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电大复变函数习题总汇与参考答案第1章复数与复变函数
一、单项选择题
1、若Z1=(ab)Z2=cd则Z1·Z2=(C)A(ac+bda)Bac-bdbC(ac-bdac+bd)Dac+bdbc-ad
2、若R0,则N(∞,R)={z(D)}A|z|RB0|z|RCR|z|+∞D|z|R
3、若z=x+iy则y=DABCD
4、若A=,则|A|=(C)A3B0C1D2
二、填空题
1、若z=x+iyw=z2=u+iv则v=(2xy)
2、复平面上满足Rez=4的点集为({z=x+iy|x=4})
3、(设E为点集,若它是开集,且是连通的,则E)称为区域
4、设z0=x0+iy0zn=xn+iynn=12……,则{zn}以zo为极限的充分必要条件是xn=x0,且yn=y0
三、计算题
1、求复数-1-i的实部、虚部、模与主辐角解Re-1-i=-1Im-1-i=-1|-1-i|=
2、写出复数-i的三角式解
3、写出复数的代数式解
4、求根式的值解
四、证明题
1、证明若,则a2+b2=1证明而
3、证明证明第2章解析函数
一、单项选择题1.若fz=x2-y2+2xyi则
2、若fz=uxy+ivxy则柯西—黎曼条件为(D)ABCD
3、若fz=z+1则fz在复平面上(C)A仅在点z=0解析B无处解析C处处解析D在z=0不解析且在z≠0解析
4、若f(z)在复平面解析,gz在复平面上连续,则fz+gz在复平面上(C)A解析B可导C连续D不连续
二、填空题
1、若fz在点a不解析,则称a为fz的奇点
2、若fz在点z=1的邻域可导,则fz在点z=1解析
3、若fz=z2+2z+1,则
4、若,则不存在
三、计算题
1、设fz=zRez求解=
2、设fz=excosy+iexsiny求解fz=excosy+iexsiny=ezz=x+iyu=excosyv=exsinyfz=u+iv∴fz在复平面解析,且=excosy+iexsiny
3、设fz=u+iv在区域G内为解析函数,且满足u=x3-3xy2,fi=0试求fz解依C-R条件有Vy=ux=3x2-3y2则V(x1y)=3x2y-y3+cc为常数故fz=x3-3xy2+i3x2y-y3+c=x3-3xy2+icx2y-y3+ic=z3+ic,为使fi=0当x=0y=1时,fi=0有f0=-i+ic=0∴c=1∴fz=Z3+i
4、设fz=u+iv在区域G内为解析函数,且满足u=2x-1yf2=-i试求fz解依C-R条件有Vy=ux=2y∴V==y2+x∴Vx=∴x=V=y2-x2+2x+cc为常数∴fz=2x-1y+iy2-x2+2x+c为使fz=-i当x=2y=0时f2=ci=-i∴c=-1∴fz=2x-1y+iy2-x2+2x-1=-z-12i
四、证明题
1、试在复平面讨论fz=iz的解析性解令fz=u+ivz=x+iy则iz=ix+iy=-y+ix∴u=-yv=x于是ux=0uy=-1Vx=1Vy=0∵ux、uy、vx在复平面内处处连接又Ux=VyUy=-Vx∴fz=iz在复平面解析
2、试证若函数fz在区域G内为解析函数,且满足条件(z)=0,z∈G,则fz在G内为常数证设fz=u+ivz=x+iyz∈G∵fz在G内解析,Ux=VyUy=-Vx又(z)=0(z)=Ux+iVxUx=0Vx=0Uy=-Vx=0Ux=Vy=0U为实常数C1,V也为实常数C2,fz=C1+iC2=Z0fz在G内为常数复变函数课程作业参考解答2第3章初等函数
一、单项选择题
1.z=A是根式函数的支点.A0B1CDi
2.z=D是函数的支点.AiB2iC-1D
03.ei=B.Ae-1+eBcos1+isin1Csin1Dcos
14.sin1=AABCD
二、填空题
1.cosi=
2.=ecos1+isin
13.lni=
4.ln1+i=k为整数.
三、计算题
1.设z=x+iy,计算.解:∴∴==
2.设z=x+iy计算.解:∵z=x+iy∴∴∴
3.求方程的解.解:∵lnz=∴由对数函数的定义有:Z=∴所给方程的解为z=i
4.求方程的解.解:∵=根据指数函数的定义有:z=n2+i或z=n1+
四、证明题
1.试证:.证明根据正弦函数及余弦正数定义有∴sin2z=2sinz·cosz
2.证明:.证明:令A=B=sinx+sin2x+…sinnx∴=∴第4章解析函数的积分理论
一、单项选择题
1.Dc为起点在0终点在1+i的直线段.A0B1C2iD21+i
2..A0B10CiD
3.AiB10C10iD
04.=A.ABCD
二、填空题
1.若与沿曲线c可积,则.
