本書是作者多年從事復(fù)變函數(shù)論雙語教學(xué)經(jīng)驗的總結(jié).其內(nèi)容設(shè)置完全適合我國現(xiàn)行高等院校(特別是師范院校)本科教學(xué)的教學(xué)目標(biāo)與課時需要.本書內(nèi)容深入淺出、層次分明,理論體系嚴(yán)謹(jǐn)、邏輯推導(dǎo)詳盡,強調(diào)“分析式”教學(xué)法,在引入概念前,加入了必要的分析與歸納總結(jié),然后提出相應(yīng)的概念;在提出問題之后,進行推理分析、增加條件,最后得到問題的答案,并把前邊的討論總結(jié)成一個定理.其次,本書配有大量圖形,幫助讀者直觀理解相應(yīng)的概念與論證思路.
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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
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