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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>
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>
33 <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>
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>
37 <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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>
40 <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>
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">&);</span>
43 <span class="special">}}</span> <span class="comment">// namespaces</span>
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>
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:
54 <div class="blockquote"><blockquote class="blockquote"><p>
55 <span class="inlinemediaobject"><img src="../../../equations/airy.svg"></span>
57 </p></blockquote></div>
59 The final <a class="link" href="../../policy.html" title="Chapter 20. 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 20. Policies: Controlling Precision, Error Handling etc">policy
62 documentation for more details</a>.
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:
69 <div class="blockquote"><blockquote class="blockquote"><p>
70 <span class="inlinemediaobject"><img src="../../../graphs/airy_bi.svg" align="middle"></span>
72 </p></blockquote></div>
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>
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.
82 In general though, the relative error is low (less than 100 ε) for <span class="emphasis"><em>x
83 > 0</em></span> while only the absolute error is low for <span class="emphasis"><em>x <
84 0</em></span> as the following error plot illustrates:
86 <div class="blockquote"><blockquote class="blockquote"><p>
87 <span class="inlinemediaobject"><img src="../../../graphs/bi_prime__double.svg" align="middle"></span>
89 </p></blockquote></div>
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>
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>.
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>
103 This function is implemented in terms of the Bessel functions using the relations:
105 <div class="blockquote"><blockquote class="blockquote"><p>
106 <span class="inlinemediaobject"><img src="../../../equations/airy_bip.svg"></span>
108 </p></blockquote></div>
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