fixed a bug where quo() would call vector::reserve() with a negative argument
[ginac.git] / ginac / inifcns_trans.cpp
1 /** @file inifcns_trans.cpp
2  *
3  *  Implementation of transcendental (and trigonometric and hyperbolic)
4  *  functions. */
5
6 /*
7  *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
8  *
9  *  This program is free software; you can redistribute it and/or modify
10  *  it under the terms of the GNU General Public License as published by
11  *  the Free Software Foundation; either version 2 of the License, or
12  *  (at your option) any later version.
13  *
14  *  This program is distributed in the hope that it will be useful,
15  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
16  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
17  *  GNU General Public License for more details.
18  *
19  *  You should have received a copy of the GNU General Public License
20  *  along with this program; if not, write to the Free Software
21  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
22  */
23
24 #include <vector>
25 #include <stdexcept>
26
27 #include "inifcns.h"
28 #include "ex.h"
29 #include "constant.h"
30 #include "numeric.h"
31 #include "power.h"
32 #include "relational.h"
33 #include "symbol.h"
34 #include "pseries.h"
35 #include "utils.h"
36
37 namespace GiNaC {
38
39 //////////
40 // exponential function
41 //////////
42
43 static ex exp_evalf(const ex & x)
44 {
45         if (is_exactly_a<numeric>(x))
46                 return exp(ex_to<numeric>(x));
47         
48         return exp(x).hold();
49 }
50
51 static ex exp_eval(const ex & x)
52 {
53         // exp(0) -> 1
54         if (x.is_zero()) {
55                 return _ex1;
56         }
57         // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
58         const ex TwoExOverPiI=(_ex2*x)/(Pi*I);
59         if (TwoExOverPiI.info(info_flags::integer)) {
60                 const numeric z = mod(ex_to<numeric>(TwoExOverPiI),_num4);
61                 if (z.is_equal(_num0))
62                         return _ex1;
63                 if (z.is_equal(_num1))
64                         return ex(I);
65                 if (z.is_equal(_num2))
66                         return _ex_1;
67                 if (z.is_equal(_num3))
68                         return ex(-I);
69         }
70         // exp(log(x)) -> x
71         if (is_ex_the_function(x, log))
72                 return x.op(0);
73         
74         // exp(float)
75         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
76                 return exp(ex_to<numeric>(x));
77         
78         return exp(x).hold();
79 }
80
81 static ex exp_deriv(const ex & x, unsigned deriv_param)
82 {
83         GINAC_ASSERT(deriv_param==0);
84
85         // d/dx exp(x) -> exp(x)
86         return exp(x);
87 }
88
89 REGISTER_FUNCTION(exp, eval_func(exp_eval).
90                        evalf_func(exp_evalf).
91                        derivative_func(exp_deriv).
92                        latex_name("\\exp"));
93
94 //////////
95 // natural logarithm
96 //////////
97
98 static ex log_evalf(const ex & x)
99 {
100         if (is_exactly_a<numeric>(x))
101                 return log(ex_to<numeric>(x));
102         
103         return log(x).hold();
104 }
105
106 static ex log_eval(const ex & x)
107 {
108         if (x.info(info_flags::numeric)) {
109                 if (x.is_zero())         // log(0) -> infinity
110                         throw(pole_error("log_eval(): log(0)",0));
111                 if (x.info(info_flags::real) && x.info(info_flags::negative))
112                         return (log(-x)+I*Pi);
113                 if (x.is_equal(_ex1))  // log(1) -> 0
114                         return _ex0;
115                 if (x.is_equal(I))       // log(I) -> Pi*I/2
116                         return (Pi*I*_num1_2);
117                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
118                         return (Pi*I*_num_1_2);
119                 // log(float)
120                 if (!x.info(info_flags::crational))
121                         return log(ex_to<numeric>(x));
122         }
123         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
124         if (is_ex_the_function(x, exp)) {
125                 const ex &t = x.op(0);
126                 if (t.info(info_flags::numeric)) {
127                         const numeric &nt = ex_to<numeric>(t);
128                         if (nt.is_real())
129                                 return t;
130                 }
131         }
132         
133         return log(x).hold();
134 }
135
136 static ex log_deriv(const ex & x, unsigned deriv_param)
137 {
138         GINAC_ASSERT(deriv_param==0);
139         
140         // d/dx log(x) -> 1/x
141         return power(x, _ex_1);
142 }
143
144 // This is a strange workaround for a compiliation problem with the try statement
145 // below.  With -O1 the exception is not caucht properly as of GCC-2.95.2, at
146 // least on i386.  Version 2.95.4 seems to have fixed this silly problem, though.
