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         BEGIN_TYPECHECK
  46                 TYPECHECK(x,numeric)
  47         END_TYPECHECK(exp(x))
  48
  49         return exp(ex_to<numeric>(x)); // -> numeric exp(numeric)
  50 }
  51
  52 static ex exp_eval(const ex & x)
  53 {
  54         // exp(0) -> 1
  55         if (x.is_zero()) {
  56                 return _ex1();
  57         }
  58         // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
  59         ex TwoExOverPiI=(_ex2()*x)/(Pi*I);
  60         if (TwoExOverPiI.info(info_flags::integer)) {
  61                 numeric z=mod(ex_to<numeric>(TwoExOverPiI),_num4());
  62                 if (z.is_equal(_num0()))
  63                         return _ex1();
  64                 if (z.is_equal(_num1()))
  65                         return ex(I);
  66                 if (z.is_equal(_num2()))
  67                         return _ex_1();
  68                 if (z.is_equal(_num3()))
  69                         return ex(-I);
  70         }
  71         // exp(log(x)) -> x
  72         if (is_ex_the_function(x, log))
  73                 return x.op(0);
  74
  75         // exp(float)
  76         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
  77                 return exp_evalf(x);
  78
  79         return exp(x).hold();
  80 }
  81
  82 static ex exp_deriv(const ex & x, unsigned deriv_param)
  83 {
  84         GINAC_ASSERT(deriv_param==0);
  85
  86         // d/dx exp(x) -> exp(x)
  87         return exp(x);
  88 }
  89
  90 REGISTER_FUNCTION(exp, eval_func(exp_eval).
  91                        evalf_func(exp_evalf).
  92                        derivative_func(exp_deriv).
  93                        latex_name("\\exp"));
  94
  95 //////////
  96 // natural logarithm
  97 //////////
  98
  99 static ex log_evalf(const ex & x)
 100 {
 101         BEGIN_TYPECHECK
 102                 TYPECHECK(x,numeric)
 103         END_TYPECHECK(log(x))
 104
 105         return log(ex_to<numeric>(x)); // -> numeric log(numeric)
 106 }
 107
 108 static ex log_eval(const ex & x)
 109 {
 110         if (x.info(info_flags::numeric)) {
 111                 if (x.is_equal(_ex0()))  // log(0) -> infinity
 112                         throw(pole_error("log_eval(): log(0)",0));
 113                 if (x.info(info_flags::real) && x.info(info_flags::negative))
 114                         return (log(-x)+I*Pi);
 115                 if (x.is_equal(_ex1()))  // log(1) -> 0
 116                         return _ex0();
 117                 if (x.is_equal(I))       // log(I) -> Pi*I/2
 118                         return (Pi*I*_num1_2());
 119                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
 120                         return (Pi*I*_num_1_2());
 121                 // log(float)
 122                 if (!x.info(info_flags::crational))
 123                         return log_evalf(x);
 124         }
 125         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
 126         if (is_ex_the_function(x, exp)) {
 127                 ex t = x.op(0);
 128                 if (t.info(info_flags::numeric)) {
 129                         numeric nt = ex_to<numeric>(t);
 130                         if (nt.is_real())
 131                                 return t;
 132                 }
 133         }
 134
 135         return log(x).hold();
 136 }
 137
 138 static ex log_deriv(const ex & x, unsigned deriv_param)
 139 {
 140         GINAC_ASSERT(deriv_param==0);
 141
 142         // d/dx log(x) -> 1/x
 143         return power(x, _ex_1());
 144 }
 145
 146 static ex log_series(const ex &arg,
 147                      const relational &rel,
 148                      int order,
 149                      unsigned options)
 150 {
 151         GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
 152         ex arg_pt;
 153         bool must_expand_arg = false;
 154         // maybe substitution of rel into arg fails because of a pole
 155         try {
 156                 arg_pt = arg.subs(rel);
 157         } catch (pole_error) {
 158                 must_expand_arg = true;
 159         }
 160         // or we are at the branch point anyways
 161         if (arg_pt.is_zero())
 162                 must_expand_arg = true;
 163
 164         if (must_expand_arg) {
 165                 // method:
 166                 // This is the branch point: Series expand the argument first, then
 167                 // trivially factorize it to isolate that part which has constant
 168                 // leading coefficient in this fashion:
 169                 //   x^n + Order(x^(n+m))  ->  x^n * (1 + Order(x^m)).
