* functions. */
/*
- * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2020 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
- * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
-#include <vector>
-#include <stdexcept>
-
#include "inifcns.h"
#include "ex.h"
#include "constant.h"
+#include "add.h"
+#include "mul.h"
#include "numeric.h"
#include "power.h"
#include "operators.h"
#include "pseries.h"
#include "utils.h"
+#include <stdexcept>
+#include <vector>
+
namespace GiNaC {
//////////
if (x.is_zero()) {
return _ex1;
}
+
// exp(n*Pi*I/2) -> {+1|+I|-1|-I}
const ex TwoExOverPiI=(_ex2*x)/(Pi*I);
if (TwoExOverPiI.info(info_flags::integer)) {
- const numeric z = mod(ex_to<numeric>(TwoExOverPiI),_num4);
- if (z.is_equal(_num0))
+ const numeric z = mod(ex_to<numeric>(TwoExOverPiI),*_num4_p);
+ if (z.is_equal(*_num0_p))
return _ex1;
- if (z.is_equal(_num1))
+ if (z.is_equal(*_num1_p))
return ex(I);
- if (z.is_equal(_num2))
+ if (z.is_equal(*_num2_p))
return _ex_1;
- if (z.is_equal(_num3))
+ if (z.is_equal(*_num3_p))
return ex(-I);
}
+
// exp(log(x)) -> x
if (is_ex_the_function(x, log))
return x.op(0);
- // exp(float)
+ // exp(float) -> float
if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
return exp(ex_to<numeric>(x));
return exp(x).hold();
}
+static ex exp_expand(const ex & arg, unsigned options)
+{
+ ex exp_arg;
+ if (options & expand_options::expand_function_args)
+ exp_arg = arg.expand(options);
+ else
+ exp_arg=arg;
+
+ if ((options & expand_options::expand_transcendental)
+ && is_exactly_a<add>(exp_arg)) {
+ exvector prodseq;
+ prodseq.reserve(exp_arg.nops());
+ for (const_iterator i = exp_arg.begin(); i != exp_arg.end(); ++i)
+ prodseq.push_back(exp(*i));
+
+ return dynallocate<mul>(prodseq).setflag(status_flags::expanded);
+ }
+
+ return exp(exp_arg).hold();
+}
+
static ex exp_deriv(const ex & x, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param==0);
return exp(x);
}
+static ex exp_real_part(const ex & x)
+{
+ return exp(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
+}
+
+static ex exp_imag_part(const ex & x)
+{
+ return exp(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
+}
+
+static ex exp_conjugate(const ex & x)
+{
+ // conjugate(exp(x))==exp(conjugate(x))
+ return exp(x.conjugate());
+}
+
+static ex exp_power(const ex & x, const ex & a)
+{
+ /*
+ * The power law (e^x)^a=e^(x*a) is used in two cases:
+ * a) a is an integer and x may be complex;
+ * b) both x and a are reals.
+ * Negative a is excluded to keep automatic simplifications like exp(x)/exp(x)=1.
+ */
+ if (a.info(info_flags::nonnegative)
+ && (a.info(info_flags::integer) || (x.info(info_flags::real) && a.info(info_flags::real))))
+ return exp(x*a);
+ else if (a.info(info_flags::negative)
+ && (a.info(info_flags::integer) || (x.info(info_flags::real) && a.info(info_flags::real))))
+ return power(exp(-x*a), _ex_1).hold();
+
+ return power(exp(x), a).hold();
+}
+
REGISTER_FUNCTION(exp, eval_func(exp_eval).
evalf_func(exp_evalf).
+ expand_func(exp_expand).
derivative_func(exp_deriv).
+ real_part_func(exp_real_part).
+ imag_part_func(exp_imag_part).
+ conjugate_func(exp_conjugate).
+ power_func(exp_power).
