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) 1999-2011 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 "inifcns.h"
  25 #include "ex.h"
  26 #include "constant.h"
  27 #include "add.h"
  28 #include "mul.h"
  29 #include "numeric.h"
  30 #include "power.h"
  31 #include "operators.h"
  32 #include "relational.h"
  33 #include "symbol.h"
  34 #include "pseries.h"
  35 #include "utils.h"
  36
  37 #include <stdexcept>
  38 #include <vector>
  39
  40 namespace GiNaC {
  41
  42 //////////
  43 // exponential function
  44 //////////
  45
  46 static ex exp_evalf(const ex & x)
  47 {
  48         if (is_exactly_a<numeric>(x))
  49                 return exp(ex_to<numeric>(x));
  50
  51         return exp(x).hold();
  52 }
  53
  54 static ex exp_eval(const ex & x)
  55 {
  56         // exp(0) -> 1
  57         if (x.is_zero()) {
  58                 return _ex1;
  59         }
  60
  61         // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
  62         const ex TwoExOverPiI=(_ex2*x)/(Pi*I);
  63         if (TwoExOverPiI.info(info_flags::integer)) {
  64                 const numeric z = mod(ex_to<numeric>(TwoExOverPiI),*_num4_p);
  65                 if (z.is_equal(*_num0_p))
  66                         return _ex1;
  67                 if (z.is_equal(*_num1_p))
  68                         return ex(I);
  69                 if (z.is_equal(*_num2_p))
  70                         return _ex_1;
  71                 if (z.is_equal(*_num3_p))
  72                         return ex(-I);
  73         }
  74
  75         // exp(log(x)) -> x
  76         if (is_ex_the_function(x, log))
  77                 return x.op(0);
  78
  79         // exp(float) -> float
  80         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
  81                 return exp(ex_to<numeric>(x));
  82
  83         return exp(x).hold();
  84 }
  85
  86 static ex exp_expand(const ex & arg, unsigned options)
  87 {
  88         ex exp_arg;
  89         if (options & expand_options::expand_function_args)
  90                 exp_arg = arg.expand(options);
  91         else
  92                 exp_arg=arg;
  93
  94         if ((options & expand_options::expand_transcendental)
  95                 && is_exactly_a<add>(exp_arg)) {
  96                 exvector prodseq;
  97                 prodseq.reserve(exp_arg.nops());
  98                 for (const_iterator i = exp_arg.begin(); i != exp_arg.end(); ++i)
  99                         prodseq.push_back(exp(*i));
 100
 101                 return (new mul(prodseq))->setflag(status_flags::dynallocated | status_flags::expanded);
 102         }
 103
 104         return exp(exp_arg).hold();
 105 }
 106
 107 static ex exp_deriv(const ex & x, unsigned deriv_param)
 108 {
 109         GINAC_ASSERT(deriv_param==0);
 110
 111         // d/dx exp(x) -> exp(x)
 112         return exp(x);
 113 }
 114
 115 static ex exp_real_part(const ex & x)
 116 {
 117         return exp(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
 118 }
 119
 120 static ex exp_imag_part(const ex & x)
 121 {
 122         return exp(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
 123 }
 124
 125 static ex exp_conjugate(const ex & x)
 126 {
 127         // conjugate(exp(x))==exp(conjugate(x))
 128         return exp(x.conjugate());
 129 }
 130
 131 REGISTER_FUNCTION(exp, eval_func(exp_eval).
 132                        evalf_func(exp_evalf).
 133                        expand_func(exp_expand).
 134                        derivative_func(exp_deriv).
 135                        real_part_func(exp_real_part).
 136                        imag_part_func(exp_imag_part).
 137                        conjugate_func(exp_conjugate).
 138                        latex_name("\\exp"));
 139
 140 //////////
 141 // natural logarithm
 142 //////////
 143
 144 static ex log_evalf(const ex & x)
 145 {
 146         if (is_exactly_a<numeric>(x))
 147                 return log(ex_to<numeric>(x));
 148
 149         return log(x).hold();
 150 }
 151
 152 static ex log_eval(const ex & x)
 153 {
 154         if (x.info(info_flags::numeric)) {
 155                 if (x.is_zero())         // log(0) -> infinity
 156                         throw(pole_error("log_eval(): log(0)",0));
 157                 if (x.info(info_flags::rational) && x.info(info_flags::negative))
 158                         return (log(-x)+I*Pi);
 159                 if (x.is_equal(_ex1))  // log(1) -> 0
 160                         return _ex0;
 161                 if (x.is_equal(I))       // log(I) -> Pi*I/2
 162                         return (Pi*I*_ex1_2);
 163                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
 164                         return (Pi*I*_ex_1_2);
 165
 166                 // log(float) -> float
 167                 if (!x.info(info_flags::crational))
 168                         return log(ex_to<numeric>(x));
 169         }
 170
 171         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
 172         if (is_ex_the_function(x, exp)) {
 173                 const ex &t = x.op(0);
 174                 if (t.info(info_flags::real))
 175                         return t;
 176         }
 177
 178         // log(p^a) -> a*log(p), if p>0 and a is real
 179         if (is_exactly_a<power>(x) && x.op(0).info(info_flags::positive) && x.op(1).info(info_flags::real)) {
 180                 return x.op(1)*log(x.op(0));
 181         }
 182
 183         return log(x).hold();
 184 }
 185
 186 static ex log_deriv(const ex & x, unsigned deriv_param)
 187 {
 188         GINAC_ASSERT(deriv_param==0);
 189
 190         // d/dx log(x) -> 1/x
 191         return power(x, _ex_1);
 192 }
 193
 194 static ex log_series(const ex &arg,
 195                      const relational &rel,
 196                      int order,
 197                      unsigned options)
 198 {
 199         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 200         ex arg_pt;
 201         bool must_expand_arg = false;
 202         // maybe substitution of rel into arg fails because of a pole
 203         try {
 204                 arg_pt = arg.subs(rel, subs_options::no_pattern);
 205         } catch (pole_error) {
 206                 must_expand_arg = true;
 207         }
 208         // or we are at the branch point anyways
 209         if (arg_pt.is_zero())
 210                 must_expand_arg = true;
 211
 212         if (must_expand_arg) {
 213                 // method:
 214                 // This is the branch point: Series expand the argument first, then
 215                 // trivially factorize it to isolate that part which has constant
 216                 // leading coefficient in this fashion:
 217                 //   x^n + x^(n+1) +...+ Order(x^(n+m))  ->  x^n * (1 + x +...+ Order(x^m)).
 218                 // Return a plain n*log(x) for the x^n part and series expand the
 219                 // other part.  Add them together and reexpand again in order to have
 220                 // one unnested pseries object.  All this also works for negative n.
