Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 36
Текст из файла (страница 36)
I1ycTL a> 0. .D:oKa3aTb,JIO:lKeHHerra16.37. I1oJiyqHTbeit dt='-!TO npH x--+ +oo cnpaBe)J.JIHBO pa3-i;:· [1+0 ( ~)) .aCHMITTOTH'-leCKOe pa3JIO:lKeHHe )J.JISI: wune2pCl.!la<Ppertel/H. npH x --+ oo<P(x) =r eit~·'16.38. Jle.MAta 3poet1u:dt=eix2(-i +~+ o(~))·2xrrycTh a;:::HO )J.HCpcpepeHQHpyel\1a Ha(0, a]sceMH npoH3BO)J.HhIMH B TO'-!Kex1,~4x> 0,xcpyttKQHsi:f(x) 6ecKoHe'-l-H o6pamaeTC.ll B HYJib BMeCTe co= a . .[(oKa3aTb,'-!TO npH /... --+ +oocnpase)J.JIHBO aCHMITTOTH'-!eCKOe pa3JIO:lKeHHef(/ xfl-lJ(x)eiA.ra dx Jo"00~k=Oa A_-(k+/J)lak'236I'.TJaea 16r.ue K03cp<jrnu11eHThI ak onpe.uemuoTrn no cpopMynaMak =1 <k> (O)ak!r (k+/J)- - e i~k+fJ- a .a16.39.
IToKa3aTh, 'ITO np11 A. --+ +oor( ~f~ f (x)e;J.x·' dx -}-1/JeiJT/ 6 •16.40. HattTH acHMrITOTH'!ecKoe pa3JIO)Kett11e np11 A. --+ +oo .umrHHTerparrail(. ,e'"'cosi0?cos-t dt .16.41. llyCThJI/ (x)= 2~ s:J( ei(xsinrp-nrp) drp(n 2 0 - uenoe <rncno) ecTh cpym<Ulll EecceJUI uenoro nopMKa. ,[(oKa3aTh, 'ITO npH x --++oo cnpaBe)lJlHBO acHMCTTOTWieCKOe npenCTaBJieHHe1,,(x)=ff1;cos( x--:J o(~}+16.42. ,[(oKa3aTh, qTO .umr cpyHKUHH Eeccem1 uenoro nop5!,UKa11 2 0 crrpasenn11so HHTerparrhHOe npencTaBJiett11e1,, (x) =_.!.._7rf:cos(xsinrp- nrp)drp.16.43. ,[(oKa3aTb, 'ITO )lJl5! cpyttKUHH l11(11x) npH11--+ oo,1l -HaTy-parrbHOe, cnpasenn11so cneny10mee acHMITTOTH'!ecKoe npe.ucTasneH11e rrpH cpHKCHpOBaHHOM1 ,,(nx) =x > 1:~ 117r"--1~ cos(narccos~-n~x~-1+Jr)+o(_!_)·x412KpoMe Toro, nnH111 (11) rrp11 n --+ oo, n - HaTypaJihHOe, cnpase.un11-so cneny10mee acHMilTOTHl!eCKOe paBeHCTBO:1 ( 11 ) -II16.44.
ITycTh a,~ -r(l/ 3)o13 I /(J2- 3 'trA. --++0( -~IJ )11•noJIO)KHTeJihHhie 411cna, y -60JlhWOe IlOJlO)KHTeJJbHOe l/HCJ10, Tl3aTb, qTo np11-113112 2-nocTaTOqHoHazypaJibHOe l!HCJIO. L(oKa-+oo cnpasenn11so ac11MrrTOTHl!eCKOe npencTaBJieH11e:METO.[(hf f10JIYYEHl15I ACl1Mf1TOTl1YECKl1X OUEHOKflyJo X er.ue-~(1-.r)-a( f'COS Joel(n/2)- 1(h)(1-1)-adt)1112X1 __dx -] ( 11 !2)- I (!U)237COS AC0A_fl+l(In A_) (a+ I)/( 2a)- cpyttKUIUI Eeccem .upo6Horo rrop5I.!J.Ka (n/2) - 1.16.45. IlycThH,C 1l (x) = _!_ J ei (.r sin :-i·:l dz- rPYHKLfUJl XaHKeJZR 1-ro7r ypo.ua mo6oro .uei:fcrnttreJihHoro (HJIH KOMrrJieKCHoro) rropMKa v. 3.uechKOHTyp y rrepexomIT 113 rroJiyrroJiochr { -rr./2 < Re z < rr./2, Im z > 0}B rroJiyrroJiocy { rr./2 < Re z < 3rr./2, Im z < 0} qepe3 roqKy z0 = rr./2 ..IJ:oKa3aTh, qro rrptt x ---+ +co crrpaBe.UJIMBO acMMIIOT11qecKoe rrpe.ucraBJieHtte-J;rr /2xH,Cl)(x) =( ei(x-vllt 2 -llt 4 )+ 0(11 x) ).16.46 .
