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) 1999-2020 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 dynallocate<mul>(prodseq).setflag(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                                 ex rest = pseries(rel, epvector{expair(-1, _ex0), expair(Order(_ex1), order)}).add_series(argser);
 246                                 ex acc = dynallocate<pseries>(rel, epvector());
 247                                 for (int i = order-1; i>0; --i) {
 248                                         epvector cterm { expair(i%2 ? _ex1/i : _ex_1/i, _ex0) };
 249                                         acc = pseries(rel, std::move(cterm)).add_series(ex_to<pseries>(acc));
 250                                         acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
 251                                 }
 252                                 return acc;
 253                         }
 254                         const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
 255                         return pseries(rel, std::move(seq)).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
 256                 } else  // it was a monomial
 257                         return pseries(rel, std::move(seq));
 258         }
 259         if (!(options & series_options::suppress_branchcut) &&
 260              arg_pt.info(info_flags::negative)) {
 261                 // method:
 262                 // This is the branch cut: assemble the primitive series manually and
 263                 // then add the corresponding complex step function.
 264                 const symbol &s = ex_to<symbol>(rel.lhs());
 265                 const ex &point = rel.rhs();
 266                 const symbol foo;
 267                 const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 268                 epvector seq;
 269                 if (order > 0) {
 270                         seq.reserve(2);
 271                         seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
 272                 }
 273                 seq.push_back(expair(Order(_ex1), order));
 274                 return series(replarg - I*Pi + pseries(rel, std::move(seq)), rel, order);
 275         }
 276         throw do_taylor();  // caught by function::series()
 277 }
 278
 279 static ex log_real_part(const ex & x)
 280 {
 281         if (x.info(info_flags::nonnegative))
 282                 return log(x).hold();
 283         return log(abs(x));
 284 }
 285
 286 static ex log_imag_part(const ex & x)
 287 {
 288         if (x.info(info_flags::nonnegative))
 289                 return 0;
 290         return atan2(GiNaC::imag_part(x), GiNaC::real_part(x));
 291 }
 292
 293 static ex log_expand(const ex & arg, unsigned options)
 294 {
 295         if ((options & expand_options::expand_transcendental)
 296                 && is_exactly_a<mul>(arg) && !arg.info(info_flags::indefinite)) {
 297                 exvector sumseq;
 298                 exvector prodseq;
 299                 sumseq.reserve(arg.nops());
 300                 prodseq.reserve(arg.nops());
 301                 bool possign=true;
 302
 303                 // searching for positive/negative factors
 304                 for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
 305                         ex e;
 306                         if (options & expand_options::expand_function_args)
 307                                 e=i->expand(options);
 308                         else
 309                                 e=*i;
 310                         if (e.info(info_flags::positive))
 311                                 sumseq.push_back(log(e));
 312                         else if (e.info(info_flags::negative)) {
 313                                 sumseq.push_back(log(-e));
 314                                 possign = !possign;
 315                         } else
 316                                 prodseq.push_back(e);
 317                 }
 318
 319                 if (sumseq.size() > 0) {
 320                         ex newarg;
 321                         if (options & expand_options::expand_function_args)
 322                                 newarg=((possign?_ex1:_ex_1)*mul(prodseq)).expand(options);
 323                         else {
 324                                 newarg=(possign?_ex1:_ex_1)*mul(prodseq);
 325                                 ex_to<basic>(newarg).setflag(status_flags::purely_indefinite);
 326                         }
 327                         return add(sumseq)+log(newarg);
 328                 } else {
 329                         if (!(options & expand_options::expand_function_args))
 330                                 ex_to<basic>(arg).setflag(status_flags::purely_indefinite);
 331                 }
 332         }
 333
 334         if (options & expand_options::expand_function_args)
 335                 return log(arg.expand(options)).hold();
 336         else
 337                 return log(arg).hold();
 338 }
 339
 340 static ex log_conjugate(const ex & x)
 341 {
 342         // conjugate(log(x))==log(conjugate(x)) unless on the branch cut which
 343         // runs along the negative real axis.
 344         if (x.info(info_flags::positive)) {
 345                 return log(x);
 346         }
 347         if (is_exactly_a<numeric>(x) &&
 348             !x.imag_part().is_zero()) {
 349                 return log(x.conjugate());
 350         }
 351         return conjugate_function(log(x)).hold();
 352 }
 353
 354 REGISTER_FUNCTION(log, eval_func(log_eval).
 355                        evalf_func(log_evalf).
