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[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) 1999-2000 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 #ifndef NO_NAMESPACE_GINAC
38 namespace GiNaC {
39 #endif // ndef NO_NAMESPACE_GINAC
40
41 //////////
42 // exponential function
43 //////////
44
45 static ex exp_evalf(const ex & x)
46 {
47         BEGIN_TYPECHECK
48                 TYPECHECK(x,numeric)
49         END_TYPECHECK(exp(x))
50         
51         return exp(ex_to_numeric(x)); // -> numeric exp(numeric)
52 }
53
54 static ex exp_eval(const ex & x)
55 {
56         // exp(0) -> 1
57         if (x.is_zero()) {
58                 return _ex1();
59         }
60         // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
61         ex TwoExOverPiI=(_ex2()*x)/(Pi*I);
62         if (TwoExOverPiI.info(info_flags::integer)) {
63                 numeric z=mod(ex_to_numeric(TwoExOverPiI),_num4());
64                 if (z.is_equal(_num0()))
65                         return _ex1();
66                 if (z.is_equal(_num1()))
67                         return ex(I);
68                 if (z.is_equal(_num2()))
69                         return _ex_1();
70                 if (z.is_equal(_num3()))
71                         return ex(-I);
72         }
73         // exp(log(x)) -> x
74         if (is_ex_the_function(x, log))
75                 return x.op(0);
76         
77         // exp(float)
78         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
79                 return exp_evalf(x);
80         
81         return exp(x).hold();
82 }
83
84 static ex exp_deriv(const ex & x, unsigned deriv_param)
85 {
86         GINAC_ASSERT(deriv_param==0);
87
88         // d/dx exp(x) -> exp(x)
89         return exp(x);
90 }
91
92 REGISTER_FUNCTION(exp, eval_func(exp_eval).
93                                            evalf_func(exp_evalf).
94                                            derivative_func(exp_deriv));
95
96 //////////
97 // natural logarithm
98 //////////
99
100 static ex log_evalf(const ex & x)
101 {
102         BEGIN_TYPECHECK
103                 TYPECHECK(x,numeric)
104         END_TYPECHECK(log(x))
105         
106         return log(ex_to_numeric(x)); // -> numeric log(numeric)
107 }
108
109 static ex log_eval(const ex & x)
110 {
111         if (x.info(info_flags::numeric)) {
112                 if (x.is_equal(_ex0()))  // log(0) -> infinity
113                         throw(pole_error("log_eval(): log(0)",0));
114                 if (x.info(info_flags::real) && x.info(info_flags::negative))
115                         return (log(-x)+I*Pi);
116                 if (x.is_equal(_ex1()))  // log(1) -> 0
117                         return _ex0();
118                 if (x.is_equal(I))       // log(I) -> Pi*I/2
119                         return (Pi*I*_num1_2());
120                 if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
121                         return (Pi*I*_num_1_2());
122                 // log(float)
123                 if (!x.info(info_flags::crational))
124                         return log_evalf(x);
125         }
126         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
127         if (is_ex_the_function(x, exp)) {
128                 ex t = x.op(0);
129                 if (t.info(info_flags::numeric)) {
130                         numeric nt = ex_to_numeric(t);
131                         if (nt.is_real())
132                                 return t;
133                 }
134         }
135         
136         return log(x).hold();
137 }
138
139 static ex log_deriv(const ex & x, unsigned deriv_param)
140 {
141         GINAC_ASSERT(deriv_param==0);
142         
143         // d/dx log(x) -> 1/x
144         return power(x, _ex_1());
145 }
146
147 static ex log_series(const ex &arg,
148                                          const relational &rel,
149                                          int order,
150                                          unsigned options)
151 {
152         GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
153         ex arg_pt;
154         bool must_expand_arg = false;
155         // maybe substitution of rel into arg fails because of a pole
156         try {
157                 arg_pt = arg.subs(rel);
158         } catch (pole_error) {
159                 must_expand_arg = true;
160         }
161         // or we are at the branch cut anyways
162         if (arg_pt.is_zero())
163                 must_expand_arg = true;
164         
165         if (must_expand_arg) {
166                 // method:
167                 // This is the branch point: Series expand the argument first, then
168                 // trivially factorize it to isolate that part which has constant
169                 // leading coefficient in this fashion:
170                 //   x^n + Order(x^(n+m))  ->  x^n * (1 + Order(x^m)).
