Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 31
Текст из файла (страница 31)
PaccMOTpeTh cpyHKU11IO u(z) = 1 - Re z - 1, KOTopa5! 5!BJJ5!eT2C51 rapMOHM'IeCKOH B Kpyre {x + y2 < x}, HerrpepbIBHOH BCIO.llY Ha2rpaH11u:e 3TOro Kpyra {x + y2x}, KpOMe TO'IKl1 (0, 0), l1 paBHOH2Hymo BCIO.llY Ha OKPY)KHOCTH {x += x}, KpOMe TO'IKl1 (0, 0). (OcB060JK,n:eHMe TOJihKO O.D:HOH TOqKM rpaHMU:hI OT ycnOBl1H .[(ttp11xneMO)KeT rrpttBeCTl1 K Hee.n1rncTBeHHOCTM 3a.n;a'IM .[(11p11xne.) y .llOBJieTBOp5!eT JIM .n;aHHa5! ciJYHKU:115! ycJIOBM5IM TeopeMbl e,n:MHCTBeHHOCTl13a.naqtt ,[(11pwxne?=l14.22.
BepHo n11 yrnep)l(.neH11e: rrycTh cpyHKU:115! u(z) 5!BJJ5!eTC5!rapMOH11qecKoi1 B BepxHeli rroJiyrrJIOCKOCTM Im z > 0 M paBHa HYJIIOsc10.n;y Ha oc11 x, T. e. u(x)= O; Tor.na u(z) = a Im z, r.ne a -HeoTp11-uaTeJihHa5! KOHCTaHTa.14.23. IlycTh ciJYHKU115! u(z) 5!BJI51eTC5! rapMOHl1'IeCKOH B o6JiaCTl1D, KOTOpa5! CMMMeTpMqHa OTHOCl1TeJlbHO OCH Ox, 11 rrycTh u(z)rrp11 z E Dn {Im z = 0). ,[(oKa3aTh, 'ITO14.24. IlycTh cpyHKU1111f(z)MOHM'IeCKMMl1, T. e.l::,.u + i l::,.v = 0,u(z)=u(z) + iv(z)=0=-u(z) rrp11 z ED.M zfiz) 5!BJI51lOTC5! rap-l::,.(xu - yv) + i !::,.(yu +xv)= 0.,[(oKa3aTb, 'ITO clJyHKUM5!fiz) 5!BJI5!eTC5! aHaJJMTMqecKOH.f APMOHJ1qECKl1E <fJYHKU111120114.24.1. I1oKa3aTb, 'ITO B npe.D.bI.D.ymeif 3a.naqe o.uHoro nepsoroycJIOBI15! He ,UOCTaTOqHo, 'lT06hr ¢YHKUI15!/(z) 6hrna aHaJIHTH'leCKOH.14.25. ITycTb KOMnneKCH03Ha<IHa5! cpyHKUI15! f(z) 5!BJI5!eTC5! rapMOH11qecKOH s 06nacT11 D.
JJ:oKa3aTb, qrn ecn11 lf{z)I0 B D, TO== const s D.f(z)14.26. JJ:oKa3aTb, qTQ cpyHKI.(I15!1u(z)r.ue(R 2- r 2 )h(z0 + Re;'!!)2Jr=-i,d<p,2;rR-+r--2Rrcos(<p-8)?0z = z0 + re;'!!, 0 < r < R,5!BJI5!eTC51 rapMOH11qecKOH B Kpyre{iz - z01 < R} H npHHHMaeT 3a,UaHHbie HenpepbIBHbie 3HaqeHI151 h(~)Ha oKpy)KHOCTH {I~ - z0 1 = R} (3a.naqa JJ:Hpmure M5! Kpyra).14.26.1. JJ:oKa3aTb, qTo ¢YHKI.(I151u(r,8)=R1 -r2?,R- + r- -2rRcos(8-<p),r~e R, q> -qrnKCHpOBaHhl, eCTb rapMOHH'leCKa5! <PYHKUI15! B Kpyrelz - z0 1 < R, z z0 + re i<J>.
