ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <vector>
  24 #include <stdexcept>
  25
  26 #include "inifcns.h"
  27 #include "ex.h"
  28 #include "constant.h"
  29 #include "lst.h"
  30 #include "matrix.h"
  31 #include "mul.h"
  32 #include "power.h"
  33 #include "operators.h"
  34 #include "relational.h"
  35 #include "pseries.h"
  36 #include "symbol.h"
  37 #include "symmetry.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 //////////
  43 // complex conjugate
  44 //////////
  45
  46 static ex conjugate_evalf(const ex & arg)
  47 {
  48         if (is_exactly_a<numeric>(arg)) {
  49                 return ex_to<numeric>(arg).conjugate();
  50         }
  51         return conjugate(arg).hold();
  52 }
  53
  54 static ex conjugate_eval(const ex & arg)
  55 {
  56         return arg.conjugate();
  57 }
  58
  59 static void conjugate_print_latex(const ex & arg, const print_context & c)
  60 {
  61         c.s << "\\bar{"; arg.print(c); c.s << "}";
  62 }
  63
  64 static ex conjugate_conjugate(const ex & arg)
  65 {
  66         return arg;
  67 }
  68
  69 REGISTER_FUNCTION(conjugate, eval_func(conjugate_eval).
  70                        evalf_func(conjugate_evalf).
  71                        print_func<print_latex>(conjugate_print_latex).
  72                        conjugate_func(conjugate_conjugate));
  73
  74 //////////
  75 // absolute value
  76 //////////
  77
  78 static ex abs_evalf(const ex & arg)
  79 {
  80         if (is_exactly_a<numeric>(arg))
  81                 return abs(ex_to<numeric>(arg));
  82
  83         return abs(arg).hold();
  84 }
  85
  86 static ex abs_eval(const ex & arg)
  87 {
  88         if (is_exactly_a<numeric>(arg))
  89                 return abs(ex_to<numeric>(arg));
  90         else
  91                 return abs(arg).hold();
  92 }
  93
  94 static void abs_print_latex(const ex & arg, const print_context & c)
  95 {
  96         c.s << "{|"; arg.print(c); c.s << "|}";
  97 }
  98
  99 static void abs_print_csrc_float(const ex & arg, const print_context & c)
 100 {
 101         c.s << "fabs("; arg.print(c); c.s << ")";
 102 }
 103
 104 static ex abs_conjugate(const ex & arg)
 105 {
 106         return abs(arg);
 107 }
 108
 109 REGISTER_FUNCTION(abs, eval_func(abs_eval).
 110                        evalf_func(abs_evalf).
 111                        print_func<print_latex>(abs_print_latex).
 112                        print_func<print_csrc_float>(abs_print_csrc_float).
 113                        print_func<print_csrc_double>(abs_print_csrc_float).
 114                        conjugate_func(abs_conjugate));
 115
 116
 117 //////////
 118 // Complex sign
 119 //////////
 120
 121 static ex csgn_evalf(const ex & arg)
 122 {
 123         if (is_exactly_a<numeric>(arg))
 124                 return csgn(ex_to<numeric>(arg));
 125
 126         return csgn(arg).hold();
 127 }
 128
 129 static ex csgn_eval(const ex & arg)
 130 {
 131         if (is_exactly_a<numeric>(arg))
 132                 return csgn(ex_to<numeric>(arg));
 133
 134         else if (is_exactly_a<mul>(arg) &&
 135                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 136                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 137                 if (oc.is_real()) {
 138                         if (oc > 0)
 139                                 // csgn(42*x) -> csgn(x)
 140                                 return csgn(arg/oc).hold();
 141                         else
 142                                 // csgn(-42*x) -> -csgn(x)
 143                                 return -csgn(arg/oc).hold();
 144                 }
 145                 if (oc.real().is_zero()) {
 146                         if (oc.imag() > 0)
 147                                 // csgn(42*I*x) -> csgn(I*x)
 148                                 return csgn(I*arg/oc).hold();
 149                         else
 150                                 // csgn(-42*I*x) -> -csgn(I*x)
 151                                 return -csgn(I*arg/oc).hold();
 152                 }
 153         }
 154
 155         return csgn(arg).hold();
 156 }
 157
 158 static ex csgn_series(const ex & arg,
 159                       const relational & rel,
 160                       int order,
 161                       unsigned options)
 162 {
 163         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 164         if (arg_pt.info(info_flags::numeric)
 165             && ex_to<numeric>(arg_pt).real().is_zero()
 166             && !(options & series_options::suppress_branchcut))
 167                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 168
 169         epvector seq;
 170         seq.push_back(expair(csgn(arg_pt), _ex0));
 171         return pseries(rel,seq);
 172 }
 173
 174 static ex csgn_conjugate(const ex& arg)
 175 {
 176         return csgn(arg);
 177 }
 178
 179 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 180                         evalf_func(csgn_evalf).