 356                        expand_func(log_expand).
 357                        derivative_func(log_deriv).
 358                        series_func(log_series).
 359                        real_part_func(log_real_part).
 360                        imag_part_func(log_imag_part).
 361                        conjugate_func(log_conjugate).
 362                        latex_name("\\ln"));
 363
 364 //////////
 365 // sine (trigonometric function)
 366 //////////
 367
 368 static ex sin_evalf(const ex & x)
 369 {
 370         if (is_exactly_a<numeric>(x))
 371                 return sin(ex_to<numeric>(x));
 372
 373         return sin(x).hold();
 374 }
 375
 376 static ex sin_eval(const ex & x)
 377 {
 378         // sin(n/d*Pi) -> { all known non-nested radicals }
 379         const ex SixtyExOverPi = _ex60*x/Pi;
 380         ex sign = _ex1;
 381         if (SixtyExOverPi.info(info_flags::integer)) {
 382                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 383                 if (z>=*_num60_p) {
 384                         // wrap to interval [0, Pi)
 385                         z -= *_num60_p;
 386                         sign = _ex_1;
 387                 }
 388                 if (z>*_num30_p) {
 389                         // wrap to interval [0, Pi/2)
 390                         z = *_num60_p-z;
 391                 }
 392                 if (z.is_equal(*_num0_p))  // sin(0)       -> 0
 393                         return _ex0;
 394                 if (z.is_equal(*_num5_p))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
 395                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 396                 if (z.is_equal(*_num6_p))  // sin(Pi/10)   -> sqrt(5)/4-1/4
 397                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 398                 if (z.is_equal(*_num10_p)) // sin(Pi/6)    -> 1/2
 399                         return sign*_ex1_2;
 400                 if (z.is_equal(*_num15_p)) // sin(Pi/4)    -> sqrt(2)/2
 401                         return sign*_ex1_2*sqrt(_ex2);
 402                 if (z.is_equal(*_num18_p)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
 403                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 404                 if (z.is_equal(*_num20_p)) // sin(Pi/3)    -> sqrt(3)/2
 405                         return sign*_ex1_2*sqrt(_ex3);
 406                 if (z.is_equal(*_num25_p)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
 407                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 408                 if (z.is_equal(*_num30_p)) // sin(Pi/2)    -> 1
 409                         return sign;
 410         }
 411
 412         if (is_exactly_a<function>(x)) {
 413                 const ex &t = x.op(0);
 414
 415                 // sin(asin(x)) -> x
 416                 if (is_ex_the_function(x, asin))
 417                         return t;
 418
 419                 // sin(acos(x)) -> sqrt(1-x^2)
 420                 if (is_ex_the_function(x, acos))
 421                         return sqrt(_ex1-power(t,_ex2));
 422
 423                 // sin(atan(x)) -> x/sqrt(1+x^2)
 424                 if (is_ex_the_function(x, atan))
 425                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 426         }
 427
 428         // sin(float) -> float
 429         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 430                 return sin(ex_to<numeric>(x));
 431
 432         // sin() is odd
 433         if (x.info(info_flags::negative))
 434                 return -sin(-x);
 435
 436         return sin(x).hold();
 437 }
 438
 439 static ex sin_deriv(const ex & x, unsigned deriv_param)
 440 {
 441         GINAC_ASSERT(deriv_param==0);
 442
 443         // d/dx sin(x) -> cos(x)
 444         return cos(x);
 445 }
 446
 447 static ex sin_real_part(const ex & x)
 448 {
 449         return cosh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 450 }
 451
 452 static ex sin_imag_part(const ex & x)
 453 {
 454         return sinh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 455 }
 456
 457 static ex sin_conjugate(const ex & x)
 458 {
 459         // conjugate(sin(x))==sin(conjugate(x))
 460         return sin(x.conjugate());
 461 }
 462
 463 REGISTER_FUNCTION(sin, eval_func(sin_eval).
 464                        evalf_func(sin_evalf).
 465                        derivative_func(sin_deriv).
 466                        real_part_func(sin_real_part).
 467                        imag_part_func(sin_imag_part).
 468                        conjugate_func(sin_conjugate).
