* library_init::library_init(): It suffices to static_cast<> the flyweights
[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) 1999-2004 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 "operators.h"
33 #include "relational.h"
34 #include "symbol.h"
35 #include "pseries.h"
36 #include "utils.h"
37
38 namespace GiNaC {
39
40 //////////
41 // exponential function
42 //////////
43
44 static ex exp_evalf(const ex & x)
45 {
46         if (is_exactly_a<numeric>(x))
47                 return exp(ex_to<numeric>(x));
48         
49         return exp(x).hold();
50 }
51
52 static ex exp_eval(const ex & x)
53 {
54         // exp(0) -> 1
55         if (x.is_zero()) {
56                 return _ex1;
57         }
58
59         // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
60         const ex TwoExOverPiI=(_ex2*x)/(Pi*I);
61         if (TwoExOverPiI.info(info_flags::integer)) {
62                 const numeric z = mod(ex_to<numeric>(TwoExOverPiI),_num4);
63                 if (z.is_equal(_num0))
64                         return _ex1;
65                 if (z.is_equal(_num1))
66                         return ex(I);
67                 if (z.is_equal(_num2))
68                         return _ex_1;
69                 if (z.is_equal(_num3))
70                         return ex(-I);
71         }
72
73         // exp(log(x)) -> x
74         if (is_ex_the_function(x, log))
75                 return x.op(0);
76         
77         // exp(float) -> float
78         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
79                 return exp(ex_to<numeric>(x));
80         
81         return exp(x).hold();
82 }
83
84 static ex exp_deriv(const ex & x, unsigned deriv_param)
85 {
86         GINAC_ASSERT(deriv_param==0);
87
88         // d/dx exp(x) -> exp(x)
89         return exp(x);
90 }
91
92 REGISTER_FUNCTION(exp, eval_func(exp_eval).
93                        evalf_func(exp_evalf).
94                        derivative_func(exp_deriv).
95                        latex_name("\\exp"));
96
97 //////////
98 // natural logarithm
99 //////////
100
101 static ex log_evalf(const ex & x)
102 {
103         if (is_exactly_a<numeric>(x))
104                 return log(ex_to<numeric>(x));
105         
106         return log(x).hold();
107 }
108
109 static ex log_eval(const ex & x)
110 {
111         if (x.info(info_flags::numeric)) {
112                 if (x.is_zero())         // 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
123                 // log(float) -> float
124                 if (!x.info(info_flags::crational))
125                         return log(ex_to<numeric>(x));
126         }
127
128         // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
129         if (is_ex_the_function(x, exp)) {
130                 const ex &t = x.op(0);
131                 if (is_a<symbol>(t) && t.info(info_flags::real)) {
132                         return t;
133                 }
134                 if (t.info(info_flags::numeric)) {
135                         const numeric &nt = ex_to<numeric>(t);
136                         if (nt.is_real())
137                                 return t;
138                 }
139         }
140         
141         return log(x).hold();
142 }
143
144 static ex log_deriv(const ex & x, unsigned deriv_param)
145 {
146         GINAC_ASSERT(deriv_param==0);
147         
148         // d/dx log(x) -> 1/x
149         return power(x, _ex_1);
150 }
151
152 static ex log_series(const ex &arg,
153                      const relational &rel,
154                      int order,
155                      unsigned options)
156 {
157         GINAC_ASSERT(is_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, subs_options::no_pattern);
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, subs_options::no_pattern);
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
282                 // sin(asin(x)) -> x
283                 if (is_ex_the_function(x, asin))
284                         return t;
285
286                 // sin(acos(x)) -> sqrt(1-x^2)
287                 if (is_ex_the_function(x, acos))
288                         return sqrt(_ex1-power(t,_ex2));
289
290                 // sin(atan(x)) -> x/sqrt(1+x^2)
291                 if (is_ex_the_function(x, atan))
292                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
293         }
294         
295         // sin(float) -> float
296         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
297                 return sin(ex_to<numeric>(x));
298
299         // sin() is odd
300         if (x.info(info_flags::negative))
301                 return -sin(-x);
302         
303         return sin(x).hold();
304 }
305
306 static ex sin_deriv(const ex & x, unsigned deriv_param)
307 {
308         GINAC_ASSERT(deriv_param==0);
309         
310         // d/dx sin(x) -> cos(x)
311         return cos(x);
312 }
313
314 REGISTER_FUNCTION(sin, eval_func(sin_eval).
315                        evalf_func(sin_evalf).
