#### 7.24.4 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 H3d. Of particular interest is the flat limit, \lambda \to \infty, \eta \to 0, \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, L^{H3d}_0(\lambda,\eta) := \sin(\lambda\eta)/(\lambda\sinh(\eta)) for \eta >= 0. In the flat limit this takes the form 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, L^{H3d}_1(\lambda,\eta) := 1/\sqrt{\lambda^2 + 1} \sin(\lambda \eta)/(\lambda \sinh(\eta)) (\coth(\eta) - \lambda \cot(\lambda\eta)) for \eta >= 0. In the flat limit this takes the form 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 \eta >= 0, l >= 0. In the flat limit this takes the form 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 L^{H3d}_l(\lambda, \eta) for 0 <= l <= lmax.

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