Convenient closed-form far-field expressions for thin circular loop antennas which possess a traveling-wave current distribution have been available for many years. However, a method for analytically evaluating the more complicated near-zone vector potential and corresponding electromagnetic field integrals associated with traveling-wave circular loop antennas has not been previously reported in the electromagnetics literature. The purpose of this paper is to introduce a systematic approach for finding exact series representations of these integrals which are valid not only in the far-field of the loop, but also in its near-field region. These mathematically exact solutions are general and can be shown to contain all of the well-known classical approximations as special cases.