Bessel functions

Bessel functions of the first kind $J_n(x)$ are defined as the solutions to the Bessel differential equation

$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+(x^2-n^2)y=0$

which are nonsingular at the origin. They are sometimes also called cylinder functions or cylindrical harmonics. The Bessel function $J_n(z)$ can also be defined by the contour integral

$J_n(z)=\frac{1}{2\pi i}\oint e^{(z/2)(t-1/t)}{t}^{-n-1}\mathrm{d}t$

See also

• Bessel Function of the First Kind -- from Wolfram MathWorld