147 // Funnily, with a simple extern declaration here it mysteriously works again.
148 #if defined(__GNUC__) && (__GNUC__==2)
149 extern "C" int putchar(int);
150 #endif
151
152 static ex log_series(const ex &arg,
153                      const relational &rel,
154                      int order,
155                      unsigned options)
156 {
157         GINAC_ASSERT(is_exactly_a<symbol>(rel.lhs()));
158         ex arg_pt;
159         bool must_expand_arg = false;
160         // maybe substitution of rel into arg fails because of a pole
161         try {
162                 arg_pt = arg.subs(rel);
163         } catch (pole_error) {
164                 must_expand_arg = true;
165         }
166         // or we are at the branch point anyways
167         if (arg_pt.is_zero())
168                 must_expand_arg = true;
169         
170         if (must_expand_arg) {
171                 // method:
172                 // This is the branch point: Series expand the argument first, then
173                 // trivially factorize it to isolate that part which has constant
174                 // leading coefficient in this fashion:
175                 //   x^n + x^(n+1) +...+ Order(x^(n+m))  ->  x^n * (1 + x +...+ Order(x^m)).
176                 // Return a plain n*log(x) for the x^n part and series expand the
177                 // other part.  Add them together and reexpand again in order to have
178                 // one unnested pseries object.  All this also works for negative n.
179                 pseries argser;          // series expansion of log's argument
180                 unsigned extra_ord = 0;  // extra expansion order
181                 do {
182                         // oops, the argument expanded to a pure Order(x^something)...
183                         argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
184                         ++extra_ord;
185                 } while (!argser.is_terminating() && argser.nops()==1);
186
187                 const symbol &s = ex_to<symbol>(rel.lhs());
188                 const ex &point = rel.rhs();
189                 const int n = argser.ldegree(s);
190                 epvector seq;
191                 // construct what we carelessly called the n*log(x) term above
192                 const ex coeff = argser.coeff(s, n);
193                 // expand the log, but only if coeff is real and > 0, since otherwise
194                 // it would make the branch cut run into the wrong direction
195                 if (coeff.info(info_flags::positive))
196                         seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
197                 else
198                         seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));
199
200                 if (!argser.is_terminating() || argser.nops()!=1) {
201                         // in this case n more (or less) terms are needed
202                         // (sadly, to generate them, we have to start from the beginning)
203                         const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
204                         return pseries(rel, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
205                 } else  // it was a monomial
206                         return pseries(rel, seq);
207         }
208         if (!(options & series_options::suppress_branchcut) &&
209              arg_pt.info(info_flags::negative)) {
210                 // method:
211                 // This is the branch cut: assemble the primitive series manually and
212                 // then add the corresponding complex step function.
213                 const symbol &s = ex_to<symbol>(rel.lhs());
214                 const ex &point = rel.rhs();
215                 const symbol foo;
216                 const ex replarg = series(log(arg), s==foo, order).subs(foo==point);
217                 epvector seq;
218                 seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
219                 seq.push_back(expair(Order(_ex1), order));
220                 return series(replarg - I*Pi + pseries(rel, seq), rel, order);
221         }
222         throw do_taylor();  // caught by function::series()
223 }
224
225 REGISTER_FUNCTION(log, eval_func(log_eval).
226                        evalf_func(log_evalf).
227                        derivative_func(log_deriv).
228                        series_func(log_series).
