Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 15
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P51)J,bl 113 AHAJ111Tl1lfECKl1X <!>YHKUI1Hnpe.ueJibHYIO ToqKy B D.11Me10w:eeZ,{f,,(z)} CXO.lll1TCR Ha MHO)J(eCTBeEcJI11noCJie.uosaTeJibHOCTbTO OHa paBHOMepHO eXO.[(HTCRBHYTPM 06JiaeT11 D.PaeeMoTpI1M cpyttKUI1IOK= {z: jz -zol< R}.f(z),attaJI11T11l.fecKy10 B tteKornpoM KpyreTaKyIO cpyHKUI1IO Ha30BeM 3!leA1eHmoA1 (3Jie-MeHT aHaJI11T11qeeKoi1 cpyttKu1111). ToqK11, Jie)J(am11e tta oKpy)J(ttocTMf = {z: lzl = R}, 6y.[(eM IIO.[(pa.3.[(eJIRTb Ha .[(Ba KJiacea:SE f, 06Jra.[(aIOIUHe OKpeCTHOCThIO1) TQqKHU~ = {z: lz -SI < p},B KOTopoi1: eymeernyeT tteKOTOpaR aHaJIMTHqecKaR cpyHKUHRf t;(z),cosrra,z:i:a10w:aR c f(z) B o6w:eH: qacTH Kpyros K H U~, tta3oBeM npa6WlbHblA1U.llJIR ,z:i:aHHOro 3JieMeHTa;2) TQqKlfoco6blA1UlfaSE f, He 06Jra.[(aIOll.l,He TaKOH OKpeCTHOeTbIO, Ha30BeMTOl.fKaMH 3JieMeHTa.Orrpe.[(eJreHHR rrpaBHJibHOH TOlfKH BbITeKaeT, qTO eeJIHSO E f -r,HaXO.llR-npaBHJibHaR TOlfKa, TO H see TOqKH ,z:i:yrtt OKPY)J(HOeTHmei1eR BHYTPH cooTBeTeTsy10w:eH: oKpecTHOCTH U ,8.1.
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.[(OKa3aTb,12""=5) L . m=I ~2 ·11lfTO Tpe6osaH11e a6coJIIOTHOH CXO.[(HMOCTH CTe-L~= 0 a 11 z B o.n:Hoii rpaHMlfHOH TOlfKe ero Kpyra11neHHOrO pH.n:aCXO.[(MMOCTM { z:lzl < R}BJielfeT 3a co6oi1: a6comornyIO M paBHO-MepHyIO CXO):(MMOCTb 3TOro pHna B 3aMKHYTOM Kpyre { z:lzl :s R}.8.9. Ilycn, {an} - MOHOTOHHO CTpeMHlllaHCH K HYJIIO IIOCJie.[(OBaTeJibHOCTb, panttyc cxonMMOCTH pHna L~= 0 a 11 z paseH 1. .[(oKa3aTh,11lfTO rnrna pHn cxonttTCH sc10ny Ha oKpy)l(HOCTM {KJIIOlfeHMeM, 6bITb MO)KeT, TOlfKH z1.=z: lzl = 1 }, 3a 11c-8.10 .
.[(oKa3aTh BTopy10 TeopeMy A6emi:.8.11. ITonh3YHCh BTopoii: TeopeMoii A6enH, noKa3aTh paseHcTBa:°"=1) L..11=1 COSIZ(/J=-ln,2sin (/JITl2 , O<m:<;;r·'I-',"'"'= sin ll(/J2) L..11=1Tl= !r -2 q; , 0 < qJ < 2;r·,105CTEDEHHhIE P.51)].bl. P.51)].bl 113 AHAJil1Tl14ECKl1X C!JYHKU:11M3)"00sin(2n + l)ip(2n + 1).L. 11 =1=n4'0<< n·<P'" 00 (- l)"+i sinnqJ = 2' I<P I< n .4 ) .L.11=18.12.qJ11Ilp1rnecrn np11Mep pacxo,D.}IW:eroc.11: p.11:.ua I~= 0 a 11 ,,~\I!o L~= 0 a 11 zrnporo cyw:ecrnyeTI~= 0 a,/' paBeH•f(z) = L~= 0 a11 zPa3JIO)KMTh B p.11:.11: Te:H:Jiopa B OKpecTHOCTH TOqK11 z= 0 3a-!i0;n Ia 11 I= 0,TOrrycTh11ecJIM11MnycTh pa.u11yc cxo,UMMOCTM;,~~0 f(z) = 08.14.1KO-118.13.
