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) 1999-2015 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 { expair(-I*csgn(arg*I)*Pi, _ex0),
 269                                expair(Order(_ex1), order) };
 270                 return series(replarg - I*Pi + pseries(rel, std::move(seq)), rel, order);
 271         }
 272         throw do_taylor();  // caught by function::series()
 273 }
 274
 275 static ex log_real_part(const ex & x)
 276 {
 277         if (x.info(info_flags::nonnegative))
 278                 return log(x).hold();
 279         return log(abs(x));
 280 }
 281
 282 static ex log_imag_part(const ex & x)
 283 {
 284         if (x.info(info_flags::nonnegative))
 285                 return 0;
 286         return atan2(GiNaC::imag_part(x), GiNaC::real_part(x));
 287 }
 288
 289 static ex log_expand(const ex & arg, unsigned options)
 290 {
 291         if ((options & expand_options::expand_transcendental)
 292                 && is_exactly_a<mul>(arg) && !arg.info(info_flags::indefinite)) {
 293                 exvector sumseq;
 294                 exvector prodseq;
 295                 sumseq.reserve(arg.nops());
 296                 prodseq.reserve(arg.nops());
 297                 bool possign=true;
 298
 299                 // searching for positive/negative factors
 300                 for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
 301                         ex e;
 302                         if (options & expand_options::expand_function_args)
 303                                 e=i->expand(options);
 304                         else
 305                                 e=*i;
 306                         if (e.info(info_flags::positive))
 307                                 sumseq.push_back(log(e));
 308                         else if (e.info(info_flags::negative)) {
 309                                 sumseq.push_back(log(-e));
 310                                 possign = !possign;
 311                         } else
 312                                 prodseq.push_back(e);
 313                 }
 314
 315                 if (sumseq.size() > 0) {
 316                         ex newarg;
 317                         if (options & expand_options::expand_function_args)
 318                                 newarg=((possign?_ex1:_ex_1)*mul(prodseq)).expand(options);
 319                         else {
 320                                 newarg=(possign?_ex1:_ex_1)*mul(prodseq);
 321                                 ex_to<basic>(newarg).setflag(status_flags::purely_indefinite);
 322                         }
 323                         return add(sumseq)+log(newarg);
 324                 } else {
 325                         if (!(options & expand_options::expand_function_args))
 326                                 ex_to<basic>(arg).setflag(status_flags::purely_indefinite);
 327                 }
 328         }
 329
 330         if (options & expand_options::expand_function_args)
 331                 return log(arg.expand(options)).hold();
 332         else
 333                 return log(arg).hold();
 334 }
 335
 336 static ex log_conjugate(const ex & x)
 337 {
 338         // conjugate(log(x))==log(conjugate(x)) unless on the branch cut which
 339         // runs along the negative real axis.
 340         if (x.info(info_flags::positive)) {
 341                 return log(x);
 342         }
 343         if (is_exactly_a<numeric>(x) &&
 344             !x.imag_part().is_zero()) {
 345                 return log(x.conjugate());
 346         }
 347         return conjugate_function(log(x)).hold();
 348 }
 349
 350 REGISTER_FUNCTION(log, eval_func(log_eval).
 351                        evalf_func(log_evalf).
 352                        expand_func(log_expand).
 353                        derivative_func(log_deriv).
 354                        series_func(log_series).
 355                        real_part_func(log_real_part).
 356                        imag_part_func(log_imag_part).
 357                        conjugate_func(log_conjugate).
 358                        latex_name("\\ln"));
 359
 360 //////////
 361 // sine (trigonometric function)
 362 //////////
 363
 364 static ex sin_evalf(const ex & x)
 365 {
 366         if (is_exactly_a<numeric>(x))
 367                 return sin(ex_to<numeric>(x));
 368
 369         return sin(x).hold();
 370 }
 371
 372 static ex sin_eval(const ex & x)
 373 {
 374         // sin(n/d*Pi) -> { all known non-nested radicals }
 375         const ex SixtyExOverPi = _ex60*x/Pi;
 376         ex sign = _ex1;
 377         if (SixtyExOverPi.info(info_flags::integer)) {
 378                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 379                 if (z>=*_num60_p) {
 380                         // wrap to interval [0, Pi)
 381                         z -= *_num60_p;
 382                         sign = _ex_1;
 383                 }
 384                 if (z>*_num30_p) {
 385                         // wrap to interval [0, Pi/2)
 386                         z = *_num60_p-z;
 387                 }
 388                 if (z.is_equal(*_num0_p))  // sin(0)       -> 0
 389                         return _ex0;
 390                 if (z.is_equal(*_num5_p))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
 391                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 392                 if (z.is_equal(*_num6_p))  // sin(Pi/10)   -> sqrt(5)/4-1/4
 393                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 394                 if (z.is_equal(*_num10_p)) // sin(Pi/6)    -> 1/2
 395                         return sign*_ex1_2;
 396                 if (z.is_equal(*_num15_p)) // sin(Pi/4)    -> sqrt(2)/2
 397                         return sign*_ex1_2*sqrt(_ex2);
 398                 if (z.is_equal(*_num18_p)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
 399                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 400                 if (z.is_equal(*_num20_p)) // sin(Pi/3)    -> sqrt(3)/2
 401                         return sign*_ex1_2*sqrt(_ex3);
 402                 if (z.is_equal(*_num25_p)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
 403                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 404                 if (z.is_equal(*_num30_p)) // sin(Pi/2)    -> 1
 405                         return sign;
 406         }
 407
 408         if (is_exactly_a<function>(x)) {
 409                 const ex &t = x.op(0);
 410
 411                 // sin(asin(x)) -> x
 412                 if (is_ex_the_function(x, asin))
 413                         return t;
 414
 415                 // sin(acos(x)) -> sqrt(1-x^2)
 416                 if (is_ex_the_function(x, acos))
 417                         return sqrt(_ex1-power(t,_ex2));
 418
 419                 // sin(atan(x)) -> x/sqrt(1+x^2)
 420                 if (is_ex_the_function(x, atan))
 421                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 422         }
 423
 424         // sin(float) -> float
 425         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 426                 return sin(ex_to<numeric>(x));
 427
 428         // sin() is odd
 429         if (x.info(info_flags::negative))
 430                 return -sin(-x);
 431
 432         return sin(x).hold();
 433 }
 434
 435 static ex sin_deriv(const ex & x, unsigned deriv_param)
 436 {
 437         GINAC_ASSERT(deriv_param==0);
 438
 439         // d/dx sin(x) -> cos(x)
 440         return cos(x);
 441 }
 442
 443 static ex sin_real_part(const ex & x)
 444 {
 445         return cosh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 446 }
 447
 448 static ex sin_imag_part(const ex & x)
 449 {
 450         return sinh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 451 }
 452
 453 static ex sin_conjugate(const ex & x)
 454 {
 455         // conjugate(sin(x))==sin(conjugate(x))
 456         return sin(x.conjugate());
