Hankel transform

Summary

The Hankel transform of order $\nu$ of a function $f$ on $(0, \infty)$ is $\int_0^\infty f(r)\, J_\nu(kr)\, r\, dr$, where $J_\nu$ is the Bessel function of the first kind. It is the radial part of the Fourier transform. The Fourier transform of a radial function in the plane is a Hankel transform of order $0$, and in higher dimensions other orders appear. With this normalization it is its own inverse. It is used to solve problems with circular or cylindrical symmetry, such as heat flow and vibrations of a drum.