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) 1999-2022 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 REGISTER_FUNCTION(exp, eval_func(exp_eval).
 150                        evalf_func(exp_evalf).
 151                        expand_func(exp_expand).
 152                        derivative_func(exp_deriv).
 153                        real_part_func(exp_real_part).
 154                        imag_part_func(exp_imag_part).
 155                        conjugate_func(exp_conjugate).
 156                        power_func(exp_power).
 157                        latex_name("\\exp"));
 158
 159 //////////
 160 // natural logarithm
 161 //////////
 162
 163 static ex log_evalf(const ex & x)
 164 {
 165         if (is_exactly_a<numeric>(x))
 166                 return log(ex_to<numeric>(x));
 167
 168         return log(x).hold();
 169 }
 170
 171 static ex log_eval(const ex & x)
 172 {
 173         if (x.info(info_flags::numeric)) {
 174                 if (x.is_zero())         // log(0) -> infinity
 175                         throw(pole_error("log_eval(): log(0)",0));
 176                 if (x.info(info_flags::rational) && x.info(info_flags::negative))
 177                         return (log(-x)+I*Pi);
 178                 if (x.is_equal(_ex1))  // log(1) -> 0
 179                         return _ex0;
 180                 if (x.is_equal(I))       // log(I) -> Pi*I/2
 181                         return (Pi*I*_ex1_2);
 182                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
 183                         return (Pi*I*_ex_1_2);
 184
 185                 // log(float) -> float
 186                 if (!x.info(info_flags::crational))
 187                         return log(ex_to<numeric>(x));
 188         }
 189
 190         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
 191         if (is_ex_the_function(x, exp)) {
 192                 const ex &t = x.op(0);
 193                 if (t.info(info_flags::real))
 194                         return t;
 195         }
 196
 197         return log(x).hold();
 198 }
 199
 200 static ex log_deriv(const ex & x, unsigned deriv_param)
 201 {
 202         GINAC_ASSERT(deriv_param==0);
 203
 204         // d/dx log(x) -> 1/x
 205         return power(x, _ex_1);
 206 }
 207
 208 static ex log_series(const ex &arg,
 209                      const relational &rel,
 210                      int order,
 211                      unsigned options)
 212 {
 213         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 214         ex arg_pt;
 215         bool must_expand_arg = false;
 216         // maybe substitution of rel into arg fails because of a pole
 217         try {
 218                 arg_pt = arg.subs(rel, subs_options::no_pattern);
 219         } catch (pole_error &) {
 220                 must_expand_arg = true;
 221         }
 222         // or we are at the branch point anyways
 223         if (arg_pt.is_zero())
 224                 must_expand_arg = true;
 225
 226         if (arg.diff(ex_to<symbol>(rel.lhs())).is_zero()) {
 227                 throw do_taylor();
 228         }
 229
 230         if (must_expand_arg) {
 231                 // method:
 232                 // This is the branch point: Series expand the argument first, then
 233                 // trivially factorize it to isolate that part which has constant
 234                 // leading coefficient in this fashion:
 235                 //   x^n + x^(n+1) +...+ Order(x^(n+m))  ->  x^n * (1 + x +...+ Order(x^m)).
 236                 // Return a plain n*log(x) for the x^n part and series expand the
 237                 // other part.  Add them together and reexpand again in order to have
 238                 // one unnested pseries object.  All this also works for negative n.
 239                 pseries argser;          // series expansion of log's argument
 240                 unsigned extra_ord = 0;  // extra expansion order
 241                 do {
 242                         // oops, the argument expanded to a pure Order(x^something)...
 243                         argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
 244                         ++extra_ord;
 245                 } while (!argser.is_terminating() && argser.nops()==1);
 246
 247                 const symbol &s = ex_to<symbol>(rel.lhs());
 248                 const ex &point = rel.rhs();
 249                 const int n = argser.ldegree(s);
 250                 epvector seq;
 251                 // construct what we carelessly called the n*log(x) term above
 252                 const ex coeff = argser.coeff(s, n);
 253                 // expand the log, but only if coeff is real and > 0, since otherwise
 254                 // it would make the branch cut run into the wrong direction
 255                 if (coeff.info(info_flags::positive))
 256                         seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
 257                 else
 258                         seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));
 259
 260                 if (!argser.is_terminating() || argser.nops()!=1) {
 261                         // in this case n more (or less) terms are needed
 262                         // (sadly, to generate them, we have to start from the beginning)
 263                         if (n == 0 && coeff == 1) {
 264                                 ex rest = pseries(rel, epvector{expair(-1, _ex0), expair(Order(_ex1), order)}).add_series(argser);
 265                                 ex acc = dynallocate<pseries>(rel, epvector());
 266                                 for (int i = order-1; i>0; --i) {
 267                                         epvector cterm { expair(i%2 ? _ex1/i : _ex_1/i, _ex0) };
 268                                         acc = pseries(rel, std::move(cterm)).add_series(ex_to<pseries>(acc));
 269                                         acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
 270                                 }
 271                                 return acc;
 272                         }
 273                         const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
 274                         return pseries(rel, std::move(seq)).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
 275                 } else  // it was a monomial
 276                         return pseries(rel, std::move(seq));
 277         }
 278         if (!(options & series_options::suppress_branchcut) &&
 279              arg_pt.info(info_flags::negative)) {
 280                 // method:
 281                 // This is the branch cut: assemble the primitive series manually and
 282                 // then add the corresponding complex step function.
