Bessel functions

Bessel functions of the first kind $J_{n}(x)$ are defined as the solutions to the Bessel differential equation

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

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

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

See also

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