The generalized dual series approach, as it is applied to the mixed boundary value problems that characterize the coupling of electromagnetic energy through apertures, is described. This approach is based upon techniques borrowed from the analysis of the Riemann-Hilbert problem of complex variable theory. It allows one to obtain essentially analytical solutions to families of canonical coupling problems in separable geometries. The generalized dual series approach is illustrated with the example of the coupling of an H-polarized plane wave to a perfectly conducting infinite cylinder with an infinite axial slot enclosing a concentric impedance cylinder. This problem encompasses the coupling to an empty cylinder as well as to a cylinder enclosing a conducting wire or cable. The solution explicitly exhibits the correct behavior near the edge of the aperture, and it can handle small to large ratios of cylinder radius to wavelength without additional special considerations. The angle of incidence is arbitrary. Results are shown for a normal incidence case and for a case in which the direction of incidence coincides with the edge of the aperture. A brief comparison with related moment method results is given.