171                 // Return a plain n*log(x) for the x^n part and series expand the
172                 // other part.  Add them together and reexpand again in order to have
173                 // one unnested pseries object.  All this also works for negative n.
174                 const pseries argser = ex_to_pseries(arg.series(rel, order, options));
175                 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
176                 const ex point = rel.rhs();
177                 const int n = argser.ldegree(*s);
178                 epvector seq;
179                 seq.push_back(expair(n*log(*s-point), _ex0()));
180                 if (!argser.is_terminating() || argser.nops()!=1) {
181                         // in this case n more terms are needed
182                         ex newarg = ex_to_pseries(arg.series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
183                         return pseries(rel, seq).add_series(ex_to_pseries(log(newarg).series(rel, order, options)));
184                 } else  // it was a monomial
185                         return pseries(rel, seq);
186         }
187         if (!(options & series_options::suppress_branchcut) &&
188                  arg_pt.info(info_flags::negative)) {
189                 // method:
190                 // This is the branch cut: assemble the primitive series manually and
191                 // then add the corresponding complex step function.
192                 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
193                 const ex point = rel.rhs();
194                 const symbol foo;
195                 ex replarg = series(log(arg), *s==foo, order, false).subs(foo==point);
196                 epvector seq;
197                 seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0()));
198                 seq.push_back(expair(Order(_ex1()), order));
199                 return series(replarg - I*Pi + pseries(rel, seq), rel, order);
200         }
201         throw do_taylor();  // caught by function::series()
202 }
203
204 REGISTER_FUNCTION(log, eval_func(log_eval).
205                                            evalf_func(log_evalf).
206                                            derivative_func(log_deriv).
207                                            series_func(log_series));
208
209 //////////
210 // sine (trigonometric function)
211 //////////
212
213 static ex sin_evalf(const ex & x)
214 {
215         BEGIN_TYPECHECK
216            TYPECHECK(x,numeric)
217         END_TYPECHECK(sin(x))
218         
219         return sin(ex_to_numeric(x)); // -> numeric sin(numeric)
220 }
221
222 static ex sin_eval(const ex & x)
223 {
224         // sin(n/d*Pi) -> { all known non-nested radicals }
225         ex SixtyExOverPi = _ex60()*x/Pi;
226         ex sign = _ex1();
227         if (SixtyExOverPi.info(info_flags::integer)) {
228                 numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
229                 if (z>=_num60()) {
230                         // wrap to interval [0, Pi)
231                         z -= _num60();
232                         sign = _ex_1();
233                 }
234                 if (z>_num30()) {
235                         // wrap to interval [0, Pi/2)
236                         z = _num60()-z;
237                 }
238                 if (z.is_equal(_num0()))  // sin(0)       -> 0
239                         return _ex0();
240                 if (z.is_equal(_num5()))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
241                         return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
242                 if (z.is_equal(_num6()))  // sin(Pi/10)   -> sqrt(5)/4-1/4
243                         return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
244                 if (z.is_equal(_num10())) // sin(Pi/6)    -> 1/2
245                         return sign*_ex1_2();
246                 if (z.is_equal(_num15())) // sin(Pi/4)    -> sqrt(2)/2
247                         return sign*_ex1_2()*power(_ex2(),_ex1_2());
248                 if (z.is_equal(_num18())) // sin(3/10*Pi) -> sqrt(5)/4+1/4
249                         return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
250                 if (z.is_equal(_num20())) // sin(Pi/3)    -> sqrt(3)/2
251                         return sign*_ex1_2()*power(_ex3(),_ex1_2());
252                 if (z.is_equal(_num25())) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
253                         return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
254                 if (z.is_equal(_num30())) // sin(Pi/2)    -> 1
255                         return sign*_ex1();
256         }
257         
258         if (is_ex_exactly_of_type(x, function)) {
259                 ex t = x.op(0);
260                 // sin(asin(x)) -> x
261                 if (is_ex_the_function(x, asin))
262                         return t;
263                 // sin(acos(x)) -> sqrt(1-x^2)
264                 if (is_ex_the_function(x, acos))
265                         return power(_ex1()-power(t,_ex2()),_ex1_2());
266                 // sin(atan(x)) -> x*(1+x^2)^(-1/2)
267                 if (is_ex_the_function(x, atan))
268                         return t*power(_ex1()+power(t,_ex2()),_ex_1_2());
269         }
270         
271         // sin(float) -> float
272         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
273                 return sin_evalf(x);
274         
275         return sin(x).hold();
276 }
277
278 static ex sin_deriv(const ex & x, unsigned deriv_param)
279 {
280         GINAC_ASSERT(deriv_param==0);
281         
282         // d/dx sin(x) -> cos(x)
283         return cos(x);
