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) 1999-2001 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 "relational.h"
  33 #include "symbol.h"
  34 #include "pseries.h"
  35 #include "utils.h"
  36
  37 namespace GiNaC {
  38
  39 //////////
  40 // exponential function
  41 //////////
  42
  43 static ex exp_evalf(const ex & x)
  44 {
  45         if (is_exactly_a<numeric>(x))
  46                 return exp(ex_to<numeric>(x));
  47
  48         return exp(x).hold();
  49 }
  50
  51 static ex exp_eval(const ex & x)
  52 {
  53         // exp(0) -> 1
  54         if (x.is_zero()) {
  55                 return _ex1;
  56         }
  57         // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
  58         const ex TwoExOverPiI=(_ex2*x)/(Pi*I);
  59         if (TwoExOverPiI.info(info_flags::integer)) {
  60                 const numeric z = mod(ex_to<numeric>(TwoExOverPiI),_num4);
  61                 if (z.is_equal(_num0))
  62                         return _ex1;
  63                 if (z.is_equal(_num1))
  64                         return ex(I);
  65                 if (z.is_equal(_num2))
  66                         return _ex_1;
  67                 if (z.is_equal(_num3))
  68                         return ex(-I);
  69         }
  70         // exp(log(x)) -> x
  71         if (is_ex_the_function(x, log))
  72                 return x.op(0);
  73
  74         // exp(float)
  75         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
  76                 return exp(ex_to<numeric>(x));
  77
  78         return exp(x).hold();
  79 }
  80
  81 static ex exp_deriv(const ex & x, unsigned deriv_param)
  82 {
  83         GINAC_ASSERT(deriv_param==0);
  84
  85         // d/dx exp(x) -> exp(x)
  86         return exp(x);
  87 }
  88
  89 REGISTER_FUNCTION(exp, eval_func(exp_eval).
  90                        evalf_func(exp_evalf).
  91                        derivative_func(exp_deriv).
  92                        latex_name("\\exp"));
  93
  94 //////////
  95 // natural logarithm
  96 //////////
  97
  98 static ex log_evalf(const ex & x)
  99 {
 100         if (is_exactly_a<numeric>(x))
 101                 return log(ex_to<numeric>(x));
 102
 103         return log(x).hold();
 104 }
 105
 106 static ex log_eval(const ex & x)
 107 {
 108         if (x.info(info_flags::numeric)) {
 109                 if (x.is_zero())         // log(0) -> infinity
 110                         throw(pole_error("log_eval(): log(0)",0));
 111                 if (x.info(info_flags::real) && x.info(info_flags::negative))
 112                         return (log(-x)+I*Pi);
 113                 if (x.is_equal(_ex1))  // log(1) -> 0
 114                         return _ex0;
 115                 if (x.is_equal(I))       // log(I) -> Pi*I/2
 116                         return (Pi*I*_num1_2);
 117                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
 118                         return (Pi*I*_num_1_2);
 119                 // log(float)
 120                 if (!x.info(info_flags::crational))
 121                         return log(ex_to<numeric>(x));
 122         }
 123         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
 124         if (is_ex_the_function(x, exp)) {
 125                 const ex &t = x.op(0);
 126                 if (t.info(info_flags::numeric)) {
 127                         const numeric &nt = ex_to<numeric>(t);
 128                         if (nt.is_real())
 129                                 return t;
 130                 }
 131         }
 132
 133         return log(x).hold();
 134 }
 135
 136 static ex log_deriv(const ex & x, unsigned deriv_param)
 137 {
 138         GINAC_ASSERT(deriv_param==0);
 139
 140         // d/dx log(x) -> 1/x
 141         return power(x, _ex_1);
 142 }
 143
 144 static ex log_series(const ex &arg,
 145                      const relational &rel,
 146                      int order,
 147                      unsigned options)
 148 {
 149         GINAC_ASSERT(is_exactly_a<symbol>(rel.lhs()));
 150         ex arg_pt;
 151         bool must_expand_arg = false;
 152         // maybe substitution of rel into arg fails because of a pole
 153         try {
 154                 arg_pt = arg.subs(rel);
 155         } catch (pole_error) {
 156                 must_expand_arg = true;
 157         }
 158         // or we are at the branch point anyways
 159         if (arg_pt.is_zero())
 160                 must_expand_arg = true;
 161
 162         if (must_expand_arg) {
 163                 // method:
 164                 // This is the branch point: Series expand the argument first, then
 165                 // trivially factorize it to isolate that part which has constant
 166                 // leading coefficient in this fashion:
 167                 //   x^n + x^(n+1) +...+ Order(x^(n+m))  ->  x^n * (1 + x +...+ Order(x^m)).
