ginac/inifcns_trans.cpp

   1 /** @file inifcns_trans.cpp
   2  *
   3  *  Implementation of transcendental (and trigonometric and hyperbolic)
   4  *  functions. */
   5
   6 /*
   7  *  GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
   8  *
   9  *  This program is free software; you can redistribute it and/or modify
  10  *  it under the terms of the GNU General Public License as published by
  11  *  the Free Software Foundation; either version 2 of the License, or
  12  *  (at your option) any later version.
  13  *
  14  *  This program is distributed in the hope that it will be useful,
  15  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  16  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  17  *  GNU General Public License for more details.
  18  *
  19  *  You should have received a copy of the GNU General Public License
  20  *  along with this program; if not, write to the Free Software
  21  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  22  */
  23
  24 #include <vector>
  25 #include <stdexcept>
  26
  27 #include "inifcns.h"
  28 #include "ex.h"
  29 #include "constant.h"
  30 #include "numeric.h"
  31 #include "power.h"
  32 #include "operators.h"
  33 #include "relational.h"
  34 #include "symbol.h"
  35 #include "pseries.h"
  36 #include "utils.h"
  37
  38 namespace GiNaC {
  39
  40 //////////
  41 // exponential function
  42 //////////
  43
  44 static ex exp_evalf(const ex & x)
  45 {
  46         if (is_exactly_a<numeric>(x))
  47                 return exp(ex_to<numeric>(x));
  48
  49         return exp(x).hold();
  50 }
  51
  52 static ex exp_eval(const ex & x)
  53 {
  54         // exp(0) -> 1
  55         if (x.is_zero()) {
  56                 return _ex1;
  57         }
  58
  59         // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
  60         const ex TwoExOverPiI=(_ex2*x)/(Pi*I);
  61         if (TwoExOverPiI.info(info_flags::integer)) {
  62                 const numeric z = mod(ex_to<numeric>(TwoExOverPiI),*_num4_p);
  63                 if (z.is_equal(*_num0_p))
  64                         return _ex1;
  65                 if (z.is_equal(*_num1_p))
  66                         return ex(I);
  67                 if (z.is_equal(*_num2_p))
  68                         return _ex_1;
  69                 if (z.is_equal(*_num3_p))
  70                         return ex(-I);
  71         }
  72
  73         // exp(log(x)) -> x
  74         if (is_ex_the_function(x, log))
  75                 return x.op(0);
  76
  77         // exp(float) -> float
  78         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
  79                 return exp(ex_to<numeric>(x));
  80
  81         return exp(x).hold();
  82 }
  83
  84 static ex exp_deriv(const ex & x, unsigned deriv_param)
  85 {
  86         GINAC_ASSERT(deriv_param==0);
  87
  88         // d/dx exp(x) -> exp(x)
  89         return exp(x);
  90 }
  91
  92 static ex exp_real_part(const ex & x)
  93 {
  94         return exp(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
  95 }
  96
  97 static ex exp_imag_part(const ex & x)
  98 {
  99         return exp(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
 100 }
 101
 102 REGISTER_FUNCTION(exp, eval_func(exp_eval).
 103                        evalf_func(exp_evalf).
 104                        derivative_func(exp_deriv).
 105                        real_part_func(exp_real_part).
 106                        imag_part_func(exp_imag_part).
 107                        latex_name("\\exp"));
 108
 109 //////////
 110 // natural logarithm
 111 //////////
 112
 113 static ex log_evalf(const ex & x)
 114 {
 115         if (is_exactly_a<numeric>(x))
 116                 return log(ex_to<numeric>(x));
 117
 118         return log(x).hold();
 119 }
 120
 121 static ex log_eval(const ex & x)
 122 {
 123         if (x.info(info_flags::numeric)) {
 124                 if (x.is_zero())         // log(0) -> infinity
 125                         throw(pole_error("log_eval(): log(0)",0));
 126                 if (x.info(info_flags::rational) && x.info(info_flags::negative))
 127                         return (log(-x)+I*Pi);
 128                 if (x.is_equal(_ex1))  // log(1) -> 0
 129                         return _ex0;
 130                 if (x.is_equal(I))       // log(I) -> Pi*I/2
 131                         return (Pi*I*_ex1_2);
 132                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
 133                         return (Pi*I*_ex_1_2);
 134
 135                 // log(float) -> float
 136                 if (!x.info(info_flags::crational))
 137                         return log(ex_to<numeric>(x));
 138         }
 139
 140         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
 141         if (is_ex_the_function(x, exp)) {
 142                 const ex &t = x.op(0);
 143                 if (t.info(info_flags::real))
 144                         return t;
 145         }
 146
 147         return log(x).hold();
 148 }
 149
 150 static ex log_deriv(const ex & x, unsigned deriv_param)
 151 {
 152         GINAC_ASSERT(deriv_param==0);
 153
 154         // d/dx log(x) -> 1/x
 155         return power(x, _ex_1);
 156 }
 157
 158 static ex log_series(const ex &arg,
 159                      const relational &rel,
 160                      int order,
 161                      unsigned options)
 162 {
 163         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 164         ex arg_pt;
 165         bool must_expand_arg = false;
 166         // maybe substitution of rel into arg fails because of a pole
 167         try {
 168                 arg_pt = arg.subs(rel, subs_options::no_pattern);
 169         } catch (pole_error) {
 170                 must_expand_arg = true;
 171         }
 172         // or we are at the branch point anyways
 173         if (arg_pt.is_zero())
 174                 must_expand_arg = true;
 175
 176         if (must_expand_arg) {
 177                 // method:
 178                 // This is the branch point: Series expand the argument first, then
 179                 // trivially factorize it to isolate that part which has constant
 180                 // leading coefficient in this fashion:
 181                 //   x^n + x^(n+1) +...+ Order(x^(n+m))  ->  x^n * (1 + x +...+ Order(x^m)).
