src/HOL/Complex.thy
 author wenzelm Tue Sep 03 01:12:40 2013 +0200 (2013-09-03) changeset 53374 a14d2a854c02 parent 53015 a1119cf551e8 child 54230 b1d955791529 permissions -rw-r--r--
tuned proofs -- clarified flow of facts wrt. calculation;
     1 (*  Title:       HOL/Complex.thy

     2     Author:      Jacques D. Fleuriot

     3     Copyright:   2001 University of Edinburgh

     4     Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4

     5 *)

     6

     7 header {* Complex Numbers: Rectangular and Polar Representations *}

     8

     9 theory Complex

    10 imports Transcendental

    11 begin

    12

    13 datatype complex = Complex real real

    14

    15 primrec Re :: "complex \<Rightarrow> real"

    16   where Re: "Re (Complex x y) = x"

    17

    18 primrec Im :: "complex \<Rightarrow> real"

    19   where Im: "Im (Complex x y) = y"

    20

    21 lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"

    22   by (induct z) simp

    23

    24 lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"

    25   by (induct x, induct y) simp

    26

    27 lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"

    28   by (induct x, induct y) simp

    29

    30

    31 subsection {* Addition and Subtraction *}

    32

    33 instantiation complex :: ab_group_add

    34 begin

    35

    36 definition complex_zero_def:

    37   "0 = Complex 0 0"

    38

    39 definition complex_add_def:

    40   "x + y = Complex (Re x + Re y) (Im x + Im y)"

    41

    42 definition complex_minus_def:

    43   "- x = Complex (- Re x) (- Im x)"

    44

    45 definition complex_diff_def:

    46   "x - (y\<Colon>complex) = x + - y"

    47

    48 lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"

    49   by (simp add: complex_zero_def)

    50

    51 lemma complex_Re_zero [simp]: "Re 0 = 0"

    52   by (simp add: complex_zero_def)

    53

    54 lemma complex_Im_zero [simp]: "Im 0 = 0"

    55   by (simp add: complex_zero_def)

    56

    57 lemma complex_add [simp]:

    58   "Complex a b + Complex c d = Complex (a + c) (b + d)"

    59   by (simp add: complex_add_def)

    60

    61 lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y"

    62   by (simp add: complex_add_def)

    63

    64 lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y"

    65   by (simp add: complex_add_def)

    66

    67 lemma complex_minus [simp]:

    68   "- (Complex a b) = Complex (- a) (- b)"

    69   by (simp add: complex_minus_def)

    70

    71 lemma complex_Re_minus [simp]: "Re (- x) = - Re x"

    72   by (simp add: complex_minus_def)

    73

    74 lemma complex_Im_minus [simp]: "Im (- x) = - Im x"

    75   by (simp add: complex_minus_def)

    76

    77 lemma complex_diff [simp]:

    78   "Complex a b - Complex c d = Complex (a - c) (b - d)"

    79   by (simp add: complex_diff_def)

    80

    81 lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y"

    82   by (simp add: complex_diff_def)

    83

    84 lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y"

    85   by (simp add: complex_diff_def)

    86

    87 instance

    88   by intro_classes (simp_all add: complex_add_def complex_diff_def)

    89

    90 end

    91

    92

    93 subsection {* Multiplication and Division *}

    94

    95 instantiation complex :: field_inverse_zero

    96 begin

    97

    98 definition complex_one_def:

    99   "1 = Complex 1 0"

   100

   101 definition complex_mult_def:

   102   "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"

   103

   104 definition complex_inverse_def:

   105   "inverse x =

   106     Complex (Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)) (- Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2))"

   107

   108 definition complex_divide_def:

   109   "x / (y\<Colon>complex) = x * inverse y"

   110

   111 lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"

   112   by (simp add: complex_one_def)

   113

   114 lemma complex_Re_one [simp]: "Re 1 = 1"

   115   by (simp add: complex_one_def)

   116

   117 lemma complex_Im_one [simp]: "Im 1 = 0"

   118   by (simp add: complex_one_def)

   119

   120 lemma complex_mult [simp]:

   121   "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"

   122   by (simp add: complex_mult_def)

   123

   124 lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"

   125   by (simp add: complex_mult_def)

   126

   127 lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"

   128   by (simp add: complex_mult_def)

