Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 49
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4) TaK KaK cpyHKU11ll f(z) 11MeeT .use oco6b1e TO'-IKH a 11 b, TO pa.uyccxo)l,11MocT11 pll,ll,a pasett R = min(Ja - iJ, Jb - iJ). o6Jiacn, CXO.UHMOCTH eCTblz - ii< R. IIpose.ueM pa3JIO){{ett11e:l =-1-(_l___l_)= _I_[ lf(z)=(z-a)(z-b)=-1a-b z-az-ba-b z-i-(a-i)I ]=z-i-(b-i)[-.1 f(z-iJ" -1-b.I f(z-iJ"]=b-1a-b z-a 11=0 a-1=1 [1.
11+111=ob-a (a-1)=I--11 =01(b-i) 11 +1Jc~-i) 11 , lz - ii < R, R = min(la - ii, lb - ii).8.54. 2) ,UaHHbIH Pll.U CXO,UHTCll pasHOMepHO Ha mo6oM KOMnaKTe KOMllJJCKCHOH nJJocKocT11, no3TOMY f(z) = I~J z I (n2+ I z /2)- 1 ecTb11enpe-pbIBHa.l! cpyttKUHll Ha scei-1 KOMnJJeKCHOH IlJIOCKocrn, HO TaK KaK Irnf(z)= 0,TO cpy1IKUH5! JCz) He 5!BJilleTC.l! aHaJJl1Tl1'-leCKOi1 HW B 0,UHOH TO'-!Ke KOMnJJeKCHOH nJIOCKOCTl1.
HanOMHHM, '-!TO cpyHKUm1JCz) aHaJIHTH'-!Ha s TO'-!Ke z0, ecJ111OHa aHaJil1Tl1'-!Ha s HeKOTOpoM: OKpeCTHOCTl1 TO'-IKl1 Zo.I'flflB(l99.7. Ilo TeopeMe JJopaHa B .uaHHOM KOJ!bUe cPYHKUH.l! fCz) MO){{eT 6bITbpa3JJO)l(etta s pR.U Jlopattaf(z) =I;=~ ak (z - a)k. Ecn11 s pR.ue JJopaHa seeK03cpcp11u11eHTbl ak np11 CTeneH.l!X k = -1, -2, ...paSHbl Hymo, TO B 3TOMPEWEHH51 H1IU14HbIX 3A.llA4322cnyciae cywecrnyeT KOHe'lttb1i1 npe.nenlimf:~a(z) = a0 11, cne.noBaTeJibHO,TO'!Ka a l!Bm1eTcl! yCTpatt11Moi1 oco6oi1 TO'IKOH (cM. 9.1). Ho 3TO nponrnope411T ycnoB1110 oTcyTCTBl1l! KOHe4ttOro npe.nena.
Attanorn'IHO, ecn11 B pl!.neak np11 CTeneHl!Xk = -1, -2, ... , OTJ1114Hb!X OT HYJil!, TO B )TOM cny4ae Jim f(z) = 00 , T. e .Jlopatta 11MeeTCl! Jl11Wb KOHe4HOe 411CJIO K03cpcpm..111eHTOB.:~aT04Ka a l!BJil!eTCl! noniocoM (cM.9.3). Ho 3TOro TaIOKe tte MO)!(eT 6bJTb Brnny ycnoB11i1 3a.na411.
TaKMM o6paJOM, cywecrnyeT 6ecKotte4HO Mttoro K03cpcp111.111eHTOBak np11 CTeneHl!x k = -1, -2, ... , OTJil14HbJX OT ttyw. A 3TO 3Ha-411T (CM. onpe,ll,enett11e), 4TO T04Ka a l!Blll!eTCl! cywecrnetttto oco6oi1 T04KOHc!JyttKU1111 fl..z).9.17. l).llJll! Bcex<l>yttKU11l! cossI;, I I < oo.s l!BJil!eTCl! uenoi1, TI03TOMYcos~= L:;=Oak~kCne.noBaTeJibHO, cpyttK11;11l! cos( l/z) .llJll! Bcexnpe.ncTaB11Ma B Bl1.llCl '\;"'okcos-=~k=-a-kzzL:~~- a_k lpa3J!O)f(ett11e Bpl!.11,. TaK11Moo,. 3 TO 3Ha'lHT, 'lTO "''t'YHKUHl!cos( l/z) 11MeeT npaa11JibHY10 qacTb pl!.lla Jlopattapl!.na JiopaHaz, 0 < lzl <a0=l 11 rnaatty!O qacTbo6paJoM, <PYHKU11l! cos(l/z) .nonyCKaeTJlopatta B OKpecTHOCTl1 TO'-l.Kl1z= 0.7) Pew11M ypaB1-1e1-111e sin z = 5.
