Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 8
Текст из файла (страница 8)
L~;J11 (z) paBHOMepHo cxo)l.HJICJI Ha MHO){{ecrneZKj(z),He06XO)l.HMO H )l.OCTaTOqHo, qTo6hIlim sup I r,, (z) I= 0.n-too:EZTeopeMa. EcJIH cpyHKUHOHaJihHaJI rrocne)l.oBaTeJihHOCTh lf,,(z)}CXO)l.HTCH Ha MHO){{eCTBef,,(z) HerrpepbIBHhl Haz paBHOMepHO K cPYHKU:HH j(z) H cPYHKU:HHZ, TOcpyHKUHHj(z) TaK){{e HerrpephIBHa Haz.TeopeMa. EcnH cpyHKUHOHaJihHhIH PM I,~; 1 /,,(z) cxo)l.HTCH Haz paBHOMepHO K cpyHKUHH f(z) H cpyHKUHH f,,(z)MHO){{eCTBepbIBHhl HaHerrpe-z, TO cyMMaj\z) TaIOKe HenpepbIBHa Ha MHO)l(eCTBe z.foBOp5IT,qTocpyHKU:HOHaJibHaHfIOCJie)l.OBaTeJihHOCThcxo)l.HTCJI paBHOMepHo BHYTPH 0611acTHcxo)l.HTC.51 Ha mo6oM KOMnaKTelf,,(z)}D, ecJIH oHa paBHOMepHoKc D.CeMettcTBO cpyHKUHH {f(z)}, onpe)l.eJieHHhIX Ha MHO)l(ecrne3hrnaeTc.H paeHocmeneHHO Henpepbl6Hbl.M HaZ,Z,Ha-ecJIH'11£>0 35(£)>0:Vzl'Z2EZ, lz1-z21<5,VfE{f(z)}~Jf(z1)- f(Z2)J<£.CeMeikTBO cpyttKU:HM {f(z)} 06pa3yeT orpaHHqeHHoe BHYTPH 06nacTHDMHO)Kecrno, ecJIH )l.JI.51 mo6oro KOMnaKTa KcD:JC(K) > O:'llf E {f(z)}, VzE K ~I f(z) I~ C.<l>yHKU:HOHaJihHOe 6eCKOHeqHoe npOH3Be)l.eHHer)l.e cpyHKUHH f,,(z) onpe)l.eJieHhI Ha MHO)l(ecrne z, J,,(z)Ha3hrnaeTC.5l cxoofl.i4uMCfl.
Ha MHO:J1CecmeeKa){{)l.OH ToqKe zE Z.f:: -1,z E Z,Z, ecJIH OHO cxo)l.HTCH B46Dzaea 4rr~=I (1 + f, , ( z ))oeCKOHe'IHOe IIp0113Be)leHHeAtepHO cxoOJlUfU.M-CJl 1-ta A11-to:J1cecmee Z,pa61W-ec;u1 nocne)loBaTeJibHOCThrr~=I (1 + f k(z))llaCTJill{HbIX IIpOH3Be,UeHHHHa MHO)KeCTBeHa3bJBaeTC5IpaBHOMepHO CXO.UHTC5Iz.,I(ocTaToqffbIHnpH3HaKpaBHOMepHOHCXO)J,HMOCTHHequoro npott3BeJJ,eHHH. Ecn11 .UJI5I mo6oro HaTypanbHorose)lJIHBO HepaseHcrno supI f,, ( z) I::::; a11u p5I)lI~=' a,,6eCKO-ncnpa-cxo)lHTCJI, To~EZZHa MHO)KeCTBe6eCKOHe'IHOe IIpOH3Be)leHJierr~=l (1 + f, , ( z )) CXO-.UHTC5I a6coJIIOTHO H paBHOMepHO, a 6eCKOHelIHOe IIpOH3Be)leHHerr~=l (I+ If,, (z) j)TaK)Ke CXO)lHTC5I paBHOMepHO.4.1. ,I(oKa3aThKp11Tep11ii Kourn pasHoMepHoii cxo)lttMocTua) cpyttKUHOHaJibHOro p5I)la,6) <liYHKUHOHaJibHOH IIOCJie)lOBaTeJibHOCTH.4.2 .
