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[ginac.git] / ginac / inifcns.cpp
1 /** @file inifcns.cpp
2  *
3  *  Implementation of GiNaC's initially known functions. */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2016 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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
21  */
22
23 #include "inifcns.h"
24 #include "ex.h"
25 #include "constant.h"
26 #include "lst.h"
27 #include "fderivative.h"
28 #include "matrix.h"
29 #include "mul.h"
30 #include "power.h"
31 #include "operators.h"
32 #include "relational.h"
33 #include "pseries.h"
34 #include "symbol.h"
35 #include "symmetry.h"
36 #include "utils.h"
37
38 #include <stdexcept>
39 #include <vector>
40
41 namespace GiNaC {
42
43 //////////
44 // complex conjugate
45 //////////
46
47 static ex conjugate_evalf(const ex & arg)
48 {
49         if (is_exactly_a<numeric>(arg)) {
50                 return ex_to<numeric>(arg).conjugate();
51         }
52         return conjugate_function(arg).hold();
53 }
54
55 static ex conjugate_eval(const ex & arg)
56 {
57         return arg.conjugate();
58 }
59
60 static void conjugate_print_latex(const ex & arg, const print_context & c)
61 {
62         c.s << "\\bar{"; arg.print(c); c.s << "}";
63 }
64
65 static ex conjugate_conjugate(const ex & arg)
66 {
67         return arg;
68 }
69
70 // If x is real then U.diff(x)-I*V.diff(x) represents both conjugate(U+I*V).diff(x) 
71 // and conjugate((U+I*V).diff(x))
72 static ex conjugate_expl_derivative(const ex & arg, const symbol & s)
73 {
74         if (s.info(info_flags::real))
75                 return conjugate(arg.diff(s));
76         else {
77                 exvector vec_arg;
78                 vec_arg.push_back(arg);
79                 return fderivative(ex_to<function>(conjugate(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
80         }
81 }
82
83 static ex conjugate_real_part(const ex & arg)
84 {
85         return arg.real_part();
86 }
87
88 static ex conjugate_imag_part(const ex & arg)
89 {
90         return -arg.imag_part();
91 }
92
93 static bool func_arg_info(const ex & arg, unsigned inf)
94 {
95         // for some functions we can return the info() of its argument
96         // (think of conjugate())
97         switch (inf) {
98                 case info_flags::polynomial:
99                 case info_flags::integer_polynomial:
100                 case info_flags::cinteger_polynomial:
101                 case info_flags::rational_polynomial:
102                 case info_flags::real:
103                 case info_flags::rational:
104                 case info_flags::integer:
105                 case info_flags::crational:
106                 case info_flags::cinteger:
107                 case info_flags::even:
108                 case info_flags::odd:
109                 case info_flags::prime:
110                 case info_flags::crational_polynomial:
111                 case info_flags::rational_function:
112                 case info_flags::algebraic:
113                 case info_flags::positive:
114                 case info_flags::negative:
115                 case info_flags::nonnegative:
116                 case info_flags::posint:
117                 case info_flags::negint:
118                 case info_flags::nonnegint:
119                 case info_flags::has_indices:
120                         return arg.info(inf);
121         }
122         return false;
123 }
124
125 static bool conjugate_info(const ex & arg, unsigned inf)
126 {
127         return func_arg_info(arg, inf);
128 }
129
130 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
131                                       evalf_func(conjugate_evalf).
132                                       expl_derivative_func(conjugate_expl_derivative).
133                                       info_func(conjugate_info).
134                                       print_func<print_latex>(conjugate_print_latex).
135                                       conjugate_func(conjugate_conjugate).
136                                       real_part_func(conjugate_real_part).
137                                       imag_part_func(conjugate_imag_part).
138                                       set_name("conjugate","conjugate"));
139
140 //////////
141 // real part
142 //////////
143
144 static ex real_part_evalf(const ex & arg)
145 {
146         if (is_exactly_a<numeric>(arg)) {
147                 return ex_to<numeric>(arg).real();
148         }
149         return real_part_function(arg).hold();
150 }
151
152 static ex real_part_eval(const ex & arg)
153 {
154         return arg.real_part();
155 }
156
157 static void real_part_print_latex(const ex & arg, const print_context & c)
158 {
159         c.s << "\\Re"; arg.print(c); c.s << "";
160 }
161
162 static ex real_part_conjugate(const ex & arg)
163 {
164         return real_part_function(arg).hold();
165 }
166
167 static ex real_part_real_part(const ex & arg)
168 {
169         return real_part_function(arg).hold();
170 }
171
172 static ex real_part_imag_part(const ex & arg)
173 {
174         return 0;
175 }
176
177 // If x is real then Re(e).diff(x) is equal to Re(e.diff(x)) 
178 static ex real_part_expl_derivative(const ex & arg, const symbol & s)
179 {
180         if (s.info(info_flags::real))
181                 return real_part_function(arg.diff(s));
182         else {
183                 exvector vec_arg;
184                 vec_arg.push_back(arg);
185                 return fderivative(ex_to<function>(real_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
186         }
187 }
188
189 REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
190                                       evalf_func(real_part_evalf).
