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复变量导引（影印版）

样章
作者：
Steven G. Krantz
定价：
99.00 元
版面字数：
350千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2022年
ISBN：
978-7-04-057022-9
物料号：
57022-00
出版时间：
2022-02-28
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
分析
暂无
  • 目录
    • 前辅文
      • 1 The Complex Plane
        • 1.1 Complex Arithmetic
          • 1.1.1 The Real Numbers
            • 1.1.2 The Complex Numbers
              • 1.1.3 Complex Conjugate
                • 1.1.4 Modulus of a Complex Number
                  • 1.1.5 The Topology of the Complex Plane
                    • 1.1.6 The Complex Numbers as a Field
                      • 1.1.7 The Fundamental Theorem of Algebra
                      • 1.2 The Exponential and Applications
                        • 1.2.1 The Exponential Function
                          • 1.2.2 The Exponential Using Power Series
                            • 1.2.3 Laws of Exponentiation
                              • 1.2.4 Polar Form of a Complex Number
                                • 1.2.5 Roots of Complex Numbers
                                  • 1.2.6 The Argument of a Complex Number
                                    • 1.2.7 Fundamental Inequalities
                                    • 1.3 Holomorphic Functions
                                      • 1.3.1 ContinuouslyDifferentiable and Ck Functions
                                        • 1.3.2 The Cauchy-Riemann Equations
                                          • 1.3.3 Derivatives
                                            • 1.3.4 Definition of Holomorphic Function
                                              • 1.3.5 The Complex Derivative
                                                • 1.3.6 Alternative Terminology for Holomorphic Functions
                                                • 1.4 Holomorphic and Harmonic Functions
                                                  • 1.4.1 Harmonic Functions
                                                    • 1.4.2 How They are Related
                                                  • 2 Complex Line Integrals
                                                    • 2.1 Real and Complex Line Integrals
                                                      • 2.1.1 Curves
                                                        • 2.1.2 Closed Curves
                                                          • 2.1.3 Differentiable and Ck Curves
                                                            • 2.1.4 Integrals on Curves
                                                              • 2.1.5 The Fundamental Theorem of Calculus along Curves
                                                                • 2.1.6 The Complex Line Integral
                                                                  • 2.1.7 Properties of Integrals
                                                                  • 2.2 Complex Differentiability and Conformality
                                                                    • 2.2.1 Limits
                                                                      • 2.2.2 Holomorphicity and the Complex Derivative
                                                                        • 2.2.3 Conformality
                                                                        • 2.3 The Cauchy Integral Formula and Theorem
                                                                          • 2.3.1 The Cauchy Integral Theorem, Basic Form
                                                                            • 2.3.2 The Cauchy Integral Formula
                                                                              • 2.3.3 More General Forms of the Cauchy Theorems
                                                                                • 2.3.4 Deformability of Curves
                                                                                • 2.4 The Limitations of the Cauchy Formula
                                                                                • 3 Applications of the Cauchy Theory
                                                                                  • 3.1 The Derivatives of a Holomorphic Function
                                                                                    • 3.1.1 A Formula for the Derivative
                                                                                      • 3.1.2 The Cauchy Estimates
                                                                                        • 3.1.3 Entire Functions and Liouville’s Theorem
                                                                                          • 3.1.4 The Fundamental Theorem of Algebra
                                                                                            • 3.1.5 Sequences of Holomorphic Functions and their Derivatives
                                                                                              • 3.1.6 The Power Series Representation of a Holomorphic Function
                                                                                              • 3.2 The Zeros of a Holomorphic Function
                                                                                                • 3.2.1 The Zero Set of a Holomorphic Function
                                                                                                  • 3.2.2 Discreteness of the Zeros of a Holomorphic Function
                                                                                                    • 3.2.3 Discrete Sets and Zero Sets
                                                                                                      • 3.2.4 Uniqueness of Analytic Continuation
                                                                                                    • 4 Laurent Series
                                                                                                      • 4.1 Behavior Near an Isolated Singularity
                                                                                                        • 4.1.1 Isolated Singularities
                                                                                                          • 4.1.2 A Holomorphic Function on a Punctured Domain
                                                                                                            • 4.1.3 Classification of Singularities
                                                                                                              • 4.1.4 Removable Singularities, Poles, and Essential Singularities
                                                                                                                • 4.1.5 The Riemann Removable Singularities Theorem
                                                                                                                  • 4.1.6 The Casorati-Weierstrass Theorem
                                                                                                                  • 4.2 Expansion around Singular Points
                                                                                                                    • 4.2.1 Laurent Series
                                                                                                                      • 4.2.2 Convergence of a Doubly Infinite Series
                                                                                                                        • 4.2.3 Annulus of Convergence
                                                                                                                          • 4.2.4 Uniqueness of the Laurent Expansion
                                                                                                                            • 4.2.5 The Cauchy Integral Formula for an Annulus
                                                                                                                              • 4.2.6 Existence of Laurent Expansions
                                                                                                                                • 4.2.7 Holomorphic Functions with Isolated Singularities
                                                                                                                                  • 4.2.8 Classification of Singularities in Terms of Laurent Series
                                                                                                                                  • 4.3 Examples of Laurent Expansions
                                                                                                                                    • 4.3.1 Principal Part of a Function
                                                                                                                                      • 4.3.2 Algorithmfor Calculating the Coefficients of the Laurent Expansion
                                                                                                                                      • 4.4 The Calculus of Residues
                                                                                                                                        • 4.4.1 Functions with Multiple Singularities
                                                                                                                                          • 4.4.2 The Residue Theorem
                                                                                                                                            • 4.4.3 Residues
                                                                                                                                              • 4.4.4 The Index orWinding Number of a Curve about a Point
                                                                                                                                                • 4.4.5 Restatement of the Residue Theorem
                                                                                                                                                  • 4.4.6 Method for Calculating Residues
                                                                                                                                                    • 4.4.7 Summary Charts of Laurent Series and Residues
                                                                                                                                                    • 4.5 Applications to Integrals
                                                                                                                                                      • 4.5.1 The Evaluation of Definite Integrals
                                                                                                                                                        • 4.5.2 A Basic Example of the Indefinite Integral
                                                                                                                                                          • 4.5.3 Complexification of the Integrand
                                                                                                                                                            • 4.5.4 An Example with a More Subtle Choice of Contour
