最新2016年电大复变函数形成性考核册参考答案

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（A,B）,Z2C,D,则Z1Z2（C）A（ACBD,A）BACBD,BC（ACBD,ACBD）DACBD,BCAD2、若R0，则N（，R）Z（D）A|Z|R3、若ZXIY,则YDABCD4、若A，则|A|（C）A3B0C1D2二、填空题1、若ZXIY,WZ2UIV,则V（2XY）2、复平面上满足REZ4的点集为（ZXIY|X4）3、（设E为点集，若它是开集，且是连通的，则E）称为区域。4、设Z0X0IY0,ZNXNIYNN1,2,，则ZN以ZO为极限的充分必要条件是XNX0，且YNY0。三、计算题1、求复数1I的实部、虚部、模与主辐角。解RE1I1IM1I1|1I|2、写出复数I的三角式。解2Z2ZIZ2IZILIMLI45|1ARCTN,12IARY在第三象限23SINCOI3、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A2B21。证明而3、证明证明IIII21311I13735SINCO3SIC27ARG4221303EWEZIII的三次根的值为BIAYIXIYIX|YIXBI2|BAI122BAYXYIRE22121ZZZRE22112121212121221ZBYAXIAYBXAIYIZXIZIBAZZ则则设E211ZZ第2章解析函数一、单项选择题1若FZX2Y22XYI,则2、若FZUX,YIVX,Y,则柯西黎曼条件为（D）ABCD3、若FZZ1,则FZ在复平面上（C）A仅在点Z0解析B无处解析C处处解析D在Z0不解析且在Z0解析4、若F（Z）在复平面解析，GZ在复平面上连续，则FZGZ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Z1的邻域可导，则FZ在点Z1解析。3、若FZZ22Z1，则4、若，则不存在。三、计算题1、设FZZREZ,求ZFYVXU且XVY且YVXU且且2ZF17ZZF1F0LIM0FFZ解2、设FZEXCOSYIEXSINY,求解FZEXCOSYIEXSINYEZ,ZXIYUEXCOSYVEXSINYFZUIVFZ在复平面解析，且EXCOSYIEXSINY3、设FZUIV在区域G内为解析函数，且满足UX33XY2，FI0,试求FZ。解依CR条件有VYUX3X23Y2则V（X1Y）3X2YY3CC为常数故FZX33XY2I3X2YY3CX33XY2ICX2YY3ICZ3IC，为使FI0,当X0,Y1时，FI0,有F0IIC0C1FZZ3I4、设FZUIV在区域G内为解析函数，且满足U2X1Y,F2I,试求FZ。解依CR条件有VYUX2Y0LIM0FFZRELIM0ZRELIM0ZRELIZFYEVXUXCOSYINIEZFXCOSZFCXQXYUYVQD663322VY2XVXXVY2X22XCC为常数FZ2X1YIY2X22XC为使FZI,当X2Y0时,F2CIIC1FZ2X1YIY2X22X1Z12I四、证明题1、试在复平面讨论FZIZ的解析性。解令FZUIVZXIY则IZIXIYYIXUYVX于是UX0UY1VX1VY0UX、UY、VX在复平面内处处连接又UXVYUYVX。FZIZ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F（Z）0，ZG，则FZ在G内为常数。证设FZUIV,ZXIY,ZGFZ在G内解析，UXVY,UYVX又F（Z）0,F（Z）UXIVXYDZXUYX2CX2UX0VX0UYVX0UXVY0U为实常数C1，V也为实常数C2，FZC1IC2Z0FZ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1ZA是根式函数NZW的支点A0B1CDI2ZD是函数ZWLN的支点AIB2IC1D03EIBAE1EBCOS1ISIN1CSIN1DCOS14SIN1AAIEII2BIEII2C1D1二、填空题1COSI21E2IE1ECOS1ISIN13LNII24LN1I241KILNK为整数三、计算题1设ZXIY，计算2ZE解XYIIYXZ2IEZ22SNCOXPXYY2ZE2X2Y2设ZXIY,计算RE1Z解ZXIY221YXIIYXSINCOS1222YXYXYXZE221SR2YX3求方程IZLN的解解LNZ2/由对数函数的定义有ZIIEI2SNCO2/所给方程的解为ZI4求方程IEZ31的解解3SINCO231IEZIS2LN根据指数函数的定义有ZN2I3/或ZN1I3四、证明题1试证ZZCOSIN2SI证明根据正弦函数及余弦正数定义有IEZIZIZ2SINCOI2IZIIZIEIZIZ2SIN2Z2SINZCOSZ2证明XNNXX2SII1SI2SIN证明令ACOCSO1BSINXSIN2XSINNXINXXIIEEBIA2121XIIZIXNEEXNIXINIEEI22SIS2SINCO2SIN1XXXNXSI2I1SII第4章解析函数的积分理论一、单项选择题1CDZ2D,C为起点在0,终点在1I的直线段A0B1C2ID21I21SINZADA0B10ICID123I35ZBDAIB10IC10ID0432SINZAACOS4IBI4CI2D2二、填空题1若ZF与XG沿曲线C可积，则CCCDZGZFDZGZF2设L为曲线C的长度,若FZ沿C可积,且在C上满足MZF,则LDZFC317IZD4E01OS2三、计算题1计算积分CZDIM,其中C为自0到2I的直线段解C的方程为10TIZTZ其次由TIYX2得TZIMDTZDCI102TI212计算积分120SINZZDZE解12IZ132SINZZDZE作区域DZ积分途径在D内被积函数的奇点Z2与Z3均不在D内,所以被积函数在D内解析由定理42得120SINZZDZE03计算积分CZCZ41,132解CDZZ132奇点Z1和Z1不在区域D,1Z内013Z的三个根2,0,32KEZIK也不在D内由定理42得CDZZ13204计算积分CE5,5Z解由定理