Value distribution theory and Teichm{ü}ller's paper "Einfache Beispiele zur Wertverteilungslehre"

Abstract

This survey will appear in Vol. VII of the Hendbook of Teichm{ü}ller theory (European Mathematical Society Publishing House, 2020). It is a commentary on Teichm{ü}ller's paper "Einfache Beispiele zur Wertverteilungslehre", published in 1944, whose English translation appears in that volume. Together with Teichm{ü}ller's paper, we survey the development of value distribution theory, in the period starting from Gauss's work on the Fundamental Theorem of Algebra and ending with the work of Teichm{ü}ller. We mention the foundational work of several mathematicians, including Picard, Laguerre, Poincar{é}, Hadamard, Borel, Montel, Valiron, and others, and we give a quick overview of the various notions introduced by Nevanlinna and some of his results on that theory.