复变量

Sectionsdenotedwithanasterisk(*)canbeeitheromittedorread

independently.

Preface

PartIFundamentalsandTechniquesofComplexFunctionTheory

1ComplexNumbersandElementaryFunctions

1.1ComplexNumbersandTheirProperties

1.2ElementaryFunctionsandStereographicProjections

1.2.1ElementaryFunctions

1.2.2StereographicProjections

1.3Limits,Continuity,andComplexDifferentiation

1.4ElementaryApplicationstoOrdinaryDifferentialEquations

2AnalyticFunctionsandIntegration

2.1AnalyticFunctions

2.1.1TheCauchy-RiemannEquations

2.1.2IdealFluidFlow

2.2MultivaluedFunctions

*2.3MoreComplicatedMultivaluedFunctionsandRiemannSurfaces

2.4ComplexIntegration

2.5CauchysTheorem

2.6CauchysIntegralFormula,ItsaGeneralizationandConsequences

2.6.1CauchysIntegralFormulaandItsDerivatives

*2.6.2Liouville,Morera,andMaximum-ModulusTheorems

*2.6.3GeneralizedCauchyFormulaandaDerivatives

*2.7TheoreticalDevelopments

3Sequences,Series,andSingularitiesofComplexFunctions

3.1DefinitionsandBasicPropertiesofComplexSequences,Series

3.2TaylorSeries

3.3LaurentSeries

*3.4TheoreticalResultsforSequencesandSeries

3.5SingularitiesofComplexFunctions

3.5.1AnalyticContinuationandNaturalBarriers

*3.6InfiniteProductsandMittag-LefflerExpansions

*3.7DifferentialEquationsintheComplexPlane:PainleveEquations

*3.8ComputationalMethods

*3.8.1LaurentSeries

*3.8.2DifferentialEquations

4ResidueCalculusandApplicationsofContourIntegration

4.1CauchyResidueTheorem

4.2EvaluationofCertainDefiniteIntegrals

4.3PrincipalValueIntegralsandIntegralswithBranchPoints

4.3.1PrincipalValueIntegrals

4.3.2IntegralswithBranchPoints

4.4TheArgumentPrinciple,RouchesTheorem

*4.5FourierandLaplaceTransforms

*4.6ApplicationsofTransformstoDifferentialEquations

PartIIApplicationsofComplexFunctionTheory

5ConformalMappingsandApplications

5.1Introduction

5.2ConformalTransformations

5.3CriticalPointsandInverseMappings

5.4PhysicalApplications

*5.5TheoreticalConsiderations-MappingTheorems

5.6TheSchwarz-ChristoffelTransformation

5.7BilinearTransformations

*5.8MappingsInvolvingCircularArcs

5.9OtherConsiderations

5.9.1RationalFunctionsoftheSecondDegree

5.9.2TheModulusofaQuadrilateral

*5.9.3ComputationalIssues

6AsymptoticEvaluationofIntegrals

6.1Introduction

6.1.1FundamentalConcepts

6.1.2ElementaryExamples

6.2LaplaceTypeIntegrals

6.2.1IntegrationbyParts

6.2.2WatsonsLemma

6.2.3LaplacesMethod

6.3FourierTypeIntegrals

6.3.1IntegrationbyParts

6.3.2AnalogofWatsonsLcmma

6.3.3TheStationaryPhaseMethod

6.4TheMethodofSteepestDescent

6.4.1LaplacesMethodforComplexContours

6.5Applications

6.6TheStokesPhenomenon

*6.6.1SmoothingofStokesDiscontinuities

6.7RelatedTechniques

*6.7.1WKBMethod

*6.7.2TheMellinTransformMethod

7Riemann-HiibertProblems

7.1Introduction

7.2CauchyTypeIntegrals

7.3ScalarRiemann-HilbertProblems

7.3.1ClosedContours

7.3.2OpenContours

7.3.3SingularIntegralEquations

7.4ApplicationsofScalarRiemann-HilbertProblems

7.4.1Riemann-HilbertProblemsontheRealAxis

7.4.2TheFourierTransform

7.4.3TheRadonTransform

*7.5MatrixRiemann-HilbertProblems

7.5.1TheRiemann-HilbertProblemforRationalMatrices

7.5.2InhomogeneousRiemann-HilbertProblemsandSingularEquations

7.5.3TheRiemann-HilbertProblemforTriangularMatrices

7.5.4SomeResultsonZeroIndices

7.6TheDBARProblem

7.6.1GeneralizedAnalyticFunctions

*7.7ApplicationsofMatrixRiemann-HilbertProblemsandProblems

AppendixAAnswerstoOdd-NumberedExercises

Bibliography

Index

　　本书是Cambridge《应用数学系列丛书》之一，内容相当精辟，巧妙地展示了复变量在数学科学中的核心地位以及其在工程和物理科学应用中的关键性作用。复变量的引入不仅增加数学理论本身的完美性，更重要的是提供了一种解决一些数学疑难问题的途径，甚至可以说是解决有些问题的唯一途径。本书的内容分为两大部分。第一部分是整个课程的引入，包括：解析函数，积分，级数和残数积分等初等理论以及一些过渡性方法：复平面的普通微分方程、数值方法等。第二部分包括保形映射，渐近扩张以及Riemann-Hilbert问题。每章节都提供了大量的应用、图例以及练习，这些可以帮助读者加深对复变量的基本概念和基本定理的理解。新版本做了全新的改进，是研究生以及分析方向本科生的理想教程。