Book:Serge Lang/Complex Analysis

Subject Matter

 * Complex Analysis

Contents
Foreword

Prerequisites

Part One: Basic Theory


 * Chapter 1: Complex Numbers and Functions


 * &sect; 1: Definition
 * &sect; 2: Polar Form
 * &sect; 3: Complex Valued Functions
 * &sect; 4: Limits and Compact Sets
 * &sect; 5: Complex Differentiability
 * &sect; 6: The Cauchy-Riemann Equations
 * &sect; 7: Angles Under Holomorphic Maps


 * Chapter 2: Power Series


 * &sect; 1: Formal Power Series
 * &sect; 2: Convergent Power Series
 * &sect; 3: Relations Between Formal and Convergent Series
 * &sect; 4: Analytic Functions
 * &sect; 5: Differentiation of Power Series
 * &sect; 6: The Inverse and Open Mapping Theorems
 * &sect; 7: The Local Maximum Modulus Principle


 * Chapter 3: Cauchy's Theorem, First Part


 * &sect; 1: Holomorphic Functions on Connected Sets
 * &sect; 2: Integrals Over Paths
 * &sect; 3: Local Primitive for a Holomorphic Function
 * &sect; 4: Another Description of the Integral Along a Path
 * &sect; 5: The Homotopy Form of Cauchy's Theorem
 * &sect; 6: Existence of Global Primitives. Definition of the Logarithm
 * &sect; 7: The Local Cauchy Formula


 * Chapter 4: Winding Numbers and Cauchy's Theorem


 * &sect; 1: The Winding Number
 * &sect; 2: The Global Cauchy Theorem
 * &sect; 3: Artin's Proof


 * Chapter 5: Applications of Cauchy's Integral Formula


 * &sect; 1: Uniform Limits of Analytic Functions
 * &sect; 2: Laurent Series
 * &sect; 3: Isolated Singularities


 * Chapter 6: Calculus of Residues


 * &sect; 1: The Residue Formula
 * &sect; 2: Evaluation of Definite Integrals


 * Chapter 7: Conformal Mappings


 * &sect; 1: Schwarz Lemma
 * &sect; 2: Analytic Automorphisms of the Disc
 * &sect; 3: The Upper Half Plane
 * &sect; 4: Other Examples
 * &sect; 5: Fractional Linear Transformations


 * Chapter 8: Harmonic Functions


 * &sect; 1: Definition
 * &sect; 2: Examples
 * &sect; 3: Basic Properties of Harmonic Functions
 * &sect; 4: The Poisson Formula
 * &sect; 5: Construction of Harmonic Functions
 * &sect; 6: Appendix. Differentiating Under the Integral Sign

Part Two: Geometric Function Theory


 * Chapter 9: Schwarz Reflection


 * &sect; 1: Schwarz Reflection (by Complex Conjugation)
 * &sect; 2: Reflection Across Analytic Arcs
 * &sect; 3: Application of Schwarz Reflection


 * Chapter 10: The Riemann Mapping Theorem


 * &sect; 1: Statement of the Theorem
 * &sect; 2: Compact Sets in Function Spaces
 * &sect; 3: Proof of the Riemann Mapping Theorem
 * &sect; 4: Behavior at the Boundary


 * Chapter 11: Analytic Continuation Along Curves


 * &sect; 1: Continuation Along a Curve
 * &sect; 2: The Dilogarithm
 * &sect; 3: Application to Picard's Theorem

Part Three: Various Analytic Topics


 * Chapter 12: Applications of the Maximum Modulus Principle and Jensen's Formula


 * &sect; 1: Jensen's Formula
 * &sect; 2: The Picard-Borel Theorem
 * &sect; 3: Bounds by the Real Part, Borel-Caratheodory Theorem
 * &sect; 4: The Use of Three Circles and the Effect of Small Derivatives
 * &sect; 5: Entire Functions with Rational Values
 * &sect; 6: The Phragmen-Lindelof and Hadamard Theorems


 * Chapter 13: Entire and Meromorphic Functions


 * &sect; 1: Infinite Products
 * &sect; 2: Weierstrass Products
 * &sect; 3: Functions of Finite Order
 * &sect; 4: Meromorphic Functions, Mittag-Leffler Theorem


 * Chapter 14: Elliptic Functions


 * &sect; 1: The Liouville Theorems
 * &sect; 2: The Weierstrass Function
 * &sect; 3: The Addition Theorem
 * &sect; 4: The Sigma and Zeta Functions


 * Chapter 15: The Gamma and Zeta Functions


 * &sect; 1: The Differentiation Lemma
 * &sect; 2: The Gamma Function
 * &sect; 3: The Lerch Formula
 * &sect; 4: Zeta Functions


 * Chapter 16: The Prime Number Theorem


 * &sect; 1: Basic Analytic Properties of the Zeta Function
 * &sect; 2: The Main Lemma and its Application
 * &sect; 3: Proof of the Main Lemma


 * Appendix


 * &sect; 1: Summation by Parts and Non-Absolute Convergence
 * &sect; 2: Difference Equations
 * &sect; 3: Analytic Differential Equations
 * &sect; 4: Fixed Points of a Fractional Linear Transformation
 * &sect; 5: Cauchy's Formula for $C^\infty$ Functions
 * &sect; 6: Cauchy's Theorem for Locally Integrable Vector Fields
 * &sect; 7: More on Cauchy-Riemann