284 }
285
286 REGISTER_FUNCTION(sin, eval_func(sin_eval).
287                                            evalf_func(sin_evalf).
288                                            derivative_func(sin_deriv));
289
290 //////////
291 // cosine (trigonometric function)
292 //////////
293
294 static ex cos_evalf(const ex & x)
295 {
296         BEGIN_TYPECHECK
297                 TYPECHECK(x,numeric)
298         END_TYPECHECK(cos(x))
299         
300         return cos(ex_to_numeric(x)); // -> numeric cos(numeric)
301 }
302
303 static ex cos_eval(const ex & x)
304 {
305         // cos(n/d*Pi) -> { all known non-nested radicals }
306         ex SixtyExOverPi = _ex60()*x/Pi;
307         ex sign = _ex1();
308         if (SixtyExOverPi.info(info_flags::integer)) {
309                 numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
310                 if (z>=_num60()) {
311                         // wrap to interval [0, Pi)
312                         z = _num120()-z;
313                 }
314                 if (z>=_num30()) {
315                         // wrap to interval [0, Pi/2)
316                         z = _num60()-z;
317                         sign = _ex_1();
318                 }
319                 if (z.is_equal(_num0()))  // cos(0)       -> 1
320                         return sign*_ex1();
321                 if (z.is_equal(_num5()))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
322                         return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
323                 if (z.is_equal(_num10())) // cos(Pi/6)    -> sqrt(3)/2
324                         return sign*_ex1_2()*power(_ex3(),_ex1_2());
325                 if (z.is_equal(_num12())) // cos(Pi/5)    -> sqrt(5)/4+1/4
326                         return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
327                 if (z.is_equal(_num15())) // cos(Pi/4)    -> sqrt(2)/2
328                         return sign*_ex1_2()*power(_ex2(),_ex1_2());
329                 if (z.is_equal(_num20())) // cos(Pi/3)    -> 1/2
330                         return sign*_ex1_2();
331                 if (z.is_equal(_num24())) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
332                         return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
333                 if (z.is_equal(_num25())) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
334                         return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
335                 if (z.is_equal(_num30())) // cos(Pi/2)    -> 0
336                         return sign*_ex0();
337         }
338         
339         if (is_ex_exactly_of_type(x, function)) {
340                 ex t = x.op(0);
341                 // cos(acos(x)) -> x
342                 if (is_ex_the_function(x, acos))
343                         return t;
344                 // cos(asin(x)) -> (1-x^2)^(1/2)
345                 if (is_ex_the_function(x, asin))
346                         return power(_ex1()-power(t,_ex2()),_ex1_2());
347                 // cos(atan(x)) -> (1+x^2)^(-1/2)
348                 if (is_ex_the_function(x, atan))
349                         return power(_ex1()+power(t,_ex2()),_ex_1_2());
350         }
351         
352         // cos(float) -> float
353         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
354                 return cos_evalf(x);
355         
356         return cos(x).hold();
357 }
358
359 static ex cos_deriv(const ex & x, unsigned deriv_param)
360 {
361         GINAC_ASSERT(deriv_param==0);
362
363         // d/dx cos(x) -> -sin(x)
364         return _ex_1()*sin(x);
365 }
366
367 REGISTER_FUNCTION(cos, eval_func(cos_eval).
368                                            evalf_func(cos_evalf).
