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) 1999-2010 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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  22  */
  23
  24 #include "inifcns.h"
  25 #include "ex.h"
  26 #include "constant.h"
  27 #include "numeric.h"
  28 #include "power.h"
  29 #include "operators.h"
  30 #include "relational.h"
  31 #include "symbol.h"
  32 #include "pseries.h"
  33 #include "utils.h"
  34
  35 #include <stdexcept>
  36 #include <vector>
  37
  38 namespace GiNaC {
  39
  40 //////////
  41 // exponential function
  42 //////////
  43
  44 static ex exp_evalf(const ex & x)
  45 {
  46         if (is_exactly_a<numeric>(x))
  47                 return exp(ex_to<numeric>(x));
  48
  49         return exp(x).hold();
  50 }
  51
  52 static ex exp_eval(const ex & x)
  53 {
  54         // exp(0) -> 1
  55         if (x.is_zero()) {
  56                 return _ex1;
  57         }
  58
  59         // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
  60         const ex TwoExOverPiI=(_ex2*x)/(Pi*I);
  61         if (TwoExOverPiI.info(info_flags::integer)) {
  62                 const numeric z = mod(ex_to<numeric>(TwoExOverPiI),*_num4_p);
  63                 if (z.is_equal(*_num0_p))
  64                         return _ex1;
  65                 if (z.is_equal(*_num1_p))
  66                         return ex(I);
  67                 if (z.is_equal(*_num2_p))
  68                         return _ex_1;
  69                 if (z.is_equal(*_num3_p))
  70                         return ex(-I);
  71         }
  72
  73         // exp(log(x)) -> x
  74         if (is_ex_the_function(x, log))
  75                 return x.op(0);
  76
  77         // exp(float) -> float
  78         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
  79                 return exp(ex_to<numeric>(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 static ex exp_real_part(const ex & x)
  93 {
  94         return exp(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
  95 }
  96
  97 static ex exp_imag_part(const ex & x)
  98 {
  99         return exp(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
 100 }
 101
 102 REGISTER_FUNCTION(exp, eval_func(exp_eval).
 103                        evalf_func(exp_evalf).
 104                        derivative_func(exp_deriv).
 105                        real_part_func(exp_real_part).
 106                        imag_part_func(exp_imag_part).
 107                        latex_name("\\exp"));
 108
 109 //////////
 110 // natural logarithm
 111 //////////
 112
 113 static ex log_evalf(const ex & x)
 114 {
 115         if (is_exactly_a<numeric>(x))
 116                 return log(ex_to<numeric>(x));
 117
 118         return log(x).hold();
 119 }
 120
 121 static ex log_eval(const ex & x)
 122 {
 123         if (x.info(info_flags::numeric)) {
 124                 if (x.is_zero())         // log(0) -> infinity
 125                         throw(pole_error("log_eval(): log(0)",0));
 126                 if (x.info(info_flags::rational) && x.info(info_flags::negative))
 127                         return (log(-x)+I*Pi);
 128                 if (x.is_equal(_ex1))  // log(1) -> 0
 129                         return _ex0;
 130                 if (x.is_equal(I))       // log(I) -> Pi*I/2
 131                         return (Pi*I*_ex1_2);
 132                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
 133                         return (Pi*I*_ex_1_2);
 134
 135                 // log(float) -> float
 136                 if (!x.info(info_flags::crational))
 137                         return log(ex_to<numeric>(x));
 138         }
 139
 140         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
 141         if (is_ex_the_function(x, exp)) {
 142                 const ex &t = x.op(0);
 143                 if (t.info(info_flags::real))
 144                         return t;
 145         }
 146
 147         return log(x).hold();
 148 }
 149
 150 static ex log_deriv(const ex & x, unsigned deriv_param)
 151 {
 152         GINAC_ASSERT(deriv_param==0);
 153
 154         // d/dx log(x) -> 1/x
 155         return power(x, _ex_1);
 156 }
 157
 158 static ex log_series(const ex &arg,
 159                      const relational &rel,
 160                      int order,
 161                      unsigned options)
 162 {
 163         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 164         ex arg_pt;
 165         bool must_expand_arg = false;
 166         // maybe substitution of rel into arg fails because of a pole
 167         try {
 168                 arg_pt = arg.subs(rel, subs_options::no_pattern);
 169         } catch (pole_error) {
 170                 must_expand_arg = true;
 171         }
 172         // or we are at the branch point anyways
 173         if (arg_pt.is_zero())
 174                 must_expand_arg = true;
 175
 176         if (must_expand_arg) {
 177                 // method:
 178                 // This is the branch point: Series expand the argument first, then
 179                 // trivially factorize it to isolate that part which has constant
 180                 // leading coefficient in this fashion:
 181                 //   x^n + x^(n+1) +...+ Order(x^(n+m))  ->  x^n * (1 + x +...+ Order(x^m)).
 182                 // Return a plain n*log(x) for the x^n part and series expand the
 183                 // other part.  Add them together and reexpand again in order to have
 184                 // one unnested pseries object.  All this also works for negative n.
 185                 pseries argser;          // series expansion of log's argument
 186                 unsigned extra_ord = 0;  // extra expansion order
 187                 do {
 188                         // oops, the argument expanded to a pure Order(x^something)...
