Compact Families of Meromorphic Functions

Abstract

In Sections 1 and 2 we treat normal families of meromorphic functions. These are families that are sequentially compact when regarded as functions with values in the extended complex plane. We give two characterizations of normal families, Marty’s theorem in Section 1 and the Zalcman lemma in Section 2. From the latter characterization we deduce Montel’s theorem on compactness of families of meromorphic functions that omit three points, and we also prove the Picard theorems. Sections 3 and 4 constitute an introduction to iteration theory and Julia sets. In Section 3 we proceed far enough into the theory to see how Montel’s theorem enters the picture and to indicate the fractal nature (self-similarity) of Julia sets. In Section 4 we relate the connectedness of Julia sets to the orbits of critical points. In Section 5 we introduce the Mandelbrot set, which has been called the “most fascinating and complicated subset of the complex plane.”