316                        derivative_func(sin_deriv).
317                        latex_name("\\sin"));
318
319 //////////
320 // cosine (trigonometric function)
321 //////////
322
323 static ex cos_evalf(const ex & x)
324 {
325         if (is_exactly_a<numeric>(x))
326                 return cos(ex_to<numeric>(x));
327         
328         return cos(x).hold();
329 }
330
331 static ex cos_eval(const ex & x)
332 {
333         // cos(n/d*Pi) -> { all known non-nested radicals }
334         const ex SixtyExOverPi = _ex60*x/Pi;
335         ex sign = _ex1;
336         if (SixtyExOverPi.info(info_flags::integer)) {
337                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num120);
338                 if (z>=_num60) {
339                         // wrap to interval [0, Pi)
340                         z = _num120-z;
341                 }
342                 if (z>=_num30) {
343                         // wrap to interval [0, Pi/2)
344                         z = _num60-z;
345                         sign = _ex_1;
346                 }
347                 if (z.is_equal(_num0))  // cos(0)       -> 1
348                         return sign;
349                 if (z.is_equal(_num5))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
350                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
351                 if (z.is_equal(_num10)) // cos(Pi/6)    -> sqrt(3)/2
352                         return sign*_ex1_2*sqrt(_ex3);
353                 if (z.is_equal(_num12)) // cos(Pi/5)    -> sqrt(5)/4+1/4
354                         return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
355                 if (z.is_equal(_num15)) // cos(Pi/4)    -> sqrt(2)/2
356                         return sign*_ex1_2*sqrt(_ex2);
357                 if (z.is_equal(_num20)) // cos(Pi/3)    -> 1/2
358                         return sign*_ex1_2;
359                 if (z.is_equal(_num24)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
360                         return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
361                 if (z.is_equal(_num25)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
362                         return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
363                 if (z.is_equal(_num30)) // cos(Pi/2)    -> 0
364                         return _ex0;
365         }
366
367         if (is_exactly_a<function>(x)) {
368                 const ex &t = x.op(0);
369
370                 // cos(acos(x)) -> x
371                 if (is_ex_the_function(x, acos))
372                         return t;
373
374                 // cos(asin(x)) -> sqrt(1-x^2)
375                 if (is_ex_the_function(x, asin))
376                         return sqrt(_ex1-power(t,_ex2));
377
378                 // cos(atan(x)) -> 1/sqrt(1+x^2)
379                 if (is_ex_the_function(x, atan))
380                         return power(_ex1+power(t,_ex2),_ex_1_2);
381         }
382         
383         // cos(float) -> float
384         if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
385                 return cos(ex_to<numeric>(x));
386         
387         // cos() is even
388         if (x.info(info_flags::negative))
389                 return cos(-x);
390         
391         return cos(x).hold();
392 }
393
394 static ex cos_deriv(const ex & x, unsigned deriv_param)
395 {
396         GINAC_ASSERT(deriv_param==0);
397
398         // d/dx cos(x) -> -sin(x)
399         return -sin(x);
400 }
401
402 REGISTER_FUNCTION(cos, eval_func(cos_eval).
403                        evalf_func(cos_evalf).
404                        derivative_func(cos_deriv).
