src/HOL/Complex.thy
author paulson <lp15@cam.ac.uk>
Thu Mar 05 17:30:29 2015 +0000 (2015-03-05)
changeset 59613 7103019278f0
parent 59000 6eb0725503fc
child 59658 0cc388370041
permissions -rw-r--r--
The function frac. Various lemmas about limits, series, the exp function, etc.
wenzelm@41959
     1
(*  Title:       HOL/Complex.thy
paulson@13957
     2
    Author:      Jacques D. Fleuriot
paulson@13957
     3
    Copyright:   2001 University of Edinburgh
paulson@14387
     4
    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
paulson@13957
     5
*)
paulson@13957
     6
wenzelm@58889
     7
section {* Complex Numbers: Rectangular and Polar Representations *}
paulson@14373
     8
nipkow@15131
     9
theory Complex
haftmann@28952
    10
imports Transcendental
nipkow@15131
    11
begin
paulson@13957
    12
hoelzl@56889
    13
text {*
blanchet@58146
    14
We use the @{text codatatype} command to define the type of complex numbers. This allows us to use
blanchet@58146
    15
@{text primcorec} to define complex functions by defining their real and imaginary result
blanchet@58146
    16
separately.
hoelzl@56889
    17
*}
paulson@14373
    18
hoelzl@56889
    19
codatatype complex = Complex (Re: real) (Im: real)
hoelzl@56889
    20
hoelzl@56889
    21
lemma complex_surj: "Complex (Re z) (Im z) = z"
hoelzl@56889
    22
  by (rule complex.collapse)
paulson@13957
    23
huffman@44065
    24
lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"
hoelzl@56889
    25
  by (rule complex.expand) simp
huffman@23125
    26
huffman@44065
    27
lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"
hoelzl@56889
    28
  by (auto intro: complex.expand)
huffman@23125
    29
huffman@23125
    30
subsection {* Addition and Subtraction *}
huffman@23125
    31
haftmann@25599
    32
instantiation complex :: ab_group_add
haftmann@25571
    33
begin
haftmann@25571
    34
hoelzl@56889
    35
primcorec zero_complex where
hoelzl@56889
    36
  "Re 0 = 0"
hoelzl@56889
    37
| "Im 0 = 0"
haftmann@25571
    38
hoelzl@56889
    39
primcorec plus_complex where
hoelzl@56889
    40
  "Re (x + y) = Re x + Re y"
hoelzl@56889
    41
| "Im (x + y) = Im x + Im y"
haftmann@25712
    42
hoelzl@56889
    43
primcorec uminus_complex where
hoelzl@56889
    44
  "Re (- x) = - Re x"
hoelzl@56889
    45
| "Im (- x) = - Im x"
huffman@23125
    46
hoelzl@56889
    47
primcorec minus_complex where
hoelzl@56889
    48
  "Re (x - y) = Re x - Re y"
hoelzl@56889
    49
| "Im (x - y) = Im x - Im y"
huffman@23125
    50
haftmann@25712
    51
instance
hoelzl@56889
    52
  by intro_classes (simp_all add: complex_eq_iff)
haftmann@25712
    53
haftmann@25712
    54
end
haftmann@25712
    55
huffman@23125
    56
subsection {* Multiplication and Division *}
huffman@23125
    57
haftmann@36409
    58
instantiation complex :: field_inverse_zero
haftmann@25571
    59
begin
haftmann@25571
    60
hoelzl@56889
    61
primcorec one_complex where
hoelzl@56889
    62
  "Re 1 = 1"
hoelzl@56889
    63
| "Im 1 = 0"
paulson@14323
    64
hoelzl@56889
    65
primcorec times_complex where
hoelzl@56889
    66
  "Re (x * y) = Re x * Re y - Im x * Im y"
hoelzl@56889
    67
| "Im (x * y) = Re x * Im y + Im x * Re y"
paulson@14323
    68
hoelzl@56889
    69
primcorec inverse_complex where
hoelzl@56889
    70
  "Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"
hoelzl@56889
    71
| "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"
paulson@14335
    72
hoelzl@56889
    73
definition "x / (y\<Colon>complex) = x * inverse y"
paulson@14335
    74
haftmann@25712
    75
instance
lp15@59613
    76
  by intro_classes
hoelzl@56889
    77
     (simp_all add: complex_eq_iff divide_complex_def
hoelzl@56889
    78
      distrib_left distrib_right right_diff_distrib left_diff_distrib
hoelzl@56889
    79
      power2_eq_square add_divide_distrib [symmetric])
paulson@14335
    80
haftmann@25712
    81
end
huffman@23125
    82
hoelzl@56889
    83
lemma Re_divide: "Re (x / y) = (Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
hoelzl@56889
    84
  unfolding divide_complex_def by (simp add: add_divide_distrib)
huffman@23125
    85
hoelzl@56889
    86
lemma Im_divide: "Im (x / y) = (Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
hoelzl@56889
    87
  unfolding divide_complex_def times_complex.sel inverse_complex.sel
hoelzl@56889
    88
  by (simp_all add: divide_simps)
huffman@23125
    89
hoelzl@56889
    90
lemma Re_power2: "Re (x ^ 2) = (Re x)^2 - (Im x)^2"
hoelzl@56889
    91
  by (simp add: power2_eq_square)
huffman@20556
    92
hoelzl@56889
    93
lemma Im_power2: "Im (x ^ 2) = 2 * Re x * Im x"
hoelzl@56889
    94
  by (simp add: power2_eq_square)
hoelzl@56889
    95
hoelzl@56889
    96
lemma Re_power_real: "Im x = 0 \<Longrightarrow> Re (x ^ n) = Re x ^ n "
huffman@44724
    97
  by (induct n) simp_all
huffman@23125
    98
hoelzl@56889
    99
lemma Im_power_real: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0"
hoelzl@56889
   100
  by (induct n) simp_all
huffman@23125
   101
huffman@23125
   102
subsection {* Scalar Multiplication *}
huffman@20556
   103
haftmann@25712
   104
instantiation complex :: real_field
haftmann@25571
   105
begin
haftmann@25571
   106
hoelzl@56889
   107
primcorec scaleR_complex where
hoelzl@56889
   108
  "Re (scaleR r x) = r * Re x"
hoelzl@56889
   109
| "Im (scaleR r x) = r * Im x"
huffman@22972
   110
haftmann@25712
   111
instance
huffman@20556
   112
proof
huffman@23125
   113
  fix a b :: real and x y :: complex
huffman@23125
   114
  show "scaleR a (x + y) = scaleR a x + scaleR a y"
webertj@49962
   115
    by (simp add: complex_eq_iff distrib_left)
huffman@23125
   116
  show "scaleR (a + b) x = scaleR a x + scaleR b x"
webertj@49962
   117
    by (simp add: complex_eq_iff distrib_right)
huffman@23125
   118
  show "scaleR a (scaleR b x) = scaleR (a * b) x"
haftmann@57512
   119
    by (simp add: complex_eq_iff mult.assoc)
huffman@23125
   120
  show "scaleR 1 x = x"
huffman@44065
   121
    by (simp add: complex_eq_iff)
huffman@23125
   122
  show "scaleR a x * y = scaleR a (x * y)"
huffman@44065
   123
    by (simp add: complex_eq_iff algebra_simps)
huffman@23125
   124
  show "x * scaleR a y = scaleR a (x * y)"
