Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 32
Текст из файла (страница 32)
CcpopMyJI11poBaThycJIOBl15I,06ecne'!11Bmorn:11ee.[(11HCT-BeHHOCTh 3a.[(a'!11 HeiiMatta .[(JI5I HeorpaH11'!eHHOtt 06JiacT11.14.46. IlOKa3aTb, 'ITOcpopMYJiaf(z) =r.uez = x + iy, (0, 0)2u(~.~J-u(O,O)+iC,2 2iE D, soccrnttas1111BaeT atta1111T11qecKy10 B o.uHo-CB5I3Hoii o6Jiacrn D cpyttKUHIO j{z) no ee .[(ettcTB11TeJihHOH qaCTH u(x, y)c TOl{HOCTb!O no l{HCTO MHHMOH KOHCTaHTbl. 3rn cpopMyJia Ha3bIBaeTC5I rjJopMyllOU I'ypca.14.47. IlOJib3Y5IChCOHa ,UJI5I Kpyra {izl < 1} rpopMylly fileap4a:f14.48. IIycTh{lzl < R}cpopMyJIOH rypca, BhIBecrn 113 cpOpMyJihI Ilyac-(z)=-1-~2Jri l~l=lcpyHKUHHf(z)(s + z).
u(s)ds + iC.(s - z) s= u(z)+iv(z)11 ttenpepbIBHa B 3aMKHYTOM Kpyre {.[(JI5I mo6oroz = ri", 0 < r < R,u(z) =v(z)cnpaBe.[(JIHBhI paBettcrna.L~=o(~J (a= .L~=o( ~atta1111T11'!Ha B Kpyrelzl :S R}. ,[(oKa3aTb, qrn11cosncp+b11 sinncp),J(-b 11 cos ncp + a 11 sin ncp),r,[(e K03cpcp11u11eHTbI all 11 bll BbJl{HCJI5IIOTC5I no cpopMyJiaMI r',,.ea =- 1; u(Re' )cosn8d8,ff11f APMOHHYECKHE207<DYHK~l111b,, =_!_J:"u(Re;0 )sinn8d8, n=l,2, ...
,7ra,, =b11Lf" v(Re;7r0)sinn8 d8,0=--1 Jof2:r v(Re iO ) cosn8 d8, n =1, 2, ... ,f7r1 f2:r v(Re iO )d8.ao = u(O) = - 1 2" u(Re iO )d8, b0 = - v(O) = - -J,2;r 02;r 014.48.1. IlycTh cpyHKU:HJI u(x, y) rapMOHHqHa B o6nacTH D. ,lJ,oKa3aTh, 'ITO B OKpecrnocTH mo6oif TOqKH (x0 , y0 ) cpyHKU:H.II u(x, y) pa3narae'.('C.II B pH.llT eifnopau(x,y)=-Iau(x-x0 l(y-y0 ) 1 ,k.1=0npHqeM p.1!.Jl cxonHTC.II a6comorno B o6nacTH:0 < R< pIx- xol + 1.Y- Yol S:: R,= p(Zo, 8D), Zo =Xo +iyo.14.48.2. ,lJ,oKa3aTb, 'ITO lJ:aCTHhie npOH3BO.llHhie rapMOHH'IeCKOHu(x, y) no x Hy TaK:>Ke .IIBJIHIOTC.II rapMOHH'IeCKHMH cpyHK-cpyHKU:HHU:H.IIMH.14.48.3.
,lJ,OKa3aTh,'ITO npO,llOJI:>KeHHe rapMOHH'IeCKOH cpyHKU:HHenHHcrneHHO, T. e. ecJIH rapMoH11qecKa5I B o6nacTH D cpyttKU:HJI u(x, y)B HeKOTOpoif nono6nacTHG C D pastta ttymo, TO u(x, y)= 0 B 06-nacTH D.14.48.4. IlycTbp.1!.llI~=o u,, (x, y)cocTOHT H3 rapMOHHqecKHXcpyHKU:HH H cxo)lHTC.II paBHOMeptto BHYTPH o6nacTH D. ,lJ,oKa3aTb,qTO1) CYMMa 3TOro p.II.lla ecTh rapMOHH'IeCKaH cpyttKUH.II B o6nacTH D;2) p.1!.ll MO:>KHO nmpcpepettu:ttposaTh noqneHHO, npH'IeM cyMMap.1Ina H3 npoH3BO.llHhIX TaK)l(e rapMOHH'!Ha B D;3) p.1!.ll H3 npott3BO.llHbIX mo6oro nopHnKa CXO.JlHTC.II paBHOMepHO BHYTPH o6rracTH D.14.48.5. IlyCThqecKHX B o6rracTHHOCThnocnenoBaTeJihHOCTh {f,,(z)} cpyHKU:HH, aHaJIHTH-D,cxonHTCJI B TO'!Ke Zo ED,a nocrrenosaTeJih-{Re f,,(z)} CXO.llHTC.II paBHOMeptto BHYTPH 3TOH o6rracrn D.I'Jzaea 14208.D:oKa3aTb, L!TO noCJJe)lOBaTeJihHOCTh {f,1(z)} CXO)ll1TC5I paBHOMepHOBHyrp11 06nacT11 D.14.49 .
