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) 1999-2011 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 "numeric.h"
  28 #include "power.h"
  29 #include "operators.h"
  30 #include "relational.h"
  31 #include "symbol.h"
  32 #include "pseries.h"
  33 #include "utils.h"
  34
  35 #include <stdexcept>
  36 #include <vector>
  37
  38 namespace GiNaC {
  39
  40 //////////
  41 // exponential function
  42 //////////
  43
  44 static ex exp_evalf(const ex & x)
  45 {
  46         if (is_exactly_a<numeric>(x))
  47                 return exp(ex_to<numeric>(x));
  48
  49         return exp(x).hold();
  50 }
  51
  52 static ex exp_eval(const ex & x)
  53 {
  54         // exp(0) -> 1
  55         if (x.is_zero()) {
  56                 return _ex1;
  57         }
  58
  59         // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
  60         const ex TwoExOverPiI=(_ex2*x)/(Pi*I);
  61         if (TwoExOverPiI.info(info_flags::integer)) {
  62                 const numeric z = mod(ex_to<numeric>(TwoExOverPiI),*_num4_p);
  63                 if (z.is_equal(*_num0_p))
  64                         return _ex1;
  65                 if (z.is_equal(*_num1_p))
  66                         return ex(I);
  67                 if (z.is_equal(*_num2_p))
  68                         return _ex_1;
  69                 if (z.is_equal(*_num3_p))
  70                         return ex(-I);
  71         }
  72
  73         // exp(log(x)) -> x
  74         if (is_ex_the_function(x, log))
  75                 return x.op(0);
  76
  77         // exp(float) -> float
  78         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
  79                 return exp(ex_to<numeric>(x));
  80
  81         return exp(x).hold();
  82 }
  83
  84 static ex exp_deriv(const ex & x, unsigned deriv_param)
  85 {
  86         GINAC_ASSERT(deriv_param==0);
  87
  88         // d/dx exp(x) -> exp(x)
  89         return exp(x);
  90 }
  91
  92 static ex exp_real_part(const ex & x)
  93 {
  94         return exp(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
  95 }
  96
  97 static ex exp_imag_part(const ex & x)
  98 {
  99         return exp(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
 100 }
 101
 102 static ex exp_conjugate(const ex & x)
 103 {
 104         // conjugate(exp(x))==exp(conjugate(x))
 105         return exp(x.conjugate());
 106 }
 107
 108 REGISTER_FUNCTION(exp, eval_func(exp_eval).
 109                        evalf_func(exp_evalf).
 110                        derivative_func(exp_deriv).
 111                        real_part_func(exp_real_part).
 112                        imag_part_func(exp_imag_part).
 113                        conjugate_func(exp_conjugate).
 114                        latex_name("\\exp"));
 115
 116 //////////
 117 // natural logarithm
 118 //////////
 119
 120 static ex log_evalf(const ex & x)
 121 {
 122         if (is_exactly_a<numeric>(x))
 123                 return log(ex_to<numeric>(x));
 124
 125         return log(x).hold();
 126 }
 127
 128 static ex log_eval(const ex & x)
 129 {
 130         if (x.info(info_flags::numeric)) {
 131                 if (x.is_zero())         // log(0) -> infinity
 132                         throw(pole_error("log_eval(): log(0)",0));
 133                 if (x.info(info_flags::rational) && x.info(info_flags::negative))
 134                         return (log(-x)+I*Pi);
 135                 if (x.is_equal(_ex1))  // log(1) -> 0
 136                         return _ex0;
 137                 if (x.is_equal(I))       // log(I) -> Pi*I/2
 138                         return (Pi*I*_ex1_2);
 139                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
 140                         return (Pi*I*_ex_1_2);
 141
 142                 // log(float) -> float
 143                 if (!x.info(info_flags::crational))
 144                         return log(ex_to<numeric>(x));
 145         }
 146
 147         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
 148         if (is_ex_the_function(x, exp)) {
 149                 const ex &t = x.op(0);
 150                 if (t.info(info_flags::real))
 151                         return t;
 152         }
 153
 154         return log(x).hold();
 155 }
 156
 157 static ex log_deriv(const ex & x, unsigned deriv_param)
 158 {
 159         GINAC_ASSERT(deriv_param==0);
 160
 161         // d/dx log(x) -> 1/x
 162         return power(x, _ex_1);
 163 }
 164
 165 static ex log_series(const ex &arg,
 166                      const relational &rel,
 167                      int order,
 168                      unsigned options)
 169 {
 170         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 171         ex arg_pt;
 172         bool must_expand_arg = false;
 173         // maybe substitution of rel into arg fails because of a pole
 174         try {
 175                 arg_pt = arg.subs(rel, subs_options::no_pattern);
 176         } catch (pole_error) {
 177                 must_expand_arg = true;
 178         }
 179         // or we are at the branch point anyways
 180         if (arg_pt.is_zero())
 181                 must_expand_arg = true;
 182
 183         if (must_expand_arg) {
 184                 // method:
 185                 // This is the branch point: Series expand the argument first, then
 186                 // trivially factorize it to isolate that part which has constant
 187                 // leading coefficient in this fashion:
 188                 //   x^n + x^(n+1) +...+ Order(x^(n+m))  ->  x^n * (1 + x +...+ Order(x^m)).
 189                 // Return a plain n*log(x) for the x^n part and series expand the
 190                 // other part.  Add them together and reexpand again in order to have
 191                 // one unnested pseries object.  All this also works for negative n.
 192                 pseries argser;          // series expansion of log's argument
 193                 unsigned extra_ord = 0;  // extra expansion order
 194                 do {
 195                         // oops, the argument expanded to a pure Order(x^something)...