 168                 // Return a plain n*log(x) for the x^n part and series expand the
 169                 // other part.  Add them together and reexpand again in order to have
 170                 // one unnested pseries object.  All this also works for negative n.
 171                 pseries argser;          // series expansion of log's argument
 172                 unsigned extra_ord = 0;  // extra expansion order
 173                 do {
 174                         // oops, the argument expanded to a pure Order(x^something)...
 175                         argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
 176                         ++extra_ord;
 177                 } while (!argser.is_terminating() && argser.nops()==1);
 178
 179                 const symbol &s = ex_to<symbol>(rel.lhs());
 180                 const ex &point = rel.rhs();
 181                 const int n = argser.ldegree(s);
 182                 epvector seq;
 183                 // construct what we carelessly called the n*log(x) term above
 184                 const ex coeff = argser.coeff(s, n);
 185                 // expand the log, but only if coeff is real and > 0, since otherwise
 186                 // it would make the branch cut run into the wrong direction
 187                 if (coeff.info(info_flags::positive))
 188                         seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
 189                 else
 190                         seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));
 191
 192                 if (!argser.is_terminating() || argser.nops()!=1) {
 193                         // in this case n more (or less) terms are needed
 194                         // (sadly, to generate them, we have to start from the beginning)
 195                         const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
 196                         return pseries(rel, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
 197                 } else  // it was a monomial
 198                         return pseries(rel, seq);
 199         }
 200         if (!(options & series_options::suppress_branchcut) &&
 201              arg_pt.info(info_flags::negative)) {
 202                 // method:
 203                 // This is the branch cut: assemble the primitive series manually and
 204                 // then add the corresponding complex step function.
 205                 const symbol &s = ex_to<symbol>(rel.lhs());
 206                 const ex &point = rel.rhs();
 207                 const symbol foo;
 208                 const ex replarg = series(log(arg), s==foo, order).subs(foo==point);
 209                 epvector seq;
 210                 seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
 211                 seq.push_back(expair(Order(_ex1), order));
 212                 return series(replarg - I*Pi + pseries(rel, seq), rel, order);
 213         }
 214         throw do_taylor();  // caught by function::series()
 215 }
 216
 217 REGISTER_FUNCTION(log, eval_func(log_eval).
 218                        evalf_func(log_evalf).
 219                        derivative_func(log_deriv).
 220                        series_func(log_series).
 221                        latex_name("\\ln"));
 222
 223 //////////
 224 // sine (trigonometric function)
 225 //////////
 226
 227 static ex sin_evalf(const ex & x)
 228 {
 229         if (is_exactly_a<numeric>(x))
 230                 return sin(ex_to<numeric>(x));
 231
 232         return sin(x).hold();
 233 }
 234
 235 static ex sin_eval(const ex & x)
 236 {
 237         // sin(n/d*Pi) -> { all known non-nested radicals }
 238         const ex SixtyExOverPi = _ex60*x/Pi;
 239         ex sign = _ex1;
 240         if (SixtyExOverPi.info(info_flags::integer)) {
 241                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num120);
 242                 if (z>=_num60) {
 243                         // wrap to interval [0, Pi)
 244                         z -= _num60;
 245                         sign = _ex_1;
 246                 }
 247                 if (z>_num30) {
 248                         // wrap to interval [0, Pi/2)
 249                         z = _num60-z;
 250                 }
 251                 if (z.is_equal(_num0))  // sin(0)       -> 0
 252                         return _ex0;
 253                 if (z.is_equal(_num5))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
 254                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 255                 if (z.is_equal(_num6))  // sin(Pi/10)   -> sqrt(5)/4-1/4
 256                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 257                 if (z.is_equal(_num10)) // sin(Pi/6)    -> 1/2
 258                         return sign*_ex1_2;
 259                 if (z.is_equal(_num15)) // sin(Pi/4)    -> sqrt(2)/2
 260                         return sign*_ex1_2*sqrt(_ex2);
 261                 if (z.is_equal(_num18)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
 262                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 263                 if (z.is_equal(_num20)) // sin(Pi/3)    -> sqrt(3)/2
 264                         return sign*_ex1_2*sqrt(_ex3);
 265                 if (z.is_equal(_num25)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
 266                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 267                 if (z.is_equal(_num30)) // sin(Pi/2)    -> 1
 268                         return sign;
 269         }
 270
 271         if (is_exactly_a<function>(x)) {
 272                 const ex &t = x.op(0);
 273                 // sin(asin(x)) -> x
 274                 if (is_ex_the_function(x, asin))
 275                         return t;
 276                 // sin(acos(x)) -> sqrt(1-x^2)
 277                 if (is_ex_the_function(x, acos))
 278                         return sqrt(_ex1-power(t,_ex2));
 279                 // sin(atan(x)) -> x/sqrt(1+x^2)
 280                 if (is_ex_the_function(x, atan))
 281                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 282         }
 283
 284         // sin(float) -> float
 285         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 286                 return sin(ex_to<numeric>(x));
 287
 288         return sin(x).hold();
 289 }
 290
 291 static ex sin_deriv(const ex & x, unsigned deriv_param)
 292 {
 293         GINAC_ASSERT(deriv_param==0);
 294
 295         // d/dx sin(x) -> cos(x)
 296         return cos(x);
