Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 23
Текст из файла (страница 23)
W2-O}lHO 113KpOMe Toro, eCJIH Wo - HeKO-TOpOe 3HalJeHHe cpyHKilHH Z a B TOlJKe Z1 · Z2, TO cymecTByIOT 3HalJeHH51:w,'ITO WoH W2 cpyHKL\HH z (/ COOTBeTCTBeHHO B TO'IKax Z1 H Z2, TaKHe,=w, . W2.I1MeHHO TaK IlOHHMaeTC51: paseHCTBO(z, .Z2) (/=z/' · z2°.11.36. )loKa3aTh,'ITO cpyttKQHRz 1111 ,n EN, B TO'IKez = re;'l',r=/=-0,HMeeT 3Ha11emrn7( ~1/u)= r l/11 e.!.c,, rp +2kirJ ,k=0, 1, ... , n -1,kB 4aCTHOCTH,11.37. ):{oKa3aTb,sernneHHR cpyHKUHH11.38.4TO TO'IKHz=0Hz= 0051:BJ151:IOTC51: TO'IKaMHz "", n 2: 2, n EN.YKa3aTh T04KH seTBJieHH51: H HX nop5!JlKH .UJIR KmK.uoi1H3 cpyHKQHH:2) [(z - l)(z + l)]w;11.39. IIocTpOHTh pHMaHosy3) z "2sin z.nosepxHOCTh cpyHKilHH zlln, n :S 2,nEN.11.40.BhllJHCJIHTh11.41.
)loKa3aTh,cpyHKUHH: F 1(z) = z11.42. ):{oKa3aTb,i ;_Ji4TOHF 2(z)onpe.uenReT.useaHaJIHTH'leCKHe=-z.4TO cpyttKUH51: cosJi.5!BJI51:eTC51: aHaJIHTHlJeCKOH,O.llH03Ha4HOH, uenoi1 cpyHKUHei1BC\{0}.11.43. ):{oKa3an., 4TO cpyHKUHR z- 1' 2 sin z 112 5!BJIReTcR uenoi1 cpyHKu11ei1.11.44. IIyen.Arctg~z=- d~1- -1+- ,,0~-r.ue HHTerp11posaH11e se.ueTCR noJII060MY nyTH, He npOXO}l51:WeMy qepe3 TO'IKH± i. ):{oKa3aTb, 4TO:I'Jwea 1114812i1+i.::1-iz'±i-JIOrapmpMH'IeCKHe TO'IKH BeTBJieHmr;1) Arctaz =-Ln--·02) TO'IKH z =3) tg (Arctg z) = z;d14) -(Arctgz)=--,.dz1 + z11.45. ITycTh f 0(z) - rnKoi1 ::rneMeHT apKrnttrettca B TOLIKe z = 0,'ITO fo(O) = 0 .
.IJ:oKa3aTb, 'ITO~ (- l) k Z2k+ Ifo(z)=L ,=o11.46. ilyCTb Arc sin z =2k+lr: ~'Jo 1 _ ~2'lzl<l.r.ue HHTerpuposaHHe se.ueT-C5! rro mo6oMy rryn1, He rrpoxonmn:eMy qepe3 TOLIKY±1. .IJ:oKa3aTh,'ITO cpyttKUH5! Arc sin z attaJIHTHLIHa B C\ { ± 1 } .11.47. I1ycn, f 0(z) - 3JieMeHT cpyHKUHH Arc sin z, KOTOpbrH 3a;:raHrro cpopMynef" ( )•Joz =arcsmz=1= ~·d~0\jl- ~2z E D = { z E C\( - co, -1] [ 1, co)}, HHTerpan 6epeTc5! rro mo6oMyrryTH, Jie:>Karn:eMy B D, sernh KOpH5! Bhr6patta rnK, 'ITO.[(oKa3aTb, 'ITO:.