 181                         series_func(csgn_series).
 182                         conjugate_func(csgn_conjugate));
 183
 184
 185 //////////
 186 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
 187 // This function is closely related to the unwinding number K, sometimes found
 188 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 189 //////////
 190
 191 static ex eta_evalf(const ex &x, const ex &y)
 192 {
 193         // It seems like we basically have to replicate the eval function here,
 194         // since the expression might not be fully evaluated yet.
 195         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 196                 return _ex0;
 197
 198         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 199                 const numeric nx = ex_to<numeric>(x);
 200                 const numeric ny = ex_to<numeric>(y);
 201                 const numeric nxy = ex_to<numeric>(x*y);
 202                 int cut = 0;
 203                 if (nx.is_real() && nx.is_negative())
 204                         cut -= 4;
 205                 if (ny.is_real() && ny.is_negative())
 206                         cut -= 4;
 207                 if (nxy.is_real() && nxy.is_negative())
 208                         cut += 4;
 209                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 210                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 211         }
 212
 213         return eta(x,y).hold();
 214 }
 215
 216 static ex eta_eval(const ex &x, const ex &y)
 217 {
 218         // trivial:  eta(x,c) -> 0  if c is real and positive
 219         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 220                 return _ex0;
 221
 222         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 223                 // don't call eta_evalf here because it would call Pi.evalf()!
 224                 const numeric nx = ex_to<numeric>(x);
 225                 const numeric ny = ex_to<numeric>(y);
 226                 const numeric nxy = ex_to<numeric>(x*y);
 227                 int cut = 0;
 228                 if (nx.is_real() && nx.is_negative())
 229                         cut -= 4;
 230                 if (ny.is_real() && ny.is_negative())
 231                         cut -= 4;
 232                 if (nxy.is_real() && nxy.is_negative())
 233                         cut += 4;
 234                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 235                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 236         }
 237
 238         return eta(x,y).hold();
 239 }
 240
 241 static ex eta_series(const ex & x, const ex & y,
 242                      const relational & rel,
 243                      int order,
 244                      unsigned options)
 245 {
 246         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 247         const ex y_pt = y.subs(rel, subs_options::no_pattern);
 248         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
 249             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
 250             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
 251                         throw (std::domain_error("eta_series(): on discontinuity"));
 252         epvector seq;
 253         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
 254         return pseries(rel,seq);
 255 }
 256
 257 static ex eta_conjugate(const ex & x, const ex & y)
 258 {
 259         return -eta(x,y);
 260 }
 261
 262 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 263                        evalf_func(eta_evalf).
 264                        series_func(eta_series).
 265                        latex_name("\\eta").
 266                        set_symmetry(sy_symm(0, 1)).