191                                       expl_derivative_func(real_part_expl_derivative).
192                                       print_func<print_latex>(real_part_print_latex).
193                                       conjugate_func(real_part_conjugate).
194                                       real_part_func(real_part_real_part).
195                                       imag_part_func(real_part_imag_part).
196                                       set_name("real_part","real_part"));
197
198 //////////
199 // imag part
200 //////////
201
202 static ex imag_part_evalf(const ex & arg)
203 {
204         if (is_exactly_a<numeric>(arg)) {
205                 return ex_to<numeric>(arg).imag();
206         }
207         return imag_part_function(arg).hold();
208 }
209
210 static ex imag_part_eval(const ex & arg)
211 {
212         return arg.imag_part();
213 }
214
215 static void imag_part_print_latex(const ex & arg, const print_context & c)
216 {
217         c.s << "\\Im"; arg.print(c); c.s << "";
218 }
219
220 static ex imag_part_conjugate(const ex & arg)
221 {
222         return imag_part_function(arg).hold();
223 }
224
225 static ex imag_part_real_part(const ex & arg)
226 {
227         return imag_part_function(arg).hold();
228 }
229
230 static ex imag_part_imag_part(const ex & arg)
231 {
232         return 0;
233 }
234
235 // If x is real then Im(e).diff(x) is equal to Im(e.diff(x)) 
236 static ex imag_part_expl_derivative(const ex & arg, const symbol & s)
237 {
238         if (s.info(info_flags::real))
239                 return imag_part_function(arg.diff(s));
240         else {
241                 exvector vec_arg;
242                 vec_arg.push_back(arg);
243                 return fderivative(ex_to<function>(imag_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
244         }
245 }
246
247 REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
248                                       evalf_func(imag_part_evalf).
249                                       expl_derivative_func(imag_part_expl_derivative).
250                                       print_func<print_latex>(imag_part_print_latex).
251                                       conjugate_func(imag_part_conjugate).
252                                       real_part_func(imag_part_real_part).
253                                       imag_part_func(imag_part_imag_part).
254                                       set_name("imag_part","imag_part"));
255
256 //////////
257 // absolute value
258 //////////
259
260 static ex abs_evalf(const ex & arg)
261 {
262         if (is_exactly_a<numeric>(arg))
263                 return abs(ex_to<numeric>(arg));
264         
265         return abs(arg).hold();
266 }
267
268 static ex abs_eval(const ex & arg)
269 {
270         if (is_exactly_a<numeric>(arg))
271                 return abs(ex_to<numeric>(arg));
272
273         if (arg.info(info_flags::nonnegative))
274                 return arg;
275
276         if (arg.info(info_flags::negative) || (-arg).info(info_flags::nonnegative))
277                 return -arg;
278
279         if (is_ex_the_function(arg, abs))
280                 return arg;
281
282         if (is_ex_the_function(arg, exp))
283                 return exp(arg.op(0).real_part());
284
285         if (is_exactly_a<power>(arg)) {
286                 const ex& base = arg.op(0);
287                 const ex& exponent = arg.op(1);
288                 if (base.info(info_flags::positive) || exponent.info(info_flags::real))
289                         return pow(abs(base), exponent.real_part());
290         }
291
292         if (is_ex_the_function(arg, conjugate_function))
293                 return abs(arg.op(0));
294
295         if (is_ex_the_function(arg, step))
296                 return arg;
297
298         return abs(arg).hold();
299 }
300
301 static ex abs_expand(const ex & arg, unsigned options)
302 {
303         if ((options & expand_options::expand_transcendental)
304                 && is_exactly_a<mul>(arg)) {
305                 exvector prodseq;
306                 prodseq.reserve(arg.nops());
307                 for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
308                         if (options & expand_options::expand_function_args)
309                                 prodseq.push_back(abs(i->expand(options)));
310                         else
311                                 prodseq.push_back(abs(*i));
312                 }
313                 return (new mul(prodseq))->setflag(status_flags::dynallocated | status_flags::expanded);
314         }
315
316         if (options & expand_options::expand_function_args)
317                 return abs(arg.expand(options)).hold();
318         else
319                 return abs(arg).hold();
320 }
321
322 static ex abs_expl_derivative(const ex & arg, const symbol & s)
323 {
324         ex diff_arg = arg.diff(s);
325         return (diff_arg*arg.conjugate()+arg*diff_arg.conjugate())/2/abs(arg);
326 }
327
328 static void abs_print_latex(const ex & arg, const print_context & c)
329 {
330         c.s << "{|"; arg.print(c); c.s << "|}";
331 }
332
333 static void abs_print_csrc_float(const ex & arg, const print_context & c)
334 {
335         c.s << "fabs("; arg.print(c); c.s << ")";
336 }
337
