GiNaC  1.8.0
inifcns_trans.cpp
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1 
6 /*
7  * GiNaC Copyright (C) 1999-2020 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  *
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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.
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21  * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
22  */
23 
24 #include "inifcns.h"
25 #include "ex.h"
26 #include "constant.h"
27 #include "add.h"
28 #include "mul.h"
29 #include "numeric.h"
30 #include "power.h"
31 #include "operators.h"
32 #include "relational.h"
33 #include "symbol.h"
34 #include "pseries.h"
35 #include "utils.h"
36 
37 #include <stdexcept>
38 #include <vector>
39 
40 namespace GiNaC {
41 
43 // exponential function
45 
46 static ex exp_evalf(const ex & x)
47 {
48  if (is_exactly_a<numeric>(x))
49  return exp(ex_to<numeric>(x));
50 
51  return exp(x).hold();
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 
61  // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
62  const ex TwoExOverPiI=(_ex2*x)/(Pi*I);
63  if (TwoExOverPiI.info(info_flags::integer)) {
64  const numeric z = mod(ex_to<numeric>(TwoExOverPiI),*_num4_p);
65  if (z.is_equal(*_num0_p))
66  return _ex1;
67  if (z.is_equal(*_num1_p))
68  return ex(I);
69  if (z.is_equal(*_num2_p))
70  return _ex_1;
71  if (z.is_equal(*_num3_p))
72  return ex(-I);
73  }
74 
75  // exp(log(x)) -> x
76  if (is_ex_the_function(x, log))
77  return x.op(0);
78 
79  // exp(float) -> float
81  return exp(ex_to<numeric>(x));
82 
83  return exp(x).hold();
84 }
85 
86 static ex exp_expand(const ex & arg, unsigned options)
87 {
88  ex exp_arg;
90  exp_arg = arg.expand(options);
91  else
92  exp_arg=arg;
93 
95  && is_exactly_a<add>(exp_arg)) {
96  exvector prodseq;
97  prodseq.reserve(exp_arg.nops());
98  for (const_iterator i = exp_arg.begin(); i != exp_arg.end(); ++i)
99  prodseq.push_back(exp(*i));
100 
101  return dynallocate<mul>(prodseq).setflag(status_flags::expanded);
102  }
103 
104  return exp(exp_arg).hold();
105 }
106 
107 static ex exp_deriv(const ex & x, unsigned deriv_param)
108 {
109  GINAC_ASSERT(deriv_param==0);
110 
111  // d/dx exp(x) -> exp(x)
112  return exp(x);
113 }
114 
115 static ex exp_real_part(const ex & x)
116 {
118 }
119 
120 static ex exp_imag_part(const ex & x)
121 {
123 }
124 
125 static ex exp_conjugate(const ex & x)
126 {
127  // conjugate(exp(x))==exp(conjugate(x))
128  return exp(x.conjugate());
129 }
130 
131 static ex exp_power(const ex & x, const ex & a)
132 {
133  /*
134  * The power law (e^x)^a=e^(x*a) is used in two cases:
135  * a) a is an integer and x may be complex;
136  * b) both x and a are reals.
137  * Negative a is excluded to keep automatic simplifications like exp(x)/exp(x)=1.
138  */
141  return exp(x*a);
142  else if (a.info(info_flags::negative)
143  && (a.info(info_flags::integer) || (x.info(info_flags::real) && a.info(info_flags::real))))
144  return power(exp(-x*a), _ex_1).hold();
145 
146  return power(exp(x), a).hold();
147 }
148 
150  evalf_func(exp_evalf).
151  expand_func(exp_expand).
152  derivative_func(exp_deriv).
153  real_part_func(exp_real_part).
154  imag_part_func(exp_imag_part).
155  conjugate_func(exp_conjugate).
156  power_func(exp_power).
157  latex_name("\\exp"));
158 
160 // natural logarithm
162 
163 static ex log_evalf(const ex & x)
164 {
165  if (is_exactly_a<numeric>(x))
166  return log(ex_to<numeric>(x));
167 
168  return log(x).hold();
169 }
170 
171 static ex log_eval(const ex & x)
172 {
173  if (x.info(info_flags::numeric)) {
174  if (x.is_zero()) // log(0) -> infinity
175  throw(pole_error("log_eval(): log(0)",0));
177  return (log(-x)+I*Pi);
178  if (x.is_equal(_ex1)) // log(1) -> 0
179  return _ex0;
180  if (x.is_equal(I)) // log(I) -> Pi*I/2
181  return (Pi*I*_ex1_2);
182  if (x.is_equal(-I)) // log(-I) -> -Pi*I/2
183  return (Pi*I*_ex_1_2);
184 
185  // log(float) -> float
187  return log(ex_to<numeric>(x));
188  }
189 
190  // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
191  if (is_ex_the_function(x, exp)) {
192  const ex &t = x.op(0);
193  if (t.info(info_flags::real))
194  return t;
195  }
196 
197  return log(x).hold();
198 }
199 
200 static ex log_deriv(const ex & x, unsigned deriv_param)
201 {
202  GINAC_ASSERT(deriv_param==0);
203 
204  // d/dx log(x) -> 1/x
205  return power(x, _ex_1);
206 }
207 
208 static ex log_series(const ex &arg,
209  const relational &rel,
210  int order,
211  unsigned options)
212 {
213  GINAC_ASSERT(is_a<symbol>(rel.lhs()));
214  ex arg_pt;
215  bool must_expand_arg = false;
216  // maybe substitution of rel into arg fails because of a pole
217  try {
218  arg_pt = arg.subs(rel, subs_options::no_pattern);
219  } catch (pole_error &) {
220  must_expand_arg = true;
221  }
222  // or we are at the branch point anyways
223  if (arg_pt.is_zero())
224  must_expand_arg = true;
225 
226  if (arg.diff(ex_to<symbol>(rel.lhs())).is_zero()) {
227  throw do_taylor();
228  }
229 
230  if (must_expand_arg) {
231  // method:
232  // This is the branch point: Series expand the argument first, then
233  // trivially factorize it to isolate that part which has constant
234  // leading coefficient in this fashion:
235  // x^n + x^(n+1) +...+ Order(x^(n+m)) -> x^n * (1 + x +...+ Order(x^m)).
236  // Return a plain n*log(x) for the x^n part and series expand the
237  // other part. Add them together and reexpand again in order to have
238  // one unnested pseries object. All this also works for negative n.
239  pseries argser; // series expansion of log's argument
240  unsigned extra_ord = 0; // extra expansion order
241  do {
242  // oops, the argument expanded to a pure Order(x^something)...
243  argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
244  ++extra_ord;
245  } while (!argser.is_terminating() && argser.nops()==1);
246 
247  const symbol &s = ex_to<symbol>(rel.lhs());
248  const ex &point = rel.rhs();
249  const int n = argser.ldegree(s);
250  epvector seq;
251  // construct what we carelessly called the n*log(x) term above
252  const ex coeff = argser.coeff(s, n);
253  // expand the log, but only if coeff is real and > 0, since otherwise
254  // it would make the branch cut run into the wrong direction
256  seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
257  else
258  seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));
259 
260  if (!argser.is_terminating() || argser.nops()!=1) {
261  // in this case n more (or less) terms are needed
262  // (sadly, to generate them, we have to start from the beginning)
263  if (n == 0 && coeff == 1) {
264  ex rest = pseries(rel, epvector{expair(-1, _ex0), expair(Order(_ex1), order)}).add_series(argser);
265  ex acc = dynallocate<pseries>(rel, epvector());
266  for (int i = order-1; i>0; --i) {
267  epvector cterm { expair(i%2 ? _ex1/i : _ex_1/i, _ex0) };
268  acc = pseries(rel, std::move(cterm)).add_series(ex_to<pseries>(acc));
269  acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
270  }
271  return acc;
272  }
273  const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
274  return pseries(rel, std::move(seq)).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
275  } else // it was a monomial
276  return pseries(rel, std::move(seq));
277  }
279  arg_pt.info(info_flags::negative)) {
280  // method:
281  // This is the branch cut: assemble the primitive series manually and
282  // then add the corresponding complex step function.
