Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 17
Текст из файла (страница 17)
ITycTh cpyHKUH}l f(z) attaJIHTHqHa B Kpyre { z: lzl < 1},lfi:z)I < 1, lzl < 1, f(O) i- 0. ITocJie,uoBaTeJihHOCTh {a 11 } - noc11e,uoBaTeJihHOCTh HyJieM: cpyHKUHH j(z), np11qeM lad ::S la2I ::S ... ::S la11I ::S .....D:oKa3aTb, qTQ p}l,U L:~=l (1- I all I) CXO,UHTC}l.8.69. .D:oKa3aThmeopeMy EnRUlKe: ,UJI}l Toro, qT06bI noCJie,uoBa-TeJibHOCTh {a II} ToqeK Kpyra { z: lzl < 1 } lad ::s la2I ::s ... , 6hrna MHO)f(eCTBOM Jie)f(all.(HX B Kpyre { lzl < 1} HyJieH aHaJIHTHqecKOH H orpaHJ-fqeHHOH B Kpyre { z: lzl < I} cpyHKU:IUf j(z), np.11qeM j(z) He ecTbTO)f(,UeCTBeHHbIH HYJib, Heo6xo,u11Mo .11 .uocTaTOY:HO, Y:T06hI cxomrncsi:PJ1.D:L~=I (1-I a,,j).8.70 .
.lJ:OKa3aTb meopeMy eOUHCmBellltoCmu: nyCTb cpyHKU:HH j(z)aHaJIHTHqHa B Kpyre { z: lzl < I} .11 orpatt.11qetta. ECJ1.11j(z) o6pamaeTcsi:BHYJihBja11 j ::S lan+d, n E N, np.11qeM psi:.uTOqKax ai, a2, ... ,L:~=l (1- I a,, I) paCXO,UHTCH, TO j(z)= 0.8.71. IlycTh cpyttKU:Hsi:j(z) attaJIHTHY:Hapatt.11Y:etta . .D:oKa3aTh, qTo ecmrj(n)B0611acT.11 Re z > 0 .11 or-= 0, n EN, TOj(z) = 0.8.72 . .D:oKa3aTb, qTO cpyHKUH}l j(z) = L:~=Oa,,z" Ha rpaHHUe KpyraCXO.D:I1MOCTH CTeneHHOro pR.ua HMeeT no KpaM:tteH: Mepe O.UHY oco6yIO ToqKy (pa.u11yc cxo,uHMOCTH cTenettttoro psi:.ua 0 < R < oo).8.73. IlycTh psi:.u j(z) = L~= 0 a,,z" HMeeT pa,uHyc cxo,u.11MocT.11 R,0 < R < oo. Pa3JIO)!(HM cpyttKUmoflz) no CTeneHRM (z - re ia), 0 <r < oo):f(z) = L~= 0 b11 (z - re;a)" ..D:oKa3aTb, '!TO ecm1 liml/~OOc.:jTbJr1= R - r, TO TOY:Ka z =Re ia 6y-.ueT oco6oii .UJIR cPYHKUHH j(z).
EcJIH )Ke Jim (11--:,00TOqKa z = Re ia-npas11JihHasr.efTb,lr 1 > R -r, TOCTEITEHHhIE PJI,Abl. PJI,Ahl Jil3 AHAJil1Tl1lJECKI1X <PYHKUI1H1138.74 . .[(oKa3aT& meopeMy IlpuHcxeuMa: ec;rn pa,D,11yc cxo,D,11MOCT11p5I,D,a f(z)= I~=o a,, z" paseH 1 11 a" ~ 0, n E N, TO ToqKa z = 1 HBJIH-eTCH oco6oii: TQqKOH )J,JI5I cPYHKU1111./lZ).8.75 . .1J:oKa3aT&, qrn cpyHKU115I j(z), npe.ucrnsrreHHaH pH.UOM1+ I~=I (z 2" In2),aHamuwqHa B Kpyre lzl <1 11 HenpepbrnHa npttlzl::;l, HO mo6a5I TOqKa OKPY)KHOCTM lzl8.76. IlyCTb b -= 1 5IBJI5IeTC5I )J,JI5I Hee oco6oi1.lbl < 1, M nyCTb c -KOMnJieKCHOe qwcrro,HaTy-pa.rrbHOe qwcrro, c ~ 2 . .[(oKa3aTb, qTO p5I,D, L~=ob" zc" MMeeT pa,D,11yccxo,D,11MOCTM R = 1 tt qrn ,D,JIH cj:>yHKUttttf(z)= L~= 0 b" zc''Ka)l(,UaH rnq-8.77 .