 221                 pseries argser;          // series expansion of log's argument
 222                 unsigned extra_ord = 0;  // extra expansion order
 223                 do {
 224                         // oops, the argument expanded to a pure Order(x^something)...
 225                         argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
 226                         ++extra_ord;
 227                 } while (!argser.is_terminating() && argser.nops()==1);
 228
 229                 const symbol &s = ex_to<symbol>(rel.lhs());
 230                 const ex &point = rel.rhs();
 231                 const int n = argser.ldegree(s);
 232                 epvector seq;
 233                 // construct what we carelessly called the n*log(x) term above
 234                 const ex coeff = argser.coeff(s, n);
 235                 // expand the log, but only if coeff is real and > 0, since otherwise
 236                 // it would make the branch cut run into the wrong direction
 237                 if (coeff.info(info_flags::positive))
 238                         seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
 239                 else
 240                         seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));
 241
 242                 if (!argser.is_terminating() || argser.nops()!=1) {
 243                         // in this case n more (or less) terms are needed
 244                         // (sadly, to generate them, we have to start from the beginning)
 245                         if (n == 0 && coeff == 1) {
 246                                 epvector epv;
 247                                 ex acc = (new pseries(rel, epv))->setflag(status_flags::dynallocated);
 248                                 epv.reserve(2);
 249                                 epv.push_back(expair(-1, _ex0));
 250                                 epv.push_back(expair(Order(_ex1), order));
 251                                 ex rest = pseries(rel, epv).add_series(argser);
 252                                 for (int i = order-1; i>0; --i) {
 253                                         epvector cterm;
 254                                         cterm.reserve(1);
 255                                         cterm.push_back(expair(i%2 ? _ex1/i : _ex_1/i, _ex0));
 256                                         acc = pseries(rel, cterm).add_series(ex_to<pseries>(acc));
 257                                         acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
 258                                 }
 259                                 return acc;
 260                         }
 261                         const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
 262                         return pseries(rel, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
 263                 } else  // it was a monomial
 264                         return pseries(rel, seq);
 265         }
 266         if (!(options & series_options::suppress_branchcut) &&
 267              arg_pt.info(info_flags::negative)) {
 268                 // method:
 269                 // This is the branch cut: assemble the primitive series manually and
 270                 // then add the corresponding complex step function.
 271                 const symbol &s = ex_to<symbol>(rel.lhs());
 272                 const ex &point = rel.rhs();
 273                 const symbol foo;
 274                 const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 275                 epvector seq;
 276                 seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
 277                 seq.push_back(expair(Order(_ex1), order));
 278                 return series(replarg - I*Pi + pseries(rel, seq), rel, order);
 279         }
 280         throw do_taylor();  // caught by function::series()
 281 }
 282
 283 static ex log_real_part(const ex & x)
 284 {
 285         if (x.info(info_flags::nonnegative))
 286                 return log(x).hold();
 287         return log(abs(x));
 288 }
 289
 290 static ex log_imag_part(const ex & x)
 291 {
 292         if (x.info(info_flags::nonnegative))
 293                 return 0;
 294         return atan2(GiNaC::imag_part(x), GiNaC::real_part(x));
 295 }
 296
 297 static ex log_expand(const ex & arg, unsigned options)
 298 {
 299         if ((options & expand_options::expand_transcendental)
 300                 && is_exactly_a<mul>(arg) && !arg.info(info_flags::indefinite)) {
 301                 exvector sumseq;
 302                 exvector prodseq;
 303                 sumseq.reserve(arg.nops());
 304                 prodseq.reserve(arg.nops());
 305                 bool possign=true;
 306
 307                 // searching for positive/negative factors
 308                 for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
 309                         ex e;
 310                         if (options & expand_options::expand_function_args)
 311                                 e=i->expand(options);
 312                         else
 313                                 e=*i;
 314                         if (e.info(info_flags::positive))
 315                                 sumseq.push_back(log(e));
 316                         else if (e.info(info_flags::negative)) {
 317                                 sumseq.push_back(log(-e));
 318                                 possign = !possign;
 319                         } else
 320                                 prodseq.push_back(e);
 321                 }
 322
 323                 if (sumseq.size() > 0) {
 324                         ex newarg;
 325                         if (options & expand_options::expand_function_args)
 326                                 newarg=((possign?_ex1:_ex_1)*mul(prodseq)).expand(options);
 327                         else {
 328                                 newarg=(possign?_ex1:_ex_1)*mul(prodseq);
 329                                 ex_to<basic>(newarg).setflag(status_flags::purely_indefinite);
 330                         }
 331                         return add(sumseq)+log(newarg);
 332                 } else {
 333                         if (!(options & expand_options::expand_function_args))
 334                                 ex_to<basic>(arg).setflag(status_flags::purely_indefinite);
 335                 }
 336         }
 337
 338         if (options & expand_options::expand_function_args)
 339                 return log(arg.expand(options)).hold();
 340         else
 341                 return log(arg).hold();
 342 }
 343
 344 static ex log_conjugate(const ex & x)
 345 {
 346         // conjugate(log(x))==log(conjugate(x)) unless on the branch cut which
 347         // runs along the negative real axis.
 348         if (x.info(info_flags::positive)) {
 349                 return log(x);
 350         }
 351         if (is_exactly_a<numeric>(x) &&
 352             !x.imag_part().is_zero()) {
 353                 return log(x.conjugate());
 354         }
 355         return conjugate_function(log(x)).hold();
 356 }
 357
 358 REGISTER_FUNCTION(log, eval_func(log_eval).
 359                        evalf_func(log_evalf).
 360                        expand_func(log_expand).
 361                        derivative_func(log_deriv).
 362                        series_func(log_series).
 363                        real_part_func(log_real_part).
 364                        imag_part_func(log_imag_part).
 365                        conjugate_func(log_conjugate).
