Application of the Second Main Theorem

Abstract

For a function meromorphic in the punctured plane z ≠ ∞ the theorem of Picard holds: the function w(z) completely omits at most two values unless it reduces to a constant ; we shall exclude this latter trivial case from future consideration. In connection with the first main theorem we have moreover found that such a function assumes almost all values a with a frequency N(r, a) that corresponds to the characteristic T(r, w). These results were sharpened for the case of a meromorphic function of finite nonintegral order to the extent that at most one value exists which is exceptional in the sense that its counting function is of lower order (class, type) than the characteristic. Even simple examples show that this no longer holds for integral orders; thus, the exponential function has two exceptional values (0, ∞), and in VIII, § 4, we have discussed a case where a lowering of the type or the class of N(r, a) occurs for two values of a. That the number of such exceptional values cannot exceed two, on the other hand, is the essential content of the extension of Picard’s theorem given by Borel [1].