 170                 // Return a plain n*log(x) for the x^n part and series expand the
 171                 // other part.  Add them together and reexpand again in order to have
 172                 // one unnested pseries object.  All this also works for negative n.
 173                 const pseries argser = ex_to<pseries>(arg.series(rel, order, options));
 174                 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
 175                 const ex point = rel.rhs();
 176                 const int n = argser.ldegree(*s);
 177                 epvector seq;
 178                 // construct what we carelessly called the n*log(x) term above
 179                 ex coeff = argser.coeff(*s, n);
 180                 // expand the log, but only if coeff is real and > 0, since otherwise
 181                 // it would make the branch cut run into the wrong direction
 182                 if (coeff.info(info_flags::positive))
 183                         seq.push_back(expair(n*log(*s-point)+log(coeff), _ex0()));
 184                 else
 185                         seq.push_back(expair(log(coeff*pow(*s-point, n)), _ex0()));
 186                 if (!argser.is_terminating() || argser.nops()!=1) {
 187                         // in this case n more terms are needed
 188                         // (sadly, to generate them, we have to start from the beginning)
 189                         ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
 190                         return pseries(rel, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
 191                 } else  // it was a monomial
 192                         return pseries(rel, seq);
 193         }
 194         if (!(options & series_options::suppress_branchcut) &&
 195              arg_pt.info(info_flags::negative)) {
 196                 // method:
 197                 // This is the branch cut: assemble the primitive series manually and
 198                 // then add the corresponding complex step function.
 199                 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
 200                 const ex point = rel.rhs();
 201                 const symbol foo;
 202                 ex replarg = series(log(arg), *s==foo, order).subs(foo==point);
 203                 epvector seq;
 204                 seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0()));
 205                 seq.push_back(expair(Order(_ex1()), order));
 206                 return series(replarg - I*Pi + pseries(rel, seq), rel, order);
 207         }
 208         throw do_taylor();  // caught by function::series()
 209 }
 210
 211 REGISTER_FUNCTION(log, eval_func(log_eval).
 212                        evalf_func(log_evalf).
 213                        derivative_func(log_deriv).
 214                        series_func(log_series).
 215                        latex_name("\\ln"));
 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                        latex_name("\\sin"));
 298
 299 //////////
 300 // cosine (trigonometric function)
 301 //////////
 302
 303 static ex cos_evalf(const ex & x)
 304 {
 305         BEGIN_TYPECHECK
 306                 TYPECHECK(x,numeric)
 307         END_TYPECHECK(cos(x))
 308
 309         return cos(ex_to<numeric>(x)); // -> numeric cos(numeric)
 310 }
 311
 312 static ex cos_eval(const ex & x)
 313 {
 314         // cos(n/d*Pi) -> { all known non-nested radicals }
 315         ex SixtyExOverPi = _ex60()*x/Pi;
 316         ex sign = _ex1();
 317         if (SixtyExOverPi.info(info_flags::integer)) {
 318                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num120());
 319                 if (z>=_num60()) {
 320                         // wrap to interval [0, Pi)
 321                         z = _num120()-z;
 322                 }
 323                 if (z>=_num30()) {
 324                         // wrap to interval [0, Pi/2)
 325                         z = _num60()-z;
 326                         sign = _ex_1();
 327                 }
 328                 if (z.is_equal(_num0()))  // cos(0)       -> 1
 329                         return sign*_ex1();
