1[section:bessel_over Bessel Function Overview] 2 3[h4 Ordinary Bessel Functions] 4 5Bessel Functions are solutions to Bessel's ordinary differential 6equation: 7 8[equation bessel1] 9 10where [nu] is the /order/ of the equation, and may be an arbitrary 11real or complex number, although integer orders are the most common occurrence. 12 13This library supports either integer or real orders. 14 15Since this is a second order differential equation, there must be two 16linearly independent solutions, the first of these is denoted J[sub v] 17and known as a Bessel function of the first kind: 18 19[equation bessel2] 20 21This function is implemented in this library as __cyl_bessel_j. 22 23The second solution is denoted either Y[sub v] or N[sub v] 24and is known as either a Bessel Function of the second kind, or as a 25Neumann function: 26 27[equation bessel3] 28 29This function is implemented in this library as __cyl_neumann. 30 31The Bessel functions satisfy the recurrence relations: 32 33[equation bessel4] 34 35[equation bessel5] 36 37Have the derivatives: 38 39[equation bessel6] 40 41[equation bessel7] 42 43Have the Wronskian relation: 44 45[equation bessel8] 46 47and the reflection formulae: 48 49[equation bessel9] 50 51[equation bessel10] 52 53 54[h4 Modified Bessel Functions] 55 56The Bessel functions are valid for complex argument /x/, and an important 57special case is the situation where /x/ is purely imaginary: giving a real 58valued result. In this case the functions are the two linearly 59independent solutions to the modified Bessel equation: 60 61[equation mbessel1] 62 63The solutions are known as the modified Bessel functions of the first and 64second kind (or occasionally as the hyperbolic Bessel functions of the first 65and second kind). They are denoted I[sub v] and K[sub v] 66respectively: 67 68[equation mbessel2] 69 70[equation mbessel3] 71 72These functions are implemented in this library as __cyl_bessel_i and 73__cyl_bessel_k respectively. 74 75The modified Bessel functions satisfy the recurrence relations: 76 77[equation mbessel4] 78 79[equation mbessel5] 80 81Have the derivatives: 82 83[equation mbessel6] 84 85[equation mbessel7] 86 87Have the Wronskian relation: 88 89[equation mbessel8] 90 91and the reflection formulae: 92 93[equation mbessel9] 94 95[equation mbessel10] 96 97[h4 Spherical Bessel Functions] 98 99When solving the Helmholtz equation in spherical coordinates by 100separation of variables, the radial equation has the form: 101 102[equation sbessel1] 103 104The two linearly independent solutions to this equation are called the 105spherical Bessel functions j[sub n] and y[sub n] and are related to the 106ordinary Bessel functions J[sub n] and Y[sub n] by: 107 108[equation sbessel2] 109 110The spherical Bessel function of the second kind y[sub n] 111is also known as the spherical Neumann function n[sub n]. 112 113These functions are implemented in this library as __sph_bessel and 114__sph_neumann. 115 116[endsect] [/section:bessel_over Bessel Function Overview] 117 118[/ 119 Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang. 120 Distributed under the Boost Software License, Version 1.0. 121 (See accompanying file LICENSE_1_0.txt or copy at 122 http://www.boost.org/LICENSE_1_0.txt). 123] 124