src/HOL/Complex.thy
 author paulson 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?]: "\Re x = Re y; Im x = Im y\ \ x = y"  hoelzl@56889  25  by (rule complex.expand) simp  huffman@23125  26 huffman@44065  27 lemma complex_eq_iff: "x = y \ Re x = Re y \ 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\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 \ 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 \ 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 \ complex"  huffman@44724  133  where "complex_of_real \ of_real"  huffman@20557  134 hoelzl@59000  135 declare [[coercion "of_real :: real \ complex"]]  hoelzl@59000  136 declare [[coercion "of_rat :: rat \ complex"]]  hoelzl@56889  137 declare [[coercion "of_int :: int \ complex"]]  hoelzl@56889  138 declare [[coercion "of_nat :: nat \ 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 ("\") 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 + \ * b"  hoelzl@57259  177  by (simp add: complex_eq_iff)  hoelzl@57259  178 hoelzl@57259  179 lemma complex_eq: "a = Re a + \ * Im a"  hoelzl@57259  180  by (simp add: complex_eq_iff)  hoelzl@57259  181 hoelzl@57259  182 lemma fun_complex_eq: "f = (\x. Re (f x) + \ * 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 \ 0"  hoelzl@56889  201  by (simp add: complex_eq_iff)  huffman@20557  202 hoelzl@56889  203 lemma complex_i_not_one [simp]: "ii \ 1"  hoelzl@56889  204  by (simp add: complex_eq_iff)  hoelzl@56889  205 hoelzl@56889  206 lemma complex_i_not_numeral [simp]: "ii \ 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 \ - numeral w"  hoelzl@56889  210  by (simp add: complex_eq_iff)  hoelzl@56889  211 hoelzl@56889  212 lemma complex_split_polar: "\r a. z = complex_of_real r * (cos a + \ * sin a)"  huffman@44827  213  by (simp add: complex_eq_iff polar_Ex)  huffman@44827  214 lp15@59613  215 lemma i_even_power [simp]: "\ ^ (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 \ real"  huffman@44724  226  where "cmod \ 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) \ (\x\S. \e>0. \y. dist y x < e \ y \ 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) \ 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) = \r\ * 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 + \ * 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 + \ * sin a)) = \r\"  hoelzl@56889  258  by (simp add: norm_mult cmod_unit_one)  huffman@22861  259 huffman@22861  260 lemma complex_Re_le_cmod: "Re x \ 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 \ 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 \ 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: "\Re x\ \ cmod x"  hoelzl@56889  271  by (simp add: norm_complex_def)  chaieb@26117  272 chaieb@26117  273 lemma abs_Im_le_cmod: "\Im x\ \ cmod x"  hoelzl@56889  274  by (simp add: norm_complex_def)  hoelzl@56889  275 hoelzl@57259  276 lemma cmod_le: "cmod z \ \Re z\ + \Im z\"  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 \ cmod z = \Re z\"  hoelzl@56889  284  by (simp add: norm_complex_def)  hoelzl@56889  285 hoelzl@56889  286 lemma cmod_eq_Im: "Re z = 0 \ cmod z = \Im z\"  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 \ 0 \ 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: "\Re z\ = cmod z \ 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 "\x::'a. x \ 0 \ P x (x\<^sup>2)" shows "P \x\ (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: "\Re z\ + \Im z\ \ 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 \ (g ---> b) F \ ((\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 \ (((\x. Re (f x)) ---> Re x) F \ ((\x. Im (f x)) ---> Im x) F)"  hoelzl@56889  354 proof safe  hoelzl@56889  355  assume "((\x. Re (f x)) ---> Re x) F" "((\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 \  hoelzl@57259  361  continuous F (\x. Re (f x)) \ continuous F (\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 \  hoelzl@57259  365  ((\x. Re (f x)) has_field_derivative (Re x)) F \  hoelzl@57259  366  ((\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 \ ((\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 \ ((\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 \ complex"  huffman@23123  381  assume X: "Cauchy X"  hoelzl@56889  382  then have "(\n. Complex (Re (X n)) (Im (X n))) ----> Complex (lim (\n. Re (X n))) (lim (\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 \ 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) = (\x\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) = (\x\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: "((\x. cnj(f x)) ---> cnj l) F \ (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: "((\x. cnj(f x)) sums cnj l) \ (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 \ (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\0 \ (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 \ 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 \ 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 \ Re (a * cnj b) > 0) \ (Im (a / b) > 0 \ 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 \ 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 \ Re (a * cnj b) > 0"  hoelzl@56889  530  and im_complex_div_gt_0: "Im (a / b) > 0 \ 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) \ 0 \ Re(a * cnj b) \ 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) \ 0 \ Im(a * cnj b) \ 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 \ 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 \ 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) \ 0 \ Re(a * cnj b) \ 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) \ 0 \ Im(a * cnj b) \ 