z = L~1) J; (z) = arcsmo2) arcsin11=0c2n)!z211+1(2n + 1)2211 (n !)2z = -i In (iz + '11- z 2 ).11.48. ilycTb D - rrnocKOCTh C c pa3pe30M no KpHsoi1, coe.uHHmoruei1 TO'IKH 0 H co.1) .IJ:oKa3aTb, 'ITO B o6nacTH D cpyHKUH5! Ln z pacrra.uaeTC5! Ha6ecKoHeLIHOe LIHCJIO peryn5!pHbIX sernei1.fk(z),k E Z, 11/1(z) - h.(z) =2kni, z ED, mef1(z),h.(z) - mo6bre nse sernH.AHAJIY!Tl14ECKOE nPO.LJ.0JI)l(EHl1E. MHOf03HA4Hb!E C!>YHKIJ;I-1111492) L(oKa3aTb, lfTO cpyHKUH.s! z a pacrra.uaeTC.s! B D Ha peryn.sipHbieBern11.
Ecn11 a - .uettcrn11TeJihHoe 11ppau110HaJihHOe lfl1CJIO mrnIm a i= 0, TO peryn.sipHbIX Bernett ClfeTHOe lfl1CJIO. Ecn11 )Ke a = p I q,r.ue p 11 q - B3al1MHO rrpOCTbie ueJibie lfl1CJia 11 q2: 1, TO B DHMeeTC.s!pOBHO q pa3JII1lfHbIX perymipHbIX Bernett Z a, 11 eCJI11 f1 (z),J2(z) - .D,Bepa3JII1lfHbie BeTB11 Z a, TOf 2(z)=e 2k"i"fi(z),zeD,keZ\{O}.11.49. IIyen, fl...z) - perymrpHa51 BeTBb cpyHKUHH Lo z B o6nacmD (cM. ll.48), rnKa.si, lfToj{l) = 0.1) Bbilfl1CJII1Tb fl...-2),f{.3),j{-4).2) Pa3JIO)!(HThf{_z) B p51.D: Tettnopa B OKpecrnocTH TOlfKH z= 3.11.50. ,UoKa3aTb, lfTO B o6nacm { z: 0 < lzl < co} HeJib3.si Bhr.D,eJIHThefi (n > 1) .pery;rnpHhie BeTBH cpyHKUHH Lo z,11.51. IIycTh D" = C \ {z EC: z =re;", 0 ~ r < 00 , 0 <a< 2n} ; TOr.ua cpyHKUH.sifzpacna.uaeTc.si Ha .D,Be peryn.sipHbre BeTBH.
IIycTb1. BhilfHCJIHTh j,,(i):j;,(z) - TaKa.si BeTBb, lfTO f,,(1)=1) n I 2 < a < 2n:; 2) 0 < a < n: I 2.11.52. IIycTb cpyHKUH51 j{z) peryn51pHa 11 OTJI11lfHa OT Hyn.si B o.nHOCB513HOtt o6nacTH D. L(oKa3aTh, lfTO cpyHKUH51F(z)=lnf(z),F(z 0 )=% (ew·o =J(z0 ))pery;rnpHa B o6nacrn D.11.53. IIyen, cpyHKUH.s! j{z) peryJI51pHa H OTJII1'IHa OT HYJI51 B o.nHOCB513HOH o6nacrn D . .IJ:oKa3aTb, lfTO cpyHKUH.siF(z) =.J f (z), F(z0)= Wa (w; = f (z 0 ))peryn.sipHa B D.11.54 . .IJ:oKaJaTb, lfTO aHaJIHT11lfeCKa51 cpyHKUH.s!.J z2-1 pacna.na-eTc.si Ha .D,Be perymrpHbie BeTBH B o6nacrn C\ {(-co, -1] U [ 1, +co)}.7- 111.55. ,UoKa3aTb, lfTO cpyHKUH.si Ln -~- pacrra.uaeTc.si B 06nacT11z+lC\{(-co, -1] U [l, +co)} Ha clfernoe lfl1CJIO peryn.sipHbIX Bernett.
3rnBern11 MO)!{HO 3a.D,aTb cpopMyJiaM11:150(I'JiaBa 11Ln.::-l) =ln z-l +2k7ri,ln(-1)=7ri,k=0,±l,±2,. ...z+lz+lk11.56. ,UoKa:Jan, qrn aHan11rnqeCKa5! cpyHKU115! .J z2 -1 pacrra.uaeTCH B 06nacT11 C\[ -1, l] Ha .use peryn5!pHbie sern11.11.57. ITycTh f(z) -.J z2OJlHa 113 perynHpHbIX sernei:i cpyHKu1111-1 (CM. 11.56), TaKaH, qrnf(2) =J3.Bb1q11cn11n 3HaqeH11e3Toi1 cpyHKU1111 .UJI5! seruecrneHtthIX z E R.11.58.
