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<title>GNU Scientific Library – Reference Manual: Radial Functions for Hyperbolic Space</title>
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<a name="Radial-Functions-for-Hyperbolic-Space"></a>
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<p>
Previous: <a href="Conical-Functions.html#Conical-Functions" accesskey="p" rel="previous">Conical Functions</a>, Up: <a href="Legendre-Functions-and-Spherical-Harmonics.html#Legendre-Functions-and-Spherical-Harmonics" accesskey="u" rel="up">Legendre Functions and Spherical Harmonics</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Radial-Functions-for-Hyperbolic-Space-1"></a>
<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 <em>H3d</em>. Of particular interest is
the flat limit, <em>\lambda \to \infty</em>, <em>\eta \to 0</em>,
<em>\lambda\eta</em> fixed.
</p>
<dl>
<dt><a name="index-gsl_005fsf_005flegendre_005fH3d_005f0"></a>Function: <em>double</em> <strong>gsl_sf_legendre_H3d_0</strong> <em>(double <var>lambda</var>, double <var>eta</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005flegendre_005fH3d_005f0_005fe"></a>Function: <em>int</em> <strong>gsl_sf_legendre_H3d_0_e</strong> <em>(double <var>lambda</var>, double <var>eta</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the zeroth radial eigenfunction of the Laplacian on the
3-dimensional hyperbolic space,
<em>L^{H3d}_0(\lambda,\eta) := \sin(\lambda\eta)/(\lambda\sinh(\eta))</em>
for <em>\eta >= 0</em>.
In the flat limit this takes the form
<em>L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005flegendre_005fH3d_005f1"></a>Function: <em>double</em> <strong>gsl_sf_legendre_H3d_1</strong> <em>(double <var>lambda</var>, double <var>eta</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005flegendre_005fH3d_005f1_005fe"></a>Function: <em>int</em> <strong>gsl_sf_legendre_H3d_1_e</strong> <em>(double <var>lambda</var>, double <var>eta</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the first radial eigenfunction of the Laplacian on
the 3-dimensional hyperbolic space,
<em>L^{H3d}_1(\lambda,\eta) := 1/\sqrt{\lambda^2 + 1} \sin(\lambda \eta)/(\lambda \sinh(\eta)) (\coth(\eta) - \lambda \cot(\lambda\eta))</em>
for <em>\eta >= 0</em>.
In the flat limit this takes the form
<em>L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005flegendre_005fH3d"></a>Function: <em>double</em> <strong>gsl_sf_legendre_H3d</strong> <em>(int <var>l</var>, double <var>lambda</var>, double <var>eta</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005flegendre_005fH3d_005fe"></a>Function: <em>int</em> <strong>gsl_sf_legendre_H3d_e</strong> <em>(int <var>l</var>, double <var>lambda</var>, double <var>eta</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the <var>l</var>-th radial eigenfunction of the
Laplacian on the 3-dimensional hyperbolic space <em>\eta >= 0</em>, <em>l >= 0</em>. In the flat limit this takes the form
<em>L^{H3d}_l(\lambda,\eta) = j_l(\lambda\eta)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005flegendre_005fH3d_005farray"></a>Function: <em>int</em> <strong>gsl_sf_legendre_H3d_array</strong> <em>(int <var>lmax</var>, double <var>lambda</var>, double <var>eta</var>, double <var>result_array</var>[])</em></dt>
<dd><p>This function computes an array of radial eigenfunctions
<em>L^{H3d}_l(\lambda, \eta)</em>
for <em>0 <= l <= lmax</em>.
</p></dd></dl>
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