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 #ifndef NO_NAMESPACE_GINAC
  38 namespace GiNaC {
  39 #endif // ndef NO_NAMESPACE_GINAC
  40
  41 //////////
  42 // exponential function
  43 //////////
  44
  45 static ex exp_evalf(const ex & x)
  46 {
  47         BEGIN_TYPECHECK
  48                 TYPECHECK(x,numeric)
  49         END_TYPECHECK(exp(x))
  50
  51         return exp(ex_to_numeric(x)); // -> numeric exp(numeric)
  52 }
  53
  54 static ex exp_eval(const ex & x)
  55 {
  56         // exp(0) -> 1
  57         if (x.is_zero()) {
  58                 return _ex1();
  59         }
  60         // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
  61         ex TwoExOverPiI=(_ex2()*x)/(Pi*I);
  62         if (TwoExOverPiI.info(info_flags::integer)) {
  63                 numeric z=mod(ex_to_numeric(TwoExOverPiI),_num4());
  64                 if (z.is_equal(_num0()))
  65                         return _ex1();
  66                 if (z.is_equal(_num1()))
  67                         return ex(I);
  68                 if (z.is_equal(_num2()))
  69                         return _ex_1();
  70                 if (z.is_equal(_num3()))
  71                         return ex(-I);
  72         }
  73         // exp(log(x)) -> x
  74         if (is_ex_the_function(x, log))
  75                 return x.op(0);
  76
  77         // exp(float)
  78         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
  79                 return exp_evalf(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
  96 //////////
  97 // natural logarithm
  98 //////////
  99
 100 static ex log_evalf(const ex & x)
 101 {
 102         BEGIN_TYPECHECK
 103                 TYPECHECK(x,numeric)
 104         END_TYPECHECK(log(x))
 105
 106         return log(ex_to_numeric(x)); // -> numeric log(numeric)
 107 }
 108
 109 static ex log_eval(const ex & x)
 110 {
 111         if (x.info(info_flags::numeric)) {
 112                 if (x.is_equal(_ex0()))  // 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                         return (log(-x)+I*Pi);
 116                 if (x.is_equal(_ex1()))  // log(1) -> 0
 117                         return _ex0();
 118                 if (x.is_equal(I))       // log(I) -> Pi*I/2
 119                         return (Pi*I*_num1_2());
 120                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
 121                         return (Pi*I*_num_1_2());
 122                 // log(float)
 123                 if (!x.info(info_flags::crational))
 124                         return log_evalf(x);
 125         }
 126         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
 127         if (is_ex_the_function(x, exp)) {
 128                 ex t = x.op(0);
 129                 if (t.info(info_flags::numeric)) {
 130                         numeric nt = ex_to_numeric(t);
 131                         if (nt.is_real())
 132                                 return t;
 133                 }
 134         }
 135
 136         return log(x).hold();
 137 }
 138
 139 static ex log_deriv(const ex & x, unsigned deriv_param)
 140 {
 141         GINAC_ASSERT(deriv_param==0);
 142
 143         // d/dx log(x) -> 1/x
 144         return power(x, _ex_1());
 145 }
 146
 147 static ex log_series(const ex &arg,
 148                      const relational &rel,
 149                      int order,
 150                      unsigned options)
 151 {
 152         GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
 153         ex arg_pt;
 154         bool must_expand_arg = false;
 155         // maybe substitution of rel into arg fails because of a pole
 156         try {
 157                 arg_pt = arg.subs(rel);
 158         } catch (pole_error) {
 159                 must_expand_arg = true;
 160         }
 161         // or we are at the branch point anyways
 162         if (arg_pt.is_zero())
 163                 must_expand_arg = true;
 164
 165         if (must_expand_arg) {
 166                 // method:
 167                 // This is the branch point: Series expand the argument first, then
 168                 // trivially factorize it to isolate that part which has constant
 169                 // leading coefficient in this fashion:
 170                 //   x^n + Order(x^(n+m))  ->  x^n * (1 + Order(x^m)).
 171                 // Return a plain n*log(x) for the x^n part and series expand the
 172                 // other part.  Add them together and reexpand again in order to have
 173                 // one unnested pseries object.  All this also works for negative n.
