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) 1999-2024 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::expanded:
 153         case info_flags::real:
 154                 return x.info(inf);
 155         case info_flags::positive:
 156         case info_flags::nonnegative:
 157                 return x.info(info_flags::real);
 158         default:
 159                 return false;
 160         }
 161 }
 162
 163 REGISTER_FUNCTION(exp, eval_func(exp_eval).
 164                        evalf_func(exp_evalf).
 165                        info_func(exp_info).
 166                        expand_func(exp_expand).
 167                        derivative_func(exp_deriv).
 168                        real_part_func(exp_real_part).
 169                        imag_part_func(exp_imag_part).
 170                        conjugate_func(exp_conjugate).
 171                        power_func(exp_power).
 172                        latex_name("\\exp"));
 173
 174 //////////
 175 // natural logarithm
 176 //////////
 177
 178 static ex log_evalf(const ex & x)
 179 {
 180         if (is_exactly_a<numeric>(x))
 181                 return log(ex_to<numeric>(x));
 182
 183         return log(x).hold();
 184 }
 185
 186 static ex log_eval(const ex & x)
 187 {
 188         if (x.info(info_flags::numeric)) {
 189                 if (x.is_zero())         // log(0) -> infinity
 190                         throw(pole_error("log_eval(): log(0)",0));
 191                 if (x.info(info_flags::rational) && x.info(info_flags::negative))
 192                         return (log(-x)+I*Pi);
 193                 if (x.is_equal(_ex1))  // log(1) -> 0
 194                         return _ex0;
 195                 if (x.is_equal(I))       // log(I) -> Pi*I/2
 196                         return (Pi*I*_ex1_2);
 197                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
 198                         return (Pi*I*_ex_1_2);
 199
 200                 // log(float) -> float
 201                 if (!x.info(info_flags::crational))
 202                         return log(ex_to<numeric>(x));
 203         }
 204
 205         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
 206         if (is_ex_the_function(x, exp)) {
 207                 const ex &t = x.op(0);
 208                 if (t.info(info_flags::real))
 209                         return t;
 210         }
 211
 212         return log(x).hold();
 213 }
 214
 215 static ex log_deriv(const ex & x, unsigned deriv_param)
 216 {
 217         GINAC_ASSERT(deriv_param==0);
 218
 219         // d/dx log(x) -> 1/x
 220         return power(x, _ex_1);
 221 }
 222
 223 static ex log_series(const ex &arg,
 224                      const relational &rel,
 225                      int order,
 226                      unsigned options)
 227 {
 228         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 229         ex arg_pt;
 230         bool must_expand_arg = false;
 231         // maybe substitution of rel into arg fails because of a pole
 232         try {
 233                 arg_pt = arg.subs(rel, subs_options::no_pattern);
 234         } catch (pole_error &) {
 235                 must_expand_arg = true;
 236         }
 237         // or we are at the branch point anyways
 238         if (arg_pt.is_zero())
 239                 must_expand_arg = true;
 240
 241         if (arg.diff(ex_to<symbol>(rel.lhs())).is_zero()) {
 242                 throw do_taylor();
 243         }
 244
 245         if (must_expand_arg) {
 246                 // method:
 247                 // This is the branch point: Series expand the argument first, then
 248                 // trivially factorize it to isolate that part which has constant
 249                 // leading coefficient in this fashion:
 250                 //   x^n + x^(n+1) +...+ Order(x^(n+m))  ->  x^n * (1 + x +...+ Order(x^m)).
 251                 // Return a plain n*log(x) for the x^n part and series expand the
 252                 // other part.  Add them together and reexpand again in order to have
 253                 // one unnested pseries object.  All this also works for negative n.
 254                 pseries argser;          // series expansion of log's argument
 255                 unsigned extra_ord = 0;  // extra expansion order
 256                 do {
 257                         // oops, the argument expanded to a pure Order(x^something)...
 258                         argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
 259                         ++extra_ord;
 260                 } while (!argser.is_terminating() && argser.nops()==1);
 261
 262                 const symbol &s = ex_to<symbol>(rel.lhs());
 263                 const ex &point = rel.rhs();
 264                 const int n = argser.ldegree(s);
 265                 epvector seq;
 266                 // construct what we carelessly called the n*log(x) term above
 267                 const ex coeff = argser.coeff(s, n);
 268                 // expand the log, but only if coeff is real and > 0, since otherwise
 269                 // it would make the branch cut run into the wrong direction
 270                 if (coeff.info(info_flags::positive))
 271                         seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
 272                 else
 273                         seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));
 274
 275                 if (!argser.is_terminating() || argser.nops()!=1) {
 276                         // in this case n more (or less) terms are needed
 277                         // (sadly, to generate them, we have to start from the beginning)
 278                         if (n == 0 && coeff == 1) {
 279                                 ex rest = pseries(rel, epvector{expair(-1, _ex0), expair(Order(_ex1), order)}).add_series(argser);
 280                                 ex acc = dynallocate<pseries>(rel, epvector());
 281                                 for (int i = order-1; i>0; --i) {
 282                                         epvector cterm { expair(i%2 ? _ex1/i : _ex_1/i, _ex0) };
 283                                         acc = pseries(rel, std::move(cterm)).add_series(ex_to<pseries>(acc));
 284                                         acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
 285                                 }
 286                                 return acc;
 287                         }
 288                         const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
 289                         return pseries(rel, std::move(seq)).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
 290                 } else  // it was a monomial
 291                         return pseries(rel, std::move(seq));
 292         }
 293         if (!(options & series_options::suppress_branchcut) &&
 294              arg_pt.info(info_flags::negative)) {
 295                 // method:
 296                 // This is the branch cut: assemble the primitive series manually and
 297                 // then add the corresponding complex step function.
