Normal families of integer translations.

摘要

Let f be a meromorphic function in the complex plane C. We consider the normality of the family of integer translations of f, ;Our question is: for a given open set G, does there exist an entire function whose integer translations forms a normal family in a neighborhood of z exactly for z in G? In the first part, we show that the necessary and sufficient condition for existence of such an entire function is that the set G is periodic with period 1. For a given open set, we construct an entire function satisfying the property.;The second part is about finitely normal family. In this case necessary and sufficient conditions are the complement of the set G has no bounded component in addition to periodicity of the set G. We prove the existence part using Arakelian's approximation theorem by entire functions on a closed subset of the complex plane C.