2 [section:bessel_over Bessel Function Overview]
4 [h4 Ordinary Bessel Functions]
6 Bessel Functions are solutions to Bessel's ordinary differential
11 where [nu][space] is the /order/ of the equation, and may be an arbitrary
12 real or complex number, although integer orders are the most common occurrence.
14 This library supports either integer or real orders.
16 Since this is a second order differential equation, there must be two
17 linearly independent solutions, the first of these is denoted J[sub v][space]
18 and known as a Bessel function of the first kind:
22 This function is implemented in this library as __cyl_bessel_j.
24 The second solution is denoted either Y[sub v][space] or N[sub v][space]
25 and is known as either a Bessel Function of the second kind, or as a
30 This function is implemented in this library as __cyl_neumann.
32 The Bessel functions satisfy the recurrence relations:
44 Have the Wronskian relation:
48 and the reflection formulae:
55 [h4 Modified Bessel Functions]
57 The Bessel functions are valid for complex argument /x/, and an important
58 special case is the situation where /x/ is purely imaginary: giving a real
59 valued result. In this case the functions are the two linearly
60 independent solutions to the modified Bessel equation:
64 The solutions are known as the modified Bessel functions of the first and
65 second kind (or occasionally as the hyperbolic Bessel functions of the first
66 and second kind). They are denoted I[sub v][space] and K[sub v][space]
73 These functions are implemented in this library as __cyl_bessel_i and
74 __cyl_bessel_k respectively.
76 The modified Bessel functions satisfy the recurrence relations:
88 Have the Wronskian relation:
92 and the reflection formulae:
98 [h4 Spherical Bessel Functions]
100 When solving the Helmholtz equation in spherical coordinates by
101 separation of variables, the radial equation has the form:
105 The two linearly independent solutions to this equation are called the
106 spherical Bessel functions j[sub n][space] and y[sub n][space], and are related to the
107 ordinary Bessel functions J[sub n][space] and Y[sub n][space] by:
111 The spherical Bessel function of the second kind y[sub n][space]
112 is also known as the spherical Neumann function n[sub n].
114 These functions are implemented in this library as __sph_bessel and
120 Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang.
121 Distributed under the Boost Software License, Version 1.0.
122 (See accompanying file LICENSE_1_0.txt or copy at
123 http://www.boost.org/LICENSE_1_0.txt).