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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 15. 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 15. 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 15. 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 15. 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 | <p> | |
56 | <span class="inlinemediaobject"><img src="../../../equations/hankel4.svg"></span> | |
57 | </p> | |
58 | <p> | |
59 | <span class="inlinemediaobject"><img src="../../../equations/hankel5.svg"></span> | |
60 | </p> | |
61 | <p> | |
62 | 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 | |
63 | type calculation rules</em></span></a> when T1 and T2 are different types. | |
64 | The functions are also optimised for the relatively common case that T1 is | |
65 | an integer. | |
66 | </p> | |
67 | <p> | |
68 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
69 | be used to control the behaviour of the function: how it handles errors, | |
70 | 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 | |
71 | documentation for more details</a>. | |
72 | </p> | |
73 | <p> | |
74 | Note that while the arguments to these functions are real values, the results | |
75 | are complex. That means that the functions can only be instantiated on types | |
76 | <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code> | |
77 | and <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>. | |
78 | The functions have also been extended to operate over the whole range of | |
79 | <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> | |
80 | and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>). | |
81 | </p> | |
82 | <h5> | |
83 | <a name="math_toolkit.hankel.sph_hankel.h2"></a> | |
84 | <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> | |
85 | </h5> | |
86 | <p> | |
87 | There are just a few spot tests to exercise all the special case handling | |
88 | - the bulk of the testing is done on the Bessel functions upon which these | |
89 | are based. | |
90 | </p> | |
91 | <h5> | |
92 | <a name="math_toolkit.hankel.sph_hankel.h3"></a> | |
93 | <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> | |
94 | </h5> | |
95 | <p> | |
96 | Refer to <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a> | |
97 | and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>. | |
98 | </p> | |
99 | <h5> | |
100 | <a name="math_toolkit.hankel.sph_hankel.h4"></a> | |
101 | <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> | |
102 | </h5> | |
103 | <p> | |
104 | These functions are trivially implemented in terms of <a class="link" href="cyl_hankel.html" title="Cyclic Hankel Functions">cyl_hankel_1</a> | |
105 | and <a class="link" href="cyl_hankel.html" title="Cyclic Hankel Functions">cyl_hankel_2</a>. | |
106 | </p> | |
107 | </div> | |
108 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
109 | <td align="left"></td> | |
110 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
111 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
112 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
113 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
114 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
115 | 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>) | |
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