Definition:Bessel Function/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): Bessel function