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PAGE第PAGE2頁(yè)(共5頁(yè))復(fù)變函數(shù)論(A)Ⅰ.ClozeTests(Points)If,then.Ifdenotesthecirclecenteredatpositivelyorientedandisapositiveinteger,then.Theradiusofconvergenceofis.Thesingularpointsofthefunctionare.,whereisapositiveinteger..Themainargumentandthemodulusofthenumberare.Thesquarerootsofare.Thedefinitionofis.Log=.Ⅱ.TrueorFalseQuestions(Points)Ifafunctionisanalyticatapoint,thenitisdifferentiableat.()Ifapointisapoleoforderof,thenisazerooforderof.()Aboundedentirefunctionmustbeaconstant.()Afunctionisanalyticapointifandonlyifwhoserealandimaginarypartsaredifferentiableat.()Ifiscontinuousontheplaneand0foreverysimpleclosedpath,thenisanentirefunction.()Ⅲ.Computations(Points)Find.Findthevalueof.Let,findtheLaurentexpansionofontheannulus.Given,where,find.5.Given,find.Ⅳ.Verifications(Points)Showthatif,thenisapolynomialoforder.Showthat,whereisthecirclecenteredatwithradius.Showthattheequationhasjusttworootsintheunitedisk復(fù)變函數(shù)論(B)Ⅰ.ClozeTests(Points)If,then.Ifdenotesthecirclecenteredatpositivelyorientedandisapositiveinteger,then.Theradiusofthepowerseriesis.Thesingularpointsofthefunctionare.,whereisapositiveinteger..Themainargumentandthemodulusofthenumberare.Thesquarerootsof1+are.Thedefinitionofis.Log=.得分評(píng)卷人Ⅱ.TrueorFalseQuestions(Points)Ifafunctionisdifferentiableatapoint,thenitiscontinuousat.()Ifapointisapoleoforderof,thenisazerooforderof.()Anentirefunctionwhichmapstheplaneintotheunitediskmustbeaconstant.()AfunctionisdifferentiableatapointifandonlyifwhoserealandimaginarypartsaredifferentiableatandtheCauchyRiemannconditionsholdthere.()Ifafunctioniscontinuousontheplaneand0foreverysimpleclosedcontour,thenisanentirefunction.()得分評(píng)卷人Ⅲ.Computations(Points)Find.Findthevalueof.Let,findtheLaurentexpansionofontheannulus.4.Given,where,find.5.Given,find.得分評(píng)卷人Ⅳ.Verifications(Points)Showthatthefunctionisanentirefunction.Showthatif,thenisapolynomialoforder.3.Showthat,whereisthecirclecenteredatwithradius.復(fù)變函數(shù)論(C)Ⅰ.ClozeTests(Points)If,then.Ifdenotesanysimpleclosedcontourandisapointinside,then,whereisaninteger.Theradiusofconvergenceofthepowerseriesis.Thesingularpointsofthefunctionare.,whereisapositiveinteger.Themainargumentandthemodulusofthenumberare.Theintegralofthefunctiononis.Thedefinitionofis.Log=.Thesolutionsoftheequationare.得分評(píng)卷人Ⅱ.TrueorFalseQuestions(Points)Ifafunctioniscontinuousatapoint,thenitisdifferentiableat.()Ifapointisapoleoforderof,thenthereisafunctionthatisanalyticatwithsuchthatonsomedeletedneighborhoodof.()Anentirefunctionwhichisidenticallyzeroonalinesegmentmustbeidenticallyzero.()AfunctionisdifferentiableonopensetifandonlyifwhoserealandimaginarypartsaredifferentiableonandtheCauchyRiemannconditionsholdon.()Ifafunctioniscontinuousontheplaneand0foreverysimpleclosedpath,thenforall.()得分評(píng)卷人Ⅲ.Computations(Points)Find.Findthevalueof.Let,findtheLaurentexpansionofontheannulus.Given,where,find.5.Find.