338 static ex abs_conjugate(const ex & arg)
339 {
340         return abs(arg).hold();
341 }
342
343 static ex abs_real_part(const ex & arg)
344 {
345         return abs(arg).hold();
346 }
347
348 static ex abs_imag_part(const ex& arg)
349 {
350         return 0;
351 }
352
353 static ex abs_power(const ex & arg, const ex & exp)
354 {
355         if ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even()) || exp.info(info_flags::even)) {
356                 if (arg.info(info_flags::real) || arg.is_equal(arg.conjugate()))
357                         return power(arg, exp);
358                 else
359                         return power(arg, exp/2)*power(arg.conjugate(), exp/2);
360         } else
361                 return power(abs(arg), exp).hold();
362 }
363
364 bool abs_info(const ex & arg, unsigned inf)
365 {
366         switch (inf) {
367                 case info_flags::integer:
368                 case info_flags::even:
369                 case info_flags::odd:
370                 case info_flags::prime:
371                         return arg.info(inf);
372                 case info_flags::nonnegint:
373                         return arg.info(info_flags::integer);
374                 case info_flags::nonnegative:
375                 case info_flags::real:
376                         return true;
377                 case info_flags::negative:
378                         return false;
379                 case info_flags::positive:
380                         return arg.info(info_flags::positive) || arg.info(info_flags::negative);
381                 case info_flags::has_indices: {
382                         if (arg.info(info_flags::has_indices))
383                                 return true;
384                         else
385                                 return false;
386                 }
387         }
388         return false;
389 }
390
391 REGISTER_FUNCTION(abs, eval_func(abs_eval).
392                        evalf_func(abs_evalf).
393                        expand_func(abs_expand).
394                        expl_derivative_func(abs_expl_derivative).
395                        info_func(abs_info).
396                        print_func<print_latex>(abs_print_latex).
397                        print_func<print_csrc_float>(abs_print_csrc_float).
398                        print_func<print_csrc_double>(abs_print_csrc_float).
399                        conjugate_func(abs_conjugate).
400                        real_part_func(abs_real_part).
401                        imag_part_func(abs_imag_part).
402                        power_func(abs_power));
403
404 //////////
405 // Step function
406 //////////
407
408 static ex step_evalf(const ex & arg)
409 {
410         if (is_exactly_a<numeric>(arg))
411                 return step(ex_to<numeric>(arg));
412         
413         return step(arg).hold();
414 }
415
416 static ex step_eval(const ex & arg)
417 {
418         if (is_exactly_a<numeric>(arg))
419                 return step(ex_to<numeric>(arg));
420         
421         else if (is_exactly_a<mul>(arg) &&
422                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
423                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
424                 if (oc.is_real()) {
425                         if (oc > 0)
426                                 // step(42*x) -> step(x)
427                                 return step(arg/oc).hold();
428                         else
429                                 // step(-42*x) -> step(-x)
430                                 return step(-arg/oc).hold();
431                 }
432                 if (oc.real().is_zero()) {
433                         if (oc.imag() > 0)
434                                 // step(42*I*x) -> step(I*x)
435                                 return step(I*arg/oc).hold();
436                         else
437                                 // step(-42*I*x) -> step(-I*x)
438                                 return step(-I*arg/oc).hold();
439                 }
440         }
441         
442         return step(arg).hold();
443 }
444
445 static ex step_series(const ex & arg,
446                       const relational & rel,
447                       int order,
448                       unsigned options)
449 {
450         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
451         if (arg_pt.info(info_flags::numeric)
452             && ex_to<numeric>(arg_pt).real().is_zero()
453             && !(options & series_options::suppress_branchcut))
454                 throw (std::domain_error("step_series(): on imaginary axis"));
455         
456         epvector seq;
457         seq.push_back(expair(step(arg_pt), _ex0));
458         return pseries(rel,seq);
459 }
460
461 static ex step_conjugate(const ex& arg)
462 {
463         return step(arg).hold();
464 }
465
466 static ex step_real_part(const ex& arg)
467 {
468         return step(arg).hold();
469 }
470
471 static ex step_imag_part(const ex& arg)
472 {
473         return 0;
474 }
475
476 REGISTER_FUNCTION(step, eval_func(step_eval).
477                         evalf_func(step_evalf).
478                         series_func(step_series).
479                         conjugate_func(step_conjugate).
480                         real_part_func(step_real_part).
