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) 1999-2023 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 static ex exp_power(const ex & x, const ex & a)
 132 {
 133         /*
 134          * The power law (e^x)^a=e^(x*a) is used in two cases:
 135          * a) a is an integer and x may be complex;
 136          * b) both x and a are reals.
 137          * Negative a is excluded to keep automatic simplifications like exp(x)/exp(x)=1.
 138          */
 139         if (a.info(info_flags::nonnegative)
 140             && (a.info(info_flags::integer) || (x.info(info_flags::real) && a.info(info_flags::real))))
 141                 return exp(x*a);
 142         else if (a.info(info_flags::negative)
 143                  && (a.info(info_flags::integer) || (x.info(info_flags::real) && a.info(info_flags::real))))
 144                 return power(exp(-x*a), _ex_1).hold();
 145
 146         return power(exp(x), a).hold();
 147 }
 148
 149 static bool exp_info(const ex & x, unsigned inf)
 150 {
 151         switch (inf) {
 152         case info_flags::numeric:
 153         case info_flags::expanded:
 154         case info_flags::real:
 155                 return x.info(inf);
 156         case info_flags::positive:
 157         case info_flags::nonnegative:
 158                 return x.info(info_flags::real);
 159         default:
 160                 return false;
 161         }
 162 }
 163
 164 REGISTER_FUNCTION(exp, eval_func(exp_eval).
 165                        evalf_func(exp_evalf).
 166                        info_func(exp_info).
 167                        expand_func(exp_expand).
 168                        derivative_func(exp_deriv).
 169                        real_part_func(exp_real_part).
 170                        imag_part_func(exp_imag_part).
 171                        conjugate_func(exp_conjugate).
 172                        power_func(exp_power).
 173                        latex_name("\\exp"));
 174
 175 //////////
 176 // natural logarithm
 177 //////////
 178
 179 static ex log_evalf(const ex & x)
 180 {
 181         if (is_exactly_a<numeric>(x))
 182                 return log(ex_to<numeric>(x));
 183
 184         return log(x).hold();
 185 }
 186
 187 static ex log_eval(const ex & x)
 188 {
 189         if (x.info(info_flags::numeric)) {
 190                 if (x.is_zero())         // log(0) -> infinity
 191                         throw(pole_error("log_eval(): log(0)",0));
 192                 if (x.info(info_flags::rational) && x.info(info_flags::negative))
 193                         return (log(-x)+I*Pi);
 194                 if (x.is_equal(_ex1))  // log(1) -> 0
 195                         return _ex0;
 196                 if (x.is_equal(I))       // log(I) -> Pi*I/2
 197                         return (Pi*I*_ex1_2);
 198                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
 199                         return (Pi*I*_ex_1_2);
 200
 201                 // log(float) -> float
 202                 if (!x.info(info_flags::crational))
 203                         return log(ex_to<numeric>(x));
 204         }
 205
 206         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
 207         if (is_ex_the_function(x, exp)) {
 208                 const ex &t = x.op(0);
 209                 if (t.info(info_flags::real))
 210                         return t;
 211         }
 212
 213         return log(x).hold();
 214 }
 215
 216 static ex log_deriv(const ex & x, unsigned deriv_param)
 217 {
 218         GINAC_ASSERT(deriv_param==0);
 219
 220         // d/dx log(x) -> 1/x
 221         return power(x, _ex_1);
 222 }
 223
 224 static ex log_series(const ex &arg,
 225                      const relational &rel,
 226                      int order,
 227                      unsigned options)
 228 {
 229         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 230         ex arg_pt;
 231         bool must_expand_arg = false;
 232         // maybe substitution of rel into arg fails because of a pole
 233         try {
 234                 arg_pt = arg.subs(rel, subs_options::no_pattern);
 235         } catch (pole_error &) {
 236                 must_expand_arg = true;
 237         }
 238         // or we are at the branch point anyways
 239         if (arg_pt.is_zero())
 240                 must_expand_arg = true;
 241
 242         if (arg.diff(ex_to<symbol>(rel.lhs())).is_zero()) {
 243                 throw do_taylor();
 244         }
 245
 246         if (must_expand_arg) {
 247                 // method:
 248                 // This is the branch point: Series expand the argument first, then
 249                 // trivially factorize it to isolate that part which has constant
 250                 // leading coefficient in this fashion:
 251                 //   x^n + x^(n+1) +...+ Order(x^(n+m))  ->  x^n * (1 + x +...+ Order(x^m)).
 252                 // Return a plain n*log(x) for the x^n part and series expand the
 253                 // other part.  Add them together and reexpand again in order to have
 254                 // one unnested pseries object.  All this also works for negative n.
 255                 pseries argser;          // series expansion of log's argument
 256                 unsigned extra_ord = 0;  // extra expansion order
 257                 do {
 258                         // oops, the argument expanded to a pure Order(x^something)...
 259                         argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
 260                         ++extra_ord;
 261                 } while (!argser.is_terminating() && argser.nops()==1);
 262
 263                 const symbol &s = ex_to<symbol>(rel.lhs());
 264                 const ex &point = rel.rhs();
 265                 const int n = argser.ldegree(s);
 266                 epvector seq;
 267                 // construct what we carelessly called the n*log(x) term above
 268                 const ex coeff = argser.coeff(s, n);
 269                 // expand the log, but only if coeff is real and > 0, since otherwise
 270                 // it would make the branch cut run into the wrong direction
 271                 if (coeff.info(info_flags::positive))
 272                         seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
 273                 else
 274                         seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));
 275
 276                 if (!argser.is_terminating() || argser.nops()!=1) {
 277                         // in this case n more (or less) terms are needed
 278                         // (sadly, to generate them, we have to start from the beginning)
 279                         if (n == 0 && coeff == 1) {
 280                                 ex rest = pseries(rel, epvector{expair(-1, _ex0), expair(Order(_ex1), order)}).add_series(argser);
 281                                 ex acc = dynallocate<pseries>(rel, epvector());
 282                                 for (int i = order-1; i>0; --i) {
 283                                         epvector cterm { expair(i%2 ? _ex1/i : _ex_1/i, _ex0) };
 284                                         acc = pseries(rel, std::move(cterm)).add_series(ex_to<pseries>(acc));
 285                                         acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
 286                                 }
 287                                 return acc;
 288                         }
 289                         const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
 290                         return pseries(rel, std::move(seq)).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
 291                 } else  // it was a monomial
 292                         return pseries(rel, std::move(seq));
 293         }
 294         if (!(options & series_options::suppress_branchcut) &&
 295              arg_pt.info(info_flags::negative)) {
 296                 // method:
 297                 // This is the branch cut: assemble the primitive series manually and
 298                 // then add the corresponding complex step function.