                                                                                                                                                              • 4.5.5 Making the Spurious Part of the Integral Disappear
                                                                                                                                                                • 4.5.6 The Use of the Logarithm
                                                                                                                                                                  • 4.5.7 Summing a Series Using Residues
                                                                                                                                                                  • 4.6 Singularities at Infinity
                                                                                                                                                                    • 4.6.1 Meromorphic Functions
                                                                                                                                                                      • 4.6.2 Definition of Meromorphic Function
                                                                                                                                                                        • 4.6.3 Examples of Meromorphic Functions
                                                                                                                                                                          • 4.6.4 Meromorphic Functions with InfinitelyMany Poles
                                                                                                                                                                            • 4.6.5 Singularities at Infinity
                                                                                                                                                                              • 4.6.6 The Laurent Expansion at Infinity
                                                                                                                                                                                • 4.6.7 Meromorphic at Infinity
                                                                                                                                                                                  • 4.6.8 Meromorphic Functions in the Extended Plane
                                                                                                                                                                                • 5 The Argument Principle
                                                                                                                                                                                  • 5.1 Counting Zeros and Poles
                                                                                                                                                                                    • 5.1.1 Local Geometric Behavior of a Holomorphic Function
                                                                                                                                                                                      • 5.1.2 Locating the Zeros of a Holomorphic Function
                                                                                                                                                                                        • 5.1.3 Zero of Order n
                                                                                                                                                                                          • 5.1.4 Counting the Zeros of a Holomorphic Function
                                                                                                                                                                                            • 5.1.5 The Argument Principle
                                                                                                                                                                                              • 5.1.6 Location of Poles
                                                                                                                                                                                                • 5.1.7 The Argument Principle for Meromorphic Functions
                                                                                                                                                                                                • 5.2 The Local Geometry of Holomorphic Functions
                                                                                                                                                                                                  • 5.2.1 The Open Mapping Theorem
                                                                                                                                                                                                  • 5.3 Further Results
                                                                                                                                                                                                    • 5.3.1 Rouch´e’s Theorem
                                                                                                                                                                                                      • 5.3.2 Typical Application of Rouch´e’s Theorem
                                                                                                                                                                                                        • 5.3.3 Rouch´e’s Theorem and the Fundamental Theorem of Algebra
                                                                                                                                                                                                          • 5.3.4 Hurwitz’s Theorem
                                                                                                                                                                                                          • 5.4 The Maximum Principle
                                                                                                                                                                                                            • 5.4.1 The Maximum Modulus Principle
                                                                                                                                                                                                              • 5.4.2 Boundary Maximum Modulus Theorem
                                                                                                                                                                                                                • 5.4.3 The Minimum Principle
                                                                                                                                                                                                                  • 5.4.4 The Maximum Principle on an Unbounded Domain
                                                                                                                                                                                                                  • 5.5 The Schwarz Lemma
                                                                                                                                                                                                                    • 5.5.1 Schwarz’s Lemma
                                                                                                                                                                                                                      • 5.5.2 The Schwarz-Pick Lemma
                                                                                                                                                                                                                    • 6 The Geometric Theory.
                                                                                                                                                                                                                      • 6.1 The Idea of a Conformal Mapping
                                                                                                                                                                                                                        • 6.1.1 Conformal Mappings
                                                                                                                                                                                                                          • 6.1.2 Conformal Self-Maps of the Plane
                                                                                                                                                                                                                          • 6.2 Linear Fractional Transformations
                                                                                                                                                                                                                            • 6.2.1 Linear Fractional Mappings
                                                                                                                                                                                                                              • 6.2.2 The Topology of the Extended Plane
                                                                                                                                                                                                                                • 6.2.3 The Riemann Sphere
                                                                                                                                                                                                                                  • 6.2.4 Conformal Self-Maps of the Riemann Sphere
                                                                                                                                                                                                                                    • 6.2.5 The Cayley Transform
                                                                                                                                                                                                                                      • 6.2.6 Generalized Circles and Lines
                                                                                                                                                                                                                                        • 6.2.7 The Cayley Transform Revisited
                                                                                                                                                                                                                                          • 6.2.8 Summary Chart of Linear Fractional Transformations