 330                 if (z.is_equal(_num5()))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 331                         return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
 332                 if (z.is_equal(_num10())) // cos(Pi/6)    -> sqrt(3)/2
 333                         return sign*_ex1_2()*power(_ex3(),_ex1_2());
 334                 if (z.is_equal(_num12())) // cos(Pi/5)    -> sqrt(5)/4+1/4
 335                         return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
 336                 if (z.is_equal(_num15())) // cos(Pi/4)    -> sqrt(2)/2
 337                         return sign*_ex1_2()*power(_ex2(),_ex1_2());
 338                 if (z.is_equal(_num20())) // cos(Pi/3)    -> 1/2
 339                         return sign*_ex1_2();
 340                 if (z.is_equal(_num24())) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 341                         return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
 342                 if (z.is_equal(_num25())) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 343                         return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
 344                 if (z.is_equal(_num30())) // cos(Pi/2)    -> 0
 345                         return sign*_ex0();
 346         }
 347
 348         if (is_ex_exactly_of_type(x, function)) {
 349                 ex t = x.op(0);
 350                 // cos(acos(x)) -> x
 351                 if (is_ex_the_function(x, acos))
 352                         return t;
 353                 // cos(asin(x)) -> (1-x^2)^(1/2)
 354                 if (is_ex_the_function(x, asin))
 355                         return power(_ex1()-power(t,_ex2()),_ex1_2());
 356                 // cos(atan(x)) -> (1+x^2)^(-1/2)
 357                 if (is_ex_the_function(x, atan))
 358                         return power(_ex1()+power(t,_ex2()),_ex_1_2());
 359         }
 360
 361         // cos(float) -> float
 362         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 363                 return cos_evalf(x);
 364
 365         return cos(x).hold();
 366 }
 367
 368 static ex cos_deriv(const ex & x, unsigned deriv_param)
 369 {
 370         GINAC_ASSERT(deriv_param==0);
 371
 372         // d/dx cos(x) -> -sin(x)
 373         return _ex_1()*sin(x);
 374 }
 375
 376 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 377                        evalf_func(cos_evalf).
 378                        derivative_func(cos_deriv).
 379                        latex_name("\\cos"));
 380
 381 //////////
 382 // tangent (trigonometric function)
 383 //////////
 384
 385 static ex tan_evalf(const ex & x)
 386 {
 387         BEGIN_TYPECHECK
 388            TYPECHECK(x,numeric)
 389         END_TYPECHECK(tan(x)) // -> numeric tan(numeric)
 390
 391         return tan(ex_to<numeric>(x));
 392 }
 393
 394 static ex tan_eval(const ex & x)
 395 {
 396         // tan(n/d*Pi) -> { all known non-nested radicals }
 397         ex SixtyExOverPi = _ex60()*x/Pi;
 398         ex sign = _ex1();
 399         if (SixtyExOverPi.info(info_flags::integer)) {
 400                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num60());
 401                 if (z>=_num60()) {
 402                         // wrap to interval [0, Pi)
 403                         z -= _num60();
 404                 }
 405                 if (z>=_num30()) {
 406                         // wrap to interval [0, Pi/2)
 407                         z = _num60()-z;
 408                         sign = _ex_1();
 409                 }
 410                 if (z.is_equal(_num0()))  // tan(0)       -> 0
 411                         return _ex0();
 412                 if (z.is_equal(_num5()))  // tan(Pi/12)   -> 2-sqrt(3)
 413                         return sign*(_ex2()-power(_ex3(),_ex1_2()));
 414                 if (z.is_equal(_num10())) // tan(Pi/6)    -> sqrt(3)/3
 415                         return sign*_ex1_3()*power(_ex3(),_ex1_2());
 416                 if (z.is_equal(_num15())) // tan(Pi/4)    -> 1
 417                         return sign*_ex1();
 418                 if (z.is_equal(_num20())) // tan(Pi/3)    -> sqrt(3)
