Book:Serge Lang/Complex Analysis
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Serge Lang: Complex Analysis
Published $\text {1977}$, 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