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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.special.bessel.sph_bessel"></a><a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds"> Spherical
Bessel Functions of the First and Second Kinds</a>
</h4></div></div></div>
<a name="math_toolkit.special.bessel.sph_bessel.synopsis"></a><h5>
<a name="id1132656"></a>
<a class="link" href="sph_bessel.html#math_toolkit.special.bessel.sph_bessel.synopsis">Synopsis</a>
</h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</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>
<a class="link" href="../../main_overview/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">template</span> <span class="special"><</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="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span>
<span class="keyword">template</span> <span class="special"><</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>
<a class="link" href="../../main_overview/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">template</span> <span class="special"><</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="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span>
</pre>
<a name="math_toolkit.special.bessel.sph_bessel.description"></a><h5>
<a name="id1133023"></a>
<a class="link" href="sph_bessel.html#math_toolkit.special.bessel.sph_bessel.description">Description</a>
</h5>
<p>
The functions <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_bessel</a>
and <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_neumann</a>
return the result of the Spherical Bessel functions of the first and second
kinds respectively:
</p>
<p>
sph_bessel(v, x) = j<sub>v</sub>(x)
</p>
<p>
sph_neumann(v, x) = y<sub>v</sub>(x) = n<sub>v</sub>(x)
</p>
<p>
where:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/sbessel2.png"></span>
</p>
<p>
The return type of these functions is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
type calculation rules</em></span></a> for the single argument type T.
</p>
<p>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Policies">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="Policies">policy documentation for more details</a>.
</p>
<p>
</p>
<p>
The functions return the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>
whenever the result is undefined or complex: this occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special"><</span> <span class="number">0</span></code>.
</p>
<p>
The j<sub>v</sub> function is cyclic like J<sub>v</sub> but differs in its behaviour at the origin:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/sph_bessel.png" align="middle"></span>
</p>
<p>
Likewise y<sub>v</sub> is also cyclic for large x, but tends to -∞
for small <span class="emphasis"><em>x</em></span>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/sph_neumann.png" align="middle"></span>
</p>
<a name="math_toolkit.special.bessel.sph_bessel.testing"></a><h5>
<a name="id1133221"></a>
<a class="link" href="sph_bessel.html#math_toolkit.special.bessel.sph_bessel.testing">Testing</a>
</h5>
<p>
There are two sets of test values: spot values calculated using <a href="http://functions.wolfram.com/" target="_top">functions.wolfram.com</a>, and
a much larger set of tests computed using a simplified version of this
implementation (with all the special case handling removed).
</p>
<a name="math_toolkit.special.bessel.sph_bessel.accuracy"></a><h5>
<a name="id1133243"></a>
<a class="link" href="sph_bessel.html#math_toolkit.special.bessel.sph_bessel.accuracy">Accuracy</a>
</h5>
<p>
Other than for some special cases, these functions are computed in terms
of <a class="link" href="bessel.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>
and <a class="link" href="bessel.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>:
refer to these functions for accuracy data.
</p>
<a name="math_toolkit.special.bessel.sph_bessel.implementation"></a><h5>
<a name="id1133269"></a>
<a class="link" href="sph_bessel.html#math_toolkit.special.bessel.sph_bessel.implementation">Implementation</a>
</h5>
<p>
Other than error handling and a couple of special cases these functions
are implemented directly in terms of their definitions:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/sbessel2.png"></span>
</p>
<p>
The special cases occur for:
</p>
<p>
j<sub>0</sub>= <a class="link" href="../sinc/sinc_pi.html" title="sinc_pi">sinc_pi</a>(x)
= sin(x) / x
</p>
<p>
and for small <span class="emphasis"><em>x < 1</em></span>, we can use the series:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/sbessel5.png"></span>
</p>
<p>
which neatly avoids the problem of calculating 0/0 that can occur with
the main definition as x → 0.
</p>
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Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani
and Thijs van den Berg<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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