src/HOL/Complex.thy
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 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@60758  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 wenzelm@60758  13 text \  wenzelm@61799  14 We use the \codatatype\ command to define the type of complex numbers. This allows us to use  wenzelm@61799  15 \primcorec\ to define complex functions by defining their real and imaginary result  blanchet@58146  16 separately.  wenzelm@60758  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 wenzelm@60758  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 wenzelm@60758  56 subsection \Multiplication and Division\  huffman@23125  57 haftmann@59867  58 instantiation complex :: field  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 wenzelm@61076  73 definition "x div (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 lp15@59862  96 lemma Re_power_real [simp]: "Im x = 0 \ Re (x ^ n) = Re x ^ n "  huffman@44724  97  by (induct n) simp_all  huffman@23125  98 lp15@59862  99 lemma Im_power_real [simp]: "Im x = 0 \ Im (x ^ n) = 0"  hoelzl@56889  100  by (induct n) simp_all  huffman@23125  101 wenzelm@60758  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 wenzelm@60758  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@61609  169 eberlm@61552  170 lemma Re_divide_of_nat: "Re (z / of_nat n) = Re z / of_nat n"  eberlm@61552  171  by (cases n) (simp_all add: Re_divide divide_simps power2_eq_square del: of_nat_Suc)  eberlm@61552  172 eberlm@61552  173 lemma Im_divide_of_nat: "Im (z / of_nat n) = Im z / of_nat n"  eberlm@61552  174  by (cases n) (simp_all add: Im_divide divide_simps power2_eq_square del: of_nat_Suc)  lp15@59613  175 lp15@60017  176 lemma of_real_Re [simp]:  lp15@60017  177  "z \ \ \ of_real (Re z) = z"  lp15@60017  178  by (auto simp: Reals_def)  lp15@60017  179 eberlm@61531  180 lemma complex_Re_fact [simp]: "Re (fact n) = fact n"  eberlm@61531  181 proof -  eberlm@61531  182  have "(fact n :: complex) = of_real (fact n)" by simp  eberlm@61531  183  also have "Re \ = fact n" by (subst Re_complex_of_real) simp_all  eberlm@61531  184  finally show ?thesis .  eberlm@61531  185 qed  eberlm@61531  186 eberlm@61531  187 lemma complex_Im_fact [simp]: "Im (fact n) = 0"  eberlm@61531  188  by (subst of_nat_fact [symmetric]) (simp only: complex_Im_of_nat)  eberlm@61531  189 eberlm@61531  190 wenzelm@60758  191 subsection \The Complex Number $i$\  hoelzl@56889  192 hoelzl@56889  193 primcorec "ii" :: complex ("\") where  hoelzl@56889  194  "Re ii = 0"  hoelzl@56889  195 | "Im ii = 1"  huffman@20557  196 hoelzl@57259  197 lemma Complex_eq[simp]: "Complex a b = a + \ * b"  hoelzl@57259  198  by (simp add: complex_eq_iff)  hoelzl@57259  199 hoelzl@57259  200 lemma complex_eq: "a = Re a + \ * Im a"  hoelzl@57259  201  by (simp add: complex_eq_iff)  hoelzl@57259  202 hoelzl@57259  203 lemma fun_complex_eq: "f = (\x. Re (f x) + \ * Im (f x))"  hoelzl@57259  204  by (simp add: fun_eq_iff complex_eq)  hoelzl@57259  205 hoelzl@56889  206 lemma i_squared [simp]: "ii * ii = -1"  hoelzl@56889  207  by (simp add: complex_eq_iff)  hoelzl@56889  208 hoelzl@56889  209 lemma power2_i [simp]: "ii\<^sup>2 = -1"  hoelzl@56889  210  by (simp add: power2_eq_square)  paulson@14377  211 hoelzl@56889  212 lemma inverse_i [simp]: "inverse ii = - ii"  hoelzl@56889  213  by (rule inverse_unique) simp  hoelzl@56889  214 hoelzl@56889  215 lemma divide_i [simp]: "x / ii = - ii * x"  hoelzl@56889  216  by (simp add: divide_complex_def)  paulson@14377  217 hoelzl@56889  218 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"  haftmann@57512  219  by (simp add: mult.assoc [symmetric])  paulson@14377  220 hoelzl@56889  221 lemma complex_i_not_zero [simp]: "ii \ 0"  hoelzl@56889  222  by (simp add: complex_eq_iff)  huffman@20557  223 hoelzl@56889  224 lemma complex_i_not_one [simp]: "ii \ 1"  hoelzl@56889  225  by (simp add: complex_eq_iff)  hoelzl@56889  226 hoelzl@56889  227 lemma complex_i_not_numeral [simp]: "ii \ numeral w"  hoelzl@56889  228  by (simp add: complex_eq_iff)  huffman@44841  229 hoelzl@56889  230 lemma complex_i_not_neg_numeral [simp]: "ii \ - numeral w"  hoelzl@56889  231  by (simp add: complex_eq_iff)  hoelzl@56889  232 hoelzl@56889  233 lemma complex_split_polar: "\r a. z = complex_of_real r * (cos a + \ * sin a)"  huffman@44827  234  by (simp add: complex_eq_iff polar_Ex)  huffman@44827  235 lp15@59613  236 lemma i_even_power [simp]: "\ ^ (n * 2) = (-1) ^ n"  lp15@59613  237  by (metis mult.commute power2_i power_mult)  lp15@59613  238 lp15@59741  239 lemma Re_ii_times [simp]: "Re (ii*z) = - Im z"  lp15@59741  240  by simp  lp15@59741  241 lp15@59741  242 lemma Im_ii_times [simp]: "Im (ii*z) = Re z"  lp15@59741  243  by simp  lp15@59741  244 lp15@59741  245 lemma ii_times_eq_iff: "ii*w = z \ w = -(ii*z)"  lp15@59741  246  by auto  lp15@59741  247 lp15@59741  248 lemma divide_numeral_i [simp]: "z / (numeral n * ii) = -(ii*z) / numeral n"  lp15@59741  249  by (metis divide_divide_eq_left divide_i mult.commute mult_minus_right)  lp15@59741  250 wenzelm@60758  251 subsection \Vector Norm\  paulson@14323  252 haftmann@25712  253 instantiation complex :: real_normed_field  haftmann@25571  254 begin  haftmann@25571  255 hoelzl@56889  256 definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"  haftmann@25571  257 huffman@44724  258 abbreviation cmod :: "complex \ real"  huffman@44724  259  where "cmod \ norm"  haftmann@25571  260 huffman@31413  261 definition complex_sgn_def:  