复变函数—课后答案习题四解答

1{}nα11i1innnα+=−2i1;2nnα−⎛⎞=+⎜⎟⎝⎠3i(1);1nnnα=−++45i/2nneπα−=i/21nnenπα−=12221i12i1i11nnnnnnnα+−==+−++2212lim1,lim011nnnnnn→∞→∞−2=−=++nαlim1nnα→∞=−2i2125nnineθα−−⎛⎞⎛⎞=+=⎜⎟⎜⎟⎝⎠⎝⎠2lim05nineθ−→∞⎛⎞=⎜⎟⎝⎠nαlim0nnα→∞=3nα{}(1)n−nα4i/2cosisin22nnnneπππα−==−nα5i/2111cosisin22nnnnennnπππα−==−11limcos0,limsin022nnnnnnππ→∞→∞==nαlim0nnα→∞=20,||1,,||1,lim1,1,||=1,1.nnαααααα→∞⎧⎪∞⎪=⎨=⎪⎪≠⎩311innn∞=∑22ilnnnn∞=∑31(6+5i)8nnn∞=∑42cosi2nnn∞=∑1icosisin22nnnππ=+1cos2nnnπ∞=∑1sin2nnnπ∞=∑1innn∞=∑i1nnn=1innn∞=∑12111(2lnnnn≥≥)3(6+5i)6188nnn⎛⎞=⎜⎜⎝⎠⎟⎟1618nn∞=⎛⎞⎜⎟⎜⎟⎝⎠∑1(6+5i)8nnn∞=∑4cosichn=nchlim02nnn→∞≠2cosi2nnn∞=∑4123Taylor0z0z1∑∞=0nnz1z1=z2(3)()zzf=,Taylor5.(02nnncz∞=−∑)0=z3=z()02nnncz∞=−∑0=zAbel220=−≥R2123=−3=z2|2|−z()02nnncz∞=−∑3=z611()npnzpn∞=∑21!nnnnzn∞=∑301)nnniz∞=∑41innnezπ∞=∑51ich(1)nnzn∞=⎛⎞−⎜⎟⎝⎠∑61lninnzn∞=⎛⎞⎜⎟⎝⎠∑11/limlim1npnnnnRan→∞→∞===22111(1)1/limlimlim01nnnnnnnnaanRaan+→∞→∞→∞++===+=;31/limlim1/|1i|1/2nnnnRa→∞→∞==+=41/lim1nnnRa→∞==511/lim1/limch1/limcos1nnnnnnniRann→∞→∞→∞⎛⎞===⎜⎟⎝⎠=61/limlim|lni|nnnnRan→∞→∞===∞;7R0nnncz∞=∑()0Rennncz∞=∑R≥Rz||z0nnncz∞=∑0nnncz∞=∑Rencc≤nRe||||nnnnczcz≤0Rennncz∞=∑()0Rennncz∞=∑Rz||R≥81limnnncc+→∞≠∞nncz∑11nnczn++∑1nnncz−∑1limnnnccρ+→∞=1/nncz∑||ρ11nnczn++∑11/(1)1/limlim1/||/(2)nnnnnnacnRacnρ+→∞→∞++===+1nnncz−∑111/limlim1/||(1)nnnnnnancRancρ+→∞→∞+===+90nnc∞=∑0nnc∞=∑0nnncz∞=∑130nnc∞=∑0nnncz∞=∑1z=Abel0nnncz∞=∑1R≥0nnc∞=∑0||nnncz∞=∑||1z=0nnncz∞=∑1R≤10nnncz∞=∑100nnncz∞=∑0zAbel0nnncz∞=∑η000|||nnnnnnccη∞∞===∑∑|z0nnncη∞=∑11z1311z+2()2211z+34sh2coszz567chz2sin2zez1zze−8z−11sin11||,11132+−+−=+zzzzz()…+−+…+−+−=+nnzzzzz3963311111||zR=12()…+−+…+−+−=+nnzzzzz1111321||z()…+−+…++−=+nnzzzz242211111||z′⎟⎠⎞⎜⎝⎛+211z()2212zz+−=()…