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) 1999-2005 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 <vector>
  25 #include <stdexcept>
  26
  27 #include "inifcns.h"
  28 #include "ex.h"
  29 #include "constant.h"
  30 #include "numeric.h"
  31 #include "power.h"
  32 #include "operators.h"
  33 #include "relational.h"
  34 #include "symbol.h"
  35 #include "pseries.h"
  36 #include "utils.h"
  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 REGISTER_FUNCTION(exp, eval_func(exp_eval).
  93                        evalf_func(exp_evalf).
  94                        derivative_func(exp_deriv).
  95                        latex_name("\\exp"));
  96
  97 //////////
  98 // natural logarithm
  99 //////////
 100
 101 static ex log_evalf(const ex & x)
 102 {
 103         if (is_exactly_a<numeric>(x))
 104                 return log(ex_to<numeric>(x));
 105
 106         return log(x).hold();
 107 }
 108
 109 static ex log_eval(const ex & x)
 110 {
 111         if (x.info(info_flags::numeric)) {
 112                 if (x.is_zero())         // log(0) -> infinity
 113                         throw(pole_error("log_eval(): log(0)",0));
 114                 if (x.info(info_flags::rational) && x.info(info_flags::negative))
 115                         return (log(-x)+I*Pi);
 116                 if (x.is_equal(_ex1))  // log(1) -> 0
 117                         return _ex0;
 118                 if (x.is_equal(I))       // log(I) -> Pi*I/2
 119                         return (Pi*I*_ex1_2);
 120                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
 121                         return (Pi*I*_ex_1_2);
 122
 123                 // log(float) -> float
 124                 if (!x.info(info_flags::crational))
 125                         return log(ex_to<numeric>(x));
 126         }
 127
 128         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
 129         if (is_ex_the_function(x, exp)) {
 130                 const ex &t = x.op(0);
 131                 if (t.info(info_flags::real))
 132                         return t;
 133         }
 134
 135         return log(x).hold();
 136 }
 137
 138 static ex log_deriv(const ex & x, unsigned deriv_param)
 139 {
 140         GINAC_ASSERT(deriv_param==0);
 141
 142         // d/dx log(x) -> 1/x
 143         return power(x, _ex_1);
 144 }
 145
 146 static ex log_series(const ex &arg,
 147                      const relational &rel,
 148                      int order,
 149                      unsigned options)
 150 {
 151         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 152         ex arg_pt;
 153         bool must_expand_arg = false;
 154         // maybe substitution of rel into arg fails because of a pole
 155         try {
 156                 arg_pt = arg.subs(rel, subs_options::no_pattern);
 157         } catch (pole_error) {
 158                 must_expand_arg = true;
 159         }
 160         // or we are at the branch point anyways
 161         if (arg_pt.is_zero())
 162                 must_expand_arg = true;
 163
 164         if (must_expand_arg) {
 165                 // method:
 166                 // This is the branch point: Series expand the argument first, then
 167                 // trivially factorize it to isolate that part which has constant
 168                 // leading coefficient in this fashion:
 169                 //   x^n + x^(n+1) +...+ Order(x^(n+m))  ->  x^n * (1 + x +...+ Order(x^m)).
 170                 // Return a plain n*log(x) for the x^n part and series expand the
 171                 // other part.  Add them together and reexpand again in order to have
 172                 // one unnested pseries object.  All this also works for negative n.
 173                 pseries argser;          // series expansion of log's argument
 174                 unsigned extra_ord = 0;  // extra expansion order
 175                 do {
 176                         // oops, the argument expanded to a pure Order(x^something)...
 177                         argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
 178                         ++extra_ord;
 179                 } while (!argser.is_terminating() && argser.nops()==1);
 180
 181                 const symbol &s = ex_to<symbol>(rel.lhs());
 182                 const ex &point = rel.rhs();
 183                 const int n = argser.ldegree(s);
 184                 epvector seq;
 185                 // construct what we carelessly called the n*log(x) term above
 186                 const ex coeff = argser.coeff(s, n);
 187                 // expand the log, but only if coeff is real and > 0, since otherwise
 188                 // it would make the branch cut run into the wrong direction
 189                 if (coeff.info(info_flags::positive))
 190                         seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
 191                 else
 192                         seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));
 193
 194                 if (!argser.is_terminating() || argser.nops()!=1) {
 195                         // in this case n more (or less) terms are needed
 196                         // (sadly, to generate them, we have to start from the beginning)
 197                         if (n == 0 && coeff == 1) {
 198                                 epvector epv;
 199                                 ex acc = (new pseries(rel, epv))->setflag(status_flags::dynallocated);
 200                                 epv.reserve(2);
 201                                 epv.push_back(expair(-1, _ex0));
 202                                 epv.push_back(expair(Order(_ex1), order));
 203                                 ex rest = pseries(rel, epv).add_series(argser);
 204                                 for (int i = order-1; i>0; --i) {
 205                                         epvector cterm;
 206                                         cterm.reserve(1);
 207                                         cterm.push_back(expair(i%2 ? _ex1/i : _ex_1/i, _ex0));
 208                                         acc = pseries(rel, cterm).add_series(ex_to<pseries>(acc));
 209                                         acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
 210                                 }
 211                                 return acc;
 212                         }
 213                         const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
 214                         return pseries(rel, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
 215                 } else  // it was a monomial
 216                         return pseries(rel, seq);
 217         }
 218         if (!(options & series_options::suppress_branchcut) &&
 219              arg_pt.info(info_flags::negative)) {
 220                 // method:
 221                 // This is the branch cut: assemble the primitive series manually and
 222                 // then add the corresponding complex step function.