 196                         argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
 197                         ++extra_ord;
 198                 } while (!argser.is_terminating() && argser.nops()==1);
 199
 200                 const symbol &s = ex_to<symbol>(rel.lhs());
 201                 const ex &point = rel.rhs();
 202                 const int n = argser.ldegree(s);
 203                 epvector seq;
 204                 // construct what we carelessly called the n*log(x) term above
 205                 const ex coeff = argser.coeff(s, n);
 206                 // expand the log, but only if coeff is real and > 0, since otherwise
 207                 // it would make the branch cut run into the wrong direction
 208                 if (coeff.info(info_flags::positive))
 209                         seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
 210                 else
 211                         seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));
 212
 213                 if (!argser.is_terminating() || argser.nops()!=1) {
 214                         // in this case n more (or less) terms are needed
 215                         // (sadly, to generate them, we have to start from the beginning)
 216                         if (n == 0 && coeff == 1) {
 217                                 epvector epv;
 218                                 ex acc = (new pseries(rel, epv))->setflag(status_flags::dynallocated);
 219                                 epv.reserve(2);
 220                                 epv.push_back(expair(-1, _ex0));
 221                                 epv.push_back(expair(Order(_ex1), order));
 222                                 ex rest = pseries(rel, epv).add_series(argser);
 223                                 for (int i = order-1; i>0; --i) {
 224                                         epvector cterm;
 225                                         cterm.reserve(1);
 226                                         cterm.push_back(expair(i%2 ? _ex1/i : _ex_1/i, _ex0));
 227                                         acc = pseries(rel, cterm).add_series(ex_to<pseries>(acc));
 228                                         acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
 229                                 }
 230                                 return acc;
 231                         }
 232                         const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
 233                         return pseries(rel, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
 234                 } else  // it was a monomial
 235                         return pseries(rel, seq);
 236         }
 237         if (!(options & series_options::suppress_branchcut) &&
 238              arg_pt.info(info_flags::negative)) {
 239                 // method:
 240                 // This is the branch cut: assemble the primitive series manually and
 241                 // then add the corresponding complex step function.
 242                 const symbol &s = ex_to<symbol>(rel.lhs());
 243                 const ex &point = rel.rhs();
 244                 const symbol foo;
 245                 const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 246                 epvector seq;
 247                 seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
 248                 seq.push_back(expair(Order(_ex1), order));
 249                 return series(replarg - I*Pi + pseries(rel, seq), rel, order);
 250         }
 251         throw do_taylor();  // caught by function::series()
 252 }
 253
 254 static ex log_real_part(const ex & x)
 255 {
 256         if (x.info(info_flags::nonnegative))
 257                 return log(x).hold();
 258         return log(abs(x));
 259 }
 260
 261 static ex log_imag_part(const ex & x)
 262 {
 263         if (x.info(info_flags::nonnegative))
 264                 return 0;
 265         return atan2(GiNaC::imag_part(x), GiNaC::real_part(x));
 266 }
 267
 268 static ex log_conjugate(const ex & x)
 269 {
 270         // conjugate(log(x))==log(conjugate(x)) unless on the branch cut which
 271         // runs along the negative real axis.
 272         if (x.info(info_flags::positive)) {
 273                 return log(x);
 274         }
 275         if (is_exactly_a<numeric>(x) &&
 276             !x.imag_part().is_zero()) {
 277                 return log(x.conjugate());
 278         }
 279         return conjugate_function(log(x)).hold();
 280 }
 281
 282 REGISTER_FUNCTION(log, eval_func(log_eval).
 283                        evalf_func(log_evalf).
 284                        derivative_func(log_deriv).
 285                        series_func(log_series).
 286                        real_part_func(log_real_part).
 287                        imag_part_func(log_imag_part).
 288                        conjugate_func(log_conjugate).
 289                        latex_name("\\ln"));
 290
 291 //////////
 292 // sine (trigonometric function)
 293 //////////
 294
 295 static ex sin_evalf(const ex & x)
 296 {
 297         if (is_exactly_a<numeric>(x))
 298                 return sin(ex_to<numeric>(x));
 299
 300         return sin(x).hold();
 301 }
 302
 303 static ex sin_eval(const ex & x)
 304 {
 305         // sin(n/d*Pi) -> { all known non-nested radicals }
 306         const ex SixtyExOverPi = _ex60*x/Pi;
 307         ex sign = _ex1;
 308         if (SixtyExOverPi.info(info_flags::integer)) {
 309                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 310                 if (z>=*_num60_p) {
 311                         // wrap to interval [0, Pi)
 312                         z -= *_num60_p;
 313                         sign = _ex_1;
 314                 }
 315                 if (z>*_num30_p) {
 316                         // wrap to interval [0, Pi/2)
 317                         z = *_num60_p-z;
 318                 }
 319                 if (z.is_equal(*_num0_p))  // sin(0)       -> 0
 320                         return _ex0;
 321                 if (z.is_equal(*_num5_p))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
 322                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 323                 if (z.is_equal(*_num6_p))  // sin(Pi/10)   -> sqrt(5)/4-1/4
 324                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 325                 if (z.is_equal(*_num10_p)) // sin(Pi/6)    -> 1/2
 326                         return sign*_ex1_2;
 327                 if (z.is_equal(*_num15_p)) // sin(Pi/4)    -> sqrt(2)/2
 328                         return sign*_ex1_2*sqrt(_ex2);
 329                 if (z.is_equal(*_num18_p)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
 330                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 331                 if (z.is_equal(*_num20_p)) // sin(Pi/3)    -> sqrt(3)/2
 332                         return sign*_ex1_2*sqrt(_ex3);
 333                 if (z.is_equal(*_num25_p)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
 334                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 335                 if (z.is_equal(*_num30_p)) // sin(Pi/2)    -> 1
 336                         return sign;
 337         }
 338
 339         if (is_exactly_a<function>(x)) {
 340                 const ex &t = x.op(0);
 341
 342                 // sin(asin(x)) -> x
 343                 if (is_ex_the_function(x, asin))
 344                         return t;
 345
 346                 // sin(acos(x)) -> sqrt(1-x^2)
 347                 if (is_ex_the_function(x, acos))
 348                         return sqrt(_ex1-power(t,_ex2));
 349
 350                 // sin(atan(x)) -> x/sqrt(1+x^2)
 351                 if (is_ex_the_function(x, atan))
 352                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 353         }
 354
 355         // sin(float) -> float
 356         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 357                 return sin(ex_to<numeric>(x));
 358
 359         // sin() is odd
 360         if (x.info(info_flags::negative))
 361                 return -sin(-x);
 362
 363         return sin(x).hold();
 364 }
 365
 366 static ex sin_deriv(const ex & x, unsigned deriv_param)
 367 {
 368         GINAC_ASSERT(deriv_param==0);
 369
 370         // d/dx sin(x) -> cos(x)
 371         return cos(x);
 372 }
 373
 374 static ex sin_real_part(const ex & x)
 375 {
 376         return cosh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 377 }
 378
 379 static ex sin_imag_part(const ex & x)
 380 {
 381         return sinh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 382 }
 383
 384 static ex sin_conjugate(const ex & x)
 385 {
 386         // conjugate(sin(x))==sin(conjugate(x))
 387         return sin(x.conjugate());
 388 }
 389
 390 REGISTER_FUNCTION(sin, eval_func(sin_eval).
 391                        evalf_func(sin_evalf).