405                        latex_name("\\cos"));
406
407 //////////
408 // tangent (trigonometric function)
409 //////////
410
411 static ex tan_evalf(const ex & x)
412 {
413         if (is_exactly_a<numeric>(x))
414                 return tan(ex_to<numeric>(x));
415         
416         return tan(x).hold();
417 }
418
419 static ex tan_eval(const ex & x)
420 {
421         // tan(n/d*Pi) -> { all known non-nested radicals }
422         const ex SixtyExOverPi = _ex60*x/Pi;
423         ex sign = _ex1;
424         if (SixtyExOverPi.info(info_flags::integer)) {
425                 numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num60);
426                 if (z>=_num60) {
427                         // wrap to interval [0, Pi)
428                         z -= _num60;
429                 }
430                 if (z>=_num30) {
431                         // wrap to interval [0, Pi/2)
432                         z = _num60-z;
433                         sign = _ex_1;
434                 }
435                 if (z.is_equal(_num0))  // tan(0)       -> 0
436                         return _ex0;
437                 if (z.is_equal(_num5))  // tan(Pi/12)   -> 2-sqrt(3)
438                         return sign*(_ex2-sqrt(_ex3));
439                 if (z.is_equal(_num10)) // tan(Pi/6)    -> sqrt(3)/3
440                         return sign*_ex1_3*sqrt(_ex3);
441                 if (z.is_equal(_num15)) // tan(Pi/4)    -> 1
442                         return sign;
443                 if (z.is_equal(_num20)) // tan(Pi/3)    -> sqrt(3)
444                         return sign*sqrt(_ex3);
445                 if (z.is_equal(_num25)) // tan(5/12*Pi) -> 2+sqrt(3)
446                         return sign*(sqrt(_ex3)+_ex2);
447                 if (z.is_equal(_num30)) // tan(Pi/2)    -> infinity
448                         throw (pole_error("tan_eval(): simple pole",1));
449         }
450
451         if (is_exactly_a<function>(x)) {
452                 const ex &t = x.op(0);
453
454                 // tan(atan(x)) -> x
455                 if (is_ex_the_function(x, atan))
456                         return t;
457
458                 // tan(asin(x)) -> x/sqrt(1+x^2)
459                 if (is_ex_the_function(x, asin))
460                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
461
462                 // tan(acos(x)) -> sqrt(1-x^2)/x
463                 if (is_ex_the_function(x, acos))
464                         return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
465         }
466         
467         // tan(float) -> float
468         if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
469                 return tan(ex_to<numeric>(x));
470         }
471         
472         // tan() is odd
473         if (x.info(info_flags::negative))
474                 return -tan(-x);
475         
476         return tan(x).hold();
477 }
478
479 static ex tan_deriv(const ex & x, unsigned deriv_param)
480 {
481         GINAC_ASSERT(deriv_param==0);
482         
483         // d/dx tan(x) -> 1+tan(x)^2;
484         return (_ex1+power(tan(x),_ex2));
485 }
486
487 static ex tan_series(const ex &x,
488                      const relational &rel,
489                      int order,
490                      unsigned options)
491 {
492         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
493         // method:
494         // Taylor series where there is no pole falls back to tan_deriv.
495         // On a pole simply expand sin(x)/cos(x).
496         const ex x_pt = x.subs(rel, subs_options::no_pattern);
497         if (!(2*x_pt/Pi).info(info_flags::odd))
498                 throw do_taylor();  // caught by function::series()
499         // if we got here we have to care for a simple pole
500         return (sin(x)/cos(x)).series(rel, order, options);
501 }
502
503 REGISTER_FUNCTION(tan, eval_func(tan_eval).
504                        evalf_func(tan_evalf).
505                        derivative_func(tan_deriv).
506                        series_func(tan_series).
507                        latex_name("\\tan"));
508
509 //////////
510 // inverse sine (arc sine)
511 //////////
512
513 static ex asin_evalf(const ex & x)
514 {
515         if (is_exactly_a<numeric>(x))
516                 return asin(ex_to<numeric>(x));
517         
518         return asin(x).hold();
519 }
520
521 static ex asin_eval(const ex & x)
522 {
523         if (x.info(info_flags::numeric)) {
524
525                 // asin(0) -> 0
526                 if (x.is_zero())
527                         return x;
528
529                 // asin(1/2) -> Pi/6
530                 if (x.is_equal(_ex1_2))
531                         return numeric(1,6)*Pi;
532
533                 // asin(1) -> Pi/2
534                 if (x.is_equal(_ex1))
535                         return _num1_2*Pi;
536
537                 // asin(-1/2) -> -Pi/6
538                 if (x.is_equal(_ex_1_2))
539                         return numeric(-1,6)*Pi;
540
541                 // asin(-1) -> -Pi/2
542                 if (x.is_equal(_ex_1))
543                         return _num_1_2*Pi;
544
545                 // asin(float) -> float
546                 if (!x.info(info_flags::crational))
547                         return asin(ex_to<numeric>(x));
548
549                 // asin() is odd
550                 if (x.info(info_flags::negative))
551                         return -asin(-x);
552         }
553         
554         return asin(x).hold();
555 }
556
557 static ex asin_deriv(const ex & x, unsigned deriv_param)
558 {
559         GINAC_ASSERT(deriv_param==0);
560         
561         // d/dx asin(x) -> 1/sqrt(1-x^2)
562         return power(1-power(x,_ex2),_ex_1_2);
563 }
564
565 REGISTER_FUNCTION(asin, eval_func(asin_eval).
566                         evalf_func(asin_evalf).
