剑桥大学――数理方法

MethodsofMathematicalPhysicsDr.M.G.Worster1Michælmas19971LATEXedbyPaulMetcalfe–[e-mail]:2.3Date:1998-07-0620:35:46+01Thefollowingpeoplehavemaintainedthesenotes.–datePaulMetcalfeContentsIntroductionv1ComplexVariables11.1Conventions...............................11.2CauchyPrincipalValue.........................11.3AnalyticContinuation..........................31.4Multivaluedfunctions..........................51.4.1Branchcutintegrals......................61.4.2Riemannsurfaces........................62SpecialFunctions72.1TheGammaFunction..........................72.2TheBetafunction............................82.3TheRiemannzetafunction.......................102.3.1Applicationstonumbertheory.................113SecondorderlinearODEs133.1MethodofFrobenius..........................143.1.1Bessel’sEquation........................153.2Classificationofequationsbysingularities...............163.2.1Equationswithnoregularsingularpoints...........163.2.2Equationswithoneregularsingularpoint...........163.2.3Equationswithtworegularsingularpoints..........173.2.4Equationswiththreeregularsingularpoints..........173.3Integralrepresentationofsolutions...................194AsymptoticExpansions234.1Motivation................................234.2Definitionsandexamples........................244.2.1Manipulations.........................254.3StokesPhenomenon...........................254.4AsymptoticApproximationofIntegrals................254.4.1Integrationbyparts.......................254.4.2Watson’sLemma........................274.4.3Laplace’sMethod........................274.4.4Themethodofstationaryphase................294.4.5MethodofSteepestDescents..................304.5Liouville-GreenFunctions.......................32iiiivCONTENTS4.5.1Connectionformulae......................335LaplaceTransforms375.1Definitionandsimpleproperties....................375.1.1AsymptoticLimits.......................385.1.2Convolutions..........................385.2Inversion................................395.3Applicationtodifferentialequations..................395.3.1Ordinarydifferentialequations.................395.3.2Partialdifferentialequations..................41IntroductionThesenotesarebasedonthecourse“MethodsofMathematicalPhysics”givenbyDr.M.G.WorsterinCambridgeintheMichælmasTerm1997.ThesetypesetnotesaretotallyunconnectedwithDr.Worster.Recommendedbookswillbediscussedattheend.Othersetsofnotesareavailablefordifferentcourses.Atthetimeoftypingthesecourseswere:ProbabilityDiscreteMathematicsAnalysisFurtherAnalysisMethodsQuantumMechanicsFluidDynamics1QuadraticMathematicsGeometryDynamicsofD.E.’sFoundationsofQMElectrodynamicsMethodsofMath.PhysFluidDynamics2Waves(etc.)StatisticalPhysicsGeneralRelativityDynamicalSystemsPhysiologicalFluidDynamicsBifurcationsinNonlinearConvectionSlowViscousFlowsTurbulenceandSelf-SimilarityAcousticsNon-NewtonianFluidsSeismicWavesTheymaybedownloadedfrom://@lists.cam.ac.uktogetacopyofthesetsyourequire.vCopyright(c)TheArchimedeans,CambridgeUniversity.Allrightsreserved.Redistributionanduseofthesenotesinelectronicorprintedform,withorwithoutmodification,arepermittedprovidedthatthefollowingconditionsaremet:1.Redistributionsoftheelectronicfilesmustretaintheabovecopyrightnotice,thislistofconditionsandthefollowingdisclaimer.2.Redistributionsinprintedformmustreproducetheabovecopyrightnotice,thislistofconditionsandthefollowingdisclaimer.3.Allmaterialsderivedfromthesenotesmustdisplaythefollowingacknowledge-ment:ThisproductincludesnotesdevelopedbyTheArchimedeans,CambridgeUniversityandtheircontributors.4.NeitherthenameofTheArchimedeansnorthenamesoftheircontributorsmaybeusedtoendorseorpromoteproductsderivedfromthesenotes.5.Neitherthesenotesnoranyderivedproductsmaybesoldonafor-profitbasis,althoughafeemayberequiredforthephysicalactofcopying.6.Youmustcauseanyeditedversionstocarryprominentnoticesstatingthatyoueditedthemandthedateofanychange.THESENOTESAREPROVIDEDBYTHEARCHIMEDEANSANDCONTRIB-UTORS“ASIS”ANDANYEXPRESSORIMPLIEDWARRANTIES,INCLUDING,BUTNOTLIMITEDTO,THEIMPLIEDWARRANTIESOFMERCHANTABIL-ITYANDFITNESSFORAPARTICULARPURPOSEAREDISCLAIMED.INNOEVENTSHALLTHEARCHIMEDEANSORCONTRIBUTORSBELIABLEFORANYDIRECT,INDIRECT,INCIDENTAL,SPECIAL,EXEMPLARY,ORCONSE-QUENTIALDAMAGESHOWEVERCAUSEDANDONANYTHEORYOFLI-ABILITY,WHETHERINCONTRACT,STRICTLIABILITY,ORTORT(INCLUD-INGNEGLIGENCEOROTHERWISE)ARISINGINANYWAYOUTOFTHEUSEOFTHESENOTES,EVENIFADVISEDOFTHEPOSSIBILITYOFSUCHDAM-AGE.Chapter1ComplexVariables1.1ConventionsVariouspeopleusedifferentmeaningsforanalytic,regular,etc.Wewillusethese:Definition.Afunctionisanalyticatapointiffthereexistsanopenneighbourhoodofthepointinwhichthefunctioniscomplexdifferentiable.ThisistrueiffthefunctionhasaTaylorexpansionaboutthatpoint.Definition.Afunctionisanalyticinadomainiffitisanalyticateverypointinthedomainandsinglevaluedinthedomain.Definition.Afunctionissingularatapointiffitisnotanalyticatthepo