OHO '.lKBHBaJieHTHO ypaBHeHmoe2iz - lOieiz - 1 =0.Pew11B ero KaK KBa.npaTHOe ypaBHeHHe OTHOCHTeJibHOeiz= (5 + 2/6 )i,eiz =ei', nonyqHM(5 - 2/6 )i.0TCIO.ll3 BbJTeKaeT, lJTO cywecrnyeT .use l10CJie,D,OBaTeJibHOCHI T04eKz;,=-iln(5+2/6)+!!..+2111r,2nEZ,z;.'=-iln(5-2/6)+!!..+2n7!, nE Z,2Ka)!(.llal! 113 KOTOpblbX eCTb peweH11e ypaBHeHl1l! sin z = 5.
KpoMe Toro, see YKaJaHHbie T04Kl1 l!BJll!IOTCl! l10JIIOCaMH nepsoro rr6pH,!l,K3 c!JyttKUHl1 l/(sin z - 5) HI z;, I, I:::;, I~+00np11n -*±oo. Cne.nosaTeJ1bHO, T04Ka z = oo He l!Blll!eTCl! !130-n11poaattttoi1 oco6oi1 TO'lKOH cpyHKU1111l/(sinz -5),T. e. 3Ta ¢YHKLU1l! He,[lonycKaeT pa3JIO)!(ett11l! B pl!.ll Jiopatta B TO'lKe z = oo.9.19. Brnny c!JopMyJI .lllll! KOJ<jJ<jJ11u11eHTOB p.S!,D,a Jiopatta 11MeeM:a,,=+J f(~)~~+I,n=0,±1,±2, ... ,-m 1:-::o/=P (z - "o)r,D,e r:Sp:S R ,TaK KaK pl!.ll cxo.n11Tcl! B 3aMKHYTOM KOJibUe no ycnos11103a.ua411. Ern11 rrOJIO)!(l1Tb p =r,TO rronyq11M ttepaseHCTBOPEWEHl15l Tl1CTl14HbIX 3A,ll,A4I a11 I= r-"27lEcm1ITOeifJ f(z 0 + reifJ)dfJJKe nonoJKHTb p323I~,.-" 1:-:ol=rmax I f(z) I= r-"M,..= R, TO attanorttlJHO nonyl!HM ttepasettcrnolf(z)J.Ja 11 l~R- MR,r)],e MR= max111:-:oJ=RCKna)J,&IBaH 3TH )],Ba ttepaBeHCTBa, nonylJHM1111M r +R-"M R )<M(-I_I+ R- 11 ) , /1 -O, +!_ , +2_ , •.• ,l<_!_(-2M = max(M,., MR)/2, lJTO 11 rpe6osanocb )J,OKa:mTb.IClr.n:e11 -9.25.
5)I1cXO)],H3H cpyttKUHSIfi.z) rrpH lzl > 2 MOJKeT 6&ITb rrpe.n:cTaBJieHaB BH)J,e pH)J,aJ (z) =_I_(25zl _ l )=l-l/z 21+4/z 22= ~[(1+1/z+1/z 4 + ... )-(1-4/z 2 +16/z 4 -64/z 6 + ... )J=5z= ~(51z 2 -15/z 4 +65/z 6 -255/z 8 + ... )= f l+ 4C-4) ·z-4Sz59) .[(attttaH cpyHKUHH fi.z) np11 0 < lzl < l MOJKeT 6bJTb rrpe.n:cTaBJieHa1121111=0BBH)J,e rrpomse)J,ettHH .n:Byx pH.n:os:-( l-lz 1)z :((-+l+z+zze· --+- =e= (i+ z +lZz2!+ ...
