您好,欢迎访问三七文档
EXERCISESofCOMPLEXANALYSISⅠ.ClozeTests1.Themodulusofthenumber512iis2.Themodulusofthenumber34iis53.zedzdz2cos4.Themainargumentandthemodulusofthenumberi1are5.Thesquarerootsof1+iare.6.Thedefinitionofzcosis.7.Log)1(i=.8.IfCdenotesthecirclecenteredat0zpositivelyorientedandnisapositiveinteger,then)(10Cndzzz.9.Theintegralofthefunction2()(1)wtttion]1,1[is.10.Thesolutionsoftheequationsin1zare.11.||1ed3zzzz12.exp(1)i13.4(1)i14.331(3)dzz015.Thesolutionsoftheequationcos0zare1(),0,1,2,2znnⅡ.TrueorFalseQuestions1.Ifafunctionfisdifferentiableatapoint0z,thenitiscontinuousat0z.(T)2.Afunctionfisdifferentiableatapoint000iyxzifandonlyifwhoserealandimaginarypartsaredifferentiableat),(00yxandtheCauchyRiemannconditionsholdthere.()3.Ifafunctionfisanalyticatapoint0z,thenitisdifferentiableat0z.()4.Afunctionfisanalyticatapoint000iyxzifandonlyifwhoserealandimaginarypartsaredifferentiableat),(00yx.()5.1212()argzzargzargz.(T)6.1122()zArgArgzArgzz.()7.sin1z.()8.22sincos1zz.9.Theexponentialfunctionezisperiodic.(T)10.Ifafunctionfisdifferentiableatapoint0z,thenitisanalyticat0z.()11.Ifvisaharmonicconjugateofuinsomedomain,thenuisaharmonicconjugateofvthere.()12.Thelogarithmicfunctionisentire.()Ⅲ.Computations1.EvaluatetheintegraldCzz,whereCisthepositivelyorientedcircle1z.Solution:Thecircle1zmayberepresentedparametricallyas(02)itzet.Consequently,20()ititCeedtzdz;Sinceititeeand()ititeie,Thismeansthat202Cidtizdz2.Evaluatetheintegral12dCzz,whereCisdenotedthesemicircularpath(0)itzetfromthepoint1ztothepoint1z.3.Find1||)2)(12(zzzzdz4.Findthevalueof2||2d()(9)zzzzizSolution:Sinceziisaninteriorto2zand3zareoutofthecircle,let2()9zfzz,wecangetthat2||2||2()()d()(9)zzfzdzzizzzizConsidingthatfisannlyticwithinandonthecircle1z,wehave2||22()5d()(9)zifizzziz5.Findthevalueof223122)1(sinzzzzdzzdzzze6.Findthevalueof3||2sind(1)zzzzSolution:Itisclearthat1zisaninteriorto2z.Let()sinfzz,since()fzisentire,wehave3||22(1)02!sind(1)zifzzz.7.GivendzzfC142)(2,where3|:|zzC,find)1(if.Solution:Let2()241g,andweknowthat()gisanentirefunction.Thusweget2241()2()CfzdigzzaccordingtotheCauchyintegralformula.Thus,2()2()2(241)fzigzi.Since()2(44)fziz,wehave(1)2(4(1)4)8fiii.8.GivendzzfC765)(2,where4|:|zzC,find)1(if.Ⅳ.Verifications1.AssumeafunctionfisanalyticthroughoutagivendomainDanditsmodulus()fzisconstantonD.ShowthatthefunctionfmustbeconstantonD.Proof:Considering()fzisconstantonD,wedivide()fzinto()0fzand()0fz.When()0fz,obviously,wegetthat0fonD.When()0fzc,thefactthat2()()fzfzctellsusthatisneverzeroonD.Hence2()()cfzfzforallzinD,Andwegetthat()fzisanalyticonDbytheassumptionthatfisanalyticonD.Thusitiseasytoshowthat()fzmustbeconstantonD.2.Showthatthefunction()3(3)fzxyiyxisentire.Proof:Thecomponentfunctionsare(,)3uxyxyand(,)3vxyyx.Because3xu,1yu,1xv,3yu,andxyuvandyxuveverywhere,itisclearthat()fzisentire.3.Showthat32(,)3uxyyxyisharmonicinsomedomainandfinditsharmonicconjugate(,)vxyandthecorrespondinganalyticfunction()fz.Proof:Since226,33xyuxyuyx,6,6xxyyuyuyand0xxyyuu,Wegetthat(,)uxyisharmonicthroughouttheentirexyplane.Sinceaharmonicconjugate(,)vxyisrelatedto(,)uxybymeansoftheCauchy-Riemannequations,xyyxuvuv.Thefirstequationtellsusthat(,)6yvxyxy,Andwefindthat2(,)63()vxyxydyxygxwhere()gxis,atpresent,anarbitraryfunctionofx.Usingthesecondequation,wehave23()xgx,thus3()Cgxx,whereCisanarbitraryrealnumber.Thefunction233vxyxCisaharmonicconjugateof(,)uxy.Thecorrespondingananlyticfunctionis32233()(3)(3)()fzyxyixyxCizC4.Showthat(,)2(1)uxyxyisharmonicinsomedomainandfinditsharmonicconjugate(,)vxyandthecorrespondinganalyticfunction()fz.Ⅴ.Findadomaininthezplanewhoseimageunderthetransformation2wzisthesquaredomaininthewplaneboundedebytheline1u,2u,1vand2v.Solution:Thetransformation2wzcanbewritten222wxyxyi,soitcanbeexpressedintheform22,2uxyvxy.Thesquaredomaininthewplaneboundedebytheline1u,2u,1vand2vis12,12uv.Sowecanget2212,122xyxyandthedomainis22{(,):12,122}xyxyxyⅥFindtheimageofthestrip0yunderthetransformationexpwz,andlabelcorrespondingportionsoftheboundaries.
本文标题:复变函数复习参考
链接地址:https://www.777doc.com/doc-2543391 .html