We present a new approach to the analysis of the field of a uniformly moving charge in medium with frequency dispersion. This is based on methods of complex function theory. First, we give the deduction of expressions for the field components directly from the causality principle. Second, the properties of integrands and integrals for the field components in the complex plane are discussed. Further, we use our approach for the case of a "left- handed" medium. The charge's field is presented as the sum of different components having clear physical interpretation: the "quasi-Coulomb" field, the wave field, and the "plasma trace". An efficient method for computing the field is developed. Additionally, the case of a boundary between an ordinary medium and a left-handed one is considered. The phenomenon of reversed Cherenkov-transition radiation is described.