 283                 const symbol &s = ex_to<symbol>(rel.lhs());
 284                 const ex &point = rel.rhs();
 285                 const symbol foo;
 286                 const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 287                 epvector seq;
 288                 if (order > 0) {
 289                         seq.reserve(2);
 290                         seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
 291                 }
 292                 seq.push_back(expair(Order(_ex1), order));
 293                 return series(replarg - I*Pi + pseries(rel, std::move(seq)), rel, order);
 294         }
 295         throw do_taylor();  // caught by function::series()
 296 }
 297
 298 static ex log_real_part(const ex & x)
 299 {
 300         if (x.info(info_flags::nonnegative))
 301                 return log(x).hold();
 302         return log(abs(x));
 303 }
 304
 305 static ex log_imag_part(const ex & x)
 306 {
 307         if (x.info(info_flags::nonnegative))
 308                 return 0;
 309         return atan2(GiNaC::imag_part(x), GiNaC::real_part(x));
 310 }
 311
 312 static ex log_expand(const ex & arg, unsigned options)
 313 {
 314         if ((options & expand_options::expand_transcendental)
 315                 && is_exactly_a<mul>(arg) && !arg.info(info_flags::indefinite)) {
 316                 exvector sumseq;
 317                 exvector prodseq;
 318                 sumseq.reserve(arg.nops());
 319                 prodseq.reserve(arg.nops());
 320                 bool possign=true;
 321
 322                 // searching for positive/negative factors
 323                 for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
 324                         ex e;
 325                         if (options & expand_options::expand_function_args)
 326                                 e=i->expand(options);
 327                         else
 328                                 e=*i;
 329                         if (e.info(info_flags::positive))
 330                                 sumseq.push_back(log(e));
 331                         else if (e.info(info_flags::negative)) {
 332                                 sumseq.push_back(log(-e));
 333                                 possign = !possign;
 334                         } else
 335                                 prodseq.push_back(e);
 336                 }
 337
 338                 if (sumseq.size() > 0) {
 339                         ex newarg;
 340                         if (options & expand_options::expand_function_args)
 341                                 newarg=((possign?_ex1:_ex_1)*mul(prodseq)).expand(options);
 342                         else {
 343                                 newarg=(possign?_ex1:_ex_1)*mul(prodseq);
 344                                 ex_to<basic>(newarg).setflag(status_flags::purely_indefinite);
 345                         }
 346                         return add(sumseq)+log(newarg);
 347                 } else {
 348                         if (!(options & expand_options::expand_function_args))
 349                                 ex_to<basic>(arg).setflag(status_flags::purely_indefinite);
 350                 }
 351         }
 352
 353         if (options & expand_options::expand_function_args)
 354                 return log(arg.expand(options)).hold();
 355         else
 356                 return log(arg).hold();
 357 }
 358
 359 static ex log_conjugate(const ex & x)
 360 {
 361         // conjugate(log(x))==log(conjugate(x)) unless on the branch cut which
 362         // runs along the negative real axis.
 363         if (x.info(info_flags::positive)) {
 364                 return log(x);
 365         }
 366         if (is_exactly_a<numeric>(x) &&
 367             !x.imag_part().is_zero()) {
 368                 return log(x.conjugate());
 369         }
 370         return conjugate_function(log(x)).hold();
 371 }
 372
 373 REGISTER_FUNCTION(log, eval_func(log_eval).
 374                        evalf_func(log_evalf).
 375                        expand_func(log_expand).
 376                        derivative_func(log_deriv).
 377                        series_func(log_series).
 378                        real_part_func(log_real_part).
 379                        imag_part_func(log_imag_part).
 380                        conjugate_func(log_conjugate).
 381                        latex_name("\\ln"));
 382
 383 //////////
 384 // sine (trigonometric function)
 385 //////////
 386
 387 static ex sin_evalf(const ex & x)
 388 {
 389         if (is_exactly_a<numeric>(x))
 390                 return sin(ex_to<numeric>(x));
 391
 392         return sin(x).hold();
 393 }
 394
 395 static ex sin_eval(const ex & x)
 396 {
 397         // sin(n/d*Pi) -> { all known non-nested radicals }
 398         const ex SixtyExOverPi = _ex60*x/Pi;
 399         ex sign = _ex1;
 400         if (SixtyExOverPi.info(info_flags::integer)) {
 401                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 402                 if (z>=*_num60_p) {
 403                         // wrap to interval [0, Pi)
 404                         z -= *_num60_p;
 405                         sign = _ex_1;
 406                 }
 407                 if (z>*_num30_p) {
 408                         // wrap to interval [0, Pi/2)
 409                         z = *_num60_p-z;
 410                 }
 411                 if (z.is_equal(*_num0_p))  // sin(0)       -> 0
 412                         return _ex0;
 413                 if (z.is_equal(*_num5_p))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
 414                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 415                 if (z.is_equal(*_num6_p))  // sin(Pi/10)   -> sqrt(5)/4-1/4
 416                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 417                 if (z.is_equal(*_num10_p)) // sin(Pi/6)    -> 1/2
 418                         return sign*_ex1_2;
 419                 if (z.is_equal(*_num15_p)) // sin(Pi/4)    -> sqrt(2)/2
 420                         return sign*_ex1_2*sqrt(_ex2);
 421                 if (z.is_equal(*_num18_p)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
 422                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 423                 if (z.is_equal(*_num20_p)) // sin(Pi/3)    -> sqrt(3)/2
 424                         return sign*_ex1_2*sqrt(_ex3);
 425                 if (z.is_equal(*_num25_p)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
 426                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 427                 if (z.is_equal(*_num30_p)) // sin(Pi/2)    -> 1
 428                         return sign;
 429         }
 430
 431         if (is_exactly_a<function>(x)) {
 432                 const ex &t = x.op(0);
 433
 434                 // sin(asin(x)) -> x
 435                 if (is_ex_the_function(x, asin))
 436                         return t;
 437
 438                 // sin(acos(x)) -> sqrt(1-x^2)
 439                 if (is_ex_the_function(x, acos))
 440                         return sqrt(_ex1-power(t,_ex2));
 441
 442                 // sin(atan(x)) -> x/sqrt(1+x^2)
 443                 if (is_ex_the_function(x, atan))
 444                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 445         }
 446
 447         // sin(float) -> float
 448         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 449                 return sin(ex_to<numeric>(x));
 450
 451         // sin() is odd
 452         if (x.info(info_flags::negative))
 453                 return -sin(-x);
 454
 455         return sin(x).hold();
 456 }
 457
 458 static ex sin_deriv(const ex & x, unsigned deriv_param)
 459 {
 460         GINAC_ASSERT(deriv_param==0);
 461
 462         // d/dx sin(x) -> cos(x)
 463         return cos(x);
 464 }
 465
 466 static ex sin_real_part(const ex & x)
 467 {
 468         return cosh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 469 }
 470
 471 static ex sin_imag_part(const ex & x)
 472 {
 473         return sinh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 474 }
 475
 476 static ex sin_conjugate(const ex & x)
 477 {
 478         // conjugate(sin(x))==sin(conjugate(x))
 479         return sin(x.conjugate());
 480 }
 481
 482 REGISTER_FUNCTION(sin, eval_func(sin_eval).
 483                        evalf_func(sin_evalf).
