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