481                         imag_part_func(step_imag_part));
482
483 //////////
484 // Complex sign
485 //////////
486
487 static ex csgn_evalf(const ex & arg)
488 {
489         if (is_exactly_a<numeric>(arg))
490                 return csgn(ex_to<numeric>(arg));
491         
492         return csgn(arg).hold();
493 }
494
495 static ex csgn_eval(const ex & arg)
496 {
497         if (is_exactly_a<numeric>(arg))
498                 return csgn(ex_to<numeric>(arg));
499         
500         else if (is_exactly_a<mul>(arg) &&
501                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
502                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
503                 if (oc.is_real()) {
504                         if (oc > 0)
505                                 // csgn(42*x) -> csgn(x)
506                                 return csgn(arg/oc).hold();
507                         else
508                                 // csgn(-42*x) -> -csgn(x)
509                                 return -csgn(arg/oc).hold();
510                 }
511                 if (oc.real().is_zero()) {
512                         if (oc.imag() > 0)
513                                 // csgn(42*I*x) -> csgn(I*x)
514                                 return csgn(I*arg/oc).hold();
515                         else
516                                 // csgn(-42*I*x) -> -csgn(I*x)
517                                 return -csgn(I*arg/oc).hold();
518                 }
519         }
520         
521         return csgn(arg).hold();
522 }
523
524 static ex csgn_series(const ex & arg,
525                       const relational & rel,
526                       int order,
527                       unsigned options)
528 {
529         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
530         if (arg_pt.info(info_flags::numeric)
531             && ex_to<numeric>(arg_pt).real().is_zero()
532             && !(options & series_options::suppress_branchcut))
533                 throw (std::domain_error("csgn_series(): on imaginary axis"));
534         
535         epvector seq;
536         seq.push_back(expair(csgn(arg_pt), _ex0));
537         return pseries(rel,seq);
538 }
539
540 static ex csgn_conjugate(const ex& arg)
541 {
542         return csgn(arg).hold();
543 }
544
545 static ex csgn_real_part(const ex& arg)
546 {
547         return csgn(arg).hold();
548 }
549
550 static ex csgn_imag_part(const ex& arg)
551 {
552         return 0;
553 }
554
555 static ex csgn_power(const ex & arg, const ex & exp)
556 {
557         if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
558                 if (ex_to<numeric>(exp).is_odd())
559                         return csgn(arg).hold();
560                 else
561                         return power(csgn(arg), _ex2).hold();
562         } else
563                 return power(csgn(arg), exp).hold();
564 }
565
566
567 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
568                         evalf_func(csgn_evalf).
569                         series_func(csgn_series).
570                         conjugate_func(csgn_conjugate).
571                         real_part_func(csgn_real_part).
572                         imag_part_func(csgn_imag_part).
573                         power_func(csgn_power));
574
575
576 //////////
577 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
578 // This function is closely related to the unwinding number K, sometimes found
579 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
580 //////////
581
582 static ex eta_evalf(const ex &x, const ex &y)
583 {
584         // It seems like we basically have to replicate the eval function here,
585         // since the expression might not be fully evaluated yet.
586         if (x.info(info_flags::positive) || y.info(info_flags::positive))
587                 return _ex0;
588
589         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
590                 const numeric nx = ex_to<numeric>(x);
591                 const numeric ny = ex_to<numeric>(y);
592                 const numeric nxy = ex_to<numeric>(x*y);
593                 int cut = 0;
594                 if (nx.is_real() && nx.is_negative())
595                         cut -= 4;
596                 if (ny.is_real() && ny.is_negative())
597                         cut -= 4;
598                 if (nxy.is_real() && nxy.is_negative())
599                         cut += 4;
600                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
601                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
602         }
603
604         return eta(x,y).hold();
605 }
606
607 static ex eta_eval(const ex &x, const ex &y)
608 {
609         // trivial:  eta(x,c) -> 0  if c is real and positive
610         if (x.info(info_flags::positive) || y.info(info_flags::positive))
611                 return _ex0;
612
613         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
614                 // don't call eta_evalf here because it would call Pi.evalf()!
615                 const numeric nx = ex_to<numeric>(x);
616                 const numeric ny = ex_to<numeric>(y);
617                 const numeric nxy = ex_to<numeric>(x*y);
618                 int cut = 0;
619                 if (nx.is_real() && nx.is_negative())
620                         cut -= 4;
621                 if (ny.is_real() && ny.is_negative())
622                         cut -= 4;
623                 if (nxy.is_real() && nxy.is_negative())
624                         cut += 4;
625                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
626                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
627         }
628         
629         return eta(x,y).hold();
630 }
631
632 static ex eta_series(const ex & x, const ex & y,
633                      const relational & rel,
634                      int order,
635                      unsigned options)
636 {
637         const ex x_pt = x.subs(rel, subs_options::no_pattern);
638         const ex y_pt = y.subs(rel, subs_options::no_pattern);
639         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
640             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
641             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
642                         throw (std::domain_error("eta_series(): on discontinuity"));
643         epvector seq;
644         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
645         return pseries(rel,seq);
646 }
647
648 static ex eta_conjugate(const ex & x, const ex & y)
649 {
650         return -eta(x, y).hold();
651 }
652
653 static ex eta_real_part(const ex & x, const ex & y)
654 {
655         return 0;
656 }
657
658 static ex eta_imag_part(const ex & x, const ex & y)
659 {
660         return -I*eta(x, y).hold();
661 }
662
663 REGISTER_FUNCTION(eta, eval_func(eta_eval).
664                        evalf_func(eta_evalf).
