Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 28
Текст из файла (страница 28)
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z,z,(a - d) 2+ +4bc = 0.= Z2, TOf)J.3 11 TOJlbKO.IJ:or<a33Tb, 4TO eCJIH HeflO)J.BH)!(Hbie TO'-IKHmrneHHOfO OT06pa)KeHJ151 pa3JIH4HbI,z,tZ2, TOz,H Z2 .upo6HO-182I'J1aea 134) )J,OKa3aTb, qTQ ecm! HenO.USJ.f>KHhie TOqKJ1 .upo6HO-JIJ1HeHHOroOTo6pa)Kemrn cosna.na10T, T. e. z 1 = z2, TO11a-d--=--+k,k=c,z 1 = - - .w- Z1 .::- 212c13.32. 1) On11caTb see .npo6Ho-mIHei1Hhie orn6pa)KeH115I sepxHei1:noJiynJIOCKOCT11 Ha SHYTpeHHOCTh e.u11H11qHoro Kpyra.2) OmrcaTb see .npo6Ho-JI11HeHHbie orn6pa)KeH115I e.n11H11qHoro Kpyra Ha ce65I.13.33 . .UoKa33Tb, 'ITO .npo6HO-JIHHeHHOe OT06paJKeHHe we.nHHHqttoro Kpyra Ha ce65I, y.uosJiernopmoruee yc11osHJIM= w(z)w(zo) = wo, arg w' (zo) =a, lzol < 1, lwol < 1, a ER,onpe.ueJIJieTCJI cpopMynoH:w-w0z- Zo a=el-ww0 1- zz013.34.
IIycTb cpyHKUHJI w = f(z) O.UHOJIHCTHa, amUIHTJ1qHa H orn16pIDKaeT e.nHHH'IHhIH Kpyr { z:rrpHqeM f(O)r.ne1.ial=lzl : : : 1} Ha e.nHHH'IHblH Kpyr { w: lwl ::::;1 }'= 0. llcrroJih3YJI JieMMY lllsapua, .UOKa3aTb, qTQ f(z) = az,13.35. llcnOJih3Y5I pe3yJihTaT rrpe.nh1.n;yrudi. 3a.na'IH, .noKa3aTb, 'ITOO.UHOJIHCTHaJI aHaJIHTHqecKa5I cpyHKUH5I, OT06pa)Ka!OruaJI e.n;HHHqHbIHKpyr Ha emrnH'-JHbIH Kpyr, JIBmieTCJI HeBblpOJK.UeHHOH .npo6HO-JlHHeHHOH cpyHKUHeH.13.36 . .UoKa3aTh, 'ITO .npo6Ho-mrneH:Hoe oTo6pa)KeHHe w =BepxHeH: rroJiynJioCKOCTH Ha ce65I, y.uosJieTBopJIIOIUee yc110BHJIMw(z 0)w(z)= w0, arg w '(z0) =a, Im zo > 0, Im wo > 0, a E R,onpe.ueJIJieTCJI cpopMy110H:w-wz-z.-----::!1- = --:!-e'a.w-woz-zo13.37 . .UoKa3aTb, 'ITO JII06oe KOHcpOpMHOe OT06pa)KeHHe BCeH:cKOMnJieKCHOH nJIOCKOCTHHa BCIO KOMITJieKCHYIO nJIOCKOCTbJI5IeTC5I .npo6HO-JIHHeHHbIM.c 5IB-13.38 .
.UoKa3aTb, 'ITO JII06oe KOHcpOpMHOe OT06paJKemre BCeHpacurnpeHHOH KOMnJieKCHOH nJIOCKOCTHKOMITJieKCHYIO IlJIOCKOCTbccHa BCJO pacumpeHHYJO5IBJI5IeTC5I .npo6tto-mrnei-1Hh!M.KOH<DOPMHbIE OTOEPA)l{EHH5118313.39. y CTaHOBl1Tb 06w:11i1 B11.U .npo6HO-Jil1HettHOfO rrpeo6pa3oBamrn:, rrepeBomrw:ero BepxmoIO rro11yn11ocKOCTh { z: Im z > 0}:I)2)3)4)HaHaHaHaBepxHIOIO no11yn11ocKoCTh { w: Im w > 0};rrpasyIO no11yrr11ocKOCTh { w: Rew> 0};HI1'.>KHIOIO rro11yrr11ocKOCTh { w: Im w < 0};JieByro rro11yrr11oc1<ocTb { w: Re w < 0}.13.39.1.
