Complex analysis, Riemann surfaces and integrable systems / Sergey M. Natanzon.
Natanzon, S. M., 1948-
2019
QA331
Available Online

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Title Complex analysis, Riemann surfaces and integrable systems / Sergey M. Natanzon.
Author Natanzon, S. M., 1948-
ISBN 9783030346409 (electronic book)
3030346404 (electronic book)
9783030346393
DOI https://doi.org/10.1007/978-3-030-34
Publication Details Cham : Springer, c2019.
Language English
Description 1 online resource (148 pages).
Item Number 10.1007/978-3-030-34
Call Number QA331
Dewey Decimal Classification 515/.9
Summary 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.
Bibliography, etc. Note Includes bibliographical references and index.
Access Note Access limited to authorized users.
Source of Description Description based on print version record.
Series Moscow lectures ; v. 3.
Available in Other Form Complex Analysis, Riemann Surfaces and Integrable Systems
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Holomorphic functions
Meromorphic functions
Riemann's theorem
Harmonic functions
Riemann surfaces and their modules
Compact Riemann surfaces and algebraic curves
Riemann-Roch theorem and theta functions
Integrable Systems
The formula for the conformal mapping of an arbitrary domain into the unit disk.

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