Introduction to advanced complex calculus

Description

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