Sound radiation from a circular duct is a classic problem. Exact solutions were previously reported for the case of a negligibly thin duct wall, using the Wiener--Hopf technique. Despite the elegance of its closed-form solutions, the numerical presentations have been limited to mere demonstrations of its capability. A computer program is not publicly available for its numerical evaluation. Such a numerical evaluation has recently become increasingly in demand not only for its own merits, but also for cross examination of the numerical results of computational aeroacoustics. These techniques are just starting to attract widespread attention as a potential tool in attacking important but unsolved aeroacoustic problems. This paper presents a comprehensive mathematical procedure for the numerical evaluation primarily for use in studies of noise emission of aircraft engines. Various mean flows are included for simulations of an aircraft jet engine exhaust and inlet, and aircraft cruise condition. Unlike previously published reports, this paper will include radiation of nonpropagating modes as well as propagating modes.