 297 }
 298
 299 REGISTER_FUNCTION(sin, eval_func(sin_eval).
 300                        evalf_func(sin_evalf).
 301                        derivative_func(sin_deriv).
 302                        latex_name("\\sin"));
 303
 304 //////////
 305 // cosine (trigonometric function)
 306 //////////
 307
 308 static ex cos_evalf(const ex & x)
 309 {
 310         if (is_exactly_a<numeric>(x))
 311                 return cos(ex_to<numeric>(x));
 312
 313         return cos(x).hold();
 314 }
 315
 316 static ex cos_eval(const ex & x)
 317 {
 318         // cos(n/d*Pi) -> { all known non-nested radicals }
 319         const ex SixtyExOverPi = _ex60*x/Pi;
 320         ex sign = _ex1;
 321         if (SixtyExOverPi.info(info_flags::integer)) {
 322                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num120);
 323                 if (z>=_num60) {
 324                         // wrap to interval [0, Pi)
 325                         z = _num120-z;
 326                 }
 327                 if (z>=_num30) {
 328                         // wrap to interval [0, Pi/2)
 329                         z = _num60-z;
 330                         sign = _ex_1;
 331                 }
 332                 if (z.is_equal(_num0))  // cos(0)       -> 1
 333                         return sign;
 334                 if (z.is_equal(_num5))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 335                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 336                 if (z.is_equal(_num10)) // cos(Pi/6)    -> sqrt(3)/2
 337                         return sign*_ex1_2*sqrt(_ex3);
 338                 if (z.is_equal(_num12)) // cos(Pi/5)    -> sqrt(5)/4+1/4
 339                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 340                 if (z.is_equal(_num15)) // cos(Pi/4)    -> sqrt(2)/2
 341                         return sign*_ex1_2*sqrt(_ex2);
 342                 if (z.is_equal(_num20)) // cos(Pi/3)    -> 1/2
 343                         return sign*_ex1_2;
 344                 if (z.is_equal(_num24)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 345                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 346                 if (z.is_equal(_num25)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 347                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 348                 if (z.is_equal(_num30)) // cos(Pi/2)    -> 0
 349                         return _ex0;
 350         }
 351
 352         if (is_exactly_a<function>(x)) {
 353                 const ex &t = x.op(0);
 354                 // cos(acos(x)) -> x
 355                 if (is_ex_the_function(x, acos))
 356                         return t;
 357                 // cos(asin(x)) -> sqrt(1-x^2)
 358                 if (is_ex_the_function(x, asin))
 359                         return sqrt(_ex1-power(t,_ex2));
 360                 // cos(atan(x)) -> 1/sqrt(1+x^2)
 361                 if (is_ex_the_function(x, atan))
 362                         return power(_ex1+power(t,_ex2),_ex_1_2);
 363         }
 364
 365         // cos(float) -> float
 366         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 367                 return cos(ex_to<numeric>(x));
 368
 369         return cos(x).hold();
 370 }
 371
 372 static ex cos_deriv(const ex & x, unsigned deriv_param)
 373 {
 374         GINAC_ASSERT(deriv_param==0);
 375
 376         // d/dx cos(x) -> -sin(x)
 377         return -sin(x);
 378 }
 379
 380 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 381                        evalf_func(cos_evalf).
 382                        derivative_func(cos_deriv).