 223                 const symbol &s = ex_to<symbol>(rel.lhs());
 224                 const ex &point = rel.rhs();
 225                 const symbol foo;
 226                 const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 227                 epvector seq;
 228                 seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
 229                 seq.push_back(expair(Order(_ex1), order));
 230                 return series(replarg - I*Pi + pseries(rel, seq), rel, order);
 231         }
 232         throw do_taylor();  // caught by function::series()
 233 }
 234
 235 REGISTER_FUNCTION(log, eval_func(log_eval).
 236                        evalf_func(log_evalf).
 237                        derivative_func(log_deriv).
 238                        series_func(log_series).
 239                        latex_name("\\ln"));
 240
 241 //////////
 242 // sine (trigonometric function)
 243 //////////
 244
 245 static ex sin_evalf(const ex & x)
 246 {
 247         if (is_exactly_a<numeric>(x))
 248                 return sin(ex_to<numeric>(x));
 249
 250         return sin(x).hold();
 251 }
 252
 253 static ex sin_eval(const ex & x)
 254 {
 255         // sin(n/d*Pi) -> { all known non-nested radicals }
 256         const ex SixtyExOverPi = _ex60*x/Pi;
 257         ex sign = _ex1;
 258         if (SixtyExOverPi.info(info_flags::integer)) {
 259                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 260                 if (z>=*_num60_p) {
 261                         // wrap to interval [0, Pi)
 262                         z -= *_num60_p;
 263                         sign = _ex_1;
 264                 }
 265                 if (z>*_num30_p) {
 266                         // wrap to interval [0, Pi/2)
 267                         z = *_num60_p-z;
 268                 }
 269                 if (z.is_equal(*_num0_p))  // sin(0)       -> 0
 270                         return _ex0;
 271                 if (z.is_equal(*_num5_p))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
 272                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 273                 if (z.is_equal(*_num6_p))  // sin(Pi/10)   -> sqrt(5)/4-1/4
 274                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 275                 if (z.is_equal(*_num10_p)) // sin(Pi/6)    -> 1/2
 276                         return sign*_ex1_2;
 277                 if (z.is_equal(*_num15_p)) // sin(Pi/4)    -> sqrt(2)/2
 278                         return sign*_ex1_2*sqrt(_ex2);
 279                 if (z.is_equal(*_num18_p)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
 280                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 281                 if (z.is_equal(*_num20_p)) // sin(Pi/3)    -> sqrt(3)/2
 282                         return sign*_ex1_2*sqrt(_ex3);
 283                 if (z.is_equal(*_num25_p)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
 284                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 285                 if (z.is_equal(*_num30_p)) // sin(Pi/2)    -> 1
 286                         return sign;
 287         }
 288
 289         if (is_exactly_a<function>(x)) {
 290                 const ex &t = x.op(0);
 291
 292                 // sin(asin(x)) -> x
 293                 if (is_ex_the_function(x, asin))
 294                         return t;
 295
 296                 // sin(acos(x)) -> sqrt(1-x^2)
 297                 if (is_ex_the_function(x, acos))
 298                         return sqrt(_ex1-power(t,_ex2));
 299
 300                 // sin(atan(x)) -> x/sqrt(1+x^2)
 301                 if (is_ex_the_function(x, atan))
 302                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 303         }
 304
 305         // sin(float) -> float
 306         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 307                 return sin(ex_to<numeric>(x));
 308
 309         // sin() is odd
 310         if (x.info(info_flags::negative))
 311                 return -sin(-x);
 312
 313         return sin(x).hold();
 314 }
 315
 316 static ex sin_deriv(const ex & x, unsigned deriv_param)
 317 {
 318         GINAC_ASSERT(deriv_param==0);
 319
 320         // d/dx sin(x) -> cos(x)
 321         return cos(x);
 322 }
 323
 324 REGISTER_FUNCTION(sin, eval_func(sin_eval).
 325                        evalf_func(sin_evalf).
 326                        derivative_func(sin_deriv).
