Contents
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
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 Differentiation
1. De nitions of complex analytic function
2. Complex di erentiation
3. The Cauchy-Riemann equations
4. Orthogonal trajectories and harmonic functions
5. A glimpse at harmonic functions
6. What is a di erential 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