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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="id1129669"></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="id1130036"></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="id1130235"></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="id1130257"></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="id1130283"></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> |
| </div> |
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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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