static ex abs_power(const ex & arg, const ex & exp)
{
if ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even()) || exp.info(info_flags::even)) {
if (arg.info(info_flags::real) || arg.is_equal(arg.conjugate()))
return pow(arg, exp);
else
return pow(arg, exp/2) * pow(arg.conjugate(), exp/2);
} else
return power(abs(arg), exp).hold();
}
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();
}
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();
}
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();
}
static ex Li2_conjugate(const ex & x)
{
// conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
// run along the positive real axis beginning at 1.
if (x.info(info_flags::negative)) {
return Li2(x).hold();
}
if (is_exactly_a<numeric>(x) &&
(!x.imag_part().is_zero() || x < *_num1_p)) {
return Li2(x.conjugate());
}
return conjugate_function(Li2(x)).hold();
}
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();
}
static ex tan_conjugate(const ex & x)
{
// conjugate(tan(x))==tan(conjugate(x))
return tan(x.conjugate());
}
static ex cos_conjugate(const ex & x)
{
// conjugate(cos(x))==cos(conjugate(x))
return cos(x.conjugate());
}
static ex sin_conjugate(const ex & x)
{
// conjugate(sin(x))==sin(conjugate(x))
return sin(x.conjugate());
}
static ex exp_conjugate(const ex & x)
{
// conjugate(exp(x))==exp(conjugate(x))
return exp(x.conjugate());
}
static ex abs_expl_derivative(const ex & arg, const symbol & s)
{
ex diff_arg = arg.diff(s);
return (diff_arg*arg.conjugate()+arg*diff_arg.conjugate())/2/abs(arg);
}