 189                         argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
 190                         ++extra_ord;
 191                 } while (!argser.is_terminating() && argser.nops()==1);
 192
 193                 const symbol &s = ex_to<symbol>(rel.lhs());
 194                 const ex &point = rel.rhs();
 195                 const int n = argser.ldegree(s);
 196                 epvector seq;
 197                 // construct what we carelessly called the n*log(x) term above
 198                 const ex coeff = argser.coeff(s, n);
 199                 // expand the log, but only if coeff is real and > 0, since otherwise
 200                 // it would make the branch cut run into the wrong direction
 201                 if (coeff.info(info_flags::positive))
 202                         seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
 203                 else
 204                         seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));
 205
 206                 if (!argser.is_terminating() || argser.nops()!=1) {
 207                         // in this case n more (or less) terms are needed
 208                         // (sadly, to generate them, we have to start from the beginning)
 209                         if (n == 0 && coeff == 1) {
 210                                 epvector epv;
 211                                 ex acc = (new pseries(rel, epv))->setflag(status_flags::dynallocated);
 212                                 epv.reserve(2);
 213                                 epv.push_back(expair(-1, _ex0));
 214                                 epv.push_back(expair(Order(_ex1), order));
 215                                 ex rest = pseries(rel, epv).add_series(argser);
 216                                 for (int i = order-1; i>0; --i) {
 217                                         epvector cterm;
 218                                         cterm.reserve(1);
 219                                         cterm.push_back(expair(i%2 ? _ex1/i : _ex_1/i, _ex0));
 220                                         acc = pseries(rel, cterm).add_series(ex_to<pseries>(acc));
 221                                         acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
 222                                 }
 223                                 return acc;
 224                         }
 225                         const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
 226                         return pseries(rel, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
 227                 } else  // it was a monomial
 228                         return pseries(rel, seq);
 229         }
 230         if (!(options & series_options::suppress_branchcut) &&
 231              arg_pt.info(info_flags::negative)) {
 232                 // method:
 233                 // This is the branch cut: assemble the primitive series manually and
 234                 // then add the corresponding complex step function.
 235                 const symbol &s = ex_to<symbol>(rel.lhs());
 236                 const ex &point = rel.rhs();
 237                 const symbol foo;
 238                 const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 239                 epvector seq;
 240                 seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
 241                 seq.push_back(expair(Order(_ex1), order));
 242                 return series(replarg - I*Pi + pseries(rel, seq), rel, order);
 243         }
 244         throw do_taylor();  // caught by function::series()
 245 }
 246
 247 static ex log_real_part(const ex & x)
 248 {
 249         if (x.info(info_flags::nonnegative))
 250                 return log(x).hold();
 251         return log(abs(x));
 252 }
 253
 254 static ex log_imag_part(const ex & x)
 255 {
 256         if (x.info(info_flags::nonnegative))
 257                 return 0;
 258         return atan2(GiNaC::imag_part(x), GiNaC::real_part(x));
 259 }
 260
 261 REGISTER_FUNCTION(log, eval_func(log_eval).
 262                        evalf_func(log_evalf).
 263                        derivative_func(log_deriv).
 264                        series_func(log_series).
 265                        real_part_func(log_real_part).
 266                        imag_part_func(log_imag_part).
 267                        latex_name("\\ln"));
 268
 269 //////////
 270 // sine (trigonometric function)
 271 //////////
 272
 273 static ex sin_evalf(const ex & x)
 274 {
 275         if (is_exactly_a<numeric>(x))
 276                 return sin(ex_to<numeric>(x));
 277
 278         return sin(x).hold();
 279 }
 280
 281 static ex sin_eval(const ex & x)
 282 {
 283         // sin(n/d*Pi) -> { all known non-nested radicals }
 284         const ex SixtyExOverPi = _ex60*x/Pi;
 285         ex sign = _ex1;
 286         if (SixtyExOverPi.info(info_flags::integer)) {
 287                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 288                 if (z>=*_num60_p) {
 289                         // wrap to interval [0, Pi)
 290                         z -= *_num60_p;
 291                         sign = _ex_1;
 292                 }
 293                 if (z>*_num30_p) {
 294                         // wrap to interval [0, Pi/2)
 295                         z = *_num60_p-z;
 296                 }
 297                 if (z.is_equal(*_num0_p))  // sin(0)       -> 0
 298                         return _ex0;
 299                 if (z.is_equal(*_num5_p))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
 300                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 301                 if (z.is_equal(*_num6_p))  // sin(Pi/10)   -> sqrt(5)/4-1/4
 302                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 303                 if (z.is_equal(*_num10_p)) // sin(Pi/6)    -> 1/2
 304                         return sign*_ex1_2;
 305                 if (z.is_equal(*_num15_p)) // sin(Pi/4)    -> sqrt(2)/2
 306                         return sign*_ex1_2*sqrt(_ex2);
 307                 if (z.is_equal(*_num18_p)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
 308                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 309                 if (z.is_equal(*_num20_p)) // sin(Pi/3)    -> sqrt(3)/2
 310                         return sign*_ex1_2*sqrt(_ex3);
 311                 if (z.is_equal(*_num25_p)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
 312                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 313                 if (z.is_equal(*_num30_p)) // sin(Pi/2)    -> 1
 314                         return sign;
 315         }
 316
 317         if (is_exactly_a<function>(x)) {
 318                 const ex &t = x.op(0);
 319
 320                 // sin(asin(x)) -> x
 321                 if (is_ex_the_function(x, asin))
 322                         return t;
 323
 324                 // sin(acos(x)) -> sqrt(1-x^2)
 325                 if (is_ex_the_function(x, acos))
 326                         return sqrt(_ex1-power(t,_ex2));
 327
 328                 // sin(atan(x)) -> x/sqrt(1+x^2)
 329                 if (is_ex_the_function(x, atan))
 330                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
 331         }
 332
 333         // sin(float) -> float
 334         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 335                 return sin(ex_to<numeric>(x));
 336
 337         // sin() is odd
 338         if (x.info(info_flags::negative))
 339                 return -sin(-x);
 340
 341         return sin(x).hold();
 342 }
 343
 344 static ex sin_deriv(const ex & x, unsigned deriv_param)
 345 {
 346         GINAC_ASSERT(deriv_param==0);
 347
 348         // d/dx sin(x) -> cos(x)
 349         return cos(x);
 350 }
 351
 352 static ex sin_real_part(const ex & x)
 353 {
 354         return cosh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 355 }
 356
 357 static ex sin_imag_part(const ex & x)
 358 {
 359         return sinh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 360 }
 361
 362 REGISTER_FUNCTION(sin, eval_func(sin_eval).
 363                        evalf_func(sin_evalf).