 383                        latex_name("\\cos"));
 384
 385 //////////
 386 // tangent (trigonometric function)
 387 //////////
 388
 389 static ex tan_evalf(const ex & x)
 390 {
 391         if (is_exactly_a<numeric>(x))
 392                 return tan(ex_to<numeric>(x));
 393
 394         return tan(x).hold();
 395 }
 396
 397 static ex tan_eval(const ex & x)
 398 {
 399         // tan(n/d*Pi) -> { all known non-nested radicals }
 400         const ex SixtyExOverPi = _ex60*x/Pi;
 401         ex sign = _ex1;
 402         if (SixtyExOverPi.info(info_flags::integer)) {
 403                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num60);
 404                 if (z>=_num60) {
 405                         // wrap to interval [0, Pi)
 406                         z -= _num60;
 407                 }
 408                 if (z>=_num30) {
 409                         // wrap to interval [0, Pi/2)
 410                         z = _num60-z;
 411                         sign = _ex_1;
 412                 }
 413                 if (z.is_equal(_num0))  // tan(0)       -> 0
 414                         return _ex0;
 415                 if (z.is_equal(_num5))  // tan(Pi/12)   -> 2-sqrt(3)
 416                         return sign*(_ex2-sqrt(_ex3));
 417                 if (z.is_equal(_num10)) // tan(Pi/6)    -> sqrt(3)/3
 418                         return sign*_ex1_3*sqrt(_ex3);
 419                 if (z.is_equal(_num15)) // tan(Pi/4)    -> 1
 420                         return sign;
 421                 if (z.is_equal(_num20)) // tan(Pi/3)    -> sqrt(3)
 422                         return sign*sqrt(_ex3);
 423                 if (z.is_equal(_num25)) // tan(5/12*Pi) -> 2+sqrt(3)
 424                         return sign*(sqrt(_ex3)+_ex2);
 425                 if (z.is_equal(_num30)) // tan(Pi/2)    -> infinity
 426                         throw (pole_error("tan_eval(): simple pole",1));
 427         }
 428
 429         if (is_exactly_a<function>(x)) {
 430                 const ex &t = x.op(0);
 431                 // tan(atan(x)) -> x
 432                 if (is_ex_the_function(x, atan))
 433                         return t;
 434                 // tan(asin(x)) -> x/sqrt(1+x^2)
 435                 if (is_ex_the_function(x, asin))
 436                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 437                 // tan(acos(x)) -> sqrt(1-x^2)/x
 438                 if (is_ex_the_function(x, acos))
 439                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
 440         }
 441
 442         // tan(float) -> float
 443         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 444                 return tan(ex_to<numeric>(x));
 445         }
 446
 447         return tan(x).hold();
 448 }
 449
 450 static ex tan_deriv(const ex & x, unsigned deriv_param)
 451 {
 452         GINAC_ASSERT(deriv_param==0);
 453
 454         // d/dx tan(x) -> 1+tan(x)^2;
 455         return (_ex1+power(tan(x),_ex2));
 456 }
 457
 458 static ex tan_series(const ex &x,
 459                      const relational &rel,
 460                      int order,
 461                      unsigned options)
 462 {
 463         GINAC_ASSERT(is_exactly_a<symbol>(rel.lhs()));
 464         // method:
 465         // Taylor series where there is no pole falls back to tan_deriv.
 466         // On a pole simply expand sin(x)/cos(x).
 467         const ex x_pt = x.subs(rel);
 468         if (!(2*x_pt/Pi).info(info_flags::odd))
 469                 throw do_taylor();  // caught by function::series()
 470         // if we got here we have to care for a simple pole
 471         return (sin(x)/cos(x)).series(rel, order+2, options);
 472 }
 473
 474 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 475                        evalf_func(tan_evalf).
 476                        derivative_func(tan_deriv).
 477                        series_func(tan_series).
 478                        latex_name("\\tan"));
 479
 480 //////////
 481 // inverse sine (arc sine)
 482 //////////
 483
 484 static ex asin_evalf(const ex & x)
 485 {
 486         if (is_exactly_a<numeric>(x))
 487                 return asin(ex_to<numeric>(x));
 488
 489         return asin(x).hold();
 490 }
 491
 492 static ex asin_eval(const ex & x)
 493 {
 494         if (x.info(info_flags::numeric)) {
 495                 // asin(0) -> 0
 496                 if (x.is_zero())
 497                         return x;
 498                 // asin(1/2) -> Pi/6
 499                 if (x.is_equal(_ex1_2))
 500                         return numeric(1,6)*Pi;
 501                 // asin(1) -> Pi/2
 502                 if (x.is_equal(_ex1))
 503                         return _num1_2*Pi;
 504                 // asin(-1/2) -> -Pi/6
 505                 if (x.is_equal(_ex_1_2))
 506                         return numeric(-1,6)*Pi;
 507                 // asin(-1) -> -Pi/2
 508                 if (x.is_equal(_ex_1))
 509                         return _num_1_2*Pi;
 510                 // asin(float) -> float
 511                 if (!x.info(info_flags::crational))
 512                         return asin(ex_to<numeric>(x));
 513         }
 514
 515         return asin(x).hold();
 516 }
 517
 518 static ex asin_deriv(const ex & x, unsigned deriv_param)
 519 {
 520         GINAC_ASSERT(deriv_param==0);
 521
 522         // d/dx asin(x) -> 1/sqrt(1-x^2)
 523         return power(1-power(x,_ex2),_ex_1_2);
 524 }
 525
 526 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 527                         evalf_func(asin_evalf).