 174                 const pseries argser = ex_to_pseries(arg.series(rel, order, options));
 175                 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
 176                 const ex point = rel.rhs();
 177                 const int n = argser.ldegree(*s);
 178                 epvector seq;
 179                 // construct what we carelessly called the n*log(x) term above
 180                 ex coeff = argser.coeff(*s, n);
 181                 // expand the log, but only if coeff is real and > 0, since otherwise
 182                 // it would make the branch cut run into the wrong direction
 183                 if (coeff.info(info_flags::positive))
 184                         seq.push_back(expair(n*log(*s-point)+log(coeff), _ex0()));
 185                 else
 186                         seq.push_back(expair(log(coeff*pow(*s-point, n)), _ex0()));
 187                 if (!argser.is_terminating() || argser.nops()!=1) {
 188                         // in this case n more terms are needed
 189                         // (sadly, to generate them, we have to start from the beginning)
 190                         ex newarg = ex_to_pseries((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
 191                         return pseries(rel, seq).add_series(ex_to_pseries(log(newarg).series(rel, order, options)));
 192                 } else  // it was a monomial
 193                         return pseries(rel, seq);
 194         }
 195         if (!(options & series_options::suppress_branchcut) &&
 196              arg_pt.info(info_flags::negative)) {
 197                 // method:
 198                 // This is the branch cut: assemble the primitive series manually and
 199                 // then add the corresponding complex step function.
 200                 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
 201                 const ex point = rel.rhs();
 202                 const symbol foo;
 203                 ex replarg = series(log(arg), *s==foo, order).subs(foo==point);
 204                 epvector seq;
 205                 seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0()));
 206                 seq.push_back(expair(Order(_ex1()), order));
 207                 return series(replarg - I*Pi + pseries(rel, seq), rel, order);
 208         }
 209         throw do_taylor();  // caught by function::series()
 210 }
 211
 212 REGISTER_FUNCTION(log, eval_func(log_eval).
 213                        evalf_func(log_evalf).
 214                        derivative_func(log_deriv).
 215                        series_func(log_series));
 216
 217 //////////
 218 // sine (trigonometric function)
 219 //////////
 220
 221 static ex sin_evalf(const ex & x)
 222 {
 223         BEGIN_TYPECHECK
 224            TYPECHECK(x,numeric)
 225         END_TYPECHECK(sin(x))
 226
 227         return sin(ex_to_numeric(x)); // -> numeric sin(numeric)
 228 }
 229
 230 static ex sin_eval(const ex & x)
 231 {
 232         // sin(n/d*Pi) -> { all known non-nested radicals }
 233         ex SixtyExOverPi = _ex60()*x/Pi;
 234         ex sign = _ex1();
 235         if (SixtyExOverPi.info(info_flags::integer)) {
 236                 numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
 237                 if (z>=_num60()) {
 238                         // wrap to interval [0, Pi)
 239                         z -= _num60();
 240                         sign = _ex_1();
 241                 }
 242                 if (z>_num30()) {
 243                         // wrap to interval [0, Pi/2)
 244                         z = _num60()-z;
 245                 }
 246                 if (z.is_equal(_num0()))  // sin(0)       -> 0
 247                         return _ex0();
 248                 if (z.is_equal(_num5()))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
 249                         return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
 250                 if (z.is_equal(_num6()))  // sin(Pi/10)   -> sqrt(5)/4-1/4
 251                         return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
 252                 if (z.is_equal(_num10())) // sin(Pi/6)    -> 1/2
 253                         return sign*_ex1_2();
 254                 if (z.is_equal(_num15())) // sin(Pi/4)    -> sqrt(2)/2
 255                         return sign*_ex1_2()*power(_ex2(),_ex1_2());
 256                 if (z.is_equal(_num18())) // sin(3/10*Pi) -> sqrt(5)/4+1/4
 257                         return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
 258                 if (z.is_equal(_num20())) // sin(Pi/3)    -> sqrt(3)/2
 259                         return sign*_ex1_2()*power(_ex3(),_ex1_2());
 260                 if (z.is_equal(_num25())) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
 261                         return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
 262                 if (z.is_equal(_num30())) // sin(Pi/2)    -> 1
 263                         return sign*_ex1();
 264         }
 265
 266         if (is_ex_exactly_of_type(x, function)) {
 267                 ex t = x.op(0);
 268                 // sin(asin(x)) -> x
 269                 if (is_ex_the_function(x, asin))
 270                         return t;
 271                 // sin(acos(x)) -> sqrt(1-x^2)
 272                 if (is_ex_the_function(x, acos))
 273                         return power(_ex1()-power(t,_ex2()),_ex1_2());
 274                 // sin(atan(x)) -> x*(1+x^2)^(-1/2)
 275                 if (is_ex_the_function(x, atan))
 276                         return t*power(_ex1()+power(t,_ex2()),_ex_1_2());
 277         }
 278
 279         // sin(float) -> float
 280         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 281                 return sin_evalf(x);
 282
 283         return sin(x).hold();
 284 }
 285
 286 static ex sin_deriv(const ex & x, unsigned deriv_param)
 287 {
 288         GINAC_ASSERT(deriv_param==0);
 289
 290         // d/dx sin(x) -> cos(x)
 291         return cos(x);
 292 }
 293
 294 REGISTER_FUNCTION(sin, eval_func(sin_eval).
 295                        evalf_func(sin_evalf).