 182                 // Return a plain n*log(x) for the x^n part and series expand the
 183                 // other part.  Add them together and reexpand again in order to have
 184                 // one unnested pseries object.  All this also works for negative n.
 185                 pseries argser;          // series expansion of log's argument
 186                 unsigned extra_ord = 0;  // extra expansion order
 187                 do {
 188                         // oops, the argument expanded to a pure Order(x^something)...
 189                         argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
 190                         ++extra_ord;
 191                 } while (!argser.is_terminating() && argser.nops()==1);
 192
 193                 const symbol &s = ex_to<symbol>(rel.lhs());
 194                 const ex &point = rel.rhs();
 195                 const int n = argser.ldegree(s);
 196                 epvector seq;
 197                 // construct what we carelessly called the n*log(x) term above
 198                 const ex coeff = argser.coeff(s, n);
 199                 // expand the log, but only if coeff is real and > 0, since otherwise
 200                 // it would make the branch cut run into the wrong direction
 201                 if (coeff.info(info_flags::positive))
 202                         seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
 203                 else
 204                         seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));
 205
 206                 if (!argser.is_terminating() || argser.nops()!=1) {
 207                         // in this case n more (or less) terms are needed
 208                         // (sadly, to generate them, we have to start from the beginning)
 209                         if (n == 0 && coeff == 1) {
 210                                 epvector epv;
 211                                 ex acc = (new pseries(rel, epv))->setflag(status_flags::dynallocated);
 212                                 epv.reserve(2);
 213                                 epv.push_back(expair(-1, _ex0));
 214                                 epv.push_back(expair(Order(_ex1), order));
 215                                 ex rest = pseries(rel, epv).add_series(argser);
 216                                 for (int i = order-1; i>0; --i) {
 217                                         epvector cterm;
 218                                         cterm.reserve(1);
 219                                         cterm.push_back(expair(i%2 ? _ex1/i : _ex_1/i, _ex0));
 220                                         acc = pseries(rel, cterm).add_series(ex_to<pseries>(acc));
 221                                         acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
 222                                 }
 223                                 return acc;
 224                         }
 225                         const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
 226                         return pseries(rel, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
 227                 } else  // it was a monomial
 228                         return pseries(rel, seq);
 229         }
 230         if (!(options & series_options::suppress_branchcut) &&
 231              arg_pt.info(info_flags::negative)) {
 232                 // method:
 233                 // This is the branch cut: assemble the primitive series manually and
 234                 // then add the corresponding complex step function.
 235                 const symbol &s = ex_to<symbol>(rel.lhs());
 236                 const ex &point = rel.rhs();
 237                 const symbol foo;
 238                 const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 239                 epvector seq;
 240                 seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
 241                 seq.push_back(expair(Order(_ex1), order));
 242                 return series(replarg - I*Pi + pseries(rel, seq), rel, order);
 243         }
 244         throw do_taylor();  // caught by function::series()
 245 }
 246
 247 static ex log_real_part(const ex & x)
 248 {
 249         return log(abs(x));
 250 }
 251
 252 static ex log_imag_part(const ex & x)
 253 {
 254         return atan2(GiNaC::imag_part(x), GiNaC::real_part(x));
 255 }
 256
 257 REGISTER_FUNCTION(log, eval_func(log_eval).
 258                        evalf_func(log_evalf).
 259                        derivative_func(log_deriv).
 260                        series_func(log_series).
 261                        real_part_func(log_real_part).
 262                        imag_part_func(log_imag_part).
 263                        latex_name("\\ln"));
 264
 265 //////////
 266 // sine (trigonometric function)
 267 //////////
 268
 269 static ex sin_evalf(const ex & x)
 270 {
 271         if (is_exactly_a<numeric>(x))
 272                 return sin(ex_to<numeric>(x));
 273
 274         return sin(x).hold();
 275 }
 276
 277 static ex sin_eval(const ex & x)
 278 {
 279         // sin(n/d*Pi) -> { all known non-nested radicals }
 280         const ex SixtyExOverPi = _ex60*x/Pi;
 281         ex sign = _ex1;
 282         if (SixtyExOverPi.info(info_flags::integer)) {
 283                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 284                 if (z>=*_num60_p) {
 285                         // wrap to interval [0, Pi)
 286                         z -= *_num60_p;
 287                         sign = _ex_1;
 288                 }
 289                 if (z>*_num30_p) {
 290                         // wrap to interval [0, Pi/2)
 291                         z = *_num60_p-z;
 292                 }
 293                 if (z.is_equal(*_num0_p))  // sin(0)       -> 0
 294                         return _ex0;
 295                 if (z.is_equal(*_num5_p))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
 296                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 297                 if (z.is_equal(*_num6_p))  // sin(Pi/10)   -> sqrt(5)/4-1/4
 298                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 299                 if (z.is_equal(*_num10_p)) // sin(Pi/6)    -> 1/2
 300                         return sign*_ex1_2;
 301                 if (z.is_equal(*_num15_p)) // sin(Pi/4)    -> sqrt(2)/2
 302                         return sign*_ex1_2*sqrt(_ex2);
 303                 if (z.is_equal(*_num18_p)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
 304                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 305                 if (z.is_equal(*_num20_p)) // sin(Pi/3)    -> sqrt(3)/2
 306                         return sign*_ex1_2*sqrt(_ex3);
 307                 if (z.is_equal(*_num25_p)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
 308                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 309                 if (z.is_equal(*_num30_p)) // sin(Pi/2)    -> 1
 310                         return sign;
 311         }
 312
 313         if (is_exactly_a<function>(x)) {
 314                 const ex &t = x.op(0);
 315
 316                 // sin(asin(x)) -> x
 317                 if (is_ex_the_function(x, asin))
 318                         return t;
 319
 320                 // sin(acos(x)) -> sqrt(1-x^2)
 321                 if (is_ex_the_function(x, acos))
 322                         return sqrt(_ex1-power(t,_ex2));
 323
 324                 // sin(atan(x)) -> x/sqrt(1+x^2)
 325                 if (is_ex_the_function(x, atan))
 326                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 327         }
 328
 329         // sin(float) -> float
 330         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 331                 return sin(ex_to<numeric>(x));
 332
 333         // sin() is odd
 334         if (x.info(info_flags::negative))
 335                 return -sin(-x);
 336
 337         return sin(x).hold();
 338 }
 339
 340 static ex sin_deriv(const ex & x, unsigned deriv_param)
 341 {
 342         GINAC_ASSERT(deriv_param==0);
 343
 344         // d/dx sin(x) -> cos(x)
 345         return cos(x);
 346 }
 347
 348 static ex sin_real_part(const ex & x)
 349 {
 350         return cosh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 351 }
 352
 353 static ex sin_imag_part(const ex & x)
 354 {
 355         return sinh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 356 }
