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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.hankel.sph_hankel"></a><a class="link" href="sph_hankel.html" title="Spherical Hankel Functions">Spherical Hankel Functions</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.hankel.sph_hankel.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.hankel.sph_hankel.synopsis"></a></span><a class="link" href="sph_hankel.html#math_toolkit.hankel.sph_hankel.synopsis">Synopsis</a>
  32       </h5>
  33 <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>
  34 <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><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="special">&gt;</span> <span class="identifier">sph_hankel_1</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span>
  35
  36 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  37 <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><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="special">&gt;</span> <span class="identifier">sph_hankel_1</span><span class="special">(</span><span class="identifier">T1</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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  38
  39 <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>
  40 <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><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="special">&gt;</span> <span class="identifier">sph_hankel_2</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span>
  41
  42 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  43 <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><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="special">&gt;</span> <span class="identifier">sph_hankel_2</span><span class="special">(</span><span class="identifier">T1</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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  44 </pre>
  45 <h5>
  46 <a name="math_toolkit.hankel.sph_hankel.h1"></a>
  47         <span class="phrase"><a name="math_toolkit.hankel.sph_hankel.description"></a></span><a class="link" href="sph_hankel.html#math_toolkit.hankel.sph_hankel.description">Description</a>
  48       </h5>
  49 <p>
  50         The functions <a class="link" href="sph_hankel.html" title="Spherical Hankel Functions">sph_hankel_1</a>
  51         and <a class="link" href="sph_hankel.html" title="Spherical Hankel Functions">sph_hankel_2</a> return
  52         the result of the <a href="http://dlmf.nist.gov/10.47#P1" target="_top">spherical Hankel
  53         functions</a> of the first and second kind respectively:
  54       </p>
  55 <div class="blockquote"><blockquote class="blockquote"><p>
  56           <span class="inlinemediaobject"><img src="../../../equations/hankel4.svg"></span>
  57
  58         </p></blockquote></div>
  59 <div class="blockquote"><blockquote class="blockquote"><p>
  60           <span class="inlinemediaobject"><img src="../../../equations/hankel5.svg"></span>
  61
  62         </p></blockquote></div>
  63 <p>
  64         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
  65         type calculation rules</em></span></a> when T1 and T2 are different types.
  66         The functions are also optimised for the relatively common case that T1 is
  67         an integer.
  68       </p>
  69 <p>
  70         The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  71         be used to control the behaviour of the function: how it handles errors,
  72         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
  73         documentation for more details</a>.
  74       </p>
  75 <p>
  76         Note that while the arguments to these functions are real values, the results
  77         are complex. That means that the functions can only be instantiated on types
  78         <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>
  79         and <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>.
  80         The functions have also been extended to operate over the whole range of
  81         <span class="emphasis"><em>v</em></span> and <span class="emphasis"><em>x</em></span> (unlike <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>
  82         and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>).
  83       </p>
  84 <h5>
  85 <a name="math_toolkit.hankel.sph_hankel.h2"></a>
  86         <span class="phrase"><a name="math_toolkit.hankel.sph_hankel.testing"></a></span><a class="link" href="sph_hankel.html#math_toolkit.hankel.sph_hankel.testing">Testing</a>
  87       </h5>
  88 <p>
  89         There are just a few spot tests to exercise all the special case handling
  90         - the bulk of the testing is done on the Bessel functions upon which these
  91         are based.
  92       </p>
  93 <h5>
  94 <a name="math_toolkit.hankel.sph_hankel.h3"></a>
  95         <span class="phrase"><a name="math_toolkit.hankel.sph_hankel.accuracy"></a></span><a class="link" href="sph_hankel.html#math_toolkit.hankel.sph_hankel.accuracy">Accuracy</a>
  96       </h5>
  97 <p>
  98         Refer to <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>
  99         and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>.
 100       </p>
 101 <h5>
 102 <a name="math_toolkit.hankel.sph_hankel.h4"></a>
 103         <span class="phrase"><a name="math_toolkit.hankel.sph_hankel.implementation"></a></span><a class="link" href="sph_hankel.html#math_toolkit.hankel.sph_hankel.implementation">Implementation</a>
 104       </h5>
 105 <p>
 106         These functions are trivially implemented in terms of <a class="link" href="cyl_hankel.html" title="Cyclic Hankel Functions">cyl_hankel_1</a>
 107         and <a class="link" href="cyl_hankel.html" title="Cyclic Hankel Functions">cyl_hankel_2</a>.
 108       </p>
 109 </div>
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 115       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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 117         Distributed under the Boost Software License, Version 1.0. (See accompanying
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