1 /** @file inifcns_trans.cpp
3 * Implementation of transcendental (and trigonometric and hyperbolic)
7 * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
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.
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.
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
32 #include "relational.h"
37 #ifndef NO_NAMESPACE_GINAC
39 #endif // ndef NO_NAMESPACE_GINAC
42 // exponential function
45 static ex exp_evalf(const ex & x)
51 return exp(ex_to_numeric(x)); // -> numeric exp(numeric)
54 static ex exp_eval(const ex & x)
60 // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
61 ex TwoExOverPiI=(_ex2()*x)/(Pi*I);
62 if (TwoExOverPiI.info(info_flags::integer)) {
63 numeric z=mod(ex_to_numeric(TwoExOverPiI),_num4());
64 if (z.is_equal(_num0()))
66 if (z.is_equal(_num1()))
68 if (z.is_equal(_num2()))
70 if (z.is_equal(_num3()))
74 if (is_ex_the_function(x, log))
78 if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
84 static ex exp_deriv(const ex & x, unsigned deriv_param)
86 GINAC_ASSERT(deriv_param==0);
88 // d/dx exp(x) -> exp(x)
92 REGISTER_FUNCTION(exp, eval_func(exp_eval).
93 evalf_func(exp_evalf).
94 derivative_func(exp_deriv));
100 static ex log_evalf(const ex & x)
104 END_TYPECHECK(log(x))
106 return log(ex_to_numeric(x)); // -> numeric log(numeric)
109 static ex log_eval(const ex & x)
111 if (x.info(info_flags::numeric)) {
112 if (x.is_equal(_ex0())) // log(0) -> infinity
113 throw(pole_error("log_eval(): log(0)",0));
114 if (x.info(info_flags::real) && x.info(info_flags::negative))
115 return (log(-x)+I*Pi);
116 if (x.is_equal(_ex1())) // log(1) -> 0
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());
123 if (!x.info(info_flags::crational))
126 // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
127 if (is_ex_the_function(x, exp)) {
129 if (t.info(info_flags::numeric)) {
130 numeric nt = ex_to_numeric(t);
136 return log(x).hold();
139 static ex log_deriv(const ex & x, unsigned deriv_param)
141 GINAC_ASSERT(deriv_param==0);
143 // d/dx log(x) -> 1/x
144 return power(x, _ex_1());
147 static ex log_series(const ex &arg,
148 const relational &rel,
152 GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
154 bool must_expand_arg = false;
155 // maybe substitution of rel into arg fails because of a pole
157 arg_pt = arg.subs(rel);
158 } catch (pole_error) {
159 must_expand_arg = true;
161 // or we are at the branch cut anyways
162 if (arg_pt.is_zero())
163 must_expand_arg = true;
165 if (must_expand_arg) {
167 // This is the branch point: Series expand the argument first, then
168 // trivially factorize it to isolate that part which has constant
169 // leading coefficient in this fashion:
170 // x^n + Order(x^(n+m)) -> x^n * (1 + Order(x^m)).
171 // Return a plain n*log(x) for the x^n part and series expand the
172 // other part. Add them together and reexpand again in order to have
173 // one unnested pseries object. All this also works for negative n.
174 const pseries argser = ex_to_pseries(arg.series(rel, order, options));
175 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
176 const ex point = rel.rhs();
177 const int n = argser.ldegree(*s);
179 seq.push_back(expair(n*log(*s-point), _ex0()));
180 if (!argser.is_terminating() || argser.nops()!=1) {
181 // in this case n more terms are needed
182 ex newarg = ex_to_pseries(arg.series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
183 return pseries(rel, seq).add_series(ex_to_pseries(log(newarg).series(rel, order, options)));
184 } else // it was a monomial
185 return pseries(rel, seq);
187 if (!(options & series_options::suppress_branchcut) &&
188 arg_pt.info(info_flags::negative)) {
190 // This is the branch cut: assemble the primitive series manually and
191 // then add the corresponding complex step function.