huffman@44065
   125
    by (simp add: complex_eq_iff algebra_simps)
huffman@20556
   126
qed
huffman@20556
   127
haftmann@25712
   128
end
haftmann@25712
   129
hoelzl@56889
   130
subsection {* Numerals, Arithmetic, and Embedding from Reals *}
paulson@14323
   131
huffman@44724
   132
abbreviation complex_of_real :: "real \<Rightarrow> complex"
huffman@44724
   133
  where "complex_of_real \<equiv> of_real"
huffman@20557
   134
hoelzl@59000
   135
declare [[coercion "of_real :: real \<Rightarrow> complex"]]
hoelzl@59000
   136
declare [[coercion "of_rat :: rat \<Rightarrow> complex"]]
hoelzl@56889
   137
declare [[coercion "of_int :: int \<Rightarrow> complex"]]
hoelzl@56889
   138
declare [[coercion "of_nat :: nat \<Rightarrow> complex"]]
hoelzl@56331
   139
hoelzl@56889
   140
lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
hoelzl@56889
   141
  by (induct n) simp_all
hoelzl@56889
   142
hoelzl@56889
   143
lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
hoelzl@56889
   144
  by (induct n) simp_all
hoelzl@56889
   145
hoelzl@56889
   146
lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
hoelzl@56889
   147
  by (cases z rule: int_diff_cases) simp
hoelzl@56889
   148
hoelzl@56889
   149
lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
hoelzl@56889
   150
  by (cases z rule: int_diff_cases) simp
hoelzl@56889
   151
hoelzl@56889
   152
lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"
hoelzl@56889
   153
  using complex_Re_of_int [of "numeral v"] by simp
hoelzl@56889
   154
hoelzl@56889
   155
lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"
hoelzl@56889
   156
  using complex_Im_of_int [of "numeral v"] by simp
huffman@20557
   157
huffman@20557
   158
lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
hoelzl@56889
   159
  by (simp add: of_real_def)
huffman@20557
   160
huffman@20557
   161
lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
hoelzl@56889
   162
  by (simp add: of_real_def)
hoelzl@56889
   163
lp15@59613
   164
lemma Re_divide_numeral [simp]: "Re (z / numeral w) = Re z / numeral w"
lp15@59613
   165
  by (simp add: Re_divide sqr_conv_mult)
lp15@59613
   166
lp15@59613
   167
lemma Im_divide_numeral [simp]: "Im (z / numeral w) = Im z / numeral w"
lp15@59613
   168
  by (simp add: Im_divide sqr_conv_mult)
lp15@59613
   169
hoelzl@56889
   170
subsection {* The Complex Number $i$ *}
hoelzl@56889
   171
hoelzl@56889
   172
primcorec "ii" :: complex  ("\<i>") where
hoelzl@56889
   173
  "Re ii = 0"
hoelzl@56889
   174
| "Im ii = 1"
huffman@20557
   175
hoelzl@57259
   176
lemma Complex_eq[simp]: "Complex a b = a + \<i> * b"
hoelzl@57259
   177
  by (simp add: complex_eq_iff)
hoelzl@57259
   178
hoelzl@57259
   179
lemma complex_eq: "a = Re a + \<i> * Im a"
hoelzl@57259
   180
  by (simp add: complex_eq_iff)
hoelzl@57259
   181
hoelzl@57259
   182
lemma fun_complex_eq: "f = (\<lambda>x. Re (f x) + \<i> * Im (f x))"
hoelzl@57259
   183
  by (simp add: fun_eq_iff complex_eq)
hoelzl@57259
   184
hoelzl@56889
   185
lemma i_squared [simp]: "ii * ii = -1"
hoelzl@56889
   186
  by (simp add: complex_eq_iff)
hoelzl@56889
   187
hoelzl@56889
   188
lemma power2_i [simp]: "ii\<^sup>2 = -1"
hoelzl@56889
   189
  by (simp add: power2_eq_square)
paulson@14377
   190
hoelzl@56889
   191
lemma inverse_i [simp]: "inverse ii = - ii"
hoelzl@56889
   192
  by (rule inverse_unique) simp
hoelzl@56889
   193
hoelzl@56889
   194
lemma divide_i [simp]: "x / ii = - ii * x"
hoelzl@56889
   195
  by (simp add: divide_complex_def)
paulson@14377
   196
hoelzl@56889
   197
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
haftmann@57512
   198
  by (simp add: mult.assoc [symmetric])
paulson@14377
   199
hoelzl@56889
   200
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
hoelzl@56889
   201
  by (simp add: complex_eq_iff)
huffman@20557
   202
hoelzl@56889
   203
lemma complex_i_not_one [simp]: "ii \<noteq> 1"
hoelzl@56889
   204
  by (simp add: complex_eq_iff)
hoelzl@56889
   205
hoelzl@56889
   206
lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"
hoelzl@56889
   207
  by (simp add: complex_eq_iff)
huffman@44841
   208
hoelzl@56889
   209
lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> - numeral w"
hoelzl@56889
   210
  by (simp add: complex_eq_iff)
hoelzl@56889
   211
hoelzl@56889
   212
lemma complex_split_polar: "\<exists>r a. z = complex_of_real r * (cos a + \<i> * sin a)"
huffman@44827
   213
  by (simp add: complex_eq_iff polar_Ex)
huffman@44827
   214
lp15@59613
   215
lemma i_even_power [simp]: "\<i> ^ (n * 2) = (-1) ^ n"
lp15@59613
   216
  by (metis mult.commute power2_i power_mult)
lp15@59613
   217
huffman@23125
   218
subsection {* Vector Norm *}
paulson@14323
   219
haftmann@25712
   220
instantiation complex :: real_normed_field
haftmann@25571
   221
begin
haftmann@25571
   222
hoelzl@56889
   223
definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
haftmann@25571
   224
huffman@44724
   225
abbreviation cmod :: "complex \<Rightarrow> real"
huffman@44724
   226
  where "cmod \<equiv> norm"
haftmann@25571
   227
huffman@31413
   228
definition complex_sgn_def:
huffman@31413
   229
  "sgn x = x /\<^sub>R cmod x"
haftmann@25571
   230
huffman@31413
   231
definition dist_complex_def:
huffman@31413
   232
  "dist x y = cmod (x - y)"
huffman@31413
   233
haftmann@37767
   234
definition open_complex_def:
huffman@31492
   235
  "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31292
   236
huffman@31413
   237
instance proof
huffman@31492
   238
  fix r :: real and x y :: complex and S :: "complex set"
huffman@23125
   239
  show "(norm x = 0) = (x = 0)"
hoelzl@56889
   240
    by (simp add: norm_complex_def complex_eq_iff)
huffman@23125
   241
  show "norm (x + y) \<le> norm x + norm y"
hoelzl@56889
   242
    by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq)
huffman@23125
   243
  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
hoelzl@56889
   244
    by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] real_sqrt_mult)
huffman@23125
   245
  show "norm (x * y) = norm x * norm y"
hoelzl@56889
   246
    by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
hoelzl@56889
