Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 4
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0603Ha-rr;=I (1 + zk) .EcJJH He cyme-CTByeT KOHe'lHOfO npene11a, OTJJJ1qHoro OT HYJlll, TO 6eCKOHe'IHOenpOH3BeL{eHtterr~=l (1 + Z,,)llBJllleTCll paCXOORUfUMCfl.EcJIH 6eCKOHe'IHOe npOH3BeL{eHtterr~=I (1 + z,,) CXOL{HTCll, TO OT-CI-OL{a c Heo6XOL{HMOCThI-O cnenyeT, qTQlim z,, = 0.11~00Ecm1 z,,,n E N, -npOH3BeL{eHtteCXO,UHTCll Pll.Unei1:cTBHTeJJhHhie '!Hcna,rr~=l (1 + Z,,)L~=l z,,Zn?: 0,To 6ecKoHeqttoeCXO,UHTCll TOf,Ua H TOJlhKO TOf,Ua, KOfL{a.EecKoHe'IHOe npOH3BeL{eHHerr~=I (1 + z,,)CXOOflUfU.MCJl, eCJJH CXO,UHTCll npOH3BeL{eH11eHa3hIBaeTCll a6co111omHOrr~=l (1 + Iz11I).A6co-JJI-OTHO CXO,UllIUeecl! rrpomse,n:eHHe llBJilleTCll CXOWllUHMCll.EecKOHe'IHOe npOH3Be)J.eHtte CXO,UHTCll a6cOJJI-OTHO TOr,n:a H TOJlhKOTorna, Korna cxonwTcll a6comoTHO pllLl2.1.,ll.oKa3aTh,'ITOL~=I z,, .eCJJH ITOCJJeL{OBaTeJJhHOCTHCXOL{llTCll, TO CXOL{llTCll ITOCJJeL{OBaTeJJhHOCTH { z,,{ z,,}H± z:,}' { z,, .
z:,}{ z:,}H21f10CJ1E)J:OBA TEJlbHOCTH, P5l)J:bl l1 f1POI13BE)J:EHH51lim(z ± z;,) = lim z ±Jim z;,, lim(z11-too11lf--tco1111-too11-700z;,) = lim z lim z;,.•11lim z;, -:t 0 ,Ec1m, KpOMe Toro, npe.unono)KHTb, 4TO•1111--too11--tooTO cxo,n:HTCR no-11-toocne.uosaTeJihHOCTh {z11Iz;,} H cnpase,n:JIHBO paseHcTBo:limz,,~-~.1lffi I - • I •, . . .~ z hmz112.2. ).:(oKa3aTb,11BaJieHTHa cxo,n:HMOCTH rrocne.uosaTeJihHOCTeil: {Re2.3. IIyen4eM.use rrocne.uosaTeJihHOCTH {lim z = lim z;, . Cne.uyeT JIH11-too11z11H {}z,,}Hz;, } cxo,n:HTCR, rrpH-0Tc10.ua, 'ITO11-toolim I z11--t-r.uez,,} 3KBH{Im z11 }.4TO CXO,UHMOCTb IIOCJie,UOBaTeJibHOCTH {11I= lim I z;, I11--tooHlim arg z1111--too= lim arg z;, ,n-tooarg z E (-rr, rr]?2.4. ).:(oKaJaTh,'ITO MR cyru.ecTBoBaHHRlim z z E R,Heo6xo-=0 ,TO cyru.ecrnyeT11-too11.UHMO H JlOCTaTO'IHO, 'IT06bI cyru.ecTBOBaJIHJim I z 1-:t 0, lim arg z -:t ±n .11--too2.5.11u--t-BepHo JIH yrnep)K.uem1e: ecJIH11lim zn--too11limarg z,,?11--too2.6.
).:(oK8.3aTb,'ITO H3 CXO.[(HMOCTH IIOCJie.[(OBaTeJibHOCTHlz11I}.{ lz11I}, soo6ru.e{ z,,}CJie,n:yeT CXOJlHMOCTb IIOCJie,UOBaTeJibHOCTH {2.7. IIoKa3aTb,'ITO H3 CXO.[(HMOCTHroBOp5!, HecnenyeT CXO.[(HMOCTh { z,,}.2.8. BKaKHX cnyqaHx cxo,n:HMOCTh rrocne.uosaTeJihHOCTH { zBHBaJieHTHa CXO)lHMOCTH IIOCJie.[(OBaTeJihHOCTH {2.9.