 298                 const symbol &s = ex_to<symbol>(rel.lhs());
 299                 const ex &point = rel.rhs();
 300                 const symbol foo;
 301                 const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 302                 epvector seq;
 303                 if (order > 0) {
 304                         seq.reserve(2);
 305                         seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
 306                 }
 307                 seq.push_back(expair(Order(_ex1), order));
 308                 return series(replarg - I*Pi + pseries(rel, std::move(seq)), rel, order);
 309         }
 310         throw do_taylor();  // caught by function::series()
 311 }
 312
 313 static ex log_real_part(const ex & x)
 314 {
 315         if (x.info(info_flags::nonnegative))
 316                 return log(x).hold();
 317         return log(abs(x));
 318 }
 319
 320 static ex log_imag_part(const ex & x)
 321 {
 322         if (x.info(info_flags::nonnegative))
 323                 return 0;
 324         return atan2(GiNaC::imag_part(x), GiNaC::real_part(x));
 325 }
 326
 327 static ex log_expand(const ex & arg, unsigned options)
 328 {
 329         if ((options & expand_options::expand_transcendental)
 330                 && is_exactly_a<mul>(arg) && !arg.info(info_flags::indefinite)) {
 331                 exvector sumseq;
 332                 exvector prodseq;
 333                 sumseq.reserve(arg.nops());
 334                 prodseq.reserve(arg.nops());
 335                 bool possign=true;
 336
 337                 // searching for positive/negative factors
 338                 for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
 339                         ex e;
 340                         if (options & expand_options::expand_function_args)
 341                                 e=i->expand(options);
 342                         else
 343                                 e=*i;
 344                         if (e.info(info_flags::positive))
 345                                 sumseq.push_back(log(e));
 346                         else if (e.info(info_flags::negative)) {
 347                                 sumseq.push_back(log(-e));
 348                                 possign = !possign;
 349                         } else
 350                                 prodseq.push_back(e);
 351                 }
 352
 353                 if (sumseq.size() > 0) {
 354                         ex newarg;
 355                         if (options & expand_options::expand_function_args)
 356                                 newarg=((possign?_ex1:_ex_1)*mul(prodseq)).expand(options);
 357                         else {
 358                                 newarg=(possign?_ex1:_ex_1)*mul(prodseq);
 359                                 ex_to<basic>(newarg).setflag(status_flags::purely_indefinite);
 360                         }
 361                         return add(sumseq)+log(newarg);
 362                 } else {
 363                         if (!(options & expand_options::expand_function_args))
 364                                 ex_to<basic>(arg).setflag(status_flags::purely_indefinite);
 365                 }
 366         }
 367
 368         if (options & expand_options::expand_function_args)
 369                 return log(arg.expand(options)).hold();
 370         else
 371                 return log(arg).hold();
 372 }
 373
 374 static ex log_conjugate(const ex & x)
 375 {
 376         // conjugate(log(x))==log(conjugate(x)) unless on the branch cut which
 377         // runs along the negative real axis.
 378         if (x.info(info_flags::positive)) {
 379                 return log(x);
 380         }
 381         if (is_exactly_a<numeric>(x) &&
 382             !x.imag_part().is_zero()) {
 383                 return log(x.conjugate());
 384         }
 385         return conjugate_function(log(x)).hold();
 386 }
 387
 388 static bool log_info(const ex & x, unsigned inf)
 389 {
 390         switch (inf) {
 391         case info_flags::expanded:
 392                 return x.info(inf);
 393         case info_flags::real:
 394                 return x.info(info_flags::positive);
 395         default:
 396                 return false;
 397         }
 398 }
 399
 400 REGISTER_FUNCTION(log, eval_func(log_eval).
 401                        evalf_func(log_evalf).
 402                        info_func(log_info).
 403                        expand_func(log_expand).
 404                        derivative_func(log_deriv).
 405                        series_func(log_series).
 406                        real_part_func(log_real_part).
 407                        imag_part_func(log_imag_part).
 408                        conjugate_func(log_conjugate).
 409                        latex_name("\\ln"));
 410
 411 //////////
 412 // sine (trigonometric function)
 413 //////////
 414
 415 static ex sin_evalf(const ex & x)
 416 {
 417         if (is_exactly_a<numeric>(x))
 418                 return sin(ex_to<numeric>(x));
 419
 420         return sin(x).hold();
 421 }
 422
 423 static ex sin_eval(const ex & x)
 424 {
 425         // sin(n/d*Pi) -> { all known non-nested radicals }
 426         const ex SixtyExOverPi = _ex60*x/Pi;
 427         ex sign = _ex1;
 428         if (SixtyExOverPi.info(info_flags::integer)) {
 429                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 430                 if (z>=*_num60_p) {
 431                         // wrap to interval [0, Pi)
 432                         z -= *_num60_p;
 433                         sign = _ex_1;
 434                 }
 435                 if (z>*_num30_p) {
 436                         // wrap to interval [0, Pi/2)
 437                         z = *_num60_p-z;
 438                 }
 439                 if (z.is_equal(*_num0_p))  // sin(0)       -> 0
 440                         return _ex0;
 441                 if (z.is_equal(*_num5_p))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
 442                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 443                 if (z.is_equal(*_num6_p))  // sin(Pi/10)   -> sqrt(5)/4-1/4
 444                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 445                 if (z.is_equal(*_num10_p)) // sin(Pi/6)    -> 1/2
 446                         return sign*_ex1_2;
 447                 if (z.is_equal(*_num15_p)) // sin(Pi/4)    -> sqrt(2)/2
 448                         return sign*_ex1_2*sqrt(_ex2);
 449                 if (z.is_equal(*_num18_p)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
 450                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 451                 if (z.is_equal(*_num20_p)) // sin(Pi/3)    -> sqrt(3)/2
 452                         return sign*_ex1_2*sqrt(_ex3);
 453                 if (z.is_equal(*_num25_p)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
 454                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 455                 if (z.is_equal(*_num30_p)) // sin(Pi/2)    -> 1
 456                         return sign;
 457         }
 458
 459         if (is_exactly_a<function>(x)) {
 460                 const ex &t = x.op(0);
 461
 462                 // sin(asin(x)) -> x
 463                 if (is_ex_the_function(x, asin))
 464                         return t;
 465
 466                 // sin(acos(x)) -> sqrt(1-x^2)
 467                 if (is_ex_the_function(x, acos))
 468                         return sqrt(_ex1-power(t,_ex2));
 469
 470                 // sin(atan(x)) -> x/sqrt(1+x^2)
 471                 if (is_ex_the_function(x, atan))
 472                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 473         }
 474
 475         // sin(float) -> float
 476         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 477                 return sin(ex_to<numeric>(x));
 478
 479         // sin() is odd
 480         if (x.info(info_flags::negative))
 481                 return -sin(-x);
 482
 483         return sin(x).hold();
 484 }
 485
 486 static ex sin_deriv(const ex & x, unsigned deriv_param)
 487 {
 488         GINAC_ASSERT(deriv_param==0);
 489
 490         // d/dx sin(x) -> cos(x)
 491         return cos(x);
 492 }
 493
 494 static ex sin_real_part(const ex & x)
 495 {
 496         return cosh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 497 }
 498
 499 static ex sin_imag_part(const ex & x)
 500 {
 501         return sinh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 502 }
 503
 504 static ex sin_conjugate(const ex & x)
 505 {
 506         // conjugate(sin(x))==sin(conjugate(x))
 507         return sin(x.conjugate());
 508 }
 509
 510 static bool trig_info(const ex & x, unsigned inf)
 511 {
 512         switch (inf) {
 513         case info_flags::expanded:
 514         case info_flags::real:
 515                 return x.info(inf);
 516         default:
 517                 return false;
 518         }
 519 }
 520
 521 REGISTER_FUNCTION(sin, eval_func(sin_eval).
 522                        evalf_func(sin_evalf).