Pa3JIO)KeHtte cpyHKI.(HH u(r, 8) B p51.D. 11MeeTBH.D,=2,112R -rR-+ r 2 - 2rRcos(8-<p)-1+2L:~_ 1 (!_) cosn(8-cp).,,_R<PyHKUHIO u(r, 8) Ha3bIBaIOT flOpOM llyacc01Ja.14.26.2. Haiirn cpyHKUI1IO, rapMOH11qecK11 conpmKeHHYIO K 51.D.PYITyaccoHa, 11 pa3JIO)KHTb ee B pRo <!Jypbe.14.27. I1oKa3aTb, qTo .D,JI51 Heorp11uaTeJibHOM rapMOHH'leCKOM BKpyre lz - zol< R cpyHKUHHR-1 z - z Iu(z) sepHbI oueHKH0- - - --u(z 0 )~ u(z) ~R+ I Z - Zo IR-1 z - z 0 Iu(z ).R-1 Z - Zo I 014.28. JJ:oKa3aTb, qTo cpyHKU115!u(z) = Imz1!rza(11)' dll-1 -111-51BJI51eTC51 rapMOHHqecKOH B sepxHeH nonyrrnocKOCTH Im z > 0 11np11H11MaeT 3a.uaHHhre orpaH11'leHHhre HenpepbIBHbie 3Ha'leH115! a(x)rrptt Im z 0 (3a.uaqa JJ:11p11xne .D.JI5! nonynnocKocT11).=202I'JZaea 1414.29.
HaH:rn pewem1e cne)lyIO!l.U1X 3a)lall L(11p11xne )lJI5! Kpyra:1) 1'1u(z)=0, lzl<l, u(z)ll:l=I=l;2) L1u( z) = 0, I z I:C 1, u( zll=t =cos cp, 0 s; cp .s; 2rr;L1u(z) =0, I z I< 1,3)4u(z)I ___ 1- 1-Isin cp , 0 <_ cp <_ 2rr,.5 + 4 cos cpL1u(z) =0, I z I< 1,))I - ·u cz l=l=t- sm 2 cp,L1u(z) = 0, x5)2o<_ cp <_ 2rr,·+ y2+ 2x < 0,= 4x3 + 6x- l.u(z)l.h/+2x=O14.30. HaH:m pewem1e cne)lyIOil.IJ-IX 3a)lall L(upuxne )lJI5! r10nyfIJ10CKOCTH:1) 1'1u(z)=0,lmz>0, u(z)lim:=o=l;Llu(z) = 0, Im z > 0,2)u(z)I1__ = - - , , z = x+ iy;tm.- 0 I+x-L1u(z) = 0, Im z > 0,3)u(z)IIm:=O={1,0,x>l,X< 1, Z = X + iy;L1u(z) = 0, Im z > 0,4)5)u(z)IxIm :=0=--1 + X2'z=x+iy·'L1u(z) = 0, Im z > 0,u(z)I Im:=0 =sgn Re z, z = x + iy.14.31. <I>opMyJ1hI 33)l3l{ 14.26 11 14.27 HOC5!T H33B3HI1e cjJopMyJ7IlyaccoHa )lll5! Kpyra 11 rronyrrJIOCKOCTI1 COOTBeTCTBeHHO.fIOMOIl.IhIO cpopMyJihI IIyaccoHa )lJ15! Kpyra pew11Th BHeWHIOIO 3a)lally L(11p11xne JlJI5! Kpyra lz - z0 1 < R: HaH:rn cpyttKUI1IO u(z), rapMOHwrnyIO so6naCTH lz - zol > R, orpatt11'IeHHY10 Ha 6ecKotte'IHOCTl1 11 np11tt11MaIOmyIO 33)l3HHbJe HenpepbIBHble 3Ha'IeHI15! h(~) Ha OKpy)!(HOCTH/~ - zo/ = R.cr APMOHWJECKI1E <PYHKUI1I120314.32.
DoKa3aTh, 'ITO BHewmmo 3a_naqy .[(11p11xJie MO)f(HO np112noMOIUH I1HBepc1111penyu11poBaTh K BHyTpeHHeH 3a_naqe.[(HpHXJie.s = zllzl14.33. DocTpOHTh perueHHe 3a_naq11 .[(ttpHXJie ,nmI BHellJHOCTHKpyra { lrl :S 1 } no KpaeBoMy ycJioB11IO u( e i<P) = cos 2cp - 1, 0 :S <p ::;211:.14.34.
