Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 12
Текст из файла (страница 12)
., a,,, HHTerparrff(z)dz = 0.IIlycTb cpyHKU11}!f(z) HenpepbIBHa B HeKOTOpOH OKpeCTHOCTl1= Zo . .LJ:oKa3aTb, '-!TOJ.11111p- 06.22.11 M(z) =l•- >ol=f'riycTb cpyHKUI151.maxlz - z0 l= •·< RI J(z)J(z)dz -_ 21fZ'f( Z ) .0J .z - zOf(z) HeIIpepbIBHa B o6Jiacrn 0 < Jz - z 0 1 < R.Il:oKa3aTb, '-!TO ecJIH lim rM(r) = 0, TO··-O~~~~JJ(z)dz = 0.lz - zol=r6.23. IlycTb!~n;, rM(r)cpyHKUH5!j(z)= 0 .
.LJ:oKa3aTb, qrnHeIIpepbIBHa:~i;,Jlz -z 0 l=r6.24.rioKa3aTb, qrnIIpHf(z)dz= 0.Jz - zol > roHMHTffPMPOBAHME CD Y HKU,l-1H. MHTErPAJ1bHA51 TEOPEMA KOW110.J' (z -=a)" dza(/_ 1,n~27ri, a E D.,, n0,- 1,n E Z,=-1,a r:/: D.,,n = -1,r.ue y - KOHTyp, orpaH11'!11Ba10m11M: KOHe'IHYIO o6nacTb3aMbIKamre75Dy, D,Dy.6.25.
fiycTb f(z) - aHaJil1Tl1'IeCKa5! cpyHKU:115! B O,UHOCB5!3HOH06nacT11 D, y - 3aMKHYTa5! Kp11Ba5! y ED. ,UoKa3aTh, 'ITO6.26. fiycTb cpyHKU:H5! f(z) HerrpepbIBHa B OKpeCTHOCTH Ha'IaJiaKOOp,UHHaT. ,UoKa3aTb, 'ITOlim fr -02Jo"J(re;" ) dcp = 27rf(O).6.27. fiycTb cpyHKU:115! f(z) aHaJIHTl1'IHa B KOHeqHoH BbIIIYKJIOHo6nacrn D 11 y.uoBnernop5!eT ycnoBHIO Refiz)?. M> 0, z ED.,UoKa3aTb, 'ITO JlJI5! n106brx ToqeK zi, z2 E D crrpaBe,UJIHBO HepaBeHCTBOEyneT JIH sepHbIM .uaHHOe HepaBeHCTBO, ecn11 ycnoB11e aHan11T11qHocT11 cpyHKU:HH 3aMeHHTb Ha ycJIOBHe HerrpepbIBHOCTH?6.28. fiycTb cl)YHKU:H5! .f(z) aHaJI11T11qHa B KOHe'IHOM BbIIIYKJIOH06nacT11 D 11 y.uoBnernop5!eT ycnOBHIO Imfiz)?.
M,UoKa3aTb, qTo JlJI5! n106brx ToqeK z 1,z2 E D> 0, z E D.crrpaBe,UJIHBO Hepa-BeHCTBOEyneT n11 sepHbIM naHHoe HepaBeHcrno, ecn11 ycnoB11e aHan11n1qHocT11 cpyHKU11H 3aMeHJ1Tb Ha ycJIOBHe HerrpepbIBHOCTl1?Dzaea 6766.29.lf(z)jDycTh <l,JyHKUmif(z) attaJrnrnqHa B BbIITYKJIOH o6nacrn2': M,D 11z ED. Cne.nyeT n11 0Tc10.na, qTQ .UJI5I n106b1x z 1, z 2 EDcrrpase,nJI11Bo ttepaBeHCTBO6.30.DycTb ¢YHKU115IttenpepbIBHa no ~ E yf(z,~) attan11mqHa no z ED np11 ~ E y 11D, r.ne y - )f(Op,nattosa, Kycoqtto-np11 z Erna,nKaH KpirnaH . .[(oKa3aTh, qrn ¢yttKU:11HF(z) =atta.n:MTMqHa B o6IIaCTMDMF'(z) =6.31. .