— Function: double gsl_sf_ellint_Kcomp (double k, gsl_mode_t mode)
— Function: int gsl_sf_ellint_Kcomp_e (double k, gsl_mode_t mode, gsl_sf_result * result)

These routines compute the complete elliptic integral K(k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.

— Function: double gsl_sf_ellint_Ecomp (double k, gsl_mode_t mode)
— Function: int gsl_sf_ellint_Ecomp_e (double k, gsl_mode_t mode, gsl_sf_result * result)

These routines compute the complete elliptic integral E(k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.

— Function: double gsl_sf_ellint_Pcomp (double k, double n, gsl_mode_t mode)
— Function: int gsl_sf_ellint_Pcomp_e (double k, double n, gsl_mode_t mode, gsl_sf_result * result)

These routines compute the complete elliptic integral \Pi(k,n) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of sign n \to -n.