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- [section:bessel_over Bessel Function Overview]
- [h4 Ordinary Bessel Functions]
- Bessel Functions are solutions to Bessel's ordinary differential
- equation:
- [equation bessel1]
- where [nu] 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]
- 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] or N[sub v]
- 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] and K[sub v]
- 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] and y[sub n] and are related to the
- ordinary Bessel functions J[sub n] and Y[sub n] by:
- [equation sbessel2]
- The spherical Bessel function of the second kind y[sub n]
- 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] [/section:bessel_over Bessel Function Overview]
- [/
- 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).
- ]
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