 299                 const symbol &s = ex_to<symbol>(rel.lhs());
 300                 const ex &point = rel.rhs();
 301                 const symbol foo;
 302                 const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 303                 epvector seq;
 304                 if (order > 0) {
 305                         seq.reserve(2);
 306                         seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
 307                 }
 308                 seq.push_back(expair(Order(_ex1), order));
 309                 return series(replarg - I*Pi + pseries(rel, std::move(seq)), rel, order);
 310         }
 311         throw do_taylor();  // caught by function::series()
 312 }
 313
 314 static ex log_real_part(const ex & x)
 315 {
 316         if (x.info(info_flags::nonnegative))
 317                 return log(x).hold();
 318         return log(abs(x));
 319 }
 320
 321 static ex log_imag_part(const ex & x)
 322 {
 323         if (x.info(info_flags::nonnegative))
 324                 return 0;
 325         return atan2(GiNaC::imag_part(x), GiNaC::real_part(x));
 326 }
 327
 328 static ex log_expand(const ex & arg, unsigned options)
 329 {
 330         if ((options & expand_options::expand_transcendental)
 331                 && is_exactly_a<mul>(arg) && !arg.info(info_flags::indefinite)) {
 332                 exvector sumseq;
 333                 exvector prodseq;
 334                 sumseq.reserve(arg.nops());
 335                 prodseq.reserve(arg.nops());
 336                 bool possign=true;
 337
 338                 // searching for positive/negative factors
 339                 for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
 340                         ex e;
 341                         if (options & expand_options::expand_function_args)
 342                                 e=i->expand(options);
 343                         else
 344                                 e=*i;
 345                         if (e.info(info_flags::positive))
 346                                 sumseq.push_back(log(e));
 347                         else if (e.info(info_flags::negative)) {
 348                                 sumseq.push_back(log(-e));
 349                                 possign = !possign;
 350                         } else
 351                                 prodseq.push_back(e);
 352                 }
 353
 354                 if (sumseq.size() > 0) {
 355                         ex newarg;
 356                         if (options & expand_options::expand_function_args)
 357                                 newarg=((possign?_ex1:_ex_1)*mul(prodseq)).expand(options);
 358                         else {
 359                                 newarg=(possign?_ex1:_ex_1)*mul(prodseq);
 360                                 ex_to<basic>(newarg).setflag(status_flags::purely_indefinite);
 361                         }
 362                         return add(sumseq)+log(newarg);
 363                 } else {
 364                         if (!(options & expand_options::expand_function_args))
 365                                 ex_to<basic>(arg).setflag(status_flags::purely_indefinite);
 366                 }
 367         }
 368
 369         if (options & expand_options::expand_function_args)
 370                 return log(arg.expand(options)).hold();
 371         else
 372                 return log(arg).hold();
 373 }
 374
 375 static ex log_conjugate(const ex & x)
 376 {
 377         // conjugate(log(x))==log(conjugate(x)) unless on the branch cut which
 378         // runs along the negative real axis.
 379         if (x.info(info_flags::positive)) {
 380                 return log(x);
 381         }
 382         if (is_exactly_a<numeric>(x) &&
 383             !x.imag_part().is_zero()) {
 384                 return log(x.conjugate());
 385         }
 386         return conjugate_function(log(x)).hold();
 387 }
 388
 389 static bool log_info(const ex & x, unsigned inf)
 390 {
 391         switch (inf) {
 392         case info_flags::numeric:
 393         case info_flags::expanded:
 394                 return x.info(inf);
 395         case info_flags::real:
 396                 return x.info(info_flags::positive);
 397         default:
 398                 return false;
 399         }
 400 }
 401
 402 REGISTER_FUNCTION(log, eval_func(log_eval).
 403                        evalf_func(log_evalf).
 404                        info_func(log_info).
 405                        expand_func(log_expand).
 406                        derivative_func(log_deriv).
 407                        series_func(log_series).
 408                        real_part_func(log_real_part).
 409                        imag_part_func(log_imag_part).
 410                        conjugate_func(log_conjugate).
 411                        latex_name("\\ln"));
 412
 413 //////////
 414 // sine (trigonometric function)
 415 //////////
 416
 417 static ex sin_evalf(const ex & x)
 418 {
 419         if (is_exactly_a<numeric>(x))
 420                 return sin(ex_to<numeric>(x));
 421
 422         return sin(x).hold();
 423 }
 424
 425 static ex sin_eval(const ex & x)
 426 {
 427         // sin(n/d*Pi) -> { all known non-nested radicals }
 428         const ex SixtyExOverPi = _ex60*x/Pi;
 429         ex sign = _ex1;
 430         if (SixtyExOverPi.info(info_flags::integer)) {
 431                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 432                 if (z>=*_num60_p) {
 433                         // wrap to interval [0, Pi)
 434                         z -= *_num60_p;
 435                         sign = _ex_1;
 436                 }
 437                 if (z>*_num30_p) {
 438                         // wrap to interval [0, Pi/2)
 439                         z = *_num60_p-z;
 440                 }
 441                 if (z.is_equal(*_num0_p))  // sin(0)       -> 0
 442                         return _ex0;
 443                 if (z.is_equal(*_num5_p))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
 444                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 445                 if (z.is_equal(*_num6_p))  // sin(Pi/10)   -> sqrt(5)/4-1/4
 446                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 447                 if (z.is_equal(*_num10_p)) // sin(Pi/6)    -> 1/2
 448                         return sign*_ex1_2;
 449                 if (z.is_equal(*_num15_p)) // sin(Pi/4)    -> sqrt(2)/2
 450                         return sign*_ex1_2*sqrt(_ex2);
 451                 if (z.is_equal(*_num18_p)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
 452                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 453                 if (z.is_equal(*_num20_p)) // sin(Pi/3)    -> sqrt(3)/2
 454                         return sign*_ex1_2*sqrt(_ex3);
 455                 if (z.is_equal(*_num25_p)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
 456                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 457                 if (z.is_equal(*_num30_p)) // sin(Pi/2)    -> 1
 458                         return sign;
 459         }
 460
 461         if (is_exactly_a<function>(x)) {
 462                 const ex &t = x.op(0);
 463
 464                 // sin(asin(x)) -> x
 465                 if (is_ex_the_function(x, asin))
 466                         return t;
 467
 468                 // sin(acos(x)) -> sqrt(1-x^2)
 469                 if (is_ex_the_function(x, acos))
 470                         return sqrt(_ex1-power(t,_ex2));
 471
 472                 // sin(atan(x)) -> x/sqrt(1+x^2)
 473                 if (is_ex_the_function(x, atan))
 474                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 475         }
 476
 477         // sin(float) -> float
 478         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 479                 return sin(ex_to<numeric>(x));
 480
 481         // sin() is odd
 482         if (x.info(info_flags::negative))
 483                 return -sin(-x);
 484
 485         return sin(x).hold();
 486 }
 487
 488 static ex sin_deriv(const ex & x, unsigned deriv_param)
 489 {
 490         GINAC_ASSERT(deriv_param==0);
 491
 492         // d/dx sin(x) -> cos(x)
 493         return cos(x);
 494 }
 495
 496 static ex sin_real_part(const ex & x)
 497 {
 498         return cosh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 499 }
 500
 501 static ex sin_imag_part(const ex & x)
 502 {
 503         return sinh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 504 }
 505
 506 static ex sin_conjugate(const ex & x)
 507 {
 508         // conjugate(sin(x))==sin(conjugate(x))
 509         return sin(x.conjugate());
 510 }
 511
 512 static bool trig_info(const ex & x, unsigned inf)
 513 {
 514         switch (inf) {
 515         case info_flags::numeric:
 516         case info_flags::expanded:
 517         case info_flags::real:
 518                 return x.info(inf);
 519         default:
 520                 return false;