 366                        latex_name("\\ln"));
 367
 368 //////////
 369 // sine (trigonometric function)
 370 //////////
 371
 372 static ex sin_evalf(const ex & x)
 373 {
 374         if (is_exactly_a<numeric>(x))
 375                 return sin(ex_to<numeric>(x));
 376
 377         return sin(x).hold();
 378 }
 379
 380 static ex sin_eval(const ex & x)
 381 {
 382         // sin(n/d*Pi) -> { all known non-nested radicals }
 383         const ex SixtyExOverPi = _ex60*x/Pi;
 384         ex sign = _ex1;
 385         if (SixtyExOverPi.info(info_flags::integer)) {
 386                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 387                 if (z>=*_num60_p) {
 388                         // wrap to interval [0, Pi)
 389                         z -= *_num60_p;
 390                         sign = _ex_1;
 391                 }
 392                 if (z>*_num30_p) {
 393                         // wrap to interval [0, Pi/2)
 394                         z = *_num60_p-z;
 395                 }
 396                 if (z.is_equal(*_num0_p))  // sin(0)       -> 0
 397                         return _ex0;
 398                 if (z.is_equal(*_num5_p))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
 399                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 400                 if (z.is_equal(*_num6_p))  // sin(Pi/10)   -> sqrt(5)/4-1/4
 401                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 402                 if (z.is_equal(*_num10_p)) // sin(Pi/6)    -> 1/2
 403                         return sign*_ex1_2;
 404                 if (z.is_equal(*_num15_p)) // sin(Pi/4)    -> sqrt(2)/2
 405                         return sign*_ex1_2*sqrt(_ex2);
 406                 if (z.is_equal(*_num18_p)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
 407                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 408                 if (z.is_equal(*_num20_p)) // sin(Pi/3)    -> sqrt(3)/2
 409                         return sign*_ex1_2*sqrt(_ex3);
 410                 if (z.is_equal(*_num25_p)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
 411                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 412                 if (z.is_equal(*_num30_p)) // sin(Pi/2)    -> 1
 413                         return sign;
 414         }
 415
 416         if (is_exactly_a<function>(x)) {
 417                 const ex &t = x.op(0);
 418
 419                 // sin(asin(x)) -> x
 420                 if (is_ex_the_function(x, asin))
 421                         return t;
 422
 423                 // sin(acos(x)) -> sqrt(1-x^2)
 424                 if (is_ex_the_function(x, acos))
 425                         return sqrt(_ex1-power(t,_ex2));
 426
 427                 // sin(atan(x)) -> x/sqrt(1+x^2)
 428                 if (is_ex_the_function(x, atan))
 429                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 430         }
 431
 432         // sin(float) -> float
 433         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 434                 return sin(ex_to<numeric>(x));
 435
 436         // sin() is odd
 437         if (x.info(info_flags::negative))
 438                 return -sin(-x);
 439
 440         return sin(x).hold();
 441 }
 442
 443 static ex sin_deriv(const ex & x, unsigned deriv_param)
 444 {
 445         GINAC_ASSERT(deriv_param==0);
 446
 447         // d/dx sin(x) -> cos(x)
 448         return cos(x);
 449 }
 450
 451 static ex sin_real_part(const ex & x)
 452 {
 453         return cosh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 454 }
 455
 456 static ex sin_imag_part(const ex & x)
 457 {
 458         return sinh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 459 }
 460
 461 static ex sin_conjugate(const ex & x)
 462 {
 463         // conjugate(sin(x))==sin(conjugate(x))
 464         return sin(x.conjugate());
 465 }
 466
 467 REGISTER_FUNCTION(sin, eval_func(sin_eval).
 468                        evalf_func(sin_evalf).
 469                        derivative_func(sin_deriv).
 470                        real_part_func(sin_real_part).
 471                        imag_part_func(sin_imag_part).
 472                        conjugate_func(sin_conjugate).
 473                        latex_name("\\sin"));
 474
 475 //////////
 476 // cosine (trigonometric function)
 477 //////////
 478
 479 static ex cos_evalf(const ex & x)
 480 {
 481         if (is_exactly_a<numeric>(x))
 482                 return cos(ex_to<numeric>(x));
 483
 484         return cos(x).hold();
 485 }
 486
 487 static ex cos_eval(const ex & x)
 488 {
 489         // cos(n/d*Pi) -> { all known non-nested radicals }
 490         const ex SixtyExOverPi = _ex60*x/Pi;
 491         ex sign = _ex1;
 492         if (SixtyExOverPi.info(info_flags::integer)) {
 493                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 494                 if (z>=*_num60_p) {
 495                         // wrap to interval [0, Pi)
 496                         z = *_num120_p-z;
 497                 }
 498                 if (z>=*_num30_p) {
 499                         // wrap to interval [0, Pi/2)
 500                         z = *_num60_p-z;
 501                         sign = _ex_1;
 502                 }
 503                 if (z.is_equal(*_num0_p))  // cos(0)       -> 1
 504                         return sign;
 505                 if (z.is_equal(*_num5_p))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 506                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 507                 if (z.is_equal(*_num10_p)) // cos(Pi/6)    -> sqrt(3)/2
 508                         return sign*_ex1_2*sqrt(_ex3);
 509                 if (z.is_equal(*_num12_p)) // cos(Pi/5)    -> sqrt(5)/4+1/4
 510                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 511                 if (z.is_equal(*_num15_p)) // cos(Pi/4)    -> sqrt(2)/2
 512                         return sign*_ex1_2*sqrt(_ex2);
 513                 if (z.is_equal(*_num20_p)) // cos(Pi/3)    -> 1/2
 514                         return sign*_ex1_2;
 515                 if (z.is_equal(*_num24_p)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 516                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 517                 if (z.is_equal(*_num25_p)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 518                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 519                 if (z.is_equal(*_num30_p)) // cos(Pi/2)    -> 0
 520                         return _ex0;
 521         }
 522
 523         if (is_exactly_a<function>(x)) {
 524                 const ex &t = x.op(0);
 525
 526                 // cos(acos(x)) -> x
 527                 if (is_ex_the_function(x, acos))
 528                         return t;
 529
 530                 // cos(asin(x)) -> sqrt(1-x^2)
 531                 if (is_ex_the_function(x, asin))
 532                         return sqrt(_ex1-power(t,_ex2));
 533
 534                 // cos(atan(x)) -> 1/sqrt(1+x^2)
 535                 if (is_ex_the_function(x, atan))
 536                         return power(_ex1+power(t,_ex2),_ex_1_2);
 537         }
 538
 539         // cos(float) -> float
 540         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 541                 return cos(ex_to<numeric>(x));
 542
 543         // cos() is even
 544         if (x.info(info_flags::negative))
 545                 return cos(-x);
 546
 547         return cos(x).hold();
 548 }
 549
 550 static ex cos_deriv(const ex & x, unsigned deriv_param)
 551 {
 552         GINAC_ASSERT(deriv_param==0);
 553
 554         // d/dx cos(x) -> -sin(x)
 555         return -sin(x);
 556 }
 557
 558 static ex cos_real_part(const ex & x)
 559 {
 560         return cosh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 561 }
 562
 563 static ex cos_imag_part(const ex & x)
 564 {
 565         return -sinh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 566 }
 567
 568 static ex cos_conjugate(const ex & x)
 569 {
 570         // conjugate(cos(x))==cos(conjugate(x))
 571         return cos(x.conjugate());
 572 }
 573
 574 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 575                        evalf_func(cos_evalf).
 576                        derivative_func(cos_deriv).