[(oKa3aTh,MHTerpanJ, J(z, ~) d~qrni +"' F(t) dtcxo,nMTC5I paBHOMepttoJ, J,'(z, ~) d(lf(z, t)j ~ F(t)ecJI11CXO.U11TC5I,noz ETOscex.UIIHMHTerpaJIz ED11i+"" J(z, t) dtD np11 ycnomrn, qTo 11HTerpanJ,Af( z, t) dt cymecrnyeT np11 JII06oM A> a.6.32.t E [a,DycTb ¢YHKUM5I+oo)f(z,M HenpepbIBHat) attaJI11TMqHa no z E D rrpH JII06oMnot E [a, +oo) np11 mo6oM z ED ..[(oKa3aTb, qrn eCJIM MHTerpaJIfB o6nacm D, TO ¢YHKU:115I F(z)=J+ov J(z, t) dtTl1 D,;F'(z) = J,,r+"' J}(z, t) dtMeptto B 06JiacT116.33.+evJ(z, t) dtCXO.U11TC5I paBHOMepHOaf{aJmT11qHa B o6Jiac-11 :noT 11HTerpaJI CXO,U11TC5I paBHO-D.HaHTl1 MHO)f(eCTBa, Ha KOTOpbIX paBHOMepHO CXO,U5ITC5I CJ1e-.ny10m11e 11HTerpaJihI:2)J+"" sin t / ez()1111dt;l1HTEfPl1POBAHl1E <DYHKUI1tl.
l1HTEfPAJ1bHA5I TEOPEMA K0111113)+"-IrJO6.34.zlulcost/ec+i"-1 eztdt; 4)j . -dt, ctc-1<0V+'=O;sf77('+in..; ezl-dt,tcc +'=0.JJ:oKa3aTh aHaJIHTHqHOCTb CJie,ayIOll.lHX cpyHKUHH B yKa3aH-HbIX o6nacrnx:Ne-tz1)l2)Jo3)11 costzdt, z r/. [0,1];z +t0- -2 dt, Rez > O;1+ tdtr+oo(t2+ 1)(1 +et')' Re z > O;0l4)6.35.+oo t sin t- 2- dt, Rez20t+z> 0.fo+ro e J(t) dt , r.ae1'JJ:m1 HHTerpana JiarrnacacpyHKUHHf(t)rrnTerpttpyeMa Ha [O, a] .z:vrn mo6oro a > 0, .l(OKa.3aTh:1) eCJIH HHTerpan CXO)J,HTCH B TOqKe zrronyrrJIOCKOCTH Re z> Re z 0, paBHOMepHO= Zo,TO OH CXO)J,HTCH Bcxo,aHTCH B yrnelarg(z - zol ~ e < 11/2;2)eCJIH HHTerpan CXO,UHTCH a6COJIIOTHO B TOqKe z= Zo,TO OHcxo)J,HTCH a6coJIIOTHO H paBHOMepHo B rronyrrnocKOCTH Re z ~ Re z 0;3)ecJIHlim ln I J(t)/.---t+ooJIIOTHOBrronyrrnocKocTHrronyrrnocKocTH Re z :'.'.
a4)ecJIHI/ t =a,Re z>aTO HHTerpan cxo,aHTCH a6co11paBHOMepHolim ln If (t) IIt = p,<p.6.36. HattTH o6nacm aHaJIHTHqHOCTH cne.aYIQll.lHX2)f(t)dtl +oo --,I f(t) l'.SM(l + tt",a > O;t +z0J:I+ooBCHKOHTO HHTerpan He cxo,aHTCH a6coJIIOT-HO HH B O,UHOH TOqKe IIOJIYIIJIOCKOCTH Re z1)BO+ c, E > 0;j(t) dtt-·) - z-"I j(t) I'.SM;cpyttKUHH:I'JraBa 678+"'-f(t) dt- 1 - , I J(t)J e+cost dt5) J-z3)-CVIC1Z, /M1::; - + t";14)-1r_e_dt:Jargl=n t+z'= {t: 0 ::;1 t I::; IT.argt = 7r /2}.6.37.