+z 11 + ... )=1122+ ... +z + ... J(!..+l+z+z2+ ... +z"+ ...2!n!zZ~Z: 11 - I= -+!+-+ ... +-+...1+ ... + ...2n.:::"..,)=,:-3.:-n+I+l+: +-+ ... +-+ ... :+ z- +-+ ... +-+ ...2!3!11!I~1I1z2!2!= -+2+(2+-):+(2+-+-)z3!n!II,,+ .. . +(2+-+ ... +--)z =2!(n+l)!l~ll= -+2+L.,(2+-+ ... +--)z.IIZn=l2!9.37. PaccMOTpHM(n+l)!cpyttKUHIOcp(z)= (z -a)"'fl.z).Tor)],a cpyttKUHHHBJrneTCH orpaHJ1LleHHOH B HeKOTOpoii OKpeCTHOCTH TOlJKHa,cp(z)nOCKOJlbKY111lcp(z)I = lz - al if{z)I < M. Cne.n:oBaTeJibHO, TOlJKa a HBJIHeTCH ycTpaHHMOHoco6oi1 TOlJKOH cpyHKUHH cp(z) 11 cyI11ecrnyeT KOHelJHblH npe.n:en (CM. 9.1 H9.2) lim(z - a)"' f(z):-Ta)],ID!=b.A3TO KaK pa3 11 03HalJaeT, LITO TOLIKacpyHKU1111.f(.Z) IIOJIIOCOM rropH)J,Ka He 6onee, LieMm.a HBJHieTCHPEWEHI151 nmVIYHbIX 3A)J,AY324I'11a6a JO10.3.
7) TaK KaK lftz)I =eReP(z)'H npH lzl =rcnpase.D,JIHBO ttepasettcrnoRe P(z) :S IPCz)I :S laol rP + ... +la,,I = rP (1a0 j +I:,!+ ... + Ir > R(£),£~; 1 J < rP(laol +s ),> 0, TO(1)jf(z)j<exp(rP(ja0j+t·), r>R(E).IlycTb arg a 0= a; Ha nyc1e arg z =-alp 11MeeM oueHKY1Re P(z) =I a0 1 rP +Re( a 1z"- + ... +a,, )::?: laol r" - la 1zlp-l+ ..
.+apl?:= r "(I aol _151_ ... _I a,,IJ > r " Cl a0I-£),:?: laol r" - CladrP-' + ... + iapj)rr"r > R 1(£). TeM caMbIMI f( z) I> exp(rP Cl a0 j-£) ) , r > R1 (£), arg z=- a.(2)p=Ha OCHOBe HepaBCHCTB ( l) 11 (2) 33KJII01..f3CM, l.fTO rropll.UOK p p, Tl1II CJ= laol·10.12. 5) Ko3cj.lcj.Jttu11eHTpJI,na R= · .. 11::1 = oo,limvlalllall=1(11In11)Illa,rro3TOMY pa,n11yc cxo,n11MOCT1111 cj.JyHKf..\HH f(z), TeM caMbIM, - f..\eJiaJI. Hcnonh3Yll·l/ """'COcj.JopMyny .nm1 Bbll.fl1CJICHl1ll rropll,nKa pp.= hm,,_,~11!1111(n I a) ln[n Inn JI-= Jimll-->~nlnnIn j l I all I' rIOJiy<Il1M=a.
TaK KaK nopHI\OK p = a> 0, TO THn BhJLIHCAA-eM rro cj.JopMyne (aep)"P =Jim(n "P~ ). YIMeeM:l/~0011(aea) a = Jim[ n 11 a 11 ( -lll-->~nlnnYITaK, rropll,nOK p= a,THnCJ=)"'al -= Jim n _!_a11 -->~l(nlnn) 11 a= 0.0.10.21. l) )lnH cj.JyHKu1111.ftz) =sin z nopHI\OK p = 1, THn CJ= 1. Ee ttyJrn:zk = ±kn, k E N. Ilpo113se.nem1e MHO)f(l1Teneii, cooTBeTCTBY!OllU1X HYJillM kn11 -kn, pasHo(1-~)e::.lk7!(1+~) e-::.lk7!= (l- ~k1lk1ll21k 1l-).Ilo:noMyPEWEH1151 THDHYHblX 3AJJ,AY.SID Z = Z eaz+bnueJia5!, rrp11 Z =?).( -?z--?=k= I 1-k-1r.SID Z _<DyHKUIUI - - -zb·a-0 ITOJI)"laeM l = e 11 SID Z = Z e ~325az+benk=l(=Z2)nk =I 1- - 2- ,=(1-Ul15! sin (z) - HeqeTHa5!, IT03TOMY e'IZ = e-az iurn mo6oro z Ek Jr-Z2 )k2:Tr2 .-<DyHK-C, IT03TOMY a=0. 0KoHqaTeJibHO rro11yqaeM pa31101KeH11esin z =zf1~={1- k~: 2 }zEC.10.38. 1) TaK KaKa'\' =f(z) =A L..
