Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 33
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I1peo6pa3osamref ttnh6epra 06naL1.aeT cneL1.y10mttMH csoiicTBaMtt:1) H 2 =-I;2) H ocymecTBmieT H30Mopcp113M lf(R) Ha lf(R), p > 1, H crrpaBe.D.JIHBO HepaBeHCTBO<f] u jPdt ~ [J Hu j Pdx~c; [Ju IPdt;3) B cnyqae p=2 HMeeT MeCTO paBeHCTBO[J Hu 1 dx =[.Iu f dt .j.[(IDI Ll.eMCTBJ1TeJibH03HaqHOH cpyHKUHttf(x), x ER, HMeIOrueH: KOMrraKTHbIM HOCHTeJib Ha R+= {x: x > 0} H 5IBJI5IIOIUeHC5I 6eCKOHeqHo.u11cpcpepeHu11pyeMOH Ha R+. orrpe.uenHM cpyHKUHIO KOMITJieKCHOrorrepeMeHHoro z no cpopMyneMf (z)=J;x:-If (x)dx.c:DyttKUH5I Mf(z) 5IBJrneTC5I attan11T11qecKoi1 Ha BceH: KOMITJieKcHoi1nJIOCKOCTH. I1peo6pa30BaHHe, ocyrueCTBJI5IeMoe 3THM HHTerpaJIOM,onEP AUHOHHOE HC4l1CJIEHl1EHa3oBeM npeo6pa3oeaHue.111 MeJ1uHa.3rn213npeo6pa30Batt11e paenpo-CTpamreTcH Ha Bee npocTpaHCTBO L\R+) KaK orpaHH'IeHHbIH JI11Hei12Hb1i1: onepaTop 113 L\R+) B L (Re z1/2). f1p11 3TOM M C)'Tb 1130MeT-=pm1 Me'/K,ll)' 3Tl1M11 npocTpaHCTBaMH 11 cnpaBe,llJil1BO paBeHCTBOf;I2f(x) l dx =fReo=inl Mf (z) f I dz 1-06parnoe npeo6pa30BaH11e Men11tta BBOlll1TCH no cpopMyJief (x) = fRe:= 112 x-:Mf(z) dz.15.1.ITycTb cpyHKUl1Hf(t) 11MeeT noKa3aTeJib eTenett11 poem s0 •,lJ.oKa3aTb, qToF(p)cxo.u11Te5! B 06JiacT11 Rep > s0 ,= f;e - pia .llJIH mo6oroexoL(HTeH paBHOMeptto B 06JiacT11 Rep15.2.ITycTb cpyHKUHHf(l)dts > s03TOT HHTerpaJI2'.: s > s0 .f(t) 11MeeT noKa3aTeJib eTenett11 poem s0 •,lJ.oKa3aTb, qTo ee 11306pa)l(ett11e JianJiaca F(p) attaJI11T11qtto B nonynJJocKocTu Rep > s 0 .15.3.
f1oKa3aTb, qTo 11306pa'IKeH11e JianJiaca F(p) -> 0 np11Rep-> + oo. B qacTHOCTH, eeJI11 F(p) attaJIHTwrna B ToqKe p = oo, TOF(p) 11MeeT B 6ecKotteqttocT11 H)'Jib.15.4.ITyc"fh cp)'HKU11Hj(t)= 0 np11 t < 0 11 np11 tteKoTOpOM p0E CJ; e-Po' f (t)dt. ,lJ.oKa3aTb, qTo .llJIH mo6oroRep> Re Po HHTerpM J; e_,,, f (t)dt exollHTeH. 3TOT HHTe-cy~ecrnyeT 1rnTerpaJIp EC:rpan, KaK OTMeqaJIQCb B Haqarre rJiaBbl, np11HHTO H33bIB3Tb npeo6pa-308GHUe.M Jlm1J1aca cPYHKU1111f(t).15.5.a" i- 0,ITycTb f(t)/1= 0 npw t < 0 11 f (!) =a,,t"+ a,,_/'- 1+ ...
+ a t + a10 ,E N, t 2'.: 0. ,lJ.oKa3aTb, 'ITO f(t) 11MeeT noKa3aTeJib cTeneHHpoem, paBHb!H H)'Jl!O.15.6.CY11TaH, YTO q))'HKUHH-op11nrnanf(t)= 0 np11 r < 0, Bb1q11e-JJl1Tb noKa3aTeJib cTenett11 pocmf(t) 11 .llOKa3aTh yTBep'IK,lleH115!:1) l-7-p- 1; 2)ea1 -7-(p-a)- 1,Rep>Rea;+ 1) , v > -1, Rep> 0 ;3) t ' + r(v,-+ipI'JZaea 15214· n!4) t " -;---p"+I '5) sinUJt+8) ch il.t +N , Rep> 0 ;,OJ, ,Rep>llmOJI;,p,Re p > IIm OJ I;p-+ OJ-6) cos OJt +7) shil.t+ilEp-+OJ 2ii., , Rep>IReill;p--k,/p -ii. 2,Rep> I Reil I-15.7. IlycTh F/p) + f/t) , Rep> sj, j=1, ...
