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Title:Normal families of integer translations
Author(s):Kim, Jeong-Heon
Doctoral Committee Chair(s):Rubel, Lee A.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:Let f be a meromorphic function in the complex plane C. We consider the normality of the family of integer translations of f, $\{ f(z + n):n = 0,\pm 1,\pm2,\...\}$. If the family is normal on a set G in C, then the set G is open and periodic with period 1, i.e., $z\pm 1\in G$ for all $z\in G$.
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.
Issue Date:1994
Rights Information:Copyright 1994 Kim, Jeong-Heon
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9503234
OCLC Identifier:(UMI)AAI9503234

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