj z * w = 2 * Re (z * cnj w)" 593 by (rule complex_eqI) auto 594 595lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2" 596 by (simp add: norm_mult power2_eq_square) 597 598lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" 599 by (simp add: norm_complex_def power2_eq_square) 600 601lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" 602 by simp 603 604lemma complex_cnj_fact [simp]: "cnj (fact n) = fact n" 605 by (subst of_nat_fact [symmetric], subst complex_cnj_of_nat) simp 606 607lemma complex_cnj_pochhammer [simp]: "cnj (pochhammer z n) = pochhammer (cnj z) n" 608 by (induct n arbitrary: z) (simp_all add: pochhammer_rec) 609 610lemma bounded_linear_cnj: "bounded_linear cnj" 611 using complex_cnj_add complex_cnj_scaleR by (rule bounded_linear_intro [where K=1]) simp 612 613lemma linear_cnj: "linear cnj" 614 using bounded_linear.linear[OF bounded_linear_cnj] . 615 616lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj] 617 and isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj] 618 and continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj] 619 and continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj] 620 and has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj] 621 622lemma lim_cnj: "((\<lambda>x. cnj(f x)) \<longlongrightarrow> cnj l) F \<longleftrightarrow> (f \<longlongrightarrow> l) F" 623 by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff) 624 625lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)" 626 by (simp add: sums_def lim_cnj cnj_sum [symmetric] del: cnj_sum) 627 628lemma differentiable_cnj_iff: 629 "(\<lambda>z. cnj (f z)) differentiable at x within A \<longleftrightarrow> f differentiable at x within A" 630proof 631 assume "(\<lambda>z. cnj (f z)) differentiable at x within A" 632 then obtain D where "((\<lambda>z. cnj (f z)) has_derivative D) (at x within A)" 633 by (auto simp: differentiable_def) 634 from has_derivative_cnj[OF this] show "f differentiable at x within A" 635 by (auto simp: differentiable_def) 636next 637 assume "f differentiable at x within A" 638 then obtain D where "(f has_derivative D) (at x within A)" 639 by (auto simp: differentiable_def) 640 from has_derivative_cnj[OF this] show "(\<lambda>z. cnj (f z)) differentiable at x within A" 641 by (auto simp: differentiable_def) 642qed 643 644lemma has_vector_derivative_cnj [derivative_intros]: 645 assumes "(f has_vector_derivative f') (at z within A)" 646 shows "((\<lambda>z. cnj (f z)) has_vector_derivative cnj f') (at z within A)" 647 using assms by (auto simp: has_vector_derivative_complex_iff intro: derivative_intros) 648 649 650subsection \<open>Basic Lemmas\<close> 651 652lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0" 653 by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff) 654 655lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0" 656 by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff) 657 658lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z" 659 by (cases z) 660 (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric] 661 simp del: of_real_power) 662 663lemma complex_div_cnj: "a / b = (a * cnj b) / (norm b)\<^sup>2" 664 using complex_norm_square by auto 665 666lemma Re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0" 667 by (auto simp add: Re_divide) 668 669lemma Im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0" 670 by (auto simp add: Im_divide) 671 672lemma 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)" 673proof (cases "b = 0") 674 case True 675 then show ?thesis by auto 676next 677 case False 678 