latex_name("\\exp"));
//////////
if (x.info(info_flags::numeric)) {
if (x.is_zero()) // log(0) -> infinity
throw(pole_error("log_eval(): log(0)",0));
- if (x.info(info_flags::real) && x.info(info_flags::negative))
+ if (x.info(info_flags::rational) && x.info(info_flags::negative))
return (log(-x)+I*Pi);
if (x.is_equal(_ex1)) // log(1) -> 0
return _ex0;
if (x.is_equal(I)) // log(I) -> Pi*I/2
- return (Pi*I*_num1_2);
+ return (Pi*I*_ex1_2);
if (x.is_equal(-I)) // log(-I) -> -Pi*I/2
- return (Pi*I*_num_1_2);
- // log(float)
+ return (Pi*I*_ex_1_2);
+
+ // log(float) -> float
if (!x.info(info_flags::crational))
return log(ex_to<numeric>(x));
}
+
// log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
if (is_ex_the_function(x, exp)) {
const ex &t = x.op(0);
- if (is_a<symbol>(t) && t.info(info_flags::real)) {
+ if (t.info(info_flags::real))
return t;
- }
- if (t.info(info_flags::numeric)) {
- const numeric &nt = ex_to<numeric>(t);
- if (nt.is_real())
- return t;
- }
}
-
+
return log(x).hold();
}
// maybe substitution of rel into arg fails because of a pole
try {
arg_pt = arg.subs(rel, subs_options::no_pattern);
- } catch (pole_error) {
+ } catch (pole_error &) {
must_expand_arg = true;
}
// or we are at the branch point anyways
if (arg_pt.is_zero())
must_expand_arg = true;
+ if (arg.diff(ex_to<symbol>(rel.lhs())).is_zero()) {
+ throw do_taylor();
+ }
+
if (must_expand_arg) {
// method:
// This is the branch point: Series expand the argument first, then
if (!argser.is_terminating() || argser.nops()!=1) {
// in this case n more (or less) terms are needed
// (sadly, to generate them, we have to start from the beginning)
+ if (n == 0 && coeff == 1) {
+ ex rest = pseries(rel, epvector{expair(-1, _ex0), expair(Order(_ex1), order)}).add_series(argser);
+ ex acc = dynallocate<pseries>(rel, epvector());
+ for (int i = order-1; i>0; --i) {
+ epvector cterm { expair(i%2 ? _ex1/i : _ex_1/i, _ex0) };
+ acc = pseries(rel, std::move(cterm)).add_series(ex_to<pseries>(acc));
+ acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
+ }
+ return acc;
+ }
const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
- return pseries(rel, s
eq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
+ return pseries(rel, s
td::move(seq)).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
} else // it was a monomial
- return pseries(rel, seq);
+ return pseries(rel, std::move(seq));
}
if (!(options & series_options::suppress_branchcut) &&
arg_pt.info(info_flags::negative)) {
const symbol foo;
const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
epvector seq;
- seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
+ if (order > 0) {
+ seq.reserve(2);
+ seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
+ }
seq.push_back(expair(Order(_ex1), order));
- return series(replarg - I*Pi + pseries(rel, seq), rel, order);
+ return series(replarg - I*Pi + pseries(rel, std::move(seq)), rel, order);
}
throw do_taylor(); // caught by function::series()
}
+static ex log_real_part(const ex & x)
+{
+ if (x.info(info_flags::nonnegative))
+ return log(x).hold();
+ return log(abs(x));
+}
+
+static ex log_imag_part(const ex & x)
+{
+ if (x.info(info_flags::nonnegative))
+ return 0;
+ return atan2(GiNaC::imag_part(x), GiNaC::real_part(x));
+}
+
+static ex log_expand(const ex & arg, unsigned options)
+{
+ if ((options & expand_options::expand_transcendental)
+ && is_exactly_a<mul>(arg) && !arg.info(info_flags::indefinite)) {
+ exvector sumseq;
+ exvector prodseq;
+ sumseq.reserve(arg.nops());
+ prodseq.reserve(arg.nops());
+ bool possign=true;
+
+ // searching for positive/negative factors
+ for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
+ ex e;
+ if (options & expand_options::expand_function_args)
+ e=i->expand(options);
+ else
+ e=*i;
+ if (e.info(info_flags::positive))
+ sumseq.push_back(log(e));
+ else if (e.info(info_flags::negative)) {
+ sumseq.push_back(log(-e));
+ possign = !possign;
+ } else
+ prodseq.push_back(e);
+ }
+
+ if (sumseq.size() > 0) {
+ ex newarg;
+ if (options & expand_options::expand_function_args)
+ newarg=((possign?_ex1:_ex_1)*mul(prodseq)).expand(options);
+ else {
+ newarg=(possign?_ex1:_ex_1)*mul(prodseq);
+ ex_to<basic>(newarg).setflag(status_flags::purely_indefinite);
+ }
+ return add(sumseq)+log(newarg);
+ } else {
+ if (!(options & expand_options::expand_function_args))
+ ex_to<basic>(arg).setflag(status_flags::purely_indefinite);
+ }
+ }
+
+ if (options & expand_options::expand_function_args)
+ return log(arg.expand(options)).hold();
+ else
+ return log(arg).hold();
+}
+
+static ex log_conjugate(const ex & x)
+{
+ // conjugate(log(x))==log(conjugate(x)) unless on the branch cut which
+ // runs along the negative real axis.
+ if (x.info(info_flags::positive)) {
+ return log(x);
+ }
+ if (is_exactly_a<numeric>(x) &&
+ !x.imag_part().is_zero()) {
+ return log(x.conjugate());
+ }
+ return conjugate_function(log(x)).hold();
+}
+
REGISTER_FUNCTION(log, eval_func(log_eval).
evalf_func(log_evalf).
+ expand_func(log_expand).
derivative_func(log_deriv).
series_func(log_series).
+ real_part_func(log_real_part).
+ imag_part_func(log_imag_part).
