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.

  • $\begingroup$ I have started to answer, but stopped and asked myself why this question is not asked at math.stackexchange? You'd get much better answer there. They even have a perfectly appropriate tag: mathematical-physics. $\endgroup$ – Misha Sep 20 '11 at 7:53
  • $\begingroup$ @Misha: I did ask it there as well under the same title. I just added that tag; thanks for the heads up. I ask it here mainly because it was motivated by a physics question, and I'm afraid it's too physics-y for mathematicians. $\endgroup$ – Victor Liu Sep 20 '11 at 10:04
  • $\begingroup$ Here's the post at math.SE. $\endgroup$ – joriki Sep 20 '11 at 13:35
  • $\begingroup$ Actually, I think you've reduced this question to pure math, there's no physics left in it ;-) although one could make a case for keeping it here... anyway, since you've already posted this on math.SE I'll just close it here. You're generally not supposed to cross-post questions to multiple sites at the same time. $\endgroup$ – David Z Sep 20 '11 at 17:06
  • $\begingroup$ Related: Plane wave expansion in cylindrical coordinates. $\endgroup$ – Emilio Pisanty Jan 15 '13 at 17:17

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