 392                        derivative_func(sin_deriv).
 393                        real_part_func(sin_real_part).
 394                        imag_part_func(sin_imag_part).
 395                        conjugate_func(sin_conjugate).
 396                        latex_name("\\sin"));
 397
 398 //////////
 399 // cosine (trigonometric function)
 400 //////////
 401
 402 static ex cos_evalf(const ex & x)
 403 {
 404         if (is_exactly_a<numeric>(x))
 405                 return cos(ex_to<numeric>(x));
 406
 407         return cos(x).hold();
 408 }
 409
 410 static ex cos_eval(const ex & x)
 411 {
 412         // cos(n/d*Pi) -> { all known non-nested radicals }
 413         const ex SixtyExOverPi = _ex60*x/Pi;
 414         ex sign = _ex1;
 415         if (SixtyExOverPi.info(info_flags::integer)) {
 416                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 417                 if (z>=*_num60_p) {
 418                         // wrap to interval [0, Pi)
 419                         z = *_num120_p-z;
 420                 }
 421                 if (z>=*_num30_p) {
 422                         // wrap to interval [0, Pi/2)
 423                         z = *_num60_p-z;
 424                         sign = _ex_1;
 425                 }
 426                 if (z.is_equal(*_num0_p))  // cos(0)       -> 1
 427                         return sign;
 428                 if (z.is_equal(*_num5_p))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 429                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 430                 if (z.is_equal(*_num10_p)) // cos(Pi/6)    -> sqrt(3)/2
 431                         return sign*_ex1_2*sqrt(_ex3);
 432                 if (z.is_equal(*_num12_p)) // cos(Pi/5)    -> sqrt(5)/4+1/4
 433                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 434                 if (z.is_equal(*_num15_p)) // cos(Pi/4)    -> sqrt(2)/2
 435                         return sign*_ex1_2*sqrt(_ex2);
 436                 if (z.is_equal(*_num20_p)) // cos(Pi/3)    -> 1/2
 437                         return sign*_ex1_2;
 438                 if (z.is_equal(*_num24_p)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 439                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 440                 if (z.is_equal(*_num25_p)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 441                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 442                 if (z.is_equal(*_num30_p)) // cos(Pi/2)    -> 0
 443                         return _ex0;
 444         }
 445
 446         if (is_exactly_a<function>(x)) {
 447                 const ex &t = x.op(0);
 448
 449                 // cos(acos(x)) -> x
 450                 if (is_ex_the_function(x, acos))
 451                         return t;
 452
 453                 // cos(asin(x)) -> sqrt(1-x^2)
 454                 if (is_ex_the_function(x, asin))
 455                         return sqrt(_ex1-power(t,_ex2));
 456
 457                 // cos(atan(x)) -> 1/sqrt(1+x^2)
 458                 if (is_ex_the_function(x, atan))
 459                         return power(_ex1+power(t,_ex2),_ex_1_2);
 460         }
 461
 462         // cos(float) -> float
 463         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 464                 return cos(ex_to<numeric>(x));
 465
 466         // cos() is even
 467         if (x.info(info_flags::negative))
 468                 return cos(-x);
 469
 470         return cos(x).hold();
 471 }
 472
 473 static ex cos_deriv(const ex & x, unsigned deriv_param)
 474 {
 475         GINAC_ASSERT(deriv_param==0);
 476
 477         // d/dx cos(x) -> -sin(x)
 478         return -sin(x);
 479 }
 480
 481 static ex cos_real_part(const ex & x)
 482 {
 483         return cosh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 484 }
 485
 486 static ex cos_imag_part(const ex & x)
 487 {
 488         return -sinh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 489 }
 490
 491 static ex cos_conjugate(const ex & x)
 492 {
 493         // conjugate(cos(x))==cos(conjugate(x))
 494         return cos(x.conjugate());
 495 }
 496
 497 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 498                        evalf_func(cos_evalf).
 499                        derivative_func(cos_deriv).
 500                        real_part_func(cos_real_part).
 501                        imag_part_func(cos_imag_part).
 502                        conjugate_func(cos_conjugate).
