The Bessel function is the solution of the Bessel differential equation:
The functions are found for systems with cylindrical symmetry.
Friedrich Wilhelm Bessel was a mathematician who lived from 1784 to 1846.
There are two linear independent solutions:
Sometimes the Bessel function is called the cylinder
function.
More common is to reserve this name for the threedimensional
cylinder function. However, the Fourier transform of this 3d function, is a Bessel
function of the first kind (of first order).
When speaking of the Bessel
function, normally the Bessel function of the first kind order
n, J_{n}(x), is meant.
It can be written as an infinite polynomial with terms derived from the gamma function:
It is found when solving wave equations. For instance, in the case of a wave equation
on a membrane ^{1)}, the solution is a Bessel function of
integer order (a). For a circular membrane the standing wave solution can be expressed as a Bessel
function, under the condition that J_{n}(R)=0, where R is the distance
from the origin to the rim of the membrane.
We see three standing wave animations: the first and second part of the order 0
function, and the first part of the order 1 function.
The Bessel function of the first kind and order 0 can occur when you wiggle an
(idealized) chain, fixed at one side:
The Bessel functions J_{n+½}(x)
are found in the definition of spherical Bessel functions.
These functions can be expressed in sine and cosine terms.
So can be found that:
and
From the Bessel function of the first kind two Kelvin
functions ber_{n}(x) and bei_{n}(x) can be derived, in the following way:
ber_{n}(x) + i bei_{n}(x) = e^{nπi}
J_{n}(x e^{  πi/4})
The Bessel function of the second kind of order n, Y_{n}(x),
is also called the Weber function or the Neumann function N_{n}(x).
it can be written as an infinite polynomial with terms derived from the gamma function:
The Bessel function of the third kind or Hankel function H_{n}(x) is a (complex)
combination of the two solutions: the real part is the Bessel function of the first kind,
the complex part the Bessel function of the second kind.
notes
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