 469                        latex_name("\\sin"));
 470
 471 //////////
 472 // cosine (trigonometric function)
 473 //////////
 474
 475 static ex cos_evalf(const ex & x)
 476 {
 477         if (is_exactly_a<numeric>(x))
 478                 return cos(ex_to<numeric>(x));
 479
 480         return cos(x).hold();
 481 }
 482
 483 static ex cos_eval(const ex & x)
 484 {
 485         // cos(n/d*Pi) -> { all known non-nested radicals }
 486         const ex SixtyExOverPi = _ex60*x/Pi;
 487         ex sign = _ex1;
 488         if (SixtyExOverPi.info(info_flags::integer)) {
 489                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 490                 if (z>=*_num60_p) {
 491                         // wrap to interval [0, Pi)
 492                         z = *_num120_p-z;
 493                 }
 494                 if (z>=*_num30_p) {
 495                         // wrap to interval [0, Pi/2)
 496                         z = *_num60_p-z;
 497                         sign = _ex_1;
 498                 }
 499                 if (z.is_equal(*_num0_p))  // cos(0)       -> 1
 500                         return sign;
 501                 if (z.is_equal(*_num5_p))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 502                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 503                 if (z.is_equal(*_num10_p)) // cos(Pi/6)    -> sqrt(3)/2
 504                         return sign*_ex1_2*sqrt(_ex3);
 505                 if (z.is_equal(*_num12_p)) // cos(Pi/5)    -> sqrt(5)/4+1/4
 506                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 507                 if (z.is_equal(*_num15_p)) // cos(Pi/4)    -> sqrt(2)/2
 508                         return sign*_ex1_2*sqrt(_ex2);
 509                 if (z.is_equal(*_num20_p)) // cos(Pi/3)    -> 1/2
 510                         return sign*_ex1_2;
 511                 if (z.is_equal(*_num24_p)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 512                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 513                 if (z.is_equal(*_num25_p)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 514                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 515                 if (z.is_equal(*_num30_p)) // cos(Pi/2)    -> 0
 516                         return _ex0;
 517         }
 518
 519         if (is_exactly_a<function>(x)) {
 520                 const ex &t = x.op(0);
 521
 522                 // cos(acos(x)) -> x
 523                 if (is_ex_the_function(x, acos))
 524                         return t;
 525
 526                 // cos(asin(x)) -> sqrt(1-x^2)
 527                 if (is_ex_the_function(x, asin))
 528                         return sqrt(_ex1-power(t,_ex2));
 529
 530                 // cos(atan(x)) -> 1/sqrt(1+x^2)
 531                 if (is_ex_the_function(x, atan))
 532                         return power(_ex1+power(t,_ex2),_ex_1_2);
 533         }
 534
 535         // cos(float) -> float
 536         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 537                 return cos(ex_to<numeric>(x));
 538
 539         // cos() is even
 540         if (x.info(info_flags::negative))
 541                 return cos(-x);
 542
 543         return cos(x).hold();
 544 }
 545
 546 static ex cos_deriv(const ex & x, unsigned deriv_param)
 547 {
 548         GINAC_ASSERT(deriv_param==0);
 549
 550         // d/dx cos(x) -> -sin(x)
 551         return -sin(x);
 552 }
 553
 554 static ex cos_real_part(const ex & x)
 555 {
 556         return cosh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 557 }
 558
 559 static ex cos_imag_part(const ex & x)
 560 {
 561         return -sinh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 562 }
 563
 564 static ex cos_conjugate(const ex & x)
 565 {
 566         // conjugate(cos(x))==cos(conjugate(x))
 567         return cos(x.conjugate());
 568 }
 569
 570 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 571                        evalf_func(cos_evalf).
 572                        derivative_func(cos_deriv).
 573                        real_part_func(cos_real_part).
 574                        imag_part_func(cos_imag_part).
 575                        conjugate_func(cos_conjugate).