 484                        derivative_func(sin_deriv).
 485                        real_part_func(sin_real_part).
 486                        imag_part_func(sin_imag_part).
 487                        conjugate_func(sin_conjugate).
 488                        latex_name("\\sin"));
 489
 490 //////////
 491 // cosine (trigonometric function)
 492 //////////
 493
 494 static ex cos_evalf(const ex & x)
 495 {
 496         if (is_exactly_a<numeric>(x))
 497                 return cos(ex_to<numeric>(x));
 498
 499         return cos(x).hold();
 500 }
 501
 502 static ex cos_eval(const ex & x)
 503 {
 504         // cos(n/d*Pi) -> { all known non-nested radicals }
 505         const ex SixtyExOverPi = _ex60*x/Pi;
 506         ex sign = _ex1;
 507         if (SixtyExOverPi.info(info_flags::integer)) {
 508                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 509                 if (z>=*_num60_p) {
 510                         // wrap to interval [0, Pi)
 511                         z = *_num120_p-z;
 512                 }
 513                 if (z>=*_num30_p) {
 514                         // wrap to interval [0, Pi/2)
 515                         z = *_num60_p-z;
 516                         sign = _ex_1;
 517                 }
 518                 if (z.is_equal(*_num0_p))  // cos(0)       -> 1
 519                         return sign;
 520                 if (z.is_equal(*_num5_p))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 521                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 522                 if (z.is_equal(*_num10_p)) // cos(Pi/6)    -> sqrt(3)/2
 523                         return sign*_ex1_2*sqrt(_ex3);
 524                 if (z.is_equal(*_num12_p)) // cos(Pi/5)    -> sqrt(5)/4+1/4
 525                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 526                 if (z.is_equal(*_num15_p)) // cos(Pi/4)    -> sqrt(2)/2
 527                         return sign*_ex1_2*sqrt(_ex2);
 528                 if (z.is_equal(*_num20_p)) // cos(Pi/3)    -> 1/2
 529                         return sign*_ex1_2;
 530                 if (z.is_equal(*_num24_p)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 531                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 532                 if (z.is_equal(*_num25_p)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 533                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 534                 if (z.is_equal(*_num30_p)) // cos(Pi/2)    -> 0
 535                         return _ex0;
 536         }
 537
 538         if (is_exactly_a<function>(x)) {
 539                 const ex &t = x.op(0);
 540
 541                 // cos(acos(x)) -> x
 542                 if (is_ex_the_function(x, acos))
 543                         return t;
 544
 545                 // cos(asin(x)) -> sqrt(1-x^2)
 546                 if (is_ex_the_function(x, asin))
 547                         return sqrt(_ex1-power(t,_ex2));
 548
 549                 // cos(atan(x)) -> 1/sqrt(1+x^2)
 550                 if (is_ex_the_function(x, atan))
 551                         return power(_ex1+power(t,_ex2),_ex_1_2);
 552         }
 553
 554         // cos(float) -> float
 555         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 556                 return cos(ex_to<numeric>(x));
 557
 558         // cos() is even
 559         if (x.info(info_flags::negative))
 560                 return cos(-x);
 561
 562         return cos(x).hold();
 563 }
 564
 565 static ex cos_deriv(const ex & x, unsigned deriv_param)
 566 {
 567         GINAC_ASSERT(deriv_param==0);
 568
 569         // d/dx cos(x) -> -sin(x)
 570         return -sin(x);
 571 }
 572
 573 static ex cos_real_part(const ex & x)
 574 {
 575         return cosh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 576 }
 577
 578 static ex cos_imag_part(const ex & x)
 579 {
 580         return -sinh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 581 }
 582
 583 static ex cos_conjugate(const ex & x)
 584 {
 585         // conjugate(cos(x))==cos(conjugate(x))
 586         return cos(x.conjugate());
 587 }
 588
 589 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 590                        evalf_func(cos_evalf).
 591                        derivative_func(cos_deriv).
 592                        real_part_func(cos_real_part).
 593                        imag_part_func(cos_imag_part).
 594                        conjugate_func(cos_conjugate).
 595                        latex_name("\\cos"));
 596
 597 //////////
 598 // tangent (trigonometric function)
 599 //////////
 600
 601 static ex tan_evalf(const ex & x)
 602 {
 603         if (is_exactly_a<numeric>(x))
 604                 return tan(ex_to<numeric>(x));
 605
 606         return tan(x).hold();
 607 }
 608
 609 static ex tan_eval(const ex & x)
 610 {
 611         // tan(n/d*Pi) -> { all known non-nested radicals }
 612         const ex SixtyExOverPi = _ex60*x/Pi;
 613         ex sign = _ex1;
 614         if (SixtyExOverPi.info(info_flags::integer)) {
 615                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num60_p);
 616                 if (z>=*_num60_p) {
 617                         // wrap to interval [0, Pi)
 618                         z -= *_num60_p;
 619                 }
 620                 if (z>=*_num30_p) {
 621                         // wrap to interval [0, Pi/2)
 622                         z = *_num60_p-z;
 623                         sign = _ex_1;
 624                 }
 625                 if (z.is_equal(*_num0_p))  // tan(0)       -> 0
 626                         return _ex0;
 627                 if (z.is_equal(*_num5_p))  // tan(Pi/12)   -> 2-sqrt(3)
 628                         return sign*(_ex2-sqrt(_ex3));
 629                 if (z.is_equal(*_num10_p)) // tan(Pi/6)    -> sqrt(3)/3
 630                         return sign*_ex1_3*sqrt(_ex3);
 631                 if (z.is_equal(*_num15_p)) // tan(Pi/4)    -> 1
 632                         return sign;
 633                 if (z.is_equal(*_num20_p)) // tan(Pi/3)    -> sqrt(3)
 634                         return sign*sqrt(_ex3);
 635                 if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3)
 636                         return sign*(sqrt(_ex3)+_ex2);
 637                 if (z.is_equal(*_num30_p)) // tan(Pi/2)    -> infinity
 638                         throw (pole_error("tan_eval(): simple pole",1));
 639         }
 640
 641         if (is_exactly_a<function>(x)) {
 642                 const ex &t = x.op(0);
 643
 644                 // tan(atan(x)) -> x
 645                 if (is_ex_the_function(x, atan))
 646                         return t;
 647
 648                 // tan(asin(x)) -> x/sqrt(1+x^2)
 649                 if (is_ex_the_function(x, asin))
 650                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 651
 652                 // tan(acos(x)) -> sqrt(1-x^2)/x
 653                 if (is_ex_the_function(x, acos))
 654                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
 655         }
 656
 657         // tan(float) -> float
 658         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 659                 return tan(ex_to<numeric>(x));
 660         }
 661
 662         // tan() is odd
 663         if (x.info(info_flags::negative))
 664                 return -tan(-x);
 665
 666         return tan(x).hold();
 667 }
 668
 669 static ex tan_deriv(const ex & x, unsigned deriv_param)
 670 {
 671         GINAC_ASSERT(deriv_param==0);
 672
 673         // d/dx tan(x) -> 1+tan(x)^2;
 674         return (_ex1+power(tan(x),_ex2));
 675 }
 676
 677 static ex tan_real_part(const ex & x)
 678 {
 679         ex a = GiNaC::real_part(x);
 680         ex b = GiNaC::imag_part(x);
 681         return tan(a)/(1+power(tan(a),2)*power(tan(b),2));
 682 }
 683
 684 static ex tan_imag_part(const ex & x)
 685 {
 686         ex a = GiNaC::real_part(x);
 687         ex b = GiNaC::imag_part(x);
 688         return tanh(b)/(1+power(tan(a),2)*power(tan(b),2));
 689 }
 690
 691 static ex tan_series(const ex &x,
 692                      const relational &rel,
 693                      int order,
 694                      unsigned options)
 695 {
 696         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 697         // method:
 698         // Taylor series where there is no pole falls back to tan_deriv.