 419                         return sign*power(_ex3(),_ex1_2());
 420                 if (z.is_equal(_num25())) // tan(5/12*Pi) -> 2+sqrt(3)
 421                         return sign*(power(_ex3(),_ex1_2())+_ex2());
 422                 if (z.is_equal(_num30())) // tan(Pi/2)    -> infinity
 423                         throw (pole_error("tan_eval(): simple pole",1));
 424         }
 425
 426         if (is_ex_exactly_of_type(x, function)) {
 427                 ex t = x.op(0);
 428                 // tan(atan(x)) -> x
 429                 if (is_ex_the_function(x, atan))
 430                         return t;
 431                 // tan(asin(x)) -> x*(1+x^2)^(-1/2)
 432                 if (is_ex_the_function(x, asin))
 433                         return t*power(_ex1()-power(t,_ex2()),_ex_1_2());
 434                 // tan(acos(x)) -> (1-x^2)^(1/2)/x
 435                 if (is_ex_the_function(x, acos))
 436                         return power(t,_ex_1())*power(_ex1()-power(t,_ex2()),_ex1_2());
 437         }
 438
 439         // tan(float) -> float
 440         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 441                 return tan_evalf(x);
 442         }
 443
 444         return tan(x).hold();
 445 }
 446
 447 static ex tan_deriv(const ex & x, unsigned deriv_param)
 448 {
 449         GINAC_ASSERT(deriv_param==0);
 450
 451         // d/dx tan(x) -> 1+tan(x)^2;
 452         return (_ex1()+power(tan(x),_ex2()));
 453 }
 454
 455 static ex tan_series(const ex &x,
 456                      const relational &rel,
 457                      int order,
 458                      unsigned options)
 459 {
 460         GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
 461         // method:
 462         // Taylor series where there is no pole falls back to tan_deriv.
 463         // On a pole simply expand sin(x)/cos(x).
 464         const ex x_pt = x.subs(rel);
 465         if (!(2*x_pt/Pi).info(info_flags::odd))
 466                 throw do_taylor();  // caught by function::series()
 467         // if we got here we have to care for a simple pole
 468         return (sin(x)/cos(x)).series(rel, order+2, options);
 469 }
 470
 471 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 472                        evalf_func(tan_evalf).
 473                        derivative_func(tan_deriv).
 474                        series_func(tan_series).
 475                        latex_name("\\tan"));
 476
 477 //////////
 478 // inverse sine (arc sine)
 479 //////////
 480
 481 static ex asin_evalf(const ex & x)
 482 {
 483         BEGIN_TYPECHECK
 484            TYPECHECK(x,numeric)
 485         END_TYPECHECK(asin(x))
 486
 487         return asin(ex_to<numeric>(x)); // -> numeric asin(numeric)
 488 }
 489
 490 static ex asin_eval(const ex & x)
 491 {
 492         if (x.info(info_flags::numeric)) {
 493                 // asin(0) -> 0
 494                 if (x.is_zero())
 495                         return x;
 496                 // asin(1/2) -> Pi/6
 497                 if (x.is_equal(_ex1_2()))
 498                         return numeric(1,6)*Pi;
 499                 // asin(1) -> Pi/2
 500                 if (x.is_equal(_ex1()))
 501                         return _num1_2()*Pi;
 502                 // asin(-1/2) -> -Pi/6
 503                 if (x.is_equal(_ex_1_2()))
 504                         return numeric(-1,6)*Pi;
 505                 // asin(-1) -> -Pi/2
 506                 if (x.is_equal(_ex_1()))
 507                         return _num_1_2()*Pi;
 508                 // asin(float) -> float
 509                 if (!x.info(info_flags::crational))
 510                         return asin_evalf(x);
 511         }
 512
 513         return asin(x).hold();
 514 }
 515
 516 static ex asin_deriv(const ex & x, unsigned deriv_param)
 517 {
 518         GINAC_ASSERT(deriv_param==0);
 519
 520         // d/dx asin(x) -> 1/sqrt(1-x^2)
 521         return power(1-power(x,_ex2()),_ex_1_2());
 522 }
 523
 524 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 525                         evalf_func(asin_evalf).
 526                         derivative_func(asin_deriv).
