===== Bessel function =====
==== Function ====
| @#FF9944: definition  | @#FF9944: $J_{}: ?\to ?$ |
| @#FF9944: definition  | @#FF9944: $J_\alpha(x) := \sum_{m=0}^\infty \dfrac{1}{\Gamma(m+0+1)\,\Gamma(m+\alpha+1)} (-1)^m{\left(\dfrac{x}{2}\right)}^{2m+\alpha}$ |

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=== 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: [[http://en.wikipedia.org/wiki/Bessel_function|Bessel function]]

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=== Requirements ===
[[Gamma function]]