   247
qed (rule complex_sgn_def dist_complex_def open_complex_def)+
huffman@20557
   248
haftmann@25712
   249
end
haftmann@25712
   250
hoelzl@56889
   251
lemma norm_ii [simp]: "norm ii = 1"
hoelzl@56889
   252
  by (simp add: norm_complex_def)
paulson@14323
   253
hoelzl@56889
   254
lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1"
hoelzl@56889
   255
  by (simp add: norm_complex_def)
hoelzl@56889
   256
hoelzl@56889
   257
lemma cmod_complex_polar: "cmod (r * (cos a + \<i> * sin a)) = \<bar>r\<bar>"
hoelzl@56889
   258
  by (simp add: norm_mult cmod_unit_one)
huffman@22861
   259
huffman@22861
   260
lemma complex_Re_le_cmod: "Re x \<le> cmod x"
hoelzl@56889
   261
  unfolding norm_complex_def
huffman@44724
   262
  by (rule real_sqrt_sum_squares_ge1)
huffman@22861
   263
huffman@44761
   264
lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
hoelzl@56889
   265
  by (rule order_trans [OF _ norm_ge_zero]) simp
huffman@22861
   266
hoelzl@56889
   267
lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \<le> cmod a"
hoelzl@56889
   268
  by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp
paulson@14323
   269
chaieb@26117
   270
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
hoelzl@56889
   271
  by (simp add: norm_complex_def)
chaieb@26117
   272
chaieb@26117
   273
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
hoelzl@56889
   274
  by (simp add: norm_complex_def)
hoelzl@56889
   275
hoelzl@57259
   276
lemma cmod_le: "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>"
hoelzl@57259
   277
  apply (subst complex_eq)
hoelzl@57259
   278
  apply (rule order_trans)
hoelzl@57259
   279
  apply (rule norm_triangle_ineq)
hoelzl@57259
   280
  apply (simp add: norm_mult)
hoelzl@57259
   281
  done
hoelzl@57259
   282
hoelzl@56889
   283
lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>"
hoelzl@56889
   284
  by (simp add: norm_complex_def)
hoelzl@56889
   285
hoelzl@56889
   286
lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>"
hoelzl@56889
   287
  by (simp add: norm_complex_def)
huffman@44724
   288
hoelzl@56889
   289
lemma cmod_power2: "cmod z ^ 2 = (Re z)^2 + (Im z)^2"
hoelzl@56889
   290
  by (simp add: norm_complex_def)
hoelzl@56889
   291
hoelzl@56889
   292
lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z = - cmod z"
hoelzl@56889
   293
  using abs_Re_le_cmod[of z] by auto
hoelzl@56889
   294
hoelzl@56889
   295
lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0"
hoelzl@56889
   296
  by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2])
hoelzl@56889
   297
     (auto simp add: norm_complex_def)
hoelzl@56369
   298
hoelzl@56369
   299
lemma abs_sqrt_wlog:
hoelzl@56369
   300
  fixes x::"'a::linordered_idom"
hoelzl@56369
   301
  assumes "\<And>x::'a. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"
hoelzl@56369
   302
by (metis abs_ge_zero assms power2_abs)
hoelzl@56369
   303
hoelzl@56369
   304
lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z"
hoelzl@56889
   305
  unfolding norm_complex_def
hoelzl@56369
   306
  apply (rule abs_sqrt_wlog [where x="Re z"])
hoelzl@56369
   307
  apply (rule abs_sqrt_wlog [where x="Im z"])
hoelzl@56369
   308
  apply (rule power2_le_imp_le)
haftmann@57512
   309
  apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])
hoelzl@56369
   310
  done
hoelzl@56369
   311
hoelzl@56369
   312
huffman@44843
   313
text {* Properties of complex signum. *}
huffman@44843
   314
huffman@44843
   315
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
haftmann@57512
   316
  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute)
huffman@44843
   317
huffman@44843
   318
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
huffman@44843
   319
  by (simp add: complex_sgn_def divide_inverse)
huffman@44843
   320
huffman@44843
   321
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
huffman@44843
   322
  by (simp add: complex_sgn_def divide_inverse)
huffman@44843
   323
paulson@14354
   324
huffman@23123
   325
subsection {* Completeness of the Complexes *}
huffman@23123
   326
huffman@44290
   327
lemma bounded_linear_Re: "bounded_linear Re"
hoelzl@56889
   328
  by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)
huffman@44290
   329
huffman@44290
   330
lemma bounded_linear_Im: "bounded_linear Im"
hoelzl@56889
   331
  by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)
huffman@23123
   332
huffman@44290
   333
lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
huffman@44290
   334
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
hoelzl@56381
   335
lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]
hoelzl@56381
   336
lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]
hoelzl@56381
   337
lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
hoelzl@56381
   338
lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
hoelzl@56381
   339
lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]
hoelzl@56381
   340
lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]
hoelzl@56381
   341
lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]
hoelzl@56381
   342
lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]
hoelzl@56381
   343
lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]
hoelzl@56381
   344
lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]
hoelzl@56381
   345
lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]
hoelzl@56381
   346
lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]
hoelzl@56369
   347
huffman@36825
   348
lemma tendsto_Complex [tendsto_intros]:
hoelzl@56889
   349
  "(f ---> a) F \<Longrightarrow> (g ---> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
hoelzl@56889
   350
  by (auto intro!: tendsto_intros)
hoelzl@56369
   351
hoelzl@56369
   352
lemma tendsto_complex_iff:
hoelzl@56369
   353
  "(f ---> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) ---> Re x) F \<and> ((\<lambda>x. Im (f x)) ---> Im x) F)"
hoelzl@56889
   354
proof safe
hoelzl@56889
   355
  assume "((\<lambda>x. Re (f x)) ---> Re x) F" "((\<lambda>x. Im (f x)) ---> Im x) F"
hoelzl@56889
   356
  from tendsto_Complex[OF this] show "(f ---> x) F"
hoelzl@56889
   357
    unfolding complex.collapse .
