123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235 |
- <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>
|