229                        latex_name("\\ln"));
230
231 //////////
232 // sine (trigonometric function)
233 //////////
234
235 static ex sin_evalf(const ex & x)
236 {
237         if (is_exactly_a<numeric>(x))
238                 return sin(ex_to<numeric>(x));
239         
240         return sin(x).hold();
241 }
242
243 static ex sin_eval(const ex & x)
244 {
245         // sin(n/d*Pi) -> { all known non-nested radicals }
246         const ex SixtyExOverPi = _ex60*x/Pi;
247         ex sign = _ex1;
248         if (SixtyExOverPi.info(info_flags::integer)) {
249                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num120);
250                 if (z>=_num60) {
251                         // wrap to interval [0, Pi)
252                         z -= _num60;
253                         sign = _ex_1;
254                 }
255                 if (z>_num30) {
256                         // wrap to interval [0, Pi/2)
257                         z = _num60-z;
258                 }
259                 if (z.is_equal(_num0))  // sin(0)       -> 0
260                         return _ex0;
261                 if (z.is_equal(_num5))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
262                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
263                 if (z.is_equal(_num6))  // sin(Pi/10)   -> sqrt(5)/4-1/4
264                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
265                 if (z.is_equal(_num10)) // sin(Pi/6)    -> 1/2
266                         return sign*_ex1_2;
267                 if (z.is_equal(_num15)) // sin(Pi/4)    -> sqrt(2)/2
268                         return sign*_ex1_2*sqrt(_ex2);
269                 if (z.is_equal(_num18)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
270                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
271                 if (z.is_equal(_num20)) // sin(Pi/3)    -> sqrt(3)/2
272                         return sign*_ex1_2*sqrt(_ex3);
273                 if (z.is_equal(_num25)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
274                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
275                 if (z.is_equal(_num30)) // sin(Pi/2)    -> 1
276                         return sign;
277         }
278         
279         if (is_exactly_a<function>(x)) {
280                 const ex &t = x.op(0);
281                 // sin(asin(x)) -> x
282                 if (is_ex_the_function(x, asin))
283                         return t;
284                 // sin(acos(x)) -> sqrt(1-x^2)
285                 if (is_ex_the_function(x, acos))
286                         return sqrt(_ex1-power(t,_ex2));
287                 // sin(atan(x)) -> x/sqrt(1+x^2)
288                 if (is_ex_the_function(x, atan))
289                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
290         }
291         
292         // sin(float) -> float
293         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
294                 return sin(ex_to<numeric>(x));
295         
296         return sin(x).hold();
297 }
298
299 static ex sin_deriv(const ex & x, unsigned deriv_param)
300 {
301         GINAC_ASSERT(deriv_param==0);
302         
303         // d/dx sin(x) -> cos(x)
304         return cos(x);
305 }
306
307 REGISTER_FUNCTION(sin, eval_func(sin_eval).
308                        evalf_func(sin_evalf).
309                        derivative_func(sin_deriv).
310                        latex_name("\\sin"));
311
312 //////////
313 // cosine (trigonometric function)
314 //////////
315
316 static ex cos_evalf(const ex & x)
317 {
318         if (is_exactly_a<numeric>(x))
319                 return cos(ex_to<numeric>(x));
320         
321         return cos(x).hold();
322 }
323
324 static ex cos_eval(const ex & x)
325 {
326         // cos(n/d*Pi) -> { all known non-nested radicals }
327         const ex SixtyExOverPi = _ex60*x/Pi;
328         ex sign = _ex1;
329         if (SixtyExOverPi.info(info_flags::integer)) {
330                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num120);
331                 if (z>=_num60) {
332                         // wrap to interval [0, Pi)
333                         z = _num120-z;
334                 }
335                 if (z>=_num30) {
336                         // wrap to interval [0, Pi/2)
337                         z = _num60-z;
338                         sign = _ex_1;
339                 }
340                 if (z.is_equal(_num0))  // cos(0)       -> 1
341                         return sign;
342                 if (z.is_equal(_num5))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
343                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
344                 if (z.is_equal(_num10)) // cos(Pi/6)    -> sqrt(3)/2
345                         return sign*_ex1_2*sqrt(_ex3);
346                 if (z.is_equal(_num12)) // cos(Pi/5)    -> sqrt(5)/4+1/4
347                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
348                 if (z.is_equal(_num15)) // cos(Pi/4)    -> sqrt(2)/2
349                         return sign*_ex1_2*sqrt(_ex2);
350                 if (z.is_equal(_num20)) // cos(Pi/3)    -> 1/2
351                         return sign*_ex1_2;
352                 if (z.is_equal(_num24)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
353                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
354                 if (z.is_equal(_num25)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
355                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
356                 if (z.is_equal(_num30)) // cos(Pi/2)    -> 0
357                         return _ex0;
358         }
359         
360         if (is_exactly_a<function>(x)) {
361                 const ex &t = x.op(0);
362                 // cos(acos(x)) -> x
363                 if (is_ex_the_function(x, acos))
364                         return t;
365                 // cos(asin(x)) -> sqrt(1-x^2)
366                 if (is_ex_the_function(x, asin))
367                         return sqrt(_ex1-power(t,_ex2));
368                 // cos(atan(x)) -> 1/sqrt(1+x^2)
369                 if (is_ex_the_function(x, atan))
370                         return power(_ex1+power(t,_ex2),_ex_1_2);
371         }
372         
373         // cos(float) -> float
374         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
375                 return cos(ex_to<numeric>(x));
376         
377         return cos(x).hold();
378 }
379
380 static ex cos_deriv(const ex & x, unsigned deriv_param)
381 {
382         GINAC_ASSERT(deriv_param==0);
383
384         // d/dx cos(x) -> -sin(x)
385         return -sin(x);
386 }
387
388 REGISTER_FUNCTION(cos, eval_func(cos_eval).
389                        evalf_func(cos_evalf).
