An introduction to classical complex analysis
An introduction to classical complex analysis
复制标题
经典复分析简介
DOI：
10.1007/978-3-0348-9374-9
复制
发表时间：
1979
影响因子：
1.4
通讯作者：
R. Burckel
中科院分区：
文献类型：
--
作者：
R. Burckel
前提和前提。-1集合论。-2代数。-3战场。-4度量空间。-5连续函数。-7微积分。-I曲线，连通性和凸性。连通性的基本结果。-2区间的连通性，曲线和凸集。-3基本连通性引理。-4分量和紧穷法。-5集的连通性。-6延拓定理。-第一章注释。-2(复数)导数和(曲线)积分。-2全纯函数和调和函数。-3曲线上的积分。-3在积分下求导。-4可微的一个有用的充分条件。-第二章注释。-3幂函数和指数函数。-1引言。-2幂函数。-3复指数函数。-4伯努利多项式，数字和函数。-5柯西定理。-6全纯对数预习。-第三章注释。-4指数和一些平面拓扑学。-1引言。-2曲线绕点。-3同伦和指数。-4连续对数的存在。-5乔丹曲线定理。-6前述技术的应用。-7开集上的连续和全纯对数。-8开集的简单连通性。-第四章的注释。-V柯西-古尔萨定理的最大值原理和局部理论的结果。-1古尔萨特引理和柯西定理。-2极大值原理。-3圆盘的狄里克莱特问题。-4幂级数展开的存在性。-5调和优化法。-6唯一性定理。-7局部理论。-第五章-VI施瓦兹引理及其许多应用的注记。-1施瓦兹引理和圆盘的共形自同构。-2圆盘到圆盘的多对一映射。-3半平面的应用，条带和环带。-4《CarathSodory，Julia，Wolff，第六章的从属注记。全纯函数的收敛序列。在H(U)中的收敛。收敛定理有界性准则的应用。描述零点。初等迭代理论。第七章的注解。第八多项式和有理逼近-Runge理论。基本积分表示定理。逼近的应用。积分表示的其他应用。一些特殊类型的逼近。-5 Carleman逼近定理。-6半平面中的调和函数。-。-9黎曼映射定理。-1导言。-2 Caratheodory和Koebe的证明。-3 Fejer和Riesz‘证明。-4约旦地区的边界行为。-5奥斯古德-泰勒-Caratheodory定理的几个应用。-7关于约旦区域和边界行为的更多内容。-7调和函数和一般的Dirichlet问题。-8狄利克莱特问题和黎曼映射定理。-第九章的注释。-X单连通性和双连通性。-1单连通性。-2双连通性。-第十章孤立奇点的注释。奇点的级数和分类。-2有理函数。-3收敛圆上的孤立奇点。-4留数定理和一些应用。-5指定主要部分-Mittag-Leffler定理。-6亚纯函数。-7环面上的泊松公式和调和函数的孤立奇点。-第十一章的注释。-12省略的值和正规族。-1对数平均和Jensen不等式。-2米兰达定理。-3米兰达的直接应用。-4正规族和Julia的推广。-5扇形极限定理。-6迭代理论的应用。-7奥斯托夫斯基对肖特基定理的证明。-第十二章注释。-名称索引。-符号索引。-级数求和。-积分求值。
0 Prerequisites and Preliminaries.- 1 Set Theory.- 2 Algebra.- 3 The Battlefield.- 4 Metric Spaces.- 5 Limsup and All That.- 6 Continuous Functions.- 7 Calculus.- I Curves, Connectedness and Convexity.- 1 Elementary Results on Connectedness.- 2 Connectedness of Intervals, Curves and Convex Sets.- 3 The Basic Connectedness Lemma.- 4 Components and Compact Exhaustions.- 5 Connectivity of a Set.- 6 Extension Theorems.- Notes to Chapter I.- II (Complex) Derivative and (Curvilinear) Integrals.- 1 Holomorphic and Harmonic Functions.- 2 Integrals along Curves.- 3 Differentiating under the Integral.- 4 A Useful Sufficient Condition for Differentiability.- Notes to Chapter II.- III Power Series and the Exponential Function.- 1 Introduction.- 2 Power Series.- 3 The Complex Exponential Function.- 4 Bernoulli Polynomials, Numbers and Functions.- 5 Cauchy's Theorem Adumbrated.- 6 Holomorphic Logarithms Previewed.- Notes to Chapter III.- IV The Index and some Plane Topology.- 1 Introduction.- 2 Curves Winding around Points.- 3 Homotopy and the Index.- 4 Existence of Continuous Logarithms.- 5 The Jordan Curve Theorem.- 6 Applications of the Foregoing Technology.- 7 Continuous and Holomorphic Logarithms in Open Sets.