 527                         latex_name("\\arcsin"));
 528
 529 //////////
 530 // inverse cosine (arc cosine)
 531 //////////
 532
 533 static ex acos_evalf(const ex & x)
 534 {
 535         BEGIN_TYPECHECK
 536            TYPECHECK(x,numeric)
 537         END_TYPECHECK(acos(x))
 538
 539         return acos(ex_to<numeric>(x)); // -> numeric acos(numeric)
 540 }
 541
 542 static ex acos_eval(const ex & x)
 543 {
 544         if (x.info(info_flags::numeric)) {
 545                 // acos(1) -> 0
 546                 if (x.is_equal(_ex1()))
 547                         return _ex0();
 548                 // acos(1/2) -> Pi/3
 549                 if (x.is_equal(_ex1_2()))
 550                         return _ex1_3()*Pi;
 551                 // acos(0) -> Pi/2
 552                 if (x.is_zero())
 553                         return _ex1_2()*Pi;
 554                 // acos(-1/2) -> 2/3*Pi
 555                 if (x.is_equal(_ex_1_2()))
 556                         return numeric(2,3)*Pi;
 557                 // acos(-1) -> Pi
 558                 if (x.is_equal(_ex_1()))
 559                         return Pi;
 560                 // acos(float) -> float
 561                 if (!x.info(info_flags::crational))
 562                         return acos_evalf(x);
 563         }
 564
 565         return acos(x).hold();
 566 }
 567
 568 static ex acos_deriv(const ex & x, unsigned deriv_param)
 569 {
 570         GINAC_ASSERT(deriv_param==0);
 571
 572         // d/dx acos(x) -> -1/sqrt(1-x^2)
 573         return _ex_1()*power(1-power(x,_ex2()),_ex_1_2());
 574 }
 575
 576 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 577                         evalf_func(acos_evalf).
 578                         derivative_func(acos_deriv).
 579                         latex_name("\\arccos"));
 580
 581 //////////
 582 // inverse tangent (arc tangent)
 583 //////////
 584
 585 static ex atan_evalf(const ex & x)
 586 {
 587         BEGIN_TYPECHECK
 588                 TYPECHECK(x,numeric)
 589         END_TYPECHECK(atan(x))
 590
 591         return atan(ex_to<numeric>(x)); // -> numeric atan(numeric)
 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_equal(_ex0()))
 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_evalf(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_ex_exactly_of_type(rel.lhs(),symbol));
 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 = static_cast<symbol *>(rel.lhs().bp);
 651                 const ex point = rel.rhs();
 652                 const symbol foo;
 653                 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         BEGIN_TYPECHECK
 680                 TYPECHECK(y,numeric)
 681                 TYPECHECK(x,numeric)
 682         END_TYPECHECK(atan2(y,x))
 683
 684         return atan(ex_to<numeric>(y),ex_to<numeric>(x)); // -> numeric atan(numeric)
 685 }
 686
 687 static ex atan2_eval(const ex & y, const ex & x)
 688 {
 689         if (y.info(info_flags::numeric) && !y.info(info_flags::crational) &&
 690                 x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 691                 return atan2_evalf(y,x);
 692         }
 693
 694         return atan2(y,x).hold();
 695 }
 696
 697 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
 698 {
 699         GINAC_ASSERT(deriv_param<2);
 700
 701         if (deriv_param==0) {
 702                 // d/dy atan(y,x)
 703                 return x*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
 704         }
 705         // d/dx atan(y,x)
 706         return -y*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
 707 }
 708
 709 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
 710                          evalf_func(atan2_evalf).