+−+−=′⎟⎠⎞⎜⎝⎛+−=+6422224321112111zzzzzz1||z=1R3,,!6!4!21cos642∞…+−+−=zzzzz+−+−=!6!4!21cos12842zzzz4+∞||z+∞=R4,||,!3!21,2sh32+∞…++++=−=−zzzzeeezzzz,||,!3!2132+∞…+−+−=−zzzzez,||,!5!3sh33+∞…+++=zzzzzR=+∞524ch1,||,2!4!zzzz=+++…+∞6,||,!3!216422+∞…++++=zzzzez,||,!5!3sin10622+∞…++−=zzzzz⎟⎟⎠⎞⎜⎜⎝⎛…++−⎟⎟⎠⎞⎜⎜⎝⎛…++++=!5!3.!3!21sin106264222zzzzzzzez,||,3642+∞…+++=zzzz;+∞=R7231,|2!3!zzzezz=++++…+∞|,2310,||1,1nnzzzzzzz∞+==−−−−…=−−∑12132310010()()112!3!2!3!nnznnnznzzzzezz∞∞++∞+==−==−+−+=−−−+∑∑∑,||1zR=18,1sin1cos1cos1sin11sin11sinzzzzzzz−+−=⎟⎠⎞⎜⎝⎛−+=−,1||,10132=…+++=−∑∞=+zzzzzzznn()()…+…+++−…+++=−33232!311sinzzzzzzzz…+++=3265zzz1||z()()…+…+++−…+++−=−432232!412111coszzzzzzzz…+−−=32211zz1||z⎟⎠⎞⎜⎝⎛…++++⎟⎠⎞⎜⎝⎛…+−−=−3232651cos2111sin11sinzzzzzz=()1||,1sin1cos651sin211cos1cos1sin32+⎟⎠⎞⎜⎝⎛−+⎟⎠⎞⎜⎝⎛−++zzzzR=112Taylor0z5111+−zz10=z2()()21++zzz20=z321z10−=z4z341−i10+=z5tanz0/4zπ=6arctanz00z=1()()211121211111−+−=+−−=+−zzzzzz1||,11132+−+−=+zzzzz()⎥⎥⎦⎤⎢⎢⎣⎡+⎟⎠⎞⎜⎝⎛−−+−⎟⎠⎞⎜⎝⎛−+−−−=+−−−112211212112111nnzzzzzz()+⎟⎠⎞⎜⎝⎛−−++⎟⎠⎞⎜⎝⎛−−−=−nnzzz211212112()()nnnnz12111−−=∑∞=−2|1|−zR=22()()112212242121+−+=⎟⎠⎞⎜⎝⎛+−+=++zzzzzzz()42114124121−+=−+=+zzz4|2|,42421412−⎥⎥⎦⎤⎢⎢⎣⎡−⎟⎠⎞⎜⎝⎛−+−−=zzz()32113123111−+=−+=+zzz=3|2|,32321312−⎥⎥⎦⎤⎢⎢⎣⎡−⎟⎠⎞⎜⎝⎛−+−−zzz()⎥⎦⎤⎢⎣⎡−−+−−=⋅222222122142zz()⎟⎟⎠⎞⎜⎜⎝⎛−−+−−−223232131zz=()()()()∑∑∞=∞=−−−−−0023213122121nnnnnnnnzz()()()()∑∑∞=+∞=+−−−−−=01012231221nnnnnnnnzz()()∑∞=++−⎟⎠⎞⎜⎝⎛−−=0112231211nnnnnz3|2|−z3=R3′⎟⎠⎞⎜⎝⎛−=zz112()()()[]+++++−=+−−=21111111zzzz1|1|+z()()++++++=−1211211nznzz()()∑∞=++=011nnzn1|1|+zR=164()[]i33i1341341−−+−−=−zz()[]i13i311+−−−=z()[]i1i31311i311+−−−−=z()[]()[]⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧++−⎟⎠⎞⎜⎝⎛−++−−+−=22i1i313i1i