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+</div>
+<div class="section">
+<div class="titlepage"><div><div><h3 class="title">
+<a name="math_toolkit.hankel.cyl_hankel"></a><a class="link" href="cyl_hankel.html" title="Cyclic Hankel Functions">Cyclic Hankel Functions</a>
+</h3></div></div></div>
+<h5>
+<a name="math_toolkit.hankel.cyl_hankel.h0"></a>
+ <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>
+ </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>
+<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</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">></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">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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
+<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</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">></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 15. Policies: Controlling Precision, Error Handling etc">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>
+<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</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">></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">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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
+<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</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">></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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
+</pre>
+<h5>
+<a name="math_toolkit.hankel.cyl_hankel.h1"></a>
+ <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>
+ </h5>
+<p>
+ The functions <a class="link" href="cyl_hankel.html" title="Cyclic Hankel Functions">cyl_hankel_1</a>
+ and <a class="link" href="cyl_hankel.html" title="Cyclic Hankel Functions">cyl_hankel_2</a> return
+ the result of the <a href="http://dlmf.nist.gov/10.2#P3" target="_top">Hankel functions</a>
+ of the first and second kind respectively:
+ </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <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>
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <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>
+ </p></blockquote></div>
+<p>
+ where:
+ </p>
+<p>
+ <span class="emphasis"><em>J<sub>v</sub>(x)</em></span> is the Bessel function of the first kind, and
+ <span class="emphasis"><em>Y<sub>v</sub>(x)</em></span> is the Bessel function of the second kind.
+ </p>
+<p>
+ 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
+ type calculation rules</em></span></a> when T1 and T2 are different types.
+ The functions are also optimised for the relatively common case that T1 is
+ an integer.
+ </p>
+<p>
+ The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
+ documentation for more details</a>.
+ </p>
+<p>
+ Note that while the arguments to these functions are real values, the results
+ are complex. That means that the functions can only be instantiated on types
+ <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>
+ and <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>.
+ The functions have also been extended to operate over the whole range of
+ <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>
+ and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>).
+ </p>
+<h5>
+<a name="math_toolkit.hankel.cyl_hankel.h2"></a>
+ <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>
+ </h5>
+<p>
+ These functions are generally more efficient than two separate calls to the
+ underlying Bessel functions as internally Bessel J and Y can be computed
+ simultaneously.
+ </p>
+<h5>
+<a name="math_toolkit.hankel.cyl_hankel.h3"></a>
+ <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>
+ </h5>
+<p>
+ There are just a few spot tests to exercise all the special case handling
+ - the bulk of the testing is done on the Bessel functions upon which these
+ are based.
+ </p>
+<h5>
+<a name="math_toolkit.hankel.cyl_hankel.h4"></a>
+ <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>
+ </h5>
+<p>
+ Refer to <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>
+ and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>.
+ </p>
+<h5>
+<a name="math_toolkit.hankel.cyl_hankel.h5"></a>
+ <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>
+ </h5>
+<p>
+ For <span class="emphasis"><em>x < 0</em></span> the following reflection formulae are used:
+ </p>
+<p>
+ <a href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/16/01/01/" target="_top">
+ <span class="inlinemediaobject"><img src="../../../equations/hankel1.svg"></span></a>
+ </p>
+<p>
+ <a href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/16/01/01/" target="_top">
+ <span class="inlinemediaobject"><img src="../../../equations/hankel2.svg"></span></a>
+ </p>
+<p>
+ <a href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/16/01/01/" target="_top">
+ <span class="inlinemediaobject"><img src="../../../equations/hankel3.svg"></span></a>
+ </p>
+<p>
+ Otherwise the implementation is trivially in terms of the Bessel J and Y
+ functions.
+ </p>
+<p>
+ Note however, that the Hankel functions compute the Bessel J and Y functions
+ simultaneously, and therefore a single Hankel function call is more efficient
+ than two Bessel function calls. The one exception is when <span class="emphasis"><em>v</em></span>
+ is a small positive integer, in which case the usual Bessel function routines
+ for integer order are used.
+ </p>
+</div>
+<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
+<td align="left"></td>
+<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
+ Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
+ Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
+ Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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>)
+ </p>
+</div></td>
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