 503                        latex_name("\\cos"));
 504
 505 //////////
 506 // tangent (trigonometric function)
 507 //////////
 508
 509 static ex tan_evalf(const ex & x)
 510 {
 511         if (is_exactly_a<numeric>(x))
 512                 return tan(ex_to<numeric>(x));
 513
 514         return tan(x).hold();
 515 }
 516
 517 static ex tan_eval(const ex & x)
 518 {
 519         // tan(n/d*Pi) -> { all known non-nested radicals }
 520         const ex SixtyExOverPi = _ex60*x/Pi;
 521         ex sign = _ex1;
 522         if (SixtyExOverPi.info(info_flags::integer)) {
 523                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num60_p);
 524                 if (z>=*_num60_p) {
 525                         // wrap to interval [0, Pi)
 526                         z -= *_num60_p;
 527                 }
 528                 if (z>=*_num30_p) {
 529                         // wrap to interval [0, Pi/2)
 530                         z = *_num60_p-z;
 531                         sign = _ex_1;
 532                 }
 533                 if (z.is_equal(*_num0_p))  // tan(0)       -> 0
 534                         return _ex0;
 535                 if (z.is_equal(*_num5_p))  // tan(Pi/12)   -> 2-sqrt(3)
 536                         return sign*(_ex2-sqrt(_ex3));
 537                 if (z.is_equal(*_num10_p)) // tan(Pi/6)    -> sqrt(3)/3
 538                         return sign*_ex1_3*sqrt(_ex3);
 539                 if (z.is_equal(*_num15_p)) // tan(Pi/4)    -> 1
 540                         return sign;
 541                 if (z.is_equal(*_num20_p)) // tan(Pi/3)    -> sqrt(3)
 542                         return sign*sqrt(_ex3);
 543                 if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3)
 544                         return sign*(sqrt(_ex3)+_ex2);
 545                 if (z.is_equal(*_num30_p)) // tan(Pi/2)    -> infinity
 546                         throw (pole_error("tan_eval(): simple pole",1));
 547         }
 548
 549         if (is_exactly_a<function>(x)) {
 550                 const ex &t = x.op(0);
 551
 552                 // tan(atan(x)) -> x
 553                 if (is_ex_the_function(x, atan))
 554                         return t;
 555
 556                 // tan(asin(x)) -> x/sqrt(1+x^2)
 557                 if (is_ex_the_function(x, asin))
 558                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 559
 560                 // tan(acos(x)) -> sqrt(1-x^2)/x
 561                 if (is_ex_the_function(x, acos))
 562                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
 563         }
 564
 565         // tan(float) -> float
 566         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 567                 return tan(ex_to<numeric>(x));
 568         }
 569
 570         // tan() is odd
 571         if (x.info(info_flags::negative))
 572                 return -tan(-x);
 573
 574         return tan(x).hold();
 575 }
 576
 577 static ex tan_deriv(const ex & x, unsigned deriv_param)
 578 {
 579         GINAC_ASSERT(deriv_param==0);
 580
 581         // d/dx tan(x) -> 1+tan(x)^2;
 582         return (_ex1+power(tan(x),_ex2));
 583 }
 584
 585 static ex tan_real_part(const ex & x)
 586 {
 587         ex a = GiNaC::real_part(x);
 588         ex b = GiNaC::imag_part(x);
 589         return tan(a)/(1+power(tan(a),2)*power(tan(b),2));
 590 }
 591
 592 static ex tan_imag_part(const ex & x)
 593 {
 594         ex a = GiNaC::real_part(x);
 595         ex b = GiNaC::imag_part(x);
 596         return tanh(b)/(1+power(tan(a),2)*power(tan(b),2));
 597 }
 598
 599 static ex tan_series(const ex &x,
 600                      const relational &rel,
 601                      int order,
 602                      unsigned options)
 603 {
 604         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 605         // method:
 606         // Taylor series where there is no pole falls back to tan_deriv.
 607         // On a pole simply expand sin(x)/cos(x).
 608         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 609         if (!(2*x_pt/Pi).info(info_flags::odd))
 610                 throw do_taylor();  // caught by function::series()
 611         // if we got here we have to care for a simple pole
 612         return (sin(x)/cos(x)).series(rel, order, options);
 613 }
 614
 615 static ex tan_conjugate(const ex & x)
 616 {
 617         // conjugate(tan(x))==tan(conjugate(x))
 618         return tan(x.conjugate());
 619 }
 620
 621 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 622                        evalf_func(tan_evalf).
 623                        derivative_func(tan_deriv).
 624                        series_func(tan_series).
 625                        real_part_func(tan_real_part).
 626                        imag_part_func(tan_imag_part).
 627                        conjugate_func(tan_conjugate).
 628                        latex_name("\\tan"));
 629
 630 //////////
 631 // inverse sine (arc sine)
 632 //////////
 633
 634 static ex asin_evalf(const ex & x)
 635 {
 636         if (is_exactly_a<numeric>(x))
 637                 return asin(ex_to<numeric>(x));
 638
 639         return asin(x).hold();
 640 }
 641
 642 static ex asin_eval(const ex & x)
 643 {
 644         if (x.info(info_flags::numeric)) {
 645
 646                 // asin(0) -> 0
 647                 if (x.is_zero())
 648                         return x;
 649
 650                 // asin(1/2) -> Pi/6
 651                 if (x.is_equal(_ex1_2))
 652                         return numeric(1,6)*Pi;
 653
 654                 // asin(1) -> Pi/2
 655                 if (x.is_equal(_ex1))
 656                         return _ex1_2*Pi;
 657
 658                 // asin(-1/2) -> -Pi/6
 659                 if (x.is_equal(_ex_1_2))
 660                         return numeric(-1,6)*Pi;
 661
 662                 // asin(-1) -> -Pi/2
 663                 if (x.is_equal(_ex_1))
 664                         return _ex_1_2*Pi;
 665
 666                 // asin(float) -> float
 667                 if (!x.info(info_flags::crational))
 668                         return asin(ex_to<numeric>(x));
 669
 670                 // asin() is odd
 671                 if (x.info(info_flags::negative))
 672                         return -asin(-x);
 673         }
 674
 675         return asin(x).hold();
 676 }
 677
 678 static ex asin_deriv(const ex & x, unsigned deriv_param)
 679 {
 680         GINAC_ASSERT(deriv_param==0);
 681
 682         // d/dx asin(x) -> 1/sqrt(1-x^2)
 683         return power(1-power(x,_ex2),_ex_1_2);
 684 }
 685
 686 static ex asin_conjugate(const ex & x)
 687 {
 688         // conjugate(asin(x))==asin(conjugate(x)) unless on the branch cuts which
 689         // run along the real axis outside the interval [-1, +1].
 690         if (is_exactly_a<numeric>(x) &&
 691             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
 692                 return asin(x.conjugate());
 693         }
 694         return conjugate_function(asin(x)).hold();
 695 }
 696
 697 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 698                         evalf_func(asin_evalf).
 699                         derivative_func(asin_deriv).
 700                         conjugate_func(asin_conjugate).
