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) 1999-2004 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., 59 Temple Place, Suite 330, Boston, MA  02111-1307  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);
  63                 if (z.is_equal(_num0))
  64                         return _ex1;
  65                 if (z.is_equal(_num1))
  66                         return ex(I);
  67                 if (z.is_equal(_num2))
  68                         return _ex_1;
  69                 if (z.is_equal(_num3))
  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::real) && x.info(info_flags::negative))
 115                 //if (x.info(info_flags::rational) && x.info(info_flags::negative))
 116                         return (log(-x)+I*Pi);
 117                 if (x.is_equal(_ex1))  // log(1) -> 0
 118                         return _ex0;
 119                 if (x.is_equal(I))       // log(I) -> Pi*I/2
 120                         return (Pi*I*_num1_2);
 121                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
 122                         return (Pi*I*_num_1_2);
 123
 124                 // log(float) -> float
 125                 if (!x.info(info_flags::crational))
 126                         return log(ex_to<numeric>(x));
 127         }
 128
 129         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
 130         if (is_ex_the_function(x, exp)) {
 131                 const ex &t = x.op(0);
 132                 if (is_a<symbol>(t) && t.info(info_flags::real)) {
 133                         return t;
 134                 }
 135                 if (t.info(info_flags::numeric)) {
 136                         const numeric &nt = ex_to<numeric>(t);
 137                         if (nt.is_real())
 138                                 return t;
 139                 }
 140         }
 141
 142         return log(x).hold();
 143 }
 144
 145 static ex log_deriv(const ex & x, unsigned deriv_param)
 146 {
 147         GINAC_ASSERT(deriv_param==0);
 148
 149         // d/dx log(x) -> 1/x
 150         return power(x, _ex_1);
 151 }
 152
 153 static ex log_series(const ex &arg,
 154                      const relational &rel,
 155                      int order,
 156                      unsigned options)
 157 {
 158         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 159         ex arg_pt;
 160         bool must_expand_arg = false;
 161         // maybe substitution of rel into arg fails because of a pole
 162         try {
 163                 arg_pt = arg.subs(rel, subs_options::no_pattern);
 164         } catch (pole_error) {
 165                 must_expand_arg = true;
 166         }
 167         // or we are at the branch point anyways
 168         if (arg_pt.is_zero())
 169                 must_expand_arg = true;
 170
 171         if (must_expand_arg) {
 172                 // method:
 173                 // This is the branch point: Series expand the argument first, then
 174                 // trivially factorize it to isolate that part which has constant
 175                 // leading coefficient in this fashion:
 176                 //   x^n + x^(n+1) +...+ Order(x^(n+m))  ->  x^n * (1 + x +...+ Order(x^m)).
 177                 // Return a plain n*log(x) for the x^n part and series expand the
 178                 // other part.  Add them together and reexpand again in order to have
 179                 // one unnested pseries object.  All this also works for negative n.
 180                 pseries argser;          // series expansion of log's argument
 181                 unsigned extra_ord = 0;  // extra expansion order
 182                 do {
 183                         // oops, the argument expanded to a pure Order(x^something)...
 184                         argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
 185                         ++extra_ord;
 186                 } while (!argser.is_terminating() && argser.nops()==1);
 187
 188                 const symbol &s = ex_to<symbol>(rel.lhs());
 189                 const ex &point = rel.rhs();
 190                 const int n = argser.ldegree(s);
 191                 epvector seq;
 192                 // construct what we carelessly called the n*log(x) term above
 193                 const ex coeff = argser.coeff(s, n);
 194                 // expand the log, but only if coeff is real and > 0, since otherwise
 195                 // it would make the branch cut run into the wrong direction
 196                 if (coeff.info(info_flags::positive))
 197                         seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
 198                 else
 199                         seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));
 200
 201                 if (!argser.is_terminating() || argser.nops()!=1) {
 202                         // in this case n more (or less) terms are needed
 203                         // (sadly, to generate them, we have to start from the beginning)
 204                         if (n == 0 && coeff == 1) {
 205                                 epvector epv;
 206                                 ex acc = (new pseries(rel, epv))->setflag(status_flags::dynallocated);
 207                                 epv.reserve(2);
 208                                 epv.push_back(expair(-1, _ex0));
 209                                 epv.push_back(expair(Order(_ex1), order));
 210                                 ex rest = pseries(rel, epv).add_series(argser);
 211                                 for (int i = order-1; i>0; --i) {
 212                                         epvector cterm;
 213                                         cterm.reserve(1);
 214                                         cterm.push_back(expair(i%2 ? _ex1/i : _ex_1/i, _ex0));
 215                                         acc = pseries(rel, cterm).add_series(ex_to<pseries>(acc));
 216                                         acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
 217                                 }
 218                                 return acc;
 219                         }
 220                         const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
 221                         return pseries(rel, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
 222                 } else  // it was a monomial
 223                         return pseries(rel, seq);
 224         }
 225         if (!(options & series_options::suppress_branchcut) &&
 226              arg_pt.info(info_flags::negative)) {
 227                 // method:
 228                 // This is the branch cut: assemble the primitive series manually and
 229                 // then add the corresponding complex step function.