hoelzl@56889
   358
qed (auto intro: tendsto_intros)
hoelzl@56369
   359
hoelzl@57259
   360
lemma continuous_complex_iff: "continuous F f \<longleftrightarrow>
hoelzl@57259
   361
    continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x))"
hoelzl@57259
   362
  unfolding continuous_def tendsto_complex_iff ..
hoelzl@57259
   363
hoelzl@57259
   364
lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \<longleftrightarrow>
hoelzl@57259
   365
    ((\<lambda>x. Re (f x)) has_field_derivative (Re x)) F \<and>
hoelzl@57259
   366
    ((\<lambda>x. Im (f x)) has_field_derivative (Im x)) F"
hoelzl@57259
   367
  unfolding has_vector_derivative_def has_field_derivative_def has_derivative_def tendsto_complex_iff
hoelzl@57259
   368
  by (simp add: field_simps bounded_linear_scaleR_left bounded_linear_mult_right)
hoelzl@57259
   369
hoelzl@57259
   370
lemma has_field_derivative_Re[derivative_intros]:
hoelzl@57259
   371
  "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Re (f x)) has_field_derivative (Re D)) F"
hoelzl@57259
   372
  unfolding has_vector_derivative_complex_iff by safe
hoelzl@57259
   373
hoelzl@57259
   374
lemma has_field_derivative_Im[derivative_intros]:
hoelzl@57259
   375
  "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Im (f x)) has_field_derivative (Im D)) F"
hoelzl@57259
   376
  unfolding has_vector_derivative_complex_iff by safe
hoelzl@57259
   377
huffman@23123
   378
instance complex :: banach
huffman@23123
   379
proof
huffman@23123
   380
  fix X :: "nat \<Rightarrow> complex"
huffman@23123
   381
  assume X: "Cauchy X"
hoelzl@56889
   382
  then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
hoelzl@56889
   383
    by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)
hoelzl@56889
   384
  then show "convergent X"
hoelzl@56889
   385
    unfolding complex.collapse by (rule convergentI)
huffman@23123
   386
qed
huffman@23123
   387
lp15@56238
   388
declare
hoelzl@56381
   389
  DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]
lp15@56238
   390
huffman@23125
   391
subsection {* Complex Conjugation *}
huffman@23125
   392
hoelzl@56889
   393
primcorec cnj :: "complex \<Rightarrow> complex" where
hoelzl@56889
   394
  "Re (cnj z) = Re z"
hoelzl@56889
   395
| "Im (cnj z) = - Im z"
huffman@23125
   396
huffman@23125
   397
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
huffman@44724
   398
  by (simp add: complex_eq_iff)
huffman@23125
   399
huffman@23125
   400
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
hoelzl@56889
   401
  by (simp add: complex_eq_iff)
huffman@23125
   402
huffman@23125
   403
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
huffman@44724
   404
  by (simp add: complex_eq_iff)
huffman@23125
   405
huffman@23125
   406
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
huffman@44724
   407
  by (simp add: complex_eq_iff)
huffman@23125
   408
hoelzl@56889
   409
lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y"
huffman@44724
   410
  by (simp add: complex_eq_iff)
huffman@23125
   411
hoelzl@56889
   412
lemma cnj_setsum [simp]: "cnj (setsum f s) = (\<Sum>x\<in>s. cnj (f x))"
hoelzl@56889
   413
  by (induct s rule: infinite_finite_induct) auto
hoelzl@56369
   414
hoelzl@56889
   415
lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y"
huffman@44724
   416
  by (simp add: complex_eq_iff)
huffman@23125
   417
hoelzl@56889
   418
lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x"
huffman@44724
   419
  by (simp add: complex_eq_iff)
huffman@23125
   420
huffman@23125
   421
lemma complex_cnj_one [simp]: "cnj 1 = 1"
huffman@44724
   422
  by (simp add: complex_eq_iff)
huffman@23125
   423
hoelzl@56889
   424
lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y"
huffman@44724
   425
  by (simp add: complex_eq_iff)
huffman@23125
   426
hoelzl@56889
   427
lemma cnj_setprod [simp]: "cnj (setprod f s) = (\<Prod>x\<in>s. cnj (f x))"
hoelzl@56889
   428
  by (induct s rule: infinite_finite_induct) auto
hoelzl@56369
   429
hoelzl@56889
   430
lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)"
hoelzl@56889
   431
  by (simp add: complex_eq_iff)
paulson@14323
   432
hoelzl@56889
   433
lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y"
hoelzl@56889
   434
  by (simp add: divide_complex_def)
huffman@23125
   435
hoelzl@56889
   436
lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n"
hoelzl@56889
   437
  by (induct n) simp_all
huffman@23125
   438
huffman@23125
   439
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
huffman@44724
   440
  by (simp add: complex_eq_iff)
huffman@23125
   441
huffman@23125
   442
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
huffman@44724
   443
  by (simp add: complex_eq_iff)
huffman@23125
   444
huffman@47108
   445
lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
huffman@47108
   446
  by (simp add: complex_eq_iff)
huffman@47108
   447
haftmann@54489
   448
lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
huffman@44724
   449
  by (simp add: complex_eq_iff)
huffman@23125
   450
hoelzl@56889
   451
lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)"
huffman@44724
   452
  by (simp add: complex_eq_iff)
huffman@23125
   453
huffman@23125
   454
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
hoelzl@56889
   455
  by (simp add: norm_complex_def)
paulson@14323
   456
huffman@23125
   457
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
huffman@44724
   458
  by (simp add: complex_eq_iff)
huffman@23125
   459
huffman@23125
   460
