Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 19
Текст из файла (страница 19)
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+11±(!_)k]· z -:t 0, kEN.z,,11PaccMorp11M ueJiyIO cpyHKU11IO f(z) rrop5!.UKa p < oo c HYJI5!MH zi, z2,. . . (z1-:f 0), lz,, lioo. IlycTh k - Ha11MeHburee ueJioe q11cJio, y.uoBJie-TBOp51:IOIUee yCJioB11IOL¢yHKU115!F(z)=~1--<oo.z,, lk+l11=! 1n:= £(.!:__,k)zll1Ha3hIBaeTC5lKGHOHU'ieCKUMnpou3eeoe11ue.111 )J.JI5! ueJioH: cpyHKu1111J(z).TeopeMa. IlycTh ueJia5l cpyHKUIDIJ(z) 11MeeT rrop5!.lJ.OK p< oo, 11 zi,z2, ... , z11 - ee HYJil1 (z 1 -:f 0), lz11 ljoo, T - rroKa3aTerrh CXO.lll1MOCTHnocJie)J.OBaTerrhHOCTH { z11 } . Tor)la KaHoHwrecKoe npomBe)leH11e F(z)ecTb ueJia5! cpyHKUH5! nop5!.UKa T, np11qeM, ecrr11L "= --<oo1 I' '~11z"TO F(z) - ueJia5! cpyHKU115! nop5!)J.Ka T KOHeqHoro THna.TeopeMa I>ope.1rn.
IlycTh f(z) - ueJia5! cpyHKU115! KOHeqHoro no-p5!.lJ.Ka p, { Z11} - ITOCJie)J.OBaTeJihHOCTh ee HyJieH: c IlOKa3aTeJieM cxo)J.11MOCTl1 T, z 1-:f 0, lz11 ljoo, k- HaHMeHhuree ueJioe 4HCJIO, Y.llOBJiernop5!IOIUee ycJioB1110"~L.., = 1111z,,f(z)1lk+i < oo.
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ITo orrpe.n:eneH11IO cpyttKU1151h( (/) )=-!· Inl(f(re;q.o))I O<Jillr-+oor,<-{/}-21l,Ha3hrnaeTC5l UflOUKampucoii pocma ¢y1-1K4uuf(z).TeopeMa Ilonua. ITycTh f(z) - u:ena5l cpyHKU:115I 3KCITOHeHU:11aJihHOro Tmrn c 11H.U11KaTpttcoi1: poem h( cp ), D - ee coITp5!)KeHHa5! .n:11arpaMMa,h(cp)=K( -K(cp) - oIToptta5l cpyHKU:115I MHO)Kecrnacp). ITycTh F(z) - u:ena5! cpyHKU115!. PH.n: B11,n:a"k~L.,= 0r.n:e /).k F(O)t:/ F(O)k!D . Tor.n:az·(z-1)· ...
·(z-k+l),=F(k)-c;F(k-l)+Cf F(k-2)+ ... +(-!l F(O), H331IBa-eTC5! u1-11nepnoJ1Rt/UOHHbl.kl pROOM Hbl011W1-la.Trnea JO126TeopeMa.na,D -I1ycn F(z) - uenaH cPYHKU115! 3KCrIOHeHu11aJJhHoro m-ee conp.5n1<eHHaH .z:umrpaMMa,nnocKocm { zE C:le z - l I < 1}. EcnHG-o6nacTb Ha KOMnneKCHOHDC G, TO cpyHKU115! F(z) pa3-naraeTC5! B HHTepnon51u110HHhIH p.51.[( HhIOTOHa=F(z)P.51.U= Lk=oL~= 1 a e-;.,,B11.ua116.k F(O)k ! z · (z -1) · ...
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EcJI11 p.51.U Bc10.n:y cxo.n:11Tc5! a6coJIIOTHO, TO noJiaraFOTa= -oo,eCJIH pH.U H11r.ue He CXO.U11TC5! a6coJIIOTHO, TO noJiaraI-OTa= +oo.Hm1<H5l5! rpaHh T MHO)Kecrna TaK11x 411cen a, qTo B nonym10c1<ocT11> a p.51.U cxo.u11TCH paBHOMepHo, Ha3hrnaeTC5! a6ci1uccoil paeHoAtepHotl cxodiwocmu pH.ua .IJ:11p11xne. CnpaBe.n:n11Bo cooTHOIDCH11eRe zC :'.ST:'.Sa.10.1..[(oKa3an, qTo .UJI.51 Toro, 4T06bI ueJiaH cpyHKUH.51 6b!JrnMHOroqJieHOM, Heo6XO.U11MO 11 .UOCTaT04HO, 4T06hr Haq11Ha5! c HeKOTOporor 2: roBbinOJIH5!JIOCb HepaBeHCTBOM,<Ar", M,=maxjf(.:)j,/:/=rr.ueA11q-IlOJ10)!(11TeJibHbie KOHCTaHTbl, He 3aBHC5!IU11e OT r.127Ll,EJibIE cDYHKLl,V!V! VI P5l)J,bl )J,V!PV!XJIE10.2. JJ:oKa3aTI,, YTO rrop.SI.!lOK p MO)!{HO BbIYHCJlHTb rro cpopMyJJe- .-lnlnM.p= 1lffix->=In rI10.3. OrrpenemITh rrop.SI.!lOK ueJJbIX cpyHKUHH:1) j(z) - MHOfOYJJeH; 2) j(z) = sin z;3) j(z) = cos z;4).f(z) = e';S)f(z) =ch z;6).f(z) =sh z;7)j(z) = eP(z) ), P(z) = a0z" + a 1z"-' + ...