 296                        derivative_func(sin_deriv));
 297
 298 //////////
 299 // cosine (trigonometric function)
 300 //////////
 301
 302 static ex cos_evalf(const ex & x)
 303 {
 304         BEGIN_TYPECHECK
 305                 TYPECHECK(x,numeric)
 306         END_TYPECHECK(cos(x))
 307
 308         return cos(ex_to_numeric(x)); // -> numeric cos(numeric)
 309 }
 310
 311 static ex cos_eval(const ex & x)
 312 {
 313         // cos(n/d*Pi) -> { all known non-nested radicals }
 314         ex SixtyExOverPi = _ex60()*x/Pi;
 315         ex sign = _ex1();
 316         if (SixtyExOverPi.info(info_flags::integer)) {
 317                 numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
 318                 if (z>=_num60()) {
 319                         // wrap to interval [0, Pi)
 320                         z = _num120()-z;
 321                 }
 322                 if (z>=_num30()) {
 323                         // wrap to interval [0, Pi/2)
 324                         z = _num60()-z;
 325                         sign = _ex_1();
 326                 }
 327                 if (z.is_equal(_num0()))  // cos(0)       -> 1
 328                         return sign*_ex1();
 329                 if (z.is_equal(_num5()))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 330                         return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
 331                 if (z.is_equal(_num10())) // cos(Pi/6)    -> sqrt(3)/2
 332                         return sign*_ex1_2()*power(_ex3(),_ex1_2());
 333                 if (z.is_equal(_num12())) // cos(Pi/5)    -> sqrt(5)/4+1/4
 334                         return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
 335                 if (z.is_equal(_num15())) // cos(Pi/4)    -> sqrt(2)/2
 336                         return sign*_ex1_2()*power(_ex2(),_ex1_2());
 337                 if (z.is_equal(_num20())) // cos(Pi/3)    -> 1/2
 338                         return sign*_ex1_2();
 339                 if (z.is_equal(_num24())) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 340                         return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
 341                 if (z.is_equal(_num25())) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 342                         return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
 343                 if (z.is_equal(_num30())) // cos(Pi/2)    -> 0
 344                         return sign*_ex0();
 345         }
 346
 347         if (is_ex_exactly_of_type(x, function)) {
 348                 ex t = x.op(0);
 349                 // cos(acos(x)) -> x
 350                 if (is_ex_the_function(x, acos))
 351                         return t;
 352                 // cos(asin(x)) -> (1-x^2)^(1/2)
 353                 if (is_ex_the_function(x, asin))
 354                         return power(_ex1()-power(t,_ex2()),_ex1_2());
 355                 // cos(atan(x)) -> (1+x^2)^(-1/2)
 356                 if (is_ex_the_function(x, atan))
 357                         return power(_ex1()+power(t,_ex2()),_ex_1_2());
 358         }
 359
 360         // cos(float) -> float
 361         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 362                 return cos_evalf(x);
 363
 364         return cos(x).hold();
 365 }
 366
 367 static ex cos_deriv(const ex & x, unsigned deriv_param)
 368 {
 369         GINAC_ASSERT(deriv_param==0);
 370
 371         // d/dx cos(x) -> -sin(x)
 372         return _ex_1()*sin(x);
 373 }
 374
 375 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 376                        evalf_func(cos_evalf).