 327                        latex_name("\\sin"));
 328
 329 //////////
 330 // cosine (trigonometric function)
 331 //////////
 332
 333 static ex cos_evalf(const ex & x)
 334 {
 335         if (is_exactly_a<numeric>(x))
 336                 return cos(ex_to<numeric>(x));
 337
 338         return cos(x).hold();
 339 }
 340
 341 static ex cos_eval(const ex & x)
 342 {
 343         // cos(n/d*Pi) -> { all known non-nested radicals }
 344         const ex SixtyExOverPi = _ex60*x/Pi;
 345         ex sign = _ex1;
 346         if (SixtyExOverPi.info(info_flags::integer)) {
 347                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 348                 if (z>=*_num60_p) {
 349                         // wrap to interval [0, Pi)
 350                         z = *_num120_p-z;
 351                 }
 352                 if (z>=*_num30_p) {
 353                         // wrap to interval [0, Pi/2)
 354                         z = *_num60_p-z;
 355                         sign = _ex_1;
 356                 }
 357                 if (z.is_equal(*_num0_p))  // cos(0)       -> 1
 358                         return sign;
 359                 if (z.is_equal(*_num5_p))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 360                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 361                 if (z.is_equal(*_num10_p)) // cos(Pi/6)    -> sqrt(3)/2
 362                         return sign*_ex1_2*sqrt(_ex3);
 363                 if (z.is_equal(*_num12_p)) // cos(Pi/5)    -> sqrt(5)/4+1/4
 364                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 365                 if (z.is_equal(*_num15_p)) // cos(Pi/4)    -> sqrt(2)/2
 366                         return sign*_ex1_2*sqrt(_ex2);
 367                 if (z.is_equal(*_num20_p)) // cos(Pi/3)    -> 1/2
 368                         return sign*_ex1_2;
 369                 if (z.is_equal(*_num24_p)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 370                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 371                 if (z.is_equal(*_num25_p)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 372                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 373                 if (z.is_equal(*_num30_p)) // cos(Pi/2)    -> 0
 374                         return _ex0;
 375         }
 376
 377         if (is_exactly_a<function>(x)) {
 378                 const ex &t = x.op(0);
 379
 380                 // cos(acos(x)) -> x
 381                 if (is_ex_the_function(x, acos))
 382                         return t;
 383
 384                 // cos(asin(x)) -> sqrt(1-x^2)
 385                 if (is_ex_the_function(x, asin))
 386                         return sqrt(_ex1-power(t,_ex2));
 387
 388                 // cos(atan(x)) -> 1/sqrt(1+x^2)
 389                 if (is_ex_the_function(x, atan))
 390                         return power(_ex1+power(t,_ex2),_ex_1_2);
 391         }
 392
 393         // cos(float) -> float
 394         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 395                 return cos(ex_to<numeric>(x));
 396
 397         // cos() is even
 398         if (x.info(info_flags::negative))
 399                 return cos(-x);
 400
 401         return cos(x).hold();
 402 }
 403
 404 static ex cos_deriv(const ex & x, unsigned deriv_param)
 405 {
 406         GINAC_ASSERT(deriv_param==0);
 407
 408         // d/dx cos(x) -> -sin(x)
 409         return -sin(x);
 410 }
 411
 412 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 413                        evalf_func(cos_evalf).
 414                        derivative_func(cos_deriv).
 415                        latex_name("\\cos"));
 416
 417 //////////
 418 // tangent (trigonometric function)
 419 //////////
 420
 421 static ex tan_evalf(const ex & x)
 422 {
 423         if (is_exactly_a<numeric>(x))
 424                 return tan(ex_to<numeric>(x));
 425
 426         return tan(x).hold();
 427 }
 428
 429 static ex tan_eval(const ex & x)
 430 {
 431         // tan(n/d*Pi) -> { all known non-nested radicals }
 432         const ex SixtyExOverPi = _ex60*x/Pi;
 433         ex sign = _ex1;
 434         if (SixtyExOverPi.info(info_flags::integer)) {
 435                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num60_p);
 436                 if (z>=*_num60_p) {
 437                         // wrap to interval [0, Pi)
 438                         z -= *_num60_p;
 439                 }
 440                 if (z>=*_num30_p) {
 441                         // wrap to interval [0, Pi/2)
 442                         z = *_num60_p-z;
 443                         sign = _ex_1;
 444                 }
 445                 if (z.is_equal(*_num0_p))  // tan(0)       -> 0
 446                         return _ex0;
 447                 if (z.is_equal(*_num5_p))  // tan(Pi/12)   -> 2-sqrt(3)
 448                         return sign*(_ex2-sqrt(_ex3));
 449                 if (z.is_equal(*_num10_p)) // tan(Pi/6)    -> sqrt(3)/3
 450                         return sign*_ex1_3*sqrt(_ex3);
 451                 if (z.is_equal(*_num15_p)) // tan(Pi/4)    -> 1
 452                         return sign;
 453                 if (z.is_equal(*_num20_p)) // tan(Pi/3)    -> sqrt(3)
 454                         return sign*sqrt(_ex3);
 455                 if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3)
 456                         return sign*(sqrt(_ex3)+_ex2);
 457                 if (z.is_equal(*_num30_p)) // tan(Pi/2)    -> infinity
 458                         throw (pole_error("tan_eval(): simple pole",1));
 459         }
 460
 461         if (is_exactly_a<function>(x)) {
 462                 const ex &t = x.op(0);
 463
 464                 // tan(atan(x)) -> x
 465                 if (is_ex_the_function(x, atan))
 466                         return t;
 467
 468                 // tan(asin(x)) -> x/sqrt(1+x^2)
 469                 if (is_ex_the_function(x, asin))
 470                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 471
 472                 // tan(acos(x)) -> sqrt(1-x^2)/x
 473                 if (is_ex_the_function(x, acos))
 474                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
 475         }
 476
 477         // tan(float) -> float
 478         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 479                 return tan(ex_to<numeric>(x));
 480         }
 481
 482         // tan() is odd
 483         if (x.info(info_flags::negative))
 484                 return -tan(-x);
 485
 486         return tan(x).hold();
 487 }
 488
 489 static ex tan_deriv(const ex & x, unsigned deriv_param)
 490 {
 491         GINAC_ASSERT(deriv_param==0);
 492
 493         // d/dx tan(x) -> 1+tan(x)^2;
 494         return (_ex1+power(tan(x),_ex2));
 495 }
 496
 497 static ex tan_series(const ex &x,
 498                      const relational &rel,
 499                      int order,
 500                      unsigned options)
 501 {
 502         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 503         // method:
 504         // Taylor series where there is no pole falls back to tan_deriv.