 357
 358 REGISTER_FUNCTION(sin, eval_func(sin_eval).
 359                        evalf_func(sin_evalf).
 360                        derivative_func(sin_deriv).
 361                        real_part_func(sin_real_part).
 362                        imag_part_func(sin_imag_part).
 363                        latex_name("\\sin"));
 364
 365 //////////
 366 // cosine (trigonometric function)
 367 //////////
 368
 369 static ex cos_evalf(const ex & x)
 370 {
 371         if (is_exactly_a<numeric>(x))
 372                 return cos(ex_to<numeric>(x));
 373
 374         return cos(x).hold();
 375 }
 376
 377 static ex cos_eval(const ex & x)
 378 {
 379         // cos(n/d*Pi) -> { all known non-nested radicals }
 380         const ex SixtyExOverPi = _ex60*x/Pi;
 381         ex sign = _ex1;
 382         if (SixtyExOverPi.info(info_flags::integer)) {
 383                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 384                 if (z>=*_num60_p) {
 385                         // wrap to interval [0, Pi)
 386                         z = *_num120_p-z;
 387                 }
 388                 if (z>=*_num30_p) {
 389                         // wrap to interval [0, Pi/2)
 390                         z = *_num60_p-z;
 391                         sign = _ex_1;
 392                 }
 393                 if (z.is_equal(*_num0_p))  // cos(0)       -> 1
 394                         return sign;
 395                 if (z.is_equal(*_num5_p))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 396                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 397                 if (z.is_equal(*_num10_p)) // cos(Pi/6)    -> sqrt(3)/2
 398                         return sign*_ex1_2*sqrt(_ex3);
 399                 if (z.is_equal(*_num12_p)) // cos(Pi/5)    -> sqrt(5)/4+1/4
 400                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 401                 if (z.is_equal(*_num15_p)) // cos(Pi/4)    -> sqrt(2)/2
 402                         return sign*_ex1_2*sqrt(_ex2);
 403                 if (z.is_equal(*_num20_p)) // cos(Pi/3)    -> 1/2
 404                         return sign*_ex1_2;
 405                 if (z.is_equal(*_num24_p)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 406                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 407                 if (z.is_equal(*_num25_p)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 408                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 409                 if (z.is_equal(*_num30_p)) // cos(Pi/2)    -> 0
 410                         return _ex0;
 411         }
 412
 413         if (is_exactly_a<function>(x)) {
 414                 const ex &t = x.op(0);
 415
 416                 // cos(acos(x)) -> x
 417                 if (is_ex_the_function(x, acos))
 418                         return t;
 419
 420                 // cos(asin(x)) -> sqrt(1-x^2)
 421                 if (is_ex_the_function(x, asin))
 422                         return sqrt(_ex1-power(t,_ex2));
 423
 424                 // cos(atan(x)) -> 1/sqrt(1+x^2)
 425                 if (is_ex_the_function(x, atan))
 426                         return power(_ex1+power(t,_ex2),_ex_1_2);
 427         }
 428
 429         // cos(float) -> float
 430         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 431                 return cos(ex_to<numeric>(x));
 432
 433         // cos() is even
 434         if (x.info(info_flags::negative))
 435                 return cos(-x);
 436
 437         return cos(x).hold();
 438 }
 439
 440 static ex cos_deriv(const ex & x, unsigned deriv_param)
 441 {
 442         GINAC_ASSERT(deriv_param==0);
 443
 444         // d/dx cos(x) -> -sin(x)
 445         return -sin(x);
 446 }
 447
 448 static ex cos_real_part(const ex & x)
 449 {
 450         return cosh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 451 }
 452
 453 static ex cos_imag_part(const ex & x)
 454 {
 455         return -sinh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 456 }
 457
 458 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 459                        evalf_func(cos_evalf).
 460                        derivative_func(cos_deriv).
 461                        real_part_func(cos_real_part).
 462                        imag_part_func(cos_imag_part).