283  const symbol &s = ex_to<symbol>(rel.lhs());
284  const ex &point = rel.rhs();
285  const symbol foo;
286  const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
287  epvector seq;
288  if (order > 0) {
289  seq.reserve(2);
290  seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
291  }
292  seq.push_back(expair(Order(_ex1), order));
293  return series(replarg - I*Pi + pseries(rel, std::move(seq)), rel, order);
294  }
295  throw do_taylor(); // caught by function::series()
296 }
297 
298 static ex log_real_part(const ex & x)
299 {
301  return log(x).hold();
302  return log(abs(x));
303 }
304 
305 static ex log_imag_part(const ex & x)
306 {
308  return 0;
309  return atan2(GiNaC::imag_part(x), GiNaC::real_part(x));
310 }
311 
312 static ex log_expand(const ex & arg, unsigned options)
313 {
315  && is_exactly_a<mul>(arg) && !arg.info(info_flags::indefinite)) {
316  exvector sumseq;
317  exvector prodseq;
318  sumseq.reserve(arg.nops());
319  prodseq.reserve(arg.nops());
320  bool possign=true;
321 
322  // searching for positive/negative factors
323  for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
324  ex e;
326  e=i->expand(options);
327  else
328  e=*i;
329  if (e.info(info_flags::positive))
330  sumseq.push_back(log(e));
331  else if (e.info(info_flags::negative)) {
332  sumseq.push_back(log(-e));
333  possign = !possign;
334  } else
335  prodseq.push_back(e);
336  }
337 
338  if (sumseq.size() > 0) {
339  ex newarg;
341  newarg=((possign?_ex1:_ex_1)*mul(prodseq)).expand(options);
342  else {
343  newarg=(possign?_ex1:_ex_1)*mul(prodseq);
344  ex_to<basic>(newarg).setflag(status_flags::purely_indefinite);
345  }
346  return add(sumseq)+log(newarg);
347  } else {
349  ex_to<basic>(arg).setflag(status_flags::purely_indefinite);
350  }
351  }
352 
354  return log(arg.expand(options)).hold();
355  else
356  return log(arg).hold();
357 }
358 
359 static ex log_conjugate(const ex & x)
360 {
361  // conjugate(log(x))==log(conjugate(x)) unless on the branch cut which
362  // runs along the negative real axis.
363  if (x.info(info_flags::positive)) {
364  return log(x);
365  }
366  if (is_exactly_a<numeric>(x) &&
367  !x.imag_part().is_zero()) {
368  return log(x.conjugate());
369  }
370  return conjugate_function(log(x)).hold();
371 }
372 
374  evalf_func(log_evalf).
375  expand_func(log_expand).
376  derivative_func(log_deriv).
377  series_func(log_series).
378  real_part_func(log_real_part).
379  imag_part_func(log_imag_part).
380  conjugate_func(log_conjugate).
381  latex_name("\\ln"));
382 
384 // sine (trigonometric function)
386 
387 static ex sin_evalf(const ex & x)
388 {
389  if (is_exactly_a<numeric>(x))
390  return sin(ex_to<numeric>(x));
391 
392  return sin(x).hold();
393 }
394 
395 static ex sin_eval(const ex & x)
396 {
397  // sin(n/d*Pi) -> { all known non-nested radicals }
398  const ex SixtyExOverPi = _ex60*x/Pi;
399  ex sign = _ex1;
400  if (SixtyExOverPi.info(info_flags::integer)) {
401  numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
402  if (z>=*_num60_p) {
403  // wrap to interval [0, Pi)
404  z -= *_num60_p;
405  sign = _ex_1;
406  }
407  if (z>*_num30_p) {
408  // wrap to interval [0, Pi/2)
409  z = *_num60_p-z;
410  }
411  if (z.is_equal(*_num0_p)) // sin(0) -> 0
412  return _ex0;
413  if (z.is_equal(*_num5_p)) // sin(Pi/12) -> sqrt(6)/4*(1-sqrt(3)/3)
414  return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
415  if (z.is_equal(*_num6_p)) // sin(Pi/10) -> sqrt(5)/4-1/4
416  return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
417  if (z.is_equal(*_num10_p)) // sin(Pi/6) -> 1/2
418  return sign*_ex1_2;
419  if (z.is_equal(*_num15_p)) // sin(Pi/4) -> sqrt(2)/2
420  return sign*_ex1_2*sqrt(_ex2);
421  if (z.is_equal(*_num18_p)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
422  return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
423  if (z.is_equal(*_num20_p)) // sin(Pi/3) -> sqrt(3)/2
424  return sign*_ex1_2*sqrt(_ex3);
425  if (z.is_equal(*_num25_p)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
426  return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
427  if (z.is_equal(*_num30_p)) // sin(Pi/2) -> 1
428  return sign;
429  }
430 
431  if (is_exactly_a<function>(x)) {
432  const ex &t = x.op(0);
433 
434  // sin(asin(x)) -> x
435  if (is_ex_the_function(x, asin))
436  return t;
437 
438  // sin(acos(x)) -> sqrt(1-x^2)
439  if (is_ex_the_function(x, acos))
440  return sqrt(_ex1-power(t,_ex2));
441 
442  // sin(atan(x)) -> x/sqrt(1+x^2)
443  if (is_ex_the_function(x, atan))
444  return t*power(_ex1+power(t,_ex2),_ex_1_2);
445  }
446 
447  // sin(float) -> float
449  return sin(ex_to<numeric>(x));
450 
451  // sin() is odd
453  return -sin(-x);
454 
455  return sin(x).hold();
456 }
457 
458 static ex sin_deriv(const ex & x, unsigned deriv_param)
459 {
460  GINAC_ASSERT(deriv_param==0);
461 
462  // d/dx sin(x) -> cos(x)
463  return cos(x);
464 }
465 
466 static ex sin_real_part(const ex & x)
467 {
469 }
470 
471 static ex sin_imag_part(const ex & x)
472 {
474 }
475 
476 static ex sin_conjugate(const ex & x)
477 {
478  // conjugate(sin(x))==sin(conjugate(x))
479  return sin(x.conjugate());
480 }
481 
483  evalf_func(sin_evalf).
484  derivative_func(sin_deriv).
485  real_part_func(sin_real_part).
486  imag_part_func(sin_imag_part).
487  conjugate_func(sin_conjugate).
488  latex_name("\\sin"));
489 
491 // cosine (trigonometric function)
493 
494 static ex cos_evalf(const ex & x)
495 {
496  if (is_exactly_a<numeric>(x))
497  return cos(ex_to<numeric>(x));
498 
499  return cos(x).hold();
500 }
501 
502 static ex cos_eval(const ex & x)
503 {
504  // cos(n/d*Pi) -> { all known non-nested radicals }
505  const ex SixtyExOverPi = _ex60*x/Pi;
506  ex sign = _ex1;
507  if (SixtyExOverPi.info(info_flags::integer)) {
508  numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
509  if (z>=*_num60_p) {
510  // wrap to interval [0, Pi)
511  z = *_num120_p-z;
512  }
513  if (z>=*_num30_p) {
514  // wrap to interval [0, Pi/2)
515  z = *_num60_p-z;
516  sign = _ex_1;
517  }
518  if (z.is_equal(*_num0_p)) // cos(0) -> 1
519  return sign;
520  if (z.is_equal(*_num5_p)) // cos(Pi/12) -> sqrt(6)/4*(1+sqrt(3)/3)
521  return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
522  if (z.is_equal(*_num10_p)) // cos(Pi/6) -> sqrt(3)/2
523  return sign*_ex1_2*sqrt(_ex3);
524  if (z.is_equal(*_num12_p)) // cos(Pi/5) -> sqrt(5)/4+1/4
525  return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
526  if (z.is_equal(*_num15_p)) // cos(Pi/4) -> sqrt(2)/2
527  return sign*_ex1_2*sqrt(_ex2);
528  if (z.is_equal(*_num20_p)) // cos(Pi/3) -> 1/2
529  return sign*_ex1_2;
530  if (z.is_equal(*_num24_p)) // cos(2/5*Pi) -> sqrt(5)/4-1/4x
531  return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
532  if (z.is_equal(*_num25_p)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
533  return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
534  if (z.is_equal(*_num30_p)) // cos(Pi/2) -> 0
535  return _ex0;
536  }
537 
538  if (is_exactly_a<function>(x)) {
539  const ex &t = x.op(0);
540 
541  // cos(acos(x)) -> x
542  if (is_ex_the_function(x, acos))
543  return t;
544 
545  // cos(asin(x)) -> sqrt(1-x^2)
546  if (is_ex_the_function(x, asin))
547  return sqrt(_ex1-power(t,_ex2));
548 
549  // cos(atan(x)) -> 1/sqrt(1+x^2)
550  if (is_ex_the_function(x, atan))
551  return power(_ex1+power(t,_ex2),_ex_1_2);
552  }
553 
554  // cos(float) -> float
556  return cos(ex_to<numeric>(x));
557 
558  // cos() is even
560  return cos(-x);
561 
562  return cos(x).hold();
563 }
564 
565 static ex cos_deriv(const ex & x, unsigned deriv_param)
566 {
567  GINAC_ASSERT(deriv_param==0);
568 
569  // d/dx cos(x) -> -sin(x)
570  return -sin(x);
571 }
572 
573 static ex cos_real_part(const ex & x)
574 {
576 }
577 
578 static ex cos_imag_part(const ex & x)
579 {
581 }
582 
583 static ex cos_conjugate(const ex & x)
584 {
585  // conjugate(cos(x))==cos(conjugate(x))
586  return cos(x.conjugate());
587 }
588 
590  evalf_func(cos_evalf).
591  derivative_func(cos_deriv).
592  real_part_func(cos_real_part).
593  imag_part_func(cos_imag_part).
594  conjugate_func(cos_conjugate).
