• Home
  • Line#
  • Scopes#
  • Navigate#
  • Raw
  • Download
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">&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 21. 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 21. 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 21. 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 21. 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 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 &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 © 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