 463                        latex_name("\\cos"));
 464
 465 //////////
 466 // tangent (trigonometric function)
 467 //////////
 468
 469 static ex tan_evalf(const ex & x)
 470 {
 471         if (is_exactly_a<numeric>(x))
 472                 return tan(ex_to<numeric>(x));
 473
 474         return tan(x).hold();
 475 }
 476
 477 static ex tan_eval(const ex & x)
 478 {
 479         // tan(n/d*Pi) -> { all known non-nested radicals }
 480         const ex SixtyExOverPi = _ex60*x/Pi;
 481         ex sign = _ex1;
 482         if (SixtyExOverPi.info(info_flags::integer)) {
 483                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num60_p);
 484                 if (z>=*_num60_p) {
 485                         // wrap to interval [0, Pi)
 486                         z -= *_num60_p;
 487                 }
 488                 if (z>=*_num30_p) {
 489                         // wrap to interval [0, Pi/2)
 490                         z = *_num60_p-z;
 491                         sign = _ex_1;
 492                 }
 493                 if (z.is_equal(*_num0_p))  // tan(0)       -> 0
 494                         return _ex0;
 495                 if (z.is_equal(*_num5_p))  // tan(Pi/12)   -> 2-sqrt(3)
 496                         return sign*(_ex2-sqrt(_ex3));
 497                 if (z.is_equal(*_num10_p)) // tan(Pi/6)    -> sqrt(3)/3
 498                         return sign*_ex1_3*sqrt(_ex3);
 499                 if (z.is_equal(*_num15_p)) // tan(Pi/4)    -> 1
 500                         return sign;
 501                 if (z.is_equal(*_num20_p)) // tan(Pi/3)    -> sqrt(3)
 502                         return sign*sqrt(_ex3);
 503                 if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3)
 504                         return sign*(sqrt(_ex3)+_ex2);
 505                 if (z.is_equal(*_num30_p)) // tan(Pi/2)    -> infinity
 506                         throw (pole_error("tan_eval(): simple pole",1));
 507         }
 508
 509         if (is_exactly_a<function>(x)) {
 510                 const ex &t = x.op(0);
 511
 512                 // tan(atan(x)) -> x
 513                 if (is_ex_the_function(x, atan))
 514                         return t;
 515
 516                 // tan(asin(x)) -> x/sqrt(1+x^2)
 517                 if (is_ex_the_function(x, asin))
 518                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 519
 520                 // tan(acos(x)) -> sqrt(1-x^2)/x
 521                 if (is_ex_the_function(x, acos))
 522                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
 523         }
 524
 525         // tan(float) -> float
 526         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 527                 return tan(ex_to<numeric>(x));
 528         }
 529
 530         // tan() is odd
 531         if (x.info(info_flags::negative))
 532                 return -tan(-x);
 533
 534         return tan(x).hold();
 535 }
 536
 537 static ex tan_deriv(const ex & x, unsigned deriv_param)
 538 {
 539         GINAC_ASSERT(deriv_param==0);
 540
 541         // d/dx tan(x) -> 1+tan(x)^2;
 542         return (_ex1+power(tan(x),_ex2));
 543 }
 544
 545 static ex tan_real_part(const ex & x)
 546 {
 547         ex a = GiNaC::real_part(x);
 548         ex b = GiNaC::imag_part(x);
 549         return tan(a)/(1+power(tan(a),2)*power(tan(b),2));
 550 }
 551
 552 static ex tan_imag_part(const ex & x)
 553 {
 554         ex a = GiNaC::real_part(x);
 555         ex b = GiNaC::imag_part(x);
 556         return tanh(b)/(1+power(tan(a),2)*power(tan(b),2));
 557 }
 558
 559 static ex tan_series(const ex &x,
 560                      const relational &rel,
 561                      int order,
 562                      unsigned options)
 563 {
 564         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 565         // method:
 566         // Taylor series where there is no pole falls back to tan_deriv.
 567         // On a pole simply expand sin(x)/cos(x).
 568         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 569         if (!(2*x_pt/Pi).info(info_flags::odd))
 570                 throw do_taylor();  // caught by function::series()
 571         // if we got here we have to care for a simple pole
 572         return (sin(x)/cos(x)).series(rel, order, options);
 573 }
 574
 575 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 576                        evalf_func(tan_evalf).
 577                        derivative_func(tan_deriv).
 578                        series_func(tan_series).
 579                        real_part_func(tan_real_part).
 580                        imag_part_func(tan_imag_part).
 581                        latex_name("\\tan"));
 582
 583 //////////
 584 // inverse sine (arc sine)
 585 //////////
 586
 587 static ex asin_evalf(const ex & x)
 588 {
 589         if (is_exactly_a<numeric>(x))
 590                 return asin(ex_to<numeric>(x));
 591
 592         return asin(x).hold();
 593 }
 594
 595 static ex asin_eval(const ex & x)
 596 {
 597         if (x.info(info_flags::numeric)) {
 598
 599                 // asin(0) -> 0
 600                 if (x.is_zero())
 601                         return x;
 602
 603                 // asin(1/2) -> Pi/6
 604                 if (x.is_equal(_ex1_2))
 605                         return numeric(1,6)*Pi;
 606
 607                 // asin(1) -> Pi/2
 608                 if (x.is_equal(_ex1))
 609                         return _ex1_2*Pi;
 610
 611                 // asin(-1/2) -> -Pi/6
 612                 if (x.is_equal(_ex_1_2))
 613                         return numeric(-1,6)*Pi;
 614
 615                 // asin(-1) -> -Pi/2
 616                 if (x.is_equal(_ex_1))
 617                         return _ex_1_2*Pi;
 618
 619                 // asin(float) -> float
 620                 if (!x.info(info_flags::crational))
 621                         return asin(ex_to<numeric>(x));
 622
 623                 // asin() is odd
 624                 if (x.info(info_flags::negative))
 625                         return -asin(-x);
 626         }
 627
 628         return asin(x).hold();
 629 }
 630
 631 static ex asin_deriv(const ex & x, unsigned deriv_param)
 632 {
 633         GINAC_ASSERT(deriv_param==0);
 634
 635         // d/dx asin(x) -> 1/sqrt(1-x^2)
 636         return power(1-power(x,_ex2),_ex_1_2);
 637 }
 638
 639 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 640                         evalf_func(asin_evalf).
