Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 46
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F 0(z)2!JieMHM-?1F,,Cz)+ L:k=17·( ~+ ... +; + ...ak -Jak) ++/~== 1, F,,(z) =2 cos(n arc cos z), 11 = 1, 2, ....17.72. Bee cpyttKL1111111Me10T 3KcnotteHL\11aJihHhIH nm cr.C_kPEllIEHIDI THIUJqHhIX 3A,II;A qI'Jlaaa 11.3.1) 11 2). IIpeo6pa1yeM Bb1paJKeH11e(l +cos qi+ i sin qi)"= (l + iP)''.C O,IJ,HOH CTOpOHhl,= °"""Ckeikrp =~k=O( l + /Cfl)"/1"'" ck"cos k qJ+16k=o·'\' " ck"sm· k(fJ= 6k=oC .UpyrOH CTOpOHhl,(1 + e;cp)"=2eirpl2 (efrp/2[+e2-irp/2 )")= 2" ei11rp12cosP...
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+z II)ir (J=ecrr11 z = I,I.(z"+-1){ e'-'(z -1) 'ecn11 z* 1.*3Ha'-!HT, ecrr11 z 1, TOcos x + r cos (x + a) + . . . + r"cos(x + na) =·. z"+l_leix(r"+·tei{n+l)a_l)(re-ia_l)=Ree'·' - - - =Rez- 1..(re'a-l)(re-ia _l ),.11+2 ei(x+11a) _ ,.11+1 ei(x+(11+l)a) _ rei(x-a) + eix=Re------~--------~2r -2rcosa+l11 2r + cos(x+na)- r"+' cos(r+ (n+l)a)- rcos(x-a) +cos ar~-2rcos a+ lPEWEHI151 TI1I1WIHbIX 3A)J,Aq3051.14, 0603Ha'il1M 11CKOMOe MHO)f(CCTBO 'iepe3 f.I. EcJI11 z 1 = z2, TO ycJios11e np11H11MaeT s11n lz - zd(l - k) = 0.
3Ha'!11T, ern11Zt =Z2 11 k 1, TO r COCTOl1T 113 0.llHOtt TO'iKI1 Z1. ECJil1 Zt = Z2 11 k = 1, TO r = c.II. IlyCTb Zt Z2. EcJil1 Im k 0, TO r - nycTOe MHO)f(CCTBO; eCJil1 kE R 11k < 0, TO f - nycTOe MHO)f(CCTBO, TaK KaKlz - zd ~ 0, k lz - z2I ~ 0***11 paBeHCTBO 06e11x qacTei1 3.llCCb HCB03MO)!(H0.Zt i= Z2ECJil1TO11r - 3TO fMTk =0, TO f COCTOl1T 113 0.llHOtt TO'iKl1 Zt.
ECJil1 Zrz E C, paBHoynaJieHHbIXOT .llByx pa3Jil1'iHbIX TO'iCKnpHMa51HaZtKOMnJieKCHOH11Z2,T. e.nJIOCKOCTl1[Zi, Z2],nepneH)ll1KYJI5ipHa51 0Tpe3KYf C,i= Z2 11k i= 1,C fm Zrrpo-XO.llllll\a51 qepe3 ero cepen11tty.EcJI11zk= Xk + iyh= x + iy (x,yzf,EXk>Yk ER) -R (k = 1, 2), aTO'!Ka 3TOtt npSIMOttTO ypaBHCHl1e ITp51MOttfMO)!(HO npe)l-Re zCTaBl1Tb B Bl1.llC(x2 -xr)(x -xo)+ (y2 -Yr)(y- Yo)= 0,r11e Xo = (xi + x 2)/2, Yo= (y 1 + y 2)/2, z 0 = x 0 + iy0cepe.n:irna OTJ'e31<a [zi, z 2].-3To ypaBHeH11e MO)f(HO 3an11can, B Bl1.llCxRe(z 2-z 1 )+ yim(z 2-z 1 )IlycTb TerrepbKaMHZtHZ2,Zr i=z 2 11TOT.JKHZ30 < k i= l .=~(I z2 l2-I z1 12 )CTPOl1M Ha rrpSIMOH, orrpeneneHHoil TO'i-H z~' KOTOphie .llCJI51T 0Tpe30KBHC!llHHM o6pa30M COOTBCTCTBCHHO B OTHO!llCHl111cZ3 E [z,, Z2], a Z4ITycTh z Eek,[zi, Z2]BHyrpeHHl1M HT.
e.lmzz[z,, Z2].r; TaK KaK z2e: r ,TO ypaBHeH11e MO)!(HO 3am1caTb B s11nelz - z 11!1z - z 2 = k.5IcHo, T.JTo OTPe3KH [z, z 3] 11[z, z 4] -611cceKTp11ch1 BHYT-peHHero11 BHelllHero yrnoBTpeyroJihHl1Ka cB TOT.JKaxZi, zHyrnoM, CT03TOMYRe zBep!ll11HaMHZ2.3Hal.JHT, 113 TOT.JKH z OTPe30Kr-[Z3,Z4] Bl1.lleH fIO.ll np51Mh!MOKPY)!(HOCTh, noCTpOeHHa51 Ha OTPe3Ke (Z3, Z4] KaK HaPEW EHJ15! THITW-!Hb!X 3A)J,A lf306i:urnMeTpe ( oKpy)f{HOCTb AnonnoH11ll).
