* functions. */
/*
- * GiNaC Copyright (C) 1999-2010 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2019 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
#include "inifcns.h"
#include "ex.h"
#include "constant.h"
+#include "add.h"
+#include "mul.h"
#include "numeric.h"
#include "power.h"
#include "operators.h"
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(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).
+ expand_func(exp_expand).
derivative_func(exp_deriv).
real_part_func(exp_real_part).
imag_part_func(exp_imag_part).
+ conjugate_func(exp_conjugate).
latex_name("\\exp"));
//////////
if (t.info(info_flags::real))
return t;
}
-
+
return log(x).hold();
}
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
// 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);
+ 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;
- 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));
+ 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()
}
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"));
//////////
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 -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 (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"));
//////////
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"));
//////////
{
if (y.is_zero()) {
- // atan(0, 0) -> 0
+ // atan2(0, 0) -> 0
if (x.is_zero())
return _ex0;
- // atan(0, x), x real and positive -> 0
+ // atan2(0, x), x real and positive -> 0
if (x.info(info_flags::positive))
return _ex0;
- // atan(0, x), x real and negative -> Pi
+ // atan2(0, x), x real and negative -> Pi
if (x.info(info_flags::negative))
return Pi;
}
if (x.is_zero()) {
- // atan(y, 0), y real and positive -> Pi/2
+ // 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 negative -> -Pi/2
- if (y.info(info_flags::real) && !y.is_zero())
+ // 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)) {
- // atan(y, y), y real and positive -> Pi/4
+ // 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 negative -> -3/4*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)) {
- // atan(y, -y), y real and positive -> 3*Pi/4
+ // 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 negative -> -Pi/4
+ // atan2(y, -y), y real and negative -> -Pi/4
if (y.info(info_flags::negative))
return _ex_1_4*Pi;
}
- // atan(float, float) -> float
+ // 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(real, real) -> atan(y/x) +/- Pi
+ // 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);
- else if (y.info(info_flags::positive))
- return atan(y/x)+Pi;
- else
- return atan(y/x)-Pi;
+
+ 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();
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(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(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 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"));
//////////
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)
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