# Definition:Bessel Function/First Kind

< Definition:Bessel Function(Redirected from Definition:Bessel Function of the First Kind)

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## Definition

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.

## Also known as

Some sources (for whatever reason) do not address Bessel functions of the second kind, and as a consequence refer to **Bessel functions of the first kind** simply as **Bessel functions**.

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

## Specific Instances

### Order $0$

\(\ds \map {J_0} x\) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {\paren {k!}^2} \paren {\dfrac x 2}^{2 k}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 1 - \dfrac {x^2} {2^2} + \dfrac {x^4} {2^2 \times 4^2} - \dfrac {x^6} {2^2 \times 4^2 \times 6^2} + \dotsb\) |

## Also see

- Series Expansion of Bessel Function of the First Kind
- Bessel Function of the First Kind of Negative Integer Order

- Definition:Modified Bessel Function of the First Kind
- Definition:Modified Bessel Function of the Second Kind

## Source of Name

This entry was named for Friedrich Wilhelm Bessel.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Bessel function**