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