 267                        conjugate_func(eta_conjugate));
 268
 269
 270 //////////
 271 // dilogarithm
 272 //////////
 273
 274 static ex Li2_evalf(const ex & x)
 275 {
 276         if (is_exactly_a<numeric>(x))
 277                 return Li2(ex_to<numeric>(x));
 278
 279         return Li2(x).hold();
 280 }
 281
 282 static ex Li2_eval(const ex & x)
 283 {
 284         if (x.info(info_flags::numeric)) {
 285                 // Li2(0) -> 0
 286                 if (x.is_zero())
 287                         return _ex0;
 288                 // Li2(1) -> Pi^2/6
 289                 if (x.is_equal(_ex1))
 290                         return power(Pi,_ex2)/_ex6;
 291                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 292                 if (x.is_equal(_ex1_2))
 293                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
 294                 // Li2(-1) -> -Pi^2/12
 295                 if (x.is_equal(_ex_1))
 296                         return -power(Pi,_ex2)/_ex12;
 297                 // Li2(I) -> -Pi^2/48+Catalan*I
 298                 if (x.is_equal(I))
 299                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
 300                 // Li2(-I) -> -Pi^2/48-Catalan*I
 301                 if (x.is_equal(-I))
 302                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
 303                 // Li2(float)
 304                 if (!x.info(info_flags::crational))
 305                         return Li2(ex_to<numeric>(x));
 306         }
 307
 308         return Li2(x).hold();
 309 }
 310
 311 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 312 {
 313         GINAC_ASSERT(deriv_param==0);
 314
 315         // d/dx Li2(x) -> -log(1-x)/x
 316         return -log(_ex1-x)/x;
 317 }
 318
 319 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 320 {
 321         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 322         if (x_pt.info(info_flags::numeric)) {
 323                 // First special case: x==0 (derivatives have poles)
 324                 if (x_pt.is_zero()) {
 325                         // method:
 326                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 327                         // simply substitute x==0.  The limit, however, exists: it is 1.
 328                         // We also know all higher derivatives' limits:
 329                         // (d/dx)^n Li2(x) == n!/n^2.
 330                         // So the primitive series expansion is
 331                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 332                         // and so on.
 333                         // We first construct such a primitive series expansion manually in
 334                         // a dummy symbol s and then insert the argument's series expansion
 335                         // for s.  Reexpanding the resulting series returns the desired
 336                         // result.
 337                         const symbol s;
 338                         ex ser;
 339                         // manually construct the primitive expansion
 340                         for (int i=1; i<order; ++i)
 341                                 ser += pow(s,i) / pow(numeric(i), _num2);
 342                         // substitute the argument's series expansion
 343                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 344                         // maybe that was terminating, so add a proper order term
 345                         epvector nseq;
 346                         nseq.push_back(expair(Order(_ex1), order));
 347                         ser += pseries(rel, nseq);
 348                         // reexpanding it will collapse the series again
 349                         return ser.series(rel, order);
 350                         // NB: Of course, this still does not allow us to compute anything
 351                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 352                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 353                         // substitution x==0.  Probably limits *are* needed for the general
 354                         // cases.  In case L'Hospital's rule is implemented for limits and
 355                         // basic::series() takes care of this, this whole block is probably
 356                         // obsolete!
 357                 }
 358                 // second special case: x==1 (branch point)
 359                 if (x_pt.is_equal(_ex1)) {
 360                         // method:
 361                         // construct series manually in a dummy symbol s
 362                         const symbol s;
 363                         ex ser = zeta(_ex2);
 364                         // manually construct the primitive expansion
 365                         for (int i=1; i<order; ++i)
 366                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 367                         // substitute the argument's series expansion
 368                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 369                         // maybe that was terminating, so add a proper order term
 370                         epvector nseq;
 371                         nseq.push_back(expair(Order(_ex1), order));
 372                         ser += pseries(rel, nseq);
 373                         // reexpanding it will collapse the series again
 374                         return ser.series(rel, order);
 375                 }
 376                 // third special case: x real, >=1 (branch cut)
 377                 if (!(options & series_options::suppress_branchcut) &&
 378                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 379                         // method:
 380                         // This is the branch cut: assemble the primitive series manually
 381                         // and then add the corresponding complex step function.
