* functions. */
/*
- * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2010 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 "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);
}
+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));
+}
+
REGISTER_FUNCTION(exp, eval_func(exp_eval).
evalf_func(exp_evalf).
derivative_func(exp_deriv).
+ real_part_func(exp_real_part).
+ imag_part_func(exp_imag_part).
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) && (ex_to<symbol>(t).get_domain() == symbol_options::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();
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) {
+ epvector epv;
+ ex acc = (new pseries(rel, epv))->setflag(status_flags::dynallocated);
+ epv.reserve(2);
+ epv.push_back(expair(-1, _ex0));
+ epv.push_back(expair(Order(_ex1), order));
+ ex rest = pseries(rel, epv).add_series(argser);
+ for (int i = order-1; i>0; --i) {
+ epvector cterm;
+ cterm.reserve(1);
+ cterm.push_back(expair(i%2 ? _ex1/i : _ex_1/i, _ex0));
+ acc = pseries(rel, 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, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
} else // it was a monomial
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));
+}
+
REGISTER_FUNCTION(log, eval_func(log_eval).
evalf_func(log_evalf).
derivative_func(log_deriv).
series_func(log_series).
+ real_part_func(log_real_part).
+ imag_part_func(log_imag_part).
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));
+}
+
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).
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));
+}
+
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).
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,
if (!(2*x_pt/Pi).info(info_flags::odd))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a simple pole
- return (sin(x)/cos(x)).series(rel, order+2, options);
+ return (sin(x)/cos(x)).series(rel, order, options);
}
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).
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();
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();
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();
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()) {
+
+ // atan(0, 0) -> 0
+ if (x.is_zero())
+ return _ex0;
+
+ // atan(0, x), x real and positive -> 0
+ if (x.info(info_flags::positive))
+ return _ex0;
+
+ // atan(0, x), x real and negative -> Pi
+ if (x.info(info_flags::negative))
+ return Pi;
}
-
- return atan2(y,x).hold();
+
+ if (x.is_zero()) {
+
+ // atan(y, 0), y real and positive -> Pi/2
+ if (y.info(info_flags::positive))
+ return _ex1_2*Pi;
+
+ // atan(y, 0), y real and negative -> -Pi/2
+ if (y.info(info_flags::real) && !y.is_zero())
+ return _ex_1_2*Pi;
+ }
+
+ if (y.is_equal(x)) {
+
+ // atan(y, y), y real and positive -> Pi/4
+ if (y.info(info_flags::positive))
+ return _ex1_4*Pi;
+
+ // atan(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)) {
+
+ // atan(y, -y), y real and positive -> 3*Pi/4
+ if (y.info(info_flags::positive))
+ return numeric(3, 4)*Pi;
+
+ // atan(y, -y), y real and negative -> -Pi/4
+ if (y.info(info_flags::negative))
+ return _ex_1_4*Pi;
+ }
+
+ // atan(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));
+
+ // atan(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);
+ else if (y.info(info_flags::positive))
+ return atan(y/x)+Pi;
+ else
+ return atan(y/x)-Pi;
+ }
+
+ return atan2(y, x).hold();
}
static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
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));
+}
+
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).
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));
+}
+
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).
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);
if (!(2*I*x_pt/Pi).info(info_flags::odd))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a simple pole
- return (sinh(x)/cosh(x)).series(rel, order+2, options);
+ 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));
}
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).
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();
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();
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();
git://www.ginac.de / ginac.git / blobdiff