 523                        info_func(trig_info).
 524                        derivative_func(sin_deriv).
 525                        real_part_func(sin_real_part).
 526                        imag_part_func(sin_imag_part).
 527                        conjugate_func(sin_conjugate).
 528                        latex_name("\\sin"));
 529
 530 //////////
 531 // cosine (trigonometric function)
 532 //////////
 533
 534 static ex cos_evalf(const ex & x)
 535 {
 536         if (is_exactly_a<numeric>(x))
 537                 return cos(ex_to<numeric>(x));
 538
 539         return cos(x).hold();
 540 }
 541
 542 static ex cos_eval(const ex & x)
 543 {
 544         // cos(n/d*Pi) -> { all known non-nested radicals }
 545         const ex SixtyExOverPi = _ex60*x/Pi;
 546         ex sign = _ex1;
 547         if (SixtyExOverPi.info(info_flags::integer)) {
 548                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 549                 if (z>=*_num60_p) {
 550                         // wrap to interval [0, Pi)
 551                         z = *_num120_p-z;
 552                 }
 553                 if (z>=*_num30_p) {
 554                         // wrap to interval [0, Pi/2)
 555                         z = *_num60_p-z;
 556                         sign = _ex_1;
 557                 }
 558                 if (z.is_equal(*_num0_p))  // cos(0)       -> 1
 559                         return sign;
 560                 if (z.is_equal(*_num5_p))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 561                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 562                 if (z.is_equal(*_num10_p)) // cos(Pi/6)    -> sqrt(3)/2
 563                         return sign*_ex1_2*sqrt(_ex3);
 564                 if (z.is_equal(*_num12_p)) // cos(Pi/5)    -> sqrt(5)/4+1/4
 565                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 566                 if (z.is_equal(*_num15_p)) // cos(Pi/4)    -> sqrt(2)/2
 567                         return sign*_ex1_2*sqrt(_ex2);
 568                 if (z.is_equal(*_num20_p)) // cos(Pi/3)    -> 1/2
 569                         return sign*_ex1_2;
 570                 if (z.is_equal(*_num24_p)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 571                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 572                 if (z.is_equal(*_num25_p)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 573                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 574                 if (z.is_equal(*_num30_p)) // cos(Pi/2)    -> 0
 575                         return _ex0;
 576         }
 577
 578         if (is_exactly_a<function>(x)) {
 579                 const ex &t = x.op(0);
 580
 581                 // cos(acos(x)) -> x
 582                 if (is_ex_the_function(x, acos))
 583                         return t;
 584
 585                 // cos(asin(x)) -> sqrt(1-x^2)
 586                 if (is_ex_the_function(x, asin))
 587                         return sqrt(_ex1-power(t,_ex2));
 588
 589                 // cos(atan(x)) -> 1/sqrt(1+x^2)
 590                 if (is_ex_the_function(x, atan))
 591                         return power(_ex1+power(t,_ex2),_ex_1_2);
 592         }
 593
 594         // cos(float) -> float
 595         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 596                 return cos(ex_to<numeric>(x));
 597
 598         // cos() is even
 599         if (x.info(info_flags::negative))
 600                 return cos(-x);
 601
 602         return cos(x).hold();
 603 }
 604
 605 static ex cos_deriv(const ex & x, unsigned deriv_param)
 606 {
 607         GINAC_ASSERT(deriv_param==0);
 608
 609         // d/dx cos(x) -> -sin(x)
 610         return -sin(x);
 611 }
 612
 613 static ex cos_real_part(const ex & x)
 614 {
 615         return cosh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 616 }
 617
 618 static ex cos_imag_part(const ex & x)
 619 {
 620         return -sinh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 621 }
 622
 623 static ex cos_conjugate(const ex & x)
 624 {
 625         // conjugate(cos(x))==cos(conjugate(x))
 626         return cos(x.conjugate());
 627 }
 628
 629 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 630                        info_func(trig_info).
 631                        evalf_func(cos_evalf).
 632                        derivative_func(cos_deriv).
 633                        real_part_func(cos_real_part).
 634                        imag_part_func(cos_imag_part).
 635                        conjugate_func(cos_conjugate).
 636                        latex_name("\\cos"));
 637
 638 //////////
 639 // tangent (trigonometric function)
 640 //////////
 641
 642 static ex tan_evalf(const ex & x)
 643 {
 644         if (is_exactly_a<numeric>(x))
 645                 return tan(ex_to<numeric>(x));
 646
 647         return tan(x).hold();
 648 }
 649
 650 static ex tan_eval(const ex & x)
 651 {
 652         // tan(n/d*Pi) -> { all known non-nested radicals }
 653         const ex SixtyExOverPi = _ex60*x/Pi;
 654         ex sign = _ex1;
 655         if (SixtyExOverPi.info(info_flags::integer)) {
 656                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num60_p);
 657                 if (z>=*_num60_p) {
 658                         // wrap to interval [0, Pi)
 659                         z -= *_num60_p;
 660                 }
 661                 if (z>=*_num30_p) {
 662                         // wrap to interval [0, Pi/2)
 663                         z = *_num60_p-z;
 664                         sign = _ex_1;
 665                 }
 666                 if (z.is_equal(*_num0_p))  // tan(0)       -> 0
 667                         return _ex0;
 668                 if (z.is_equal(*_num5_p))  // tan(Pi/12)   -> 2-sqrt(3)
 669                         return sign*(_ex2-sqrt(_ex3));
 670                 if (z.is_equal(*_num10_p)) // tan(Pi/6)    -> sqrt(3)/3
 671                         return sign*_ex1_3*sqrt(_ex3);
 672                 if (z.is_equal(*_num15_p)) // tan(Pi/4)    -> 1
 673                         return sign;
 674                 if (z.is_equal(*_num20_p)) // tan(Pi/3)    -> sqrt(3)
 675                         return sign*sqrt(_ex3);
 676                 if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3)
 677                         return sign*(sqrt(_ex3)+_ex2);
 678                 if (z.is_equal(*_num30_p)) // tan(Pi/2)    -> infinity
 679                         throw (pole_error("tan_eval(): simple pole",1));
 680         }
 681
 682         if (is_exactly_a<function>(x)) {
 683                 const ex &t = x.op(0);
 684
 685                 // tan(atan(x)) -> x
 686                 if (is_ex_the_function(x, atan))
 687                         return t;
 688
 689                 // tan(asin(x)) -> x/sqrt(1+x^2)
 690                 if (is_ex_the_function(x, asin))
 691                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 692
 693                 // tan(acos(x)) -> sqrt(1-x^2)/x
 694                 if (is_ex_the_function(x, acos))
 695                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
 696         }
 697
 698         // tan(float) -> float
 699         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 700                 return tan(ex_to<numeric>(x));
 701         }
 702
 703         // tan() is odd
 704         if (x.info(info_flags::negative))
 705                 return -tan(-x);
 706
 707         return tan(x).hold();
 708 }
 709
 710 static ex tan_deriv(const ex & x, unsigned deriv_param)
 711 {
 712         GINAC_ASSERT(deriv_param==0);
 713
 714         // d/dx tan(x) -> 1+tan(x)^2;
 715         return (_ex1+power(tan(x),_ex2));
 716 }
 717
 718 static ex tan_real_part(const ex & x)
 719 {
 720         ex a = GiNaC::real_part(x);
 721         ex b = GiNaC::imag_part(x);
 722         return tan(a)/(1+power(tan(a),2)*power(tan(b),2));
 723 }
 724
 725 static ex tan_imag_part(const ex & x)
 726 {
 727         ex a = GiNaC::real_part(x);
 728         ex b = GiNaC::imag_part(x);
 729         return tanh(b)/(1+power(tan(a),2)*power(tan(b),2));
 730 }
 731
 732 static ex tan_series(const ex &x,
 733                      const relational &rel,
 734                      int order,
 735                      unsigned options)
 736 {
 737         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 738         // method:
 739         // Taylor series where there is no pole falls back to tan_deriv.
