单复变函数论(第三版)(英文版)

　　Preface to the Third Edition
　　Preface to the Second Edition
　　Preface to the First Edition
　　Acknowledgments
　　Chapter 1． Fundamental Concepts
　　1．1． Elementary Properties of the Complex Numbers
　　1．2． Further Properties of the Complex Numbers
　　1．3． Complex Polynomials
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　　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。
　　《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。

　　Preface to the Third Edition
　　Preface to the Second Edition
　　Preface to the First Edition
　　Acknowledgments
　　Chapter 1． Fundamental Concepts
　　1．1． Elementary Properties of the Complex Numbers
　　1．2． Further Properties of the Complex Numbers
　　1．3． Complex Polynomials
　　1．4． Holomorphic Functions， the Cauchy-Riemann Equations， and Harmonic Functions
　　1．5． Real and Holomorphic Antiderivatives
　　Exercises
　　Chapter 2． Complex Line Integrals
　　2．1． Real and Complex Line Integrals
　　2．2． Complex Differentiability and Conformality
　　2．3． Antiderivatives Revisited
　　2．4． The Cauchy Integral Formula and the Cauchy Integral Theorem
　　2．5． The Cauchy Integral Formula： Some Examples
　　2．6． An Introduction to the Cauchy Integral Theorem and the Cauchy Integral Formula for More General Curves
　　Exercises
　　Chapter 3． Applications of the Cauchy Integral
　　3．1． Differentiability Properties of Holomorphic Functions
　　3．2． Complex Power Series
　　3．3． The Power Series Expansion for a Holomorphic Function
　　3．4． The Cauchy Estimates and Liouville's Theorem
　　3．5． Uniform Limits of Holomorphic Functions
　　3．6． The Zeros of a Holomorphic Function
　　Exercises
　　Chapter 4． Meromorphic Functions and Residues
　　4．1． The Behavior of a Holomorphic Function Near an Isolated Singularity
　　4．2． Expansion around Singular Points
　　4．3． Existence of Laurent Expansions
　　4．4． Examples of Laurent Expansions
　　4．5． The Calculus of Residues
　　4．6． Applications of the Calculus of Residues to the Calculation of Definite Integrals and Sums
　　4．7． Meromorphic Functions and Singularities at Infinity
　　Exercises
　　Chapter 5． The Zeros of a Holomorphic Function
　　5．1． Counting Zeros and Poles
　　5．2． The Local Geometry of Holomorphic Functions
　　5．3． Further Results on the Zeros of Holomorphic Functions
　　5．4． The Maximum Modulus Principle
　　5．5． The Schwarz Lemma
　　Exercises
　　Chapter 6． Holomorphic Functions as Geometric Mappings
　　6．1． Biholomorphic Mappings of the Complex Plane to Itself
　　6．2． Biholomorphic Mappings of the Unit Disc to Itself
　　6．3． Linear Fractional Transformations
　　6．4． The Riemann Mapping Theorem： Statement and Idea of Proof
　　6．5． Normal Families
　　6．6． Holomorphically Simply Connected Domains
　　6．7． The Proof of the Analytic Form of the Riemann Mapping Theorem
　　Exercises
　　Chapter 7． Harmonic Functions
　　7．1． Basic Properties of Harmonic Functions
　　7．2． The Maximum Principle and the Mean Value Property
　　7．3． The Poisson Integral Formula
　　7．4． Regularity of Harmonic Functions
　　7．5． The Schwarz Reflection Principle
　　7．6． Harnack's Principle
　　7．7． The Dirichlet Problem and Subharmonic Functions
　　7．8． The Perrbn Method and the Solution of the Dirichlet Problem
　　7．9． Conformal Mappings of Annuli
　　Exercises
　　Chapter 8． Infinite Series and Products
　　8．1． Basic Concepts Concerning Infinite Sums and Products
　　8．2． The Weierstrass Factorization Theorem
　　8．3． The Theorems of Weierstrass and Mittag-Leffler： Interpolation Problems
　　Exercises
　　Chapter 9． Applications of Infinite Sums and Products
　　9．1． Jensen's Formula and an Introduction to Blaschke Products
　　9．2． The Hadamard Gap Theorem
　　9．3． Entire Functions of Finite Order
　　Exercises
　　Chapter 10． Analytic Continuation
　　10．1． Definition of an Analytic Function Element
　　10．2． Analytic Continuation along a Curve
　　10．3． The Monodromy Theorem
　　10．4． The Idea of a Riemann Surface
　　10．5． The Elliptic Modular Function and Picard's Theorem
　　10．6． Elliptic Functions
　　Exercises
　　Chapter 11． Topology
　　11．1． Multiply Connected Domains
　　11．2． The Cauchy Integral Formula for Multiply Connected Domains
　　11．3． Holomorphic Simple Connectivity and Topological Simple Connectivity
　　11．4． Simple Connectivity and Connectedness of the Complement
　　11．5． Multiply Connected Domains Revisited
　　Exercises
　　Chapter 12． Rational Approximation Theory
　　12．1． Runge's Theorem
　　12．2． Mergelyan's Theorem
　　12．3． Some Remarks about Analytic Capacity
　　Exercises
　　Chapter 13． Special Classes of Holomorphic Functions
　　13．1． Schlicht Functions and the Bieberbach Conjecture
　　13．2． Continuity to the Boundary of Conformal Mappings
　　13．3． Hardy Spaces
　　13．4． Boundary Behavior of Functions in Hardy Classes
　　[An Optional Section for Those Who Know
　　Elementary Measure Theory]
　　Exercises
　　Chapter 14． Hilbert Spaces of Holomorphic Functions， the Bergman Kernel， and Biholomorphic Mappings
　　14．1． The Geometry of Hilbert Space
　　14．2． Orthonormal Systems in Hilbert Space
　　14．3． The Bergman Kernel
　　14．4． Bell's Condition R
　　14．5， Smoothness to the Boundary of Conformal Mappings
　　Exercises
　　Chapter 15． Special Functions
　　15．1． The Gamma and Beta Functions
　　15．2． The Riemann Zeta Function
　　Exercises
　　Chapter 16． The Prime Number Theorem
　　16．0． Introduction
　　16．1． Complex Analysis and the Prime Number Theorem
　　16．2． Precise Connections to Complex Analysis
　　16．3． Proof of the Integral Theorem
　　Exercises
　　APPENDIX A： Real Analysis
　　APPENDIX B： The Statement and Proof of Goursat's Theorem
　　References
　　Index
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　　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。
　　《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。