mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2024-11-18 01:52:05 -05:00
125 lines
3.0 KiB
Plaintext
125 lines
3.0 KiB
Plaintext
|
|
||
|
|
[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).
|
||
|
|
]
|