3131i311zz()[]1i1i313+−−z()()[]nnnnzzi1i31334101+−−=−∑∞=+()3103i31|i1|=−+−z310=R5352tan,||3152zzzzzπ=+++441tan()tan1tan()zzzππ+−=−−2444tan[1tan()](1tan()tan())zzzzπππ=+−+−+−+23812()2()(),4434zzzπππ+−+−+−+4Rπ=621(arctan)'1zz=+24211,1zzzz||1=−+−[phone]arctan(1)(1)12nzznnnnzzdzzdzzn1+∞∞===−=−++∑∑∫∫||1z1R=13||zR(,)RR−()fzz()fzz()(0)!nnfcn=()fz(,)RR−()(0)nf141()cos()fzzz=+z201cos(2cos)cos,(0,1,2,)2ncndnπθθθπ==∫±±1()cos()fzzz=+0z=0||z+∞1cos()nnnzcz∞=−∞+=∑z71||1cos()1,(0,1,2,,0)2nnzrzzcdznizπ+=+r==±±+∞∫vnc1r=ii221i00||11cos()11cos()1cos(2cos)cos2i22nnnzzeezcdzdzeθθππθndθθθπππ−+=++===∫∫∫vθ20icos(2cos)sin2ndπθθθπ−∫201cos(2cos)cos2ndπθθθπ∫20cos(2cos)sinndπθθθ∫cos(2cos)sinndππθθθ−∫cos(2cos)sinnθθθ15231zzzzz=+++−21111zzzz=+++−011zzzz+=−−2111zz++++230zzz+++=z||||1z1z16Laurent1()()2112−+zz2||1z2()211zz−1|1|0,1||0−zz3()()+∞−−−−|2|1,1|1|0,211zzzz411ze−+∞||1z521(izz−)i6z−11sin+∞−|1|0z7(1)(2),3||4,4||(3)(4)zzzzzz−−+−−∞1251152151)2)(1(1222−++−++−=−+zzzzzz211101111152111151)2)(1(122222zzzzzzzz−−+−+⋅−=−+8()()∑∑∑∞=∞=∞=−−−−⋅−=000212222101115211151nnnnnnnnzzzzzz()()∑∑∑∞=∞=+∞=+−−−−−=00)1(2012210111521151nnnnnnnnnzzz−−−−−−−++=80402010115115215115232234zzzzzzz2||1z21||0z()()2221111+++++=−nzzzzzz()()++++++=nznzzz132112()++++++=−11321nznzz()∑∞−=+=12nnzn1|1|0−z()()()()()()∑∞=−−−=−+−=−022211111111111nnnzzzzzz()()nnnz112−−=∑∞−=31|1|0−z()111111121)2)(1(1−−−−=−−−=−−zzzzzz()()∑∞=−−−−=−−−−−=011111111nnzzzz()∑∞−=−−=11nnz+∞−|2|1z()121211121)2)(1(1+−−−=−−−=−−zzzzzz2111212z1−+−−−=zz()()nnnzzz21121210−−−−−=∑∞=()()∑∞=++−−+−=01121121nnnzz()()∑∞=−−+−=121121nnnzz()()∑∞=−−−=21211nnnz4+∞||1z⎟⎠⎞⎜⎝⎛+++−=⎟⎠⎞⎜⎝⎛+++−=⎟⎠⎞⎜⎝⎛−−=−322111111111111zzzzzzzzz9+⎟⎠⎞⎜⎝⎛+++−⎟⎠⎞⎜⎝⎛++++⎟⎠⎞⎜⎝⎛+++−=