 230                 const symbol &s = ex_to<symbol>(rel.lhs());
 231                 const ex &point = rel.rhs();
 232                 const symbol foo;
 233                 const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 234                 epvector seq;
 235                 seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
 236                 seq.push_back(expair(Order(_ex1), order));
 237                 return series(replarg - I*Pi + pseries(rel, seq), rel, order);
 238         }
 239         throw do_taylor();  // caught by function::series()
 240 }
 241
 242 REGISTER_FUNCTION(log, eval_func(log_eval).
 243                        evalf_func(log_evalf).
 244                        derivative_func(log_deriv).
 245                        series_func(log_series).
 246                        latex_name("\\ln"));
 247
 248 //////////
 249 // sine (trigonometric function)
 250 //////////
 251
 252 static ex sin_evalf(const ex & x)
 253 {
 254         if (is_exactly_a<numeric>(x))
 255                 return sin(ex_to<numeric>(x));
 256
 257         return sin(x).hold();
 258 }
 259
 260 static ex sin_eval(const ex & x)
 261 {
 262         // sin(n/d*Pi) -> { all known non-nested radicals }
 263         const ex SixtyExOverPi = _ex60*x/Pi;
 264         ex sign = _ex1;
 265         if (SixtyExOverPi.info(info_flags::integer)) {
 266                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num120);
 267                 if (z>=_num60) {
 268                         // wrap to interval [0, Pi)
 269                         z -= _num60;
 270                         sign = _ex_1;
 271                 }
 272                 if (z>_num30) {
 273                         // wrap to interval [0, Pi/2)
 274                         z = _num60-z;
 275                 }
 276                 if (z.is_equal(_num0))  // sin(0)       -> 0
 277                         return _ex0;
 278                 if (z.is_equal(_num5))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
 279                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 280                 if (z.is_equal(_num6))  // sin(Pi/10)   -> sqrt(5)/4-1/4
 281                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 282                 if (z.is_equal(_num10)) // sin(Pi/6)    -> 1/2
 283                         return sign*_ex1_2;
 284                 if (z.is_equal(_num15)) // sin(Pi/4)    -> sqrt(2)/2
 285                         return sign*_ex1_2*sqrt(_ex2);
 286                 if (z.is_equal(_num18)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
 287                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 288                 if (z.is_equal(_num20)) // sin(Pi/3)    -> sqrt(3)/2
 289                         return sign*_ex1_2*sqrt(_ex3);
 290                 if (z.is_equal(_num25)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
 291                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 292                 if (z.is_equal(_num30)) // sin(Pi/2)    -> 1
 293                         return sign;
 294         }
 295
 296         if (is_exactly_a<function>(x)) {
 297                 const ex &t = x.op(0);
 298
 299                 // sin(asin(x)) -> x
 300                 if (is_ex_the_function(x, asin))
 301                         return t;
 302
 303                 // sin(acos(x)) -> sqrt(1-x^2)
 304                 if (is_ex_the_function(x, acos))
 305                         return sqrt(_ex1-power(t,_ex2));
 306
 307                 // sin(atan(x)) -> x/sqrt(1+x^2)
 308                 if (is_ex_the_function(x, atan))
 309                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 310         }
 311
 312         // sin(float) -> float
 313         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 314                 return sin(ex_to<numeric>(x));
 315
 316         // sin() is odd
 317         if (x.info(info_flags::negative))
 318                 return -sin(-x);
 319
 320         return sin(x).hold();
 321 }
 322
 323 static ex sin_deriv(const ex & x, unsigned deriv_param)
 324 {
 325         GINAC_ASSERT(deriv_param==0);
 326
 327         // d/dx sin(x) -> cos(x)
 328         return cos(x);
 329 }
 330
 331 REGISTER_FUNCTION(sin, eval_func(sin_eval).
 332                        evalf_func(sin_evalf).
 333                        derivative_func(sin_deriv).
