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首页 04 数学补充——函数论基础.pdf

04 数学补充——函数论基础.pdf

04 数学补充——函数论基础.pdf

上传者: zenger2017 2018-03-03 评分 0 0 0 0 0 0 暂无简介 简介 举报

简介:本文档为《04 数学补充——函数论基础pdf》,可适用于自然科学领域,主题内容包含ElementaryConsiderationsonFunctionTheoryInpotentialtheoryaswellaslateronin符等。

ElementaryConsiderationsonFunctionTheoryInpotentialtheoryaswellaslateroninthecomputationofcomplicatedintegrals,weneedmethodscomingfromfunctiontheory,ie,fromthetheoryoffunctionswithcomplexargumentsForthesereasons,wewillarmourselveswiththemostimportanttoolsinthisbranchofmathematicsComplexnumbersComplexnumberscanbewrittenintheform:ReImzxiyziz,:i()wherexandyarerealnumbersThesetofallzdenotedbycanberepresentedbyaplane:everypointoftheplanecorrespondstoacomplexnumberduetotheisomorphismBesides()thereisanalternativerepresentation||izze()with||zxy,tanyxItcorrespondstotheintroductionofpolarcoordinatesinthecomplexplane(seeFigure)Obviously,acomparisonof()and()yields||cosxz,||sinyzInthearithmeticaloperations,iistreatedlikeanalgebraicquantity,eg,()()()()xiyxiyxxiyy()()()xiyxiyxxyyixyxy()()()()()xiyxiyxiyxxyyixyxyxiyxiyxiyxyTwocomplexnumbersareequaltoeachotherifandonlyiftheirrealparts,aswellasimaginaryparts,areequaltoeachotherSinceiisdefinedbythe(manyvalued)rootfunctiontheexponentiallawcannotbetransferredthatis,therootcannotbesolved,asthefollowingexampledemonstrates:iiOneisonlyallowedtorewriteibytakingitspowers,eg,iiiiiInadditiontotheknownbasicarithmeticoperations,animportantoperationisthecomplexconjugation,indicatedbyanasterisk:():zxiyxiy()OnemaybeconvincedeasilythatRe()zxzz,Im()zyzzi,||zxyzz()Example:ComputingwithcomplexnumbersWecompute()()ii:iiiiiiiiCauchyRiemanndifferentialequationsThesignificantstatementsoffunctiontheoryrefertofunctions()fzdefinedonanopenandconnectedsubset(domain)Gof:():fzGThenotionscontinuityanddifferentiabilityofcomplexfunctionsareintroducedanalogouslytotherealcase,butwithafundamentaladditionalrequirement:Continuity()fziscalledcontinuousinzGifforanythereis()suchthatfor||zz,|()()|fzfzDifferentiability()fziscalledcomplexdifferentiable,holomorphic,regular,oranalyticinzGifthelimit()()()limzzzzfzfzdfzdzzz()existsandisuniqueHere,tobeuniquemeansthatthelimitingvalueisindependentofthepathalongwhichoneapproachesthepointzAtfirst,thereisaninfinitenumberofpossibilities,buttheymaybereducedtotwoindependentones:thetwolimitsinthedirectionofthecoordinateaxesWithzzz,onecanrewriteequation():()()()limzzzfzzfzdfzdzz()Now,weset(i)zxand(ii)ziyandobtain(i)()()limxfzxfzdffdzxx(ii)()()limyfziyfzdffdziyiyTherequirementthatthelimitingvaluehastobeindependentofthedirectionleadstoffxiy()Since()fzthedecomposition()(,)(,)fzuxyivxywith(,),(,)uxyvxy()isalwayspossibleSubstitutingequation()into(),acomparisonoftherealandtheimaginarypartleadstotheCauchyRiemanndifferentialequations:(,)(,)uxyvxyxy,(,)(,)uxyvxyyx()Hence,afunctionisdifferentiableinthecomplexdomainifitisdifferentiableintherealdomainandsatisfiestheequations()Theadditionalrequirement()isthereasonformanycuriouspropertiesofholomorphicfunctions,aswillbeshownnowBytheway,theCauchyRiemanndifferentialequationsmaybesummarizedalsointheshortform()iuivxy()Example:HolomorphicfunctionsArethefunctions()fzz,()Refzzholomorphic(),uxuvuvfzzvyxyyx()fzisholomorphicbecausetheCauchyRiemanndifferentialequations()arefulfilled()Reuxuvfzzvxy()fzisnotholomorphicbecausetheCauchyRiemannequationsarenotfulfilledanywhereLineintegralsLet():fzGbeafunctioncontinuousalongthepathCinG(seeFigure)Tocalculatetheintegral()Cfzdz,definedasthelimitingvalueofasum,likeintherealdomain,onehastoparametrizethepathCthatis,onedetermines:,zabCsuchthattoany,tabtherecorrespondsuniquelya()ztCIf()()zazb,thenthepathisclosedThen,thelineintegralisgivenby()(())()bCafzdzfztztdt()Thefollowingstatementsarevalid:()()CCfzdzfzdz()and()CfzdzMl()whereCisthepathtraversedintheoppositesense,Misthemaximumof|()|fzalongC,andlisthelengthofCTheproofsof()and()proceedanalogouslytotherealdomainTheyaresoobviousthatweomitthemhereExample:IntegrationinthecomplexplaneLet()fzzandCbetheunitcircleabouttheorigin(Figure)LettheparametrizationofCbegivenby()cossinzttit,:tAccordingto(),sincos()(cossin)cossincossinCtitfzdzdtitdtidtitittitthatisCdzizCauchy#sintegraltheoremLet()fzbeafunctionholomorphicinthecomplexdomainG,,zzGThen,thelineintegral()CfzdzhasthesamevalueforanpathsfromztozinG(seeFigure)Thisstatement,calledCauchy#sintegraltheorem,canbeformulatedequivalently:foranyclosedpathinG(Figure)()Cfzdz()ifthedomainenclosedbyCbelongsentirelytotheregionofregularitySinceaccordingtothefirststatement()()CCfzdzfzdz()fromequation()follows()ExampledemonstratesthenecessitythatanypointhastobelongtothedomainofregularityWeproveCauchy#sintegraltheoremintheformofequation()Onehastoshowthattheclosedlineintegral()Cfzdzvanishesifitisextendedoveranarbitrary,closedpathenclosingadomaininwhich()fzisregularToexaminethis,equation()iswritteninthefollowingway:()()()()()CCCCfzdzuivdxidyudxvdyiudyvdxTherealpart,aswellastheimaginarypart,ofthisintegralcanberegardedasascalarproductofthetwodimensionalvectors(,,)uvRand(,,)vuIinthe,xyplane,with(,,)ddxdydzrHence,wecanwrite()CCCfzdzdidRrIrAccordingtoStoke#sintegraltheoremwecantransformtheclosedlineintegralsintosurfaceintegrals:curl,,(,,)yyxxCFFFRRRRdddxdydxdyxyxyRrRFandcurl,,(,,)yyxxCFFFIIIIdddxdydxdyxyxyIrIFHereinthelast,wehaveusedthesecondandthefirst,respectively,ofequations(),and,hence,theregularityofthefunction()fz:yxRRvuxyxy,yxIIuvxyxyCauchy#sintegralformulaNow,wewillderiveCauchy#sintegralformulaasthemostimportantconsequenceofCauchy#stheorem:iffisregularinGandCisaclosedpathinGwhoseinteriorregionCGliesentirelyinG,thenforallCzG:()()Cffzdiz()Fortheproof,wewrite()()()()CCCffzffzdddizizizThefirstintegralontherighthandsideyields()()()()CCCfzfzdfzddfziziziz()whereCisaconcentriccircleaboutz,andthecorrespondingintegralissolvedasinExampleThetransitionfromCtoCisallowedaccordingtoequation