.UoKa3aTh, qro rrptt A-+ +coJ~-~ (1e;;··:dx-[:rr(2-..fi.))112e-J.c.fi-1)(2..fi.-2r;. .+ x- )''-116.47 . .IJ:oKa3aTh, qro rrptt HarypanhHOM nJ +~+;_e2- (1 +z )-"dz- ( 7r 12 e )---00+1112L·----+11+co(2 enI ) 1112 .16.48. IlycTh cpyttKUIDifi:z) attan11rnqHa rrptt Re z:::; a, a> 0, KpOMeKOHe'IHOro 'IMCJia oco6hrx roqeK z 1, z2, .
.. , z,,, Re zk <a, k = 1, 2, ... , n,11fi:z) ---+ 0 rrptt z -+co, Re z:::; a . .IJ:oKa3aTh, qTo rrp11 t---+ +co crrpaBe.uJIMBO aCMMIITOTHqecKoe rrpe.ucraBJieHMe1- - _J'x+i~ e=r f (z)dz res[e=r f (z)].2:rrz a-1=Lr.ue BbI'leTbI 6epyrcH rro BCeM oco6hrM ToqKaM, .!l.Jl5I KOTOphrx Re z :::=: 0.16.49. Hccne.uoBaTh rrptt t---+ +co 11 rrpttn---+ co ac11Mrrror11qecKoerr0Be.uett11e MHTerpana1--_f2ma+i=.a-i=e:' dz,z-(z +a)·1,a> O,Rea > 0.16.50. Haffrn ac11Mmor11qecKoe pa31Io)!{eHtte rrptt t---+ +co MHTerpana12n1--.r)le-!;, > 0,fa+i=.a-i=ecn11 z > 0.e=t .JZ d,z-+ w-1z, a> 0, co > 0.238I'lla6a1616.51. ,UoKa3aTb, LJTO .nm1 cpyttKu1111 5eccem1 J11 (z) uenoro nop5!.UKa11 KOMTIJieKCHoro apryMeHTa z cnpaBe.[lJil1BO 11HTerpaJibHOe npe.nCTaBJieH11e1r e-"<~-_!_)2~ ):'-11-l d J=_27ri J1~l=l"'"'J ( ) =-zII16.52.
Mcxo.n51 113 11HTerpanbHOro npe.ncrnsnett115! .nm1 J 11 (z) 113npe.nhl.Llyw:ew 3a,uaLJ11 .noKa3aTh, LJTO nptt z--> ro, jarg zi ~ n - £, £ > 0,l 1 (z)= fnz [ cos(z-nn 12 -n I 4) +o( ~)sin o( ~)cos Jz+z16.53. c/Jop.Mylla llyaccorw. IIycTh cpyttKUIDI j(x) ttenpepbIBHO .nttcpcpepeHU:11pyeMa nptt x E R, a6comorno 11HTerp11pyeMa Ha R, P5I.D:L~=-~f(n) cxo.nHTrn 11 P5I.D: L~=-~f'(n+x) CXO.UHTC5! paBHOMepttonptt 0 ~ x ~ 1. ,UoKa3aTb, LJTO cnpase.nnrrsa cne,n:yIOW:a5! cpopMynacyMMttposamrn IIyaccotta:"L~=-~f Cn) ="L~=-~f~ e-iZmlX J Cx)dx.16.54. HaiiT11 ac11MnTOTttLJecKoe nose.nett11e rrp11 x · --> +ro <J:>yttKU1111~f(x) =(-1)"L11=- ..f122 + x216.55.
