Complex Analysis, Riemann Surfaces and Integrable Systems

✍ Scribed by Sergey M. Natanzon

• Publisher: Springer
• Year: 2019
• Tongue: English
• Leaves: 149
• Series: Moscow Lectures, Volume 3
• Category: Library

✦ Table of Contents

Preface to the Book Series Moscow Lectures
Introduction
Contents
1 Holomorphic Functions
1.1 Complex Derivative
1.2 The Differential of a Complex Function
1.3 Holomorphic Functions
1.4 Complex Integration
1.5 Cauchy's Theorem
1.6 Antiderivative
1.7 Cauchy's Integral Formula
1.8 Taylor Series Expansion
1.9 A Criterion for a Function to Be Holomorphic
1.10 Weierstrass' Theorem
2 Meromorphic Functions
2.1 Functions Holomorphic on a Ring: Laurent Series
2.2 Isolated Singularities
2.3 Residues and Principal Value Integrals
2.4 The Argument Principle
2.5 Topological Properties of Meromorphic Functions
3 Riemann Mapping Theorem
3.1 Continuous Functionals on Compact Families of Functions
3.2 Hurwitz' Theorem and Univalent Functions
3.3 Analytic Continuation
3.4 Riemann Mapping Theorem
3.5 Automorphisms of Simply Connected Domains
3.6 Carathéodory's Theorem
4 Harmonic Functions
4.1 Holomorphic and Harmonic Functions
4.2 Integral Formulas
4.3 Green's Function
4.4 Dirichlet Problem
5 Riemann Surfaces and Their Modules
5.1 Riemann Surfaces
5.2 Riemann Surfaces of Analytic Functions
5.3 Uniformization
5.4 Moduli of Compact Riemann Surfaces of Genus 1
5.5 Automorphisms of the Upper Half-plane
5.6 Types of Riemann Surfaces
5.7 Sequential Sets of Automorphisms
5.8 The Geometry of Fuchsian Groups
5.9 Sequential Sets of Types (0,3,0), (0,2,1), and (0,1,2)
5.10 Sequential Sets of Type (1,1,0)
5.11 Fricke–Klein–Teichmüller Type Spaces
5.12 The Moduli Space Mg,k,m
6 Compact Riemann Surfaces
6.1 The Riemann–Hurwitz Formula
6.2 Meromorphic Functions and Differentials
6.3 Plane Algebraic Curves
6.4 The Field of Algebraic Functions
6.5 Periods of Holomorphic Differentials
6.6 Riemann's Bilinear Relations
7 The Riemann–Roch Theorem and Theta Functions
7.1 Divisors
7.2 The Riemann–Roch Theorem
7.3 Weierstrass Points
7.4 Abelian Tori and Theta Functions
7.5 Abel's Theorem
7.6 Jacobi Inversion Problem
8 Integrable Systems
8.1 Formal Exponentials
8.2 The KP Hierarchy
8.3 The n-KdV Hierarchy
Examples
8.4 Baker–Akhiezer Functions
8.5 Normalized Baker–Akhiezer Functions
8.6 Algebro-Geometric Solutions of the KP and n-KdV Equations
9 Formula for a Conformal Map from an Arbitrary Domain onto Disk
9.1 The Space of Simply Connected Domains
9.2 Conformal Maps and Integrable Systems
9.3 Formal Solutions to the Dispersionless 2D Toda Hierarhy
9.4 Proof of the Theorem on Symmetric Solutions
9.5 Effectivization of Riemann's Theorem
References
Index