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) 1999-2003 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         // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
  59         const ex TwoExOverPiI=(_ex2*x)/(Pi*I);
  60         if (TwoExOverPiI.info(info_flags::integer)) {
  61                 const numeric z = mod(ex_to<numeric>(TwoExOverPiI),_num4);
  62                 if (z.is_equal(_num0))
  63                         return _ex1;
  64                 if (z.is_equal(_num1))
  65                         return ex(I);
  66                 if (z.is_equal(_num2))
  67                         return _ex_1;
  68                 if (z.is_equal(_num3))
  69                         return ex(-I);
  70         }
  71         // exp(log(x)) -> x
  72         if (is_ex_the_function(x, log))
  73                 return x.op(0);
  74
  75         // exp(float)
  76         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
  77                 return exp(ex_to<numeric>(x));
  78
  79         return exp(x).hold();
  80 }
  81
  82 static ex exp_deriv(const ex & x, unsigned deriv_param)
  83 {
  84         GINAC_ASSERT(deriv_param==0);
  85
  86         // d/dx exp(x) -> exp(x)
  87         return exp(x);
  88 }
  89
  90 REGISTER_FUNCTION(exp, eval_func(exp_eval).
  91                        evalf_func(exp_evalf).
  92                        derivative_func(exp_deriv).
  93                        latex_name("\\exp"));
  94
  95 //////////
  96 // natural logarithm
  97 //////////
  98
  99 static ex log_evalf(const ex & x)
 100 {
 101         if (is_exactly_a<numeric>(x))
 102                 return log(ex_to<numeric>(x));
 103
 104         return log(x).hold();
 105 }
 106
 107 static ex log_eval(const ex & x)
 108 {
 109         if (x.info(info_flags::numeric)) {
 110                 if (x.is_zero())         // log(0) -> infinity
 111                         throw(pole_error("log_eval(): log(0)",0));
 112                 if (x.info(info_flags::real) && x.info(info_flags::negative))
 113                         return (log(-x)+I*Pi);
 114                 if (x.is_equal(_ex1))  // log(1) -> 0
 115                         return _ex0;
 116                 if (x.is_equal(I))       // log(I) -> Pi*I/2
 117                         return (Pi*I*_num1_2);
 118                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
 119                         return (Pi*I*_num_1_2);
 120                 // log(float)
 121                 if (!x.info(info_flags::crational))
 122                         return log(ex_to<numeric>(x));
 123         }
 124         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
 125         if (is_ex_the_function(x, exp)) {
 126                 const ex &t = x.op(0);
 127                 if (t.info(info_flags::numeric)) {
 128                         const numeric &nt = ex_to<numeric>(t);
 129                         if (nt.is_real())
 130                                 return t;
 131                 }
 132         }
 133
 134         return log(x).hold();
 135 }
 136
 137 static ex log_deriv(const ex & x, unsigned deriv_param)
 138 {
 139         GINAC_ASSERT(deriv_param==0);
 140
 141         // d/dx log(x) -> 1/x
 142         return power(x, _ex_1);
 143 }
 144
 145 static ex log_series(const ex &arg,
 146                      const relational &rel,
 147                      int order,
 148                      unsigned options)
 149 {
 150         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 151         ex arg_pt;
 152         bool must_expand_arg = false;
 153         // maybe substitution of rel into arg fails because of a pole
 154         try {
 155                 arg_pt = arg.subs(rel, subs_options::no_pattern);
 156         } catch (pole_error) {
 157                 must_expand_arg = true;
 158         }
 159         // or we are at the branch point anyways
 160         if (arg_pt.is_zero())
 161                 must_expand_arg = true;
 162
 163         if (must_expand_arg) {
 164                 // method:
 165                 // This is the branch point: Series expand the argument first, then
 166                 // trivially factorize it to isolate that part which has constant
 167                 // leading coefficient in this fashion:
 168                 //   x^n + x^(n+1) +...+ Order(x^(n+m))  ->  x^n * (1 + x +...+ Order(x^m)).
 169                 // Return a plain n*log(x) for the x^n part and series expand the
 170                 // other part.  Add them together and reexpand again in order to have
 171                 // one unnested pseries object.  All this also works for negative n.
 172                 pseries argser;          // series expansion of log's argument
 173                 unsigned extra_ord = 0;  // extra expansion order
 174                 do {
 175                         // oops, the argument expanded to a pure Order(x^something)...