 521         }
 522 }
 523
 524 REGISTER_FUNCTION(sin, eval_func(sin_eval).
 525                        evalf_func(sin_evalf).
 526                        info_func(trig_info).
 527                        derivative_func(sin_deriv).
 528                        real_part_func(sin_real_part).
 529                        imag_part_func(sin_imag_part).
 530                        conjugate_func(sin_conjugate).
 531                        latex_name("\\sin"));
 532
 533 //////////
 534 // cosine (trigonometric function)
 535 //////////
 536
 537 static ex cos_evalf(const ex & x)
 538 {
 539         if (is_exactly_a<numeric>(x))
 540                 return cos(ex_to<numeric>(x));
 541
 542         return cos(x).hold();
 543 }
 544
 545 static ex cos_eval(const ex & x)
 546 {
 547         // cos(n/d*Pi) -> { all known non-nested radicals }
 548         const ex SixtyExOverPi = _ex60*x/Pi;
 549         ex sign = _ex1;
 550         if (SixtyExOverPi.info(info_flags::integer)) {
 551                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 552                 if (z>=*_num60_p) {
 553                         // wrap to interval [0, Pi)
 554                         z = *_num120_p-z;
 555                 }
 556                 if (z>=*_num30_p) {
 557                         // wrap to interval [0, Pi/2)
 558                         z = *_num60_p-z;
 559                         sign = _ex_1;
 560                 }
 561                 if (z.is_equal(*_num0_p))  // cos(0)       -> 1
 562                         return sign;
 563                 if (z.is_equal(*_num5_p))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 564                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 565                 if (z.is_equal(*_num10_p)) // cos(Pi/6)    -> sqrt(3)/2
 566                         return sign*_ex1_2*sqrt(_ex3);
 567                 if (z.is_equal(*_num12_p)) // cos(Pi/5)    -> sqrt(5)/4+1/4
 568                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 569                 if (z.is_equal(*_num15_p)) // cos(Pi/4)    -> sqrt(2)/2
 570                         return sign*_ex1_2*sqrt(_ex2);
 571                 if (z.is_equal(*_num20_p)) // cos(Pi/3)    -> 1/2
 572                         return sign*_ex1_2;
 573                 if (z.is_equal(*_num24_p)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 574                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 575                 if (z.is_equal(*_num25_p)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 576                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 577                 if (z.is_equal(*_num30_p)) // cos(Pi/2)    -> 0
 578                         return _ex0;
 579         }
 580
 581         if (is_exactly_a<function>(x)) {
 582                 const ex &t = x.op(0);
 583
 584                 // cos(acos(x)) -> x
 585                 if (is_ex_the_function(x, acos))
 586                         return t;
 587
 588                 // cos(asin(x)) -> sqrt(1-x^2)
 589                 if (is_ex_the_function(x, asin))
 590                         return sqrt(_ex1-power(t,_ex2));
 591
 592                 // cos(atan(x)) -> 1/sqrt(1+x^2)
 593                 if (is_ex_the_function(x, atan))
 594                         return power(_ex1+power(t,_ex2),_ex_1_2);
 595         }
 596
 597         // cos(float) -> float
 598         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 599                 return cos(ex_to<numeric>(x));
 600
 601         // cos() is even
 602         if (x.info(info_flags::negative))
 603                 return cos(-x);
 604
 605         return cos(x).hold();
 606 }
 607
 608 static ex cos_deriv(const ex & x, unsigned deriv_param)
 609 {
 610         GINAC_ASSERT(deriv_param==0);
 611
 612         // d/dx cos(x) -> -sin(x)
 613         return -sin(x);
 614 }
 615
 616 static ex cos_real_part(const ex & x)
 617 {
 618         return cosh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 619 }
 620
 621 static ex cos_imag_part(const ex & x)
 622 {
 623         return -sinh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 624 }
 625
 626 static ex cos_conjugate(const ex & x)
 627 {
 628         // conjugate(cos(x))==cos(conjugate(x))
 629         return cos(x.conjugate());
 630 }
 631
 632 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 633                        info_func(trig_info).
 634                        evalf_func(cos_evalf).
 635                        derivative_func(cos_deriv).
 636                        real_part_func(cos_real_part).
 637                        imag_part_func(cos_imag_part).
 638                        conjugate_func(cos_conjugate).