 364                        derivative_func(sin_deriv).
 365                        real_part_func(sin_real_part).
 366                        imag_part_func(sin_imag_part).
 367                        latex_name("\\sin"));
 368
 369 //////////
 370 // cosine (trigonometric function)
 371 //////////
 372
 373 static ex cos_evalf(const ex & x)
 374 {
 375         if (is_exactly_a<numeric>(x))
 376                 return cos(ex_to<numeric>(x));
 377
 378         return cos(x).hold();
 379 }
 380
 381 static ex cos_eval(const ex & x)
 382 {
 383         // cos(n/d*Pi) -> { all known non-nested radicals }
 384         const ex SixtyExOverPi = _ex60*x/Pi;
 385         ex sign = _ex1;
 386         if (SixtyExOverPi.info(info_flags::integer)) {
 387                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
 388                 if (z>=*_num60_p) {
 389                         // wrap to interval [0, Pi)
 390                         z = *_num120_p-z;
 391                 }
 392                 if (z>=*_num30_p) {
 393                         // wrap to interval [0, Pi/2)
 394                         z = *_num60_p-z;
 395                         sign = _ex_1;
 396                 }
 397                 if (z.is_equal(*_num0_p))  // cos(0)       -> 1
 398                         return sign;
 399                 if (z.is_equal(*_num5_p))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
 400                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
 401                 if (z.is_equal(*_num10_p)) // cos(Pi/6)    -> sqrt(3)/2
 402                         return sign*_ex1_2*sqrt(_ex3);
 403                 if (z.is_equal(*_num12_p)) // cos(Pi/5)    -> sqrt(5)/4+1/4
 404                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
 405                 if (z.is_equal(*_num15_p)) // cos(Pi/4)    -> sqrt(2)/2
 406                         return sign*_ex1_2*sqrt(_ex2);
 407                 if (z.is_equal(*_num20_p)) // cos(Pi/3)    -> 1/2
 408                         return sign*_ex1_2;
 409                 if (z.is_equal(*_num24_p)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
 410                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
 411                 if (z.is_equal(*_num25_p)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
 412                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
 413                 if (z.is_equal(*_num30_p)) // cos(Pi/2)    -> 0
 414                         return _ex0;
 415         }
 416
 417         if (is_exactly_a<function>(x)) {
 418                 const ex &t = x.op(0);
 419
 420                 // cos(acos(x)) -> x
 421                 if (is_ex_the_function(x, acos))
 422                         return t;
 423
 424                 // cos(asin(x)) -> sqrt(1-x^2)
 425                 if (is_ex_the_function(x, asin))
 426                         return sqrt(_ex1-power(t,_ex2));
 427
 428                 // cos(atan(x)) -> 1/sqrt(1+x^2)
 429                 if (is_ex_the_function(x, atan))
 430                         return power(_ex1+power(t,_ex2),_ex_1_2);
 431         }
 432
 433         // cos(float) -> float
 434         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
 435                 return cos(ex_to<numeric>(x));
 436
 437         // cos() is even
 438         if (x.info(info_flags::negative))
 439                 return cos(-x);
 440
 441         return cos(x).hold();
 442 }
 443
 444 static ex cos_deriv(const ex & x, unsigned deriv_param)
 445 {
 446         GINAC_ASSERT(deriv_param==0);
 447
 448         // d/dx cos(x) -> -sin(x)
 449         return -sin(x);
 450 }
 451
 452 static ex cos_real_part(const ex & x)
 453 {
 454         return cosh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
 455 }
 456
 457 static ex cos_imag_part(const ex & x)
 458 {
 459         return -sinh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
 460 }
 461
 462 REGISTER_FUNCTION(cos, eval_func(cos_eval).
 463                        evalf_func(cos_evalf).
 464                        derivative_func(cos_deriv).
 465                        real_part_func(cos_real_part).
 466                        imag_part_func(cos_imag_part).