 382                         const symbol &s = ex_to<symbol>(rel.lhs());
 383                         const ex point = rel.rhs();
 384                         const symbol foo;
 385                         epvector seq;
 386                         // zeroth order term:
 387                         seq.push_back(expair(Li2(x_pt), _ex0));
 388                         // compute the intermediate terms:
 389                         ex replarg = series(Li2(x), s==foo, order);
 390                         for (size_t i=1; i<replarg.nops()-1; ++i)
 391                                 seq.push_back(expair((replarg.op(i)/power(s-foo,i)).series(foo==point,1,options).op(0).subs(foo==s, subs_options::no_pattern),i));
 392                         // append an order term:
 393                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
 394                         return pseries(rel, seq);
 395                 }
 396         }
 397         // all other cases should be safe, by now:
 398         throw do_taylor();  // caught by function::series()
 399 }
 400
 401 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 402                        evalf_func(Li2_evalf).
 403                        derivative_func(Li2_deriv).
 404                        series_func(Li2_series).
 405                        latex_name("\\mbox{Li}_2"));
 406
 407 //////////
 408 // trilogarithm
 409 //////////
 410
 411 static ex Li3_eval(const ex & x)
 412 {
 413         if (x.is_zero())
 414                 return x;
 415         return Li3(x).hold();
 416 }
 417
 418 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 419                        latex_name("\\mbox{Li}_3"));
 420
 421 //////////
 422 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
 423 //////////
 424
 425 static ex zetaderiv_eval(const ex & n, const ex & x)
 426 {
 427         if (n.info(info_flags::numeric)) {
 428                 // zetaderiv(0,x) -> zeta(x)
 429                 if (n.is_zero())
 430                         return zeta(x);
 431         }
 432
 433         return zetaderiv(n, x).hold();
 434 }
 435
 436 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
 437 {
 438         GINAC_ASSERT(deriv_param<2);
 439
 440         if (deriv_param==0) {
 441                 // d/dn zeta(n,x)
 442                 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
 443         }
 444         // d/dx psi(n,x)
 445         return zetaderiv(n+1,x);
 446 }
 447
 448 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
 449                                  derivative_func(zetaderiv_deriv).
 450                                  latex_name("\\zeta^\\prime"));
 451
 452 //////////
 453 // factorial
 454 //////////
 455
 456 static ex factorial_evalf(const ex & x)
 457 {
 458         return factorial(x).hold();
 459 }
 460
 461 static ex factorial_eval(const ex & x)
 462 {
 463         if (is_exactly_a<numeric>(x))
 464                 return factorial(ex_to<numeric>(x));
 465         else
 466                 return factorial(x).hold();
 467 }
 468
 469 static ex factorial_conjugate(const ex & x)
 470 {
 471         return factorial(x);
 472 }
 473
 474 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 475                              evalf_func(factorial_evalf).
 476                              conjugate_func(factorial_conjugate));
 477
 478 //////////
 479 // binomial
 480 //////////
 481
 482 static ex binomial_evalf(const ex & x, const ex & y)
 483 {
 484         return binomial(x, y).hold();
 485 }
 486
 487 static ex binomial_eval(const ex & x, const ex &y)
 488 {
 489         if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y))
 490                 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 491         else
 492                 return binomial(x, y).hold();
 493 }
 494
 495 // At the moment the numeric evaluation of a binomail function always
 496 // gives a real number, but if this would be implemented using the gamma
 497 // function, also complex conjugation should be changed (or rather, deleted).
 498 static ex binomial_conjugate(const ex & x, const ex & y)
 499 {
 500         return binomial(x,y);
 501 }
 502
 503 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 504                             evalf_func(binomial_evalf).