.[{oKa3aTbnpH3HaK Bei1epwrpacca paBHOMepHOH cxo)l11Mo-CTH <PYHKUHOHaJibHOro p5I,Ua.4.3 . .[{OKa3aTbIIp113HaK.[{upuxnepaBHOMepHOHCXO)lHMOCTH<PYHKUHOHaJibHOro p5I.ua.4.3.1. IlpuBeCTHpuxne ycnosueIIpHMep, IIOKa3b1Ba10mui1, lITO B IIpH3HaKe .[{H-lim u11 ( z) = 05IBJI5IeTC5I cymecrneHHhIM.n --+ (XJ4.4. L{oKa3aTbnptt3HaK A6emr paBHOMepHoi1 cxo)lHMOCTH <PYHK-UHOHaJihHOro p5I.ua.4.4.1. Ilp11secT11JI5I YCJIOBHe -npuMep, IIOKa3b1Ba10mui1, qrn B nptt3HaKe A6e-lim V11 ( Z)eCTb orpaHHlleHHa5I <i>YHKUH5I Ha MHO)KeCT-n - rxiBez - 5IBJI5IeTC5I cymecTBeHHbIM.4.4.2.p5I)laBepHo JIH yrnep)K)leHue: .UJIJI paBHOMepHOH CXO.UHMOCTHI~=' u,, ( z) v,, ( z)I ~= ' u,, ( z )Ha MHO)KecrneZ.uocTaTOlIHO, lIT06hr p5IJlpasHoMepHo cxO)lHJIC5I Ha MHO)KecrneZK cyMMe, KOTO-pa5I orpaHHlleHa Ha )TOM MHO)Kecrne, a IIOCJie.UOBaTeJibHOCTb { V11(Z)}6hIJia IIOCJie)lOBaTeJibHOCTblO c paBHOMepHO orpaHHl!eHHbIM H3MeHeHHeM .47ITOCJ1E.[\OBATEJ1bHOCTI1, P51.[\bl, ITPOI138E.[\EHI1514.5.JJ:oKa3aTh, qrnlim sup Ir;, (z) I= 0 - Heo6xo,z:u1Moe 11 ,n:ocrn"--+oo.:EZrnqHoe ycJIOBHe, qrnfa1 cpyHKU:HOHaJibHbIH p5I,n: HJIH cpyHKU:HOHaJibHa5I IIOCJie,n:oBaTeJibHOCTb CXOJJ:HJIHCb paBHOMepHO.4.6.JJ:oKa3aTb, qTQ eCJIH cpyHKU:HOHaJibHa5I IIOCJie,n:oBaTeJibHOCTbZ,j,,(z), cocrnBJieHHa5I 113 HerrpepbIBHhIX cpyHKU:HH Ha MHO)Kecrnecxo,n:HTC5I paBHOMepHo Kf{z) Ha MHO)KecrneTO cpyHKU:H5I j{z) He-Z.rrpephIBHa Ha4.
7.Z,HattTH MHO)Kecrna, Ha KOTOpbIX paBHOMepHO CXOJJ:5ITC5I CJie-,n:y10rn.11e IIOCJie,n:oBaTeJihHOCTH:3) { si:nz }·4.8.HOCTbIlpH KaKHX ,n:ettCTBHTeJihHbIX 3HaqeHmIX a IIOCJie,n:oBaTeJib-f,, (z) = na ze-n::1) CXOJJ:HTC5I;2) CXOJJ:HTC5I paBHOMepHO Ha o6JiaCTH CXOJJ:HMOCTH.4.9.JJ:oKa3aTb, qTo IIOCJie,n:oBaTeJibHOCTb0<MepHO CXOJJ:HTC5I B yrne Jarg zJ :::; a,4.10.JJ:oKa3aTh, qrn ecJIH pH,n: ,L~JHa MHO)KecrneZ,4.11. HattTHl)L=11=1a<10)smnz. 2) L~ cosnz.'11=1'11f,, (z),z EZ.3) L~=I shnz ;11(-1)"z+nL...in=l= 0.o6JiacTb cxo,n:HMOCTH CJie,n:y10rn.ttx pH,n:oB:L11=I-;"'=paBHO-K cpyHKU:HHj{z)TO cxo,n:HTCH paBHOMepHo H pH,n: ,L~=I"'=~n/4,f,, (z) I cxo,n:11TcH paBHOMepHo11e-11=·'412• • IToKa3aTh,+z1 ) .
9) L=L=(z;, +z" ,11)"'=ne-qTo p5I,n:L 11=11=111=11112)=·'=1,.6) .L..11=1_1__8).L..,n=l1"'=II5) .Lru=l_l_z_u .'7)!,, (z) = nze- 11'='1n-(z-zll)+z111) .-n1 ( z +z11 ,"'=II1 11L-t11=l e= ",'r,n:ecxo,n:HThCH rrp11 Bcex z, He Jie)Kanrnx Ha OKpy)KHOCTH.z = ei11 ,"I z I = 1.6y,n:eT48I'JIQ8Q4n-14.13. IloKa3aTJ>, qTo cyMMa p51.na s( z)pasHa1rrpH2(1- z)'I z I< 14•14• IT OKa3aTb, qTO p51.[(1z(l - z) 2H. zL -11=1 2" sm211- (1- zll z)(1- z"+1)="L...11=1rrpHI z I> 1.6y.neT a6COJIIOTHO cxo-.[(HTJ>C51 rrptt scex z E C.4.15.
IlycTb I c I < 1. IloKa3aTb, qrn pH.n L~=i c" e": a6comoTHO2CXO.[(HTC51 rrptt scex z Ec.4.16. IlOKa3aTb, qTQ p.Hna-bz+--z 2 + ... +2!(a-b)(a-2b) ... (a-(n-l)b) ,,I I< I b 1- 1.z + ...n!a6co1110rno cxo.nttTC51 nptt z4.17.11oKa:3aTb, 'ITOpH~ " - ,L..J,,:11-Inzz" - ( 1+ 1/ n)a6comOTHOCXO~HTCHI I< 1.rrptt z4.18 . .[{oKa3aTb, qro rnrrepreoMerpttqecKttH: p51.naba(a + l)b(b + 1) ,z· + ...l+-z+c1·2·c(c+l)+a(a+l) ... (a+(n-l)·b(b+l) ... (b+(n-1)) ,,z + ... ,n!c(c + 1) ... (c + (n -1))I I<r.ue c i= 0, -1, .. ., -(n - 1), ..
., CXO.[(HTC51 a6co1110rno rrptt z1,pacxonttTCH rrptt z1, a rrpH z1 Heo6xo.nttMoe H .nocraToqHoeycJIOBHe a6COJIIOTHOH CXO.[(HMOCTH - 3TOI I>I I=Re (a+ b - c) < 0.4.19. IloKa3aTb, qTo pH.n 2:~= 1I(-1)"-I,z- +ncxo.nttTC.H rrpttIz I <1, HOHe a6COJIIOTHO.4.20.
Orrpe.ne11ttT1> 0611acT1> a6comoTHoH: cxonttMOCTH pH.nos (p .neH:CTBHTeJibHOe qHCJIO):1)L~= 1 sin nz ;n2)L~=i cos nz ;113)L~= 1 sin :,1z ;nDOCJIE.[lOBA TEJibHOCTH, P5!.[lbl, DPOH3BE.[lEHH5I7) "- z" ..L.Jn=In '4.21.8) "1 ..L.i11=1 1 + z" ,491) ;9) ".L.i11=l 21 ( z +--;;IInz(p -Halim: o6naCTh paBHOMepHoii CXO.ZUfMOCrn: p-H.UOB.ueii-CTBHTeJibHOe q11cJIO ):l) "- sinnz. 2) "- cosnz.L...11=1L...u=In 'n '4) .L.i11=I" - e-11: •'7) ""'-15) "- -1? ( Z + Z-11) ;.L.i11=ln-6) " - (-1)" .z"9) , - 2" "·II.
8) ""'-.L.i11=1 1+ z" ,4.22.BeZ.IIycTb p-H,uZ.L...i11=lz+n'z 'L~=l f,, (z) cxo,u11Tc.H paBHOMepHo Ha MHO:JKeCT-f,, (z) Icxo.u11TC$l Ha MHO-Z?4.23. IIycTbBe..L.i11=1 z + z" 'Cne.uyeT JIH oTcFO,ua, qTo p-H,u I~J:JKecTBeL...11=1p$i.U I~=I11 cxo,u11TC$i pj{.u0Tc10,ua, qrn pH,uMHO:>KecTBef,, ( z)cxo.u11TC$l paBHOMepHo Ha MHO:JKecT-I~J f,, (z) II~=il f,, (z) IHa 3TOM MHO:JKecTBe. Cne.uyeT JIH6y.ueT cxo.u11TbCH paBHOMepHo HaZ?4.24. I1p11Becrn:1rrp11Mep p.H,ua L~= f,,(z), KOTOpbIH cxo,uHTCH a6-COJIFOTHO, HO He paBHOMepHO Ha MHO:>KeCTBe4.25. ,UoKa3aTb,z.z2qTQ p.H.U " -.L.,,,,=He paBHOMepHo Ha MHO)l(ecTBe {02(1 + z)"CXO,UHTC.H a6cOJifOTHO, HOz E C : Iarg z I: : ; tr I 4} .Dzaea 450= {z: I z I < 1, larg zl :::; n:/4}4.26 .
.[(oKa3aTh, LITO Ha MHO)l(ecrne Z~(-l)" z 2511l~P~ 11=0 (1+ 2 2 )"L ~=o lCX01ll1TC51 a6comOTHO 11 paBHOMepHO, a p511lz2 II I(1 + z 2 )" IHa MHO)l(ecrne Z paBHOMepHo cxon11TbCR He6yneT.4.27. Onpenen11Tho6nacTbcyru:ecrnosaH115!L(JIHcpyHKu:1111\)kf (z) = L~=t (1 +11 11ccnenosaTh ee tta tterrpephIBHOCTh.4.28 . .[(oKa3aTh, qTQ L(JI51 TOro qrn6b1 110CJieL(OBaTeJihHOCTh {f,,(z)}tterrpepbIBHhIX cpyttKU:11tt pasttoMepHo cxon11nacb Ha KOMrraKTe K KcpyHKU:1111 f(z), Heo6XOL(HMO 11 L(OCTaTOqHo, qTo6hI IlOCJieL(OBaTeJibHOCTb {f,,(z)} CXOL(l1JiaCh BO scex TOqKax 3TOro MHO)l(ecrna:limf,,( z) = f(z),zE K, 11 ecn11 z0npeneJibHa.R ToqKa K, TO LtJIH-n-+ oomo6ott nocnenosaTeJibHOCTl1 { z11 }lim f,, ( z ) =Jj~ OQf (z0 ),z11K, lim z11 =E11 -tooz0 ,cnpasenn11so.4.29.
OnpeneJIHTh o6nacTh cyru:ecrnosatt11H cpyHKU:HHfz~~( z) = ~ 11=12n +z211 HCCJie,[(OBaTh ee Ha HenpephIBHOCTb.4.30. Onpenen11Th o6nacTh cxon11MocT11 pHnaL~=i (nze- 11:-(n - l)ze-<11 -t>: )11 ttattTH ero cyMMy. IIoKa3aTb, qTo cyMMa p51na ecTb cpyHKU:H51 HenpepbIBHaR, HO PM Ha o6nacm CXOL(HMOCTH CX01ll1TC51 HepaBHOMepHO.4.31. Orrpe,n:eJIHTh o6nacTh cyru:ecrnosaHHH 8-cpyttKU:HH8( z) = ~ ~~11=-00e- 1111' : .IIoKa3aTh, qTo Ha o6nacTH cyruecrnosaHHH 8(z) ecTh cPYHKUHH ttenpepbIBHa51.4.32.
OnpenenHTb o6naCTh cyru:ecrnosaH115! cyMMhI pHna~~~ 11=1sinnz2".nOCJIE,l:\OBATEJibHOCTl1, p5!.[:\bl. nPOl13BE,l:\EHl15J514.33. )loKa:Jan, HenpepbIBHOCTb cyMMbI p.sr.n:a B yKa:JaHHbIX 06nacrnx:1)I~=I n <n+z )'1- (-1)"2) L11=I--,z-n3)z * 1, 2, ... ;L~= 1 (2(z,;n)J,'°'-4 ) L..i11=l sinnzI ,ll.5)z;t-1,-2,. ..
;lzl<l;Iz I< oo;L-11=1 shn!nz , IZ I< oo;6) L~= 1 n e1111\Re z < 0;- -11 1l31l37r1l--<argz<-.L 11 1e- '4-<araz<e4 '4427)=4.34. IloKa3aTb, qro ceMeiicrso <PYHKUHH {cos (a I (1 - z))}, a .n:eiicTBHTeJJbHOe 'IHCJJO, 06pa3yer orpamrqeHHoe BHYTPH o6nacTHD = {I z I < 1 } , Ho HeorpaH11qeHHOe B o6nacTH D ceMeiicrno.4.35. IlpHBeCTH npHMep paBHOCTeneHHO HenpepbIBHOro Ha 06JiaCTH D, HO HeorpaH11qeHHOro BlfYTPH D ceMeiicrna cpyttKUHM.4.36. IlpHBeCTH npHMep ceMeHCTBa <PYHKUHM, orpaHHqeHHOroBttyrpH o6JiaCTH D, HO He paBHOCTeneHHo HenpepbIBHoro Ha o6JiaCTH D.4.37.
BepHo JIH yrnepJK.n:eHHe: mo6oe ceMeiicrno HenpepbIBHbIXcpyHKUHH, orpaHHqeHHOe BHYTPH o6nacrH D, .srsn.sreTc.sr paBHOCTeneHHO Henpep1>IBH1>IM Ha mo6oM KOMnaKTe KC D.4.38. IlycTb ceMeHCTBO <PYHKUHH paBHOCTeneHHO HenpepbIBHO Hao6nacTH D H orpaHH'leHo BHYTPH D. Cne.n:yer JIH orc10.n:a:1) H3 .n:aHHoro ceMdicrna MOJKHO Bb16parr, cxo.n:.srmy10c.sr no.n:nocne.n:osaTeJI1>HOCTb cpyHKUHH Ha o6JiaCTH D;2) H3 .n:aHHOro ceMeHCTBa MO)l(HO BbI6paTb paBHOMepHO CXO.ll.R:IUYIOC.R: no.n:nocne.n:osaTeJJbHOCTb cpyHKUHH Ha o6Jiacnr D.4.39.
)loKa3aTb, 'ITO Ha MHOJKecrne Z 6ecKoHeqHoe npo113se.n:eHHe rr~=I (1 + f,, (z)) CXO.llHTC.R: paBHOMepHO H a6COJIIOTHO, eCJIH ,UJI.R:I'Jiaea 452scexzEZcnpase.n:mrno HepaseHCTBOIJ,,(z) I :S a,,, n EN,l1 p51,n:L:~=' a11 cxo,n:ttTC51. ,[(oKa3aTb, qTo 6ecKoHeqHoe rrpoIBBe,n:eHtten~=I (1+ I!,, (z) DCXOJUfTC.sI TaIOKe paBHOMepHO Ha MHO)l<'.eCTBez.4.40 . .lJ:oKa3aTh, qTo ecm1 p.si:,n:hI :L;~=' f,, (z) 11 :L;~J f,, (z) 1 cxo.n:.si:Tc.si: pasttoMepHo Ha MHO)l<'.ecrne Z, TO 6ecKoHeqHoe npo113se,n:eH11e2Il~=I (1 + !,, (z)) TaK)l<'.e paBHOMepHO CXOAHTC.sI Ha MHO)l<'.eCTBe4.41. IIycTh p.si:.n: :L~=I a11z.cxo,n:ttTC51.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