 711                          derivative_func(atan2_deriv));
 712
 713 //////////
 714 // hyperbolic sine (trigonometric function)
 715 //////////
 716
 717 static ex sinh_evalf(const ex & x)
 718 {
 719         BEGIN_TYPECHECK
 720            TYPECHECK(x,numeric)
 721         END_TYPECHECK(sinh(x))
 722
 723         return sinh(ex_to<numeric>(x)); // -> numeric sinh(numeric)
 724 }
 725
 726 static ex sinh_eval(const ex & x)
 727 {
 728         if (x.info(info_flags::numeric)) {
 729                 if (x.is_zero())  // sinh(0) -> 0
 730                         return _ex0();
 731                 if (!x.info(info_flags::crational))  // sinh(float) -> float
 732                         return sinh_evalf(x);
 733         }
 734
 735         if ((x/Pi).info(info_flags::numeric) &&
 736                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
 737                 return I*sin(x/I);
 738
 739         if (is_ex_exactly_of_type(x, function)) {
 740                 ex t = x.op(0);
 741                 // sinh(asinh(x)) -> x
 742                 if (is_ex_the_function(x, asinh))
 743                         return t;
 744                 // sinh(acosh(x)) -> (x-1)^(1/2) * (x+1)^(1/2)
 745                 if (is_ex_the_function(x, acosh))
 746                         return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2());
 747                 // sinh(atanh(x)) -> x*(1-x^2)^(-1/2)
 748                 if (is_ex_the_function(x, atanh))
 749                         return t*power(_ex1()-power(t,_ex2()),_ex_1_2());
 750         }
 751
 752         return sinh(x).hold();
 753 }
 754
 755 static ex sinh_deriv(const ex & x, unsigned deriv_param)
 756 {
 757         GINAC_ASSERT(deriv_param==0);
 758
 759         // d/dx sinh(x) -> cosh(x)
 760         return cosh(x);
 761 }
 762
 763 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
 764                         evalf_func(sinh_evalf).
 765                         derivative_func(sinh_deriv).
 766                         latex_name("\\sinh"));
 767
 768 //////////
 769 // hyperbolic cosine (trigonometric function)
 770 //////////
 771
 772 static ex cosh_evalf(const ex & x)
 773 {
 774         BEGIN_TYPECHECK
 775            TYPECHECK(x,numeric)
 776         END_TYPECHECK(cosh(x))
 777
 778         return cosh(ex_to<numeric>(x)); // -> numeric cosh(numeric)
 779 }
 780
 781 static ex cosh_eval(const ex & x)
 782 {
 783         if (x.info(info_flags::numeric)) {
 784                 if (x.is_zero())  // cosh(0) -> 1
 785                         return _ex1();
 786                 if (!x.info(info_flags::crational))  // cosh(float) -> float
 787                         return cosh_evalf(x);
 788         }
 789
 790         if ((x/Pi).info(info_flags::numeric) &&
 791                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
 792                 return cos(x/I);
 793
 794         if (is_ex_exactly_of_type(x, function)) {
 795                 ex t = x.op(0);
 796                 // cosh(acosh(x)) -> x
 797                 if (is_ex_the_function(x, acosh))
 798                         return t;
 799                 // cosh(asinh(x)) -> (1+x^2)^(1/2)
 800                 if (is_ex_the_function(x, asinh))
 801                         return power(_ex1()+power(t,_ex2()),_ex1_2());
 802                 // cosh(atanh(x)) -> (1-x^2)^(-1/2)
 803                 if (is_ex_the_function(x, atanh))
 804                         return power(_ex1()-power(t,_ex2()),_ex_1_2());
 805         }
 806
 807         return cosh(x).hold();
 808 }
 809
 810 static ex cosh_deriv(const ex & x, unsigned deriv_param)
 811 {
 812         GINAC_ASSERT(deriv_param==0);
 813
 814         // d/dx cosh(x) -> sinh(x)
 815         return sinh(x);
 816 }
 817
 818 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
 819                         evalf_func(cosh_evalf).
 820                         derivative_func(cosh_deriv).
