本书是作者多年从事复变函数论双语教学经验的总结.其内容设置完全适合我国现行高等院校(特别是师范院校)本科教学的教学目标与课时需要.本书内容深入浅出、层次分明,理论体系严谨、逻辑推导详尽,强调“分析式”教学法,在引入概念前,加入了必要的分析与归纳总结,然后提出相应的概念;在提出问题之后,进行推理分析、增加条件,最后得到问题的答案,并把前边的讨论总结成一个定理.其次,本书配有大量图形,帮助读者直观理解相应的概念与论证思路.
更多科学出版社服务，请扫码获取。
Contents
Preface
Chapter 1 Complex Number Field 1
1.1 Addition and Multiplication 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 Cauchy Integral Theorem and its Extensions 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
当前图书最新荐购
