# Definition:Modified Bessel Function

## Definition

The modified Bessel functions are solutions to Bessel's modified 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 modified Bessel functions of the first kind $I_n$

and:

the modified Bessel functions of the second kind $K_n$.

### Modified Bessel Function of the First Kind

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

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

### Modified Bessel Function of the Second Kind

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

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

## Order of Modified Bessel Function

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

## Also known as

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

## Also see

• Results about modified Bessel functions can be found here.

## Source of Name

This entry was named for Friedrich Wilhelm Bessel.

## Historical Note

Bessel functions were introduced by Friedrich Wilhelm Bessel in $1824$.

They arise in many problems of physics and engineering.

However, despite bearing 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.