665                        series_func(eta_series).
666                        latex_name("\\eta").
667                        set_symmetry(sy_symm(0, 1)).
668                        conjugate_func(eta_conjugate).
669                        real_part_func(eta_real_part).
670                        imag_part_func(eta_imag_part));
671
672
673 //////////
674 // dilogarithm
675 //////////
676
677 static ex Li2_evalf(const ex & x)
678 {
679         if (is_exactly_a<numeric>(x))
680                 return Li2(ex_to<numeric>(x));
681         
682         return Li2(x).hold();
683 }
684
685 static ex Li2_eval(const ex & x)
686 {
687         if (x.info(info_flags::numeric)) {
688                 // Li2(0) -> 0
689                 if (x.is_zero())
690                         return _ex0;
691                 // Li2(1) -> Pi^2/6
692                 if (x.is_equal(_ex1))
693                         return power(Pi,_ex2)/_ex6;
694                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
695                 if (x.is_equal(_ex1_2))
696                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
697                 // Li2(-1) -> -Pi^2/12
698                 if (x.is_equal(_ex_1))
699                         return -power(Pi,_ex2)/_ex12;
700                 // Li2(I) -> -Pi^2/48+Catalan*I
701                 if (x.is_equal(I))
702                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
703                 // Li2(-I) -> -Pi^2/48-Catalan*I
704                 if (x.is_equal(-I))
705                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
706                 // Li2(float)
707                 if (!x.info(info_flags::crational))
708                         return Li2(ex_to<numeric>(x));
709         }
710         
711         return Li2(x).hold();
712 }
713
714 static ex Li2_deriv(const ex & x, unsigned deriv_param)
715 {
716         GINAC_ASSERT(deriv_param==0);
717         
718         // d/dx Li2(x) -> -log(1-x)/x
719         return -log(_ex1-x)/x;
720 }
721
722 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
723 {
724         const ex x_pt = x.subs(rel, subs_options::no_pattern);
725         if (x_pt.info(info_flags::numeric)) {
726                 // First special case: x==0 (derivatives have poles)
727                 if (x_pt.is_zero()) {
728                         // method:
729                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot 
730                         // simply substitute x==0.  The limit, however, exists: it is 1.
731                         // We also know all higher derivatives' limits:
732                         // (d/dx)^n Li2(x) == n!/n^2.
733                         // So the primitive series expansion is
734                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
735                         // and so on.
736                         // We first construct such a primitive series expansion manually in
737                         // a dummy symbol s and then insert the argument's series expansion
738                         // for s.  Reexpanding the resulting series returns the desired
739                         // result.
740                         const symbol s;
741                         ex ser;
742                         // manually construct the primitive expansion
743                         for (int i=1; i<order; ++i)
744                                 ser += pow(s,i) / pow(numeric(i), *_num2_p);
745                         // substitute the argument's series expansion
746                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
747                         // maybe that was terminating, so add a proper order term
748                         epvector nseq;
749                         nseq.push_back(expair(Order(_ex1), order));
750                         ser += pseries(rel, nseq);
751                         // reexpanding it will collapse the series again
752                         return ser.series(rel, order);
753                         // NB: Of course, this still does not allow us to compute anything
754                         // like sin(Li2(x)).series(x==0,2), since then this code here is
755                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
756                         // substitution x==0.  Probably limits *are* needed for the general
757                         // cases.  In case L'Hospital's rule is implemented for limits and
758                         // basic::series() takes care of this, this whole block is probably
759                         // obsolete!
760                 }
761                 // second special case: x==1 (branch point)
762                 if (x_pt.is_equal(_ex1)) {
763                         // method:
764                         // construct series manually in a dummy symbol s
765                         const symbol s;
766                         ex ser = zeta(_ex2);
767                         // manually construct the primitive expansion
768                         for (int i=1; i<order; ++i)
769                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
770                         // substitute the argument's series expansion
771                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
772                         // maybe that was terminating, so add a proper order term
773                         epvector nseq;
774                         nseq.push_back(expair(Order(_ex1), order));
775                         ser += pseries(rel, nseq);
776                         // reexpanding it will collapse the series again
777                         return ser.series(rel, order);
778                 }
779                 // third special case: x real, >=1 (branch cut)
780                 if (!(options & series_options::suppress_branchcut) &&
781                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
782                         // method:
783                         // This is the branch cut: assemble the primitive series manually
784                         // and then add the corresponding complex step function.
785                         const symbol &s = ex_to<symbol>(rel.lhs());
786                         const ex point = rel.rhs();
787                         const symbol foo;
788                         epvector seq;
789                         // zeroth order term:
790                         seq.push_back(expair(Li2(x_pt), _ex0));
791                         // compute the intermediate terms:
792                         ex replarg = series(Li2(x), s==foo, order);
793                         for (size_t i=1; i<replarg.nops()-1; ++i)
794                                 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));
795                         // append an order term:
796                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
797                         return pseries(rel, seq);
798                 }
799         }
800         // all other cases should be safe, by now:
801         throw do_taylor();  // caught by function::series()
802 }
803
804 static ex Li2_conjugate(const ex & x)
805 {
806         // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
807         // run along the positive real axis beginning at 1.