Pa3JIQ)f<I1Th B p5!,a Jlopatta cpyttKUHfO f(z) (cM. 11.57) BOKpeCTHOCTl1 TOqK11 Z 00.=1- z11.59. ,UoKa:JaTb, qTO aHan11T11qecKa5! cpyHKUH5! Ln - - pacrra1+ z.uaeTcH Ha cqeTHoe q11cno peryn5!pHbIX serneif fk(z), k E Z, B 06nacT11C\[ -1, 1]. 3T11 sern11 3a.ua10Tc5! cpopMynaM11:f, ( z) = lnl l - zl +l+ziD.arg _ l-z+' l+zilm z0 + 2k7ri, k= 0,±1,±2, ... ,r.ue y - rrpOCTa5! 3aMKHyra5! Kp11Ba5! c HaqanoM B TOqKe z o, Jie)R'.aUJa5!B 06nacT11 C\[ -1, 1].1-z11.60. ITycTb f(z) - pery1rnpHa5! sernh cpyttKU1111 Ln-- (cM.l+z11.59), TaKa5!, qTO f(O) = 0. Bb1q11cn11Tb 3HaqeH115! f(z) )lJIH .uei1:crn11TeJibHhIX 11 q11CTO MHl1MbIX 3Ha 'IeH11i1: z.11.61. Pa3JIO)R'.11Th B p5!.U JlopaHa B 0KpecrHocr11 roqK11 zcpyHKU11IO j( Z) (CM.
11. 60) .= oo11.62. ,UoKa3aTh, 'ITO aHan11r11qecKaH cpyHKU115! ~ z /(1- z) pacrra.uaercH B 0611acr11 C \ [0, 1] Ha rp11 peryn5!pHb1e sern11.11.63. ITycrb .uaH anre6patt'IecK11i1 rron11HOM~,(z) =z 1, z 2, ... , z11-a(z- z,)(z- z2 ) •.. (z- z,), a+= 0,I I·pa3JIH'IHhie ero KOpH11, 11 R >max zk1) ,UOKa3aTb, 'ITO aHaJ111TJ1qeCKa5! cpyHKU115! ~~I (z) pacrra.uaeTC5!iz/peryJIHpHbIX Bernett B KOJibUe { z: R << 00}.2) ITycn D - BHeurnocTb 06oe.u11Hett115! orpe3KOB, coe.u11H5!f0UJHX cpHKC11pOBaHHYIO TOqKy Zo c TOqKaM11 Zi, Z2, ...
, Zn. ,UoKa3aTb,HaII151AHAJIIHl1YECKOE OPOLJ,OJDKEHl1E. MHOf03HAYHblE C!JYHKU1111~ P,, (::)qTo aHaJI11rnqecKa5! cpyHKU115!pacrra.naeTc5! BDHanpery-JI5!pHhIX BerneH:.11.64. LJ:oKa3aTh,qrn ¢YHKUtt5!C \ {[- 2, - l] U [1, 2]}06JiacT11~(z 2l)(z-24)-pacrra.naeTc5! BHa ,nBe peryJI5!pHhie BeTB11.z 1, z 2, ... , z,, - pa3Jil1qHhie TO'iKH 113 C, D - ITJIOCKOCTh c pa3pe3aMH B,LlOJih rrpOCThIX He nepeceKaIOll.{11XC5! Kp11Bh!X yj,11.65. ITycTbcoe.nttH5!IOW:l1X TOqK11 Z2j-J, Z2j . LJ:oKa3aTh, qTo aHaJI11T11qeCKa5! ¢YHKUH5!Jez - z, )(z -::2 ) ••. (z -z 2 ,,) pacrra.naeTc5! BDHa ,nBe peryJI5!p-Hh1e BeTBl1.11.66.