 740         // On a pole simply expand sin(x)/cos(x).
 741         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 742         if (!(2*x_pt/Pi).info(info_flags::odd))
 743                 throw do_taylor();  // caught by function::series()
 744         // if we got here we have to care for a simple pole
 745         return (sin(x)/cos(x)).series(rel, order, options);
 746 }
 747
 748 static ex tan_conjugate(const ex & x)
 749 {
 750         // conjugate(tan(x))==tan(conjugate(x))
 751         return tan(x.conjugate());
 752 }
 753
 754 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 755                        evalf_func(tan_evalf).
 756                        info_func(trig_info).
 757                        derivative_func(tan_deriv).
 758                        series_func(tan_series).
 759                        real_part_func(tan_real_part).
 760                        imag_part_func(tan_imag_part).
 761                        conjugate_func(tan_conjugate).
 762                        latex_name("\\tan"));
 763
 764 //////////
 765 // inverse sine (arc sine)
 766 //////////
 767
 768 static ex asin_evalf(const ex & x)
 769 {
 770         if (is_exactly_a<numeric>(x))
 771                 return asin(ex_to<numeric>(x));
 772
 773         return asin(x).hold();
 774 }
 775
 776 static ex asin_eval(const ex & x)
 777 {
 778         if (x.info(info_flags::numeric)) {
 779
 780                 // asin(0) -> 0
 781                 if (x.is_zero())
 782                         return x;
 783
 784                 // asin(1/2) -> Pi/6
 785                 if (x.is_equal(_ex1_2))
 786                         return numeric(1,6)*Pi;
 787
 788                 // asin(1) -> Pi/2
 789                 if (x.is_equal(_ex1))
 790                         return _ex1_2*Pi;
 791
 792                 // asin(-1/2) -> -Pi/6
 793                 if (x.is_equal(_ex_1_2))
 794                         return numeric(-1,6)*Pi;
 795
 796                 // asin(-1) -> -Pi/2
 797                 if (x.is_equal(_ex_1))
 798                         return _ex_1_2*Pi;
 799
 800                 // asin(float) -> float
 801                 if (!x.info(info_flags::crational))
 802                         return asin(ex_to<numeric>(x));
 803
 804                 // asin() is odd
 805                 if (x.info(info_flags::negative))
 806                         return -asin(-x);
 807         }
 808
 809         return asin(x).hold();
 810 }
 811
 812 static ex asin_deriv(const ex & x, unsigned deriv_param)
 813 {
 814         GINAC_ASSERT(deriv_param==0);
 815
 816         // d/dx asin(x) -> 1/sqrt(1-x^2)
 817         return power(1-power(x,_ex2),_ex_1_2);
 818 }
 819
 820 static ex asin_conjugate(const ex & x)
 821 {
 822         // conjugate(asin(x))==asin(conjugate(x)) unless on the branch cuts which
 823         // run along the real axis outside the interval [-1, +1].
 824         if (is_exactly_a<numeric>(x) &&
 825             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
 826                 return asin(x.conjugate());
 827         }
 828         return conjugate_function(asin(x)).hold();
 829 }
 830
 831 static bool asin_info(const ex & x, unsigned inf)
 832 {
 833         switch (inf) {
 834         case info_flags::expanded:
 835                 return x.info(inf);
 836         default:
 837                 return false;
 838         }
 839 }
 840
 841 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 842                         evalf_func(asin_evalf).
 843                         info_func(asin_info).
 844                         derivative_func(asin_deriv).
 845                         conjugate_func(asin_conjugate).
 846                         latex_name("\\arcsin"));
 847
 848 //////////
 849 // inverse cosine (arc cosine)
 850 //////////
 851
 852 static ex acos_evalf(const ex & x)
 853 {
 854         if (is_exactly_a<numeric>(x))
 855                 return acos(ex_to<numeric>(x));
 856
 857         return acos(x).hold();
 858 }
 859
 860 static ex acos_eval(const ex & x)
 861 {
 862         if (x.info(info_flags::numeric)) {
 863
 864                 // acos(1) -> 0
 865                 if (x.is_equal(_ex1))
 866                         return _ex0;
 867
 868                 // acos(1/2) -> Pi/3
 869                 if (x.is_equal(_ex1_2))
 870                         return _ex1_3*Pi;
 871
 872                 // acos(0) -> Pi/2
 873                 if (x.is_zero())
 874                         return _ex1_2*Pi;
 875
 876                 // acos(-1/2) -> 2/3*Pi
 877                 if (x.is_equal(_ex_1_2))
 878                         return numeric(2,3)*Pi;
 879
 880                 // acos(-1) -> Pi
 881                 if (x.is_equal(_ex_1))
 882                         return Pi;
 883
 884                 // acos(float) -> float
 885                 if (!x.info(info_flags::crational))
 886                         return acos(ex_to<numeric>(x));
 887
 888                 // acos(-x) -> Pi-acos(x)
 889                 if (x.info(info_flags::negative))
 890                         return Pi-acos(-x);
 891         }
 892
 893         return acos(x).hold();
 894 }
 895
 896 static ex acos_deriv(const ex & x, unsigned deriv_param)
 897 {
 898         GINAC_ASSERT(deriv_param==0);
 899
 900         // d/dx acos(x) -> -1/sqrt(1-x^2)
 901         return -power(1-power(x,_ex2),_ex_1_2);
 902 }
 903
 904 static ex acos_conjugate(const ex & x)
 905 {
 906         // conjugate(acos(x))==acos(conjugate(x)) unless on the branch cuts which
 907         // run along the real axis outside the interval [-1, +1].