 505         // On a pole simply expand sin(x)/cos(x).
 506         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 507         if (!(2*x_pt/Pi).info(info_flags::odd))
 508                 throw do_taylor();  // caught by function::series()
 509         // if we got here we have to care for a simple pole
 510         return (sin(x)/cos(x)).series(rel, order, options);
 511 }
 512
 513 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 514                        evalf_func(tan_evalf).
 515                        derivative_func(tan_deriv).
 516                        series_func(tan_series).
 517                        latex_name("\\tan"));
 518
 519 //////////
 520 // inverse sine (arc sine)
 521 //////////
 522
 523 static ex asin_evalf(const ex & x)
 524 {
 525         if (is_exactly_a<numeric>(x))
 526                 return asin(ex_to<numeric>(x));
 527
 528         return asin(x).hold();
 529 }
 530
 531 static ex asin_eval(const ex & x)
 532 {
 533         if (x.info(info_flags::numeric)) {
 534
 535                 // asin(0) -> 0
 536                 if (x.is_zero())
 537                         return x;
 538
 539                 // asin(1/2) -> Pi/6
 540                 if (x.is_equal(_ex1_2))
 541                         return numeric(1,6)*Pi;
 542
 543                 // asin(1) -> Pi/2
 544                 if (x.is_equal(_ex1))
 545                         return _ex1_2*Pi;
 546
 547                 // asin(-1/2) -> -Pi/6
 548                 if (x.is_equal(_ex_1_2))
 549                         return numeric(-1,6)*Pi;
 550
 551                 // asin(-1) -> -Pi/2
 552                 if (x.is_equal(_ex_1))
 553                         return _ex_1_2*Pi;
 554
 555                 // asin(float) -> float
 556                 if (!x.info(info_flags::crational))
 557                         return asin(ex_to<numeric>(x));
 558
 559                 // asin() is odd
 560                 if (x.info(info_flags::negative))
 561                         return -asin(-x);
 562         }
 563
 564         return asin(x).hold();
 565 }
 566
 567 static ex asin_deriv(const ex & x, unsigned deriv_param)
 568 {
 569         GINAC_ASSERT(deriv_param==0);
 570
 571         // d/dx asin(x) -> 1/sqrt(1-x^2)
 572         return power(1-power(x,_ex2),_ex_1_2);
 573 }
 574
 575 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 576                         evalf_func(asin_evalf).
 577                         derivative_func(asin_deriv).
 578                         latex_name("\\arcsin"));
 579
 580 //////////
 581 // inverse cosine (arc cosine)
 582 //////////
 583
 584 static ex acos_evalf(const ex & x)
 585 {
 586         if (is_exactly_a<numeric>(x))
 587                 return acos(ex_to<numeric>(x));
 588
 589         return acos(x).hold();
 590 }
 591
 592 static ex acos_eval(const ex & x)
 593 {
 594         if (x.info(info_flags::numeric)) {
 595
 596                 // acos(1) -> 0
 597                 if (x.is_equal(_ex1))
 598                         return _ex0;
 599
 600                 // acos(1/2) -> Pi/3
 601                 if (x.is_equal(_ex1_2))
 602                         return _ex1_3*Pi;
 603
 604                 // acos(0) -> Pi/2
 605                 if (x.is_zero())
 606                         return _ex1_2*Pi;
 607
 608                 // acos(-1/2) -> 2/3*Pi
 609                 if (x.is_equal(_ex_1_2))
 610                         return numeric(2,3)*Pi;
 611
 612                 // acos(-1) -> Pi
 613                 if (x.is_equal(_ex_1))
 614                         return Pi;
 615
 616                 // acos(float) -> float
 617                 if (!x.info(info_flags::crational))
 618                         return acos(ex_to<numeric>(x));
 619
 620                 // acos(-x) -> Pi-acos(x)
 621                 if (x.info(info_flags::negative))
 622                         return Pi-acos(-x);
 623         }
 624
 625         return acos(x).hold();
 626 }
 627
 628 static ex acos_deriv(const ex & x, unsigned deriv_param)
 629 {
 630         GINAC_ASSERT(deriv_param==0);
 631
 632         // d/dx acos(x) -> -1/sqrt(1-x^2)
 633         return -power(1-power(x,_ex2),_ex_1_2);
 634 }
 635
 636 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 637                         evalf_func(acos_evalf).
