復(fù)變函數(shù)復(fù)習(xí)資料

PAGE第PAGE２頁(共5頁)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=.得分評卷人Ⅱ.TrueorFalseQuestions(Points)Ifafunctionisdifferentiableatapoint，thenitiscontinuousat.（）Ifapointisapoleoforderof，thenisazerooforderof.（）Anentirefunctionwhichmapstheplaneintotheunitediskmustbeaconstant.（）AfunctionisdifferentiableatapointifandonlyifwhoserealandimaginarypartsaredifferentiableatandtheCauchyRiemannconditionsholdthere.（）Ifafunctioniscontinuousontheplaneand0foreverysimpleclosedcontour,thenisanentirefunction.()得分評卷人Ⅲ.Computations(Points)Find.Findthevalueof.Let，findtheLaurentexpansionofontheannulus.4.Given，where，find.5.Given，find.得分評卷人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.得分評卷人Ⅱ.TrueorFalseQuestions(Points)Ifafunctioniscontinuousatapoint，thenitisdifferentiableat.（）Ifapointisapoleoforderof，thenthereisafunctionthatisanalyticatwithsuchthatonsomedeletedneighborhoodof.（）Anentirefunctionwhichisidenticallyzeroonalinesegmentmustbeidenticallyzero.（）AfunctionisdifferentiableonopensetifandonlyifwhoserealandimaginarypartsaredifferentiableonandtheCauchyRiemannconditionsholdon.（）Ifafunctioniscontinuousontheplaneand0foreverysimpleclosedpath,thenforall.()得分評卷人Ⅲ.Computations(Points)Find.Findthevalueof.Let，findtheLaurentexpansionofontheannulus.Given，where，find.5.Find.得分評卷人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