 639                        latex_name("\\cos"));
 640
 641 //////////
 642 // tangent (trigonometric function)
 643 //////////
 644
 645 static ex tan_evalf(const ex & x)
 646 {
 647         if (is_exactly_a<numeric>(x))
 648                 return tan(ex_to<numeric>(x));
 649
 650         return tan(x).hold();
 651 }
 652
 653 static ex tan_eval(const ex & x)
 654 {
 655         // tan(n/d*Pi) -> { all known non-nested radicals }
 656         const ex SixtyExOverPi = _ex60*x/Pi;
 657         ex sign = _ex1;
 658         if (SixtyExOverPi.info(info_flags::integer)) {
 659                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num60_p);
 660                 if (z>=*_num60_p) {
 661                         // wrap to interval [0, Pi)
 662                         z -= *_num60_p;
 663                 }
 664                 if (z>=*_num30_p) {
 665                         // wrap to interval [0, Pi/2)
 666                         z = *_num60_p-z;
 667                         sign = _ex_1;
 668                 }
 669                 if (z.is_equal(*_num0_p))  // tan(0)       -> 0
 670                         return _ex0;
 671                 if (z.is_equal(*_num5_p))  // tan(Pi/12)   -> 2-sqrt(3)
 672                         return sign*(_ex2-sqrt(_ex3));
 673                 if (z.is_equal(*_num10_p)) // tan(Pi/6)    -> sqrt(3)/3
 674                         return sign*_ex1_3*sqrt(_ex3);
 675                 if (z.is_equal(*_num15_p)) // tan(Pi/4)    -> 1
 676                         return sign;
 677                 if (z.is_equal(*_num20_p)) // tan(Pi/3)    -> sqrt(3)
 678                         return sign*sqrt(_ex3);
 679                 if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3)
 680                         return sign*(sqrt(_ex3)+_ex2);
 681                 if (z.is_equal(*_num30_p)) // tan(Pi/2)    -> infinity
 682                         throw (pole_error("tan_eval(): simple pole",1));
 683         }
 684
 685         if (is_exactly_a<function>(x)) {
 686                 const ex &t = x.op(0);
 687
 688                 // tan(atan(x)) -> x
 689                 if (is_ex_the_function(x, atan))
 690                         return t;
 691
 692                 // tan(asin(x)) -> x/sqrt(1+x^2)
 693                 if (is_ex_the_function(x, asin))
 694                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 695
 696                 // tan(acos(x)) -> sqrt(1-x^2)/x
 697                 if (is_ex_the_function(x, acos))
 698                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
 699         }
 700
 701         // tan(float) -> float
 702         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 703                 return tan(ex_to<numeric>(x));
 704         }
 705
 706         // tan() is odd
 707         if (x.info(info_flags::negative))
 708                 return -tan(-x);
 709
 710         return tan(x).hold();
 711 }
 712
 713 static ex tan_deriv(const ex & x, unsigned deriv_param)
 714 {
 715         GINAC_ASSERT(deriv_param==0);
 716
 717         // d/dx tan(x) -> 1+tan(x)^2;
 718         return (_ex1+power(tan(x),_ex2));
 719 }
 720
 721 static ex tan_real_part(const ex & x)
 722 {
 723         ex a = GiNaC::real_part(x);
 724         ex b = GiNaC::imag_part(x);
 725         return tan(a)/(1+power(tan(a),2)*power(tan(b),2));
 726 }
 727
 728 static ex tan_imag_part(const ex & x)
 729 {
 730         ex a = GiNaC::real_part(x);
 731         ex b = GiNaC::imag_part(x);
 732         return tanh(b)/(1+power(tan(a),2)*power(tan(b),2));
 733 }
 734
 735 static ex tan_series(const ex &x,
 736                      const relational &rel,
 737                      int order,
 738                      unsigned options)
 739 {
 740         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 741         // method:
 742         // Taylor series where there is no pole falls back to tan_deriv.
 743         // On a pole simply expand sin(x)/cos(x).
 744         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 745         if (!(2*x_pt/Pi).info(info_flags::odd))
 746                 throw do_taylor();  // caught by function::series()
 747         // if we got here we have to care for a simple pole
 748         return (sin(x)/cos(x)).series(rel, order, options);
 749 }
 750
 751 static ex tan_conjugate(const ex & x)
 752 {
 753         // conjugate(tan(x))==tan(conjugate(x))
 754         return tan(x.conjugate());
 755 }
 756
 757 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 758                        evalf_func(tan_evalf).
 759                        info_func(trig_info).
 760                        derivative_func(tan_deriv).
 761                        series_func(tan_series).
 762                        real_part_func(tan_real_part).
 763                        imag_part_func(tan_imag_part).
 764                        conjugate_func(tan_conjugate).
 765                        latex_name("\\tan"));
 766
 767 //////////
 768 // inverse sine (arc sine)
 769 //////////
 770
 771 static ex asin_evalf(const ex & x)
 772 {
 773         if (is_exactly_a<numeric>(x))
 774                 return asin(ex_to<numeric>(x));
 775
 776         return asin(x).hold();
 777 }
 778
 779 static ex asin_eval(const ex & x)
 780 {
 781         if (x.info(info_flags::numeric)) {
 782
 783                 // asin(0) -> 0
 784                 if (x.is_zero())
 785                         return x;
 786
 787                 // asin(1/2) -> Pi/6
 788                 if (x.is_equal(_ex1_2))
 789                         return numeric(1,6)*Pi;
 790
 791                 // asin(1) -> Pi/2
 792                 if (x.is_equal(_ex1))
 793                         return _ex1_2*Pi;
 794
 795                 // asin(-1/2) -> -Pi/6
 796                 if (x.is_equal(_ex_1_2))
 797                         return numeric(-1,6)*Pi;
 798
 799                 // asin(-1) -> -Pi/2
 800                 if (x.is_equal(_ex_1))
 801                         return _ex_1_2*Pi;
 802
 803                 // asin(float) -> float
 804                 if (!x.info(info_flags::crational))
 805                         return asin(ex_to<numeric>(x));
 806
 807                 // asin() is odd
 808                 if (x.info(info_flags::negative))
 809                         return -asin(-x);
 810         }
 811
 812         return asin(x).hold();
 813 }
 814
 815 static ex asin_deriv(const ex & x, unsigned deriv_param)
 816 {
 817         GINAC_ASSERT(deriv_param==0);
 818
 819         // d/dx asin(x) -> 1/sqrt(1-x^2)
 820         return power(1-power(x,_ex2),_ex_1_2);
 821 }
 822
 823 static ex asin_conjugate(const ex & x)
 824 {
 825         // conjugate(asin(x))==asin(conjugate(x)) unless on the branch cuts which
 826         // run along the real axis outside the interval [-1, +1].
 827         if (is_exactly_a<numeric>(x) &&
 828             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
 829                 return asin(x.conjugate());
 830         }
 831         return conjugate_function(asin(x)).hold();
 832 }
 833
 834 static bool asin_info(const ex & x, unsigned inf)
 835 {
 836         switch (inf) {
 837         case info_flags::numeric:
 838         case info_flags::expanded:
 839                 return x.info(inf);
 840         default:
 841                 return false;
 842         }
 843 }
 844
 845 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 846                         evalf_func(asin_evalf).
 847                         info_func(asin_info).
 848                         derivative_func(asin_deriv).
 849                         conjugate_func(asin_conjugate).