 577                        real_part_func(cos_real_part).
 578                        imag_part_func(cos_imag_part).
 579                        conjugate_func(cos_conjugate).
 580                        latex_name("\\cos"));
 581
 582 //////////
 583 // tangent (trigonometric function)
 584 //////////
 585
 586 static ex tan_evalf(const ex & x)
 587 {
 588         if (is_exactly_a<numeric>(x))
 589                 return tan(ex_to<numeric>(x));
 590
 591         return tan(x).hold();
 592 }
 593
 594 static ex tan_eval(const ex & x)
 595 {
 596         // tan(n/d*Pi) -> { all known non-nested radicals }
 597         const ex SixtyExOverPi = _ex60*x/Pi;
 598         ex sign = _ex1;
 599         if (SixtyExOverPi.info(info_flags::integer)) {
 600                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num60_p);
 601                 if (z>=*_num60_p) {
 602                         // wrap to interval [0, Pi)
 603                         z -= *_num60_p;
 604                 }
 605                 if (z>=*_num30_p) {
 606                         // wrap to interval [0, Pi/2)
 607                         z = *_num60_p-z;
 608                         sign = _ex_1;
 609                 }
 610                 if (z.is_equal(*_num0_p))  // tan(0)       -> 0
 611                         return _ex0;
 612                 if (z.is_equal(*_num5_p))  // tan(Pi/12)   -> 2-sqrt(3)
 613                         return sign*(_ex2-sqrt(_ex3));
 614                 if (z.is_equal(*_num10_p)) // tan(Pi/6)    -> sqrt(3)/3
 615                         return sign*_ex1_3*sqrt(_ex3);
 616                 if (z.is_equal(*_num15_p)) // tan(Pi/4)    -> 1
 617                         return sign;
 618                 if (z.is_equal(*_num20_p)) // tan(Pi/3)    -> sqrt(3)
 619                         return sign*sqrt(_ex3);
 620                 if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3)
 621                         return sign*(sqrt(_ex3)+_ex2);
 622                 if (z.is_equal(*_num30_p)) // tan(Pi/2)    -> infinity
 623                         throw (pole_error("tan_eval(): simple pole",1));
 624         }
 625
 626         if (is_exactly_a<function>(x)) {
 627                 const ex &t = x.op(0);
 628
 629                 // tan(atan(x)) -> x
 630                 if (is_ex_the_function(x, atan))
 631                         return t;
 632
 633                 // tan(asin(x)) -> x/sqrt(1+x^2)
 634                 if (is_ex_the_function(x, asin))
 635                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 636
 637                 // tan(acos(x)) -> sqrt(1-x^2)/x
 638                 if (is_ex_the_function(x, acos))
 639                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
 640         }
 641
 642         // tan(float) -> float
 643         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 644                 return tan(ex_to<numeric>(x));
 645         }
 646
 647         // tan() is odd
 648         if (x.info(info_flags::negative))
 649                 return -tan(-x);
 650
 651         return tan(x).hold();
 652 }
 653
 654 static ex tan_deriv(const ex & x, unsigned deriv_param)
 655 {
 656         GINAC_ASSERT(deriv_param==0);
 657
 658         // d/dx tan(x) -> 1+tan(x)^2;
 659         return (_ex1+power(tan(x),_ex2));
 660 }
 661
 662 static ex tan_real_part(const ex & x)
 663 {
 664         ex a = GiNaC::real_part(x);
 665         ex b = GiNaC::imag_part(x);
 666         return tan(a)/(1+power(tan(a),2)*power(tan(b),2));
 667 }
 668
 669 static ex tan_imag_part(const ex & x)
 670 {
 671         ex a = GiNaC::real_part(x);
 672         ex b = GiNaC::imag_part(x);
 673         return tanh(b)/(1+power(tan(a),2)*power(tan(b),2));
 674 }
 675
 676 static ex tan_series(const ex &x,
 677                      const relational &rel,
 678                      int order,
 679                      unsigned options)
 680 {
 681         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 682         // method:
 683         // Taylor series where there is no pole falls back to tan_deriv.
 684         // On a pole simply expand sin(x)/cos(x).
 685         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 686         if (!(2*x_pt/Pi).info(info_flags::odd))
 687                 throw do_taylor();  // caught by function::series()
 688         // if we got here we have to care for a simple pole
 689         return (sin(x)/cos(x)).series(rel, order, options);
 690 }
 691
 692 static ex tan_conjugate(const ex & x)
 693 {
 694         // conjugate(tan(x))==tan(conjugate(x))
 695         return tan(x.conjugate());
 696 }
 697
 698 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 699                        evalf_func(tan_evalf).
 700                        derivative_func(tan_deriv).
 701                        series_func(tan_series).
 702                        real_part_func(tan_real_part).
 703                        imag_part_func(tan_imag_part).
 704                        conjugate_func(tan_conjugate).
 705                        latex_name("\\tan"));
 706
 707 //////////
 708 // inverse sine (arc sine)
 709 //////////
 710
 711 static ex asin_evalf(const ex & x)
 712 {
 713         if (is_exactly_a<numeric>(x))
 714                 return asin(ex_to<numeric>(x));
 715
 716         return asin(x).hold();
 717 }
 718
 719 static ex asin_eval(const ex & x)
 720 {
 721         if (x.info(info_flags::numeric)) {
 722
 723                 // asin(0) -> 0
 724                 if (x.is_zero())
 725                         return x;
 726
 727                 // asin(1/2) -> Pi/6
 728                 if (x.is_equal(_ex1_2))
 729                         return numeric(1,6)*Pi;
 730
 731                 // asin(1) -> Pi/2
 732                 if (x.is_equal(_ex1))
 733                         return _ex1_2*Pi;
 734
 735                 // asin(-1/2) -> -Pi/6
 736                 if (x.is_equal(_ex_1_2))
 737                         return numeric(-1,6)*Pi;
 738
 739                 // asin(-1) -> -Pi/2
 740                 if (x.is_equal(_ex_1))
 741                         return _ex_1_2*Pi;
 742
 743                 // asin(float) -> float
 744                 if (!x.info(info_flags::crational))
 745                         return asin(ex_to<numeric>(x));
 746
 747                 // asin() is odd
 748                 if (x.info(info_flags::negative))
 749                         return -asin(-x);
 750         }
 751
 752         return asin(x).hold();
 753 }
 754
 755 static ex asin_deriv(const ex & x, unsigned deriv_param)
 756 {
 757         GINAC_ASSERT(deriv_param==0);
 758
 759         // d/dx asin(x) -> 1/sqrt(1-x^2)
 760         return power(1-power(x,_ex2),_ex_1_2);
 761 }
 762
 763 static ex asin_conjugate(const ex & x)
 764 {
 765         // conjugate(asin(x))==asin(conjugate(x)) unless on the branch cuts which
 766         // run along the real axis outside the interval [-1, +1].