595  latex_name("\\cos"));
596 
598 // tangent (trigonometric function)
600 
601 static ex tan_evalf(const ex & x)
602 {
603  if (is_exactly_a<numeric>(x))
604  return tan(ex_to<numeric>(x));
605 
606  return tan(x).hold();
607 }
608 
609 static ex tan_eval(const ex & x)
610 {
611  // tan(n/d*Pi) -> { all known non-nested radicals }
612  const ex SixtyExOverPi = _ex60*x/Pi;
613  ex sign = _ex1;
614  if (SixtyExOverPi.info(info_flags::integer)) {
615  numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num60_p);
616  if (z>=*_num60_p) {
617  // wrap to interval [0, Pi)
618  z -= *_num60_p;
619  }
620  if (z>=*_num30_p) {
621  // wrap to interval [0, Pi/2)
622  z = *_num60_p-z;
623  sign = _ex_1;
624  }
625  if (z.is_equal(*_num0_p)) // tan(0) -> 0
626  return _ex0;
627  if (z.is_equal(*_num5_p)) // tan(Pi/12) -> 2-sqrt(3)
628  return sign*(_ex2-sqrt(_ex3));
629  if (z.is_equal(*_num10_p)) // tan(Pi/6) -> sqrt(3)/3
630  return sign*_ex1_3*sqrt(_ex3);
631  if (z.is_equal(*_num15_p)) // tan(Pi/4) -> 1
632  return sign;
633  if (z.is_equal(*_num20_p)) // tan(Pi/3) -> sqrt(3)
634  return sign*sqrt(_ex3);
635  if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3)
636  return sign*(sqrt(_ex3)+_ex2);
637  if (z.is_equal(*_num30_p)) // tan(Pi/2) -> infinity
638  throw (pole_error("tan_eval(): simple pole",1));
639  }
640 
641  if (is_exactly_a<function>(x)) {
642  const ex &t = x.op(0);
643 
644  // tan(atan(x)) -> x
645  if (is_ex_the_function(x, atan))
646  return t;
647 
648  // tan(asin(x)) -> x/sqrt(1+x^2)
649  if (is_ex_the_function(x, asin))
650  return t*power(_ex1-power(t,_ex2),_ex_1_2);
651 
652  // tan(acos(x)) -> sqrt(1-x^2)/x
653  if (is_ex_the_function(x, acos))
654  return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
655  }
656 
657  // tan(float) -> float
659  return tan(ex_to<numeric>(x));
660  }
661 
662  // tan() is odd
664  return -tan(-x);
665 
666  return tan(x).hold();
667 }
668 
669 static ex tan_deriv(const ex & x, unsigned deriv_param)
670 {
671  GINAC_ASSERT(deriv_param==0);
672 
673  // d/dx tan(x) -> 1+tan(x)^2;
674  return (_ex1+power(tan(x),_ex2));
675 }
676 
677 static ex tan_real_part(const ex & x)
678 {
679  ex a = GiNaC::real_part(x);
680  ex b = GiNaC::imag_part(x);
681  return tan(a)/(1+power(tan(a),2)*power(tan(b),2));
682 }
683 
684 static ex tan_imag_part(const ex & x)
685 {
686  ex a = GiNaC::real_part(x);
687  ex b = GiNaC::imag_part(x);
688  return tanh(b)/(1+power(tan(a),2)*power(tan(b),2));
689 }
690 
691 static ex tan_series(const ex &x,
692  const relational &rel,
693  int order,
694  unsigned options)
695 {
696  GINAC_ASSERT(is_a<symbol>(rel.lhs()));
697  // method:
698  // Taylor series where there is no pole falls back to tan_deriv.
699  // On a pole simply expand sin(x)/cos(x).
700  const ex x_pt = x.subs(rel, subs_options::no_pattern);
701  if (!(2*x_pt/Pi).info(info_flags::odd))
702  throw do_taylor(); // caught by function::series()
703  // if we got here we have to care for a simple pole
704  return (sin(x)/cos(x)).series(rel, order, options);
705 }
706 
707 static ex tan_conjugate(const ex & x)
708 {
709  // conjugate(tan(x))==tan(conjugate(x))
710  return tan(x.conjugate());
711 }
712 
714  evalf_func(tan_evalf).
715  derivative_func(tan_deriv).
716  series_func(tan_series).
717  real_part_func(tan_real_part).
718  imag_part_func(tan_imag_part).
719  conjugate_func(tan_conjugate).
720  latex_name("\\tan"));
721 
723 // inverse sine (arc sine)
725 
726 static ex asin_evalf(const ex & x)
727 {
728  if (is_exactly_a<numeric>(x))
729  return asin(ex_to<numeric>(x));
730 
731  return asin(x).hold();
732 }
733 
734 static ex asin_eval(const ex & x)
735 {
736  if (x.info(info_flags::numeric)) {
737 
738  // asin(0) -> 0
739  if (x.is_zero())
740  return x;
741 
742  // asin(1/2) -> Pi/6
743  if (x.is_equal(_ex1_2))
744  return numeric(1,6)*Pi;
745 
746  // asin(1) -> Pi/2
747  if (x.is_equal(_ex1))
748  return _ex1_2*Pi;
749 
750  // asin(-1/2) -> -Pi/6
751  if (x.is_equal(_ex_1_2))
752  return numeric(-1,6)*Pi;
753 
754  // asin(-1) -> -Pi/2
755  if (x.is_equal(_ex_1))
756  return _ex_1_2*Pi;
757 
758  // asin(float) -> float
760  return asin(ex_to<numeric>(x));
761 
762  // asin() is odd
764  return -asin(-x);
765  }
766 
767  return asin(x).hold();
768 }
769 
770 static ex asin_deriv(const ex & x, unsigned deriv_param)
771 {
772  GINAC_ASSERT(deriv_param==0);
773 
774  // d/dx asin(x) -> 1/sqrt(1-x^2)
775  return power(1-power(x,_ex2),_ex_1_2);
776 }
777 
778 static ex asin_conjugate(const ex & x)
779 {
780  // conjugate(asin(x))==asin(conjugate(x)) unless on the branch cuts which
781  // run along the real axis outside the interval [-1, +1].
782  if (is_exactly_a<numeric>(x) &&
783  (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
784  return asin(x.conjugate());
785  }
786  return conjugate_function(asin(x)).hold();
787 }
788 
790  evalf_func(asin_evalf).
791  derivative_func(asin_deriv).
792  conjugate_func(asin_conjugate).
793  latex_name("\\arcsin"));
794 
796 // inverse cosine (arc cosine)
798 
799 static ex acos_evalf(const ex & x)
800 {
801  if (is_exactly_a<numeric>(x))
802  return acos(ex_to<numeric>(x));
803 
804  return acos(x).hold();
805 }
806 
807 static ex acos_eval(const ex & x)
808 {
809  if (x.info(info_flags::numeric)) {
810 
811  // acos(1) -> 0
812  if (x.is_equal(_ex1))
813  return _ex0;
814 
815  // acos(1/2) -> Pi/3
816  if (x.is_equal(_ex1_2))
817  return _ex1_3*Pi;
818 
819  // acos(0) -> Pi/2
820  if (x.is_zero())
821  return _ex1_2*Pi;
822 
823  // acos(-1/2) -> 2/3*Pi
824  if (x.is_equal(_ex_1_2))
825  return numeric(2,3)*Pi;
826 
827  // acos(-1) -> Pi
828  if (x.is_equal(_ex_1))
829  return Pi;
830 
831  // acos(float) -> float
833  return acos(ex_to<numeric>(x));
834 
835  // acos(-x) -> Pi-acos(x)
837  return Pi-acos(-x);
838  }
839 
840  return acos(x).hold();
841 }
842 
843 static ex acos_deriv(const ex & x, unsigned deriv_param)
844 {
845  GINAC_ASSERT(deriv_param==0);
846 
847  // d/dx acos(x) -> -1/sqrt(1-x^2)
848  return -power(1-power(x,_ex2),_ex_1_2);
849 }
850 
851 static ex acos_conjugate(const ex & x)
852 {
853  // conjugate(acos(x))==acos(conjugate(x)) unless on the branch cuts which
854  // run along the real axis outside the interval [-1, +1].
855  if (is_exactly_a<numeric>(x) &&
856  (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
857  return acos(x.conjugate());
858  }
859  return conjugate_function(acos(x)).hold();
860 }
861 
863  evalf_func(acos_evalf).
864  derivative_func(acos_deriv).
865  conjugate_func(acos_conjugate).