 528                         derivative_func(asin_deriv).
 529                         latex_name("\\arcsin"));
 530
 531 //////////
 532 // inverse cosine (arc cosine)
 533 //////////
 534
 535 static ex acos_evalf(const ex & x)
 536 {
 537         if (is_exactly_a<numeric>(x))
 538                 return acos(ex_to<numeric>(x));
 539
 540         return acos(x).hold();
 541 }
 542
 543 static ex acos_eval(const ex & x)
 544 {
 545         if (x.info(info_flags::numeric)) {
 546                 // acos(1) -> 0
 547                 if (x.is_equal(_ex1))
 548                         return _ex0;
 549                 // acos(1/2) -> Pi/3
 550                 if (x.is_equal(_ex1_2))
 551                         return _ex1_3*Pi;
 552                 // acos(0) -> Pi/2
 553                 if (x.is_zero())
 554                         return _ex1_2*Pi;
 555                 // acos(-1/2) -> 2/3*Pi
 556                 if (x.is_equal(_ex_1_2))
 557                         return numeric(2,3)*Pi;
 558                 // acos(-1) -> Pi
 559                 if (x.is_equal(_ex_1))
 560                         return Pi;
 561                 // acos(float) -> float
 562                 if (!x.info(info_flags::crational))
 563                         return acos(ex_to<numeric>(x));
 564         }
 565
 566         return acos(x).hold();
 567 }
 568
 569 static ex acos_deriv(const ex & x, unsigned deriv_param)
 570 {
 571         GINAC_ASSERT(deriv_param==0);
 572
 573         // d/dx acos(x) -> -1/sqrt(1-x^2)
 574         return -power(1-power(x,_ex2),_ex_1_2);
 575 }
 576
 577 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 578                         evalf_func(acos_evalf).
 579                         derivative_func(acos_deriv).
 580                         latex_name("\\arccos"));
 581
 582 //////////
 583 // inverse tangent (arc tangent)
 584 //////////
 585
 586 static ex atan_evalf(const ex & x)
 587 {
 588         if (is_exactly_a<numeric>(x))
 589                 return atan(ex_to<numeric>(x));
 590
 591         return atan(x).hold();
 592 }
 593
 594 static ex atan_eval(const ex & x)
 595 {
 596         if (x.info(info_flags::numeric)) {
 597                 // atan(0) -> 0
 598                 if (x.is_zero())
 599                         return _ex0;
 600                 // atan(1) -> Pi/4
 601                 if (x.is_equal(_ex1))
 602                         return _ex1_4*Pi;
 603                 // atan(-1) -> -Pi/4
 604                 if (x.is_equal(_ex_1))
 605                         return _ex_1_4*Pi;
 606                 if (x.is_equal(I) || x.is_equal(-I))
 607                         throw (pole_error("atan_eval(): logarithmic pole",0));
 608                 // atan(float) -> float
 609                 if (!x.info(info_flags::crational))
 610                         return atan(ex_to<numeric>(x));
 611         }
 612
 613         return atan(x).hold();
 614 }
 615
 616 static ex atan_deriv(const ex & x, unsigned deriv_param)
 617 {
 618         GINAC_ASSERT(deriv_param==0);
 619
 620         // d/dx atan(x) -> 1/(1+x^2)
 621         return power(_ex1+power(x,_ex2), _ex_1);
 622 }
 623
 624 static ex atan_series(const ex &arg,
 625                       const relational &rel,
 626                       int order,
 627                       unsigned options)
 628 {
 629         GINAC_ASSERT(is_exactly_a<symbol>(rel.lhs()));
 630         // method:
 631         // Taylor series where there is no pole or cut falls back to atan_deriv.
 632         // There are two branch cuts, one runnig from I up the imaginary axis and
 633         // one running from -I down the imaginary axis.  The points I and -I are
 634         // poles.
 635         // On the branch cuts and the poles series expand
 636         //     (log(1+I*x)-log(1-I*x))/(2*I)
 637         // instead.
 638         const ex arg_pt = arg.subs(rel);
 639         if (!(I*arg_pt).info(info_flags::real))
 640                 throw do_taylor();     // Re(x) != 0
 641         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
 642                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
 643         // care for the poles, using the defining formula for atan()...
