# Definition:Bessel Function

## Definition

The **Bessel functions** are solutions to Bessel's equation:

- $x^2 \dfrac {\d^2 y} {\d x^2} + x \dfrac {\d y} {\d x} + \paren {x^2 - n^2} y = 0$

These solutions have two main classes:

- the Bessel functions of the first kind $J_n$

and:

- the Bessel functions of the second kind $Y_n$.

### Bessel Function of the First Kind

A **Bessel function of the first kind of order $n$** is a Bessel function which is non-singular at the origin

It is usually denoted $\map {J_n} x$, where $x$ is the dependent variable of the instance of **Bessel's equation** to which $\map {J_n} x$ forms a solution.

### Bessel Function of the Second Kind

A **Bessel function of the second kind of order $n$** is a Bessel function which is singular at the origin.

It is usually denoted $\map {Y_n} x$, where $x$ is the dependent variable of the instance of **Bessel's equation** to which $\map {Y_n} x$ forms a solution.

## Order of Bessel Function

The parameter $n$ is known as the **order** of the Bessel function.

## Also known as

Some sources use $p$ to denote the order of the Bessel function.

## Also see

- Results about
**Bessel functions**can be found here.

## Source of Name

This entry was named for Friedrich Wilhelm Bessel.

## Historical Note

Despite the fact that the Bessel functions bears the name of Friedrich Wilhelm Bessel, they were first studied by Leonhard Paul Euler.

He encountered them during his study of the vibrations of a stretched circular membrane.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Bessel functions** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Bessel functions** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Bessel function**