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Complex Analysis, Riemann Surfaces and Integrable Systems - Sergey M. Natanzon

Complex Analysis, Riemann Surfaces and Integrable Systems

Buch | Hardcover
XIII, 139 Seiten
2020 | 1st ed. 2019
Springer International Publishing (Verlag)
978-3-030-34639-3 (ISBN)
CHF 97,35 inkl. MwSt

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.


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.

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.

Erscheinungsdatum
Reihe/Serie Moscow Lectures
Übersetzer Natalia Tsilevich
Zusatzinfo XIII, 139 p. 22 illus.
Verlagsort Cham
Sprache englisch
Original-Titel ,
Maße 155 x 235 mm
Gewicht 391 g
Themenwelt Mathematik / Informatik Mathematik Analysis
Schlagworte Abel theorem • algebraic curves • algebro-geometric solutions of KP • Baker-Akhiezer function • conformal mappings to disk • dispersionless 2D Toda hierarchy • Fuchsian Groups • Harmonic Functions • Kadomtsev-Petviashvili (KP) hierarchy • meromorphic functions • Moduli of Riemann surfaces • Riemann-Roch theorem • Riemann Surfaces • Riemann theorem • theta function • Weierstrass points
ISBN-10 3-030-34639-0 / 3030346390
ISBN-13 978-3-030-34639-3 / 9783030346393
Zustand Neuware
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