 457 }
 458
 459 REGISTER_FUNCTION(sin, eval_func(sin_eval).
 460                        evalf_func(sin_evalf).
 461                        derivative_func(sin_deriv).
 462                        real_part_func(sin_real_part).
 463                        imag_part_func(sin_imag_part).
 464                        conjugate_func(sin_conjugate).
 465                        latex_name("\\sin"));
 466
 467 //////////
 468 // cosine (trigonometric function)
 469 //////////
 470
 471 static ex cos_evalf(const ex & x)
 472 {
 473         if (is_exactly_a<numeric>(x))
 474                 return cos(ex_to<numeric>(x));
 475
 476         return cos(x).hold();
 477 }
 478
 479 static ex cos_eval(const ex & x)
 480 {
 481         // cos(n/d*Pi) -> { all known non-nested radicals }
 482         const ex SixtyExOverPi = _ex60*x/Pi;
 483         ex sign = _ex1;
 484         if (SixtyExOverPi.info(info_flags::integer)) {
 485                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 486                 if (z>=*_num60_p) {
 487                         // wrap to interval [0, Pi)
 488                         z = *_num120_p-z;
 489                 }
 490                 if (z>=*_num30_p) {
 491                         // wrap to interval [0, Pi/2)
 492                         z = *_num60_p-z;
 493                         sign = _ex_1;
 494                 }
 495                 if (z.is_equal(*_num0_p))  // cos(0)       -> 1
 496                         return sign;
 497                 if (z.is_equal(*_num5_p))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 498                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 499                 if (z.is_equal(*_num10_p)) // cos(Pi/6)    -> sqrt(3)/2
 500                         return sign*_ex1_2*sqrt(_ex3);
 501                 if (z.is_equal(*_num12_p)) // cos(Pi/5)    -> sqrt(5)/4+1/4
 502                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 503                 if (z.is_equal(*_num15_p)) // cos(Pi/4)    -> sqrt(2)/2
 504                         return sign*_ex1_2*sqrt(_ex2);
 505                 if (z.is_equal(*_num20_p)) // cos(Pi/3)    -> 1/2
 506                         return sign*_ex1_2;
 507                 if (z.is_equal(*_num24_p)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 508                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 509                 if (z.is_equal(*_num25_p)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 510                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 511                 if (z.is_equal(*_num30_p)) // cos(Pi/2)    -> 0
 512                         return _ex0;
 513         }
 514
 515         if (is_exactly_a<function>(x)) {
 516                 const ex &t = x.op(0);
 517
 518                 // cos(acos(x)) -> x
 519                 if (is_ex_the_function(x, acos))
 520                         return t;
 521
 522                 // cos(asin(x)) -> sqrt(1-x^2)
 523                 if (is_ex_the_function(x, asin))
 524                         return sqrt(_ex1-power(t,_ex2));
 525
 526                 // cos(atan(x)) -> 1/sqrt(1+x^2)
 527                 if (is_ex_the_function(x, atan))
 528                         return power(_ex1+power(t,_ex2),_ex_1_2);
 529         }
 530
 531         // cos(float) -> float
 532         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 533                 return cos(ex_to<numeric>(x));
 534
 535         // cos() is even
 536         if (x.info(info_flags::negative))
 537                 return cos(-x);
 538
 539         return cos(x).hold();
 540 }
 541
 542 static ex cos_deriv(const ex & x, unsigned deriv_param)
 543 {
 544         GINAC_ASSERT(deriv_param==0);
 545
 546         // d/dx cos(x) -> -sin(x)
 547         return -sin(x);
 548 }
 549
 550 static ex cos_real_part(const ex & x)
 551 {
 552         return cosh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 553 }
 554
 555 static ex cos_imag_part(const ex & x)
 556 {
 557         return -sinh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 558 }
 559
 560 static ex cos_conjugate(const ex & x)
 561 {
 562         // conjugate(cos(x))==cos(conjugate(x))
 563         return cos(x.conjugate());
 564 }
 565
 566 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 567                        evalf_func(cos_evalf).
 568                        derivative_func(cos_deriv).
 569                        real_part_func(cos_real_part).
 570                        imag_part_func(cos_imag_part).
