# 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 (Neumann Function)

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.

### Bessel Function of the Third Kind (Hankel Function)

A **Hankel function** is the sum of **Bessel functions** in either of the following two ways:

### Hankel Function of the First Kind

The **Hankel function of the first kind** is defined as:

- $\map {H_n^{\paren 1} } z = \map {J_n} z + i \map {Y_n} z$

where:

- $\map {J_n} z$ denotes the Bessel function of the first kind of order $n$
- $\map {Y_n} z$ denotes the Bessel function of the second kind of order $n$.

### Hankel Function of the Second Kind

The **Hankel function of the second kind** is defined as:

- $\map {H_n^{\paren 2} } z = \map {J_n} z - i \map {Y_n} z$

where:

- $\map {J_n} z$ denotes the Bessel function of the first kind of order $n$
- $\map {Y_n} z$ denotes the Bessel function of the second kind of order $n$.

### 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 Bessel Function

The parameter $n$ is known as 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

**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.

## Sources

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