 850                         latex_name("\\arcsin"));
 851
 852 //////////
 853 // inverse cosine (arc cosine)
 854 //////////
 855
 856 static ex acos_evalf(const ex & x)
 857 {
 858         if (is_exactly_a<numeric>(x))
 859                 return acos(ex_to<numeric>(x));
 860
 861         return acos(x).hold();
 862 }
 863
 864 static ex acos_eval(const ex & x)
 865 {
 866         if (x.info(info_flags::numeric)) {
 867
 868                 // acos(1) -> 0
 869                 if (x.is_equal(_ex1))
 870                         return _ex0;
 871
 872                 // acos(1/2) -> Pi/3
 873                 if (x.is_equal(_ex1_2))
 874                         return _ex1_3*Pi;
 875
 876                 // acos(0) -> Pi/2
 877                 if (x.is_zero())
 878                         return _ex1_2*Pi;
 879
 880                 // acos(-1/2) -> 2/3*Pi
 881                 if (x.is_equal(_ex_1_2))
 882                         return numeric(2,3)*Pi;
 883
 884                 // acos(-1) -> Pi
 885                 if (x.is_equal(_ex_1))
 886                         return Pi;
 887
 888                 // acos(float) -> float
 889                 if (!x.info(info_flags::crational))
 890                         return acos(ex_to<numeric>(x));
 891
 892                 // acos(-x) -> Pi-acos(x)
 893                 if (x.info(info_flags::negative))
 894                         return Pi-acos(-x);
 895         }
 896
 897         return acos(x).hold();
 898 }
 899
 900 static ex acos_deriv(const ex & x, unsigned deriv_param)
 901 {
 902         GINAC_ASSERT(deriv_param==0);
 903
 904         // d/dx acos(x) -> -1/sqrt(1-x^2)
 905         return -power(1-power(x,_ex2),_ex_1_2);
 906 }
 907
 908 static ex acos_conjugate(const ex & x)
 909 {
 910         // conjugate(acos(x))==acos(conjugate(x)) unless on the branch cuts which
 911         // run along the real axis outside the interval [-1, +1].
 912         if (is_exactly_a<numeric>(x) &&
 913             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
 914                 return acos(x.conjugate());
 915         }
 916         return conjugate_function(acos(x)).hold();
 917 }
 918
 919 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 920                         evalf_func(acos_evalf).
 921                         info_func(asin_info). // Flags of acos are shared with asin functions
 922                         derivative_func(acos_deriv).
 923                         conjugate_func(acos_conjugate).
 924                         latex_name("\\arccos"));
 925
 926 //////////
 927 // inverse tangent (arc tangent)
 928 //////////
 929
 930 static ex atan_evalf(const ex & x)
 931 {
 932         if (is_exactly_a<numeric>(x))
 933                 return atan(ex_to<numeric>(x));
 934
 935         return atan(x).hold();
 936 }
 937
 938 static ex atan_eval(const ex & x)
 939 {
 940         if (x.info(info_flags::numeric)) {
 941
 942                 // atan(0) -> 0
 943                 if (x.is_zero())
 944                         return _ex0;
 945
 946                 // atan(1) -> Pi/4
 947                 if (x.is_equal(_ex1))
 948                         return _ex1_4*Pi;
 949
 950                 // atan(-1) -> -Pi/4
 951                 if (x.is_equal(_ex_1))
 952                         return _ex_1_4*Pi;
 953
 954                 if (x.is_equal(I) || x.is_equal(-I))
 955                         throw (pole_error("atan_eval(): logarithmic pole",0));
 956
 957                 // atan(float) -> float
 958                 if (!x.info(info_flags::crational))
 959                         return atan(ex_to<numeric>(x));
 960
 961                 // atan() is odd
 962                 if (x.info(info_flags::negative))
 963                         return -atan(-x);
 964         }
 965
 966         return atan(x).hold();
 967 }
 968
 969 static ex atan_deriv(const ex & x, unsigned deriv_param)
 970 {
 971         GINAC_ASSERT(deriv_param==0);
 972
 973         // d/dx atan(x) -> 1/(1+x^2)
 974         return power(_ex1+power(x,_ex2), _ex_1);
 975 }
 976
 977 static ex atan_series(const ex &arg,
 978                       const relational &rel,
 979                       int order,
 980                       unsigned options)
 981 {
 982         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 983         // method:
 984         // Taylor series where there is no pole or cut falls back to atan_deriv.
 985         // There are two branch cuts, one runnig from I up the imaginary axis and
 986         // one running from -I down the imaginary axis.  The points I and -I are
 987         // poles.
 988         // On the branch cuts and the poles series expand
 989         //     (log(1+I*x)-log(1-I*x))/(2*I)
 990         // instead.
 991         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 992         if (!(I*arg_pt).info(info_flags::real))
 993                 throw do_taylor();     // Re(x) != 0
 994         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
 995                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
 996         // care for the poles, using the defining formula for atan()...
 997         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
 998                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
 999         if (!(options & series_options::suppress_branchcut)) {
1000                 // method:
1001                 // This is the branch cut: assemble the primitive series manually and
1002                 // then add the corresponding complex step function.
1003                 const symbol &s = ex_to<symbol>(rel.lhs());
1004                 const ex &point = rel.rhs();
1005                 const symbol foo;
1006                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1007                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
1008                 if ((I*arg_pt)<_ex0)
1009                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
1010                 else
1011                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
1012                 epvector seq;
1013                 if (order > 0) {
1014                         seq.reserve(2);
1015                         seq.push_back(expair(Order0correction, _ex0));
1016                 }
1017                 seq.push_back(expair(Order(_ex1), order));
1018                 return series(replarg - pseries(rel, std::move(seq)), rel, order);
1019         }
1020         throw do_taylor();
1021 }
1022
1023 static ex atan_conjugate(const ex & x)
1024 {
1025         // conjugate(atan(x))==atan(conjugate(x)) unless on the branch cuts which
1026         // run along the imaginary axis outside the interval [-I, +I].
1027         if (x.info(info_flags::real))
1028                 return atan(x);
1029         if (is_exactly_a<numeric>(x)) {
1030                 const numeric x_re = ex_to<numeric>(x.real_part());
1031                 const numeric x_im = ex_to<numeric>(x.imag_part());
1032                 if (!x_re.is_zero() ||
1033                     (x_im > *_num_1_p && x_im < *_num1_p))
1034                         return atan(x.conjugate());
1035         }
1036         return conjugate_function(atan(x)).hold();
1037 }
1038
1039 static bool atan_info(const ex & x, unsigned inf)
1040 {
1041         switch (inf) {
1042         case info_flags::numeric:
1043         case info_flags::expanded:
1044         case info_flags::real:
1045                 return x.info(inf);
1046         case info_flags::positive:
1047         case info_flags::negative:
1048         case info_flags::nonnegative:
1049                 return x.info(info_flags::real) && x.info(inf);
1050         default:
1051                 return false;
1052         }
1053 }
1054
1055 REGISTER_FUNCTION(atan, eval_func(atan_eval).
1056                         evalf_func(atan_evalf).
1057                         info_func(atan_info).
1058                         derivative_func(atan_deriv).
1059                         series_func(atan_series).
1060                         conjugate_func(atan_conjugate).