Hai1:rn 06w:11i1: B11.U .npo6Ho-1111Hei1Horo orn6pa)l.(eHmI,rrepeBO)UIW:ero BepXHIOIO rro11yrrnoCKOCTb Ha BHyrpeHHOCTb Kpyralwl<R.13.39.2. HaH:T11 06w:11H: s11.n .npo6tto-mrneH:ttoro 0T06pa)f(ett11H,rrepeBO.U5!W:ero Kpyr lzl < R Ha Kpyr lwl < R.13.40. Hai1:rn rrpeo6pa30Batt11e w = w(z), orn6pa)l.(aIOw:ee OTKpbIThIH Kpyr { z: lzl < R} Ha npasyIO no11yn11oc1<ocTb { w: Re w > 0}, rnKoe, 'iTO w(R) = 0, w(- R) = oo, w(O) = 1. YcTaHOBl1Th o6pa3hI nonyOKPY)l.(I-IOCTei1 {z: lzl = r, 0 < r < R, Im z > O}.13.41. Hai1T11 .npo6tto-1111ttei1:ttoe 0T06pa)l(ett11e, nepeso.n51mec:1) TO'iKl1 z = -1, i, l+i COOTBeTCTBeHHO B TO'iKl1 w = 0, 2i, 1- i;2) TO'iKl1 z = -1, i' 1+i COOTBeTCTBeI-IHO B TO'iKl1 w = i, oo, 1;3) TO'iKl1 z = -1, oo, i COOTBeTCTBeHHO B TO'iKl1 w = i, I, l+ i;4) TO'iKH z = -1, oo, i COOTBeTCTBeHHO B TO'iKl1 w =oo, i, 1;5) TO'iKl1 z = -1, oo, i COOTBeTCTBeHHO B TO'iKl1 w = 0, oo, I.13.42.
Hai1T11 .npo6tto-1111ttei1:ttoe 0To6pa)l(ett11e, TaKoe, 'iTO1) TO'iKl1 1 11 i HenO.UBH)l.(Hhl, a TO'iKa 0 nepexO.UHT B TO'iKY - l;l 25 3 .2) TO'iKH - H Heno.nsmKHhI, a TO'iKa - + - I nepexo.u1u B oo.24 413.43. Haifrn .npo6Ho-n11tteH:Hoe oT06paJK:ett11e w = w(z), TaKoe,'iTO TO'iKa i HBJIHeTCH .UBOMHOH HeITO.UBH)l(ll.OH TO'iKOH, a TO'iI<a z = 1rrepeXO.U11T B 00.13.44.
Haifnr TO'iKl1, CHMMeTpHLJHble T0 1-IKe 2 + i OTHOCHTeJibI-10OKPY)l(HOCTH:3) {z: lz -il=3}.1) {z: lzl= l}; 2) {z: lzl=3};13.45. HaHTH C11MMeTpH'iHhlH o6pa3 OTHOCHTeJibI-10 emrnH'i!IOHOKPY)l(HOCT11 rne.nyIOW:HX JJHHHH:1l){z:lzl=T };2)z:lz-1J=l};3){z:Imz=2};4) {z: lz- zol = lzoJ};6) {z: IRe zl2-5) {z: Jz- zuJ=~I zoi2-I, I z:0 I> I;IIm zl~ =1 };7) rpaH11ua TpeyronhH11Ka c Bepw11Ha~rn B TO'iKaxI'Jzaea 13184Z1, Z1, Z313.46.Im z > 0}E C\{O},Zk*z,,k, l= 1, 2, 3, k*l.0T06pa311Th KOHCpopMHO BepXHKHO rronyrnIOCKOCTh { z:Ha e)lHIDf'IHhIH Kpyr { w:1) w(i)jwj~1}TaK, GT06o1:=0, arg w '(i) =-11: I 2;2) w(2i) = 0, arg w '(2i) = O;3) w(a +bi)= 0,argw '(a+ bi)= 0, b > 0.13.47. 