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.