1061                         latex_name("\\arctan"));
1062
1063 //////////
1064 // inverse tangent (atan2(y,x))
1065 //////////
1066
1067 static ex atan2_evalf(const ex &y, const ex &x)
1068 {
1069         if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
1070                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
1071
1072         return atan2(y, x).hold();
1073 }
1074
1075 static ex atan2_eval(const ex & y, const ex & x)
1076 {
1077         if (y.is_zero()) {
1078
1079                 // atan2(0, 0) -> 0
1080                 if (x.is_zero())
1081                         return _ex0;
1082
1083                 // atan2(0, x), x real and positive -> 0
1084                 if (x.info(info_flags::positive))
1085                         return _ex0;
1086
1087                 // atan2(0, x), x real and negative -> Pi
1088                 if (x.info(info_flags::negative))
1089                         return Pi;
1090         }
1091
1092         if (x.is_zero()) {
1093
1094                 // atan2(y, 0), y real and positive -> Pi/2
1095                 if (y.info(info_flags::positive))
1096                         return _ex1_2*Pi;
1097
1098                 // atan2(y, 0), y real and negative -> -Pi/2
1099                 if (y.info(info_flags::negative))
1100                         return _ex_1_2*Pi;
1101         }
1102
1103         if (y.is_equal(x)) {
1104
1105                 // atan2(y, y), y real and positive -> Pi/4
1106                 if (y.info(info_flags::positive))
1107                         return _ex1_4*Pi;
1108
1109                 // atan2(y, y), y real and negative -> -3/4*Pi
1110                 if (y.info(info_flags::negative))
1111                         return numeric(-3, 4)*Pi;
1112         }
1113
1114         if (y.is_equal(-x)) {
1115
1116                 // atan2(y, -y), y real and positive -> 3*Pi/4
1117                 if (y.info(info_flags::positive))
1118                         return numeric(3, 4)*Pi;
1119
1120                 // atan2(y, -y), y real and negative -> -Pi/4
1121                 if (y.info(info_flags::negative))
1122                         return _ex_1_4*Pi;
1123         }
1124
1125         // atan2(float, float) -> float
1126         if (is_a<numeric>(y) && !y.info(info_flags::crational) &&
1127             is_a<numeric>(x) && !x.info(info_flags::crational))
1128                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
1129
1130         // atan2(real, real) -> atan(y/x) +/- Pi
1131         if (y.info(info_flags::real) && x.info(info_flags::real)) {
1132                 if (x.info(info_flags::positive))
1133                         return atan(y/x);
1134
1135                 if (x.info(info_flags::negative)) {
1136                         if (y.info(info_flags::positive))
1137                                 return atan(y/x)+Pi;
1138                         if (y.info(info_flags::negative))
1139                                 return atan(y/x)-Pi;
1140                 }
1141         }
1142
1143         return atan2(y, x).hold();
1144 }
1145
1146 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
1147 {
1148         GINAC_ASSERT(deriv_param<2);
1149
1150         if (deriv_param==0) {
1151                 // d/dy atan2(y,x)
1152                 return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
1153         }
1154         // d/dx atan2(y,x)
1155         return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
1156 }
1157
1158 static bool atan2_info(const ex & y, const ex & x, unsigned inf)
1159 {
1160         switch (inf) {
1161         case info_flags::numeric:
1162         case info_flags::expanded:
1163         case info_flags::real:
1164                 return y.info(inf) && x.info(inf);
1165         case info_flags::positive:
1166         case info_flags::negative:
1167         case info_flags::nonnegative:
1168                 return y.info(info_flags::real) && x.info(info_flags::real)
1169                         && y.info(inf);
1170         default:
1171                 return false;
1172         }
1173 }
1174
1175 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
1176                          evalf_func(atan2_evalf).
1177                          info_func(atan2_info).
1178                          evalf_func(atan2_evalf).
1179                          derivative_func(atan2_deriv));
1180
1181 //////////
1182 // hyperbolic sine (trigonometric function)
1183 //////////
1184
1185 static ex sinh_evalf(const ex & x)
1186 {
1187         if (is_exactly_a<numeric>(x))
1188                 return sinh(ex_to<numeric>(x));
1189
1190         return sinh(x).hold();
1191 }
1192
1193 static ex sinh_eval(const ex & x)
1194 {
1195         if (x.info(info_flags::numeric)) {
1196
1197                 // sinh(0) -> 0
1198                 if (x.is_zero())
1199                         return _ex0;
1200
1201                 // sinh(float) -> float
1202                 if (!x.info(info_flags::crational))
1203                         return sinh(ex_to<numeric>(x));
1204
1205                 // sinh() is odd
1206                 if (x.info(info_flags::negative))
1207                         return -sinh(-x);
1208         }
1209
1210         if ((x/Pi).info(info_flags::numeric) &&
1211                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
1212                 return I*sin(x/I);
1213
1214         if (is_exactly_a<function>(x)) {
1215                 const ex &t = x.op(0);
1216
1217                 // sinh(asinh(x)) -> x
1218                 if (is_ex_the_function(x, asinh))
1219                         return t;
1220
1221                 // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
1222                 if (is_ex_the_function(x, acosh))
1223                         return sqrt(t-_ex1)*sqrt(t+_ex1);
1224
1225                 // sinh(atanh(x)) -> x/sqrt(1-x^2)
1226                 if (is_ex_the_function(x, atanh))
1227                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
1228         }
1229
1230         return sinh(x).hold();
1231 }
1232
1233 static ex sinh_deriv(const ex & x, unsigned deriv_param)
1234 {
1235         GINAC_ASSERT(deriv_param==0);
1236
1237         // d/dx sinh(x) -> cosh(x)
1238         return cosh(x);
1239 }
1240
1241 static ex sinh_real_part(const ex & x)
1242 {
1243         return sinh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
1244 }
1245
1246 static ex sinh_imag_part(const ex & x)
1247 {
1248         return cosh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
1249 }
1250
1251 static ex sinh_conjugate(const ex & x)
1252 {
1253         // conjugate(sinh(x))==sinh(conjugate(x))
1254         return sinh(x.conjugate());
1255 }
1256
1257 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
1258                         evalf_func(sinh_evalf).
1259                         info_func(atan_info). // Flags of sinh are shared with atan functions
1260                         derivative_func(sinh_deriv).
1261                         real_part_func(sinh_real_part).
1262                         imag_part_func(sinh_imag_part).
1263                         conjugate_func(sinh_conjugate).