2.设L为曲线c的长度若fz沿c可积且在c上满足则.
3.
4.
三、计算题
1.计算积分其中c为自0到2+i的直线段.解:c的方程为:其次由得∴==
2.计算积分.解:=作区域D:积分途径在D内被积函数的奇点Z=2与Z=3均不在D内所以被积函数在D内解析.由定理
4.2得:=
03.计算积分.解:∵奇点z=1和z=-1不在区域D内的三个根也不在D内∴由定理
4.2得=
04.计算积分.解:由定理
4.6得
四、证明题
1.计算积分,并由此证明.证明∵在圆域|z|≤1内解析∴=另一方面在圆|z|=∴=实部和虚部为0====∵=0∴∴而为偶函数∴0==∴复变函数课程作业参考解答3第5章解析函数的幂级数表示
一、单项选择题
1.幂级数的收敛半径等于BA0B1C2D
32.点z=-1是fz=rB级零点.A1B2C3D
53.级数的收敛圆为D.A|z-1|3B|z|3C|z-1|1D|z|
14.设fz在点a解析点b是fz的奇点中离点a最近的奇点于是使fz=成立的收敛圆的半径等于C.Aa+b+1Bb-a+1C|a-b|D|a+b|
二、填空题
1.级数1+z+的收敛圆R=+.即整个复平面.
2.若fz=k为常数则z=mm=0……为fz的1级零点.
3.幂有数的收敛半径等于
0.
4.z=0是fz=ez-1的1级零点.
三、计算题
1.将函数fz=在点z=0展开幂级数.解:fz==-
2.将函数fz=1-z-2在点z=0展开成幂级数.解而1-z-1==3.将函数fz=z+2-1在点z=1展开成幂级数.解fz=z+2-1==4.将函数fz=ez在点z=1展开成幂级数.解fz=ezfn=ez =
四、证明题1.证明:1-ei2z=-2isinzeiz证eiz=cosz+isinze-iz=cos-isinzeiz-e-iz=2isinz-2isinz=-eiz-e-iz=eiz-e-iz-2isinzeiz =e-iz-eizeiz=e0-e2iz=1-e2iz2.试用解析函数的唯一性定理证明等式:cos2z=cos2z-sin2z证
①f1z=cos2z则f1z复平面G解析设f2z=cosz-sin2z,则f2z也在整个复平面G解析
②取E=K为实数轴则E在G内有聚点.
③当E为实数时知cos2z=cos2z-sin2z即f1z=f2z由解析函数唯一性定理由以上三条知f1z=f2z成立即cos2z=cos2z-sin2z第6章解析函数的罗朗级数表示
一、单项选择题1.函数fz=在点z=2的去心邻域D内可展成罗朗级数.A0B0C1D02.设点为fz的孤立奇点,若=c,则点为fz的C.A本性奇点B极点C可去奇点D解析点3.若点为函数fz的孤立奇点,则点为fz的极点的充分必要条件是D.Afz=cBfz=Cfz=cDfz=4.若点为函数fz的孤立奇点,则点为fz的本性奇点的充要条件是(B).Afz=cBfz不存在Cfz=cDfZ=
二、填空题1.设为函数fz在点的罗朗级数称为该级数的主要部分.
2.设点为函数fz的奇点若fz在点的某个某个去心邻域内解析则称点为fz的孤立奇点.
3.若fz=,则点z=0为fz的0级极点.不是极点若fz=则z=0为fz的一个极点.
4.若fz=sin-1则点z=0为fz非孤立奇点.
三、计算题1.将函数fz=z-2-1在点z=0的去心邻域展成罗朗级数.解:fz==-=-2.将函数fz=在点z=1的去心邻域展成罗朗级数.解:fz=3.试求函数fz=z-3·sinz3的有限奇点并判定奇点的类别.解:解析无奇点fz的有限奇点为z=
0.并且为3阶极点.4.试求函数fz=[z]-1的有限奇点,并判定奇点的类别.解:fz的m阶奇点即的阶零点而零点为z=0,z=1,z=-1,且均为1阶零点的有限奇点为z=0,z=1z=-1且均为1阶极点.