369                                            derivative_func(cos_deriv));
370
371 //////////
372 // tangent (trigonometric function)
373 //////////
374
375 static ex tan_evalf(const ex & x)
376 {
377         BEGIN_TYPECHECK
378            TYPECHECK(x,numeric)
379         END_TYPECHECK(tan(x)) // -> numeric tan(numeric)
380         
381         return tan(ex_to_numeric(x));
382 }
383
384 static ex tan_eval(const ex & x)
385 {
386         // tan(n/d*Pi) -> { all known non-nested radicals }
387         ex SixtyExOverPi = _ex60()*x/Pi;
388         ex sign = _ex1();
389         if (SixtyExOverPi.info(info_flags::integer)) {
390                 numeric z = mod(ex_to_numeric(SixtyExOverPi),_num60());
391                 if (z>=_num60()) {
392                         // wrap to interval [0, Pi)
393                         z -= _num60();
394                 }
395                 if (z>=_num30()) {
396                         // wrap to interval [0, Pi/2)
397                         z = _num60()-z;
398                         sign = _ex_1();
399                 }
400                 if (z.is_equal(_num0()))  // tan(0)       -> 0
401                         return _ex0();
402                 if (z.is_equal(_num5()))  // tan(Pi/12)   -> 2-sqrt(3)
403                         return sign*(_ex2()-power(_ex3(),_ex1_2()));
404                 if (z.is_equal(_num10())) // tan(Pi/6)    -> sqrt(3)/3
405                         return sign*_ex1_3()*power(_ex3(),_ex1_2());
406                 if (z.is_equal(_num15())) // tan(Pi/4)    -> 1
407                         return sign*_ex1();
408                 if (z.is_equal(_num20())) // tan(Pi/3)    -> sqrt(3)
409                         return sign*power(_ex3(),_ex1_2());
410                 if (z.is_equal(_num25())) // tan(5/12*Pi) -> 2+sqrt(3)
411                         return sign*(power(_ex3(),_ex1_2())+_ex2());
412                 if (z.is_equal(_num30())) // tan(Pi/2)    -> infinity
413                         throw (pole_error("tan_eval(): simple pole",1));
414         }
415         
416         if (is_ex_exactly_of_type(x, function)) {
417                 ex t = x.op(0);
418                 // tan(atan(x)) -> x
419                 if (is_ex_the_function(x, atan))
420                         return t;
421                 // tan(asin(x)) -> x*(1+x^2)^(-1/2)
422                 if (is_ex_the_function(x, asin))
423                         return t*power(_ex1()-power(t,_ex2()),_ex_1_2());
424                 // tan(acos(x)) -> (1-x^2)^(1/2)/x
425                 if (is_ex_the_function(x, acos))
426                         return power(t,_ex_1())*power(_ex1()-power(t,_ex2()),_ex1_2());
427         }
428         
429         // tan(float) -> float
430         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
431                 return tan_evalf(x);
432         }
433         
434         return tan(x).hold();
435 }
436
437 static ex tan_deriv(const ex & x, unsigned deriv_param)
438 {
439         GINAC_ASSERT(deriv_param==0);
440         
441         // d/dx tan(x) -> 1+tan(x)^2;
442         return (_ex1()+power(tan(x),_ex2()));
443 }
444
445 static ex tan_series(const ex &x,
446                                          const relational &rel,
447                                          int order,
448                                          unsigned options)
449 {
450         // method:
451         // Taylor series where there is no pole falls back to tan_deriv.
452         // On a pole simply expand sin(x)/cos(x).
453         const ex x_pt = x.subs(rel);
454         if (!(2*x_pt/Pi).info(info_flags::odd))
455                 throw do_taylor();  // caught by function::series()
456         // if we got here we have to care for a simple pole
457         return (sin(x)/cos(x)).series(rel, order+2);
458 }
459
460 REGISTER_FUNCTION(tan, eval_func(tan_eval).
461                                            evalf_func(tan_evalf).
462                                            derivative_func(tan_deriv).