 699         // On a pole simply expand sin(x)/cos(x).
 700         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 701         if (!(2*x_pt/Pi).info(info_flags::odd))
 702                 throw do_taylor();  // caught by function::series()
 703         // if we got here we have to care for a simple pole
 704         return (sin(x)/cos(x)).series(rel, order, options);
 705 }
 706
 707 static ex tan_conjugate(const ex & x)
 708 {
 709         // conjugate(tan(x))==tan(conjugate(x))
 710         return tan(x.conjugate());
 711 }
 712
 713 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 714                        evalf_func(tan_evalf).
 715                        derivative_func(tan_deriv).
 716                        series_func(tan_series).
 717                        real_part_func(tan_real_part).
 718                        imag_part_func(tan_imag_part).
 719                        conjugate_func(tan_conjugate).
 720                        latex_name("\\tan"));
 721
 722 //////////
 723 // inverse sine (arc sine)
 724 //////////
 725
 726 static ex asin_evalf(const ex & x)
 727 {
 728         if (is_exactly_a<numeric>(x))
 729                 return asin(ex_to<numeric>(x));
 730
 731         return asin(x).hold();
 732 }
 733
 734 static ex asin_eval(const ex & x)
 735 {
 736         if (x.info(info_flags::numeric)) {
 737
 738                 // asin(0) -> 0
 739                 if (x.is_zero())
 740                         return x;
 741
 742                 // asin(1/2) -> Pi/6
 743                 if (x.is_equal(_ex1_2))
 744                         return numeric(1,6)*Pi;
 745
 746                 // asin(1) -> Pi/2
 747                 if (x.is_equal(_ex1))
 748                         return _ex1_2*Pi;
 749
 750                 // asin(-1/2) -> -Pi/6
 751                 if (x.is_equal(_ex_1_2))
 752                         return numeric(-1,6)*Pi;
 753
 754                 // asin(-1) -> -Pi/2
 755                 if (x.is_equal(_ex_1))
 756                         return _ex_1_2*Pi;
 757
 758                 // asin(float) -> float
 759                 if (!x.info(info_flags::crational))
 760                         return asin(ex_to<numeric>(x));
 761
 762                 // asin() is odd
 763                 if (x.info(info_flags::negative))
 764                         return -asin(-x);
 765         }
 766
 767         return asin(x).hold();
 768 }
 769
 770 static ex asin_deriv(const ex & x, unsigned deriv_param)
 771 {
 772         GINAC_ASSERT(deriv_param==0);
 773
 774         // d/dx asin(x) -> 1/sqrt(1-x^2)
 775         return power(1-power(x,_ex2),_ex_1_2);
 776 }
 777
 778 static ex asin_conjugate(const ex & x)
 779 {
 780         // conjugate(asin(x))==asin(conjugate(x)) unless on the branch cuts which
 781         // run along the real axis outside the interval [-1, +1].
 782         if (is_exactly_a<numeric>(x) &&
 783             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
 784                 return asin(x.conjugate());
 785         }
 786         return conjugate_function(asin(x)).hold();
 787 }
 788
 789 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 790                         evalf_func(asin_evalf).
 791                         derivative_func(asin_deriv).
 792                         conjugate_func(asin_conjugate).
 793                         latex_name("\\arcsin"));
 794
 795 //////////
 796 // inverse cosine (arc cosine)
 797 //////////
 798
 799 static ex acos_evalf(const ex & x)
 800 {
 801         if (is_exactly_a<numeric>(x))
 802                 return acos(ex_to<numeric>(x));
 803
 804         return acos(x).hold();
 805 }
 806
 807 static ex acos_eval(const ex & x)
 808 {
 809         if (x.info(info_flags::numeric)) {
 810
 811                 // acos(1) -> 0
 812                 if (x.is_equal(_ex1))
 813                         return _ex0;
 814
 815                 // acos(1/2) -> Pi/3
 816                 if (x.is_equal(_ex1_2))
 817                         return _ex1_3*Pi;
 818
 819                 // acos(0) -> Pi/2
 820                 if (x.is_zero())
 821                         return _ex1_2*Pi;
 822
 823                 // acos(-1/2) -> 2/3*Pi
 824                 if (x.is_equal(_ex_1_2))
 825                         return numeric(2,3)*Pi;
 826
 827                 // acos(-1) -> Pi
 828                 if (x.is_equal(_ex_1))
 829                         return Pi;
 830
 831                 // acos(float) -> float
 832                 if (!x.info(info_flags::crational))
 833                         return acos(ex_to<numeric>(x));
 834
 835                 // acos(-x) -> Pi-acos(x)
 836                 if (x.info(info_flags::negative))
 837                         return Pi-acos(-x);
 838         }
 839
 840         return acos(x).hold();
 841 }
 842
 843 static ex acos_deriv(const ex & x, unsigned deriv_param)
 844 {
 845         GINAC_ASSERT(deriv_param==0);
 846
 847         // d/dx acos(x) -> -1/sqrt(1-x^2)
 848         return -power(1-power(x,_ex2),_ex_1_2);
 849 }
 850
 851 static ex acos_conjugate(const ex & x)
 852 {
 853         // conjugate(acos(x))==acos(conjugate(x)) unless on the branch cuts which
 854         // run along the real axis outside the interval [-1, +1].
