Bessel Function: Simple Definition, Characteristics

What is the Bessel Function?

Bessel functions (named after the astronomer F.W. Bessel) are solutions to differential equations:

x²y”” + xy′ + (x² – y²)y = 0

Where:

n is a non-negative real number.

Function values don’t usually have to be calculated by hand; They can be found in many tables (like these Bessel tables).

The solutions are called Bessel functions of order n or—less commonly—cylindrical functions of order n. They are one of the most widely used functions in applied mathematics and are popular in problems that involve circular or cylindrical symmetry, so are sometimes called cylinder functions. They are also important in the study of wave propagation.

Solutions to Bessel’s Equation

Bessel’s equation is a second-order differential equation with two linearly independent solutions:

Bessel function of the first kind,
Bessel function of the second kind.

Bessel Function of the first kind

Bessel functions of the first kind (sometimes called ordinary Bessel functions), are denoted by J_n(x), where n is the order.

Bessel Function of the second kind

The second solution (Y_v or N_v) is called a Bessel Function of the second kind and is denoted by n_n(x). It can also be expressed as a Neumann function:

Uses

A large number of fields use Bessel functions, including:

Acoustic theory,
Electric field theory,
Hydrodynamics,
Nuclear Physics,
Radio Physics.

History

Although the functions are named after Bessel (1824), they appear in much earlier work, including:

Euler’s 1760’s work on vibrations of a stretched membrane,
Fourier’s 1822 theory of heat flow in spherical bodies.

Bernoulli (1703) solved a differential equation by an infinite series, which is largely regarded as the first time the functions appeared in print. It was Bessel, however, who studied the functions in detail while investigating the elliptic motion of planets.

References

Bessel, F. (1825). Uber die Berechnung der geo-graphischen Längen und Breiten aus geodätischen Vermessungen (The calculation of longitude and latitude from geodesic measurements), Astronomische Nachrichten, 4, 241-254.
Dublin Institute of Technology. Table of Bessel Functions. Retrieved 1/2/2017 from: http://www.electronics.dit.ie/staff/akelly/bessel-tables.pdf.
Euler, L. (1766). De motu vibratorio tympanorum, Novi Commentarii academiae scientiarum Petropolitanae. 10, 1766, pp. 243-260.
Fourier, M. 1822. Theorie Analytique De La Chaleur.