 334                        latex_name("\\sin"));
 335
 336 //////////
 337 // cosine (trigonometric function)
 338 //////////
 339
 340 static ex cos_evalf(const ex & x)
 341 {
 342         if (is_exactly_a<numeric>(x))
 343                 return cos(ex_to<numeric>(x));
 344
 345         return cos(x).hold();
 346 }
 347
 348 static ex cos_eval(const ex & x)
 349 {
 350         // cos(n/d*Pi) -> { all known non-nested radicals }
 351         const ex SixtyExOverPi = _ex60*x/Pi;
 352         ex sign = _ex1;
 353         if (SixtyExOverPi.info(info_flags::integer)) {
 354                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num120);
 355                 if (z>=_num60) {
 356                         // wrap to interval [0, Pi)
 357                         z = _num120-z;
 358                 }
 359                 if (z>=_num30) {
 360                         // wrap to interval [0, Pi/2)
 361                         z = _num60-z;
 362                         sign = _ex_1;
 363                 }
 364                 if (z.is_equal(_num0))  // cos(0)       -> 1
 365                         return sign;
 366                 if (z.is_equal(_num5))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 367                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 368                 if (z.is_equal(_num10)) // cos(Pi/6)    -> sqrt(3)/2
 369                         return sign*_ex1_2*sqrt(_ex3);
 370                 if (z.is_equal(_num12)) // cos(Pi/5)    -> sqrt(5)/4+1/4
 371                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 372                 if (z.is_equal(_num15)) // cos(Pi/4)    -> sqrt(2)/2
 373                         return sign*_ex1_2*sqrt(_ex2);
 374                 if (z.is_equal(_num20)) // cos(Pi/3)    -> 1/2
 375                         return sign*_ex1_2;
 376                 if (z.is_equal(_num24)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 377                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 378                 if (z.is_equal(_num25)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 379                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 380                 if (z.is_equal(_num30)) // cos(Pi/2)    -> 0
 381                         return _ex0;
 382         }
 383
 384         if (is_exactly_a<function>(x)) {
 385                 const ex &t = x.op(0);
 386
 387                 // cos(acos(x)) -> x
 388                 if (is_ex_the_function(x, acos))
 389                         return t;
 390
 391                 // cos(asin(x)) -> sqrt(1-x^2)
 392                 if (is_ex_the_function(x, asin))
 393                         return sqrt(_ex1-power(t,_ex2));
 394
 395                 // cos(atan(x)) -> 1/sqrt(1+x^2)
 396                 if (is_ex_the_function(x, atan))
 397                         return power(_ex1+power(t,_ex2),_ex_1_2);
 398         }
 399
 400         // cos(float) -> float
 401         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 402                 return cos(ex_to<numeric>(x));
 403
 404         // cos() is even
 405         if (x.info(info_flags::negative))
 406                 return cos(-x);
 407
 408         return cos(x).hold();
 409 }
 410
 411 static ex cos_deriv(const ex & x, unsigned deriv_param)
 412 {
 413         GINAC_ASSERT(deriv_param==0);
 414
 415         // d/dx cos(x) -> -sin(x)
 416         return -sin(x);
 417 }
 418
 419 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 420                        evalf_func(cos_evalf).
 421                        derivative_func(cos_deriv).
 422                        latex_name("\\cos"));
 423
 424 //////////
 425 // tangent (trigonometric function)
 426 //////////
 427
 428 static ex tan_evalf(const ex & x)
 429 {
 430         if (is_exactly_a<numeric>(x))
 431                 return tan(ex_to<numeric>(x));
 432
 433         return tan(x).hold();
 434 }
 435
 436 static ex tan_eval(const ex & x)
 437 {
 438         // tan(n/d*Pi) -> { all known non-nested radicals }
 439         const ex SixtyExOverPi = _ex60*x/Pi;
 440         ex sign = _ex1;
 441         if (SixtyExOverPi.info(info_flags::integer)) {
 442                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num60);
 443                 if (z>=_num60) {
 444                         // wrap to interval [0, Pi)
 445                         z -= _num60;
 446                 }
 447                 if (z>=_num30) {
 448                         // wrap to interval [0, Pi/2)
 449                         z = _num60-z;
 450                         sign = _ex_1;
 451                 }
 452                 if (z.is_equal(_num0))  // tan(0)       -> 0
 453                         return _ex0;
 454                 if (z.is_equal(_num5))  // tan(Pi/12)   -> 2-sqrt(3)
 455                         return sign*(_ex2-sqrt(_ex3));
 456                 if (z.is_equal(_num10)) // tan(Pi/6)    -> sqrt(3)/3
 457                         return sign*_ex1_3*sqrt(_ex3);
 458                 if (z.is_equal(_num15)) // tan(Pi/4)    -> 1
 459                         return sign;
 460                 if (z.is_equal(_num20)) // tan(Pi/3)    -> sqrt(3)
 461                         return sign*sqrt(_ex3);
 462                 if (z.is_equal(_num25)) // tan(5/12*Pi) -> 2+sqrt(3)
 463                         return sign*(sqrt(_ex3)+_ex2);
 464                 if (z.is_equal(_num30)) // tan(Pi/2)    -> infinity
 465                         throw (pole_error("tan_eval(): simple pole",1));
 466         }
 467
 468         if (is_exactly_a<function>(x)) {
 469                 const ex &t = x.op(0);
 470
 471                 // tan(atan(x)) -> x
 472                 if (is_ex_the_function(x, atan))
 473                         return t;
 474
 475                 // tan(asin(x)) -> x/sqrt(1+x^2)
 476                 if (is_ex_the_function(x, asin))
 477                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 478
 479                 // tan(acos(x)) -> sqrt(1-x^2)/x
 480                 if (is_ex_the_function(x, acos))
 481                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
 482         }
 483
 484         // tan(float) -> float
 485         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 486                 return tan(ex_to<numeric>(x));
 487         }
 488
 489         // tan() is odd
 490         if (x.info(info_flags::negative))
 491                 return -tan(-x);
 492
 493         return tan(x).hold();
 494 }
 495
 496 static ex tan_deriv(const ex & x, unsigned deriv_param)
 497 {
 498         GINAC_ASSERT(deriv_param==0);
 499
 500         // d/dx tan(x) -> 1+tan(x)^2;
 501         return (_ex1+power(tan(x),_ex2));
 502 }
 503
 504 static ex tan_series(const ex &x,
 505                      const relational &rel,
 506                      int order,
 507                      unsigned options)
 508 {
 509         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 510         // method:
 511         // Taylor series where there is no pole falls back to tan_deriv.
