Complex Analysis

Cover
Title Page
Copyright Page
Dedication
Preface
Table of Contents
Introduction
FIRST PART
Chapter I The Complex Plane and Elementary Functions
1. Complex Numbers
2. Polar Representation
3. Stereographic Projection
4. The Square and Square Root Functions
5. The Exponential Function
6. The Logarithm Function
7. Power Functions and Phase Factors
8. Trigonometric and Hyperbolic Functions
Chapter II Analytic Functions
1. Review of Basic Analysis
2. Analytic Functions
3. The Cauchy-Riemann Equations
4. Inverse Mappings and the Jacobian
5. Harmonic Functions
6. Conformal Mappings
7. Fractional Linear Transformations
Chapter III Line Integrals and Harmonic Functions
1. Line Integrals and Green's Theorem
2. Independence of Path
3. Harmonic Conjugates
4. The Mean Value Property
5. The Maximum Principle
6. Applications to Fluid Dynamics
7. Other Applications to Physics
Chapter IV Complex Integration and Analyticity
1. Complex Line Integrals
2. Fundamental Theorem of Calculus for Analytic Functions
3. Cauchy's Theorem
4. The Cauchy Integral Formula
5. Liouville's Theorem
6. Morera's Theorem
7. Goursat's Theorem
8. Complex Notation and Pompeiu's Formula
Chapter V Power Series
1. Infinite Series
2. Sequences and Series of Functions
3. Power Series
4. Power Series Expansion of an Analytic Function
5. Power Series Expansion at Infinity
6. Manipulation of Power Series
7. The Zeros of an Analytic Function
8. Analytic Continuation
Chapter VI Laurent Series and Isolated Singularities
1. The Laurent Decomposition
2. Isolated Singularities of an Analytic Function
3. Isolated Singularity at Infinity
4. Partial Fractions Decomposition
5. Periodic Functions
6. Fourier Series
Chapter VII The Residue Calculus
1. The Residue Theorem
2. Integrals Featuring Rational Functions
3. Integrals of Trigonometric Functions
4. Integrands with Branch Points
5. Fractional Residues
6. Principal Values
7. Jordan's Lemma
8. Exterior Domains
SECOND PART
Chapter VIII The Logarithmic Integral
1. The Argument Principle
2. Rouche's Theorem
3. Hurwitz's Theorem
4. Open Mapping and Inverse Function Theorems
5. Critical Points
6. Winding Numbers
7. The Jump Theorem for Cauchy Integrals
8. Simply Connected Domains
Chapter IX The Schwarz Lemma and Hyperbolic Geometry
1. The Schwarz Lemma
2. Conformal Self-Maps of the Unit Disk
3. Hyperbolic Geometry
Chapter X Harmonic Functions and the Reflection Principle
1. The Poisson Integral Formula
2. Characterization of Harmonic Functions
3. The Schwarz Reflection Principle
Chapter XI Conformal Mapping
1. Mappings to the Unit Disk and Upper Half-Plane
2. The Riemann Mapping Theorem
3. The Schwarz-Christoffel Formula
4. Return to Fluid Dynamics
5. Compactness of Families of Functions
6. Proof of the Riemann Mapping Theorem
THIRD PART
Chapter XII Compact Families of Meromorphic Functions
1. Marty's Theorem
2. Theorems of Montel and Picard
3. Julia Sets
4. Connectedness of Julia Sets
5. The Mandelbrot Set
Chapter XIII Approximation Theorems
1. Runge's Theorem
2. The Mittag-Leffler Theorem
3. Infinite Products
4. The Weierstrass Product Theorem
Chapter XIV Some Special Functions
1. The Gamma Function
2. Laplace Transforms
3. The Zeta Function
4. Dirichlet Series
5. The Prime Number Theorem
Chapter XV The Dirichlet Problem
1. Green's Formulae
2. Subharmonic Functions
3. Compactness of Families of Harmonic Functions
4. The Perron Method
5. The Riemann Mapping Theorem Revisited
6. Green's Function for Domains with Analytic Boundary
7. Green's Function for General Domains
Chapter XVI Riemann Surfaces
1. Abstract Riemann Surfaces
2. Harmonic Functions on a Riemann Surface
3. Green's Function of a Surface
4. Symmetry of Green's Function
5. Bipolar Green's Function
6. The Uniformization Theorem
7. Covering Surfaces
Hints and Solutions for Selected Exercises
References
List of Symbols
Index