Bessel functions
From SklogWiki
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
\[J_n (z) = \frac{1}{2 \pi i} \oint e^{(z/2)(t-1/t)}t^{-n-1}{\rm d}t\]