得分評(píng)卷人Ⅳ.Verifications(Points)Showthat,whereisthecirclecenteredatwithradius.2.Supposethatisanalyticandisaconstantonadomainadomain,provethatforsomeconstantandall.3.Showthattheequationhasjustthreerootsintheunitedisk.《復(fù)變函數(shù)論》試題(D)Ⅰ.ClozeTests(Points)If,then.Ifdenotesthecirclecenteredatpositivelyorientedandisapositiveinteger,then.Theradiusofthepowerseriesis.Thesingularpointsofthefunctionare.,whereisapositiveinteger..Themainargumentandthemodulusofthenumberare.Thesquarerootsof1+are.Thedefinitionofis.Log=.Ⅱ.TrueorFalseQuestions(Points)Ifafunctionisdifferentiableatapoint,thenitisanalyticat.()Ifapointisapoleoforderof,thenisazerooforderof.()Aboundedentirefunctionmustbeaconstant.()AfunctionisanalyticapointifandonlyifwhoserealandimaginarypartsaredifferentiableandtheCauchyRiemannconditionsholdinaneighborhoodof.()Ifafunctioniscontinuousontheplaneand0foreverysimpleclosedcontour,thenisanentirefunction.()Ⅲ.Computations(Points)Find.Findthevalueof.Let,findtheLaurentexpansionofontheannulus.Given,where,find.Given,find.Ⅳ.Proving(Points)Showthatif,thenisapolynomialoforder.Showthat,whereisthecirclecenteredatwithradius.Showthattheequationhasjusttworootsintheunitedisk.《復(fù)變函數(shù)論》試題(E)Ⅰ.ClozeTests(Points)If,then.Ifdenotesthecirclecenteredatandisaninteger,then.Theradiusofthepowerseriesis.Thesingularpointsofthefunctionare.,whereisapositiveinteger..Themainargumentandthemodulusofthenumberare.Thesquarerootsofare.Thedefinitionofis.Log=.Ⅱ.TrueorFalseQuestions(Points)Ifafunctionisdifferentiableatapoint,thenitiscontinuousat.()Ifapointisazerooforderof,thenisapoleoforderof.()Thereisanon-constantentirefunctionwhichmapstheplaneintothedisk.()AfunctionisdifferentiableatapointifandonlyifwhoserealandimaginarypartsaredifferentiableatandtheCauchyRiemannconditionsholdthere.()Ifafunctioniscontinuousontheplaneand0foreverysimpleclosedcontour,thenitisanentirefunction.()Ⅲ.Computations(Points)Findtheintegral,whereisthecircle.Findthevalueof.Let,findtheLaurentexpansionofontheannulus.Given,where,find.Given,find.Ⅳ.Proving(Points)Showthat,whereisthecirclecenteredatwithradius.Supposethatisanentirefunctionandthereisaconstantandapositiveintegersuchthat.Provethatforsomeconstants,andallintheplane.3·Showthattheequationhasjustthreerootsintheunitedisk2005-2006學(xué)年第一學(xué)期期末考試2003級(jí)數(shù)學(xué)與應(yīng)用數(shù)學(xué)專業(yè)《復(fù)變函數(shù)論》試題(C)Ⅰ.ClozeTests(Points)If,then.Ifdenotesanysimpleclosedcontourandisapointinside,then,whereisaninteger.Theradiusofthepowerseriesis.Thesingularpointsofthefunctionare.,whereisapositiveinteger.Themainargumentandthemodulusofthenumberare.Theintegralofthefunctiononis.Thedefinitionofis.Log=.Thesolutionsoftheequationare.Ⅱ.TrueorFalseQuestions(Points)Ifafunctioniscontinuousatapoint,thenitisdifferentiableat.()Ifapointisapoleoforderof,thenthereisanalyticfunctionatwithsuchthatonsomedeletedneighborhoodof.()Anentirefunctionwhichisidenticallyzeroontherealaxismustbezero.()AfunctionisdifferentiableonadomainifandonlyifwhoserealandimaginarypartsaredifferentiableonandtheCauchyRiemannconditionsholdon.()Ifafunctioniscontinuousontheplaneand0foreverysimpl

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