866  latex_name("\\arccos"));
867 
869 // inverse tangent (arc tangent)
871 
872 static ex atan_evalf(const ex & x)
873 {
874  if (is_exactly_a<numeric>(x))
875  return atan(ex_to<numeric>(x));
876 
877  return atan(x).hold();
878 }
879 
880 static ex atan_eval(const ex & x)
881 {
882  if (x.info(info_flags::numeric)) {
883 
884  // atan(0) -> 0
885  if (x.is_zero())
886  return _ex0;
887 
888  // atan(1) -> Pi/4
889  if (x.is_equal(_ex1))
890  return _ex1_4*Pi;
891 
892  // atan(-1) -> -Pi/4
893  if (x.is_equal(_ex_1))
894  return _ex_1_4*Pi;
895 
896  if (x.is_equal(I) || x.is_equal(-I))
897  throw (pole_error("atan_eval(): logarithmic pole",0));
898 
899  // atan(float) -> float
901  return atan(ex_to<numeric>(x));
902 
903  // atan() is odd
905  return -atan(-x);
906  }
907 
908  return atan(x).hold();
909 }
910 
911 static ex atan_deriv(const ex & x, unsigned deriv_param)
912 {
913  GINAC_ASSERT(deriv_param==0);
914 
915  // d/dx atan(x) -> 1/(1+x^2)
916  return power(_ex1+power(x,_ex2), _ex_1);
917 }
918 
919 static ex atan_series(const ex &arg,
920  const relational &rel,
921  int order,
922  unsigned options)
923 {
924  GINAC_ASSERT(is_a<symbol>(rel.lhs()));
925  // method:
926  // Taylor series where there is no pole or cut falls back to atan_deriv.
927  // There are two branch cuts, one runnig from I up the imaginary axis and
928  // one running from -I down the imaginary axis. The points I and -I are
929  // poles.
930  // On the branch cuts and the poles series expand
931  // (log(1+I*x)-log(1-I*x))/(2*I)
932  // instead.
933  const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
934  if (!(I*arg_pt).info(info_flags::real))
935  throw do_taylor(); // Re(x) != 0
936  if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
937  throw do_taylor(); // Re(x) == 0, but abs(x)<1
938  // care for the poles, using the defining formula for atan()...
939  if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
940  return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
942  // method:
943  // This is the branch cut: assemble the primitive series manually and
944  // then add the corresponding complex step function.
945  const symbol &s = ex_to<symbol>(rel.lhs());
946  const ex &point = rel.rhs();
947  const symbol foo;
948  const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
949  ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
950  if ((I*arg_pt)<_ex0)
951  Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
952  else
953  Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
954  epvector seq;
955  if (order > 0) {
956  seq.reserve(2);
957  seq.push_back(expair(Order0correction, _ex0));
958  }
959  seq.push_back(expair(Order(_ex1), order));
960  return series(replarg - pseries(rel, std::move(seq)), rel, order);
961  }
962  throw do_taylor();
963 }
964 
965 static ex atan_conjugate(const ex & x)
966 {
967  // conjugate(atan(x))==atan(conjugate(x)) unless on the branch cuts which
968  // run along the imaginary axis outside the interval [-I, +I].
969  if (x.info(info_flags::real))
970  return atan(x);
971  if (is_exactly_a<numeric>(x)) {
972  const numeric x_re = ex_to<numeric>(x.real_part());
973  const numeric x_im = ex_to<numeric>(x.imag_part());
974  if (!x_re.is_zero() ||
975  (x_im > *_num_1_p && x_im < *_num1_p))
976  return atan(x.conjugate());
977  }
978  return conjugate_function(atan(x)).hold();
979 }
980 
982  evalf_func(atan_evalf).
983  derivative_func(atan_deriv).
984  series_func(atan_series).
985  conjugate_func(atan_conjugate).
986  latex_name("\\arctan"));
987 
989 // inverse tangent (atan2(y,x))
991 
992 static ex atan2_evalf(const ex &y, const ex &x)
993 {
994  if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
995  return atan(ex_to<numeric>(y), ex_to<numeric>(x));
996 
997  return atan2(y, x).hold();
998 }
999 
1000 static ex atan2_eval(const ex & y, const ex & x)
1001 {
1002  if (y.is_zero()) {
1003 
1004  // atan2(0, 0) -> 0
1005  if (x.is_zero())
1006  return _ex0;
1007 
1008  // atan2(0, x), x real and positive -> 0
1010  return _ex0;
1011 
1012  // atan2(0, x), x real and negative -> Pi
1014  return Pi;
1015  }
1016 
1017  if (x.is_zero()) {
1018 
1019  // atan2(y, 0), y real and positive -> Pi/2
1020  if (y.info(info_flags::positive))
1021  return _ex1_2*Pi;
1022 
1023  // atan2(y, 0), y real and negative -> -Pi/2
1024  if (y.info(info_flags::negative))
1025  return _ex_1_2*Pi;
1026  }
1027 
1028  if (y.is_equal(x)) {
1029 
1030  // atan2(y, y), y real and positive -> Pi/4
1031  if (y.info(info_flags::positive))
1032  return _ex1_4*Pi;
1033 
1034  // atan2(y, y), y real and negative -> -3/4*Pi
1035  if (y.info(info_flags::negative))
1036  return numeric(-3, 4)*Pi;
1037  }
1038 
1039  if (y.is_equal(-x)) {
1040 
1041  // atan2(y, -y), y real and positive -> 3*Pi/4
1042  if (y.info(info_flags::positive))
1043  return numeric(3, 4)*Pi;
1044 
1045  // atan2(y, -y), y real and negative -> -Pi/4
1046  if (y.info(info_flags::negative))
1047  return _ex_1_4*Pi;
1048  }
1049 
1050  // atan2(float, float) -> float
1051  if (is_a<numeric>(y) && !y.info(info_flags::crational) &&
1052  is_a<numeric>(x) && !x.info(info_flags::crational))
1053  return atan(ex_to<numeric>(y), ex_to<numeric>(x));
1054 
1055  // atan2(real, real) -> atan(y/x) +/- Pi
1058  return atan(y/x);
1059 
1060  if (x.info(info_flags::negative)) {
1061  if (y.info(info_flags::positive))
1062  return atan(y/x)+Pi;
1063  if (y.info(info_flags::negative))
1064  return atan(y/x)-Pi;
1065  }
1066  }
1067 
1068  return atan2(y, x).hold();
1069 }
1070 
1071 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
1072 {
1073  GINAC_ASSERT(deriv_param<2);
1074 
1075  if (deriv_param==0) {
1076  // d/dy atan2(y,x)
1077  return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
1078  }
1079  // d/dx atan2(y,x)
1080  return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
1081 }
1082 
1083 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
1084  evalf_func(atan2_evalf).
1085  derivative_func(atan2_deriv));
1086 
1088 // hyperbolic sine (trigonometric function)
1090 
1091 static ex sinh_evalf(const ex & x)
1092 {
1093  if (is_exactly_a<numeric>(x))
1094  return sinh(ex_to<numeric>(x));
1095 
1096  return sinh(x).hold();
1097 }
1098 
1099 static ex sinh_eval(const ex & x)
1100 {
1101  if (x.info(info_flags::numeric)) {
1102 
1103  // sinh(0) -> 0
1104  if (x.is_zero())
1105  return _ex0;
1106 
1107  // sinh(float) -> float
1109  return sinh(ex_to<numeric>(x));
1110 
1111  // sinh() is odd
1113  return -sinh(-x);
1114  }
1115 
1116  if ((x/Pi).info(info_flags::numeric) &&
1117  ex_to<numeric>(x/Pi).real().is_zero()) // sinh(I*x) -> I*sin(x)
1118  return I*sin(x/I);
1119 
1120  if (is_exactly_a<function>(x)) {
1121  const ex &t = x.op(0);
1122 
1123  // sinh(asinh(x)) -> x
1124  if (is_ex_the_function(x, asinh))
1125  return t;
1126 
1127  // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
1128  if (is_ex_the_function(x, acosh))
1129  return sqrt(t-_ex1)*sqrt(t+_ex1);
1130 
1131  // sinh(atanh(x)) -> x/sqrt(1-x^2)
1132  if (is_ex_the_function(x, atanh))
1133  return t*power(_ex1-power(t,_ex2),_ex_1_2);
1134  }
1135 
1136  return sinh(x).hold();
1137 }
1138 
1139 static ex sinh_deriv(const ex & x, unsigned deriv_param)
1140 {
1141  GINAC_ASSERT(deriv_param==0);
1142 
1143  // d/dx sinh(x) -> cosh(x)
1144  return cosh(x);
1145 }
1146 
1147 static ex sinh_real_part(const ex & x)
1148 {
1150 }
1151 
1152 static ex sinh_imag_part(const ex & x)
1153 {
1155 }
1156 
1157 static ex sinh_conjugate(const ex & x)
1158 {
1159  // conjugate(sinh(x))==sinh(conjugate(x))
1160  return sinh(x.conjugate());
1161 }
1162 
1164  evalf_func(sinh_evalf).
1165  derivative_func(sinh_deriv).
1166  real_part_func(sinh_real_part).
1167  imag_part_func(sinh_imag_part).
1168  conjugate_func(sinh_conjugate).