 644         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
 645                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
 646         if (!(options & series_options::suppress_branchcut)) {
 647                 // method:
 648                 // This is the branch cut: assemble the primitive series manually and
 649                 // then add the corresponding complex step function.
 650                 const symbol &s = ex_to<symbol>(rel.lhs());
 651                 const ex &point = rel.rhs();
 652                 const symbol foo;
 653                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point);
 654                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
 655                 if ((I*arg_pt)<_ex0)
 656                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
 657                 else
 658                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
 659                 epvector seq;
 660                 seq.push_back(expair(Order0correction, _ex0));
 661                 seq.push_back(expair(Order(_ex1), order));
 662                 return series(replarg - pseries(rel, seq), rel, order);
 663         }
 664         throw do_taylor();
 665 }
 666
 667 REGISTER_FUNCTION(atan, eval_func(atan_eval).
 668                         evalf_func(atan_evalf).
 669                         derivative_func(atan_deriv).
 670                         series_func(atan_series).
 671                         latex_name("\\arctan"));
 672
 673 //////////
 674 // inverse tangent (atan2(y,x))
 675 //////////
 676
 677 static ex atan2_evalf(const ex &y, const ex &x)
 678 {
 679         if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
 680                 return atan2(ex_to<numeric>(y), ex_to<numeric>(x));
 681
 682         return atan2(y, x).hold();
 683 }
 684
 685 static ex atan2_eval(const ex & y, const ex & x)
 686 {
 687         if (y.info(info_flags::numeric) && !y.info(info_flags::crational) &&
 688                 x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 689                 return atan2_evalf(y,x);
 690         }
 691
 692         return atan2(y,x).hold();
 693 }
 694
 695 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
 696 {
 697         GINAC_ASSERT(deriv_param<2);
 698
 699         if (deriv_param==0) {
 700                 // d/dy atan(y,x)
 701                 return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
 702         }
 703         // d/dx atan(y,x)
 704         return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
 705 }
 706
 707 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
 708                          evalf_func(atan2_evalf).
 709                          derivative_func(atan2_deriv));
 710
 711 //////////
 712 // hyperbolic sine (trigonometric function)
 713 //////////
 714
 715 static ex sinh_evalf(const ex & x)
 716 {
 717         if (is_exactly_a<numeric>(x))
 718                 return sinh(ex_to<numeric>(x));
 719
 720         return sinh(x).hold();
 721 }
 722
 723 static ex sinh_eval(const ex & x)
 724 {
 725         if (x.info(info_flags::numeric)) {
 726                 if (x.is_zero())  // sinh(0) -> 0
 727                         return _ex0;
 728                 if (!x.info(info_flags::crational))  // sinh(float) -> float
 729                         return sinh(ex_to<numeric>(x));
 730         }
 731
 732         if ((x/Pi).info(info_flags::numeric) &&
 733                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
 734                 return I*sin(x/I);
 735
 736         if (is_exactly_a<function>(x)) {
 737                 const ex &t = x.op(0);
 738                 // sinh(asinh(x)) -> x
 739                 if (is_ex_the_function(x, asinh))
 740                         return t;
 741                 // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
 742                 if (is_ex_the_function(x, acosh))
 743                         return sqrt(t-_ex1)*sqrt(t+_ex1);
 744                 // sinh(atanh(x)) -> x/sqrt(1-x^2)
 745                 if (is_ex_the_function(x, atanh))
 746                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 747         }
 748
 749         return sinh(x).hold();
 750 }
 751
 752 static ex sinh_deriv(const ex & x, unsigned deriv_param)
 753 {
 754         GINAC_ASSERT(deriv_param==0);
 755
 756         // d/dx sinh(x) -> cosh(x)
 757         return cosh(x);
 758 }
 759
 760 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
 761                         evalf_func(sinh_evalf).
 762                         derivative_func(sinh_deriv).
