×

Value distribution of meromorphic functions. Transl. from the Russian by Mikhail Ostrovskii. With an appendix by Alexandre Eremenko and James K. Langley. (English) Zbl 1152.30026

Translations of Mathematical Monographs 236. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4265-2/hbk). xv, 488 p. (2008).
I was waiting for the appearance of the translation of the Russian monograph originally published in 1970, which contains a comprehensive exposition of the Nevanlinna theory of meromorphic functions of one complex variable. The monograph was written by Goldberg and Ostrovskii. I believe that this book is one of the most important expositions for the study of the Nevanlinna theory. Furthermore, there are the remarkable strong points “self-contained”, “full detail” and “examples”.
This book is a self-contained exposition for the Nevanlinna theory aiming to help beginning mathematicians. The only prerequisite for understanding the material of this book is an undergraduate course in the theory of functions of one complex variable. The authors pay great attention to examples of functions with “exotic” properties. Without them the reader will get a restricted image of the theory under consideration.
Eremenko and Langley added references and footnotes related to recent achievements in the topics considered in the original edition. They gave a survey of the results obtained since 1970 in a separate new appendix, Appendix B, at the end of the book.
The present book consists of seven chapters, Appendix A, Notes and Appendix B.
The value distribution theory of meromorphic functions occupies one of the central places in complex analysis. Numerous works are devoted to studying its connections with other areas of mathematics extending its inferences to wider classes and also its applications, mainly to the analytic theory of differential equations. In the present book, the related topics are left aside, and the main attention is concentrated on inner problems of the value distribution theory.
In the first three chapters, the results of the Nevanlinna theory are considered in detail. Chapters 4 and 5 are devoted to the detailed study of deficiencies, value distribution, and asymptotic properties of meromorphic functions. Chapter 6 deals with the study of the value distribution with respect to arguments (not only with respect to moduli as in classical Nevanlinna theory). In chapter 7, applications of Riemann surfaces to value distribution are discussed. The authors added Appendix A, in which they consider the magnitude of type for entire functions. All of the facts related to the history of the problems under consideration along with references on the subject are placed in the notes.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30D30 Meromorphic functions of one complex variable (general theory)
30D20 Entire functions of one complex variable (general theory)
30D15 Special classes of entire functions of one complex variable and growth estimates
PDFBibTeX XMLCite