 638                         derivative_func(acos_deriv).
 639                         latex_name("\\arccos"));
 640
 641 //////////
 642 // inverse tangent (arc tangent)
 643 //////////
 644
 645 static ex atan_evalf(const ex & x)
 646 {
 647         if (is_exactly_a<numeric>(x))
 648                 return atan(ex_to<numeric>(x));
 649
 650         return atan(x).hold();
 651 }
 652
 653 static ex atan_eval(const ex & x)
 654 {
 655         if (x.info(info_flags::numeric)) {
 656
 657                 // atan(0) -> 0
 658                 if (x.is_zero())
 659                         return _ex0;
 660
 661                 // atan(1) -> Pi/4
 662                 if (x.is_equal(_ex1))
 663                         return _ex1_4*Pi;
 664
 665                 // atan(-1) -> -Pi/4
 666                 if (x.is_equal(_ex_1))
 667                         return _ex_1_4*Pi;
 668
 669                 if (x.is_equal(I) || x.is_equal(-I))
 670                         throw (pole_error("atan_eval(): logarithmic pole",0));
 671
 672                 // atan(float) -> float
 673                 if (!x.info(info_flags::crational))
 674                         return atan(ex_to<numeric>(x));
 675
 676                 // atan() is odd
 677                 if (x.info(info_flags::negative))
 678                         return -atan(-x);
 679         }
 680
 681         return atan(x).hold();
 682 }
 683
 684 static ex atan_deriv(const ex & x, unsigned deriv_param)
 685 {
 686         GINAC_ASSERT(deriv_param==0);
 687
 688         // d/dx atan(x) -> 1/(1+x^2)
 689         return power(_ex1+power(x,_ex2), _ex_1);
 690 }
 691
 692 static ex atan_series(const ex &arg,
 693                       const relational &rel,
 694                       int order,
 695                       unsigned options)
 696 {
 697         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 698         // method:
 699         // Taylor series where there is no pole or cut falls back to atan_deriv.
 700         // There are two branch cuts, one runnig from I up the imaginary axis and
 701         // one running from -I down the imaginary axis.  The points I and -I are
 702         // poles.
 703         // On the branch cuts and the poles series expand
 704         //     (log(1+I*x)-log(1-I*x))/(2*I)
 705         // instead.
 706         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 707         if (!(I*arg_pt).info(info_flags::real))
 708                 throw do_taylor();     // Re(x) != 0
 709         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
 710                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
 711         // care for the poles, using the defining formula for atan()...
 712         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
 713                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
 714         if (!(options & series_options::suppress_branchcut)) {
 715                 // method:
 716                 // This is the branch cut: assemble the primitive series manually and
 717                 // then add the corresponding complex step function.
 718                 const symbol &s = ex_to<symbol>(rel.lhs());
 719                 const ex &point = rel.rhs();
 720                 const symbol foo;
 721                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 722                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
 723                 if ((I*arg_pt)<_ex0)
 724                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
 725                 else
 726                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
 727                 epvector seq;
 728                 seq.push_back(expair(Order0correction, _ex0));
 729                 seq.push_back(expair(Order(_ex1), order));
 730                 return series(replarg - pseries(rel, seq), rel, order);
 731         }
 732         throw do_taylor();
 733 }
 734
 735 REGISTER_FUNCTION(atan, eval_func(atan_eval).
 736                         evalf_func(atan_evalf).
 737                         derivative_func(atan_deriv).
 738                         series_func(atan_series).
