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<title>Airy Bi' Function</title>
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<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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/airy.svg"></span>
</p></blockquote></div>
<p>
The final <a class="link" href="../../policy.html" title="Chapter 22. 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 22. 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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/airy_bi.svg" align="middle"></span>
</p></blockquote></div>
<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> as the following error plot illustrates:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/bi_prime__double.svg" align="middle"></span>
</p></blockquote></div>
<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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/airy_bip.svg"></span>
</p></blockquote></div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>)
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