Hai1:rtt acttMITTOTHLJecKoe nose.nem1e rrptt x --> +oo cpyttKUtttte-n2x.f( x)=""oo~n=-oo16.56. ,UoKa3aTb, LJTO nptt xI~crll=I-->+ro1YIe- 11 - ln=--lnx+-+O("x).22r.ne y - rrocT05!HHa5! 3H:nepa.16.57. ,UoKa3aTh, LJTO np11 £--> +O1I~ -=~-_!__+o[_!__e-J?]1 + t:2t: 211=1211E.I'Raea 173JIEMEHThl TEOPHH IIPHliJIH)KEHHHIIycThX - BeIUecrnettttoe 11n11 KOMrrneKcHoe HOpMwposaHHOeB 3TOM rnase MhI 6y.n.eM paccMaTPI1BaTh TOJihKO 3a-npocTpattcrno..n.aqy o np116JIH)!(ett1111 cj:>HKCHposaHHoro JJieMeHTasaHHhIM MHO)!(eCTBOMcFx E Xcj:>11Kc11po-X.Hawryttiuu.M npu6J1u:>1CeHuv.1 3JieMeHTa x EXMHO)!(eCTBOMF Ha-3hIBaJOT BeJI11q11HyE(x, F)=infllx - u.11·ueF3aqaCTyIO MHO)!(ecrna F - no.unpocTpattcrna (KotteqttoMepHbie HJIH6ecKOHeqHOMepHbie), a TaIOKe BhmyKJibie 3aMKHYThie MHO)!(eCTBa.Ecnu cyll(ecTnyeT 3JieMeHT u 0 EF, peaJIH3YIOIUHH yKa3aHHYIOToqHyIO HH)!(HlOlO rpaHb, T. e.
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Ba)!{HhIM sonpocoM .srnm1eTC5! 113y'-!emre xapaKTep11cT11'-!ecK11xcsoi1crn Ha11ny1m1ero ::ineMeHrn 11 cnoco6 ero nocTpoeH11.si.B npocrpaHcrne C[a, b] cpyttKu11i1 J, HenpephIBHbIX Ha orpe3Ke[a, b], 6y.lleM paccMaTp11BaTh HOpMyl\Jllc =sup If (t) j.a$t$bB rrpocTpaHcrne Lp[a, b] cpyHKUHtt .f. l1HTerp11pyeMhIX no Jie6eryco CTerretthIO p ~ 1, 6y.ueM paccMaTp11saTh ttopMy11!1/p=cC11ct)I" dt)'.'".Ecm1 cpyttKu1111 2n-rrep110.u11'-!ecK11e, TO paccMarp11saeM coorneTcrnettHo HOpMbIl!Jllc = sup If (t) j,0:St:S21l"l!JllP =(I:JTlf(t)IP dt)"P,p;?:l.(pITycTh X - npocrpattcrno C-HerrpephIBHhIX 11n11 rrpocTpattcrno LP1) 2n-rrep110,ll11'-!eCKl1X seruecrneHH03HaqHhIX 111111 KOMIIJleKCH0-~3Ha'-!HhIX cpyttKu11i1, aII · II - ttopMa BX.Mooylle.M Henpepbl6Hocmu cpyttKl..(1111 f E X B rrpocTpaHcrne X Ha3hIBaIOT cpyHKU11!0w(f, t) =supjjJ(x+ u)- f(x)llx,t 2': 0.iu!S1IIycTb X - n11Hei1Hoe HOpM11posattHoe rrpocrpaHCTBO tta.u noneMP, P = R 11n11 C, c HOpMoi1 II · II.
MHO)!{ecrno ::ineMeHTOB ~c X tta3hrsa10T 3aMKHYThIM B X, ecn11 .unR n106oro ::ineMeHrn f E X 11 .un.si n106oro £ > 0 Hai1.uyTc.si cp 1,••• ,cp 11 E~ 11 TaK11e '-!11cna a 1, ••• ,a11 E P, '-!TOAnze6paul/eCKlW 110.r1uH0Jlto,11 O.llHOtt nepeMeHHOtt z E R (11n11 z EC) c .uei1crn11TeJJhHbIM11 (11J111 KOMrrneKCHhIM11) K03cpqmu11eHrnM11 a 0 ,0, Ha3bJBalOT cpyHKUIHO11P11(z) =a11 z + a11-1 z11-' + ... a, z + ao.ai, ... , all CTefleHH il, all :;z::3JIEMEHTbl TEOPl111 f1Pl15Jll1)!{EHl1H241Tpu20HoMempul/eCKlW noJil1HOMOM tta3bIBalDT QJYHKU111D s11nat,,(z) = a 0 + a 1 cos x+ b 1 sin x + a 2 cos 2.x + b2 sin 2x + ... ++ a,, cos nx + b,, sin 1u.Pa71UOHGJlbHOU opo6b10 Ha3bIBalDT cpyHKIJ,11!0, paBHYID OTHOWeHl1lO LJ.BYX aJire6pawreCKl1X Il0Jll1HOMOB=P,, (z)R(z)Q111- 1(z),P11 (z) = a,, zII + a11-1 z11-1 + ... a 1z + a0, a117=Q111 (z) = b111 zIll + a 111_ 1 z111-I + ...
a 1 z + ao, b1110,7=0•TeopeMhI BeH:epurrpacca o np116JI11)1(ett1111 cpyHKU11H noJI11HoMaM11COCT05IT B TOM, qTQ aJire6pa11qecK11e (Tp11roHOMeTpWieCK11e) IlOJ111HOMhl o6pa3yIOT 3aMKHYTbie MHO)l(eCTBa B npocTpaHCTBaxc11LP,p ~ 1.Kp1uep11H: 3aMKttyrocT11 MHO)l(ecrna B JI11HeHHOM HOpMHposaH-J naeT cJie.UyI01Ua5I TeopeMa.MHo)l(ecrno J 3aMKHYTO B X TornaHOM npocTpaHcrneTeopeMa.11 TOJihKO Tor.ua,KOf,Ua BC5IKl1H Jil1HeHHhIH HenpepbIBHhIH cpyHKUl10HaJIconp5I)l(eHHOe npocTpaHcrno K6oM seKTOpey E J, cp(y) = 0,flpOCTpaHCTBOU110HaJIOB<px•X),06par.ua10r.u11H:cH B HYJih Ha· JII0-TO)l(LJ.ecrneHHO paseH HYJIIO.COCTOHT 113 JIHHeHHblX HenpepbIBHhIX QJYHK-C HOpMOHIll<JJll = sup <p (x)llxllx SIIIycTh cpyHKU1111 fix) 11p(x)<p E X* (X*-g(x)I·HenpephIBHhI Ha oTpe3Ke[a, b],cpyHKU11HHMeeT BHLI.( )px( ) =sxb,,_,,x"_,.
+ ... + b,x + b0a _Px111-p1110 :S µ :S111 ,0 :S v :Sn, a07=0,( ) B(x)=sx-A( x')'+ ... +a1x+a0MHOroYJieHhIA(x)11B(x)He HMeIOT06m11x .ueJI11TeJieH, KpOMe MHoroqJieHOB ttyJiesoH: CTeneH11. IIycThp -MHO)l(eCTBO cpyHKUHHHhl,µ, v, a0 , ai, ... , a 111 ,p(x),,UJ15I KOTOpb!Xs(x),a TaK)l(eb0 , bi. ... , b,, -npo113BOJihHhie.ITycTbE(f,P)= inf sup lf(x)-p(x)Ip(x'JE P a S.r!: b111, 1l cp11KC11pOBa-I'Jiaea 17242Lfe6bnUe6aDo meopeMeMHO:>KeCTBO p 3a.uaeTC51 MHO:>KeCTBOM cyme-CTBOBamrn: H e)J,HHCTBeHHOCTH H p * (x)E p5!BJrneTC5! HaHJI)l1IlllHMpaBHOMepHbIM rrp1161111:>KemrnM .UJI5! cpyHKUHHTOJibKO Tor.ua, Kor.ua ':111CJIO TO':leKNfix) E C[a, b]113 0Tpe3Ka [a,b],rnr.ua 11B KOTOpbIXfix) - p"(x) npHHHMaeT c ':lepe.uyIOI.UHMHC5! 3HaKaM11 3Hasup If (x)- p ''' (x) I He MeHbrne, ':leM m + n - min(µ, v)+2pa3HOCTb':leHHepa3; ern11 )Kep(x)= 0, TON~ n + 2E(f , P)11= sup IJ (x)- p(x) j.o5x5bEc1111 s(x)= 1, m = 0,TO rro11y':laeM meopeMy Lfe6bzu1eea o HauJZy'i-uteM pa6HOMepHOM npu6JZU:JICeHUU Henpepbl6HOU rjJyHKL/UU QJZ2e6pau'i€CKU.MU /10JlUHOAW.MU.
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