463                                            series_func(tan_series));
464
465 //////////
466 // inverse sine (arc sine)
467 //////////
468
469 static ex asin_evalf(const ex & x)
470 {
471         BEGIN_TYPECHECK
472            TYPECHECK(x,numeric)
473         END_TYPECHECK(asin(x))
474         
475         return asin(ex_to_numeric(x)); // -> numeric asin(numeric)
476 }
477
478 static ex asin_eval(const ex & x)
479 {
480         if (x.info(info_flags::numeric)) {
481                 // asin(0) -> 0
482                 if (x.is_zero())
483                         return x;
484                 // asin(1/2) -> Pi/6
485                 if (x.is_equal(_ex1_2()))
486                         return numeric(1,6)*Pi;
487                 // asin(1) -> Pi/2
488                 if (x.is_equal(_ex1()))
489                         return _num1_2()*Pi;
490                 // asin(-1/2) -> -Pi/6
491                 if (x.is_equal(_ex_1_2()))
492                         return numeric(-1,6)*Pi;
493                 // asin(-1) -> -Pi/2
494                 if (x.is_equal(_ex_1()))
495                         return _num_1_2()*Pi;
496                 // asin(float) -> float
497                 if (!x.info(info_flags::crational))
498                         return asin_evalf(x);
499         }
500         
501         return asin(x).hold();
502 }
503
504 static ex asin_deriv(const ex & x, unsigned deriv_param)
505 {
506         GINAC_ASSERT(deriv_param==0);
507         
508         // d/dx asin(x) -> 1/sqrt(1-x^2)
509         return power(1-power(x,_ex2()),_ex_1_2());
510 }
511
512 REGISTER_FUNCTION(asin, eval_func(asin_eval).
513                                                 evalf_func(asin_evalf).
514                                                 derivative_func(asin_deriv));
515
516 //////////
517 // inverse cosine (arc cosine)
518 //////////
519
520 static ex acos_evalf(const ex & x)
521 {
522         BEGIN_TYPECHECK
523            TYPECHECK(x,numeric)
524         END_TYPECHECK(acos(x))
525         
526         return acos(ex_to_numeric(x)); // -> numeric acos(numeric)
527 }
528
529 static ex acos_eval(const ex & x)
530 {
531         if (x.info(info_flags::numeric)) {
532                 // acos(1) -> 0
533                 if (x.is_equal(_ex1()))
534                         return _ex0();
535                 // acos(1/2) -> Pi/3
536                 if (x.is_equal(_ex1_2()))
537                         return _ex1_3()*Pi;
538                 // acos(0) -> Pi/2
539                 if (x.is_zero())
540                         return _ex1_2()*Pi;
541                 // acos(-1/2) -> 2/3*Pi
542                 if (x.is_equal(_ex_1_2()))
543                         return numeric(2,3)*Pi;
544                 // acos(-1) -> Pi
545                 if (x.is_equal(_ex_1()))
546                         return Pi;
547                 // acos(float) -> float
548                 if (!x.info(info_flags::crational))
549                         return acos_evalf(x);
550         }
551         
552         return acos(x).hold();
553 }
554
555 static ex acos_deriv(const ex & x, unsigned deriv_param)
556 {
557         GINAC_ASSERT(deriv_param==0);
558         
559         // d/dx acos(x) -> -1/sqrt(1-x^2)
560         return _ex_1()*power(1-power(x,_ex2()),_ex_1_2());
561 }
562
563 REGISTER_FUNCTION(acos, eval_func(acos_eval).
564                                                 evalf_func(acos_evalf).
565                                                 derivative_func(acos_deriv));
566
567 //////////
568 // inverse tangent (arc tangent)
569 //////////
570
571 static ex atan_evalf(const ex & x)
572 {
573         BEGIN_TYPECHECK
574                 TYPECHECK(x,numeric)
575         END_TYPECHECK(atan(x))
576         
577         return atan(ex_to_numeric(x)); // -> numeric atan(numeric)
578 }
579
580 static ex atan_eval(const ex & x)
581 {
582         if (x.info(info_flags::numeric)) {
583                 // atan(0) -> 0
584                 if (x.is_equal(_ex0()))
585                         return _ex0();
586                 // atan(float) -> float
587                 if (!x.info(info_flags::crational))
588                         return atan_evalf(x);
589         }
590         
591         return atan(x).hold();
592 }    
593
594 static ex atan_deriv(const ex & x, unsigned deriv_param)
595 {
596         GINAC_ASSERT(deriv_param==0);
597
598         // d/dx atan(x) -> 1/(1+x^2)
599         return power(_ex1()+power(x,_ex2()), _ex_1());
600 }
601
602 REGISTER_FUNCTION(atan, eval_func(atan_eval).
603                                                 evalf_func(atan_evalf).
