Bessel 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

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