huffman@31413  262  "sgn x = x /\<^sub>R cmod x"  haftmann@25571  263 huffman@31413  264 definition dist_complex_def:  huffman@31413  265  "dist x y = cmod (x - y)"  huffman@31413  266 haftmann@37767  267 definition open_complex_def:  huffman@31492  268  "open (S :: complex set) \ (\x\S. \e>0. \y. dist y x < e \ y \ S)"  huffman@31292  269 huffman@31413  270 instance proof  huffman@31492  271  fix r :: real and x y :: complex and S :: "complex set"  huffman@23125  272  show "(norm x = 0) = (x = 0)"  hoelzl@56889  273  by (simp add: norm_complex_def complex_eq_iff)  huffman@23125  274  show "norm (x + y) \ norm x + norm y"  hoelzl@56889  275  by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq)  huffman@23125  276  show "norm (scaleR r x) = \r\ * norm x"  hoelzl@56889  277  by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] real_sqrt_mult)  huffman@23125  278  show "norm (x * y) = norm x * norm y"  hoelzl@56889  279  by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] power2_eq_square algebra_simps)  hoelzl@56889  280 qed (rule complex_sgn_def dist_complex_def open_complex_def)+  huffman@20557  281 haftmann@25712  282 end  haftmann@25712  283 hoelzl@56889  284 lemma norm_ii [simp]: "norm ii = 1"  hoelzl@56889  285  by (simp add: norm_complex_def)  paulson@14323  286 hoelzl@56889  287 lemma cmod_unit_one: "cmod (cos a + \ * sin a) = 1"  hoelzl@56889  288  by (simp add: norm_complex_def)  hoelzl@56889  289 hoelzl@56889  290 lemma cmod_complex_polar: "cmod (r * (cos a + \ * sin a)) = \r\"  hoelzl@56889  291  by (simp add: norm_mult cmod_unit_one)  huffman@22861  292 huffman@22861  293 lemma complex_Re_le_cmod: "Re x \ cmod x"  hoelzl@56889  294  unfolding norm_complex_def  huffman@44724  295  by (rule real_sqrt_sum_squares_ge1)  huffman@22861  296 huffman@44761  297 lemma complex_mod_minus_le_complex_mod: "- cmod x \ cmod x"  hoelzl@56889  298  by (rule order_trans [OF _ norm_ge_zero]) simp  huffman@22861  299 hoelzl@56889  300 lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \ cmod a"  hoelzl@56889  301  by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp  paulson@14323  302 chaieb@26117  303 lemma abs_Re_le_cmod: "\Re x\ \ cmod x"  hoelzl@56889  304  by (simp add: norm_complex_def)  chaieb@26117  305 chaieb@26117  306 lemma abs_Im_le_cmod: "\Im x\ \ cmod x"  hoelzl@56889  307  by (simp add: norm_complex_def)  hoelzl@56889  308 hoelzl@57259  309 lemma cmod_le: "cmod z \ \Re z\ + \Im z\"  hoelzl@57259  310  apply (subst complex_eq)  hoelzl@57259  311  apply (rule order_trans)  hoelzl@57259  312  apply (rule norm_triangle_ineq)  hoelzl@57259  313  apply (simp add: norm_mult)  hoelzl@57259  314  done  hoelzl@57259  315 hoelzl@56889  316 lemma cmod_eq_Re: "Im z = 0 \ cmod z = \Re z\"  hoelzl@56889  317  by (simp add: norm_complex_def)  hoelzl@56889  318 hoelzl@56889  319 lemma cmod_eq_Im: "Re z = 0 \ cmod z = \Im z\"  hoelzl@56889  320  by (simp add: norm_complex_def)  huffman@44724  321 hoelzl@56889  322 lemma cmod_power2: "cmod z ^ 2 = (Re z)^2 + (Im z)^2"  hoelzl@56889  323  by (simp add: norm_complex_def)  hoelzl@56889  324 hoelzl@56889  325 lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \ 0 \ Re z = - cmod z"  hoelzl@56889  326  using abs_Re_le_cmod[of z] by auto  hoelzl@56889  327 hoelzl@56889  328 lemma Im_eq_0: "\Re z\ = cmod z \ Im z = 0"  hoelzl@56889  329  by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2])  hoelzl@56889  330  (auto simp add: norm_complex_def)  hoelzl@56369  331 hoelzl@56369  332 lemma abs_sqrt_wlog:  hoelzl@56369  333  fixes x::"'a::linordered_idom"  hoelzl@56369  334  assumes "\x::'a. x \ 0 \ P x (x\<^sup>2)" shows "P \x\ (x\<^sup>2)"  hoelzl@56369  335 by (metis abs_ge_zero assms power2_abs)  hoelzl@56369  336 hoelzl@56369  337 lemma complex_abs_le_norm: "\Re z\ + \Im z\ \ sqrt 2 * norm z"  hoelzl@56889  338  unfolding norm_complex_def  hoelzl@56369  339  apply (rule abs_sqrt_wlog [where x="Re z"])  hoelzl@56369  340  apply (rule abs_sqrt_wlog [where x="Im z"])  hoelzl@56369  341  apply (rule power2_le_imp_le)  haftmann@57512  342  apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])  hoelzl@56369  343  done  hoelzl@56369  344 lp15@59741  345 lemma complex_unit_circle: "z \ 0 \ (Re z / cmod z)\<^sup>2 + (Im z / cmod z)\<^sup>2 = 1"  lp15@59741  346  by (simp add: norm_complex_def divide_simps complex_eq_iff)  lp15@59741  347 hoelzl@56369  348 wenzelm@60758  349 text \Properties of complex signum.\  huffman@44843  350 huffman@44843  351 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"  haftmann@57512  352  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute)  huffman@44843  353 huffman@44843  354 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"  huffman@44843  355  by (simp add: complex_sgn_def divide_inverse)  huffman@44843  356 huffman@44843  357 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"  huffman@44843  358  by (simp add: complex_sgn_def divide_inverse)  huffman@44843  359 paulson@14354  360 wenzelm@60758  361 subsection \Completeness of the Complexes\  huffman@23123  362 huffman@44290  363 lemma bounded_linear_Re: "bounded_linear Re"  hoelzl@56889  364  by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)  huffman@44290  365 huffman@44290  366 lemma bounded_linear_Im: "bounded_linear Im"  hoelzl@56889  367  by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)  huffman@23123  368 huffman@44290  369 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]  huffman@44290  370 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]  