**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).

Table of contents |

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:

- electromagnetic waves in a cylindrical waveguide
- heat conduction in a cylindrical object.
- modes of vibration of a thin circular (or annular) membrane.

These are perhaps the most commonly used forms of the Bessel functions.

- Bessel functions of the first kind,
*J*_{α}(*x*), are solutions of Bessel's differential equation which are finite at*x*= 0 for α an integer or α non-negative. (The specific choice and normalization of*J*_{α}are defined by its properties below.) - Bessel functions of the second kind,
*Y*_{α}(*x*), are solutions which are singular (infinite) at*x*= 0.

For *integer* order *n*, *J*_{n} 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*.

Plot of three Bessel functions of the first kind:

Plot of three Bessel functions of the second kind:

Another important formulation of the two linearly independent solutions to Bessel's equation are the **Hankel functions** *H*_{α}^{(1)}(*x*) and *H*_{α}^{(2)}(*x*), defined by:

For integer order α = *n*, *J*_{n} is often defined via a Laurent series for a generating function:

The functions *J*_{α}, *Y*_{α}, *H*_{α}^{(1)}, and *H*_{α}^{(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.)

- George B. Arfken and Hans J. Weber,
*Mathematical Methods for Physicists*(Harcourt: San Diego, 2001). - Frank Bowman,
*Introduction to Bessel Functions*(Dover: New York, 1958). - M. Abramowitz and I. A. Stegun, eds.,
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*(Dover: New York, 1972).