On limit functions and their natural boundaries in infinite compositions of entire functions

• Published in: Complex variables and elliptic equations 2007-09, Vol.52 (9), p.807-825
• Main Author: Maalouf, Ramez N.
• Format: Article
• Language: English
• Subjects: AMS Subject Classifications; Boundary behaviour of analytic functions; Composition of entire functions; Entire functions; Iteration of analytic functions
• ISSN: 1747-6933; 1747-6941
• DOI: 10.1080/17476930701627561

In this article, we consider infinite sequences {Φ n } of entire functions in the complex plane C defined as compositions of the form where each f n , n = 1, 2, ... , is an entire function, and the limit functions Φ of such sequences. Under reasonable conditions on the sequence {f n }, and for the cases where Φ exists and is a nonconstant analytic function, one finds that the boundary of the domain where {Φ n } converges to Φ is in fact the natural boundary of Φ, and that this boundary satisfies certain "expansion" properties when considered under the composition of the f n 's. We also consider the case of constant limit functions Φ. In the final section we discuss the connection between the coefficients of a power series representation of a nonconstant limit Φ and the sequence {a n } of a one parameter family of entire functions f n (z) = f (a n,z ), whose composition as in ( 1 ) converges in some domain to Φ.