DycTh D - orpatt11qeHHa5I o6JiaCTh Ha KOMnJieKCHOH rrJiocKOCTH, rpaHHQa KOTOpoH: 5IBJI5IeTC5I KycoqHo-rna,nKOH KpHBOH )IfopnaHa . .[(oKa3an cJieny10ruyro cpopMyJiy fp11tta:ffo (vl'iu - ul'iv) dxdy =fa v ~: - u0 (i: )idzJ,rne cpyHKUHH u H v nBa)f(nhr HerrpepbIBHO n11cpcpepeHQHpyeMhI B 06JiaCTH-D H HerrpephIBHO ri:mpqH'peHUHpyeMhI B D ' asontta5I nolHH1'1[lTT'HH (14.35. ·; ,c, ~'UThHl' LH;.'H HOpMa'lH K rpamni:e"Tor \yHl<llH,(-r)1ana - rrpOH3-av o6Jiacrn: D.-.sreTcsr rapMOH«qecKOH BonttoCB5!3Hoi1: 06Jiacn1D L =.. ro - a u TOJihKO Torna, Karnart-'Yydu ldzJ=Oanno mo6oMy 3aMKHYTOMY KOHTYPY y, Jie)f(arueMy B D.14.36.
IIpoBep11Th cneny10ru11e paBettcrna:l) _l J;log-1-dB=flogfh,ls-zl2n2)JzJ>R,l1ogR-1, lzJ:s;R;11-JrIoa--dr- dT] =tr R2 Jrsl<RIs- z I "'e1Jzl>R,Joa-"'J zl'=~2ii ( 1 -I z-1-)lo -+0 R2R203)1f"_,, ~Joa-ds = { cp,ari "'I s- z I-cp,'I zJ:s;R,Imz > 0,Imz <0,= s+ iT];I'.wea 14204cp, 0 < cp < 11:,rneTQqKJ1 Z, a- yron, non KOTopr,rM Bi-i:neH OTPe3oKa14) ~ v(s)-log--ldslanyaan -rne -[-a, a} 113S= ~+i 11;Is- z I= f v(s)darg(s-z),y~rrpOH3BOJ{HaH no HanpaBJieHHIO BHeurne11 HOpMaJI11 K 3aMK-y, a cpyHKUHH v(i;;) Herrpepr,rnHa Ha y. Ecn11 y - OKIsl = R}, To rrpenwlymaH cpopMyJia rrp1106perneT BHJlHYTOMY KOHTYPYPY:>KHOCTb { (,:s- -112Rf'' v(Re;e)i_logd8 = r " v(Re; 8 )d arg(I;;- z).odrI Z I r=RJo14.37.
IlycTbcpyHKUHHf(z) HBJrneTCH HenpepbIBHOH Ha KpHBOH'Y·Ll.oKa3aTb, qrn cpyHKUHHu(z)=1f µ(s)log--ldslIs-z IypurjJA1Ul/eCKU.M nomeHifUaJIO.M'Y ( cpyHKUHH u(z) Ha3hIBaeTCH JI02ac ITJIOTHOCTblO µ ) . .ll:OKa3aTb TaK>Ke,qTO eCJIH y- OKpy:>KHOCTh 11 µ= 1, TO u(z) =canst BHyrp11HBJIHeTCH rapMOHHqecKOH BHe14.38. IlycTh D -'Y·orpaH11qeHHaH OJlHOCBH3HaH o6naCTh 11 rrycThcpyttKUHH W(z, /;;), z, (, E D, rrp11 Ka:lKJlOM (, E D KaK cpyHKUHH z KOHcpopMHo orn6pa:>KaeT o6nacThOT06pa:>KaeTCH B TO'IKYw = 0,UHH f p11Ha 3a,lla'IH Ll,11p11xneD Ha Kpyr lwl < 1, rrpHqeM TO'-!Ka z = (,T. e.
W(t;,, (,) = 0. Ll.oKa3aTb, qTo cpyHK-mmorreparnpa JJarnraca B 06nacT11HMeeT BH,ll1G(z, /;;)=-log IW(z,2nIlpH 3TOM, ecnH W(z) HHe o6Jiacrn D Ha Kpyrs) 1.KaKOe-H116y,llh KOHcpopMttoe orn6pa:>Ke-lwl < 1, TO cpyHKUHH W(z, (,) HMeeT BH.U·W(z,i;;)= W(z)-W(s)_1-W(z)W(I;;)14.39. IloJib3YHCbcpopMyJIOH fp1rna H3 3a.n:aq1114.34, .UOKa3aTb,qTo ecJI11 u(z)- rapMOHH'-!ecKaH cpyHKU:HH s 3aMbJKatt1111 D orpaH11'IeHHOH 06nacT11paBeHCTBOD, To nmr n1060H' TO'IKM z E D cnpase.D.J111BODr APMOHl14ECKl1E ct>YHKU1111u(z)=r.ue -a-an2051 rh {a1ograu(s)}n'fav u(O-a;:;--logra;- ldsl,2vnpott3BO.UHaH no ttanpasJieHttIO BHenrnett ttopMaJIH K rpa-HHUe o6JiaCTH D,r = Is - zl,sE oD.14.40.
3a.uaqa ,UttpttxJie B o6Jiacrn D .UJIH ypastteHHH IlyaccoHa·COCTOHT B TOM, '1T06hI HaHTH cpyHKQHIO u(z), Y.UOBJieTBopmoruyroYCJIOBH5!M:fl.u(z) F(z),zt=. D,=u(z)l~EdD = <p(z), Z EaD,r.ue F(z), cp(z) - 3a.uattHhie ttenpephIBHhie cpyttKUHH.,UoKa3aTh, 'ITO cpyttKUH5! u(z), y.uosJieTBop5!IOil{a5! Bhilllenepeq11cJieHHhIM ycJIOBH5!M, MO)f(eT 6hITh Hai1.ueHa no cpopMyJieu(z)=ffDG(z,s)F(s)d~d11+rf-'fao aG(z,s)an cp(~)id~i,r.ue G(z, t;,) - cpyttKQH.SI !ptttta 3a.uaq11 ,U11p11XJie .UJI5! oneparnpa JianJiaca B 06JiacT11 D.14.41. ,UoKa3aTb, 'lTO He06XO.UHMhIM ycJiomreM pa3pellIHMOCTH3a.uaqrr HeiiMatta .UJI5! onepaTopa JianJiaca B o6JiaCTH D HBJI5leTc51ycJIOBHe~av \Jf(s)ds = o,r.ue \j/(t;,) - rpaHH'lHOe ycJIOBHe .UJI5! HOpMaJihHOH np0113BO.UHOH HCKOMOH cpyHKQHl:f.14.42.
,UOKa3aTb emrncTBeHHOCTh c TO'lHOCThIO .uo a.u.uttTHBHOMnocT05!HHOi1: 3a.uaq11 HeiiMaHa .UJI5! orpatt11qeHHOM o6JiaCTH.14.43. HaiiTtt pellleHtte cJie.uyromeii 3a.uaq11 Hei1Matta:fl.u(z) = 0,I z I< 1,aucz)I = <p (8) ,ar r;Iz = re iO , o<8<2_ _ n.14.44. ,UoKa3aTb, 'ITO <l>YHKQH5! u(z) BH.Ua1 f~ <r(ll) log z -11 dll + C(x),u(z) = - -n-II206I'Jwea 14r.[(e z=x+ iy,cp(11) HenpephrnHa Ha R 11 np11 11 --t oocp(11) = oc1111- I-£), 5IBJI5IeTC5I perneH11eM CJie-a cpyHKU115Iy.[(OBJieTBOp5IeT YCJIOBl1!0mm sepxtteH: 110Jiy1111ocKocrn:t.u(z) = 0, Im z > 0,.UyffirueH: 3a.uaq11 HeH:MaHadu'dul-dll=cp(x),-oo<x<oo,=--dyIm :=0r:r=Ocp(x)d.x =0.14.45.
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