 377                        derivative_func(cos_deriv));
 378
 379 //////////
 380 // tangent (trigonometric function)
 381 //////////
 382
 383 static ex tan_evalf(const ex & x)
 384 {
 385         BEGIN_TYPECHECK
 386            TYPECHECK(x,numeric)
 387         END_TYPECHECK(tan(x)) // -> numeric tan(numeric)
 388
 389         return tan(ex_to_numeric(x));
 390 }
 391
 392 static ex tan_eval(const ex & x)
 393 {
 394         // tan(n/d*Pi) -> { all known non-nested radicals }
 395         ex SixtyExOverPi = _ex60()*x/Pi;
 396         ex sign = _ex1();
 397         if (SixtyExOverPi.info(info_flags::integer)) {
 398                 numeric z = mod(ex_to_numeric(SixtyExOverPi),_num60());
 399                 if (z>=_num60()) {
 400                         // wrap to interval [0, Pi)
 401                         z -= _num60();
 402                 }
 403                 if (z>=_num30()) {
 404                         // wrap to interval [0, Pi/2)
 405                         z = _num60()-z;
 406                         sign = _ex_1();
 407                 }
 408                 if (z.is_equal(_num0()))  // tan(0)       -> 0
 409                         return _ex0();
 410                 if (z.is_equal(_num5()))  // tan(Pi/12)   -> 2-sqrt(3)
 411                         return sign*(_ex2()-power(_ex3(),_ex1_2()));
 412                 if (z.is_equal(_num10())) // tan(Pi/6)    -> sqrt(3)/3
 413                         return sign*_ex1_3()*power(_ex3(),_ex1_2());
 414                 if (z.is_equal(_num15())) // tan(Pi/4)    -> 1
 415                         return sign*_ex1();
 416                 if (z.is_equal(_num20())) // tan(Pi/3)    -> sqrt(3)
 417                         return sign*power(_ex3(),_ex1_2());
 418                 if (z.is_equal(_num25())) // tan(5/12*Pi) -> 2+sqrt(3)
 419                         return sign*(power(_ex3(),_ex1_2())+_ex2());
 420                 if (z.is_equal(_num30())) // tan(Pi/2)    -> infinity
 421                         throw (pole_error("tan_eval(): simple pole",1));
 422         }
 423
 424         if (is_ex_exactly_of_type(x, function)) {
 425                 ex t = x.op(0);
 426                 // tan(atan(x)) -> x
 427                 if (is_ex_the_function(x, atan))
 428                         return t;
 429                 // tan(asin(x)) -> x*(1+x^2)^(-1/2)
 430                 if (is_ex_the_function(x, asin))
 431                         return t*power(_ex1()-power(t,_ex2()),_ex_1_2());
 432                 // tan(acos(x)) -> (1-x^2)^(1/2)/x
 433                 if (is_ex_the_function(x, acos))
 434                         return power(t,_ex_1())*power(_ex1()-power(t,_ex2()),_ex1_2());
 435         }
 436
 437         // tan(float) -> float
 438         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 439                 return tan_evalf(x);
 440         }
 441
 442         return tan(x).hold();
 443 }
 444
 445 static ex tan_deriv(const ex & x, unsigned deriv_param)
 446 {
 447         GINAC_ASSERT(deriv_param==0);
 448
 449         // d/dx tan(x) -> 1+tan(x)^2;
 450         return (_ex1()+power(tan(x),_ex2()));
 451 }
 452
 453 static ex tan_series(const ex &x,
 454                      const relational &rel,
 455                      int order,
 456                      unsigned options)
 457 {
 458         GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
 459         // method:
 460         // Taylor series where there is no pole falls back to tan_deriv.
 461         // On a pole simply expand sin(x)/cos(x).
 462         const ex x_pt = x.subs(rel);
 463         if (!(2*x_pt/Pi).info(info_flags::odd))
 464                 throw do_taylor();  // caught by function::series()
 465         // if we got here we have to care for a simple pole
 466         return (sin(x)/cos(x)).series(rel, order+2, options);
 467 }
 468
 469 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 470                        evalf_func(tan_evalf).
 471                        derivative_func(tan_deriv).
 472                        series_func(tan_series));
 473
 474 //////////
 475 // inverse sine (arc sine)
 476 //////////
 477
 478 static ex asin_evalf(const ex & x)
 479 {
 480         BEGIN_TYPECHECK
 481            TYPECHECK(x,numeric)
 482         END_TYPECHECK(asin(x))
 483
 484         return asin(ex_to_numeric(x)); // -> numeric asin(numeric)
 485 }
 486
 487 static ex asin_eval(const ex & x)
 488 {
 489         if (x.info(info_flags::numeric)) {
 490                 // asin(0) -> 0
 491                 if (x.is_zero())
 492                         return x;
 493                 // asin(1/2) -> Pi/6
 494                 if (x.is_equal(_ex1_2()))
 495                         return numeric(1,6)*Pi;
 496                 // asin(1) -> Pi/2
 497                 if (x.is_equal(_ex1()))
 498                         return _num1_2()*Pi;
 499                 // asin(-1/2) -> -Pi/6
 500                 if (x.is_equal(_ex_1_2()))
 501                         return numeric(-1,6)*Pi;
 502                 // asin(-1) -> -Pi/2
 503                 if (x.is_equal(_ex_1()))
 504                         return _num_1_2()*Pi;
 505                 // asin(float) -> float
 506                 if (!x.info(info_flags::crational))
 507                         return asin_evalf(x);
 508         }
 509
 510         return asin(x).hold();
 511 }
 512
 513 static ex asin_deriv(const ex & x, unsigned deriv_param)
 514 {
 515         GINAC_ASSERT(deriv_param==0);
 516
 517         // d/dx asin(x) -> 1/sqrt(1-x^2)
 518         return power(1-power(x,_ex2()),_ex_1_2());
 519 }
 520
 521 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 522                         evalf_func(asin_evalf).