1264                         latex_name("\\sinh"));
1265
1266 //////////
1267 // hyperbolic cosine (trigonometric function)
1268 //////////
1269
1270 static ex cosh_evalf(const ex & x)
1271 {
1272         if (is_exactly_a<numeric>(x))
1273                 return cosh(ex_to<numeric>(x));
1274
1275         return cosh(x).hold();
1276 }
1277
1278 static ex cosh_eval(const ex & x)
1279 {
1280         if (x.info(info_flags::numeric)) {
1281
1282                 // cosh(0) -> 1
1283                 if (x.is_zero())
1284                         return _ex1;
1285
1286                 // cosh(float) -> float
1287                 if (!x.info(info_flags::crational))
1288                         return cosh(ex_to<numeric>(x));
1289
1290                 // cosh() is even
1291                 if (x.info(info_flags::negative))
1292                         return cosh(-x);
1293         }
1294
1295         if ((x/Pi).info(info_flags::numeric) &&
1296                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
1297                 return cos(x/I);
1298
1299         if (is_exactly_a<function>(x)) {
1300                 const ex &t = x.op(0);
1301
1302                 // cosh(acosh(x)) -> x
1303                 if (is_ex_the_function(x, acosh))
1304                         return t;
1305
1306                 // cosh(asinh(x)) -> sqrt(1+x^2)
1307                 if (is_ex_the_function(x, asinh))
1308                         return sqrt(_ex1+power(t,_ex2));
1309
1310                 // cosh(atanh(x)) -> 1/sqrt(1-x^2)
1311                 if (is_ex_the_function(x, atanh))
1312                         return power(_ex1-power(t,_ex2),_ex_1_2);
1313         }
1314
1315         return cosh(x).hold();
1316 }
1317
1318 static ex cosh_deriv(const ex & x, unsigned deriv_param)
1319 {
1320         GINAC_ASSERT(deriv_param==0);
1321
1322         // d/dx cosh(x) -> sinh(x)
1323         return sinh(x);
1324 }
1325
1326 static ex cosh_real_part(const ex & x)
1327 {
1328         return cosh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
1329 }
1330
1331 static ex cosh_imag_part(const ex & x)
1332 {
1333         return sinh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
1334 }
1335
1336 static ex cosh_conjugate(const ex & x)
1337 {
1338         // conjugate(cosh(x))==cosh(conjugate(x))
1339         return cosh(x.conjugate());
1340 }
1341
1342 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
1343                         evalf_func(cosh_evalf).
1344                         info_func(exp_info). // Flags of cosh are shared with exp functions
1345                         derivative_func(cosh_deriv).
1346                         real_part_func(cosh_real_part).
1347                         imag_part_func(cosh_imag_part).
1348                         conjugate_func(cosh_conjugate).
1349                         latex_name("\\cosh"));
1350
1351 //////////
1352 // hyperbolic tangent (trigonometric function)
1353 //////////
1354
1355 static ex tanh_evalf(const ex & x)
1356 {
1357         if (is_exactly_a<numeric>(x))
1358                 return tanh(ex_to<numeric>(x));
1359
1360         return tanh(x).hold();
1361 }
1362
1363 static ex tanh_eval(const ex & x)
1364 {
1365         if (x.info(info_flags::numeric)) {
1366
1367                 // tanh(0) -> 0
1368                 if (x.is_zero())
1369                         return _ex0;
1370
1371                 // tanh(float) -> float
1372                 if (!x.info(info_flags::crational))
1373                         return tanh(ex_to<numeric>(x));
1374
1375                 // tanh() is odd
1376                 if (x.info(info_flags::negative))
1377                         return -tanh(-x);
1378         }
1379
1380         if ((x/Pi).info(info_flags::numeric) &&
1381                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
1382                 return I*tan(x/I);
1383
1384         if (is_exactly_a<function>(x)) {
1385                 const ex &t = x.op(0);
1386
1387                 // tanh(atanh(x)) -> x
1388                 if (is_ex_the_function(x, atanh))
1389                         return t;
1390
1391                 // tanh(asinh(x)) -> x/sqrt(1+x^2)
1392                 if (is_ex_the_function(x, asinh))
1393                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
1394
1395                 // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
1396                 if (is_ex_the_function(x, acosh))
1397                         return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
1398         }
1399
1400         return tanh(x).hold();
1401 }
1402
1403 static ex tanh_deriv(const ex & x, unsigned deriv_param)
1404 {
1405         GINAC_ASSERT(deriv_param==0);
1406
1407         // d/dx tanh(x) -> 1-tanh(x)^2
1408         return _ex1-power(tanh(x),_ex2);
1409 }
1410
1411 static ex tanh_series(const ex &x,
1412                       const relational &rel,
1413                       int order,
1414                       unsigned options)
1415 {
1416         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1417         // method:
1418         // Taylor series where there is no pole falls back to tanh_deriv.
1419         // On a pole simply expand sinh(x)/cosh(x).
1420         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1421         if (!(2*I*x_pt/Pi).info(info_flags::odd))
1422                 throw do_taylor();  // caught by function::series()
1423         // if we got here we have to care for a simple pole
1424         return (sinh(x)/cosh(x)).series(rel, order, options);
1425 }
1426
1427 static ex tanh_real_part(const ex & x)
1428 {
1429         ex a = GiNaC::real_part(x);
1430         ex b = GiNaC::imag_part(x);
1431         return tanh(a)/(1+power(tanh(a),2)*power(tan(b),2));
1432 }
1433
1434 static ex tanh_imag_part(const ex & x)
1435 {
1436         ex a = GiNaC::real_part(x);
1437         ex b = GiNaC::imag_part(x);
1438         return tan(b)/(1+power(tanh(a),2)*power(tan(b),2));
1439 }
1440
1441 static ex tanh_conjugate(const ex & x)
1442 {
1443         // conjugate(tanh(x))==tanh(conjugate(x))
1444         return tanh(x.conjugate());
1445 }
1446
1447 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
1448                         evalf_func(tanh_evalf).
1449                         info_func(atan_info). // Flags of tanh are shared with atan functions
1450                         derivative_func(tanh_deriv).
1451                         series_func(tanh_series).
1452                         real_part_func(tanh_real_part).
1453                         imag_part_func(tanh_imag_part).
1454                         conjugate_func(tanh_conjugate).
1455                         latex_name("\\tanh"));
1456
1457 //////////
1458 // inverse hyperbolic sine (trigonometric function)
1459 //////////
1460
1461 static ex asinh_evalf(const ex & x)
1462 {
1463         if (is_exactly_a<numeric>(x))
1464                 return asinh(ex_to<numeric>(x));
1465
1466         return asinh(x).hold();
1467 }
1468
1469 static ex asinh_eval(const ex & x)
1470 {
1471         if (x.info(info_flags::numeric)) {
1472
1473                 // asinh(0) -> 0
1474                 if (x.is_zero())
1475                         return _ex0;
1476
1477                 // asinh(float) -> float
1478                 if (!x.info(info_flags::crational))
1479                         return asinh(ex_to<numeric>(x));
1480
1481                 // asinh() is odd
1482                 if (x.info(info_flags::negative))
1483                         return -asinh(-x);
1484         }
1485
1486         return asinh(x).hold();
1487 }
1488
1489 static ex asinh_deriv(const ex & x, unsigned deriv_param)
1490 {
1491         GINAC_ASSERT(deriv_param==0);
1492
1493         // d/dx asinh(x) -> 1/sqrt(1+x^2)
1494         return power(_ex1+power(x,_ex2),_ex_1_2);
1495 }
1496
1497 static ex asinh_conjugate(const ex & x)
1498 {
1499         // conjugate(asinh(x))==asinh(conjugate(x)) unless on the branch cuts which
1500         // run along the imaginary axis outside the interval [-I, +I].