11 =0 -IIz11 , z E C,1kA'\' += aTO y(t) =A L..k =O k+l = - - .t1i.t-a10.51. 1) IlepBbIM cnoco6 peweH115!. TaK KaK J:IJI5! cjJyHKl.(1111 e' cpyHKL\1151,accou1111posaHHal! no 6opemo, eCTh y(t) = l/(t - 1), TO conp5!)!(eHHa5! ):l11arpaMMa D = ( 1 }. OnopHal! cjJyttKU11l! K D ecTb cjJyttKUH5! K( cp) = cos cp =K(-cp). IlmToMy l1H.lll1KaTp11ca poem h(cp) = K(-cp) =cos cp.BTOpoi1 cnoco6 peweHirn . 110 orrpeJieJieHlno, HH.ll11KaTP11ca poem <PYHKl.(1111 J(z) B&1q11cm1eTCl! no ¢opMy11eI()i rp- .-ln/f(rei(/!)/ O'~ rpr= 1llTI~2Jr.r~oo)],JUI cpyHKU1111f(z)= e', z = rh(rp) =Jim.:--tooiP, 11MeeM-lo Ieriq> IrIn ercosrp=Jimrr---'too=COS(/) .10.66. 1) TaK KaK 06m11i1: q11ett p5!.lla./( )/-112+xn2Jii.a11 ( z) = e-112 · e11 2Jii.z , z = x + iy,TO a11 z = e.Heo6xo,[l11Moe yc11os11e CXOJ:ll1MOCTl1 p5!.lla - 3TO a11 (z) -> 0, II03TOMYrrp11 x > 0 p5!.ll pacXO.lll1TC5!.
Ec1111 x '.S 0, TO .llaHH&1i1: p5!.ll CXO.lll1TC5! a6comoTHO. IfraK, p5!.ll CXO.ll11TCl! Ha 3aMKHYTOi1 rro11yrr11ocKocT11 Re z '.S 0.10.72. 5) 3arr11weM ,LlaHHbIM pl!J:l B Bl1,[le I ;= 1a 11 e-A,,z, r,[le a11 = (-1) 11/11,A.11 = In n. IlycTb z = x + iy. PaccM0Tp11M .llei1cTB11TeJI&Hoe 3Haqett11e z = x,--(-1)11l=n11TOr.lla a 11 e A,.- = - - · ---X- . Pl!.ll In(-1)11=1--..1-· CXO.lJl1TCl! rrp11 x > -1 11 pac-n·nXO.lll1TCH np11 x '.S -1, rro3TOMY a6cu11cca CXO.lll1MOCT11 c = -1. TaK KaK rrp11x2: -1 + E, E > 0, p5!.ll CXO.lll1TC5! paBHOMepHO, a rrp11 x < -1 - E p5!.ll pacxo.lll1TC5!, TO a6cu11cca paBHOMepHOM CXO.lll1MOCTl1 r = -1. lfccJie.llyeM P5!.ll HaPEW EHl15f THCTWIHbIX 3A)J,A 4326x = 0HMeeM \ a e-A,,: \=~a6comorny1D cxo.nHMOCTb. TipHx > 0 p51.n L~= 1 \a 1 e-A,,:\ = L~=I~pacxo.nHTC51, a rrpHcxo.nHTC51, TOn·na6cuHcca a6COJIJQTHOH CXO.UHMOCTH aI-haK, c = r = -1, a = 0.I~=I ~H p51.n11= 0.I'JiaBa J121111.2.
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