# Definition:Bessel Function

## Definition

The Bessel functions are solutions to Bessel's Equation:

$x^2 \dfrac{\mathrm d^2 y} {\mathrm d x^2} + x \dfrac{\mathrm d y} {\mathrm d x} + \left({x^2 - p^2}\right) y = 0$

These solutions have two main classes:

the Bessel functions of the first kind $J_p$

and:

the Bessel functions of the second kind $Y_p$.

### Bessel Function of the First Kind

A Bessel function of the first kind is a Bessel function which is nonsingular at the origin.

### Bessel Function of the Second Kind

A Bessel function of the second kind is a Bessel function which is singular at the origin.

## 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.