then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2" 679 by (simp add: complex_eq_iff sum_power2_gt_zero_iff) 680 then show ?thesis 681 by (simp add: Re_divide Im_divide zero_less_divide_iff) 682qed 683 684lemma Re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0" 685 and Im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0" 686 using complex_div_gt_0 by auto 687 688lemma Re_complex_div_ge_0: "Re (a / b) \<ge> 0 \<longleftrightarrow> Re (a * cnj b) \<ge> 0" 689 by (metis le_less Re_complex_div_eq_0 Re_complex_div_gt_0) 690 691lemma Im_complex_div_ge_0: "Im (a / b) \<ge> 0 \<longleftrightarrow> Im (a * cnj b) \<ge> 0" 692 by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 le_less) 693 694lemma Re_complex_div_lt_0: "Re (a / b) < 0 \<longleftrightarrow> Re (a * cnj b) < 0" 695 by (metis less_asym neq_iff Re_complex_div_eq_0 Re_complex_div_gt_0) 696 697lemma Im_complex_div_lt_0: "Im (a / b) < 0 \<longleftrightarrow> Im (a * cnj b) < 0" 698 by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 less_asym neq_iff) 699 700lemma Re_complex_div_le_0: "Re (a / b) \<le> 0 \<longleftrightarrow> Re (a * cnj b) \<le> 0" 701 by (metis not_le Re_complex_div_gt_0) 702 703lemma Im_complex_div_le_0: "Im (a / b) \<le> 0 \<longleftrightarrow> Im (a * cnj b) \<le> 0" 704 by (metis Im_complex_div_gt_0 not_le) 705 706lemma Re_divide_of_real [simp]: "Re (z / of_real r) = Re z / r" 707 by (simp add: Re_divide power2_eq_square) 708 709lemma Im_divide_of_real [simp]: "Im (z / of_real r) = Im z / r" 710 by (simp add: Im_divide power2_eq_square) 711 712lemma Re_divide_Reals [simp]: "r \<in> \<real> \<Longrightarrow> Re (z / r) = Re z / Re r" 713 by (metis Re_divide_of_real of_real_Re) 714 715lemma Im_divide_Reals [simp]: "r \<in> \<real> \<Longrightarrow> Im (z / r) = Im z / Re r" 716 by (metis Im_divide_of_real of_real_Re) 717 718lemma Re_sum[simp]: "Re (sum f s) = (\<Sum>x\<in>s. Re (f x))" 719 by (induct s rule: infinite_finite_induct) auto 720 721lemma Im_sum[simp]: "Im (sum f s) = (\<Sum>x\<in>s. Im(f x))" 722 by (induct s rule: infinite_finite_induct) auto 723 724lemma sums_complex_iff: "f sums x \<longleftrightarrow> ((\<lambda>x. Re (f x)) sums Re x) \<and> ((\<lambda>x. Im (f x)) sums Im x)" 725 unfolding sums_def tendsto_complex_iff Im_sum Re_sum .. 726 727lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and> summable (\<lambda>x. Im (f x))" 728 unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel) 729 730lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f" 731 unfolding summable_complex_iff by simp 732 733lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))" 734 unfolding summable_complex_iff by blast 735 736lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))" 737 unfolding summable_complex_iff by blast 738 739lemma complex_is_Nat_iff: "z \<in> \<nat> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_nat i)" 740 by (auto simp: Nats_def complex_eq_iff) 741 742lemma complex_is_Int_iff: "z \<in> \<int> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_int i)" 743 by (auto simp: Ints_def complex_eq_iff) 744 745lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0" 746 by (auto simp: Reals_def complex_eq_iff) 747 748lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z" 749 by (auto simp: complex_is_Real_iff complex_eq_iff) 750 751lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm z = \<bar>Re z\<bar>" 752 by (simp add: complex_is_Real_iff norm_complex_def) 753 754lemma Re_Reals_divide: "r \<in> \<real> \<Longrightarrow> Re (r / z) = Re r * Re z / (norm z)\<^sup>2" 755 by (simp add: Re_divide complex_is_Real_iff cmod_power2) 