Normal Families and Value Distribution

Normal families, that is relatively compact sets of analytic or of meromorphic functions on a domain, are introduced. The relevant topology of uniform (spherical) convergence on compact subsets is introduced and the convergence of sequences of analytic and meromorphic functions is discussed. A version of the Arzelà-Ascoli Theorem is proven and its consequences for analytic and meromorphic functions, namely Montel’s Theorem and Marty’s Theorem, are obtained. Zalcman’s Lemma, which gives a criterion for a family to not be normal, is proven. The chapter ends with a proof of Montel’s Second Theorem and then the Great Picard Theorem on value distribution.