567                         derivative_func(asin_deriv).
568                         latex_name("\\arcsin"));
569
570 //////////
571 // inverse cosine (arc cosine)
572 //////////
573
574 static ex acos_evalf(const ex & x)
575 {
576         if (is_exactly_a<numeric>(x))
577                 return acos(ex_to<numeric>(x));
578         
579         return acos(x).hold();
580 }
581
582 static ex acos_eval(const ex & x)
583 {
584         if (x.info(info_flags::numeric)) {
585
586                 // acos(1) -> 0
587                 if (x.is_equal(_ex1))
588                         return _ex0;
589
590                 // acos(1/2) -> Pi/3
591                 if (x.is_equal(_ex1_2))
592                         return _ex1_3*Pi;
593
594                 // acos(0) -> Pi/2
595                 if (x.is_zero())
596                         return _ex1_2*Pi;
597
598                 // acos(-1/2) -> 2/3*Pi
599                 if (x.is_equal(_ex_1_2))
600                         return numeric(2,3)*Pi;
601
602                 // acos(-1) -> Pi
603                 if (x.is_equal(_ex_1))
604                         return Pi;
605
606                 // acos(float) -> float
607                 if (!x.info(info_flags::crational))
608                         return acos(ex_to<numeric>(x));
609
610                 // acos(-x) -> Pi-acos(x)
611                 if (x.info(info_flags::negative))
612                         return Pi-acos(-x);
613         }
614         
615         return acos(x).hold();
616 }
617
618 static ex acos_deriv(const ex & x, unsigned deriv_param)
619 {
620         GINAC_ASSERT(deriv_param==0);
621         
622         // d/dx acos(x) -> -1/sqrt(1-x^2)
623         return -power(1-power(x,_ex2),_ex_1_2);
624 }
625
626 REGISTER_FUNCTION(acos, eval_func(acos_eval).
627                         evalf_func(acos_evalf).
628                         derivative_func(acos_deriv).
629                         latex_name("\\arccos"));
630
631 //////////
632 // inverse tangent (arc tangent)
633 //////////
634
635 static ex atan_evalf(const ex & x)
636 {
637         if (is_exactly_a<numeric>(x))
638                 return atan(ex_to<numeric>(x));
639         
640         return atan(x).hold();
641 }
642
643 static ex atan_eval(const ex & x)
644 {
645         if (x.info(info_flags::numeric)) {
646
647                 // atan(0) -> 0
648                 if (x.is_zero())
649                         return _ex0;
650
651                 // atan(1) -> Pi/4
652                 if (x.is_equal(_ex1))
653                         return _ex1_4*Pi;
654
655                 // atan(-1) -> -Pi/4
656                 if (x.is_equal(_ex_1))
657                         return _ex_1_4*Pi;
658
659                 if (x.is_equal(I) || x.is_equal(-I))
660                         throw (pole_error("atan_eval(): logarithmic pole",0));
661
662                 // atan(float) -> float
663                 if (!x.info(info_flags::crational))
664                         return atan(ex_to<numeric>(x));
665
666                 // atan() is odd
667                 if (x.info(info_flags::negative))
668                         return -atan(-x);
669         }
670         
671         return atan(x).hold();
672 }
673
674 static ex atan_deriv(const ex & x, unsigned deriv_param)
675 {
676         GINAC_ASSERT(deriv_param==0);
677
678         // d/dx atan(x) -> 1/(1+x^2)
679         return power(_ex1+power(x,_ex2), _ex_1);
680 }
681
682 static ex atan_series(const ex &arg,
683                       const relational &rel,
684                       int order,
685                       unsigned options)
686 {
687         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
688         // method:
689         // Taylor series where there is no pole or cut falls back to atan_deriv.
690         // There are two branch cuts, one runnig from I up the imaginary axis and
691         // one running from -I down the imaginary axis.  The points I and -I are
692         // poles.