()Thesecondintegralvanishesduetothecontinuityoff:onechoosestheradiusrofCsosmallthatforanyalways|()()|ffzThenaccordingto()()()()()CCffzffzddrizizrCauchy#sintegralformula()maybeunderstoodalsointhefollowingwayLetuswriteequation()intheform()()Cfzfzdzizz()Againwehaveassumedthedifferentiability(regularity)offinthedomainwiththeboundaryCWeexpand()fzinapowerseriesaboutz:()()()()()()!zzfzfzzzfzfzandrealizethatin()()()CCCfzfzdzdzfzdzizzizzi()!Czzfzdzi()allhighertermsvanishdueto()OnlythefirsttermcontainsanintegrandthatisnotdifferentiableatzzWeconsiderthisinmoredetail:()()()CCCfzfzfzdzdzdzizzizzizz()TheintegraloverCcanbeextendedalonganypathCaboutzsincethedifferenceoftheintegrationsovertwodistinctintegrationpathsCandCiszero(comparethistoFigure):CCCCdzdzdzzzzzzz,becausethefunction()zzisregularintheshadowedinteriorregionenclosedbythepathCCmeanstheintegrationpathindicatedinthefigureforwardandbackwardalongaconnectinglinebetweenCandC,whichofcourse,yieldnocontributionNowwecancalculate()easilybychoosingCtobesimplyacircleaboutz,asinFigure::iCzzre,:Weobtain()()()iiCfzfzdzredzizzizrez()()iiiifzfzirediredireire()()fzdfz()Substituting()into(),wecanimmediatelyobtainCauchy#sintegralformula()Justassimplywecanprove()!()()()nnCnfzfzdzizz()PowerseriesThesurprisingstatementoftheintegralformula(),namely,thatitissufficienttoknowafunctionalongaclosedpathtodetermineanyfunctionvalueintheinterior,isstillsurpassedbytheprincipleofanalyticcontinuationAtfirst,weintroduceapowerseriesby()()nnnPzazz()whoseradiusofconvergenceisgivenbytheCauchyHadamardformulalim||nnnra()HerelimmeansthelimessuperiorofthesequenceThevalidityofequation()isseenimmediatelystartingfromtheknownroottestfortheconvergenceofaseries:Aseriesnaisconvergentiflim||nnaForthepowerseries()thismeanslim||||||lim||nnnnnnnazzzzaor||lim||nnnzzaleadingdirectlytotheradiusofconvergence()SeeFigureTheidentitytheoremforpowerseriesstates:Iftwopowerseries()()nnnPzazz,()()nnnPzbzzcoincideinzforasequence{}lzwiththelimitingpointzthatis,if()lPz()lPzforalll,thenPandPareidenticalReally,thismeansnothingmorethanapowerserieswhichbecomeszerowithalimitingpointatinfinitelymanydistinctpointshastovanishidenticallyTheTaylor#stheoremstates:LetfbeholomorphicinG,zGThen,intheneighborhoodofzthereisauniquerepresentationoffbyapowerseries()()nnnfzazz()and()()!nnafzn()Thepowerseriesisconvergentwithinthelargestcircle(radiusr)aboutzcontainingonlypointsofGFromtheidentitytheoremforpowerseriesandTaylor#stheorem,theidentitytheoremforholomorphicfunctionsfollows:LetfandfbetwoholomorphicfunctionsinGandzGFurthermore,letffforanarbitraryneighborhoodofz,orforasequenceaccumulatinginz,orforanarbitrarypathsegmentemergingfromzThen,ffintheentiredomainGTheproofcanbegivenwiththeaidofthesocalledchainofcircleprocedure:Atfirst,oneexpandsfandfaboutzAccordingtoTaylor#stheoremthepowerseriesexpansionoffandfisuniqueandconvergentinthelargestcircleaboutzinsideGThen,fromtheidentitytheoremforpowerserieswehaveffforallzinsidethecircleofconvergenceNowonechoosesanewpointzpossiblyclosetotheperipheryofthefirstcircleandexpandsagainBythesameargument,theidentityoffandfisvalidinthenewcircleofconvergenceInthismannertheentiredomainGmaybecoveredbycircleswithinwhichffAdirectconclusionoftheidentitytheoremofholomorphicfunctionsistheprincipleofanalyticcontinuation:IffisholomorphicinGandGG,thenthereexistsonlyasingleholomorphiccontinuationoffintheentiredomainG(SeeFigure)Thechainofcirclesproceduredescribedaboveshowsinprinciple,thewayinwhichtheanalyticcontinuationofthefunctionmaybearchievedintermsofasuccessivepowerseriesexpansionLaurentseriesNowweinvestigateinwhichwaynonholomorphicfunctionsmayberepresentedbyseries(SeeFigure)Letusconsiderafunction()fzaboutwhoseregularityintheshadowedregion(inFigure)nothingcanbestatedFurthermore,itisassumedthatthetwoconcentriccirclesKandK(radii,rr)aboutzandtheregionofthecirclesbetweenthembelongtotheholomorphicdomainof()fzIntroducinganauxiliarypathSandusingwiththeintegralformula(),onecanshowatoncethatforanypointzwithinacircularring,thatis,foranypointzwith||rzzrr:()()()()KKKKffffzdddiziziz()ThecutSistraversedtwiceindifferentdirections,sothatthesecontributionscanceleachotherTheinnerringistraversedinamathematicallynegativesenseHence,thesecondtermontherighthandsideofequation()hasanegativesignIfliesontheoutercircleK,onecanwrite()()()()()nnnzzzzzzzzzzzbecausezzrqzrandnqqqforqIfliesonK,then()()()()()nnnzzzzzzzzzzzzsincenowzzrzrorzrzzrSubstitutingbothexpressionsintothecorrespondingintegrals(),oneobtains:()()()()()()()nnnnnnKKfzzfzfzddizizz()()()()nnnaaazzaazzazzzzzz()withthedefinition()()nnCfadiz()whereCmaybeequaltoKorK,oranyothercurveenclosingtheshadowedregionandlyinginsidethecircularringEquations()and()arecalledLaurent#sexpansionof()fzfortheregionofthecircularringIfzisapointforwhich()fzisnotholomorphicborwhichaholomorphicneighborhoodexists,onetalksaboutanisolatedsingularityIfzistheonlysingularityof()fzinsideC,then()Cafdi()iscalledtheresidueof()fzatzIftheseries()terminatesatfinitenegativen,thatis,ifnnaaornma()forallm,onetalksaboutapoleoftheordernatzziscalledanessentialsingularityifnOtherwise,ziscalledanonessentialsingularityResiduetheoremLet()fzbeafunctionregularinG,CaclosedpathwithinG,intheinteriorregionofwhichthereisafinitenumberofisolatedsingularpointsof()fzThen()ResiduesinCCfzdziG()Theproofoftheresiduetheorem()followsfromthedefinitionoftheresidues()andthealreadyfrequentlyuseddecomposition()()()NCCCfzdzfzdzfzdzToanswerthequestionofhowtocalculateresidues,weconsiderthefunction()fzwithapoleoftheordernatzThen,theLaurent#sserieshastheform()()()nnaafzaazzzzzz()Thecoefficienta,theresidue,isobtainedby(()())()!nnnzzdazzfzndz()Example:CalculationofaresidueLet()()zfzzAtz,()fzhasapoleoftheorder,hence,accordingto(),()()!