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| 23 | <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> | |
| 24 | </div> | |
| 25 | <div class="section"> | |
| 26 | <div class="titlepage"><div><div><h3 class="title"> | |
| 27 | <a name="math_toolkit.bessel.bessel_over"></a><a class="link" href="bessel_over.html" title="Bessel Function Overview">Bessel Function Overview</a> | |
| 28 | </h3></div></div></div> | |
| 29 | <h5> | |
| 30 | <a 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 | <p> | |
| 38 | <span class="inlinemediaobject"><img src="../../../equations/bessel1.svg"></span> | |
| 39 | </p> | |
| 40 | <p> | |
| 41 | where ν   is the <span class="emphasis"><em>order</em></span> of the equation, and may be an arbitrary | |
| 42 | real or complex number, although integer orders are the most common occurrence. | |
| 43 | </p> | |
| 44 | <p> | |
| 45 | This library supports either integer or real orders. | |
| 46 | </p> | |
| 47 | <p> | |
| 48 | Since this is a second order differential equation, there must be two linearly | |
| 49 | independent solutions, the first of these is denoted J<sub>v</sub>   | |
| 50 | and known as a Bessel | |
| 51 | function of the first kind: | |
| 52 | </p> | |
| 53 | <p> | |
| 54 | <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span> | |
| 55 | </p> | |
| 56 | <p> | |
| 57 | 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>. | |
| 58 | </p> | |
| 59 | <p> | |
| 60 | The second solution is denoted either Y<sub>v</sub>   or N<sub>v</sub>   | |
| 61 | and is known as either a Bessel | |
| 62 | Function of the second kind, or as a Neumann function: | |
| 63 | </p> | |
| 64 | <p> | |
| 65 | <span class="inlinemediaobject"><img src="../../../equations/bessel3.svg"></span> | |
| 66 | </p> | |
| 67 | <p> | |
| 68 | 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>. | |
| 69 | </p> | |
| 70 | <p> | |
| 71 | The Bessel functions satisfy the recurrence relations: | |
| 72 | </p> | |
| 73 | <p> | |
| 74 | <span class="inlinemediaobject"><img src="../../../equations/bessel4.svg"></span> | |
| 75 | </p> | |
| 76 | <p> | |
| 77 | <span class="inlinemediaobject"><img src="../../../equations/bessel5.svg"></span> | |
| 78 | </p> | |
| 79 | <p> | |
| 80 | Have the derivatives: | |
| 81 | </p> | |
| 82 | <p> | |
| 83 | <span class="inlinemediaobject"><img src="../../../equations/bessel6.svg"></span> | |
| 84 | </p> | |
| 85 | <p> | |
| 86 | <span class="inlinemediaobject"><img src="../../../equations/bessel7.svg"></span> | |
| 87 | </p> | |
| 88 | <p> | |
| 89 | Have the Wronskian relation: | |
| 90 | </p> | |
| 91 | <p> | |
| 92 | <span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span> | |
| 93 | </p> | |
| 94 | <p> | |
| 95 | and the reflection formulae: | |
| 96 | </p> | |
| 97 | <p> | |
| 98 | <span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span> | |
| 99 | </p> | |
| 100 | <p> | |
| 101 | <span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span> | |
| 102 | </p> | |
| 103 | <h5> | |
| 104 | <a name="math_toolkit.bessel.bessel_over.h1"></a> | |
| 105 | <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 | |
| 106 | Bessel Functions</a> | |
| 107 | </h5> | |
| 108 | <p> | |
| 109 | The Bessel functions are valid for complex argument <span class="emphasis"><em>x</em></span>, | |
| 110 | and an important special case is the situation where <span class="emphasis"><em>x</em></span> | |
| 111 | is purely imaginary: giving a real valued result. In this case the functions | |
| 112 | are the two linearly independent solutions to the modified Bessel equation: | |
| 113 | </p> | |
| 114 | <p> | |
| 115 | <span class="inlinemediaobject"><img src="../../../equations/mbessel1.svg"></span> | |
| 116 | </p> | |
| 117 | <p> | |
