/// Complete elliptic integrals of the second kind.
double
comp_ellint_2(double k)
{
const gsl_mode_t mode = GSL_PREC_DOUBLE;
gsl_sf_result result;
int stat = gsl_sf_ellint_Ecomp_e(k, mode, &result);
if (stat != GSL_SUCCESS)
{
std::ostringstream msg("Error in ellint_2:");
msg << " k=" << k;
throw std::runtime_error(msg.str());
}
else
return result.val;
}
/* [Carlson, Numer. Math. 33 (1979) 1, (4.2)] */
int
gsl_sf_ellint_E_e(double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
{
const double sin_phi = sin(phi);
const double sin2_phi = sin_phi * sin_phi;
const double x = 1.0 - sin2_phi;
const double y = 1.0 - k*k*sin2_phi;
if(x < GSL_DBL_EPSILON) {
return gsl_sf_ellint_Ecomp_e(k, mode, result);
}
else {
gsl_sf_result rf;
gsl_sf_result rd;
const double sin3_phi = sin2_phi * sin_phi;
const int rfstatus = gsl_sf_ellint_RF_e(x, y, 1.0, mode, &rf);
const int rdstatus = gsl_sf_ellint_RD_e(x, y, 1.0, mode, &rd);
result->val = sin_phi * rf.val - k*k/3.0 * sin3_phi * rd.val;
result->err = GSL_DBL_EPSILON * fabs(sin_phi * rf.val);
result->err += fabs(sin_phi*rf.err);
result->err += k*k/3.0 * GSL_DBL_EPSILON * fabs(sin3_phi * rd.val);
result->err += k*k/3.0 * fabs(sin3_phi*rd.err);
return GSL_ERROR_SELECT_2(rfstatus, rdstatus);
}
}
double gsl_sf_ellint_Ecomp(double k, gsl_mode_t mode)
{
EVAL_RESULT(gsl_sf_ellint_Ecomp_e(k, mode, &result));
}
/**
* C++ version of gsl_sf_ellint_Ecomp_e().
* Legendre form of complete elliptic integrals
*
* E(k) = Integral[ Sqrt[1 - k^2 Sin[t]^2], {t, 0, Pi/2}]
*
* @param k A real number
* @param mode The mode
* @param result The result as a @c gsl::sf::result object
* @return GSL_SUCCESS or GSL_EDOM
*/
inline int Ecomp_e( double k, mode_t mode, result& result ){
return gsl_sf_ellint_Ecomp_e( k, mode, &result ); }