 467                        latex_name("\\cos"));
 468
 469 //////////
 470 // tangent (trigonometric function)
 471 //////////
 472
 473 static ex tan_evalf(const ex & x)
 474 {
 475         if (is_exactly_a<numeric>(x))
 476                 return tan(ex_to<numeric>(x));
 477
 478         return tan(x).hold();
 479 }
 480
 481 static ex tan_eval(const ex & x)
 482 {
 483         // tan(n/d*Pi) -> { all known non-nested radicals }
 484         const ex SixtyExOverPi = _ex60*x/Pi;
 485         ex sign = _ex1;
 486         if (SixtyExOverPi.info(info_flags::integer)) {
 487                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num60_p);
 488                 if (z>=*_num60_p) {
 489                         // wrap to interval [0, Pi)
 490                         z -= *_num60_p;
 491                 }
 492                 if (z>=*_num30_p) {
 493                         // wrap to interval [0, Pi/2)
 494                         z = *_num60_p-z;
 495                         sign = _ex_1;
 496                 }
 497                 if (z.is_equal(*_num0_p))  // tan(0)       -> 0
 498                         return _ex0;
 499                 if (z.is_equal(*_num5_p))  // tan(Pi/12)   -> 2-sqrt(3)
 500                         return sign*(_ex2-sqrt(_ex3));
 501                 if (z.is_equal(*_num10_p)) // tan(Pi/6)    -> sqrt(3)/3
 502                         return sign*_ex1_3*sqrt(_ex3);
 503                 if (z.is_equal(*_num15_p)) // tan(Pi/4)    -> 1
 504                         return sign;
 505                 if (z.is_equal(*_num20_p)) // tan(Pi/3)    -> sqrt(3)
 506                         return sign*sqrt(_ex3);
 507                 if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3)
 508                         return sign*(sqrt(_ex3)+_ex2);
 509                 if (z.is_equal(*_num30_p)) // tan(Pi/2)    -> infinity
 510                         throw (pole_error("tan_eval(): simple pole",1));
 511         }
 512
 513         if (is_exactly_a<function>(x)) {
 514                 const ex &t = x.op(0);
 515
 516                 // tan(atan(x)) -> x
 517                 if (is_ex_the_function(x, atan))
 518                         return t;
 519
 520                 // tan(asin(x)) -> x/sqrt(1+x^2)
 521                 if (is_ex_the_function(x, asin))
 522                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 523
 524                 // tan(acos(x)) -> sqrt(1-x^2)/x
 525                 if (is_ex_the_function(x, acos))
 526                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
 527         }
 528
 529         // tan(float) -> float
 530         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
 531                 return tan(ex_to<numeric>(x));
 532         }
 533
 534         // tan() is odd
 535         if (x.info(info_flags::negative))
 536                 return -tan(-x);
 537
 538         return tan(x).hold();
 539 }
 540
 541 static ex tan_deriv(const ex & x, unsigned deriv_param)
 542 {
 543         GINAC_ASSERT(deriv_param==0);
 544
 545         // d/dx tan(x) -> 1+tan(x)^2;
 546         return (_ex1+power(tan(x),_ex2));
 547 }
 548
 549 static ex tan_real_part(const ex & x)
 550 {
 551         ex a = GiNaC::real_part(x);
 552         ex b = GiNaC::imag_part(x);
 553         return tan(a)/(1+power(tan(a),2)*power(tan(b),2));
 554 }
 555
 556 static ex tan_imag_part(const ex & x)
 557 {
 558         ex a = GiNaC::real_part(x);
 559         ex b = GiNaC::imag_part(x);
 560         return tanh(b)/(1+power(tan(a),2)*power(tan(b),2));
 561 }
 562
 563 static ex tan_series(const ex &x,
 564                      const relational &rel,
 565                      int order,
 566                      unsigned options)
 567 {
 568         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 569         // method:
 570         // Taylor series where there is no pole falls back to tan_deriv.
 571         // On a pole simply expand sin(x)/cos(x).
 572         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 573         if (!(2*x_pt/Pi).info(info_flags::odd))
 574                 throw do_taylor();  // caught by function::series()
 575         // if we got here we have to care for a simple pole
 576         return (sin(x)/cos(x)).series(rel, order, options);
 577 }
 578
 579 REGISTER_FUNCTION(tan, eval_func(tan_eval).
 580                        evalf_func(tan_evalf).
 581                        derivative_func(tan_deriv).
 582                        series_func(tan_series).
 583                        real_part_func(tan_real_part).
 584                        imag_part_func(tan_imag_part).
 585                        latex_name("\\tan"));
 586
 587 //////////
 588 // inverse sine (arc sine)
 589 //////////
 590
 591 static ex asin_evalf(const ex & x)
 592 {
 593         if (is_exactly_a<numeric>(x))
 594                 return asin(ex_to<numeric>(x));
 595
 596         return asin(x).hold();
 597 }
 598
 599 static ex asin_eval(const ex & x)
 600 {
 601         if (x.info(info_flags::numeric)) {
 602
 603                 // asin(0) -> 0
 604                 if (x.is_zero())
 605                         return x;
 606
 607                 // asin(1/2) -> Pi/6
 608                 if (x.is_equal(_ex1_2))
 609                         return numeric(1,6)*Pi;
 610
 611                 // asin(1) -> Pi/2
 612                 if (x.is_equal(_ex1))
 613                         return _ex1_2*Pi;
 614
 615                 // asin(-1/2) -> -Pi/6
 616                 if (x.is_equal(_ex_1_2))
 617                         return numeric(-1,6)*Pi;
 618
 619                 // asin(-1) -> -Pi/2
 620                 if (x.is_equal(_ex_1))
 621                         return _ex_1_2*Pi;
 622
 623                 // asin(float) -> float
 624                 if (!x.info(info_flags::crational))
 625                         return asin(ex_to<numeric>(x));
 626
 627                 // asin() is odd
 628                 if (x.info(info_flags::negative))
 629                         return -asin(-x);
 630         }
 631
 632         return asin(x).hold();
 633 }
 634
 635 static ex asin_deriv(const ex & x, unsigned deriv_param)
 636 {
 637         GINAC_ASSERT(deriv_param==0);
 638
 639         // d/dx asin(x) -> 1/sqrt(1-x^2)
 640         return power(1-power(x,_ex2),_ex_1_2);
 641 }
 642
 643 REGISTER_FUNCTION(asin, eval_func(asin_eval).
 644                         evalf_func(asin_evalf).
 645                         derivative_func(asin_deriv).