693         // On the branch cuts and the poles series expand
694         //     (log(1+I*x)-log(1-I*x))/(2*I)
695         // instead.
696         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
697         if (!(I*arg_pt).info(info_flags::real))
698                 throw do_taylor();     // Re(x) != 0
699         if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
700                 throw do_taylor();     // Re(x) == 0, but abs(x)<1
701         // care for the poles, using the defining formula for atan()...
702         if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
703                 return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
704         if (!(options & series_options::suppress_branchcut)) {
705                 // method:
706                 // This is the branch cut: assemble the primitive series manually and
707                 // then add the corresponding complex step function.
708                 const symbol &s = ex_to<symbol>(rel.lhs());
709                 const ex &point = rel.rhs();
710                 const symbol foo;
711                 const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
712                 ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
713                 if ((I*arg_pt)<_ex0)
714                         Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
715                 else
716                         Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
717                 epvector seq;
718                 seq.push_back(expair(Order0correction, _ex0));
719                 seq.push_back(expair(Order(_ex1), order));
720                 return series(replarg - pseries(rel, seq), rel, order);
721         }
722         throw do_taylor();
723 }
724
725 REGISTER_FUNCTION(atan, eval_func(atan_eval).
726                         evalf_func(atan_evalf).
727                         derivative_func(atan_deriv).
728                         series_func(atan_series).
729                         latex_name("\\arctan"));
730
731 //////////
732 // inverse tangent (atan2(y,x))
733 //////////
734
735 static ex atan2_evalf(const ex &y, const ex &x)
736 {
737         if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
738                 return atan(ex_to<numeric>(y), ex_to<numeric>(x));
739         
740         return atan2(y, x).hold();
741 }
742
743 static ex atan2_eval(const ex & y, const ex & x)
744 {
745         if (y.info(info_flags::numeric) && x.info(info_flags::numeric)) {
746
747                 if (y.is_zero()) {
748
749                         // atan(0, 0) -> 0
750                         if (x.is_zero())
751                                 return _ex0;
752
753                         // atan(0, x), x real and positive -> 0
754                         if (x.info(info_flags::positive))
755                                 return _ex0;
756
757                         // atan(0, x), x real and negative -> -Pi
758                         if (x.info(info_flags::negative))
759                                 return _ex_1*Pi;
760                 }
761
762                 if (x.is_zero()) {
763
764                         // atan(y, 0), y real and positive -> Pi/2
765                         if (y.info(info_flags::positive))
766                                 return _ex1_2*Pi;
767
768                         // atan(y, 0), y real and negative -> -Pi/2
769                         if (y.info(info_flags::negative))
770                                 return _ex_1_2*Pi;
771                 }
772
773                 if (y.is_equal(x)) {
774
775                         // atan(y, y), y real and positive -> Pi/4
776                         if (y.info(info_flags::positive))
777                                 return _ex1_4*Pi;
778
779                         // atan(y, y), y real and negative -> -3/4*Pi
780                         if (y.info(info_flags::negative))
781                                 return numeric(-3, 4)*Pi;
782                 }
783
784                 if (y.is_equal(-x)) {
785
786                         // atan(y, -y), y real and positive -> 3*Pi/4
787                         if (y.info(info_flags::positive))
788                                 return numeric(3, 4)*Pi;
789
790                         // atan(y, -y), y real and negative -> -Pi/4
791                         if (y.info(info_flags::negative))
792                                 return _ex_1_4*Pi;
793                 }
794
795                 // atan(float, float) -> float
796                 if (!y.info(info_flags::crational) && !x.info(info_flags::crational))
797                         return atan(ex_to<numeric>(y), ex_to<numeric>(x));
798
799                 // atan(real, real) -> atan(y/x) +/- Pi
800                 if (y.info(info_flags::real) && x.info(info_flags::real)) {
801                         if (x.info(info_flags::positive))
802                                 return atan(y/x);
803                         else if(y.info(info_flags::positive))
804                                 return atan(y/x)+Pi;
805                         else
806                                 return atan(y/x)-Pi;
807                 }
808         }
809
810         return atan2(y, x).hold();
811 }    
812
813 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
814 {
815         GINAC_ASSERT(deriv_param<2);
816         
817         if (deriv_param==0) {
818                 // d/dy atan(y,x)
819                 return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
820         }
821         // d/dx atan(y,x)
822         return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
823 }
824
825 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
826                          evalf_func(atan2_evalf).