808         if (x.info(info_flags::negative)) {
809                 return Li2(x).hold();
810         }
811         if (is_exactly_a<numeric>(x) &&
812             (!x.imag_part().is_zero() || x < *_num1_p)) {
813                 return Li2(x.conjugate());
814         }
815         return conjugate_function(Li2(x)).hold();
816 }
817
818 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
819                        evalf_func(Li2_evalf).
820                        derivative_func(Li2_deriv).
821                        series_func(Li2_series).
822                        conjugate_func(Li2_conjugate).
823                        latex_name("\\mathrm{Li}_2"));
824
825 //////////
826 // trilogarithm
827 //////////
828
829 static ex Li3_eval(const ex & x)
830 {
831         if (x.is_zero())
832                 return x;
833         return Li3(x).hold();
834 }
835
836 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
837                        latex_name("\\mathrm{Li}_3"));
838
839 //////////
840 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
841 //////////
842
843 static ex zetaderiv_eval(const ex & n, const ex & x)
844 {
845         if (n.info(info_flags::numeric)) {
846                 // zetaderiv(0,x) -> zeta(x)
847                 if (n.is_zero())
848                         return zeta(x).hold();
849         }
850         
851         return zetaderiv(n, x).hold();
852 }
853
854 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
855 {
856         GINAC_ASSERT(deriv_param<2);
857         
858         if (deriv_param==0) {
859                 // d/dn zeta(n,x)
860                 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
861         }
862         // d/dx psi(n,x)
863         return zetaderiv(n+1,x);
864 }
865
866 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
867                                  derivative_func(zetaderiv_deriv).
868                                  latex_name("\\zeta^\\prime"));
869
870 //////////
871 // factorial
872 //////////
873
874 static ex factorial_evalf(const ex & x)
875 {
876         return factorial(x).hold();
877 }
878
879 static ex factorial_eval(const ex & x)
880 {
881         if (is_exactly_a<numeric>(x))
882                 return factorial(ex_to<numeric>(x));
883         else
884                 return factorial(x).hold();
885 }
886
887 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
888 {
889         if (is_exactly_a<symbol>(x) ||
890             is_exactly_a<constant>(x) ||
891                 is_exactly_a<function>(x)) {
892                 x.print(c); c.s << "!";
893         } else {
894                 c.s << "("; x.print(c); c.s << ")!";
895         }
896 }
897
898 static ex factorial_conjugate(const ex & x)
899 {
900         return factorial(x).hold();
901 }
902
903 static ex factorial_real_part(const ex & x)
904 {
905         return factorial(x).hold();
906 }
907
908 static ex factorial_imag_part(const ex & x)
909 {
910         return 0;
911 }
912
913 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
914                              evalf_func(factorial_evalf).
915                              print_func<print_dflt>(factorial_print_dflt_latex).
916                              print_func<print_latex>(factorial_print_dflt_latex).
917                              conjugate_func(factorial_conjugate).
918                              real_part_func(factorial_real_part).
919                              imag_part_func(factorial_imag_part));
920
921 //////////
922 // binomial
923 //////////
924
925 static ex binomial_evalf(const ex & x, const ex & y)
926 {
927         return binomial(x, y).hold();
928 }
929
930 static ex binomial_sym(const ex & x, const numeric & y)
931 {
932         if (y.is_integer()) {
933                 if (y.is_nonneg_integer()) {
934                         const unsigned N = y.to_int();
935                         if (N == 0) return _ex1;
936                         if (N == 1) return x;
937                         ex t = x.expand();
938                         for (unsigned i = 2; i <= N; ++i)
939                                 t = (t * (x + i - y - 1)).expand() / i;
940                         return t;
941                 } else
942                         return _ex0;
943         }
944
945         return binomial(x, y).hold();
946 }
947
948 static ex binomial_eval(const ex & x, const ex &y)
949 {
950         if (is_exactly_a<numeric>(y)) {
951                 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
952                         return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
953                 else
954                         return binomial_sym(x, ex_to<numeric>(y));
955         } else
956                 return binomial(x, y).hold();
957 }
958
959 // At the moment the numeric evaluation of a binomial function always
960 // gives a real number, but if this would be implemented using the gamma
961 // function, also complex conjugation should be changed (or rather, deleted).
962 static ex binomial_conjugate(const ex & x, const ex & y)
963 {
964         return binomial(x,y).hold();
965 }
966
967 static ex binomial_real_part(const ex & x, const ex & y)
968 {
969         return binomial(x,y).hold();
970 }
971
972 static ex binomial_imag_part(const ex & x, const ex & y)
973 {
974         return 0;
975 }
976
977 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
978                             evalf_func(binomial_evalf).
