Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation:
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).
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2 Definitions 4 Properties 5 References |
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+1/2.) For example:
Applications
Bessel functions also have useful properties for other problems, such as signal processing (e.g. see FM synthesis or Kaiser window).
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.
For integer order n, Jn and J-n are not linearly independent:
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.
Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions Hα(1)(x) and Hα(2)(x), defined by:
The Bessel functions are valid even for complex arguments x, and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions of the first and second kind, and are defined by:
When solving for separable solutions of Laplace's equation in spherical coordinates, the radial equation has the form:
The Bessel functions have the following asymptotic forms. For small arguments 0 < x << 1, one obtains:
For integer order α = n, Jn is often defined via a Laurent series for a generating function:
Definitions
Bessel functions of the first and second kind
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.
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).
Plot of three Bessel functions of the first kind: J0, J1, and J2.
Plot of three Bessel functions of the second kind: Y0, Y1, and Y2.Hankel functions
where i is the imaginary unit. (The Hankel functions express inward- and outward-propagating cylindrical wave solutions of the cylindrical wave equation.)Modified Bessel functions
These are chosen to be real-valued for real arguments x. They are the two linearly independent solutions to the modified Bessel's equation:
Unlike the ordinary Bessel functions, which are oscillating, Iα and Kα are exponentially growing and decaying functions, respectively. Like the ordinary Bessel function Jα, the function Iα goes to zero at x=0 for α > 0 and is finite at x=0 for α=0. Analogously, Kα diverges at x=0.Spherical Bessel functions
The two linearly independent solutions to this equation are called the spherical Bessel functions jn and yn (also denoted nn), and are related to the ordinary Bessel functions Jα and Yα by:
There are also spherical analogues of the Hankel functions:
In fact, there are simple closed-form expressions for the Bessel functions of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions. In particular, for non-negative integers n:
and hn(2) is the complex-conjugate of this (for real x). (!! is the double factorial.) It follows, for example, that j0(x) = sin(x)/x and y0(x) = -cos(x)/x, and so on.Asymptotic Forms
where α is non-negative and Γ denotes the Gamma function. For large arguments x >> 1, they become:
Asymptotic forms for the other types of Bessel function follow straightforwardly from the above relations. For example, for large x >> 1, the modified Bessel functions become:Properties
an approach used by P. A. Hansen in 1843. (This can be generalized to non-integer order by contour integration or other methods.) Another important relation for integer orders is the Jacobi-Anger identity:
which is used to expand a plane wave as a sum of cylindrical waves.
The functions Jα, Yα, Hα(1), and Hα(2) all satisfy the recurrence relations:
Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by x, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that:
Another orthogonality relation is the closure equation:
Another important consequence of the Hermitian nature of Bessel's equations involves the Wronskian of the solutions:
(There are a large number of other known integrals and identities that are not reproduced here, but which can be found in the references.)