   129

   130 lemma complex_inverse [simp]:

   131   "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"

   132   by (simp add: complex_inverse_def)

   133

   134 lemma complex_Re_inverse:

   135   "Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"

   136   by (simp add: complex_inverse_def)

   137

   138 lemma complex_Im_inverse:

   139   "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"

   140   by (simp add: complex_inverse_def)

   141

   142 instance

   143   by intro_classes (simp_all add: complex_mult_def

   144     distrib_left distrib_right right_diff_distrib left_diff_distrib

   145     complex_inverse_def complex_divide_def

   146     power2_eq_square add_divide_distrib [symmetric]

   147     complex_eq_iff)

   148

   149 end

   150

   151

   152 subsection {* Numerals and Arithmetic *}

   153

   154 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"

   155   by (induct n) simp_all

   156

   157 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"

   158   by (induct n) simp_all

   159

   160 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"

   161   by (cases z rule: int_diff_cases) simp

   162

   163 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"

   164   by (cases z rule: int_diff_cases) simp

   165

   166 lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"

   167   using complex_Re_of_int [of "numeral v"] by simp

   168

   169 lemma complex_Re_neg_numeral [simp]: "Re (neg_numeral v) = neg_numeral v"

   170   using complex_Re_of_int [of "neg_numeral v"] by simp

   171

   172 lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"

   173   using complex_Im_of_int [of "numeral v"] by simp

   174

   175 lemma complex_Im_neg_numeral [simp]: "Im (neg_numeral v) = 0"

   176   using complex_Im_of_int [of "neg_numeral v"] by simp

   177

   178 lemma Complex_eq_numeral [simp]:

   179   "(Complex a b = numeral w) = (a = numeral w \<and> b = 0)"

   180   by (simp add: complex_eq_iff)

   181

   182 lemma Complex_eq_neg_numeral [simp]:

   183   "(Complex a b = neg_numeral w) = (a = neg_numeral w \<and> b = 0)"

   184   by (simp add: complex_eq_iff)

   185

   186

   187 subsection {* Scalar Multiplication *}

   188

   189 instantiation complex :: real_field

   190 begin

   191

   192 definition complex_scaleR_def:

   193   "scaleR r x = Complex (r * Re x) (r * Im x)"

   194

   195 lemma complex_scaleR [simp]:

   196   "scaleR r (Complex a b) = Complex (r * a) (r * b)"

   197   unfolding complex_scaleR_def by simp

   198

   199 lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"

   200   unfolding complex_scaleR_def by simp

   201

   202 lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"

   203   unfolding complex_scaleR_def by simp

   204

   205 instance

   206 proof

   207   fix a b :: real and x y :: complex

   208   show "scaleR a (x + y) = scaleR a x + scaleR a y"

   209     by (simp add: complex_eq_iff distrib_left)

   210   show "scaleR (a + b) x = scaleR a x + scaleR b x"

   211     by (simp add: complex_eq_iff distrib_right)

   212   show "scaleR a (scaleR b x) = scaleR (a * b) x"

   213     by (simp add: complex_eq_iff mult_assoc)

   214   show "scaleR 1 x = x"

   215     by (simp add: complex_eq_iff)

   216   show "scaleR a x * y = scaleR a (x * y)"

   217     by (simp add: complex_eq_iff algebra_simps)

   218   show "x * scaleR a y = scaleR a (x * y)"

   219     by (simp add: complex_eq_iff algebra_simps)

   220 qed

   221

   222 end

   223

   224

   225 subsection{* Properties of Embedding from Reals *}

   226

   227 abbreviation complex_of_real :: "real \<Rightarrow> complex"

   228   where "complex_of_real \<equiv> of_real"

   229

   230 lemma complex_of_real_def: "complex_of_real r = Complex r 0"

   231   by (simp add: of_real_def complex_scaleR_def)

   232

   233 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"

   234   by (simp add: complex_of_real_def)

   235

   236 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"

   237   by (simp add: complex_of_real_def)

   238

   239 lemma Complex_add_complex_of_real [simp]:

   240   shows "Complex x y + complex_of_real r = Complex (x+r) y"

   241   by (simp add: complex_of_real_def)

   242

   243 lemma complex_of_real_add_Complex [simp]:

   244   shows "complex_of_real r + Complex x y = Complex (r+x) y"