 855         if (is_exactly_a<numeric>(x) &&
 856             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
 857                 return acos(x.conjugate());
 858         }
 859         return conjugate_function(acos(x)).hold();
 860 }
 861
 862 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 863                         evalf_func(acos_evalf).
 864                         derivative_func(acos_deriv).
 865                         conjugate_func(acos_conjugate).
 866                         latex_name("\\arccos"));
 867
 868 //////////
 869 // inverse tangent (arc tangent)
 870 //////////
 871
 872 static ex atan_evalf(const ex & x)
 873 {
 874         if (is_exactly_a<numeric>(x))
 875                 return atan(ex_to<numeric>(x));
 876
 877         return atan(x).hold();
 878 }
 879
 880 static ex atan_eval(const ex & x)
 881 {
 882         if (x.info(info_flags::numeric)) {
 883
 884                 // atan(0) -> 0
 885                 if (x.is_zero())
 886                         return _ex0;
 887
 888                 // atan(1) -> Pi/4
 889                 if (x.is_equal(_ex1))
 890                         return _ex1_4*Pi;
 891
 892                 // atan(-1) -> -Pi/4
 893                 if (x.is_equal(_ex_1))
 894                         return _ex_1_4*Pi;
 895
 896                 if (x.is_equal(I) || x.is_equal(-I))
 897                         throw (pole_error("atan_eval(): logarithmic pole",0));
 898
 899                 // atan(float) -> float
 900                 if (!x.info(info_flags::crational))
 901                         return atan(ex_to<numeric>(x));
 902
 903                 // atan() is odd
 904                 if (x.info(info_flags::negative))
 905                         return -atan(-x);
 906         }
 907
 908         return atan(x).hold();
 909 }
 910
 911 static ex atan_deriv(const ex & x, unsigned deriv_param)
 912 {
 913         GINAC_ASSERT(deriv_param==0);
 914
 915         // d/dx atan(x) -> 1/(1+x^2)
 916         return power(_ex1+power(x,_ex2), _ex_1);
 917 }
 918
 919 static ex atan_series(const ex &arg,
 920                       const relational &rel,
 921                       int order,
 922                       unsigned options)
 923 {
 924         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 925         // method:
 926         // Taylor series where there is no pole or cut falls back to atan_deriv.
 927         // There are two branch cuts, one runnig from I up the imaginary axis and
 928         // one running from -I down the imaginary axis.  The points I and -I are
 929         // poles.
 930         // On the branch cuts and the poles series expand
 931         //     (log(1+I*x)-log(1-I*x))/(2*I)
 932         // instead.
 933         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 934         if (!(I*arg_pt).info(info_flags::real))
 935                 throw do_taylor();     // Re(x) != 0
 936         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
 937                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
 938         // care for the poles, using the defining formula for atan()...
 939         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
 940                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
 941         if (!(options & series_options::suppress_branchcut)) {
 942                 // method:
 943                 // This is the branch cut: assemble the primitive series manually and
 944                 // then add the corresponding complex step function.
 945                 const symbol &s = ex_to<symbol>(rel.lhs());
 946                 const ex &point = rel.rhs();
 947                 const symbol foo;
 948                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 949                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
 950                 if ((I*arg_pt)<_ex0)
 951                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
 952                 else
 953                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
 954                 epvector seq;
 955                 if (order > 0) {
 956                         seq.reserve(2);
 957                         seq.push_back(expair(Order0correction, _ex0));
 958                 }
 959                 seq.push_back(expair(Order(_ex1), order));
 960                 return series(replarg - pseries(rel, std::move(seq)), rel, order);
 961         }
 962         throw do_taylor();
 963 }
 964
 965 static ex atan_conjugate(const ex & x)
 966 {
 967         // conjugate(atan(x))==atan(conjugate(x)) unless on the branch cuts which
 968         // run along the imaginary axis outside the interval [-I, +I].
 969         if (x.info(info_flags::real))
 970                 return atan(x);
 971         if (is_exactly_a<numeric>(x)) {
 972                 const numeric x_re = ex_to<numeric>(x.real_part());
 973                 const numeric x_im = ex_to<numeric>(x.imag_part());
 974                 if (!x_re.is_zero() ||
 975                     (x_im > *_num_1_p && x_im < *_num1_p))
 976                         return atan(x.conjugate());
 977         }
 978         return conjugate_function(atan(x)).hold();
 979 }
 980
 981 REGISTER_FUNCTION(atan, eval_func(atan_eval).
 982                         evalf_func(atan_evalf).
 983                         derivative_func(atan_deriv).
 984                         series_func(atan_series).
 985                         conjugate_func(atan_conjugate).