                                                                                                                                                                                                                                          • 6.3 The Riemann Mapping Theorem
                                                                                                                                                                                                                                            • 6.3.1 The Concept of Homeomorphism
                                                                                                                                                                                                                                              • 6.3.2 The Riemann Mapping Theorem
                                                                                                                                                                                                                                                • 6.3.3 The Riemann Mapping Theorem: Second Formulation
                                                                                                                                                                                                                                                • 6.4 Conformal Mappings of Annuli
                                                                                                                                                                                                                                                  • 6.4.1 A Riemann Mapping Theorem for Annuli
                                                                                                                                                                                                                                                    • 6.4.2 Conformal Equivalence of Annuli
                                                                                                                                                                                                                                                      • 6.4.3 Classification of Planar Domains
                                                                                                                                                                                                                                                    • 7 Harmonic Functions
                                                                                                                                                                                                                                                      • 7.1 Basic Properties of Harmonic Functions
                                                                                                                                                                                                                                                        • 7.1.1 The Laplace Equation
                                                                                                                                                                                                                                                          • 7.1.2 Definition of Harmonic Function
                                                                                                                                                                                                                                                            • 7.1.3 Real- and Complex-Valued Harmonic Functions
                                                                                                                                                                                                                                                              • 7.1.4 Harmonic Functions as the Real Parts of Holomorphic Functions
                                                                                                                                                                                                                                                                • 7.1.5 Smoothness of Harmonic Functions
                                                                                                                                                                                                                                                                • 7.2 The Maximum Principle and the Mean Value Property
                                                                                                                                                                                                                                                                  • 7.2.1 The Maximum Principle for Harmonic Functions
                                                                                                                                                                                                                                                                    • 7.2.2 The Minimum Principle for Harmonic Functions
                                                                                                                                                                                                                                                                      • 7.2.3 The Boundary Maximum and Minimum Principles
                                                                                                                                                                                                                                                                        • 7.2.4 The Mean Value Property
                                                                                                                                                                                                                                                                          • 7.2.5 Boundary Uniqueness for Harmonic Functions
                                                                                                                                                                                                                                                                          • 7.3 The Poisson Integral Formula
                                                                                                                                                                                                                                                                            • 7.3.1 The Poisson Integral
                                                                                                                                                                                                                                                                              • 7.3.2 The Poisson Kernel
                                                                                                                                                                                                                                                                                • 7.3.3 The Dirichlet Problem
                                                                                                                                                                                                                                                                                  • 7.3.4 The Solution of the Dirichlet Problem on the Disc
                                                                                                                                                                                                                                                                                    • 7.3.5 The Dirichlet Problem on a General Disc
                                                                                                                                                                                                                                                                                    • 7.4 Regularity of Harmonic Functions
                                                                                                                                                                                                                                                                                      • 7.4.1 The Mean Value Property on Circles
                                                                                                                                                                                                                                                                                        • 7.4.2 The Limit of a Sequence of Harmonic Functions
                                                                                                                                                                                                                                                                                        • 7.5 The Schwarz Reflection Principle
                                                                                                                                                                                                                                                                                          • 7.5.1 Reflection of Harmonic Functions
                                                                                                                                                                                                                                                                                            • 7.5.2 Schwarz Reflection Principle for Harmonic Functions
                                                                                                                                                                                                                                                                                              • 7.5.3 The Schwarz Reflection Principle for Holomorphic Functions
                                                                                                                                                                                                                                                                                                • 7.5.4 More General Versions of the Schwarz Reflection Principle
                                                                                                                                                                                                                                                                                                • 7.6 Harnack’s Principle
                                                                                                                                                                                                                                                                                                  • 7.6.1 The Harnack Inequality
                                                                                                                                                                                                                                                                                                    • 7.6.2 Harnack’s Principle