192 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
193 const ex point = rel.rhs();
195 ex replarg = series(log(arg), *s==foo, order, false).subs(foo==point);
197 seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0()));
198 seq.push_back(expair(Order(_ex1()), order));
199 return series(replarg - I*Pi + pseries(rel, seq), rel, order);
201 throw do_taylor(); // caught by function::series()
204 REGISTER_FUNCTION(log, eval_func(log_eval).
205 evalf_func(log_evalf).
206 derivative_func(log_deriv).
207 series_func(log_series));
210 // sine (trigonometric function)
213 static ex sin_evalf(const ex & x)
217 END_TYPECHECK(sin(x))
219 return sin(ex_to_numeric(x)); // -> numeric sin(numeric)
222 static ex sin_eval(const ex & x)
224 // sin(n/d*Pi) -> { all known non-nested radicals }
225 ex SixtyExOverPi = _ex60()*x/Pi;
227 if (SixtyExOverPi.info(info_flags::integer)) {
228 numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
230 // wrap to interval [0, Pi)
235 // wrap to interval [0, Pi/2)
238 if (z.is_equal(_num0())) // sin(0) -> 0
240 if (z.is_equal(_num5())) // sin(Pi/12) -> sqrt(6)/4*(1-sqrt(3)/3)
241 return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
242 if (z.is_equal(_num6())) // sin(Pi/10) -> sqrt(5)/4-1/4
243 return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
244 if (z.is_equal(_num10())) // sin(Pi/6) -> 1/2
245 return sign*_ex1_2();
246 if (z.is_equal(_num15())) // sin(Pi/4) -> sqrt(2)/2
247 return sign*_ex1_2()*power(_ex2(),_ex1_2());
248 if (z.is_equal(_num18())) // sin(3/10*Pi) -> sqrt(5)/4+1/4
249 return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
250 if (z.is_equal(_num20())) // sin(Pi/3) -> sqrt(3)/2
251 return sign*_ex1_2()*power(_ex3(),_ex1_2());
252 if (z.is_equal(_num25())) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
253 return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
254 if (z.is_equal(_num30())) // sin(Pi/2) -> 1
258 if (is_ex_exactly_of_type(x, function)) {
261 if (is_ex_the_function(x, asin))
263 // sin(acos(x)) -> sqrt(1-x^2)
264 if (is_ex_the_function(x, acos))
265 return power(_ex1()-power(t,_ex2()),_ex1_2());
266 // sin(atan(x)) -> x*(1+x^2)^(-1/2)
267 if (is_ex_the_function(x, atan))
268 return t*power(_ex1()+power(t,_ex2()),_ex_1_2());
271 // sin(float) -> float
272 if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
275 return sin(x).hold();
278 static ex sin_deriv(const ex & x, unsigned deriv_param)
280 GINAC_ASSERT(deriv_param==0);
282 // d/dx sin(x) -> cos(x)
286 REGISTER_FUNCTION(sin, eval_func(sin_eval).
287 evalf_func(sin_evalf).
288 derivative_func(sin_deriv));
291 // cosine (trigonometric function)
294 static ex cos_evalf(const ex & x)
298 END_TYPECHECK(cos(x))
300 return cos(ex_to_numeric(x)); // -> numeric cos(numeric)
303 static ex cos_eval(const ex & x)
305 // cos(n/d*Pi) -> { all known non-nested radicals }
306 ex SixtyExOverPi = _ex60()*x/Pi;
308 if (SixtyExOverPi.info(info_flags::integer)) {
309 numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
311 // wrap to interval [0, Pi)
315 // wrap to interval [0, Pi/2)
319 if (z.is_equal(_num0())) // cos(0) -> 1
321 if (z.is_equal(_num5())) // cos(Pi/12) -> sqrt(6)/4*(1+sqrt(3)/3)
322 return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
323 if (z.is_equal(_num10())) // cos(Pi/6) -> sqrt(3)/2
324 return sign*_ex1_2()*power(_ex3(),_ex1_2());
325 if (z.is_equal(_num12())) // cos(Pi/5) -> sqrt(5)/4+1/4
326 return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
327 if (z.is_equal(_num15())) // cos(Pi/4) -> sqrt(2)/2
328 return sign*_ex1_2()*power(_ex2(),_ex1_2());
329 if (z.is_equal(_num20())) // cos(Pi/3) -> 1/2
330 return sign*_ex1_2();
331 if (z.is_equal(_num24())) // cos(2/5*Pi) -> sqrt(5)/4-1/4x
332 return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
333 if (z.is_equal(_num25())) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
334 return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
335 if (z.is_equal(_num30())) // cos(Pi/2) -> 0
339 if (is_ex_exactly_of_type(x, function)) {
342 if (is_ex_the_function(x, acos))
344 // cos(asin(x)) -> (1-x^2)^(1/2)
345 if (is_ex_the_function(x, asin))
346 return power(_ex1()-power(t,_ex2()),_ex1_2());
347 // cos(atan(x)) -> (1+x^2)^(-1/2)
348 if (is_ex_the_function(x, atan))
349 return power(_ex1()+power(t,_ex2()),_ex_1_2());
352 // cos(float) -> float
353 if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
356 return cos(x).hold();
359 static ex cos_deriv(const ex & x, unsigned deriv_param)
361 GINAC_ASSERT(deriv_param==0);
363 // d/dx cos(x) -> -sin(x)
364 return _ex_1()*sin(x);
