加入VIP
  • 专属下载特权
  • 现金文档折扣购买
  • VIP免费专区
  • 千万文档免费下载

上传资料

关闭

关闭

关闭

封号提示

内容

首页 复分析可视化方法[英文]

复分析可视化方法[英文].pdf

复分析可视化方法[英文]

右眼狂跳2013
2014-03-19 0人阅读 举报 0 0 暂无简介

简介:本文档为《复分析可视化方法[英文]pdf》,可适用于高等教育领域

PrefaceAcknowledgementsContentsGeometryandComplexArithmeticIntroductionHistoricalSkentchBombelli´s"WildThought"SometerminologyandnotationPracticeEquivalenceofSymbolicandgeometricarithmeticEuler´sFormulaIntroductionMovingparticleargumentPowerseriesargumentSineandcosineintermsofEuler´sformulaSomeapplicationsIntroductionTrigonometryGeometryCalculusAlgebraVectorialoperationsTransformationsandEuclideangeometryGeometrythroughtheeyesofFelixKleinClassifyingmotionsThreereflectionstheoremSimilaritiesandComplexarithmeticSpatialcomplexnumersExcercisesComplexfunctionsastransformationsIntroductionPolynomalsPositiveIntegerPowersCubicsrevisited*CassinianCurves*PowerseriesThemysteryofrealpowerseriesThediscofconvergenceApproximatingapowerserieswithapolynomialUniquenessManipulatingpowerseriesFindingtheradiusofconvergenceFourierseries*TheexponentialfunctionPowerseriesapproachThegeometryofthemappingAnotherapproachCosineandsineDefinitionsandidentitiesRelationtohyperbolicfunctionsThegeometryofthemappingMultifunctionsExample:FractionalpowersSinglevaluedbranchesofamultifunctionRelevancetopowerseriesAnexamplewithtwobranchpointsThelogarithmfunctionInverseoftheexponentialfunctionThelogarithmicpowerseriesGeneralpowersAveragingovercircles*ThecentroidAveragingoverregularpolygonsAveragingovercirclesExercisesMöbiusTransformationsandInversionIntroductionDefinitionofMöbiustransformationsConnectionwithEinstein´stheoryofrelativity*DecompositionintosimpletransformationsInversionPreliminarydefinitionsandfactsPreservationofcirclesConstructionusingorthogonalcirclesPreservationofanglesPreservationofsymmetryInversioninasphereThreeillustrativeapplicationsofinversionAproblemontouchingcirclesQuadrilateralswithorthogonaldiagonalsPtolemy´stheoremTheriemannsphereThepointatinfinityStereograficprojectionTransferringcomplexfunctionstothesphereBehaviouroffunctionsatinfinityStereographicformulaeMöbiustransformations:BasicresultsPreservationofcircles,anglesandsymmetryNonuniquenessofthecoefficientsThegrouppropertyFixedpointsFixedpointsatinfinityThecrossratioMöbiustransformationsasmatrices*EvidenceofalinkwithlinearalgebraTheexplanation:HomogeneouscoordinatesEigenvectorsandeigenvaluesRotationsofthesphereVisualizationandclassificationThemainideaElliptic,hiperbolic,andloxodromictypesLocalgeometricinerpretationofthemultiplerParabolictransformationsComputingthemultiplerEingenvalueinterpretationofthemultiplerDecompositionintoorreflectionsIntroductionEllipticcaseHyperboliccaseParaboliccaseSummaryAutomorphismsoftheunitdiscCountingderreesoffreedomFindingtheformulaviathesymmetryprincipieInterpretingtheformulageometricallyIntroductiontoRiemann´sMappingTheoremExercisesDifferentiation:theamplitwistconceptIntroductionApuzzlingphenomenonLocaldescriptionofmappingsintheplaneIntroductionThejacobianmatrixTheamplitwistconceptThecomplexdireivativeasamplitwistTherealderivativereexaminedThecomplexderivativeAnalyticfunctionsAbriefsummarySomesimpleexamplesConformal=analyticIntroductionConformalitythroughoutaregionConformalityandtheRiemannsphereCriticalpointsDegreesofcrushingBreakdownofconformalityBranchpointsTheCauchyRiemannequationsIntroductionThegeometryoflineartransformationsTheCauchyRiemannequationsExercisesFurthergeometryofdifferentiationCauchyRiemannrevealedIntroductionThecartesianformThepolarformAnintimationofrigidityVisualdifferentiationoflog(z)RulesofdifferentiationCompositionInversefunctionsAdditionandmultiplicationPolynomials,powerseries,andrationalfunctionsPolynomialsPowerseriesRationalfunctionsVisualdifferentiationofthepowerfunctionVisualdifferentiationofexp(z)GeometricsolutionofE´=EAnapplicationfohigherderivates:curvature*IntroductionAnalytictransformationofcurvatureComplexcurvatureCelestialmechanics*CentralforcefieldsTwokindsofellipticalorbitChangingthefirstintothesecondThegeometryofforceAnexplanationTheKasnerArnold´stheoremAnaliticcontinuation*IntroductionRigidityUniquenessPreservationofindentitiesAnalyticcontinuationviareflectionsExercisesNonEuclideangeometryIntroductionTheparallelaxiomSomefactsfromnoneuclideangeometryGeometryonacurvedsurfaceIntrinsicversusextrinsicgeometryGaussiancurvatureSurfacesofconstantcurvatureTheconnectionwithMöbiustransformationsSphericalgeometryTheangularexcessofasphericaltriangleMotionsofthesphereAconformalmapofthesphereSpatialrotationsasMöbiustransformationsSpatialRotationsandquaternionsHiperbolicgeometryThetractixandthepseudosphereTheconstantcurvatureofthepseudosphereAconformalmapofthepseudosphereBeltrami´shiperbolicplaneHiperboliclinesandreflectionsTheBolyaiLobachevskyformulaThethreetypesofdirectmotionDecompositionintotworeflectionsTheangularexcessofahiperbolictriangleThePoincarediscMotionsofthePoincarédiscThehemispheremodelandhyperbolicspaceExercisesWindingnumbersandtopologyWindingnumberDefinitionWhatdoes"inside"meanFindingwindingnumbersquicklyHopf´sdegreetheoremTheresultLoopsasmappingsofthecircle*Theexplanation*PolynomialsandtheargumentprincipieAtopologicalargumentprincipie*CountingpreimagesalgebraicallyCountingpreimagesgeometricallyTopologicalcharacteristicsofanalyticityAtopologicalargumentprincipieTwoexamplesRouché´stheoremTheresultThefundamentaltheoremofalgebraBrouwer´sfixedpointtheorem*MaximaandminimaMaximummodulustheoremRelatedresultsTheSchwarzPicklemma*Schwarz´slemmaLiouville´stheoremPick´sresultThegeneralizedargumentprincipleRationalfunctionsPolesandessentialsingularitiesTheexplanation*ExercisesComplexintegration:Cauchy´stheoremIntroductionTherealintegralTheRiemannsumThetrapezoidalruleGeometricestimationoferrorsThecomplexintegralComplexRiemannsumsVisualTechniqueAusefulinequalityRulesofintegrationComplexinversionAcirculararcGeneralloopsWindingnumberConjugationIntroductionAreainterpretationGeneralloopsPowerfunctionsIntegrationalongacirculararcComplexinversionasalimitingcaseGeneralcontoursandthedeformationtheoremAfurtherextensionofthetheoremResiduesTheexponentialmappingThefundamentaltheoremIntroductionAnexampleThefundamentaltheoremTheintegralasantiderivateLogaritmasintegralParametricevaluationCauchy´stheoremSomepreliminariesTheexplanationThegeneralCauchytheoremTheresultTheexplanationAsimplerexplanationThegeneralformulaofcontourintegrationExercisesCauchy´sformulaanditsapplicationsCauchy´sFormulaIntroductionFirstexplanationGauss´meanvaluetheoremGeneralCauchyformulaInfinitedifferentiabilityandTaylorseriesInfinitydifferentiabilityTaylorseriesCalculusofresiduesLaurentseriescentredatapoleAformulaforcalculatingresiduesApplicationtorealintegralsCalculatingresiduesusingtaylorseriesApplicationtosummationofseriesAnnularLaurentseriesAnexampleLaurent´stheoremExercisesVectorfields:physicsandtopologyVectorfieldsComplexfunctionsasvectorfieldsPhysicalvectorfieldsFlowsandforcefieldsSourcesandsinksWindingnumbersandvectorfields*TheindexofasingularpointTheindexaccordingtoPoincaréTheindextheoremFlowsonclosedsurfaces*FormulationofthePoincaréHopftheoremDefiningtheindexonasurfaceAnexplanationfothePoincaréHopftheoremExercisesVectorfieldsandcomplexintegrationFluxandworkFluxWorkLocalfluxandlocalworkDivergenceandcrulingeometricform*DivergencefreeandcrulfreevectorfieldsComplexintegrationintermsofvectorfieldsThePólyavectorfieldCauchy´stheoremExample:AreaasfluxExample:WindingnumberasfluxLocalbehaviourofvectorfields*Cauchy´sformulaPositivepowersNegativepowersandmultipolesMultipolesatinfinityLaurent´sseriesasamultipoleexpansionThecomplexpotentialIntroductionThestreamfunctionThegradientfieldThepotentialfunctionThecomplexpotentialfunctionExamplesExercisesFlowsandharmonicfunctionsHarmonicdualsDualflowsHarmonicdualsConformalinverianceConformalinvarianceofharmonicityConformalinvarianceoftheLaplacianThemeaningfotheLaplacianApowerfulcomputationaltoolThecomplexcurvaturerevisited*SomegeometryofharmonicequipotentialsThecurvatureofharmonicequipotentialsFurthercomplexcurvaturecalculationsFurthergeometryofthecomplexcurvatureFlowaroundanoblstacleIntroductionAnexampleThemetothofimagesMappingoneflowontoanotherThephysicsofRiemann´smappingtheoremIntroductionExteriormappingsandflowsroundobstaclesInteriormappingsanddipolesInteriormappings,vortices,andsourcesAnexample:automorphismsofthediscGreen´sfunctionDirichlet´sproblemIntroductionSchwarz´sinterpretationDirichlet´sproblemforthediscTheinterpretationsofNeumannandBöcherGreengeneralformulaExercisesReferencesIndex

用户评价(0)

关闭

新课改视野下建构高中语文教学实验成果报告(32KB)

抱歉,积分不足下载失败,请稍后再试!

提示

试读已结束,如需要继续阅读或者下载,敬请购买!

文档小程序码

使用微信“扫一扫”扫码寻找文档

1

打开微信

2

扫描小程序码

3

发布寻找信息

4

等待寻找结果

我知道了
评分:

/49

复分析可视化方法[英文]

仅供在线阅读

VIP

在线
客服

免费
邮箱

爱问共享资料服务号

扫描关注领取更多福利