/* psi(z) for large |z| in the right half-plane; [Abramowitz + Stegun, 6.3.18] */
static
gsl_complex
psi_complex_asymp(gsl_complex z)
{
/* coefficients in the asymptotic expansion for large z;
* let w = z^(-2) and write the expression in the form
*
* ln(z) - 1/(2z) - 1/12 w (1 + c1 w + c2 w + c3 w + ... )
*/
static const double c1 = -0.1;
static const double c2 = 1.0/21.0;
static const double c3 = -0.05;
gsl_complex zi = gsl_complex_inverse(z);
gsl_complex w = gsl_complex_mul(zi, zi);
gsl_complex cs;
/* Horner method evaluation of term in parentheses */
gsl_complex sum;
sum = gsl_complex_mul_real(w, c3/c2);
sum = gsl_complex_add_real(sum, 1.0);
sum = gsl_complex_mul_real(sum, c2/c1);
sum = gsl_complex_mul(sum, w);
sum = gsl_complex_add_real(sum, 1.0);
sum = gsl_complex_mul_real(sum, c1);
sum = gsl_complex_mul(sum, w);
sum = gsl_complex_add_real(sum, 1.0);
/* correction added to log(z) */
cs = gsl_complex_mul(sum, w);
cs = gsl_complex_mul_real(cs, -1.0/12.0);
cs = gsl_complex_add(cs, gsl_complex_mul_real(zi, -0.5));
return gsl_complex_add(gsl_complex_log(z), cs);
}
gsl_complex gsl_complex_log_b(gsl_complex a, gsl_complex b)
{
return gsl_complex_div(gsl_complex_log(a), gsl_complex_log(b));
}
gsl_complex gsl_complex_log10(gsl_complex a)
{ /* z = log10(a) */
return gsl_complex_mul_real(gsl_complex_log(a), 1 / log(10.));
}
/** Logarithm of a complex number (base e)
\ingroup complex
\param[in] z Complex number
\return \f$ \log z \f$*/
complex log(const complex& z)
{
return complex(gsl_complex_log(z.as_gsl_type()));
}
