libs/math/doc/html/math_toolkit/hypergeometric/hypergeometric_2f0.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Hypergeometric 2F0</title>
   5 <link rel="stylesheet" href="../../math.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
   7 <link rel="home" href="../../index.html" title="Math Toolkit 2.11.0">
   8 <link rel="up" href="../hypergeometric.html" title="Hypergeometric Functions">
   9 <link rel="prev" href="hypergeometric_0f1.html" title="Hypergeometric 0F1">
  10 <link rel="next" href="hypergeometric_1f1.html" title="Hypergeometric 1F1">
  11 </head>
  12 <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
  13 <table cellpadding="2" width="100%"><tr>
  14 <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
  15 <td align="center"><a href="../../../../../../index.html">Home</a></td>
  16 <td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
  17 <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
  18 <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
  19 <td align="center"><a href="../../../../../../more/index.htm">More</a></td>
  20 </tr></table>
  21 <hr>
  22 <div class="spirit-nav">
  23 <a accesskey="p" href="hypergeometric_0f1.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../hypergeometric.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="hypergeometric_1f1.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.hypergeometric.hypergeometric_2f0"></a><a class="link" href="hypergeometric_2f0.html" title="Hypergeometric 2F0">Hypergeometric
  28       <sub>2</sub><span class="emphasis"><em>F</em></span><sub>0</sub></a>
  29 </h3></div></div></div>
  30 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">hypergeometric_2F0</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  31
  32 <span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>
  33
  34 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
  35 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">hypergeometric_2F0</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a1</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">a2</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">);</span>
  36
  37 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  38 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">hypergeometric_2F0</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a1</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">a2</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  39
  40 <span class="special">}}</span>
  41 </pre>
  42 <h5>
  43 <a name="math_toolkit.hypergeometric.hypergeometric_2f0.h0"></a>
  44         <span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_2f0.description"></a></span><a class="link" href="hypergeometric_2f0.html#math_toolkit.hypergeometric.hypergeometric_2f0.description">Description</a>
  45       </h5>
  46 <p>
  47         The function <code class="computeroutput"><span class="identifier">hypergeometric_2F0</span></code>
  48         returns the result of
  49       </p>
  50 <div class="blockquote"><blockquote class="blockquote"><p>
  51           <span class="inlinemediaobject"><img src="../../../equations/hyper_2f0.svg"></span>
  52
  53         </p></blockquote></div>
  54 <p>
  55         The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
  56         type calculation rules</em></span></a> when <code class="computeroutput"><span class="identifier">T1</span></code>
  57         and <code class="computeroutput"><span class="identifier">T2</span></code> are different types.
  58       </p>
  59 <p>
  60         The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  61         be used to control the behaviour of the function: how it handles errors,
  62         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
  63         documentation for more details</a>.
  64       </p>
  65 <p>
  66         The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
  67         whenever the result is undefined or complex. The valid domain for this function
  68         occurs only when one of <code class="computeroutput"><span class="identifier">a1</span></code>
  69         or <code class="computeroutput"><span class="identifier">a2</span></code> is a negative integer:
  70         ie the polynomial case.
  71       </p>
  72 <h5>
  73 <a name="math_toolkit.hypergeometric.hypergeometric_2f0.h1"></a>
  74         <span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_2f0.implementation"></a></span><a class="link" href="hypergeometric_2f0.html#math_toolkit.hypergeometric.hypergeometric_2f0.implementation">Implementation</a>
  75       </h5>
  76 <p>
  77         When <code class="computeroutput"><span class="identifier">a1</span> <span class="special">==</span>
  78         <span class="identifier">a2</span> <span class="special">-</span>
  79         <span class="number">0.5</span></code> then the function is implemented
  80         in terms of the Hermite polynomial:
  81       </p>
  82 <div class="blockquote"><blockquote class="blockquote"><p>
  83           <span class="inlinemediaobject"><img src="../../../equations/hyper_2f0_hermite.svg"></span>
  84
  85         </p></blockquote></div>
  86 <p>
  87         When both <code class="computeroutput"><span class="identifier">a1</span></code> and <code class="computeroutput"><span class="identifier">a2</span></code> are integers then the function is implemented
  88         in terms of the associated-Laguerre polynomial:
  89       </p>
  90 <div class="blockquote"><blockquote class="blockquote"><p>
  91           <span class="inlinemediaobject"><img src="../../../equations/hyper_2f0_laguerre.svg"></span>
  92
  93         </p></blockquote></div>
  94 <p>
  95         If the defining series is divergent, we use the continued fraction
  96       </p>
  97 <div class="blockquote"><blockquote class="blockquote"><p>
  98           <span class="inlinemediaobject"><img src="../../../equations/hyper_2f0_cf.svg"></span>
  99
 100         </p></blockquote></div>
 101 <p>
 102         Otherwise we use the defining series.
 103       </p>
 104 </div>
 105 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 106 <td align="left"></td>
 107 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 108       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 109       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 110       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 111       Daryle Walker and Xiaogang Zhang<p>
 112         Distributed under the Boost Software License, Version 1.0. (See accompanying
 113         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
 114       </p>
 115 </div></td>
 116 </tr></table>
 117 <hr>
 118 <div class="spirit-nav">
 119 <a accesskey="p" href="hypergeometric_0f1.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../hypergeometric.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="hypergeometric_1f1.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
 120 </div>
 121 </body>
 122 </html>