| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> |
| <title>Bessel Function Overview</title> |
| <link rel="stylesheet" href="../../math.css" type="text/css"> |
| <meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> |
| <link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> |
| <link rel="up" href="../bessel.html" title="Bessel Functions"> |
| <link rel="prev" href="../bessel.html" title="Bessel Functions"> |
| <link rel="next" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds"> |
| </head> |
| <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> |
| <table cellpadding="2" width="100%"><tr> |
| <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> |
| <td align="center"><a href="../../../../../../index.html">Home</a></td> |
| <td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> |
| <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> |
| <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> |
| <td align="center"><a href="../../../../../../more/index.htm">More</a></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <a accesskey="p" href="../bessel.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../bessel.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="bessel_first.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.bessel.bessel_over"></a><a class="link" href="bessel_over.html" title="Bessel Function Overview">Bessel Function Overview</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.bessel.bessel_over.h0"></a> |
| <span class="phrase"><a name="math_toolkit.bessel.bessel_over.ordinary_bessel_functions"></a></span><a class="link" href="bessel_over.html#math_toolkit.bessel.bessel_over.ordinary_bessel_functions">Ordinary |
| Bessel Functions</a> |
| </h5> |
| <p> |
| Bessel Functions are solutions to Bessel's ordinary differential equation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel1.svg"></span> |
| </p> |
| <p> |
| where ν   is the <span class="emphasis"><em>order</em></span> of the equation, and may be an arbitrary |
| real or complex number, although integer orders are the most common occurrence. |
| </p> |
| <p> |
| This library supports either integer or real orders. |
| </p> |
| <p> |
| Since this is a second order differential equation, there must be two linearly |
| independent solutions, the first of these is denoted J<sub>v</sub>   |
| and known as a Bessel |
| function of the first kind: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span> |
| </p> |
| <p> |
| This function is implemented in this library as <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>. |
| </p> |
| <p> |
| The second solution is denoted either Y<sub>v</sub>   or N<sub>v</sub>   |
| and is known as either a Bessel |
| Function of the second kind, or as a Neumann function: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel3.svg"></span> |
| </p> |
| <p> |
| This function is implemented in this library as <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>. |
| </p> |
| <p> |
| The Bessel functions satisfy the recurrence relations: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel4.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel5.svg"></span> |
| </p> |
| <p> |
| Have the derivatives: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel6.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel7.svg"></span> |
| </p> |
| <p> |
| Have the Wronskian relation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span> |
| </p> |
| <p> |
| and the reflection formulae: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.bessel.bessel_over.h1"></a> |
| <span class="phrase"><a name="math_toolkit.bessel.bessel_over.modified_bessel_functions"></a></span><a class="link" href="bessel_over.html#math_toolkit.bessel.bessel_over.modified_bessel_functions">Modified |
| Bessel Functions</a> |
| </h5> |
| <p> |
| The Bessel functions are valid for complex argument <span class="emphasis"><em>x</em></span>, |
| and an important special case is the situation where <span class="emphasis"><em>x</em></span> |
| is purely imaginary: giving a real valued result. In this case the functions |
| are the two linearly independent solutions to the modified Bessel equation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/mbessel1.svg"></span> |
| </p> |
| <p> |
| The solutions are known as the modified Bessel functions of the first and |
| second kind (or occasionally as the hyperbolic Bessel functions of the first |
| and second kind). They are denoted I<sub>v</sub>   and K<sub>v</sub>   |
| respectively: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/mbessel2.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/mbessel3.svg"></span> |
| </p> |
| <p> |
| These functions are implemented in this library as <a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a> |
| and <a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_k</a> respectively. |
| </p> |
| <p> |
| The modified Bessel functions satisfy the recurrence relations: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/mbessel4.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/mbessel5.svg"></span> |
| </p> |
| <p> |
| Have the derivatives: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/mbessel6.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/mbessel7.svg"></span> |
| </p> |
| <p> |
| Have the Wronskian relation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/mbessel8.svg"></span> |
| </p> |
| <p> |
| and the reflection formulae: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/mbessel9.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/mbessel10.svg"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.bessel.bessel_over.h2"></a> |
| <span class="phrase"><a name="math_toolkit.bessel.bessel_over.spherical_bessel_functions"></a></span><a class="link" href="bessel_over.html#math_toolkit.bessel.bessel_over.spherical_bessel_functions">Spherical |
| Bessel Functions</a> |
| </h5> |
| <p> |
| When solving the Helmholtz equation in spherical coordinates by separation |
| of variables, the radial equation has the form: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/sbessel1.svg"></span> |
| </p> |
| <p> |
| The two linearly independent solutions to this equation are called the spherical |
| Bessel functions j<sub>n</sub>   and y<sub>n</sub>  , and are related to the ordinary Bessel functions |
| J<sub>n</sub>   and Y<sub>n</sub>   by: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span> |
| </p> |
| <p> |
| The spherical Bessel function of the second kind y<sub>n</sub>   |
| is also known as the spherical |
| Neumann function n<sub>n</sub>. |
| </p> |
| <p> |
| These functions are implemented in this library as <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_bessel</a> |
| and <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_neumann</a>. |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| 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>) |
| </p> |
| </div></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <a accesskey="p" href="../bessel.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../bessel.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="bessel_first.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| </body> |
| </html> |