756 757lemma Im_Reals_divide: "r \<in> \<real> \<Longrightarrow> Im (r / z) = -Re r * Im z / (norm z)\<^sup>2" 758 by (simp add: Im_divide complex_is_Real_iff cmod_power2) 759 760lemma series_comparison_complex: 761 fixes f:: "nat \<Rightarrow> 'a::banach" 762 assumes sg: "summable g" 763 and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0" 764 and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)" 765 shows "summable f" 766proof - 767 have g: "\<And>n. cmod (g n) = Re (g n)" 768 using assms by (metis abs_of_nonneg in_Reals_norm) 769 show ?thesis 770 apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N]) 771 using sg 772 apply (auto simp: summable_def) 773 apply (rule_tac x = "Re s" in exI) 774 apply (auto simp: g sums_Re) 775 apply (metis fg g) 776 done 777qed 778 779 780subsection \<open>Polar Form for Complex Numbers\<close> 781 782lemma complex_unimodular_polar: 783 assumes "norm z = 1" 784 obtains t where "0 \<le> t" "t < 2 * pi" "z = Complex (cos t) (sin t)" 785 by (metis cmod_power2 one_power2 complex_surj sincos_total_2pi [of "Re z" "Im z"] assms) 786 787 788subsubsection \<open>$\cos \theta + i \sin \theta$\<close> 789 790primcorec cis :: "real \<Rightarrow> complex" 791 where 792 "Re (cis a) = cos a" 793 | "Im (cis a) = sin a" 794 795lemma cis_zero [simp]: "cis 0 = 1" 796 by (simp add: complex_eq_iff) 797 798lemma norm_cis [simp]: "norm (cis a) = 1" 799 by (simp add: norm_complex_def) 800 801lemma sgn_cis [simp]: "sgn (cis a) = cis a" 802 by (simp add: sgn_div_norm) 803 804lemma cis_2pi [simp]: "cis (2 * pi) = 1" 805 by (simp add: cis.ctr complex_eq_iff) 806 807lemma cis_neq_zero [simp]: "cis a \<noteq> 0" 808 by (metis norm_cis norm_zero zero_neq_one) 809 810lemma cis_cnj: "cnj (cis t) = cis (-t)" 811 by (simp add: complex_eq_iff) 812 813lemma cis_mult: "cis a * cis b = cis (a + b)" 814 by (simp add: complex_eq_iff cos_add sin_add) 815 816lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" 817 by (induct n) (simp_all add: algebra_simps cis_mult) 818 819lemma cis_inverse [simp]: "inverse (cis a) = cis (- a)" 820 by (simp add: complex_eq_iff) 821 822lemma cis_divide: "cis a / cis b = cis (a - b)" 823 by (simp add: divide_complex_def cis_mult) 824 825lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re (cis a ^ n)" 826 by (auto simp add: DeMoivre) 827 828lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im (cis a ^ n)" 829 by (auto simp add: DeMoivre) 830 831lemma cis_pi [simp]: "cis pi = -1" 832 by (simp add: complex_eq_iff) 833 834lemma cis_pi_half[simp]: "cis (pi / 2) = \<i>" 835 by (simp add: cis.ctr complex_eq_iff) 836 837lemma cis_minus_pi_half[simp]: "cis (-(pi / 2)) = -\<i>" 838 by (simp add: cis.ctr complex_eq_iff) 839 840lemma cis_multiple_2pi[simp]: "n \<in> \<int> \<Longrightarrow> cis (2 * pi * n) = 1" 841 by (auto elim!: Ints_cases simp: cis.ctr one_complex.ctr) 842 843 844subsubsection \<open>$r(\cos \theta + i \sin \theta)$\<close> 845 846definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex" 847 where "rcis r a = complex_of_real r * cis a" 848 849lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" 850 by (simp add: rcis_def) 851 852lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" 853 by (simp add: rcis_def) 854 855lemma rcis_Ex: "\<exists>r a. z = rcis r a" 856 by (simp add: complex_eq_iff polar_Ex) 857 858lemma complex_mod_rcis [simp]: "cmod (rcis r a) = \<bar>r\<bar>" 859 by (simp add: rcis_def norm_mult) 860 861lemma cis_rcis_eq: "cis a = rcis 1 a" 862 by (simp add: rcis_def) 863 864lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1 * r2) (a + b)" 