 767         if (is_exactly_a<numeric>(x) &&
 768             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
 769                 return asin(x.conjugate());
 770         }
 771         return conjugate_function(asin(x)).hold();
 772 }
 773
 774 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 775                         evalf_func(asin_evalf).
 776                         derivative_func(asin_deriv).
 777                         conjugate_func(asin_conjugate).
 778                         latex_name("\\arcsin"));
 779
 780 //////////
 781 // inverse cosine (arc cosine)
 782 //////////
 783
 784 static ex acos_evalf(const ex & x)
 785 {
 786         if (is_exactly_a<numeric>(x))
 787                 return acos(ex_to<numeric>(x));
 788
 789         return acos(x).hold();
 790 }
 791
 792 static ex acos_eval(const ex & x)
 793 {
 794         if (x.info(info_flags::numeric)) {
 795
 796                 // acos(1) -> 0
 797                 if (x.is_equal(_ex1))
 798                         return _ex0;
 799
 800                 // acos(1/2) -> Pi/3
 801                 if (x.is_equal(_ex1_2))
 802                         return _ex1_3*Pi;
 803
 804                 // acos(0) -> Pi/2
 805                 if (x.is_zero())
 806                         return _ex1_2*Pi;
 807
 808                 // acos(-1/2) -> 2/3*Pi
 809                 if (x.is_equal(_ex_1_2))
 810                         return numeric(2,3)*Pi;
 811
 812                 // acos(-1) -> Pi
 813                 if (x.is_equal(_ex_1))
 814                         return Pi;
 815
 816                 // acos(float) -> float
 817                 if (!x.info(info_flags::crational))
 818                         return acos(ex_to<numeric>(x));
 819
 820                 // acos(-x) -> Pi-acos(x)
 821                 if (x.info(info_flags::negative))
 822                         return Pi-acos(-x);
 823         }
 824
 825         return acos(x).hold();
 826 }
 827
 828 static ex acos_deriv(const ex & x, unsigned deriv_param)
 829 {
 830         GINAC_ASSERT(deriv_param==0);
 831
 832         // d/dx acos(x) -> -1/sqrt(1-x^2)
 833         return -power(1-power(x,_ex2),_ex_1_2);
 834 }
 835
 836 static ex acos_conjugate(const ex & x)
 837 {
 838         // conjugate(acos(x))==acos(conjugate(x)) unless on the branch cuts which
 839         // run along the real axis outside the interval [-1, +1].
 840         if (is_exactly_a<numeric>(x) &&
 841             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
 842                 return acos(x.conjugate());
 843         }
 844         return conjugate_function(acos(x)).hold();
 845 }
 846
 847 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 848                         evalf_func(acos_evalf).
 849                         derivative_func(acos_deriv).
 850                         conjugate_func(acos_conjugate).
 851                         latex_name("\\arccos"));
 852
 853 //////////
 854 // inverse tangent (arc tangent)
 855 //////////
 856
 857 static ex atan_evalf(const ex & x)
 858 {
 859         if (is_exactly_a<numeric>(x))
 860                 return atan(ex_to<numeric>(x));
 861
 862         return atan(x).hold();
 863 }
 864
 865 static ex atan_eval(const ex & x)
 866 {
 867         if (x.info(info_flags::numeric)) {
 868
 869                 // atan(0) -> 0
 870                 if (x.is_zero())
 871                         return _ex0;
 872
 873                 // atan(1) -> Pi/4
 874                 if (x.is_equal(_ex1))
 875                         return _ex1_4*Pi;
 876
 877                 // atan(-1) -> -Pi/4
 878                 if (x.is_equal(_ex_1))
 879                         return _ex_1_4*Pi;
 880
 881                 if (x.is_equal(I) || x.is_equal(-I))
 882                         throw (pole_error("atan_eval(): logarithmic pole",0));
 883
 884                 // atan(float) -> float
 885                 if (!x.info(info_flags::crational))
 886                         return atan(ex_to<numeric>(x));
 887
 888                 // atan() is odd
 889                 if (x.info(info_flags::negative))
 890                         return -atan(-x);
 891         }
 892
 893         return atan(x).hold();
 894 }
 895
 896 static ex atan_deriv(const ex & x, unsigned deriv_param)
 897 {
 898         GINAC_ASSERT(deriv_param==0);
 899
 900         // d/dx atan(x) -> 1/(1+x^2)
 901         return power(_ex1+power(x,_ex2), _ex_1);
 902 }
 903
 904 static ex atan_series(const ex &arg,
 905                       const relational &rel,
 906                       int order,
 907                       unsigned options)
 908 {
 909         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 910         // method:
 911         // Taylor series where there is no pole or cut falls back to atan_deriv.
 912         // There are two branch cuts, one runnig from I up the imaginary axis and
 913         // one running from -I down the imaginary axis.  The points I and -I are
 914         // poles.
 915         // On the branch cuts and the poles series expand
 916         //     (log(1+I*x)-log(1-I*x))/(2*I)
 917         // instead.
 918         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 919         if (!(I*arg_pt).info(info_flags::real))
 920                 throw do_taylor();     // Re(x) != 0
 921         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
 922                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
 923         // care for the poles, using the defining formula for atan()...
 924         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
 925                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
 926         if (!(options & series_options::suppress_branchcut)) {
 927                 // method:
 928                 // This is the branch cut: assemble the primitive series manually and
 929                 // then add the corresponding complex step function.
 930                 const symbol &s = ex_to<symbol>(rel.lhs());
 931                 const ex &point = rel.rhs();
 932                 const symbol foo;
 933                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 934                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
 935                 if ((I*arg_pt)<_ex0)
 936                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
 937                 else
 938                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
 939                 epvector seq;
 940                 seq.push_back(expair(Order0correction, _ex0));
 941                 seq.push_back(expair(Order(_ex1), order));
 942                 return series(replarg - pseries(rel, seq), rel, order);
 943         }
 944         throw do_taylor();
 945 }
 946
 947 static ex atan_conjugate(const ex & x)
 948 {
 949         // conjugate(atan(x))==atan(conjugate(x)) unless on the branch cuts which
 950         // run along the imaginary axis outside the interval [-I, +I].
 951         if (x.info(info_flags::real))
 952                 return atan(x);
 953         if (is_exactly_a<numeric>(x)) {
 954                 const numeric x_re = ex_to<numeric>(x.real_part());
 955                 const numeric x_im = ex_to<numeric>(x.imag_part());
 956                 if (!x_re.is_zero() ||
 957                     (x_im > *_num_1_p && x_im < *_num1_p))
 958                         return atan(x.conjugate());
 959         }
 960         return conjugate_function(atan(x)).hold();
 961 }
 962
 963 REGISTER_FUNCTION(atan, eval_func(atan_eval).
 964                         evalf_func(atan_evalf).
