This is an old revision of the document!
Gamma function
Function
| definition | $J_{}: ?\to ?$ |
| definition | $J_\alpha(x) := \sum_{m=0}^\infty \dfrac{1}{\Gamma(m+0+1)\,\Gamma(m+\alpha+1)} (-1)^m{\left(\dfrac{x}{2}\right)}^{2m+\alpha}$ |
Discussion
The Bessel functions are basically the angle part of a fouriertransform of radial functions in ${\mathbb R}^n$, $\int_\text{angles}{\mathrm e}^{i\langle k,x\rangle}$.
They solve
$x^2 \dfrac{d^2 y}{dx^2} + x \dfrac{dy}{dx} + (x^2 - \alpha^2)y = 0$
Theorems
| $J_n (x) = \frac{1}{2 \pi} {\int_{-\pi}^\pi} \,{\mathrm e}^{i(n \tau - x \sin(\tau))} \,{\mathrm d}\tau$ |
|---|
| $J_\alpha(x) = \dfrac{(\frac{x}{2})^\alpha} {\Gamma(\alpha+1)} {}_0F_1 (\alpha+1 - {\tfrac{x^2}{4}})$ |
Reference
Wikipedia: Bessel function
Requirements