 176                         argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
 177                         ++extra_ord;
 178                 } while (!argser.is_terminating() && argser.nops()==1);
 179
 180                 const symbol &s = ex_to<symbol>(rel.lhs());
 181                 const ex &point = rel.rhs();
 182                 const int n = argser.ldegree(s);
 183                 epvector seq;
 184                 // construct what we carelessly called the n*log(x) term above
 185                 const ex coeff = argser.coeff(s, n);
 186                 // expand the log, but only if coeff is real and > 0, since otherwise
 187                 // it would make the branch cut run into the wrong direction
 188                 if (coeff.info(info_flags::positive))
 189                         seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
 190                 else
 191                         seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));
 192
 193                 if (!argser.is_terminating() || argser.nops()!=1) {
 194                         // in this case n more (or less) terms are needed
 195                         // (sadly, to generate them, we have to start from the beginning)
 196                         const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
 197                         return pseries(rel, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
 198                 } else  // it was a monomial
 199                         return pseries(rel, seq);
 200         }
 201         if (!(options & series_options::suppress_branchcut) &&
 202              arg_pt.info(info_flags::negative)) {
 203                 // method:
 204                 // This is the branch cut: assemble the primitive series manually and
 205                 // then add the corresponding complex step function.
 206                 const symbol &s = ex_to<symbol>(rel.lhs());
 207                 const ex &point = rel.rhs();
 208                 const symbol foo;
 209                 const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 210                 epvector seq;
 211                 seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
 212                 seq.push_back(expair(Order(_ex1), order));
 213                 return series(replarg - I*Pi + pseries(rel, seq), rel, order);
 214         }
 215         throw do_taylor();  // caught by function::series()
 216 }
 217
 218 REGISTER_FUNCTION(log, eval_func(log_eval).
 219                        evalf_func(log_evalf).
 220                        derivative_func(log_deriv).
 221                        series_func(log_series).
 222                        latex_name("\\ln"));
 223
 224 //////////
 225 // sine (trigonometric function)
 226 //////////
 227
 228 static ex sin_evalf(const ex & x)
 229 {
 230         if (is_exactly_a<numeric>(x))
 231                 return sin(ex_to<numeric>(x));
 232
 233         return sin(x).hold();
 234 }
 235
 236 static ex sin_eval(const ex & x)
 237 {
 238         // sin(n/d*Pi) -> { all known non-nested radicals }
 239         const ex SixtyExOverPi = _ex60*x/Pi;
 240         ex sign = _ex1;
 241         if (SixtyExOverPi.info(info_flags::integer)) {
 242                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num120);
 243                 if (z>=_num60) {
 244                         // wrap to interval [0, Pi)
 245                         z -= _num60;
 246                         sign = _ex_1;
 247                 }
 248                 if (z>_num30) {
 249                         // wrap to interval [0, Pi/2)
 250                         z = _num60-z;
 251                 }
 252                 if (z.is_equal(_num0))  // sin(0)       -> 0
 253                         return _ex0;
 254                 if (z.is_equal(_num5))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
 255                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 256                 if (z.is_equal(_num6))  // sin(Pi/10)   -> sqrt(5)/4-1/4
 257                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 258                 if (z.is_equal(_num10)) // sin(Pi/6)    -> 1/2
 259                         return sign*_ex1_2;
 260                 if (z.is_equal(_num15)) // sin(Pi/4)    -> sqrt(2)/2
 261                         return sign*_ex1_2*sqrt(_ex2);
 262                 if (z.is_equal(_num18)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
 263                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 264                 if (z.is_equal(_num20)) // sin(Pi/3)    -> sqrt(3)/2
 265                         return sign*_ex1_2*sqrt(_ex3);
 266                 if (z.is_equal(_num25)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
 267                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 268                 if (z.is_equal(_num30)) // sin(Pi/2)    -> 1
 269                         return sign;
 270         }
 271
 272         if (is_exactly_a<function>(x)) {
 273                 const ex &t = x.op(0);
 274                 // sin(asin(x)) -> x
 275                 if (is_ex_the_function(x, asin))
 276                         return t;
 277                 // sin(acos(x)) -> sqrt(1-x^2)
 278                 if (is_ex_the_function(x, acos))
 279                         return sqrt(_ex1-power(t,_ex2));
 280                 // sin(atan(x)) -> x/sqrt(1+x^2)
 281                 if (is_ex_the_function(x, atan))
 282                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 283         }
 284
 285         // sin(float) -> float
 286         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 287                 return sin(ex_to<numeric>(x));
 288
 289         return sin(x).hold();
 290 }
 291
 292 static ex sin_deriv(const ex & x, unsigned deriv_param)
 293 {
 294         GINAC_ASSERT(deriv_param==0);
 295
 296         // d/dx sin(x) -> cos(x)
 297         return cos(x);
 298 }
 299
 300 REGISTER_FUNCTION(sin, eval_func(sin_eval).
 301                        evalf_func(sin_evalf).
