1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Bessel Function Overview</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../bessel.html" title="Bessel Functions"> 9<link rel="prev" href="../bessel.html" title="Bessel Functions"> 10<link rel="next" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a 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name="math_toolkit.bessel.bessel_over.h0"></a> 31 <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 32 Bessel Functions</a> 33 </h5> 34<p> 35 Bessel Functions are solutions to Bessel's ordinary differential equation: 36 </p> 37<div class="blockquote"><blockquote class="blockquote"><p> 38 <span class="inlinemediaobject"><img src="../../../equations/bessel1.svg"></span> 39 40 </p></blockquote></div> 41<p> 42 where ν is the <span class="emphasis"><em>order</em></span> of the equation, and may be an arbitrary 43 real or complex number, although integer orders are the most common occurrence. 44 </p> 45<p> 46 This library supports either integer or real orders. 47 </p> 48<p> 49 Since this is a second order differential equation, there must be two linearly 50 independent solutions, the first of these is denoted J<sub>v</sub> 51and known as a Bessel 52 function of the first kind: 53 </p> 54<div class="blockquote"><blockquote class="blockquote"><p> 55 <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span> 56 57 </p></blockquote></div> 58<p> 59 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>. 60 </p> 61<p> 62 The second solution is denoted either Y<sub>v</sub> or N<sub>v</sub> 63and is known as either a Bessel 64 Function of the second kind, or as a Neumann function: 65 </p> 66<div class="blockquote"><blockquote class="blockquote"><p> 67 <span class="inlinemediaobject"><img src="../../../equations/bessel3.svg"></span> 68 69 </p></blockquote></div> 70<p> 71 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>. 72 </p> 73<p> 74 The Bessel functions satisfy the recurrence relations: 75 </p> 76<div class="blockquote"><blockquote class="blockquote"><p> 77 <span class="inlinemediaobject"><img src="../../../equations/bessel4.svg"></span> 78 79 </p></blockquote></div> 80<div class="blockquote"><blockquote class="blockquote"><p> 81 <span class="inlinemediaobject"><img src="../../../equations/bessel5.svg"></span> 82 83 </p></blockquote></div> 84<p> 85 Have the derivatives: 86 </p> 87<div class="blockquote"><blockquote class="blockquote"><p> 88 <span class="inlinemediaobject"><img src="../../../equations/bessel6.svg"></span> 89 90 </p></blockquote></div> 91<div class="blockquote"><blockquote class="blockquote"><p> 92 <span class="inlinemediaobject"><img src="../../../equations/bessel7.svg"></span> 93 94 </p></blockquote></div> 95<p> 96 Have the Wronskian relation: 97 </p> 98<div class="blockquote"><blockquote class="blockquote"><p> 99 <span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span> 100 101 </p></blockquote></div> 102<p> 103 and the reflection formulae: 104 </p> 105<div class="blockquote"><blockquote class="blockquote"><p> 106 <span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span> 107 108 </p></blockquote></div> 109<div class="blockquote"><blockquote class="blockquote"><p> 110 <span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span> 111 112 </p></blockquote></div> 113<h5> 114<a name="math_toolkit.bessel.bessel_over.h1"></a> 115 <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 116 Bessel Functions</a> 117 </h5> 118<p> 119 The Bessel functions are valid for complex argument <span class="emphasis"><em>x</em></span>, 120 and an important special case is the situation where <span class="emphasis"><em>x</em></span> 121 is purely imaginary: giving a real valued result. In this case the functions 122 are the two linearly independent solutions to the modified Bessel equation: 123 </p> 124<div class="blockquote"><blockquote class="blockquote"><p> 125 <span class="inlinemediaobject"><img src="../../../equations/mbessel1.svg"></span> 126 127 </p></blockquote></div> 128<p> 129 The solutions are known as the modified Bessel functions of the first and 130 second kind (or occasionally as the hyperbolic Bessel functions of the first 131 and second kind). They are denoted I<sub>v</sub> and K<sub>v</sub> 132respectively: 133 </p> 134<div class="blockquote"><blockquote class="blockquote"><p> 135 <span class="inlinemediaobject"><img src="../../../equations/mbessel2.svg"></span> 136 137 </p></blockquote></div> 138<div class="blockquote"><blockquote class="blockquote"><p> 139 <span class="inlinemediaobject"><img src="../../../equations/mbessel3.svg"></span> 140 141 </p></blockquote></div> 142<p> 143 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> 144 and <a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_k</a> respectively. 145 </p> 146<p> 147 The modified Bessel functions satisfy the recurrence relations: 148 </p> 149<div class="blockquote"><blockquote class="blockquote"><p> 150 <span class="inlinemediaobject"><img src="../../../equations/mbessel4.svg"></span> 151 152 </p></blockquote></div> 153<div class="blockquote"><blockquote class="blockquote"><p> 154 <span class="inlinemediaobject"><img src="../../../equations/mbessel5.svg"></span> 155 156 </p></blockquote></div> 157<p> 158 Have the derivatives: 159 </p> 160<div class="blockquote"><blockquote class="blockquote"><p> 161 <span class="inlinemediaobject"><img src="../../../equations/mbessel6.svg"></span> 162 163 </p></blockquote></div> 164<div class="blockquote"><blockquote class="blockquote"><p> 165 <span class="inlinemediaobject"><img src="../../../equations/mbessel7.svg"></span> 166 167 </p></blockquote></div> 168<p> 169 Have the Wronskian relation: 170 </p> 171<div class="blockquote"><blockquote class="blockquote"><p> 172 <span class="inlinemediaobject"><img src="../../../equations/mbessel8.svg"></span> 173 174 </p></blockquote></div> 175<p> 176 and the reflection formulae: 177 </p> 178<div class="blockquote"><blockquote class="blockquote"><p> 179 <span class="inlinemediaobject"><img src="../../../equations/mbessel9.svg"></span> 180 181 </p></blockquote></div> 182<div class="blockquote"><blockquote class="blockquote"><p> 183 <span class="inlinemediaobject"><img src="../../../equations/mbessel10.svg"></span> 184 185 </p></blockquote></div> 186<h5> 187<a name="math_toolkit.bessel.bessel_over.h2"></a> 188 <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 189 Bessel Functions</a> 190 </h5> 191<p> 192 When solving the Helmholtz equation in spherical coordinates by separation 193 of variables, the radial equation has the form: 194 </p> 195<div class="blockquote"><blockquote class="blockquote"><p> 196 <span class="inlinemediaobject"><img src="../../../equations/sbessel1.svg"></span> 197 198 </p></blockquote></div> 199<p> 200 The two linearly independent solutions to this equation are called the spherical 201 Bessel functions j<sub>n</sub> and y<sub>n</sub> and are related to the ordinary Bessel functions 202 J<sub>n</sub> and Y<sub>n</sub> by: 203 </p> 204<div class="blockquote"><blockquote class="blockquote"><p> 205 <span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span> 206 207 </p></blockquote></div> 208<p> 209 The spherical Bessel function of the second kind y<sub>n</sub> 210is also known as the spherical 211 Neumann function n<sub>n</sub>. 212 </p> 213<p> 214 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> 215 and <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_neumann</a>. 216 </p> 217</div> 218<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 219<td align="left"></td> 220<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 221 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 222 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 223 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 224 Daryle Walker and Xiaogang Zhang<p> 225 Distributed under the Boost Software License, Version 1.0. (See accompanying 226 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>) 227 </p> 228</div></td> 229</tr></table> 230<hr> 231<div class="spirit-nav"> 232<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> 233</div> 234</body> 235</html> 236