 701                         latex_name("\\arcsin"));
 702
 703 //////////
 704 // inverse cosine (arc cosine)
 705 //////////
 706
 707 static ex acos_evalf(const ex & x)
 708 {
 709         if (is_exactly_a<numeric>(x))
 710                 return acos(ex_to<numeric>(x));
 711
 712         return acos(x).hold();
 713 }
 714
 715 static ex acos_eval(const ex & x)
 716 {
 717         if (x.info(info_flags::numeric)) {
 718
 719                 // acos(1) -> 0
 720                 if (x.is_equal(_ex1))
 721                         return _ex0;
 722
 723                 // acos(1/2) -> Pi/3
 724                 if (x.is_equal(_ex1_2))
 725                         return _ex1_3*Pi;
 726
 727                 // acos(0) -> Pi/2
 728                 if (x.is_zero())
 729                         return _ex1_2*Pi;
 730
 731                 // acos(-1/2) -> 2/3*Pi
 732                 if (x.is_equal(_ex_1_2))
 733                         return numeric(2,3)*Pi;
 734
 735                 // acos(-1) -> Pi
 736                 if (x.is_equal(_ex_1))
 737                         return Pi;
 738
 739                 // acos(float) -> float
 740                 if (!x.info(info_flags::crational))
 741                         return acos(ex_to<numeric>(x));
 742
 743                 // acos(-x) -> Pi-acos(x)
 744                 if (x.info(info_flags::negative))
 745                         return Pi-acos(-x);
 746         }
 747
 748         return acos(x).hold();
 749 }
 750
 751 static ex acos_deriv(const ex & x, unsigned deriv_param)
 752 {
 753         GINAC_ASSERT(deriv_param==0);
 754
 755         // d/dx acos(x) -> -1/sqrt(1-x^2)
 756         return -power(1-power(x,_ex2),_ex_1_2);
 757 }
 758
 759 static ex acos_conjugate(const ex & x)
 760 {
 761         // conjugate(acos(x))==acos(conjugate(x)) unless on the branch cuts which
 762         // run along the real axis outside the interval [-1, +1].
 763         if (is_exactly_a<numeric>(x) &&
 764             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
 765                 return acos(x.conjugate());
 766         }
 767         return conjugate_function(acos(x)).hold();
 768 }
 769
 770 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 771                         evalf_func(acos_evalf).
 772                         derivative_func(acos_deriv).
 773                         conjugate_func(acos_conjugate).
 774                         latex_name("\\arccos"));
 775
 776 //////////
 777 // inverse tangent (arc tangent)
 778 //////////
 779
 780 static ex atan_evalf(const ex & x)
 781 {
 782         if (is_exactly_a<numeric>(x))
 783                 return atan(ex_to<numeric>(x));
 784
 785         return atan(x).hold();
 786 }
 787
 788 static ex atan_eval(const ex & x)
 789 {
 790         if (x.info(info_flags::numeric)) {
 791
 792                 // atan(0) -> 0
 793                 if (x.is_zero())
 794                         return _ex0;
 795
 796                 // atan(1) -> Pi/4
 797                 if (x.is_equal(_ex1))
 798                         return _ex1_4*Pi;
 799
 800                 // atan(-1) -> -Pi/4
 801                 if (x.is_equal(_ex_1))
 802                         return _ex_1_4*Pi;
 803
 804                 if (x.is_equal(I) || x.is_equal(-I))
 805                         throw (pole_error("atan_eval(): logarithmic pole",0));
 806
 807                 // atan(float) -> float
 808                 if (!x.info(info_flags::crational))
 809                         return atan(ex_to<numeric>(x));
 810
 811                 // atan() is odd
 812                 if (x.info(info_flags::negative))
 813                         return -atan(-x);
 814         }
 815
 816         return atan(x).hold();
 817 }
 818
 819 static ex atan_deriv(const ex & x, unsigned deriv_param)
 820 {
 821         GINAC_ASSERT(deriv_param==0);
 822
 823         // d/dx atan(x) -> 1/(1+x^2)
 824         return power(_ex1+power(x,_ex2), _ex_1);
 825 }
 826
 827 static ex atan_series(const ex &arg,
 828                       const relational &rel,
 829                       int order,
 830                       unsigned options)
 831 {
 832         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 833         // method:
 834         // Taylor series where there is no pole or cut falls back to atan_deriv.
 835         // There are two branch cuts, one runnig from I up the imaginary axis and
 836         // one running from -I down the imaginary axis.  The points I and -I are
 837         // poles.
 838         // On the branch cuts and the poles series expand
 839         //     (log(1+I*x)-log(1-I*x))/(2*I)
 840         // instead.
 841         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 842         if (!(I*arg_pt).info(info_flags::real))
 843                 throw do_taylor();     // Re(x) != 0
 844         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
 845                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
 846         // care for the poles, using the defining formula for atan()...
 847         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
 848                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
 849         if (!(options & series_options::suppress_branchcut)) {
 850                 // method:
 851                 // This is the branch cut: assemble the primitive series manually and
 852                 // then add the corresponding complex step function.
 853                 const symbol &s = ex_to<symbol>(rel.lhs());
 854                 const ex &point = rel.rhs();
 855                 const symbol foo;
 856                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 857                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
 858                 if ((I*arg_pt)<_ex0)
 859                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
 860                 else
 861                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
 862                 epvector seq;
 863                 seq.push_back(expair(Order0correction, _ex0));
 864                 seq.push_back(expair(Order(_ex1), order));
 865                 return series(replarg - pseries(rel, seq), rel, order);
 866         }
 867         throw do_taylor();
 868 }
 869
 870 static ex atan_conjugate(const ex & x)
 871 {
 872         // conjugate(atan(x))==atan(conjugate(x)) unless on the branch cuts which
 873         // run along the imaginary axis outside the interval [-I, +I].
 874         if (x.info(info_flags::real))
 875                 return atan(x);
 876         if (is_exactly_a<numeric>(x)) {
 877                 const numeric x_re = ex_to<numeric>(x.real_part());
 878                 const numeric x_im = ex_to<numeric>(x.imag_part());
 879                 if (!x_re.is_zero() ||
 880                     (x_im > *_num_1_p && x_im < *_num1_p))
 881                         return atan(x.conjugate());
 882         }
 883         return conjugate_function(atan(x)).hold();
 884 }
 885
 886 REGISTER_FUNCTION(atan, eval_func(atan_eval).
 887                         evalf_func(atan_evalf).
 888                         derivative_func(atan_deriv).
 889                         series_func(atan_series).
 890                         conjugate_func(atan_conjugate).