390                        derivative_func(cos_deriv).
391                        latex_name("\\cos"));
392
393 //////////
394 // tangent (trigonometric function)
395 //////////
396
397 static ex tan_evalf(const ex & x)
398 {
399         if (is_exactly_a<numeric>(x))
400                 return tan(ex_to<numeric>(x));
401         
402         return tan(x).hold();
403 }
404
405 static ex tan_eval(const ex & x)
406 {
407         // tan(n/d*Pi) -> { all known non-nested radicals }
408         const ex SixtyExOverPi = _ex60*x/Pi;
409         ex sign = _ex1;
410         if (SixtyExOverPi.info(info_flags::integer)) {
411                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num60);
412                 if (z>=_num60) {
413                         // wrap to interval [0, Pi)
414                         z -= _num60;
415                 }
416                 if (z>=_num30) {
417                         // wrap to interval [0, Pi/2)
418                         z = _num60-z;
419                         sign = _ex_1;
420                 }
421                 if (z.is_equal(_num0))  // tan(0)       -> 0
422                         return _ex0;
423                 if (z.is_equal(_num5))  // tan(Pi/12)   -> 2-sqrt(3)
424                         return sign*(_ex2-sqrt(_ex3));
425                 if (z.is_equal(_num10)) // tan(Pi/6)    -> sqrt(3)/3
426                         return sign*_ex1_3*sqrt(_ex3);
427                 if (z.is_equal(_num15)) // tan(Pi/4)    -> 1
428                         return sign;
429                 if (z.is_equal(_num20)) // tan(Pi/3)    -> sqrt(3)
430                         return sign*sqrt(_ex3);
431                 if (z.is_equal(_num25)) // tan(5/12*Pi) -> 2+sqrt(3)
432                         return sign*(sqrt(_ex3)+_ex2);
433                 if (z.is_equal(_num30)) // tan(Pi/2)    -> infinity
434                         throw (pole_error("tan_eval(): simple pole",1));
435         }
436         
437         if (is_exactly_a<function>(x)) {
438                 const ex &t = x.op(0);
439                 // tan(atan(x)) -> x
440                 if (is_ex_the_function(x, atan))
441                         return t;
442                 // tan(asin(x)) -> x/sqrt(1+x^2)
443                 if (is_ex_the_function(x, asin))
444                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
445                 // tan(acos(x)) -> sqrt(1-x^2)/x
446                 if (is_ex_the_function(x, acos))
447                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
448         }
449         
450         // tan(float) -> float
451         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
452                 return tan(ex_to<numeric>(x));
453         }
454         
455         return tan(x).hold();
456 }
457
458 static ex tan_deriv(const ex & x, unsigned deriv_param)
459 {
460         GINAC_ASSERT(deriv_param==0);
461         
462         // d/dx tan(x) -> 1+tan(x)^2;
463         return (_ex1+power(tan(x),_ex2));
464 }
465
466 static ex tan_series(const ex &x,
467                      const relational &rel,
468                      int order,
469                      unsigned options)
470 {
471         GINAC_ASSERT(is_exactly_a<symbol>(rel.lhs()));
472         // method:
473         // Taylor series where there is no pole falls back to tan_deriv.
474         // On a pole simply expand sin(x)/cos(x).
475         const ex x_pt = x.subs(rel);
476         if (!(2*x_pt/Pi).info(info_flags::odd))
477                 throw do_taylor();  // caught by function::series()
478         // if we got here we have to care for a simple pole
479         return (sin(x)/cos(x)).series(rel, order+2, options);
480 }
481
482 REGISTER_FUNCTION(tan, eval_func(tan_eval).
483                        evalf_func(tan_evalf).
484                        derivative_func(tan_deriv).
485                        series_func(tan_series).