YpasHeH11e 3TOH OKPY)f{HOCTH ecTblz -zol = R,rJJ,ez0 -cepe.nHHa orpe1Ka[z3 , z4],a R - nonos11Ha .UJII1HhI 3TOro0Tpe1Ka.z0 H R '-!epe1 .naHHbie 1aJJ,a'-!H. ITo onpe.neneHH!O TO'IeK z3 H z4z3 = (z 1 + kz 2)/(l + k), z4 = (z 1- kz 2)/(l - k). 0Tc!OJJ,azo = (Z3 + Z4)/2 = (z1 - k2z2)/(l - k\ R = lz1 - zol = k2 lz2 - zd Ill - k21.Bb1pa3HMnony'-IHMPerneHHe 3TOH 1aJJ,a'-!H MO)f{HO nony'IHTb cTaHJJ.apTHbIM 06pa10M. IIycTbzk = xk +iyb xbYk E R (k = 1, 2), z = x + iy, x, )' E R. TorJJ,a ypasHeHHe3a)],a'-!!1 npHH!1MaeT BH)],1.17.Bhrne.neM <PopMyJibJ, 1aJJ,aro11.1He CTepeorpa<PH'-!eCKY!O npoeKl.IHIO KOMnneKCHOH nnocKocrnCHa ccpepy P11MaHaS,11 cpopMyJihI o6parnoro orn-6pa)f{eHHll.ITycThz= x + iy,zC.
BseJJ,eM s rrpoCTpaHcTBe rrpllMoyronhHYIO JJ,eOs cosnaJJ,aeT c ochIO Re z, ocb OriEKaprnsy c11cTeMy KoopJJ.11HaT - ocbcosrra.uaeT c och!OIm z,rreHJJ.11KynllpHa I1JIOCKOCTHochc,OCHOs rrepOs, 011,(N(0. 0. 1)Os 06pa1yroT npasy10 rpoH.Ky.To'-!Ka A(1;, ~, ri) - CTepeorpacjmqecKallrrpoeKl.111ll TO'-IKI1zE CHa ccpepy P11MaHaAKoopJJ,11HaThI TO'-IKHHaxoJJ,HMS.KaKyKOOpJ],HHaTbl TO'-IKH nepeceqeHHll ccpepblP11MaHa, onpeJJ,eJilleMoi1 ypasttem1eM22s + 11 +cs - 112)2= c112) 2,• Z = X+ iyc rrpl!MOH, rrpoxOJJ.llll.leli qepe1 TO'-IKH N, AHz, 1aJJ,asaeMott ypasHeHHeMCs - O)l(x 3Ha'-!HT,TO'-!Kas* lAx=~(lO)J(y- O)(s *=Cs - l)/(O -1).s- (,), y = T]/(l - (,),1). 3Ha'-!eHHe = l 03HaqaeT, 'ITON (cesepHbIH rronroc c<PepbI S).
TaK KaK npHcosnaJJ,aeT c TO'IKOHHMeeMlzl 2 =x 2 + y2TO= c11 -O)= (s2 + TJ 2 )/(l -2l - s = 1/(1 + lzl )Hs)2S= xl(l += s(l -s)/(l - si2lzl ), T]=2lzl ),yl( l += ~(l -1.24.3) ITycTb z = x + iy, TOrJJ,aIIz- i 12I x2< (I+lzl2)·(l+li12) =22s),2s = lzl /(l + lzl ).+ ( y -1) 2I+x2+ y2 ,2PEWEH!15l TYIIU14Hb!X 3AJJ,A4222T. e. x + (y- 1) > 1 + x + /rroJiynJIOCKOCTb Im z < 0.HJIH -2y > 0. 3Ha'fHT, y307< 0. 11CKOMOe fMT -1.25.8) TaK KaK z = i +2iP, TO2lzl = cos <p + (1 +sincp)2 = 2 + 2 sin<p = 2 (sin (<p/2) +cos (<p/2))2 .ITycTb a - onHo 113 3HaqeHHH apryMeHTa z. Tornataa=l+sintpl+tg(tp/2)= cos(tp/2)+sin(tp/2) =-___;:___cos tp<>T.