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) = (\x\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) = (\x\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 \ ((\x. Re (f x)) sums Re x) \ ((\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 \ summable (\x. Re (f x)) \ summable (\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 (\n. complex_of_real (f n)) \ summable f"  hoelzl@56369  564  unfolding summable_complex_iff by simp  hoelzl@56369  565 hoelzl@56369  566 lemma summable_Re: "summable f \ summable (\x. Re (f x))"  hoelzl@56369  567  unfolding summable_complex_iff by blast  hoelzl@56369  568 hoelzl@56369  569 lemma summable_Im: "summable f \ summable (\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 \ \ \ 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 \ \ \ 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 \ \ \ 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 \ 'a::banach"  hoelzl@56369  583  assumes sg: "summable g"  hoelzl@56369  584  and "\n. g n \ \" "\n. Re (g n) \ 0"  hoelzl@56369  585  and fg: "\n. n \ N \ norm(f n) \ norm(g n)"  hoelzl@56369  586  shows "summable f"  hoelzl@56369  587 proof -  hoelzl@56369  588  have g: "\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 = "\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 \ 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 \ 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 \ real \ 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: "\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 \ 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 \ complex"  huffman@44291  685  where "expi \ exp"  huffman@44291  686 hoelzl@56889  687 lemma cis_conv_exp: "cis b = exp (\ * b)"  hoelzl@56889  688 proof -  hoelzl@56889  689  { fix n :: nat  hoelzl@56889  690  have "\ ^ n = fact n *\<^sub>R (cos_coeff n + \ * 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 "(\ * complex_of_real b) ^ n /\<^sub>R fact n =  hoelzl@56889  696  of_real (cos_coeff n * b^n) + \ * 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: "\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 \ real" where  huffman@44844  724  "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \ -pi < a \ a \ 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 \ pi"  huffman@44844  731  shows "arg z = x"  huffman@44844  732 proof -  huffman@44844  733  from assms have "z \ 0" by auto  huffman@44844  734  have "(SOME a. sgn z = cis a \ -pi < a \ a \ pi) = x"  huffman@44844  735  proof  huffman@44844  736  fix a def d \ "a - x"  huffman@44844  737  assume a: "sgn z = cis a \ - pi < a \ a \ pi"  huffman@44844  738  from a assms have "- (2*pi) < d \ 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 \ 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 \ 0" shows "sgn z = cis (arg z) \ -pi < arg z \ arg z \ 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 \ 0" by auto  huffman@44844  758  def b \ "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 \ 0  hoelzl@56889  761  by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff)  huffman@44844  762  have cis_2pi_nat: "\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: "\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 \ "b - 2*pi * of_int $$b - pi) / (2*pi)\"  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 \ c \ 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 "\a. sgn z = cis a \ -pi < a \ a \ pi" by fast  huffman@44844  775 qed  huffman@44844  776 huffman@44844  777 lemma arg_bounded: "- pi < arg z \ arg z \ 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 \ 0 \ 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 \ 0 \ Re y = 0 \ 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 \ 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 \ Re x \ 0 \ 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 \ Re x \ 0 \ csqrt x = \ * sqrt \Re x\"  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 \ = (1 +$$ / 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 \ 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 "\Re z\ \ 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 \ 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 \ 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) \ Re (csqrt z) = 0 \ 0 \ 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 \ 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 \ (Re b = 0 \ 0 \ Im b)"  hoelzl@56889  843  shows "csqrt (b^2) = b"  hoelzl@56889  844 proof -  hoelzl@56889  845  have "csqrt (b^2) = b \ 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) \ -b \ 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 \ (Im x = 0 \ 0 \ Re x)"  hoelzl@56889  855  shows "csqrt (- x) = \ * csqrt x"  hoelzl@56889  856 proof -  hoelzl@56889  857  have "csqrt ((\ * csqrt x)^2) = \ * csqrt x"  hoelzl@56889  858  proof (rule csqrt_square)  hoelzl@56889  859  have "Im (csqrt x) \ 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 (\ * csqrt x) \ Re (\ * csqrt x) = 0 \ 0 \ Im (\ * csqrt x)"  hoelzl@56889  862  by (auto simp add: Re_csqrt simp del: csqrt.simps)  hoelzl@56889  863  qed  hoelzl@56889  864  also have "(\ * 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 \ a = 0 \ 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 \ a = 1 \ b = 0"  hoelzl@56889  884  and Complex_eq_neg_1: "Complex a b = - 1 \ a = - 1 \ 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 \ a = numeral w \ b = 0"  hoelzl@56889  888  and Complex_eq_neg_numeral: "Complex a b = - numeral w \ a = - numeral w \ 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 \ 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 \ \"  hoelzl@56889  908  by (metis Reals_of_real complex_of_real_def)  huffman@44065  909 paulson@13957  910 end