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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;15.&#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;15.&#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;15.&#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;15.&#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="emphasis"><em>cyl_hankel_1(v, x) = H<sub>v</sub><sup>(1)</sup>(x) = J<sub>v</sub>(x) + i Y<sub>v</sub>(x)</em></span>
  57         </p></blockquote></div>
  58 <div class="blockquote"><blockquote class="blockquote"><p>
  59           <span class="emphasis"><em>cyl_hankel_2(v, x) = H<sub>v</sub><sup>(2)</sup>(x) = J<sub>v</sub>(x) - i Y<sub>v</sub>(x)</em></span>
  60         </p></blockquote></div>
  61 <p>
  62         where:
  63       </p>
  64 <p>
  65         <span class="emphasis"><em>J<sub>v</sub>(x)</em></span> is the Bessel function of the first kind, and
  66         <span class="emphasis"><em>Y<sub>v</sub>(x)</em></span> is the Bessel function of the second kind.
  67       </p>
  68 <p>
  69         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
  70         type calculation rules</em></span></a> when T1 and T2 are different types.
  71         The functions are also optimised for the relatively common case that T1 is
  72         an integer.
  73       </p>
  74 <p>
  75         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  76         be used to control the behaviour of the function: how it handles errors,
  77         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
  78         documentation for more details</a>.
  79       </p>
  80 <p>
  81         Note that while the arguments to these functions are real values, the results
  82         are complex. That means that the functions can only be instantiated on types
  83         <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>
  84         and <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>.
  85         The functions have also been extended to operate over the whole range of
  86         <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>
  87         and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>).
  88       </p>
  89 <h5>
  90 <a name="math_toolkit.hankel.cyl_hankel.h2"></a>
  91         <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>
  92       </h5>
  93 <p>
  94         These functions are generally more efficient than two separate calls to the
  95         underlying Bessel functions as internally Bessel J and Y can be computed
  96         simultaneously.
  97       </p>
  98 <h5>
  99 <a name="math_toolkit.hankel.cyl_hankel.h3"></a>
 100         <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>
 101       </h5>
 102 <p>
 103         There are just a few spot tests to exercise all the special case handling
 104         - the bulk of the testing is done on the Bessel functions upon which these
 105         are based.
 106       </p>
 107 <h5>
 108 <a name="math_toolkit.hankel.cyl_hankel.h4"></a>
 109         <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>
 110       </h5>
 111 <p>
 112         Refer to <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>
 113         and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>.
 114       </p>
 115 <h5>
 116 <a name="math_toolkit.hankel.cyl_hankel.h5"></a>
 117         <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>
 118       </h5>
 119 <p>
 120         For <span class="emphasis"><em>x &lt; 0</em></span> the following reflection formulae are used:
 121       </p>
 122 <p>
 123         <a href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/16/01/01/" target="_top">
 124         <span class="inlinemediaobject"><img src="../../../equations/hankel1.svg"></span></a>
 125       </p>
 126 <p>
 127         <a href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/16/01/01/" target="_top">
 128         <span class="inlinemediaobject"><img src="../../../equations/hankel2.svg"></span></a>
 129       </p>
 130 <p>
 131         <a href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/16/01/01/" target="_top">
 132         <span class="inlinemediaobject"><img src="../../../equations/hankel3.svg"></span></a>
 133       </p>
 134 <p>
 135         Otherwise the implementation is trivially in terms of the Bessel J and Y
 136         functions.
 137       </p>
 138 <p>
 139         Note however, that the Hankel functions compute the Bessel J and Y functions
 140         simultaneously, and therefore a single Hankel function call is more efficient
 141         than two Bessel function calls. The one exception is when <span class="emphasis"><em>v</em></span>
 142         is a small positive integer, in which case the usual Bessel function routines
 143         for integer order are used.
 144       </p>
 145 </div>
 146 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 147 <td align="left"></td>
 148 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 149       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 150       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 151       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 152         Distributed under the Boost Software License, Version 1.0. (See accompanying
 153         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>)
 154       </p>
 155 </div></td>
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