 641                         derivative_func(asin_deriv).
 642                         latex_name("\\arcsin"));
 643
 644 //////////
 645 // inverse cosine (arc cosine)
 646 //////////
 647
 648 static ex acos_evalf(const ex & x)
 649 {
 650         if (is_exactly_a<numeric>(x))
 651                 return acos(ex_to<numeric>(x));
 652
 653         return acos(x).hold();
 654 }
 655
 656 static ex acos_eval(const ex & x)
 657 {
 658         if (x.info(info_flags::numeric)) {
 659
 660                 // acos(1) -> 0
 661                 if (x.is_equal(_ex1))
 662                         return _ex0;
 663
 664                 // acos(1/2) -> Pi/3
 665                 if (x.is_equal(_ex1_2))
 666                         return _ex1_3*Pi;
 667
 668                 // acos(0) -> Pi/2
 669                 if (x.is_zero())
 670                         return _ex1_2*Pi;
 671
 672                 // acos(-1/2) -> 2/3*Pi
 673                 if (x.is_equal(_ex_1_2))
 674                         return numeric(2,3)*Pi;
 675
 676                 // acos(-1) -> Pi
 677                 if (x.is_equal(_ex_1))
 678                         return Pi;
 679
 680                 // acos(float) -> float
 681                 if (!x.info(info_flags::crational))
 682                         return acos(ex_to<numeric>(x));
 683
 684                 // acos(-x) -> Pi-acos(x)
 685                 if (x.info(info_flags::negative))
 686                         return Pi-acos(-x);
 687         }
 688
 689         return acos(x).hold();
 690 }
 691
 692 static ex acos_deriv(const ex & x, unsigned deriv_param)
 693 {
 694         GINAC_ASSERT(deriv_param==0);
 695
 696         // d/dx acos(x) -> -1/sqrt(1-x^2)
 697         return -power(1-power(x,_ex2),_ex_1_2);
 698 }
 699
 700 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 701                         evalf_func(acos_evalf).
 702                         derivative_func(acos_deriv).
 703                         latex_name("\\arccos"));
 704
 705 //////////
 706 // inverse tangent (arc tangent)
 707 //////////
 708
 709 static ex atan_evalf(const ex & x)
 710 {
 711         if (is_exactly_a<numeric>(x))
 712                 return atan(ex_to<numeric>(x));
 713
 714         return atan(x).hold();
 715 }
 716
 717 static ex atan_eval(const ex & x)
 718 {
 719         if (x.info(info_flags::numeric)) {
 720
 721                 // atan(0) -> 0
 722                 if (x.is_zero())
 723                         return _ex0;
 724
 725                 // atan(1) -> Pi/4
 726                 if (x.is_equal(_ex1))
 727                         return _ex1_4*Pi;
 728
 729                 // atan(-1) -> -Pi/4
 730                 if (x.is_equal(_ex_1))
 731                         return _ex_1_4*Pi;
 732
 733                 if (x.is_equal(I) || x.is_equal(-I))
 734                         throw (pole_error("atan_eval(): logarithmic pole",0));
 735
 736                 // atan(float) -> float
 737                 if (!x.info(info_flags::crational))
 738                         return atan(ex_to<numeric>(x));
 739
 740                 // atan() is odd
 741                 if (x.info(info_flags::negative))
 742                         return -atan(-x);
 743         }
 744
 745         return atan(x).hold();
 746 }
 747
 748 static ex atan_deriv(const ex & x, unsigned deriv_param)
 749 {
 750         GINAC_ASSERT(deriv_param==0);
 751
 752         // d/dx atan(x) -> 1/(1+x^2)
 753         return power(_ex1+power(x,_ex2), _ex_1);
 754 }
 755
 756 static ex atan_series(const ex &arg,
 757                       const relational &rel,
 758                       int order,
 759                       unsigned options)
 760 {
 761         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 762         // method:
 763         // Taylor series where there is no pole or cut falls back to atan_deriv.
 764         // There are two branch cuts, one runnig from I up the imaginary axis and
 765         // one running from -I down the imaginary axis.  The points I and -I are
 766         // poles.
 767         // On the branch cuts and the poles series expand
 768         //     (log(1+I*x)-log(1-I*x))/(2*I)
 769         // instead.
 770         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 771         if (!(I*arg_pt).info(info_flags::real))
 772                 throw do_taylor();     // Re(x) != 0
 773         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
 774                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
 775         // care for the poles, using the defining formula for atan()...
 776         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
 777                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
 778         if (!(options & series_options::suppress_branchcut)) {
 779                 // method:
 780                 // This is the branch cut: assemble the primitive series manually and
 781                 // then add the corresponding complex step function.
 782                 const symbol &s = ex_to<symbol>(rel.lhs());
 783                 const ex &point = rel.rhs();
 784                 const symbol foo;
 785                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 786                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
 787                 if ((I*arg_pt)<_ex0)
 788                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
 789                 else
 790                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
 791                 epvector seq;
 792                 seq.push_back(expair(Order0correction, _ex0));
 793                 seq.push_back(expair(Order(_ex1), order));
 794                 return series(replarg - pseries(rel, seq), rel, order);
 795         }
 796         throw do_taylor();
 797 }
 798
 799 REGISTER_FUNCTION(atan, eval_func(atan_eval).
 800                         evalf_func(atan_evalf).
 801                         derivative_func(atan_deriv).
 802                         series_func(atan_series).