 523                         derivative_func(asin_deriv));
 524
 525 //////////
 526 // inverse cosine (arc cosine)
 527 //////////
 528
 529 static ex acos_evalf(const ex & x)
 530 {
 531         BEGIN_TYPECHECK
 532            TYPECHECK(x,numeric)
 533         END_TYPECHECK(acos(x))
 534
 535         return acos(ex_to_numeric(x)); // -> numeric acos(numeric)
 536 }
 537
 538 static ex acos_eval(const ex & x)
 539 {
 540         if (x.info(info_flags::numeric)) {
 541                 // acos(1) -> 0
 542                 if (x.is_equal(_ex1()))
 543                         return _ex0();
 544                 // acos(1/2) -> Pi/3
 545                 if (x.is_equal(_ex1_2()))
 546                         return _ex1_3()*Pi;
 547                 // acos(0) -> Pi/2
 548                 if (x.is_zero())
 549                         return _ex1_2()*Pi;
 550                 // acos(-1/2) -> 2/3*Pi
 551                 if (x.is_equal(_ex_1_2()))
 552                         return numeric(2,3)*Pi;
 553                 // acos(-1) -> Pi
 554                 if (x.is_equal(_ex_1()))
 555                         return Pi;
 556                 // acos(float) -> float
 557                 if (!x.info(info_flags::crational))
 558                         return acos_evalf(x);
 559         }
 560
 561         return acos(x).hold();
 562 }
 563
 564 static ex acos_deriv(const ex & x, unsigned deriv_param)
 565 {
 566         GINAC_ASSERT(deriv_param==0);
 567
 568         // d/dx acos(x) -> -1/sqrt(1-x^2)
 569         return _ex_1()*power(1-power(x,_ex2()),_ex_1_2());
 570 }
 571
 572 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 573                         evalf_func(acos_evalf).
 574                         derivative_func(acos_deriv));
 575
 576 //////////
 577 // inverse tangent (arc tangent)
 578 //////////
 579
 580 static ex atan_evalf(const ex & x)
 581 {
 582         BEGIN_TYPECHECK
 583                 TYPECHECK(x,numeric)
 584         END_TYPECHECK(atan(x))
 585
 586         return atan(ex_to_numeric(x)); // -> numeric atan(numeric)
 587 }
 588
 589 static ex atan_eval(const ex & x)
 590 {
 591         if (x.info(info_flags::numeric)) {
 592                 // atan(0) -> 0
 593                 if (x.is_equal(_ex0()))
 594                         return _ex0();
 595                 // atan(1) -> Pi/4
 596                 if (x.is_equal(_ex1()))
 597                         return _ex1_4()*Pi;
 598                 // atan(-1) -> -Pi/4
 599                 if (x.is_equal(_ex_1()))
 600                         return _ex_1_4()*Pi;
 601                 if (x.is_equal(I) || x.is_equal(-I))
 602                         throw (pole_error("atan_eval(): logarithmic pole",0));
 603                 // atan(float) -> float
 604                 if (!x.info(info_flags::crational))
 605                         return atan_evalf(x);
 606         }
 607
 608         return atan(x).hold();
 609 }
 610
 611 static ex atan_deriv(const ex & x, unsigned deriv_param)
 612 {
 613         GINAC_ASSERT(deriv_param==0);
 614
 615         // d/dx atan(x) -> 1/(1+x^2)
 616         return power(_ex1()+power(x,_ex2()), _ex_1());
 617 }
 618
 619 static ex atan_series(const ex &arg,
 620                       const relational &rel,
 621                       int order,
 622                       unsigned options)
 623 {
 624         GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
 625         // method:
 626         // Taylor series where there is no pole or cut falls back to atan_deriv.