 908         if (is_exactly_a<numeric>(x) &&
 909             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
 910                 return acos(x.conjugate());
 911         }
 912         return conjugate_function(acos(x)).hold();
 913 }
 914
 915 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 916                         evalf_func(acos_evalf).
 917                         info_func(asin_info). // Flags of acos are shared with asin functions
 918                         derivative_func(acos_deriv).
 919                         conjugate_func(acos_conjugate).
 920                         latex_name("\\arccos"));
 921
 922 //////////
 923 // inverse tangent (arc tangent)
 924 //////////
 925
 926 static ex atan_evalf(const ex & x)
 927 {
 928         if (is_exactly_a<numeric>(x))
 929                 return atan(ex_to<numeric>(x));
 930
 931         return atan(x).hold();
 932 }
 933
 934 static ex atan_eval(const ex & x)
 935 {
 936         if (x.info(info_flags::numeric)) {
 937
 938                 // atan(0) -> 0
 939                 if (x.is_zero())
 940                         return _ex0;
 941
 942                 // atan(1) -> Pi/4
 943                 if (x.is_equal(_ex1))
 944                         return _ex1_4*Pi;
 945
 946                 // atan(-1) -> -Pi/4
 947                 if (x.is_equal(_ex_1))
 948                         return _ex_1_4*Pi;
 949
 950                 if (x.is_equal(I) || x.is_equal(-I))
 951                         throw (pole_error("atan_eval(): logarithmic pole",0));
 952
 953                 // atan(float) -> float
 954                 if (!x.info(info_flags::crational))
 955                         return atan(ex_to<numeric>(x));
 956
 957                 // atan() is odd
 958                 if (x.info(info_flags::negative))
 959                         return -atan(-x);
 960         }
 961
 962         return atan(x).hold();
 963 }
 964
 965 static ex atan_deriv(const ex & x, unsigned deriv_param)
 966 {
 967         GINAC_ASSERT(deriv_param==0);
 968
 969         // d/dx atan(x) -> 1/(1+x^2)
 970         return power(_ex1+power(x,_ex2), _ex_1);
 971 }
 972
 973 static ex atan_series(const ex &arg,
 974                       const relational &rel,
 975                       int order,
 976                       unsigned options)
 977 {
 978         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 979         // method:
 980         // Taylor series where there is no pole or cut falls back to atan_deriv.
 981         // There are two branch cuts, one runnig from I up the imaginary axis and
 982         // one running from -I down the imaginary axis.  The points I and -I are
 983         // poles.
 984         // On the branch cuts and the poles series expand
 985         //     (log(1+I*x)-log(1-I*x))/(2*I)
 986         // instead.
 987         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 988         if (!(I*arg_pt).info(info_flags::real))
 989                 throw do_taylor();     // Re(x) != 0
 990         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
 991                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
 992         // care for the poles, using the defining formula for atan()...
 993         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
 994                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
 995         if (!(options & series_options::suppress_branchcut)) {
 996                 // method:
 997                 // This is the branch cut: assemble the primitive series manually and
 998                 // then add the corresponding complex step function.
 999                 const symbol &s = ex_to<symbol>(rel.lhs());
1000                 const ex &point = rel.rhs();
1001                 const symbol foo;
1002                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1003                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
1004                 if ((I*arg_pt)<_ex0)
1005                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
1006                 else
1007                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
1008                 epvector seq;
1009                 if (order > 0) {
1010                         seq.reserve(2);
1011                         seq.push_back(expair(Order0correction, _ex0));
1012                 }
1013                 seq.push_back(expair(Order(_ex1), order));
1014                 return series(replarg - pseries(rel, std::move(seq)), rel, order);
1015         }
1016         throw do_taylor();
1017 }
1018
1019 static ex atan_conjugate(const ex & x)
1020 {
1021         // conjugate(atan(x))==atan(conjugate(x)) unless on the branch cuts which
1022         // run along the imaginary axis outside the interval [-I, +I].
1023         if (x.info(info_flags::real))
1024                 return atan(x);
1025         if (is_exactly_a<numeric>(x)) {
1026                 const numeric x_re = ex_to<numeric>(x.real_part());
1027                 const numeric x_im = ex_to<numeric>(x.imag_part());
1028                 if (!x_re.is_zero() ||
1029                     (x_im > *_num_1_p && x_im < *_num1_p))
1030                         return atan(x.conjugate());
1031         }
1032         return conjugate_function(atan(x)).hold();
1033 }
1034
1035 static bool atan_info(const ex & x, unsigned inf)
1036 {
1037         switch (inf) {
1038         case info_flags::expanded:
1039         case info_flags::real:
1040                 return x.info(inf);
1041         case info_flags::positive:
1042         case info_flags::negative:
1043         case info_flags::nonnegative:
1044                 return x.info(info_flags::real) && x.info(inf);
1045         default:
1046                 return false;
1047         }
1048 }
1049
1050 REGISTER_FUNCTION(atan, eval_func(atan_eval).
1051                         evalf_func(atan_evalf).
1052                         info_func(atan_info).
1053                         derivative_func(atan_deriv).
1054                         series_func(atan_series).
1055                         conjugate_func(atan_conjugate).