979                             conjugate_func(binomial_conjugate).
980                             real_part_func(binomial_real_part).
981                             imag_part_func(binomial_imag_part));
982
983 //////////
984 // Order term function (for truncated power series)
985 //////////
986
987 static ex Order_eval(const ex & x)
988 {
989         if (is_exactly_a<numeric>(x)) {
990                 // O(c) -> O(1) or 0
991                 if (!x.is_zero())
992                         return Order(_ex1).hold();
993                 else
994                         return _ex0;
995         } else if (is_exactly_a<mul>(x)) {
996                 const mul &m = ex_to<mul>(x);
997                 // O(c*expr) -> O(expr)
998                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
999                         return Order(x / m.op(m.nops() - 1)).hold();
1000         }
1001         return Order(x).hold();
1002 }
1003
1004 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
1005 {
1006         // Just wrap the function into a pseries object
1007         epvector new_seq;
1008         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1009         const symbol &s = ex_to<symbol>(r.lhs());
1010         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
1011         return pseries(r, new_seq);
1012 }
1013
1014 static ex Order_conjugate(const ex & x)
1015 {
1016         return Order(x).hold();
1017 }
1018
1019 static ex Order_real_part(const ex & x)
1020 {
1021         return Order(x).hold();
1022 }
1023
1024 static ex Order_imag_part(const ex & x)
1025 {
1026         if(x.info(info_flags::real))
1027                 return 0;
1028         return Order(x).hold();
1029 }
1030
1031 static ex Order_expl_derivative(const ex & arg, const symbol & s)
1032 {
1033         return Order(arg.diff(s));
1034 }
1035
1036 REGISTER_FUNCTION(Order, eval_func(Order_eval).
1037                          series_func(Order_series).
1038                          latex_name("\\mathcal{O}").
1039                          expl_derivative_func(Order_expl_derivative).
1040                          conjugate_func(Order_conjugate).
1041                          real_part_func(Order_real_part).
1042                          imag_part_func(Order_imag_part));
1043
1044 //////////
1045 // Solve linear system
1046 //////////
1047
1048 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
1049 {
1050         // solve a system of linear equations
1051         if (eqns.info(info_flags::relation_equal)) {
1052                 if (!symbols.info(info_flags::symbol))
1053                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
1054                 const ex sol = lsolve(lst(eqns),lst(symbols));
1055                 
1056                 GINAC_ASSERT(sol.nops()==1);
1057                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
1058                 
1059                 return sol.op(0).op(1); // return rhs of first solution
1060         }
1061         
1062         // syntax checks
1063         if (!eqns.info(info_flags::list)) {
1064                 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
1065         }
1066         for (size_t i=0; i<eqns.nops(); i++) {
1067                 if (!eqns.op(i).info(info_flags::relation_equal)) {
1068                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
1069                 }
1070         }
1071         if (!symbols.info(info_flags::list)) {
1072                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
1073         }
1074         for (size_t i=0; i<symbols.nops(); i++) {
1075                 if (!symbols.op(i).info(info_flags::symbol)) {
1076                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
1077                 }
1078         }
1079         
1080         // build matrix from equation system
1081         matrix sys(eqns.nops(),symbols.nops());
1082         matrix rhs(eqns.nops(),1);
1083         matrix vars(symbols.nops(),1);
1084         
1085         for (size_t r=0; r<eqns.nops(); r++) {
1086                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
1087                 ex linpart = eq;
1088                 for (size_t c=0; c<symbols.nops(); c++) {
1089                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
1090                         linpart -= co*symbols.op(c);
1091                         sys(r,c) = co;
1092                 }
1093                 linpart = linpart.expand();
1094                 rhs(r,0) = -linpart;
1095         }
1096         
1097         // test if system is linear and fill vars matrix
1098         for (size_t i=0; i<symbols.nops(); i++) {
1099                 vars(i,0) = symbols.op(i);
1100                 if (sys.has(symbols.op(i)))
1101                         throw(std::logic_error("lsolve: system is not linear"));
1102                 if (rhs.has(symbols.op(i)))
1103                         throw(std::logic_error("lsolve: system is not linear"));
1104         }
1105         
1106         matrix solution;
1107         try {
1108                 solution = sys.solve(vars,rhs,options);
1109         } catch (const std::runtime_error & e) {
1110                 // Probably singular matrix or otherwise overdetermined system:
1111                 // It is consistent to return an empty list
1112                 return lst();
1113         }
1114         GINAC_ASSERT(solution.cols()==1);
1115         GINAC_ASSERT(solution.rows()==symbols.nops());
1116         
1117         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
1118         lst sollist;
1119         for (size_t i=0; i<symbols.nops(); i++)
1120                 sollist.append(symbols.op(i)==solution(i,0));
1121         
1122         return sollist;
1123 }
1124
1125 //////////
1126 // Find real root of f(x) numerically
1127 //////////
1128
1129 const numeric
1130 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
1131 {
1132         if (!x1.is_real() || !x2.is_real()) {
1133                 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
1134         }
1135         if (x1==x2) {
1136                 throw std::runtime_error("fsolve(): vanishing interval");
1137         }
1138         // xx[0] == left interval limit, xx[1] == right interval limit.
