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<title>Airy Ai' Function</title>
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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.airy.aip"></a><a class="link" href="aip.html" title="Airy Ai' Function">Airy Ai' Function</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.airy.aip.h0"></a>
<span class="phrase"><a name="math_toolkit.airy.aip.synopsis"></a></span><a class="link" href="aip.html#math_toolkit.airy.aip.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_ai_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_ai_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.aip.h1"></a>
<span class="phrase"><a name="math_toolkit.airy.aip.description"></a></span><a class="link" href="aip.html#math_toolkit.airy.aip.description">Description</a>
</h5>
<p>
The function <a class="link" href="aip.html" title="Airy Ai' Function">airy_ai_prime</a> calculates
the Airy function Ai' which is the derivative of the first solution to the
differential equation:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/airy.png"></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 zero:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/airy_aip.png" align="middle"></span>
</p>
<h5>
<a name="math_toolkit.airy.aip.h2"></a>
<span class="phrase"><a name="math_toolkit.airy.aip.accuracy"></a></span><a class="link" href="aip.html#math_toolkit.airy.aip.accuracy">Accuracy</a>
</h5>
<p>
This function is implemented entirely in terms of the Bessel functions <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a> and <a class="link" href="../bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_k</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.aip.h3"></a>
<span class="phrase"><a name="math_toolkit.airy.aip.testing"></a></span><a class="link" href="aip.html#math_toolkit.airy.aip.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.aip.h4"></a>
<span class="phrase"><a name="math_toolkit.airy.aip.implementation"></a></span><a class="link" href="aip.html#math_toolkit.airy.aip.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_aip.png"></span>
</p>
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<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
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