| |
| [section:bessel_over Bessel Function Overview] |
| |
| [h4 Ordinary Bessel Functions] |
| |
| Bessel Functions are solutions to Bessel's ordinary differential |
| equation: |
| |
| [equation bessel1] |
| |
| where [nu][space] is the /order/ of the equation, and may be an arbitrary |
| real or complex number, although integer orders are the most common occurrence. |
| |
| This library supports either integer or real orders. |
| |
| Since this is a second order differential equation, there must be two |
| linearly independent solutions, the first of these is denoted J[sub v][space] |
| and known as a Bessel function of the first kind: |
| |
| [equation bessel2] |
| |
| This function is implemented in this library as __cyl_bessel_j. |
| |
| The second solution is denoted either Y[sub v][space] or N[sub v][space] |
| and is known as either a Bessel Function of the second kind, or as a |
| Neumann function: |
| |
| [equation bessel3] |
| |
| This function is implemented in this library as __cyl_neumann. |
| |
| The Bessel functions satisfy the recurrence relations: |
| |
| [equation bessel4] |
| |
| [equation bessel5] |
| |
| Have the derivatives: |
| |
| [equation bessel6] |
| |
| [equation bessel7] |
| |
| Have the Wronskian relation: |
| |
| [equation bessel8] |
| |
| and the reflection formulae: |
| |
| [equation bessel9] |
| |
| [equation bessel10] |
| |
| |
| [h4 Modified Bessel Functions] |
| |
| The Bessel functions are valid for complex argument /x/, and an important |
| special case is the situation where /x/ is purely imaginary: giving a real |
| valued result. In this case the functions are the two linearly |
| independent solutions to the modified Bessel equation: |
| |
| [equation mbessel1] |
| |
| The solutions are known as the modified Bessel functions of the first and |
| second kind (or occasionally as the hyperbolic Bessel functions of the first |
| and second kind). They are denoted I[sub v][space] and K[sub v][space] |
| respectively: |
| |
| [equation mbessel2] |
| |
| [equation mbessel3] |
| |
| These functions are implemented in this library as __cyl_bessel_i and |
| __cyl_bessel_k respectively. |
| |
| The modified Bessel functions satisfy the recurrence relations: |
| |
| [equation mbessel4] |
| |
| [equation mbessel5] |
| |
| Have the derivatives: |
| |
| [equation mbessel6] |
| |
| [equation mbessel7] |
| |
| Have the Wronskian relation: |
| |
| [equation mbessel8] |
| |
| and the reflection formulae: |
| |
| [equation mbessel9] |
| |
| [equation mbessel10] |
| |
| [h4 Spherical Bessel Functions] |
| |
| When solving the Helmholtz equation in spherical coordinates by |
| separation of variables, the radial equation has the form: |
| |
| [equation sbessel1] |
| |
| The two linearly independent solutions to this equation are called the |
| spherical Bessel functions j[sub n][space] and y[sub n][space], and are related to the |
| ordinary Bessel functions J[sub n][space] and Y[sub n][space] by: |
| |
| [equation sbessel2] |
| |
| The spherical Bessel function of the second kind y[sub n][space] |
| is also known as the spherical Neumann function n[sub n]. |
| |
| These functions are implemented in this library as __sph_bessel and |
| __sph_neumann. |
| |
| [endsect] |
| |
| [/ |
| Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang. |
| Distributed under the Boost Software License, Version 1.0. |
| (See accompanying file LICENSE_1_0.txt or copy at |
| http://www.boost.org/LICENSE_1_0.txt). |
| ] |