单复变函数论
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作者Robert E. Greene 等
出版社高等教育出版社
ISBN9787040469073
出版时间2017-02
版次1
装帧平装
开本16开
纸张胶版纸
页数504页
定价199元
上书时间2024-03-26
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基本信息
书名:单复变函数论
定价：199元
作者:Robert E. Greene 等
出版社：高等教育出版社
出版日期：2017-02-01
ISBN：9787040469073
字数：
页码：504
版次：1
装帧：平装
开本：16开
商品重量：
编辑推荐
《单复变函数论(第三版)(英文版)》是一本很吸引人且现代的复分析导引，可用作研究生一年级的复分析教材，它反映了作者们作为数学家和写作者的专业素质。
内容提要
复分析是数学核心的学科之一，不但自身引人入胜、丰富多彩，而且在多种其他数学学科（纯数学和应用数学）中都非常有用。《单复变函数论(第三版)(英文版)》的与众不同之处在于它从多变量实微积分中直接发展出复变量。每一个新概念引进时，它总对应了实分析和微积分中相应的概念，《单复变函数论(第三版)(英文版)》配有丰富的例题和习题来印证此点。作者有条不紊地将分析从拓扑中分离出来，从柯西定理的证明中可见一斑。《单复变函数论(第三版)(英文版)》分几章讨论专题，如对特殊函数的完整处理、素数定理和Bergman核。作者还处理了Hp空间，以及共形映射边界光滑性的Painleve定理。
目录
Preface to the Third EditionPreface to the Second EditionPreface to the First EditionAcknowledgmentsChapter 1.Fundamental Concepts1.1.Elementary Properties of the Complex Numbers1.2.Further Properties of the Complex Numbers1.3.Complex Polynomials1.4.Holomorphic Functions， the Cauchy—Riemann Equations，and Harmonic Functions1.5.Real and Holomorphic AntiderivativesExercisesChapter 2.Complex Line Integrals2.1.Real and Complex Line Integrals2.2.Complex Differentiability and Conformality2.3.Antiderivatives Revisited2.4.The Cauchy Integral Formula and the Cauchy Integral Theorem2.5.The Cauchy Integral Formula： Some Examples2.6.An Introduction to the Cauchy Integral Theorem and the Cauchy Integral Formula for More General CurvesExercisesChapter 3.Applications of the Cauchy Integral3.1.Differentiability Properties of Holomorphic Functions3.2.Complex Power Series3.3.The Power Series Expansion for a Holomorphic Function3.4.The Cauchy Estimates and Liouville's Theorem3.5.Uniform Limits of Holomorphic Functions3.6.The Zeros of a Holomorphic FunctionExercisesChapter 4.Meromorphic Functions and Residues4.1.The Behavior of a Holomorphic Function Near an Isolated Singularity4.2.Expansion around Singular Points4.3.Estence of Laurent Expansions4.4.Examples of Laurent Expansions4.5.The Calculus of Residues4.6.Applications of the Calculus of Residues to the Calculation of Definite Integrals and Sums4.7.Meromorphic Functions and Singularities at InfinityExercisesChapter 5.The Zeros of a Holomorphic Function5.1.Counting Zeros and Poles5.2.The Local Geometry of Holomorphic Functions5.3.Further Results on the Zeros of Holomorphic Functions5.4.The Mamum Modulus Principle5.5.The Schwarz LemmaExercisesChapter 6.Holomorphic Functions as Geometric Mappings6.1.Biholomorphic Mappings of the Complex Plao Itself6.2.Biholomorphic Mappings of the Unit Disc to Itself6.3.Linear Fractional Transformations6.4.The Riemann Mapping Theorem： Statement and Idea of Proof6.5.Normal Families6.6.Holomorphically Simply Connected Domains6.7.The Proof of the Analytic Form of the Riemann Mapping TheoremExercisesChapter 7.Harmonic Functions7.1.Basic Properties of Harmonic Functions7.2.The Mamum Principle and the Mean Value Property7.3.The Poisson Integral Formula7.4.Regularity of Harmonic Functions7.5.The Schwarz Reflection Principle7.6.Harnack's Principle7.7.The Dirichlet Problem and Subharmonic Functions7.8.The Perron Method and the Solution of the Dirichlet Problem7.9.Conformal Mappings of AnnuliExercisesChapter 8.Infinite Series and Products8.1.Basic Concepts Concerning Infinite Sums and Products8.2.The Weierstrass Factorization Theorem8.3.The Theorems of Weierstrass and Mittag—Leffler：Interpolation ProblemsExercisesChapter 9.Applications of Infinite Sums and Products9.1.Jensen's Formula and an Introduction to Blaschke Products9.2.The Hadamard Gap Theorem9.3.Entire Functions of Finite OrderExercisesChapter 10.Analyt.ic Continuation10.1.Definition of an Analytic Function Element10.2.Analytic Continuation along a Curve10.3.The Monodromy Theorem10.4.The Idea of a Riemann Surface10.5.The Elliptic Modular Function and Picard's Theorem10.6.EllipticFunctionsExercisesChapterll.Topology11.1.Multiply Connected Domains11.2.The Cauchy Integral Formula for Multiply Connected Domains11.3.Holomorphic Simple Connectivity and Topological Simple Connectivity11.4.Simple Connectivity and Connectedness of the Complement11.5.Multiply Connected Domains RevisitedExercisesChapter 12.Rational Appromation Theory12.1.Runge's Theorem12.2.Mergelyan's Theorem12.3.Some Remarks about Analytic CapacityExercisesChapter 13.Special Classes of Holomorphic Functions13.1.Schlicht Functions and the Bieberbach Conjecture13.2.Continuity to the Boundary of Conformal Mappings13.3.Hardy Spaces13.4.Boundary Behavior of Functions in Hardy Classes （An Optional Section for Those Who Know Elementary Measure Theory）ExercisesChapter 14.Hilbert Spaces of Holomorphic Functions， the Bergman Kernel， and Biholomorphic Mappings14.1.The Geometry of Hilbert Space14.2.Orthonormal Systems in Hilbert Space，14.3.The Bergman Kernel14.4.Bell's Condition R14.5.Smoothness to the Boundary of Conformal MappingsExercisesChapter 15.Special Functions15.1.The Gamma and Beta Functions15.2.The Riemann Zeta FunctionExercisesChapter 16.The Prime Number Theorem16.0.Introduction16.1.Complex Analysis and the Prime Number Theorem16.2.Precise Connections to Complex Analysis16.3.Proof of the Integral TheoremExercisesAPPENDIX A： Real AnalysisAPPENDIX B： The Statement and Proof of Goursat's TheoremReferencesIndex
作者介绍
作者：（美国）罗伯特·格林（Robert E.Greene） （美国）斯蒂芬·克兰茨（Steven G.Krantz）
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