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Chapter 1 Complex Number Field 1

1.1 Addition and Multiplicatio 1

1.2 Basic Algebraic Properties 3

1.3 Further Properties 6

1.4 Moduli of Complex Numbers 8

1.5 Conjugates of Complex Numbers 12

1.6 Arguments of Complex Numbers 15

1.7 Arguments of Products and Quotients 18

1.8 Roots of Complex Numbers 22

1.9 Examples of Roots 24

1.10 Domains and Regions in the Complex Plane 28

Chapter 2 Complex Variable Functions 33

2.1 Complex Variable Functions 33

2.2 Functions as Mappings 36

2.3 The Exponential Function and its Mapping Properties 40

2.4 Limits of Sequences and Functions 42

2.5 Properties of Limits 45

2.6 Limits Involving the Infinity 47

2.7 Continuous Functions 50

2.8 Differentiable Functions 52

2.9 Differentiation Formulas 54

2.10 A Characterization of Differentiability 57

2.11 Cauchy-Riemann Equations in Polar Coordinates 62

2.12 Analytic Functions 65

Chapter 3 Elementary Functions 69

3.1 The Exponential Function 69

3.2 Trigonometric Functions 71

3.3 The Logarithmic Function 73

3.4 Branches of Logarithms 76

3.5 Complex Power Functions 79

Chapter 4 Integral Theory of Complex Functions 82

4.1 Definite Integrals 82

4.2 Path Integrals 87

4.3 Computation and Estimation of Integrals 91

4.4 CauchyIntegralTheoremanditsExtensions 97

4.5 Proof of Cauchy Integral Theorem 105

4.6 Cauchy Integral Formula 110

4.7 Cauchy Integral Formula for Derivatives 113

4.8 Liouville’s Theorem and Maximum Modulus Principle 120

Chapter 5 Taylor Series and Laurent Series 126

5.1 Convergence of Series 126

5.2 Taylor Series 129

5.3 Laurent Series 136

5.4 Absolute and Uniform Convergence of Power Series 144

5.5 Properties of Sums of Power Series 148

5.6 Uniqueness of Series Representations 154

Chapter 6 Singular Points and Zeros of Analytic Functions 159

6.1 Singular Points 159

6.2 Behavior of a Function Near Isolated Singular Points 164

6.3 Residues of Functions 169

6.4 Zeros of Analytic Functions 178

6.5 Zeros and Poles 182

6.6 Argument Principle 186

6.7 Rouche’s Theorem 191

Chapter 7 Conformal Mappings 196

7.1 Concepts and Examples 196

7.2 Unilateral Functions 201

7.3 Local Inverses 204

7.4 Affine Transformations 208

7.5 The Reciprocal Transformation 210

7.6 Fractional Linear Transformations 215

7.7 Cross Ratios 217

7.8 Mappings of the Upper Half Plane 222

Bibliography 229

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