~IDI ttHTerparra Jlarrnaca (CM. 6.35) ttaH:Ttt o6nacTh cyruecTBoBattmr I1 o6JJaCTb attar:IHTWIHOCTH, eCJJH QJYHKUHRj{Z) eCTh:011)1;2)t",nEN;3)e5) cos co t ;6) sh co t ;7) ch co t;l 11 cos co t,9)co*4)sincot;;at .8) e sm co t;0 - ;:i:eHCTBHTeJlbHOe qttcJIO.6.38. HaH:rn rrepBoo6pa.3Hhie JJ:IDI cpyHKuttH::l)sinaz;2)cosaz;003)e'12 ;4)chaz;05)shaz;06) e ' sin bz; 7) e ' cos bz; 8) e ' ch bz; 9) e ' sh bz(at 0,b t 0,at b).6.39. ,Ll;oKa'.3aTh, 'ITO Heo6xo;:i:ttMhIM H )J,OCTaTO'IHhIM ycJIOBHeMcymecTBOBaHHR rrepBoo6pa.3HOH MH HerrpepbIBHOH QJYHKUHH j{z) Bo6JiacTtt D HBJIHeTcH paBeHcrno ttymo HHTerpaJia OT cpyHKUHI.f j{z) rromo6oMy 3aMKHyTOMY )l(Op;:i:aHOBY crrpHMJIHeMoMy KOHTYPY y E D.6.40.
,Ll;OKa.3aTh, qTQ CJJe;:i:yIOIUtte cpyHKUI.fH He HMeIOT rrepB006pa.3HbIX B YKa3aHHhIX o6Jiacrnx:3) (sin z)- 1, 0 <I z I< n;14) (cos z-1)- , 05) P(z) I Q(z) B o6JiaCTH I z I< R, lz -<I z I< 2n;a11>11, j = 1, 2, ... , 11, r;:i:eP(z) tt Q(z) - MHoroqJJeHhI, He HMeIOrutte o6mHX MHQ)l(ttTeJieH:, a 1, a 2,... , a11-Bee HYJIH Q(z), rrpw-IeM 06JiacTH lz - a11<'1•j=1, 2, ... ,11,rrpHHaJJ:]]e)!(aT o6JiaCTH I z I< R.6.41. IIycTh cpyHKU1urf(z, t) aHaJittTttqHa B OJJ:HOCBH3HOH o6JiacTtt,co;:i:ep)!(aweii TO'IKY z 00. ,Ll;oKa3aTb, qTo cpyHKUI.fH HMeeT B yKa3aHHOH o6JiaCTH rrepBoo6pa3HYIO TOr;:i:a I1 TOJihKO Tor;:i:a, Kor;:i:a=limzf(z)=O ..::~ool1HTEfPl1POBAHl1E <PYHKIJJ1H.
l1HTEfPAJibHA51 TEOPEMA KOWl1796.42. ITycTb D - OJlHOCB5!3Ha5! o6nacTh, He conep)l(arua5! ToqKyz = 0. ,UoKa3aTb, qTO cpyHKU115! Inz = f:dtIt5!BmreTC5! nepBoo6pa3HOi1)lJI5! cpyHKU1111 z-' B o6nacm D.6.43. ,UoKa3aTh, 4TO ecn11 IlYTb l1HTerp11poBaHl15! He npoXO)ll1Tqepe3 ttaqarro Koop.u11ttaT, TOf:-d~ =1nI~r. + l·2k1l z = re irp+ 1m'f'''m 't'nrnBHoe 3Haqemre apryMeHTa, k - ueJioe q11cJio, YKa3brnaroruee,CKOJibKO pa3 IIYTb 11HTerpI1pOBaHH5! o6XO)ll1T Haqarro KOOp,U11HaT.6.44. ITycTb D - O)lHOCB5!3Ha5! o6nacTb, He co)lep)l(arua51 ToqeKz = ± i. ,UoKa3aTb, qTQ B o6JiaCTI1 D cpyHKUI15! arctg z = r:d~2Jo 1+ ~5!BJI5!eTC5! nepBoo6pa3HOH .UJI5! cpyHKUI111f(z)= 1 I (1 + z\6.45.