 512         // On a pole simply expand sin(x)/cos(x).
 513         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 514         if (!(2*x_pt/Pi).info(info_flags::odd))
 515                 throw do_taylor();  // caught by function::series()
 516         // if we got here we have to care for a simple pole
 517         return (sin(x)/cos(x)).series(rel, order, options);
 518 }
 519
 520 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 521                        evalf_func(tan_evalf).
 522                        derivative_func(tan_deriv).
 523                        series_func(tan_series).
 524                        latex_name("\\tan"));
 525
 526 //////////
 527 // inverse sine (arc sine)
 528 //////////
 529
 530 static ex asin_evalf(const ex & x)
 531 {
 532         if (is_exactly_a<numeric>(x))
 533                 return asin(ex_to<numeric>(x));
 534
 535         return asin(x).hold();
 536 }
 537
 538 static ex asin_eval(const ex & x)
 539 {
 540         if (x.info(info_flags::numeric)) {
 541
 542                 // asin(0) -> 0
 543                 if (x.is_zero())
 544                         return x;
 545
 546                 // asin(1/2) -> Pi/6
 547                 if (x.is_equal(_ex1_2))
 548                         return numeric(1,6)*Pi;
 549
 550                 // asin(1) -> Pi/2
 551                 if (x.is_equal(_ex1))
 552                         return _num1_2*Pi;
 553
 554                 // asin(-1/2) -> -Pi/6
 555                 if (x.is_equal(_ex_1_2))
 556                         return numeric(-1,6)*Pi;
 557
 558                 // asin(-1) -> -Pi/2
 559                 if (x.is_equal(_ex_1))
 560                         return _num_1_2*Pi;
 561
 562                 // asin(float) -> float
 563                 if (!x.info(info_flags::crational))
 564                         return asin(ex_to<numeric>(x));
 565
 566                 // asin() is odd
 567                 if (x.info(info_flags::negative))
 568                         return -asin(-x);
 569         }
 570
 571         return asin(x).hold();
 572 }
 573
 574 static ex asin_deriv(const ex & x, unsigned deriv_param)
 575 {
 576         GINAC_ASSERT(deriv_param==0);
 577
 578         // d/dx asin(x) -> 1/sqrt(1-x^2)
 579         return power(1-power(x,_ex2),_ex_1_2);
 580 }
 581
 582 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 583                         evalf_func(asin_evalf).
 584                         derivative_func(asin_deriv).
 585                         latex_name("\\arcsin"));
 586
 587 //////////
 588 // inverse cosine (arc cosine)
 589 //////////
 590
 591 static ex acos_evalf(const ex & x)
 592 {
 593         if (is_exactly_a<numeric>(x))
 594                 return acos(ex_to<numeric>(x));
 595
 596         return acos(x).hold();
 597 }
 598
 599 static ex acos_eval(const ex & x)
 600 {
 601         if (x.info(info_flags::numeric)) {
 602
 603                 // acos(1) -> 0
 604                 if (x.is_equal(_ex1))
 605                         return _ex0;
 606
 607                 // acos(1/2) -> Pi/3
 608                 if (x.is_equal(_ex1_2))
 609                         return _ex1_3*Pi;
 610
 611                 // acos(0) -> Pi/2
 612                 if (x.is_zero())
 613                         return _ex1_2*Pi;
 614
 615                 // acos(-1/2) -> 2/3*Pi
 616                 if (x.is_equal(_ex_1_2))
 617                         return numeric(2,3)*Pi;
 618
 619                 // acos(-1) -> Pi
 620                 if (x.is_equal(_ex_1))
 621                         return Pi;
 622
 623                 // acos(float) -> float
 624                 if (!x.info(info_flags::crational))
 625                         return acos(ex_to<numeric>(x));
 626
 627                 // acos(-x) -> Pi-acos(x)
 628                 if (x.info(info_flags::negative))
 629                         return Pi-acos(-x);
 630         }
 631
 632         return acos(x).hold();
 633 }
 634
 635 static ex acos_deriv(const ex & x, unsigned deriv_param)
 636 {
 637         GINAC_ASSERT(deriv_param==0);
 638
 639         // d/dx acos(x) -> -1/sqrt(1-x^2)
 640         return -power(1-power(x,_ex2),_ex_1_2);
 641 }
 642
 643 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 644                         evalf_func(acos_evalf).
