236 lines
11 KiB
HTML
236 lines
11 KiB
HTML
<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 V1.79.1">
|
|
<link rel="home" href="../../index.html" title="Math Toolkit 2.11.0">
|
|
<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>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel1.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<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>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<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>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel3.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<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>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel4.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel5.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
Have the derivatives:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel6.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel7.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
Have the Wronskian relation:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
and the reflection formulae:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<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>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel1.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<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>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel2.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel3.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<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>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel4.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel5.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
Have the derivatives:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel6.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel7.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
Have the Wronskian relation:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel8.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
and the reflection formulae:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel9.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel10.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<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>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/sbessel1.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<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>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<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 © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>
|