1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 The GSL Team.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3 or
any later version published by the Free Software Foundation; with no
Invariant Sections and no cover texts. A copy of the license is
included in the section entitled "GNU Free Documentation License". -->
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>GNU Scientific Library – Reference Manual: Radial Functions for Hyperbolic Space</title>
<meta name="description" content="GNU Scientific Library – Reference Manual: Radial Functions for Hyperbolic Space">
<meta name="keywords" content="GNU Scientific Library – Reference Manual: Radial Functions for Hyperbolic Space">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<link href="index.html#Top" rel="start" title="Top">
<link href="Function-Index.html#Function-Index" rel="index" title="Function Index">
<link href="Legendre-Functions-and-Spherical-Harmonics.html#Legendre-Functions-and-Spherical-Harmonics" rel="up" title="Legendre Functions and Spherical Harmonics">
<link href="Logarithm-and-Related-Functions.html#Logarithm-and-Related-Functions" rel="next" title="Logarithm and Related Functions">
<link href="Conical-Functions.html#Conical-Functions" rel="previous" title="Conical Functions">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.indentedblock {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
div.smalllisp {margin-left: 3.2em}
kbd {font-style:oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nocodebreak {white-space:nowrap}
span.nolinebreak {white-space:nowrap}
span.roman {font-family:serif; font-weight:normal}
span.sansserif {font-family:sans-serif; font-weight:normal}
ul.no-bullet {list-style: none}
-->
</style>
</head>
<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Radial-Functions-for-Hyperbolic-Space"></a>
<div class="header">
<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>
</div>
<hr>
<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>
</body>
</html>
|