486                        latex_name("\\tan"));
487
488 //////////
489 // inverse sine (arc sine)
490 //////////
491
492 static ex asin_evalf(const ex & x)
493 {
494         if (is_exactly_a<numeric>(x))
495                 return asin(ex_to<numeric>(x));
496         
497         return asin(x).hold();
498 }
499
500 static ex asin_eval(const ex & x)
501 {
502         if (x.info(info_flags::numeric)) {
503                 // asin(0) -> 0
504                 if (x.is_zero())
505                         return x;
506                 // asin(1/2) -> Pi/6
507                 if (x.is_equal(_ex1_2))
508                         return numeric(1,6)*Pi;
509                 // asin(1) -> Pi/2
510                 if (x.is_equal(_ex1))
511                         return _num1_2*Pi;
512                 // asin(-1/2) -> -Pi/6
513                 if (x.is_equal(_ex_1_2))
514                         return numeric(-1,6)*Pi;
515                 // asin(-1) -> -Pi/2
516                 if (x.is_equal(_ex_1))
517                         return _num_1_2*Pi;
518                 // asin(float) -> float
519                 if (!x.info(info_flags::crational))
520                         return asin(ex_to<numeric>(x));
521         }
522         
523         return asin(x).hold();
524 }
525
526 static ex asin_deriv(const ex & x, unsigned deriv_param)
527 {
528         GINAC_ASSERT(deriv_param==0);
529         
530         // d/dx asin(x) -> 1/sqrt(1-x^2)
531         return power(1-power(x,_ex2),_ex_1_2);
532 }
533
534 REGISTER_FUNCTION(asin, eval_func(asin_eval).
535                         evalf_func(asin_evalf).
536                         derivative_func(asin_deriv).
537                         latex_name("\\arcsin"));
538
539 //////////
540 // inverse cosine (arc cosine)
541 //////////
542
543 static ex acos_evalf(const ex & x)
544 {
545         if (is_exactly_a<numeric>(x))
546                 return acos(ex_to<numeric>(x));
547         
548         return acos(x).hold();
549 }
550
551 static ex acos_eval(const ex & x)
552 {
553         if (x.info(info_flags::numeric)) {
554                 // acos(1) -> 0
555                 if (x.is_equal(_ex1))
556                         return _ex0;
557                 // acos(1/2) -> Pi/3
558                 if (x.is_equal(_ex1_2))
559                         return _ex1_3*Pi;
560                 // acos(0) -> Pi/2
561                 if (x.is_zero())
562                         return _ex1_2*Pi;
563                 // acos(-1/2) -> 2/3*Pi
564                 if (x.is_equal(_ex_1_2))
565                         return numeric(2,3)*Pi;
566                 // acos(-1) -> Pi
567                 if (x.is_equal(_ex_1))
568                         return Pi;
569                 // acos(float) -> float
570                 if (!x.info(info_flags::crational))
571                         return acos(ex_to<numeric>(x));
572         }
573         
574         return acos(x).hold();
575 }
576
577 static ex acos_deriv(const ex & x, unsigned deriv_param)
578 {
579         GINAC_ASSERT(deriv_param==0);
580         
581         // d/dx acos(x) -> -1/sqrt(1-x^2)
582         return -power(1-power(x,_ex2),_ex_1_2);
583 }
584
585 REGISTER_FUNCTION(acos, eval_func(acos_eval).
586                         evalf_func(acos_evalf).
587                         derivative_func(acos_deriv).
588                         latex_name("\\arccos"));
589
590 //////////
591 // inverse tangent (arc tangent)
592 //////////
593
594 static ex atan_evalf(const ex & x)
595 {
596         if (is_exactly_a<numeric>(x))
597                 return atan(ex_to<numeric>(x));
598         
599         return atan(x).hold();
600 }
601
602 static ex atan_eval(const ex & x)
603 {
604         if (x.info(info_flags::numeric)) {
605                 // atan(0) -> 0
606                 if (x.is_zero())
607                         return _ex0;
608                 // atan(1) -> Pi/4
609                 if (x.is_equal(_ex1))
610                         return _ex1_4*Pi;
611                 // atan(-1) -> -Pi/4
612                 if (x.is_equal(_ex_1))
613                         return _ex_1_4*Pi;
614                 if (x.is_equal(I) || x.is_equal(-I))
615                         throw (pole_error("atan_eval(): logarithmic pole",0));
616                 // atan(float) -> float
617                 if (!x.info(info_flags::crational))
618                         return atan(ex_to<numeric>(x));
619         }
620         
621         return atan(x).hold();
622 }
623
624 static ex atan_deriv(const ex & x, unsigned deriv_param)
625 {
626         GINAC_ASSERT(deriv_param==0);
627
628         // d/dx atan(x) -> 1/(1+x^2)
629         return power(_ex1+power(x,_ex2), _ex_1);
630 }
631
632 static ex atan_series(const ex &arg,
633                       const relational &rel,
634                       int order,
635                       unsigned options)
636 {
637         GINAC_ASSERT(is_exactly_a<symbol>(rel.lhs()));
638         // method:
639         // Taylor series where there is no pole or cut falls back to atan_deriv.