 571                        conjugate_func(cos_conjugate).
 572                        latex_name("\\cos"));
 573
 574 //////////
 575 // tangent (trigonometric function)
 576 //////////
 577
 578 static ex tan_evalf(const ex & x)
 579 {
 580         if (is_exactly_a<numeric>(x))
 581                 return tan(ex_to<numeric>(x));
 582
 583         return tan(x).hold();
 584 }
 585
 586 static ex tan_eval(const ex & x)
 587 {
 588         // tan(n/d*Pi) -> { all known non-nested radicals }
 589         const ex SixtyExOverPi = _ex60*x/Pi;
 590         ex sign = _ex1;
 591         if (SixtyExOverPi.info(info_flags::integer)) {
 592                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num60_p);
 593                 if (z>=*_num60_p) {
 594                         // wrap to interval [0, Pi)
 595                         z -= *_num60_p;
 596                 }
 597                 if (z>=*_num30_p) {
 598                         // wrap to interval [0, Pi/2)
 599                         z = *_num60_p-z;
 600                         sign = _ex_1;
 601                 }
 602                 if (z.is_equal(*_num0_p))  // tan(0)       -> 0
 603                         return _ex0;
 604                 if (z.is_equal(*_num5_p))  // tan(Pi/12)   -> 2-sqrt(3)
 605                         return sign*(_ex2-sqrt(_ex3));
 606                 if (z.is_equal(*_num10_p)) // tan(Pi/6)    -> sqrt(3)/3
 607                         return sign*_ex1_3*sqrt(_ex3);
 608                 if (z.is_equal(*_num15_p)) // tan(Pi/4)    -> 1
 609                         return sign;
 610                 if (z.is_equal(*_num20_p)) // tan(Pi/3)    -> sqrt(3)
 611                         return sign*sqrt(_ex3);
 612                 if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3)
 613                         return sign*(sqrt(_ex3)+_ex2);
 614                 if (z.is_equal(*_num30_p)) // tan(Pi/2)    -> infinity
 615                         throw (pole_error("tan_eval(): simple pole",1));
 616         }
 617
 618         if (is_exactly_a<function>(x)) {
 619                 const ex &t = x.op(0);
 620
 621                 // tan(atan(x)) -> x
 622                 if (is_ex_the_function(x, atan))
 623                         return t;
 624
 625                 // tan(asin(x)) -> x/sqrt(1+x^2)
 626                 if (is_ex_the_function(x, asin))
 627                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 628
 629                 // tan(acos(x)) -> sqrt(1-x^2)/x
 630                 if (is_ex_the_function(x, acos))
 631                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
 632         }
 633
 634         // tan(float) -> float
 635         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 636                 return tan(ex_to<numeric>(x));
 637         }
 638
 639         // tan() is odd
 640         if (x.info(info_flags::negative))
 641                 return -tan(-x);
 642
 643         return tan(x).hold();
 644 }
 645
 646 static ex tan_deriv(const ex & x, unsigned deriv_param)
 647 {
 648         GINAC_ASSERT(deriv_param==0);
 649
 650         // d/dx tan(x) -> 1+tan(x)^2;
 651         return (_ex1+power(tan(x),_ex2));
 652 }
 653
 654 static ex tan_real_part(const ex & x)
 655 {
 656         ex a = GiNaC::real_part(x);
 657         ex b = GiNaC::imag_part(x);
 658         return tan(a)/(1+power(tan(a),2)*power(tan(b),2));
 659 }
 660
 661 static ex tan_imag_part(const ex & x)
 662 {
 663         ex a = GiNaC::real_part(x);
 664         ex b = GiNaC::imag_part(x);
 665         return tanh(b)/(1+power(tan(a),2)*power(tan(b),2));
 666 }
 667
 668 static ex tan_series(const ex &x,
 669                      const relational &rel,
 670                      int order,
 671                      unsigned options)
 672 {
 673         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 674         // method:
 675         // Taylor series where there is no pole falls back to tan_deriv.
 676         // On a pole simply expand sin(x)/cos(x).
 677         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 678         if (!(2*x_pt/Pi).info(info_flags::odd))
 679                 throw do_taylor();  // caught by function::series()
 680         // if we got here we have to care for a simple pole
 681         return (sin(x)/cos(x)).series(rel, order, options);
 682 }
 683
 684 static ex tan_conjugate(const ex & x)
 685 {
 686         // conjugate(tan(x))==tan(conjugate(x))
 687         return tan(x.conjugate());
 688 }
 689
 690 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 691                        evalf_func(tan_evalf).
 692                        derivative_func(tan_deriv).
 693                        series_func(tan_series).
 694                        real_part_func(tan_real_part).
 695                        imag_part_func(tan_imag_part).
 696                        conjugate_func(tan_conjugate).