865 by (simp add: rcis_def cis_mult) 866 867lemma rcis_zero_mod [simp]: "rcis 0 a = 0" 868 by (simp add: rcis_def) 869 870lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" 871 by (simp add: rcis_def) 872 873lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0" 874 by (simp add: rcis_def) 875 876lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" 877 by (simp add: rcis_def power_mult_distrib DeMoivre) 878 879lemma rcis_inverse: "inverse(rcis r a) = rcis (1 / r) (- a)" 880 by (simp add: divide_inverse rcis_def) 881 882lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1 / r2) (a - b)" 883 by (simp add: rcis_def cis_divide [symmetric]) 884 885 886subsubsection \<open>Complex exponential\<close> 887 888lemma exp_Reals_eq: 889 assumes "z \<in> \<real>" 890 shows "exp z = of_real (exp (Re z))" 891 using assms by (auto elim!: Reals_cases simp: exp_of_real) 892 893lemma cis_conv_exp: "cis b = exp (\<i> * b)" 894proof - 895 have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n = 896 of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)" 897 for n :: nat 898 proof - 899 have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)" 900 by (induct n) 901 (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps 902 power2_eq_square add_nonneg_eq_0_iff) 903 then show ?thesis 904 by (simp add: field_simps) 905 qed 906 then show ?thesis 907 using sin_converges [of b] cos_converges [of b] 908 by (auto simp add: Complex_eq cis.ctr exp_def simp del: of_real_mult 909 intro!: sums_unique sums_add sums_mult sums_of_real) 910qed 911 912lemma exp_eq_polar: "exp z = exp (Re z) * cis (Im z)" 913 unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp 914 by (cases z) (simp add: Complex_eq) 915 916lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)" 917 unfolding exp_eq_polar by simp 918 919lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)" 920 unfolding exp_eq_polar by simp 921 922lemma norm_cos_sin [simp]: "norm (Complex (cos t) (sin t)) = 1" 923 by (simp add: norm_complex_def) 924 925lemma norm_exp_eq_Re [simp]: "norm (exp z) = exp (Re z)" 926 by (simp add: cis.code cmod_complex_polar exp_eq_polar Complex_eq) 927 928lemma complex_exp_exists: "\<exists>a r. z = complex_of_real r * exp a" 929 apply (insert rcis_Ex [of z]) 930 apply (auto simp add: exp_eq_polar rcis_def mult.assoc [symmetric]) 931 apply (rule_tac x = "\<i> * complex_of_real a" in exI) 932 apply auto 933 done 934 935lemma exp_pi_i [simp]: "exp (of_real pi * \<i>) = -1" 936 by (metis cis_conv_exp cis_pi mult.commute) 937 938lemma exp_pi_i' [simp]: "exp (\<i> * of_real pi) = -1" 939 using cis_conv_exp cis_pi by auto 940 941lemma exp_two_pi_i [simp]: "exp (2 * of_real pi * \<i>) = 1" 942 by (simp add: exp_eq_polar complex_eq_iff) 943 944lemma exp_two_pi_i' [simp]: "exp (\<i> * (of_real pi * 2)) = 1" 945 by (metis exp_two_pi_i mult.commute) 946 947lemma continuous_on_cis [continuous_intros]: 948 "continuous_on A f \<Longrightarrow> continuous_on A (\<lambda>x. cis (f x))" 949 by (auto simp: cis_conv_exp intro!: continuous_intros) 950 951 952subsubsection \<open>Complex argument\<close> 953 954definition arg :: "complex \<Rightarrow> real" 955 where "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> - pi < a \<and> a \<le> pi))" 956 957lemma arg_zero: "arg 0 = 0" 958 by (simp add: arg_def) 959 960lemma arg_unique: 961 assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi" 962 shows "arg z = x" 963proof - 964 from assms have "z \<noteq> 0" by auto 965 have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x" 966 proof 967 fix a 968 define d where "d = a - x" 969 assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi" 970 from a assms have "- (2*pi) < d \<and> d < 2*pi" 971 unfolding d_def by simp 972 moreover 973 from a assms have "cos a = cos x" and "sin a = sin x" 974 by (simp_all add: complex_eq_iff) 975 then have cos: "cos d = 1" 976 by (simp add: d_def cos_diff) 977 moreover from cos have "sin d = 0" 978 by (rule cos_one_sin_zero) 979 ultimately have "d = 0" 980 by (auto simp: sin_zero_iff elim!: evenE dest!: less_2_cases) 981 then show "a = x" 982 by (simp add: d_def) 983 qed (simp add: assms del: Re_sgn Im_sgn) 984 with \<open>z \<noteq> 0\<close> show "arg z = x" 985 by (simp add: arg_def) 986qed 987 988lemma arg_correct: 989 assumes "z \<noteq> 0" 990 shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi" 991proof (simp add: arg_def assms, rule someI_ex) 992 obtain r a where z: "z = rcis r a" 993 using rcis_Ex by fast 994 with assms have "r \<noteq> 0" by auto 995 define b where "b = (if 0 < r then a else a + pi)" 996 have b: "sgn z = cis b" 997 using \<open>r \<noteq> 0\<close> by (simp add: z b_def rcis_def of_real_def sgn_scaleR sgn_if complex_eq_iff) 998 have cis_2pi_nat: "cis (2 * pi * real_of_nat n) = 1" for n 999 by (induct n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff) 1000 have cis_2pi_int: "cis (2 * pi * real_of_int x) = 1" for x 1001 by (cases x rule: int_diff_cases) 1002 (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat) 1003 define c where "c = b - 2 * pi * of_int \<lceil>(b - pi) / (2 * pi)\<rceil>" 1004 have "sgn z = cis c" 1005 by (simp add: b c_def cis_divide [symmetric] cis_2pi_int) 1006 moreover have "- pi < c \<and> c \<le> pi" 1007 using ceiling_correct [of "(b - pi) / (2*pi)"] 1008 by (simp add: c_def less_divide_eq divide_le_eq algebra_simps del: le_of_int_ceiling) 1009 ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" 1010 by fast 1011qed 1012 1013lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi" 1014 by (cases "z = 0") (simp_all add: arg_zero arg_correct) 1015 1016lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z" 1017 by (simp add: arg_correct) 1018 1019lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z" 1020 by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def) 1021 1022lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0" 1023 using cis_arg [of y] by (simp add: complex_eq_iff) 1024 1025subsection \<open>Complex n-th roots\<close> 1026 1027lemma bij_betw_roots_unity: 1028 assumes "n > 0" 1029 shows "bij_betw (\<lambda>k. cis (2 * pi * real k / real n)) {..<n} {z. z ^ n = 1}" 1030 (is "bij_betw ?f _ _") 1031 unfolding bij_betw_def 1032proof (intro conjI) 1033 show inj: "inj_on ?f {..<n}" unfolding inj_on_def 1034 proof (safe, goal_cases) 1035 case (1 k l) 1036 hence kl: "k < n" "l < n" by simp_all 1037 from 1 have "1 = ?f k / ?f l" by simp 1038 also have "\<dots> = cis (2*pi*(real k - real l)/n)" 1039 using assms by (simp add: field_simps cis_divide) 1040 finally have "cos (2*pi*(real k - real l) / n) = 1" 1041 by (simp add: complex_eq_iff) 1042 then obtain m :: int where "2 * pi * (real k - real l) / real n = real_of_int m * 2 * pi" 1043 by (subst (asm) cos_one_2pi_int) blast 1044 hence "real_of_int (int k - int l) = real_of_int (m * int n)" 1045 unfolding of_int_diff of_int_mult using assms 1046 by (simp add: nonzero_divide_eq_eq) 1047 also note of_int_eq_iff 1048 finally have *: "abs m * n = abs (int k - int l)" by (simp add: abs_mult) 1049 also have "\<dots> < int n" using kl by linarith 1050 finally have "m = 0" using assms by simp 1051 with * show "k = l" by simp 1052 qed 1053 1054 have subset: "?f ` {..