Ahlfors’ contribution to the theory of meromorphic functions

This is an expanded version of one of the Lectures in memory of Lars Ahlfors in Haifa in 1996. Some mistakes are corrected and references added. This article is an exposition for non-specialists of Ahlfors’ work in the theory of meromorphic functions. When the domain is not specified we mean meromorphic functions in the complex plane C. The theory of meromorphic functions probably begins with the book by Briot and Bouquet [18] where the terms “pole”, “essential singularity” and “meromorphic” were introduced and what is known now as the Casorati– Weierstrass Theorem was stated for the first time. A major discovery was Picard’s theorem (1879) which says that a meromorphic function omitting three values in the extended complex plane C̄ is constant. The modern theory of meromorphic functions begins with attempts to give an “elementary proof” of this theorem. These attempts culminated in R. Nevanlinna’s theory which was published first in 1925. Nevanlinna’s books [46] and [47] were very influential and shaped much of the research in function theory in this century. Nevanlinna theory, also known as value distribution theory, was considered as one of the most important areas of research in 1930-40, so it is not surprising that Ahlfors started his career with work in this subject (besides he was ∗Supported by US-Israel Binational Science Foundation and by NSF Grant DMS9800084