1169  latex_name("\\sinh"));
1170 
1172 // hyperbolic cosine (trigonometric function)
1174 
1175 static ex cosh_evalf(const ex & x)
1176 {
1177  if (is_exactly_a<numeric>(x))
1178  return cosh(ex_to<numeric>(x));
1179 
1180  return cosh(x).hold();
1181 }
1182 
1183 static ex cosh_eval(const ex & x)
1184 {
1185  if (x.info(info_flags::numeric)) {
1186 
1187  // cosh(0) -> 1
1188  if (x.is_zero())
1189  return _ex1;
1190 
1191  // cosh(float) -> float
1193  return cosh(ex_to<numeric>(x));
1194 
1195  // cosh() is even
1197  return cosh(-x);
1198  }
1199 
1200  if ((x/Pi).info(info_flags::numeric) &&
1201  ex_to<numeric>(x/Pi).real().is_zero()) // cosh(I*x) -> cos(x)
1202  return cos(x/I);
1203 
1204  if (is_exactly_a<function>(x)) {
1205  const ex &t = x.op(0);
1206 
1207  // cosh(acosh(x)) -> x
1208  if (is_ex_the_function(x, acosh))
1209  return t;
1210 
1211  // cosh(asinh(x)) -> sqrt(1+x^2)
1212  if (is_ex_the_function(x, asinh))
1213  return sqrt(_ex1+power(t,_ex2));
1214 
1215  // cosh(atanh(x)) -> 1/sqrt(1-x^2)
1216  if (is_ex_the_function(x, atanh))
1217  return power(_ex1-power(t,_ex2),_ex_1_2);
1218  }
1219 
1220  return cosh(x).hold();
1221 }
1222 
1223 static ex cosh_deriv(const ex & x, unsigned deriv_param)
1224 {
1225  GINAC_ASSERT(deriv_param==0);
1226 
1227  // d/dx cosh(x) -> sinh(x)
1228  return sinh(x);
1229 }
1230 
1231 static ex cosh_real_part(const ex & x)
1232 {
1234 }
1235 
1236 static ex cosh_imag_part(const ex & x)
1237 {
1239 }
1240 
1241 static ex cosh_conjugate(const ex & x)
1242 {
1243  // conjugate(cosh(x))==cosh(conjugate(x))
1244  return cosh(x.conjugate());
1245 }
1246 
1248  evalf_func(cosh_evalf).
1249  derivative_func(cosh_deriv).
1250  real_part_func(cosh_real_part).
1251  imag_part_func(cosh_imag_part).
1252  conjugate_func(cosh_conjugate).
1253  latex_name("\\cosh"));
1254 
1256 // hyperbolic tangent (trigonometric function)
1258 
1259 static ex tanh_evalf(const ex & x)
1260 {
1261  if (is_exactly_a<numeric>(x))
1262  return tanh(ex_to<numeric>(x));
1263 
1264  return tanh(x).hold();
1265 }
1266 
1267 static ex tanh_eval(const ex & x)
1268 {
1269  if (x.info(info_flags::numeric)) {
1270 
1271  // tanh(0) -> 0
1272  if (x.is_zero())
1273  return _ex0;
1274 
1275  // tanh(float) -> float
1277  return tanh(ex_to<numeric>(x));
1278 
1279  // tanh() is odd
1281  return -tanh(-x);
1282  }
1283 
1284  if ((x/Pi).info(info_flags::numeric) &&
1285  ex_to<numeric>(x/Pi).real().is_zero()) // tanh(I*x) -> I*tan(x);
1286  return I*tan(x/I);
1287 
1288  if (is_exactly_a<function>(x)) {
1289  const ex &t = x.op(0);
1290 
1291  // tanh(atanh(x)) -> x
1292  if (is_ex_the_function(x, atanh))
1293  return t;
1294 
1295  // tanh(asinh(x)) -> x/sqrt(1+x^2)
1296  if (is_ex_the_function(x, asinh))
1297  return t*power(_ex1+power(t,_ex2),_ex_1_2);
1298 
1299  // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
1300  if (is_ex_the_function(x, acosh))
1301  return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
1302  }
1303 
1304  return tanh(x).hold();
1305 }
1306 
1307 static ex tanh_deriv(const ex & x, unsigned deriv_param)
1308 {
1309  GINAC_ASSERT(deriv_param==0);
1310 
1311  // d/dx tanh(x) -> 1-tanh(x)^2
1312  return _ex1-power(tanh(x),_ex2);
1313 }
1314 
1315 static ex tanh_series(const ex &x,
1316  const relational &rel,
1317  int order,
1318  unsigned options)
1319 {
1320  GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1321  // method:
1322  // Taylor series where there is no pole falls back to tanh_deriv.
1323  // On a pole simply expand sinh(x)/cosh(x).
1324  const ex x_pt = x.subs(rel, subs_options::no_pattern);
1325  if (!(2*I*x_pt/Pi).info(info_flags::odd))
1326  throw do_taylor(); // caught by function::series()
1327  // if we got here we have to care for a simple pole
1328  return (sinh(x)/cosh(x)).series(rel, order, options);
1329 }
1330 
1331 static ex tanh_real_part(const ex & x)
1332 {
1333  ex a = GiNaC::real_part(x);
1334  ex b = GiNaC::imag_part(x);
1335  return tanh(a)/(1+power(tanh(a),2)*power(tan(b),2));
1336 }
1337 
1338 static ex tanh_imag_part(const ex & x)
1339 {
1340  ex a = GiNaC::real_part(x);
1341  ex b = GiNaC::imag_part(x);
1342  return tan(b)/(1+power(tanh(a),2)*power(tan(b),2));
1343 }
1344 
1345 static ex tanh_conjugate(const ex & x)
1346 {
1347  // conjugate(tanh(x))==tanh(conjugate(x))
1348  return tanh(x.conjugate());
1349 }
1350 
1352  evalf_func(tanh_evalf).
1353  derivative_func(tanh_deriv).
1354  series_func(tanh_series).
1355  real_part_func(tanh_real_part).
1356  imag_part_func(tanh_imag_part).
1357  conjugate_func(tanh_conjugate).
1358  latex_name("\\tanh"));
1359 
1361 // inverse hyperbolic sine (trigonometric function)
1363 
1364 static ex asinh_evalf(const ex & x)
1365 {
1366  if (is_exactly_a<numeric>(x))
1367  return asinh(ex_to<numeric>(x));
1368 
1369  return asinh(x).hold();
1370 }
1371 
1372 static ex asinh_eval(const ex & x)
1373 {
1374  if (x.info(info_flags::numeric)) {
1375 
1376  // asinh(0) -> 0
1377  if (x.is_zero())
1378  return _ex0;
1379 
1380  // asinh(float) -> float
1382  return asinh(ex_to<numeric>(x));
1383 
1384  // asinh() is odd
1386  return -asinh(-x);
1387  }
1388 
1389  return asinh(x).hold();
1390 }
1391 
1392 static ex asinh_deriv(const ex & x, unsigned deriv_param)
1393 {
1394  GINAC_ASSERT(deriv_param==0);
1395 
1396  // d/dx asinh(x) -> 1/sqrt(1+x^2)
1397  return power(_ex1+power(x,_ex2),_ex_1_2);
1398 }
1399 
1400 static ex asinh_conjugate(const ex & x)
1401 {
1402  // conjugate(asinh(x))==asinh(conjugate(x)) unless on the branch cuts which
1403  // run along the imaginary axis outside the interval [-I, +I].
1404  if (x.info(info_flags::real))
1405  return asinh(x);
1406  if (is_exactly_a<numeric>(x)) {
1407  const numeric x_re = ex_to<numeric>(x.real_part());
1408  const numeric x_im = ex_to<numeric>(x.imag_part());
1409  if (!x_re.is_zero() ||
1410  (x_im > *_num_1_p && x_im < *_num1_p))
1411  return asinh(x.conjugate());
1412  }
1413  return conjugate_function(asinh(x)).hold();
1414 }
1415 
1417  evalf_func(asinh_evalf).
1418  derivative_func(asinh_deriv).
1419  conjugate_func(asinh_conjugate));
1420 
1422 // inverse hyperbolic cosine (trigonometric function)
1424 
1425 static ex acosh_evalf(const ex & x)
1426 {
1427  if (is_exactly_a<numeric>(x))
1428  return acosh(ex_to<numeric>(x));
1429 
1430  return acosh(x).hold();
1431 }
1432 
1433 static ex acosh_eval(const ex & x)
1434 {
1435  if (x.info(info_flags::numeric)) {
1436 
1437  // acosh(0) -> Pi*I/2
1438  if (x.is_zero())
1439  return Pi*I*numeric(1,2);
1440 
1441  // acosh(1) -> 0
1442  if (x.is_equal(_ex1))
1443  return _ex0;
1444 
1445  // acosh(-1) -> Pi*I
1446  if (x.is_equal(_ex_1))
1447  return Pi*I;
1448 
1449  // acosh(float) -> float
1451  return acosh(ex_to<numeric>(x));
1452 
1453  // acosh(-x) -> Pi*I-acosh(x)
1455  return Pi*I-acosh(-x);
1456  }
1457 
1458  return acosh(x).hold();
1459 }
1460 
1461 static ex acosh_deriv(const ex & x, unsigned deriv_param)
1462 {
1463  GINAC_ASSERT(deriv_param==0);
1464 
1465  // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
1466  return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
1467 }
1468 
1469 static ex acosh_conjugate(const ex & x)
1470 {
1471  // conjugate(acosh(x))==acosh(conjugate(x)) unless on the branch cut
1472  // which runs along the real axis from +1 to -inf.