hoelzl@56381  371 lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]  hoelzl@56381  372 lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]  hoelzl@56381  373 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]  hoelzl@56381  374 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]  hoelzl@56381  375 lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]  hoelzl@56381  376 lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]  hoelzl@56381  377 lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]  hoelzl@56381  378 lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]  hoelzl@56381  379 lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]  hoelzl@56381  380 lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]  hoelzl@56381  381 lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]  hoelzl@56381  382 lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]  hoelzl@56369  383 huffman@36825  384 lemma tendsto_Complex [tendsto_intros]:  hoelzl@56889  385  "(f ---> a) F \ (g ---> b) F \ ((\x. Complex (f x) (g x)) ---> Complex a b) F"  hoelzl@56889  386  by (auto intro!: tendsto_intros)  hoelzl@56369  387 hoelzl@56369  388 lemma tendsto_complex_iff:  hoelzl@56369  389  "(f ---> x) F \ (((\x. Re (f x)) ---> Re x) F \ ((\x. Im (f x)) ---> Im x) F)"  hoelzl@56889  390 proof safe  hoelzl@56889  391  assume "((\x. Re (f x)) ---> Re x) F" "((\x. Im (f x)) ---> Im x) F"  hoelzl@56889  392  from tendsto_Complex[OF this] show "(f ---> x) F"  hoelzl@56889  393  unfolding complex.collapse .  hoelzl@56889  394 qed (auto intro: tendsto_intros)  hoelzl@56369  395 hoelzl@57259  396 lemma continuous_complex_iff: "continuous F f \  hoelzl@57259  397  continuous F (\x. Re (f x)) \ continuous F (\x. Im (f x))"  hoelzl@57259  398  unfolding continuous_def tendsto_complex_iff ..  hoelzl@57259  399 hoelzl@57259  400 lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \  hoelzl@57259  401  ((\x. Re (f x)) has_field_derivative (Re x)) F \  hoelzl@57259  402  ((\x. Im (f x)) has_field_derivative (Im x)) F"  hoelzl@57259  403  unfolding has_vector_derivative_def has_field_derivative_def has_derivative_def tendsto_complex_iff  hoelzl@57259  404  by (simp add: field_simps bounded_linear_scaleR_left bounded_linear_mult_right)  hoelzl@57259  405 hoelzl@57259  406 lemma has_field_derivative_Re[derivative_intros]:  hoelzl@57259  407  "(f has_vector_derivative D) F \ ((\x. Re (f x)) has_field_derivative (Re D)) F"  hoelzl@57259  408  unfolding has_vector_derivative_complex_iff by safe  hoelzl@57259  409 hoelzl@57259  410 lemma has_field_derivative_Im[derivative_intros]:  hoelzl@57259  411  "(f has_vector_derivative D) F \ ((\x. Im (f x)) has_field_derivative (Im D)) F"  hoelzl@57259  412  unfolding has_vector_derivative_complex_iff by safe  hoelzl@57259  413 huffman@23123  414 instance complex :: banach  huffman@23123  415 proof  huffman@23123  416  fix X :: "nat \ complex"  huffman@23123  417  assume X: "Cauchy X"  hoelzl@56889  418  then have "(\n. Complex (Re (X n)) (Im (X n))) ----> Complex (lim (\n. Re (X n))) (lim (\n. Im (X n)))"  hoelzl@56889  419  by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)  hoelzl@56889  420  then show "convergent X"  hoelzl@56889  421  unfolding complex.collapse by (rule convergentI)  huffman@23123  422 qed  huffman@23123  423 lp15@56238  424 declare  hoelzl@56381  425  DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]  lp15@56238  426 wenzelm@60758  427 subsection \Complex Conjugation\  huffman@23125  428 hoelzl@56889  429 primcorec cnj :: "complex \ complex" where  hoelzl@56889  430  "Re (cnj z) = Re z"  hoelzl@56889  431 | "Im (cnj z) = - Im z"  huffman@23125  432 huffman@23125  433 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"  huffman@44724  434  by (simp add: complex_eq_iff)  huffman@23125  435 huffman@23125  436 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"  hoelzl@56889  437  by (simp add: complex_eq_iff)  huffman@23125  438 huffman@23125  439 lemma complex_cnj_zero [simp]: "cnj 0 = 0"  huffman@44724  440  by (simp add: complex_eq_iff)  huffman@23125  441 huffman@23125  442 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"  huffman@44724  443  by (simp add: complex_eq_iff)  huffman@23125  444 hoelzl@56889  445 lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y"  huffman@44724  446  by (simp add: complex_eq_iff)  huffman@23125  447 hoelzl@56889  448 lemma cnj_setsum [simp]: "cnj (setsum f s) = (\x\s. cnj (f x))"  hoelzl@56889  449  by (induct s rule: infinite_finite_induct) auto  hoelzl@56369  450 hoelzl@56889  451 lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y"  huffman@44724  452  by (simp add: complex_eq_iff)  huffman@23125  453 hoelzl@56889  454 lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x"  huffman@44724  455  by (simp add: complex_eq_iff)  huffman@23125  456 huffman@23125  457 lemma complex_cnj_one [simp]: "cnj 1 = 1"  huffman@44724  458  by (simp add: complex_eq_iff)  huffman@23125  459 hoelzl@56889  460 lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y"  huffman@44724  461  by (simp add: complex_eq_iff)  huffman@23125  462 hoelzl@56889  463 lemma cnj_setprod [simp]: "cnj (setprod f s) = (\x\s. cnj (f x))"  hoelzl@56889  464  by (induct s rule: infinite_finite_induct) auto  hoelzl@56369  465 hoelzl@56889  466 lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)"  hoelzl@56889  467  by (simp add: complex_eq_iff)  paulson@14323  468 hoelzl@56889  469 lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y"  