University of Illinois Urbana-Champaign

Normal families of integer translations

Kim, Jeong-Heon

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https://hdl.handle.net/2142/22458

Description

Title
Normal families of integer translations
Author(s)
Kim, Jeong-Heon
Issue Date
1994
Doctoral Committee Chair(s)
Rubel, Lee A.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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.
Type of Resource
text
Permalink
http://hdl.handle.net/2142/22458
Copyright and License Information
Copyright 1994 Kim, Jeong-Heon

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