Value distribution theory and Teichmüller’s paper "Einfache Beispiele zur Wertverteilungslehre"

Abstract

This survey will appear in Vol. VII of the Handbook of Teichmuller theory (European Mathematical Society Publishing House, 2020). It is a commentary on Teichmuller's paper "Einfache Beispiele zur Wertverteilungslehre", published in 1944, whose English translation appears in that volume. Together with Teichmuller'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 Teichmuller. We mention the foundational work of several mathematicians , including Picard, Laguerre, Poincare, 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.