# Book:Serge Lang/Complex Analysis

## Serge Lang: Complex Analysis

Published $\text {1977}$, Springer

ISBN 0-387-98592-1.

### Contents

Foreword

Prerequisites

Part One: Basic Theory

Chapter 1: Complex Numbers and Functions
§ 1: Definition
§ 2: Polar Form
§ 3: Complex Valued Functions
§ 4: Limits and Compact Sets
§ 5: Complex Differentiability
§ 6: The Cauchy-Riemann Equations
§ 7: Angles Under Holomorphic Maps
Chapter 2: Power Series
§ 1: Formal Power Series
§ 2: Convergent Power Series
§ 3: Relations Between Formal and Convergent Series
§ 4: Analytic Functions
§ 5: Differentiation of Power Series
§ 6: The Inverse and Open Mapping Theorems
§ 7: The Local Maximum Modulus Principle
Chapter 3: Cauchy's Theorem, First Part
§ 1: Holomorphic Functions on Connected Sets
§ 2: Integrals Over Paths
§ 3: Local Primitive for a Holomorphic Function
§ 4: Another Description of the Integral Along a Path
§ 5: The Homotopy Form of Cauchy's Theorem
§ 6: Existence of Global Primitives. Definition of the Logarithm
§ 7: The Local Cauchy Formula
Chapter 4: Winding Numbers and Cauchy's Theorem
§ 1: The Winding Number
§ 2: The Global Cauchy Theorem
§ 3: Artin's Proof
Chapter 5: Applications of Cauchy's Integral Formula
§ 1: Uniform Limits of Analytic Functions
§ 2: Laurent Series
§ 3: Isolated Singularities
Chapter 6: Calculus of Residues
§ 1: The Residue Formula
§ 2: Evaluation of Definite Integrals
Chapter 7: Conformal Mappings
§ 1: Schwarz Lemma
§ 2: Analytic Automorphisms of the Disc
§ 3: The Upper Half Plane
§ 4: Other Examples
§ 5: Fractional Linear Transformations
Chapter 8: Harmonic Functions
§ 1: Definition
§ 2: Examples
§ 3: Basic Properties of Harmonic Functions
§ 4: The Poisson Formula
§ 5: Construction of Harmonic Functions
§ 6: Appendix. Differentiating Under the Integral Sign

Part Two: Geometric Function Theory

Chapter 9: Schwarz Reflection
§ 1: Schwarz Reflection (by Complex Conjugation)
§ 2: Reflection Across Analytic Arcs
§ 3: Application of Schwarz Reflection
Chapter 10: The Riemann Mapping Theorem
§ 1: Statement of the Theorem
§ 2: Compact Sets in Function Spaces
§ 3: Proof of the Riemann Mapping Theorem
§ 4: Behavior at the Boundary
Chapter 11: Analytic Continuation Along Curves
§ 1: Continuation Along a Curve
§ 2: The Dilogarithm
§ 3: Application to Picard's Theorem

Part Three: Various Analytic Topics

Chapter 12: Applications of the Maximum Modulus Principle and Jensen's Formula
§ 1: Jensen's Formula
§ 2: The Picard-Borel Theorem
§ 3: Bounds by the Real Part, Borel-Caratheodory Theorem
§ 4: The Use of Three Circles and the Effect of Small Derivatives
§ 5: Entire Functions with Rational Values
§ 6: The Phragmen-Lindelof and Hadamard Theorems
Chapter 13: Entire and Meromorphic Functions
§ 1: Infinite Products
§ 2: Weierstrass Products
§ 3: Functions of Finite Order
§ 4: Meromorphic Functions, Mittag-Leffler Theorem
Chapter 14: Elliptic Functions
§ 1: The Liouville Theorems
§ 2: The Weierstrass Function
§ 4: The Sigma and Zeta Functions
Chapter 15: The Gamma and Zeta Functions
§ 1: The Differentiation Lemma
§ 2: The Gamma Function
§ 3: The Lerch Formula
§ 4: Zeta Functions
Chapter 16: The Prime Number Theorem
§ 1: Basic Analytic Properties of the Zeta Function
§ 2: The Main Lemma and its Application
§ 3: Proof of the Main Lemma
Appendix
§ 1: Summation by Parts and Non-Absolute Convergence
§ 2: Difference Equations
§ 3: Analytic Differential Equations
§ 4: Fixed Points of a Fractional Linear Transformation
§ 5: Cauchy's Formula for $C^\infty$ Functions
§ 6: Cauchy's Theorem for Locally Integrable Vector Fields
§ 7: More on Cauchy-Riemann