 986                         latex_name("\\arctan"));
 987
 988 //////////
 989 // inverse tangent (atan2(y,x))
 990 //////////
 991
 992 static ex atan2_evalf(const ex &y, const ex &x)
 993 {
 994         if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
 995                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 996
 997         return atan2(y, x).hold();
 998 }
 999
1000 static ex atan2_eval(const ex & y, const ex & x)
1001 {
1002         if (y.is_zero()) {
1003
1004                 // atan2(0, 0) -> 0
1005                 if (x.is_zero())
1006                         return _ex0;
1007
1008                 // atan2(0, x), x real and positive -> 0
1009                 if (x.info(info_flags::positive))
1010                         return _ex0;
1011
1012                 // atan2(0, x), x real and negative -> Pi
1013                 if (x.info(info_flags::negative))
1014                         return Pi;
1015         }
1016
1017         if (x.is_zero()) {
1018
1019                 // atan2(y, 0), y real and positive -> Pi/2
1020                 if (y.info(info_flags::positive))
1021                         return _ex1_2*Pi;
1022
1023                 // atan2(y, 0), y real and negative -> -Pi/2
1024                 if (y.info(info_flags::negative))
1025                         return _ex_1_2*Pi;
1026         }
1027
1028         if (y.is_equal(x)) {
1029
1030                 // atan2(y, y), y real and positive -> Pi/4
1031                 if (y.info(info_flags::positive))
1032                         return _ex1_4*Pi;
1033
1034                 // atan2(y, y), y real and negative -> -3/4*Pi
1035                 if (y.info(info_flags::negative))
1036                         return numeric(-3, 4)*Pi;
1037         }
1038
1039         if (y.is_equal(-x)) {
1040
1041                 // atan2(y, -y), y real and positive -> 3*Pi/4
1042                 if (y.info(info_flags::positive))
1043                         return numeric(3, 4)*Pi;
1044
1045                 // atan2(y, -y), y real and negative -> -Pi/4
1046                 if (y.info(info_flags::negative))
1047                         return _ex_1_4*Pi;
1048         }
1049
1050         // atan2(float, float) -> float
1051         if (is_a<numeric>(y) && !y.info(info_flags::crational) &&
1052             is_a<numeric>(x) && !x.info(info_flags::crational))
1053                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
1054
1055         // atan2(real, real) -> atan(y/x) +/- Pi
1056         if (y.info(info_flags::real) && x.info(info_flags::real)) {
1057                 if (x.info(info_flags::positive))
1058                         return atan(y/x);
1059
1060                 if (x.info(info_flags::negative)) {
1061                         if (y.info(info_flags::positive))
1062                                 return atan(y/x)+Pi;
1063                         if (y.info(info_flags::negative))
1064                                 return atan(y/x)-Pi;
1065                 }
1066         }
1067
1068         return atan2(y, x).hold();
1069 }
1070
1071 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
1072 {
1073         GINAC_ASSERT(deriv_param<2);
1074
1075         if (deriv_param==0) {
1076                 // d/dy atan2(y,x)
1077                 return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
1078         }
1079         // d/dx atan2(y,x)
1080         return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
1081 }
1082
1083 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
1084                          evalf_func(atan2_evalf).
1085                          derivative_func(atan2_deriv));
1086
1087 //////////
1088 // hyperbolic sine (trigonometric function)
1089 //////////
1090
1091 static ex sinh_evalf(const ex & x)
1092 {
1093         if (is_exactly_a<numeric>(x))
1094                 return sinh(ex_to<numeric>(x));
1095
1096         return sinh(x).hold();
1097 }
1098
1099 static ex sinh_eval(const ex & x)
1100 {
1101         if (x.info(info_flags::numeric)) {
1102
1103                 // sinh(0) -> 0
1104                 if (x.is_zero())
1105                         return _ex0;
1106
1107                 // sinh(float) -> float
1108                 if (!x.info(info_flags::crational))
1109                         return sinh(ex_to<numeric>(x));
1110
1111                 // sinh() is odd
1112                 if (x.info(info_flags::negative))
1113                         return -sinh(-x);
1114         }
1115
1116         if ((x/Pi).info(info_flags::numeric) &&
1117                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
1118                 return I*sin(x/I);
1119
1120         if (is_exactly_a<function>(x)) {
1121                 const ex &t = x.op(0);
1122
1123                 // sinh(asinh(x)) -> x
1124                 if (is_ex_the_function(x, asinh))
1125                         return t;
1126
1127                 // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
1128                 if (is_ex_the_function(x, acosh))
1129                         return sqrt(t-_ex1)*sqrt(t+_ex1);
1130
1131                 // sinh(atanh(x)) -> x/sqrt(1-x^2)
1132                 if (is_ex_the_function(x, atanh))
1133                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
1134         }
1135
1136         return sinh(x).hold();
1137 }
1138
1139 static ex sinh_deriv(const ex & x, unsigned deriv_param)
1140 {
1141         GINAC_ASSERT(deriv_param==0);
1142
1143         // d/dx sinh(x) -> cosh(x)
1144         return cosh(x);
1145 }
1146
1147 static ex sinh_real_part(const ex & x)
1148 {
1149         return sinh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
1150 }
1151
1152 static ex sinh_imag_part(const ex & x)
1153 {
1154         return cosh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
1155 }
1156
1157 static ex sinh_conjugate(const ex & x)
1158 {
1159         // conjugate(sinh(x))==sinh(conjugate(x))
1160         return sinh(x.conjugate());
1161 }
1162
1163 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
1164                         evalf_func(sinh_evalf).
1165                         derivative_func(sinh_deriv).
1166                         real_part_func(sinh_real_part).
1167                         imag_part_func(sinh_imag_part).
1168                         conjugate_func(sinh_conjugate).
1169                         latex_name("\\sinh"));
1170
1171 //////////
1172 // hyperbolic cosine (trigonometric function)
1173 //////////
1174
1175 static ex cosh_evalf(const ex & x)
1176 {
1177         if (is_exactly_a<numeric>(x))
1178                 return cosh(ex_to<numeric>(x));
1179
1180         return cosh(x).hold();
1181 }
1182
1183 static ex cosh_eval(const ex & x)
1184 {
1185         if (x.info(info_flags::numeric)) {
1186
1187                 // cosh(0) -> 1
1188                 if (x.is_zero())
1189                         return _ex1;
1190
1191                 // cosh(float) -> float
1192                 if (!x.info(info_flags::crational))
1193                         return cosh(ex_to<numeric>(x));
1194
1195                 // cosh() is even
1196                 if (x.info(info_flags::negative))
1197                         return cosh(-x);
1198         }
1199
1200         if ((x/Pi).info(info_flags::numeric) &&
1201                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
1202                 return cos(x/I);
1203
1204         if (is_exactly_a<function>(x)) {
1205                 const ex &t = x.op(0);
1206
1207                 // cosh(acosh(x)) -> x
1208                 if (is_ex_the_function(x, acosh))
1209                         return t;
1210
1211                 // cosh(asinh(x)) -> sqrt(1+x^2)
1212                 if (is_ex_the_function(x, asinh))
1213                         return sqrt(_ex1+power(t,_ex2));
1214
1215                 // cosh(atanh(x)) -> 1/sqrt(1-x^2)
1216                 if (is_ex_the_function(x, atanh))
1217                         return power(_ex1-power(t,_ex2),_ex_1_2);
1218         }
1219
1220         return cosh(x).hold();
1221 }
1222
1223 static ex cosh_deriv(const ex & x, unsigned deriv_param)
1224 {
1225         GINAC_ASSERT(deriv_param==0);
1226
1227         // d/dx cosh(x) -> sinh(x)
1228         return sinh(x);
1229 }
1230
1231 static ex cosh_real_part(const ex & x)
1232 {
1233         return cosh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
1234 }
1235
1236 static ex cosh_imag_part(const ex & x)
1237 {
1238         return sinh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
1239 }
1240
1241 static ex cosh_conjugate(const ex & x)
1242 {
1243         // conjugate(cosh(x))==cosh(conjugate(x))
1244         return cosh(x.conjugate());
1245 }
1246
1247 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
1248                         evalf_func(cosh_evalf).