 891                         latex_name("\\arctan"));
 892
 893 //////////
 894 // inverse tangent (atan2(y,x))
 895 //////////
 896
 897 static ex atan2_evalf(const ex &y, const ex &x)
 898 {
 899         if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
 900                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 901
 902         return atan2(y, x).hold();
 903 }
 904
 905 static ex atan2_eval(const ex & y, const ex & x)
 906 {
 907         if (y.is_zero()) {
 908
 909                 // atan2(0, 0) -> 0
 910                 if (x.is_zero())
 911                         return _ex0;
 912
 913                 // atan2(0, x), x real and positive -> 0
 914                 if (x.info(info_flags::positive))
 915                         return _ex0;
 916
 917                 // atan2(0, x), x real and negative -> Pi
 918                 if (x.info(info_flags::negative))
 919                         return Pi;
 920         }
 921
 922         if (x.is_zero()) {
 923
 924                 // atan2(y, 0), y real and positive -> Pi/2
 925                 if (y.info(info_flags::positive))
 926                         return _ex1_2*Pi;
 927
 928                 // atan2(y, 0), y real and negative -> -Pi/2
 929                 if (y.info(info_flags::negative))
 930                         return _ex_1_2*Pi;
 931         }
 932
 933         if (y.is_equal(x)) {
 934
 935                 // atan2(y, y), y real and positive -> Pi/4
 936                 if (y.info(info_flags::positive))
 937                         return _ex1_4*Pi;
 938
 939                 // atan2(y, y), y real and negative -> -3/4*Pi
 940                 if (y.info(info_flags::negative))
 941                         return numeric(-3, 4)*Pi;
 942         }
 943
 944         if (y.is_equal(-x)) {
 945
 946                 // atan2(y, -y), y real and positive -> 3*Pi/4
 947                 if (y.info(info_flags::positive))
 948                         return numeric(3, 4)*Pi;
 949
 950                 // atan2(y, -y), y real and negative -> -Pi/4
 951                 if (y.info(info_flags::negative))
 952                         return _ex_1_4*Pi;
 953         }
 954
 955         // atan2(float, float) -> float
 956         if (is_a<numeric>(y) && !y.info(info_flags::crational) &&
 957             is_a<numeric>(x) && !x.info(info_flags::crational))
 958                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 959
 960         // atan2(real, real) -> atan(y/x) +/- Pi
 961         if (y.info(info_flags::real) && x.info(info_flags::real)) {
 962                 if (x.info(info_flags::positive))
 963                         return atan(y/x);
 964
 965                 if (x.info(info_flags::negative)) {
 966                         if (y.info(info_flags::positive))
 967                                 return atan(y/x)+Pi;
 968                         if (y.info(info_flags::negative))
 969                                 return atan(y/x)-Pi;
 970                 }
 971         }
 972
 973         return atan2(y, x).hold();
 974 }
 975
 976 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
 977 {
 978         GINAC_ASSERT(deriv_param<2);
 979
 980         if (deriv_param==0) {
 981                 // d/dy atan2(y,x)
 982                 return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
 983         }
 984         // d/dx atan2(y,x)
 985         return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
 986 }
 987
 988 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
 989                          evalf_func(atan2_evalf).
 990                          derivative_func(atan2_deriv));
 991
 992 //////////
 993 // hyperbolic sine (trigonometric function)
 994 //////////
 995
 996 static ex sinh_evalf(const ex & x)
 997 {
 998         if (is_exactly_a<numeric>(x))
 999                 return sinh(ex_to<numeric>(x));
1000
1001         return sinh(x).hold();
1002 }
1003
1004 static ex sinh_eval(const ex & x)
1005 {
1006         if (x.info(info_flags::numeric)) {
1007
1008                 // sinh(0) -> 0
1009                 if (x.is_zero())
1010                         return _ex0;
1011
1012                 // sinh(float) -> float
1013                 if (!x.info(info_flags::crational))
1014                         return sinh(ex_to<numeric>(x));
1015
1016                 // sinh() is odd
1017                 if (x.info(info_flags::negative))
1018                         return -sinh(-x);
1019         }
1020
1021         if ((x/Pi).info(info_flags::numeric) &&
1022                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
1023                 return I*sin(x/I);
1024
1025         if (is_exactly_a<function>(x)) {
1026                 const ex &t = x.op(0);
1027
1028                 // sinh(asinh(x)) -> x
1029                 if (is_ex_the_function(x, asinh))
1030                         return t;
1031
1032                 // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
1033                 if (is_ex_the_function(x, acosh))
1034                         return sqrt(t-_ex1)*sqrt(t+_ex1);
1035
1036                 // sinh(atanh(x)) -> x/sqrt(1-x^2)
1037                 if (is_ex_the_function(x, atanh))
1038                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
1039         }
1040
1041         return sinh(x).hold();
1042 }
1043
1044 static ex sinh_deriv(const ex & x, unsigned deriv_param)
1045 {
1046         GINAC_ASSERT(deriv_param==0);
1047
1048         // d/dx sinh(x) -> cosh(x)
1049         return cosh(x);
1050 }
1051
1052 static ex sinh_real_part(const ex & x)
1053 {
1054         return sinh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
1055 }
1056
1057 static ex sinh_imag_part(const ex & x)
1058 {
1059         return cosh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
1060 }
1061
1062 static ex sinh_conjugate(const ex & x)
1063 {
1064         // conjugate(sinh(x))==sinh(conjugate(x))
1065         return sinh(x.conjugate());
1066 }
1067
1068 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
1069                         evalf_func(sinh_evalf).
1070                         derivative_func(sinh_deriv).
1071                         real_part_func(sinh_real_part).
1072                         imag_part_func(sinh_imag_part).
1073                         conjugate_func(sinh_conjugate).
1074                         latex_name("\\sinh"));
1075
1076 //////////
1077 // hyperbolic cosine (trigonometric function)
1078 //////////
1079
1080 static ex cosh_evalf(const ex & x)
1081 {
1082         if (is_exactly_a<numeric>(x))
1083                 return cosh(ex_to<numeric>(x));
1084
1085         return cosh(x).hold();
1086 }
1087
1088 static ex cosh_eval(const ex & x)
1089 {
1090         if (x.info(info_flags::numeric)) {
1091
1092                 // cosh(0) -> 1
1093                 if (x.is_zero())
1094                         return _ex1;
1095
1096                 // cosh(float) -> float
1097                 if (!x.info(info_flags::crational))
1098                         return cosh(ex_to<numeric>(x));
1099
1100                 // cosh() is even
1101                 if (x.info(info_flags::negative))
1102                         return cosh(-x);
1103         }
1104
1105         if ((x/Pi).info(info_flags::numeric) &&
1106                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
1107                 return cos(x/I);
1108
1109         if (is_exactly_a<function>(x)) {
1110                 const ex &t = x.op(0);
1111
1112                 // cosh(acosh(x)) -> x
1113                 if (is_ex_the_function(x, acosh))
1114                         return t;
1115
1116                 // cosh(asinh(x)) -> sqrt(1+x^2)
1117                 if (is_ex_the_function(x, asinh))
1118                         return sqrt(_ex1+power(t,_ex2));
1119
1120                 // cosh(atanh(x)) -> 1/sqrt(1-x^2)
1121                 if (is_ex_the_function(x, atanh))
1122                         return power(_ex1-power(t,_ex2),_ex_1_2);
1123         }
1124
1125         return cosh(x).hold();
1126 }
1127
1128 static ex cosh_deriv(const ex & x, unsigned deriv_param)
1129 {
1130         GINAC_ASSERT(deriv_param==0);
1131
1132         // d/dx cosh(x) -> sinh(x)
1133         return sinh(x);
1134 }
1135
1136 static ex cosh_real_part(const ex & x)
1137 {
1138         return cosh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
1139 }
1140
1141 static ex cosh_imag_part(const ex & x)
1142 {
1143         return sinh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
1144 }
1145
1146 static ex cosh_conjugate(const ex & x)
1147 {
1148         // conjugate(cosh(x))==cosh(conjugate(x))
1149         return cosh(x.conjugate());
1150 }
1151
1152 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
1153                         evalf_func(cosh_evalf).