827                          derivative_func(atan2_deriv));
828
829 //////////
830 // hyperbolic sine (trigonometric function)
831 //////////
832
833 static ex sinh_evalf(const ex & x)
834 {
835         if (is_exactly_a<numeric>(x))
836                 return sinh(ex_to<numeric>(x));
837         
838         return sinh(x).hold();
839 }
840
841 static ex sinh_eval(const ex & x)
842 {
843         if (x.info(info_flags::numeric)) {
844
845                 // sinh(0) -> 0
846                 if (x.is_zero())
847                         return _ex0;        
848
849                 // sinh(float) -> float
850                 if (!x.info(info_flags::crational))
851                         return sinh(ex_to<numeric>(x));
852
853                 // sinh() is odd
854                 if (x.info(info_flags::negative))
855                         return -sinh(-x);
856         }
857         
858         if ((x/Pi).info(info_flags::numeric) &&
859                 ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
860                 return I*sin(x/I);
861         
862         if (is_exactly_a<function>(x)) {
863                 const ex &t = x.op(0);
864
865                 // sinh(asinh(x)) -> x
866                 if (is_ex_the_function(x, asinh))
867                         return t;
868
869                 // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
870                 if (is_ex_the_function(x, acosh))
871                         return sqrt(t-_ex1)*sqrt(t+_ex1);
872
873                 // sinh(atanh(x)) -> x/sqrt(1-x^2)
874                 if (is_ex_the_function(x, atanh))
875                         return t*power(_ex1-power(t,_ex2),_ex_1_2);
876         }
877         
878         return sinh(x).hold();
879 }
880
881 static ex sinh_deriv(const ex & x, unsigned deriv_param)
882 {
883         GINAC_ASSERT(deriv_param==0);
884         
885         // d/dx sinh(x) -> cosh(x)
886         return cosh(x);
887 }
888
889 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
890                         evalf_func(sinh_evalf).
891                         derivative_func(sinh_deriv).
892                         latex_name("\\sinh"));
893
894 //////////
895 // hyperbolic cosine (trigonometric function)
896 //////////
897
898 static ex cosh_evalf(const ex & x)
899 {
900         if (is_exactly_a<numeric>(x))
901                 return cosh(ex_to<numeric>(x));
902         
903         return cosh(x).hold();
904 }
905
906 static ex cosh_eval(const ex & x)
907 {
908         if (x.info(info_flags::numeric)) {
909
910                 // cosh(0) -> 1
911                 if (x.is_zero())
912                         return _ex1;
913
914                 // cosh(float) -> float
915                 if (!x.info(info_flags::crational))
916                         return cosh(ex_to<numeric>(x));
917
918                 // cosh() is even
919                 if (x.info(info_flags::negative))
920                         return cosh(-x);
921         }
922         
923         if ((x/Pi).info(info_flags::numeric) &&
924                 ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
925                 return cos(x/I);
926         
927         if (is_exactly_a<function>(x)) {
928                 const ex &t = x.op(0);
929
930                 // cosh(acosh(x)) -> x
931                 if (is_ex_the_function(x, acosh))
932                         return t;
933
934                 // cosh(asinh(x)) -> sqrt(1+x^2)
935                 if (is_ex_the_function(x, asinh))
936                         return sqrt(_ex1+power(t,_ex2));
937
938                 // cosh(atanh(x)) -> 1/sqrt(1-x^2)
939                 if (is_ex_the_function(x, atanh))
940                         return power(_ex1-power(t,_ex2),_ex_1_2);
941         }
942         
943         return cosh(x).hold();
944 }
945
946 static ex cosh_deriv(const ex & x, unsigned deriv_param)
947 {
948         GINAC_ASSERT(deriv_param==0);
949         
950         // d/dx cosh(x) -> sinh(x)
951         return sinh(x);
952 }
953
954 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
955                         evalf_func(cosh_evalf).
956                         derivative_func(cosh_deriv).
