复变函数及应用（英文版·第9版）
Preface
1 Complex Numbers
Sums and Products
Basic Algebraic Properties
Further Algebraic Properties
Vectors and Moduli
Triangle Inequality
Complex Conjugates
Exponential Form
Products and Powers in Exponential …
查看完整
1 Complex Numbers
Sums and Products
Basic Algebraic Properties
Further Algebraic Properties
Vectors and Moduli
Triangle Inequality
Complex Conjugates
Exponential Form
Products and Powers in Exponential …
查看完整
James Ward Brown，密歇根大学迪尔伯恩分校数学系荣休教授，美国数学会会士，入编《美国名人录》和《世界名人录》。1964年于密歇根大学获得博士学位，1971年至2011年任密歇根大学教授，并于 1976年获得密歇根大学杰出教学奖。
Ruel V. Churchill(1899—1987 )生前是密歇根大学知名教授，于芝加哥大学取得理学学士学位，于密歇根大学取得物理学硕士及数学博士学位，自1922年起在密歇根大学执教。他从20世纪40年代开始 编写一系列经典教材，除本书外，还与James Ward Rrown合著有《Fourier Series and Boundary Value Problems》。
Ruel V. Churchill(1899—1987 )生前是密歇根大学知名教授，于芝加哥大学取得理学学士学位，于密歇根大学取得物理学硕士及数学博士学位，自1922年起在密歇根大学执教。他从20世纪40年代开始 编写一系列经典教材，除本书外，还与James Ward Rrown合著有《Fourier Series and Boundary Value Problems》。
《华章数学原版精品系列：复变函数及应用（英文版·第9版）》阐述了复变函数的理论及应用，还介绍了留数及保形映射理论在物理、流体及热传导等边值问题中的应用。这本经典的教材对复变函数的教学影响深远，被美国斯坦福大学、加州理工学院、加州大学伯克利分校、佐治亚理工学院、普度大学、达特茅斯学院、南加州大学等众多名校采用。新版对原有内容进行了重新组织，增加了更现代的示例和应用，更加方便教学。
Preface
1 Complex Numbers
Sums and Products
Basic Algebraic Properties
Further Algebraic Properties
Vectors and Moduli
Triangle Inequality
Complex Conjugates
Exponential Form
Products and Powers in Exponential Form
Arguments of Products and Quotients
Roots of Complex Numbers
Examples
Regions in the Complex Plane
2 Analytic Functions
Functions and Mappings
The Mapping w = zz
Limits
Theorems on Limits
Limits Involving the Point at Infinity
Continuity
Derivatives
Rules for Differentiation
Cauchy-Riemann Equations
Examples
Sufficient Conditions for Differentiability
Polar Coordinates
Analytic Functions
Further Examples
Harmonic Functions
Uniquely Determined Analytic Functions
Reflection Principle
3 Elementary Functions
The Exponential Function
The Logarithmic Function
Examples
Branches and Derivatives of Logarithms
Some Identities Involving Logarithms
The Power Function
Examples
The Trigonometric Functions sin z and cos z
Zeros and Singularities of Trigonometric Functions
Hyperbolic Functions
Inverse Trigonometric and Hyperbolic Functions
4 Integrals
Derivatives of Functions w (t)
Definite Integrals of Functions w (t)
Contours
Contour Integrals
Some Examples
Examples Involving Branch Cuts
Upper Bounds for Moduli of Contour Integrals
Antiderivatives
Proof of the Theorem
Cauchy-Goursat Theorem
Proof of the Theorem
Simply Connected Domains
Multiply Connected Domains
Cauchy Integral Formula
An Extension of the Cauchy Integral Formula
Verification of the Extension
Some Consequences of the Extension
Liouville's Theorem and the Fundamental Theorem of Algebra
Maximum Modulus Principle
5 Series
Convergence of Sequences
Convergence of Series
Taylor Series
Proof of Taylor's Theorem
Examples
Negative Powers of (z - z0)
Laurent Series
Proof of Laurent's Theorem
Examples
Absolute and Uniform Convergence of Power Series
Continuity of Sums of Power Series
Integration and Differentiation of Power Series
Uniqueness of Series Representations
Multiplication and Division of Power Series
6 Residues and Poles
Isolated Singular Points
Residues
Cauchy's Residue Theorem
Residue at Infinity
The Three Types of Isolated Singular Points
Examples
Residues at Poles
Examples
Zeros of Analytic Functions
