The Riemann Surface of a Univalent Function

Abstract

Until now we have investigated single-valued functions w = w(z) for |z| < R ≦ ∞ without in general bothering with the image in the w-plane generated by the given correspondence z → w. To be sure, in Chapter I we thoroughly investigated the Riemann surface associated with the special automorphic mapping function w(z; a ₁,..., a _q ) (the universal covering surface for the plane punctured at q points) ; and we have in the course of our discussion repeatedly made use of a more general concept, the universal covering surface of an arbitrary schlicht region in the w-plane. But even these surfaces should be considered as still very special, since the associated mapping functions are automorphic. Of course, automorphic functions and the regularly ramified Riemann surfaces associated with them very often provide the most interesting examples for the general value distribution theory. Nevertheless, it will be useful to become acquainted with a few general properties of surfaces generated by an arbitrary single-valued meromorphic function as the image of the disk |z| < R ≦ ∞. On such simply connected surfaces the inverse function z of w is well-defined and univalent, and our problem is thus to examine analytic functions characterized by these properties.