### Legendre Form of Incomplete Elliptic Integrals

Function: double gsl_sf_ellint_F (double phi, double k, gsl_mode_t mode)
Function: int gsl_sf_ellint_F_e (double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the incomplete elliptic integral F(\phi,k) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_E (double phi, double k, gsl_mode_t mode)
Function: int gsl_sf_ellint_E_e (double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the incomplete elliptic integral E(\phi,k) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_P (double phi, double k, double n, gsl_mode_t mode)
Function: int gsl_sf_ellint_P_e (double phi, double k, double n, gsl_mode_t mode, gsl_sf_result * result)
  These routines compute the incomplete elliptic integral P(\phi,k,n) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_D (double phi, double k, double n, gsl_mode_t mode)
Function: int gsl_sf_ellint_D_e (double phi, double k, double n, gsl_mode_t mode, gsl_sf_result * result)
  These functions compute the incomplete elliptic integral D(\phi,k,n) which is defined through the Carlson form RD(x,y,z) by the following relation,

      D(\phi,k,n) = RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).