1473  if (is_exactly_a<numeric>(x) &&
1474  (!x.imag_part().is_zero() || x > *_num1_p)) {
1475  return acosh(x.conjugate());
1476  }
1477  return conjugate_function(acosh(x)).hold();
1478 }
1479 
1481  evalf_func(acosh_evalf).
1482  derivative_func(acosh_deriv).
1483  conjugate_func(acosh_conjugate));
1484 
1486 // inverse hyperbolic tangent (trigonometric function)
1488 
1489 static ex atanh_evalf(const ex & x)
1490 {
1491  if (is_exactly_a<numeric>(x))
1492  return atanh(ex_to<numeric>(x));
1493 
1494  return atanh(x).hold();
1495 }
1496 
1497 static ex atanh_eval(const ex & x)
1498 {
1499  if (x.info(info_flags::numeric)) {
1500 
1501  // atanh(0) -> 0
1502  if (x.is_zero())
1503  return _ex0;
1504 
1505  // atanh({+|-}1) -> throw
1506  if (x.is_equal(_ex1) || x.is_equal(_ex_1))
1507  throw (pole_error("atanh_eval(): logarithmic pole",0));
1508 
1509  // atanh(float) -> float
1511  return atanh(ex_to<numeric>(x));
1512 
1513  // atanh() is odd
1515  return -atanh(-x);
1516  }
1517 
1518  return atanh(x).hold();
1519 }
1520 
1521 static ex atanh_deriv(const ex & x, unsigned deriv_param)
1522 {
1523  GINAC_ASSERT(deriv_param==0);
1524 
1525  // d/dx atanh(x) -> 1/(1-x^2)
1526  return power(_ex1-power(x,_ex2),_ex_1);
1527 }
1528 
1529 static ex atanh_series(const ex &arg,
1530  const relational &rel,
1531  int order,
1532  unsigned options)
1533 {
1534  GINAC_ASSERT(is_a<symbol>(rel.lhs()));
1535  // method:
1536  // Taylor series where there is no pole or cut falls back to atanh_deriv.
1537  // There are two branch cuts, one runnig from 1 up the real axis and one
1538  // one running from -1 down the real axis. The points 1 and -1 are poles
1539  // On the branch cuts and the poles series expand
1540  // (log(1+x)-log(1-x))/2
1541  // instead.
1542  const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
1543  if (!(arg_pt).info(info_flags::real))
1544  throw do_taylor(); // Im(x) != 0
1545  if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
1546  throw do_taylor(); // Im(x) == 0, but abs(x)<1
1547  // care for the poles, using the defining formula for atanh()...
1548  if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
1549  return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
1550  // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
1552  // method:
1553  // This is the branch cut: assemble the primitive series manually and
1554  // then add the corresponding complex step function.
1555  const symbol &s = ex_to<symbol>(rel.lhs());
1556  const ex &point = rel.rhs();
1557  const symbol foo;
1558  const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
1559  ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
1560  if (arg_pt<_ex0)
1561  Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
1562  else
1563  Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
1564  epvector seq;
1565  if (order > 0) {
1566  seq.reserve(2);
1567  seq.push_back(expair(Order0correction, _ex0));
1568  }
1569  seq.push_back(expair(Order(_ex1), order));
1570  return series(replarg - pseries(rel, std::move(seq)), rel, order);
1571  }
1572  throw do_taylor();
1573 }
1574 
1575 static ex atanh_conjugate(const ex & x)
1576 {
1577  // conjugate(atanh(x))==atanh(conjugate(x)) unless on the branch cuts which
1578  // run along the real axis outside the interval [-1, +1].
1579  if (is_exactly_a<numeric>(x) &&
1580  (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
1581  return atanh(x.conjugate());
1582  }
1583  return conjugate_function(atanh(x)).hold();
1584 }
1585 
1587  evalf_func(atanh_evalf).
1588  derivative_func(atanh_deriv).
1589  series_func(atanh_series).
1590  conjugate_func(atanh_conjugate));
1591 
1592 
1593 } // namespace GiNaC
GiNaC::acosh_deriv
static ex acosh_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:1461
inifcns.h
Interface to GiNaC's initially known functions.
GiNaC::asinh_evalf
static ex asinh_evalf(const ex &x)
Definition: inifcns_trans.cpp:1364
GiNaC::_num3_p
const numeric * _num3_p
Definition: utils.cpp:200
GiNaC::tanh_series
static ex tanh_series(const ex &x, const relational &rel, int order, unsigned options)
Definition: inifcns_trans.cpp:1315
GiNaC::epvector
std::vector< expair > epvector
expair-vector
Definition: expairseq.h:33
GiNaC::ex::expand
ex expand(unsigned options=0) const
Definition: ex.cpp:73
GiNaC::info_flags::real
@ real
Definition: flags.h:221
GiNaC::const_iterator
Definition: ex.h:370
GiNaC::log_evalf
static ex log_evalf(const ex &x)
Definition: inifcns_trans.cpp:163
GiNaC::tan_deriv
static ex tan_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:669
constant.h
Interface to GiNaC's constant types and some special constants.
GiNaC::acosh_evalf
static ex acosh_evalf(const ex &x)
Definition: inifcns_trans.cpp:1425
GiNaC::real_part
ex real_part(const ex &thisex)
Definition: ex.h:721
GiNaC::log_expand
static ex log_expand(const ex &arg, unsigned options)
Definition: inifcns_trans.cpp:312
GiNaC::ex::end
const_iterator end() const noexcept
Definition: ex.h:652
GiNaC::pseries::is_terminating
bool is_terminating() const
Returns true if there is no order term, i.e.
Definition: pseries.cpp:586
GiNaC::cosh_deriv
static ex cosh_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:1223
GiNaC::atan
const numeric atan(const numeric &x)
Numeric arcustangent.
Definition: numeric.cpp:1508
GiNaC::info_flags::integer
@ integer
Definition: flags.h:223
GiNaC::atan_conjugate
static ex atan_conjugate(const ex &x)
Definition: inifcns_trans.cpp:965
GiNaC::atanh_deriv
static ex atanh_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:1521
GiNaC::info_flags::odd
@ odd
Definition: flags.h:233
GiNaC::info_flags::negative
@ negative
Definition: flags.h:227
mul.h
Interface to GiNaC's products of expressions.
GiNaC::pseries
This class holds a extended truncated power series (positive and negative integer powers).
Definition: pseries.h:36
GiNaC::cos_imag_part
static ex cos_imag_part(const ex &x)
Definition: inifcns_trans.cpp:578
GiNaC::tan_series
static ex tan_series(const ex &x, const relational &rel, int order, unsigned options)
Definition: inifcns_trans.cpp:691
GiNaC::cosh_conjugate
static ex cosh_conjugate(const ex &x)
Definition: inifcns_trans.cpp:1241
GiNaC::status_flags::expanded
@ expanded
.expand(0) has already done its job (other expand() options ignore this flag)
Definition: flags.h:204
GiNaC::log_eval
static ex log_eval(const ex &x)
Definition: inifcns_trans.cpp:171
GiNaC::_num10_p
const numeric * _num10_p
Definition: utils.cpp:228
GiNaC::sin_eval
static ex sin_eval(const ex &x)
Definition: inifcns_trans.cpp:395
GiNaC::cos_deriv
static ex cos_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:565
GiNaC::_ex0
const ex _ex0
Definition: utils.cpp:177
GiNaC::relational
This class holds a relation consisting of two expressions and a logical relation between them.
Definition: relational.h:35
numeric.h
Makes the interface to the underlying bignum package available.
GiNaC::_ex1_3
const ex _ex1_3
Definition: utils.cpp:185
GiNaC::_ex_1_4
const ex _ex_1_4
Definition: utils.cpp:172
GiNaC::do_taylor
Exception class thrown by classes which provide their own series expansion to signal that ordinary Ta...
Definition: function.h:668
GiNaC::exp_eval
static ex exp_eval(const ex &x)
Definition: inifcns_trans.cpp:54
GiNaC::exp_conjugate
static ex exp_conjugate(const ex &x)
Definition: inifcns_trans.cpp:125
GiNaC::_ex5
const ex _ex5
Definition: utils.cpp:209
GiNaC::ex::subs
ex subs(const exmap &m, unsigned options=0) const
Definition: ex.h:826
GiNaC::_num5_p
const numeric * _num5_p
Definition: utils.cpp:208
GiNaC::log_deriv
static ex log_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:200
add.h
Interface to GiNaC's sums of expressions.
GiNaC::atan2_eval
static ex atan2_eval(const ex &y, const ex &x)
Definition: inifcns_trans.cpp:1000
GiNaC::_num4_p
const numeric * _num4_p
Definition: utils.cpp:204
GiNaC::exp_evalf
static ex exp_evalf(const ex &x)
Definition: inifcns_trans.cpp:46
GiNaC::atan_evalf
static ex atan_evalf(const ex &x)
Definition: inifcns_trans.cpp:872
GiNaC::ex::conjugate
ex conjugate() const
Definition: ex.h:146
power.h
Interface to GiNaC's symbolic exponentiation (basis^exponent).
