Integral Representations of the Green's Function for the Helmholtz Equation in Parallel Plate, Rectangular Waveguide and Resonator

Integral representations of the Green's function for the Helmholtz equation in the parallel plate, rectangular waveguide and resonator domains is obtained by the summation of series of the mirror reflections and separation of variables methods. Integral representations offer the advantage in comparison to the series in the possibility of wide variety of transformations that give convenient for numerical calculations expressions. A unified formula for all three cases that contains elliptical theta-function is obtained. Earlier the same results for a simpler plane case were obtained by Titchmarch