1249                         derivative_func(cosh_deriv).
1250                         real_part_func(cosh_real_part).
1251                         imag_part_func(cosh_imag_part).
1252                         conjugate_func(cosh_conjugate).
1253                         latex_name("\\cosh"));
1254
1255 //////////
1256 // hyperbolic tangent (trigonometric function)
1257 //////////
1258
1259 static ex tanh_evalf(const ex & x)
1260 {
1261         if (is_exactly_a<numeric>(x))
1262                 return tanh(ex_to<numeric>(x));
1263
1264         return tanh(x).hold();
1265 }
1266
1267 static ex tanh_eval(const ex & x)
1268 {
1269         if (x.info(info_flags::numeric)) {
1270
1271                 // tanh(0) -> 0
1272                 if (x.is_zero())
1273                         return _ex0;
1274
1275                 // tanh(float) -> float
1276                 if (!x.info(info_flags::crational))
1277                         return tanh(ex_to<numeric>(x));
1278
1279                 // tanh() is odd
1280                 if (x.info(info_flags::negative))
1281                         return -tanh(-x);
1282         }
1283
1284         if ((x/Pi).info(info_flags::numeric) &&
1285                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
1286                 return I*tan(x/I);
1287
1288         if (is_exactly_a<function>(x)) {
1289                 const ex &t = x.op(0);
1290
1291                 // tanh(atanh(x)) -> x
1292                 if (is_ex_the_function(x, atanh))
1293                         return t;
1294
1295                 // tanh(asinh(x)) -> x/sqrt(1+x^2)
1296                 if (is_ex_the_function(x, asinh))
1297                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
1298
1299                 // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
1300                 if (is_ex_the_function(x, acosh))
1301                         return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
1302         }
1303
1304         return tanh(x).hold();
1305 }
1306
1307 static ex tanh_deriv(const ex & x, unsigned deriv_param)
1308 {
1309         GINAC_ASSERT(deriv_param==0);
1310
1311         // d/dx tanh(x) -> 1-tanh(x)^2
1312         return _ex1-power(tanh(x),_ex2);
1313 }
1314
1315 static ex tanh_series(const ex &x,
1316                       const relational &rel,
1317                       int order,
1318                       unsigned options)
1319 {
1320         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1321         // method:
1322         // Taylor series where there is no pole falls back to tanh_deriv.
1323         // On a pole simply expand sinh(x)/cosh(x).
1324         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1325         if (!(2*I*x_pt/Pi).info(info_flags::odd))
1326                 throw do_taylor();  // caught by function::series()
1327         // if we got here we have to care for a simple pole
1328         return (sinh(x)/cosh(x)).series(rel, order, options);
1329 }
1330
1331 static ex tanh_real_part(const ex & x)
1332 {
1333         ex a = GiNaC::real_part(x);
1334         ex b = GiNaC::imag_part(x);
1335         return tanh(a)/(1+power(tanh(a),2)*power(tan(b),2));
1336 }
1337
1338 static ex tanh_imag_part(const ex & x)
1339 {
1340         ex a = GiNaC::real_part(x);
1341         ex b = GiNaC::imag_part(x);
1342         return tan(b)/(1+power(tanh(a),2)*power(tan(b),2));
1343 }
1344
1345 static ex tanh_conjugate(const ex & x)
1346 {
1347         // conjugate(tanh(x))==tanh(conjugate(x))
1348         return tanh(x.conjugate());
1349 }
1350
1351 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
1352                         evalf_func(tanh_evalf).
1353                         derivative_func(tanh_deriv).
1354                         series_func(tanh_series).
1355                         real_part_func(tanh_real_part).
1356                         imag_part_func(tanh_imag_part).
1357                         conjugate_func(tanh_conjugate).
1358                         latex_name("\\tanh"));
1359
1360 //////////
1361 // inverse hyperbolic sine (trigonometric function)
1362 //////////
1363
1364 static ex asinh_evalf(const ex & x)
1365 {
1366         if (is_exactly_a<numeric>(x))
1367                 return asinh(ex_to<numeric>(x));
1368
1369         return asinh(x).hold();
1370 }
1371
1372 static ex asinh_eval(const ex & x)
1373 {
1374         if (x.info(info_flags::numeric)) {
1375
1376                 // asinh(0) -> 0
1377                 if (x.is_zero())
1378                         return _ex0;
1379
1380                 // asinh(float) -> float
1381                 if (!x.info(info_flags::crational))
1382                         return asinh(ex_to<numeric>(x));
1383
1384                 // asinh() is odd
1385                 if (x.info(info_flags::negative))
1386                         return -asinh(-x);
1387         }
1388
1389         return asinh(x).hold();
1390 }
1391
1392 static ex asinh_deriv(const ex & x, unsigned deriv_param)
1393 {
1394         GINAC_ASSERT(deriv_param==0);
1395
1396         // d/dx asinh(x) -> 1/sqrt(1+x^2)
1397         return power(_ex1+power(x,_ex2),_ex_1_2);
1398 }
1399
1400 static ex asinh_conjugate(const ex & x)
1401 {
1402         // conjugate(asinh(x))==asinh(conjugate(x)) unless on the branch cuts which
1403         // run along the imaginary axis outside the interval [-I, +I].