+ conjugate_func(log_conjugate).
latex_name("\\ln"));
//////////
const ex SixtyExOverPi = _ex60*x/Pi;
ex sign = _ex1;
if (SixtyExOverPi.info(info_flags::integer)) {
- numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num120);
- if (z>=_num60) {
+ numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
+ if (z>=*_num60_p) {
// wrap to interval [0, Pi)
- z -= _num60;
+ z -= *_num60_p;
sign = _ex_1;
}
- if (z>_num30) {
+ if (z>*_num30_p) {
// wrap to interval [0, Pi/2)
- z = _num60-z;
+ z = *_num60_p-z;
}
- if (z.is_equal(_num0)) // sin(0) -> 0
+ if (z.is_equal(*_num0_p)) // sin(0) -> 0
return _ex0;
- if (z.is_equal(_num5)) // sin(Pi/12) -> sqrt(6)/4*(1-sqrt(3)/3)
+ if (z.is_equal(*_num5_p)) // sin(Pi/12) -> sqrt(6)/4*(1-sqrt(3)/3)
return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
- if (z.is_equal(_num6)) // sin(Pi/10) -> sqrt(5)/4-1/4
+ if (z.is_equal(*_num6_p)) // sin(Pi/10) -> sqrt(5)/4-1/4
return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
- if (z.is_equal(_num10)) // sin(Pi/6) -> 1/2
+ if (z.is_equal(*_num10_p)) // sin(Pi/6) -> 1/2
return sign*_ex1_2;
- if (z.is_equal(_num15)) // sin(Pi/4) -> sqrt(2)/2
+ if (z.is_equal(*_num15_p)) // sin(Pi/4) -> sqrt(2)/2
return sign*_ex1_2*sqrt(_ex2);
- if (z.is_equal(_num18)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
+ if (z.is_equal(*_num18_p)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
- if (z.is_equal(_num20)) // sin(Pi/3) -> sqrt(3)/2
+ if (z.is_equal(*_num20_p)) // sin(Pi/3) -> sqrt(3)/2
return sign*_ex1_2*sqrt(_ex3);
- if (z.is_equal(_num25)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
+ if (z.is_equal(*_num25_p)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
- if (z.is_equal(_num30)) // sin(Pi/2) -> 1
+ if (z.is_equal(*_num30_p)) // sin(Pi/2) -> 1
return sign;
}
-
+
if (is_exactly_a<function>(x)) {
const ex &t = x.op(0);
+
// sin(asin(x)) -> x
if (is_ex_the_function(x, asin))
return t;
+
// sin(acos(x)) -> sqrt(1-x^2)
if (is_ex_the_function(x, acos))
return sqrt(_ex1-power(t,_ex2));
+
// sin(atan(x)) -> x/sqrt(1+x^2)
if (is_ex_the_function(x, atan))
return t*power(_ex1+power(t,_ex2),_ex_1_2);
// sin(float) -> float
if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
return sin(ex_to<numeric>(x));
+
+ // sin() is odd
+ if (x.info(info_flags::negative))
+ return -sin(-x);
return sin(x).hold();
}
return cos(x);
}
+static ex sin_real_part(const ex & x)
+{
+ return cosh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
+}
+
+static ex sin_imag_part(const ex & x)
+{
+ return sinh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
+}
+
+static ex sin_conjugate(const ex & x)
+{
+ // conjugate(sin(x))==sin(conjugate(x))
+ return sin(x.conjugate());
+}
+
REGISTER_FUNCTION(sin, eval_func(sin_eval).
evalf_func(sin_evalf).
derivative_func(sin_deriv).
+ real_part_func(sin_real_part).
+ imag_part_func(sin_imag_part).
+ conjugate_func(sin_conjugate).
