1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Cyclic Hankel Functions</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../hankel.html" title="Hankel Functions"> 9<link rel="prev" href="../hankel.html" title="Hankel Functions"> 10<link rel="next" href="sph_hankel.html" title="Spherical Hankel Functions"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="../hankel.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../hankel.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="sph_hankel.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 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"><</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> 34<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> 35 36<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 37<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 38 39<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> 40<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> 41 42<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 43<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</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 21. 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 21. 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 < 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 © 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 Råde, 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> 170</div></td> 171</tr></table> 172<hr> 173<div class="spirit-nav"> 174<a accesskey="p" href="../hankel.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../hankel.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="sph_hankel.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 175</div> 176</body> 177</html> 178