 739                         latex_name("\\arctan"));
 740
 741 //////////
 742 // inverse tangent (atan2(y,x))
 743 //////////
 744
 745 static ex atan2_evalf(const ex &y, const ex &x)
 746 {
 747         if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
 748                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 749
 750         return atan2(y, x).hold();
 751 }
 752
 753 static ex atan2_eval(const ex & y, const ex & x)
 754 {
 755         if (y.info(info_flags::numeric) && x.info(info_flags::numeric)) {
 756
 757                 if (y.is_zero()) {
 758
 759                         // atan(0, 0) -> 0
 760                         if (x.is_zero())
 761                                 return _ex0;
 762
 763                         // atan(0, x), x real and positive -> 0
 764                         if (x.info(info_flags::positive))
 765                                 return _ex0;
 766
 767                         // atan(0, x), x real and negative -> -Pi
 768                         if (x.info(info_flags::negative))
 769                                 return _ex_1*Pi;
 770                 }
 771
 772                 if (x.is_zero()) {
 773
 774                         // atan(y, 0), y real and positive -> Pi/2
 775                         if (y.info(info_flags::positive))
 776                                 return _ex1_2*Pi;
 777
 778                         // atan(y, 0), y real and negative -> -Pi/2
 779                         if (y.info(info_flags::negative))
 780                                 return _ex_1_2*Pi;
 781                 }
 782
 783                 if (y.is_equal(x)) {
 784
 785                         // atan(y, y), y real and positive -> Pi/4
 786                         if (y.info(info_flags::positive))
 787                                 return _ex1_4*Pi;
 788
 789                         // atan(y, y), y real and negative -> -3/4*Pi
 790                         if (y.info(info_flags::negative))
 791                                 return numeric(-3, 4)*Pi;
 792                 }
 793
 794                 if (y.is_equal(-x)) {
 795
 796                         // atan(y, -y), y real and positive -> 3*Pi/4
 797                         if (y.info(info_flags::positive))
 798                                 return numeric(3, 4)*Pi;
 799
 800                         // atan(y, -y), y real and negative -> -Pi/4
 801                         if (y.info(info_flags::negative))
 802                                 return _ex_1_4*Pi;
 803                 }
 804
 805                 // atan(float, float) -> float
 806                 if (!y.info(info_flags::crational) && !x.info(info_flags::crational))
 807                         return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 808
 809                 // atan(real, real) -> atan(y/x) +/- Pi
 810                 if (y.info(info_flags::real) && x.info(info_flags::real)) {
 811                         if (x.info(info_flags::positive))
 812                                 return atan(y/x);
 813                         else if(y.info(info_flags::positive))
 814                                 return atan(y/x)+Pi;
 815                         else
 816                                 return atan(y/x)-Pi;
 817                 }
 818         }
 819
 820         return atan2(y, x).hold();
 821 }
 822
 823 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
 824 {
 825         GINAC_ASSERT(deriv_param<2);
 826
 827         if (deriv_param==0) {
 828                 // d/dy atan(y,x)
 829                 return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
 830         }
 831         // d/dx atan(y,x)
 832         return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
 833 }
 834
 835 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
 836                          evalf_func(atan2_evalf).
 837                          derivative_func(atan2_deriv));
 838
 839 //////////
 840 // hyperbolic sine (trigonometric function)
 841 //////////
 842
 843 static ex sinh_evalf(const ex & x)
 844 {
 845         if (is_exactly_a<numeric>(x))
 846                 return sinh(ex_to<numeric>(x));
 847
 848         return sinh(x).hold();
 849 }
 850
 851 static ex sinh_eval(const ex & x)
 852 {
 853         if (x.info(info_flags::numeric)) {
 854
 855                 // sinh(0) -> 0
 856                 if (x.is_zero())
 857                         return _ex0;
 858
 859                 // sinh(float) -> float
 860                 if (!x.info(info_flags::crational))
 861                         return sinh(ex_to<numeric>(x));
 862
 863                 // sinh() is odd
 864                 if (x.info(info_flags::negative))
 865                         return -sinh(-x);
 866         }
 867
 868         if ((x/Pi).info(info_flags::numeric) &&
 869                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
 870                 return I*sin(x/I);
 871
 872         if (is_exactly_a<function>(x)) {
 873                 const ex &t = x.op(0);
 874
 875                 // sinh(asinh(x)) -> x
 876                 if (is_ex_the_function(x, asinh))
 877                         return t;
 878
 879                 // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
 880                 if (is_ex_the_function(x, acosh))
 881                         return sqrt(t-_ex1)*sqrt(t+_ex1);
 882
 883                 // sinh(atanh(x)) -> x/sqrt(1-x^2)
 884                 if (is_ex_the_function(x, atanh))
 885                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 886         }
 887
 888         return sinh(x).hold();
 889 }
 890
 891 static ex sinh_deriv(const ex & x, unsigned deriv_param)
 892 {
 893         GINAC_ASSERT(deriv_param==0);
 894
 895         // d/dx sinh(x) -> cosh(x)
 896         return cosh(x);
 897 }
 898
 899 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
 900                         evalf_func(sinh_evalf).
 901                         derivative_func(sinh_deriv).