<n} \<subseteq> {z. z ^ n = 1}" 1055 proof safe 1056 fix k :: nat 1057 have "cis (2 * pi * real k / real n) ^ n = cis (2 * pi) ^ k" 1058 using assms by (simp add: DeMoivre mult_ac) 1059 also have "cis (2 * pi) = 1" by (simp add: complex_eq_iff) 1060 finally show "?f k ^ n = 1" by simp 1061 qed 1062 1063 have "n = card {..<n}" by simp 1064 also from assms and subset have "\<dots> \<le> card {z::complex. z ^ n = 1}" 1065 by (intro card_inj_on_le[OF inj]) (auto simp: finite_roots_unity) 1066 finally have card: "card {z::complex. z ^ n = 1} = n" 1067 using assms by (intro antisym card_roots_unity) auto 1068 1069 have "card (?f ` {..<n}) = card {z::complex. z ^ n = 1}" 1070 using card inj by (subst card_image) auto 1071 with subset and assms show "?f ` {..<n} = {z::complex. z ^ n = 1}" 1072 by (intro card_subset_eq finite_roots_unity) auto 1073qed 1074 1075lemma card_roots_unity_eq: 1076 assumes "n > 0" 1077 shows "card {z::complex. z ^ n = 1} = n" 1078 using bij_betw_same_card [OF bij_betw_roots_unity [OF assms]] by simp 1079 1080lemma bij_betw_nth_root_unity: 1081 fixes c :: complex and n :: nat 1082 assumes c: "c \<noteq> 0" and n: "n > 0" 1083 defines "c' \<equiv> root n (norm c) * cis (arg c / n)" 1084 shows "bij_betw (\<lambda>z. c' * z) {z. z ^ n = 1} {z. z ^ n = c}" 1085proof - 1086 have "c' ^ n = of_real (root n (norm c) ^ n) * cis (arg c)" 1087 unfolding of_real_power using n by (simp add: c'_def power_mult_distrib DeMoivre) 1088 also from n have "root n (norm c) ^ n = norm c" by simp 1089 also from c have "of_real \<dots> * cis (arg c) = c" by (simp add: cis_arg Complex.sgn_eq) 1090 finally have [simp]: "c' ^ n = c" . 1091 1092 show ?thesis unfolding bij_betw_def inj_on_def 1093 proof safe 1094 fix z :: complex assume "z ^ n = 1" 1095 hence "(c' * z) ^ n = c' ^ n" by (simp add: power_mult_distrib) 1096 also have "c' ^ n = of_real (root n (norm c) ^ n) * cis (arg c)" 1097 unfolding of_real_power using n by (simp add: c'_def power_mult_distrib DeMoivre) 1098 also from n have "root n (norm c) ^ n = norm c" by simp 1099 also from c have "\<dots> * cis (arg c) = c" by (simp add: cis_arg Complex.sgn_eq) 1100 finally show "(c' * z) ^ n = c" . 1101 next 1102 fix z assume z: "c = z ^ n" 1103 define z' where "z' = z / c'" 1104 from c and n have "c' \<noteq> 0" by (auto simp: c'_def) 1105 with n c have "z = c' * z'" and "z' ^ n = 1" 1106 by (auto simp: z'_def power_divide z) 1107 thus "z \<in> (\<lambda>z. c' * z) ` {z. z ^ n = 1}" by blast 1108 qed (insert c n, auto simp: c'_def) 1109qed 1110 1111lemma finite_nth_roots [intro]: 1112 assumes "n > 0" 1113 shows "finite {z::complex. z ^ n = c}" 1114proof (cases "c = 0") 1115 case True 1116 with assms have "{z::complex. z ^ n = c} = {0}" by auto 1117 thus ?thesis by simp 1118next 1119 case False 1120 from assms have "finite {z::complex. z ^ n = 1}" by (intro finite_roots_unity) simp_all 1121 also have "?this \<longleftrightarrow> ?thesis" 1122 by (rule bij_betw_finite, rule bij_betw_nth_root_unity) fact+ 1123 finally show ?thesis . 1124qed 1125 1126lemma card_nth_roots: 1127 assumes "c \<noteq> 0" "n > 0" 1128 shows "card {z::complex. z ^ n = c} = n" 1129proof - 1130 have "card {z. z ^ n = c} = card {z::complex. z ^ n = 1}" 1131 by (rule sym, rule bij_betw_same_card, rule bij_betw_nth_root_unity) fact+ 1132 also have "\<dots> = n" by (rule card_roots_unity_eq) fact+ 1133 finally show ?thesis . 