                                                                                                                                                                                                                                                                                                    • 7.7 The Dirichlet Problem
                                                                                                                                                                                                                                                                                                      • 7.7.1 The Dirichlet Problem
                                                                                                                                                                                                                                                                                                        • 7.7.2 Conditions for Solving the Dirichlet Problem
                                                                                                                                                                                                                                                                                                          • 7.7.3 Motivation for Subharmonic Functions
                                                                                                                                                                                                                                                                                                            • 7.7.4 Definition of Subharmonic Function
                                                                                                                                                                                                                                                                                                              • 7.7.5 Other Characterizations of Subharmonic Functions
                                                                                                                                                                                                                                                                                                                • 7.7.6 The Maximum Principle
                                                                                                                                                                                                                                                                                                                  • 7.7.7 Lack of A Minimum Principle
                                                                                                                                                                                                                                                                                                                    • 7.7.8 Basic Properties of Subharmonic Functions
                                                                                                                                                                                                                                                                                                                      • 7.7.9 The Concept of a Barrier
                                                                                                                                                                                                                                                                                                                      • 7.8 The General Solution of the Dirichlet Problem
                                                                                                                                                                                                                                                                                                                        • 7.8.1 Enunciation of the Solution of the Dirichlet Problem
                                                                                                                                                                                                                                                                                                                      • 8 Infinite Series and Products
                                                                                                                                                                                                                                                                                                                        • 8.1 Basic Concepts
                                                                                                                                                                                                                                                                                                                          • 8.1.1 Uniform Convergence of a Sequence
                                                                                                                                                                                                                                                                                                                            • 8.1.2 The Cauchy Condition for a Sequence of Functions
                                                                                                                                                                                                                                                                                                                              • 8.1.3 Normal Convergence of a Sequence
                                                                                                                                                                                                                                                                                                                                • 8.1.4 Normal Convergence of a Series
                                                                                                                                                                                                                                                                                                                                  • 8.1.5 The Cauchy Condition for a Series
                                                                                                                                                                                                                                                                                                                                    • 8.1.6 The Concept of an Infinite Product
                                                                                                                                                                                                                                                                                                                                      • 8.1.7 Infinite Products of Scalars
                                                                                                                                                                                                                                                                                                                                        • 8.1.8 Partial Products
                                                                                                                                                                                                                                                                                                                                          • 8.1.9 Convergence of an Infinite Product
                                                                                                                                                                                                                                                                                                                                            • 8.1.10 The Value of an Infinite Product
                                                                                                                                                                                                                                                                                                                                              • 8.1.11 Products That Are Disallowed
                                                                                                                                                                                                                                                                                                                                                • 8.1.12 Condition for Convergence of an Infinite Product
                                                                                                                                                                                                                                                                                                                                                  • 8.1.13 Infinite Products of Holomorphic Functions
                                                                                                                                                                                                                                                                                                                                                    • 8.1.14 Vanishing of an Infinite Product
                                                                                                                                                                                                                                                                                                                                                      • 8.1.15 Uniform Convergence of an Infinite Product of Functions
                                                                                                                                                                                                                                                                                                                                                        • 8.1.16 Condition for the Uniform Convergence of an Infinite Product of Functions
                                                                                                                                                                                                                                                                                                                                                        • 8.2 The Weierstrass Factorization Theorem
                                                                                                                                                                                                                                                                                                                                                          • 8.2.1 Prologue
