Bessel functions

From SklogWiki
Jump to: navigation, search

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

x^2 \frac{d^2y}{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

Applications in statistical mechanics[edit]

See also[edit]