 505                             conjugate_func(binomial_conjugate));
 506
 507 //////////
 508 // Order term function (for truncated power series)
 509 //////////
 510
 511 static ex Order_eval(const ex & x)
 512 {
 513         if (is_exactly_a<numeric>(x)) {
 514                 // O(c) -> O(1) or 0
 515                 if (!x.is_zero())
 516                         return Order(_ex1).hold();
 517                 else
 518                         return _ex0;
 519         } else if (is_exactly_a<mul>(x)) {
 520                 const mul &m = ex_to<mul>(x);
 521                 // O(c*expr) -> O(expr)
 522                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
 523                         return Order(x / m.op(m.nops() - 1)).hold();
 524         }
 525         return Order(x).hold();
 526 }
 527
 528 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 529 {
 530         // Just wrap the function into a pseries object
 531         epvector new_seq;
 532         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 533         const symbol &s = ex_to<symbol>(r.lhs());
 534         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
 535         return pseries(r, new_seq);
 536 }
 537
 538 static ex Order_conjugate(const ex & x)
 539 {
 540         return Order(x);
 541 }
 542
 543 // Differentiation is handled in function::derivative because of its special requirements
 544
 545 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 546                          series_func(Order_series).
 547                          latex_name("\\mathcal{O}").
 548                          conjugate_func(Order_conjugate));
 549
 550 //////////
 551 // Solve linear system
 552 //////////
 553
 554 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 555 {
 556         // solve a system of linear equations
 557         if (eqns.info(info_flags::relation_equal)) {
 558                 if (!symbols.info(info_flags::symbol))
 559                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 560                 const ex sol = lsolve(lst(eqns),lst(symbols));
 561
 562                 GINAC_ASSERT(sol.nops()==1);
 563                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
 564
 565                 return sol.op(0).op(1); // return rhs of first solution
 566         }
 567
 568         // syntax checks
 569         if (!eqns.info(info_flags::list)) {
 570                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
 571         }
 572         for (size_t i=0; i<eqns.nops(); i++) {
 573                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 574                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 575                 }
 576         }
 577         if (!symbols.info(info_flags::list)) {
 578                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
 579         }
 580         for (size_t i=0; i<symbols.nops(); i++) {
 581                 if (!symbols.op(i).info(info_flags::symbol)) {
 582                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 583                 }
 584         }
 585
 586         // build matrix from equation system
 587         matrix sys(eqns.nops(),symbols.nops());
 588         matrix rhs(eqns.nops(),1);
 589         matrix vars(symbols.nops(),1);
 590
 591         for (size_t r=0; r<eqns.nops(); r++) {
 592                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
 593                 ex linpart = eq;
 594                 for (size_t c=0; c<symbols.nops(); c++) {
 595                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
 596                         linpart -= co*symbols.op(c);
 597                         sys(r,c) = co;
 598                 }
 599                 linpart = linpart.expand();
 600                 rhs(r,0) = -linpart;
 601         }
 602
 603         // test if system is linear and fill vars matrix
 604         for (size_t i=0; i<symbols.nops(); i++) {
 605                 vars(i,0) = symbols.op(i);
 606                 if (sys.has(symbols.op(i)))
 607                         throw(std::logic_error("lsolve: system is not linear"));
 608                 if (rhs.has(symbols.op(i)))
 609                         throw(std::logic_error("lsolve: system is not linear"));
 610         }
 611
 612         matrix solution;
 613         try {
 614                 solution = sys.solve(vars,rhs,options);
 615         } catch (const std::runtime_error & e) {
 616                 // Probably singular matrix or otherwise overdetermined system:
 617                 // It is consistent to return an empty list
 618                 return lst();
 619         }
 620         GINAC_ASSERT(solution.cols()==1);
 621         GINAC_ASSERT(solution.rows()==symbols.nops());
 622
 623         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
 624         lst sollist;
 625         for (size_t i=0; i<symbols.nops(); i++)
 626                 sollist.append(symbols.op(i)==solution(i,0));
 627
 628         return sollist;
 629 }
 630
 631 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
 632  * for static lib (so ginsh will see them). */
 633 unsigned force_include_tgamma = tgamma_SERIAL::serial;
 634 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
 635
 636 } // namespace GiNaC