| 118 | The solutions are known as the modified Bessel functions of the first and | |
| 119 | second kind (or occasionally as the hyperbolic Bessel functions of the first | |
| 120 | and second kind). They are denoted I<sub>v</sub>   and K<sub>v</sub>   | |
| 121 | respectively: | |
| 122 | </p> | |
| 123 | <p> | |
| 124 | <span class="inlinemediaobject"><img src="../../../equations/mbessel2.svg"></span> | |
| 125 | </p> | |
| 126 | <p> | |
| 127 | <span class="inlinemediaobject"><img src="../../../equations/mbessel3.svg"></span> | |
| 128 | </p> | |
| 129 | <p> | |
| 130 | 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> | |
| 131 | and <a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_k</a> respectively. | |
| 132 | </p> | |
| 133 | <p> | |
| 134 | The modified Bessel functions satisfy the recurrence relations: | |
| 135 | </p> | |
| 136 | <p> | |
| 137 | <span class="inlinemediaobject"><img src="../../../equations/mbessel4.svg"></span> | |
| 138 | </p> | |
| 139 | <p> | |
| 140 | <span class="inlinemediaobject"><img src="../../../equations/mbessel5.svg"></span> | |
| 141 | </p> | |
| 142 | <p> | |
| 143 | Have the derivatives: | |
| 144 | </p> | |
| 145 | <p> | |
| 146 | <span class="inlinemediaobject"><img src="../../../equations/mbessel6.svg"></span> | |
| 147 | </p> | |
| 148 | <p> | |
| 149 | <span class="inlinemediaobject"><img src="../../../equations/mbessel7.svg"></span> | |
| 150 | </p> | |
| 151 | <p> | |
| 152 | Have the Wronskian relation: | |
| 153 | </p> | |
| 154 | <p> | |
| 155 | <span class="inlinemediaobject"><img src="../../../equations/mbessel8.svg"></span> | |
| 156 | </p> | |
| 157 | <p> | |
| 158 | and the reflection formulae: | |
| 159 | </p> | |
| 160 | <p> | |
| 161 | <span class="inlinemediaobject"><img src="../../../equations/mbessel9.svg"></span> | |
| 162 | </p> | |
| 163 | <p> | |
| 164 | <span class="inlinemediaobject"><img src="../../../equations/mbessel10.svg"></span> | |
| 165 | </p> | |
| 166 | <h5> | |
| 167 | <a name="math_toolkit.bessel.bessel_over.h2"></a> | |
| 168 | <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 | |
| 169 | Bessel Functions</a> | |
| 170 | </h5> | |
| 171 | <p> | |
| 172 | When solving the Helmholtz equation in spherical coordinates by separation | |
| 173 | of variables, the radial equation has the form: | |
| 174 | </p> | |
| 175 | <p> | |
| 176 | <span class="inlinemediaobject"><img src="../../../equations/sbessel1.svg"></span> | |
| 177 | </p> | |
| 178 | <p> | |
| 179 | The two linearly independent solutions to this equation are called the spherical | |
| 180 | Bessel functions j<sub>n</sub>   and y<sub>n</sub>  , and are related to the ordinary Bessel functions | |
| 181 | J<sub>n</sub>   and Y<sub>n</sub>   by: | |
| 182 | </p> | |
| 183 | <p> | |
| 184 | <span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span> | |
| 185 | </p> | |
| 186 | <p> | |
| 187 | The spherical Bessel function of the second kind y<sub>n</sub>   | |
| 188 | is also known as the spherical | |
| 189 | Neumann function n<sub>n</sub>. | |
| 190 | </p> | |
| 191 | <p> | |
| 192 | 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> | |
| 193 | and <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_neumann</a>. | |
| 194 | </p> | |
| 195 | </div> | |
| 196 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 197 | <td align="left"></td> | |
| 198 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 199 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 200 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 201 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 202 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 203 | 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>) | |
| 204 | </p> | |
| 205 | </div></td> | |
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