).:(OK8.3aTb,lz,,I}?11 } 3K-'ITO CXO,U5IIUHeC5I IIOCJie,UOBaTeJibHOCTH orpaHHlleHhI.2.10. ).:(oKa3aTb,'ITOH3orpaHH'IeHHOttJlOCJie.[(OBaTeJibHOCTHMO)KHO BhI,UeJIHTh CXOJl5IIUYIOC5I JlO,UJlOCJie,UOBaTeJibHOCTh.2.11. ).:(oKa3aTb,'ITO 113 HeorpaHH4eHHOH IIOCJie.[(OBaTeJihHOCTHMO)KHO BhI,UeJIHTh 6eCKOHe4HO 60JihIIIYIO JlO,UIIOCJie,UOBaTeJihHOCTb.22I'Jwea22.12.
IIpirnecrn: rrp11Mep 6ecKoHeqHo 6oJiblllOH rrocJie.uosaTeJibHOCTH { Z11}, .umr KOTOpOH HH fIOCJie,UOBaTeJibHOCTb {Re Z11}, HH rmCJie,UOBaTeJibHOCTb {Im z11 } He 5rBm1eTCH 6ecKoHeqHo 6oJihlllOH.2.13. ,UoKa3aTb, qTO eCJIH fIOCJie,UOBaTeJibHOCTb {z11} - 6eCKOHeqHO 6oJI&IllaH, a rrocJie.uosaTeJibHOCTb {BaTeJibHOCTb { z,,z;, } orpaH11qeHa, TO rrocne.uo-± z:. } 5IBJI5leTC5l 6eCKOHeqHQ 60JiblllOH.2.14.
,UoKa3aTb, qTo lim z11 =00Tor.ua 11 TOJibKO Tor.ua, Kor.ua11-toolimz,~ =0.111-too2.15. ,UoKa3aTb, qTQ eCJIHlim z,, =oo ,a Jim z:, 7' 0 , TO lim z,, z:, = lim ~ =11-too11->oo11->oo11->oozlloo •2.16. ,UoKa3aTb, qTo fIOCJie.UOBaTeJibHOCTb c orpaHifqeHHbIMMettem1eM HBJrneTCH CXO)J;Hlll,eikH.If3-2.17. IIp11necrH rrpHMep cxonmueH:cH rrocJienonarenbHOCTH c HeorpaHifqeHHhIM H3MeHeHHeM.2.18. IIycrb rrocnenonareJibHOCTb { z11 } TaKOBa, qrolim(z11-z,,_,) = 0,11-tOQlim I z,, I= a.n-tooCJie,n;yer JIH orcro.ua, 'iTO cyiuecrnyer Jim z,, ?114002.19. IIycrb rroCJie.n;osareJibHOCTh { a11 }orpaHH'ieHa HZ11= ao+ a, z + ... + a"z", lzl < 1.,UoKa3aTb, qTo cymecTByeT lim z,,.fl-tOO2.20. ,UoKa:mrh, qTOlim(l + zl n)"11-tg.g= e' (cosy+ isin y), z = x+ iy.TTqTo rrocnenonaTeJibHOCTb z =2•21 • µ.OKa3aTh,II= 3, 4, ... , a Zi. Z2 - 3anaHHbie '!HCJla, CXO,D.HTCHIfZ11-1+ Z11-'2rne nHaHTH ee rrpenen.2.22.