LJ:oKa3aTh,C\[O, 1].qTo ¢YHKU115!(z) =~ z(z -1) 3peryJI5!pHa BBhJqHCJil1Thj(-l),f'(-1),f"(-l).11.67. LJ:oKa3aTh,qTo B 06JiaCT11 { z:peryJI5!pHhie Bern11 cpyHKU11H11.68. ,lJ.oKa3aTb,nopR,nKa1)fTlqTo TOqKH0~zlzl<00l HeJih35! BhI,LleJIHTh2-1).11 00 5!BJI5!IOTC5! TOqKaMH BeTBJieHl15!aHaJIHTHqecKHX cpyHKU11i1::,,~ ;!.;./Ln (z +l <z2) efi. + e:;11.69. )J.oKa3aTb,3)qTQ TO'iKH01·1+efi.'4)efi.sinz.11 00 RBJI5!IOTCR JIOrap11cpM11qecKHMl1TOqKaMH BeTBJieHH5! aHaJIHTHqecKHX cl>YHKUHH:Ln 7l1) z + Ln z; 2) --- ; 3) - ; 4)z-111.70.Lnz1; 5) e:Ln z.Lnz+lITycTh ¢YHKUHR j(z), He paBHaR TO)!(,necrneHHO HYJIIO, JIH-60 peryJIRpHa B TO'iKe a 11j(a)=0, JI1160 HMeeT nomoc B TOqKe a.)J.oKa3an,, qTo:1) TOqKa a 5!BJI5!eTCR JIOrapmpM11qecKOH TOqKQH BeTBJieHlrnclJyttKUHH Ln/(z);2)TOqKaa5!BJI5!eTCR ToqKoH: BeTBJieHH5! rropR,nKancpyHKUHH~ f(z).11.71.
,lJ.oKa3aTb, 'iTO eCJIH cpym<UH51 /(z) attamITH'iHa BCIO.LlY B c11 /(z) f 0 .UJI5! JII06oro z E C, TO /(z) = e<p<z>, r.ne <p(z) aHaJIHTH'iHaBCIO.llY BC.I'JzaGa 1115211.72. ITycTbTO'lKa z= Zo -TO'lKa BeTBJiemrn KOHe'lHOfO rrop51.U-Ka .IJ.Jl51 cpyHKUHH f{z) 11 g(z). ,L{oKa'.3aTh, 'ITO .ll.Jl51 cpyHKU11HTO'lKa z= Zo 5!BJrneTC5! 11Jll1 TO'lKOHf ±g, f gBeTBJieHI151 KOHe'lHOfO rrop51.IJ.K3,HJil1 1130JI11p0B3HHOH oco6oi1: TO'lKOH O.UH03Ha'lHOro xapaKTepa.11.73. ,[{OKa3aTb,{ z: lzl <1}.Z:::::'ITO p51.U> 1}11 { z: lzl100I["= 1- z"+1-)1- z"I --B o6Jiacrnxrrpe.ucrnsmreT co6oi1: .use aHaJil1TH'leCKHecpyHKUHH, He 5!BJl5!JOIUHeC51 aHaJIHTH'leCKHM rrpo.uomKeHHeM .upyr.upyra.11.74.
,[loKa3aTh,'ITO:.J1) Arc cos z = iLn (z + z 2 -1);2) Arc sin z = -i Ln (iz + .J1- z2 );11 +iz3) Arctgz=-Ln--.2i11.75. BhI.UeJIHThl-izO.UH03Ha'lHhie sern11 MHOf03Ha'lHhIX cpyHKUHH ByKa3aHHOH o6JiaCTH D:21) Ln(l - z ), D {z: z=2)2~l+z ,D=3) Ln [ l -};{z:zf.i,-i,oo };z: l· = {z: z f.I+z11.76.f. 1, -1, ooDITycTb cpyHKU1151Ey.ueT JIM s 3TOH o6Jiacrnf(z)i, -i, 1, -1}.D, f(z)aHaJI11TH'lHa BD MHOf03Ha'lHa5! cpyHKU1151f. 0,zE D..Jf (z) 11MeTbO.UH03Ha'lHbie aHaJil1TH'leCKl1e BeTBH?11.77. ITycTb D z0.1J.HOCB513Ha5! o6JiaCTh, He co.uep)!(arua51 TO'leK= 0 11 z = oo, HO TO'lKa z = 1 ED. CKOJihKO pa3Jil1'lHhIX serneH:f{z)MO)!(HO BhI.UeJil1Tb .IJ.Jl51 MHOf03Ha'lHbIX cpyHKU11H:1) (z - 1) Ln z,j{l)= O;2)z',j{l)= 1;3)z',f'(l)= 1.11.78.