640         // There are two branch cuts, one runnig from I up the imaginary axis and
641         // one running from -I down the imaginary axis.  The points I and -I are
642         // poles.
643         // On the branch cuts and the poles series expand
644         //     (log(1+I*x)-log(1-I*x))/(2*I)
645         // instead.
646         const ex arg_pt = arg.subs(rel);
647         if (!(I*arg_pt).info(info_flags::real))
648                 throw do_taylor();     // Re(x) != 0
649         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
650                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
651         // care for the poles, using the defining formula for atan()...
652         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
653                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
654         if (!(options & series_options::suppress_branchcut)) {
655                 // method:
656                 // This is the branch cut: assemble the primitive series manually and
657                 // then add the corresponding complex step function.
658                 const symbol &s = ex_to<symbol>(rel.lhs());
659                 const ex &point = rel.rhs();
660                 const symbol foo;
661                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point);
662                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
663                 if ((I*arg_pt)<_ex0)
664                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
665                 else
666                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
667                 epvector seq;
668                 seq.push_back(expair(Order0correction, _ex0));
669                 seq.push_back(expair(Order(_ex1), order));
670                 return series(replarg - pseries(rel, seq), rel, order);
671         }
672         throw do_taylor();
673 }
674
675 REGISTER_FUNCTION(atan, eval_func(atan_eval).
676                         evalf_func(atan_evalf).
677                         derivative_func(atan_deriv).
678                         series_func(atan_series).
679                         latex_name("\\arctan"));
680
681 //////////
682 // inverse tangent (atan2(y,x))
683 //////////
684
685 static ex atan2_evalf(const ex &y, const ex &x)
686 {
687         if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
688                 return atan2(ex_to<numeric>(y), ex_to<numeric>(x));
689         
690         return atan2(y, x).hold();
691 }
692
693 static ex atan2_eval(const ex & y, const ex & x)
694 {
695         if (y.info(info_flags::numeric) && !y.info(info_flags::crational) &&
696                 x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
697                 return atan2_evalf(y,x);
698         }
699         
700         return atan2(y,x).hold();
701 }    
702
703 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
704 {
705         GINAC_ASSERT(deriv_param<2);
706         
707         if (deriv_param==0) {
708                 // d/dy atan(y,x)
709                 return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
710         }
711         // d/dx atan(y,x)
712         return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
713 }
714
715 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
716                          evalf_func(atan2_evalf).
717                          derivative_func(atan2_deriv));
718
719 //////////
720 // hyperbolic sine (trigonometric function)
721 //////////
722
723 static ex sinh_evalf(const ex & x)
724 {
725         if (is_exactly_a<numeric>(x))
726                 return sinh(ex_to<numeric>(x));
727         
728         return sinh(x).hold();
729 }
730
731 static ex sinh_eval(const ex & x)
732 {
733         if (x.info(info_flags::numeric)) {
734                 if (x.is_zero())  // sinh(0) -> 0
735                         return _ex0;        
736                 if (!x.info(info_flags::crational))  // sinh(float) -> float
737                         return sinh(ex_to<numeric>(x));
738         }
739         
740         if ((x/Pi).info(info_flags::numeric) &&
741                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
742                 return I*sin(x/I);
743         
744         if (is_exactly_a<function>(x)) {
745                 const ex &t = x.op(0);
746                 // sinh(asinh(x)) -> x
747                 if (is_ex_the_function(x, asinh))
748                         return t;
749                 // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
750                 if (is_ex_the_function(x, acosh))
751                         return sqrt(t-_ex1)*sqrt(t+_ex1);
752                 // sinh(atanh(x)) -> x/sqrt(1-x^2)
753                 if (is_ex_the_function(x, atanh))
754                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
755         }
756         
757         return sinh(x).hold();
758 }
759
760 static ex sinh_deriv(const ex & x, unsigned deriv_param)
761 {
762         GINAC_ASSERT(deriv_param==0);
763         
764         // d/dx sinh(x) -> cosh(x)
765         return cosh(x);
766 }
767
768 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
769                         evalf_func(sinh_evalf).
770                         derivative_func(sinh_deriv).
