This is a brief textbook on complex analysis intended for the students of upper undergraduate or beginning graduate level. The author stresses the aspects of complex analysis that are most important for the student planning to study algebraic geometry and related topics. The exposition is rigorous but elementary: abstract notions are introduced only if they are really indispensable. This approach provides a motivation for the reader to digest more abstract definitions (e.g., those of sheaves or line bundles, which are not mentioned in the book) when he/she is ready for that level of abstraction indeed. In the chapter on Riemann surfaces, several key results on compact Riemann surfaces are stated and proved in the first nontrivial case, i.e. that of elliptic curves.
Author(s): Serge Lvovski
Series: Moscow Lectures 6
Edition: 1
Publisher: Springer
Year: 2020
Language: English
Pages: 257
Tags: Complex Analysis
Preface to the Book Series Moscow Lectures
Preface
Contents
Chapter 1 Preliminaries
1.1 Absolute and Uniform Convergence
1.2 Open, Closed, Compact, Connected Sets
1.3 Power Series
1.4 The Exponential Function
1.5 Necessary Background From Multivariable Analysis
1.6 Linear Fractional Transformations
Exercises
Chapter 2 Derivatives of Complex Functions
2.1 Inverse Functions, Roots, Logarithms
2.2 The Cauchy–Riemann Equations
Exercises
Chapter 3 A Tutorial on Conformal Maps
3.1 Linear Fractional Transformations
3.2 More Complicated Maps
Exercises
Chapter 4 Complex Integrals
4.1 Basic Definitions
4.2 The Index of a Curve Around a Point
Exercises
Chapter 5 Cauchy’s Theorem and Its Corollaries
5.1 Cauchy’s Theorem
5.2 Cauchy’s Formula and Analyticity of Holomorphic Functions
5.3 Infinite Differentiability. Term-By-Term Differentiability
Exercises
Chapter 6 Homotopy and Analytic Continuation
6.1 Homotopy of Paths
6.2 Analytic Continuation
6.3 Cauchy’s Theorem Revisited
6.4 Indices of Curves Revisited
Exercises
Chapter 7 Laurent Series and Isolated Singularities
7.1 The Multiplicity of a Zero
7.2 Laurent Series
7.3 Isolated Singularities
7.4 The Point ∞ as an Isolated Singularity
Exercises
Chapter 8 Residues
8.1 Basic Definitions
8.2 The Argument Principle
8.3 Computing Integrals
Exercise
Chapter 9 Local Properties of Holomorphic Functions
9.1 The Open Mapping Theorem
9.2 Ramification
9.3 The Maximum Modulus Principle and Its Corollaries
9.4 Bloch’s Theorem
Exercises
Chapter 10 Conformal Maps. Part 1
10.1 Holomorphic Functions on Subsets of the Riemann Sphere
10.2 The Reflection Principle
10.3 Mapping the Upper Half-Plane onto a Rectangle
10.4 Carathéodory’s Theorem
10.5 Quasiconformal Maps
Exercises
Chapter 11 Infinite Sums and Products
11.1 The Cotangent as an Infinite Sum
11.2 Elliptic Functions
11.3 Infinite Products
11.4 The Mittag-Leffler andWeierstrass Theorems
11.5 Blaschke Products
Exercises
Chapter 12 Conformal Maps. Part 2
12.1 The Riemann Mapping Theorem: the Statement and a Sketch of the Proof
12.2 The Riemann Mapping Theorem: Justifications
12.3 The Schwarz–Christoffel Formula
12.4 The Hyperbolic Metric
Exercises
Chapter 13 A Thing or Two About Riemann Surfaces
13.1 Definitions, Simplest Examples, General Facts
13.2 The Riemann Surface of an Algebraic Function
13.3 Genus; the Riemann–Hurwitz Formula
13.4 Differential Forms and Residues
13.5 On Riemann’s Existence Theorem
13.6 On the Field of Meromorphic Functions
13.7 On the Riemann–Roch Theorem
13.8 On Abel’s Theorem
Exercises
References
Index