lemma complex_cnj_i [simp]: "cnj ii = - ii"
huffman@44724
   461
  by (simp add: complex_eq_iff)
huffman@23125
   462
huffman@23125
   463
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
huffman@44724
   464
  by (simp add: complex_eq_iff)
huffman@23125
   465
huffman@23125
   466
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
huffman@44724
   467
  by (simp add: complex_eq_iff)
paulson@14354
   468
wenzelm@53015
   469
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
huffman@44724
   470
  by (simp add: complex_eq_iff power2_eq_square)
huffman@23125
   471
wenzelm@53015
   472
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"
huffman@44724
   473
  by (simp add: norm_mult power2_eq_square)
huffman@23125
   474
huffman@44827
   475
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
hoelzl@56889
   476
  by (simp add: norm_complex_def power2_eq_square)
huffman@44827
   477
huffman@44827
   478
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
huffman@44827
   479
  by simp
huffman@44827
   480
huffman@44290
   481
lemma bounded_linear_cnj: "bounded_linear cnj"
huffman@44127
   482
  using complex_cnj_add complex_cnj_scaleR
huffman@44127
   483
  by (rule bounded_linear_intro [where K=1], simp)
paulson@14354
   484
hoelzl@56381
   485
lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]
hoelzl@56381
   486
lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]
hoelzl@56381
   487
lemmas continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]
hoelzl@56381
   488
lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]
hoelzl@56381
   489
lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]
huffman@44290
   490
hoelzl@56369
   491
lemma lim_cnj: "((\<lambda>x. cnj(f x)) ---> cnj l) F \<longleftrightarrow> (f ---> l) F"
hoelzl@56889
   492
  by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)
hoelzl@56369
   493
hoelzl@56369
   494
lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
hoelzl@56889
   495
  by (simp add: sums_def lim_cnj cnj_setsum [symmetric] del: cnj_setsum)
hoelzl@56369
   496
paulson@14354
   497
lp15@55734
   498
subsection{*Basic Lemmas*}
lp15@55734
   499
lp15@55734
   500
lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"
hoelzl@56889
   501
  by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)
lp15@55734
   502
lp15@55734
   503
lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"
hoelzl@56889
   504
  by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)
lp15@55734
   505
lp15@55734
   506
lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"
hoelzl@56889
   507
by (cases z)
hoelzl@56889
   508
   (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric]
hoelzl@56889
   509
         simp del: of_real_power)
lp15@55734
   510
hoelzl@56889
   511
lemma re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0"
hoelzl@56889
   512
  by (auto simp add: Re_divide)
lp15@59613
   513
hoelzl@56889
   514
lemma im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0"
hoelzl@56889
   515
  by (auto simp add: Im_divide)
hoelzl@56889
   516
lp15@59613
   517
lemma complex_div_gt_0:
hoelzl@56889
   518
  "(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)"
hoelzl@56889
   519
proof cases
hoelzl@56889
   520
  assume "b = 0" then show ?thesis by auto
lp15@55734
   521
next
hoelzl@56889
   522
  assume "b \<noteq> 0"
hoelzl@56889
   523
  then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2"
hoelzl@56889
   524
    by (simp add: complex_eq_iff sum_power2_gt_zero_iff)
hoelzl@56889
   525
  then show ?thesis
hoelzl@56889
   526
    by (simp add: Re_divide Im_divide zero_less_divide_iff)
lp15@55734
   527
qed
lp15@55734
   528
hoelzl@56889
   529
lemma re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0"
hoelzl@56889
   530
  and im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0"
hoelzl@56889
   531
  using complex_div_gt_0 by auto
lp15@55734
   532
lp15@55734
   533
lemma re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0"
lp15@55734
   534
  by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0)
lp15@55734
   535
lp15@55734
   536
lemma im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0"
lp15@55734
   537
  by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less)
lp15@55734
   538
lp15@55734
   539
lemma re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0"
boehmes@55759
   540
  by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0)
lp15@55734
   541
lp15@55734
   542
lemma im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0"
lp15@55734
   543
  by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff)
lp15@55734
   544
lp15@55734
   545
lemma re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0"
lp15@55734
   546
  by (metis not_le re_complex_div_gt_0)
lp15@55734
   547
lp15@55734
   548
lemma im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0"
lp15@55734
   549
  by (metis im_complex_div_gt_0 not_le)
lp15@55734
   550
hoelzl@56889
   551
lemma Re_setsum[simp]: "Re (setsum f s) = (\<Sum>x\<in>s. Re (f x))"
hoelzl@56369
   552
  by (induct s rule: infinite_finite_induct) auto
lp15@55734
   553
hoelzl@56889
   554
lemma Im_setsum[simp]: "Im (setsum f s) = (\<Sum>x\<in>s. Im(f x))"
hoelzl@56369
   555
  by (induct s rule: infinite_finite_induct) auto
hoelzl@56369
   556
hoelzl@56369
   557
lemma sums_complex_iff: "f sums x \<longleftrightarrow> ((\<lambda>x. Re (f x)) sums Re x) \<and> ((\<lambda>x. Im (f x)) sums Im x)"
hoelzl@56369
   558
  unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..
