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Chapter 1 Complex Numbers 1

1.1 Complex Numbers 1

Exercises for 1.1 5

Answers or Hints for Exercises 1.1 6

1.2 Moduli and Conjugates 6

Exercises for 1.2 9

Answers or Hints for Exercises 1.2 10

1.3 Exponential Form 10

Exercises for 1.3 13

Answers or Hints for Exercises 1.3 14

1.4 Powers and Roots 14

Exercises for 1.4 17

Answers or Hints for Exercises 1.4 19

1.5 Geometrically Application of Complex Numbers 20

Exercises for 1.5 23

1.6 Plane Topology 23

Exercises for 1.6 25

Answers or Hints for Exercises 1.6 26

1.7 Curves 27

Chapter 2 Analytic Functions 30

2.1 Complex-valued Functions of a Complex Variable 30

Exercises for 2.1 35

Answers or Hints for Exercises 2.1 37

2.2 Limits and Continuity 37

Exercises for 2.2 43

Answers or Hints for Exercises 2.2 44

2.3 The Extended Plane and Infinity 44

Exercises for 2.3 47

Answers or Hints for Exercises 2.3 47

2.4 Complex Differentiability 48

Exercises for 2.4 54

Answers or Hints for Exercises 2.4 56

2.5 Analytic Functions 57

Exercises for 2.5 60

Answers or Hints for Exercises 2.5 61

2.6 Laplace’ s Equation and Harmonic Conjugates 62

Exercises for 2.6 66

Answers or Hints for Exercises 2.6 67

Chapter 3 Elementary Functions 69

3.1 The Exponential Functions 69

Exercises for 3.1 71

Answers or Hints for Exercises 3.1 72

3.2 Linear Fractional Transformations 73

Exercises for 3.2 81

Answers or Hints for Exercises 3.2 82

3.3 Trigonometric Functions 83

Exercises for 3.3 85

Answers or Hints for Exercises 3.3 87

3.4 The Radical Functions 87

Exercises for 3.4 91

Answers or Hints for Exercises 3.4 92

3.5 The Logarithm Function 92

Exercises for 3.5 95

Answers or Hints for Exercises 3.5 97

3.6 Complex Exponents 97

Exercises for 3.6 101

Answers or Hints for Exercises 3.6 102

3.7 Inverse Trigonometric and Hyperbolic Functions 103

Exercises for 3.7 104

Answers or Hints for Exercises 3.7 105

Chapter 4 Complex Integrals 107

4.1 Contour Integrals and Its Simple Properties 107

Exercise for 4.1 114

Answers or Hints for Exercises 4.1 117

4.2 Antiderivatives 118

Exercises for 4.2 124

Answers or Hints for Exercises 4.2 126

4.3 Cauchy Theorem 126

Exercises for 4.3 134

Answers or Hints for Exercises 4.3 135

4.4 Cauchy Integral Formula 136

Exercises for 4.4 146

Answers or Hints for Exercises 4.4 148

4.5 Maximum Modulus Principle 149

Exercises for 4.5 153

Answers or Hints for Exercises 4.5 153

Chapter 5 Power Series 155

5.1 Complex Sequences， Series and Their Basic Properties 155

Exercises for 5.1 158

Answers or Hints for Exercises 5.1 160

5.2 Series of Complex Functions and Its Basic Properties 160

Exercises for 5.2 165

Answers or Hints for Exercises 5.2 166

5.3 Power Series 167

Exercises for 5.3 172

Answers or Hints for Exercises 5.3 173

5.4 Taylor Series for Analytic Functions 173

Exercises for 5.4 180

Answers or Hints for Exercises 5.4 183

5.5 Manipulation of Power Series 184

Exercises for 5.5 187

Answers or Hints for Exercises 5.5 188

5.6 The Zeros of Analytic Functions 189

Exercises for 5.6 194

Answers or Hints for Exercises 5.6 196

Chapter 6 Laurent Series and Isolated Singularities 197

6.1 Laurent Decomposition 197

Exercises for 6.1 202

Answers or Hints for Exercises 6.1 205

6.2 Isolated Singular Point and Its Types 209

Exercises for 6.2 221

Answers or Hints for Exercises 6.2 223

6.3 Isolated Singularity at Infinity 224

Exercises for 6.3 228

Answers or Hints for Exercises 6.3 228

6.4 Entire Functions and Meromorphic Functions 230

Exercises for 6.4 233

Answers or Hints for Exercises 6.4 234

Chapter 7 Residue 236

7.1 Residue and Cauchy Residue Theorem 236

Exercises for 7.1 244

Answers or Hints for Exercises 7.1 246

7.2 The Argument Principle，Rouche’s Theorem 247

Exercises for 7.2 254

Answers or Hints for Exercises 7.2 255

Chapter 8 Evaluation of Real Integrals 257

8.1 Integrals of Trigonometric Functions 257

Exercises for 8.1 260

Answers or Hints for Exercises 8.1 262

8.2 Rational Functions over the Real Line 263

Exercises for 8.2 268

Answers or Hints for Exercises 8.2 270

8.3 Rational and Trigonometric Functions over the Real Line， 272

Exercises for 8.3 275

Answers or Hints for Exercises 8.3 276

8.4 Principal Value Integrals，Indentation Round a Singularity 277

Exercises for 8.4 288

Answers or Hints for Exercises 8.4 288

8.5 Integrals with Branch Points 289

Exercises for 8.5 297

Answers or Hints for Exercises 8.5 297

参考文献 299