 576                        latex_name("\\cos"));
 577
 578 //////////
 579 // tangent (trigonometric function)
 580 //////////
 581
 582 static ex tan_evalf(const ex & x)
 583 {
 584         if (is_exactly_a<numeric>(x))
 585                 return tan(ex_to<numeric>(x));
 586
 587         return tan(x).hold();
 588 }
 589
 590 static ex tan_eval(const ex & x)
 591 {
 592         // tan(n/d*Pi) -> { all known non-nested radicals }
 593         const ex SixtyExOverPi = _ex60*x/Pi;
 594         ex sign = _ex1;
 595         if (SixtyExOverPi.info(info_flags::integer)) {
 596                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num60_p);
 597                 if (z>=*_num60_p) {
 598                         // wrap to interval [0, Pi)
 599                         z -= *_num60_p;
 600                 }
 601                 if (z>=*_num30_p) {
 602                         // wrap to interval [0, Pi/2)
 603                         z = *_num60_p-z;
 604                         sign = _ex_1;
 605                 }
 606                 if (z.is_equal(*_num0_p))  // tan(0)       -> 0
 607                         return _ex0;
 608                 if (z.is_equal(*_num5_p))  // tan(Pi/12)   -> 2-sqrt(3)
 609                         return sign*(_ex2-sqrt(_ex3));
 610                 if (z.is_equal(*_num10_p)) // tan(Pi/6)    -> sqrt(3)/3
 611                         return sign*_ex1_3*sqrt(_ex3);
 612                 if (z.is_equal(*_num15_p)) // tan(Pi/4)    -> 1
 613                         return sign;
 614                 if (z.is_equal(*_num20_p)) // tan(Pi/3)    -> sqrt(3)
 615                         return sign*sqrt(_ex3);
 616                 if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3)
 617                         return sign*(sqrt(_ex3)+_ex2);
 618                 if (z.is_equal(*_num30_p)) // tan(Pi/2)    -> infinity
 619                         throw (pole_error("tan_eval(): simple pole",1));
 620         }
 621
 622         if (is_exactly_a<function>(x)) {
 623                 const ex &t = x.op(0);
 624
 625                 // tan(atan(x)) -> x
 626                 if (is_ex_the_function(x, atan))
 627                         return t;
 628
 629                 // tan(asin(x)) -> x/sqrt(1+x^2)
 630                 if (is_ex_the_function(x, asin))
 631                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 632
 633                 // tan(acos(x)) -> sqrt(1-x^2)/x
 634                 if (is_ex_the_function(x, acos))
 635                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
 636         }
 637
 638         // tan(float) -> float
 639         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 640                 return tan(ex_to<numeric>(x));
 641         }
 642
 643         // tan() is odd
 644         if (x.info(info_flags::negative))
 645                 return -tan(-x);
 646
 647         return tan(x).hold();
 648 }
 649
 650 static ex tan_deriv(const ex & x, unsigned deriv_param)
 651 {
 652         GINAC_ASSERT(deriv_param==0);
 653
 654         // d/dx tan(x) -> 1+tan(x)^2;
 655         return (_ex1+power(tan(x),_ex2));
 656 }
 657
 658 static ex tan_real_part(const ex & x)
 659 {
 660         ex a = GiNaC::real_part(x);
 661         ex b = GiNaC::imag_part(x);
 662         return tan(a)/(1+power(tan(a),2)*power(tan(b),2));
 663 }
 664
 665 static ex tan_imag_part(const ex & x)
 666 {
 667         ex a = GiNaC::real_part(x);
 668         ex b = GiNaC::imag_part(x);
 669         return tanh(b)/(1+power(tan(a),2)*power(tan(b),2));
 670 }
 671
 672 static ex tan_series(const ex &x,
 673                      const relational &rel,
 674                      int order,
 675                      unsigned options)
 676 {
 677         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 678         // method:
 679         // Taylor series where there is no pole falls back to tan_deriv.
 680         // On a pole simply expand sin(x)/cos(x).
 681         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 682         if (!(2*x_pt/Pi).info(info_flags::odd))
 683                 throw do_taylor();  // caught by function::series()
 684         // if we got here we have to care for a simple pole
 685         return (sin(x)/cos(x)).series(rel, order, options);
 686 }
 687
 688 static ex tan_conjugate(const ex & x)
 689 {
 690         // conjugate(tan(x))==tan(conjugate(x))
 691         return tan(x.conjugate());
 692 }
 693
 694 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 695                        evalf_func(tan_evalf).
 696                        derivative_func(tan_deriv).
 697                        series_func(tan_series).
 698                        real_part_func(tan_real_part).
 699                        imag_part_func(tan_imag_part).
 700                        conjugate_func(tan_conjugate).
