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for an arbitrary real number α (the order). The most common and important special case is where α is an integer n.
Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two orders (e.g. so that the Bessel functions are mostly analytic functions of α).
Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates, and Bessel functions are therefore especially important for many problems of wave propagation , static potentials, and so on. (For cylindrical problems, one obtains Bessel functions of integer order α = n; for spherical problems, one obtains half integer orders α = n+½.) For example:
Bessel functions also have useful properties for other problems, such as signal processing (e.g. see FM synthesis or Kaiser windowThe Kaiser window is a nearly optimal window function w used for digital signal processing, and is defined by the formula: : where I is the zeroth order modified Bessel function of the first kind, α is an arbitrary real number that determines the sh).
Since this is a second-order differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient, and the different variations are described below.
These are perhaps the most commonly used forms of the Bessel functions.
Yα(x) is sometimes also called the Neumann function, and is occasionally denoted instead by Nα(x). It is related to Jα(x) by:
where the case of integer α is handled by taking the limit.
For integer order n, Jn and J-n are not linearly independent:
in which case Yn is needed to provide the second linearly independent solution of Bessel's equation. In contrast, for non-integer order, Jα and J-α are linearly independent, and Yα is redundant (as is clear from its definition above).
The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportional to 1/√x (see also their asymptotic forms, below), although their roots are not generally periodic except asymptotically for large x.
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