1056                         latex_name("\\arctan"));
1057
1058 //////////
1059 // inverse tangent (atan2(y,x))
1060 //////////
1061
1062 static ex atan2_evalf(const ex &y, const ex &x)
1063 {
1064         if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
1065                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
1066
1067         return atan2(y, x).hold();
1068 }
1069
1070 static ex atan2_eval(const ex & y, const ex & x)
1071 {
1072         if (y.is_zero()) {
1073
1074                 // atan2(0, 0) -> 0
1075                 if (x.is_zero())
1076                         return _ex0;
1077
1078                 // atan2(0, x), x real and positive -> 0
1079                 if (x.info(info_flags::positive))
1080                         return _ex0;
1081
1082                 // atan2(0, x), x real and negative -> Pi
1083                 if (x.info(info_flags::negative))
1084                         return Pi;
1085         }
1086
1087         if (x.is_zero()) {
1088
1089                 // atan2(y, 0), y real and positive -> Pi/2
1090                 if (y.info(info_flags::positive))
1091                         return _ex1_2*Pi;
1092
1093                 // atan2(y, 0), y real and negative -> -Pi/2
1094                 if (y.info(info_flags::negative))
1095                         return _ex_1_2*Pi;
1096         }
1097
1098         if (y.is_equal(x)) {
1099
1100                 // atan2(y, y), y real and positive -> Pi/4
1101                 if (y.info(info_flags::positive))
1102                         return _ex1_4*Pi;
1103
1104                 // atan2(y, y), y real and negative -> -3/4*Pi
1105                 if (y.info(info_flags::negative))
1106                         return numeric(-3, 4)*Pi;
1107         }
1108
1109         if (y.is_equal(-x)) {
1110
1111                 // atan2(y, -y), y real and positive -> 3*Pi/4
1112                 if (y.info(info_flags::positive))
1113                         return numeric(3, 4)*Pi;
1114
1115                 // atan2(y, -y), y real and negative -> -Pi/4
1116                 if (y.info(info_flags::negative))
1117                         return _ex_1_4*Pi;
1118         }
1119
1120         // atan2(float, float) -> float
1121         if (is_a<numeric>(y) && !y.info(info_flags::crational) &&
1122             is_a<numeric>(x) && !x.info(info_flags::crational))
1123                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
1124
1125         // atan2(real, real) -> atan(y/x) +/- Pi
1126         if (y.info(info_flags::real) && x.info(info_flags::real)) {
1127                 if (x.info(info_flags::positive))
1128                         return atan(y/x);
1129
1130                 if (x.info(info_flags::negative)) {
1131                         if (y.info(info_flags::positive))
1132                                 return atan(y/x)+Pi;
1133                         if (y.info(info_flags::negative))
1134                                 return atan(y/x)-Pi;
1135                 }
1136         }
1137
1138         return atan2(y, x).hold();
1139 }
1140
1141 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
1142 {
1143         GINAC_ASSERT(deriv_param<2);
1144
1145         if (deriv_param==0) {
1146                 // d/dy atan2(y,x)
1147                 return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
1148         }
1149         // d/dx atan2(y,x)
1150         return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
1151 }
1152
1153 static bool atan2_info(const ex & y, const ex & x, unsigned inf)
1154 {
1155         switch (inf) {
1156         case info_flags::expanded:
1157         case info_flags::real:
1158                 return y.info(inf) && x.info(inf);
1159         case info_flags::positive:
1160         case info_flags::negative:
1161         case info_flags::nonnegative:
1162                 return y.info(info_flags::real) && x.info(info_flags::real)
1163                         && y.info(inf);
1164         default:
1165                 return false;
1166         }
1167 }
1168
1169 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
1170                          evalf_func(atan2_evalf).
1171                          info_func(atan2_info).
1172                          evalf_func(atan2_evalf).
1173                          derivative_func(atan2_deriv));
1174
1175 //////////
1176 // hyperbolic sine (trigonometric function)
1177 //////////
1178
1179 static ex sinh_evalf(const ex & x)
1180 {
1181         if (is_exactly_a<numeric>(x))
1182                 return sinh(ex_to<numeric>(x));
1183
1184         return sinh(x).hold();
1185 }
1186
1187 static ex sinh_eval(const ex & x)
1188 {
1189         if (x.info(info_flags::numeric)) {
1190
1191                 // sinh(0) -> 0
1192                 if (x.is_zero())
1193                         return _ex0;
1194
1195                 // sinh(float) -> float
1196                 if (!x.info(info_flags::crational))
1197                         return sinh(ex_to<numeric>(x));
1198
1199                 // sinh() is odd
1200                 if (x.info(info_flags::negative))
1201                         return -sinh(-x);
1202         }
1203
1204         if ((x/Pi).info(info_flags::numeric) &&
1205                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
1206                 return I*sin(x/I);
1207
1208         if (is_exactly_a<function>(x)) {
1209                 const ex &t = x.op(0);
1210
1211                 // sinh(asinh(x)) -> x
1212                 if (is_ex_the_function(x, asinh))
1213                         return t;
1214
1215                 // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
1216                 if (is_ex_the_function(x, acosh))
1217                         return sqrt(t-_ex1)*sqrt(t+_ex1);
1218
1219                 // sinh(atanh(x)) -> x/sqrt(1-x^2)
1220                 if (is_ex_the_function(x, atanh))
1221                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
1222         }
1223
1224         return sinh(x).hold();
1225 }
1226
1227 static ex sinh_deriv(const ex & x, unsigned deriv_param)
1228 {
1229         GINAC_ASSERT(deriv_param==0);
1230
1231         // d/dx sinh(x) -> cosh(x)
1232         return cosh(x);
1233 }
1234
1235 static ex sinh_real_part(const ex & x)
1236 {
1237         return sinh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
1238 }
1239
1240 static ex sinh_imag_part(const ex & x)
1241 {
1242         return cosh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
1243 }
1244
1245 static ex sinh_conjugate(const ex & x)
1246 {
1247         // conjugate(sinh(x))==sinh(conjugate(x))
1248         return sinh(x.conjugate());
1249 }
1250
1251 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
1252                         evalf_func(sinh_evalf).
1253                         info_func(atan_info). // Flags of sinh are shared with atan functions
1254                         derivative_func(sinh_deriv).
1255                         real_part_func(sinh_real_part).
1256                         imag_part_func(sinh_imag_part).
1257                         conjugate_func(sinh_conjugate).