 645                         derivative_func(acos_deriv).
 646                         latex_name("\\arccos"));
 647
 648 //////////
 649 // inverse tangent (arc tangent)
 650 //////////
 651
 652 static ex atan_evalf(const ex & x)
 653 {
 654         if (is_exactly_a<numeric>(x))
 655                 return atan(ex_to<numeric>(x));
 656
 657         return atan(x).hold();
 658 }
 659
 660 static ex atan_eval(const ex & x)
 661 {
 662         if (x.info(info_flags::numeric)) {
 663
 664                 // atan(0) -> 0
 665                 if (x.is_zero())
 666                         return _ex0;
 667
 668                 // atan(1) -> Pi/4
 669                 if (x.is_equal(_ex1))
 670                         return _ex1_4*Pi;
 671
 672                 // atan(-1) -> -Pi/4
 673                 if (x.is_equal(_ex_1))
 674                         return _ex_1_4*Pi;
 675
 676                 if (x.is_equal(I) || x.is_equal(-I))
 677                         throw (pole_error("atan_eval(): logarithmic pole",0));
 678
 679                 // atan(float) -> float
 680                 if (!x.info(info_flags::crational))
 681                         return atan(ex_to<numeric>(x));
 682
 683                 // atan() is odd
 684                 if (x.info(info_flags::negative))
 685                         return -atan(-x);
 686         }
 687
 688         return atan(x).hold();
 689 }
 690
 691 static ex atan_deriv(const ex & x, unsigned deriv_param)
 692 {
 693         GINAC_ASSERT(deriv_param==0);
 694
 695         // d/dx atan(x) -> 1/(1+x^2)
 696         return power(_ex1+power(x,_ex2), _ex_1);
 697 }
 698
 699 static ex atan_series(const ex &arg,
 700                       const relational &rel,
 701                       int order,
 702                       unsigned options)
 703 {
 704         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 705         // method:
 706         // Taylor series where there is no pole or cut falls back to atan_deriv.
 707         // There are two branch cuts, one runnig from I up the imaginary axis and
 708         // one running from -I down the imaginary axis.  The points I and -I are
 709         // poles.
 710         // On the branch cuts and the poles series expand
 711         //     (log(1+I*x)-log(1-I*x))/(2*I)
 712         // instead.
 713         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 714         if (!(I*arg_pt).info(info_flags::real))
 715                 throw do_taylor();     // Re(x) != 0
 716         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
 717                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
 718         // care for the poles, using the defining formula for atan()...
 719         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
 720                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
 721         if (!(options & series_options::suppress_branchcut)) {
 722                 // method:
 723                 // This is the branch cut: assemble the primitive series manually and
 724                 // then add the corresponding complex step function.
 725                 const symbol &s = ex_to<symbol>(rel.lhs());
 726                 const ex &point = rel.rhs();
 727                 const symbol foo;
 728                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 729                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
 730                 if ((I*arg_pt)<_ex0)
 731                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
 732                 else
 733                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
 734                 epvector seq;
 735                 seq.push_back(expair(Order0correction, _ex0));
 736                 seq.push_back(expair(Order(_ex1), order));
 737                 return series(replarg - pseries(rel, seq), rel, order);
 738         }
 739         throw do_taylor();
 740 }
 741
 742 REGISTER_FUNCTION(atan, eval_func(atan_eval).
 743                         evalf_func(atan_evalf).
 744                         derivative_func(atan_deriv).
 745                         series_func(atan_series).