0T06pa311Th KOHQJOpMHO BepXHlOI-0 rronyrrnoCKOCTh { z:Im z > 0} Ha OTKphIThIB Kpyr { w: jw - w 0 j < R} TaK, qrnfa1 TOGKa irrepeurna B ueHTp Kpyra, a rrpOI13BO)lHa5I B 3TOH TOGKe 6orna ITOJ10)KJ1TeJlhHOH.13.48.0T06pa311Th KOHQJOpMHO OTKphlThIB Kpyr {rrpaByIO rroJiyrrnocKOCTh { wargw'(O)13.49.Im z > 0}Rew > 0}TaK, qrnfa1< 2} Haw(O) = 1,= 11: I 2.0T06pa311Th KOHQJOPMHO BepxHI-OI-0 rronyrrnOCKOCTh { z:Ha HIDKHfOlO IlOJIYIIJIOCKOCTh TaK, GT06hlw(z 0 )13.50.E C:z: lzl= 20 , arg w'(z0 ) = -tr/2, Im z0 > 0.0To6pa311ThKpyra { z: jzj< 1}KOHQJOpMHOBHyTpeHHOCThe)lHHHGHOroHa BttyTPeHHOCTb e)l11HI1GHoro Kpyra { w:lwl < 1}TaK, qT06hI1) w(T 1)= 0,argw'(2- 1)12) w(r i) = 0, arg w'(T3) w(O)4) w(zo)13.51.= 0,=zo,w(zo)Kpyra { z:w'(zo) =a, a ER.arg=jwj < R}1vo,arglzl <w'(zo)=a,jw - 2ij < 2}0To6pa311Thlzl < 1 }r}HaTaK, GTOfaIlzol < r, lwol < R.TaK, qT0601KOHcpopMHOw(2)= i, argBHyTpeHHOCTh21< 1}w'(2) = 0.0T06pa311Th KOHcpopMHO OTKphIThlH Kpyr { z: jz -Ha OTKphIThrn Kpyr { 1v:13.53.= -11: I 2;0T06pa311Th KOHQJOPMHO OTKphIThll Kpyr {z :OTKphIThIH Kpyr { w:13.52.w'(O)arg= O;i) = 11: I 2;1e)lHHHGHOroHa BttyTPeHHOCTh e)lHHHGHOfO Kpyra { w:lwl < 1 }KOHC!JOPMHb!E OTOEP A)!{EHI151185z z lzd, lz2I < 1, neperrurn B TOqK11 ±a, 0 < a < l.TaK, qT06b1 TOqKM 1 11 2,Onpenen11Tba.13.54.
Halim: 06ru11ii s11.u npo6Ho-n11HeiiHoro rrpeo6pa1osaH115fww(z), oT06pa)KaIOruero OTKphIThlli Kpyr< R} Ha OTKphIThlliKpyr { w:< R} TaK, 'IT06or=lwl1) w(a) =0, lal < R;{z: lzl2) w(a)=b, lal < R, lbl < R;3) w(±R)=± R.13.55. IIocTpo11Tb oT06pa)KeH11e BHyTpeHHOCTH e,UHHHqHoroKpyra {< 1} Ha seyrpeHHOCTh e,1J.11HWIHoro Kpyra { w:< 1} ,rrpH KOTOpOM rrpoo6pa3 ueHTpa H3XO,Ul1TC5f Ha nettCTBHTeJibHOH OCH,z: lzla nyra1)lwl0 ::; t.p '.S rr./2 e,UHHWIHOH OKPY)KHOCTH OTo6paJKaeTc5f B .uyry0 ::;e ::;rr./2;2) 0 ::;e ::;n/3;3) n/2 ::;e ::; 7 n/6.13.56 . .n:oKa3aTh, qTO HeBblpO)K,UeHHOe ,Up06HO-JIHHeHHOe rrpeo6pa30BaHHe w = (az + b) I (cz + d) MO)KHO rrpHBecrn: K BM.UY w =(az +~)I ( yz + 8), r.ueI: ;1 =1. EcnH (a+ 8) ER, TO rrpeo6pa3o-laBaHHe w = w(z) Ha3oIBaIOT 3JIJiunmuttecKu.M rrpH+ 81 < 2, 2unep60J1uttecKUM - rrptt+ 81 > 2 H 11apa60J1uttecKUM - rrpH + 81 = 2.