957                         latex_name("\\cosh"));
958
959 //////////
960 // hyperbolic tangent (trigonometric function)
961 //////////
962
963 static ex tanh_evalf(const ex & x)
964 {
965         if (is_exactly_a<numeric>(x))
966                 return tanh(ex_to<numeric>(x));
967         
968         return tanh(x).hold();
969 }
970
971 static ex tanh_eval(const ex & x)
972 {
973         if (x.info(info_flags::numeric)) {
974
975                 // tanh(0) -> 0
976                 if (x.is_zero())
977                         return _ex0;
978
979                 // tanh(float) -> float
980                 if (!x.info(info_flags::crational))
981                         return tanh(ex_to<numeric>(x));
982
983                 // tanh() is odd
984                 if (x.info(info_flags::negative))
985                         return -tanh(-x);
986         }
987         
988         if ((x/Pi).info(info_flags::numeric) &&
989                 ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
990                 return I*tan(x/I);
991         
992         if (is_exactly_a<function>(x)) {
993                 const ex &t = x.op(0);
994
995                 // tanh(atanh(x)) -> x
996                 if (is_ex_the_function(x, atanh))
997                         return t;
998
999                 // tanh(asinh(x)) -> x/sqrt(1+x^2)
1000                 if (is_ex_the_function(x, asinh))
1001                         return t*power(_ex1+power(t,_ex2),_ex_1_2);
1002
1003                 // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
1004                 if (is_ex_the_function(x, acosh))
1005                         return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
1006         }
1007         
1008         return tanh(x).hold();
1009 }
1010
1011 static ex tanh_deriv(const ex & x, unsigned deriv_param)
1012 {
1013         GINAC_ASSERT(deriv_param==0);
1014         
1015         // d/dx tanh(x) -> 1-tanh(x)^2
1016         return _ex1-power(tanh(x),_ex2);
1017 }
1018
1019 static ex tanh_series(const ex &x,
1020                       const relational &rel,
1021                       int order,
1022                       unsigned options)
1023 {
1024         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1025         // method:
1026         // Taylor series where there is no pole falls back to tanh_deriv.
1027         // On a pole simply expand sinh(x)/cosh(x).
1028         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1029         if (!(2*I*x_pt/Pi).info(info_flags::odd))
1030                 throw do_taylor();  // caught by function::series()
1031         // if we got here we have to care for a simple pole
1032         return (sinh(x)/cosh(x)).series(rel, order, options);
1033 }
1034
1035 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
1036                         evalf_func(tanh_evalf).
1037                         derivative_func(tanh_deriv).
1038                         series_func(tanh_series).
1039                         latex_name("\\tanh"));
1040
1041 //////////
1042 // inverse hyperbolic sine (trigonometric function)
1043 //////////
1044
1045 static ex asinh_evalf(const ex & x)
1046 {
1047         if (is_exactly_a<numeric>(x))
1048                 return asinh(ex_to<numeric>(x));
1049         
1050         return asinh(x).hold();
1051 }
1052
1053 static ex asinh_eval(const ex & x)
1054 {
1055         if (x.info(info_flags::numeric)) {
1056
1057                 // asinh(0) -> 0
1058                 if (x.is_zero())
1059                         return _ex0;
1060
1061                 // asinh(float) -> float
1062                 if (!x.info(info_flags::crational))
1063                         return asinh(ex_to<numeric>(x));
1064
1065                 // asinh() is odd
1066                 if (x.info(info_flags::negative))
1067                         return -asinh(-x);
1068         }
1069         
1070         return asinh(x).hold();
1071 }
1072
1073 static ex asinh_deriv(const ex & x, unsigned deriv_param)
1074 {
1075         GINAC_ASSERT(deriv_param==0);
1076         
1077         // d/dx asinh(x) -> 1/sqrt(1+x^2)
1078         return power(_ex1+power(x,_ex2),_ex_1_2);
1079 }
1080
1081 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
1082                          evalf_func(asinh_evalf).