   245   by (simp add: complex_of_real_def)

   246

   247 lemma Complex_mult_complex_of_real:

   248   shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"

   249   by (simp add: complex_of_real_def)

   250

   251 lemma complex_of_real_mult_Complex:

   252   shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"

   253   by (simp add: complex_of_real_def)

   254

   255 lemma complex_eq_cancel_iff2 [simp]:

   256   shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"

   257   by (simp add: complex_of_real_def)

   258

   259 lemma complex_split_polar:

   260      "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"

   261   by (simp add: complex_eq_iff polar_Ex)

   262

   263

   264 subsection {* Vector Norm *}

   265

   266 instantiation complex :: real_normed_field

   267 begin

   268

   269 definition complex_norm_def:

   270   "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"

   271

   272 abbreviation cmod :: "complex \<Rightarrow> real"

   273   where "cmod \<equiv> norm"

   274

   275 definition complex_sgn_def:

   276   "sgn x = x /\<^sub>R cmod x"

   277

   278 definition dist_complex_def:

   279   "dist x y = cmod (x - y)"

   280

   281 definition open_complex_def:

   282   "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"

   283

   284 lemmas cmod_def = complex_norm_def

   285

   286 lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"

   287   by (simp add: complex_norm_def)

   288

   289 instance proof

   290   fix r :: real and x y :: complex and S :: "complex set"

   291   show "(norm x = 0) = (x = 0)"

   292     by (induct x) simp

   293   show "norm (x + y) \<le> norm x + norm y"

   294     by (induct x, induct y)

   295        (simp add: real_sqrt_sum_squares_triangle_ineq)

   296   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"

   297     by (induct x)

   298        (simp add: power_mult_distrib distrib_left [symmetric] real_sqrt_mult)

   299   show "norm (x * y) = norm x * norm y"

   300     by (induct x, induct y)

   301        (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)

   302   show "sgn x = x /\<^sub>R cmod x"

   303     by (rule complex_sgn_def)

   304   show "dist x y = cmod (x - y)"

   305     by (rule dist_complex_def)

   306   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"

   307     by (rule open_complex_def)

   308 qed

   309

   310 end

   311

   312 lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1"

   313   by simp

   314

   315 lemma cmod_complex_polar:

   316   "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"

   317   by (simp add: norm_mult)

   318

   319 lemma complex_Re_le_cmod: "Re x \<le> cmod x"

   320   unfolding complex_norm_def

   321   by (rule real_sqrt_sum_squares_ge1)

   322

   323 lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"

   324   by (rule order_trans [OF _ norm_ge_zero], simp)

   325

   326 lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \<le> cmod a"

   327   by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)

   328

   329 lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"

   330   by (cases x) simp

   331

   332 lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"

   333   by (cases x) simp

   334

   335 text {* Properties of complex signum. *}

   336

   337 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"

   338   by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)

   339

   340 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"

   341   by (simp add: complex_sgn_def divide_inverse)

   342

   343 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"

   344   by (simp add: complex_sgn_def divide_inverse)

   345

   346

   347 subsection {* Completeness of the Complexes *}

   348

   349 lemma bounded_linear_Re: "bounded_linear Re"

   350   by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)

   351

   352 lemma bounded_linear_Im: "bounded_linear Im"

   353   by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)

   354

   355 lemmas tendsto_Re [tendsto_intros] =

   356   bounded_linear.tendsto [OF bounded_linear_Re]

   357

   358 lemmas tendsto_Im [tendsto_intros] =

   359   bounded_linear.tendsto [OF bounded_linear_Im]

   360

   361 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]

   362 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]

   363 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]

   364 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]

   365

   366 lemma tendsto_Complex [tendsto_intros]:

   367   assumes "(f ---> a) F" and "(g ---> b) F"

   368   shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"

   369 proof (rule tendstoI)

   370   fix r :: real assume "0 < r"

   371   hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)

   372   have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F"

   373     using (f ---> a) F and 0 < r / sqrt 2 by (rule tendstoD)

   374   moreover

   375   have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F"

   376     using (g ---> b) F and 0 < r / sqrt 2 by (rule tendstoD)

   377   ultimately

   378   show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F"

   379     by (rule eventually_elim2)

   380        (simp add: dist_norm real_sqrt_sum_squares_less)