+a", ao -:;r:. O;8) f(z)= e sin';= e cos';9) fiz)ll)f(z) = e: cos z;10) fiz) = e: sin z;12)f(z) = e·-.10.4. OrrpenemITb nm ueJJbIX cpyHKUHH, paccMoTpeHHhIX B rrpe.!lhI.!lymeH: 3anaYe.10.5. )J:oKa3aTh, YTO o6o6m.eHHa.SI rrmepreoMerp11qecKa.SI ¢YHKUH.S!pF" (a 1 , a 1 ,. . ., a,,, /31 , /31 ,. . ., /3l/, z)=="'=(a1 )11 ·····(a,, )11 .
zL..11=0 (/3,)(/Jq ),,1 " · ••• •11n.''rne (a)11 =a (a+ 1) -. . .- (a+ n), (a) 0 = 1, napaMeTphI a 1, a 2,. .. a",~ 2 ,. . ., ~q - rrpoH3BOJlhHhre KOMrrJJeKCHbre YHCJJa, ~j -:;r:.p"O,j = 1, 2,. . ., q,.SIBJJ.SieTC.SI ueJJoH cpyHKu11ei1, ecJJ11 q ~ p. Hai1T11 ee nop.SI.!lOK.10.6. JJ:oKa3aTb, YTO cpyHKUH.S!F(z)= L~= 1 q"1k > 1, Iq I< 1,e;"',.S!BJl.S!eTC.SI ueJJOH, HaHTH ee rrop.SI.!lOK.10.7. ITycTb a> 0.
)J:oKa3aTb, •no cpyHKUH.SI"'=11- L.. = f(l +an)F(-) '-0~ecTb ueJJa.SI cpyHKUH.SI rrop.SinKa l/a.10.8. JJ:oKa3aTh, YTO THO CT ueJJOH cpyHKUHH nop.SI.!lKa p MO)!{HOBbIYHCJlHTb no cpopMyJJe-lnMcr=lim--,..r-+oorP10.9. ITycTh P(z) - MHOfOYJJeH. )J:oKa3aTb:1) ecJJH j(z) - ueJJa.SI cpyHKUH.SI rrop.SinKa p, 0f(z)·P(z) HMeeT nop.SI.!lOK p;~ p ~ oo,TOT!laea 101282) ecm1 f(z) - uenaR cpyHKUH5I rrop5I.UKa p, 0 < p < oo, nma CT,0~CT~oo, TO cpyHKUH5I f(z)-P(z) HMeeT THIT CT rrpw rrop5I,UKe p.10.10 . .UoKa3aTb, qTo ecm1 cpyHKUiur.f(z) = L~= 0 a11 z11-u;eJia5I, TOrrop5I.UOK p no.ucqwTnmaeTc5I no cpopMyJie- .- nlnnp= 1i m - - ,,_.= Iln Ia II1110.10.1.
.UoK333Tb, qTO cpyHKUHRj(z) = 1 I l(z), r.ue f(z) - raMMaQJYHKUH5I 3.Hnepa, eCTb uena5I cpyHKUH5I. I10K333Tb, qTO nop5I.UOKp = 1, a THn CT= 00.10.11 . .UoKa3aTh, qTO ecJIH cpyHKUH5I .f(z) = L~= 0 a11 z - uenaRcpyHKUH5I nop5I,UKa p, 0 < p < 00, TO ee THn CT onpe.ueJI5IeTC5I no cpop11MyJie10.12 . .UoKa3aTb, qTo cne.uymruwe <PYHKUHH ueni,ie, HaHTH nopR,n:oK H THn cpyHKUHH:1)L ~ _(A_z_)"_rcan + 1) ,A> 0,a> 0;n=O2):L:~=,(~J;11/a3)L~=' (l~in5)1 JL== 2 ( nlnn7)"=Jz", a> O;11II6)~t+a , a > O;I::,11/a11L,,11=011z ,a>O;ch.[,;, z" .n.'10.13.