 646                         latex_name("\\arcsin"));
 647
 648 //////////
 649 // inverse cosine (arc cosine)
 650 //////////
 651
 652 static ex acos_evalf(const ex & x)
 653 {
 654         if (is_exactly_a<numeric>(x))
 655                 return acos(ex_to<numeric>(x));
 656
 657         return acos(x).hold();
 658 }
 659
 660 static ex acos_eval(const ex & x)
 661 {
 662         if (x.info(info_flags::numeric)) {
 663
 664                 // acos(1) -> 0
 665                 if (x.is_equal(_ex1))
 666                         return _ex0;
 667
 668                 // acos(1/2) -> Pi/3
 669                 if (x.is_equal(_ex1_2))
 670                         return _ex1_3*Pi;
 671
 672                 // acos(0) -> Pi/2
 673                 if (x.is_zero())
 674                         return _ex1_2*Pi;
 675
 676                 // acos(-1/2) -> 2/3*Pi
 677                 if (x.is_equal(_ex_1_2))
 678                         return numeric(2,3)*Pi;
 679
 680                 // acos(-1) -> Pi
 681                 if (x.is_equal(_ex_1))
 682                         return Pi;
 683
 684                 // acos(float) -> float
 685                 if (!x.info(info_flags::crational))
 686                         return acos(ex_to<numeric>(x));
 687
 688                 // acos(-x) -> Pi-acos(x)
 689                 if (x.info(info_flags::negative))
 690                         return Pi-acos(-x);
 691         }
 692
 693         return acos(x).hold();
 694 }
 695
 696 static ex acos_deriv(const ex & x, unsigned deriv_param)
 697 {
 698         GINAC_ASSERT(deriv_param==0);
 699
 700         // d/dx acos(x) -> -1/sqrt(1-x^2)
 701         return -power(1-power(x,_ex2),_ex_1_2);
 702 }
 703
 704 REGISTER_FUNCTION(acos, eval_func(acos_eval).
 705                         evalf_func(acos_evalf).
 706                         derivative_func(acos_deriv).
 707                         latex_name("\\arccos"));
 708
 709 //////////
 710 // inverse tangent (arc tangent)
 711 //////////
 712
 713 static ex atan_evalf(const ex & x)
 714 {
 715         if (is_exactly_a<numeric>(x))
 716                 return atan(ex_to<numeric>(x));
 717
 718         return atan(x).hold();
 719 }
 720
 721 static ex atan_eval(const ex & x)
 722 {
 723         if (x.info(info_flags::numeric)) {
 724
 725                 // atan(0) -> 0
 726                 if (x.is_zero())
 727                         return _ex0;
 728
 729                 // atan(1) -> Pi/4
 730                 if (x.is_equal(_ex1))
 731                         return _ex1_4*Pi;
 732
 733                 // atan(-1) -> -Pi/4
 734                 if (x.is_equal(_ex_1))
 735                         return _ex_1_4*Pi;
 736
 737                 if (x.is_equal(I) || x.is_equal(-I))
 738                         throw (pole_error("atan_eval(): logarithmic pole",0));
 739
 740                 // atan(float) -> float
 741                 if (!x.info(info_flags::crational))
 742                         return atan(ex_to<numeric>(x));
 743
 744                 // atan() is odd
 745                 if (x.info(info_flags::negative))
 746                         return -atan(-x);
 747         }
 748
 749         return atan(x).hold();
 750 }
 751
 752 static ex atan_deriv(const ex & x, unsigned deriv_param)
 753 {
 754         GINAC_ASSERT(deriv_param==0);
 755
 756         // d/dx atan(x) -> 1/(1+x^2)
 757         return power(_ex1+power(x,_ex2), _ex_1);
 758 }
 759
 760 static ex atan_series(const ex &arg,
 761                       const relational &rel,
 762                       int order,
 763                       unsigned options)
 764 {
 765         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 766         // method:
 767         // Taylor series where there is no pole or cut falls back to atan_deriv.
 768         // There are two branch cuts, one runnig from I up the imaginary axis and
 769         // one running from -I down the imaginary axis.  The points I and -I are
 770         // poles.
 771         // On the branch cuts and the poles series expand
 772         //     (log(1+I*x)-log(1-I*x))/(2*I)
 773         // instead.
 774         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 775         if (!(I*arg_pt).info(info_flags::real))
 776                 throw do_taylor();     // Re(x) != 0
 777         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
 778                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
 779         // care for the poles, using the defining formula for atan()...
 780         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
 781                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
 782         if (!(options & series_options::suppress_branchcut)) {
 783                 // method:
 784                 // This is the branch cut: assemble the primitive series manually and
 785                 // then add the corresponding complex step function.
 786                 const symbol &s = ex_to<symbol>(rel.lhs());
 787                 const ex &point = rel.rhs();
 788                 const symbol foo;
 789                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
 790                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
 791                 if ((I*arg_pt)<_ex0)
 792                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
 793                 else
 794                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
 795                 epvector seq;
 796                 seq.push_back(expair(Order0correction, _ex0));
 797                 seq.push_back(expair(Order(_ex1), order));
 798                 return series(replarg - pseries(rel, seq), rel, order);
 799         }
 800         throw do_taylor();
 801 }
 802
 803 REGISTER_FUNCTION(atan, eval_func(atan_eval).
 804                         evalf_func(atan_evalf).
 805                         derivative_func(atan_deriv).
 806                         series_func(atan_series).
