John P. D''Angelo, University of Illinois, Urbana, IL

Preface
Chapter 1. From the Real Numbers to the Complex Numbers
1. Introduction
2. Number systems
3. Inequalities and ordered elds
4. The complex numbers
5. Alternative denitions of C
6. A glimpse at metric spaces

Chapter 2. Complex Numbers
1. Complex conjugation
2. Existence of square roots
3. Limits
4. Convergent innite series
5. Uniform convergence and consequences
6. The unit circle and trigonometry
7. The geometry of addition and multiplication
8. Logarithms

Chapter 3. Complex Numbers and Geometry
1. Lines, circles, and balls
2. Analytic geometry
3. Quadratic polynomials
4. Linear fractional transformations
5. The Riemann sphere

Chapter 4. Power Series Expansions
1. Geometric series
2. The radius of convergence
3. Generating functions
4. Fibonacci numbers
5. An application of power series
6. Rationality

Chapter 5. Complex Dierentiation
1. Denitions of complex analytic function
2. Complex dierentiation
3. The Cauchy-Riemann equations
4. Orthogonal trajectories and harmonic functions
5. A glimpse at harmonic functions
6. What is a dierential form?

Chapter 6. Complex Integration
1. Complex-valued functions
2. Line integrals
3. Goursat's proof
4. The Cauchy integral formula
5. A return to the denition of complex analytic function

Chapter 7. Applications of Complex Integration
1. Singularities and residues
2. Evaluating real integrals using complex variables methods
3. Fourier transforms
4. The Gamma function

Chapter 8. Additional Topics
1. The minimum-maximum theorem
2. The fundamental theorem of algebra
3. Winding numbers, zeroes, and poles
4. Pythagorean triples
5. Elementary mappings
6. Quaternions
7. Higher-dimensional complex analysis

Further reading
Bibliography
Index