 746                         latex_name("\\arctan"));
 747
 748 //////////
 749 // inverse tangent (atan2(y,x))
 750 //////////
 751
 752 static ex atan2_evalf(const ex &y, const ex &x)
 753 {
 754         if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
 755                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 756
 757         return atan2(y, x).hold();
 758 }
 759
 760 static ex atan2_eval(const ex & y, const ex & x)
 761 {
 762         if (y.info(info_flags::numeric) && x.info(info_flags::numeric)) {
 763
 764                 if (y.is_zero()) {
 765
 766                         // atan(0, 0) -> 0
 767                         if (x.is_zero())
 768                                 return _ex0;
 769
 770                         // atan(0, x), x real and positive -> 0
 771                         if (x.info(info_flags::positive))
 772                                 return _ex0;
 773
 774                         // atan(0, x), x real and negative -> -Pi
 775                         if (x.info(info_flags::negative))
 776                                 return _ex_1*Pi;
 777                 }
 778
 779                 if (x.is_zero()) {
 780
 781                         // atan(y, 0), y real and positive -> Pi/2
 782                         if (y.info(info_flags::positive))
 783                                 return _ex1_2*Pi;
 784
 785                         // atan(y, 0), y real and negative -> -Pi/2
 786                         if (y.info(info_flags::negative))
 787                                 return _ex_1_2*Pi;
 788                 }
 789
 790                 if (y.is_equal(x)) {
 791
 792                         // atan(y, y), y real and positive -> Pi/4
 793                         if (y.info(info_flags::positive))
 794                                 return _ex1_4*Pi;
 795
 796                         // atan(y, y), y real and negative -> -3/4*Pi
 797                         if (y.info(info_flags::negative))
 798                                 return numeric(-3, 4)*Pi;
 799                 }
 800
 801                 if (y.is_equal(-x)) {
 802
 803                         // atan(y, -y), y real and positive -> 3*Pi/4
 804                         if (y.info(info_flags::positive))
 805                                 return numeric(3, 4)*Pi;
 806
 807                         // atan(y, -y), y real and negative -> -Pi/4
 808                         if (y.info(info_flags::negative))
 809                                 return _ex_1_4*Pi;
 810                 }
 811
 812                 // atan(float, float) -> float
 813                 if (!y.info(info_flags::crational) && !x.info(info_flags::crational))
 814                         return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 815
 816                 // atan(real, real) -> atan(y/x) +/- Pi
 817                 if (y.info(info_flags::real) && x.info(info_flags::real)) {
 818                         if (x.info(info_flags::positive))
 819                                 return atan(y/x);
 820                         else if(y.info(info_flags::positive))
 821                                 return atan(y/x)+Pi;
 822                         else
 823                                 return atan(y/x)-Pi;
 824                 }
 825         }
 826
 827         return atan2(y, x).hold();
 828 }
 829
 830 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
 831 {
 832         GINAC_ASSERT(deriv_param<2);
 833
 834         if (deriv_param==0) {
 835                 // d/dy atan(y,x)
 836                 return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
 837         }
 838         // d/dx atan(y,x)
 839         return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
 840 }
 841
 842 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
 843                          evalf_func(atan2_evalf).
 844                          derivative_func(atan2_deriv));
 845
 846 //////////
 847 // hyperbolic sine (trigonometric function)
 848 //////////
 849
 850 static ex sinh_evalf(const ex & x)
 851 {
 852         if (is_exactly_a<numeric>(x))
 853                 return sinh(ex_to<numeric>(x));
 854
 855         return sinh(x).hold();
 856 }
 857
 858 static ex sinh_eval(const ex & x)
 859 {
 860         if (x.info(info_flags::numeric)) {
 861
 862                 // sinh(0) -> 0
 863                 if (x.is_zero())
 864                         return _ex0;
 865
 866                 // sinh(float) -> float
 867                 if (!x.info(info_flags::crational))
 868                         return sinh(ex_to<numeric>(x));
 869
 870                 // sinh() is odd
 871                 if (x.info(info_flags::negative))
 872                         return -sinh(-x);
 873         }
 874
 875         if ((x/Pi).info(info_flags::numeric) &&
 876                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
 877                 return I*sin(x/I);
 878
 879         if (is_exactly_a<function>(x)) {
 880                 const ex &t = x.op(0);
 881
 882                 // sinh(asinh(x)) -> x
 883                 if (is_ex_the_function(x, asinh))
 884                         return t;
 885
 886                 // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
 887                 if (is_ex_the_function(x, acosh))
 888                         return sqrt(t-_ex1)*sqrt(t+_ex1);
 889
 890                 // sinh(atanh(x)) -> x/sqrt(1-x^2)
 891                 if (is_ex_the_function(x, atanh))
 892                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 893         }
 894
 895         return sinh(x).hold();
 896 }
 897
 898 static ex sinh_deriv(const ex & x, unsigned deriv_param)
 899 {
 900         GINAC_ASSERT(deriv_param==0);
 901
 902         // d/dx sinh(x) -> cosh(x)
 903         return cosh(x);
 904 }
 905
 906 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
 907                         evalf_func(sinh_evalf).
 908                         derivative_func(sinh_deriv).
