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