,[lorrycKaIOT Jll1 CJie.UyJOIU11e MHOf03H3'lHbJe cpyHKU1111 Bbl.ueJieH11e O.UH03Ha'lHhIX attaJIMT11'lec1<11x BeTBei1 B o6JiaCTM D:1)~ ,D= {z: l </z/<oo};2) 2Ln(z +1)-Ln (z-i), D= {z:1 < /z/ < oo};AHAJU1Tl1'-IECKOE f1PO.[(OJDKEHI1E. MHOf03HA'-!HbIE ct>YHKUI1l13)4)153~' D= {z:Rez>O};~(z2-l)(z 2 -4), D={z:Rez>O,lz-31>5/2}.11.79. Pa3JIO)KHTb B p5I.U Teif11opa B oKpecTHOCTH TO'IKH z = 0 o6eBeTBH MHOro3Ha'IHOH cpyttKU:HH:1)J1- z2;11.80. Pa3JIO)KHTb s p5I.U Teif11opa s oKpecTHOCTH TO'IKH z = 0cpyHKU:HH'.2) In(l + z).1) arctg z;11.81. Pa3JIO)KHTb s p5I.U Jiopatta ITO cTeITeH5IM z cpyttKU:HIO:z+az-af(z) =In--, In2a > 0.11.82. Pa3JIO)KHTb see attaJIHTH'!eCKHe sernH yKa3attHOH cpyttKU:HH B p5I.U JiopaHa ITO CTeITeH5IM z:21) Ln (z-l)(z- ),(z+l)(z+2)(z + 1)2) Ln ,,D={z: l<lzl<2};2D={z:lzl>2}.z- +411.83. ITycTb ./{z) - attaJIHTH'!ecKa5I seTBb cpyHKU:HH ~ z /(1- z) sC\[O, 1], TaKa5I, '!To./{1/2 + i) > 0.
Pa3JIO)l(HTb./{z) B p5I.U Jiopatta ITOCTeITeH5IM z B o6JiaCTH { z: lzl > 1}.11.84. HaifTH oco6bre TO'IKH scex attaJIHTH'!eCKHX sernew cpym<u:HH:l) Arcctg7rJZ.JZ'2)JZ .shJZ11.85. ,[(oITycKaIOT JIH YKa3aHHbie cpyttKU:HH pa3JIO)KeHHe s p5I.UJiopaHa B OKpeCTHOCTH TO'IKH z = a:1) In z, a= 0;2) In (1/(z - 1)), a f:. oo, a= oo;3) In ((z - l)/(z + i)), a= oo; 4) z05)~ z(z -1) 2 , a= oo;7) arcsin z, a= 0;= ealnz;6).JI+J;,,8),J7f /2-arcsin z, a= 1.a= 0;154TwBa 1111.86.KYDycTb[-1, l] .D -KOMnJieKcHa51 nJiocKOCTb c pa3pe30M no oTpe3-DoKa3aTb, qTo yKa3aHHbie cpyHKU1111 aHa1111rnqHbI BonpeneJil1Tb xapaKTep H3011HpOBaHHOH ToqK11 BeTBJieHl151 z1)~z(z"-1);11.87.2)Ln(z+~~~-1);Hz+~.2z3) LnW(z)I1p11 KaK11x 3Ha'!eHl15!X z 3Ha'!eHH5!D,=oo:Ha scex ee 1111crnxp11M3HOBOH IlOBepXHOCTl1 H31l Z-IlJIOCKOCTblO omrnaKOBbI:1) W( z)=(z 2 -9).fz; 2) W(z)=sinz+(z 2 +4)Lnz;223) W(z)=sinz+(z +4) Lnz.11.88.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
