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4<title>Airy Bi' Function</title>
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26<div class="titlepage"><div><div><h3 class="title">
27<a name="math_toolkit.airy.bip"></a><a class="link" href="bip.html" title="Airy Bi' Function">Airy Bi' Function</a>
28</h3></div></div></div>
29<h5>
30<a name="math_toolkit.airy.bip.h0"></a>
31        <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>
32      </h5>
33<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>
34</pre>
35<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>
36
37 <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>
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">airy_bi_prime</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
39
40 <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>
41 <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">&amp;);</span>
42
43<span class="special">}}</span> <span class="comment">// namespaces</span>
44</pre>
45<h5>
46<a name="math_toolkit.airy.bip.h1"></a>
47        <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>
48      </h5>
49<p>
50        The function <a class="link" href="bip.html" title="Airy Bi' Function">airy_bi_prime</a> calculates
51        the Airy function Bi' which is the derivative of the second solution to the
52        differential equation:
53      </p>
54<div class="blockquote"><blockquote class="blockquote"><p>
55          <span class="inlinemediaobject"><img src="../../../equations/airy.svg"></span>
56
57        </p></blockquote></div>
58<p>
59        The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
60        be used to control the behaviour of the function: how it handles errors,
61        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
62        documentation for more details</a>.
63      </p>
64<p>
65        The following graph illustrates how this function changes as <span class="emphasis"><em>x</em></span>
66        changes: for negative <span class="emphasis"><em>x</em></span> the function is cyclic, while
67        for positive <span class="emphasis"><em>x</em></span> the value tends to infinity:
68      </p>
69<div class="blockquote"><blockquote class="blockquote"><p>
70          <span class="inlinemediaobject"><img src="../../../graphs/airy_bi.svg" align="middle"></span>
71
72        </p></blockquote></div>
73<h5>
74<a name="math_toolkit.airy.bip.h2"></a>
75        <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>
76      </h5>
77<p>
78        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>
79        - refer to those functions for detailed accuracy information.
80      </p>
81<p>
82        In general though, the relative error is low (less than 100 ε) for <span class="emphasis"><em>x
83        &gt; 0</em></span> while only the absolute error is low for <span class="emphasis"><em>x &lt;
84        0</em></span> as the following error plot illustrates:
85      </p>
86<div class="blockquote"><blockquote class="blockquote"><p>
87          <span class="inlinemediaobject"><img src="../../../graphs/bi_prime__double.svg" align="middle"></span>
88
89        </p></blockquote></div>
90<h5>
91<a name="math_toolkit.airy.bip.h3"></a>
92        <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>
93      </h5>
94<p>
95        Since this function is implemented in terms of other special functions, there
96        are only a few basic sanity checks, using test values from <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>.
97      </p>
98<h5>
99<a name="math_toolkit.airy.bip.h4"></a>
100        <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>
101      </h5>
102<p>
103        This function is implemented in terms of the Bessel functions using the relations:
104      </p>
105<div class="blockquote"><blockquote class="blockquote"><p>
106          <span class="inlinemediaobject"><img src="../../../equations/airy_bip.svg"></span>
107
108        </p></blockquote></div>
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113      Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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115      Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
116      Daryle Walker and Xiaogang Zhang<p>
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