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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.hankel.cyl_hankel"></a><a class="link" href="cyl_hankel.html" title="Cyclic Hankel Functions">Cyclic Hankel Functions</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.hankel.cyl_hankel.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.hankel.cyl_hankel.synopsis"></a></span><a class="link" href="cyl_hankel.html#math_toolkit.hankel.cyl_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">cyl_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">cyl_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">cyl_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">cyl_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.cyl_hankel.h1"></a>
  47         <span class="phrase"><a name="math_toolkit.hankel.cyl_hankel.description"></a></span><a class="link" href="cyl_hankel.html#math_toolkit.hankel.cyl_hankel.description">Description</a>
  48       </h5>
  49 <p>
  50         The functions <a class="link" href="cyl_hankel.html" title="Cyclic Hankel Functions">cyl_hankel_1</a>
  51         and <a class="link" href="cyl_hankel.html" title="Cyclic Hankel Functions">cyl_hankel_2</a> return
  52         the result of the <a href="http://dlmf.nist.gov/10.2#P3" target="_top">Hankel functions</a>
  53         of the first and second kind respectively:
  54       </p>
  55 <div class="blockquote"><blockquote class="blockquote"><p>
  56           <span class="serif_italic"><span class="emphasis"><em>cyl_hankel_1(v, x) = H<sub>v</sub><sup>(1)</sup>(x) = J<sub>v</sub>(x)
  57           + i Y<sub>v</sub>(x)</em></span></span>
  58         </p></blockquote></div>
  59 <div class="blockquote"><blockquote class="blockquote"><p>
  60           <span class="serif_italic"><span class="emphasis"><em>cyl_hankel_2(v, x) = H<sub>v</sub><sup>(2)</sup>(x) = J<sub>v</sub>(x)
  61           - i Y<sub>v</sub>(x)</em></span></span>
  62         </p></blockquote></div>
  63 <p>
  64         where:
  65       </p>
  66 <p>
  67         <span class="emphasis"><em>J<sub>v</sub>(x)</em></span> is the Bessel function of the first kind, and
  68         <span class="emphasis"><em>Y<sub>v</sub>(x)</em></span> is the Bessel function of the second kind.
  69       </p>
  70 <p>
  71         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
  72         type calculation rules</em></span></a> when T1 and T2 are different types.
  73         The functions are also optimised for the relatively common case that T1 is
  74         an integer.
  75       </p>
  76 <p>
  77         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
  78         be used to control the behaviour of the function: how it handles errors,
  79         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
  80         documentation for more details</a>.
  81       </p>
  82 <p>
  83         Note that while the arguments to these functions are real values, the results
  84         are complex. That means that the functions can only be instantiated on types
  85         <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>
  86         and <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>.
  87         The functions have also been extended to operate over the whole range of
  88         <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>
  89         and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>).
  90       </p>
  91 <h5>
  92 <a name="math_toolkit.hankel.cyl_hankel.h2"></a>
  93         <span class="phrase"><a name="math_toolkit.hankel.cyl_hankel.performance"></a></span><a class="link" href="cyl_hankel.html#math_toolkit.hankel.cyl_hankel.performance">Performance</a>
  94       </h5>
  95 <p>
  96         These functions are generally more efficient than two separate calls to the
  97         underlying Bessel functions as internally Bessel J and Y can be computed
  98         simultaneously.
  99       </p>
 100 <h5>
 101 <a name="math_toolkit.hankel.cyl_hankel.h3"></a>
 102         <span class="phrase"><a name="math_toolkit.hankel.cyl_hankel.testing"></a></span><a class="link" href="cyl_hankel.html#math_toolkit.hankel.cyl_hankel.testing">Testing</a>
 103       </h5>
 104 <p>
 105         There are just a few spot tests to exercise all the special case handling
 106         - the bulk of the testing is done on the Bessel functions upon which these
 107         are based.
 108       </p>
 109 <h5>
 110 <a name="math_toolkit.hankel.cyl_hankel.h4"></a>
 111         <span class="phrase"><a name="math_toolkit.hankel.cyl_hankel.accuracy"></a></span><a class="link" href="cyl_hankel.html#math_toolkit.hankel.cyl_hankel.accuracy">Accuracy</a>
 112       </h5>
 113 <p>
 114         Refer to <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>
 115         and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>.
 116       </p>
 117 <h5>
 118 <a name="math_toolkit.hankel.cyl_hankel.h5"></a>
 119         <span class="phrase"><a name="math_toolkit.hankel.cyl_hankel.implementation"></a></span><a class="link" href="cyl_hankel.html#math_toolkit.hankel.cyl_hankel.implementation">Implementation</a>
 120       </h5>
 121 <p>
 122         For <span class="emphasis"><em>x &lt; 0</em></span> the following reflection formulae are used:
 123       </p>
 124 <p>
 125         <a href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/16/01/01/" target="_top">
 126         <div class="blockquote"><blockquote class="blockquote"><p>
 127             <span class="inlinemediaobject"><img src="../../../equations/hankel1.svg"></span>
 128
 129           </p></blockquote></div>
 130         </a>
 131       </p>
 132 <p>
 133         <a href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/16/01/01/" target="_top">
 134         <div class="blockquote"><blockquote class="blockquote"><p>
 135             <span class="inlinemediaobject"><img src="../../../equations/hankel2.svg"></span>
 136
 137           </p></blockquote></div>
 138         </a>
 139       </p>
 140 <p>
 141         <a href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/16/01/01/" target="_top">
 142         <div class="blockquote"><blockquote class="blockquote"><p>
 143             <span class="inlinemediaobject"><img src="../../../equations/hankel3.svg"></span>
 144
 145           </p></blockquote></div>
 146         </a>
 147       </p>
 148 <p>
 149         Otherwise the implementation is trivially in terms of the Bessel J and Y
 150         functions.
 151       </p>
 152 <p>
 153         Note however, that the Hankel functions compute the Bessel J and Y functions
 154         simultaneously, and therefore a single Hankel function call is more efficient
 155         than two Bessel function calls. The one exception is when <span class="emphasis"><em>v</em></span>
 156         is a small positive integer, in which case the usual Bessel function routines
 157         for integer order are used.
 158       </p>
 159 </div>
 160 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 161 <td align="left"></td>
 162 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 163       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 164       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 165       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 166       Daryle Walker and Xiaogang Zhang<p>
 167         Distributed under the Boost Software License, Version 1.0. (See accompanying
 168         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>)
 169       </p>
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