771                         latex_name("\\sinh"));
772
773 //////////
774 // hyperbolic cosine (trigonometric function)
775 //////////
776
777 static ex cosh_evalf(const ex & x)
778 {
779         if (is_exactly_a<numeric>(x))
780                 return cosh(ex_to<numeric>(x));
781         
782         return cosh(x).hold();
783 }
784
785 static ex cosh_eval(const ex & x)
786 {
787         if (x.info(info_flags::numeric)) {
788                 if (x.is_zero())  // cosh(0) -> 1
789                         return _ex1;
790                 if (!x.info(info_flags::crational))  // cosh(float) -> float
791                         return cosh(ex_to<numeric>(x));
792         }
793         
794         if ((x/Pi).info(info_flags::numeric) &&
795                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
796                 return cos(x/I);
797         
798         if (is_exactly_a<function>(x)) {
799                 const ex &t = x.op(0);
800                 // cosh(acosh(x)) -> x
801                 if (is_ex_the_function(x, acosh))
802                         return t;
803                 // cosh(asinh(x)) -> sqrt(1+x^2)
804                 if (is_ex_the_function(x, asinh))
805                         return sqrt(_ex1+power(t,_ex2));
806                 // cosh(atanh(x)) -> 1/sqrt(1-x^2)
807                 if (is_ex_the_function(x, atanh))
808                         return power(_ex1-power(t,_ex2),_ex_1_2);
809         }
810         
811         return cosh(x).hold();
812 }
813
814 static ex cosh_deriv(const ex & x, unsigned deriv_param)
815 {
816         GINAC_ASSERT(deriv_param==0);
817         
818         // d/dx cosh(x) -> sinh(x)
819         return sinh(x);
820 }
821
822 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
823                         evalf_func(cosh_evalf).
824                         derivative_func(cosh_deriv).
825                         latex_name("\\cosh"));
826
827 //////////
828 // hyperbolic tangent (trigonometric function)
829 //////////
830
831 static ex tanh_evalf(const ex & x)
832 {
833         if (is_exactly_a<numeric>(x))
834                 return tanh(ex_to<numeric>(x));
835         
836         return tanh(x).hold();
837 }
838
839 static ex tanh_eval(const ex & x)
840 {
841         if (x.info(info_flags::numeric)) {
842                 if (x.is_zero())  // tanh(0) -> 0
843                         return _ex0;
844                 if (!x.info(info_flags::crational))  // tanh(float) -> float
845                         return tanh(ex_to<numeric>(x));
846         }
847         
848         if ((x/Pi).info(info_flags::numeric) &&
849                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
850                 return I*tan(x/I);
851         
852         if (is_exactly_a<function>(x)) {
853                 const ex &t = x.op(0);
854                 // tanh(atanh(x)) -> x
855                 if (is_ex_the_function(x, atanh))
856                         return t;
857                 // tanh(asinh(x)) -> x/sqrt(1+x^2)
858                 if (is_ex_the_function(x, asinh))
859                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
860                 // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
861                 if (is_ex_the_function(x, acosh))
862                         return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
863         }
864         
865         return tanh(x).hold();
866 }
867
868 static ex tanh_deriv(const ex & x, unsigned deriv_param)
869 {
870         GINAC_ASSERT(deriv_param==0);
871         
872         // d/dx tanh(x) -> 1-tanh(x)^2
873         return _ex1-power(tanh(x),_ex2);
874 }
875
876 static ex tanh_series(const ex &x,
877                       const relational &rel,
878                       int order,
879                       unsigned options)
880 {
881         GINAC_ASSERT(is_exactly_a<symbol>(rel.lhs()));
882         // method:
883         // Taylor series where there is no pole falls back to tanh_deriv.
884         // On a pole simply expand sinh(x)/cosh(x).
885         const ex x_pt = x.subs(rel);
886         if (!(2*I*x_pt/Pi).info(info_flags::odd))
887                 throw do_taylor();  // caught by function::series()
888         // if we got here we have to care for a simple pole
889         return (sinh(x)/cosh(x)).series(rel, order+2, options);
890 }
891
892 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
893                         evalf_func(tanh_evalf).
894                         derivative_func(tanh_deriv).
895                         series_func(tanh_series).
896                         latex_name("\\tanh"));
897
898 //////////
899 // inverse hyperbolic sine (trigonometric function)
900 //////////
901
902 static ex asinh_evalf(const ex & x)
903 {
904         if (is_exactly_a<numeric>(x))
905                 return asinh(ex_to<numeric>(x));
906         
907         return asinh(x).hold();
908 }
909
910 static ex asinh_eval(const ex & x)
911 {
912         if (x.info(info_flags::numeric)) {
913                 // asinh(0) -> 0
914                 if (x.is_zero())
915                         return _ex0;
916                 // asinh(float) -> float
917                 if (!x.info(info_flags::crational))
918                         return asinh(ex_to<numeric>(x));
919         }
920         
921         return asinh(x).hold();
922 }
923
924 static ex asinh_deriv(const ex & x, unsigned deriv_param)
925 {
926         GINAC_ASSERT(deriv_param==0);
927         
928         // d/dx asinh(x) -> 1/sqrt(1+x^2)
929         return power(_ex1+power(x,_ex2),_ex_1_2);
930 }
931
932 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
933                          evalf_func(asinh_evalf).