latex_name("\\sin"));
//////////
const ex SixtyExOverPi = _ex60*x/Pi;
ex sign = _ex1;
if (SixtyExOverPi.info(info_flags::integer)) {
- numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num120);
- if (z>=_num60) {
+ numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
+ if (z>=*_num60_p) {
// wrap to interval [0, Pi)
- z = _num120-z;
+ z = *_num120_p-z;
}
- if (z>=_num30) {
+ if (z>=*_num30_p) {
// wrap to interval [0, Pi/2)
- z = _num60-z;
+ z = *_num60_p-z;
sign = _ex_1;
}
- if (z.is_equal(_num0)) // cos(0) -> 1
+ if (z.is_equal(*_num0_p)) // cos(0) -> 1
return sign;
- if (z.is_equal(_num5)) // cos(Pi/12) -> sqrt(6)/4*(1+sqrt(3)/3)
+ if (z.is_equal(*_num5_p)) // cos(Pi/12) -> sqrt(6)/4*(1+sqrt(3)/3)
return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
- if (z.is_equal(_num10)) // cos(Pi/6) -> sqrt(3)/2
+ if (z.is_equal(*_num10_p)) // cos(Pi/6) -> sqrt(3)/2
return sign*_ex1_2*sqrt(_ex3);
- if (z.is_equal(_num12)) // cos(Pi/5) -> sqrt(5)/4+1/4
+ if (z.is_equal(*_num12_p)) // cos(Pi/5) -> sqrt(5)/4+1/4
return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
- if (z.is_equal(_num15)) // cos(Pi/4) -> sqrt(2)/2
+ if (z.is_equal(*_num15_p)) // cos(Pi/4) -> sqrt(2)/2
return sign*_ex1_2*sqrt(_ex2);
- if (z.is_equal(_num20)) // cos(Pi/3) -> 1/2
+ if (z.is_equal(*_num20_p)) // cos(Pi/3) -> 1/2
return sign*_ex1_2;
- if (z.is_equal(_num24)) // cos(2/5*Pi) -> sqrt(5)/4-1/4x
+ if (z.is_equal(*_num24_p)) // cos(2/5*Pi) -> sqrt(5)/4-1/4x
return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
- if (z.is_equal(_num25)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
+ if (z.is_equal(*_num25_p)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
- if (z.is_equal(_num30)) // cos(Pi/2) -> 0
+ if (z.is_equal(*_num30_p)) // cos(Pi/2) -> 0
return _ex0;
}
-
+
if (is_exactly_a<function>(x)) {
const ex &t = x.op(0);
+
// cos(acos(x)) -> x
if (is_ex_the_function(x, acos))
return t;
+
// cos(asin(x)) -> sqrt(1-x^2)
if (is_ex_the_function(x, asin))
return sqrt(_ex1-power(t,_ex2));
+
// cos(atan(x)) -> 1/sqrt(1+x^2)
if (is_ex_the_function(x, atan))
return power(_ex1+power(t,_ex2),_ex_1_2);
if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
return cos(ex_to<numeric>(x));
+ // cos() is even
+ if (x.info(info_flags::negative))
+ return cos(-x);
+
return cos(x).hold();
}
return -sin(x);
}
+static ex cos_real_part(const ex & x)
+{
+ return cosh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
+}
+
+static ex cos_imag_part(const ex & x)
+{
+ return -sinh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
+}
+
+static ex cos_conjugate(const ex & x)
+{
+ // conjugate(cos(x))==cos(conjugate(x))
+ return cos(x.conjugate());
+}
+
REGISTER_FUNCTION(cos, eval_func(cos_eval).
evalf_func(cos_evalf).
derivative_func(cos_deriv).
+ real_part_func(cos_real_part).
+ imag_part_func(cos_imag_part).
+ conjugate_func(cos_conjugate).
latex_name("\\cos"));
//////////
const ex SixtyExOverPi = _ex60*x/Pi;
ex sign = _ex1;
if (SixtyExOverPi.info(info_flags::integer)) {
- numeric z = mod(ex_to<numeric>(SixtyExOverPi),_num60);
- if (z>=_num60) {
+ numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num60_p);
+ if (z>=*_num60_p) {
// wrap to interval [0, Pi)
- z -= _num60;
+ z -= *_num60_p;
}
- if (z>=_num30) {
+ if (z>=*_num30_p) {
// wrap to interval [0, Pi/2)
- z = _num60-z;
+ z = *_num60_p-z;
sign = _ex_1;
}
- if (z.is_equal(_num0)) // tan(0) -> 0
+ if (z.is_equal(*_num0_p)) // tan(0) -> 0
return _ex0;
- if (z.is_equal(_num5)) // tan(Pi/12) -> 2-sqrt(3)
+ if (z.is_equal(*_num5_p)) // tan(Pi/12) -> 2-sqrt(3)
return sign*(_ex2-sqrt(_ex3));
- if (z.is_equal(_num10)) // tan(Pi/6) -> sqrt(3)/3
+ if (z.is_equal(*_num10_p)) // tan(Pi/6) -> sqrt(3)/3
return sign*_ex1_3*sqrt(_ex3);
- if (z.is_equal(_num15)) // tan(Pi/4) -> 1
+ if (z.is_equal(*_num15_p)) // tan(Pi/4) -> 1
return sign;
- if (z.is_equal(_num20)) // tan(Pi/3) -> sqrt(3)
+ if (z.is_equal(*_num20_p)) // tan(Pi/3) -> sqrt(3)
return sign*sqrt(_ex3);
- if (z.is_equal(_num25)) // tan(5/12*Pi) -> 2+sqrt(3)
+ if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3)
return sign*(sqrt(_ex3)+_ex2);
- if (z.is_equal(_num30)) // tan(Pi/2) -> infinity
+ if (z.is_equal(*_num30_p)) // tan(Pi/2) -> infinity
throw (pole_error("tan_eval(): simple pole",1));
}
-
+
if (is_exactly_a<function>(x)) {
const ex &t = x.op(0);
+
// tan(atan(x)) -> x
if (is_ex_the_function(x, atan))
return t;
+
// tan(asin(x)) -> x/sqrt(1+x^2)
if (is_ex_the_function(x, asin))
return t*power(_ex1-power(t,_ex2),_ex_1_2);
+
// tan(acos(x)) -> sqrt(1-x^2)/x
if (is_ex_the_function(x, acos))
return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
return tan(ex_to<numeric>(x));
}
+ // tan() is odd
+ if (x.info(info_flags::negative))
+ return -tan(-x);
+
return tan(x).hold();
}
return (_ex1+power(tan(x),_ex2));
}
+static ex tan_real_part(const ex & x)
+{
+ ex a = GiNaC::real_part(x);
+ ex b = GiNaC::imag_part(x);
+ return tan(a)/(1+power(tan(a),2)*power(tan(b),2));
+}
+
+static ex tan_imag_part(const ex & x)
+{
+ ex a = GiNaC::real_part(x);
+ ex b = GiNaC::imag_part(x);
+ return tanh(b)/(1+power(tan(a),2)*power(tan(b),2));
+}
+
static ex tan_series(const ex &x,
const relational &rel,
int order,
return (sin(x)/cos(x)).series(rel, order, options);
}
+static ex tan_conjugate(const ex & x)
+{
+ // conjugate(tan(x))==tan(conjugate(x))
+ return tan(x.conjugate());
+}
+
REGISTER_FUNCTION(tan, eval_func(tan_eval).
