Radial Functions for Hyperbolic Space

The following spherical functions are specializations of Legendre functions which give the regular eigenfunctions of the Laplacian on a 3-dimensional hyperbolic space @math{H3d}. Of particular interest is the flat limit, @math{\lambda \to \infty}, @math{\eta \to 0}, @math{\lambda\eta} fixed.

Function: double gsl_sf_legendre_H3d_0 (double lambda, double eta)

Function: int gsl_sf_legendre_H3d_0_e (double lambda, double eta, gsl_sf_result * result)

These routines compute the zeroth radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space, @math{L^{H3d}_0(\lambda,\eta) := \sin(\lambda\eta)/(\lambda\sinh(\eta))} for @c{$\eta \ge 0$} @math{\eta >= 0}. In the flat limit this takes the form @math{L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta)}

Function: double gsl_sf_legendre_H3d_1 (double lambda, double eta)

Function: int gsl_sf_legendre_H3d_1_e (double lambda, double eta, gsl_sf_result * result)

These routines compute the first radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space, @math{L^{H3d}_1(\lambda,\eta) := 1/\sqrt{\lambda^2 + 1} \sin(\lambda \eta)/(\lambda \sinh(\eta)) (\coth(\eta) - \lambda \cot(\lambda\eta))} for @c{$\eta \ge 0$} @math{\eta >= 0}. In the flat limit this takes the form @math{L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta)}.

Function: double gsl_sf_legendre_H3d (int l, double lambda, double eta)

Function: int gsl_sf_legendre_H3d_e (int l, double lambda, double eta, gsl_sf_result * result)

These routines compute the l'th radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space @c{$\eta \ge 0$} @math{\eta >= 0}, @c{$l \ge 0$} @math{l >= 0}. In the flat limit this takes the form @math{L^{H3d}_l(\lambda,\eta) = j_l(\lambda\eta)}.

Function: int gsl_sf_legendre_H3d_array (int lmax, double lambda, double eta, double result_array[])

This function computes an array of radial eigenfunctions @math{L^{H3d}_l(\lambda, \eta)} for @c{$0 \le l \le lmax$} @math{0 <= l <= lmax}.