ON THE THEOREMS OF GROSS AND IVERSEN

摘要

This paper, a contribution to the theory of analytic functions, is concerned with the theory of cluster sets, where the topological properties of meromorphic functions are separated from the analytic properties. The principal results are generalizations of theorems of Gross and Iver-sen, asserting the inclusion of the frontier of the cluster set of a function f at P in the boundary cluster set of f at P. The generalization is obtained by the introduction of a refined boundary cluster set relative to a set of measure 0 on the unit circle. Every value of the cluster set minus the modified boundary cluster set is assumed infinitely often near P, with the exception of a set of logarithmic capacity 0. The results include a generalization of Picard's theorem to non-isolated essential singularities.