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<title>Bessel Function Overview</title>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.bessel.bessel_over"></a><a class="link" href="bessel_over.html" title="Bessel Function Overview">Bessel Function Overview</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.bessel.bessel_over.h0"></a>
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<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
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Bessel Functions</a>
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</h5>
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<p>
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Bessel Functions are solutions to Bessel's ordinary differential equation:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/bessel1.svg"></span>
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</p>
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<p>
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where ν   is the <span class="emphasis"><em>order</em></span> of the equation, and may be an arbitrary
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real or complex number, although integer orders are the most common occurrence.
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</p>
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<p>
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This library supports either integer or real orders.
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</p>
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<p>
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Since this is a second order differential equation, there must be two linearly
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independent solutions, the first of these is denoted J<sub>v</sub>  
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and known as a Bessel
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function of the first kind:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
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</p>
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<p>
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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>.
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</p>
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<p>
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The second solution is denoted either Y<sub>v</sub>   or N<sub>v</sub>  
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and is known as either a Bessel
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Function of the second kind, or as a Neumann function:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/bessel3.svg"></span>
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</p>
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<p>
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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>.
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</p>
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<p>
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The Bessel functions satisfy the recurrence relations:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/bessel4.svg"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/bessel5.svg"></span>
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</p>
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<p>
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Have the derivatives:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/bessel6.svg"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/bessel7.svg"></span>
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</p>
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<p>
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Have the Wronskian relation:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span>
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</p>
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<p>
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and the reflection formulae:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span>
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</p>
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<h5>
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<a name="math_toolkit.bessel.bessel_over.h1"></a>
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<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
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Bessel Functions</a>
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</h5>
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<p>
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The Bessel functions are valid for complex argument <span class="emphasis"><em>x</em></span>,
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and an important special case is the situation where <span class="emphasis"><em>x</em></span>
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is purely imaginary: giving a real valued result. In this case the functions
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are the two linearly independent solutions to the modified Bessel equation:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/mbessel1.svg"></span>
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</p>
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<p>
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The solutions are known as the modified Bessel functions of the first and
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second kind (or occasionally as the hyperbolic Bessel functions of the first
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and second kind). They are denoted I<sub>v</sub>   and K<sub>v</sub>  
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respectively:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/mbessel2.svg"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/mbessel3.svg"></span>
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</p>
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<p>
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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>
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and <a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_k</a> respectively.
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</p>
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<p>
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The modified Bessel functions satisfy the recurrence relations:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/mbessel4.svg"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/mbessel5.svg"></span>
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</p>
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<p>
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Have the derivatives:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/mbessel6.svg"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/mbessel7.svg"></span>
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</p>
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<p>
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Have the Wronskian relation:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/mbessel8.svg"></span>
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</p>
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<p>
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and the reflection formulae:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/mbessel9.svg"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/mbessel10.svg"></span>
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</p>
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<h5>
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<a name="math_toolkit.bessel.bessel_over.h2"></a>
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<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
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Bessel Functions</a>
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</h5>
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<p>
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When solving the Helmholtz equation in spherical coordinates by separation
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of variables, the radial equation has the form:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/sbessel1.svg"></span>
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</p>
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<p>
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The two linearly independent solutions to this equation are called the spherical
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Bessel functions j<sub>n</sub>   and y<sub>n</sub>  , and are related to the ordinary Bessel functions
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J<sub>n</sub>   and Y<sub>n</sub>   by:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span>
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</p>
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<p>
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The spherical Bessel function of the second kind y<sub>n</sub>  
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is also known as the spherical
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Neumann function n<sub>n</sub>.
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</p>
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<p>
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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>
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and <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_neumann</a>.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
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Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
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Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
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Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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</p>
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</div></td>
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</tr></table>
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<hr>
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<div class="spirit-nav">
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<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>
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