1139         // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
1140         // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
1141         numeric xx[2] = { x1<x2 ? x1 : x2,
1142                           x1<x2 ? x2 : x1 };
1143         ex f;
1144         if (is_a<relational>(f_in)) {
1145                 f = f_in.lhs()-f_in.rhs();
1146         } else {
1147                 f = f_in;
1148         }
1149         const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
1150                             f.subs(x==xx[1]).evalf() };
1151         if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
1152                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1153         }
1154         numeric fx[2] = { ex_to<numeric>(fx_[0]),
1155                           ex_to<numeric>(fx_[1]) };
1156         if (!fx[0].is_real() || !fx[1].is_real()) {
1157                 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
1158         }
1159         if (fx[0]*fx[1]>=0) {
1160                 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1161         }
1162
1163         // The Newton-Raphson method has quadratic convergence!  Simply put, it
1164         // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
1165         const ex ff = normal(-f/f.diff(x));
1166         int side = 0;  // Start at left interval limit.
1167         numeric xxprev;
1168         numeric fxprev;
1169         do {
1170                 xxprev = xx[side];
1171                 fxprev = fx[side];
1172                 ex dx_ = ff.subs(x == xx[side]).evalf();
1173                 if (!is_a<numeric>(dx_))
1174                         throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1175                 xx[side] += ex_to<numeric>(dx_);
1176                 // Now check if Newton-Raphson method shot out of the interval 
1177                 bool bad_shot = (side == 0 && xx[0] < xxprev) || 
1178                                 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1179                 if (!bad_shot) {
1180                         // Compute f(x) only if new x is inside the interval.
1181                         // The function might be difficult to compute numerically
1182                         // or even ill defined outside the interval. Also it's
1183                         // a small optimization. 
1184                         ex f_x = f.subs(x == xx[side]).evalf();
1185                         if (!is_a<numeric>(f_x))
1186                                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1187                         fx[side] = ex_to<numeric>(f_x);
1188                 }
1189                 if (bad_shot) {
1190                         // Oops, Newton-Raphson method shot out of the interval.
1191                         // Restore, and try again with the other side instead!
1192                         xx[side] = xxprev;
1193                         fx[side] = fxprev;
1194                         side = !side;
1195                         xxprev = xx[side];
1196                         fxprev = fx[side];
1197
1198                         ex dx_ = ff.subs(x == xx[side]).evalf();
1199                         if (!is_a<numeric>(dx_))
1200                                 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1201                         xx[side] += ex_to<numeric>(dx_);
1202
1203                         ex f_x = f.subs(x==xx[side]).evalf();
1204                         if (!is_a<numeric>(f_x))
1205                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1206                         fx[side] = ex_to<numeric>(f_x);
1207                 }
1208                 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1209                         // Oops, the root isn't bracketed any more.
1210                         // Restore, and perform a bisection!
1211                         xx[side] = xxprev;
1212                         fx[side] = fxprev;
1213
1214                         // Ah, the bisection! Bisections converge linearly. Unfortunately,
1215                         // they occur pretty often when Newton-Raphson arrives at an x too
1216                         // close to the result on one side of the interval and
1217                         // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1218                         // precision errors! Recall that this function does not have a
1219                         // precision goal as one of its arguments but instead relies on
1220                         // x converging to a fixed point. We speed up the (safe but slow)
1221                         // bisection method by mixing in a dash of the (unsafer but faster)
1222                         // secant method: Instead of splitting the interval at the
1223                         // arithmetic mean (bisection), we split it nearer to the root as
1224                         // determined by the secant between the values xx[0] and xx[1].
1225                         // Don't set the secant_weight to one because that could disturb
1226                         // the convergence in some corner cases!
1227                         static const double secant_weight = 0.984375;  // == 63/64 < 1
1228                         numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1229                             + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1230                         ex fxmid_ = f.subs(x == xxmid).evalf();
1231                         if (!is_a<numeric>(fxmid_))
1232                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1233                         numeric fxmid = ex_to<numeric>(fxmid_);
1234                         if (fxmid.is_zero()) {
1235                                 // Luck strikes...
1236                                 return xxmid;
1237                         }
1238                         if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1239                                 side = !side;
1240                         }
1241                         xxprev = xx[side];
1242                         fxprev = fx[side];
1243                         xx[side] = xxmid;
1244                         fx[side] = fxmid;
1245                 }
1246         } while (xxprev!=xx[side]);
1247         return xxprev;
1248 }
1249
1250
1251 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1252  * for static lib (so ginsh will see them). */
1253 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1254 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1255
1256 } // namespace GiNaC