- 复分析入门(英文)/他山之石系列/国外优秀数学著作原版系列
- - 作者：(美)O.卡鲁斯·麦基希|责编:聂兆慈//李兰静
  - 出版社：哈尔滨工业大学
  - ISBN：9787560393155
  - 出版日期：2021/03/01
  - 页数：529
- 售价：19.2

内容大纲
本书主要包括流体流动，集、函数和可视化，实线与平面，开放集、开映射和连通集，域的定义，全纯函数的基本性质，柯西定理，孤立奇点，剩余定理和论点原则，边界值问题，共形映射方法，黎曼曲面的集成，有理函数在直线上的积分等内容。本书适合作为数学研究生学习复分析第一门课程的参考，使学生回顾并积极利用他们的微积分背景知识。书中有很多图形，可以提供有用的例子，也可以向读者展示重要的思想。本书是为了适应和遵循学生们以前学过的知识而设计的。
作者介绍
目录
Preface
Symbols and Terms
Preliminaries
1.1 Preview
  A It Takes Two Harmonic Functions
  B Heat Flow
  C A Geometric Rule
  D Electrostatics
  E Fluid Flow
  F One Model Many Applications
  Exercises
1.2 Sets, Functions, and Visualization
  A Terminology and Notation for Sets
  B Terminology and Notation for Functions
  C Functions from R to R
  D Functions from R2 to R
  E Functions from R2 to R2
  Exercises
1.3 Structures on R2, and Linear Maps from R2 to R2
  A The Real Line and the Plane
  B Polar Coordinates in the Plane
  C When Is a Mapping M : R2 → R2 Linear?
  D Visualizing Nonsingular Linear Mappings
  E The Determinant of a Two-by-Two Matrix
  F Pure Magnifications, Rotations, and
  Conjugation
  G Conformal Linear Mappings
  Exercises
1.4 Open Sets, Open Mappings, Connected Sets
  A Distance, Interior, Boundary, Openness
  B Continuity in Terms of Open Sets
  C Open Mappings
  D Connected Sets
  Exercises
1.5 A Review of Some Calculus
  A Integration Theory for Real- Valued Functions
  B Improper Integrals, Principal Values
  C Partial Derivatives
  D Divergence and Curl
  Exercises
1.6 Harmonic Functions
  A The Geometry of Laplace's Equation
  B The Geometry of the Cauchy-Riemann
  Equations
  C The Mean Value Property
  D Changing Variables in a Dirichtet or Neumann
Problem
Exercises
2 Basic Tools
2.1 The Complex Plane

  A The Definition of a Field
  B Complex Multiplication
  C Powers and Roots
  D Conjugation
  E Quotients of Complex Numbers
  F When Is a Mapping L : C → C Linear?
  G Complex Equations for Lines and Circles
  H The Reciprocal Map, and Reflection in the
  Unit Circle
  I Reflections in Lines and Circles
  Exercises
2.2 Visualizing Powers, Exponential, Logarithm, and Sine
  A Powers ofz
  B Exponential and Logarithms
  C Sin z
  D The Cosine and Sine, and the Hyperbolic
  Cosine and Sine
  Exercises
2.3 Differentiability
  A Differentiability at a Point
  B Differentiability in the Complex Sense:
  Holomorphy
  C Finding Derivatives
  D Picturing the Local Behavior of Holomorphic
  Mappings
  Exercises
2.4 Sequences, Compactness, Convergence
  A Sequences of Complex Numbers
  B The Limit Superior of a Sequence of Reals
  C Implications of Compactness
  D Sequences of Functions
  Exercises
2.5 Integrals Over Curves, Paths, and Contours
  A Integrals of Complex-Valued Functions
  B Curves
  C Paths
  D Pathwise Connected Sets
  E Independence of Path and Morera's Theorem
  F Goursat's Lemma
  G The Winding Number
  H Green's Theorem
  I Irrotational and Incompressible Fluid Flow
  J Contours
  Exercises
2.6 Power Series
  A Infinite Series
  B The Geometric Series
  C An Improved Root Test
  D Power Series and the Cauchy-Hadamard
  Theorem

  E Uniqueness of the Power Series Representation
  F Integrals That Give Rise to Power Series
  Exercises
3 The Cauchy Theory
3.1 Fundamental Properties of Holomorphic Functions
  A Integral and Series Representations
  B Eight Ways to Say "'Holomorphic"
  C Determinism
  D Liouville's Theorem
  E The Fundamental Theorem of Algebra
  F Subuniform Convergence Preserves
  Holomorphy
  Exercises
3.2 Cauchy's Theorem
  A Cerny's 1976 Proof
  B Simply Connected Sets
  C Subuniform Boundedness, Subuniform
  Convergence
3.3 lsolated Singularities
  A The Laurent Series Representation on an Annulus
  B Behavior Near an Isolated Singularity in the Plane
  C Examples: Classifying Singularities, Finding Residues
  D Behavior Near a Singularity at Infinity
  E A Digression: Picard'sGreat Theorem
  Exercises
3.4 The Residue Theorem and the Argument Principle
  A Meromorphic Functions and the Extended
  Plane
  B The Residue Theorem
  C Multiplicity and Valence
  D Valence.for a Rational Function
  E The Argument Principle: Integrals That Count
  Exercises
3.5 Mapping Properties
  Exercises
3.6 The Riemann Sphere
  Exercises
4 The Residue Calculus
4.1 Integrals of Trigonometric Functions Over a Compact lnterval
  Exercises
4.2 Estimating Complex Integrals
  Exercises
4.3 Integrals of Rational Functions Over the Line
  Exercises
4.4 Integrals Involving the Exponential
  A Integrals Giving Fourier Transforms
  Exercises
4.5 Integrals Involving a Logarithm
  Exercises
4.6 Integration on a Riemann Surface

  A Mellin Transforms
  Exercises
4. 7 The Complex Inversion Formula for the Laplace Transform
  Exercises
5 Boundary Value Problems
5.1 Examples
  A Easy Problems
  B The Conformal Mapping Method
  Exercises
5.2 The Mobius Maps
  Exercises
5.3 Electric Fields
  A A Point Charge in 3-Space
  B Uniform Charge on One or More Long Wires
  C Examples with Bounded Potentials
  Exercises
5.4 Steady Flow of a Perfect Fluid
  Exercises
5.5 Using the Poisson Integral to Obtain Solutions
  A The Poisson Integral on a Disk
  B Solutions on the Disk by the Poisson Integral
  C Geometry of the Poisson Integral
  D Harmonic Functions and the Mean Value Property
  E The Neumann Problem on a Disk
  F The Poisson Integral on a Half-Plane, and on Other Domains
  Exercises
5.6 When Is the Solution Unique?
  Exercises
5.7 The Schwarz Reflection Principle
5.8 Schwarz-Christoffel Formulas
  A Triangles
  B Rectangles and Other Polygons
  C Generalized Polygons
  Exercises
6 Lagniappe
6.1 Dixon's 1971ProofofCauchy's Theorem
6.2 Runge's Theorem
  Exercises
6.3 The Riemann Mapping Theorem
  Exercises
6.4 The Osgood-Taylor-Carath~odory Theorem
  References
  Index

内容大纲

作者介绍

目录

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