IToKa3aTb, qTo ecn11 IJYTb He npoxon11T qepe3 TO~y +i, TO' -_ + k1l, k - 0, +_ l,+- 2, . . . 'i1__!JJ_1 +~- 41l-0rne qJ1CJIO k 3aBl1Cl1T OT Bl1)la IlYTl1.6.46. ITycTb cpyHKUI15!f(z) aHarr11T11qHa B nonoce 0 <Im z <a 11 B3TOH 06nacT11 lim f (z):;:-70011HTerpana f_:f(z)dze::= 0.,UoKa3aTb, qTo 113 cyruecrnoBaHH5!cJienyeT TaK)l(e cyruecrnoBaH11e HHTerpaJiaf(z)dz, a E (-a, a) 1111HTerparr He 3aBJ1Cl1T OT a.6.47. ITycTb cpyHKUI15! f(z) aHa.JIHTJ1qHa B 06JiacT11 larg zl < a 11lim z f (z) = 0, lim z f (z) = 0, larg zl <a. ,UoKa3aTb, qTO 113 cyruecT:-+o:-+=BOBaHJ15! HHTerpaJiar'°j (x) dx CJie)lyeT cyrueCTBOBaHl1e I1HTerpanaL g:=a f (z)dz, a E (-a, a) , I111HTerparr He 3aBHCl1T OT a.6.48.
,[(oKa3aTb J/e)1wy JKopoaHa:1) ecn11j(z) - tterrpepbrnHa5! cpyHKUI15! B o6Jiacm lzl ::'."'.:Ro, Im z ::'."'.:a,limf(z)=O , TO .u1rn mo6oro /11 > 0 limf. e;111: f( z) dz =O, r.ue YR-:~ooR-+oo YR.uyra oKp~Hocm lzl = R, Jie)!(arua51 B paccMarp11BaeMoii o6Jiacm;I'J1aea 6802) Ecm1fiz) - tterrpephrnttasr cpyttKUI151. B rroJiymmcKocm Re z 2'. 0,lim f (z) = 0, TO .UJIH mo6oro t < 0,:~oome YR- .uyra oKp)')Kttocrn Iz I= R,6.49. Bb1q11CJil1Tb 11HTerpanpan ITyaccoHaJ:Re z 2'. o.f; e-x' cos 2bx dx' 11CITOJib3YH MHTer-e'' dt =.Jii.6.50. Bhlq11cn11Tb wa11e2paJ1bZ <PpeHenfl:f0+- cosfa- sin x dx.x 2 dx ,2r~ sinx6.51.
BL1q11cJIMTb uHme2paJ1 ,l{upUX!le: J, - - dx.x0.6.52. Bhl'IMCJIMTh MHTerpan-sm111+10xixdx, n E N.6.53. BL1q11cJI11Tb MHTerpan22r-sin "ax-sin "/3x dJ,,flX,x0aJJ=F-0•6.54. Bh1q11cJIMTh MHTerpanJ;( Jdx,si:xIZEN.6.55. I1oKa3aTb, qTO rrp11 a > 0r- cosx 1 __ ..!!_ _",, c.x e .J,0x- +a-2a6.56.
I1oKa3aTh, qrn rrp11 a > 0, b > 0i-cos2ax~cos2bx dx=tr(b-a) .0x6.57. IloKa3aTb, qTo rrp11 a> 0, b > 0ro- e-ay_ye-bybJ, ---dy=ln-.a11HTErPl1POBAHl1E <DYHKU11H. 11HTErPAJ1bHA5l TEOPEMA KOWl16.58.81Bh1q0cJIHTh HHTerpaJI22r-cos "ax-cos "/Jx dJ,x0·Rx, ap* O.6.59. Bh1q0cJIHTh HHTerpaJI- cos 2"+ 1ax - cos 2"+ 1/Jxdx, a(Ji:-0.0xf6.60.
IIoKa3aTh,qTQ npHb > 0, m ;:: 06.61. IIoKa3aTh,qTO nptta > 0, k ~ 0r-xsin ax dx- 1 -ka,,--n:e .x- + k2J,06.62. IIoKa3aTh,qTQ nptt1_x(xsinm~+ a-)026.63. IIycTh Re z > 0.fa > 0, m > 0dx=_!!_- 1re22a"(m+~).4a 3aIIoKa3aTh, qTQ-e- 1 -e-1:Jo6.64.4111tdt =In z.Bh1q0cJIHTh HHTerpaJI6.65. Bh1q0cJIHTh HHTerpaJIr- e"cosbx sin (a sin bx) dx.Jox6.66.