604                                                 derivative_func(atan_deriv));
605
606 //////////
607 // inverse tangent (atan2(y,x))
608 //////////
609
610 static ex atan2_evalf(const ex & y, const ex & x)
611 {
612         BEGIN_TYPECHECK
613                 TYPECHECK(y,numeric)
614                 TYPECHECK(x,numeric)
615         END_TYPECHECK(atan2(y,x))
616         
617         return atan(ex_to_numeric(y),ex_to_numeric(x)); // -> numeric atan(numeric)
618 }
619
620 static ex atan2_eval(const ex & y, const ex & x)
621 {
622         if (y.info(info_flags::numeric) && !y.info(info_flags::crational) &&
623                 x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
624                 return atan2_evalf(y,x);
625         }
626         
627         return atan2(y,x).hold();
628 }    
629
630 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
631 {
632         GINAC_ASSERT(deriv_param<2);
633         
634         if (deriv_param==0) {
635                 // d/dy atan(y,x)
636                 return x*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
637         }
638         // d/dx atan(y,x)
639         return -y*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
640 }
641
642 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
643                                                  evalf_func(atan2_evalf).
644                                                  derivative_func(atan2_deriv));
645
646 //////////
647 // hyperbolic sine (trigonometric function)
648 //////////
649
650 static ex sinh_evalf(const ex & x)
651 {
652         BEGIN_TYPECHECK
653            TYPECHECK(x,numeric)
654         END_TYPECHECK(sinh(x))
655         
656         return sinh(ex_to_numeric(x)); // -> numeric sinh(numeric)
657 }
658
659 static ex sinh_eval(const ex & x)
660 {
661         if (x.info(info_flags::numeric)) {
662                 if (x.is_zero())  // sinh(0) -> 0
663                         return _ex0();        
664                 if (!x.info(info_flags::crational))  // sinh(float) -> float
665                         return sinh_evalf(x);
666         }
667         
668         if ((x/Pi).info(info_flags::numeric) &&
669                 ex_to_numeric(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
670                 return I*sin(x/I);
671         
672         if (is_ex_exactly_of_type(x, function)) {
673                 ex t = x.op(0);
674                 // sinh(asinh(x)) -> x
675                 if (is_ex_the_function(x, asinh))
676                         return t;
677                 // sinh(acosh(x)) -> (x-1)^(1/2) * (x+1)^(1/2)
678                 if (is_ex_the_function(x, acosh))
679                         return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2());
680                 // sinh(atanh(x)) -> x*(1-x^2)^(-1/2)
681                 if (is_ex_the_function(x, atanh))
682                         return t*power(_ex1()-power(t,_ex2()),_ex_1_2());
683         }
684         
685         return sinh(x).hold();
686 }
687
688 static ex sinh_deriv(const ex & x, unsigned deriv_param)
689 {
690         GINAC_ASSERT(deriv_param==0);
691         
692         // d/dx sinh(x) -> cosh(x)
693         return cosh(x);
694 }
695
696 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
697                                                 evalf_func(sinh_evalf).
