复分析

'1GeometryandCompleXArIthmetIc

ⅠIntroductIon

ⅡEuler’sFormula

ⅢSomeApplIcatIons

ⅣTransformatIonsandEuclIdeanGeometry*

ⅤEXercIses

2CompleXFunctIonsasTransformatIons

ⅠIntroductIon

ⅡPolynomIals

ⅢPowerSerIes

ⅣTheEXponentIalFunctIon

ⅤCosIneandSIne

ⅥMultIfunctIons

ⅦTheLogarIthmFunctIon

ⅧAVeragIngoVerCIrcles*

ⅨEXercIses

3M?bIusTransformatIonsandInVersIon

ⅠIntroductIon

ⅡInVersIon

ⅢThreeIllustrativeApplIcatIonsofInVersIon

ⅣTheRIemannSphere

ⅤM?bIusTransformatIons:BasIcResults

ⅥM?bIusTransformatIonsasMatrIces*

ⅦVisualIzatIonandClassIfIcatIon*

ⅧDecomposItIonInto2or4ReflectIons*

ⅨAutomorphIsmsoftheUnItDIsc*

ⅩEXercIses

4DIfferentIatIon:TheAmplItwIstConcept

ⅠIntroductIon

ⅡAPuzzlIngPhenomenon

ⅢLocalDescrIptIonofMappIngsInthePlane

ⅣTheCompleXDerivativeasAmplItwIst

ⅤSomeSImpleEXamples

ⅥConformal=AnalytIc

ⅦCrItIcalPoInts

ⅧTheCauchy-RIemannEquatIons

ⅨEXercIses

5FurtherGeometryofDIfferentIatIon

ⅠCauchy-RIemannReVealed

ⅡAnIntImatIonofRIgIdIty

ⅢVisualDIfferentIatIonoflog(z)

ⅣRulesofDIfferentIatIon

ⅤPolynomIals,PowerSerIes,andRatIonalFunc-tIons

ⅥVisualDIfferentIatIonofthePowerFunctIon

ⅦVisualDIfferentIatIonofeXp(z)231

ⅧGeometrIcSolutIonofE’=E

ⅨAnApplIcatIonofHIgherDerivatives:CurVa-ture*

ⅩCelestIalMechanIcs*

ⅪAnalytIcContInuatIon*

ⅫEXercIses6Non-EuclIdeanGeometry*

ⅡIntroductIon

ⅡSpherIcalGeometry

ⅢHyperbolIcGeometry

ⅣEXercIses

7WIndIngNumbersandTopology

ⅠWIndIngNumber

ⅡHopf’sDegreeTheorem

ⅢPolynomIalsandtheArgumentPrIncIple

ⅣATopologIcalArgumentPrIncIple*

ⅤRouché’sTheorem

ⅥMaXImaandMInIma

ⅦTheSchwarz-PIckLemma*

ⅧTheGeneralIzedArgumentPrIncIple

ⅨEXercIses

8CompleXIntegratIon:Cauchy’sTheorem

ⅡntroductIon

ⅡTheRealIntegral

ⅢTheCompleXIntegral

ⅣCompleXInVersIon

ⅤConjugatIon

ⅥPowerFunctIons

ⅦTheEXponentIalMappIng

ⅧTheFundamentalTheorem

ⅨParametrIcEValuatIon

ⅩCauchy’sTheorem

ⅪTheGeneralCauchyTheorem

ⅫTheGeneralFormulaofContourIntegratIon

ⅫEXercIses

9Cauchy’sFormulaandItsApplIcatIons

ⅠCauchy’sFormula

ⅡInfInIteDIfferentIabIlItyandTaylorSerIes

ⅢCalculusofResIdues

ⅣAnnularLaurentSerIes

ⅤEXercIses

10VectorFIelds:PhysIcsandTopology

ⅠVectorFIelds

ⅡWIndIngNumbersandVectorFIelds*

ⅢFlowsonClosedSurfaces*

ⅣEXercIses

11VectorFIeldsandCompleXIntegratIon

ⅠFluXandWork

ⅡCompleXIntegratIonInTermsofVectorFIelds

ⅢTheCompleXPotentIal

ⅣEXercIses

12FlowsandHarmonIcFunctIons

ⅠHarmonIcDuals

ⅡConformalInVarIance

ⅢAPowerfulComputatIonalTool

ⅣTheCompleXCurVatureReVIsIted*ⅤFlowAroundanObstacle

ⅥThePhysIcsofRIemann’sMappIngTheorem

ⅦDirichlet’sProblem

ⅧExercIses

References

IndeX

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　　本书是复分析领域近年来较有影响的一本著作。作者用丰富的图例展示各种概念、定理和证明思路，十分便于读者理解，充分揭示了复分析的数学之美。书中讲述的内容有几何、复变函数变换、默比乌斯变换、微分、非欧几何、复积分、柯西公式、向量场、复积分、调和函数等。
　　本书是复分析领域近年来较有影响的一本著作。作者用丰富的图例展示各种概念、定理和证明思路，十分便于读者理解，充分揭示了复分析的数学之美。书中讲述的内容有几何、复变函数变换、默比乌斯变换、微分、非欧几何、复积分、柯西公式、向量场、复积分、调和函数等。本书可作为大学本科、研究生的复分析课程教材或参考书。
作者简介：
　　TristanNeedham，旧金山大学教授系教授，理学院副院长。牛津大学博士，导师为RogerPenrose（与霍金齐名的英国物理学家）。因本书被美国数学会授予CarlB.Allendoerfer奖。他的研究领域包括几何、复分析、数学史、广义相对论。