Properties of functions which uniformize certain classes of simply connected Riemann surfaces
Curtis, Howard B., Jr
Doctor of Philosophy
Three classes of simply connected Riemann surfaces are considered in the following study. They are designated as Class III, Class IV, and Class V in keeping with the notation of Classes I, II, and III already considered by Taylor. Class III is a subclass of Class IV; it is shown only that the surfaces of Class III are parabolic, while here the forms of the uniformizing function and its derivative are found for Class III. For Classes IV and V the forms of the uniformizing function and its derivative are obtained, and the type of the surface is determined. Finally, a proof is given to show that every mapping function of the form obtained for Class V defines a surface of this class. The principal method used in the proofs is that of approximating the given surface by a sequence of compact surfaces.