EcJIH )Ke Im (a+ 8) i- 0, TO npeo6pa1osaHtte wHa3oIBaIOT flOKcoopoMuttecKUM.13.57. HattTM 06m11i1 stt.n; rrapa6onwqecKoro rrpeo6pa3oBaHH5f OTKpbIToro Kpyra {< R} Ha OTKpbIThrH Kpyr { w:< R}' rrpH KOTOpOM TQqKa R OCTaeTC5f HenO,UBl1)KHOH.laz: lzl= w(z)lalwl13.58 . .z:i:oKa3aTb, qTO rrpH rnnep6onwqecKOM rrpeo6pa1osamm:1) JII06a5f OKp~HOCTb, npOXOMmM qepe3 .use HenO,UBH)KHbieTOqK11, nepexo,n:HT caMa B ce65f, H HanpaBJieHHe o6xona coxpaH5leTc5f;2) 11106a5f OKPY)KHOCTb, opTOroHaJibHa5f K OKPY)KHOCT5fM, npoxo.U5f1UHM qepe3 Heno.UBH)l(Hbie ToqKH, rrepexo,n:HT B oKpy)KHOCTb, 0611a.n;aIOIUYIO TeM )Ke CBOHCTBOM.13.59 . .z:i:oKa3aTb, qTo nptt 3JIJIHrrTHqecKoM rrpeo6pa3osamrn:1) JII06a5f OKpy:>KHOCTb, opTOrOHaJibH35f K OKpy)KHOCT5fM, npoxo.Ll5flll.11M qepe3 HeIIO,UBH:>KHhie TOqKJ1, rrepeXO,UHT caMa B ce65f c coxpaHeHHeM Harrpas11eH115f o6xo.n;a;2) .uyra oKpy)KHOcrn:, coe.u11H5fIOI1W5f Heno.usttJKHhie TO'-IKH, nepexonHT B .n;yry OKPY)KHOCT11, coe.n;11H5fI01UYIO Herrons11)1(Hbre TOqK1111 06pa1yIOruyIO yro11 a c nepsoii .n;yroii, a= arg k (cM.
13.30).I'.rzaBa 1318613.60. LI.oKa3aTb, lJTO np11 JIOKCO)lpOMl1lJeCKOM npeo6pa30BaHl1H:1) BbinOJIH5UOTC51 CBOHCTBa 13.58 2) H 13.59 2);2) He cywecrnyeT Heno)lBmKHbIX OKPY)l<HOCTei1, ecnH TOJibKOarg k f. 11: (cM. 13.30); ecn11 )!(e arg k 11:, TO oKpy)!(HOcrn, npoXO)l51lllHe qepe3 HenO)lBmKHble TOlJKH, rrepeXO)l5!T caMH B ce65! c 113MeHe1meM HarrpasneHH5! o6xo)la.=w=e;;. -::,-al-a-::.'a=lale;a,ial<l,MO)KeT 6bJTb TOJibKO n1160 3JIJil1ITTHlJeCKHM, JIH60 napa6oJIHlJeCKHM,1rn60 rV1rrep60JI11l!eCKl1M.
BhrnCHl1Th, npw KaK11x 3HalJeHH51X a H A.HMeeT MeCTO Ka)K)lh!H 113 yKa3aHHhIX cJiyqaeB.13.62. Ll.oKa3aTb, lJTO KOHcpOpMHOe orn6pa)KeHHe KOHUeHTpHLJeCKOro KOJibUa { z: ,., < lzl < r2} Ha KOHUeHTpHLieCKOe KOJlbUO{ w: R 1 < lwl < R1} cywecrnyeT TOr)la H TOJihKO TOr)la, KOr)la r 1/r2 =R ii R2 • IlpH BhinOJIHCH1111 3Toro ycnOBH51 BC51KOe KOHcjlopMHOe oTo6pa)KeHHe 51BJI51eTC51 OT06pa)KeHHeM BH)la w = az HJil1 w = az -I H O)lH03Hal.JHO orrpe)leJI5!eTC51 3a)laHHeM O)lHOH rraphl COOTBeTCTBYIOWMX.upyr .upyry rpattl1lJHbIX TOlJeK.13.63. Orn6pa3HTh KOttcpopMHO:1) KOJihUO { z: 2 < lzl < 5} Ha KOJibUO { w: 4 < lwl < 10} TaK,lJT06bI w(S) = -4;2) KOJlbUO { z: 1 < lz - 2ilTaK, 'lT06hI w(O) -1 - 2i.=< 2} Ha KOJihUO {iv: 2 < lw - 3 + 2il < 4}13.64. Orn6pa3MTh KOHcpopMHO:1) o6nacTb {z: Re z > O} \ {z: lz - hi< R}, h > R, Ha KOJihUO{ w: p < lwl < 1} TaK, lJT06bJ MHHMa5! OCb rrepeurna B e)lHHHlJHYIO OKPY)KHOCTb { w: lwl=1}; onpe)leJIHTh p;2) o6nacTb {z: Re z > 0} \ {z: lz - hi< 1}, h > 1, Ha KOJihUO{iv: 1 < lwl < r}; onpe)lem1Tb h.13.65.