evalf_func(tan_evalf).
derivative_func(tan_deriv).
series_func(tan_series).
+ real_part_func(tan_real_part).
+ imag_part_func(tan_imag_part).
+ conjugate_func(tan_conjugate).
latex_name("\\tan"));
//////////
static ex asin_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
+
// asin(0) -> 0
if (x.is_zero())
return x;
+
// asin(1/2) -> Pi/6
if (x.is_equal(_ex1_2))
return numeric(1,6)*Pi;
+
// asin(1) -> Pi/2
if (x.is_equal(_ex1))
- return _num1_2*Pi;
+ return _ex1_2*Pi;
+
// asin(-1/2) -> -Pi/6
if (x.is_equal(_ex_1_2))
return numeric(-1,6)*Pi;
+
// asin(-1) -> -Pi/2
if (x.is_equal(_ex_1))
- return _num_1_2*Pi;
+ return _ex_1_2*Pi;
+
// asin(float) -> float
if (!x.info(info_flags::crational))
return asin(ex_to<numeric>(x));
+
+ // asin() is odd
+ if (x.info(info_flags::negative))
+ return -asin(-x);
}
return asin(x).hold();
return power(1-power(x,_ex2),_ex_1_2);
}
+static ex asin_conjugate(const ex & x)
+{
+ // conjugate(asin(x))==asin(conjugate(x)) unless on the branch cuts which
+ // run along the real axis outside the interval [-1, +1].
+ if (is_exactly_a<numeric>(x) &&
+ (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
+ return asin(x.conjugate());
+ }
+ return conjugate_function(asin(x)).hold();
+}
+
REGISTER_FUNCTION(asin, eval_func(asin_eval).
evalf_func(asin_evalf).
derivative_func(asin_deriv).
+ conjugate_func(asin_conjugate).
latex_name("\\arcsin"));
//////////
static ex acos_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
+
// acos(1) -> 0
if (x.is_equal(_ex1))
return _ex0;
+
// acos(1/2) -> Pi/3
if (x.is_equal(_ex1_2))
return _ex1_3*Pi;
+
// acos(0) -> Pi/2
if (x.is_zero())
return _ex1_2*Pi;
+
// acos(-1/2) -> 2/3*Pi
if (x.is_equal(_ex_1_2))
return numeric(2,3)*Pi;
+
// acos(-1) -> Pi
if (x.is_equal(_ex_1))
return Pi;
+
// acos(float) -> float
if (!x.info(info_flags::crational))
return acos(ex_to<numeric>(x));
+
+ // acos(-x) -> Pi-acos(x)
+ if (x.info(info_flags::negative))
+ return Pi-acos(-x);
}
return acos(x).hold();
return -power(1-power(x,_ex2),_ex_1_2);
}
+static ex acos_conjugate(const ex & x)
+{
+ // conjugate(acos(x))==acos(conjugate(x)) unless on the branch cuts which
+ // run along the real axis outside the interval [-1, +1].
+ if (is_exactly_a<numeric>(x) &&
+ (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
+ return acos(x.conjugate());
+ }
+ return conjugate_function(acos(x)).hold();
+}
+
REGISTER_FUNCTION(acos, eval_func(acos_eval).
evalf_func(acos_evalf).
derivative_func(acos_deriv).