 697                        latex_name("\\tan"));
 698
 699 //////////
 700 // inverse sine (arc sine)
 701 //////////
 702
 703 static ex asin_evalf(const ex & x)
 704 {
 705         if (is_exactly_a<numeric>(x))
 706                 return asin(ex_to<numeric>(x));
 707
 708         return asin(x).hold();
 709 }
 710
 711 static ex asin_eval(const ex & x)
 712 {
 713         if (x.info(info_flags::numeric)) {
 714
 715                 // asin(0) -> 0
 716                 if (x.is_zero())
 717                         return x;
 718
 719                 // asin(1/2) -> Pi/6
 720                 if (x.is_equal(_ex1_2))
 721                         return numeric(1,6)*Pi;
 722
 723                 // asin(1) -> Pi/2
 724                 if (x.is_equal(_ex1))
 725                         return _ex1_2*Pi;
 726
 727                 // asin(-1/2) -> -Pi/6
 728                 if (x.is_equal(_ex_1_2))
 729                         return numeric(-1,6)*Pi;
 730
 731                 // asin(-1) -> -Pi/2
 732                 if (x.is_equal(_ex_1))
 733                         return _ex_1_2*Pi;
 734
 735                 // asin(float) -> float
 736                 if (!x.info(info_flags::crational))
 737                         return asin(ex_to<numeric>(x));
 738
 739                 // asin() is odd
 740                 if (x.info(info_flags::negative))
 741                         return -asin(-x);
 742         }
 743
 744         return asin(x).hold();
 745 }
 746
 747 static ex asin_deriv(const ex & x, unsigned deriv_param)
 748 {
 749         GINAC_ASSERT(deriv_param==0);
 750
 751         // d/dx asin(x) -> 1/sqrt(1-x^2)
 752         return power(1-power(x,_ex2),_ex_1_2);
 753 }
 754
 755 static ex asin_conjugate(const ex & x)
 756 {
 757         // conjugate(asin(x))==asin(conjugate(x)) unless on the branch cuts which
 758         // run along the real axis outside the interval [-1, +1].
 759         if (is_exactly_a<numeric>(x) &&
 760             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
 761                 return asin(x.conjugate());
 762         }
 763         return conjugate_function(asin(x)).hold();
 764 }
 765
 766 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 767                         evalf_func(asin_evalf).
 768                         derivative_func(asin_deriv).
 769                         conjugate_func(asin_conjugate).
 770                         latex_name("\\arcsin"));
 771
 772 //////////
 773 // inverse cosine (arc cosine)
 774 //////////
 775
 776 static ex acos_evalf(const ex & x)
 777 {
 778         if (is_exactly_a<numeric>(x))
 779                 return acos(ex_to<numeric>(x));
 780
 781         return acos(x).hold();
 782 }
 783
 784 static ex acos_eval(const ex & x)
 785 {
 786         if (x.info(info_flags::numeric)) {
 787
 788                 // acos(1) -> 0
 789                 if (x.is_equal(_ex1))
 790                         return _ex0;
 791
 792                 // acos(1/2) -> Pi/3
 793                 if (x.is_equal(_ex1_2))
 794                         return _ex1_3*Pi;
 795
 796                 // acos(0) -> Pi/2
 797                 if (x.is_zero())
 798                         return _ex1_2*Pi;
 799
 800                 // acos(-1/2) -> 2/3*Pi
 801                 if (x.is_equal(_ex_1_2))
 802                         return numeric(2,3)*Pi;
 803
 804                 // acos(-1) -> Pi
 805                 if (x.is_equal(_ex_1))
 806                         return Pi;
 807
 808                 // acos(float) -> float
 809                 if (!x.info(info_flags::crational))
 810                         return acos(ex_to<numeric>(x));
 811
 812                 // acos(-x) -> Pi-acos(x)
 813                 if (x.info(info_flags::negative))
 814                         return Pi-acos(-x);
 815         }
 816
 817         return acos(x).hold();
 818 }
 819
 820 static ex acos_deriv(const ex & x, unsigned deriv_param)
 821 {
 822         GINAC_ASSERT(deriv_param==0);
 823
 824         // d/dx acos(x) -> -1/sqrt(1-x^2)
 825         return -power(1-power(x,_ex2),_ex_1_2);
 826 }
 827
 828 static ex acos_conjugate(const ex & x)
 829 {
 830         // conjugate(acos(x))==acos(conjugate(x)) unless on the branch cuts which
 831         // run along the real axis outside the interval [-1, +1].
 832         if (is_exactly_a<numeric>(x) &&
 833             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
 834                 return acos(x.conjugate());
 835         }
 836         return conjugate_function(acos(x)).hold();
 837 }
 838
 839 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 840                         evalf_func(acos_evalf).
 841                         derivative_func(acos_deriv).
 842                         conjugate_func(acos_conjugate).
 843                         latex_name("\\arccos"));
 844
 845 //////////
 846 // inverse tangent (arc tangent)
 847 //////////
 848
 849 static ex atan_evalf(const ex & x)
 850 {
 851         if (is_exactly_a<numeric>(x))
 852                 return atan(ex_to<numeric>(x));
 853
 854         return atan(x).hold();
 855 }
 856
 857 static ex atan_eval(const ex & x)
 858 {
 859         if (x.info(info_flags::numeric)) {
 860
 861                 // atan(0) -> 0
 862                 if (x.is_zero())
 863                         return _ex0;
 864
 865                 // atan(1) -> Pi/4
 866                 if (x.is_equal(_ex1))
 867                         return _ex1_4*Pi;
 868
 869                 // atan(-1) -> -Pi/4
 870                 if (x.is_equal(_ex_1))
 871                         return _ex_1_4*Pi;
 872
 873                 if (x.is_equal(I) || x.is_equal(-I))
 874                         throw (pole_error("atan_eval(): logarithmic pole",0));
 875
 876                 // atan(float) -> float
 877                 if (!x.info(info_flags::crational))
 878                         return atan(ex_to<numeric>(x));
 879
 880                 // atan() is odd
 881                 if (x.info(info_flags::negative))
 882                         return -atan(-x);
 883         }
 884
 885         return atan(x).hold();
 886 }
 887
 888 static ex atan_deriv(const ex & x, unsigned deriv_param)
 889 {
 890         GINAC_ASSERT(deriv_param==0);
 891
 892         // d/dx atan(x) -> 1/(1+x^2)
 893         return power(_ex1+power(x,_ex2), _ex_1);
 894 }
 895
 896 static ex atan_series(const ex &arg,
 897                       const relational &rel,
 898                       int order,
 899                       unsigned options)
 900 {
 901         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 902         // method:
 903         // Taylor series where there is no pole or cut falls back to atan_deriv.