IIpeo6pa3oBami:eM<PypheJ(q)cpyHKUHHHHTerpaJI~f e_;, .; f(x)dx:= f(q).".J2TC -f(x)Ha3hrnaeTc~I'Jzaea 682.[loKa3aTb, qTo rrpeo6pa30Bam1eM <Pypbe cpyttKr.urnf(x) =e-"x', a> 0,5IBJU!eTC5I cpyHKUH5If(f) =-1-e-~' '4" .J2:iCpaBHHTh c rrpHMepoM 6.49 .I'!laaa 7IIHTErPAJihHAJI <I>OPMYJIA KOIIIII.IIHTErPAJI TIIIIA KOIIIIIIIycTb D - orpaHHqeHHa5I o6nacn Ha C c KycoqHo-rna.uKow rpaHHU:eH l (D MO)KeT 6bITb H MHOI'OCB5!3HOH 06JiaCTblO ), cpyHKU:H.5! .f(z)aHaJIHTMqHa B D II HerrpepbIBHa B DHHTerpa111Ha5I cpopMyJia Korrrn=Dur._ l ~ J(;)d; ={f(z),27ri r0,;-zToma crrpaBe,UJIHBaZED,zE C\D.Ec1111: 0611acTb D .5IBJI.5IeTC.5I O.UHOCB.5I3HOH M 3aMKHYThIH KOHTYP)Jfop.uaHa y co.uep:)l<MTC.ll B D, TO TaK>Ke crrpase.u1111:sa cpopMyJia Korrrn_1_~ J(i;)di; ={f(z),27ri r r; - z0,zE inty,z E !it int yrrpH yc.rroBMH, qTo cpyHKU:H5Ij(z) aHarrHTH"'IHa B 0611acT11: D.IlyCTb l - KycoqHO-rJia,UKa.5! KpJIBa.5! )l(op,UaHa (f - He 06.5I3aTeJioHO 3aMKHYTa.5I KpHBa.5!) H 3a,UaHHa.5I Ha l cpyHKU:H.5! f(~) .5IBJI.5IeTC.5IHerrpepornHoH: .
.[(11.5! 11106oro ztf. rorrpe.ue1111:M cpyHKU:HIO F(z) nocpopMy11eF(z)=-l ~'J(;)d;2nirr;- z '.5IBJI.lIIOIUYIOC5l 0.llH03HaqHOH cpyHKUHeH. 3zy cpyttKU:HIO H33bIBaIOTumne2paJ10J11 muna Koiuu. MttTerpa111tta5! cpopMy11a Korrrn .umaHaJIMTMqecKOH cpyHKUHM .5IBJI5IeTC5I qaCTHhIM CJiyqaeM MHTerpaJia TMrraKorru:1. CrrpaBenmrna c11e.uy10ma51 TeopeMa.TeopeMa. B .JII06oH: TQqKe z ticrHHTerpaJI THIIa Konrn 5IBJT5IeTC.lfaHaJIHTJi!qecKOH cPYHKL{Ji!eH, KOTOpa5I HMeeT rrpOH3BOL(Hhie mo6ororrop5I,llKa 11 crrpasen1111sa cpopMyJiaFr">(z)=~thJ(;)d;27ri 'fr er;- zr+''n=l 2 ....' '84Jib :noi1 TeopeMhI CJie;::i:yeT, qrn aHaJJ11T11qecKa5I B 06JJacT11 D cpyHKU115I5IBJJ5IeTC5I 6ecKoHeqHo ;::i:11¢¢epeHu11pyeMoi1 B o6mcrnD,11 crrpa-se;::i:mrna cpopMyJJaf <">(z) =~,+, J(;)d;2Jri 'fr ci; - z)"+I 'r;::i:e z -rrpo113BOJlbHa5I TOqKa B o6JJaCTHTaKOB, qrn intD H 3aMKH)'TbIH KOHT)'p yy c D 11 z E int y.EecKoHeqHa5I;::i:11¢cpepeHu11pyeMOCTbaHaJJHT11qecKoi1cpyHKUH11II03BOJJ5IeT ;::i:oKa3aTb TeopeMy Mopepa, 5IBJJfilOIUYJOC5I B HeKOTopoMCMbICJJe o6paIUeHHeM TeopeMbI Korrm.TeopeMa.
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