 909                         latex_name("\\sinh"));
 910
 911 //////////
 912 // hyperbolic cosine (trigonometric function)
 913 //////////
 914
 915 static ex cosh_evalf(const ex & x)
 916 {
 917         if (is_exactly_a<numeric>(x))
 918                 return cosh(ex_to<numeric>(x));
 919
 920         return cosh(x).hold();
 921 }
 922
 923 static ex cosh_eval(const ex & x)
 924 {
 925         if (x.info(info_flags::numeric)) {
 926
 927                 // cosh(0) -> 1
 928                 if (x.is_zero())
 929                         return _ex1;
 930
 931                 // cosh(float) -> float
 932                 if (!x.info(info_flags::crational))
 933                         return cosh(ex_to<numeric>(x));
 934
 935                 // cosh() is even
 936                 if (x.info(info_flags::negative))
 937                         return cosh(-x);
 938         }
 939
 940         if ((x/Pi).info(info_flags::numeric) &&
 941                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
 942                 return cos(x/I);
 943
 944         if (is_exactly_a<function>(x)) {
 945                 const ex &t = x.op(0);
 946
 947                 // cosh(acosh(x)) -> x
 948                 if (is_ex_the_function(x, acosh))
 949                         return t;
 950
 951                 // cosh(asinh(x)) -> sqrt(1+x^2)
 952                 if (is_ex_the_function(x, asinh))
 953                         return sqrt(_ex1+power(t,_ex2));
 954
 955                 // cosh(atanh(x)) -> 1/sqrt(1-x^2)
 956                 if (is_ex_the_function(x, atanh))
 957                         return power(_ex1-power(t,_ex2),_ex_1_2);
 958         }
 959
 960         return cosh(x).hold();
 961 }
 962
 963 static ex cosh_deriv(const ex & x, unsigned deriv_param)
 964 {
 965         GINAC_ASSERT(deriv_param==0);
 966
 967         // d/dx cosh(x) -> sinh(x)
 968         return sinh(x);
 969 }
 970
 971 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
 972                         evalf_func(cosh_evalf).
 973                         derivative_func(cosh_deriv).
 974                         latex_name("\\cosh"));
 975
 976 //////////
 977 // hyperbolic tangent (trigonometric function)
 978 //////////
 979
 980 static ex tanh_evalf(const ex & x)
 981 {
 982         if (is_exactly_a<numeric>(x))
 983                 return tanh(ex_to<numeric>(x));
 984
 985         return tanh(x).hold();
 986 }
 987
 988 static ex tanh_eval(const ex & x)
 989 {
 990         if (x.info(info_flags::numeric)) {
 991
 992                 // tanh(0) -> 0
 993                 if (x.is_zero())
 994                         return _ex0;
 995
 996                 // tanh(float) -> float
 997                 if (!x.info(info_flags::crational))
 998                         return tanh(ex_to<numeric>(x));
 999
1000                 // tanh() is odd
1001                 if (x.info(info_flags::negative))
1002                         return -tanh(-x);
1003         }
1004
1005         if ((x/Pi).info(info_flags::numeric) &&
1006                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
1007                 return I*tan(x/I);
1008
1009         if (is_exactly_a<function>(x)) {
1010                 const ex &t = x.op(0);
1011
1012                 // tanh(atanh(x)) -> x
1013                 if (is_ex_the_function(x, atanh))
1014                         return t;
1015
1016                 // tanh(asinh(x)) -> x/sqrt(1+x^2)
1017                 if (is_ex_the_function(x, asinh))
1018                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
1019
1020                 // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
1021                 if (is_ex_the_function(x, acosh))
1022                         return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
1023         }
1024
1025         return tanh(x).hold();
1026 }
1027
1028 static ex tanh_deriv(const ex & x, unsigned deriv_param)
1029 {
1030         GINAC_ASSERT(deriv_param==0);
1031
1032         // d/dx tanh(x) -> 1-tanh(x)^2
1033         return _ex1-power(tanh(x),_ex2);
1034 }
1035
1036 static ex tanh_series(const ex &x,
1037                       const relational &rel,
1038                       int order,
1039                       unsigned options)
1040 {
1041         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1042         // method:
1043         // Taylor series where there is no pole falls back to tanh_deriv.
1044         // On a pole simply expand sinh(x)/cosh(x).
1045         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1046         if (!(2*I*x_pt/Pi).info(info_flags::odd))
1047                 throw do_taylor();  // caught by function::series()
1048         // if we got here we have to care for a simple pole
1049         return (sinh(x)/cosh(x)).series(rel, order, options);
1050 }
1051
1052 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
1053                         evalf_func(tanh_evalf).
1054                         derivative_func(tanh_deriv).
1055                         series_func(tanh_series).
1056                         latex_name("\\tanh"));
1057
1058 //////////
1059 // inverse hyperbolic sine (trigonometric function)
1060 //////////
1061
1062 static ex asinh_evalf(const ex & x)
1063 {
1064         if (is_exactly_a<numeric>(x))
1065                 return asinh(ex_to<numeric>(x));
1066
1067         return asinh(x).hold();
1068 }
1069
1070 static ex asinh_eval(const ex & x)
1071 {
1072         if (x.info(info_flags::numeric)) {
1073
1074                 // asinh(0) -> 0
1075                 if (x.is_zero())
1076                         return _ex0;
1077
1078                 // asinh(float) -> float
1079                 if (!x.info(info_flags::crational))
1080                         return asinh(ex_to<numeric>(x));
1081
1082                 // asinh() is odd
1083                 if (x.info(info_flags::negative))
1084                         return -asinh(-x);
1085         }
1086
1087         return asinh(x).hold();
1088 }
1089
1090 static ex asinh_deriv(const ex & x, unsigned deriv_param)
1091 {
1092         GINAC_ASSERT(deriv_param==0);
1093
1094         // d/dx asinh(x) -> 1/sqrt(1+x^2)
1095         return power(_ex1+power(x,_ex2),_ex_1_2);
1096 }
1097
1098 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
1099                          evalf_func(asinh_evalf).