 302                        derivative_func(sin_deriv).
 303                        latex_name("\\sin"));
 304
 305 //////////
 306 // cosine (trigonometric function)
 307 //////////
 308
 309 static ex cos_evalf(const ex & x)
 310 {
 311         if (is_exactly_a<numeric>(x))
 312                 return cos(ex_to<numeric>(x));
 313
 314         return cos(x).hold();
 315 }
 316
 317 static ex cos_eval(const ex & x)
 318 {
 319         // cos(n/d*Pi) -> { all known non-nested radicals }
 320         const ex SixtyExOverPi = _ex60*x/Pi;
 321         ex sign = _ex1;
 322         if (SixtyExOverPi.info(info_flags::integer)) {
 323                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num120);
 324                 if (z>=_num60) {
 325                         // wrap to interval [0, Pi)
 326                         z = _num120-z;
 327                 }
 328                 if (z>=_num30) {
 329                         // wrap to interval [0, Pi/2)
 330                         z = _num60-z;
 331                         sign = _ex_1;
 332                 }
 333                 if (z.is_equal(_num0))  // cos(0)       -> 1
 334                         return sign;
 335                 if (z.is_equal(_num5))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 336                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 337                 if (z.is_equal(_num10)) // cos(Pi/6)    -> sqrt(3)/2
 338                         return sign*_ex1_2*sqrt(_ex3);
 339                 if (z.is_equal(_num12)) // cos(Pi/5)    -> sqrt(5)/4+1/4
 340                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 341                 if (z.is_equal(_num15)) // cos(Pi/4)    -> sqrt(2)/2
 342                         return sign*_ex1_2*sqrt(_ex2);
 343                 if (z.is_equal(_num20)) // cos(Pi/3)    -> 1/2
 344                         return sign*_ex1_2;
 345                 if (z.is_equal(_num24)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 346                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 347                 if (z.is_equal(_num25)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 348                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 349                 if (z.is_equal(_num30)) // cos(Pi/2)    -> 0
 350                         return _ex0;
 351         }
 352
 353         if (is_exactly_a<function>(x)) {
 354                 const ex &t = x.op(0);
 355                 // cos(acos(x)) -> x
 356                 if (is_ex_the_function(x, acos))
 357                         return t;
 358                 // cos(asin(x)) -> sqrt(1-x^2)
 359                 if (is_ex_the_function(x, asin))
 360                         return sqrt(_ex1-power(t,_ex2));
 361                 // cos(atan(x)) -> 1/sqrt(1+x^2)
 362                 if (is_ex_the_function(x, atan))
 363                         return power(_ex1+power(t,_ex2),_ex_1_2);
 364         }
 365
 366         // cos(float) -> float
 367         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 368                 return cos(ex_to<numeric>(x));
 369
 370         return cos(x).hold();
 371 }
 372
 373 static ex cos_deriv(const ex & x, unsigned deriv_param)
 374 {
 375         GINAC_ASSERT(deriv_param==0);
 376
 377         // d/dx cos(x) -> -sin(x)
 378         return -sin(x);
 379 }
 380
 381 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 382                        evalf_func(cos_evalf).
 383                        derivative_func(cos_deriv).
