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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.bessel.sph_bessel"></a><a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">Spherical Bessel Functions
  28       of the First and Second Kinds</a>
  29 </h3></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.bessel.sph_bessel.h0"></a>
  32         <span class="phrase"><a name="math_toolkit.bessel.sph_bessel.synopsis"></a></span><a class="link" href="sph_bessel.html#math_toolkit.bessel.sph_bessel.synopsis">Synopsis</a>
  33       </h5>
  34 <p>
  35         <code class="computeroutput"><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">bessel</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></code>
  36       </p>
  37 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">sph_bessel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><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">sph_bessel</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  42
  43 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
  44 <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">sph_neumann</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span>
  45
  46 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  47 <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">sph_neumann</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  48 </pre>
  49 <h5>
  50 <a name="math_toolkit.bessel.sph_bessel.h1"></a>
  51         <span class="phrase"><a name="math_toolkit.bessel.sph_bessel.description"></a></span><a class="link" href="sph_bessel.html#math_toolkit.bessel.sph_bessel.description">Description</a>
  52       </h5>
  53 <p>
  54         The functions <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_bessel</a>
  55         and <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_neumann</a> return
  56         the result of the Spherical Bessel functions of the first and second kinds
  57         respectively:
  58       </p>
  59 <p>
  60         sph_bessel(v, x) = j<sub>v</sub>(x)
  61       </p>
  62 <p>
  63         sph_neumann(v, x) = y<sub>v</sub>(x) = n<sub>v</sub>(x)
  64       </p>
  65 <p>
  66         where:
  67       </p>
  68 <p>
  69         <span class="inlinemediaobject"><img src="../../../equations/sbessel2.png"></span>
  70       </p>
  71 <p>
  72         The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
  73         type calculation rules</em></span></a> for the single argument type T.
  74       </p>
  75 <p>
  76         The final <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  77         be used to control the behaviour of the function: how it handles errors,
  78         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">policy
  79         documentation for more details</a>.
  80       </p>
  81 <p>
  82         The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
  83         whenever the result is undefined or complex: this occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special">&lt;</span> <span class="number">0</span></code>.
  84       </p>
  85 <p>
  86         The j<sub>v</sub> &#160; function is cyclic like J<sub>v</sub> &#160; but differs in its behaviour at the origin:
  87       </p>
  88 <p>
  89         <span class="inlinemediaobject"><img src="../../../graphs/sph_bessel.png" align="middle"></span>
  90       </p>
  91 <p>
  92         Likewise y<sub>v</sub> &#160; is also cyclic for large x, but tends to -&#8734; &#160;
  93 for small <span class="emphasis"><em>x</em></span>:
  94       </p>
  95 <p>
  96         <span class="inlinemediaobject"><img src="../../../graphs/sph_neumann.png" align="middle"></span>
  97       </p>
  98 <h5>
  99 <a name="math_toolkit.bessel.sph_bessel.h2"></a>
 100         <span class="phrase"><a name="math_toolkit.bessel.sph_bessel.testing"></a></span><a class="link" href="sph_bessel.html#math_toolkit.bessel.sph_bessel.testing">Testing</a>
 101       </h5>
 102 <p>
 103         There are two sets of test values: spot values calculated using <a href="http://functions.wolfram.com/" target="_top">functions.wolfram.com</a>,
 104         and a much larger set of tests computed using a simplified version of this
 105         implementation (with all the special case handling removed).
 106       </p>
 107 <h5>
 108 <a name="math_toolkit.bessel.sph_bessel.h3"></a>
 109         <span class="phrase"><a name="math_toolkit.bessel.sph_bessel.accuracy"></a></span><a class="link" href="sph_bessel.html#math_toolkit.bessel.sph_bessel.accuracy">Accuracy</a>
 110       </h5>
 111 <p>
 112         Other than for some special cases, these functions are computed in terms
 113         of <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a> and
 114         <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>: refer
 115         to these functions for accuracy data.
 116       </p>
 117 <h5>
 118 <a name="math_toolkit.bessel.sph_bessel.h4"></a>
 119         <span class="phrase"><a name="math_toolkit.bessel.sph_bessel.implementation"></a></span><a class="link" href="sph_bessel.html#math_toolkit.bessel.sph_bessel.implementation">Implementation</a>
 120       </h5>
 121 <p>
 122         Other than error handling and a couple of special cases these functions are
 123         implemented directly in terms of their definitions:
 124       </p>
 125 <p>
 126         <span class="inlinemediaobject"><img src="../../../equations/sbessel2.png"></span>
 127       </p>
 128 <p>
 129         The special cases occur for:
 130       </p>
 131 <p>
 132         j<sub>0</sub> &#160;= <a class="link" href="../sinc/sinc_pi.html" title="sinc_pi">sinc_pi</a>(x) = sin(x)
 133         / x
 134       </p>
 135 <p>
 136         and for small <span class="emphasis"><em>x &lt; 1</em></span>, we can use the series:
 137       </p>
 138 <p>
 139         <span class="inlinemediaobject"><img src="../../../equations/sbessel5.png"></span>
 140       </p>
 141 <p>
 142         which neatly avoids the problem of calculating 0/0 that can occur with the
 143         main definition as x &#8594; 0.
 144       </p>
 145 </div>
 146 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 147 <td align="left"></td>
 148 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 149       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 150       Holin, Bruno Lalande, John Maddock, Johan R&#229;de, Gautam Sewani, Benjamin Sobotta,
 151       Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 152         Distributed under the Boost Software License, Version 1.0. (See accompanying
 153         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>)
 154       </p>
 155 </div></td>
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