| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> |
| <title>Airy Bi' Function</title> |
| <link rel="stylesheet" href="../../math.css" type="text/css"> |
| <meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> |
| <link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> |
| <link rel="up" href="../airy.html" title="Airy Functions"> |
| <link rel="prev" href="aip.html" title="Airy Ai' Function"> |
| <link rel="next" href="../ellint.html" title="Elliptic Integrals"> |
| </head> |
| <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> |
| <table cellpadding="2" width="100%"><tr> |
| <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> |
| <td align="center"><a href="../../../../../../index.html">Home</a></td> |
| <td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> |
| <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> |
| <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> |
| <td align="center"><a href="../../../../../../more/index.htm">More</a></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <a accesskey="p" href="aip.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../airy.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="../ellint.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.airy.bip"></a><a class="link" href="bip.html" title="Airy Bi' Function">Airy Bi' Function</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.airy.bip.h0"></a> |
| <span class="phrase"><a name="math_toolkit.airy.bip.synopsis"></a></span><a class="link" href="bip.html#math_toolkit.airy.bip.synopsis">Synopsis</a> |
| </h5> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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">airy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| </pre> |
| <pre class="programlisting"><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> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <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">airy_bi_prime</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span> |
| <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">airy_bi_prime</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&);</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.airy.bip.h1"></a> |
| <span class="phrase"><a name="math_toolkit.airy.bip.description"></a></span><a class="link" href="bip.html#math_toolkit.airy.bip.description">Description</a> |
| </h5> |
| <p> |
| The function <a class="link" href="bip.html" title="Airy Bi' Function">airy_bi_prime</a> calculates |
| the Airy function Bi' which is the derivative of the second solution to the |
| differential equation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/airy.svg"></span> |
| </p> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can |
| be used to control the behaviour of the function: how it handles errors, |
| what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy |
| documentation for more details</a>. |
| </p> |
| <p> |
| The following graph illustrates how this function changes as <span class="emphasis"><em>x</em></span> |
| changes: for negative <span class="emphasis"><em>x</em></span> the function is cyclic, while |
| for positive <span class="emphasis"><em>x</em></span> the value tends to infinity: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/airy_bi.svg" align="middle"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.airy.bip.h2"></a> |
| <span class="phrase"><a name="math_toolkit.airy.bip.accuracy"></a></span><a class="link" href="bip.html#math_toolkit.airy.bip.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| This function is implemented entirely in terms of the Bessel functions <a class="link" href="../bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a> and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a> |
| - refer to those functions for detailed accuracy information. |
| </p> |
| <p> |
| In general though, the relative error is low (less than 100 ε) for <span class="emphasis"><em>x |
| > 0</em></span> while only the absolute error is low for <span class="emphasis"><em>x < |
| 0</em></span>. |
| </p> |
| <h5> |
| <a name="math_toolkit.airy.bip.h3"></a> |
| <span class="phrase"><a name="math_toolkit.airy.bip.testing"></a></span><a class="link" href="bip.html#math_toolkit.airy.bip.testing">Testing</a> |
| </h5> |
| <p> |
| Since this function is implemented in terms of other special functions, there |
| are only a few basic sanity checks, using test values from <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>. |
| </p> |
| <h5> |
| <a name="math_toolkit.airy.bip.h4"></a> |
| <span class="phrase"><a name="math_toolkit.airy.bip.implementation"></a></span><a class="link" href="bip.html#math_toolkit.airy.bip.implementation">Implementation</a> |
| </h5> |
| <p> |
| This function is implemented in terms of the Bessel functions using the relations: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/airy_bip.svg"></span> |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
| </div></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <a accesskey="p" href="aip.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../airy.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="../ellint.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| </body> |
| </html> |