# Book:Serge Lang/Complex Analysis

## Serge Lang: *Complex Analysis*

Published $1977$, **Springer**

- ISBN 0-387-98592-1.

### Subject Matter

### 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
- § 3: The Addition Theorem
- § 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