I1ycrb KOMfIJieKCHOe q11cJIO z i- 0. ,UJIH KaKHX z cymecTByeTlimz,,,ecJIH11->~z11+1=_!_(z2 II +zII1)'zI =z?•ITOCJIE,LlOBATEJibHOCTI1, P51.[lbl l1 ITPOI13BE,LlEHI151232.23. ,ll.Jrn KaKHX KOMIIJieKCHb!X z CXO,ll.5ITC5I IIOCJie,D.oBaTeJibHO{z,,}:crn1) Z114)= z";2 ) z,,kz"z =-·f'II3) z11 = zII · nK, k - ueJioe;= "'\'"L..k=o zk ;5)n."'\'" zz" = L..k=r k_2·2.24. IIyCTb lim z,, = a i- oo.
,ll.oKa3aTb, qrn11-?oo. Z + z, + ... + Z,' =a.11m 1 1111-?ooIIoKa3aTb, qTo B cJiyqaea= ooyrnep)K.uemi:e, Boo6me roBop5I, He-Beptto.2.25. IIycTb lim z,, =11-?00a i- oo, Ak>0,k = 1, 2, ... , liml::~=r,1.,k =+ 00 •11-7 00. Al .::I + A, Z, + ... + A z- " "-1., +A., + ... +-1..hm=a.2.26. IlpH KaKHX .ueliCTBHTeJibHbIX 3HaqeHH5IX cp cymecrnyeTlime;"'P?2.27.
IIp1rnecTHMHO)KeCTBO ToqeK:1) KOHeqtto;rrp11Meprrorne.uosaTeJihHOCTH,2) cqerno;,D.JIHKOTopoM:3) ttecqerno.2.28. IIp11secT11 rrpttMep rrocJie.uosaTeJibHOCTH, ,D.JI5! KOTopowMHO)KeCTBO rrpe.ueJibHbIX TO'leK ecTb:1) rrpHMa.H; 2) OKp)')KHOCTh { JzJ=R};3) 3aMKHyrhlli Kpyr { JzJ ::::: R};4) rro.i:IyrrnocKocTh {Rez2::0},{Imz2::0};5) BC5! KOMIIJieKCHaJI IIJIOCKOCTb.2.29. HaliTH rrpe.ueJihHhie ToqKH MHO)Kecrn:1) {z",n=l, 2, ...},z cpI1KCHpOBaHO;2){lzl< 1};113) {Rez- =canst} ;4) {rmz- =const};8) {Im;:- 1 <l}; 9) {a<argz<,8,-1l<a<,8<1l}.Diaea 2242.30. )JoKa3aTh,4TOPH.ll L~=J z,, CXO./lHTCSI TOr.ua 11 TOJJhKO rnr.ua,Kor.ua O./lHOBpeMeHHO CXO.llSITCSI pSI.llhI2.31. )JoKa3aTb,L~=i Re z,, , L~= 1 Im z,, .4TO 113 a6comOTHOH CXO,lll1MOCTH pSI,lla rne.uyeTCXO./lHMOCTb pSI,lla.2.32. )JoKa3aTh,'-!TO .UJJSI TOro '-IT06h1 pSI.ll:L~=i z,,cxo.u1111cSI a6co-JIIOTHO, Heo6XO,lll1MO 11 .llOCTaTO'-IHO, '-IT06hI CXO.llHJIHCb pSI.llbII~= 1 I Re z,,2.33.