()zzdzdazzdzzdzTheresiduetheoremisofgreatimportanceinthesolutionofrealdefiniteintegrals,asdemonstratedinthenextexampleExample:ComputationoftheintegralWehavetocomputedxIxAtfirst,theintegralistransferredintothecomplexintegralCdzIzwhereCisthepathindicatedinFigureThen,sclimlimRRIIIwherescIistheintegralalongthesemicircleThetransitionfromtheintegralItotheintegralIismeaningfulonlyiflimRII,thatis,iftheintegralscIovertheinfinitelydistantsemicirclevanishesAscanbeseenfromzizizi()()fzhasonlyonepoleziofthefirstorder,withtheresiduei,intheinteriorregionofCHence,limRIii()TheintegralovertheinfinitelydistantsemicirclescImaybeestimatedusingequation():scsc||max||idzIRzRe()sothatsclimRIInsummary,oneobtainsdxIx()ApplicationsinelectrostaticsAnyanalyticfunction()(,)(,)fzuxyivxyobeysLaplaceequationintwodimensions,becauseduetotheCauchyRiemanndifferentialequations(),uvvuuuxxyyxyxy()andvuuvvvxxyyxyxy()Thus,onehasalso()(,)fzfxyxy()If()fzisacomplexsolutionoftheLaplaceequation,thenalso()fz,thecomplexconjugateof()fz,issuchasolution,andtherefore,soisthesumofbothHence,bothof()()(,)fxiyfxiyuxyand()()(,)fxiyfxiyvxyi()aresolutionsofLaplace#sequation,asweknowalreadyfrom()Anyanalyticfunction,duetotheknowledgejustdeducedintermsofequations()and(),correspondstoasolutionofatwodimensionalboundaryproblemoftheDirichlettypeTounderstandthisevenbetter,weconsidertwocurvesofconstantpotentials(,)uxyinthe,xyplane,(,)uxyCand(,)uxyC()asillustratedinFigureNowthecurvescharacterizedbyconstant(,)uxyarealwaysorthogonaltothefamilyofcurvesgivenbyconstant(,)vxy,sinceuvuvvvvvuvxxyyyxxy()Theorthogonalityfollowsimmediatelyfrom()andfromthefactthatgraduandgradvareperpendiculartothecurvesofconstantuandv,respectivelyThisweknowalreadyfromourgeneralknowledgeongradientsbutwemayalsorepeatitrapidly:consideringtwoneighboringpointsxanddxxintheplane,()()uuduududxdyudxyxxxx()Ifdxliesinthetangenttothecurve,then()uxisnotallowedtochangebecausethecurveisdeterminedby()constuxHence,alongthecurvewehavegrad()()()ududuxxxxx()sothateverythingisclearNowweassumethat(,)uxyistheelectrostaticpotentialbelongingtoaplanepotentialproblemThen,theelectricfieldintensitycanbeobtainedfromtheregularfunction()fzwhoserealpartis(,)uxyNamely,xydfuvuuiiEiEdzxxxy()Hence,thecomponentsofthefieldintensityxEandyEareidenticalwiththerealandimaginarypartof()dfdzOnecalls()(,)(,)fzuxyivxythecomplexpotential,(,)uxythepotential,and(,)vxythestreamfunctionTheelectricfieldgraduEpointseverywhereinthedirectionof(,)constvxyObviously,thesearethefieldlinesoftheproblemInthecontextdiscussedhere(electrostatics)thephysicalmeaningofthestreamfunctionistorepresentameasureofthechargedensityontheboundarysurfacesiftheboundarysurfaceisaconductorInthiscase,thepotentialoftheboundarysurfaceis(,)constuxyandthelines(,)constvxyareperpendiculartotheboundarysurfaceChoosingthenormalvectortotheboundarysurface(,)xynnntobedirectedoutwardand(,)xyttt(,)yxnntolietangentiallyintheboundarysurface(compareFigure),thenthesurfacechargeisxyxyuuvvunnnnxyyxEnnyxvvvttvyxtt()ThechargeperunitoflengthdQdl(lmeansthecylindriccoordinateperpendiculartothe,xyplane)betweenandontheboundarycurve(theimageoftheboundarysurfaceinthe,xyplane)is()dQvdtvvdlt()Example:PotentialofachargedwireWeconsiderthecomplexfunction()lnqfzzL()andintroducecylindricpolarcoordinatesbysettingizreNowwemustnotethatthefunctionlnlnzriin,n()ismanyvaluedbecauseinfinitelymanyvaluesin()yieldthesame()inzreInordertoclarifythis,wedifferentiate()forfixednandrealizethatthederivativelndzdzdoesnotdependonnItexistseverywhereexceptattheorigin(r)andatinfinity(r)InmanysimilarcasesitisthebesttoavoidthemanyvaluednessfromthebeginningSo,wemaydefinethelogarithmicfunctioninauniquemannerbyprescribingImlnz()Thederivativeofthelnzfunctiondefineduniquelyinthiswaydoesexisteverywhereexceptatr,randalongthecutsandWiththisdescription,function()reads()(ln)qfzriL()sothatlnqurL()istheelectricpotentialandqvL()isthestreamfunctionBecauseisusuallydefinedinthecounterclockwisesense(Figure),thatis,oppositethetangentvectortinthefigure,thetotalchargeperunitoflengthis()()dQqvvdlL()accordingtoequation()Thus,thepotential()referstoawirewiththelinechargedensityqLristhedistancefromthewireExample:ThepotentialattheedgeofaparallelplatecapacitorLetusconsiderthefunction()(,)(,)fzuxyivxydefinedimplicitlybythefollowingequations:()()ifzizifzeoriuvixyiuve()Hence,sinvxueu,cosvyveu()WeconsideraregioninspaceboundedbytheequipotentialsurfacesuanduAccordingtoequation(),foruweobtainx,vyve()Thefunction()yvhasanextremumforvdyedv()givingvNowitisclearthatvyveisalwayspositiveandhasaminimumwithvandyThereforetheboundarycurves(surfaces)udeterminetheplatesofacapacitorasrepresentedinFigureForv(andthereforey),oneobtainsfrom()xu,yv()implyingthattheequipotentialcurves(surfaces)areverticalstraightlinesTheirdistancefromthexdirectionisproportionaltouThiscorrespondsperfectlytothesituationintheinteriorofacapacitor,farfromtheedgeIfvoneobtainsfrom()sinvxeu,cosvyeu()ifneithersinunorcosuvanishThen,tanxuy()andtheequipotentialcurves(surfaces)becomestraightlinesagainFrom()onecanseethaturepresentstheentireyaxisForuonehasyv()sothattheequationfortheequipotentialcurvereadsvxe()ThiscurvenevergoesintoastraightlineasinthecaseoftheotherequipotentialcurvesonbothsidesofthatbelongingtoExample:PotentialofachargedwireinfrontofaconductingsurfaceAsthelastexampleofthesolutionofthepotentialproblembymeansoffunctiontheorywediscussthepotentialofalinecharge(chargedwire)inthevincinityofasurfaceasrepresentedinFigureThelinechargedensity(chargeperunitoflength)isdenotedbyqlLetthewirecrossthe,xyplaneatxyc(compareFigure),andlettheequationforthehyperbolicsurfacebexycWeassumethesurfacetobegroundedInthisproblemitissuitabletointroducethevariableszzxyixyxxy,yxy()Thenthecurvexycandthepointxycarerepresentedbyycand,xyc,respectivelySo,inthe,xyplanetheproblemisreducedtotheproblemofapointchargeqlatzicinfrontoftheplaneycwhichwillbesolvedwiththeimagechargeqlatzic(seeExample)Hence,thecomplexpotential(seeExample)is()ln()ln()lnqqqzicfzzicziclllzic

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