367 REGISTER_FUNCTION(cos, eval_func(cos_eval).
368 evalf_func(cos_evalf).
369 derivative_func(cos_deriv));
372 // tangent (trigonometric function)
375 static ex tan_evalf(const ex & x)
379 END_TYPECHECK(tan(x)) // -> numeric tan(numeric)
381 return tan(ex_to_numeric(x));
384 static ex tan_eval(const ex & x)
386 // tan(n/d*Pi) -> { all known non-nested radicals }
387 ex SixtyExOverPi = _ex60()*x/Pi;
389 if (SixtyExOverPi.info(info_flags::integer)) {
390 numeric z = mod(ex_to_numeric(SixtyExOverPi),_num60());
392 // wrap to interval [0, Pi)
396 // wrap to interval [0, Pi/2)
400 if (z.is_equal(_num0())) // tan(0) -> 0
402 if (z.is_equal(_num5())) // tan(Pi/12) -> 2-sqrt(3)
403 return sign*(_ex2()-power(_ex3(),_ex1_2()));
404 if (z.is_equal(_num10())) // tan(Pi/6) -> sqrt(3)/3
405 return sign*_ex1_3()*power(_ex3(),_ex1_2());
406 if (z.is_equal(_num15())) // tan(Pi/4) -> 1
408 if (z.is_equal(_num20())) // tan(Pi/3) -> sqrt(3)
409 return sign*power(_ex3(),_ex1_2());
410 if (z.is_equal(_num25())) // tan(5/12*Pi) -> 2+sqrt(3)
411 return sign*(power(_ex3(),_ex1_2())+_ex2());
412 if (z.is_equal(_num30())) // tan(Pi/2) -> infinity
413 throw (pole_error("tan_eval(): simple pole",1));
416 if (is_ex_exactly_of_type(x, function)) {
419 if (is_ex_the_function(x, atan))
421 // tan(asin(x)) -> x*(1+x^2)^(-1/2)
422 if (is_ex_the_function(x, asin))
423 return t*power(_ex1()-power(t,_ex2()),_ex_1_2());
424 // tan(acos(x)) -> (1-x^2)^(1/2)/x
425 if (is_ex_the_function(x, acos))
426 return power(t,_ex_1())*power(_ex1()-power(t,_ex2()),_ex1_2());
429 // tan(float) -> float
430 if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
434 return tan(x).hold();
437 static ex tan_deriv(const ex & x, unsigned deriv_param)
439 GINAC_ASSERT(deriv_param==0);
441 // d/dx tan(x) -> 1+tan(x)^2;
442 return (_ex1()+power(tan(x),_ex2()));
445 static ex tan_series(const ex &x,
446 const relational &rel,
451 // Taylor series where there is no pole falls back to tan_deriv.
452 // On a pole simply expand sin(x)/cos(x).
453 const ex x_pt = x.subs(rel);
454 if (!(2*x_pt/Pi).info(info_flags::odd))
455 throw do_taylor(); // caught by function::series()
456 // if we got here we have to care for a simple pole
457 return (sin(x)/cos(x)).series(rel, order+2);
460 REGISTER_FUNCTION(tan, eval_func(tan_eval).
461 evalf_func(tan_evalf).