lp15@59613
   559
hoelzl@56369
   560
lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and>  summable (\<lambda>x. Im (f x))"
hoelzl@56889
   561
  unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel)
hoelzl@56369
   562
hoelzl@56369
   563
lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f"
hoelzl@56369
   564
  unfolding summable_complex_iff by simp
hoelzl@56369
   565
hoelzl@56369
   566
lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))"
hoelzl@56369
   567
  unfolding summable_complex_iff by blast
hoelzl@56369
   568
hoelzl@56369
   569
lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))"
hoelzl@56369
   570
  unfolding summable_complex_iff by blast
lp15@56217
   571
hoelzl@56889
   572
lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
hoelzl@56889
   573
  by (auto simp: Reals_def complex_eq_iff)
lp15@55734
   574
lp15@55734
   575
lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z"
hoelzl@56889
   576
  by (auto simp: complex_is_Real_iff complex_eq_iff)
lp15@55734
   577
lp15@55734
   578
lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm(z) = abs(Re z)"
hoelzl@56889
   579
  by (simp add: complex_is_Real_iff norm_complex_def)
hoelzl@56369
   580
hoelzl@56369
   581
lemma series_comparison_complex:
hoelzl@56369
   582
  fixes f:: "nat \<Rightarrow> 'a::banach"
hoelzl@56369
   583
  assumes sg: "summable g"
hoelzl@56369
   584
     and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"
hoelzl@56369
   585
     and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"
hoelzl@56369
   586
  shows "summable f"
hoelzl@56369
   587
proof -
hoelzl@56369
   588
  have g: "\<And>n. cmod (g n) = Re (g n)" using assms
hoelzl@56369
   589
    by (metis abs_of_nonneg in_Reals_norm)
hoelzl@56369
   590
  show ?thesis
hoelzl@56369
   591
    apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])
hoelzl@56369
   592
    using sg
hoelzl@56369
   593
    apply (auto simp: summable_def)
hoelzl@56369
   594
    apply (rule_tac x="Re s" in exI)
hoelzl@56369
   595
    apply (auto simp: g sums_Re)
hoelzl@56369
   596
    apply (metis fg g)
hoelzl@56369
   597
    done
hoelzl@56369
   598
qed
lp15@55734
   599
paulson@14323
   600
subsection{*Finally! Polar Form for Complex Numbers*}
paulson@14323
   601
huffman@44827
   602
subsubsection {* $\cos \theta + i \sin \theta$ *}
huffman@20557
   603
hoelzl@56889
   604
primcorec cis :: "real \<Rightarrow> complex" where
hoelzl@56889
   605
  "Re (cis a) = cos a"
hoelzl@56889
   606
| "Im (cis a) = sin a"
huffman@44827
   607
huffman@44827
   608
lemma cis_zero [simp]: "cis 0 = 1"
hoelzl@56889
   609
  by (simp add: complex_eq_iff)
huffman@44827
   610
huffman@44828
   611
lemma norm_cis [simp]: "norm (cis a) = 1"
hoelzl@56889
   612
  by (simp add: norm_complex_def)
huffman@44828
   613
huffman@44828
   614
lemma sgn_cis [simp]: "sgn (cis a) = cis a"
huffman@44828
   615
  by (simp add: sgn_div_norm)
huffman@44828
   616
huffman@44828
   617
lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
huffman@44828
   618
  by (metis norm_cis norm_zero zero_neq_one)
huffman@44828
   619
huffman@44827
   620
lemma cis_mult: "cis a * cis b = cis (a + b)"
hoelzl@56889
   621
  by (simp add: complex_eq_iff cos_add sin_add)
huffman@44827
   622
huffman@44827
   623
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
huffman@44827
   624
  by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
huffman@44827
   625
huffman@44827
   626
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
hoelzl@56889
   627
  by (simp add: complex_eq_iff)
huffman@44827
   628
huffman@44827
   629
lemma cis_divide: "cis a / cis b = cis (a - b)"
hoelzl@56889
   630
  by (simp add: divide_complex_def cis_mult)
huffman@44827
   631
huffman@44827
   632
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
huffman@44827
   633
  by (auto simp add: DeMoivre)
huffman@44827
   634
huffman@44827
   635
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
huffman@44827
   636
  by (auto simp add: DeMoivre)
huffman@44827
   637
hoelzl@56889
   638
lemma cis_pi: "cis pi = -1"
hoelzl@56889
   639
  by (simp add: complex_eq_iff)
hoelzl@56889
   640
huffman@44827
   641
subsubsection {* $r(\cos \theta + i \sin \theta)$ *}
huffman@44715
   642
hoelzl@56889
   643
definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex" where
huffman@20557
   644
  "rcis r a = complex_of_real r * cis a"
huffman@20557
   645
huffman@44827
   646
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
huffman@44828
   647
  by (simp add: rcis_def)
huffman@44827
   648
huffman@44827
   649
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
huffman@44828
   650
  by (simp add: rcis_def)
huffman@44827
   651
huffman@44827
   652
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
huffman@44828
   653
  by (simp add: complex_eq_iff polar_Ex)
huffman@44827
   654
huffman@44827
   655
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
huffman@44828
   656
  by (simp add: rcis_def norm_mult)
huffman@44827
   657
huffman@44827
   658
lemma cis_rcis_eq: "cis a = rcis 1 a"
huffman@44827
   659
  by (simp add: rcis_def)
huffman@44827
   660
huffman@44827
   661
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
huffman@44828
   662
  by (simp add: rcis_def cis_mult)
huffman@44827
   663
huffman@44827
   664
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
huffman@44827
   665
  by (simp add: rcis_def)
huffman@44827
   666
huffman@44827
   667
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
huffman@44827
   668
  by (simp add: rcis_def)
huffman@44827
   669
huffman@44828
   670
lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
huffman@44828
   671
  by (simp add: rcis_def)
huffman@44828
   672
huffman@44827
   673
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
huffman@44827
   674
  by (simp add: rcis_def power_mult_distrib DeMoivre)
huffman@44827
   675
huffman@44827
   676
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
huffman@44827
   677
  by (simp add: divide_inverse rcis_def)
huffman@44827
   678
huffman@44827
   679
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
huffman@44828
   680
  by (simp add: rcis_def cis_divide [symmetric])
huffman@44827
   681
huffman@44827
   682
subsubsection {* Complex exponential *}
huffman@44827
   683
huffman@44291
   684
abbreviation expi :: "complex \<Rightarrow> complex"
huffman@44291
   685
  where "expi \<equiv> exp"
huffman@44291
   686
hoelzl@56889
   687
lemma cis_conv_exp: "cis b = exp (\<i> * b)"
hoelzl@56889
   688
proof -
hoelzl@56889
   689
  { fix n :: nat
