Entire and meromorphic functions. With assistance from James E. Colliander
type: Item
instance of: scholarly article
title: Entire and meromorphic functions. With assistance from James E. Colliander
zbMATH Open document ID: 0859.30001
author: Lee A. Rubel
published in: Universitext
publication date: 1996-05-12T00:00:00Z
review text: This book by the late Professor Rubel deals with many important topics from the theory of entire and meromorphic functions. There are 24 chapters, loosely connected, but the emphasis throughout is on the Nevanlinna theory with various applications. Frequently, the material is taken from earlier papers of the author and its collaborators (Taylor, Malliavin). -- Among the better known subjects are the following. Jensen's theorem, the first fundamental theorem and applications, Cartan's formulation of the characteristic and the Ahlfors-Shimizu characteristic. Borel's lemma follows, leading to estimates of \(\log M(r)\) via \(T(r)\) excluding sets of finite logarithmic length. Basic facts of the Wiman-Valiron theory (maximal term and index, Newton polygon) are next. Later, the second fundamental theorem and some applications are given includig functions that share distinct values. For example, if \(f\) is entire and \(f\) and \(f'\) share two distinct values, then \(f=f'\). Functions \(f\) of exponential type, their indicator diagram and its relation to the singularities of the Borel transform \(\phi\) of \(f\) are studied, and various generalizations of Carlson's identity theorem are given. -- Some less known topics are treated in a long chapter ``A Fourier series method''. Given \(f\) meromorphic in \(\mathbb{C}\), the Fourier coefficients \[ c_k(r,f)= {1\over 2\pi} \int^{2\pi}_0[\log|f(re^{\text{it}})|]e^{-ikt}dt \] are associated, and their growth is connected with the growth of \(f\). If \(\lambda(r)\nearrow\), the class \(\Lambda\) of functions \(f\) meromorphic in \(\mathbb{C}\) with \(T(r,f)\leq A\lambda(Br)\) (with constants \(A\) and \(B\)) and the class \(\Lambda_E\) of entire functions \(f\) with \(\log M(r)\leq A\lambda(Br)\) are studied including their zero sets and pole sets. It is shown that each meromorphic function \(f\) is \(f_1/f_2\) where \(T(r,f_k)\leq AT(Br,f)\) for \(k=1,2\). As an application of the Nevanlinna theory a lemma of Hiromi and Ozawa …
Mathematics Subject Classification ID: 30-01
Mathematics Subject Classification ID: 30D20
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 30D30
zbMATH DE Number: 877625
zbMATH Keywords: meromorphic functions
zbMATH Keywords: Nevanlinna theory
zbMATH Keywords: entire function
reviewed by: Dieter Gaier
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 877625