698                                                 derivative_func(sinh_deriv));
699
700 //////////
701 // hyperbolic cosine (trigonometric function)
702 //////////
703
704 static ex cosh_evalf(const ex & x)
705 {
706         BEGIN_TYPECHECK
707            TYPECHECK(x,numeric)
708         END_TYPECHECK(cosh(x))
709         
710         return cosh(ex_to_numeric(x)); // -> numeric cosh(numeric)
711 }
712
713 static ex cosh_eval(const ex & x)
714 {
715         if (x.info(info_flags::numeric)) {
716                 if (x.is_zero())  // cosh(0) -> 1
717                         return _ex1();
718                 if (!x.info(info_flags::crational))  // cosh(float) -> float
719                         return cosh_evalf(x);
720         }
721         
722         if ((x/Pi).info(info_flags::numeric) &&
723                 ex_to_numeric(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
724                 return cos(x/I);
725         
726         if (is_ex_exactly_of_type(x, function)) {
727                 ex t = x.op(0);
728                 // cosh(acosh(x)) -> x
729                 if (is_ex_the_function(x, acosh))
730                         return t;
731                 // cosh(asinh(x)) -> (1+x^2)^(1/2)
732                 if (is_ex_the_function(x, asinh))
733                         return power(_ex1()+power(t,_ex2()),_ex1_2());
734                 // cosh(atanh(x)) -> (1-x^2)^(-1/2)
735                 if (is_ex_the_function(x, atanh))
736                         return power(_ex1()-power(t,_ex2()),_ex_1_2());
737         }
738         
739         return cosh(x).hold();
740 }
741
742 static ex cosh_deriv(const ex & x, unsigned deriv_param)
743 {
744         GINAC_ASSERT(deriv_param==0);
745         
746         // d/dx cosh(x) -> sinh(x)
747         return sinh(x);
748 }
749
750 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
751                                                 evalf_func(cosh_evalf).
752                                                 derivative_func(cosh_deriv));
753
754
755 //////////
756 // hyperbolic tangent (trigonometric function)
757 //////////
758
759 static ex tanh_evalf(const ex & x)
760 {
761         BEGIN_TYPECHECK
762            TYPECHECK(x,numeric)
763         END_TYPECHECK(tanh(x))
764         
765         return tanh(ex_to_numeric(x)); // -> numeric tanh(numeric)
766 }
767
768 static ex tanh_eval(const ex & x)
769 {
770         if (x.info(info_flags::numeric)) {
771                 if (x.is_zero())  // tanh(0) -> 0
772                         return _ex0();
773                 if (!x.info(info_flags::crational))  // tanh(float) -> float
774                         return tanh_evalf(x);
775         }
776         
777         if ((x/Pi).info(info_flags::numeric) &&
778                 ex_to_numeric(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
779                 return I*tan(x/I);
780         
781         if (is_ex_exactly_of_type(x, function)) {
782                 ex t = x.op(0);
783                 // tanh(atanh(x)) -> x
784                 if (is_ex_the_function(x, atanh))
785                         return t;
786                 // tanh(asinh(x)) -> x*(1+x^2)^(-1/2)
787                 if (is_ex_the_function(x, asinh))
788                         return t*power(_ex1()+power(t,_ex2()),_ex_1_2());
789                 // tanh(acosh(x)) -> (x-1)^(1/2)*(x+1)^(1/2)/x
790                 if (is_ex_the_function(x, acosh))
791                         return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2())*power(t,_ex_1());
792         }
793         
794         return tanh(x).hold();
795 }
796
797 static ex tanh_deriv(const ex & x, unsigned deriv_param)
798 {
799         GINAC_ASSERT(deriv_param==0);
800         
801         // d/dx tanh(x) -> 1-tanh(x)^2
802         return _ex1()-power(tanh(x),_ex2());
803 }
804
805 static ex tanh_series(const ex &x,
806                                           const relational &rel,
807                                           int order,
808                                           unsigned options)
809 {
810         // method:
811         // Taylor series where there is no pole falls back to tanh_deriv.
812         // On a pole simply expand sinh(x)/cosh(x).
813         const ex x_pt = x.subs(rel);
814         if (!(2*I*x_pt/Pi).info(info_flags::odd))
815                 throw do_taylor();  // caught by function::series()
816         // if we got here we have to care for a simple pole
817         return (sinh(x)/cosh(x)).series(rel, order+2);
818 }
819
820 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
821                                                 evalf_func(tanh_evalf).
822                                                 derivative_func(tanh_deriv).