1100                          derivative_func(asinh_deriv));
1101
1102 //////////
1103 // inverse hyperbolic cosine (trigonometric function)
1104 //////////
1105
1106 static ex acosh_evalf(const ex & x)
1107 {
1108         if (is_exactly_a<numeric>(x))
1109                 return acosh(ex_to<numeric>(x));
1110
1111         return acosh(x).hold();
1112 }
1113
1114 static ex acosh_eval(const ex & x)
1115 {
1116         if (x.info(info_flags::numeric)) {
1117
1118                 // acosh(0) -> Pi*I/2
1119                 if (x.is_zero())
1120                         return Pi*I*numeric(1,2);
1121
1122                 // acosh(1) -> 0
1123                 if (x.is_equal(_ex1))
1124                         return _ex0;
1125
1126                 // acosh(-1) -> Pi*I
1127                 if (x.is_equal(_ex_1))
1128                         return Pi*I;
1129
1130                 // acosh(float) -> float
1131                 if (!x.info(info_flags::crational))
1132                         return acosh(ex_to<numeric>(x));
1133
1134                 // acosh(-x) -> Pi*I-acosh(x)
1135                 if (x.info(info_flags::negative))
1136                         return Pi*I-acosh(-x);
1137         }
1138
1139         return acosh(x).hold();
1140 }
1141
1142 static ex acosh_deriv(const ex & x, unsigned deriv_param)
1143 {
1144         GINAC_ASSERT(deriv_param==0);
1145
1146         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
1147         return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
1148 }
1149
1150 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
1151                          evalf_func(acosh_evalf).
1152                          derivative_func(acosh_deriv));
1153
1154 //////////
1155 // inverse hyperbolic tangent (trigonometric function)
1156 //////////
1157
1158 static ex atanh_evalf(const ex & x)
1159 {
1160         if (is_exactly_a<numeric>(x))
1161                 return atanh(ex_to<numeric>(x));
1162
1163         return atanh(x).hold();
1164 }
1165
1166 static ex atanh_eval(const ex & x)
1167 {
1168         if (x.info(info_flags::numeric)) {
1169
1170                 // atanh(0) -> 0
1171                 if (x.is_zero())
1172                         return _ex0;
1173
1174                 // atanh({+|-}1) -> throw
1175                 if (x.is_equal(_ex1) || x.is_equal(_ex_1))
1176                         throw (pole_error("atanh_eval(): logarithmic pole",0));
1177
1178                 // atanh(float) -> float
1179                 if (!x.info(info_flags::crational))
1180                         return atanh(ex_to<numeric>(x));
1181
1182                 // atanh() is odd
1183                 if (x.info(info_flags::negative))
1184                         return -atanh(-x);
1185         }
1186
1187         return atanh(x).hold();
1188 }
1189
1190 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1191 {
1192         GINAC_ASSERT(deriv_param==0);
1193
1194         // d/dx atanh(x) -> 1/(1-x^2)
1195         return power(_ex1-power(x,_ex2),_ex_1);
1196 }
1197
1198 static ex atanh_series(const ex &arg,
1199                        const relational &rel,
1200                        int order,
1201                        unsigned options)
1202 {
1203         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1204         // method:
1205         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1206         // There are two branch cuts, one runnig from 1 up the real axis and one
1207         // one running from -1 down the real axis.  The points 1 and -1 are poles
1208         // On the branch cuts and the poles series expand
1209         //     (log(1+x)-log(1-x))/2
1210         // instead.
1211         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
1212         if (!(arg_pt).info(info_flags::real))
1213                 throw do_taylor();     // Im(x) != 0
1214         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1215                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1216         // care for the poles, using the defining formula for atanh()...
1217         if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1218                 return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1219         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1220         if (!(options & series_options::suppress_branchcut)) {
1221                 // method:
1222                 // This is the branch cut: assemble the primitive series manually and
1223                 // then add the corresponding complex step function.
1224                 const symbol &s = ex_to<symbol>(rel.lhs());
1225                 const ex &point = rel.rhs();
1226                 const symbol foo;
1227                 const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1228                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1229                 if (arg_pt<_ex0)
1230                         Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1231                 else
1232                         Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1233                 epvector seq;
1234                 seq.push_back(expair(Order0correction, _ex0));
1235                 seq.push_back(expair(Order(_ex1), order));
1236                 return series(replarg - pseries(rel, seq), rel, order);
1237         }
1238         throw do_taylor();
1239 }
1240
1241 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1242                          evalf_func(atanh_evalf).
1243                          derivative_func(atanh_deriv).
1244                          series_func(atanh_series));
1245
1246
1247 } // namespace GiNaC