 627         // There are two branch cuts, one runnig from I up the imaginary axis and
 628         // one running from -I down the imaginary axis.  The points I and -I are
 629         // poles.
 630         // On the branch cuts and the poles series expand
 631         //     (log(1+I*x)-log(1-I*x))/(2*I)
 632         // instead.
 633         const ex arg_pt = arg.subs(rel);
 634         if (!(I*arg_pt).info(info_flags::real))
 635                 throw do_taylor();     // Re(x) != 0
 636         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1())
 637                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
 638         // care for the poles, using the defining formula for atan()...
 639         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
 640                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
 641         if (!(options & series_options::suppress_branchcut)) {
 642                 // method:
 643                 // This is the branch cut: assemble the primitive series manually and
 644                 // then add the corresponding complex step function.
 645                 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
 646                 const ex point = rel.rhs();
 647                 const symbol foo;
 648                 ex replarg = series(atan(arg), *s==foo, order).subs(foo==point);
 649                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2();
 650                 if ((I*arg_pt)<_ex0())
 651                         Order0correction += log((I*arg_pt+_ex_1())/(I*arg_pt+_ex1()))*I*_ex_1_2();
 652                 else
 653                         Order0correction += log((I*arg_pt+_ex1())/(I*arg_pt+_ex_1()))*I*_ex1_2();
 654                 epvector seq;
 655                 seq.push_back(expair(Order0correction, _ex0()));
 656                 seq.push_back(expair(Order(_ex1()), order));
 657                 return series(replarg - pseries(rel, seq), rel, order);
 658         }
 659         throw do_taylor();
 660 }
 661
 662 REGISTER_FUNCTION(atan, eval_func(atan_eval).
 663                         evalf_func(atan_evalf).
 664                         derivative_func(atan_deriv).
 665                         series_func(atan_series));
 666
 667 //////////
 668 // inverse tangent (atan2(y,x))
 669 //////////
 670
 671 static ex atan2_evalf(const ex & y, const ex & x)
 672 {
 673         BEGIN_TYPECHECK
 674                 TYPECHECK(y,numeric)
 675                 TYPECHECK(x,numeric)
 676         END_TYPECHECK(atan2(y,x))
 677
 678         return atan(ex_to_numeric(y),ex_to_numeric(x)); // -> numeric atan(numeric)
 679 }
 680
 681 static ex atan2_eval(const ex & y, const ex & x)
 682 {
 683         if (y.info(info_flags::numeric) && !y.info(info_flags::crational) &&
 684                 x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 685                 return atan2_evalf(y,x);
 686         }
 687
 688         return atan2(y,x).hold();
 689 }
 690
 691 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
 692 {
 693         GINAC_ASSERT(deriv_param<2);
 694
 695         if (deriv_param==0) {
 696                 // d/dy atan(y,x)
 697                 return x*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
 698         }
 699         // d/dx atan(y,x)
 700         return -y*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
 701 }
 702
 703 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
 704                          evalf_func(atan2_evalf).
 705                          derivative_func(atan2_deriv));
 706
 707 //////////
 708 // hyperbolic sine (trigonometric function)
 709 //////////
 710
 711 static ex sinh_evalf(const ex & x)
 712 {
 713         BEGIN_TYPECHECK
 714            TYPECHECK(x,numeric)
 715         END_TYPECHECK(sinh(x))
 716
 717         return sinh(ex_to_numeric(x)); // -> numeric sinh(numeric)
 718 }
 719
 720 static ex sinh_eval(const ex & x)
 721 {
 722         if (x.info(info_flags::numeric)) {
 723                 if (x.is_zero())  // sinh(0) -> 0
 724                         return _ex0();
 725                 if (!x.info(info_flags::crational))  // sinh(float) -> float
 726                         return sinh_evalf(x);
 727         }
 728
 729         if ((x/Pi).info(info_flags::numeric) &&
 730                 ex_to_numeric(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
 731                 return I*sin(x/I);
 732
 733         if (is_ex_exactly_of_type(x, function)) {
 734                 ex t = x.op(0);
 735                 // sinh(asinh(x)) -> x
 736                 if (is_ex_the_function(x, asinh))
 737                         return t;
 738                 // sinh(acosh(x)) -> (x-1)^(1/2) * (x+1)^(1/2)
 739                 if (is_ex_the_function(x, acosh))
 740                         return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2());
 741                 // sinh(atanh(x)) -> x*(1-x^2)^(-1/2)
 742                 if (is_ex_the_function(x, atanh))
 743                         return t*power(_ex1()-power(t,_ex2()),_ex_1_2());
 744         }
 745
 746         return sinh(x).hold();
 747 }
 748
 749 static ex sinh_deriv(const ex & x, unsigned deriv_param)
 750 {
 751         GINAC_ASSERT(deriv_param==0);
 752
 753         // d/dx sinh(x) -> cosh(x)
 754         return cosh(x);
 755 }
 756
 757 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
 758                         evalf_func(sinh_evalf).