                                                                                                                                                                                                                                                                                                                                                            • 8.2.2 Weierstrass Factors
                                                                                                                                                                                                                                                                                                                                                              • 8.2.3 Convergence of theWeierstrass Product
                                                                                                                                                                                                                                                                                                                                                                • 8.2.4 Existence of an Entire Function with Prescribed Zeros
                                                                                                                                                                                                                                                                                                                                                                  • 8.2.5 TheWeierstrass Factorization Theorem
                                                                                                                                                                                                                                                                                                                                                                  • 8.3 Weierstrass and Mittag-Leffler Theorems
                                                                                                                                                                                                                                                                                                                                                                    • 8.3.1 The Concept of Weierstrass’s Theorem
                                                                                                                                                                                                                                                                                                                                                                      • 8.3.2 Weierstrass’s Theorem
                                                                                                                                                                                                                                                                                                                                                                        • 8.3.3 Construction of a Discrete Set
                                                                                                                                                                                                                                                                                                                                                                          • 8.3.4 Domains of Existence for Holomorphic Functions
                                                                                                                                                                                                                                                                                                                                                                            • 8.3.5 The Field Generated by the Ring of Holomorphic Functions
                                                                                                                                                                                                                                                                                                                                                                              • 8.3.6 The Mittag-Leffler Theorem
                                                                                                                                                                                                                                                                                                                                                                                • 8.3.7 Prescribing Principal Parts
                                                                                                                                                                                                                                                                                                                                                                                • 8.4 Normal Families
                                                                                                                                                                                                                                                                                                                                                                                  • 8.4.1 Normal Convergence
                                                                                                                                                                                                                                                                                                                                                                                    • 8.4.2 Normal Families
                                                                                                                                                                                                                                                                                                                                                                                      • 8.4.3 Montel’s Theorem, First Version
                                                                                                                                                                                                                                                                                                                                                                                        • 8.4.4 Montel’s Theorem, Second Version
                                                                                                                                                                                                                                                                                                                                                                                          • 8.4.5 Examples of Normal Families
                                                                                                                                                                                                                                                                                                                                                                                        • 9 Analytic Continuation
                                                                                                                                                                                                                                                                                                                                                                                          • 9.1 Definition of an Analytic Function Element
                                                                                                                                                                                                                                                                                                                                                                                            • 9.1.1 Continuation of Holomorphic Functions
                                                                                                                                                                                                                                                                                                                                                                                              • 9.1.2 Examples of Analytic Continuation
                                                                                                                                                                                                                                                                                                                                                                                                • 9.1.3 Function Elements
                                                                                                                                                                                                                                                                                                                                                                                                  • 9.1.4 Direct Analytic Continuation
                                                                                                                                                                                                                                                                                                                                                                                                    • 9.1.5 Analytic Continuation of a Function
                                                                                                                                                                                                                                                                                                                                                                                                      • 9.1.6 Global Analytic Functions
                                                                                                                                                                                                                                                                                                                                                                                                        • 9.1.7 An Example of Analytic Continuation
                                                                                                                                                                                                                                                                                                                                                                                                        • 9.2 Analytic Continuation along a Curve
