Introduction to advanced complex calculus

Title page
Preface
Chapter 1. Numbers and Convergence
1.1 Introduction
1.2 Complex numbers
1.3 The imaginary unit
1.4 Properties of complex numbers
1.5 Sequences of complex numbers
Exercises
Chapter 2. Topological Preliminaries
2.1 Introduction
2.2 Point sets
2.3 Continuous curves
2.4 Functions of bounded variation
2.5 Jordan's theorem
Exercises
Chapter 3. Functions of a Complex Variable
3.1 Introduction
3.2 Functions
3.3 Continuity
3.4 The elementary functions
3.5 Extension to the complex plane
3.6 Multi-valued real functions
3.7 Riemann surfaces
3.8 Differentiability
3.9 The Cauchy-Riemann conditions
3.10 Analyticity
3.11 Mappings
3.12 Laplace's equation
Exercises
Chapter 4. Contour Integrals
4.1 Introduction
4.2 Contour integrals
4.3 Integrability of a continuous function
4.4 The Cauchy integral theorem
4.5 The Cauchy integral formula
Exercises
Chapter 5. Sequences and Series
5.1 Introduction
5.2 Series of complex numbers
5.3 Sequences and series of functions
5.4 Sequences of analytic functions
5.5 Power series
5.6 Taylor's expansion
5.7 The root test
5.8 Laurent series
5.9 Classification of singularities
Exercises
Chapter 6. The Calculus of Residues
6.1 Introduction
6.2 Theorems on residues
6.3 Integrals involving rational functions
6.4 Further applications
6.5 Integration around branch points
Exercises
Chapter 7. Some Properties of Analytic Functions
7.1 Introduction
7.2 A residue theorem
7.3 The fundamental theorem of algebra
7.4 Rational functions
7.5 The maximum modulus theorem
7.6 Analytic continuation
Exercises
Chapter 8. Conformai Mapping
8.1 Introduction
8.2 The fundamental property
8.3 Some simple examples
8.4 The inverse transformation
8.5 The bilinear transformation
8.6 Elliptic integral
8.7 The Schwarz-Christoffel transformation
Exercises
Chapter 9. The Method of Laplace Integrais
9.1 Introduction
9.2 The adjoint operator
9.3 Laplace integrals
9.4 Linear coefficients
9.5 Examples of contour integrals as solutions
9.6 Repeated linear factors
9.7 Constant coefficients
Exercises
References
Index