+ conjugate_func(acos_conjugate).
latex_name("\\arccos"));
//////////
static ex atan_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
+
// atan(0) -> 0
if (x.is_zero())
return _ex0;
+
// atan(1) -> Pi/4
if (x.is_equal(_ex1))
return _ex1_4*Pi;
+
// atan(-1) -> -Pi/4
if (x.is_equal(_ex_1))
return _ex_1_4*Pi;
+
if (x.is_equal(I) || x.is_equal(-I))
throw (pole_error("atan_eval(): logarithmic pole",0));
+
// atan(float) -> float
if (!x.info(info_flags::crational))
return atan(ex_to<numeric>(x));
+
+ // atan() is odd
+ if (x.info(info_flags::negative))
+ return -atan(-x);
}
return atan(x).hold();
else
Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
epvector seq;
- seq.push_back(expair(Order0correction, _ex0));
+ if (order > 0) {
+ seq.reserve(2);
+ seq.push_back(expair(Order0correction, _ex0));
+ }
seq.push_back(expair(Order(_ex1), order));
- return series(replarg - pseries(rel, seq), rel, order);
+ return series(replarg - pseries(rel, std::move(seq)), rel, order);
}
throw do_taylor();
}
+static ex atan_conjugate(const ex & x)
+{
+ // conjugate(atan(x))==atan(conjugate(x)) unless on the branch cuts which
+ // run along the imaginary axis outside the interval [-I, +I].
+ if (x.info(info_flags::real))
+ return atan(x);
+ if (is_exactly_a<numeric>(x)) {
+ const numeric x_re = ex_to<numeric>(x.real_part());
+ const numeric x_im = ex_to<numeric>(x.imag_part());
+ if (!x_re.is_zero() ||
+ (x_im > *_num_1_p && x_im < *_num1_p))
+ return atan(x.conjugate());
+ }
+ return conjugate_function(atan(x)).hold();
+}
+
REGISTER_FUNCTION(atan, eval_func(atan_eval).
evalf_func(atan_evalf).
derivative_func(atan_deriv).
series_func(atan_series).
+ conjugate_func(atan_conjugate).
latex_name("\\arctan"));
//////////
static ex atan2_evalf(const ex &y, const ex &x)
{
if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
- return atan2(ex_to<numeric>(y), ex_to<numeric>(x));
+ return atan(ex_to<numeric>(y), ex_to<numeric>(x));
return atan2(y, x).hold();
}
static ex atan2_eval(const ex & y, const ex & x)
{
- if (y.info(info_flags::numeric) && !y.info(info_flags::crational) &&
- x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
- return atan2_evalf(y,x);
+ if (y.is_zero()) {
+
+ // atan2(0, 0) -> 0
+ if (x.is_zero())
+ return _ex0;
+
+ // atan2(0, x), x real and positive -> 0
+ if (x.info(info_flags::positive))
+ return _ex0;
+
+ // atan2(0, x), x real and negative -> Pi
+ if (x.info(info_flags::negative))
+ return Pi;
}
-
- return atan2(y,x).hold();
+
+ if (x.is_zero()) {
+
+ // atan2(y, 0), y real and positive -> Pi/2
+ if (y.info(info_flags::positive))
+ return _ex1_2*Pi;
+
+ // atan2(y, 0), y real and negative -> -Pi/2
+ if (y.info(info_flags::negative))
+ return _ex_1_2*Pi;
+ }
+
+ if (y.is_equal(x)) {
+
+ // atan2(y, y), y real and positive -> Pi/4
+ if (y.info(info_flags::positive))
+ return _ex1_4*Pi;
+
+ // atan2(y, y), y real and negative -> -3/4*Pi
+ if (y.info(info_flags::negative))
+ return numeric(-3, 4)*Pi;
+ }
+
+ if (y.is_equal(-x)) {
+
+ // atan2(y, -y), y real and positive -> 3*Pi/4
+ if (y.info(info_flags::positive))
+ return numeric(3, 4)*Pi;
+
+ // atan2(y, -y), y real and negative -> -Pi/4
+ if (y.info(info_flags::negative))
+ return _ex_1_4*Pi;
+ }
+
+ // atan2(float, float) -> float
+ if (is_a<numeric>(y) && !y.info(info_flags::crational) &&
+ is_a<numeric>(x) && !x.info(info_flags::crational))
+ return atan(ex_to<numeric>(y), ex_to<numeric>(x));
+
+ // atan2(real, real) -> atan(y/x) +/- Pi
+ if (y.info(info_flags::real) && x.info(info_flags::real)) {
+ if (x.info(info_flags::positive))
+ return atan(y/x);
+
+ if (x.info(info_flags::negative)) {
+ if (y.info(info_flags::positive))
+ return atan(y/x)+Pi;
+ if (y.info(info_flags::negative))
+ return atan(y/x)-Pi;
+ }
+ }
+
+ return atan2(y, x).hold();
}
static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
GINAC_ASSERT(deriv_param<2);
if (deriv_param==0) {
- // d/dy atan(y,x)
+ // d/dy atan2(y,x)
return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
}
- // d/dx atan(y,x)
+ // d/dx atan2(y,x)
return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
}
static ex sinh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
- if (x.is_zero()) // sinh(0) -> 0
+
+ // sinh(0) -> 0
+ if (x.is_zero())
return _ex0;
- if (!x.info(info_flags::crational)) // sinh(float) -> float
+
+ // sinh(float) -> float
+ if (!x.info(info_flags::crational))
return sinh(ex_to<numeric>(x));
+
+ // sinh() is odd
+ if (x.info(info_flags::negative))
+ return -sinh(-x);
}
if ((x/Pi).info(info_flags::numeric) &&
if (is_exactly_a<function>(x)) {
const ex &t = x.op(0);
+
// sinh(asinh(x)) -> x
if (is_ex_the_function(x, asinh))
return t;
+
// sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
if (is_ex_the_function(x, acosh))
return sqrt(t-_ex1)*sqrt(t+_ex1);
+
// sinh(atanh(x)) -> x/sqrt(1-x^2)
if (is_ex_the_function(x, atanh))
return t*power(_ex1-power(t,_ex2),_ex_1_2);
return cosh(x);
}
+static ex sinh_real_part(const ex & x)
+{
+ return sinh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
+}
+
+static ex sinh_imag_part(const ex & x)
+{
+ return cosh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
+}
+
+static ex sinh_conjugate(const ex & x)
+{
+ // conjugate(sinh(x))==sinh(conjugate(x))
+ return sinh(x.conjugate());
+}
+
REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
evalf_func(sinh_evalf).