.D:oKa3aTb, L!TO )lJI5I rapMOHHL!eCKOH B Kpyre { izl < R} H HenpepbIBHOH B 3aMKHYTOM Kpyre { izl :S R} cpyHKUI111 u(z) cnpaBe.nn11BO paBeHCTBOJtl<R(u; + u ~ )dxdy =trL,~= 1 n(a,~ + b,~),r.ne K03cpcp11u11eHThIa11 11 b11 onpe.neneHhI B 3a,naqe 14.48.14.50. JleBa5I Liacn paBeHcrna B npe.nh1.nyruei1 3a,naLie Ha3hrnaeTC5Iurtme2paJlOM /{upux!le . .D:oKa3aTh, L!TO <PYHKUH5I. '8(!.....J"u(re;e ) = L~ smn.11 =1n2Rizl5IBJI5IeTC5I rapMOHHLieCKOH B Kpyre {< R}, np11H11MaeT HenpephIBHhie 3HaL1eH115I u(Re ;o) Ha rpaH11ue 3TOro Kpyra H o6na.naeT 6ecKoHeLIHbIM 11HTerpaJIOM .D:HpHXJle.14.51. I1ycTh <p(t,) - HenpepbIBHa5I .neHCTBI1TeJibHa5I <PYHKUI15I HarpaH11ue iJD 06nacT11 D.
I1ycTb u(z) - mo6a5I HenpephIBHO .n11<P<PepeHu11pyeMa5I B 06nacT11 D <PYHKUI15I, cosna.naIOmaH Ha 8D c <PYHKuHeH: cp(t,) . .D:oKa3aT&, LITO ecm1 cymecrnyeT xorn 6n1 o.nHa TaKa5I<PYHKUI15I u(z), )lJl5I KOTOpOH 11HTerpan ,r:i:11p11xne KOHeLieH,D(u)= ffv (u;+ u;.)dxdy < oo ,TO HaHMeHblllee 3HaL1eH11e D(u) )lOCTHraeTc5I Ha rapMOHHLieCKOH B06nacn1 D <PYHKu;1111, KOTopaH Ha rpaH11ue 06nacT11 D coBnanaeT ccp(t,), T. e. 5IBJl5IeTC5I pellleH11eM 3anaLIH ,r:i:11p11xne.14.52 .
.D:oKa3aTb meope.Aty XaprtaKa: npe.nen y6hrnaIOmeH: nocneJlOBaTeJlbHOG-T11 rapMoH11L1ecK11x B 06nacr11 D <PYHKU:HH 5IBJl5IeTC5Im160 rapMOH11L1ecKoi1 B D <PYHKU11ei1, n1160 TO)K)lecrneHHO paseH - oo.14.53 . .D:oKa3aT& eme o.nHy reopeMy XapHaKa: nycTh <PYHKUHH~ 0 , k = 1, 2, ... , u 5IBJl5IIOTC5I rapMoH11LieCKI1MH B 06nacT11 D;uk(z)Tor.na ecn11 p5IJJ L,~= 1 uk (z) cxon11Tc5I xorn 6i,1 s onHoi1 TOLIKe z E D ,TO OH CXO)ll1TC5I paBHOMepHo BHyrp11 D.14.54. BepHo n11 yrnep)KJleH11e, LITO ecn11 p5I.ll L, ~= 1 uk ( z) H3 cpyttKu:11i1, KOTOpbie rapMOHI14Hhl B D 11 HenpepbIBHhl B15'CXO.UHTC5I pas-f APMOHHY:ECK!1EHOMepHO B<DYHK~l1!1209J5' TO OH CXO,llJ1TC51 paBHOMepHO Ha rpamme fJD o6JiaC'm D?I1Meer JIH 3ttaqett11e orpaH11'-!ettHocTb 06Jiacr11 D 11JI11 yrnep)K.ueH11ecrrpaBe.UJIHBO .UJIH mo6oH: o6Jiacrn D?14.55.
Bhm11caTb rrptt/1= 1, 2, 3, 42apM0Hu1JecKue no!luHOAtbt==P11(X, y), q,i(x, y), orrpe,n:eJIHeMbie paBeHCTBOM zp 11 + iq,,, z x + iy.3arr11caTb B o6Il(eM BJ1,Ue P11(X, y) l1 q11(X, y) B IIOJI51pHOH Cl1CTeMe KO11op.uirnaT.14.56. IIycTb V(z) (z = x + iy) - rapMOHttlleCKa51 l1 orpaH11'-!eHHa5!cpyHKUl151 B o6JiaCTl1 { z: Im z > 0}. CrrpaBe,UJil1BbI JIH yrnep)K.uemrn:a) .UJIH Ka)K.uoro t E R cyll(ecrnyerV(z)=V(t);lim:-->1.lm:>O=_!_r Iz -yV(t), dt, y > 0t( cpaBHHTb c 3a,n:aqei1: 14.24).6) V(z)Jl-001-14.57. Dyen.
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