 821                         latex_name("\\cosh"));
 822
 823 //////////
 824 // hyperbolic tangent (trigonometric function)
 825 //////////
 826
 827 static ex tanh_evalf(const ex & x)
 828 {
 829         BEGIN_TYPECHECK
 830            TYPECHECK(x,numeric)
 831         END_TYPECHECK(tanh(x))
 832
 833         return tanh(ex_to<numeric>(x)); // -> numeric tanh(numeric)
 834 }
 835
 836 static ex tanh_eval(const ex & x)
 837 {
 838         if (x.info(info_flags::numeric)) {
 839                 if (x.is_zero())  // tanh(0) -> 0
 840                         return _ex0();
 841                 if (!x.info(info_flags::crational))  // tanh(float) -> float
 842                         return tanh_evalf(x);
 843         }
 844
 845         if ((x/Pi).info(info_flags::numeric) &&
 846                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
 847                 return I*tan(x/I);
 848
 849         if (is_ex_exactly_of_type(x, function)) {
 850                 ex t = x.op(0);
 851                 // tanh(atanh(x)) -> x
 852                 if (is_ex_the_function(x, atanh))
 853                         return t;
 854                 // tanh(asinh(x)) -> x*(1+x^2)^(-1/2)
 855                 if (is_ex_the_function(x, asinh))
 856                         return t*power(_ex1()+power(t,_ex2()),_ex_1_2());
 857                 // tanh(acosh(x)) -> (x-1)^(1/2)*(x+1)^(1/2)/x
 858                 if (is_ex_the_function(x, acosh))
 859                         return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2())*power(t,_ex_1());
 860         }
 861
 862         return tanh(x).hold();
 863 }
 864
 865 static ex tanh_deriv(const ex & x, unsigned deriv_param)
 866 {
 867         GINAC_ASSERT(deriv_param==0);
 868
 869         // d/dx tanh(x) -> 1-tanh(x)^2
 870         return _ex1()-power(tanh(x),_ex2());
 871 }
 872
 873 static ex tanh_series(const ex &x,
 874                       const relational &rel,
 875                       int order,
 876                       unsigned options)
 877 {
 878         GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
 879         // method:
 880         // Taylor series where there is no pole falls back to tanh_deriv.
 881         // On a pole simply expand sinh(x)/cosh(x).
 882         const ex x_pt = x.subs(rel);
 883         if (!(2*I*x_pt/Pi).info(info_flags::odd))
 884                 throw do_taylor();  // caught by function::series()
 885         // if we got here we have to care for a simple pole
 886         return (sinh(x)/cosh(x)).series(rel, order+2, options);
 887 }
 888
 889 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
 890                         evalf_func(tanh_evalf).
 891                         derivative_func(tanh_deriv).
 892                         series_func(tanh_series).
 893                         latex_name("\\tanh"));
 894
 895 //////////
 896 // inverse hyperbolic sine (trigonometric function)
 897 //////////
 898
 899 static ex asinh_evalf(const ex & x)
 900 {
 901         BEGIN_TYPECHECK
 902            TYPECHECK(x,numeric)
 903         END_TYPECHECK(asinh(x))
 904
 905         return asinh(ex_to<numeric>(x)); // -> numeric asinh(numeric)
 906 }
 907
 908 static ex asinh_eval(const ex & x)
 909 {
 910         if (x.info(info_flags::numeric)) {
 911                 // asinh(0) -> 0
 912                 if (x.is_zero())
 913                         return _ex0();
 914                 // asinh(float) -> float
 915                 if (!x.info(info_flags::crational))
 916                         return asinh_evalf(x);
 917         }
 918
 919         return asinh(x).hold();
 920 }
 921
 922 static ex asinh_deriv(const ex & x, unsigned deriv_param)
 923 {
 924         GINAC_ASSERT(deriv_param==0);
 925
 926         // d/dx asinh(x) -> 1/sqrt(1+x^2)
 927         return power(_ex1()+power(x,_ex2()),_ex_1_2());
 928 }
 929
 930 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
 931                          evalf_func(asinh_evalf).
 932                          derivative_func(asinh_deriv));
 933
 934 //////////
 935 // inverse hyperbolic cosine (trigonometric function)
 936 //////////
 937
 938 static ex acosh_evalf(const ex & x)
 939 {
 940         BEGIN_TYPECHECK
 941            TYPECHECK(x,numeric)
 942         END_TYPECHECK(acosh(x))
 943
 944         return acosh(ex_to<numeric>(x)); // -> numeric acosh(numeric)
 945 }
 946
 947 static ex acosh_eval(const ex & x)
 948 {
 949         if (x.info(info_flags::numeric)) {
 950                 // acosh(0) -> Pi*I/2
 951                 if (x.is_zero())
 952                         return Pi*I*numeric(1,2);
 953                 // acosh(1) -> 0
 954                 if (x.is_equal(_ex1()))
 955                         return _ex0();
 956                 // acosh(-1) -> Pi*I
 957                 if (x.is_equal(_ex_1()))
 958                         return Pi*I;
 959                 // acosh(float) -> float
 960                 if (!x.info(info_flags::crational))
 961                         return acosh_evalf(x);
 962         }
 963
 964         return acosh(x).hold();
 965 }
 966
 967 static ex acosh_deriv(const ex & x, unsigned deriv_param)
 968 {
 969         GINAC_ASSERT(deriv_param==0);
 970
 971         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
 972         return power(x+_ex_1(),_ex_1_2())*power(x+_ex1(),_ex_1_2());
 973 }
 974
 975 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
 976                          evalf_func(acosh_evalf).