 759                         derivative_func(sinh_deriv));
 760
 761 //////////
 762 // hyperbolic cosine (trigonometric function)
 763 //////////
 764
 765 static ex cosh_evalf(const ex & x)
 766 {
 767         BEGIN_TYPECHECK
 768            TYPECHECK(x,numeric)
 769         END_TYPECHECK(cosh(x))
 770
 771         return cosh(ex_to_numeric(x)); // -> numeric cosh(numeric)
 772 }
 773
 774 static ex cosh_eval(const ex & x)
 775 {
 776         if (x.info(info_flags::numeric)) {
 777                 if (x.is_zero())  // cosh(0) -> 1
 778                         return _ex1();
 779                 if (!x.info(info_flags::crational))  // cosh(float) -> float
 780                         return cosh_evalf(x);
 781         }
 782
 783         if ((x/Pi).info(info_flags::numeric) &&
 784                 ex_to_numeric(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
 785                 return cos(x/I);
 786
 787         if (is_ex_exactly_of_type(x, function)) {
 788                 ex t = x.op(0);
 789                 // cosh(acosh(x)) -> x
 790                 if (is_ex_the_function(x, acosh))
 791                         return t;
 792                 // cosh(asinh(x)) -> (1+x^2)^(1/2)
 793                 if (is_ex_the_function(x, asinh))
 794                         return power(_ex1()+power(t,_ex2()),_ex1_2());
 795                 // cosh(atanh(x)) -> (1-x^2)^(-1/2)
 796                 if (is_ex_the_function(x, atanh))
 797                         return power(_ex1()-power(t,_ex2()),_ex_1_2());
 798         }
 799
 800         return cosh(x).hold();
 801 }
 802
 803 static ex cosh_deriv(const ex & x, unsigned deriv_param)
 804 {
 805         GINAC_ASSERT(deriv_param==0);
 806
 807         // d/dx cosh(x) -> sinh(x)
 808         return sinh(x);
 809 }
 810
 811 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
 812                         evalf_func(cosh_evalf).
 813                         derivative_func(cosh_deriv));
 814
 815
 816 //////////
 817 // hyperbolic tangent (trigonometric function)
 818 //////////
 819
 820 static ex tanh_evalf(const ex & x)
 821 {
 822         BEGIN_TYPECHECK
 823            TYPECHECK(x,numeric)
 824         END_TYPECHECK(tanh(x))
 825
 826         return tanh(ex_to_numeric(x)); // -> numeric tanh(numeric)
 827 }
 828
 829 static ex tanh_eval(const ex & x)
 830 {
 831         if (x.info(info_flags::numeric)) {
 832                 if (x.is_zero())  // tanh(0) -> 0
 833                         return _ex0();
 834                 if (!x.info(info_flags::crational))  // tanh(float) -> float
 835                         return tanh_evalf(x);
 836         }
 837
 838         if ((x/Pi).info(info_flags::numeric) &&
 839                 ex_to_numeric(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
 840                 return I*tan(x/I);
 841
 842         if (is_ex_exactly_of_type(x, function)) {
 843                 ex t = x.op(0);
 844                 // tanh(atanh(x)) -> x
 845                 if (is_ex_the_function(x, atanh))
 846                         return t;
 847                 // tanh(asinh(x)) -> x*(1+x^2)^(-1/2)
 848                 if (is_ex_the_function(x, asinh))
 849                         return t*power(_ex1()+power(t,_ex2()),_ex_1_2());
 850                 // tanh(acosh(x)) -> (x-1)^(1/2)*(x+1)^(1/2)/x
 851                 if (is_ex_the_function(x, acosh))
 852                         return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2())*power(t,_ex_1());
 853         }
 854
 855         return tanh(x).hold();
 856 }
 857
 858 static ex tanh_deriv(const ex & x, unsigned deriv_param)
 859 {
 860         GINAC_ASSERT(deriv_param==0);
 861
 862         // d/dx tanh(x) -> 1-tanh(x)^2
 863         return _ex1()-power(tanh(x),_ex2());
 864 }
 865
 866 static ex tanh_series(const ex &x,
 867                       const relational &rel,
 868                       int order,
 869                       unsigned options)
 870 {
 871         GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
 872         // method:
 873         // Taylor series where there is no pole falls back to tanh_deriv.