1154                         derivative_func(cosh_deriv).
1155                         real_part_func(cosh_real_part).
1156                         imag_part_func(cosh_imag_part).
1157                         conjugate_func(cosh_conjugate).
1158                         latex_name("\\cosh"));
1159
1160 //////////
1161 // hyperbolic tangent (trigonometric function)
1162 //////////
1163
1164 static ex tanh_evalf(const ex & x)
1165 {
1166         if (is_exactly_a<numeric>(x))
1167                 return tanh(ex_to<numeric>(x));
1168
1169         return tanh(x).hold();
1170 }
1171
1172 static ex tanh_eval(const ex & x)
1173 {
1174         if (x.info(info_flags::numeric)) {
1175
1176                 // tanh(0) -> 0
1177                 if (x.is_zero())
1178                         return _ex0;
1179
1180                 // tanh(float) -> float
1181                 if (!x.info(info_flags::crational))
1182                         return tanh(ex_to<numeric>(x));
1183
1184                 // tanh() is odd
1185                 if (x.info(info_flags::negative))
1186                         return -tanh(-x);
1187         }
1188
1189         if ((x/Pi).info(info_flags::numeric) &&
1190                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
1191                 return I*tan(x/I);
1192
1193         if (is_exactly_a<function>(x)) {
1194                 const ex &t = x.op(0);
1195
1196                 // tanh(atanh(x)) -> x
1197                 if (is_ex_the_function(x, atanh))
1198                         return t;
1199
1200                 // tanh(asinh(x)) -> x/sqrt(1+x^2)
1201                 if (is_ex_the_function(x, asinh))
1202                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
1203
1204                 // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
1205                 if (is_ex_the_function(x, acosh))
1206                         return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
1207         }
1208
1209         return tanh(x).hold();
1210 }
1211
1212 static ex tanh_deriv(const ex & x, unsigned deriv_param)
1213 {
1214         GINAC_ASSERT(deriv_param==0);
1215
1216         // d/dx tanh(x) -> 1-tanh(x)^2
1217         return _ex1-power(tanh(x),_ex2);
1218 }
1219
1220 static ex tanh_series(const ex &x,
1221                       const relational &rel,
1222                       int order,
1223                       unsigned options)
1224 {
1225         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1226         // method:
1227         // Taylor series where there is no pole falls back to tanh_deriv.
1228         // On a pole simply expand sinh(x)/cosh(x).
1229         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1230         if (!(2*I*x_pt/Pi).info(info_flags::odd))
1231                 throw do_taylor();  // caught by function::series()
1232         // if we got here we have to care for a simple pole
1233         return (sinh(x)/cosh(x)).series(rel, order, options);
1234 }
1235
1236 static ex tanh_real_part(const ex & x)
1237 {
1238         ex a = GiNaC::real_part(x);
1239         ex b = GiNaC::imag_part(x);
1240         return tanh(a)/(1+power(tanh(a),2)*power(tan(b),2));
1241 }
1242
1243 static ex tanh_imag_part(const ex & x)
1244 {
1245         ex a = GiNaC::real_part(x);
1246         ex b = GiNaC::imag_part(x);
1247         return tan(b)/(1+power(tanh(a),2)*power(tan(b),2));
1248 }
1249
1250 static ex tanh_conjugate(const ex & x)
1251 {
1252         // conjugate(tanh(x))==tanh(conjugate(x))
1253         return tanh(x.conjugate());
1254 }
1255
1256 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
1257                         evalf_func(tanh_evalf).
1258                         derivative_func(tanh_deriv).
1259                         series_func(tanh_series).
1260                         real_part_func(tanh_real_part).
1261                         imag_part_func(tanh_imag_part).
1262                         conjugate_func(tanh_conjugate).
1263                         latex_name("\\tanh"));
1264
1265 //////////
1266 // inverse hyperbolic sine (trigonometric function)
1267 //////////
1268
1269 static ex asinh_evalf(const ex & x)
1270 {
1271         if (is_exactly_a<numeric>(x))
1272                 return asinh(ex_to<numeric>(x));
1273
1274         return asinh(x).hold();
1275 }
1276
1277 static ex asinh_eval(const ex & x)
1278 {
1279         if (x.info(info_flags::numeric)) {
1280
1281                 // asinh(0) -> 0
1282                 if (x.is_zero())
1283                         return _ex0;
1284
1285                 // asinh(float) -> float
1286                 if (!x.info(info_flags::crational))
1287                         return asinh(ex_to<numeric>(x));
1288
1289                 // asinh() is odd
1290                 if (x.info(info_flags::negative))
1291                         return -asinh(-x);
1292         }
1293
1294         return asinh(x).hold();
1295 }
1296
1297 static ex asinh_deriv(const ex & x, unsigned deriv_param)
1298 {
1299         GINAC_ASSERT(deriv_param==0);
1300
1301         // d/dx asinh(x) -> 1/sqrt(1+x^2)
1302         return power(_ex1+power(x,_ex2),_ex_1_2);
1303 }
1304
1305 static ex asinh_conjugate(const ex & x)
1306 {
1307         // conjugate(asinh(x))==asinh(conjugate(x)) unless on the branch cuts which
1308         // run along the imaginary axis outside the interval [-I, +I].