.UoK333Tb, qTQ eCJIH cpyHKUH5I f(z) - ueJia5I, TO nop5I.UOK HTHn cpyHKUHH/(z) paBeH nop.51.UKY 11 mny ee npOH3BO.[(HOH.10.14. IIycTb / 1(z) - uena.51 cpyHKUH5I nopR.UKa p, /2(z) - ueJia5IcpyHKU115I rrop5I,n:Ka MeHbllie p . .UoKa33Tb, qTQ cpyHKUHH /1(z)f2(z)· Hj,(z) + fi(z) HMeIOT nop5I.UOK p.10.15. IIycTh cpyHKUHH/1(z) w/2(z) - ueni,1e cpyHKUHH o.ri:Horo nopR.ri:Ka H THnOB CT1 H CT2 COOTBeTCTBeHHO. IIycTb CT1 < CJ2. 1ho MO)KHOCK333Tb 0 rrop5I.UKe H mne cpyHKUHH /1(z)f2(z)· H /1(Z) + /2(z)?'129UEJibIE <!>YHKUYIH YI P51.[\bl .[(YIPI1XJIE10.16.
TiycTh cpyHKQ1111 f 1(z) 11 fi(z) - ueJihre cpyHKu1111 O.llHOrorrop5I.llKa 11 0.llHOro T11rra. "llTO MO)!(HO CKa3aTh 0 rrop5I.llKe 11 Tl1rrecpyHKu11H:j,(z)f2(z)· 11 f1(z)10.17.+ f2(z)?KaKOB rrop5I.llOK "''t'YHKU1111f(z) = L~= 0 a,,z"P5I.llOK cpyttKIJJm10.18. IlycTh f(z) -F(z)ecTb= "=~n=O Ia,, Ip z, , ecJI11rro-p?u:eJia5I cpyHKU:115I KOHeqHOro rrop5I.llKaer 6eCKOHeqHO MHOfO HyJieH Zi. Z2, ... , z,,, ... (z,p 1111Me-t 0), lz11 lioo. ,aoKa-3aTb, qTo ITOKa3aTeJib r ITOCJie,[(OBaTeJibHOCTl1 { z,,} MeHbUie 11Jil1 paBeH10.19.f(z) -TiycThu:eJia5I cpyttKU:11H rropH.n:Kap, 0 < p < oo,p.KO-HeqHoro Tl1rra <J, 11 y Hee 11MeeTC5I 6eCKOHeqHo MHOro HyJieH { z,,}'z,:f 0,lz,, lioo.
,aoKa3aTh, qTo-.- nbm-- ~ a-ep.,,_,=I zlllp10.20. IloKa3aTb,qTo KaHoH11qecKoe rrpo113se,[(ett11eF(z)=Il~=i £(3-,k)z,,Y.llOBJieTBopHeT ycJIOBl15IM:1) 5IBJI5IeTC5I u:eJIOH cpyttKU:11ei1:;2) MMeeT rrop5I.llOK p3)eCJIM=r;1=L 11= I 7 Ir <1--oo,TOnm KOHeqeH.~II10.21. Pa3JJO)!(l1ThB 6ecKoHeqttoerrpo113senemiecJie.ay10urnecpyHKU11M:1) sin z; 2) cos z; 3) sh z; 4) ch z; 5) e 0 z - ebz; 6) ch z - cos z.10.22. Hai1:T11HYJIM cpyttKU:M11e"- -111 rroKa3aTb, qro 0H11 He11MeIOT KOHeqHoro ITOKa3aTeJIH CXO.lll1MOCTl1.10.23. IloKa3aTb,qrn cpyttKU115I EecceJI5I=J (7)=" 11,.
~(-l)" z211+1·~ = 0 2 211 +'· n !r(v + n + 1)'v>-1'5IBJI5IeTC5I ueJIOH, HaHTl1 rrop5I.llOK 11 Tl1IT cpyHKUM11. ,aoKa3aTh, qToJ,.(z) 11MeeT 6ecKotteqtto Mttoro ttyJieii, 11 see ee ttymr BeHHbie qHcna.sew,ecT-I'Jzaea 1013010.24. )J;oKa3aTb, qTQ cpyHKUH5! F ( z) = rr:=I (1 + z I e") - ueJia5!, aee rrop5!.L\OK p = 0.10.25. )J;oKa3aTb, qTQ cpyHKU115! F (z) = rr:=I (1+zI11(5) - uena5!,ee rrop5!.L\OK p = 1 I a, cr> 1.10.26. IlyCTb ueJia5! cpyHKUH5! f(z) HMeeT rrop5!.L\OK p, 0 < p < 00.)J;oKa3aT1, qTo ecn11 p He u:enoe, TO rrocneL(oBaTeJI1HocT1 A-ToqeK11MeeT rroKa3aTeJib CXOL\HMOCTH LA = p rrp11 mo6oM A.
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