   381 qed

   382

   383 instance complex :: banach

   384 proof

   385   fix X :: "nat \<Rightarrow> complex"

   386   assume X: "Cauchy X"

   387   from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"

   388     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)

   389   from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"

   390     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)

   391   have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"

   392     using tendsto_Complex [OF 1 2] by simp

   393   thus "convergent X"

   394     by (rule convergentI)

   395 qed

   396

   397

   398 subsection {* The Complex Number $i$ *}

   399

   400 definition "ii" :: complex  ("\<i>")

   401   where i_def: "ii \<equiv> Complex 0 1"

   402

   403 lemma complex_Re_i [simp]: "Re ii = 0"

   404   by (simp add: i_def)

   405

   406 lemma complex_Im_i [simp]: "Im ii = 1"

   407   by (simp add: i_def)

   408

   409 lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"

   410   by (simp add: i_def)

   411

   412 lemma norm_ii [simp]: "norm ii = 1"

   413   by (simp add: i_def)

   414

   415 lemma complex_i_not_zero [simp]: "ii \<noteq> 0"

   416   by (simp add: complex_eq_iff)

   417

   418 lemma complex_i_not_one [simp]: "ii \<noteq> 1"

   419   by (simp add: complex_eq_iff)

   420

   421 lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"

   422   by (simp add: complex_eq_iff)

   423

   424 lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> neg_numeral w"

   425   by (simp add: complex_eq_iff)

   426

   427 lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"

   428   by (simp add: complex_eq_iff)

   429

   430 lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"

   431   by (simp add: complex_eq_iff)

   432

   433 lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"

   434   by (simp add: i_def complex_of_real_def)

   435

   436 lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"

   437   by (simp add: i_def complex_of_real_def)

   438

   439 lemma i_squared [simp]: "ii * ii = -1"

   440   by (simp add: i_def)

   441

   442 lemma power2_i [simp]: "ii\<^sup>2 = -1"

   443   by (simp add: power2_eq_square)

   444

   445 lemma inverse_i [simp]: "inverse ii = - ii"

   446   by (rule inverse_unique, simp)

   447

   448 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"

   449   by (simp add: mult_assoc [symmetric])

   450

   451

   452 subsection {* Complex Conjugation *}

   453

   454 definition cnj :: "complex \<Rightarrow> complex" where

   455   "cnj z = Complex (Re z) (- Im z)"

   456

   457 lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"

   458   by (simp add: cnj_def)

   459

   460 lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"

   461   by (simp add: cnj_def)

   462

   463 lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"

   464   by (simp add: cnj_def)

   465

   466 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"

   467   by (simp add: complex_eq_iff)

   468

   469 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"

   470   by (simp add: cnj_def)

   471

   472 lemma complex_cnj_zero [simp]: "cnj 0 = 0"

   473   by (simp add: complex_eq_iff)

   474

   475 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"

   476   by (simp add: complex_eq_iff)

   477

   478 lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"

   479   by (simp add: complex_eq_iff)

   480

   481 lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"

   482   by (simp add: complex_eq_iff)

   483

   484 lemma complex_cnj_minus: "cnj (- x) = - cnj x"

   485   by (simp add: complex_eq_iff)

   486

   487 lemma complex_cnj_one [simp]: "cnj 1 = 1"

   488   by (simp add: complex_eq_iff)

   489

   490 lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"

   491   by (simp add: complex_eq_iff)

   492

   493 lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"

   494   by (simp add: complex_inverse_def)

   495

   496 lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"

   497   by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)

   498

   499 lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"

   500   by (induct n, simp_all add: complex_cnj_mult)

   501

   502 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"

   503   by (simp add: complex_eq_iff)

   504

   505 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"

   506   by (simp add: complex_eq_iff)

   507

   508 lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"

   509   by (simp add: complex_eq_iff)

   510

   511 lemma complex_cnj_neg_numeral [simp]: "cnj (neg_numeral w) = neg_numeral w"

   512   by (simp add: complex_eq_iff)

   513

   514 lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"

   515   by (simp add: complex_eq_iff)

   516

   517 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"

   518   by (simp add: complex_norm_def)

   519

   520 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"

   521   by (simp add: complex_eq_iff)

   522

   523 lemma complex_cnj_i [simp]: "cnj ii = - ii"