1501         if (x.info(info_flags::real))
1502                 return asinh(x);
1503         if (is_exactly_a<numeric>(x)) {
1504                 const numeric x_re = ex_to<numeric>(x.real_part());
1505                 const numeric x_im = ex_to<numeric>(x.imag_part());
1506                 if (!x_re.is_zero() ||
1507                     (x_im > *_num_1_p && x_im < *_num1_p))
1508                         return asinh(x.conjugate());
1509         }
1510         return conjugate_function(asinh(x)).hold();
1511 }
1512
1513 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
1514                          evalf_func(asinh_evalf).
1515                          info_func(atan_info). // Flags of asinh are shared with atan functions
1516                          derivative_func(asinh_deriv).
1517                          conjugate_func(asinh_conjugate));
1518
1519 //////////
1520 // inverse hyperbolic cosine (trigonometric function)
1521 //////////
1522
1523 static ex acosh_evalf(const ex & x)
1524 {
1525         if (is_exactly_a<numeric>(x))
1526                 return acosh(ex_to<numeric>(x));
1527
1528         return acosh(x).hold();
1529 }
1530
1531 static ex acosh_eval(const ex & x)
1532 {
1533         if (x.info(info_flags::numeric)) {
1534
1535                 // acosh(0) -> Pi*I/2
1536                 if (x.is_zero())
1537                         return Pi*I*numeric(1,2);
1538
1539                 // acosh(1) -> 0
1540                 if (x.is_equal(_ex1))
1541                         return _ex0;
1542
1543                 // acosh(-1) -> Pi*I
1544                 if (x.is_equal(_ex_1))
1545                         return Pi*I;
1546
1547                 // acosh(float) -> float
1548                 if (!x.info(info_flags::crational))
1549                         return acosh(ex_to<numeric>(x));
1550
1551                 // acosh(-x) -> Pi*I-acosh(x)
1552                 if (x.info(info_flags::negative))
1553                         return Pi*I-acosh(-x);
1554         }
1555
1556         return acosh(x).hold();
1557 }
1558
1559 static ex acosh_deriv(const ex & x, unsigned deriv_param)
1560 {
1561         GINAC_ASSERT(deriv_param==0);
1562
1563         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
1564         return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
1565 }
1566
1567 static ex acosh_conjugate(const ex & x)
1568 {
1569         // conjugate(acosh(x))==acosh(conjugate(x)) unless on the branch cut
1570         // which runs along the real axis from +1 to -inf.
1571         if (is_exactly_a<numeric>(x) &&
1572             (!x.imag_part().is_zero() || x > *_num1_p)) {
1573                 return acosh(x.conjugate());
1574         }
1575         return conjugate_function(acosh(x)).hold();
1576 }
1577
1578 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
1579                          evalf_func(acosh_evalf).
1580                          info_func(asin_info). // Flags of acosh are shared with asin functions
1581                          derivative_func(acosh_deriv).
1582                          conjugate_func(acosh_conjugate));
1583
1584 //////////
1585 // inverse hyperbolic tangent (trigonometric function)
1586 //////////
1587
1588 static ex atanh_evalf(const ex & x)
1589 {
1590         if (is_exactly_a<numeric>(x))
1591                 return atanh(ex_to<numeric>(x));
1592
1593         return atanh(x).hold();
1594 }
1595
1596 static ex atanh_eval(const ex & x)
1597 {
1598         if (x.info(info_flags::numeric)) {
1599
1600                 // atanh(0) -> 0
1601                 if (x.is_zero())
1602                         return _ex0;
1603
1604                 // atanh({+|-}1) -> throw
1605                 if (x.is_equal(_ex1) || x.is_equal(_ex_1))
1606                         throw (pole_error("atanh_eval(): logarithmic pole",0));
1607
1608                 // atanh(float) -> float
1609                 if (!x.info(info_flags::crational))
1610                         return atanh(ex_to<numeric>(x));
1611
1612                 // atanh() is odd
1613                 if (x.info(info_flags::negative))
1614                         return -atanh(-x);
1615         }
1616
1617         return atanh(x).hold();
1618 }
1619
1620 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1621 {
1622         GINAC_ASSERT(deriv_param==0);
1623
1624         // d/dx atanh(x) -> 1/(1-x^2)
1625         return power(_ex1-power(x,_ex2),_ex_1);
1626 }
1627
1628 static ex atanh_series(const ex &arg,
1629                        const relational &rel,
1630                        int order,
1631                        unsigned options)
1632 {
1633         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1634         // method:
1635         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1636         // There are two branch cuts, one runnig from 1 up the real axis and one
1637         // one running from -1 down the real axis.  The points 1 and -1 are poles
1638         // On the branch cuts and the poles series expand
1639         //     (log(1+x)-log(1-x))/2
1640         // instead.
1641         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
1642         if (!(arg_pt).info(info_flags::real))
1643                 throw do_taylor();     // Im(x) != 0
1644         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1645                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1646         // care for the poles, using the defining formula for atanh()...
1647         if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1648                 return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1649         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1650         if (!(options & series_options::suppress_branchcut)) {
1651                 // method:
1652                 // This is the branch cut: assemble the primitive series manually and
1653                 // then add the corresponding complex step function.
1654                 const symbol &s = ex_to<symbol>(rel.lhs());
1655                 const ex &point = rel.rhs();
1656                 const symbol foo;
1657                 const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1658                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1659                 if (arg_pt<_ex0)
1660                         Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1661                 else
1662                         Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1663                 epvector seq;
1664                 if (order > 0) {
1665                         seq.reserve(2);
1666                         seq.push_back(expair(Order0correction, _ex0));
1667                 }
1668                 seq.push_back(expair(Order(_ex1), order));
1669                 return series(replarg - pseries(rel, std::move(seq)), rel, order);
1670         }
1671         throw do_taylor();
1672 }
1673
1674 static ex atanh_conjugate(const ex & x)
1675 {
1676         // conjugate(atanh(x))==atanh(conjugate(x)) unless on the branch cuts which
1677         // run along the real axis outside the interval [-1, +1].
1678         if (is_exactly_a<numeric>(x) &&
1679             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
1680                 return atanh(x.conjugate());
1681         }
1682         return conjugate_function(atanh(x)).hold();
1683 }
1684
1685 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1686                          evalf_func(atanh_evalf).
1687                          info_func(asin_info). // Flags of atanh are shared with asin functions
1688                          derivative_func(atanh_deriv).
1689                          series_func(atanh_series).
1690                          conjugate_func(atanh_conjugate));
1691
1692
1693 } // namespace GiNaC