 904         // There are two branch cuts, one runnig from I up the imaginary axis and
 905         // one running from -I down the imaginary axis.  The points I and -I are
 906         // poles.
 907         // On the branch cuts and the poles series expand
 908         //     (log(1+I*x)-log(1-I*x))/(2*I)
 909         // instead.
 910         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 911         if (!(I*arg_pt).info(info_flags::real))
 912                 throw do_taylor();     // Re(x) != 0
 913         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
 914                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
 915         // care for the poles, using the defining formula for atan()...
 916         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
 917                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
 918         if (!(options & series_options::suppress_branchcut)) {
 919                 // method:
 920                 // This is the branch cut: assemble the primitive series manually and
 921                 // then add the corresponding complex step function.
 922                 const symbol &s = ex_to<symbol>(rel.lhs());
 923                 const ex &point = rel.rhs();
 924                 const symbol foo;
 925                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 926                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
 927                 if ((I*arg_pt)<_ex0)
 928                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
 929                 else
 930                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
 931                 epvector seq { expair(Order0correction, _ex0),
 932                                expair(Order(_ex1), order) };
 933                 return series(replarg - pseries(rel, std::move(seq)), rel, order);
 934         }
 935         throw do_taylor();
 936 }
 937
 938 static ex atan_conjugate(const ex & x)
 939 {
 940         // conjugate(atan(x))==atan(conjugate(x)) unless on the branch cuts which
 941         // run along the imaginary axis outside the interval [-I, +I].
 942         if (x.info(info_flags::real))
 943                 return atan(x);
 944         if (is_exactly_a<numeric>(x)) {
 945                 const numeric x_re = ex_to<numeric>(x.real_part());
 946                 const numeric x_im = ex_to<numeric>(x.imag_part());
 947                 if (!x_re.is_zero() ||
 948                     (x_im > *_num_1_p && x_im < *_num1_p))
 949                         return atan(x.conjugate());
 950         }
 951         return conjugate_function(atan(x)).hold();
 952 }
 953
 954 REGISTER_FUNCTION(atan, eval_func(atan_eval).
 955                         evalf_func(atan_evalf).
 956                         derivative_func(atan_deriv).
 957                         series_func(atan_series).
 958                         conjugate_func(atan_conjugate).
 959                         latex_name("\\arctan"));
 960
 961 //////////
 962 // inverse tangent (atan2(y,x))
 963 //////////
 964
 965 static ex atan2_evalf(const ex &y, const ex &x)
 966 {
 967         if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
 968                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 969
 970         return atan2(y, x).hold();
 971 }
 972
 973 static ex atan2_eval(const ex & y, const ex & x)
 974 {
 975         if (y.is_zero()) {
 976
 977                 // atan2(0, 0) -> 0
 978                 if (x.is_zero())
 979                         return _ex0;
 980
 981                 // atan2(0, x), x real and positive -> 0
 982                 if (x.info(info_flags::positive))
 983                         return _ex0;
 984
 985                 // atan2(0, x), x real and negative -> Pi
 986                 if (x.info(info_flags::negative))
 987                         return Pi;
 988         }
 989
 990         if (x.is_zero()) {
 991
 992                 // atan2(y, 0), y real and positive -> Pi/2
 993                 if (y.info(info_flags::positive))
 994                         return _ex1_2*Pi;
 995
 996                 // atan2(y, 0), y real and negative -> -Pi/2
 997                 if (y.info(info_flags::negative))
 998                         return _ex_1_2*Pi;
 999         }
1000
1001         if (y.is_equal(x)) {
1002
1003                 // atan2(y, y), y real and positive -> Pi/4
1004                 if (y.info(info_flags::positive))
1005                         return _ex1_4*Pi;
1006
1007                 // atan2(y, y), y real and negative -> -3/4*Pi
1008                 if (y.info(info_flags::negative))
1009                         return numeric(-3, 4)*Pi;
1010         }
1011
1012         if (y.is_equal(-x)) {
1013
1014                 // atan2(y, -y), y real and positive -> 3*Pi/4
1015                 if (y.info(info_flags::positive))
1016                         return numeric(3, 4)*Pi;
1017
1018                 // atan2(y, -y), y real and negative -> -Pi/4
1019                 if (y.info(info_flags::negative))
1020                         return _ex_1_4*Pi;
1021         }
1022
1023         // atan2(float, float) -> float
1024         if (is_a<numeric>(y) && !y.info(info_flags::crational) &&
1025             is_a<numeric>(x) && !x.info(info_flags::crational))
1026                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
1027
1028         // atan2(real, real) -> atan(y/x) +/- Pi
1029         if (y.info(info_flags::real) && x.info(info_flags::real)) {
1030                 if (x.info(info_flags::positive))
1031                         return atan(y/x);
1032
1033                 if (x.info(info_flags::negative)) {
1034                         if (y.info(info_flags::positive))
1035                                 return atan(y/x)+Pi;
1036                         if (y.info(info_flags::negative))
1037                                 return atan(y/x)-Pi;
1038                 }
1039         }
1040
1041         return atan2(y, x).hold();
1042 }
1043
1044 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
1045 {
1046         GINAC_ASSERT(deriv_param<2);
1047
1048         if (deriv_param==0) {
1049                 // d/dy atan2(y,x)
1050                 return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
1051         }
1052         // d/dx atan2(y,x)
1053         return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
1054 }
1055
1056 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
1057                          evalf_func(atan2_evalf).
