Properties of functions which uniformize certain classes of simply connected Riemann surfaces

[摘要] 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.

[发布机构] Rice University