Orn6pa311Th KOHcpopMtto:1) 3KCUeHTpHcieCKOe l<OJibUO{z: lzl < r, lz - 1/21 < 1} Ha KOHUeHTpHl!eCKOe KOJihUO { w: 1 < jivl < p}, orrpe.ueJIHTb p.2) 31<CUeHTpH4eCKOe KOJibUO{ z: iz - 31 <9, lz - 81 < 16} Im KOHueHTp11c1eCKOe KOJihUO { w: 1 < lwl < p}, onpe)leJI11Th p;13.66. Halh11 rpynnhJ mrnetttthlX npeo6pa3oBaHHtt, coornercrsy10ru11x rrpH cTepeorpacj)114ecKoi1 rrpoeI<u1111 spaweH1110 ccpepbl P11Matta:KOH<DOPMHb!E OTOEP A}!{EHH5I1871) BOKpyr BepTHKaJ!bHOfO ,nHaMerpa;2) BOKpyr JlHaMerpa, rraparIJieJibHOfO .uei1CTBHTeJibHOH om;3) soKpyr .UHaMeTpa, rraparIJieJihHoro MHttMoi1 octt;4) BOKpyr mrnMeTpa, cTepeorpacp11qecKa5! rrpoeKUHH o.uHoro 113KOHUOB KOTOporo eCTb TO'-!Ka a.13.67.
,UoKa3aTb,'-ITO rpyrrrra JIHHei1HhIX rrpeo6pa3osaH11i1, cooT-BeTcrny10mttx sparueHttIO ccpepbI PttMaHa tt rrepeso,UHm11x ToqKtt coacTepeorpacptt'-IeCKHMH ITpoeKUHHMH11b .upyr B .npyra,oITpe.uenHeT-CH COOTHOilleHHeMw-bI+bwiaI z-al----=e - - - .13.68. ,UoKa3aTb,l+az'-ITO cpyHKUHH w= z2 KOHcpopMHOcsepXHIOIO (HH:>KHIOIO) IIOJIYITJIOCKOCTb Ha ITJIOCKOCTboro6pa:>KaeTc pa3pe30MITO tteoTp11uaTe.flnHoi1 qacrH .uei1crs11TeJinHOH octt. Hai1T11 o6pa3hIITPH 3TOM OT06pa1KeHmI:1) nyqei1: arg z =a;3) ITpHMbIX Re z13.69 . .D;oKa3aTb,= c;2) OKPYIKHOCTeH4) ITp5!MhIX Im zlzl = r;= c.w = .Fzc c pa3pe30M ITO HeorpuuareJibHOH'ITO perymrpttbIC BCTBH cpyttKIJ;HHcpopMHO OT06pa1KaIOT ITJIOCKOCThKOH-qacTH .uei1cTmneJihHOH OCH COOTBeTCTBeHHO Ha BepXHIOIO 11 HH)!{HfOIO rronymIOCKOCTH.13.70.HaHTH KompopMHhIH o6pa3 rrnocKOCTHCc pa3pe30M rroHeITOJIO)l(l1TeJibHOH qaCTH .uei1CTBHTeJibHOH OCH, OITpe.uenHeMbIH peryn5IpHhIMH BeTB5IMH cpyHKUHH w =13.71.Fz.Orn6pa3HTb KOHcpopMHO BHyTpeHHOCTh KaKoif-n1160 sernHpaBHOCTOpOHHei1 rnrrep60JihI Ha sepXHIOIO rronyrrJIOCKOCTb.13.72.Oro6pa3HTh KOHcj:iopMHO BHernHOCTh rrapa6onr,1 Ha sepx-HIOIO ITOJIYITJIOCKOCTH.13.73.
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