1058                          derivative_func(atan2_deriv));
1059
1060 //////////
1061 // hyperbolic sine (trigonometric function)
1062 //////////
1063
1064 static ex sinh_evalf(const ex & x)
1065 {
1066         if (is_exactly_a<numeric>(x))
1067                 return sinh(ex_to<numeric>(x));
1068
1069         return sinh(x).hold();
1070 }
1071
1072 static ex sinh_eval(const ex & x)
1073 {
1074         if (x.info(info_flags::numeric)) {
1075
1076                 // sinh(0) -> 0
1077                 if (x.is_zero())
1078                         return _ex0;
1079
1080                 // sinh(float) -> float
1081                 if (!x.info(info_flags::crational))
1082                         return sinh(ex_to<numeric>(x));
1083
1084                 // sinh() is odd
1085                 if (x.info(info_flags::negative))
1086                         return -sinh(-x);
1087         }
1088
1089         if ((x/Pi).info(info_flags::numeric) &&
1090                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
1091                 return I*sin(x/I);
1092
1093         if (is_exactly_a<function>(x)) {
1094                 const ex &t = x.op(0);
1095
1096                 // sinh(asinh(x)) -> x
1097                 if (is_ex_the_function(x, asinh))
1098                         return t;
1099
1100                 // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
1101                 if (is_ex_the_function(x, acosh))
1102                         return sqrt(t-_ex1)*sqrt(t+_ex1);
1103
1104                 // sinh(atanh(x)) -> x/sqrt(1-x^2)
1105                 if (is_ex_the_function(x, atanh))
1106                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
1107         }
1108
1109         return sinh(x).hold();
1110 }
1111
1112 static ex sinh_deriv(const ex & x, unsigned deriv_param)
1113 {
1114         GINAC_ASSERT(deriv_param==0);
1115
1116         // d/dx sinh(x) -> cosh(x)
1117         return cosh(x);
1118 }
1119
1120 static ex sinh_real_part(const ex & x)
1121 {
1122         return sinh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
1123 }
1124
1125 static ex sinh_imag_part(const ex & x)
1126 {
1127         return cosh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
1128 }
1129
1130 static ex sinh_conjugate(const ex & x)
1131 {
1132         // conjugate(sinh(x))==sinh(conjugate(x))
1133         return sinh(x.conjugate());
1134 }
1135
1136 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
1137                         evalf_func(sinh_evalf).
1138                         derivative_func(sinh_deriv).
1139                         real_part_func(sinh_real_part).
1140                         imag_part_func(sinh_imag_part).
1141                         conjugate_func(sinh_conjugate).
1142                         latex_name("\\sinh"));
1143
1144 //////////
1145 // hyperbolic cosine (trigonometric function)
1146 //////////
1147
1148 static ex cosh_evalf(const ex & x)
1149 {
1150         if (is_exactly_a<numeric>(x))
1151                 return cosh(ex_to<numeric>(x));
1152
1153         return cosh(x).hold();
1154 }
1155
1156 static ex cosh_eval(const ex & x)
1157 {
1158         if (x.info(info_flags::numeric)) {
1159
1160                 // cosh(0) -> 1
1161                 if (x.is_zero())
1162                         return _ex1;
1163
1164                 // cosh(float) -> float
1165                 if (!x.info(info_flags::crational))
1166                         return cosh(ex_to<numeric>(x));
1167
1168                 // cosh() is even
1169                 if (x.info(info_flags::negative))
1170                         return cosh(-x);
1171         }
1172
1173         if ((x/Pi).info(info_flags::numeric) &&
1174                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
1175                 return cos(x/I);
1176
1177         if (is_exactly_a<function>(x)) {
1178                 const ex &t = x.op(0);
1179
1180                 // cosh(acosh(x)) -> x
1181                 if (is_ex_the_function(x, acosh))
1182                         return t;
1183
1184                 // cosh(asinh(x)) -> sqrt(1+x^2)
1185                 if (is_ex_the_function(x, asinh))
1186                         return sqrt(_ex1+power(t,_ex2));
1187
1188                 // cosh(atanh(x)) -> 1/sqrt(1-x^2)
1189                 if (is_ex_the_function(x, atanh))
1190                         return power(_ex1-power(t,_ex2),_ex_1_2);
1191         }
1192
1193         return cosh(x).hold();
1194 }
1195
1196 static ex cosh_deriv(const ex & x, unsigned deriv_param)
1197 {
1198         GINAC_ASSERT(deriv_param==0);
1199
1200         // d/dx cosh(x) -> sinh(x)
1201         return sinh(x);
1202 }
1203
1204 static ex cosh_real_part(const ex & x)
1205 {
1206         return cosh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
1207 }
1208
1209 static ex cosh_imag_part(const ex & x)
1210 {
1211         return sinh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
1212 }
1213
1214 static ex cosh_conjugate(const ex & x)
1215 {
1216         // conjugate(cosh(x))==cosh(conjugate(x))
1217         return cosh(x.conjugate());
1218 }
1219
1220 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
1221                         evalf_func(cosh_evalf).
1222                         derivative_func(cosh_deriv).
1223                         real_part_func(cosh_real_part).
1224                         imag_part_func(cosh_imag_part).
1225                         conjugate_func(cosh_conjugate).
