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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

• Computational implementation of integral equations

See also

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