 803                         latex_name("\\arctan"));
 804
 805 //////////
 806 // inverse tangent (atan2(y,x))
 807 //////////
 808
 809 static ex atan2_evalf(const ex &y, const ex &x)
 810 {
 811         if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
 812                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 813
 814         return atan2(y, x).hold();
 815 }
 816
 817 static ex atan2_eval(const ex & y, const ex & x)
 818 {
 819         if (y.info(info_flags::numeric) && x.info(info_flags::numeric)) {
 820
 821                 if (y.is_zero()) {
 822
 823                         // atan(0, 0) -> 0
 824                         if (x.is_zero())
 825                                 return _ex0;
 826
 827                         // atan(0, x), x real and positive -> 0
 828                         if (x.info(info_flags::positive))
 829                                 return _ex0;
 830
 831                         // atan(0, x), x real and negative -> -Pi
 832                         if (x.info(info_flags::negative))
 833                                 return _ex_1*Pi;
 834                 }
 835
 836                 if (x.is_zero()) {
 837
 838                         // atan(y, 0), y real and positive -> Pi/2
 839                         if (y.info(info_flags::positive))
 840                                 return _ex1_2*Pi;
 841
 842                         // atan(y, 0), y real and negative -> -Pi/2
 843                         if (y.info(info_flags::negative))
 844                                 return _ex_1_2*Pi;
 845                 }
 846
 847                 if (y.is_equal(x)) {
 848
 849                         // atan(y, y), y real and positive -> Pi/4
 850                         if (y.info(info_flags::positive))
 851                                 return _ex1_4*Pi;
 852
 853                         // atan(y, y), y real and negative -> -3/4*Pi
 854                         if (y.info(info_flags::negative))
 855                                 return numeric(-3, 4)*Pi;
 856                 }
 857
 858                 if (y.is_equal(-x)) {
 859
 860                         // atan(y, -y), y real and positive -> 3*Pi/4
 861                         if (y.info(info_flags::positive))
 862                                 return numeric(3, 4)*Pi;
 863
 864                         // atan(y, -y), y real and negative -> -Pi/4
 865                         if (y.info(info_flags::negative))
 866                                 return _ex_1_4*Pi;
 867                 }
 868
 869                 // atan(float, float) -> float
 870                 if (!y.info(info_flags::crational) && !x.info(info_flags::crational))
 871                         return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 872
 873                 // atan(real, real) -> atan(y/x) +/- Pi
 874                 if (y.info(info_flags::real) && x.info(info_flags::real)) {
 875                         if (x.info(info_flags::positive))
 876                                 return atan(y/x);
 877                         else if(y.info(info_flags::positive))
 878                                 return atan(y/x)+Pi;
 879                         else
 880                                 return atan(y/x)-Pi;
 881                 }
 882         }
 883
 884         return atan2(y, x).hold();
 885 }
 886
 887 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
 888 {
 889         GINAC_ASSERT(deriv_param<2);
 890
 891         if (deriv_param==0) {
 892                 // d/dy atan(y,x)
 893                 return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
 894         }
 895         // d/dx atan(y,x)
 896         return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
 897 }
 898
 899 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
 900                          evalf_func(atan2_evalf).
 901                          derivative_func(atan2_deriv));
 902
 903 //////////
 904 // hyperbolic sine (trigonometric function)
 905 //////////
 906
 907 static ex sinh_evalf(const ex & x)
 908 {
 909         if (is_exactly_a<numeric>(x))
 910                 return sinh(ex_to<numeric>(x));
 911
 912         return sinh(x).hold();
 913 }
 914
 915 static ex sinh_eval(const ex & x)
 916 {
 917         if (x.info(info_flags::numeric)) {
 918
 919                 // sinh(0) -> 0
 920                 if (x.is_zero())
 921                         return _ex0;
 922
 923                 // sinh(float) -> float
 924                 if (!x.info(info_flags::crational))
 925                         return sinh(ex_to<numeric>(x));
 926
 927                 // sinh() is odd
 928                 if (x.info(info_flags::negative))
 929                         return -sinh(-x);
 930         }
 931
 932         if ((x/Pi).info(info_flags::numeric) &&
 933                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
 934                 return I*sin(x/I);
 935
 936         if (is_exactly_a<function>(x)) {
 937                 const ex &t = x.op(0);
 938
 939                 // sinh(asinh(x)) -> x
 940                 if (is_ex_the_function(x, asinh))
 941                         return t;
 942
 943                 // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
 944                 if (is_ex_the_function(x, acosh))
 945                         return sqrt(t-_ex1)*sqrt(t+_ex1);
 946
 947                 // sinh(atanh(x)) -> x/sqrt(1-x^2)
 948                 if (is_ex_the_function(x, atanh))
 949                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 950         }
 951
 952         return sinh(x).hold();
 953 }
 954
 955 static ex sinh_deriv(const ex & x, unsigned deriv_param)
 956 {
 957         GINAC_ASSERT(deriv_param==0);
 958
 959         // d/dx sinh(x) -> cosh(x)
 960         return cosh(x);
 961 }
 962
 963 static ex sinh_real_part(const ex & x)
 964 {
 965         return sinh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
 966 }
 967
 968 static ex sinh_imag_part(const ex & x)
 969 {
 970         return cosh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
 971 }
 972
 973 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
 974                         evalf_func(sinh_evalf).
 975                         derivative_func(sinh_deriv).
 976                         real_part_func(sinh_real_part).
 977                         imag_part_func(sinh_imag_part).