462 derivative_func(tan_deriv).
463 series_func(tan_series));
466 // inverse sine (arc sine)
469 static ex asin_evalf(const ex & x)
473 END_TYPECHECK(asin(x))
475 return asin(ex_to_numeric(x)); // -> numeric asin(numeric)
478 static ex asin_eval(const ex & x)
480 if (x.info(info_flags::numeric)) {
485 if (x.is_equal(_ex1_2()))
486 return numeric(1,6)*Pi;
488 if (x.is_equal(_ex1()))
490 // asin(-1/2) -> -Pi/6
491 if (x.is_equal(_ex_1_2()))
492 return numeric(-1,6)*Pi;
494 if (x.is_equal(_ex_1()))
495 return _num_1_2()*Pi;
496 // asin(float) -> float
497 if (!x.info(info_flags::crational))
498 return asin_evalf(x);
501 return asin(x).hold();
504 static ex asin_deriv(const ex & x, unsigned deriv_param)
506 GINAC_ASSERT(deriv_param==0);
508 // d/dx asin(x) -> 1/sqrt(1-x^2)
509 return power(1-power(x,_ex2()),_ex_1_2());
512 REGISTER_FUNCTION(asin, eval_func(asin_eval).
513 evalf_func(asin_evalf).
514 derivative_func(asin_deriv));
517 // inverse cosine (arc cosine)
520 static ex acos_evalf(const ex & x)
524 END_TYPECHECK(acos(x))
526 return acos(ex_to_numeric(x)); // -> numeric acos(numeric)
529 static ex acos_eval(const ex & x)
531 if (x.info(info_flags::numeric)) {
533 if (x.is_equal(_ex1()))
536 if (x.is_equal(_ex1_2()))
541 // acos(-1/2) -> 2/3*Pi
542 if (x.is_equal(_ex_1_2()))
543 return numeric(2,3)*Pi;
545 if (x.is_equal(_ex_1()))
547 // acos(float) -> float
548 if (!x.info(info_flags::crational))
549 return acos_evalf(x);
552 return acos(x).hold();
555 static ex acos_deriv(const ex & x, unsigned deriv_param)
557 GINAC_ASSERT(deriv_param==0);
559 // d/dx acos(x) -> -1/sqrt(1-x^2)
560 return _ex_1()*power(1-power(x,_ex2()),_ex_1_2());
563 REGISTER_FUNCTION(acos, eval_func(acos_eval).
564 evalf_func(acos_evalf).
565 derivative_func(acos_deriv));
568 // inverse tangent (arc tangent)
571 static ex atan_evalf(const ex & x)
575 END_TYPECHECK(atan(x))
577 return atan(ex_to_numeric(x)); // -> numeric atan(numeric)
580 static ex atan_eval(const ex & x)
582 if (x.info(info_flags::numeric)) {
584 if (x.is_equal(_ex0()))
586 // atan(float) -> float
587 if (!x.info(info_flags::crational))
588 return atan_evalf(x);
591 return atan(x).hold();
594 static ex atan_deriv(const ex & x, unsigned deriv_param)
596 GINAC_ASSERT(deriv_param==0);
598 // d/dx atan(x) -> 1/(1+x^2)
599 return power(_ex1()+power(x,_ex2()), _ex_1());
602 REGISTER_FUNCTION(atan, eval_func(atan_eval).
603 evalf_func(atan_evalf).
604 derivative_func(atan_deriv));
607 // inverse tangent (atan2(y,x))
610 static ex atan2_evalf(const ex & y, const ex & x)
615 END_TYPECHECK(atan2(y,x))
617 return atan(ex_to_numeric(y),ex_to_numeric(x)); // -> numeric atan(numeric)
620 static ex atan2_eval(const ex & y, const ex & x)
622 if (y.info(info_flags::numeric) && !y.info(info_flags::crational) &&
623 x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
624 return atan2_evalf(y,x);
627 return atan2(y,x).hold();
630 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
632 GINAC_ASSERT(deriv_param<2);
634 if (deriv_param==0) {
636 return x*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
639 return -y*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
642 REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
643 evalf_func(atan2_evalf).
