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| <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 14. 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 14. 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 14. 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 14. 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 14. 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 14. 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> |
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| Holin, Bruno Lalande, John Maddock, 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>) |
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