This book centers on normal families of holomorphic and meromorphic functions and also normal functions. The authors treat one complex variable, several complex variables, and infinitely many complex variables (i.e., Hilbert space).
The theory of normal families is more than 100 years old. It has played a seminal role in the function theory of complex variables. It was used in the first rigorous proof of the Riemann mapping theorem. It is used to study automorphism groups of domains, geometric analysis, and partial differential equations.
The theory of normal families led to the idea, in 1957, of normal functions as developed by Lehto and Virtanen. This is the natural class of functions for treating the Lindelof principle. The latter is a key idea in the boundary behavior of holomorphic functions.
This book treats normal families, normal functions, the Lindelof principle, and other related ideas. Both the analytic and the geometric approaches to the subject area are offered. The authors include many incisive examples.
The book could be used as the text for a graduate research seminar. It would also be useful reading for established researchers and for budding complex analysts.
Table of Contents:
1. Introduction. 2. A Glimpse of Normal Families. 3. Normal Families in Cn. 4. Normal Functions in Cn. 5. A Geometric Approach to the Theory of Normal Families. 6. Some Classical Theorems. 7. Normal Families of Holomorphic Functions. 8. Spaces that Omit the Values 0 and 1. 9. Concluding Remarks.
About the Author :
Peter V. Dovbush Dr. habil., Associate Professor, in Moldova State University, Institute of Mathematics and Computer Science. He received his Ph.D. in Lomonosov Moscow State University in 1983 and Doctor of Sciences in 2003. He has published over over 50 scholarly articles.
Steven G. Krantz is a Professor of Mathematics at Washington University in St. Louis. He has previously taught at UCLA, Princeton University, and Penn State University. He received his Ph.D. from Princeton University in 1974. Krantz has directed 20 Ph.D. students and 8 Masters students. He has published over 130 books and over 300 scholarly articles. He is the holder of the Chauvenet Prize and the Beckenbach Book Award and the Kemper Prize. He is a Fellow of the American Mathematical Society.