Zeros and Poles
Behavior of Functions Near Isolated Singular Points
7 Applications of Residues
Evaluation of Improper Integrals
Example
Improper Integrals from Fourier Analysis
Jordan's Lemma
An Indented Path
An Indentation Around a Branch Point
Integration Along a Branch Cut
Definite Integrals Involving Sines and Cosines
Argument Principle
Rouche's Theorem
Inverse Laplace Transforms
Mapping by Elementary Functions
Linear Transformations
The Transformation w = 1/z
Mappings by 1/z
Linear Fractional Transformations
An Implicit Form
Mappings of the Upper Half Plane
Examples
Mappings by the Exponential Function
Mapping Vertical Line Segments by w=sin z
Mapping Horizontal Line Segments by w=sin z
Some Related Mappings
Mappings by z2
Mappings by Branches of z1/2
Square Roots of Polynomials
Riemann Surfaces
Surfaces for Related Functions
9 Conformal Mapping
Preservation of Angles and Scale Factors
Further Examples
Local Inverses
Harmonic Conjugates
Transformations of Harmonic Functions
Transformations of Boundary Conditions
10 Applications of Conformal Mapping
Steady Temperatures
Steady Temperatures in a Half Plane
A Related Problem
Temperatures in a Quadrant
Electrostatic Potential
Examples
Two-Dimensional Fluid Flow
The Stream Function
Flows Around a Comer and Around a Cylinder
11 The Schwarz-Christoffel Transformation
Mapping the Real Axis onto a Polygon
Schwarz-Christoffel Transformation
Triangles and Rectangles
Degenerate Polygons
Fluid Flow in a Channel through a Slit
Flow in a Channel with an Offset
Electrostatic Potential about an Edge of a Conducting Plate
12 Integral Formulas of the Poisson Type
Poisson Integral Formula
Dirichlet Problem for a Disk
Examples
Related Boundary Value Problems
Schwarz Integral Formula
Dirichlet Problem for a Half Plane
Neumann Problems
Appendixes
Bibliography
Table of Transformations of Regions
Index
^ 收 起
1 Complex Numbers
Sums and Products
Basic Algebraic Properties
Further Algebraic Properties
Vectors and Moduli
Triangle Inequality
Complex Conjugates
Exponential Form
Products and Powers in Exponential Form
Arguments of Products and Quotients
Roots of Complex Numbers
Examples
Regions in the Complex Plane
2 Analytic Functions
Functions and Mappings
The Mapping w = zz
Limits
Theorems on Limits
Limits Involving the Point at Infinity
Continuity
Derivatives
Rules for Differentiation
Cauchy-Riemann Equations
Examples
Sufficient Conditions for Differentiability
Polar Coordinates
Analytic Functions
Further Examples
Harmonic Functions
Uniquely Determined Analytic Functions
Reflection Principle
3 Elementary Functions
The Exponential Function
The Logarithmic Function
Examples
Branches and Derivatives of Logarithms
Some Identities Involving Logarithms
The Power Function
Examples
The Trigonometric Functions sin z and cos z
Zeros and Singularities of Trigonometric Functions
Hyperbolic Functions
Inverse Trigonometric and Hyperbolic Functions
4 Integrals
Derivatives of Functions w (t)
Definite Integrals of Functions w (t)
Contours
Contour Integrals
Some Examples
Examples Involving Branch Cuts
Upper Bounds for Moduli of Contour Integrals
Antiderivatives
Proof of the Theorem
Cauchy-Goursat Theorem
Proof of the Theorem
Simply Connected Domains
Multiply Connected Domains
Cauchy Integral Formula
An Extension of the Cauchy Integral Formula
Verification of the Extension