1309         if (x.info(info_flags::real))
1310                 return asinh(x);
1311         if (is_exactly_a<numeric>(x)) {
1312                 const numeric x_re = ex_to<numeric>(x.real_part());
1313                 const numeric x_im = ex_to<numeric>(x.imag_part());
1314                 if (!x_re.is_zero() ||
1315                     (x_im > *_num_1_p && x_im < *_num1_p))
1316                         return asinh(x.conjugate());
1317         }
1318         return conjugate_function(asinh(x)).hold();
1319 }
1320
1321 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
1322                          evalf_func(asinh_evalf).
1323                          derivative_func(asinh_deriv).
1324                          conjugate_func(asinh_conjugate));
1325
1326 //////////
1327 // inverse hyperbolic cosine (trigonometric function)
1328 //////////
1329
1330 static ex acosh_evalf(const ex & x)
1331 {
1332         if (is_exactly_a<numeric>(x))
1333                 return acosh(ex_to<numeric>(x));
1334
1335         return acosh(x).hold();
1336 }
1337
1338 static ex acosh_eval(const ex & x)
1339 {
1340         if (x.info(info_flags::numeric)) {
1341
1342                 // acosh(0) -> Pi*I/2
1343                 if (x.is_zero())
1344                         return Pi*I*numeric(1,2);
1345
1346                 // acosh(1) -> 0
1347                 if (x.is_equal(_ex1))
1348                         return _ex0;
1349
1350                 // acosh(-1) -> Pi*I
1351                 if (x.is_equal(_ex_1))
1352                         return Pi*I;
1353
1354                 // acosh(float) -> float
1355                 if (!x.info(info_flags::crational))
1356                         return acosh(ex_to<numeric>(x));
1357
1358                 // acosh(-x) -> Pi*I-acosh(x)
1359                 if (x.info(info_flags::negative))
1360                         return Pi*I-acosh(-x);
1361         }
1362
1363         return acosh(x).hold();
1364 }
1365
1366 static ex acosh_deriv(const ex & x, unsigned deriv_param)
1367 {
1368         GINAC_ASSERT(deriv_param==0);
1369
1370         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
1371         return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
1372 }
1373
1374 static ex acosh_conjugate(const ex & x)
1375 {
1376         // conjugate(acosh(x))==acosh(conjugate(x)) unless on the branch cut
1377         // which runs along the real axis from +1 to -inf.
1378         if (is_exactly_a<numeric>(x) &&
1379             (!x.imag_part().is_zero() || x > *_num1_p)) {
1380                 return acosh(x.conjugate());
1381         }
1382         return conjugate_function(acosh(x)).hold();
1383 }
1384
1385 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
1386                          evalf_func(acosh_evalf).
1387                          derivative_func(acosh_deriv).
1388                          conjugate_func(acosh_conjugate));
1389
1390 //////////
1391 // inverse hyperbolic tangent (trigonometric function)
1392 //////////
1393
1394 static ex atanh_evalf(const ex & x)
1395 {
1396         if (is_exactly_a<numeric>(x))
1397                 return atanh(ex_to<numeric>(x));
1398
1399         return atanh(x).hold();
1400 }
1401
1402 static ex atanh_eval(const ex & x)
1403 {
1404         if (x.info(info_flags::numeric)) {
1405
1406                 // atanh(0) -> 0
1407                 if (x.is_zero())
1408                         return _ex0;
1409
1410                 // atanh({+|-}1) -> throw
1411                 if (x.is_equal(_ex1) || x.is_equal(_ex_1))
1412                         throw (pole_error("atanh_eval(): logarithmic pole",0));
1413
1414                 // atanh(float) -> float
1415                 if (!x.info(info_flags::crational))
1416                         return atanh(ex_to<numeric>(x));
1417
1418                 // atanh() is odd
1419                 if (x.info(info_flags::negative))
1420                         return -atanh(-x);
1421         }
1422
1423         return atanh(x).hold();
1424 }
1425
1426 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1427 {
1428         GINAC_ASSERT(deriv_param==0);
1429
1430         // d/dx atanh(x) -> 1/(1-x^2)
1431         return power(_ex1-power(x,_ex2),_ex_1);
1432 }
1433
1434 static ex atanh_series(const ex &arg,
1435                        const relational &rel,
1436                        int order,
1437                        unsigned options)
1438 {
1439         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1440         // method:
1441         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1442         // There are two branch cuts, one runnig from 1 up the real axis and one
1443         // one running from -1 down the real axis.  The points 1 and -1 are poles
1444         // On the branch cuts and the poles series expand
1445         //     (log(1+x)-log(1-x))/2
1446         // instead.
1447         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
1448         if (!(arg_pt).info(info_flags::real))
1449                 throw do_taylor();     // Im(x) != 0
1450         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1451                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1452         // care for the poles, using the defining formula for atanh()...
1453         if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1454                 return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1455         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1456         if (!(options & series_options::suppress_branchcut)) {
1457                 // method:
1458                 // This is the branch cut: assemble the primitive series manually and
1459                 // then add the corresponding complex step function.
1460                 const symbol &s = ex_to<symbol>(rel.lhs());
1461                 const ex &point = rel.rhs();
1462                 const symbol foo;
1463                 const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1464                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1465                 if (arg_pt<_ex0)
1466                         Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1467                 else
1468                         Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1469                 epvector seq;
1470                 seq.push_back(expair(Order0correction, _ex0));
1471                 seq.push_back(expair(Order(_ex1), order));
1472                 return series(replarg - pseries(rel, seq), rel, order);
1473         }
1474         throw do_taylor();
1475 }
1476
1477 static ex atanh_conjugate(const ex & x)
1478 {
1479         // conjugate(atanh(x))==atanh(conjugate(x)) unless on the branch cuts which
1480         // run along the real axis outside the interval [-1, +1].
1481         if (is_exactly_a<numeric>(x) &&
1482             (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
1483                 return atanh(x.conjugate());
1484         }
1485         return conjugate_function(atanh(x)).hold();
1486 }
1487
1488 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1489                          evalf_func(atanh_evalf).
1490                          derivative_func(atanh_deriv).
1491                          series_func(atanh_series).
1492                          conjugate_func(atanh_conjugate));
1493
1494
1495 } // namespace GiNaC