644 derivative_func(atan2_deriv));
647 // hyperbolic sine (trigonometric function)
650 static ex sinh_evalf(const ex & x)
654 END_TYPECHECK(sinh(x))
656 return sinh(ex_to_numeric(x)); // -> numeric sinh(numeric)
659 static ex sinh_eval(const ex & x)
661 if (x.info(info_flags::numeric)) {
662 if (x.is_zero()) // sinh(0) -> 0
664 if (!x.info(info_flags::crational)) // sinh(float) -> float
665 return sinh_evalf(x);
668 if ((x/Pi).info(info_flags::numeric) &&
669 ex_to_numeric(x/Pi).real().is_zero()) // sinh(I*x) -> I*sin(x)
672 if (is_ex_exactly_of_type(x, function)) {
674 // sinh(asinh(x)) -> x
675 if (is_ex_the_function(x, asinh))
677 // sinh(acosh(x)) -> (x-1)^(1/2) * (x+1)^(1/2)
678 if (is_ex_the_function(x, acosh))
679 return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2());
680 // sinh(atanh(x)) -> x*(1-x^2)^(-1/2)
681 if (is_ex_the_function(x, atanh))
682 return t*power(_ex1()-power(t,_ex2()),_ex_1_2());
685 return sinh(x).hold();
688 static ex sinh_deriv(const ex & x, unsigned deriv_param)
690 GINAC_ASSERT(deriv_param==0);
692 // d/dx sinh(x) -> cosh(x)
696 REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
697 evalf_func(sinh_evalf).
698 derivative_func(sinh_deriv));
701 // hyperbolic cosine (trigonometric function)
704 static ex cosh_evalf(const ex & x)
708 END_TYPECHECK(cosh(x))
710 return cosh(ex_to_numeric(x)); // -> numeric cosh(numeric)
713 static ex cosh_eval(const ex & x)
715 if (x.info(info_flags::numeric)) {
716 if (x.is_zero()) // cosh(0) -> 1
718 if (!x.info(info_flags::crational)) // cosh(float) -> float
719 return cosh_evalf(x);
722 if ((x/Pi).info(info_flags::numeric) &&
723 ex_to_numeric(x/Pi).real().is_zero()) // cosh(I*x) -> cos(x)
726 if (is_ex_exactly_of_type(x, function)) {
728 // cosh(acosh(x)) -> x
729 if (is_ex_the_function(x, acosh))
731 // cosh(asinh(x)) -> (1+x^2)^(1/2)
732 if (is_ex_the_function(x, asinh))
733 return power(_ex1()+power(t,_ex2()),_ex1_2());
734 // cosh(atanh(x)) -> (1-x^2)^(-1/2)
735 if (is_ex_the_function(x, atanh))
736 return power(_ex1()-power(t,_ex2()),_ex_1_2());
739 return cosh(x).hold();
742 static ex cosh_deriv(const ex & x, unsigned deriv_param)
744 GINAC_ASSERT(deriv_param==0);
746 // d/dx cosh(x) -> sinh(x)
750 REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
751 evalf_func(cosh_evalf).
752 derivative_func(cosh_deriv));
756 // hyperbolic tangent (trigonometric function)
759 static ex tanh_evalf(const ex & x)
763 END_TYPECHECK(tanh(x))
765 return tanh(ex_to_numeric(x)); // -> numeric tanh(numeric)
768 static ex tanh_eval(const ex & x)
770 if (x.info(info_flags::numeric)) {
771 if (x.is_zero()) // tanh(0) -> 0
773 if (!x.info(info_flags::crational)) // tanh(float) -> float
774 return tanh_evalf(x);
777 if ((x/Pi).info(info_flags::numeric) &&
778 ex_to_numeric(x/Pi).real().is_zero()) // tanh(I*x) -> I*tan(x);
781 if (is_ex_exactly_of_type(x, function)) {
783 // tanh(atanh(x)) -> x
784 if (is_ex_the_function(x, atanh))
786 // tanh(asinh(x)) -> x*(1+x^2)^(-1/2)
787 if (is_ex_the_function(x, asinh))
788 return t*power(_ex1()+power(t,_ex2()),_ex_1_2());
789 // tanh(acosh(x)) -> (x-1)^(1/2)*(x+1)^(1/2)/x
790 if (is_ex_the_function(x, acosh))
791 return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2())*power(t,_ex_1());
794 return tanh(x).hold();
797 static ex tanh_deriv(const ex & x, unsigned deriv_param)
799 GINAC_ASSERT(deriv_param==0);
801 // d/dx tanh(x) -> 1-tanh(x)^2
802 return _ex1()-power(tanh(x),_ex2());
805 static ex tanh_series(const ex &x,
806 const relational &rel,
811 // Taylor series where there is no pole falls back to tanh_deriv.
