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年于芬兰著名的土尔库大学获得博士学位。期间他还师从著名数学家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 lnvariants等。
目錄：
PrefaceCHAPTER 1 COMPLEX NUMBERS11 The Algebra of Complex Numbers11.1 Arithmetic Operations11.2 Square Roots31.3 Justification41.4 Conjugation， Absolute Value61.5 Inequalities92 The Geometric Representation of Complex Numbers122.1 Geometric Addition and Multiplication122.2 The Binomial Equation152.3 Analytic Geometry172.4 The Spherical Representation18CHAPTER 2 COMPLEX FUNCTIONS211 Introduction to the Concept of Aaalytic Function211.1 Limits and Continuity221.2 Aaalytic Functions241.3 Polynomials281.4 Rational Functions302 Elementary Theory of Power Serices 332.1 Sequences332.2 Serues352.3 Uniform Convergence352.4 Power Series382.5 Abel‘s Limit Theorem413 The Exponential and Trigonometric Functions423.1 The Exponential423.2 The Trigonometric Functions433.3 The Periodicity443.4 The Logarithm46CHAPTER 3 ANALYTIC FUNCTIONS AS MAPPINGS491 Elementary Point Set Topology501.1 Sets and Elements501.2 Metric Spaces511.3 Connectedness541.4 Connectedness591.5 Continuous Functions631.6 Topoliogical Spaces 662 Conformality2.1 Arcs and Closed Curves672.2 Analytic Function in Regions692.3 Conformal Mapping732.4 Length and Area753 Linear Transformations763.1 The Linear Group763.2 The Cross Ratio783.3 Symmetry803.4 Oriented Circles833.5 Families of Circles844 Elementary Conformal Mappings894.1 The Use of Level Curves894.2 A Survey of Elementary Mappings934.3 Elementary Riemann Surfaces 97CHAPTER 4 COMPLEX INTEGRATION1011 Fundamental Theorems1011.1 Line Integrals1011.2 Rectifiable Arcs1041.3 Line Integrals as Functions of Ares1051.4 Cauchy’s Theorem for a Recatangle1091.5 Cauchy‘s Theorem in a Disk1122 Cauchy’s Integral Formula1142.1 The Index of a Point with Respect to a Closed Curve1142.2 The Integral Formula1182.3 Higher Dervatives1203 Local Properties of Aaalytic Functions1243.1 Removable Singularites. Taylor‘s Theorem1243.2 Zeros and Poles1263.3 The Local Mapping1303.4 The Mazimum Principle1334 The General Form of Cauchy’s Theorem1374.1 Chains and Cycles 1374.2 Siple Connectivity1384.3 Homology1414.4 The General Statement of Cauchy‘s Theorem1414.5 Proof of Cauchy’s Theorem1424.6 Locally Exact Differentials1444.7 Multiply Connected Regions1465 The Calculus of Residues1485.1 The Residue Theorem1485.2 The Argument Principle1525.3 Evaluation of Definite Integrals1546 Harmonic Functions1626.1 Definition and Basic Properties1626.2 The Mean-value Property1656.3 Poisson‘s Formula1686.4 Schwarz’s Theorem 1686.5 The Reflection Principle172CHAPTER 5 SERIES AND PRODUCT DEVELOPMENTS1751 Power Serices Expansions1751.1 Weierstrass‘s Theorem1751.2 The Taylor Series1791.3 The Laurent Series1842 Partial Fractions and Factorzation1872.1 Partial Fractions1872.2 Infinite Products1912.3 Canonical Products 1932.4 The Gamma Function1982.5 Stirling’s Formula2013 Entire Functions2063.1 Jensen‘s Formula2073.2 Hadamard’s Theorem2084 The Riemann Zeta Function2124.1 The Product Development2134.2 Extension of (s）to the Whole Plane2144.3 The Functioal Equation2164.4 The Zeros of the Zeta Functaion2185 Normal Families 2195.1 Equicontinuity2195.2 Normality and Compactness2205.3 Arzela‘s Theorem2225.4 Families of Analytic Functions2235.5 The Claaical Definition225CHAPTER 6 CONFORMAL MAPPUNG. DIRICHLET’S PROBLEM2291 The Riemann Mapping Throrem2291.1 Statement and Proof2291.2 Boundary Behavior2321.3 Use of the Reflection Principle2331.4 Analytic Arcs2342 Conformal Mapping of Polygons2352.1 The Behavior at an Angle 2352.2 The Schwarz-Christoffel Formula2362.3 Mapping on a Rectangle2382.4 The Triangle Functions of Schwarz2413 A Closer Look at Harmonic Functions2413.1 Functions with the Mean-value Property2423.2 Harnack‘s Principle 2434 The Dirichlet Problem2454.1 Subharmonic Functions2454.2 Solution of Dirchlet’s Problem2485 Canonical Mappings of Multiply Connected Regions2515.1 Harmonic Measures2525.2 Greens
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