hoelzl@56889
   690
    have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)"
hoelzl@56889
   691
      by (induct n)
hoelzl@56889
   692
         (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps
hoelzl@56889
   693
                        power2_eq_square real_of_nat_Suc add_nonneg_eq_0_iff
hoelzl@56889
   694
                        real_of_nat_def[symmetric])
hoelzl@56889
   695
    then have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n =
hoelzl@56889
   696
        of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)"
hoelzl@56889
   697
      by (simp add: field_simps) }
hoelzl@56889
   698
  then show ?thesis
hoelzl@56889
   699
    by (auto simp add: cis.ctr exp_def simp del: of_real_mult
hoelzl@56889
   700
             intro!: sums_unique sums_add sums_mult sums_of_real sin_converges cos_converges)
huffman@44291
   701
qed
huffman@44291
   702
hoelzl@56889
   703
lemma expi_def: "expi z = exp (Re z) * cis (Im z)"
hoelzl@56889
   704
  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by (cases z) simp
huffman@20557
   705
huffman@44828
   706
lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
huffman@44828
   707
  unfolding expi_def by simp
huffman@44828
   708
huffman@44828
   709
lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
huffman@44828
   710
  unfolding expi_def by simp
huffman@44828
   711
paulson@14374
   712
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
paulson@14373
   713
apply (insert rcis_Ex [of z])
haftmann@57512
   714
apply (auto simp add: expi_def rcis_def mult.assoc [symmetric])
paulson@14334
   715
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
paulson@14323
   716
done
paulson@14323
   717
paulson@14387
   718
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
hoelzl@56889
   719
  by (simp add: expi_def complex_eq_iff)
paulson@14387
   720
huffman@44844
   721
subsubsection {* Complex argument *}
huffman@44844
   722
huffman@44844
   723
definition arg :: "complex \<Rightarrow> real" where
huffman@44844
   724
  "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"
huffman@44844
   725
huffman@44844
   726
lemma arg_zero: "arg 0 = 0"
huffman@44844
   727
  by (simp add: arg_def)
huffman@44844
   728
huffman@44844
   729
lemma arg_unique:
huffman@44844
   730
  assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
huffman@44844
   731
  shows "arg z = x"
huffman@44844
   732
proof -
huffman@44844
   733
  from assms have "z \<noteq> 0" by auto
huffman@44844
   734
  have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
huffman@44844
   735
  proof
huffman@44844
   736
    fix a def d \<equiv> "a - x"
huffman@44844
   737
    assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
huffman@44844
   738
    from a assms have "- (2*pi) < d \<and> d < 2*pi"
huffman@44844
   739
      unfolding d_def by simp
huffman@44844
   740
    moreover from a assms have "cos a = cos x" and "sin a = sin x"
huffman@44844
   741
      by (simp_all add: complex_eq_iff)
wenzelm@53374
   742
    hence cos: "cos d = 1" unfolding d_def cos_diff by simp
wenzelm@53374
   743
    moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)
huffman@44844
   744
    ultimately have "d = 0"
haftmann@58709
   745
      unfolding sin_zero_iff
haftmann@58740
   746
      by (auto elim!: evenE dest!: less_2_cases)
huffman@44844
   747
    thus "a = x" unfolding d_def by simp
huffman@44844
   748
  qed (simp add: assms del: Re_sgn Im_sgn)
huffman@44844
   749
  with `z \<noteq> 0` show "arg z = x"
huffman@44844
   750
    unfolding arg_def by simp
huffman@44844
   751
qed
huffman@44844
   752
huffman@44844
   753
lemma arg_correct:
huffman@44844
   754
  assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
huffman@44844
   755
proof (simp add: arg_def assms, rule someI_ex)
huffman@44844
   756
  obtain r a where z: "z = rcis r a" using rcis_Ex by fast
huffman@44844
   757
  with assms have "r \<noteq> 0" by auto
huffman@44844
   758
  def b \<equiv> "if 0 < r then a else a + pi"
huffman@44844
   759
  have b: "sgn z = cis b"
huffman@44844
   760
    unfolding z b_def rcis_def using `r \<noteq> 0`
hoelzl@56889
   761
    by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff)
huffman@44844
   762
  have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
hoelzl@56889
   763
    by (induct_tac n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)
huffman@44844
   764
  have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
hoelzl@56889
   765
    by (case_tac x rule: int_diff_cases)
hoelzl@56889
   766
       (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
huffman@44844
   767
  def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
huffman@44844
   768
  have "sgn z = cis c"
huffman@44844
   769
    unfolding b c_def
huffman@44844
   770
    by (simp add: cis_divide [symmetric] cis_2pi_int)
huffman@44844
   771
  moreover have "- pi < c \<and> c \<le> pi"
huffman@44844
   772
    using ceiling_correct [of "(b - pi) / (2*pi)"]
huffman@44844
   773
    by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)
huffman@44844
   774
  ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
huffman@44844
   775
qed
huffman@44844
   776
huffman@44844
   777
lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
hoelzl@56889
   778
  by (cases "z = 0") (simp_all add: arg_zero arg_correct)
huffman@44844
   779
huffman@44844
   780
lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
huffman@44844
   781
  by (simp add: arg_correct)
huffman@44844
   782
huffman@44844
   783
lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
hoelzl@56889
   784
  by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
hoelzl@56889
   785
hoelzl@56889
   786
lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0"
hoelzl@56889
   787
  using cis_arg [of y] by (simp add: complex_eq_iff)
hoelzl@56889
   788
hoelzl@56889
   789
subsection {* Square root of complex numbers *}
hoelzl@56889
   790
hoelzl@56889
   791
primcorec csqrt :: "complex \<Rightarrow> complex" where
hoelzl@56889
   792
  "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"
hoelzl@56889
   793
| "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)"
hoelzl@56889
   794
hoelzl@56889
   795
lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)"
hoelzl@56889
   796
  by (simp add: complex_eq_iff norm_complex_def)
hoelzl@56889
   797
hoelzl@56889
   798
lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>"
hoelzl@56889
   799
  by (simp add: complex_eq_iff norm_complex_def)
hoelzl@56889
   800
hoelzl@56889
   801