 807                         latex_name("\\arctan"));
 808
 809 //////////
 810 // inverse tangent (atan2(y,x))
 811 //////////
 812
 813 static ex atan2_evalf(const ex &y, const ex &x)
 814 {
 815         if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
 816                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 817
 818         return atan2(y, x).hold();
 819 }
 820
 821 static ex atan2_eval(const ex & y, const ex & x)
 822 {
 823         if (y.is_zero()) {
 824
 825                 // atan2(0, 0) -> 0
 826                 if (x.is_zero())
 827                         return _ex0;
 828
 829                 // atan2(0, x), x real and positive -> 0
 830                 if (x.info(info_flags::positive))
 831                         return _ex0;
 832
 833                 // atan2(0, x), x real and negative -> Pi
 834                 if (x.info(info_flags::negative))
 835                         return Pi;
 836         }
 837
 838         if (x.is_zero()) {
 839
 840                 // atan2(y, 0), y real and positive -> Pi/2
 841                 if (y.info(info_flags::positive))
 842                         return _ex1_2*Pi;
 843
 844                 // atan2(y, 0), y real and negative -> -Pi/2
 845                 if (y.info(info_flags::negative))
 846                         return _ex_1_2*Pi;
 847         }
 848
 849         if (y.is_equal(x)) {
 850
 851                 // atan2(y, y), y real and positive -> Pi/4
 852                 if (y.info(info_flags::positive))
 853                         return _ex1_4*Pi;
 854
 855                 // atan2(y, y), y real and negative -> -3/4*Pi
 856                 if (y.info(info_flags::negative))
 857                         return numeric(-3, 4)*Pi;
 858         }
 859
 860         if (y.is_equal(-x)) {
 861
 862                 // atan2(y, -y), y real and positive -> 3*Pi/4
 863                 if (y.info(info_flags::positive))
 864                         return numeric(3, 4)*Pi;
 865
 866                 // atan2(y, -y), y real and negative -> -Pi/4
 867                 if (y.info(info_flags::negative))
 868                         return _ex_1_4*Pi;
 869         }
 870
 871         // atan2(float, float) -> float
 872         if (is_a<numeric>(y) && !y.info(info_flags::crational) &&
 873             is_a<numeric>(x) && !x.info(info_flags::crational))
 874                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 875
 876         // atan2(real, real) -> atan(y/x) +/- Pi
 877         if (y.info(info_flags::real) && x.info(info_flags::real)) {
 878                 if (x.info(info_flags::positive))
 879                         return atan(y/x);
 880
 881                 if (x.info(info_flags::negative)) {
 882                         if (y.info(info_flags::positive))
 883                                 return atan(y/x)+Pi;
 884                         if (y.info(info_flags::negative))
 885                                 return atan(y/x)-Pi;
 886                 }
 887         }
 888
 889         return atan2(y, x).hold();
 890 }
 891
 892 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
 893 {
 894         GINAC_ASSERT(deriv_param<2);
 895
 896         if (deriv_param==0) {
 897                 // d/dy atan2(y,x)
 898                 return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
 899         }
 900         // d/dx atan2(y,x)
 901         return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
 902 }
 903
 904 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
 905                          evalf_func(atan2_evalf).
 906                          derivative_func(atan2_deriv));
 907
 908 //////////
 909 // hyperbolic sine (trigonometric function)
 910 //////////
 911
 912 static ex sinh_evalf(const ex & x)
 913 {
 914         if (is_exactly_a<numeric>(x))
 915                 return sinh(ex_to<numeric>(x));
 916
 917         return sinh(x).hold();
 918 }
 919
 920 static ex sinh_eval(const ex & x)
 921 {
 922         if (x.info(info_flags::numeric)) {
 923
 924                 // sinh(0) -> 0
 925                 if (x.is_zero())
 926                         return _ex0;
 927
 928                 // sinh(float) -> float
 929                 if (!x.info(info_flags::crational))
 930                         return sinh(ex_to<numeric>(x));
 931
 932                 // sinh() is odd
 933                 if (x.info(info_flags::negative))
 934                         return -sinh(-x);
 935         }
 936
 937         if ((x/Pi).info(info_flags::numeric) &&
 938                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
 939                 return I*sin(x/I);
 940
 941         if (is_exactly_a<function>(x)) {
 942                 const ex &t = x.op(0);
 943
 944                 // sinh(asinh(x)) -> x
 945                 if (is_ex_the_function(x, asinh))
 946                         return t;
 947
 948                 // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
 949                 if (is_ex_the_function(x, acosh))
 950                         return sqrt(t-_ex1)*sqrt(t+_ex1);
 951
 952                 // sinh(atanh(x)) -> x/sqrt(1-x^2)
 953                 if (is_ex_the_function(x, atanh))
 954                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
 955         }
 956
 957         return sinh(x).hold();
 958 }
 959
 960 static ex sinh_deriv(const ex & x, unsigned deriv_param)
 961 {
 962         GINAC_ASSERT(deriv_param==0);
 963
 964         // d/dx sinh(x) -> cosh(x)
 965         return cosh(x);
 966 }
 967
 968 static ex sinh_real_part(const ex & x)
 969 {
 970         return sinh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
 971 }
 972
 973 static ex sinh_imag_part(const ex & x)
 974 {
 975         return cosh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
 976 }
 977
 978 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
 979                         evalf_func(sinh_evalf).
 980                         derivative_func(sinh_deriv).
 981                         real_part_func(sinh_real_part).
 982                         imag_part_func(sinh_imag_part).