   524   by (simp add: complex_eq_iff)

   525

   526 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"

   527   by (simp add: complex_eq_iff)

   528

   529 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"

   530   by (simp add: complex_eq_iff)

   531

   532 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"

   533   by (simp add: complex_eq_iff power2_eq_square)

   534

   535 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"

   536   by (simp add: norm_mult power2_eq_square)

   537

   538 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"

   539   by (simp add: cmod_def power2_eq_square)

   540

   541 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"

   542   by simp

   543

   544 lemma bounded_linear_cnj: "bounded_linear cnj"

   545   using complex_cnj_add complex_cnj_scaleR

   546   by (rule bounded_linear_intro [where K=1], simp)

   547

   548 lemmas tendsto_cnj [tendsto_intros] =

   549   bounded_linear.tendsto [OF bounded_linear_cnj]

   550

   551 lemmas isCont_cnj [simp] =

   552   bounded_linear.isCont [OF bounded_linear_cnj]

   553

   554

   555 subsection{*Finally! Polar Form for Complex Numbers*}

   556

   557 subsubsection {* $\cos \theta + i \sin \theta$ *}

   558

   559 definition cis :: "real \<Rightarrow> complex" where

   560   "cis a = Complex (cos a) (sin a)"

   561

   562 lemma Re_cis [simp]: "Re (cis a) = cos a"

   563   by (simp add: cis_def)

   564

   565 lemma Im_cis [simp]: "Im (cis a) = sin a"

   566   by (simp add: cis_def)

   567

   568 lemma cis_zero [simp]: "cis 0 = 1"

   569   by (simp add: cis_def)

   570

   571 lemma norm_cis [simp]: "norm (cis a) = 1"

   572   by (simp add: cis_def)

   573

   574 lemma sgn_cis [simp]: "sgn (cis a) = cis a"

   575   by (simp add: sgn_div_norm)

   576

   577 lemma cis_neq_zero [simp]: "cis a \<noteq> 0"

   578   by (metis norm_cis norm_zero zero_neq_one)

   579

   580 lemma cis_mult: "cis a * cis b = cis (a + b)"

   581   by (simp add: cis_def cos_add sin_add)

   582

   583 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"

   584   by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)

   585

   586 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"

   587   by (simp add: cis_def)

   588

   589 lemma cis_divide: "cis a / cis b = cis (a - b)"

   590   by (simp add: complex_divide_def cis_mult diff_minus)

   591

   592 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"

   593   by (auto simp add: DeMoivre)

   594

   595 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"

   596   by (auto simp add: DeMoivre)

   597

   598 subsubsection {* $r(\cos \theta + i \sin \theta)$ *}

   599

   600 definition rcis :: "[real, real] \<Rightarrow> complex" where

   601   "rcis r a = complex_of_real r * cis a"

   602

   603 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"

   604   by (simp add: rcis_def)

   605

   606 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"

   607   by (simp add: rcis_def)

   608

   609 lemma rcis_Ex: "\<exists>r a. z = rcis r a"

   610   by (simp add: complex_eq_iff polar_Ex)

   611

   612 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"

   613   by (simp add: rcis_def norm_mult)

   614

   615 lemma cis_rcis_eq: "cis a = rcis 1 a"

   616   by (simp add: rcis_def)

   617

   618 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"

   619   by (simp add: rcis_def cis_mult)

   620

   621 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"

   622   by (simp add: rcis_def)

   623

   624 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"

   625   by (simp add: rcis_def)

   626

   627 lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"

   628   by (simp add: rcis_def)

   629

   630 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"

   631   by (simp add: rcis_def power_mult_distrib DeMoivre)

   632

   633 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"

   634   by (simp add: divide_inverse rcis_def)

   635

   636 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"

   637   by (simp add: rcis_def cis_divide [symmetric])

   638

   639 subsubsection {* Complex exponential *}

   640

   641 abbreviation expi :: "complex \<Rightarrow> complex"

   642   where "expi \<equiv> exp"

   643

   644 lemma cis_conv_exp: "cis b = exp (Complex 0 b)"

   645 proof (rule complex_eqI)

   646   { fix n have "Complex 0 b ^ n =

   647     real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"

   648       apply (induct n)

   649       apply (simp add: cos_coeff_def sin_coeff_def)

   650       apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)

