Definition:Bessel Function/First Kind

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.

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.

$\ds \map {J_0} x$

$=$

$\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {\paren {k!}^2} \paren {\dfrac x 2}^{2 k}$

$\ds $

$=$

$\ds 1 - \dfrac {x^2} {2^2} + \dfrac {x^4} {2^2 \times 4^2} - \dfrac {x^6} {2^2 \times 4^2 \times 6^2} + \dotsb$

This entry was named for Friedrich Wilhelm Bessel.