 384                        latex_name("\\cos"));
 385
 386 //////////
 387 // tangent (trigonometric function)
 388 //////////
 389
 390 static ex tan_evalf(const ex & x)
 391 {
 392         if (is_exactly_a<numeric>(x))
 393                 return tan(ex_to<numeric>(x));
 394
 395         return tan(x).hold();
 396 }
 397
 398 static ex tan_eval(const ex & x)
 399 {
 400         // tan(n/d*Pi) -> { all known non-nested radicals }
 401         const ex SixtyExOverPi = _ex60*x/Pi;
 402         ex sign = _ex1;
 403         if (SixtyExOverPi.info(info_flags::integer)) {
 404                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num60);
 405                 if (z>=_num60) {
 406                         // wrap to interval [0, Pi)
 407                         z -= _num60;
 408                 }
 409                 if (z>=_num30) {
 410                         // wrap to interval [0, Pi/2)
 411                         z = _num60-z;
 412                         sign = _ex_1;
 413                 }
 414                 if (z.is_equal(_num0))  // tan(0)       -> 0
 415                         return _ex0;
 416                 if (z.is_equal(_num5))  // tan(Pi/12)   -> 2-sqrt(3)
 417                         return sign*(_ex2-sqrt(_ex3));
 418                 if (z.is_equal(_num10)) // tan(Pi/6)    -> sqrt(3)/3
 419                         return sign*_ex1_3*sqrt(_ex3);
 420                 if (z.is_equal(_num15)) // tan(Pi/4)    -> 1
 421                         return sign;
 422                 if (z.is_equal(_num20)) // tan(Pi/3)    -> sqrt(3)
 423                         return sign*sqrt(_ex3);
 424                 if (z.is_equal(_num25)) // tan(5/12*Pi) -> 2+sqrt(3)
 425                         return sign*(sqrt(_ex3)+_ex2);
 426                 if (z.is_equal(_num30)) // tan(Pi/2)    -> infinity
 427                         throw (pole_error("tan_eval(): simple pole",1));
 428         }
 429
 430         if (is_exactly_a<function>(x)) {
 431                 const ex &t = x.op(0);
 432                 // tan(atan(x)) -> x
 433                 if (is_ex_the_function(x, atan))
 434                         return t;
 435                 // tan(asin(x)) -> x/sqrt(1+x^2)
 436                 if (is_ex_the_function(x, asin))
 437                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 438                 // tan(acos(x)) -> sqrt(1-x^2)/x
 439                 if (is_ex_the_function(x, acos))
 440                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
 441         }
 442
 443         // tan(float) -> float
 444         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 445                 return tan(ex_to<numeric>(x));
 446         }
 447
 448         return tan(x).hold();
 449 }
 450
 451 static ex tan_deriv(const ex & x, unsigned deriv_param)
 452 {
 453         GINAC_ASSERT(deriv_param==0);
 454
 455         // d/dx tan(x) -> 1+tan(x)^2;
 456         return (_ex1+power(tan(x),_ex2));
 457 }
 458
 459 static ex tan_series(const ex &x,
 460                      const relational &rel,
 461                      int order,
 462                      unsigned options)
 463 {
 464         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 465         // method:
 466         // Taylor series where there is no pole falls back to tan_deriv.
 467         // On a pole simply expand sin(x)/cos(x).
 468         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 469         if (!(2*x_pt/Pi).info(info_flags::odd))
 470                 throw do_taylor();  // caught by function::series()
 471         // if we got here we have to care for a simple pole
 472         return (sin(x)/cos(x)).series(rel, order+2, options);
 473 }
 474
 475 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 476                        evalf_func(tan_evalf).
 477                        derivative_func(tan_deriv).
 478                        series_func(tan_series).
 479                        latex_name("\\tan"));
 480
 481 //////////
 482 // inverse sine (arc sine)
 483 //////////
 484
 485 static ex asin_evalf(const ex & x)
 486 {
 487         if (is_exactly_a<numeric>(x))
 488                 return asin(ex_to<numeric>(x));
 489
 490         return asin(x).hold();
 491 }
 492
 493 static ex asin_eval(const ex & x)
 494 {
 495         if (x.info(info_flags::numeric)) {
 496                 // asin(0) -> 0
 497                 if (x.is_zero())
 498                         return x;
 499                 // asin(1/2) -> Pi/6
 500                 if (x.is_equal(_ex1_2))
 501                         return numeric(1,6)*Pi;
 502                 // asin(1) -> Pi/2
 503                 if (x.is_equal(_ex1))
 504                         return _num1_2*Pi;
 505                 // asin(-1/2) -> -Pi/6
 506                 if (x.is_equal(_ex_1_2))
 507                         return numeric(-1,6)*Pi;
 508                 // asin(-1) -> -Pi/2
 509                 if (x.is_equal(_ex_1))
 510                         return _num_1_2*Pi;
 511                 // asin(float) -> float
 512                 if (!x.info(info_flags::crational))
 513                         return asin(ex_to<numeric>(x));
 514         }
 515
 516         return asin(x).hold();
 517 }
 518
 519 static ex asin_deriv(const ex & x, unsigned deriv_param)
 520 {
 521         GINAC_ASSERT(deriv_param==0);
 522
 523         // d/dx asin(x) -> 1/sqrt(1-x^2)
 524         return power(1-power(x,_ex2),_ex_1_2);
 525 }
 526
 527 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 528                         evalf_func(asin_evalf).
