Preface ix

1 The Complex Numbers 1

1.1 Why? 1

1.2 The Algebra of Complex Numbers 3

1.3 The Geometry of the Complex Plane 7

1.4 The Topology of the Complex Plane 9

1.5 The Extended Complex Plane 16

1.6 Complex Sequences 18

1.7 Complex Series 24

2 Complex Functions and Mappings 29

2.1 Continuous Functions 29

2.2 Uniform Convergence 34

2.3 Power Series 38

2.4 Elementary Functions and Euler’s Formula 43

2.5 Continuous Functions as Mappings 50

2.6 Linear Fractional Transformations 53

2.7 Derivatives 64

2.8 The Calculus of Real Variable Functions 70

2.9 Contour Integrals 75

3 Analytic Functions 87

3.1 The Principle of Analyticity 87

3.2 Differentiable Functions are Analytic 89

3.3 Consequences of Goursat’s Theorem 100

3.4 The Zeros of Analytic Functions 104

3.5 The Open Mapping Theorem and Maximum Principle 108

3.6 The Cauchy–Riemann Equations 113

3.7 Conformal Mapping and Local Univalence 117

4 Cauchy’s Integral Theory 127

4.1 The Index of a Closed Contour 127

4.2 The Cauchy Integral Formula 133

4.3 Cauchy’s Theorem 139

5 The Residue Theorem 145

5.1 Laurent Series 145

5.2 Classification of Singularities 152

5.3 Residues 158

5.4 Evaluation of Real Integrals 165

5.5 The Laplace Transform 174

6 Harmonic Functions and Fourier Series 183

6.1 Harmonic Functions 183

6.2 The Poisson Integral Formula 191

6.3 Further Connections to Analytic Functions 201

6.4 Fourier Series 210

Epilogue 227

A Sets and Functions 239

B Topics from Advanced Calculus 247

References 255

Index 257