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A new method is presented for Fourier decomposition of the Helmholtz GreenFunction in cylindrical coordinates, which is equivalent to obtaining thesolution of the Helmholtz equation for a general ring source. The Fouriercoefficients of the Helmholtz Green function are split into their halfadvanced+half retarded and half advanced-half retarded components. Closed formsolutions are given for these components in terms of a Horn function and aKampe de Feriet function, respectively. The systems of partial differentialequations associated with these two-dimensional hypergeometric functions areused to construct a fourth-order ordinary differential equation which bothcomponents satisfy. A second fourth-order ordinary differential equation forthe general Fourier coefficent is derived from an integral representation ofthe coefficient, and both differential equations are shown to be equivalent.Series solutions for the various Fourier coefficients are also given, mostly interms of Legendre functions and Bessel/Hankel functions. These are derived fromthe closed form hypergeometric solutions or an integral representation, orboth. Numerical calculations comparing different methods of calculating theFourier coefficients are presented.

Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function