   651       done } note * = this

   652   show "Re (cis b) = Re (exp (Complex 0 b))"

   653     unfolding exp_def cis_def cos_def

   654     by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],

   655       simp add: * mult_assoc [symmetric])

   656   show "Im (cis b) = Im (exp (Complex 0 b))"

   657     unfolding exp_def cis_def sin_def

   658     by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],

   659       simp add: * mult_assoc [symmetric])

   660 qed

   661

   662 lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"

   663   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp

   664

   665 lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"

   666   unfolding expi_def by simp

   667

   668 lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"

   669   unfolding expi_def by simp

   670

   671 lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"

   672 apply (insert rcis_Ex [of z])

   673 apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])

   674 apply (rule_tac x = "ii * complex_of_real a" in exI, auto)

   675 done

   676

   677 lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"

   678   by (simp add: expi_def cis_def)

   679

   680 subsubsection {* Complex argument *}

   681

   682 definition arg :: "complex \<Rightarrow> real" where

   683   "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"

   684

   685 lemma arg_zero: "arg 0 = 0"

   686   by (simp add: arg_def)

   687

   688 lemma of_nat_less_of_int_iff: (* TODO: move *)

   689   "(of_nat n :: 'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"

   690   by (metis of_int_of_nat_eq of_int_less_iff)

   691

   692 lemma real_of_nat_less_numeral_iff [simp]: (* TODO: move *)

   693   "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"

   694   using of_nat_less_of_int_iff [of n "numeral w", where 'a=real]

   695   by (simp add: real_of_nat_def zless_nat_eq_int_zless [symmetric])

   696

   697 lemma arg_unique:

   698   assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"

   699   shows "arg z = x"

   700 proof -

   701   from assms have "z \<noteq> 0" by auto

   702   have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"

   703   proof

   704     fix a def d \<equiv> "a - x"

   705     assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"

   706     from a assms have "- (2*pi) < d \<and> d < 2*pi"

   707       unfolding d_def by simp

   708     moreover from a assms have "cos a = cos x" and "sin a = sin x"

   709       by (simp_all add: complex_eq_iff)

   710     hence cos: "cos d = 1" unfolding d_def cos_diff by simp

   711     moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)

   712     ultimately have "d = 0"

   713       unfolding sin_zero_iff even_mult_two_ex

   714       by (auto simp add: numeral_2_eq_2 less_Suc_eq)

   715     thus "a = x" unfolding d_def by simp

   716   qed (simp add: assms del: Re_sgn Im_sgn)

   717   with z \<noteq> 0 show "arg z = x"

   718     unfolding arg_def by simp

   719 qed

   720

   721 lemma arg_correct:

   722   assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"

   723 proof (simp add: arg_def assms, rule someI_ex)

   724   obtain r a where z: "z = rcis r a" using rcis_Ex by fast

   725   with assms have "r \<noteq> 0" by auto

   726   def b \<equiv> "if 0 < r then a else a + pi"

   727   have b: "sgn z = cis b"

   728     unfolding z b_def rcis_def using r \<noteq> 0

   729     by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def)

   730   have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"

   731     by (induct_tac n, simp_all add: distrib_left cis_mult [symmetric],

   732       simp add: cis_def)

   733   have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"

   734     by (case_tac x rule: int_diff_cases,

   735       simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)

   736   def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"

   737   have "sgn z = cis c"

   738     unfolding b c_def

   739     by (simp add: cis_divide [symmetric] cis_2pi_int)

   740   moreover have "- pi < c \<and> c \<le> pi"

   741     using ceiling_correct [of "(b - pi) / (2*pi)"]

   742     by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)

   743   ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast

   744 qed

   745

   746 lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"

   747   by (cases "z = 0", simp_all add: arg_zero arg_correct)

   748

   749 lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"

   750   by (simp add: arg_correct)

   751

   752 lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"

   753   by (cases "z = 0", simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)

   754

   755 lemma cos_arg_i_mult_zero [simp]:

   756      "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"

   757   using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff)

   758

   759 text {* Legacy theorem names *}

   760

   761 lemmas expand_complex_eq = complex_eq_iff

   762 lemmas complex_Re_Im_cancel_iff = complex_eq_iff

   763 lemmas complex_equality = complex_eqI

   764

   765 end