 529                         derivative_func(asin_deriv).
 530                         latex_name("\\arcsin"));
 531
 532 //////////
 533 // inverse cosine (arc cosine)
 534 //////////
 535
 536 static ex acos_evalf(const ex & x)
 537 {
 538         if (is_exactly_a<numeric>(x))
 539                 return acos(ex_to<numeric>(x));
 540
 541         return acos(x).hold();
 542 }
 543
 544 static ex acos_eval(const ex & x)
 545 {
 546         if (x.info(info_flags::numeric)) {
 547                 // acos(1) -> 0
 548                 if (x.is_equal(_ex1))
 549                         return _ex0;
 550                 // acos(1/2) -> Pi/3
 551                 if (x.is_equal(_ex1_2))
 552                         return _ex1_3*Pi;
 553                 // acos(0) -> Pi/2
 554                 if (x.is_zero())
 555                         return _ex1_2*Pi;
 556                 // acos(-1/2) -> 2/3*Pi
 557                 if (x.is_equal(_ex_1_2))
 558                         return numeric(2,3)*Pi;
 559                 // acos(-1) -> Pi
 560                 if (x.is_equal(_ex_1))
 561                         return Pi;
 562                 // acos(float) -> float
 563                 if (!x.info(info_flags::crational))
 564                         return acos(ex_to<numeric>(x));
 565         }
 566
 567         return acos(x).hold();
 568 }
 569
 570 static ex acos_deriv(const ex & x, unsigned deriv_param)
 571 {
 572         GINAC_ASSERT(deriv_param==0);
 573
 574         // d/dx acos(x) -> -1/sqrt(1-x^2)
 575         return -power(1-power(x,_ex2),_ex_1_2);
 576 }
 577
 578 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 579                         evalf_func(acos_evalf).
 580                         derivative_func(acos_deriv).
 581                         latex_name("\\arccos"));
 582
 583 //////////
 584 // inverse tangent (arc tangent)
 585 //////////
 586
 587 static ex atan_evalf(const ex & x)
 588 {
 589         if (is_exactly_a<numeric>(x))
 590                 return atan(ex_to<numeric>(x));
 591
 592         return atan(x).hold();
 593 }
 594
 595 static ex atan_eval(const ex & x)
 596 {
 597         if (x.info(info_flags::numeric)) {
 598                 // atan(0) -> 0
 599                 if (x.is_zero())
 600                         return _ex0;
 601                 // atan(1) -> Pi/4
 602                 if (x.is_equal(_ex1))
 603                         return _ex1_4*Pi;
 604                 // atan(-1) -> -Pi/4
 605                 if (x.is_equal(_ex_1))
 606                         return _ex_1_4*Pi;
 607                 if (x.is_equal(I) || x.is_equal(-I))
 608                         throw (pole_error("atan_eval(): logarithmic pole",0));
 609                 // atan(float) -> float
 610                 if (!x.info(info_flags::crational))
 611                         return atan(ex_to<numeric>(x));
 612         }
 613
 614         return atan(x).hold();
 615 }
 616
 617 static ex atan_deriv(const ex & x, unsigned deriv_param)
 618 {
 619         GINAC_ASSERT(deriv_param==0);
 620
 621         // d/dx atan(x) -> 1/(1+x^2)
 622         return power(_ex1+power(x,_ex2), _ex_1);
 623 }
 624
 625 static ex atan_series(const ex &arg,
 626                       const relational &rel,
 627                       int order,
 628                       unsigned options)
 629 {
 630         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 631         // method:
 632         // Taylor series where there is no pole or cut falls back to atan_deriv.
 633         // There are two branch cuts, one runnig from I up the imaginary axis and
 634         // one running from -I down the imaginary axis.  The points I and -I are
 635         // poles.
 636         // On the branch cuts and the poles series expand
 637         //     (log(1+I*x)-log(1-I*x))/(2*I)
 638         // instead.
 639         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 640         if (!(I*arg_pt).info(info_flags::real))
 641                 throw do_taylor();     // Re(x) != 0
 642         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
 643                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
 644         // care for the poles, using the defining formula for atan()...
 645         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
 646                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
 647         if (!(options & series_options::suppress_branchcut)) {
 648                 // method:
 649                 // This is the branch cut: assemble the primitive series manually and
 650                 // then add the corresponding complex step function.
