The Bessel function is the solution of the Bessel differential equation:
The functions are found for systems with cylindrical symmetry.
There are two linear independent solutions:
Sometimes the Bessel function is called the cylinder
When speaking of the Bessel function, normally the Bessel function of the first kind order n, Jn(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 Jn(R)=0, where R is the distance
from the origin to the rim of the membrane.
The Bessel functions Jn+½(x)
are found in the definition of spherical Bessel functions.
From the Bessel function of the first kind two Kelvin functions bern(x) and bein(x) can be derived, in the following way:
bern(x) + i bein(x) = enπi Jn(x e - πi/4)
The Bessel function of the second kind of order n, Yn(x),
is also called the Weber function or the Neumann function Nn(x).
The Bessel function of the third kind or Hankel function Hn(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.