derivative_func(sinh_deriv).
+ real_part_func(sinh_real_part).
+ imag_part_func(sinh_imag_part).
+ conjugate_func(sinh_conjugate).
latex_name("\\sinh"));
//////////
static ex cosh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
- if (x.is_zero()) // cosh(0) -> 1
+
+ // cosh(0) -> 1
+ if (x.is_zero())
return _ex1;
- if (!x.info(info_flags::crational)) // cosh(float) -> float
+
+ // cosh(float) -> float
+ if (!x.info(info_flags::crational))
return cosh(ex_to<numeric>(x));
+
+ // cosh() is even
+ if (x.info(info_flags::negative))
+ return cosh(-x);
}
if ((x/Pi).info(info_flags::numeric) &&
if (is_exactly_a<function>(x)) {
const ex &t = x.op(0);
+
// cosh(acosh(x)) -> x
if (is_ex_the_function(x, acosh))
return t;
+
// cosh(asinh(x)) -> sqrt(1+x^2)
if (is_ex_the_function(x, asinh))
return sqrt(_ex1+power(t,_ex2));
+
// cosh(atanh(x)) -> 1/sqrt(1-x^2)
if (is_ex_the_function(x, atanh))
return power(_ex1-power(t,_ex2),_ex_1_2);
return sinh(x);
}
+static ex cosh_real_part(const ex & x)
+{
+ return cosh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
+}
+
+static ex cosh_imag_part(const ex & x)
+{
+ return sinh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
+}
+
+static ex cosh_conjugate(const ex & x)
+{
+ // conjugate(cosh(x))==cosh(conjugate(x))
+ return cosh(x.conjugate());
+}
+
REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
evalf_func(cosh_evalf).
derivative_func(cosh_deriv).
+ real_part_func(cosh_real_part).
+ imag_part_func(cosh_imag_part).
+ conjugate_func(cosh_conjugate).
latex_name("\\cosh"));
//////////
static ex tanh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
- if (x.is_zero()) // tanh(0) -> 0
+
+ // tanh(0) -> 0
+ if (x.is_zero())
return _ex0;
- if (!x.info(info_flags::crational)) // tanh(float) -> float
+
+ // tanh(float) -> float
+ if (!x.info(info_flags::crational))
return tanh(ex_to<numeric>(x));
+
+ // tanh() is odd
+ if (x.info(info_flags::negative))
+ return -tanh(-x);
}
if ((x/Pi).info(info_flags::numeric) &&
if (is_exactly_a<function>(x)) {
const ex &t = x.op(0);
+
// tanh(atanh(x)) -> x
if (is_ex_the_function(x, atanh))
return t;
+
// tanh(asinh(x)) -> x/sqrt(1+x^2)
if (is_ex_the_function(x, asinh))
return t*power(_ex1+power(t,_ex2),_ex_1_2);
+
// tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
if (is_ex_the_function(x, acosh))
return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
return (sinh(x)/cosh(x)).series(rel, order, options);
}
+static ex tanh_real_part(const ex & x)
+{
+ ex a = GiNaC::real_part(x);
+ ex b = GiNaC::imag_part(x);
+ return tanh(a)/(1+power(tanh(a),2)*power(tan(b),2));
+}
+
+static ex tanh_imag_part(const ex & x)
+{
+ ex a = GiNaC::real_part(x);
+ ex b = GiNaC::imag_part(x);
+ return tan(b)/(1+power(tanh(a),2)*power(tan(b),2));
+}
+
+static ex tanh_conjugate(const ex & x)
+{
+ // conjugate(tanh(x))==tanh(conjugate(x))
+ return tanh(x.conjugate());
+}
+
REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
evalf_func(tanh_evalf).
derivative_func(tanh_deriv).
series_func(tanh_series).
+ real_part_func(tanh_real_part).
+ imag_part_func(tanh_imag_part).
