Authors:
- Using basic tools from the first year of university studies, the book leads a reader to the impressive achievements of mathematics of the 21st century
- Studying the book, the reader will get acquainted with analytical and harmonic functions, as well as with the main results of the theory of Riemann surfaces. The reader will also get acquainted with the modern use of these results for solving classical problems of practical importance. These applications are based on the theory of integrable systems, which is also discussed in the book
- Practical all the statements are given in the book with full proofs
Part of the book series: Moscow Lectures (ML, volume 3)
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Table of contents (9 chapters)
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Front Matter
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Back Matter
About this book
This book is devoted to classical and modern achievements in complex analysis. In order to benefit most from it, a first-year university background is sufficient; all other statements and proofs are provided.
We begin with a brief but fairly complete course on the theory of holomorphic, meromorphic, and harmonic functions. We then present a uniformization theory, and discuss a representation of the moduli space of Riemann surfaces of a fixed topological type as a factor space of a contracted space by a discrete group. Next, we consider compact Riemann surfaces and prove the classical theorems of Riemann-Roch, Abel, Weierstrass, etc. We also construct theta functions that are very important for a range of applications.
After that, we turn to modern applications of this theory. First, we build the (important for mathematics and mathematical physics) Kadomtsev-Petviashvili hierarchy and use validated results to arrive at important solutions to these differential equations. We subsequently use the theory of harmonic functions and the theory of differential hierarchies to explicitly construct a conformal mapping that translates an arbitrary contractible domain into a standard disk – a classical problem that has important applications in hydrodynamics, gas dynamics, etc.
The book is based on numerous lecture courses given by the author at the Independent University of Moscow and at the Mathematics Department of the Higher School of Economics.
Keywords
- meromorphic functions
- Riemann theorem
- harmonic functions
- Fuchsian groups
- Riemann surfaces
- moduli of Riemann surfaces
- algebraic curves
- Riemann-Roch theorem
- Weierstrass points
- Abel theorem
- theta function
- Baker-Akhiezer function
- Kadomtsev-Petviashvili (KP) hierarchy
- algebro-geometric solutions of KP
- dispersionless 2D Toda hierarchy
- conformal mappings to disk
Authors and Affiliations
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HSE University, Moscow, Russia
Sergey M. Natanzon
About the author
Sergey M. Natanzon is a professor of mathematics at the NRU Higher School of Economics since 2008, and a professor of mathematics at the Independent University of Moscow since 1991.
Bibliographic Information
Book Title: Complex Analysis, Riemann Surfaces and Integrable Systems
Authors: Sergey M. Natanzon
Translated by: Natalia Tsilevich
Series Title: Moscow Lectures
DOI: https://doi.org/10.1007/978-3-030-34640-9
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2019
Hardcover ISBN: 978-3-030-34639-3Published: 30 January 2020
Softcover ISBN: 978-3-030-34642-3Published: 30 January 2021
eBook ISBN: 978-3-030-34640-9Published: 03 January 2020
Series ISSN: 2522-0314
Series E-ISSN: 2522-0322
Edition Number: 1
Number of Pages: XIII, 139
Number of Illustrations: 22 b/w illustrations
Topics: Analysis