GiNaC::acos_conjugate
static ex acos_conjugate(const ex &x)
Definition: inifcns_trans.cpp:851
GiNaC::acos_eval
static ex acos_eval(const ex &x)
Definition: inifcns_trans.cpp:807
GiNaC::expand_options::expand_transcendental
@ expand_transcendental
expands transcendental functions like log and exp
Definition: flags.h:35
GiNaC::exvector
std::vector< ex > exvector
Definition: basic.h:46
GiNaC::ex::nops
size_t nops() const
Definition: ex.h:135
GiNaC::asin_deriv
static ex asin_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:770
GiNaC::series
ex series(const ex &thisex, const ex &r, int order, unsigned options=0)
Definition: ex.h:781
GiNaC::power
This class holds a two-component object, a basis and and exponent representing exponentiation.
Definition: power.h:39
GiNaC::asin
const numeric asin(const numeric &x)
Numeric inverse sine (trigonometric function).
Definition: numeric.cpp:1488
GiNaC::sinh_conjugate
static ex sinh_conjugate(const ex &x)
Definition: inifcns_trans.cpp:1157
GiNaC::cos_evalf
static ex cos_evalf(const ex &x)
Definition: inifcns_trans.cpp:494
GiNaC::info_flags::nonnegative
@ nonnegative
Definition: flags.h:228
GiNaC::tanh_conjugate
static ex tanh_conjugate(const ex &x)
Definition: inifcns_trans.cpp:1345
GiNaC::_ex1
const ex _ex1
Definition: utils.cpp:193
GiNaC::atanh_series
static ex atanh_series(const ex &arg, const relational &rel, int order, unsigned options)
Definition: inifcns_trans.cpp:1529
GiNaC::cosh_evalf
static ex cosh_evalf(const ex &x)
Definition: inifcns_trans.cpp:1175
relational.h
Interface to relations between expressions.
GiNaC::info_flags::positive
@ positive
Definition: flags.h:226
options
unsigned options
Definition: factor.cpp:2480
GiNaC::ex::is_equal
bool is_equal(const ex &other) const
Definition: ex.h:345
GiNaC::log_series
static ex log_series(const ex &arg, const relational &rel, int order, unsigned options)
Definition: inifcns_trans.cpp:208
GiNaC::tan_eval
static ex tan_eval(const ex &x)
Definition: inifcns_trans.cpp:609
GiNaC::add
Sum of expressions.
Definition: add.h:32
GiNaC::sin_conjugate
static ex sin_conjugate(const ex &x)
Definition: inifcns_trans.cpp:476
GiNaC::sqrt
const numeric sqrt(const numeric &x)
Numeric square root.
Definition: numeric.cpp:2475
GiNaC::acosh
const numeric acosh(const numeric &x)
Numeric inverse hyperbolic cosine (trigonometric function).
Definition: numeric.cpp:1590
GiNaC::_num30_p
const numeric * _num30_p
Definition: utils.cpp:260
GiNaC
Definition: add.cpp:38
GiNaC::_ex6
const ex _ex6
Definition: utils.cpp:213
GiNaC::sinh_deriv
static ex sinh_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:1139
GiNaC::tanh
const numeric tanh(const numeric &x)
Numeric hyperbolic tangent (trigonometric function).
Definition: numeric.cpp:1572
GiNaC::acosh_conjugate
static ex acosh_conjugate(const ex &x)
Definition: inifcns_trans.cpp:1469
GiNaC::_num120_p
const numeric * _num120_p
Definition: utils.cpp:272
GiNaC::asinh
const numeric asinh(const numeric &x)
Numeric inverse hyperbolic sine (trigonometric function).
Definition: numeric.cpp:1581
GiNaC::acosh_eval
static ex acosh_eval(const ex &x)
Definition: inifcns_trans.cpp:1433
x
ex x
Definition: factor.cpp:1641
GiNaC::ex::op
ex op(size_t i) const
Definition: ex.h:136
GiNaC::ex::info
bool info(unsigned inf) const
Definition: ex.h:132
utils.h
Interface to several small and furry utilities needed within GiNaC but not of any interest to the use...
GiNaC::sin
const numeric sin(const numeric &x)
Numeric sine (trigonometric function).
Definition: numeric.cpp:1461
GiNaC::ex
Lightweight wrapper for GiNaC's symbolic objects.
Definition: ex.h:72
GiNaC::ex::begin
const_iterator begin() const noexcept
Definition: ex.h:647
GiNaC::numeric::is_zero
bool is_zero() const
True if object is zero.
Definition: numeric.cpp:1129
GiNaC::csgn
int csgn(const numeric &x)
Definition: numeric.h:260
GiNaC::_ex2
const ex _ex2
Definition: utils.cpp:197
GiNaC::tan_conjugate
static ex tan_conjugate(const ex &x)
Definition: inifcns_trans.cpp:707
GiNaC::_num6_p
const numeric * _num6_p
Definition: utils.cpp:212
GiNaC::relational::lhs
ex lhs() const
Definition: relational.h:79
GiNaC::atanh_conjugate
static ex atanh_conjugate(const ex &x)
Definition: inifcns_trans.cpp:1575
GiNaC::ex::imag_part
ex imag_part() const
Definition: ex.h:148
GiNaC::cosh
const numeric cosh(const numeric &x)
Numeric hyperbolic cosine (trigonometric function).
Definition: numeric.cpp:1563
GiNaC::exp
const numeric exp(const numeric &x)
Exponential function.
Definition: numeric.cpp:1439
GiNaC::ex::diff
ex diff(const symbol &s, unsigned nth=1) const
Compute partial derivative of an expression.
Definition: ex.cpp:86
GiNaC::cos
const numeric cos(const numeric &x)
Numeric cosine (trigonometric function).
Definition: numeric.cpp:1470
GiNaC::tan_imag_part
static ex tan_imag_part(const ex &x)
Definition: inifcns_trans.cpp:684
GiNaC::atanh_eval
static ex atanh_eval(const ex &x)
Definition: inifcns_trans.cpp:1497
GiNaC::expand
ex expand(const ex &thisex, unsigned options=0)
Definition: ex.h:715
GiNaC::info_flags::rational
@ rational
Definition: flags.h:222
ex.h
Interface to GiNaC's light-weight expression handles.
GiNaC::basic::hold
const basic & hold() const
Stop further evaluation.
Definition: basic.cpp:887
GiNaC::I
const numeric I
Imaginary unit.
Definition: numeric.cpp:1433
GiNaC::pseries::add_series
ex add_series(const pseries &other) const
Add one series object to another, producing a pseries object that represents the sum.
Definition: pseries.cpp:681
symbol.h
Interface to GiNaC's symbolic objects.
GiNaC::sinh_eval
static ex sinh_eval(const ex &x)
Definition: inifcns_trans.cpp:1099
GiNaC::_ex3
const ex _ex3
Definition: utils.cpp:201
GiNaC::atan2_deriv
static ex atan2_deriv(const ex &y, const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:1071
GiNaC::atan_eval
static ex atan_eval(const ex &x)
Definition: inifcns_trans.cpp:880
GiNaC::ex::real_part
ex real_part() const
Definition: ex.h:147
GiNaC::_num20_p
const numeric * _num20_p
Definition: utils.cpp:248
GiNaC::acos
const numeric acos(const numeric &x)
Numeric inverse cosine (trigonometric function).
Definition: numeric.cpp:1497
GiNaC::exp_imag_part
static ex exp_imag_part(const ex &x)
Definition: inifcns_trans.cpp:120
GiNaC::cosh_imag_part
static ex cosh_imag_part(const ex &x)
Definition: inifcns_trans.cpp:1236
GiNaC::sin_evalf
static ex sin_evalf(const ex &x)
Definition: inifcns_trans.cpp:387
GiNaC::tanh_real_part
static ex tanh_real_part(const ex &x)
Definition: inifcns_trans.cpp:1331
GiNaC::sinh_real_part
static ex sinh_real_part(const ex &x)
Definition: inifcns_trans.cpp:1147
GiNaC::atanh_evalf
static ex atanh_evalf(const ex &x)
Definition: inifcns_trans.cpp:1489
GiNaC::info_flags::numeric
@ numeric
Definition: flags.h:220
GiNaC::Pi
const constant Pi("Pi", PiEvalf, "\\pi", domain::positive)
Pi.
Definition: constant.h:82
GiNaC::_ex1_4
const ex _ex1_4
Definition: utils.cpp:181
GiNaC::basic::setflag
const basic & setflag(unsigned f) const
Set some status_flags.
Definition: basic.h:288
GiNaC::cosh_real_part
static ex cosh_real_part(const ex &x)
Definition: inifcns_trans.cpp:1231
GiNaC::_num_1_p
const numeric * _num_1_p
Definition: utils.cpp:159
GiNaC::pseries::nops
size_t nops() const override
Return the number of operands including a possible order term.