1083                          derivative_func(asinh_deriv));
1084
1085 //////////
1086 // inverse hyperbolic cosine (trigonometric function)
1087 //////////
1088
1089 static ex acosh_evalf(const ex & x)
1090 {
1091         if (is_exactly_a<numeric>(x))
1092                 return acosh(ex_to<numeric>(x));
1093         
1094         return acosh(x).hold();
1095 }
1096
1097 static ex acosh_eval(const ex & x)
1098 {
1099         if (x.info(info_flags::numeric)) {
1100
1101                 // acosh(0) -> Pi*I/2
1102                 if (x.is_zero())
1103                         return Pi*I*numeric(1,2);
1104
1105                 // acosh(1) -> 0
1106                 if (x.is_equal(_ex1))
1107                         return _ex0;
1108
1109                 // acosh(-1) -> Pi*I
1110                 if (x.is_equal(_ex_1))
1111                         return Pi*I;
1112
1113                 // acosh(float) -> float
1114                 if (!x.info(info_flags::crational))
1115                         return acosh(ex_to<numeric>(x));
1116
1117                 // acosh(-x) -> Pi*I-acosh(x)
1118                 if (x.info(info_flags::negative))
1119                         return Pi*I-acosh(-x);
1120         }
1121         
1122         return acosh(x).hold();
1123 }
1124
1125 static ex acosh_deriv(const ex & x, unsigned deriv_param)
1126 {
1127         GINAC_ASSERT(deriv_param==0);
1128         
1129         // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
1130         return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
1131 }
1132
1133 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
1134                          evalf_func(acosh_evalf).
1135                          derivative_func(acosh_deriv));
1136
1137 //////////
1138 // inverse hyperbolic tangent (trigonometric function)
1139 //////////
1140
1141 static ex atanh_evalf(const ex & x)
1142 {
1143         if (is_exactly_a<numeric>(x))
1144                 return atanh(ex_to<numeric>(x));
1145         
1146         return atanh(x).hold();
1147 }
1148
1149 static ex atanh_eval(const ex & x)
1150 {
1151         if (x.info(info_flags::numeric)) {
1152
1153                 // atanh(0) -> 0
1154                 if (x.is_zero())
1155                         return _ex0;
1156
1157                 // atanh({+|-}1) -> throw
1158                 if (x.is_equal(_ex1) || x.is_equal(_ex_1))
1159                         throw (pole_error("atanh_eval(): logarithmic pole",0));
1160
1161                 // atanh(float) -> float
1162                 if (!x.info(info_flags::crational))
1163                         return atanh(ex_to<numeric>(x));
1164
1165                 // atanh() is odd
1166                 if (x.info(info_flags::negative))
1167                         return -atanh(-x);
1168         }
1169         
1170         return atanh(x).hold();
1171 }
1172
1173 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1174 {
1175         GINAC_ASSERT(deriv_param==0);
1176         
1177         // d/dx atanh(x) -> 1/(1-x^2)
1178         return power(_ex1-power(x,_ex2),_ex_1);
1179 }
1180
1181 static ex atanh_series(const ex &arg,
1182                        const relational &rel,
1183                        int order,
1184                        unsigned options)
1185 {
1186         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1187         // method:
1188         // Taylor series where there is no pole or cut falls back to atanh_deriv.
1189         // There are two branch cuts, one runnig from 1 up the real axis and one
1190         // one running from -1 down the real axis.  The points 1 and -1 are poles
1191         // On the branch cuts and the poles series expand
1192         //     (log(1+x)-log(1-x))/2
1193         // instead.
1194         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
1195         if (!(arg_pt).info(info_flags::real))
1196                 throw do_taylor();     // Im(x) != 0
1197         if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1198                 throw do_taylor();     // Im(x) == 0, but abs(x)<1
1199         // care for the poles, using the defining formula for atanh()...
1200         if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1201                 return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1202         // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1203         if (!(options & series_options::suppress_branchcut)) {
1204                 // method:
1205                 // This is the branch cut: assemble the primitive series manually and
1206                 // then add the corresponding complex step function.
1207                 const symbol &s = ex_to<symbol>(rel.lhs());
1208                 const ex &point = rel.rhs();
1209                 const symbol foo;
1210                 const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1211                 ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1212                 if (arg_pt<_ex0)
1213                         Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1214                 else
1215                         Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1216                 epvector seq;
1217                 seq.push_back(expair(Order0correction, _ex0));
1218                 seq.push_back(expair(Order(_ex1), order));
1219                 return series(replarg - pseries(rel, seq), rel, order);
1220         }
1221         throw do_taylor();
1222 }
1223
1224 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
1225                          evalf_func(atanh_evalf).
1226                          derivative_func(atanh_deriv).
1227                          series_func(atanh_series));
1228
1229
1230 } // namespace GiNaC