Some Consequences of the Extension
Liouville's Theorem and the Fundamental Theorem of Algebra
Maximum Modulus Principle
5 Series
Convergence of Sequences
Convergence of Series
Taylor Series
Proof of Taylor's Theorem
Examples
Negative Powers of (z - z0)
Laurent Series
Proof of Laurent's Theorem
Examples
Absolute and Uniform Convergence of Power Series
Continuity of Sums of Power Series
Integration and Differentiation of Power Series
Uniqueness of Series Representations
Multiplication and Division of Power Series
6 Residues and Poles
Isolated Singular Points
Residues
Cauchy's Residue Theorem
Residue at Infinity
The Three Types of Isolated Singular Points
Examples
Residues at Poles
Examples
Zeros of Analytic Functions
Zeros and Poles
Behavior of Functions Near Isolated Singular Points
7 Applications of Residues
Evaluation of Improper Integrals
Example
Improper Integrals from Fourier Analysis
Jordan's Lemma
An Indented Path
An Indentation Around a Branch Point
Integration Along a Branch Cut
Definite Integrals Involving Sines and Cosines
Argument Principle
Rouche's Theorem
Inverse Laplace Transforms
Mapping by Elementary Functions
Linear Transformations
The Transformation w = 1/z
Mappings by 1/z
Linear Fractional Transformations
An Implicit Form
Mappings of the Upper Half Plane
Examples
Mappings by the Exponential Function
Mapping Vertical Line Segments by w=sin z
Mapping Horizontal Line Segments by w=sin z
Some Related Mappings
Mappings by z2
Mappings by Branches of z1/2
Square Roots of Polynomials
Riemann Surfaces
Surfaces for Related Functions
9 Conformal Mapping
Preservation of Angles and Scale Factors
Further Examples
Local Inverses
Harmonic Conjugates
Transformations of Harmonic Functions
Transformations of Boundary Conditions
10 Applications of Conformal Mapping
Steady Temperatures
Steady Temperatures in a Half Plane
A Related Problem
Temperatures in a Quadrant
Electrostatic Potential
Examples
Two-Dimensional Fluid Flow
The Stream Function
Flows Around a Comer and Around a Cylinder
11 The Schwarz-Christoffel Transformation
Mapping the Real Axis onto a Polygon
Schwarz-Christoffel Transformation
Triangles and Rectangles
Degenerate Polygons
Fluid Flow in a Channel through a Slit
Flow in a Channel with an Offset
Electrostatic Potential about an Edge of a Conducting Plate
12 Integral Formulas of the Poisson Type
Poisson Integral Formula
Dirichlet Problem for a Disk
Examples
Related Boundary Value Problems
Schwarz Integral Formula
Dirichlet Problem for a Half Plane
Neumann Problems
Appendixes
Bibliography
Table of Transformations of Regions
Index
^ 收 起
James Ward Brown，密歇根大学迪尔伯恩分校数学系荣休教授，美国数学会会士，入编《美国名人录》和《世界名人录》。1964年于密歇根大学获得博士学位，1971年至2011年任密歇根大学教授，并于 1976年获得密歇根大学杰出教学奖。
Ruel V. Churchill(1899—1987 )生前是密歇根大学知名教授，于芝加哥大学取得理学学士学位，于密歇根大学取得物理学硕士及数学博士学位，自1922年起在密歇根大学执教。他从20世纪40年代开始 编写一系列经典教材，除本书外，还与James Ward Rrown合著有《Fourier Series and Boundary Value Problems》。
Ruel V. Churchill(1899—1987 )生前是密歇根大学知名教授，于芝加哥大学取得理学学士学位，于密歇根大学取得物理学硕士及数学博士学位，自1922年起在密歇根大学执教。他从20世纪40年代开始 编写一系列经典教材，除本书外，还与James Ward Rrown合著有《Fourier Series and Boundary Value Problems》。
《华章数学原版精品系列：复变函数及应用（英文版·第9版）》阐述了复变函数的理论及应用，还介绍了留数及保形映射理论在物理、流体及热传导等边值问题中的应用。这本经典的教材对复变函数的教学影响深远，被美国斯坦福大学、加州理工学院、加州大学伯克利分校、佐治亚理工学院、普度大学、达特茅斯学院、南加州大学等众多名校采用。新版对原有内容进行了重新组织，增加了更现代的示例和应用，更加方便教学。
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