 978                         latex_name("\\sinh"));
 979
 980 //////////
 981 // hyperbolic cosine (trigonometric function)
 982 //////////
 983
 984 static ex cosh_evalf(const ex & x)
 985 {
 986         if (is_exactly_a<numeric>(x))
 987                 return cosh(ex_to<numeric>(x));
 988
 989         return cosh(x).hold();
 990 }
 991
 992 static ex cosh_eval(const ex & x)
 993 {
 994         if (x.info(info_flags::numeric)) {
 995
 996                 // cosh(0) -> 1
 997                 if (x.is_zero())
 998                         return _ex1;
 999
1000                 // cosh(float) -> float
1001                 if (!x.info(info_flags::crational))
1002                         return cosh(ex_to<numeric>(x));
1003
1004                 // cosh() is even
1005                 if (x.info(info_flags::negative))
1006                         return cosh(-x);
1007         }
1008
1009         if ((x/Pi).info(info_flags::numeric) &&
1010                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
1011                 return cos(x/I);
1012
1013         if (is_exactly_a<function>(x)) {
1014                 const ex &t = x.op(0);
1015
1016                 // cosh(acosh(x)) -> x
1017                 if (is_ex_the_function(x, acosh))
1018                         return t;
1019
1020                 // cosh(asinh(x)) -> sqrt(1+x^2)
1021                 if (is_ex_the_function(x, asinh))
1022                         return sqrt(_ex1+power(t,_ex2));
1023
1024                 // cosh(atanh(x)) -> 1/sqrt(1-x^2)
1025                 if (is_ex_the_function(x, atanh))
1026                         return power(_ex1-power(t,_ex2),_ex_1_2);
1027         }
1028
1029         return cosh(x).hold();
1030 }
1031
1032 static ex cosh_deriv(const ex & x, unsigned deriv_param)
1033 {
1034         GINAC_ASSERT(deriv_param==0);
1035
1036         // d/dx cosh(x) -> sinh(x)
1037         return sinh(x);
1038 }
1039
1040 static ex cosh_real_part(const ex & x)
1041 {
1042         return cosh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
1043 }
1044
1045 static ex cosh_imag_part(const ex & x)
1046 {
1047         return sinh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
1048 }
1049
1050 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
1051                         evalf_func(cosh_evalf).
1052                         derivative_func(cosh_deriv).
1053                         real_part_func(cosh_real_part).
1054                         imag_part_func(cosh_imag_part).
1055                         latex_name("\\cosh"));
1056
1057 //////////
1058 // hyperbolic tangent (trigonometric function)
1059 //////////
1060
1061 static ex tanh_evalf(const ex & x)
1062 {
1063         if (is_exactly_a<numeric>(x))
1064                 return tanh(ex_to<numeric>(x));
1065
1066         return tanh(x).hold();
1067 }
1068
1069 static ex tanh_eval(const ex & x)
1070 {
1071         if (x.info(info_flags::numeric)) {
1072
1073                 // tanh(0) -> 0
1074                 if (x.is_zero())
1075                         return _ex0;
1076
1077                 // tanh(float) -> float
1078                 if (!x.info(info_flags::crational))
1079                         return tanh(ex_to<numeric>(x));
1080
1081                 // tanh() is odd
1082                 if (x.info(info_flags::negative))
1083                         return -tanh(-x);
1084         }
1085
1086         if ((x/Pi).info(info_flags::numeric) &&
1087                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
1088                 return I*tan(x/I);
1089
1090         if (is_exactly_a<function>(x)) {
1091                 const ex &t = x.op(0);
1092
1093                 // tanh(atanh(x)) -> x
1094                 if (is_ex_the_function(x, atanh))
1095                         return t;
1096
1097                 // tanh(asinh(x)) -> x/sqrt(1+x^2)
1098                 if (is_ex_the_function(x, asinh))
1099                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
1100
1101                 // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
1102                 if (is_ex_the_function(x, acosh))
1103                         return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
1104         }
1105
1106         return tanh(x).hold();
1107 }
1108
1109 static ex tanh_deriv(const ex & x, unsigned deriv_param)
1110 {
1111         GINAC_ASSERT(deriv_param==0);
1112
1113         // d/dx tanh(x) -> 1-tanh(x)^2
1114         return _ex1-power(tanh(x),_ex2);
1115 }
1116
1117 static ex tanh_series(const ex &x,
1118                       const relational &rel,
1119                       int order,
1120                       unsigned options)
1121 {
1122         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1123         // method:
1124         // Taylor series where there is no pole falls back to tanh_deriv.
1125         // On a pole simply expand sinh(x)/cosh(x).
1126         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1127         if (!(2*I*x_pt/Pi).info(info_flags::odd))
1128                 throw do_taylor();  // caught by function::series()
1129         // if we got here we have to care for a simple pole
1130         return (sinh(x)/cosh(x)).series(rel, order, options);
1131 }
1132
1133 static ex tanh_real_part(const ex & x)
1134 {
1135         ex a = GiNaC::real_part(x);
1136         ex b = GiNaC::imag_part(x);
1137         return tanh(a)/(1+power(tanh(a),2)*power(tan(b),2));
1138 }
1139
1140 static ex tanh_imag_part(const ex & x)
1141 {
1142         ex a = GiNaC::real_part(x);
1143         ex b = GiNaC::imag_part(x);
1144         return tan(b)/(1+power(tanh(a),2)*power(tan(b),2));
1145 }
1146
1147 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
1148                         evalf_func(tanh_evalf).
1149                         derivative_func(tanh_deriv).
1150                         series_func(tanh_series).
1151                         real_part_func(tanh_real_part).
1152                         imag_part_func(tanh_imag_part).