 701                        latex_name("\\tan"));
 702
 703 //////////
 704 // inverse sine (arc sine)
 705 //////////
 706
 707 static ex asin_evalf(const ex & x)
 708 {
 709         if (is_exactly_a<numeric>(x))
 710                 return asin(ex_to<numeric>(x));
 711
 712         return asin(x).hold();
 713 }
 714
 715 static ex asin_eval(const ex & x)
 716 {
 717         if (x.info(info_flags::numeric)) {
 718
 719                 // asin(0) -> 0
 720                 if (x.is_zero())
 721                         return x;
 722
 723                 // asin(1/2) -> Pi/6
 724                 if (x.is_equal(_ex1_2))
 725                         return numeric(1,6)*Pi;
 726
 727                 // asin(1) -> Pi/2
 728                 if (x.is_equal(_ex1))
 729                         return _ex1_2*Pi;
 730
 731                 // asin(-1/2) -> -Pi/6
 732                 if (x.is_equal(_ex_1_2))
 733                         return numeric(-1,6)*Pi;
 734
 735                 // asin(-1) -> -Pi/2
 736                 if (x.is_equal(_ex_1))
 737                         return _ex_1_2*Pi;
 738
 739                 // asin(float) -> float
 740                 if (!x.info(info_flags::crational))
 741                         return asin(ex_to<numeric>(x));
 742
 743                 // asin() is odd
 744                 if (x.info(info_flags::negative))
 745                         return -asin(-x);
 746         }
 747
 748         return asin(x).hold();
 749 }
 750
 751 static ex asin_deriv(const ex & x, unsigned deriv_param)
 752 {
 753         GINAC_ASSERT(deriv_param==0);
 754
 755         // d/dx asin(x) -> 1/sqrt(1-x^2)
 756         return power(1-power(x,_ex2),_ex_1_2);
 757 }
 758
 759 static ex asin_conjugate(const ex & x)
 760 {
 761         // conjugate(asin(x))==asin(conjugate(x)) unless on the branch cuts which
 762         // run along the real axis outside the interval [-1, +1].
 763         if (is_exactly_a<numeric>(x) &&
 764             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
 765                 return asin(x.conjugate());
 766         }
 767         return conjugate_function(asin(x)).hold();
 768 }
 769
 770 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 771                         evalf_func(asin_evalf).
 772                         derivative_func(asin_deriv).
 773                         conjugate_func(asin_conjugate).
 774                         latex_name("\\arcsin"));
 775
 776 //////////
 777 // inverse cosine (arc cosine)
 778 //////////
 779
 780 static ex acos_evalf(const ex & x)
 781 {
 782         if (is_exactly_a<numeric>(x))
 783                 return acos(ex_to<numeric>(x));
 784
 785         return acos(x).hold();
 786 }
 787
 788 static ex acos_eval(const ex & x)
 789 {
 790         if (x.info(info_flags::numeric)) {
 791
 792                 // acos(1) -> 0
 793                 if (x.is_equal(_ex1))
 794                         return _ex0;
 795
 796                 // acos(1/2) -> Pi/3
 797                 if (x.is_equal(_ex1_2))
 798                         return _ex1_3*Pi;
 799
 800                 // acos(0) -> Pi/2
 801                 if (x.is_zero())
 802                         return _ex1_2*Pi;
 803
 804                 // acos(-1/2) -> 2/3*Pi
 805                 if (x.is_equal(_ex_1_2))
 806                         return numeric(2,3)*Pi;
 807
 808                 // acos(-1) -> Pi
 809                 if (x.is_equal(_ex_1))
 810                         return Pi;
 811
 812                 // acos(float) -> float
 813                 if (!x.info(info_flags::crational))
 814                         return acos(ex_to<numeric>(x));
 815
 816                 // acos(-x) -> Pi-acos(x)
 817                 if (x.info(info_flags::negative))
 818                         return Pi-acos(-x);
 819         }
 820
 821         return acos(x).hold();
 822 }
 823
 824 static ex acos_deriv(const ex & x, unsigned deriv_param)
 825 {
 826         GINAC_ASSERT(deriv_param==0);
 827
 828         // d/dx acos(x) -> -1/sqrt(1-x^2)
 829         return -power(1-power(x,_ex2),_ex_1_2);
 830 }
 831
 832 static ex acos_conjugate(const ex & x)
 833 {
 834         // conjugate(acos(x))==acos(conjugate(x)) unless on the branch cuts which
 835         // run along the real axis outside the interval [-1, +1].
 836         if (is_exactly_a<numeric>(x) &&
 837             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
 838                 return acos(x.conjugate());
 839         }
 840         return conjugate_function(acos(x)).hold();
 841 }
 842
 843 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 844                         evalf_func(acos_evalf).
