Complex Analysis, Riemann Surfaces and Integrable Systems

Preface to the Book Series Moscow Lectures......Page 6
Introduction......Page 9
Contents......Page 12
1.1 Complex Derivative......Page 13
1.2 The Differential of a Complex Function......Page 15
1.3 Holomorphic Functions......Page 17
1.4 Complex Integration......Page 18
1.5 Cauchy's Theorem......Page 20
1.6 Antiderivative......Page 22
1.7 Cauchy's Integral Formula......Page 23
1.8 Taylor Series Expansion......Page 25
1.9 A Criterion for a Function to Be Holomorphic......Page 27
1.10 Weierstrass' Theorem......Page 28
2.1 Functions Holomorphic on a Ring: Laurent Series......Page 30
2.2 Isolated Singularities......Page 32
2.3 Residues and Principal Value Integrals......Page 34
2.4 The Argument Principle......Page 36
2.5 Topological Properties of Meromorphic Functions......Page 38
3.1 Continuous Functionals on Compact Families of Functions......Page 40
3.2 Hurwitz' Theorem and Univalent Functions......Page 42
3.3 Analytic Continuation......Page 43
3.4 Riemann Mapping Theorem......Page 44
3.5 Automorphisms of Simply Connected Domains......Page 45
3.6 Carathéodory's Theorem......Page 46
4.1 Holomorphic and Harmonic Functions......Page 47
4.2 Integral Formulas......Page 49
4.3 Green's Function......Page 51
4.4 Dirichlet Problem......Page 52
5.1 Riemann Surfaces......Page 55
5.2 Riemann Surfaces of Analytic Functions......Page 56
5.3 Uniformization......Page 57
5.5 Automorphisms of the Upper Half-plane......Page 59
5.6 Types of Riemann Surfaces......Page 60
5.7 Sequential Sets of Automorphisms......Page 61
5.8 The Geometry of Fuchsian Groups......Page 65
5.9 Sequential Sets of Types (0,3,0), (0,2,1), and (0,1,2)......Page 70
5.10 Sequential Sets of Type (1,1,0)......Page 73
5.11 Fricke–Klein–Teichmüller Type Spaces......Page 75
5.12 The Moduli Space Mg,k,m......Page 76
6.1 The Riemann–Hurwitz Formula......Page 79
6.2 Meromorphic Functions and Differentials......Page 81
6.3 Plane Algebraic Curves......Page 82
6.4 The Field of Algebraic Functions......Page 84
6.5 Periods of Holomorphic Differentials......Page 87
6.6 Riemann's Bilinear Relations......Page 90
7.1 Divisors......Page 93
7.2 The Riemann–Roch Theorem......Page 94
7.3 Weierstrass Points......Page 98
7.4 Abelian Tori and Theta Functions......Page 100
7.5 Abel's Theorem......Page 105
7.6 Jacobi Inversion Problem......Page 107
8.1 Formal Exponentials......Page 112
8.2 The KP Hierarchy......Page 116
8.3 The n-KdV Hierarchy......Page 119
8.4 Baker–Akhiezer Functions......Page 120
8.5 Normalized Baker–Akhiezer Functions......Page 123
8.6 Algebro-Geometric Solutions of the KP and n-KdV Equations......Page 126
9.1 The Space of Simply Connected Domains......Page 128
9.2 Conformal Maps and Integrable Systems......Page 130
9.3 Formal Solutions to the Dispersionless 2D Toda Hierarhy......Page 134
9.4 Proof of the Theorem on Symmetric Solutions......Page 136
9.5 Effectivization of Riemann's Theorem......Page 142
References......Page 144
Index......Page 146