1404         if (x.info(info_flags::real))
1405                 return asinh(x);
1406         if (is_exactly_a<numeric>(x)) {
1407                 const numeric x_re = ex_to<numeric>(x.real_part());
1408                 const numeric x_im = ex_to<numeric>(x.imag_part());
1409                 if (!x_re.is_zero() ||
1410                     (x_im > *_num_1_p && x_im < *_num1_p))
1411                         return asinh(x.conjugate());
1412         }
1413         return conjugate_function(asinh(x)).hold();
1414 }
1415
1416 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
1417                          evalf_func(asinh_evalf).
1418                          derivative_func(asinh_deriv).
1419                          conjugate_func(asinh_conjugate));
1420
1421 //////////
1422 // inverse hyperbolic cosine (trigonometric function)
1423 //////////
1424
1425 static ex acosh_evalf(const ex & x)
1426 {
1427         if (is_exactly_a<numeric>(x))
1428                 return acosh(ex_to<numeric>(x));
1429
1430         return acosh(x).hold();
1431 }
1432
1433 static ex acosh_eval(const ex & x)
1434 {
1435         if (x.info(info_flags::numeric)) {
1436
1437                 // acosh(0) -> Pi*I/2
1438                 if (x.is_zero())
1439                         return Pi*I*numeric(1,2);
1440
1441                 // acosh(1) -> 0
1442                 if (x.is_equal(_ex1))
1443                         return _ex0;
1444
1445                 // acosh(-1) -> Pi*I
1446                 if (x.is_equal(_ex_1))
1447                         return Pi*I;
1448
1449                 // acosh(float) -> float
1450                 if (!x.info(info_flags::crational))
1451                         return acosh(ex_to<numeric>(x));
1452
1453                 // acosh(-x) -> Pi*I-acosh(x)
1454                 if (x.info(info_flags::negative))
1455                         return Pi*I-acosh(-x);
1456         }
1457
1458         return acosh(x).hold();
1459 }
1460
1461 static ex acosh_deriv(const ex & x, unsigned deriv_param)
1462 {
1463         GINAC_ASSERT(deriv_param==0);
1464
1465         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
1466         return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
1467 }
1468
1469 static ex acosh_conjugate(const ex & x)
1470 {
1471         // conjugate(acosh(x))==acosh(conjugate(x)) unless on the branch cut
1472         // which runs along the real axis from +1 to -inf.
1473         if (is_exactly_a<numeric>(x) &&
1474             (!x.imag_part().is_zero() || x > *_num1_p)) {
1475                 return acosh(x.conjugate());
1476         }
1477         return conjugate_function(acosh(x)).hold();
1478 }
1479
1480 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
1481                          evalf_func(acosh_evalf).
1482                          derivative_func(acosh_deriv).
1483                          conjugate_func(acosh_conjugate));
1484
1485 //////////
1486 // inverse hyperbolic tangent (trigonometric function)
1487 //////////
1488
1489 static ex atanh_evalf(const ex & x)
1490 {
1491         if (is_exactly_a<numeric>(x))
1492                 return atanh(ex_to<numeric>(x));
1493
1494         return atanh(x).hold();
1495 }
1496
1497 static ex atanh_eval(const ex & x)
1498 {
1499         if (x.info(info_flags::numeric)) {
1500
1501                 // atanh(0) -> 0
1502                 if (x.is_zero())
1503                         return _ex0;
1504
1505                 // atanh({+|-}1) -> throw
1506                 if (x.is_equal(_ex1) || x.is_equal(_ex_1))
1507                         throw (pole_error("atanh_eval(): logarithmic pole",0));
1508
1509                 // atanh(float) -> float
1510                 if (!x.info(info_flags::crational))
1511                         return atanh(ex_to<numeric>(x));
1512
1513                 // atanh() is odd
1514                 if (x.info(info_flags::negative))
1515                         return -atanh(-x);
1516         }
1517
1518         return atanh(x).hold();
1519 }
1520
1521 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1522 {
1523         GINAC_ASSERT(deriv_param==0);
1524
1525         // d/dx atanh(x) -> 1/(1-x^2)
1526         return power(_ex1-power(x,_ex2),_ex_1);
1527 }
1528
1529 static ex atanh_series(const ex &arg,
1530                        const relational &rel,
1531                        int order,
1532                        unsigned options)
1533 {
1534         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1535         // method:
1536         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1537         // There are two branch cuts, one runnig from 1 up the real axis and one
1538         // one running from -1 down the real axis.  The points 1 and -1 are poles
1539         // On the branch cuts and the poles series expand
1540         //     (log(1+x)-log(1-x))/2
1541         // instead.
1542         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
1543         if (!(arg_pt).info(info_flags::real))
1544                 throw do_taylor();     // Im(x) != 0
1545         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1546                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1547         // care for the poles, using the defining formula for atanh()...
1548         if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1549                 return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1550         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1551         if (!(options & series_options::suppress_branchcut)) {
1552                 // method:
1553                 // This is the branch cut: assemble the primitive series manually and
1554                 // then add the corresponding complex step function.
1555                 const symbol &s = ex_to<symbol>(rel.lhs());
1556                 const ex &point = rel.rhs();
1557                 const symbol foo;
1558                 const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1559                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1560                 if (arg_pt<_ex0)
1561                         Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1562                 else
1563                         Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1564                 epvector seq;
1565                 if (order > 0) {
1566                         seq.reserve(2);
1567                         seq.push_back(expair(Order0correction, _ex0));
1568                 }
1569                 seq.push_back(expair(Order(_ex1), order));
1570                 return series(replarg - pseries(rel, std::move(seq)), rel, order);
1571         }
1572         throw do_taylor();
1573 }
1574
1575 static ex atanh_conjugate(const ex & x)
1576 {
1577         // conjugate(atanh(x))==atanh(conjugate(x)) unless on the branch cuts which
1578         // run along the real axis outside the interval [-1, +1].
1579         if (is_exactly_a<numeric>(x) &&
1580             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
1581                 return atanh(x.conjugate());
1582         }
1583         return conjugate_function(atanh(x)).hold();
1584 }
1585
1586 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1587                          evalf_func(atanh_evalf).
1588                          derivative_func(atanh_deriv).
1589                          series_func(atanh_series).
1590                          conjugate_func(atanh_conjugate));
1591
1592
1593 } // namespace GiNaC