 845                         derivative_func(acos_deriv).
 846                         conjugate_func(acos_conjugate).
 847                         latex_name("\\arccos"));
 848
 849 //////////
 850 // inverse tangent (arc tangent)
 851 //////////
 852
 853 static ex atan_evalf(const ex & x)
 854 {
 855         if (is_exactly_a<numeric>(x))
 856                 return atan(ex_to<numeric>(x));
 857
 858         return atan(x).hold();
 859 }
 860
 861 static ex atan_eval(const ex & x)
 862 {
 863         if (x.info(info_flags::numeric)) {
 864
 865                 // atan(0) -> 0
 866                 if (x.is_zero())
 867                         return _ex0;
 868
 869                 // atan(1) -> Pi/4
 870                 if (x.is_equal(_ex1))
 871                         return _ex1_4*Pi;
 872
 873                 // atan(-1) -> -Pi/4
 874                 if (x.is_equal(_ex_1))
 875                         return _ex_1_4*Pi;
 876
 877                 if (x.is_equal(I) || x.is_equal(-I))
 878                         throw (pole_error("atan_eval(): logarithmic pole",0));
 879
 880                 // atan(float) -> float
 881                 if (!x.info(info_flags::crational))
 882                         return atan(ex_to<numeric>(x));
 883
 884                 // atan() is odd
 885                 if (x.info(info_flags::negative))
 886                         return -atan(-x);
 887         }
 888
 889         return atan(x).hold();
 890 }
 891
 892 static ex atan_deriv(const ex & x, unsigned deriv_param)
 893 {
 894         GINAC_ASSERT(deriv_param==0);
 895
 896         // d/dx atan(x) -> 1/(1+x^2)
 897         return power(_ex1+power(x,_ex2), _ex_1);
 898 }
 899
 900 static ex atan_series(const ex &arg,
 901                       const relational &rel,
 902                       int order,
 903                       unsigned options)
 904 {
 905         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 906         // method:
 907         // Taylor series where there is no pole or cut falls back to atan_deriv.
 908         // There are two branch cuts, one runnig from I up the imaginary axis and
 909         // one running from -I down the imaginary axis.  The points I and -I are
 910         // poles.
 911         // On the branch cuts and the poles series expand
 912         //     (log(1+I*x)-log(1-I*x))/(2*I)
 913         // instead.
 914         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 915         if (!(I*arg_pt).info(info_flags::real))
 916                 throw do_taylor();     // Re(x) != 0
 917         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
 918                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
 919         // care for the poles, using the defining formula for atan()...
 920         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
 921                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
 922         if (!(options & series_options::suppress_branchcut)) {
 923                 // method:
 924                 // This is the branch cut: assemble the primitive series manually and
 925                 // then add the corresponding complex step function.
 926                 const symbol &s = ex_to<symbol>(rel.lhs());
 927                 const ex &point = rel.rhs();
 928                 const symbol foo;
 929                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 930                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
 931                 if ((I*arg_pt)<_ex0)
 932                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
 933                 else
 934                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
 935                 epvector seq;
 936                 if (order > 0) {
 937                         seq.reserve(2);
 938                         seq.push_back(expair(Order0correction, _ex0));
 939                 }
 940                 seq.push_back(expair(Order(_ex1), order));
 941                 return series(replarg - pseries(rel, std::move(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                 if (order > 0) {
1547                         seq.reserve(2);
1548                         seq.push_back(expair(Order0correction, _ex0));
1549                 }
1550                 seq.push_back(expair(Order(_ex1), order));
1551                 return series(replarg - pseries(rel, std::move(seq)), rel, order);
1552         }
1553         throw do_taylor();
1554 }
1555
1556 static ex atanh_conjugate(const ex & x)
1557 {
1558         // conjugate(atanh(x))==atanh(conjugate(x)) unless on the branch cuts which
1559         // run along the real axis outside the interval [-1, +1].
1560         if (is_exactly_a<numeric>(x) &&
1561             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
1562                 return atanh(x.conjugate());
1563         }
1564         return conjugate_function(atanh(x)).hold();
1565 }
1566
1567 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1568                          evalf_func(atanh_evalf).
1569                          derivative_func(atanh_deriv).
1570                          series_func(atanh_series).
1571                          conjugate_func(atanh_conjugate));
1572
1573
1574 } // namespace GiNaC