/// Dilogarithm function.
double
dilog(double x)
{
gsl_sf_result result;
int stat = gsl_sf_dilog_e(x, &result);
if (stat != GSL_SUCCESS)
{
std::ostringstream msg("Error in dilog:");
msg << " x=" << x;
throw std::runtime_error(msg.str());
}
else
return result.val;
}
double gsl_sf_dilog(const double x)
{
EVAL_RESULT(gsl_sf_dilog_e(x, &result));
}
int
gsl_sf_complex_dilog_xy_e(
const double x,
const double y,
gsl_sf_result * real_dl,
gsl_sf_result * imag_dl
)
{
const double zeta2 = M_PI*M_PI/6.0;
const double r2 = x*x + y*y;
if(y == 0.0)
{
if(x >= 1.0)
{
imag_dl->val = -M_PI * log(x);
imag_dl->err = 2.0 * GSL_DBL_EPSILON * fabs(imag_dl->val);
}
else
{
imag_dl->val = 0.0;
imag_dl->err = 0.0;
}
return gsl_sf_dilog_e(x, real_dl);
}
else if(fabs(r2 - 1.0) < GSL_DBL_EPSILON)
{
/* Lewin A.2.4.1 and A.2.4.2 */
const double theta = atan2(y, x);
const double term1 = theta*theta/4.0;
const double term2 = M_PI*fabs(theta)/2.0;
real_dl->val = zeta2 + term1 - term2;
real_dl->err = 2.0 * GSL_DBL_EPSILON * (zeta2 + term1 + term2);
return gsl_sf_clausen_e(theta, imag_dl);
}
else if(r2 < 1.0)
{
return dilogc_unitdisk(x, y, real_dl, imag_dl);
}
else
{
/* Reduce argument to unit disk. */
const double r = sqrt(r2);
const double x_tmp = x/r2;
const double y_tmp = -y/r2;
/* const double r_tmp = 1.0/r; */
gsl_sf_result result_re_tmp, result_im_tmp;
const int stat_dilog =
dilogc_unitdisk(x_tmp, y_tmp, &result_re_tmp, &result_im_tmp);
/* Unwind the inversion.
*
* Li_2(z) + Li_2(1/z) = -zeta(2) - 1/2 ln(-z)^2
*/
const double theta = atan2(y, x);
const double theta_abs = fabs(theta);
const double theta_sgn = ( theta < 0.0 ? -1.0 : 1.0 );
const double ln_minusz_re = log(r);
const double ln_minusz_im = theta_sgn * (theta_abs - M_PI);
const double lmz2_re = ln_minusz_re*ln_minusz_re - ln_minusz_im*ln_minusz_im;
const double lmz2_im = 2.0*ln_minusz_re*ln_minusz_im;
real_dl->val = -result_re_tmp.val - 0.5 * lmz2_re - zeta2;
real_dl->err = result_re_tmp.err + 2.0*GSL_DBL_EPSILON*(0.5 * fabs(lmz2_re) + zeta2);
imag_dl->val = -result_im_tmp.val - 0.5 * lmz2_im;
imag_dl->err = result_im_tmp.err + 2.0*GSL_DBL_EPSILON*fabs(lmz2_im);
return stat_dilog;
}
}
