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<title>Radial Functions for Hyperbolic Space - GNU Scientific Library -- Reference Manual</title>
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<a name="Radial-Functions-for-Hyperbolic-Space"></a>
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<h4 class="subsection">7.24.4 Radial Functions for Hyperbolic Space</h4>
<p>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.
<div class="defun">
— Function: double <b>gsl_sf_legendre_H3d_0</b> (<var>double lambda, double eta</var>)<var><a name="index-gsl_005fsf_005flegendre_005fH3d_005f0-719"></a></var><br>
— Function: int <b>gsl_sf_legendre_H3d_0_e</b> (<var>double lambda, double eta, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005flegendre_005fH3d_005f0_005fe-720"></a></var><br>
<blockquote><p>These routines compute the zeroth radial eigenfunction of the Laplacian on the
3-dimensional hyperbolic space,
<!-- {$$L^{H3d}_0(\lambda,\eta) := {\sin(\lambda\eta) \over \lambda\sinh(\eta)}$$} -->
L^{H3d}_0(\lambda,\eta) := \sin(\lambda\eta)/(\lambda\sinh(\eta))
for <!-- {$\eta \ge 0$} -->
\eta >= 0.
In the flat limit this takes the form
<!-- {$L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta)$} -->
L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta).
<!-- Exceptional Return Values: GSL_EDOM -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_legendre_H3d_1</b> (<var>double lambda, double eta</var>)<var><a name="index-gsl_005fsf_005flegendre_005fH3d_005f1-721"></a></var><br>
— Function: int <b>gsl_sf_legendre_H3d_1_e</b> (<var>double lambda, double eta, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005flegendre_005fH3d_005f1_005fe-722"></a></var><br>
<blockquote><p>These routines compute the first radial eigenfunction of the Laplacian on
the 3-dimensional hyperbolic space,
<!-- {$$L^{H3d}_1(\lambda,\eta) := {1\over\sqrt{\lambda^2 + 1}} {\left(\sin(\lambda \eta)\over \lambda \sinh(\eta)\right)} \left(\coth(\eta) - \lambda \cot(\lambda\eta)\right)$$} -->
L^{H3d}_1(\lambda,\eta) := 1/\sqrt{\lambda^2 + 1} \sin(\lambda \eta)/(\lambda \sinh(\eta)) (\coth(\eta) - \lambda \cot(\lambda\eta))
for <!-- {$\eta \ge 0$} -->
\eta >= 0.
In the flat limit this takes the form
<!-- {$L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta)$} -->
L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta).
<!-- Exceptional Return Values: GSL_EDOM -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_legendre_H3d</b> (<var>int l, double lambda, double eta</var>)<var><a name="index-gsl_005fsf_005flegendre_005fH3d-723"></a></var><br>
— Function: int <b>gsl_sf_legendre_H3d_e</b> (<var>int l, double lambda, double eta, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005flegendre_005fH3d_005fe-724"></a></var><br>
<blockquote><p>These routines compute the <var>l</var>-th radial eigenfunction of the
Laplacian on the 3-dimensional hyperbolic space <!-- {$\eta \ge 0$} -->
\eta >= 0, <!-- {$l \ge 0$} -->
l >= 0. In the flat limit this takes the form
<!-- {$L^{H3d}_l(\lambda,\eta) = j_l(\lambda\eta)$} -->
L^{H3d}_l(\lambda,\eta) = j_l(\lambda\eta).
<!-- Exceptional Return Values: GSL_EDOM -->
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_sf_legendre_H3d_array</b> (<var>int lmax, double lambda, double eta, double result_array</var>[])<var><a name="index-gsl_005fsf_005flegendre_005fH3d_005farray-725"></a></var><br>
<blockquote><p>This function computes an array of radial eigenfunctions
<!-- {$L^{H3d}_l( \lambda, \eta)$} -->
L^{H3d}_l(\lambda, \eta)
for <!-- {$0 \le l \le lmax$} -->
0 <= l <= lmax.
<!-- Exceptional Return Values: -->
</p></blockquote></div>
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