 902                         latex_name("\\sinh"));
 903
 904 //////////
 905 // hyperbolic cosine (trigonometric function)
 906 //////////
 907
 908 static ex cosh_evalf(const ex & x)
 909 {
 910         if (is_exactly_a<numeric>(x))
 911                 return cosh(ex_to<numeric>(x));
 912
 913         return cosh(x).hold();
 914 }
 915
 916 static ex cosh_eval(const ex & x)
 917 {
 918         if (x.info(info_flags::numeric)) {
 919
 920                 // cosh(0) -> 1
 921                 if (x.is_zero())
 922                         return _ex1;
 923
 924                 // cosh(float) -> float
 925                 if (!x.info(info_flags::crational))
 926                         return cosh(ex_to<numeric>(x));
 927
 928                 // cosh() is even
 929                 if (x.info(info_flags::negative))
 930                         return cosh(-x);
 931         }
 932
 933         if ((x/Pi).info(info_flags::numeric) &&
 934                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
 935                 return cos(x/I);
 936
 937         if (is_exactly_a<function>(x)) {
 938                 const ex &t = x.op(0);
 939
 940                 // cosh(acosh(x)) -> x
 941                 if (is_ex_the_function(x, acosh))
 942                         return t;
 943
 944                 // cosh(asinh(x)) -> sqrt(1+x^2)
 945                 if (is_ex_the_function(x, asinh))
 946                         return sqrt(_ex1+power(t,_ex2));
 947
 948                 // cosh(atanh(x)) -> 1/sqrt(1-x^2)
 949                 if (is_ex_the_function(x, atanh))
 950                         return power(_ex1-power(t,_ex2),_ex_1_2);
 951         }
 952
 953         return cosh(x).hold();
 954 }
 955
 956 static ex cosh_deriv(const ex & x, unsigned deriv_param)
 957 {
 958         GINAC_ASSERT(deriv_param==0);
 959
 960         // d/dx cosh(x) -> sinh(x)
 961         return sinh(x);
 962 }
 963
 964 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
 965                         evalf_func(cosh_evalf).
 966                         derivative_func(cosh_deriv).
 967                         latex_name("\\cosh"));
 968
 969 //////////
 970 // hyperbolic tangent (trigonometric function)
 971 //////////
 972
 973 static ex tanh_evalf(const ex & x)
 974 {
 975         if (is_exactly_a<numeric>(x))
 976                 return tanh(ex_to<numeric>(x));
 977
 978         return tanh(x).hold();
 979 }
 980
 981 static ex tanh_eval(const ex & x)
 982 {
 983         if (x.info(info_flags::numeric)) {
 984
 985                 // tanh(0) -> 0
 986                 if (x.is_zero())
 987                         return _ex0;
 988
 989                 // tanh(float) -> float
 990                 if (!x.info(info_flags::crational))
 991                         return tanh(ex_to<numeric>(x));
 992
 993                 // tanh() is odd
 994                 if (x.info(info_flags::negative))
 995                         return -tanh(-x);
 996         }
 997
 998         if ((x/Pi).info(info_flags::numeric) &&
 999                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
1000                 return I*tan(x/I);
1001
1002         if (is_exactly_a<function>(x)) {
1003                 const ex &t = x.op(0);
1004
1005                 // tanh(atanh(x)) -> x
1006                 if (is_ex_the_function(x, atanh))
1007                         return t;
1008
1009                 // tanh(asinh(x)) -> x/sqrt(1+x^2)
1010                 if (is_ex_the_function(x, asinh))
1011                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
1012
1013                 // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
1014                 if (is_ex_the_function(x, acosh))
1015                         return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
1016         }
1017
1018         return tanh(x).hold();
1019 }
1020
1021 static ex tanh_deriv(const ex & x, unsigned deriv_param)
1022 {
1023         GINAC_ASSERT(deriv_param==0);
1024
1025         // d/dx tanh(x) -> 1-tanh(x)^2
1026         return _ex1-power(tanh(x),_ex2);
1027 }
1028
1029 static ex tanh_series(const ex &x,
1030                       const relational &rel,
1031                       int order,
1032                       unsigned options)
1033 {
1034         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1035         // method:
1036         // Taylor series where there is no pole falls back to tanh_deriv.
1037         // On a pole simply expand sinh(x)/cosh(x).
1038         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1039         if (!(2*I*x_pt/Pi).info(info_flags::odd))
1040                 throw do_taylor();  // caught by function::series()
1041         // if we got here we have to care for a simple pole
1042         return (sinh(x)/cosh(x)).series(rel, order, options);
1043 }
1044
1045 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
1046                         evalf_func(tanh_evalf).
1047                         derivative_func(tanh_deriv).
1048                         series_func(tanh_series).
1049                         latex_name("\\tanh"));
1050
1051 //////////
1052 // inverse hyperbolic sine (trigonometric function)
1053 //////////
1054
1055 static ex asinh_evalf(const ex & x)
1056 {
1057         if (is_exactly_a<numeric>(x))
1058                 return asinh(ex_to<numeric>(x));
1059
1060         return asinh(x).hold();
1061 }
1062
1063 static ex asinh_eval(const ex & x)
1064 {
1065         if (x.info(info_flags::numeric)) {
1066
1067                 // asinh(0) -> 0
1068                 if (x.is_zero())
1069                         return _ex0;
1070
1071                 // asinh(float) -> float
1072                 if (!x.info(info_flags::crational))
1073                         return asinh(ex_to<numeric>(x));
1074
1075                 // asinh() is odd
1076                 if (x.info(info_flags::negative))
1077                         return -asinh(-x);
1078         }
1079
1080         return asinh(x).hold();
1081 }
1082
1083 static ex asinh_deriv(const ex & x, unsigned deriv_param)
1084 {
1085         GINAC_ASSERT(deriv_param==0);
1086
1087         // d/dx asinh(x) -> 1/sqrt(1+x^2)
1088         return power(_ex1+power(x,_ex2),_ex_1_2);
1089 }
1090
1091 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
1092                          evalf_func(asinh_evalf).
