* functions. */
/*
- * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2011 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
* 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 {
//////////
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());
+}
+
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).
+ conjugate_func(exp_conjugate).
latex_name("\\exp"));
//////////
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_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).
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"));
//////////
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"));
//////////
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"));
//////////
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"));
//////////
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"));
//////////
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"));
//////////
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_eval(const ex & y, const ex & x)
{
- if (y.info(info_flags::numeric) && x.info(info_flags::numeric)) {
+ if (y.is_zero()) {
- if (y.is_zero()) {
+ // atan2(0, 0) -> 0
+ if (x.is_zero())
+ return _ex0;
- // atan(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;
- // atan(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;
+ }
- // atan(0, x), x real and negative -> -Pi
- if (x.info(info_flags::negative))
- return _ex_1*Pi;
- }
+ if (x.is_zero()) {
- if (x.is_zero()) {
+ // atan2(y, 0), y real and positive -> Pi/2
+ if (y.info(info_flags::positive))
+ return _ex1_2*Pi;
- // atan(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;
+ }
- // atan(y, 0), y real and negative -> -Pi/2
- if (y.info(info_flags::negative))
- return _ex_1_2*Pi;
- }
+ if (y.is_equal(x)) {
- if (y.is_equal(x)) {
+ // atan2(y, y), y real and positive -> Pi/4
+ if (y.info(info_flags::positive))
+ return _ex1_4*Pi;
- // atan(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;
+ }
- // 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)) {
- 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;
- // atan(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;
+ }
- // atan(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));
- // atan(float, float) -> float
- if (!y.info(info_flags::crational) && !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);
- // 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))
+ if (x.info(info_flags::negative)) {
+ if (y.info(info_flags::positive))
return atan(y/x)+Pi;
- else
+ if (y.info(info_flags::negative))
return atan(y/x)-Pi;
}
}
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);
}
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"));
//////////
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"));
//////////
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(tan(x))==tan(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"));
//////////
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)
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)
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