lemma csqrt_0 [simp]: "csqrt 0 = 0"
hoelzl@56889
   802
  by simp
hoelzl@56889
   803
hoelzl@56889
   804
lemma csqrt_1 [simp]: "csqrt 1 = 1"
hoelzl@56889
   805
  by simp
hoelzl@56889
   806
hoelzl@56889
   807
lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2"
hoelzl@56889
   808
  by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt)
huffman@44844
   809
hoelzl@56889
   810
lemma power2_csqrt[algebra]: "(csqrt z)\<^sup>2 = z"
hoelzl@56889
   811
proof cases
hoelzl@56889
   812
  assume "Im z = 0" then show ?thesis
hoelzl@56889
   813
    using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"]
hoelzl@56889
   814
    by (cases "0::real" "Re z" rule: linorder_cases)
hoelzl@56889
   815
       (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)
hoelzl@56889
   816
next
hoelzl@56889
   817
  assume "Im z \<noteq> 0"
hoelzl@56889
   818
  moreover
hoelzl@56889
   819
  have "cmod z * cmod z - Re z * Re z = Im z * Im z"
hoelzl@56889
   820
    by (simp add: norm_complex_def power2_eq_square)
hoelzl@56889
   821
  moreover
hoelzl@56889
   822
  have "\<bar>Re z\<bar> \<le> cmod z"
hoelzl@56889
   823
    by (simp add: norm_complex_def)
hoelzl@56889
   824
  ultimately show ?thesis
hoelzl@56889
   825
    by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq
hoelzl@56889
   826
                  field_simps real_sqrt_mult[symmetric] real_sqrt_divide)
hoelzl@56889
   827
qed
hoelzl@56889
   828
hoelzl@56889
   829
lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
hoelzl@56889
   830
  by auto (metis power2_csqrt power_eq_0_iff)
hoelzl@56889
   831
hoelzl@56889
   832
lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
hoelzl@56889
   833
  by auto (metis power2_csqrt power2_eq_1_iff)
hoelzl@56889
   834
hoelzl@56889
   835
lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)"
hoelzl@56889
   836
  by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)
hoelzl@56889
   837
hoelzl@56889
   838
lemma Re_csqrt: "0 \<le> Re (csqrt z)"
hoelzl@56889
   839
  by (metis csqrt_principal le_less)
hoelzl@56889
   840
hoelzl@56889
   841
lemma csqrt_square:
hoelzl@56889
   842
  assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)"
hoelzl@56889
   843
  shows "csqrt (b^2) = b"
hoelzl@56889
   844
proof -
hoelzl@56889
   845
  have "csqrt (b^2) = b \<or> csqrt (b^2) = - b"
hoelzl@56889
   846
    unfolding power2_eq_iff[symmetric] by (simp add: power2_csqrt)
hoelzl@56889
   847
  moreover have "csqrt (b^2) \<noteq> -b \<or> b = 0"
hoelzl@56889
   848
    using csqrt_principal[of "b ^ 2"] assms by (intro disjCI notI) (auto simp: complex_eq_iff)
hoelzl@56889
   849
  ultimately show ?thesis
hoelzl@56889
   850
    by auto
hoelzl@56889
   851
qed
hoelzl@56889
   852
lp15@59613
   853
lemma csqrt_minus [simp]:
hoelzl@56889
   854
  assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)"
hoelzl@56889
   855
  shows "csqrt (- x) = \<i> * csqrt x"
hoelzl@56889
   856
proof -
hoelzl@56889
   857
  have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x"
hoelzl@56889
   858
  proof (rule csqrt_square)
hoelzl@56889
   859
    have "Im (csqrt x) \<le> 0"
hoelzl@56889
   860
      using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod)
hoelzl@56889
   861
    then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)"
hoelzl@56889
   862
      by (auto simp add: Re_csqrt simp del: csqrt.simps)
hoelzl@56889
   863
  qed
hoelzl@56889
   864
  also have "(\<i> * csqrt x)^2 = - x"
hoelzl@56889
   865
    by (simp add: power2_csqrt power_mult_distrib)
hoelzl@56889
   866
  finally show ?thesis .
hoelzl@56889
   867
qed
huffman@44844
   868
huffman@44065
   869
text {* Legacy theorem names *}
huffman@44065
   870
huffman@44065
   871
lemmas expand_complex_eq = complex_eq_iff
huffman@44065
   872
lemmas complex_Re_Im_cancel_iff = complex_eq_iff
huffman@44065
   873
lemmas complex_equality = complex_eqI
hoelzl@56889
   874
lemmas cmod_def = norm_complex_def
hoelzl@56889
   875
lemmas complex_norm_def = norm_complex_def
hoelzl@56889
   876
lemmas complex_divide_def = divide_complex_def
hoelzl@56889
   877
hoelzl@56889
   878
lemma legacy_Complex_simps:
hoelzl@56889
   879
  shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
hoelzl@56889
   880
    and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"
hoelzl@56889
   881
    and complex_minus: "- (Complex a b) = Complex (- a) (- b)"
hoelzl@56889
   882
    and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"
hoelzl@56889
   883
    and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"
hoelzl@56889
   884
    and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"
hoelzl@56889
   885
    and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
hoelzl@56889
   886
    and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"
hoelzl@56889
   887
    and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"
hoelzl@56889
   888
    and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"
hoelzl@56889
   889
    and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"
hoelzl@56889
   890
    and Complex_eq_i: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
hoelzl@56889
   891
    and i_mult_Complex: "ii * Complex a b = Complex (- b) a"
hoelzl@56889
   892
    and Complex_mult_i: "Complex a b * ii = Complex (- b) a"
hoelzl@56889
   893
    and i_complex_of_real: "ii * complex_of_real r = Complex 0 r"
hoelzl@56889
   894
    and complex_of_real_i: "complex_of_real r * ii = Complex 0 r"
hoelzl@56889
   895
    and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"
hoelzl@56889
   896
    and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"
hoelzl@56889
   897
    and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
hoelzl@56889
   898
    and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
hoelzl@56889
   899
    and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
hoelzl@56889
   900
    and complex_cn: "cnj (Complex a b) = Complex a (- b)"
hoelzl@56889
   901
    and Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"
hoelzl@56889
   902
    and Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"
hoelzl@56889
   903
    and complex_of_real_def: "complex_of_real r = Complex r 0"
hoelzl@56889
   904
    and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"
hoelzl@56889
   905
  by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide del: Complex_eq)
hoelzl@56889
   906
hoelzl@56889
   907
lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"
hoelzl@56889
   908
  by (metis Reals_of_real complex_of_real_def)
huffman@44065
   909
paulson@13957
   910
end