 965                         derivative_func(atan_deriv).
 966                         series_func(atan_series).
 967                         conjugate_func(atan_conjugate).
 968                         latex_name("\\arctan"));
 969
 970 //////////
 971 // inverse tangent (atan2(y,x))
 972 //////////
 973
 974 static ex atan2_evalf(const ex &y, const ex &x)
 975 {
 976         if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
 977                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 978
 979         return atan2(y, x).hold();
 980 }
 981
 982 static ex atan2_eval(const ex & y, const ex & x)
 983 {
 984         if (y.is_zero()) {
 985
 986                 // atan2(0, 0) -> 0
 987                 if (x.is_zero())
 988                         return _ex0;
 989
 990                 // atan2(0, x), x real and positive -> 0
 991                 if (x.info(info_flags::positive))
 992                         return _ex0;
 993
 994                 // atan2(0, x), x real and negative -> Pi
 995                 if (x.info(info_flags::negative))
 996                         return Pi;
 997         }
 998
 999         if (x.is_zero()) {
1000
1001                 // atan2(y, 0), y real and positive -> Pi/2
1002                 if (y.info(info_flags::positive))
1003                         return _ex1_2*Pi;
1004
1005                 // atan2(y, 0), y real and negative -> -Pi/2
1006                 if (y.info(info_flags::negative))
1007                         return _ex_1_2*Pi;
1008         }
1009
1010         if (y.is_equal(x)) {
1011
1012                 // atan2(y, y), y real and positive -> Pi/4
1013                 if (y.info(info_flags::positive))
1014                         return _ex1_4*Pi;
1015
1016                 // atan2(y, y), y real and negative -> -3/4*Pi
1017                 if (y.info(info_flags::negative))
1018                         return numeric(-3, 4)*Pi;
1019         }
1020
1021         if (y.is_equal(-x)) {
1022
1023                 // atan2(y, -y), y real and positive -> 3*Pi/4
1024                 if (y.info(info_flags::positive))
1025                         return numeric(3, 4)*Pi;
1026
1027                 // atan2(y, -y), y real and negative -> -Pi/4
1028                 if (y.info(info_flags::negative))
1029                         return _ex_1_4*Pi;
1030         }
1031
1032         // atan2(float, float) -> float
1033         if (is_a<numeric>(y) && !y.info(info_flags::crational) &&
1034             is_a<numeric>(x) && !x.info(info_flags::crational))
1035                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
1036
1037         // atan2(real, real) -> atan(y/x) +/- Pi
1038         if (y.info(info_flags::real) && x.info(info_flags::real)) {
1039                 if (x.info(info_flags::positive))
1040                         return atan(y/x);
1041
1042                 if (x.info(info_flags::negative)) {
1043                         if (y.info(info_flags::positive))
1044                                 return atan(y/x)+Pi;
1045                         if (y.info(info_flags::negative))
1046                                 return atan(y/x)-Pi;
1047                 }
1048         }
1049
1050         return atan2(y, x).hold();
1051 }
1052
1053 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
1054 {
1055         GINAC_ASSERT(deriv_param<2);
1056
1057         if (deriv_param==0) {
1058                 // d/dy atan2(y,x)
1059                 return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
1060         }
1061         // d/dx atan2(y,x)
1062         return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
1063 }
1064
1065 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
1066                          evalf_func(atan2_evalf).
1067                          derivative_func(atan2_deriv));
1068
1069 //////////
1070 // hyperbolic sine (trigonometric function)
1071 //////////
1072
1073 static ex sinh_evalf(const ex & x)
1074 {
1075         if (is_exactly_a<numeric>(x))
1076                 return sinh(ex_to<numeric>(x));
1077
1078         return sinh(x).hold();
1079 }
1080
1081 static ex sinh_eval(const ex & x)
1082 {
1083         if (x.info(info_flags::numeric)) {
1084
1085                 // sinh(0) -> 0
1086                 if (x.is_zero())
1087                         return _ex0;
1088
1089                 // sinh(float) -> float
1090                 if (!x.info(info_flags::crational))
1091                         return sinh(ex_to<numeric>(x));
1092
1093                 // sinh() is odd
1094                 if (x.info(info_flags::negative))
1095                         return -sinh(-x);
1096         }
1097
1098         if ((x/Pi).info(info_flags::numeric) &&
1099                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
1100                 return I*sin(x/I);
1101
1102         if (is_exactly_a<function>(x)) {
1103                 const ex &t = x.op(0);
1104
1105                 // sinh(asinh(x)) -> x
1106                 if (is_ex_the_function(x, asinh))
1107                         return t;
1108
1109                 // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
1110                 if (is_ex_the_function(x, acosh))
1111                         return sqrt(t-_ex1)*sqrt(t+_ex1);
1112
1113                 // sinh(atanh(x)) -> x/sqrt(1-x^2)
1114                 if (is_ex_the_function(x, atanh))
1115                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
1116         }
1117
1118         return sinh(x).hold();
1119 }
1120
1121 static ex sinh_deriv(const ex & x, unsigned deriv_param)
1122 {
1123         GINAC_ASSERT(deriv_param==0);
1124
1125         // d/dx sinh(x) -> cosh(x)
1126         return cosh(x);
1127 }
1128
1129 static ex sinh_real_part(const ex & x)
1130 {
1131         return sinh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
1132 }
1133
1134 static ex sinh_imag_part(const ex & x)
1135 {
1136         return cosh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
1137 }
1138
1139 static ex sinh_conjugate(const ex & x)
1140 {
1141         // conjugate(sinh(x))==sinh(conjugate(x))
1142         return sinh(x.conjugate());
1143 }
1144
1145 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
1146                         evalf_func(sinh_evalf).
1147                         derivative_func(sinh_deriv).
1148                         real_part_func(sinh_real_part).
1149                         imag_part_func(sinh_imag_part).
1150                         conjugate_func(sinh_conjugate).
