Exact, closed-form field expressions are derived for the canonical problem of electromagnetic radiation by a semi-infinite traveling-wave current filament in free space. In order to avoid time consuming numerical integration, the fields are represented in terms of complementary incomplete Lipschitz-Hankel integrals, whose series expansions allow for accurate and efficient computation of the field components everywhere in space. The field expressions are also shown to reduce to elementary functions in many special cases, e.g., in the far-zone, in the vicinity of the current filament, and when the propagation constant along the wire is that of free space. Representative plots for the electric field components are presented for values of the propagation constant corresponding to both fast and slow-wave propagation. Since a superposition of two semi-infinite current filaments yields a finite-length current filament, the method and the results obtained in this paper can be readily applied to a number of problems in electromagnetics.