Definition: pseries.cpp:296
GiNaC::relational::rhs
ex rhs() const
Definition: relational.h:80
GiNaC::subs_options::no_pattern
@ no_pattern
disable pattern matching
Definition: flags.h:51
GiNaC::pseries::ldegree
int ldegree(const ex &s) const override
Return degree of lowest power of the series.
Definition: pseries.cpp:335
GiNaC::_num18_p
const numeric * _num18_p
Definition: utils.cpp:244
GiNaC::REGISTER_FUNCTION
REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval). evalf_func(conjugate_evalf). expl_derivative_func(conjugate_expl_derivative). info_func(conjugate_info). print_func< print_latex >(conjugate_print_latex). conjugate_func(conjugate_conjugate). real_part_func(conjugate_real_part). imag_part_func(conjugate_imag_part). set_name("conjugate","conjugate"))
GiNaC::sinh
const numeric sinh(const numeric &x)
Numeric hyperbolic sine (trigonometric function).
Definition: numeric.cpp:1554
GiNaC::cosh_eval
static ex cosh_eval(const ex &x)
Definition: inifcns_trans.cpp:1183
GiNaC::pole_error
Exception class thrown when a singularity is encountered.
Definition: numeric.h:70
GiNaC::tanh_evalf
static ex tanh_evalf(const ex &x)
Definition: inifcns_trans.cpp:1259
GiNaC::tan_real_part
static ex tan_real_part(const ex &x)
Definition: inifcns_trans.cpp:677
GiNaC::ex::series
ex series(const ex &r, int order, unsigned options=0) const
Compute the truncated series expansion of an expression.
Definition: pseries.cpp:1259
GiNaC::imag_part
ex imag_part(const ex &thisex)
Definition: ex.h:724
GiNaC::real
const numeric real(const numeric &x)
Definition: numeric.h:311
GiNaC::coeff
ex coeff(const ex &thisex, const ex &s, int n=1)
Definition: ex.h:742
GiNaC::numeric::is_equal
bool is_equal(const numeric &other) const
Definition: numeric.cpp:1122
GiNaC::mod
const numeric mod(const numeric &a, const numeric &b)
Modulus (in positive representation).
Definition: numeric.cpp:2328
GiNaC::_ex1_2
const ex _ex1_2
Definition: utils.cpp:189
GiNaC::atan_deriv
static ex atan_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:911
GiNaC::status_flags::purely_indefinite
@ purely_indefinite
Definition: flags.h:211
GiNaC::symbol
Basic CAS symbol.
Definition: symbol.h:39
n
size_t n
Definition: factor.cpp:1463
GiNaC::cos_real_part
static ex cos_real_part(const ex &x)
Definition: inifcns_trans.cpp:573
GiNaC::pow
const numeric pow(const numeric &x, const numeric &y)
Definition: numeric.h:251
pseries.h
Interface to class for extended truncated power series.
GiNaC::cos_eval
static ex cos_eval(const ex &x)
Definition: inifcns_trans.cpp:502
GiNaC::_num12_p
const numeric * _num12_p
Definition: utils.cpp:236
GiNaC::sin_real_part
static ex sin_real_part(const ex &x)
Definition: inifcns_trans.cpp:466
GiNaC::is_zero
bool is_zero(const ex &thisex)
Definition: ex.h:820
GiNaC::_num15_p
const numeric * _num15_p
Definition: utils.cpp:240
GiNaC::expair
A pair of expressions.
Definition: expair.h:38
GiNaC::_num0_p
const numeric * _num0_p
Definition: utils.cpp:175
GiNaC::exp_expand
static ex exp_expand(const ex &arg, unsigned options)
Definition: inifcns_trans.cpp:86
GiNaC::tan_evalf
static ex tan_evalf(const ex &x)
Definition: inifcns_trans.cpp:601
GiNaC::exp_real_part
static ex exp_real_part(const ex &x)
Definition: inifcns_trans.cpp:115
GiNaC::_ex60
const ex _ex60
Definition: utils.cpp:269
GiNaC::_ex_1_3
const ex _ex_1_3
Definition: utils.cpp:168
GiNaC::ex::is_zero
bool is_zero() const
Definition: ex.h:213
GiNaC::tanh_eval
static ex tanh_eval(const ex &x)
Definition: inifcns_trans.cpp:1267
is_ex_the_function
#define is_ex_the_function(OBJ, FUNCNAME)
Definition: function.h:765
GiNaC::exp_deriv
static ex exp_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:107
GiNaC::sin_imag_part
static ex sin_imag_part(const ex &x)
Definition: inifcns_trans.cpp:471
GiNaC::log
const numeric log(const numeric &x)
Natural logarithm.
Definition: numeric.cpp:1450
GiNaC::expand_options::expand_function_args
@ expand_function_args
expands the arguments of functions
Definition: flags.h:33
GiNaC::info_flags::crational
@ crational
Definition: flags.h:224
GiNaC::log_conjugate
static ex log_conjugate(const ex &x)
Definition: inifcns_trans.cpp:359
GiNaC::tan
const numeric tan(const numeric &x)
Numeric tangent (trigonometric function).
Definition: numeric.cpp:1479
GiNaC::abs
const numeric abs(const numeric &x)
Absolute value.
Definition: numeric.cpp:2315
GiNaC::sin_deriv
static ex sin_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:458
GiNaC::pseries::coeff
ex coeff(const ex &s, int n=1) const override
Return coefficient of degree n in power series if s is the expansion variable.
Definition: pseries.cpp:357
GiNaC::log_real_part
static ex log_real_part(const ex &x)
Definition: inifcns_trans.cpp:298
GiNaC::atan2_evalf
static ex atan2_evalf(const ex &y, const ex &x)
Definition: inifcns_trans.cpp:992
GiNaC::_num1_p
const numeric * _num1_p
Definition: utils.cpp:192
GiNaC::asinh_deriv
static ex asinh_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:1392
GiNaC::asinh_conjugate
static ex asinh_conjugate(const ex &x)
Definition: inifcns_trans.cpp:1400
GiNaC::_ex_1_2
const ex _ex_1_2
Definition: utils.cpp:164
GiNaC::sinh_evalf
static ex sinh_evalf(const ex &x)
Definition: inifcns_trans.cpp:1091
GiNaC::asin_eval
static ex asin_eval(const ex &x)
Definition: inifcns_trans.cpp:734
GiNaC::tanh_imag_part
static ex tanh_imag_part(const ex &x)
Definition: inifcns_trans.cpp:1338
GiNaC::_num60_p
const numeric * _num60_p
Definition: utils.cpp:268
GiNaC::_num24_p
const numeric * _num24_p
Definition: utils.cpp:252
GiNaC::asin_conjugate
static ex asin_conjugate(const ex &x)
Definition: inifcns_trans.cpp:778
GiNaC::acos_deriv
static ex acos_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:843
GiNaC::_num25_p
const numeric * _num25_p
Definition: utils.cpp:256
order
int order
Definition: integration_kernel.cpp:248
operators.h
Interface to GiNaC's overloaded operators.
GiNaC::tanh_deriv
static ex tanh_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns_trans.cpp:1307
GiNaC::_ex_1
const ex _ex_1
Definition: utils.cpp:160
GiNaC::numeric
This class is a wrapper around CLN-numbers within the GiNaC class hierarchy.
Definition: numeric.h:82
GINAC_ASSERT
#define GINAC_ASSERT(X)
Assertion macro for checking invariances.
Definition: assertion.h:33
GiNaC::_num2_p
const numeric * _num2_p
Definition: utils.cpp:196
GiNaC::cos_conjugate
static ex cos_conjugate(const ex &x)
Definition: inifcns_trans.cpp:583
GiNaC::series_options::suppress_branchcut
@ suppress_branchcut
Suppress branch cuts in series expansion.
Definition: flags.h:83
GiNaC::acos_evalf
static ex acos_evalf(const ex &x)
Definition: inifcns_trans.cpp:799
GiNaC::atanh
const numeric atanh(const numeric &x)
Numeric inverse hyperbolic tangent (trigonometric function).
Definition: numeric.cpp:1599
GiNaC::atan_series
static ex atan_series(const ex &arg, const relational &rel, int order, unsigned options)
Definition: inifcns_trans.cpp:919
GiNaC::log_imag_part
static ex log_imag_part(const ex &x)
Definition: inifcns_trans.cpp:305
GiNaC::asinh_eval
static ex asinh_eval(const ex &x)
Definition: inifcns_trans.cpp:1372
GiNaC::mul
Product of expressions.
Definition: mul.h:32
GiNaC::exp_power
static ex exp_power(const ex &x, const ex &a)
Definition: inifcns_trans.cpp:131
GiNaC::asin_evalf
static ex asin_evalf(const ex &x)
Definition: inifcns_trans.cpp:726
GiNaC::sinh_imag_part
static ex sinh_imag_part(const ex &x)
Definition: inifcns_trans.cpp:1152
GiNaC::info_flags::indefinite
@ indefinite
Definition: flags.h:273

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