 651                 const symbol &s = ex_to<symbol>(rel.lhs());
 652                 const ex &point = rel.rhs();
 653                 const symbol foo;
 654                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 655                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
 656                 if ((I*arg_pt)<_ex0)
 657                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
 658                 else
 659                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
 660                 epvector seq;
 661                 seq.push_back(expair(Order0correction, _ex0));
 662                 seq.push_back(expair(Order(_ex1), order));
 663                 return series(replarg - pseries(rel, seq), rel, order);
 664         }
 665         throw do_taylor();
 666 }
 667
 668 REGISTER_FUNCTION(atan, eval_func(atan_eval).
 669                         evalf_func(atan_evalf).
 670                         derivative_func(atan_deriv).
 671                         series_func(atan_series).
 672                         latex_name("\\arctan"));
 673
 674 //////////
 675 // inverse tangent (atan2(y,x))
 676 //////////
 677
 678 static ex atan2_evalf(const ex &y, const ex &x)
 679 {
 680         if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
 681                 return atan2(ex_to<numeric>(y), ex_to<numeric>(x));
 682
 683         return atan2(y, x).hold();
 684 }
 685
 686 static ex atan2_eval(const ex & y, const ex & x)
 687 {
 688         if (y.info(info_flags::numeric) && !y.info(info_flags::crational) &&
 689                 x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 690                 return atan2_evalf(y,x);
 691         }
 692
 693         return atan2(y,x).hold();
 694 }
 695
 696 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
 697 {
 698         GINAC_ASSERT(deriv_param<2);
 699
 700         if (deriv_param==0) {
 701                 // d/dy atan(y,x)
 702                 return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
 703         }
 704         // d/dx atan(y,x)
 705         return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
 706 }
 707
 708 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
 709                          evalf_func(atan2_evalf).
 710                          derivative_func(atan2_deriv));
 711
 712 //////////
 713 // hyperbolic sine (trigonometric function)
 714 //////////
 715
 716 static ex sinh_evalf(const ex & x)
 717 {
 718         if (is_exactly_a<numeric>(x))
 719                 return sinh(ex_to<numeric>(x));
 720
 721         return sinh(x).hold();
 722 }
 723
 724 static ex sinh_eval(const ex & x)
 725 {
 726         if (x.info(info_flags::numeric)) {
 727                 if (x.is_zero())  // sinh(0) -> 0
 728                         return _ex0;
 729                 if (!x.info(info_flags::crational))  // sinh(float) -> float
 730                         return sinh(ex_to<numeric>(x));
 731         }
 732
 733         if ((x/Pi).info(info_flags::numeric) &&
 734                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
 735                 return I*sin(x/I);
 736
 737         if (is_exactly_a<function>(x)) {
 738                 const ex &t = x.op(0);
 739                 // sinh(asinh(x)) -> x
 740                 if (is_ex_the_function(x, asinh))
 741                         return t;
 742                 // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
 743                 if (is_ex_the_function(x, acosh))
 744                         return sqrt(t-_ex1)*sqrt(t+_ex1);
 745                 // sinh(atanh(x)) -> x/sqrt(1-x^2)
 746                 if (is_ex_the_function(x, atanh))
 747                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 748         }
 749
 750         return sinh(x).hold();
 751 }
 752
 753 static ex sinh_deriv(const ex & x, unsigned deriv_param)
 754 {
 755         GINAC_ASSERT(deriv_param==0);
 756
 757         // d/dx sinh(x) -> cosh(x)
 758         return cosh(x);
 759 }
 760
 761 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
 762                         evalf_func(sinh_evalf).
 763                         derivative_func(sinh_deriv).