 983                         latex_name("\\sinh"));
 984
 985 //////////
 986 // hyperbolic cosine (trigonometric function)
 987 //////////
 988
 989 static ex cosh_evalf(const ex & x)
 990 {
 991         if (is_exactly_a<numeric>(x))
 992                 return cosh(ex_to<numeric>(x));
 993
 994         return cosh(x).hold();
 995 }
 996
 997 static ex cosh_eval(const ex & x)
 998 {
 999         if (x.info(info_flags::numeric)) {
1000
1001                 // cosh(0) -> 1
1002                 if (x.is_zero())
1003                         return _ex1;
1004
1005                 // cosh(float) -> float
1006                 if (!x.info(info_flags::crational))
1007                         return cosh(ex_to<numeric>(x));
1008
1009                 // cosh() is even
1010                 if (x.info(info_flags::negative))
1011                         return cosh(-x);
1012         }
1013
1014         if ((x/Pi).info(info_flags::numeric) &&
1015                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
1016                 return cos(x/I);
1017
1018         if (is_exactly_a<function>(x)) {
1019                 const ex &t = x.op(0);
1020
1021                 // cosh(acosh(x)) -> x
1022                 if (is_ex_the_function(x, acosh))
1023                         return t;
1024
1025                 // cosh(asinh(x)) -> sqrt(1+x^2)
1026                 if (is_ex_the_function(x, asinh))
1027                         return sqrt(_ex1+power(t,_ex2));
1028
1029                 // cosh(atanh(x)) -> 1/sqrt(1-x^2)
1030                 if (is_ex_the_function(x, atanh))
1031                         return power(_ex1-power(t,_ex2),_ex_1_2);
1032         }
1033
1034         return cosh(x).hold();
1035 }
1036
1037 static ex cosh_deriv(const ex & x, unsigned deriv_param)
1038 {
1039         GINAC_ASSERT(deriv_param==0);
1040
1041         // d/dx cosh(x) -> sinh(x)
1042         return sinh(x);
1043 }
1044
1045 static ex cosh_real_part(const ex & x)
1046 {
1047         return cosh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
1048 }
1049
1050 static ex cosh_imag_part(const ex & x)
1051 {
1052         return sinh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
1053 }
1054
1055 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
1056                         evalf_func(cosh_evalf).
1057                         derivative_func(cosh_deriv).
1058                         real_part_func(cosh_real_part).
1059                         imag_part_func(cosh_imag_part).
1060                         latex_name("\\cosh"));
1061
1062 //////////
1063 // hyperbolic tangent (trigonometric function)
1064 //////////
1065
1066 static ex tanh_evalf(const ex & x)
1067 {
1068         if (is_exactly_a<numeric>(x))
1069                 return tanh(ex_to<numeric>(x));
1070
1071         return tanh(x).hold();
1072 }
1073
1074 static ex tanh_eval(const ex & x)
1075 {
1076         if (x.info(info_flags::numeric)) {
1077
1078                 // tanh(0) -> 0
1079                 if (x.is_zero())
1080                         return _ex0;
1081
1082                 // tanh(float) -> float
1083                 if (!x.info(info_flags::crational))
1084                         return tanh(ex_to<numeric>(x));
1085
1086                 // tanh() is odd
1087                 if (x.info(info_flags::negative))
1088                         return -tanh(-x);
1089         }
1090
1091         if ((x/Pi).info(info_flags::numeric) &&
1092                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
1093                 return I*tan(x/I);
1094
1095         if (is_exactly_a<function>(x)) {
1096                 const ex &t = x.op(0);
1097
1098                 // tanh(atanh(x)) -> x
1099                 if (is_ex_the_function(x, atanh))
1100                         return t;
1101
1102                 // tanh(asinh(x)) -> x/sqrt(1+x^2)
1103                 if (is_ex_the_function(x, asinh))
1104                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
1105
1106                 // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
1107                 if (is_ex_the_function(x, acosh))
1108                         return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
1109         }
1110
1111         return tanh(x).hold();
1112 }
1113
1114 static ex tanh_deriv(const ex & x, unsigned deriv_param)
1115 {
1116         GINAC_ASSERT(deriv_param==0);
1117
1118         // d/dx tanh(x) -> 1-tanh(x)^2
1119         return _ex1-power(tanh(x),_ex2);
1120 }
1121
1122 static ex tanh_series(const ex &x,
1123                       const relational &rel,
1124                       int order,
1125                       unsigned options)
1126 {
1127         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1128         // method:
1129         // Taylor series where there is no pole falls back to tanh_deriv.
1130         // On a pole simply expand sinh(x)/cosh(x).
1131         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1132         if (!(2*I*x_pt/Pi).info(info_flags::odd))
1133                 throw do_taylor();  // caught by function::series()
1134         // if we got here we have to care for a simple pole
1135         return (sinh(x)/cosh(x)).series(rel, order, options);
1136 }
1137
1138 static ex tanh_real_part(const ex & x)
1139 {
1140         ex a = GiNaC::real_part(x);
1141         ex b = GiNaC::imag_part(x);
1142         return tanh(a)/(1+power(tanh(a),2)*power(tan(b),2));
1143 }
1144
1145 static ex tanh_imag_part(const ex & x)
1146 {
1147         ex a = GiNaC::real_part(x);
1148         ex b = GiNaC::imag_part(x);
1149         return tan(b)/(1+power(tanh(a),2)*power(tan(b),2));
1150 }
1151
1152 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
1153                         evalf_func(tanh_evalf).
1154                         derivative_func(tanh_deriv).
1155                         series_func(tanh_series).
1156                         real_part_func(tanh_real_part).
1157                         imag_part_func(tanh_imag_part).