,[(oKa3aTh meope.My Tay6epa:p.11:.uammI~= 0 a 11 =0..11:aHHhie cpyHKUMM M HaHTM pa.11:11yc CXOJJ:MMOCTM IlOJiyqeHHhIX p.11:.11:os:1)(1-z)- 2 ; 2)(l+z)- 3 ; 3)(a2 +z 2)- 1,a*0;4)(1+z2)- 1;5) c1 - z 2)- 1; 6)o+ z32)- ;7) [(1 - z 2)(4 + z 2)r 1•8.15. Pa3JIO)l{HTh B p.11:A TeH:Jiopa B oKpecTHOCTM z= oo<l>YHKUHHH3 npeJJ:hIJJ:YW:eH 3a.11:aqH H YKa3aTh o6JiaCTH CXOJJ:HMOCTM IIOJiyqeHHhIX PHJJ:OB.8.16. Pa3JIO)KMThs p.11:.11: TettJiopa B oKpecTHOCTH ToqKH z=0 CJie-JJ:YIOW:He cpyHKUHH H HaHTH pa.11:ttyc CXOJJ:HMOCTH IIOJiyqeHHOfO p,lf.[{a:1) e z;2) sin z;6) sin 2 z; 7) cos 2 z;10)f: e~' d~;11)3) cos z;4) sh z;8) e z cos z;9) e z sin z;fo' si~~ d;; 12)f 1:;;;'05) ch z;13) r=d;, .Jo 1+ ;-8.17.
Pa3JIO)KMTh s pHJJ:hI TettJiopa rro cTerreHHM (z - z 0), lzol > 1,CJieny10w:11e cpyHKUHH H yKa3aTh o6JiaCTH CXOJJ:HMOCTH 3THX p.11:.11:os:1)(1-z)- 2;8.18.2)(z 2 -l)- 1;3)(z2 +1)- 1•HattTH pa.11:11yc cxo.11:11MOCTH 11 yKa3aTb o6JiaCTh cxon11MOCTHp.11:na TettJiopa s oKpecTHOCTH z= i JJ:JI.11: cJieny10w:Hx cpyHKUMH:2l)(z3-l)-2;2 )ezsinz;4) [(z-a)(z-b)r1•3)Z(1 + z)2 'a*i,b *i,a*b.T1aea 81068.19.
JJ:oKa3aTb, LITO K03cpcp11u11eHThI a,, pa3JJO)!(ettm1 B pRJl Tei1nopa cpyttKUHH f(z) = 1/(1-z-z )= L:~= 0 a,,z" yJlOBJiernopmoT co2=OTHorneHHIO a,, a,,_ 1 + a ,, _2 • Hai1T11 a JI 11 paJl11yc CXOJlI1MOCTI1 p51Jla. qI1CJia a JI Ha3bJBaIOTC51 'IUCJWJ\IU C/Ju60HQ'l'IU.8.20. IlycTbf(z)= L:~= 0 a11 z11, lzl < R.a) I1oKa3aTb, '-ITO ecn11 np11 KaKoM-JI1160 r, 0 <so Kow11 rrepexomu B pasettcrno, TO f(z) = a k z k.6) ITycTh s n. a) np11 HeKOTopoMTerpaJibHoe paBeHCTBOr < R, HepaseHCT-r, 0 < r < R, BhIITOJIHReTcR HH-1 (JT·o ,2-J,- I f(re' ) 1- dB= max I f(r) I .2Jr Ol~l=rCneJlyeT n11 OTCIOJla, '-ITO f(z)=a kz \k E N?8.21.
JJ:oKa3aTb, '-ITO ecn11 cpyHKUHR f(z)np11 lzl= L:~= 0 a11 z" orpatt11Lietta< 1, TO L:~=ol aJ CXOJll1TC5!.8.22 . .D:oKa3aTb,p51Jl L:~=olqTo ecJIH ,11,mr cpyHKQHHlzl <f(z) = L~= 0 a,,z'',1,a,,12 CXOJlI1TC51, TO cnpaBejlJII1BO paBeHCTBOi r2JTI,~,,.1Z"Jo f(r,rp)- f(p,rp) 1-drp=,f:ol a 11 l-(r"-p"t, r,p < 1.28.23. JJ:oKa3aTb, 4TO ecn11 cpyHKUI151 f(z) aHaJII1TH4Ha B Kpyre{ z: lzl < 1} ,f(O)= 0, TO pRJlL:~=J (z') CXOJlHTCR JlJIR lzl < 1 11 rrpen-CTaBJIReT BHYTPH yKa3aHHoro Kpyra attan11T11qecKyIO cpyttKUHIO.8.24.
IlyCTb cpyHKUI151 f(z) aHaJIHTH'-IHa B OKpeCTHOCTI1 TO'-IKH2I1 f(O) = 0. JJ:oKa3aTb, '-ITO eCJIH f(z) = z + f(z ) ' TOz=0f(z)= L:~=o z2".8.25. IlycTb cpyHKUI1I1 f(z) I1 g(z) aHaJIHTH'-IHbl B OKpeCTHOCTHT04KH z=0H g(::) =f (z) . JJ:oKa3aTb, '-ITO eCJIH f(z)1-az= L:~ oa ::"'11="g( z) = "'~L...,;,,= 0 b11 :::" , TO bo = ao, bJl+I =a bJI + aJl+I·8.26.
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