hoelzl@56889  470  by (simp add: divide_complex_def)  huffman@23125  471 hoelzl@56889  472 lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n"  hoelzl@56889  473  by (induct n) simp_all  huffman@23125  474 huffman@23125  475 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"  huffman@44724  476  by (simp add: complex_eq_iff)  huffman@23125  477 huffman@23125  478 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"  huffman@44724  479  by (simp add: complex_eq_iff)  huffman@23125  480 huffman@47108  481 lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"  huffman@47108  482  by (simp add: complex_eq_iff)  huffman@47108  483 haftmann@54489  484 lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"  huffman@44724  485  by (simp add: complex_eq_iff)  huffman@23125  486 hoelzl@56889  487 lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)"  huffman@44724  488  by (simp add: complex_eq_iff)  huffman@23125  489 huffman@23125  490 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"  hoelzl@56889  491  by (simp add: norm_complex_def)  paulson@14323  492 huffman@23125  493 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"  huffman@44724  494  by (simp add: complex_eq_iff)  huffman@23125  495 huffman@23125  496 lemma complex_cnj_i [simp]: "cnj ii = - ii"  huffman@44724  497  by (simp add: complex_eq_iff)  huffman@23125  498 huffman@23125  499 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"  huffman@44724  500  by (simp add: complex_eq_iff)  huffman@23125  501 huffman@23125  502 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"  huffman@44724  503  by (simp add: complex_eq_iff)  paulson@14354  504 wenzelm@53015  505 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"  huffman@44724  506  by (simp add: complex_eq_iff power2_eq_square)  huffman@23125  507 wenzelm@53015  508 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"  huffman@44724  509  by (simp add: norm_mult power2_eq_square)  huffman@23125  510 huffman@44827  511 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"  hoelzl@56889  512  by (simp add: norm_complex_def power2_eq_square)  huffman@44827  513 huffman@44827  514 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"  huffman@44827  515  by simp  huffman@44827  516 eberlm@61531  517 lemma complex_cnj_fact [simp]: "cnj (fact n) = fact n"  eberlm@61531  518  by (subst of_nat_fact [symmetric], subst complex_cnj_of_nat) simp  eberlm@61531  519 eberlm@61531  520 lemma complex_cnj_pochhammer [simp]: "cnj (pochhammer z n) = pochhammer (cnj z) n"  eberlm@61531  521  by (induction n arbitrary: z) (simp_all add: pochhammer_rec)  eberlm@61531  522 huffman@44290  523 lemma bounded_linear_cnj: "bounded_linear cnj"  huffman@44127  524  using complex_cnj_add complex_cnj_scaleR  huffman@44127  525  by (rule bounded_linear_intro [where K=1], simp)  paulson@14354  526 hoelzl@56381  527 lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]  hoelzl@56381  528 lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]  hoelzl@56381  529 lemmas continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]  hoelzl@56381  530 lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]  hoelzl@56381  531 lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]  huffman@44290  532 hoelzl@56369  533 lemma lim_cnj: "((\x. cnj(f x)) ---> cnj l) F \ (f ---> l) F"  hoelzl@56889  534  by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)  hoelzl@56369  535 hoelzl@56369  536 lemma sums_cnj: "((\x. cnj(f x)) sums cnj l) \ (f sums l)"  hoelzl@56889  537  by (simp add: sums_def lim_cnj cnj_setsum [symmetric] del: cnj_setsum)  hoelzl@56369  538 paulson@14354  539 wenzelm@60758  540 subsection\Basic Lemmas\  lp15@55734  541 lp15@55734  542 lemma complex_eq_0: "z=0 \ (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"  hoelzl@56889  543  by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)  lp15@55734  544 lp15@55734  545 lemma complex_neq_0: "z\0 \ (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"  hoelzl@56889  546  by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)  lp15@55734  547 lp15@55734  548 lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"  hoelzl@56889  549 by (cases z)  hoelzl@56889  550  (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric]  hoelzl@56889  551  simp del: of_real_power)  lp15@55734  552 paulson@61104  553 lemma complex_div_cnj: "a / b = (a * cnj b) / (norm b)^2"  paulson@61104  554  using complex_norm_square by auto  paulson@61104  555 lp15@59741  556 lemma Re_complex_div_eq_0: "Re (a / b) = 0 \ Re (a * cnj b) = 0"  hoelzl@56889  557  by (auto simp add: Re_divide)  lp15@59613  558 lp15@59741  559 lemma Im_complex_div_eq_0: "Im (a / b) = 0 \ Im (a * cnj b) = 0"  hoelzl@56889  560  by (auto simp add: Im_divide)  hoelzl@56889  561 lp15@59613  562 lemma complex_div_gt_0:  hoelzl@56889  563  "(Re (a / b) > 0 \ Re (a * cnj b) > 0) \ (Im (a / b) > 0 \ Im (a * cnj b) > 0)"  hoelzl@56889  564 proof cases  hoelzl@56889  565  assume "b = 0" then show ?thesis by auto  lp15@55734  566 next  hoelzl@56889  567  assume "b \ 0"  hoelzl@56889  568  then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2"  hoelzl@56889  569  by (simp add: complex_eq_iff sum_power2_gt_zero_iff)  hoelzl@56889  570  then show ?thesis  hoelzl@56889  571  by (simp add: Re_divide Im_divide zero_less_divide_iff)  lp15@55734  572 qed  lp15@55734  573 lp15@59741  574 lemma Re_complex_div_gt_0: "Re (a / b) > 0 \ Re (a * cnj b) > 0"  lp15@59741  575  and Im_complex_div_gt_0: "Im (a / b) > 0 \ Im (a * cnj b) > 0"  hoelzl@56889  576  using complex_div_gt_0 by auto  lp15@55734  577 lp15@59741  578 lemma Re_complex_div_ge_0: "Re(a / b) \ 0 \ Re(a * cnj b) \ 0"  lp15@59741  579  by (metis le_less Re_complex_div_eq_0 Re_complex_div_gt_0)  lp15@55734  580 lp15@59741  581 lemma Im_complex_div_ge_0: "Im(a / b) \ 0 \ Im(a * cnj b) \ 0"  lp15@59741  582  by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 le_less)  lp15@55734  583 lp15@59741  584 lemma Re_complex_div_lt_0: "Re(a / b) < 0 \ Re(a * cnj b) < 0"  lp15@59741  585  by (metis less_asym neq_iff Re_complex_div_eq_0 Re_complex_div_gt_0)  lp15@55734  586 lp15@59741  587 lemma Im_complex_div_lt_0: "Im(a / b) < 0 \ Im(a * cnj b) < 0"  lp15@59741  588  by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 less_asym neq_iff)  lp15@55734  589 lp15@59741  590 lemma Re_complex_div_le_0: "Re(a / b) \ 0 \ Re(a * cnj b) \ 0"  lp15@59741  591  by (metis not_le Re_complex_div_gt_0)  lp15@55734  592 lp15@59741  593 lemma Im_complex_div_le_0: "Im(a / b) \ 0 \ Im(a * cnj b) \ 0"  lp15@59741  594  by (metis Im_complex_div_gt_0 not_le)  lp15@55734  595 paulson@61104  596 lemma Re_divide_of_real [simp]: "Re (z / of_real r) = Re z / r"  paulson@61104  597  by (simp add: Re_divide power2_eq_square)  paulson@61104  598 paulson@61104  599 lemma Im_divide_of_real [simp]: "Im (z / of_real r) = Im z / r"  paulson@61104  600  by (simp add: Im_divide power2_eq_square)  paulson@61104  601 paulson@61104  602 lemma Re_divide_Reals: "r \ Reals \ Re (z / r) = Re z / Re r"  paulson@61104  603  by (metis Re_divide_of_real of_real_Re)  paulson@61104  604 paulson@61104  605 lemma Im_divide_Reals: "r \ Reals \ Im (z / r) = Im z / Re r"  paulson@61104  606  by (metis Im_divide_of_real of_real_Re)  paulson@61104  607 hoelzl@56889  608 lemma Re_setsum[simp]: "Re (setsum f s) = (\x\s. Re (f x))"  hoelzl@56369  609  by (induct s rule: infinite_finite_induct) auto  lp15@55734  610 hoelzl@56889  611 lemma Im_setsum[simp]: "Im (setsum f s) = (\x\s. Im(f x))"  hoelzl@56369  612  by (induct s rule: infinite_finite_induct) auto  hoelzl@56369  613 hoelzl@56369  614 lemma sums_complex_iff: "f sums x \ ((\x. Re (f x)) sums Re x) \ ((\x. Im (f x)) sums Im x)"  hoelzl@56369  615  unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..  lp15@59613  616 hoelzl@56369  617 lemma summable_complex_iff: "summable f \ summable (\x. Re (f x)) \ summable (\x. Im (f x))"  hoelzl@56889  618  unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel)  hoelzl@56369  619 hoelzl@56369  620 lemma summable_complex_of_real [simp]: "summable (\n. complex_of_real (f n)) \ summable f"  hoelzl@56369  621  unfolding summable_complex_iff by simp  hoelzl@56369  622 hoelzl@56369  623 lemma summable_Re: "summable f \ summable (\x. Re (f x))"  hoelzl@56369  624  unfolding summable_complex_iff by blast  hoelzl@56369  625 hoelzl@56369  626 lemma summable_Im: "summable f \ summable (\x. Im (f x))"  hoelzl@56369  627  unfolding summable_complex_iff by blast  lp15@56217  628 paulson@61104  629 lemma complex_is_Nat_iff: "z \ \ \ Im z = 0 \ (\i. Re z = of_nat i)"  paulson@61104  630  by (auto simp: Nats_def complex_eq_iff)  paulson@61104  631 paulson@61104  632 lemma complex_is_Int_iff: "z \ \ \ Im z = 0 \ (\i. Re z = of_int i)"  paulson@61104  633  by (auto simp: Ints_def complex_eq_iff)  paulson@61104  634 hoelzl@56889  635 lemma complex_is_Real_iff: "z \ \ \ Im z = 0"  hoelzl@56889  636  by (auto simp: Reals_def complex_eq_iff)  lp15@55734  637 lp15@55734  638 lemma Reals_cnj_iff: "z \ \ \ cnj z = z"  hoelzl@56889  639  by (auto simp: complex_is_Real_iff complex_eq_iff)  lp15@55734  640 lp15@55734  641 lemma in_Reals_norm: "z \ \ \ norm(z) = abs(Re z)"  hoelzl@56889  642  by (simp add: complex_is_Real_iff norm_complex_def)  hoelzl@56369  643 hoelzl@56369  644 lemma series_comparison_complex:  hoelzl@56369  645  fixes f:: "nat \ 'a::banach"  hoelzl@56369  646  assumes sg: "summable g"  hoelzl@56369  647  and "\n. g n \ \" "\n. Re (g n) \ 0"  hoelzl@56369  648  and fg: "\n. n \ N \ norm(f n) \ norm(g n)"  hoelzl@56369  649  shows "summable f"  hoelzl@56369  650 proof -  hoelzl@56369  651  have g: "\n. cmod (g n) = Re (g n)" using assms  hoelzl@56369  652  by (metis abs_of_nonneg in_Reals_norm)  hoelzl@56369  653  show ?thesis  hoelzl@56369  654  apply (rule summable_comparison_test' [where g = "\n. norm (g n)" and N=N])  hoelzl@56369  655  using sg  hoelzl@56369  656  apply (auto simp: summable_def)  hoelzl@56369  657  apply (rule_tac x="Re s" in exI)  hoelzl@56369  658  apply (auto simp: g sums_Re)  hoelzl@56369  659  apply (metis fg g)  hoelzl@56369  660  done  hoelzl@56369  661 qed  lp15@55734  662 wenzelm@60758  663 subsection\Polar Form for Complex Numbers\  lp15@59746  664 lp15@59746  665 lemma complex_unimodular_polar: "(norm z = 1) \ \x. z = Complex (cos x) (sin x)"  lp15@59746  666  using sincos_total_2pi [of "Re z" "Im z"]  lp15@59746  667  by auto (metis cmod_power2 complex_eq power_one)  paulson@14323  668 wenzelm@60758  669 subsubsection \$\cos \theta + i \sin \theta$\  huffman@20557  670 hoelzl@56889  671 primcorec cis :: "real \ complex" where  hoelzl@56889  672  "Re (cis a) = cos a"  hoelzl@56889  673 | "Im (cis a) = sin a"  huffman@44827  674 huffman@44827  675 lemma cis_zero [simp]: "cis 0 = 1"  hoelzl@56889  676  by (simp add: complex_eq_iff)  huffman@44827  677 huffman@44828  678 lemma norm_cis [simp]: "norm (cis a) = 1"  hoelzl@56889  679  by (simp add: norm_complex_def)  huffman@44828  680 huffman@44828  681 lemma sgn_cis [simp]: "sgn (cis a) = cis a"  huffman@44828  682  by (simp add: sgn_div_norm)  huffman@44828  683 huffman@44828  684 lemma cis_neq_zero [simp]: "cis a \ 0"  huffman@44828  685  by (metis norm_cis norm_zero zero_neq_one)  huffman@44828  686 huffman@44827  687 lemma cis_mult: "cis a * cis b = cis (a + b)"  hoelzl@56889  688  by (simp add: complex_eq_iff cos_add sin_add)  huffman@44827  689 huffman@44827  690 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"  lp15@61609  691  by (induct n, simp_all add: of_nat_Suc algebra_simps cis_mult)  huffman@44827  692 huffman@44827  693 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"  hoelzl@56889  694  by (simp add: complex_eq_iff)  huffman@44827  695 huffman@44827  696 lemma cis_divide: "cis a / cis b = cis (a - b)"  hoelzl@56889  697  by (simp add: divide_complex_def cis_mult)  huffman@44827  698 huffman@44827  699 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"  huffman@44827  700  by (auto simp add: DeMoivre)  huffman@44827  701 huffman@44827  702 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"  huffman@44827  703  by (auto simp add: DeMoivre)  huffman@44827  704 hoelzl@56889  705 lemma cis_pi: "cis pi = -1"  hoelzl@56889  706  by (simp add: complex_eq_iff)  hoelzl@56889  707 wenzelm@60758  708 subsubsection \$r(\cos \theta + i \sin \theta)$\  huffman@44715  709 hoelzl@56889  710 definition rcis :: "real \ real \ complex" where  huffman@20557  711  "rcis r a = complex_of_real r * cis a"  huffman@20557  712 huffman@44827  713 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"  huffman@44828  714  by (simp add: rcis_def)  huffman@44827  715 huffman@44827  716 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"  huffman@44828  717  by (simp add: rcis_def)  huffman@44827  718 huffman@44827  719 lemma rcis_Ex: "\r a. z = rcis r a"  huffman@44828  720  by (simp add: complex_eq_iff polar_Ex)  huffman@44827  721 huffman@44827  722 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"  huffman@44828  723  by (simp add: rcis_def norm_mult)  huffman@44827  724 huffman@44827  725 lemma cis_rcis_eq: "cis a = rcis 1 a"  huffman@44827  726  by (simp add: rcis_def)  huffman@44827  727 huffman@44827  728 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"  huffman@44828  729  by (simp add: rcis_def cis_mult)  huffman@44827  730 huffman@44827  731 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"  huffman@44827  732  by (simp add: rcis_def)  huffman@44827  733 huffman@44827  734 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"  huffman@44827  735  by (simp add: rcis_def)  huffman@44827  736 huffman@44828  737 lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \ r = 0"  huffman@44828  738  by (simp add: rcis_def)  huffman@44828  739 huffman@44827  740 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"  huffman@44827  741  by (simp add: rcis_def power_mult_distrib DeMoivre)  huffman@44827  742 huffman@44827  743 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"  huffman@44827  744  by (simp add: divide_inverse rcis_def)  huffman@44827  745 huffman@44827  746 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"  huffman@44828  747  by (simp add: rcis_def cis_divide [symmetric])  huffman@44827  748 wenzelm@60758  749 subsubsection \Complex exponential\  huffman@44827  750 hoelzl@56889  751 lemma cis_conv_exp: "cis b = exp (\ * b)"  hoelzl@56889  752 proof -  hoelzl@56889  753  { fix n :: nat  hoelzl@56889  754  have "\ ^ n = fact n *\<^sub>R (cos_coeff n + \ * sin_coeff n)"  hoelzl@56889  755  by (induct n)  hoelzl@56889  756  (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps  lp15@61609  757  power2_eq_square of_nat_Suc add_nonneg_eq_0_iff)  hoelzl@56889  758  then have "(\ * complex_of_real b) ^ n /\<^sub>R fact n =  hoelzl@56889  759  of_real (cos_coeff n * b^n) + \ * of_real (sin_coeff n * b^n)"  hoelzl@56889  760  by (simp add: field_simps) }  lp15@59658  761  then show ?thesis using sin_converges [of b] cos_converges [of b]  hoelzl@56889  762  by (auto simp add: cis.ctr exp_def simp del: of_real_mult  lp15@59658  763  intro!: sums_unique sums_add sums_mult sums_of_real)  huffman@44291  764 qed  huffman@44291  765 lp15@61762  766 lemma exp_eq_polar: "exp z = exp (Re z) * cis (Im z)"  hoelzl@56889  767  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by (cases z) simp  huffman@20557  768 huffman@44828  769 lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"  lp15@61762  770  unfolding exp_eq_polar by simp  huffman@44828  771 huffman@44828  772 lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"  lp15@61762  773  unfolding exp_eq_polar by simp  huffman@44828  774 lp15@59746  775 lemma norm_cos_sin [simp]: "norm (Complex (cos t) (sin t)) = 1"  lp15@59746  776  by (simp add: norm_complex_def)  lp15@59746  777 lp15@59746  778 lemma norm_exp_eq_Re [simp]: "norm (exp z) = exp (Re z)"  lp15@61762  779  by (simp add: cis.code cmod_complex_polar exp_eq_polar)  lp15@59746  780 lp15@61762  781 lemma complex_exp_exists: "\a r. z = complex_of_real r * exp a"  lp15@59746  782  apply (insert rcis_Ex [of z])  lp15@61762  783  apply (auto simp add: exp_eq_polar rcis_def mult.assoc [symmetric])  lp15@59746  784  apply (rule_tac x = "ii * complex_of_real a" in exI, auto)  lp15@59746  785  done  paulson@14323  786 lp15@61762  787 lemma exp_two_pi_i [simp]: "exp(2 * complex_of_real pi * ii) = 1"  lp15@61762  788  by (simp add: exp_eq_polar complex_eq_iff)  paulson@14387  789 wenzelm@60758  790 subsubsection \Complex argument\  