Overview
- Provides a self-contained introduction to Nevanlinna theory and normal families
- Collects recent results on first-order and Painlevé differential equations, and 2D Hamiltonian systems
- Presents applications of Nevanlinna theory to the value distribution of differential polynomials and uniqueness of meromorphic functions
Part of the book series: Universitext (UTX)
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Table of contents (6 chapters)
Keywords
- Nevanlinna theory
- applications of Nevanlinna theory
- uniqueness theorem complex analysis
- Yosida Class
- Zalcman rescaling method
- Zalcman lemma
- algebraic functions
- normal families
- ordinary differential equations
- algebraic differential equations
- asymptotic expansion
- Yosida rescaling
- Painlevé transcendents
- Painlevé property
- Malmquist theorem
- Bloch's principle
- MSC (2010): 30D35, 30D45, 34M05, 34M55, 30D05, 34E05, 34M40
About this book
Following a comprehensive treatment of Nevanlinna’s theory of value distribution, the author presents advances made since Hayman’s work on the value distribution of differential polynomials and illustrates how value- and pair-sharing problems are linked to algebraic curves and Briot–Bouquet differential equations. In addition to discussing classical applications of Nevanlinna theory, the book outlines state-of-the-art research, such as the effect of the Yosida and Zalcman–Pang method of re-scaling to algebraic differential equations, and presents the Painlevé–Yosida theorem, which relates Painlevé transcendents and solutions to selected 2D Hamiltonian systems to certain Yosida classes of meromorphic functions.
Aimed at graduate students interested in recent developments in the field and researchers working on related problems, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations will also be of interest to complex analysts looking for an introduction to various topics in the subject area. With examples, exercises and proofs seamlessly intertwined with the body of the text, this book is particularly suitable for the more advanced reader.
Reviews
“The list of references contains more than 200 items including very recent results of the author and other. … I recommend this book to any person who is interested in complex analysis, in particular, in value distribution theory and complex differential equations.” (Igor Chyzhykov, zbMATH, 2018)
Authors and Affiliations
About the author
Bibliographic Information
Book Title: Nevanlinna Theory, Normal Families, and Algebraic Differential Equations
Authors: Norbert Steinmetz
Series Title: Universitext
DOI: https://doi.org/10.1007/978-3-319-59800-0
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG 2017
Softcover ISBN: 978-3-319-59799-7Published: 07 August 2017
eBook ISBN: 978-3-319-59800-0Published: 24 July 2017
Series ISSN: 0172-5939
Series E-ISSN: 2191-6675
Edition Number: 1
Number of Pages: XVIII, 235
Number of Illustrations: 8 b/w illustrations
Topics: Functions of a Complex Variable, Ordinary Differential Equations, Difference and Functional Equations