1151                         latex_name("\\sinh"));
1152
1153 //////////
1154 // hyperbolic cosine (trigonometric function)
1155 //////////
1156
1157 static ex cosh_evalf(const ex & x)
1158 {
1159         if (is_exactly_a<numeric>(x))
1160                 return cosh(ex_to<numeric>(x));
1161
1162         return cosh(x).hold();
1163 }
1164
1165 static ex cosh_eval(const ex & x)
1166 {
1167         if (x.info(info_flags::numeric)) {
1168
1169                 // cosh(0) -> 1
1170                 if (x.is_zero())
1171                         return _ex1;
1172
1173                 // cosh(float) -> float
1174                 if (!x.info(info_flags::crational))
1175                         return cosh(ex_to<numeric>(x));
1176
1177                 // cosh() is even
1178                 if (x.info(info_flags::negative))
1179                         return cosh(-x);
1180         }
1181
1182         if ((x/Pi).info(info_flags::numeric) &&
1183                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
1184                 return cos(x/I);
1185
1186         if (is_exactly_a<function>(x)) {
1187                 const ex &t = x.op(0);
1188
1189                 // cosh(acosh(x)) -> x
1190                 if (is_ex_the_function(x, acosh))
1191                         return t;
1192
1193                 // cosh(asinh(x)) -> sqrt(1+x^2)
1194                 if (is_ex_the_function(x, asinh))
1195                         return sqrt(_ex1+power(t,_ex2));
1196
1197                 // cosh(atanh(x)) -> 1/sqrt(1-x^2)
1198                 if (is_ex_the_function(x, atanh))
1199                         return power(_ex1-power(t,_ex2),_ex_1_2);
1200         }
1201
1202         return cosh(x).hold();
1203 }
1204
1205 static ex cosh_deriv(const ex & x, unsigned deriv_param)
1206 {
1207         GINAC_ASSERT(deriv_param==0);
1208
1209         // d/dx cosh(x) -> sinh(x)
1210         return sinh(x);
1211 }
1212
1213 static ex cosh_real_part(const ex & x)
1214 {
1215         return cosh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
1216 }
1217
1218 static ex cosh_imag_part(const ex & x)
1219 {
1220         return sinh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
1221 }
1222
1223 static ex cosh_conjugate(const ex & x)
1224 {
1225         // conjugate(cosh(x))==cosh(conjugate(x))
1226         return cosh(x.conjugate());
1227 }
1228
1229 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
1230                         evalf_func(cosh_evalf).
1231                         derivative_func(cosh_deriv).
1232                         real_part_func(cosh_real_part).
1233                         imag_part_func(cosh_imag_part).
1234                         conjugate_func(cosh_conjugate).
1235                         latex_name("\\cosh"));
1236
1237 //////////
1238 // hyperbolic tangent (trigonometric function)
1239 //////////
1240
1241 static ex tanh_evalf(const ex & x)
1242 {
1243         if (is_exactly_a<numeric>(x))
1244                 return tanh(ex_to<numeric>(x));
1245
1246         return tanh(x).hold();
1247 }
1248
1249 static ex tanh_eval(const ex & x)
1250 {
1251         if (x.info(info_flags::numeric)) {
1252
1253                 // tanh(0) -> 0
1254                 if (x.is_zero())
1255                         return _ex0;
1256
1257                 // tanh(float) -> float
1258                 if (!x.info(info_flags::crational))
1259                         return tanh(ex_to<numeric>(x));
1260
1261                 // tanh() is odd
1262                 if (x.info(info_flags::negative))
1263                         return -tanh(-x);
1264         }
1265
1266         if ((x/Pi).info(info_flags::numeric) &&
1267                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
1268                 return I*tan(x/I);
1269
1270         if (is_exactly_a<function>(x)) {
1271                 const ex &t = x.op(0);
1272
1273                 // tanh(atanh(x)) -> x
1274                 if (is_ex_the_function(x, atanh))
1275                         return t;
1276
1277                 // tanh(asinh(x)) -> x/sqrt(1+x^2)
1278                 if (is_ex_the_function(x, asinh))
1279                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
1280
1281                 // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
1282                 if (is_ex_the_function(x, acosh))
1283                         return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
1284         }
1285
1286         return tanh(x).hold();
1287 }
1288
1289 static ex tanh_deriv(const ex & x, unsigned deriv_param)
1290 {
1291         GINAC_ASSERT(deriv_param==0);
1292
1293         // d/dx tanh(x) -> 1-tanh(x)^2
1294         return _ex1-power(tanh(x),_ex2);
1295 }
1296
1297 static ex tanh_series(const ex &x,
1298                       const relational &rel,
1299                       int order,
1300                       unsigned options)
1301 {
1302         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1303         // method:
1304         // Taylor series where there is no pole falls back to tanh_deriv.
1305         // On a pole simply expand sinh(x)/cosh(x).
1306         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1307         if (!(2*I*x_pt/Pi).info(info_flags::odd))
1308                 throw do_taylor();  // caught by function::series()
1309         // if we got here we have to care for a simple pole
1310         return (sinh(x)/cosh(x)).series(rel, order, options);
1311 }
1312
1313 static ex tanh_real_part(const ex & x)
1314 {
1315         ex a = GiNaC::real_part(x);
1316         ex b = GiNaC::imag_part(x);
1317         return tanh(a)/(1+power(tanh(a),2)*power(tan(b),2));
1318 }
1319
1320 static ex tanh_imag_part(const ex & x)
1321 {
1322         ex a = GiNaC::real_part(x);
1323         ex b = GiNaC::imag_part(x);
1324         return tan(b)/(1+power(tanh(a),2)*power(tan(b),2));
1325 }
1326
1327 static ex tanh_conjugate(const ex & x)
1328 {
1329         // conjugate(tanh(x))==tanh(conjugate(x))
1330         return tanh(x.conjugate());
1331 }
1332
1333 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
1334                         evalf_func(tanh_evalf).
1335                         derivative_func(tanh_deriv).
1336                         series_func(tanh_series).
1337                         real_part_func(tanh_real_part).
1338                         imag_part_func(tanh_imag_part).
1339                         conjugate_func(tanh_conjugate).
1340                         latex_name("\\tanh"));
1341
1342 //////////
1343 // inverse hyperbolic sine (trigonometric function)
1344 //////////
1345
1346 static ex asinh_evalf(const ex & x)
1347 {
1348         if (is_exactly_a<numeric>(x))
1349                 return asinh(ex_to<numeric>(x));
1350
1351         return asinh(x).hold();
1352 }
1353
1354 static ex asinh_eval(const ex & x)
1355 {
1356         if (x.info(info_flags::numeric)) {
1357
1358                 // asinh(0) -> 0
1359                 if (x.is_zero())
1360                         return _ex0;
1361
1362                 // asinh(float) -> float
1363                 if (!x.info(info_flags::crational))
1364                         return asinh(ex_to<numeric>(x));
1365
1366                 // asinh() is odd
1367                 if (x.info(info_flags::negative))
1368                         return -asinh(-x);
1369         }
1370
1371         return asinh(x).hold();
1372 }
1373
1374 static ex asinh_deriv(const ex & x, unsigned deriv_param)
1375 {
1376         GINAC_ASSERT(deriv_param==0);
1377
1378         // d/dx asinh(x) -> 1/sqrt(1+x^2)
1379         return power(_ex1+power(x,_ex2),_ex_1_2);
1380 }
1381
1382 static ex asinh_conjugate(const ex & x)
1383 {
1384         // conjugate(asinh(x))==asinh(conjugate(x)) unless on the branch cuts which
1385         // run along the imaginary axis outside the interval [-I, +I].