.[(oKa3aTb, qrn ecrr11 cj:>yHKUM5I./lz)= L~=0 a,,z"aHa.rrwmqHaB Kpyre { z : lzl < R} M Re fiz) S u, TO Ia,, Is2(u-Rea)Ka oKpy)KHOCTM { z : lzl= 1 } 5IB.IDieTC5I oco6oii:.R0° , n;::: 1.8.78. IlycT& cPYHKU115I fiz) aHa.rrttTttqHa B Kpyre lzl S Ro 11 n(r) qwcrro Hyrreii: fiz) B Kpyre JzJ < r <Ro, fiO) = 1. IloKman, qrn rnr.uan(r) S In M(er), er< Ro, M (r) =max J f (z) 1l=l=r8.79.
IlycTb cj:>yHKUM5I fiz) aHa.rrwrnqHa B Kpyre JzJ < R tt B :noMKpyre Re fiz) :::; A (R); noKmaTb, qrn TOr,D,a2rM (r)s (A(R)- Ref (0))--+ If (0) J, 0 < r < R.R-r8.80. ITycTh cj:>yHKUtt5If(z) aHa.rrttT11qHa B Kpyre lzl S R,fiO)=1ttj(z) -::f. 0 nptt z E {izl < R}; noKa3an, qrn B 3TOM crryqae-2rIn I f(z) l;:::--InM(r), I z Is r< R.R-r8.81.
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(12-k+l) a,, (z-z 0)"-k, kEN,rrocnentt11i1 p.Hn cxon11TC}l paBHOMeptto Ha mo6oM KOMnaKTe, np11ttan11e)l<arueM KOJJLuy { z: 0< lz - z ol < R}.P51.[l JlOPAHA. l130Jll1POBAHHbIE OCOEbIE T04Kl1 <DYHKUHH115ITycTb cpyttKurrn f(z) aHaJI11rn 1rna B oKpecrnocrn TO'lKl1 z 0, 3a 11cKJHO'leH11eM, 6bITb MO)!(eT, caMOH 3TOH TO'lKl1. Tor.ua B HeKOTOpOMKOJlbue { z: 0 < lz - z al < R} cpyHKUHIO fl.z) MO)!(HO pa3JlO)!(HTb B p51.UJiopaHaf(z)= L~=~a,Jz- z1111 )•=EcJm K03cpcp11u11eHTbI a11 = 0 np11 n -1, -2, ... , TO TO'lKY z 0 Ha3bIBaIOT ycmpaHu.Mou oco6ou nio'tKou.
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EcJIH TOqKa a 5IBJI5IeTC5I IIOJIIOCOM nop5I,UKa k .UJI5I cpyHKUHH fiz), TO OHa5IBJI5IeTC5I HyJieM nop5I,nKa k .UJI5I <PYHKUHH 1 I f(z).9. 7. IIycTh 4>YHKUH5I flz) aHaJIHTwrna B KOJihue { z: 0 < jz - al < r}.)J,oKa3aTh, 'ITO ecJIH He cymecrnyeT limf (z), HH KOHe'Itt.b1if, HH 6ec-9.6. IIycTbc}lyttKUH5Ij(z) aHaJIHTH'IHa B KOJibUe {z:)J,oKa3aTh, 'ITO ecJIH TO'IKaaecTb HYJih rrop5I.UKa:-4aKOHe'IHbIH, TO TO'IKa a 5IBJI5IeTC5I cymecrneHHO oco6oif" ,UJI5Ij(Z).9.8. )J,oKa3aTbTeopeMy CoxouKoro-Beifepnnpacca .UJI5I QeJioif<PYHKI.(HH, HCIIOJib3Y5I TeopeMy JlHyBHJIJI5I.9.9. )J,oKa3aTh Ma!ly10 meopeAty IluKapa:ueJia5I cpym<umrfiz),oT-JIH'IHa5I OT IIOCT05IHHOH, rrpHHHMaeT JII06oe KOHe'IHOe KOMfIJieKCHOe3Ha'IeHHe, 3a HCKJIJ0qem1eM, 6hITb MO)!(eT, 0,L(HOfO.9.10.
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