huffman@44844  791 huffman@44844  792 definition arg :: "complex \ real" where  huffman@44844  793  "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \ -pi < a \ a \ pi))"  huffman@44844  794 huffman@44844  795 lemma arg_zero: "arg 0 = 0"  huffman@44844  796  by (simp add: arg_def)  huffman@44844  797 huffman@44844  798 lemma arg_unique:  huffman@44844  799  assumes "sgn z = cis x" and "-pi < x" and "x \ pi"  huffman@44844  800  shows "arg z = x"  huffman@44844  801 proof -  huffman@44844  802  from assms have "z \ 0" by auto  huffman@44844  803  have "(SOME a. sgn z = cis a \ -pi < a \ a \ pi) = x"  huffman@44844  804  proof  huffman@44844  805  fix a def d \ "a - x"  huffman@44844  806  assume a: "sgn z = cis a \ - pi < a \ a \ pi"  huffman@44844  807  from a assms have "- (2*pi) < d \ d < 2*pi"  huffman@44844  808  unfolding d_def by simp  huffman@44844  809  moreover from a assms have "cos a = cos x" and "sin a = sin x"  huffman@44844  810  by (simp_all add: complex_eq_iff)  wenzelm@53374  811  hence cos: "cos d = 1" unfolding d_def cos_diff by simp  wenzelm@53374  812  moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)  huffman@44844  813  ultimately have "d = 0"  haftmann@58709  814  unfolding sin_zero_iff  haftmann@58740  815  by (auto elim!: evenE dest!: less_2_cases)  huffman@44844  816  thus "a = x" unfolding d_def by simp  huffman@44844  817  qed (simp add: assms del: Re_sgn Im_sgn)  wenzelm@60758  818  with \z \ 0\ show "arg z = x"  huffman@44844  819  unfolding arg_def by simp  huffman@44844  820 qed  huffman@44844  821 huffman@44844  822 lemma arg_correct:  huffman@44844  823  assumes "z \ 0" shows "sgn z = cis (arg z) \ -pi < arg z \ arg z \ pi"  huffman@44844  824 proof (simp add: arg_def assms, rule someI_ex)  huffman@44844  825  obtain r a where z: "z = rcis r a" using rcis_Ex by fast  huffman@44844  826  with assms have "r \ 0" by auto  huffman@44844  827  def b \ "if 0 < r then a else a + pi"  huffman@44844  828  have b: "sgn z = cis b"  wenzelm@60758  829  unfolding z b_def rcis_def using \r \ 0\  hoelzl@56889  830  by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff)  huffman@44844  831  have cis_2pi_nat: "\n. cis (2 * pi * real_of_nat n) = 1"  hoelzl@56889  832  by (induct_tac n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)  huffman@44844  833  have cis_2pi_int: "\x. cis (2 * pi * real_of_int x) = 1"  hoelzl@56889  834  by (case_tac x rule: int_diff_cases)  hoelzl@56889  835  (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)  huffman@44844  836  def c \ "b - 2*pi * of_int $$b - pi) / (2*pi)\"  huffman@44844  837  have "sgn z = cis c"  huffman@44844  838  unfolding b c_def  huffman@44844  839  by (simp add: cis_divide [symmetric] cis_2pi_int)  huffman@44844  840  moreover have "- pi < c \ c \ pi"  huffman@44844  841  using ceiling_correct [of "(b - pi) / (2*pi)"]  lp15@61649  842  by (simp add: c_def less_divide_eq divide_le_eq algebra_simps del: le_of_int_ceiling)  huffman@44844  843  ultimately show "\a. sgn z = cis a \ -pi < a \ a \ pi" by fast  huffman@44844  844 qed  huffman@44844  845 huffman@44844  846 lemma arg_bounded: "- pi < arg z \ arg z \ pi"  hoelzl@56889  847  by (cases "z = 0") (simp_all add: arg_zero arg_correct)  huffman@44844  848 huffman@44844  849 lemma cis_arg: "z \ 0 \ cis (arg z) = sgn z"  huffman@44844  850  by (simp add: arg_correct)  huffman@44844  851 huffman@44844  852 lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"  hoelzl@56889  853  by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)  hoelzl@56889  854 hoelzl@56889  855 lemma cos_arg_i_mult_zero [simp]: "y \ 0 \ Re y = 0 \ cos (arg y) = 0"  hoelzl@56889  856  using cis_arg [of y] by (simp add: complex_eq_iff)  hoelzl@56889  857 wenzelm@60758  858 subsection \Square root of complex numbers\  hoelzl@56889  859 hoelzl@56889  860 primcorec csqrt :: "complex \ complex" where  hoelzl@56889  861  "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"  hoelzl@56889  862 | "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)"  hoelzl@56889  863 hoelzl@56889  864 lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \ Re x \ 0 \ csqrt x = sqrt (Re x)"  hoelzl@56889  865  by (simp add: complex_eq_iff norm_complex_def)  hoelzl@56889  866 hoelzl@56889  867 lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \ Re x \ 0 \ csqrt x = \ * sqrt \Re x\"  hoelzl@56889  868  by (simp add: complex_eq_iff norm_complex_def)  hoelzl@56889  869 lp15@59862  870 lemma of_real_sqrt: "x \ 0 \ of_real (sqrt x) = csqrt (of_real x)"  lp15@59862  871  by (simp add: complex_eq_iff norm_complex_def)  lp15@59862  872 hoelzl@56889  873 lemma csqrt_0 [simp]: "csqrt 0 = 0"  hoelzl@56889  874  by simp  hoelzl@56889  875 hoelzl@56889  876 lemma csqrt_1 [simp]: "csqrt 1 = 1"  hoelzl@56889  877  by simp  hoelzl@56889  878 hoelzl@56889  879 lemma csqrt_ii [simp]: "csqrt \ = (1 +$$ / sqrt 2"  hoelzl@56889  880  by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt)  huffman@44844  881 lp15@59741  882 lemma power2_csqrt[simp,algebra]: "(csqrt z)\<^sup>2 = z"  hoelzl@56889  883 proof cases  hoelzl@56889  884  assume "Im z = 0" then show ?thesis  hoelzl@56889  885  using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"]  hoelzl@56889  886  by (cases "0::real" "Re z" rule: linorder_cases)  hoelzl@56889  887  (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)  hoelzl@56889  888 next  hoelzl@56889  889  assume "Im z \ 0"  hoelzl@56889  890  moreover  hoelzl@56889  891  have "cmod z * cmod z - Re z * Re z = Im z * Im z"  hoelzl@56889  892  by (simp add: norm_complex_def power2_eq_square)  hoelzl@56889  893  moreover  hoelzl@56889  894  have "\Re z\ \ cmod z"  hoelzl@56889  895  by (simp add: norm_complex_def)  hoelzl@56889  896  ultimately show ?thesis  hoelzl@56889  897  by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq  hoelzl@56889  898  field_simps real_sqrt_mult[symmetric] real_sqrt_divide)  hoelzl@56889  899 qed  hoelzl@56889  900 hoelzl@56889  901 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \ z = 0"  hoelzl@56889  902  by auto (metis power2_csqrt power_eq_0_iff)  hoelzl@56889  903 hoelzl@56889  904 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \ z = 1"  hoelzl@56889  905  by auto (metis power2_csqrt power2_eq_1_iff)  hoelzl@56889  906 hoelzl@56889  907 lemma csqrt_principal: "0 < Re (csqrt z) \ Re (csqrt z) = 0 \ 0 \ Im (csqrt z)"  hoelzl@56889  908  by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)  hoelzl@56889  909 hoelzl@56889  910 lemma Re_csqrt: "0 \ Re (csqrt z)"  hoelzl@56889  911  by (metis csqrt_principal le_less)  hoelzl@56889  912 hoelzl@56889  913 lemma csqrt_square:  hoelzl@56889  914  assumes "0 < Re b \ (Re b = 0 \ 0 \ Im b)"  hoelzl@56889  915  shows "csqrt (b^2) = b"  hoelzl@56889  916 proof -  hoelzl@56889  917  have "csqrt (b^2) = b \ csqrt (b^2) = - b"  hoelzl@56889  918  unfolding power2_eq_iff[symmetric] by (simp add: power2_csqrt)  hoelzl@56889  919  moreover have "csqrt (b^2) \ -b \ b = 0"  hoelzl@56889  920  using csqrt_principal[of "b ^ 2"] assms by (intro disjCI notI) (auto simp: complex_eq_iff)  hoelzl@56889  921  ultimately show ?thesis  hoelzl@56889  922  by auto  hoelzl@56889  923 qed  hoelzl@56889  924 lp15@59746  925 lemma csqrt_unique:  lp15@59746  926  "w^2 = z \ (0 < Re w \ Re w = 0 \ 0 \ Im w) \ csqrt z = w"  lp15@59746  927  by (auto simp: csqrt_square)  lp15@59746  928 lp15@59613  929 lemma csqrt_minus [simp]:  hoelzl@56889  930  assumes "Im x < 0 \ (Im x = 0 \ 0 \ Re x)"  hoelzl@56889  931  shows "csqrt (- x) = \ * csqrt x"  hoelzl@56889  932 proof -  hoelzl@56889  933  have "csqrt ((\ * csqrt x)^2) = \ * csqrt x"  hoelzl@56889  934  proof (rule csqrt_square)  hoelzl@56889  935  have "Im (csqrt x) \ 0"  hoelzl@56889  936  using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod)  hoelzl@56889  937  then show "0 < Re (\ * csqrt x) \ Re (\ * csqrt x) = 0 \ 0 \ Im (\ * csqrt x)"  hoelzl@56889  938  by (auto simp add: Re_csqrt simp del: csqrt.simps)  hoelzl@56889  939  qed  hoelzl@56889  940  also have "(\ * csqrt x)^2 = - x"  lp15@59746  941  by (simp add: power_mult_distrib)  hoelzl@56889  942  finally show ?thesis .  hoelzl@56889  943 qed  huffman@44844  944 wenzelm@60758  945 text \Legacy theorem names\  huffman@44065  946 huffman@44065  947 lemmas expand_complex_eq = complex_eq_iff  huffman@44065  948 lemmas complex_Re_Im_cancel_iff = complex_eq_iff  huffman@44065  949 lemmas complex_equality = complex_eqI  hoelzl@56889  950 lemmas cmod_def = norm_complex_def  hoelzl@56889  951 lemmas complex_norm_def = norm_complex_def  hoelzl@56889  952 lemmas complex_divide_def = divide_complex_def  hoelzl@56889  953 hoelzl@56889  954 lemma legacy_Complex_simps:  hoelzl@56889  955  shows Complex_eq_0: "Complex a b = 0 \ a = 0 \ b = 0"  hoelzl@56889  956  and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"  hoelzl@56889  957  and complex_minus: "- (Complex a b) = Complex (- a) (- b)"  hoelzl@56889  958  and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"  hoelzl@56889  959  and Complex_eq_1: "Complex a b = 1 \ a = 1 \ b = 0"  hoelzl@56889  960  and Complex_eq_neg_1: "Complex a b = - 1 \ a = - 1 \ b = 0"  hoelzl@56889  961  and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"  hoelzl@56889  962  and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"  hoelzl@56889  963  and Complex_eq_numeral: "Complex a b = numeral w \ a = numeral w \ b = 0"  hoelzl@56889  964  and Complex_eq_neg_numeral: "Complex a b = - numeral w \ a = - numeral w \ b = 0"  hoelzl@56889  965  and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"  hoelzl@56889  966  and Complex_eq_i: "(Complex x y = ii) = (x = 0 \ y = 1)"  hoelzl@56889  967  and i_mult_Complex: "ii * Complex a b = Complex (- b) a"  hoelzl@56889  968  and Complex_mult_i: "Complex a b * ii = Complex (- b) a"  hoelzl@56889  969  and i_complex_of_real: "ii * complex_of_real r = Complex 0 r"  hoelzl@56889  970  and complex_of_real_i: "complex_of_real r * ii = Complex 0 r"  hoelzl@56889  971  and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"  hoelzl@56889  972  and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"  hoelzl@56889  973  and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"  hoelzl@56889  974  and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"  hoelzl@56889  975  and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"  hoelzl@56889  976  and complex_cn: "cnj (Complex a b) = Complex a (- b)"  hoelzl@56889  977  and Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"  hoelzl@56889  978  and Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"  hoelzl@56889  979  and complex_of_real_def: "complex_of_real r = Complex r 0"  hoelzl@56889  980  and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"  hoelzl@56889  981  by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide del: Complex_eq)  hoelzl@56889  982 hoelzl@56889  983 lemma Complex_in_Reals: "Complex x 0 \ \"  hoelzl@56889  984  by (metis Reals_of_real complex_of_real_def)  huffman@44065  985 paulson@13957  986 end