1258                         latex_name("\\sinh"));
1259
1260 //////////
1261 // hyperbolic cosine (trigonometric function)
1262 //////////
1263
1264 static ex cosh_evalf(const ex & x)
1265 {
1266         if (is_exactly_a<numeric>(x))
1267                 return cosh(ex_to<numeric>(x));
1268
1269         return cosh(x).hold();
1270 }
1271
1272 static ex cosh_eval(const ex & x)
1273 {
1274         if (x.info(info_flags::numeric)) {
1275
1276                 // cosh(0) -> 1
1277                 if (x.is_zero())
1278                         return _ex1;
1279
1280                 // cosh(float) -> float
1281                 if (!x.info(info_flags::crational))
1282                         return cosh(ex_to<numeric>(x));
1283
1284                 // cosh() is even
1285                 if (x.info(info_flags::negative))
1286                         return cosh(-x);
1287         }
1288
1289         if ((x/Pi).info(info_flags::numeric) &&
1290                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
1291                 return cos(x/I);
1292
1293         if (is_exactly_a<function>(x)) {
1294                 const ex &t = x.op(0);
1295
1296                 // cosh(acosh(x)) -> x
1297                 if (is_ex_the_function(x, acosh))
1298                         return t;
1299
1300                 // cosh(asinh(x)) -> sqrt(1+x^2)
1301                 if (is_ex_the_function(x, asinh))
1302                         return sqrt(_ex1+power(t,_ex2));
1303
1304                 // cosh(atanh(x)) -> 1/sqrt(1-x^2)
1305                 if (is_ex_the_function(x, atanh))
1306                         return power(_ex1-power(t,_ex2),_ex_1_2);
1307         }
1308
1309         return cosh(x).hold();
1310 }
1311
1312 static ex cosh_deriv(const ex & x, unsigned deriv_param)
1313 {
1314         GINAC_ASSERT(deriv_param==0);
1315
1316         // d/dx cosh(x) -> sinh(x)
1317         return sinh(x);
1318 }
1319
1320 static ex cosh_real_part(const ex & x)
1321 {
1322         return cosh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
1323 }
1324
1325 static ex cosh_imag_part(const ex & x)
1326 {
1327         return sinh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
1328 }
1329
1330 static ex cosh_conjugate(const ex & x)
1331 {
1332         // conjugate(cosh(x))==cosh(conjugate(x))
1333         return cosh(x.conjugate());
1334 }
1335
1336 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
1337                         evalf_func(cosh_evalf).
1338                         info_func(exp_info). // Flags of cosh are shared with exp functions
1339                         derivative_func(cosh_deriv).
1340                         real_part_func(cosh_real_part).
1341                         imag_part_func(cosh_imag_part).
1342                         conjugate_func(cosh_conjugate).
1343                         latex_name("\\cosh"));
1344
1345 //////////
1346 // hyperbolic tangent (trigonometric function)
1347 //////////
1348
1349 static ex tanh_evalf(const ex & x)
1350 {
1351         if (is_exactly_a<numeric>(x))
1352                 return tanh(ex_to<numeric>(x));
1353
1354         return tanh(x).hold();
1355 }
1356
1357 static ex tanh_eval(const ex & x)
1358 {
1359         if (x.info(info_flags::numeric)) {
1360
1361                 // tanh(0) -> 0
1362                 if (x.is_zero())
1363                         return _ex0;
1364
1365                 // tanh(float) -> float
1366                 if (!x.info(info_flags::crational))
1367                         return tanh(ex_to<numeric>(x));
1368
1369                 // tanh() is odd
1370                 if (x.info(info_flags::negative))
1371                         return -tanh(-x);
1372         }
1373
1374         if ((x/Pi).info(info_flags::numeric) &&
1375                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
1376                 return I*tan(x/I);
1377
1378         if (is_exactly_a<function>(x)) {
1379                 const ex &t = x.op(0);
1380
1381                 // tanh(atanh(x)) -> x
1382                 if (is_ex_the_function(x, atanh))
1383                         return t;
1384
1385                 // tanh(asinh(x)) -> x/sqrt(1+x^2)
1386                 if (is_ex_the_function(x, asinh))
1387                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
1388
1389                 // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
1390                 if (is_ex_the_function(x, acosh))
1391                         return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
1392         }
1393
1394         return tanh(x).hold();
1395 }
1396
1397 static ex tanh_deriv(const ex & x, unsigned deriv_param)
1398 {
1399         GINAC_ASSERT(deriv_param==0);
1400
1401         // d/dx tanh(x) -> 1-tanh(x)^2
1402         return _ex1-power(tanh(x),_ex2);
1403 }
1404
1405 static ex tanh_series(const ex &x,
1406                       const relational &rel,
1407                       int order,
1408                       unsigned options)
1409 {
1410         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1411         // method:
1412         // Taylor series where there is no pole falls back to tanh_deriv.
1413         // On a pole simply expand sinh(x)/cosh(x).
1414         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1415         if (!(2*I*x_pt/Pi).info(info_flags::odd))
1416                 throw do_taylor();  // caught by function::series()
1417         // if we got here we have to care for a simple pole
1418         return (sinh(x)/cosh(x)).series(rel, order, options);
1419 }
1420
1421 static ex tanh_real_part(const ex & x)
1422 {
1423         ex a = GiNaC::real_part(x);
1424         ex b = GiNaC::imag_part(x);
1425         return tanh(a)/(1+power(tanh(a),2)*power(tan(b),2));
1426 }
1427
1428 static ex tanh_imag_part(const ex & x)
1429 {
1430         ex a = GiNaC::real_part(x);
1431         ex b = GiNaC::imag_part(x);
1432         return tan(b)/(1+power(tanh(a),2)*power(tan(b),2));
1433 }
1434
1435 static ex tanh_conjugate(const ex & x)
1436 {
1437         // conjugate(tanh(x))==tanh(conjugate(x))
1438         return tanh(x.conjugate());
1439 }
1440
1441 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
1442                         evalf_func(tanh_evalf).
1443                         info_func(atan_info). // Flags of tanh are shared with atan functions
1444                         derivative_func(tanh_deriv).
1445                         series_func(tanh_series).
1446                         real_part_func(tanh_real_part).
1447                         imag_part_func(tanh_imag_part).
1448                         conjugate_func(tanh_conjugate).
1449                         latex_name("\\tanh"));
1450
1451 //////////
1452 // inverse hyperbolic sine (trigonometric function)
1453 //////////
1454
1455 static ex asinh_evalf(const ex & x)
1456 {
1457         if (is_exactly_a<numeric>(x))
1458                 return asinh(ex_to<numeric>(x));
1459
1460         return asinh(x).hold();
1461 }
1462
1463 static ex asinh_eval(const ex & x)
1464 {
1465         if (x.info(info_flags::numeric)) {
1466
1467                 // asinh(0) -> 0
1468                 if (x.is_zero())
1469                         return _ex0;
1470
1471                 // asinh(float) -> float
1472                 if (!x.info(info_flags::crational))
1473                         return asinh(ex_to<numeric>(x));
1474
1475                 // asinh() is odd
1476                 if (x.info(info_flags::negative))
1477                         return -asinh(-x);
1478         }
1479
1480         return asinh(x).hold();
1481 }
1482
1483 static ex asinh_deriv(const ex & x, unsigned deriv_param)
1484 {
1485         GINAC_ASSERT(deriv_param==0);
1486
1487         // d/dx asinh(x) -> 1/sqrt(1+x^2)
1488         return power(_ex1+power(x,_ex2),_ex_1_2);
1489 }
1490
1491 static ex asinh_conjugate(const ex & x)
1492 {
1493         // conjugate(asinh(x))==asinh(conjugate(x)) unless on the branch cuts which
1494         // run along the imaginary axis outside the interval [-I, +I].