                                                                                                                                                                                                                                                                                                                                                                                                          • 9.2.1 Continuation on a Curve
                                                                                                                                                                                                                                                                                                                                                                                                            • 9.2.2 Uniqueness of Continuation along a Curve
                                                                                                                                                                                                                                                                                                                                                                                                            • 9.3 The Monodromy Theorem
                                                                                                                                                                                                                                                                                                                                                                                                              • 9.3.1 Unambiguity of Analytic Continuation
                                                                                                                                                                                                                                                                                                                                                                                                                • 9.3.2 The Concept of Homotopy
                                                                                                                                                                                                                                                                                                                                                                                                                  • 9.3.3 Fixed Endpoint Homotopy
                                                                                                                                                                                                                                                                                                                                                                                                                    • 9.3.4 Unrestricted Continuation
                                                                                                                                                                                                                                                                                                                                                                                                                      • 9.3.5 The Monodromy Theorem
                                                                                                                                                                                                                                                                                                                                                                                                                        • 9.3.6 Monodromy and Globally Defined Analytic Functions
                                                                                                                                                                                                                                                                                                                                                                                                                        • 9.4 The Idea of a Riemann Surface
                                                                                                                                                                                                                                                                                                                                                                                                                          • 9.4.1 What is a Riemann Surface?
                                                                                                                                                                                                                                                                                                                                                                                                                            • 9.4.2 Examples of Riemann Surfaces
                                                                                                                                                                                                                                                                                                                                                                                                                              • 9.4.3 The Riemann Surface for the Square Root Function
                                                                                                                                                                                                                                                                                                                                                                                                                                • 9.4.4 Holomorphic Functions on a Riemann Surface
                                                                                                                                                                                                                                                                                                                                                                                                                                  • 9.4.5 The Riemann Surface for the Logarithm
                                                                                                                                                                                                                                                                                                                                                                                                                                    • 9.4.6 Riemann Surfaces in General
                                                                                                                                                                                                                                                                                                                                                                                                                                    • 9.5 Picard’s Theorems
                                                                                                                                                                                                                                                                                                                                                                                                                                      • 9.5.1 Value Distribution for Entire Functions
                                                                                                                                                                                                                                                                                                                                                                                                                                        • 9.5.2 Picard’s Little Theorem
                                                                                                                                                                                                                                                                                                                                                                                                                                          • 9.5.3 Picard’s Great Theorem
                                                                                                                                                                                                                                                                                                                                                                                                                                            • 9.5.4 The Little Theorem, the Great Theorem, and the Casorati-Weierstrass Theorem
                                                                                                                                                                                                                                                                                                                                                                                                                                          • Glossary of Terms from Complex Variable Theory and Analysis
                                                                                                                                                                                                                                                                                                                                                                                                                                            • Bibliography
                                                                                                                                                                                                                                                                                                                                                                                                                                              • Index
                                                                                                                                                                                                                                                                                                                                                                                                                                                • About the Author

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