 764                         latex_name("\\sinh"));
 765
 766 //////////
 767 // hyperbolic cosine (trigonometric function)
 768 //////////
 769
 770 static ex cosh_evalf(const ex & x)
 771 {
 772         if (is_exactly_a<numeric>(x))
 773                 return cosh(ex_to<numeric>(x));
 774
 775         return cosh(x).hold();
 776 }
 777
 778 static ex cosh_eval(const ex & x)
 779 {
 780         if (x.info(info_flags::numeric)) {
 781                 if (x.is_zero())  // cosh(0) -> 1
 782                         return _ex1;
 783                 if (!x.info(info_flags::crational))  // cosh(float) -> float
 784                         return cosh(ex_to<numeric>(x));
 785         }
 786
 787         if ((x/Pi).info(info_flags::numeric) &&
 788                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
 789                 return cos(x/I);
 790
 791         if (is_exactly_a<function>(x)) {
 792                 const ex &t = x.op(0);
 793                 // cosh(acosh(x)) -> x
 794                 if (is_ex_the_function(x, acosh))
 795                         return t;
 796                 // cosh(asinh(x)) -> sqrt(1+x^2)
 797                 if (is_ex_the_function(x, asinh))
 798                         return sqrt(_ex1+power(t,_ex2));
 799                 // cosh(atanh(x)) -> 1/sqrt(1-x^2)
 800                 if (is_ex_the_function(x, atanh))
 801                         return power(_ex1-power(t,_ex2),_ex_1_2);
 802         }
 803
 804         return cosh(x).hold();
 805 }
 806
 807 static ex cosh_deriv(const ex & x, unsigned deriv_param)
 808 {
 809         GINAC_ASSERT(deriv_param==0);
 810
 811         // d/dx cosh(x) -> sinh(x)
 812         return sinh(x);
 813 }
 814
 815 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
 816                         evalf_func(cosh_evalf).
 817                         derivative_func(cosh_deriv).
 818                         latex_name("\\cosh"));
 819
 820 //////////
 821 // hyperbolic tangent (trigonometric function)
 822 //////////
 823
 824 static ex tanh_evalf(const ex & x)
 825 {
 826         if (is_exactly_a<numeric>(x))
 827                 return tanh(ex_to<numeric>(x));
 828
 829         return tanh(x).hold();
 830 }
 831
 832 static ex tanh_eval(const ex & x)
 833 {
 834         if (x.info(info_flags::numeric)) {
 835                 if (x.is_zero())  // tanh(0) -> 0
 836                         return _ex0;
 837                 if (!x.info(info_flags::crational))  // tanh(float) -> float
 838                         return tanh(ex_to<numeric>(x));
 839         }
 840
 841         if ((x/Pi).info(info_flags::numeric) &&
 842                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
 843                 return I*tan(x/I);
 844
 845         if (is_exactly_a<function>(x)) {
 846                 const ex &t = x.op(0);
 847                 // tanh(atanh(x)) -> x
 848                 if (is_ex_the_function(x, atanh))
 849                         return t;
 850                 // tanh(asinh(x)) -> x/sqrt(1+x^2)
 851                 if (is_ex_the_function(x, asinh))
 852                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 853                 // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
 854                 if (is_ex_the_function(x, acosh))
 855                         return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
 856         }
 857
 858         return tanh(x).hold();
 859 }
 860
 861 static ex tanh_deriv(const ex & x, unsigned deriv_param)
 862 {
 863         GINAC_ASSERT(deriv_param==0);
 864
 865         // d/dx tanh(x) -> 1-tanh(x)^2
 866         return _ex1-power(tanh(x),_ex2);
 867 }
 868
 869 static ex tanh_series(const ex &x,
 870                       const relational &rel,
 871                       int order,
 872                       unsigned options)
 873 {
 874         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 875         // method:
 876         // Taylor series where there is no pole falls back to tanh_deriv.
 877         // On a pole simply expand sinh(x)/cosh(x).
 878         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 879         if (!(2*I*x_pt/Pi).info(info_flags::odd))
 880                 throw do_taylor();  // caught by function::series()
 881         // if we got here we have to care for a simple pole
 882         return (sinh(x)/cosh(x)).series(rel, order+2, options);
 883 }
 884
 885 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
 886                         evalf_func(tanh_evalf).
 887                         derivative_func(tanh_deriv).
 888                         series_func(tanh_series).
 889                         latex_name("\\tanh"));
 890
 891 //////////
 892 // inverse hyperbolic sine (trigonometric function)
 893 //////////
 894
 895 static ex asinh_evalf(const ex & x)
 896 {
 897         if (is_exactly_a<numeric>(x))
 898                 return asinh(ex_to<numeric>(x));
 899
 900         return asinh(x).hold();
 901 }
 902
 903 static ex asinh_eval(const ex & x)
 904 {
 905         if (x.info(info_flags::numeric)) {
 906                 // asinh(0) -> 0
 907                 if (x.is_zero())
 908                         return _ex0;
 909                 // asinh(float) -> float
 910                 if (!x.info(info_flags::crational))
 911                         return asinh(ex_to<numeric>(x));
 912         }
 913
 914         return asinh(x).hold();
 915 }
 916
 917 static ex asinh_deriv(const ex & x, unsigned deriv_param)
 918 {
 919         GINAC_ASSERT(deriv_param==0);
 920
 921         // d/dx asinh(x) -> 1/sqrt(1+x^2)
 922         return power(_ex1+power(x,_ex2),_ex_1_2);
 923 }
 924
 925 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
 926                          evalf_func(asinh_evalf).
