Calculus > Bessel Function

## What is the Bessel Function?

Bessel functions (named after the astronomer F.W. Bessel) are one of the most widely used functions in applied mathematics, appearing in problems that involve circular or cylindrical symmetry. They are the solutions to the differential equation:

**x**

^{2}y”” + xy’ + (x^{2}– y^{2})y = 0Where:

*n*is a non-negative real number.

The solutions are called

*Bessel functions of order n*or — less commonly —

*cylindrical functions of order n.*

Function values can be found in many mathematical tables (like these bessel tables).

## Solutions to Bessel’s Equation

Bessel’s equation is a second-order differential equation with two linearly independent solutions: a Bessel function of the first kind and a 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,

**Note**: *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 and 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.

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