 763                         latex_name("\\sinh"));
 764
 765 //////////
 766 // hyperbolic cosine (trigonometric function)
 767 //////////
 768
 769 static ex cosh_evalf(const ex & x)
 770 {
 771         if (is_exactly_a<numeric>(x))
 772                 return cosh(ex_to<numeric>(x));
 773
 774         return cosh(x).hold();
 775 }
 776
 777 static ex cosh_eval(const ex & x)
 778 {
 779         if (x.info(info_flags::numeric)) {
 780                 if (x.is_zero())  // cosh(0) -> 1
 781                         return _ex1;
 782                 if (!x.info(info_flags::crational))  // cosh(float) -> float
 783                         return cosh(ex_to<numeric>(x));
 784         }
 785
 786         if ((x/Pi).info(info_flags::numeric) &&
 787                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
 788                 return cos(x/I);
 789
 790         if (is_exactly_a<function>(x)) {
 791                 const ex &t = x.op(0);
 792                 // cosh(acosh(x)) -> x
 793                 if (is_ex_the_function(x, acosh))
 794                         return t;
 795                 // cosh(asinh(x)) -> sqrt(1+x^2)
 796                 if (is_ex_the_function(x, asinh))
 797                         return sqrt(_ex1+power(t,_ex2));
 798                 // cosh(atanh(x)) -> 1/sqrt(1-x^2)
 799                 if (is_ex_the_function(x, atanh))
 800                         return power(_ex1-power(t,_ex2),_ex_1_2);
 801         }
 802
 803         return cosh(x).hold();
 804 }
 805
 806 static ex cosh_deriv(const ex & x, unsigned deriv_param)
 807 {
 808         GINAC_ASSERT(deriv_param==0);
 809
 810         // d/dx cosh(x) -> sinh(x)
 811         return sinh(x);
 812 }
 813
 814 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
 815                         evalf_func(cosh_evalf).
 816                         derivative_func(cosh_deriv).
 817                         latex_name("\\cosh"));
 818
 819 //////////
 820 // hyperbolic tangent (trigonometric function)
 821 //////////
 822
 823 static ex tanh_evalf(const ex & x)
 824 {
 825         if (is_exactly_a<numeric>(x))
 826                 return tanh(ex_to<numeric>(x));
 827
 828         return tanh(x).hold();
 829 }
 830
 831 static ex tanh_eval(const ex & x)
 832 {
 833         if (x.info(info_flags::numeric)) {
 834                 if (x.is_zero())  // tanh(0) -> 0
 835                         return _ex0;
 836                 if (!x.info(info_flags::crational))  // tanh(float) -> float
 837                         return tanh(ex_to<numeric>(x));
 838         }
 839
 840         if ((x/Pi).info(info_flags::numeric) &&
 841                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
 842                 return I*tan(x/I);
 843
 844         if (is_exactly_a<function>(x)) {
 845                 const ex &t = x.op(0);
 846                 // tanh(atanh(x)) -> x
 847                 if (is_ex_the_function(x, atanh))
 848                         return t;
 849                 // tanh(asinh(x)) -> x/sqrt(1+x^2)
 850                 if (is_ex_the_function(x, asinh))
 851                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 852                 // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
 853                 if (is_ex_the_function(x, acosh))
 854                         return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
 855         }
 856
 857         return tanh(x).hold();
 858 }
 859
 860 static ex tanh_deriv(const ex & x, unsigned deriv_param)
 861 {
 862         GINAC_ASSERT(deriv_param==0);
 863
 864         // d/dx tanh(x) -> 1-tanh(x)^2
 865         return _ex1-power(tanh(x),_ex2);
 866 }
 867
 868 static ex tanh_series(const ex &x,
 869                       const relational &rel,
 870                       int order,
 871                       unsigned options)
 872 {
 873         GINAC_ASSERT(is_exactly_a<symbol>(rel.lhs()));
 874         // method:
 875         // Taylor series where there is no pole falls back to tanh_deriv.
 876         // On a pole simply expand sinh(x)/cosh(x).
 877         const ex x_pt = x.subs(rel);
 878         if (!(2*I*x_pt/Pi).info(info_flags::odd))
 879                 throw do_taylor();  // caught by function::series()
 880         // if we got here we have to care for a simple pole
 881         return (sinh(x)/cosh(x)).series(rel, order+2, options);
 882 }
 883
 884 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
 885                         evalf_func(tanh_evalf).
 886                         derivative_func(tanh_deriv).
 887                         series_func(tanh_series).
 888                         latex_name("\\tanh"));
 889
 890 //////////
 891 // inverse hyperbolic sine (trigonometric function)
 892 //////////
 893
 894 static ex asinh_evalf(const ex & x)
 895 {
 896         if (is_exactly_a<numeric>(x))
 897                 return asinh(ex_to<numeric>(x));
 898
 899         return asinh(x).hold();
 900 }
 901
 902 static ex asinh_eval(const ex & x)
 903 {
 904         if (x.info(info_flags::numeric)) {
 905                 // asinh(0) -> 0
 906                 if (x.is_zero())
 907                         return _ex0;
 908                 // asinh(float) -> float
 909                 if (!x.info(info_flags::crational))
 910                         return asinh(ex_to<numeric>(x));
 911         }
 912
 913         return asinh(x).hold();
 914 }
 915
 916 static ex asinh_deriv(const ex & x, unsigned deriv_param)
 917 {
 918         GINAC_ASSERT(deriv_param==0);
 919
 920         // d/dx asinh(x) -> 1/sqrt(1+x^2)
 921         return power(_ex1+power(x,_ex2),_ex_1_2);
 922 }
 923
 924 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
 925                          evalf_func(asinh_evalf).
