Entire and Meromorphic Functions / Edition 1

by Lee A. Rubel

Overview

This book is an introduction to the theory of entire and meromorphic functions intended for advanced graduate students in mathematics and for professional mathematicians. The book provides a clear treatment of the Nevanlinna theory of value distribution of meromorphic functions, and presentation of the Rubel-Taylor Fourier Series method for meromorphic functions and the Miles theorem on efficient quotient representation. It has a concise but complete treatment of the Polya theory of the Borel transform and the conjugate indicator diagram. It contains some of Buck's results on integer-valued entire functions, and closes with the Malliavin-Rubel uniqueness theorem. The approach gets to the heart of the matter without excessive scholarly detours. It prepares the reader for further study of the vast literature on the subject, which is one of the cornerstones of complex analysis.

Product Details

ISBN-13:	9780387945101
Publisher:	Springer New York
Publication date:	02/28/1996
Series:	Universitext
Edition description:	Softcover reprint of the original 1st ed. 1996
Pages:	188
Product dimensions:	6.10(w) x 9.25(h) x (d)

Table of Contents

* Introduction
• The Riemann-Stieltjes Integral
• Jensen’s Theorem and Applications
• The First Fundamental Theorem of Nevanlinna Theory
• Elementary Properties of T(r,f)
• The Cartan Formulation of the Characteristic
• The Poisson-Jensen Formula
• Applications of T(r)
• A Lemma of Borel and Some Applications
• The Maximum Term of an Entire Function
• Relation Between the Growth of an Entire Function and the Size of Its Taylor Coefficients
• Carleman’s Theorem
• A Fourier Series Method
• The Miles-Rubel-Taylor Theorem on Quotient Representations of Meromorphic Functions
• Canonical Products
• Formal Power Series
• Picard’s Theorem and the Second Fundamental Theorem
• A Proof of the Second Fundamental Theorem
• 'Two Constant' Theorems and the Phragmén-Lindelöf Theorems
• The Pólya Representation Theorem
• Integer-Valued Entire Functions
• On Small Entire Functions of Exponential-Type with Given Zeros
• The First-Order Theory of the Ring of All Entire Functions
• Identities of Exponential Functions
• References
• Index

Editorial Reviews

From the reviews:

"This book grew out of lectures from courses given at the University of Illinois in 1993, and earlier. It is written at a level which assumes only a reasonable first course in complex analysis and knowledge of some standard results about the representation of linear functionals....The book is divided into 24 relatively short chapters, and, although some of the later material depends on earlier material, the reader who already has some knowledge of the general subject from other sources should be able to go directly to one of the later sections of the book without undue difficulties.

This book is a very welcome addition to the resources available for learning about entire and meromorphic functions." (P. Lappan, Mathematical Reviews)

From the Publisher