1226                         latex_name("\\cosh"));
1227
1228 //////////
1229 // hyperbolic tangent (trigonometric function)
1230 //////////
1231
1232 static ex tanh_evalf(const ex & x)
1233 {
1234         if (is_exactly_a<numeric>(x))
1235                 return tanh(ex_to<numeric>(x));
1236
1237         return tanh(x).hold();
1238 }
1239
1240 static ex tanh_eval(const ex & x)
1241 {
1242         if (x.info(info_flags::numeric)) {
1243
1244                 // tanh(0) -> 0
1245                 if (x.is_zero())
1246                         return _ex0;
1247
1248                 // tanh(float) -> float
1249                 if (!x.info(info_flags::crational))
1250                         return tanh(ex_to<numeric>(x));
1251
1252                 // tanh() is odd
1253                 if (x.info(info_flags::negative))
1254                         return -tanh(-x);
1255         }
1256
1257         if ((x/Pi).info(info_flags::numeric) &&
1258                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
1259                 return I*tan(x/I);
1260
1261         if (is_exactly_a<function>(x)) {
1262                 const ex &t = x.op(0);
1263
1264                 // tanh(atanh(x)) -> x
1265                 if (is_ex_the_function(x, atanh))
1266                         return t;
1267
1268                 // tanh(asinh(x)) -> x/sqrt(1+x^2)
1269                 if (is_ex_the_function(x, asinh))
1270                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
1271
1272                 // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
1273                 if (is_ex_the_function(x, acosh))
1274                         return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
1275         }
1276
1277         return tanh(x).hold();
1278 }
1279
1280 static ex tanh_deriv(const ex & x, unsigned deriv_param)
1281 {
1282         GINAC_ASSERT(deriv_param==0);
1283
1284         // d/dx tanh(x) -> 1-tanh(x)^2
1285         return _ex1-power(tanh(x),_ex2);
1286 }
1287
1288 static ex tanh_series(const ex &x,
1289                       const relational &rel,
1290                       int order,
1291                       unsigned options)
1292 {
1293         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1294         // method:
1295         // Taylor series where there is no pole falls back to tanh_deriv.
1296         // On a pole simply expand sinh(x)/cosh(x).
1297         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1298         if (!(2*I*x_pt/Pi).info(info_flags::odd))
1299                 throw do_taylor();  // caught by function::series()
1300         // if we got here we have to care for a simple pole
1301         return (sinh(x)/cosh(x)).series(rel, order, options);
1302 }
1303
1304 static ex tanh_real_part(const ex & x)
1305 {
1306         ex a = GiNaC::real_part(x);
1307         ex b = GiNaC::imag_part(x);
1308         return tanh(a)/(1+power(tanh(a),2)*power(tan(b),2));
1309 }
1310
1311 static ex tanh_imag_part(const ex & x)
1312 {
1313         ex a = GiNaC::real_part(x);
1314         ex b = GiNaC::imag_part(x);
1315         return tan(b)/(1+power(tanh(a),2)*power(tan(b),2));
1316 }
1317
1318 static ex tanh_conjugate(const ex & x)
1319 {
1320         // conjugate(tanh(x))==tanh(conjugate(x))
1321         return tanh(x.conjugate());
1322 }
1323
1324 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
1325                         evalf_func(tanh_evalf).
1326                         derivative_func(tanh_deriv).
1327                         series_func(tanh_series).
1328                         real_part_func(tanh_real_part).
1329                         imag_part_func(tanh_imag_part).
1330                         conjugate_func(tanh_conjugate).
1331                         latex_name("\\tanh"));
1332
1333 //////////
1334 // inverse hyperbolic sine (trigonometric function)
1335 //////////
1336
1337 static ex asinh_evalf(const ex & x)
1338 {
1339         if (is_exactly_a<numeric>(x))
1340                 return asinh(ex_to<numeric>(x));
1341
1342         return asinh(x).hold();
1343 }
1344
1345 static ex asinh_eval(const ex & x)
1346 {
1347         if (x.info(info_flags::numeric)) {
1348
1349                 // asinh(0) -> 0
1350                 if (x.is_zero())
1351                         return _ex0;
1352
1353                 // asinh(float) -> float
1354                 if (!x.info(info_flags::crational))
1355                         return asinh(ex_to<numeric>(x));
1356
1357                 // asinh() is odd
1358                 if (x.info(info_flags::negative))
1359                         return -asinh(-x);
1360         }
1361
1362         return asinh(x).hold();
1363 }
1364
1365 static ex asinh_deriv(const ex & x, unsigned deriv_param)
1366 {
1367         GINAC_ASSERT(deriv_param==0);
1368
1369         // d/dx asinh(x) -> 1/sqrt(1+x^2)
1370         return power(_ex1+power(x,_ex2),_ex_1_2);
1371 }
1372
1373 static ex asinh_conjugate(const ex & x)
1374 {
1375         // conjugate(asinh(x))==asinh(conjugate(x)) unless on the branch cuts which
1376         // run along the imaginary axis outside the interval [-I, +I].
1377         if (x.info(info_flags::real))
1378                 return asinh(x);
1379         if (is_exactly_a<numeric>(x)) {
1380                 const numeric x_re = ex_to<numeric>(x.real_part());
1381                 const numeric x_im = ex_to<numeric>(x.imag_part());
1382                 if (!x_re.is_zero() ||
1383                     (x_im > *_num_1_p && x_im < *_num1_p))
1384                         return asinh(x.conjugate());
1385         }
1386         return conjugate_function(asinh(x)).hold();
1387 }
1388
1389 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
1390                          evalf_func(asinh_evalf).
1391                          derivative_func(asinh_deriv).