46得0452ZCZEIDE1I四、证明题1计算积分12ZD，并由此证明0COS45210DN证明F在圆域|Z|1内解析12ZD10Z另一方面,在圆|Z|2SINCOZ12ZDI2SINCOD实部和虚部为0DIICSNCO2SNOIDSINCOS412INZI5DIDCOS4521COS45IN21Z00SIN0COS452D而1为偶函数0DCOS452010COS4520D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0NZ的收敛半径等于BA0B1C2D32点Z1是FZ51052ZZRB级零点A1B2C3D53级数0NZ的收敛圆为DA|Z1|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FZ0NNZC成立的收敛圆的半径等于CAAB1BBA1C|AB|D|AB|二、填空题1级数1Z2NZZ的收敛圆R即整个复平面2若FZSIKK为常数,则ZMM0,2,1为FZ的1级零点3幂有数0NZ的收敛半径等于04Z0是FZEZ1的1级零点三、计算题1将函数FZ12Z在点Z0展开幂级数解FZ216321310ZZZZN000612163NNNNZZZ1Z2将函数FZ1Z2在点Z0展开成幂级数解121FZZ而1Z101NZ0012NNZ0NZ1Z3将函数FZZ21在点Z1展开成幂级数解FZZ21311312ZZZ00313NNNZZ4将函数FZEZ在点Z1展开成幂级数解FZEZFNEZEFN1NNZE001N四、证明题1证明1EI2Z2ISINZEIZ证EIZCOSZISINZEIZCOSISINZEIZEIZ2ISINZ2ISINZEIZEIZEIZEIZ2ISINZEIZEIZEIZEIZE0E2IZ1E2IZ2试用解析函数的唯一性定理证明等式COS2ZCOS2ZSIN2Z证F1ZCOS2Z,则F1Z复平面G解析设F2ZCOSZSIN2，则F2Z也在整个复平面G解析取EK为实数轴,则E在G内有聚点当E为实数时,知COS2ZCOS2ZSIN2Z,即F1ZF2Z由解析函数唯一性定理,由以上三条知F1ZF2ZGZ成立即COS2ZCOS2ZSIN2Z第6章解析函数的罗朗级数表示一、单项选择题1函数FZ2312Z在点Z2的去心邻域D内可展成罗朗级数A0ZB051ZC1D02Z2设点为FZ的孤立奇点，若ZIIMFC，则点为FZ的CA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DAZIIMFFZCBZIIMFZCZFZCDZFZ4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）AZIIMFZCBZIIMFZ不存在CZFZCDZFZ二、填空题1设NNZC为函数FZ在点的罗朗级数,称NNAZC1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Z内解析,则称点为FZ的孤立奇点3若FZZE14，则点Z0为FZ的0级极点不是极点,若FZZE14则Z0为FZ的一个极点4若FZSIN21,则点Z0为FZ非孤立奇点三、计算题1将函数FZZ21在点Z0的去心邻域展成罗朗级数解FZ21ZZNNNZZZ212121002将函数FZ2在点的去心邻域展成罗朗级数解FZ121112ZZZZ3试求函数FZZ3SINZ3的有限奇点,并判定奇点的类别解3SINZ解析,无奇点,FZ的有限奇点为Z0并且为3阶极点4试求函数FZZ21Z1的有限奇点，并判定奇点的类别解FZ的M阶奇点即ZF的阶零点,而112ZZZF零点为Z0，Z1，Z1，且均为1阶零点。12ZZF的有限奇点为Z0，Z1,Z1且均为1阶极点四、证明题1设FZ138ZE,试证Z0为FZ的6级极点证要证Z0为FZ的6级极点,只需证Z0为1813GEZF的6阶零点即可而1321881333NZZZEZF8Z323Z8Z61133NZZ令Z322则0Z为1ZF的6阶零点Z0为FZ的6级极点请您删除一下内容，O_O谢谢2016年中央电大期末复习考试小抄大全，电大期末考试必备小抄，电大考试必过小抄BASKETBALLCANMAKEATRUECLAIMTOBEINGTHEONLYMAJORSPORTTHATISANAMERICANINVENTIONFROMHIGHSCHOOLTOTHEPROFESSIONALLEVEL,BASKETBALLATTRACTSALARGEFOLLOWINGFORLIVEGAMESASWELLASTELEVISIONCOVERAGEOFEVENTSLIKETHENATIONALCOLLEGIATEATHLETICASSOCIATIONNCAAANNUALTOURNAMENTANDTHENATIONALBASKETBALLASSOCIATIONNBAANDWOMENSNATIONALBASKETBALLASSOCIATIONWNBAPLAYOFFSANDITHASALSOMADEAMERICANHEROESOUTOFITSPLAYERANDCOACHLEGENDSLIKEMICHAELJORDAN,LARRYBIRD,EARVIN“MAGIC“JOHNSON,SHERYLSWOOPES,ANDOTHERGREATPLAYERSATTHEHEARTOFTHEGAMEISTHEPLAYINGSPACEANDTHEEQUIPMENTTHESPACEISARECTANGULAR,INDOORCOURTTHEPRINCIPALPIECESOFEQUIPMENTARETHETWOELEVATEDBASKETS,ONEATEACHENDINTHELONGDIRECTIONOFTHECOURT,ANDTHEBASKETBALLITSELFTHEBALLISSPHERICALINSHAPEANDISINFLATEDBASKETBALLSRANGEINSIZEFROM28530IN7276CMINCIRCUMFERENCE,ANDINWEIGHTF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