823                                                 series_func(tanh_series));
824
825 //////////
826 // inverse hyperbolic sine (trigonometric function)
827 //////////
828
829 static ex asinh_evalf(const ex & x)
830 {
831         BEGIN_TYPECHECK
832            TYPECHECK(x,numeric)
833         END_TYPECHECK(asinh(x))
834         
835         return asinh(ex_to_numeric(x)); // -> numeric asinh(numeric)
836 }
837
838 static ex asinh_eval(const ex & x)
839 {
840         if (x.info(info_flags::numeric)) {
841                 // asinh(0) -> 0
842                 if (x.is_zero())
843                         return _ex0();
844                 // asinh(float) -> float
845                 if (!x.info(info_flags::crational))
846                         return asinh_evalf(x);
847         }
848         
849         return asinh(x).hold();
850 }
851
852 static ex asinh_deriv(const ex & x, unsigned deriv_param)
853 {
854         GINAC_ASSERT(deriv_param==0);
855         
856         // d/dx asinh(x) -> 1/sqrt(1+x^2)
857         return power(_ex1()+power(x,_ex2()),_ex_1_2());
858 }
859
860 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
861                                                  evalf_func(asinh_evalf).
862                                                  derivative_func(asinh_deriv));
863
864 //////////
865 // inverse hyperbolic cosine (trigonometric function)
866 //////////
867
868 static ex acosh_evalf(const ex & x)
869 {
870         BEGIN_TYPECHECK
871            TYPECHECK(x,numeric)
872         END_TYPECHECK(acosh(x))
873         
874         return acosh(ex_to_numeric(x)); // -> numeric acosh(numeric)
875 }
876
877 static ex acosh_eval(const ex & x)
878 {
879         if (x.info(info_flags::numeric)) {
880                 // acosh(0) -> Pi*I/2
881                 if (x.is_zero())
882                         return Pi*I*numeric(1,2);
883                 // acosh(1) -> 0
884                 if (x.is_equal(_ex1()))
885                         return _ex0();
886                 // acosh(-1) -> Pi*I
887                 if (x.is_equal(_ex_1()))
888                         return Pi*I;
889                 // acosh(float) -> float
890                 if (!x.info(info_flags::crational))
891                         return acosh_evalf(x);
892         }
893         
894         return acosh(x).hold();
895 }
896
897 static ex acosh_deriv(const ex & x, unsigned deriv_param)
898 {
899         GINAC_ASSERT(deriv_param==0);
900         
901         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
902         return power(x+_ex_1(),_ex_1_2())*power(x+_ex1(),_ex_1_2());
903 }
904
905 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
906                                                  evalf_func(acosh_evalf).
907                                                  derivative_func(acosh_deriv));
908
909 //////////
910 // inverse hyperbolic tangent (trigonometric function)
911 //////////
912
913 static ex atanh_evalf(const ex & x)
914 {
915         BEGIN_TYPECHECK
916            TYPECHECK(x,numeric)
917         END_TYPECHECK(atanh(x))
918         
919         return atanh(ex_to_numeric(x)); // -> numeric atanh(numeric)
920 }
921
922 static ex atanh_eval(const ex & x)
923 {
924         if (x.info(info_flags::numeric)) {
925                 // atanh(0) -> 0
926                 if (x.is_zero())
927                         return _ex0();
928                 // atanh({+|-}1) -> throw
929                 if (x.is_equal(_ex1()) || x.is_equal(_ex_1()))
930                         throw (pole_error("atanh_eval(): logarithmic pole",0));
931                 // atanh(float) -> float
932                 if (!x.info(info_flags::crational))
933                         return atanh_evalf(x);
934         }
935         
936         return atanh(x).hold();
937 }
938
939 static ex atanh_deriv(const ex & x, unsigned deriv_param)
940 {
941         GINAC_ASSERT(deriv_param==0);
942         
943         // d/dx atanh(x) -> 1/(1-x^2)
944         return power(_ex1()-power(x,_ex2()),_ex_1());
945 }
946
947 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
948                                                  evalf_func(atanh_evalf).
949                                                  derivative_func(atanh_deriv));
950
951 #ifndef NO_NAMESPACE_GINAC
952 } // namespace GiNaC
953 #endif // ndef NO_NAMESPACE_GINAC