)JoKa3aTh,IL~= 1 I Imz,, I·Hqrn ecJI11 pSI.ll cxo.u11TCSI a6coJIIOTHO, TO npoH3-BOJlhHOe 113MeHeHHe nopSI.llKa CJle.llOBaHHSI qJieHOB pSI.ua He BJlHSieT Haero CXO,llHMOCTb H cyMMy.2.34. )JoKa3aTb,4TO ec1111 pSI.ll cxo.u11TCSI, HO He a6comoTHO, TOMO)KHO TaK 113MeHMTb nop.H,llOK CJie,llOBaHM.H '-!JieHOB p.H,lla, '-!TO BHOBbnoJiyqeHHbIH pSI.ll 6y.ueT pacxOJJ..HIUHMCSI.2.35.BepHO JIH yrsep)l{)leHHe: nycTb p.sr.u CXO.llHTC.sI, HO He a6co-JIIOTHO; Tor.ua ,/lJISI mo6oroaEpSI.ua, '-!TO OH 6y.ueT CXO./ll1TbCSI2.36. )JoKa3aTb,cMO)l(HO TaK nepeCTaBHTb qJieHbIKa .4TO ec1111 p.HJl CXO.llHTCSI, HO He a6COJ1IOTHO, TOMO)KHO 6e3 nepecTaHOBKH TaK crpynn11poBaTb '-IJieHbI p.H.lla, '-!TOBHOBb noIIyqeHHhIH p.H.ll 6y.ueT CXO./lHTbCSI a6coJIIOTHO.2.37. )JoKa3aTh,rnez;,=:L;:·~~14TO ec1111 pSI.llzk , { p,,}-:L~=r z,,L~=J z;,:L~= 1 z;, ,npo113BOJihHaSI cTporo B03pacrn10ma.H no-cne.uosaTeJJbHOCTh HaTypMhHhIX '-mce11 cPR):(cxo.u11TCSI, TO pSillHa3brnaeTc.HPi= 1, cxon11Tc.H.pRooM, noJLy'lertHbIM c noAtOUfbTO 2pyn-nupo8KU 'l!leuoe ucxoouozo paoa 6e3 H3Mettemrn ttx CJie,[(oBamrn.2.38.IIp11secT11np11Meppacxonmueroc.HpSI):(a,JlJISIKOToporoHaH:.ueTCSI HeKOTOpaSI rpynnHpOBKa ero qJieHOB, SIBJISIIOIUaSICSI cxo,[(.HIUMMC.H p.H,[(OM.2.39.
)JoKa3aT1>,no11HeHhI yc110BHSI:1) Jim z,, = O;11-+oo'-!TO p.H):(L~=J z,,6y):(eT cxo,[(HIUMMC.H, ecJitt BbI-ITOCJlE..QOBATEJlhHOCTl1, P5l..Qbl l1 ITPOI13BE..QEHI15l252) HeKOTOpa5I rpyrrmrpOBKa p5I,Ua (CM. 3a,Ua':ly 2.37) 5IBJrneTC5Icxo.u5Illl.HMC5I p5I.UOM H rrpH 3TOM sup(p11 +1 - p 11 )<00 •2.40.
,[(oKa3aTb rrpH3HaK Koum.2.41. ,[(oKa3aTb rrpmHaK ,[(anaM6epa.2.42. IIycTb lim J z,,+ 1 I z,,11........:,00J= q > 1.Cne.uyer JIH orc10.ua, qTo p5I.UI~=i Z11 6y.ueT pacxo,D;5Illl.HMC5I?2.43. ,[(oKa3aTb npu3HaK Paa6e: ecJIH lim n(i z,, I z,,+ 1 I-1) = p , TO11~ 00rrpH p> 1 p5I.U L~=I z11C5I a6coJIIOTHO. IlpH pcxo.u11Tc5I a6coJIIOTHO, a rrpH p= 1 p5I.U< 1 He cxo,D;HT-L~=I z11 MO)l(eT KaK a6coJIIOTHO cxo-.UHThC5I, TaK H a6COJIIOTHO paCXO,ll;HTbC5I.2.44.
,[(oKa3aTb npu3HaK I'aycca: ecJIHI z/I I Z11+I I= A+µ/ 11+8/l ln 1+e'c > 0, Ie,, 1~ C,ro rrpH A. > l p5I.U L:~=I z11 cxo,UHTC5I a6comortto, npH A. < 1 - He cxo-=JlHTC5I a6COJUOTHO; rrpH /...1 H µ > 1 - CXO,UHTC5I a6COJIIOTHO, a rrpH1 H µ < 1 - He CXOJlHTC5I a6cOJIIOTHO./. . =2.45. ,[(oKa'.3aTh mo:J1coecmeo A6eJUlr.ue1 :S 111 < 11, Sk= L~=t a1,S0 = 0, {ak}, {bk} - nocnenosaTeJibHO-CTH KOMIUieKCHhIX ':IHCeJI. B npyrott <t>opMe TO)K)l:ecrno A6emr HMeer BHJl2::~= 1, aA = I:~:~ (bk- bk+1 )ck+ b1/q ,n:re ck= ap+1 +... + ak.2.46.
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