 927                          derivative_func(asinh_deriv));
 928
 929 //////////
 930 // inverse hyperbolic cosine (trigonometric function)
 931 //////////
 932
 933 static ex acosh_evalf(const ex & x)
 934 {
 935         if (is_exactly_a<numeric>(x))
 936                 return acosh(ex_to<numeric>(x));
 937
 938         return acosh(x).hold();
 939 }
 940
 941 static ex acosh_eval(const ex & x)
 942 {
 943         if (x.info(info_flags::numeric)) {
 944                 // acosh(0) -> Pi*I/2
 945                 if (x.is_zero())
 946                         return Pi*I*numeric(1,2);
 947                 // acosh(1) -> 0
 948                 if (x.is_equal(_ex1))
 949                         return _ex0;
 950                 // acosh(-1) -> Pi*I
 951                 if (x.is_equal(_ex_1))
 952                         return Pi*I;
 953                 // acosh(float) -> float
 954                 if (!x.info(info_flags::crational))
 955                         return acosh(ex_to<numeric>(x));
 956         }
 957
 958         return acosh(x).hold();
 959 }
 960
 961 static ex acosh_deriv(const ex & x, unsigned deriv_param)
 962 {
 963         GINAC_ASSERT(deriv_param==0);
 964
 965         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
 966         return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
 967 }
 968
 969 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
 970                          evalf_func(acosh_evalf).
 971                          derivative_func(acosh_deriv));
 972
 973 //////////
 974 // inverse hyperbolic tangent (trigonometric function)
 975 //////////
 976
 977 static ex atanh_evalf(const ex & x)
 978 {
 979         if (is_exactly_a<numeric>(x))
 980                 return atanh(ex_to<numeric>(x));
 981
 982         return atanh(x).hold();
 983 }
 984
 985 static ex atanh_eval(const ex & x)
 986 {
 987         if (x.info(info_flags::numeric)) {
 988                 // atanh(0) -> 0
 989                 if (x.is_zero())
 990                         return _ex0;
 991                 // atanh({+|-}1) -> throw
 992                 if (x.is_equal(_ex1) || x.is_equal(_ex_1))
 993                         throw (pole_error("atanh_eval(): logarithmic pole",0));
 994                 // atanh(float) -> float
 995                 if (!x.info(info_flags::crational))
 996                         return atanh(ex_to<numeric>(x));
 997         }
 998
 999         return atanh(x).hold();
1000 }
1001
1002 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1003 {
1004         GINAC_ASSERT(deriv_param==0);
1005
1006         // d/dx atanh(x) -> 1/(1-x^2)
1007         return power(_ex1-power(x,_ex2),_ex_1);
1008 }
1009
1010 static ex atanh_series(const ex &arg,
1011                        const relational &rel,
1012                        int order,
1013                        unsigned options)
1014 {
1015         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1016         // method:
1017         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1018         // There are two branch cuts, one runnig from 1 up the real axis and one
1019         // one running from -1 down the real axis.  The points 1 and -1 are poles
1020         // On the branch cuts and the poles series expand
1021         //     (log(1+x)-log(1-x))/2
1022         // instead.
1023         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
1024         if (!(arg_pt).info(info_flags::real))
1025                 throw do_taylor();     // Im(x) != 0
1026         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1027                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1028         // care for the poles, using the defining formula for atanh()...
1029         if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1030                 return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1031         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1032         if (!(options & series_options::suppress_branchcut)) {
1033                 // method:
1034                 // This is the branch cut: assemble the primitive series manually and
1035                 // then add the corresponding complex step function.
1036                 const symbol &s = ex_to<symbol>(rel.lhs());
1037                 const ex &point = rel.rhs();
1038                 const symbol foo;
1039                 const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1040                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1041                 if (arg_pt<_ex0)
1042                         Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1043                 else
1044                         Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1045                 epvector seq;
1046                 seq.push_back(expair(Order0correction, _ex0));
1047                 seq.push_back(expair(Order(_ex1), order));
1048                 return series(replarg - pseries(rel, seq), rel, order);
1049         }
1050         throw do_taylor();
1051 }
1052
1053 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1054                          evalf_func(atanh_evalf).
1055                          derivative_func(atanh_deriv).
1056                          series_func(atanh_series));
1057
1058
1059 } // namespace GiNaC