 926                          derivative_func(asinh_deriv));
 927
 928 //////////
 929 // inverse hyperbolic cosine (trigonometric function)
 930 //////////
 931
 932 static ex acosh_evalf(const ex & x)
 933 {
 934         if (is_exactly_a<numeric>(x))
 935                 return acosh(ex_to<numeric>(x));
 936
 937         return acosh(x).hold();
 938 }
 939
 940 static ex acosh_eval(const ex & x)
 941 {
 942         if (x.info(info_flags::numeric)) {
 943                 // acosh(0) -> Pi*I/2
 944                 if (x.is_zero())
 945                         return Pi*I*numeric(1,2);
 946                 // acosh(1) -> 0
 947                 if (x.is_equal(_ex1))
 948                         return _ex0;
 949                 // acosh(-1) -> Pi*I
 950                 if (x.is_equal(_ex_1))
 951                         return Pi*I;
 952                 // acosh(float) -> float
 953                 if (!x.info(info_flags::crational))
 954                         return acosh(ex_to<numeric>(x));
 955         }
 956
 957         return acosh(x).hold();
 958 }
 959
 960 static ex acosh_deriv(const ex & x, unsigned deriv_param)
 961 {
 962         GINAC_ASSERT(deriv_param==0);
 963
 964         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
 965         return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
 966 }
 967
 968 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
 969                          evalf_func(acosh_evalf).
 970                          derivative_func(acosh_deriv));
 971
 972 //////////
 973 // inverse hyperbolic tangent (trigonometric function)
 974 //////////
 975
 976 static ex atanh_evalf(const ex & x)
 977 {
 978         if (is_exactly_a<numeric>(x))
 979                 return atanh(ex_to<numeric>(x));
 980
 981         return atanh(x).hold();
 982 }
 983
 984 static ex atanh_eval(const ex & x)
 985 {
 986         if (x.info(info_flags::numeric)) {
 987                 // atanh(0) -> 0
 988                 if (x.is_zero())
 989                         return _ex0;
 990                 // atanh({+|-}1) -> throw
 991                 if (x.is_equal(_ex1) || x.is_equal(_ex_1))
 992                         throw (pole_error("atanh_eval(): logarithmic pole",0));
 993                 // atanh(float) -> float
 994                 if (!x.info(info_flags::crational))
 995                         return atanh(ex_to<numeric>(x));
 996         }
 997
 998         return atanh(x).hold();
 999 }
1000
1001 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1002 {
1003         GINAC_ASSERT(deriv_param==0);
1004
1005         // d/dx atanh(x) -> 1/(1-x^2)
1006         return power(_ex1-power(x,_ex2),_ex_1);
1007 }
1008
1009 static ex atanh_series(const ex &arg,
1010                        const relational &rel,
1011                        int order,
1012                        unsigned options)
1013 {
1014         GINAC_ASSERT(is_exactly_a<symbol>(rel.lhs()));
1015         // method:
1016         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1017         // There are two branch cuts, one runnig from 1 up the real axis and one
1018         // one running from -1 down the real axis.  The points 1 and -1 are poles
1019         // On the branch cuts and the poles series expand
1020         //     (log(1+x)-log(1-x))/2
1021         // instead.
1022         const ex arg_pt = arg.subs(rel);
1023         if (!(arg_pt).info(info_flags::real))
1024                 throw do_taylor();     // Im(x) != 0
1025         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1026                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1027         // care for the poles, using the defining formula for atanh()...
1028         if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1029                 return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1030         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1031         if (!(options & series_options::suppress_branchcut)) {
1032                 // method:
1033                 // This is the branch cut: assemble the primitive series manually and
1034                 // then add the corresponding complex step function.
1035                 const symbol &s = ex_to<symbol>(rel.lhs());
1036                 const ex &point = rel.rhs();
1037                 const symbol foo;
1038                 const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point);
1039                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1040                 if (arg_pt<_ex0)
1041                         Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1042                 else
1043                         Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1044                 epvector seq;
1045                 seq.push_back(expair(Order0correction, _ex0));
1046                 seq.push_back(expair(Order(_ex1), order));
1047                 return series(replarg - pseries(rel, seq), rel, order);
1048         }
1049         throw do_taylor();
1050 }
1051
1052 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1053                          evalf_func(atanh_evalf).
1054                          derivative_func(atanh_deriv).
1055                          series_func(atanh_series));
1056
1057
1058 } // namespace GiNaC