1134qed 1135 1136lemma sum_roots_unity: 1137 assumes "n > 1" 1138 shows "\<Sum>{z::complex. z ^ n = 1} = 0" 1139proof - 1140 define \<omega> where "\<omega> = cis (2 * pi / real n)" 1141 have [simp]: "\<omega> \<noteq> 1" 1142 proof 1143 assume "\<omega> = 1" 1144 with assms obtain k :: int where "2 * pi / real n = 2 * pi * of_int k" 1145 by (auto simp: \<omega>_def complex_eq_iff cos_one_2pi_int) 1146 with assms have "real n * of_int k = 1" by (simp add: field_simps) 1147 also have "real n * of_int k = of_int (int n * k)" by simp 1148 also have "1 = (of_int 1 :: real)" by simp 1149 also note of_int_eq_iff 1150 finally show False using assms by (auto simp: zmult_eq_1_iff) 1151 qed 1152 1153 have "(\<Sum>z | z ^ n = 1. z :: complex) = (\<Sum>k<n. cis (2 * pi * real k / real n))" 1154 using assms by (intro sum.reindex_bij_betw [symmetric] bij_betw_roots_unity) auto 1155 also have "\<dots> = (\<Sum>k<n. \<omega> ^ k)" 1156 by (intro sum.cong refl) (auto simp: \<omega>_def DeMoivre mult_ac) 1157 also have "\<dots> = (\<omega> ^ n - 1) / (\<omega> - 1)" 1158 by (subst geometric_sum) auto 1159 also have "\<omega> ^ n - 1 = cis (2 * pi) - 1" using assms by (auto simp: \<omega>_def DeMoivre) 1160 also have "\<dots> = 0" by (simp add: complex_eq_iff) 1161 finally show ?thesis by simp 1162qed 1163 1164lemma sum_nth_roots: 1165 assumes "n > 1" 1166 shows "\<Sum>{z::complex. z ^ n = c} = 0" 1167proof (cases "c = 0") 1168 case True 1169 with assms have "{z::complex. z ^ n = c} = {0}" by auto 1170 also have "\<Sum>\<dots> = 0" by simp 1171 finally show ?thesis . 1172next 1173 case False 1174 define c' where "c' = root n (norm c) * cis (arg c / n)" 1175 from False and assms have "(\<Sum>{z. z ^ n = c}) = (\<Sum>z | z ^ n = 1. c' * z)" 1176 by (subst sum.reindex_bij_betw [OF bij_betw_nth_root_unity, symmetric]) 1177 (auto simp: sum_distrib_left finite_roots_unity c'_def) 1178 also from assms have "\<dots> = 0" 1179 by (simp add: sum_distrib_left [symmetric] sum_roots_unity) 1180 finally show ?thesis . 1181qed 1182 1183subsection \<open>Square root of complex numbers\<close> 1184 1185primcorec csqrt :: "complex \<Rightarrow> complex" 1186 where 1187 "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)" 1188 | "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)" 1189 1190lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)" 1191 by (simp add: complex_eq_iff norm_complex_def) 1192 1193lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>" 1194 by (simp add: complex_eq_iff norm_complex_def) 1195 1196lemma of_real_sqrt: "x \<ge> 0 \<Longrightarrow> of_real (sqrt x) = csqrt (of_real x)" 1197 by (simp add: complex_eq_iff norm_complex_def) 1198 1199lemma csqrt_0 [simp]: "csqrt 0 = 0" 1200 by simp 1201 1202lemma csqrt_1 [simp]: "csqrt 1 = 1" 1203 by simp 1204 1205lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2" 1206 by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt) 1207 1208lemma power2_csqrt[simp,algebra]: "(csqrt z)\<^sup>2 = z" 1209proof (cases "Im z = 0") 1210 case True 1211 then show ?thesis 1212 using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"] 1213 by (cases "0::real" "Re z" rule: linorder_cases) 1214 (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re) 1215next 1216 case False 1217 moreover have "cmod z * cmod z - Re z * Re z = Im z * Im z" 1218 by (simp add: norm_complex_def power2_eq_square) 1219 moreover have "\<bar>Re z\<bar> \<le> cmod z" 1220 by (simp add: norm_complex_def) 1221 ultimately show ?thesis 1222 by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq 1223 field_simps real_sqrt_mult[symmetric] real_sqrt_divide) 1224qed 1225 1226lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0" 1227 by auto (metis power2_csqrt power_eq_0_iff) 1228 