1386         if (x.info(info_flags::real))
1387                 return asinh(x);
1388         if (is_exactly_a<numeric>(x)) {
1389                 const numeric x_re = ex_to<numeric>(x.real_part());
1390                 const numeric x_im = ex_to<numeric>(x.imag_part());
1391                 if (!x_re.is_zero() ||
1392                     (x_im > *_num_1_p && x_im < *_num1_p))
1393                         return asinh(x.conjugate());
1394         }
1395         return conjugate_function(asinh(x)).hold();
1396 }
1397
1398 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
1399                          evalf_func(asinh_evalf).
1400                          derivative_func(asinh_deriv).
1401                          conjugate_func(asinh_conjugate));
1402
1403 //////////
1404 // inverse hyperbolic cosine (trigonometric function)
1405 //////////
1406
1407 static ex acosh_evalf(const ex & x)
1408 {
1409         if (is_exactly_a<numeric>(x))
1410                 return acosh(ex_to<numeric>(x));
1411
1412         return acosh(x).hold();
1413 }
1414
1415 static ex acosh_eval(const ex & x)
1416 {
1417         if (x.info(info_flags::numeric)) {
1418
1419                 // acosh(0) -> Pi*I/2
1420                 if (x.is_zero())
1421                         return Pi*I*numeric(1,2);
1422
1423                 // acosh(1) -> 0
1424                 if (x.is_equal(_ex1))
1425                         return _ex0;
1426
1427                 // acosh(-1) -> Pi*I
1428                 if (x.is_equal(_ex_1))
1429                         return Pi*I;
1430
1431                 // acosh(float) -> float
1432                 if (!x.info(info_flags::crational))
1433                         return acosh(ex_to<numeric>(x));
1434
1435                 // acosh(-x) -> Pi*I-acosh(x)
1436                 if (x.info(info_flags::negative))
1437                         return Pi*I-acosh(-x);
1438         }
1439
1440         return acosh(x).hold();
1441 }
1442
1443 static ex acosh_deriv(const ex & x, unsigned deriv_param)
1444 {
1445         GINAC_ASSERT(deriv_param==0);
1446
1447         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
1448         return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
1449 }
1450
1451 static ex acosh_conjugate(const ex & x)
1452 {
1453         // conjugate(acosh(x))==acosh(conjugate(x)) unless on the branch cut
1454         // which runs along the real axis from +1 to -inf.
1455         if (is_exactly_a<numeric>(x) &&
1456             (!x.imag_part().is_zero() || x > *_num1_p)) {
1457                 return acosh(x.conjugate());
1458         }
1459         return conjugate_function(acosh(x)).hold();
1460 }
1461
1462 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
1463                          evalf_func(acosh_evalf).
1464                          derivative_func(acosh_deriv).
1465                          conjugate_func(acosh_conjugate));
1466
1467 //////////
1468 // inverse hyperbolic tangent (trigonometric function)
1469 //////////
1470
1471 static ex atanh_evalf(const ex & x)
1472 {
1473         if (is_exactly_a<numeric>(x))
1474                 return atanh(ex_to<numeric>(x));
1475
1476         return atanh(x).hold();
1477 }
1478
1479 static ex atanh_eval(const ex & x)
1480 {
1481         if (x.info(info_flags::numeric)) {
1482
1483                 // atanh(0) -> 0
1484                 if (x.is_zero())
1485                         return _ex0;
1486
1487                 // atanh({+|-}1) -> throw
1488                 if (x.is_equal(_ex1) || x.is_equal(_ex_1))
1489                         throw (pole_error("atanh_eval(): logarithmic pole",0));
1490
1491                 // atanh(float) -> float
1492                 if (!x.info(info_flags::crational))
1493                         return atanh(ex_to<numeric>(x));
1494
1495                 // atanh() is odd
1496                 if (x.info(info_flags::negative))
1497                         return -atanh(-x);
1498         }
1499
1500         return atanh(x).hold();
1501 }
1502
1503 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1504 {
1505         GINAC_ASSERT(deriv_param==0);
1506
1507         // d/dx atanh(x) -> 1/(1-x^2)
1508         return power(_ex1-power(x,_ex2),_ex_1);
1509 }
1510
1511 static ex atanh_series(const ex &arg,
1512                        const relational &rel,
1513                        int order,
1514                        unsigned options)
1515 {
1516         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1517         // method:
1518         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1519         // There are two branch cuts, one runnig from 1 up the real axis and one
1520         // one running from -1 down the real axis.  The points 1 and -1 are poles
1521         // On the branch cuts and the poles series expand
1522         //     (log(1+x)-log(1-x))/2
1523         // instead.
1524         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
1525         if (!(arg_pt).info(info_flags::real))
1526                 throw do_taylor();     // Im(x) != 0
1527         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1528                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1529         // care for the poles, using the defining formula for atanh()...
1530         if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1531                 return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1532         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1533         if (!(options & series_options::suppress_branchcut)) {
1534                 // method:
1535                 // This is the branch cut: assemble the primitive series manually and
1536                 // then add the corresponding complex step function.
1537                 const symbol &s = ex_to<symbol>(rel.lhs());
1538                 const ex &point = rel.rhs();
1539                 const symbol foo;
1540                 const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1541                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1542                 if (arg_pt<_ex0)
1543                         Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1544                 else
1545                         Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1546                 epvector seq;
1547                 seq.push_back(expair(Order0correction, _ex0));
1548                 seq.push_back(expair(Order(_ex1), order));
1549                 return series(replarg - pseries(rel, seq), rel, order);
1550         }
1551         throw do_taylor();
1552 }
1553
1554 static ex atanh_conjugate(const ex & x)
1555 {
1556         // conjugate(atanh(x))==atanh(conjugate(x)) unless on the branch cuts which
1557         // run along the real axis outside the interval [-1, +1].
1558         if (is_exactly_a<numeric>(x) &&
1559             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
1560                 return atanh(x.conjugate());
1561         }
1562         return conjugate_function(atanh(x)).hold();
1563 }
1564
1565 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1566                          evalf_func(atanh_evalf).
1567                          derivative_func(atanh_deriv).
1568                          series_func(atanh_series).
1569                          conjugate_func(atanh_conjugate));
1570
1571
1572 } // namespace GiNaC