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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<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>
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221      Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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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
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