934                          derivative_func(asinh_deriv));
935
936 //////////
937 // inverse hyperbolic cosine (trigonometric function)
938 //////////
939
940 static ex acosh_evalf(const ex & x)
941 {
942         if (is_exactly_a<numeric>(x))
943                 return acosh(ex_to<numeric>(x));
944         
945         return acosh(x).hold();
946 }
947
948 static ex acosh_eval(const ex & x)
949 {
950         if (x.info(info_flags::numeric)) {
951                 // acosh(0) -> Pi*I/2
952                 if (x.is_zero())
953                         return Pi*I*numeric(1,2);
954                 // acosh(1) -> 0
955                 if (x.is_equal(_ex1))
956                         return _ex0;
957                 // acosh(-1) -> Pi*I
958                 if (x.is_equal(_ex_1))
959                         return Pi*I;
960                 // acosh(float) -> float
961                 if (!x.info(info_flags::crational))
962                         return acosh(ex_to<numeric>(x));
963         }
964         
965         return acosh(x).hold();
966 }
967
968 static ex acosh_deriv(const ex & x, unsigned deriv_param)
969 {
970         GINAC_ASSERT(deriv_param==0);
971         
972         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
973         return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
974 }
975
976 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
977                          evalf_func(acosh_evalf).
978                          derivative_func(acosh_deriv));
979
980 //////////
981 // inverse hyperbolic tangent (trigonometric function)
982 //////////
983
984 static ex atanh_evalf(const ex & x)
985 {
986         if (is_exactly_a<numeric>(x))
987                 return atanh(ex_to<numeric>(x));
988         
989         return atanh(x).hold();
990 }
991
992 static ex atanh_eval(const ex & x)
993 {
994         if (x.info(info_flags::numeric)) {
995                 // atanh(0) -> 0
996                 if (x.is_zero())
997                         return _ex0;
998                 // atanh({+|-}1) -> throw
999                 if (x.is_equal(_ex1) || x.is_equal(_ex_1))
1000                         throw (pole_error("atanh_eval(): logarithmic pole",0));
1001                 // atanh(float) -> float
1002                 if (!x.info(info_flags::crational))
1003                         return atanh(ex_to<numeric>(x));
1004         }
1005         
1006         return atanh(x).hold();
1007 }
1008
1009 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1010 {
1011         GINAC_ASSERT(deriv_param==0);
1012         
1013         // d/dx atanh(x) -> 1/(1-x^2)
1014         return power(_ex1-power(x,_ex2),_ex_1);
1015 }
1016
1017 static ex atanh_series(const ex &arg,
1018                        const relational &rel,
1019                        int order,
1020                        unsigned options)
1021 {
1022         GINAC_ASSERT(is_exactly_a<symbol>(rel.lhs()));
1023         // method:
1024         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1025         // There are two branch cuts, one runnig from 1 up the real axis and one
1026         // one running from -1 down the real axis.  The points 1 and -1 are poles
1027         // On the branch cuts and the poles series expand
1028         //     (log(1+x)-log(1-x))/2
1029         // instead.
1030         const ex arg_pt = arg.subs(rel);
1031         if (!(arg_pt).info(info_flags::real))
1032                 throw do_taylor();     // Im(x) != 0
1033         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1034                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1035         // care for the poles, using the defining formula for atanh()...
1036         if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1037                 return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1038         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1039         if (!(options & series_options::suppress_branchcut)) {
1040                 // method:
1041                 // This is the branch cut: assemble the primitive series manually and
1042                 // then add the corresponding complex step function.
1043                 const symbol &s = ex_to<symbol>(rel.lhs());
1044                 const ex &point = rel.rhs();
1045                 const symbol foo;
1046                 const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point);
1047                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1048                 if (arg_pt<_ex0)
1049                         Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1050                 else
1051                         Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1052                 epvector seq;
1053                 seq.push_back(expair(Order0correction, _ex0));
1054                 seq.push_back(expair(Order(_ex1), order));
1055                 return series(replarg - pseries(rel, seq), rel, order);
1056         }
1057         throw do_taylor();
1058 }
1059
1060 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1061                          evalf_func(atanh_evalf).
1062                          derivative_func(atanh_deriv).
1063                          series_func(atanh_series));
1064
1065
1066 } // namespace GiNaC