812 // On a pole simply expand sinh(x)/cosh(x).
813 const ex x_pt = x.subs(rel);
814 if (!(2*I*x_pt/Pi).info(info_flags::odd))
815 throw do_taylor(); // caught by function::series()
816 // if we got here we have to care for a simple pole
817 return (sinh(x)/cosh(x)).series(rel, order+2);
820 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
821 evalf_func(tanh_evalf).
822 derivative_func(tanh_deriv).
823 series_func(tanh_series));
826 // inverse hyperbolic sine (trigonometric function)
829 static ex asinh_evalf(const ex & x)
833 END_TYPECHECK(asinh(x))
835 return asinh(ex_to_numeric(x)); // -> numeric asinh(numeric)
838 static ex asinh_eval(const ex & x)
840 if (x.info(info_flags::numeric)) {
844 // asinh(float) -> float
845 if (!x.info(info_flags::crational))
846 return asinh_evalf(x);
849 return asinh(x).hold();
852 static ex asinh_deriv(const ex & x, unsigned deriv_param)
854 GINAC_ASSERT(deriv_param==0);
856 // d/dx asinh(x) -> 1/sqrt(1+x^2)
857 return power(_ex1()+power(x,_ex2()),_ex_1_2());
860 REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
861 evalf_func(asinh_evalf).
862 derivative_func(asinh_deriv));
865 // inverse hyperbolic cosine (trigonometric function)
868 static ex acosh_evalf(const ex & x)
872 END_TYPECHECK(acosh(x))
874 return acosh(ex_to_numeric(x)); // -> numeric acosh(numeric)
877 static ex acosh_eval(const ex & x)
879 if (x.info(info_flags::numeric)) {
880 // acosh(0) -> Pi*I/2
882 return Pi*I*numeric(1,2);
884 if (x.is_equal(_ex1()))
887 if (x.is_equal(_ex_1()))
889 // acosh(float) -> float
890 if (!x.info(info_flags::crational))
891 return acosh_evalf(x);
894 return acosh(x).hold();
897 static ex acosh_deriv(const ex & x, unsigned deriv_param)
899 GINAC_ASSERT(deriv_param==0);
901 // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
902 return power(x+_ex_1(),_ex_1_2())*power(x+_ex1(),_ex_1_2());
905 REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
906 evalf_func(acosh_evalf).
907 derivative_func(acosh_deriv));
910 // inverse hyperbolic tangent (trigonometric function)
913 static ex atanh_evalf(const ex & x)
917 END_TYPECHECK(atanh(x))
919 return atanh(ex_to_numeric(x)); // -> numeric atanh(numeric)
922 static ex atanh_eval(const ex & x)
924 if (x.info(info_flags::numeric)) {
928 // atanh({+|-}1) -> throw
929 if (x.is_equal(_ex1()) || x.is_equal(_ex_1()))
930 throw (pole_error("atanh_eval(): logarithmic pole",0));
931 // atanh(float) -> float
932 if (!x.info(info_flags::crational))
933 return atanh_evalf(x);
936 return atanh(x).hold();
939 static ex atanh_deriv(const ex & x, unsigned deriv_param)
941 GINAC_ASSERT(deriv_param==0);
943 // d/dx atanh(x) -> 1/(1-x^2)
944 return power(_ex1()-power(x,_ex2()),_ex_1());
947 REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
948 evalf_func(atanh_evalf).
949 derivative_func(atanh_deriv));
951 #ifndef NO_NAMESPACE_GINAC
953 #endif // ndef NO_NAMESPACE_GINAC