复分析(英文版·原书第3版·典藏版)

复分析研究复自变量复值函数，是数学的重要分支之一，同时在数学的其他分支（如微分方程、积分方程、概率论、数论等）以及自然科学的其他领域（如空气动力学、流体力学、电学、热学、理论物理等）都有着重要的应用。 虽然本书的诞生是20世纪50年代的事情，但是，深贯其中的严谨的学术风范以及针对不同时代所做出的切实改进使得它历久弥新，成为复分析领域历经考验的一本经典教材。本书作者在数学分析领域声名卓著，多次荣获国际大奖，这也是本书始终保持旺盛生命力的原因之一。

复分析研究复自变量复值函数，是数学的重要分支之一，同时在数学的其他分支（如微分方程、积分方程、概率论、数论等）以及自然科学的其他领域（如空气动力学、流体力学、电学、热学、理论物理等）都有着重要的应用。虽然本书的诞生是20世纪50年代的事情，但是，深贯其中的严谨的学术风范以及针对不同时代所做出的切实改进使得它历久弥新，成为复分析领域历经考验的一本经典教材。本书作者在数学分析领域声名卓著，多次荣获国际大奖，这也是本书始终保持旺盛生命力的原因之一。

拉尔斯·V.阿尔福斯（Lars V.Ahlfors）生前是哈佛大学数学教授。他于1924年进入赫尔辛基大学学习，并在1930年于芬兰有名的土尔库大学获得博士学位。期间他还师从有名数学家Nevanlinna共同进行研究工作。1936年荣获菲尔茨奖。第二次世界大战结束后，他辗转到哈佛大学从事教学工作。1953年当选为美国国家科学院院士。他又于1968年和1981年分别荣获Vihuri奖和沃尔夫奖。他的著述很多，除本书外，还著有Riemann Surfaces和Conformal Invariants等。

Preface
CHAPTER 1 COMPLEX NUMBERS
1 The Algebra of Complex Numbers
1.1 Arithmetic Operations
1.2 Square Roots
1.3 Justification
1.4 Conjugation, Absolute Value
1.5 Inequalities
2 The Geometric Representation of Complex Numbers
2.1 Geometric Addition and Multiplication
2.2 The Binomial Equation
2.3 Analytic Geometry
2.4 The Spherical Representation
CHAPTER 2 COMPLEX FUNCTIONS
1 Introduction to the Concept of Aaalytic Function
1.1 Limits and Continuity
1.2 Aaalytic Functions
1.3 Polynomials
1.4 Rational Functions
2 Elementary Theory of Power Serices
2.1 Sequences
2.2 Serues
2.3 Uniform Convergence
2.4 Power Series
2.5 Abel's Limit Theorem
3 The Exponential and Trigonometric Functions
3.1 The Exponential
3.2 The Trigonometric Functions
3.3 The Periodicity
3.4 The Logarithm
CHAPTER 3 ANALYTIC FUNCTIONS AS MAPPINGS
1 Elementary Point Set Topology
1.1 Sets and Elements
1.2 Metric Spaces
1.3 Connectedness
1.4 Connectedness
1.5 Continuous Functions
1.6 Topoliogical Spaces
2 Conformality
2.1 Arcs and Closed Curves
2.2 Analytic Function in Regions
2.3 Conformal Mapping
2.4 Length and Area
3 Linear Transformations
3.1 The Linear Group
3.2 The Cross Ratio
3.3 Symmetry
3.4 Oriented Circles
3.5 Families of Circles
4 Elementary Conformal Mappings
4.1 The Use of Level Curves
4.2 A Survey of Elementary Mappings
4.3 Elementary Riemann Surfaces
CHAPTER 4 COMPLEX INTEGRATION
1 Fundamental Theorems
1.1 Line Integrals
1.2 Rectifiable Arcs
1.3 Line Integrals as Functions of Ares
1.4 Cauchy's Theorem for a Recatangle
1.5 Cauchy's Theorem in a Disk
2 Cauchy's Integral Formula
2.1 The Index of a Point with Respect to a Closed Curve
2.2 The Integral Formula
2.3 Higher Dervatives
3 Local Properties of Aaalytic Functions
3.1 Removable Singularites. Taylor's Theorem
3.2 Zeros and Poles
3.3 The Local Mapping
3.4 The Mazimum Principle
4 The General Form of Cauchy's Theorem
4.1 Chains and Cycles
4.2 Siple Connectivity
4.3 Homology
4.4 The General Statement of Cauchy's Theorem
4.5 Proof of Cauchy's Theorem
4.6 Locally Exact Differentials
4.7 Multiply Connected Regions
5 The Calculus of Residues
5.1 The Residue Theorem
5.2 The Argument Principle
5.3 Evaluation of Definite Integrals
6 Harmonic Functions
6.1 Definition and Basic Properties
6.2 The Mean-value Property
6.3 Poisson's Formula
6.4 Schwarz's Theorem
6.5 The Reflection Principle
CHAPTER 5 SERIES AND PRODUCT DEVELOPMENTS
1 Power Serices Expansions
1.1 Weierstrass's Theorem
1.2 The Taylor Series
1.3 The Laurent Series
2 Partial Fractions and Factorzation
2.1 Partial Fractions
2.2 Infinite Products
2.3 Canonical Products
2.4 The Gamma Function
2.5 Stirling's Formula
3 Entire Functions
3.1 Jensen's Formula
3.2 Hadamard's Theorem
4 The Riemann Zeta Function
4.1 The Product Development
4.2 Extension of (s）to the Whole Plane
4.3 The Functioal Equation
4.4 The Zeros of the Zeta Functaion
5 Normal Families
5.1 Equicontinuity
5.2 Normality and Compactness
5.3 Arzela's Theorem
5.4 Families of Analytic Functions
5.5 The Claaical Definition
CHAPTER 6 CONFORMAL MAPPUNG. DIRICHLET'S PROBLEM
1 The Riemann Mapping Throrem
1.1 Statement and Proof
1.2 Boundary Behavior
1.3 Use of the Reflection Principle
1.4 Analytic Arcs
2 Conformal Mapping of Polygons
2.1 The Behavior at an Angle
2.2 The Schwarz-Christoffel Formula
2.3 Mapping on a Rectangle
2.4 The Triangle Functions of Schwarz
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