e. a= arctgcos ( tp I 2) - sin (<PI 2)1- tg ( tp/2) 'l+tcr(cp/2)"'(cM. 3anaYy 1.12). 3ttaq11T,1-tg(cp/2).Jz =±J2jsin(tp/2)+cos(tp/2)leia/ 2 , rne2a= arctgl+tg(tp/ ).1-tg(tp/2)c-1.36. 6) ,L{oKa3aTeJibCTBO. ITycTb Z1> ••• Z11 EKopmr MHOfO'fJieHaf(z),T. e.f(z) = (z - z 1) · ••• (z - z11 ), rnrnaf'(z)/f(z) = l/(z - z 1) + ... + l/(z - z11 ).ITycTb CUE C TaKoe, qTO f'(cu) = 0, /(cu) ot 0, 11 cu He rrp1rnanJie)!rnT BbIrryKnoi1 o6oJIO'fKe T04eK z,, .... z...
Torna qepe3 T04KY (J) MO)!(HO rrpoBeCTHrrpHMyl{), He rrepeceKal{)myw BbIITYKJIYIO o6oJIOYKY YKa3aHHbIX TO'feK. ITo3TOMY BeKTOpbl (J) - Z1> •.. , (J) - Z11 Jie)!(aT B 01lHOH ITOJIYIIJIOCKOCTH, orrpeneJieHHOH 3TOH rrpHMoH.. TaK KaKz, , (zotO),z I z 1l-TO B 01lHOH IIOJIYIIJIOCKOCTH Jie)!(aT BeKTOpbl l/( (J)3TOMYf'(w) I f(w)'ITO HeB03MQ)!(H0. 3Ha4HT,(!)= l/(w -z I), ... ' 11( (!)--z,i), II0-z 1) + ...
+ ll(w- z11 ) t 0,np11Ha.LJ)1e)!(HT BblrryKJIOH o6oJIO'fKe KOpHettfiz).F!laea 22.20. <DyHKQHIO ez orrpeneJIHeM KaK ez = e'" (cosy+ i sin y). TaK KaKZ=Xr.D.e tg <P,, =+.1y ,zx.yx,-y2]1/2TO 1+ - = 1+ - + l - = (1 + -) + 2nnn11.n----1....!!!.__, n 2: 110 .
Tor)J.al+x/11[(1 ~)" [(1+=11,L{JIH .D.OKa3aTeJibCTBa HY)!(HO rroKa3aTb, 'fTOie rp" ,1112+ .::_)\11y: ]11einrp".PEWEHl151 Tl1Ill14HbIX 3A.lJ:A4308'[Jim (1 + .::)- +n--4-oo/1[[ncos (nip11 ) = e-' cosy,IC. <]"'2 sin (nipJim (1 + .::)- +11 -->~<]"/2n-= e"' sin y.)11. ]"'2 =limexp[~ln(l+x/n) 2 +y"!n"]=2lim (1+.::)-+Y:11-+oonn-11-+oon. n-'?xX-11n-[,,_,_ 2 (-=exp lim- -=-+-.,+1'.,+o(l/n 2 ))]"/2 =e'limcos(nip,,) = Iimcos(nZ) =cosy, Jimsin(nip,,) = Jimsin(nX) =sin y,1111-+oon11-+00II-JOO11-400TaK KaK rrpw n -;)oo WMeeM tg<p 11 - <p11 - yin, n<p112.55.
,I(aHHhIH p}l)J, CXOJJ:JHC}l TOr)J,apm1a1111( ?)--;)y.TOJihKO TOr)J,a, KOr)J,a CXO)J,}ITC}l )J,Ba°"-L..11:!_IfJ11•PH.U (I) cxonwTcH rrpH ex> 0 no rrp113HaKy Jlei16HWL(a 11 pacxon11rcH rrpw ex s 0,TaK KaK He Bh!ITOJIHlleTCH He06XO)J,11MOe ycnoBwe CXO)J,11MOCTl1 PH.Ila - o6ll\11M •rnett pH)J,a )J,OJI)!(eH crpeMHThCH K ttymo. PH.U (2) cxon11TcH np11 ~ > 1, anp11 ~ s 1 - pacxo)J,HTCH. TaK KaK12i=.!r+i-1l=(-1 +-1)a,/3n2a2pl1111''TO a6comOTHO pHll CXOLI11TCH npH min( ex,~)> 1.2.58. 4) IlyCTh ak =(211 -1) !z"1+ z", · - - , Torna(n !t2a 11 +1 1= 4n +2n·lzl· l+lzl"l+lzl"+1l a" (n+l)2EcJT11 lzl < I,TOIimla"+ 1 I= 4 lzl; ecnw lzl > 1,a,,11-tooTOIim,a"+1 I= 4. Ilo3TOMYa,,11--+oonp11 lzl < 1A pH.ll cxo)J,11TCH a6comorno no rrpH3HaKy ,I(aJTaM6epa.
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