Hankel Transform

The Hankel transform of a function \(f: \mathbb{R}_{\ge 0} \rightarrow \mathbb{R}\) is given by

\[\begin{align} F_\nu(k) = \int_0^\infty f(r) J_\nu(kr) \,r\,\mathrm{d}r, \end{align}\]

This makes sense for all compactly supported functions, for example, when \(\nu > -1/2\).

The inverse Hankel transform of \(F_\nu(k)\) is defined as

\[\begin{align} f(r) = \int_0^\infty F_\nu(k) J_\nu(kr) \,k\,\mathrm{d}k, \end{align}\]

This transform is closely related to the Fourier transform of radial functions.