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