四、证明题1.设fz=试证z=0为fz的6级极点.证:要证z=0为fz的6级极点只需证z=0为的6阶零点即可.而=8z3=8z6令则为的6阶零点z=0为fz的6级极点.Visa-freepolicybringsChengdubiztourismboost.MakingnationalheadlinesseveraltimesChengdus72-hourvisa-freepolicyhasattractedwideattentionfrombothChineseandforeignexpertsandbusinessmensinceittookeffectonSept1lastyear.Theprogrampermitscitizensfrom51countriesandregionsincludingtheUnitedStatesAustraliaCanadaandJapanwhohavevalidvisasandflightticketstoathirdcountrytospendthreedaysinthecity.ThecapitalofSichuanprovinceisthefirstcityinthewesternregionofChinatoofferforeigntouristsathree-dayvisaandthefourthnationwidetoadoptthepolicyfollowingShanghaiBeijingandGuangzhou.LiZhiyongdeputydeanofthetourisminstituteatSichuanUniversitysaidthemovecontributestoalargeincreaseinthenumberofoverseastouristsandraisesthecityslevelofinternationalization.ThepolicywillalsobringdirecteconomicrevenueLisaid.Chengduhasmanyculturallegaciesandisalsoaparadiseforpandaloverswiththeworldslargestbreedingandresearchcenter.Threedaysarelongenoughforforeignvisitorstovisitthoseiconictouristspotshenoted.ThecityishometotheremainsoftheJinshacivilizationthatdatesbackmorethan3000yearsaswellastheQingchengMountainsandtheDujiangyanirrigationsystem.QingchenghaslongbeenrecognizedasthebirthplaceofTaoismChinasancientindigenousreligionwhileDujiangyanisconsideredtobetheoldestfunctioningwater-controlprojectintheworld.Chengdurankedthirdintouristfacilitiesmanagementandservicesamong60Chinesecitiesinacustomersatisfactionsurveyreleasedlastyear.ButLiaddedthateffortsarestillneededtodevelopmoretourismproductsimproveEnglishservicesandprovideaccuratetranslationoftrafficsignsandscenicbillboards.ZhaoYunchairwomanofBritishChamberofCommerceSouthwestChinatoldChinaDailythathiscolleaguesfoundthepolicyveryconvenient.ABritishclientonceflewhereandstayedforjustonedaytocheckherorderedgoodsshesaid.ZhaowasborninShanxiprovincebutshehaslivedinChengduformorethan10years.Mylifewaslikearunningracemovingfromplacetoplace.IalsolivedinBeijingandShanghaibeforeshesaid.ButChengduisaplacethatyouneverwanttoleaveoncesettlingdown.Itisnowmysecondhometownshesaid.Iftheenvironmentisfurtherimprovedthecitywillattractmorepeopletovisitandlivewiththe72-hourvisa-freepolicyandcompellingconditionsintransportationcultureclimateandcuisinehesaid.Foreignersalsogavepositivefeedbackonthepolicy.AspokesmanfromDellIncsaidthecompanyhasaglobalhubofoperationinChengdusothethree-dayvisahasanimmediateandpositiveinfluenceonthecompanysbusinessdevelopment.RudyButtignolpresidentofthepublicbroadcastingcompanyinBritishColumbiaCanadasaidhisworkrequiresfrequenttraveltoChengduandthepolicymakesthetripseasier.Datafromthecityspublicsecuritybureaushowssome100foreignvisitorsenjoyedthe72-hourpolicybytheendofMarchmostofthemfromtheUnitedStatestheUnitedKingdomandGermany.Chengdualsoreportedrobustgrowthinitsoveralltouristindustrylastyear.Officialstatisticsshowthatitreceivedsome150milliontouristslastyearanincreaseof28percentfrom
2012.Around
1.7millioncamefromabroadanincreaseof12percent.Totalrevenuefromtourismsurpassed133billionyuan$
21.7billion.DuringhisvisittoKazakhstaninSeptemberChinesePresidentXiJinpingproposedthatChinaandCentralAsiajoinhandstobuildaSilkRoadeconomicbelttoboostcooperation.TheideahasbeenwidelyechoedinCentralAsiancountriesbecominganencouragingblueprintforChineseareasalongtheSilkRoadthathaslinkedAsiaandEuropeformorethan2000years.InthenextthreeweeksChinaDailyreporterswilltravelthroughthebeltinChinaandinKazakhstanUzbekistanandTurkey.Theywillshowtheprogressandexpectationsofthecountriesbusinessesandpeoplesontheroute.Shaanxi-thestartoftheancientSilkRoad-haspositioneditselfasthenewstartingpointforthedevelopmentoftheSilkRoadEconomicBeltwhichwillstrengthenChinascooperationwithCentralAsiancountriesaseniorofficialsaid.ShaanxiGovernorLouQinjiansaidtheprovinceisfreshrichanduniqueasitwaswhenitanchoredoneendoftheancientSilkRoad.ItisthebestoptionforaccommodatingindustrialtransfersfromEastChinaortheworldatlargehesaidonWednesdayinXian.Louheldajointinterviewwith27mediaincludingChinaDailythefirstinaseriesofinterviewsentitledChineseMediaAlongtheSilkRoad.TheinterviewswillbeinShaanxiandGansuprovincesandtheXinjiangUygurautonomousregionaswellasKazakhstanUzbekistanandTurkey.ThemediagroupheldthefirstinterviewonWednesdaymorninginXianthestartingpointoftheancientSilkRoadatradechannelestablishedmorethan2000yearsagolinkingChinaCentralAsiaandEurope.。