1153                         latex_name("\\tanh"));
1154
1155 //////////
1156 // inverse hyperbolic sine (trigonometric function)
1157 //////////
1158
1159 static ex asinh_evalf(const ex & x)
1160 {
1161         if (is_exactly_a<numeric>(x))
1162                 return asinh(ex_to<numeric>(x));
1163
1164         return asinh(x).hold();
1165 }
1166
1167 static ex asinh_eval(const ex & x)
1168 {
1169         if (x.info(info_flags::numeric)) {
1170
1171                 // asinh(0) -> 0
1172                 if (x.is_zero())
1173                         return _ex0;
1174
1175                 // asinh(float) -> float
1176                 if (!x.info(info_flags::crational))
1177                         return asinh(ex_to<numeric>(x));
1178
1179                 // asinh() is odd
1180                 if (x.info(info_flags::negative))
1181                         return -asinh(-x);
1182         }
1183
1184         return asinh(x).hold();
1185 }
1186
1187 static ex asinh_deriv(const ex & x, unsigned deriv_param)
1188 {
1189         GINAC_ASSERT(deriv_param==0);
1190
1191         // d/dx asinh(x) -> 1/sqrt(1+x^2)
1192         return power(_ex1+power(x,_ex2),_ex_1_2);
1193 }
1194
1195 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
1196                          evalf_func(asinh_evalf).
1197                          derivative_func(asinh_deriv));
1198
1199 //////////
1200 // inverse hyperbolic cosine (trigonometric function)
1201 //////////
1202
1203 static ex acosh_evalf(const ex & x)
1204 {
1205         if (is_exactly_a<numeric>(x))
1206                 return acosh(ex_to<numeric>(x));
1207
1208         return acosh(x).hold();
1209 }
1210
1211 static ex acosh_eval(const ex & x)
1212 {
1213         if (x.info(info_flags::numeric)) {
1214
1215                 // acosh(0) -> Pi*I/2
1216                 if (x.is_zero())
1217                         return Pi*I*numeric(1,2);
1218
1219                 // acosh(1) -> 0
1220                 if (x.is_equal(_ex1))
1221                         return _ex0;
1222
1223                 // acosh(-1) -> Pi*I
1224                 if (x.is_equal(_ex_1))
1225                         return Pi*I;
1226
1227                 // acosh(float) -> float
1228                 if (!x.info(info_flags::crational))
1229                         return acosh(ex_to<numeric>(x));
1230
1231                 // acosh(-x) -> Pi*I-acosh(x)
1232                 if (x.info(info_flags::negative))
1233                         return Pi*I-acosh(-x);
1234         }
1235
1236         return acosh(x).hold();
1237 }
1238
1239 static ex acosh_deriv(const ex & x, unsigned deriv_param)
1240 {
1241         GINAC_ASSERT(deriv_param==0);
1242
1243         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
1244         return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
1245 }
1246
1247 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
1248                          evalf_func(acosh_evalf).
1249                          derivative_func(acosh_deriv));
1250
1251 //////////
1252 // inverse hyperbolic tangent (trigonometric function)
1253 //////////
1254
1255 static ex atanh_evalf(const ex & x)
1256 {
1257         if (is_exactly_a<numeric>(x))
1258                 return atanh(ex_to<numeric>(x));
1259
1260         return atanh(x).hold();
1261 }
1262
1263 static ex atanh_eval(const ex & x)
1264 {
1265         if (x.info(info_flags::numeric)) {
1266
1267                 // atanh(0) -> 0
1268                 if (x.is_zero())
1269                         return _ex0;
1270
1271                 // atanh({+|-}1) -> throw
1272                 if (x.is_equal(_ex1) || x.is_equal(_ex_1))
1273                         throw (pole_error("atanh_eval(): logarithmic pole",0));
1274
1275                 // atanh(float) -> float
1276                 if (!x.info(info_flags::crational))
1277                         return atanh(ex_to<numeric>(x));
1278
1279                 // atanh() is odd
1280                 if (x.info(info_flags::negative))
1281                         return -atanh(-x);
1282         }
1283
1284         return atanh(x).hold();
1285 }
1286
1287 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1288 {
1289         GINAC_ASSERT(deriv_param==0);
1290
1291         // d/dx atanh(x) -> 1/(1-x^2)
1292         return power(_ex1-power(x,_ex2),_ex_1);
1293 }
1294
1295 static ex atanh_series(const ex &arg,
1296                        const relational &rel,
1297                        int order,
1298                        unsigned options)
1299 {
1300         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1301         // method:
1302         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1303         // There are two branch cuts, one runnig from 1 up the real axis and one
1304         // one running from -1 down the real axis.  The points 1 and -1 are poles
1305         // On the branch cuts and the poles series expand
1306         //     (log(1+x)-log(1-x))/2
1307         // instead.
1308         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
1309         if (!(arg_pt).info(info_flags::real))
1310                 throw do_taylor();     // Im(x) != 0
1311         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1312                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1313         // care for the poles, using the defining formula for atanh()...
1314         if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1315                 return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1316         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1317         if (!(options & series_options::suppress_branchcut)) {
1318                 // method:
1319                 // This is the branch cut: assemble the primitive series manually and
1320                 // then add the corresponding complex step function.
1321                 const symbol &s = ex_to<symbol>(rel.lhs());
1322                 const ex &point = rel.rhs();
1323                 const symbol foo;
1324                 const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1325                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1326                 if (arg_pt<_ex0)
1327                         Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1328                 else
1329                         Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1330                 epvector seq;
1331                 seq.push_back(expair(Order0correction, _ex0));
1332                 seq.push_back(expair(Order(_ex1), order));
1333                 return series(replarg - pseries(rel, seq), rel, order);
1334         }
1335         throw do_taylor();
1336 }
1337
1338 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1339                          evalf_func(atanh_evalf).
1340                          derivative_func(atanh_deriv).
1341                          series_func(atanh_series));
1342
1343
1344 } // namespace GiNaC