- 8 Simple Connectivity for Open Sets.- Notes to Chapter IV.- V Consequences of the Cauchy-Goursat Theorem-Maximum Principles and the Local Theory.- 1 Goursat's Lemma and Cauchy's Theorem for Starlike Regions.- 2 Maximum Principles.- 3 The Dirichlet Problem for Disks.- 4 Existence of Power Series Expansions.- 5 Harmonic Majorization.- 6 Uniqueness Theorems.- 7 Local Theory.- Notes to Chapter V.- VI Schwarz' Lemma and its Many Applications.- 1 Schwarz' Lemma and the Conformal Automorphisms of Disks.- 2 Many-to-one Maps of Disks onto Disks.- 3 Applications to Half-planes, Strips and Annuli.- 4 The Theorem of CarathSodory, Julia, Wolff, et al.- 5 Subordination.- Notes to Chapter VI.- VII Convergent Sequences of Holomorphic Functions.- 1 Convergence in H(U).- 2 Applications of the Convergence Theorems Boundedness Criteria.- 3 Prescribing Zeros.- 4 Elementary Iteration Theory.- Notes to Chapter VII.- VIII Polynomial and Rational Approximation-Runge Theory.- 1 The Basic Integral Representation Theorem.- 2 Applications to Approximation.- 3 Other Applications of the Integral Representation.- 4 Some Special Kinds of Approximation.- 5 Carleman's Approximation Theorem.- 6 Harmonic Functions in a Half-plane.- Notes to Chapter VIII.- IX The Riemann Mapping Theorem.- 1 Introduction.- 2 The Proof of Caratheodory and Koebe.- 3 Fejer and Riesz' Proof.- 4 Boundary Behavior for Jordan Regions.- 5 A Few Applications of the Osgood-Taylor-Caratheodory Theorem.- 6 More on Jordan Regions and Boundary Behavior.- 7 Harmonic Functions and the General Dirichlet Problem.- 8 The Dirichlet Problem and the Riemann Mapping Theorem.- Notes to Chapter IX.- X Simple and Double Connectivity.- 1 Simple Connectivity.- 2 Double Connectivity.- Notes to Chapter X.- XI Isolated Singularities.- 1 Laurent Series and Classification of Singularities.- 2 Rational Functions.- 3 Isolated Singularities on the Circle of Convergence.- 4 The Residue Theorem and Some Applications.- 5 Specifying Principal Parts-Mittag-Leffler's Theorem.- 6 Meromorphic Functions.- 7 Poisson's Formula in an Annulus and Isolated Singularities of Harmonic Functions.- Notes to Chapter XI.- XII Omitted Values and Normal Families.- 1 Logarithmic Means and Jensen's Inequality.- 2 Miranda's Theorem.- 3 Immediate Applications of Miranda.- 4 Normal Families and Julia's Extension of Picard's Great Theorem.- 5 Sectorial Limit Theorems.- 6 Applications to Iteration Theory.- 7 Ostrowski's Proof of Schottky's Theorem.- Notes to Chapter XII.- Name Index.- Symbol Index.- Series Summed.- Integrals Evaluated.