1495         if (x.info(info_flags::real))
1496                 return asinh(x);
1497         if (is_exactly_a<numeric>(x)) {
1498                 const numeric x_re = ex_to<numeric>(x.real_part());
1499                 const numeric x_im = ex_to<numeric>(x.imag_part());
1500                 if (!x_re.is_zero() ||
1501                     (x_im > *_num_1_p && x_im < *_num1_p))
1502                         return asinh(x.conjugate());
1503         }
1504         return conjugate_function(asinh(x)).hold();
1505 }
1506
1507 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
1508                          evalf_func(asinh_evalf).
1509                          info_func(atan_info). // Flags of asinh are shared with atan functions
1510                          derivative_func(asinh_deriv).
1511                          conjugate_func(asinh_conjugate));
1512
1513 //////////
1514 // inverse hyperbolic cosine (trigonometric function)
1515 //////////
1516
1517 static ex acosh_evalf(const ex & x)
1518 {
1519         if (is_exactly_a<numeric>(x))
1520                 return acosh(ex_to<numeric>(x));
1521
1522         return acosh(x).hold();
1523 }
1524
1525 static ex acosh_eval(const ex & x)
1526 {
1527         if (x.info(info_flags::numeric)) {
1528
1529                 // acosh(0) -> Pi*I/2
1530                 if (x.is_zero())
1531                         return Pi*I*numeric(1,2);
1532
1533                 // acosh(1) -> 0
1534                 if (x.is_equal(_ex1))
1535                         return _ex0;
1536
1537                 // acosh(-1) -> Pi*I
1538                 if (x.is_equal(_ex_1))
1539                         return Pi*I;
1540
1541                 // acosh(float) -> float
1542                 if (!x.info(info_flags::crational))
1543                         return acosh(ex_to<numeric>(x));
1544
1545                 // acosh(-x) -> Pi*I-acosh(x)
1546                 if (x.info(info_flags::negative))
1547                         return Pi*I-acosh(-x);
1548         }
1549
1550         return acosh(x).hold();
1551 }
1552
1553 static ex acosh_deriv(const ex & x, unsigned deriv_param)
1554 {
1555         GINAC_ASSERT(deriv_param==0);
1556
1557         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
1558         return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
1559 }
1560
1561 static ex acosh_conjugate(const ex & x)
1562 {
1563         // conjugate(acosh(x))==acosh(conjugate(x)) unless on the branch cut
1564         // which runs along the real axis from +1 to -inf.
1565         if (is_exactly_a<numeric>(x) &&
1566             (!x.imag_part().is_zero() || x > *_num1_p)) {
1567                 return acosh(x.conjugate());
1568         }
1569         return conjugate_function(acosh(x)).hold();
1570 }
1571
1572 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
1573                          evalf_func(acosh_evalf).
1574                          info_func(asin_info). // Flags of acosh are shared with asin functions
1575                          derivative_func(acosh_deriv).
1576                          conjugate_func(acosh_conjugate));
1577
1578 //////////
1579 // inverse hyperbolic tangent (trigonometric function)
1580 //////////
1581
1582 static ex atanh_evalf(const ex & x)
1583 {
1584         if (is_exactly_a<numeric>(x))
1585                 return atanh(ex_to<numeric>(x));
1586
1587         return atanh(x).hold();
1588 }
1589
1590 static ex atanh_eval(const ex & x)
1591 {
1592         if (x.info(info_flags::numeric)) {
1593
1594                 // atanh(0) -> 0
1595                 if (x.is_zero())
1596                         return _ex0;
1597
1598                 // atanh({+|-}1) -> throw
1599                 if (x.is_equal(_ex1) || x.is_equal(_ex_1))
1600                         throw (pole_error("atanh_eval(): logarithmic pole",0));
1601
1602                 // atanh(float) -> float
1603                 if (!x.info(info_flags::crational))
1604                         return atanh(ex_to<numeric>(x));
1605
1606                 // atanh() is odd
1607                 if (x.info(info_flags::negative))
1608                         return -atanh(-x);
1609         }
1610
1611         return atanh(x).hold();
1612 }
1613
1614 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1615 {
1616         GINAC_ASSERT(deriv_param==0);
1617
1618         // d/dx atanh(x) -> 1/(1-x^2)
1619         return power(_ex1-power(x,_ex2),_ex_1);
1620 }
1621
1622 static ex atanh_series(const ex &arg,
1623                        const relational &rel,
1624                        int order,
1625                        unsigned options)
1626 {
1627         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1628         // method:
1629         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1630         // There are two branch cuts, one runnig from 1 up the real axis and one
1631         // one running from -1 down the real axis.  The points 1 and -1 are poles
1632         // On the branch cuts and the poles series expand
1633         //     (log(1+x)-log(1-x))/2
1634         // instead.
1635         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
1636         if (!(arg_pt).info(info_flags::real))
1637                 throw do_taylor();     // Im(x) != 0
1638         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1639                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1640         // care for the poles, using the defining formula for atanh()...
1641         if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1642                 return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1643         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1644         if (!(options & series_options::suppress_branchcut)) {
1645                 // method:
1646                 // This is the branch cut: assemble the primitive series manually and
1647                 // then add the corresponding complex step function.
1648                 const symbol &s = ex_to<symbol>(rel.lhs());
1649                 const ex &point = rel.rhs();
1650                 const symbol foo;
1651                 const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1652                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1653                 if (arg_pt<_ex0)
1654                         Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1655                 else
1656                         Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1657                 epvector seq;
1658                 if (order > 0) {
1659                         seq.reserve(2);
1660                         seq.push_back(expair(Order0correction, _ex0));
1661                 }
1662                 seq.push_back(expair(Order(_ex1), order));
1663                 return series(replarg - pseries(rel, std::move(seq)), rel, order);
1664         }
1665         throw do_taylor();
1666 }
1667
1668 static ex atanh_conjugate(const ex & x)
1669 {
1670         // conjugate(atanh(x))==atanh(conjugate(x)) unless on the branch cuts which
1671         // run along the real axis outside the interval [-1, +1].
1672         if (is_exactly_a<numeric>(x) &&
1673             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
1674                 return atanh(x.conjugate());
1675         }
1676         return conjugate_function(atanh(x)).hold();
1677 }
1678
1679 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1680                          evalf_func(atanh_evalf).
1681                          info_func(asin_info). // Flags of atanh are shared with asin functions
1682                          derivative_func(atanh_deriv).
1683                          series_func(atanh_series).
1684                          conjugate_func(atanh_conjugate));
1685
1686
1687 } // namespace GiNaC