 874         // On a pole simply expand sinh(x)/cosh(x).
 875         const ex x_pt = x.subs(rel);
 876         if (!(2*I*x_pt/Pi).info(info_flags::odd))
 877                 throw do_taylor();  // caught by function::series()
 878         // if we got here we have to care for a simple pole
 879         return (sinh(x)/cosh(x)).series(rel, order+2, options);
 880 }
 881
 882 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
 883                         evalf_func(tanh_evalf).
 884                         derivative_func(tanh_deriv).
 885                         series_func(tanh_series));
 886
 887 //////////
 888 // inverse hyperbolic sine (trigonometric function)
 889 //////////
 890
 891 static ex asinh_evalf(const ex & x)
 892 {
 893         BEGIN_TYPECHECK
 894            TYPECHECK(x,numeric)
 895         END_TYPECHECK(asinh(x))
 896
 897         return asinh(ex_to_numeric(x)); // -> numeric asinh(numeric)
 898 }
 899
 900 static ex asinh_eval(const ex & x)
 901 {
 902         if (x.info(info_flags::numeric)) {
 903                 // asinh(0) -> 0
 904                 if (x.is_zero())
 905                         return _ex0();
 906                 // asinh(float) -> float
 907                 if (!x.info(info_flags::crational))
 908                         return asinh_evalf(x);
 909         }
 910
 911         return asinh(x).hold();
 912 }
 913
 914 static ex asinh_deriv(const ex & x, unsigned deriv_param)
 915 {
 916         GINAC_ASSERT(deriv_param==0);
 917
 918         // d/dx asinh(x) -> 1/sqrt(1+x^2)
 919         return power(_ex1()+power(x,_ex2()),_ex_1_2());
 920 }
 921
 922 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
 923                          evalf_func(asinh_evalf).
 924                          derivative_func(asinh_deriv));
 925
 926 //////////
 927 // inverse hyperbolic cosine (trigonometric function)
 928 //////////
 929
 930 static ex acosh_evalf(const ex & x)
 931 {
 932         BEGIN_TYPECHECK
 933            TYPECHECK(x,numeric)
 934         END_TYPECHECK(acosh(x))
 935
 936         return acosh(ex_to_numeric(x)); // -> numeric acosh(numeric)
 937 }
 938
 939 static ex acosh_eval(const ex & x)
 940 {
 941         if (x.info(info_flags::numeric)) {
 942                 // acosh(0) -> Pi*I/2
 943                 if (x.is_zero())
 944                         return Pi*I*numeric(1,2);
 945                 // acosh(1) -> 0
 946                 if (x.is_equal(_ex1()))
 947                         return _ex0();
 948                 // acosh(-1) -> Pi*I
 949                 if (x.is_equal(_ex_1()))
 950                         return Pi*I;
 951                 // acosh(float) -> float
 952                 if (!x.info(info_flags::crational))
 953                         return acosh_evalf(x);
 954         }
 955
 956         return acosh(x).hold();
 957 }
 958
 959 static ex acosh_deriv(const ex & x, unsigned deriv_param)
 960 {
 961         GINAC_ASSERT(deriv_param==0);
 962
 963         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
 964         return power(x+_ex_1(),_ex_1_2())*power(x+_ex1(),_ex_1_2());
 965 }
 966
 967 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
 968                          evalf_func(acosh_evalf).
 969                          derivative_func(acosh_deriv));
 970
 971 //////////
 972 // inverse hyperbolic tangent (trigonometric function)
 973 //////////
 974
 975 static ex atanh_evalf(const ex & x)
 976 {
 977         BEGIN_TYPECHECK
 978            TYPECHECK(x,numeric)
 979         END_TYPECHECK(atanh(x))
 980
 981         return atanh(ex_to_numeric(x)); // -> numeric atanh(numeric)
 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_evalf(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_ex_exactly_of_type(rel.lhs(),symbol));
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 = static_cast<symbol *>(rel.lhs().bp);
1036                 const ex point = rel.rhs();
1037                 const symbol foo;
1038                 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 #ifndef NO_NAMESPACE_GINAC
1059 } // namespace GiNaC
1060 #endif // ndef NO_NAMESPACE_GINAC