1093                          derivative_func(asinh_deriv));
1094
1095 //////////
1096 // inverse hyperbolic cosine (trigonometric function)
1097 //////////
1098
1099 static ex acosh_evalf(const ex & x)
1100 {
1101         if (is_exactly_a<numeric>(x))
1102                 return acosh(ex_to<numeric>(x));
1103
1104         return acosh(x).hold();
1105 }
1106
1107 static ex acosh_eval(const ex & x)
1108 {
1109         if (x.info(info_flags::numeric)) {
1110
1111                 // acosh(0) -> Pi*I/2
1112                 if (x.is_zero())
1113                         return Pi*I*numeric(1,2);
1114
1115                 // acosh(1) -> 0
1116                 if (x.is_equal(_ex1))
1117                         return _ex0;
1118
1119                 // acosh(-1) -> Pi*I
1120                 if (x.is_equal(_ex_1))
1121                         return Pi*I;
1122
1123                 // acosh(float) -> float
1124                 if (!x.info(info_flags::crational))
1125                         return acosh(ex_to<numeric>(x));
1126
1127                 // acosh(-x) -> Pi*I-acosh(x)
1128                 if (x.info(info_flags::negative))
1129                         return Pi*I-acosh(-x);
1130         }
1131
1132         return acosh(x).hold();
1133 }
1134
1135 static ex acosh_deriv(const ex & x, unsigned deriv_param)
1136 {
1137         GINAC_ASSERT(deriv_param==0);
1138
1139         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
1140         return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
1141 }
1142
1143 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
1144                          evalf_func(acosh_evalf).
1145                          derivative_func(acosh_deriv));
1146
1147 //////////
1148 // inverse hyperbolic tangent (trigonometric function)
1149 //////////
1150
1151 static ex atanh_evalf(const ex & x)
1152 {
1153         if (is_exactly_a<numeric>(x))
1154                 return atanh(ex_to<numeric>(x));
1155
1156         return atanh(x).hold();
1157 }
1158
1159 static ex atanh_eval(const ex & x)
1160 {
1161         if (x.info(info_flags::numeric)) {
1162
1163                 // atanh(0) -> 0
1164                 if (x.is_zero())
1165                         return _ex0;
1166
1167                 // atanh({+|-}1) -> throw
1168                 if (x.is_equal(_ex1) || x.is_equal(_ex_1))
1169                         throw (pole_error("atanh_eval(): logarithmic pole",0));
1170
1171                 // atanh(float) -> float
1172                 if (!x.info(info_flags::crational))
1173                         return atanh(ex_to<numeric>(x));
1174
1175                 // atanh() is odd
1176                 if (x.info(info_flags::negative))
1177                         return -atanh(-x);
1178         }
1179
1180         return atanh(x).hold();
1181 }
1182
1183 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1184 {
1185         GINAC_ASSERT(deriv_param==0);
1186
1187         // d/dx atanh(x) -> 1/(1-x^2)
1188         return power(_ex1-power(x,_ex2),_ex_1);
1189 }
1190
1191 static ex atanh_series(const ex &arg,
1192                        const relational &rel,
1193                        int order,
1194                        unsigned options)
1195 {
1196         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1197         // method:
1198         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1199         // There are two branch cuts, one runnig from 1 up the real axis and one
1200         // one running from -1 down the real axis.  The points 1 and -1 are poles
1201         // On the branch cuts and the poles series expand
1202         //     (log(1+x)-log(1-x))/2
1203         // instead.
1204         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
1205         if (!(arg_pt).info(info_flags::real))
1206                 throw do_taylor();     // Im(x) != 0
1207         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1208                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1209         // care for the poles, using the defining formula for atanh()...
1210         if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1211                 return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1212         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1213         if (!(options & series_options::suppress_branchcut)) {
1214                 // method:
1215                 // This is the branch cut: assemble the primitive series manually and
1216                 // then add the corresponding complex step function.
1217                 const symbol &s = ex_to<symbol>(rel.lhs());
1218                 const ex &point = rel.rhs();
1219                 const symbol foo;
1220                 const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1221                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1222                 if (arg_pt<_ex0)
1223                         Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1224                 else
1225                         Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1226                 epvector seq;
1227                 seq.push_back(expair(Order0correction, _ex0));
1228                 seq.push_back(expair(Order(_ex1), order));
1229                 return series(replarg - pseries(rel, seq), rel, order);
1230         }
1231         throw do_taylor();
1232 }
1233
1234 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1235                          evalf_func(atanh_evalf).
1236                          derivative_func(atanh_deriv).
1237                          series_func(atanh_series));
1238
1239
1240 } // namespace GiNaC