 977                          derivative_func(acosh_deriv));
 978
 979 //////////
 980 // inverse hyperbolic tangent (trigonometric function)
 981 //////////
 982
 983 static ex atanh_evalf(const ex & x)
 984 {
 985         BEGIN_TYPECHECK
 986            TYPECHECK(x,numeric)
 987         END_TYPECHECK(atanh(x))
 988
 989         return atanh(ex_to<numeric>(x)); // -> numeric atanh(numeric)
 990 }
 991
 992 static ex atanh_eval(const ex & x)
 993 {
 994         if (x.info(info_flags::numeric)) {
 995                 // atanh(0) -> 0
 996                 if (x.is_zero())
 997                         return _ex0();
 998                 // atanh({+|-}1) -> throw
 999                 if (x.is_equal(_ex1()) || x.is_equal(_ex_1()))
1000                         throw (pole_error("atanh_eval(): logarithmic pole",0));
1001                 // atanh(float) -> float
1002                 if (!x.info(info_flags::crational))
1003                         return atanh_evalf(x);
1004         }
1005
1006         return atanh(x).hold();
1007 }
1008
1009 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1010 {
1011         GINAC_ASSERT(deriv_param==0);
1012
1013         // d/dx atanh(x) -> 1/(1-x^2)
1014         return power(_ex1()-power(x,_ex2()),_ex_1());
1015 }
1016
1017 static ex atanh_series(const ex &arg,
1018                        const relational &rel,
1019                        int order,
1020                        unsigned options)
1021 {
1022         GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
1023         // method:
1024         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1025         // There are two branch cuts, one runnig from 1 up the real axis and one
1026         // one running from -1 down the real axis.  The points 1 and -1 are poles
1027         // On the branch cuts and the poles series expand
1028         //     (log(1+x)-log(1-x))/2
1029         // instead.
1030         const ex arg_pt = arg.subs(rel);
1031         if (!(arg_pt).info(info_flags::real))
1032                 throw do_taylor();     // Im(x) != 0
1033         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1())
1034                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1035         // care for the poles, using the defining formula for atanh()...
1036         if (arg_pt.is_equal(_ex1()) || arg_pt.is_equal(_ex_1()))
1037                 return ((log(_ex1()+arg)-log(_ex1()-arg))*_ex1_2()).series(rel, order, options);
1038         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1039         if (!(options & series_options::suppress_branchcut)) {
1040                 // method:
1041                 // This is the branch cut: assemble the primitive series manually and
1042                 // then add the corresponding complex step function.
1043                 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
1044                 const ex point = rel.rhs();
1045                 const symbol foo;
1046                 ex replarg = series(atanh(arg), *s==foo, order).subs(foo==point);
1047                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2();
1048                 if (arg_pt<_ex0())
1049                         Order0correction += log((arg_pt+_ex_1())/(arg_pt+_ex1()))*_ex1_2();
1050                 else
1051                         Order0correction += log((arg_pt+_ex1())/(arg_pt+_ex_1()))*_ex_1_2();
1052                 epvector seq;
1053                 seq.push_back(expair(Order0correction, _ex0()));
1054                 seq.push_back(expair(Order(_ex1()), order));
1055                 return series(replarg - pseries(rel, seq), rel, order);
1056         }
1057         throw do_taylor();
1058 }
1059
1060 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1061                          evalf_func(atanh_evalf).
1062                          derivative_func(atanh_deriv).
1063                          series_func(atanh_series));
1064
1065
1066 } // namespace GiNaC