The resolvent method, one of the most powerful methods of solving nonstationary problems in quantum mechanics, is presented. The use of this method has been hindered by its reputation for excessive complexity. In the proposed form, the resolvent method is no more complex than common perturbation theory. One of the attractive features of the method is the possibility of widespread use of the theory of functions of a complex variable. The main ideas of the method, which was initially developed for systems with a discrete spectrum, are generalized for systems with a continuous spectrum. The radiation from atoms in a waveguide and in free space is examined as examples of this method.