+ conjugate_func(tanh_conjugate).
latex_name("\\tanh"));
//////////
static ex asinh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
+
// asinh(0) -> 0
if (x.is_zero())
return _ex0;
+
// asinh(float) -> float
if (!x.info(info_flags::crational))
return asinh(ex_to<numeric>(x));
+
+ // asinh() is odd
+ if (x.info(info_flags::negative))
+ return -asinh(-x);
}
return asinh(x).hold();
return power(_ex1+power(x,_ex2),_ex_1_2);
}
+static ex asinh_conjugate(const ex & x)
+{
+ // conjugate(asinh(x))==asinh(conjugate(x)) unless on the branch cuts which
+ // run along the imaginary axis outside the interval [-I, +I].
+ if (x.info(info_flags::real))
+ return asinh(x);
+ if (is_exactly_a<numeric>(x)) {
+ const numeric x_re = ex_to<numeric>(x.real_part());
+ const numeric x_im = ex_to<numeric>(x.imag_part());
+ if (!x_re.is_zero() ||
+ (x_im > *_num_1_p && x_im < *_num1_p))
+ return asinh(x.conjugate());
+ }
+ return conjugate_function(asinh(x)).hold();
+}
+
REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
evalf_func(asinh_evalf).
- derivative_func(asinh_deriv));
+ derivative_func(asinh_deriv).
+ conjugate_func(asinh_conjugate));
//////////
// inverse hyperbolic cosine (trigonometric function)
static ex acosh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
+
// acosh(0) -> Pi*I/2
if (x.is_zero())
return Pi*I*numeric(1,2);
+
// acosh(1) -> 0
if (x.is_equal(_ex1))
return _ex0;
+
// acosh(-1) -> Pi*I
if (x.is_equal(_ex_1))
return Pi*I;
+
// acosh(float) -> float
if (!x.info(info_flags::crational))
return acosh(ex_to<numeric>(x));
+
+ // acosh(-x) -> Pi*I-acosh(x)
+ if (x.info(info_flags::negative))
+ return Pi*I-acosh(-x);
}
return acosh(x).hold();
return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
}
+static ex acosh_conjugate(const ex & x)
+{
+ // conjugate(acosh(x))==acosh(conjugate(x)) unless on the branch cut
+ // which runs along the real axis from +1 to -inf.
+ if (is_exactly_a<numeric>(x) &&
+ (!x.imag_part().is_zero() || x > *_num1_p)) {
+ return acosh(x.conjugate());
+ }
+ return conjugate_function(acosh(x)).hold();
+}
+
REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
evalf_func(acosh_evalf).
- derivative_func(acosh_deriv));
+ derivative_func(acosh_deriv).
+ conjugate_func(acosh_conjugate));
//////////
// inverse hyperbolic tangent (trigonometric function)
static ex atanh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
+
// atanh(0) -> 0
if (x.is_zero())
return _ex0;
+
// atanh({+|-}1) -> throw
if (x.is_equal(_ex1) || x.is_equal(_ex_1))
throw (pole_error("atanh_eval(): logarithmic pole",0));
+
// atanh(float) -> float
if (!x.info(info_flags::crational))
return atanh(ex_to<numeric>(x));
+
+ // atanh() is odd
+ if (x.info(info_flags::negative))
+ return -atanh(-x);
}
return atanh(x).hold();
return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
// ...and the branch cuts (the discontinuity at the cut being just I*Pi)
if (!(options & series_options::suppress_branchcut)) {
- // method:
- // This is the branch cut: assemble the primitive series manually and
- // then add the corresponding complex step function.
- const symbol &s = ex_to<symbol>(rel.lhs());
- const ex &point = rel.rhs();
- const symbol foo;
- const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
+ // method:
+ // This is the branch cut: assemble the primitive series manually and
+ // then add the corresponding complex step function.
+ const symbol &s = ex_to<symbol>(rel.lhs());
+ const ex &point = rel.rhs();
+ const symbol foo;
+ const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
if (arg_pt<_ex0)
Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
else
Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
- epvector seq;
- seq.push_back(expair(Order0correction, _ex0));
- seq.push_back(expair(Order(_ex1), order));
- return series(replarg - pseries(rel, seq), rel, order);
+ epvector seq;
+ if (order > 0) {
+ seq.reserve(2);
+ seq.push_back(expair(Order0correction, _ex0));
+ }
+ seq.push_back(expair(Order(_ex1), order));
+ return series(replarg - pseries(rel, std::move(seq)), rel, order);
}
throw do_taylor();
}
+static ex atanh_conjugate(const ex & x)
+{
+ // conjugate(atanh(x))==atanh(conjugate(x)) unless on the branch cuts which
+ // run along the real axis outside the interval [-1, +1].
+ if (is_exactly_a<numeric>(x) &&
+ (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
+ return atanh(x.conjugate());
+ }
+ return conjugate_function(atanh(x)).hold();
+}
+
REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
evalf_func(atanh_evalf).
derivative_func(atanh_deriv).
- series_func(atanh_series));
+ series_func(atanh_series).
+ conjugate_func(atanh_conjugate));
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff