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
|
<!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, 2014, 2015, 2016 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 the
Invariant Sections being "GNU General Public License" and "Free Software
Needs Free Documentation", the Front-Cover text being "A GNU Manual",
and with the Back-Cover Text being (a) (see below). A copy of the
license is included in the section entitled "GNU Free Documentation
License".
(a) The Back-Cover Text is: "You have the freedom to copy and modify this
GNU Manual." -->
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>GNU Scientific Library – Reference Manual: The Landau Distribution</title>
<meta name="description" content="GNU Scientific Library – Reference Manual: The Landau Distribution">
<meta name="keywords" content="GNU Scientific Library – Reference Manual: The Landau Distribution">
<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="Random-Number-Distributions.html#Random-Number-Distributions" rel="up" title="Random Number Distributions">
<link href="The-Levy-alpha_002dStable-Distributions.html#The-Levy-alpha_002dStable-Distributions" rel="next" title="The Levy alpha-Stable Distributions">
<link href="The-Rayleigh-Tail-Distribution.html#The-Rayleigh-Tail-Distribution" rel="previous" title="The Rayleigh Tail Distribution">
<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="The-Landau-Distribution"></a>
<div class="header">
<p>
Next: <a href="The-Levy-alpha_002dStable-Distributions.html#The-Levy-alpha_002dStable-Distributions" accesskey="n" rel="next">The Levy alpha-Stable Distributions</a>, Previous: <a href="The-Rayleigh-Tail-Distribution.html#The-Rayleigh-Tail-Distribution" accesskey="p" rel="previous">The Rayleigh Tail Distribution</a>, Up: <a href="Random-Number-Distributions.html#Random-Number-Distributions" accesskey="u" rel="up">Random Number Distributions</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="The-Landau-Distribution-1"></a>
<h3 class="section">20.12 The Landau Distribution</h3>
<dl>
<dt><a name="index-gsl_005fran_005flandau"></a>Function: <em>double</em> <strong>gsl_ran_landau</strong> <em>(const gsl_rng * <var>r</var>)</em></dt>
<dd><a name="index-Landau-distribution"></a>
<p>This function returns a random variate from the Landau distribution. The
probability distribution for Landau random variates is defined
analytically by the complex integral,
</p>
<div class="example">
<pre class="example">p(x) = (1/(2 \pi i)) \int_{c-i\infty}^{c+i\infty} ds exp(s log(s) + x s)
</pre></div>
<p>For numerical purposes it is more convenient to use the following
equivalent form of the integral,
</p>
<div class="example">
<pre class="example">p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).
</pre></div>
</dd></dl>
<dl>
<dt><a name="index-gsl_005fran_005flandau_005fpdf"></a>Function: <em>double</em> <strong>gsl_ran_landau_pdf</strong> <em>(double <var>x</var>)</em></dt>
<dd><p>This function computes the probability density <em>p(x)</em> at <var>x</var>
for the Landau distribution using an approximation to the formula given
above.
</p></dd></dl>
<br>
</body>
</html>
|
