It is well know that planewaves are a complete basis for solutions to the wave equation. Let us assume a 2D space, and at fixed temporal frequency, the equation reduces to the Helmholtz equation. In cylindrical coordinates, the most appropriate solutions are the two kinds of Hankel functions, representing outgoing and incoming wave solutions. Actually, the Hankel functions should be multiplied by $e^{i m \theta}$ to produce cylindrical harmonics, which are a complete basis. My question is this:
If cylindrical harmonics are a complete basis, is there a closed form expression relating them to planewaves?
I know that 1st kind Bessel functions $J_m$ have a planewave decomposition by way of the Jacobi-Anger identity. However, a Hankel function's real part is a bessel function while its imaginary part is a 2nd kind Bessel function (Neumann function) with a singularity at the origin. I can't find an analogous expression for expression Neumann functions in terms of planewaves.
