In this paper it is shown how the Coulomb wave functions, commonly used in the description of a Coulomb field surrounding a nucleus, can be used in the description of electromagnetic fields that are symmetric with respect of [Formula: see text] inside a paraboloidal waveguide. The Abraham potentials Q and U, which are useful in describing fields with rational symmetry, are used to simplify the problem. It is shown that these potentials must satisfy a partial differential equation that when separated yields the Coulomb wave equation of order L = 0. Electromagnetic fields due to simple source distributions inside the paraboloid are expanded in terms of these functions. Specifically, solutions for current-loop sources located in the focal plane of the paraboloid are obtained. The case where the wall of the paraboloidal waveguide is assumed to be perfectly conducting is treated as well as the case where the wall has finite impedance. The finite paraboloid is also considered, and the far field is formulated using Huygen's principle. It is found that for the finite surface-impedance case, the far-field pattern due to a current loop operating at 100 MHz in the focal plane of a paraboloidal reflector of 1 m focal length is different from the perfectly conducting case. Specifically, the pattern seems to be more omnidirectional for the impedance case than for the perfectly conducting case. Numerical results are presented for relevant aspects of the problem.
