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<div class="titlepage"><div><div><h3 class="title">
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<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
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of the First and Second Kinds</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.bessel.sph_bessel.h0"></a>
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<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>
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</h5>
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<p>
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<code class="computeroutput"><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">bessel</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></code>
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</p>
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<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>
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<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>
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<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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<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>
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<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>
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<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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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</pre>
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<h5>
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<a name="math_toolkit.bessel.sph_bessel.h1"></a>
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<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>
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</h5>
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<p>
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The functions <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_bessel</a>
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and <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_neumann</a> return
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the result of the Spherical Bessel functions of the first and second kinds
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respectively:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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sph_bessel(v, x) = j<sub>v</sub>(x)
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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sph_neumann(v, x) = y<sub>v</sub>(x) = n<sub>v</sub>(x)
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</p></blockquote></div>
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<p>
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where:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span>
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</p></blockquote></div>
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<p>
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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
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type calculation rules</em></span></a> for the single argument type T.
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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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
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documentation for more details</a>.
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</p>
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<p>
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The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
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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>.
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</p>
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<p>
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The j<sub>v</sub> function is cyclic like J<sub>v</sub> but differs in its behaviour at the origin:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/sph_bessel.svg" align="middle"></span>
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</p></blockquote></div>
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<p>
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Likewise y<sub>v</sub> is also cyclic for large x, but tends to -∞
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for small <span class="emphasis"><em>x</em></span>:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/sph_neumann.svg" align="middle"></span>
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</p></blockquote></div>
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<h5>
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<a name="math_toolkit.bessel.sph_bessel.h2"></a>
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<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>
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</h5>
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<p>
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There are two sets of test values: spot values calculated using <a href="http://functions.wolfram.com/" target="_top">functions.wolfram.com</a>,
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and a much larger set of tests computed using a simplified version of this
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implementation (with all the special case handling removed).
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</p>
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<h5>
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<a name="math_toolkit.bessel.sph_bessel.h3"></a>
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<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>
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</h5>
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<div class="table">
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<a name="math_toolkit.bessel.sph_bessel.table_sph_bessel"></a><p class="title"><b>Table 8.48. Error rates for sph_bessel</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for sph_bessel">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5150<br> Sun Solaris<br> long double
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</p>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 14.1<br> Win32<br> double
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</p>
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</th>
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</tr></thead>
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<tbody><tr>
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<td>
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<p>
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Bessel j: Random Data
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 243ε (Mean = 13.3ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 1.91e+06ε (Mean = 1.09e+05ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.978ε (Mean = 0.0445ε)</span><br>
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<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.79e+03ε (Mean = 107ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 243ε (Mean = 33.7ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 245ε (Mean = 16.3ε)</span>
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</p>
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</td>
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</tr></tbody>
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</table></div>
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</div>
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<br class="table-break"><div class="table">
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<a name="math_toolkit.bessel.sph_bessel.table_sph_neumann"></a><p class="title"><b>Table 8.49. Error rates for sph_neumann</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for sph_neumann">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5150<br> Sun Solaris<br> long double
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</p>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 14.1<br> Win32<br> double
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</p>
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</th>
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</tr></thead>
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<tbody><tr>
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<td>
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<p>
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y: Random Data
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 234ε (Mean = 19.5ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 1.6e+06ε (Mean = 1.4e+05ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.995ε (Mean = 0.0665ε)</span><br>
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<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 8.5e+04ε (Mean = 5.33e+03ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 234ε (Mean = 19.8ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 281ε (Mean = 31.1ε)</span>
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</p>
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</td>
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</tr></tbody>
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</table></div>
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</div>
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<br class="table-break"><h5>
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<a name="math_toolkit.bessel.sph_bessel.h4"></a>
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<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>
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</h5>
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<p>
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Other than error handling and a couple of special cases these functions are
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implemented directly in terms of their definitions:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span>
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</p></blockquote></div>
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<p>
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The special cases occur for:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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j<sub>0</sub>= <a class="link" href="../sinc/sinc_pi.html" title="sinc_pi">sinc_pi</a>(x) = sin(x)
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/ x
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</p></blockquote></div>
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<p>
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and for small <span class="emphasis"><em>x < 1</em></span>, we can use the series:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/sbessel5.svg"></span>
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</p></blockquote></div>
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<p>
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which neatly avoids the problem of calculating 0/0 that can occur with the
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main definition as x → 0.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
|
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
|
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Daryle Walker and Xiaogang Zhang<p>
|
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Distributed under the Boost Software License, Version 1.0. (See accompanying
|
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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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</p>
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</div></td>
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</tr></table>
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