1158                         latex_name("\\tanh"));
1159
1160 //////////
1161 // inverse hyperbolic sine (trigonometric function)
1162 //////////
1163
1164 static ex asinh_evalf(const ex & x)
1165 {
1166         if (is_exactly_a<numeric>(x))
1167                 return asinh(ex_to<numeric>(x));
1168
1169         return asinh(x).hold();
1170 }
1171
1172 static ex asinh_eval(const ex & x)
1173 {
1174         if (x.info(info_flags::numeric)) {
1175
1176                 // asinh(0) -> 0
1177                 if (x.is_zero())
1178                         return _ex0;
1179
1180                 // asinh(float) -> float
1181                 if (!x.info(info_flags::crational))
1182                         return asinh(ex_to<numeric>(x));
1183
1184                 // asinh() is odd
1185                 if (x.info(info_flags::negative))
1186                         return -asinh(-x);
1187         }
1188
1189         return asinh(x).hold();
1190 }
1191
1192 static ex asinh_deriv(const ex & x, unsigned deriv_param)
1193 {
1194         GINAC_ASSERT(deriv_param==0);
1195
1196         // d/dx asinh(x) -> 1/sqrt(1+x^2)
1197         return power(_ex1+power(x,_ex2),_ex_1_2);
1198 }
1199
1200 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
1201                          evalf_func(asinh_evalf).
1202                          derivative_func(asinh_deriv));
1203
1204 //////////
1205 // inverse hyperbolic cosine (trigonometric function)
1206 //////////
1207
1208 static ex acosh_evalf(const ex & x)
1209 {
1210         if (is_exactly_a<numeric>(x))
1211                 return acosh(ex_to<numeric>(x));
1212
1213         return acosh(x).hold();
1214 }
1215
1216 static ex acosh_eval(const ex & x)
1217 {
1218         if (x.info(info_flags::numeric)) {
1219
1220                 // acosh(0) -> Pi*I/2
1221                 if (x.is_zero())
1222                         return Pi*I*numeric(1,2);
1223
1224                 // acosh(1) -> 0
1225                 if (x.is_equal(_ex1))
1226                         return _ex0;
1227
1228                 // acosh(-1) -> Pi*I
1229                 if (x.is_equal(_ex_1))
1230                         return Pi*I;
1231
1232                 // acosh(float) -> float
1233                 if (!x.info(info_flags::crational))
1234                         return acosh(ex_to<numeric>(x));
1235
1236                 // acosh(-x) -> Pi*I-acosh(x)
1237                 if (x.info(info_flags::negative))
1238                         return Pi*I-acosh(-x);
1239         }
1240
1241         return acosh(x).hold();
1242 }
1243
1244 static ex acosh_deriv(const ex & x, unsigned deriv_param)
1245 {
1246         GINAC_ASSERT(deriv_param==0);
1247
1248         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
1249         return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
1250 }
1251
1252 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
1253                          evalf_func(acosh_evalf).
1254                          derivative_func(acosh_deriv));
1255
1256 //////////
1257 // inverse hyperbolic tangent (trigonometric function)
1258 //////////
1259
1260 static ex atanh_evalf(const ex & x)
1261 {
1262         if (is_exactly_a<numeric>(x))
1263                 return atanh(ex_to<numeric>(x));
1264
1265         return atanh(x).hold();
1266 }
1267
1268 static ex atanh_eval(const ex & x)
1269 {
1270         if (x.info(info_flags::numeric)) {
1271
1272                 // atanh(0) -> 0
1273                 if (x.is_zero())
1274                         return _ex0;
1275
1276                 // atanh({+|-}1) -> throw
1277                 if (x.is_equal(_ex1) || x.is_equal(_ex_1))
1278                         throw (pole_error("atanh_eval(): logarithmic pole",0));
1279
1280                 // atanh(float) -> float
1281                 if (!x.info(info_flags::crational))
1282                         return atanh(ex_to<numeric>(x));
1283
1284                 // atanh() is odd
1285                 if (x.info(info_flags::negative))
1286                         return -atanh(-x);
1287         }
1288
1289         return atanh(x).hold();
1290 }
1291
1292 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1293 {
1294         GINAC_ASSERT(deriv_param==0);
1295
1296         // d/dx atanh(x) -> 1/(1-x^2)
1297         return power(_ex1-power(x,_ex2),_ex_1);
1298 }
1299
1300 static ex atanh_series(const ex &arg,
1301                        const relational &rel,
1302                        int order,
1303                        unsigned options)
1304 {
1305         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1306         // method:
1307         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1308         // There are two branch cuts, one runnig from 1 up the real axis and one
1309         // one running from -1 down the real axis.  The points 1 and -1 are poles
1310         // On the branch cuts and the poles series expand
1311         //     (log(1+x)-log(1-x))/2
1312         // instead.
1313         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
1314         if (!(arg_pt).info(info_flags::real))
1315                 throw do_taylor();     // Im(x) != 0
1316         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1317                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1318         // care for the poles, using the defining formula for atanh()...
1319         if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1320                 return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1321         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1322         if (!(options & series_options::suppress_branchcut)) {
1323                 // method:
1324                 // This is the branch cut: assemble the primitive series manually and
1325                 // then add the corresponding complex step function.
1326                 const symbol &s = ex_to<symbol>(rel.lhs());
1327                 const ex &point = rel.rhs();
1328                 const symbol foo;
1329                 const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1330                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1331                 if (arg_pt<_ex0)
1332                         Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1333                 else
1334                         Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1335                 epvector seq;
1336                 seq.push_back(expair(Order0correction, _ex0));
1337                 seq.push_back(expair(Order(_ex1), order));
1338                 return series(replarg - pseries(rel, seq), rel, order);
1339         }
1340         throw do_taylor();
1341 }
1342
1343 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1344                          evalf_func(atanh_evalf).
1345                          derivative_func(atanh_deriv).
1346                          series_func(atanh_series));
1347
1348
1349 } // namespace GiNaC