1229lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1" 1230 by auto (metis power2_csqrt power2_eq_1_iff) 1231 1232lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)" 1233 by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0) 1234 1235lemma Re_csqrt: "0 \<le> Re (csqrt z)" 1236 by (metis csqrt_principal le_less) 1237 1238lemma csqrt_square: 1239 assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)" 1240 shows "csqrt (b^2) = b" 1241proof - 1242 have "csqrt (b^2) = b \<or> csqrt (b^2) = - b" 1243 by (simp add: power2_eq_iff[symmetric]) 1244 moreover have "csqrt (b^2) \<noteq> -b \<or> b = 0" 1245 using csqrt_principal[of "b ^ 2"] assms 1246 by (intro disjCI notI) (auto simp: complex_eq_iff) 1247 ultimately show ?thesis 1248 by auto 1249qed 1250 1251lemma csqrt_unique: "w\<^sup>2 = z \<Longrightarrow> 0 < Re w \<or> Re w = 0 \<and> 0 \<le> Im w \<Longrightarrow> csqrt z = w" 1252 by (auto simp: csqrt_square) 1253 1254lemma csqrt_minus [simp]: 1255 assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)" 1256 shows "csqrt (- x) = \<i> * csqrt x" 1257proof - 1258 have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x" 1259 proof (rule csqrt_square) 1260 have "Im (csqrt x) \<le> 0" 1261 using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod) 1262 then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)" 1263 by (auto simp add: Re_csqrt simp del: csqrt.simps) 1264 qed 1265 also have "(\<i> * csqrt x)^2 = - x" 1266 by (simp add: power_mult_distrib) 1267 finally show ?thesis . 1268qed 1269 1270 1271text \<open>Legacy theorem names\<close> 1272 1273lemmas cmod_def = norm_complex_def 1274 1275lemma legacy_Complex_simps: 1276 shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" 1277 and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)" 1278 and complex_minus: "- (Complex a b) = Complex (- a) (- b)" 1279 and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)" 1280 and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0" 1281 and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0" 1282 and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)" 1283 and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))" 1284 and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0" 1285 and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0" 1286 and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)" 1287 and Complex_eq_i: "Complex x y = \<i> \<longleftrightarrow> x = 0 \<and> y = 1" 1288 and i_mult_Complex: "\<i> * Complex a b = Complex (- b) a" 1289 and Complex_mult_i: "Complex a b * \<i> = Complex (- b) a" 1290 and i_complex_of_real: "\<i> * complex_of_real r = Complex 0 r" 1291 and complex_of_real_i: "complex_of_real r * \<i> = Complex 0 r" 1292 and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y" 1293 and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y" 1294 and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)" 1295 and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)" 1296 and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa \<and> y = 0)" 1297 and complex_cnj: "cnj (Complex a b) = Complex a (- b)" 1298 and Complex_sum': "sum (\<lambda>x. Complex (f x) 0) s = Complex (sum f s) 0" 1299 and Complex_sum: "Complex (sum f s) 0 = sum (\<lambda>x. Complex (f x) 0) s" 1300 and complex_of_real_def: "complex_of_real r = Complex r 0" 1301 and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)" 1302 by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide) 1303 1304lemma Complex_in_Reals: "Complex x 0 \<in> \<real>" 1305 by (metis Reals_of_real complex_of_real_def) 1306 1307end 1308