复变函数及应用（英文版·第9版）

Preface1　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 Plane2　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 Principle3　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 Functions4　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 Principle5　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 Series6　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 Points7　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 Functions9　Conformal Mapping　Preservation of Angles and Scale Factors　Further Examples　Local Inverses　Harmonic Conjugates　Transformations of Harmonic Functions　Transformations of Boundary Conditions10　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 Cylinder11　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 Plate12　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 ProblemsAppendixesBibliographyTable of Transformations of RegionsIndex