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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.airy.ai"></a><a class="link" href="ai.html" title="Airy Ai Function">Airy Ai Function</a>
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<h5>
<a name="math_toolkit.airy.ai.h0"></a>
<span class="phrase"><a name="math_toolkit.airy.ai.synopsis"></a></span><a class="link" href="ai.html#math_toolkit.airy.ai.synopsis">Synopsis</a>
</h5>
<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">airy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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</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">&lt;</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">&gt;</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</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">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<h5>
<a name="math_toolkit.airy.ai.h1"></a>
<span class="phrase"><a name="math_toolkit.airy.ai.description"></a></span><a class="link" href="ai.html#math_toolkit.airy.ai.description">Description</a>
</h5>
<p>
The function <a class="link" href="ai.html" title="Airy Ai Function">airy_ai</a> calculates
the Airy function Ai which is the first solution to the differential equation:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/airy.svg"></span>
</p>
<p>
See Weisstein, Eric W. "Airy Functions." From MathWorld--A Wolfram
Web Resource. <a href="http://mathworld.wolfram.com/AiryFunctions.html" target="_top">http://mathworld.wolfram.com/AiryFunctions.html</a>;
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;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&#160;15.&#160;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_ai.svg" align="middle"></span>
</p>
<h5>
<a name="math_toolkit.airy.ai.h2"></a>
<span class="phrase"><a name="math_toolkit.airy.ai.accuracy"></a></span><a class="link" href="ai.html#math_toolkit.airy.ai.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 &#949;) for <span class="emphasis"><em>x
&gt; 0</em></span> while only the absolute error is low for <span class="emphasis"><em>x &lt;
0</em></span>.
</p>
<h5>
<a name="math_toolkit.airy.ai.h3"></a>
<span class="phrase"><a name="math_toolkit.airy.ai.testing"></a></span><a class="link" href="ai.html#math_toolkit.airy.ai.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">Wolfram
Airy Functions</a>.
</p>
<h5>
<a name="math_toolkit.airy.ai.h4"></a>
<span class="phrase"><a name="math_toolkit.airy.ai.implementation"></a></span><a class="link" href="ai.html#math_toolkit.airy.ai.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_ai.svg"></span>
</p>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, 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>)
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