1392                          conjugate_func(asinh_conjugate));
1393
1394 //////////
1395 // inverse hyperbolic cosine (trigonometric function)
1396 //////////
1397
1398 static ex acosh_evalf(const ex & x)
1399 {
1400         if (is_exactly_a<numeric>(x))
1401                 return acosh(ex_to<numeric>(x));
1402
1403         return acosh(x).hold();
1404 }
1405
1406 static ex acosh_eval(const ex & x)
1407 {
1408         if (x.info(info_flags::numeric)) {
1409
1410                 // acosh(0) -> Pi*I/2
1411                 if (x.is_zero())
1412                         return Pi*I*numeric(1,2);
1413
1414                 // acosh(1) -> 0
1415                 if (x.is_equal(_ex1))
1416                         return _ex0;
1417
1418                 // acosh(-1) -> Pi*I
1419                 if (x.is_equal(_ex_1))
1420                         return Pi*I;
1421
1422                 // acosh(float) -> float
1423                 if (!x.info(info_flags::crational))
1424                         return acosh(ex_to<numeric>(x));
1425
1426                 // acosh(-x) -> Pi*I-acosh(x)
1427                 if (x.info(info_flags::negative))
1428                         return Pi*I-acosh(-x);
1429         }
1430
1431         return acosh(x).hold();
1432 }
1433
1434 static ex acosh_deriv(const ex & x, unsigned deriv_param)
1435 {
1436         GINAC_ASSERT(deriv_param==0);
1437
1438         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
1439         return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
1440 }
1441
1442 static ex acosh_conjugate(const ex & x)
1443 {
1444         // conjugate(acosh(x))==acosh(conjugate(x)) unless on the branch cut
1445         // which runs along the real axis from +1 to -inf.
1446         if (is_exactly_a<numeric>(x) &&
1447             (!x.imag_part().is_zero() || x > *_num1_p)) {
1448                 return acosh(x.conjugate());
1449         }
1450         return conjugate_function(acosh(x)).hold();
1451 }
1452
1453 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
1454                          evalf_func(acosh_evalf).
1455                          derivative_func(acosh_deriv).
1456                          conjugate_func(acosh_conjugate));
1457
1458 //////////
1459 // inverse hyperbolic tangent (trigonometric function)
1460 //////////
1461
1462 static ex atanh_evalf(const ex & x)
1463 {
1464         if (is_exactly_a<numeric>(x))
1465                 return atanh(ex_to<numeric>(x));
1466
1467         return atanh(x).hold();
1468 }
1469
1470 static ex atanh_eval(const ex & x)
1471 {
1472         if (x.info(info_flags::numeric)) {
1473
1474                 // atanh(0) -> 0
1475                 if (x.is_zero())
1476                         return _ex0;
1477
1478                 // atanh({+|-}1) -> throw
1479                 if (x.is_equal(_ex1) || x.is_equal(_ex_1))
1480                         throw (pole_error("atanh_eval(): logarithmic pole",0));
1481
1482                 // atanh(float) -> float
1483                 if (!x.info(info_flags::crational))
1484                         return atanh(ex_to<numeric>(x));
1485
1486                 // atanh() is odd
1487                 if (x.info(info_flags::negative))
1488                         return -atanh(-x);
1489         }
1490
1491         return atanh(x).hold();
1492 }
1493
1494 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1495 {
1496         GINAC_ASSERT(deriv_param==0);
1497
1498         // d/dx atanh(x) -> 1/(1-x^2)
1499         return power(_ex1-power(x,_ex2),_ex_1);
1500 }
1501
1502 static ex atanh_series(const ex &arg,
1503                        const relational &rel,
1504                        int order,
1505                        unsigned options)
1506 {
1507         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1508         // method:
1509         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1510         // There are two branch cuts, one runnig from 1 up the real axis and one
1511         // one running from -1 down the real axis.  The points 1 and -1 are poles
1512         // On the branch cuts and the poles series expand
1513         //     (log(1+x)-log(1-x))/2
1514         // instead.
1515         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
1516         if (!(arg_pt).info(info_flags::real))
1517                 throw do_taylor();     // Im(x) != 0
1518         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1519                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1520         // care for the poles, using the defining formula for atanh()...
1521         if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1522                 return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1523         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1524         if (!(options & series_options::suppress_branchcut)) {
1525                 // method:
1526                 // This is the branch cut: assemble the primitive series manually and
1527                 // then add the corresponding complex step function.
1528                 const symbol &s = ex_to<symbol>(rel.lhs());
1529                 const ex &point = rel.rhs();
1530                 const symbol foo;
1531                 const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1532                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1533                 if (arg_pt<_ex0)
1534                         Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1535                 else
1536                         Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1537                 epvector seq { expair(Order0correction, _ex0),
1538                                expair(Order(_ex1), order) };
1539                 return series(replarg - pseries(rel, std::move(seq)), rel, order);
1540         }
1541         throw do_taylor();
1542 }
1543
1544 static ex atanh_conjugate(const ex & x)
1545 {
1546         // conjugate(atanh(x))==atanh(conjugate(x)) unless on the branch cuts which
1547         // run along the real axis outside the interval [-1, +1].
1548         if (is_exactly_a<numeric>(x) &&
1549             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
1550                 return atanh(x.conjugate());
1551         }
1552         return conjugate_function(atanh(x)).hold();
1553 }
1554
1555 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1556                          evalf_func(atanh_evalf).
1557                          derivative_func(atanh_deriv).
1558                          series_func(atanh_series).
1559                          conjugate_func(atanh_conjugate));
1560
1561
1562 } // namespace GiNaC