1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Spherical 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="cyl_hankel.html" title="Cyclic Hankel Functions"> 10<link rel="next" href="../airy.html" title="Airy 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="cyl_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="../airy.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.sph_hankel"></a><a class="link" href="sph_hankel.html" title="Spherical Hankel Functions">Spherical Hankel Functions</a> 28</h3></div></div></div> 29<h5> 30<a name="math_toolkit.hankel.sph_hankel.h0"></a> 31 <span class="phrase"><a name="math_toolkit.hankel.sph_hankel.synopsis"></a></span><a class="link" href="sph_hankel.html#math_toolkit.hankel.sph_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">sph_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">sph_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">sph_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">sph_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.sph_hankel.h1"></a> 47 <span class="phrase"><a name="math_toolkit.hankel.sph_hankel.description"></a></span><a class="link" href="sph_hankel.html#math_toolkit.hankel.sph_hankel.description">Description</a> 48 </h5> 49<p> 50 The functions <a class="link" href="sph_hankel.html" title="Spherical Hankel Functions">sph_hankel_1</a> 51 and <a class="link" href="sph_hankel.html" title="Spherical Hankel Functions">sph_hankel_2</a> return 52 the result of the <a href="http://dlmf.nist.gov/10.47#P1" target="_top">spherical Hankel 53 functions</a> of the first and second kind respectively: 54 </p> 55<div class="blockquote"><blockquote class="blockquote"><p> 56 <span class="inlinemediaobject"><img src="../../../equations/hankel4.svg"></span> 57 58 </p></blockquote></div> 59<div class="blockquote"><blockquote class="blockquote"><p> 60 <span class="inlinemediaobject"><img src="../../../equations/hankel5.svg"></span> 61 62 </p></blockquote></div> 63<p> 64 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 65 type calculation rules</em></span></a> when T1 and T2 are different types. 66 The functions are also optimised for the relatively common case that T1 is 67 an integer. 68 </p> 69<p> 70 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 71 be used to control the behaviour of the function: how it handles errors, 72 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 73 documentation for more details</a>. 74 </p> 75<p> 76 Note that while the arguments to these functions are real values, the results 77 are complex. That means that the functions can only be instantiated on types 78 <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code> 79 and <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>. 80 The functions have also been extended to operate over the whole range of 81 <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> 82 and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>). 83 </p> 84<h5> 85<a name="math_toolkit.hankel.sph_hankel.h2"></a> 86 <span class="phrase"><a name="math_toolkit.hankel.sph_hankel.testing"></a></span><a class="link" href="sph_hankel.html#math_toolkit.hankel.sph_hankel.testing">Testing</a> 87 </h5> 88<p> 89 There are just a few spot tests to exercise all the special case handling 90 - the bulk of the testing is done on the Bessel functions upon which these 91 are based. 92 </p> 93<h5> 94<a name="math_toolkit.hankel.sph_hankel.h3"></a> 95 <span class="phrase"><a name="math_toolkit.hankel.sph_hankel.accuracy"></a></span><a class="link" href="sph_hankel.html#math_toolkit.hankel.sph_hankel.accuracy">Accuracy</a> 96 </h5> 97<p> 98 Refer to <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a> 99 and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>. 100 </p> 101<h5> 102<a name="math_toolkit.hankel.sph_hankel.h4"></a> 103 <span class="phrase"><a name="math_toolkit.hankel.sph_hankel.implementation"></a></span><a class="link" href="sph_hankel.html#math_toolkit.hankel.sph_hankel.implementation">Implementation</a> 104 </h5> 105<p> 106 These functions are trivially implemented in terms of <a class="link" href="cyl_hankel.html" title="Cyclic Hankel Functions">cyl_hankel_1</a> 107 and <a class="link" href="cyl_hankel.html" title="Cyclic Hankel Functions">cyl_hankel_2</a>. 108 </p> 109</div> 110<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 111<td align="left"></td> 112<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 113 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 114 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 115 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 116 Daryle Walker and Xiaogang Zhang<p> 117 Distributed under the Boost Software License, Version 1.0. (See accompanying 118 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>) 119 </p> 120</div></td> 121</tr></table> 122<hr> 123<div class="spirit-nav"> 124<a accesskey="p" href="cyl_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="../airy.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 125</div> 126</body> 127</html> 128