,11 ..D:oKa3aTb, 1ITO"L,~=tajF/p) + "L,~= 1 aJ/t) , Rep> maxsrl $j$113TO CBOHCTBO 11306pa)l(eHJrn Ha3bIBaeTCH JlU!-leUl-lOC/11b/O.15.8. ITycTb F(p) -o- f(t), Rep> s 0 ; ,ll,OKa:mTh ,LJ:TO TOr,ll,a±F(~)+ f(at) , a>O,Rep>as0•15.9. IlycTh F(p) -:- f(t) , Rep > s 0. .D:oKa3aTb reopeMy 3ana3L{hIBaHl1H, T. e.t<T, T>0,t >T.15.10. IlycT.b f' (t) 11 f(t) 11MeIOT noKa3aTeJih cTerreH11 poem s0 11f(t)-:- F(p) , Rep> s0 . .D:oKa3aT.b, 11roJ'(t) + pF(p)- f (0), Rep> s0 .15.11.
IlycTh f <"l (!) , ... ,f (I) (t),f (t) 11MeIOT noKa3aTeJih cTenett11poems0 11 f(t) -:- F(p ), Rep > s0 . .D:oKa3aTb, 'ITO1pp:.co)}·1f <"i (r) +{F(p)- J(O) _ 1 <>;0) - ... - 1 <"-p15.12. IlycThj(t)-:- F(p), Rep> s0 . .D:oKa3aT.b, 11rn:OCTEPAUHOHHOEHC4HCflEHHE21511) f~f(T)dT+p- F(p),Rep>s0 ;2) f~dt,f~' dt1 ..•f~"-' f (t,)dt,,Ji (t) + F; (p ),15.13. IlycTb+p-" F(p), Rep> s0 .Rep> s,; / 2(t) + F1 (p ), Rep> s2.,ll.oKa3aTb, <ITOf~ J; (T)f2 (t - r)clT + F; (p )F1 (p), Rep> max(s" s 2 ).15.14.
IlycTb F(p)--:-- fit), Rep> s0. ,ll.oKa3aTb, qrn:1) F 1 (p)--:-- - tj(t), Rep> so;2) P")(p)--:-- (- l)"t''J(t), Rep> so.15.15. IlycTb cpyHKll)111 f(t) 11 t - '.f(t)Rep > s0 . ,ll.oKa3aTb, qroHMeIOT fIOKa3aTeJib CTenem1poem s0 ,r1f(t)+f=F(q)dq, Rep>s0 •p15.16. ITycn.f(t)...;...
F(p), Rep>s 0 • .D:oKa3UTb, 'ITO:F(p +A)+ e-h f (t), Rep> s 0 - Re A..15.17. IlyCTbf1(t) -:- F(p), Rep >So,CTeneHH pocTa s, 11 Sz,cpyHKU:H11CTBeHHO fIOKa3aTeJIH1 fa+i=- . a-;"' F; (z)F2 (p - z)dz +2mHMeIOT COOTBeT-J; (t)/1 (t),a> s 1, Rep >s2 +a.15.18. CtmraH, qTQ cpyHKU:HH-OpHrHHa.Jl f(t)r.[(e3aTI, cne.L\yIOurne yrnep)!(.L\eH11H:1) t " e"' + (n! )"+' , R e p > Rea;p-a.2)tsmOJt+2pOJ,, , , Rep>IImOJI;(p-+OJ-)-3) tcosOJt+ p:-OJ: ,Rep>IImOJI;(p-+OJ-)= 0 np11t < 0,.L\OKa-Dwea 152164) /''sin wt+, , Rep> Rei!+ IImm I;m,(p -At+ m-.p-A, Rep> Rei!.+ IImm I;(p-i!t+m 2sin mt 7lp .6) - - + - -arctg-,Rep>IIrnml;t2m5) eA' cos wt+7) Isin mt I :e-m8) __,_.J7ii.??m?p-+ m1~p+a'cth p7l, Rep> !Imm I;2mRep>-a .15.19. IIycTb <l>(p) =~re -z2dz - TaK'\f Jl 0OULU60K.l)if;yHKLJU51.,I(oKa3aTb, qTo:e_0111+ ~ eP1 1<401)(i-<I>(;:)}2) e-2"Ji /(Jill+ )p e"' [I -<I>[ Jp)1P3)Ha3nIBaeMa~Fap~p+a+4) p-le -afP +l<l>(h/ra);1-<l>(_!!:_)·2-Ji15.20.
IIycTn F(p) - cpyHKUI1~, amumT11qecKa~ B TOqKe pee JIOpaHOBCKOe pa3JIO:JKeHtte B 6eCKOHeqHOCTI1 I1MeeT BI1}.J,F( )P'\'~= L..i11=1c11p-11= oo,11·,I(oKa3aTh, qTo cpyHKUI1~ F(p) eCTn 11306pa:JKeH11e opttrnHana{'\'~ ~t"j(t) = L..iu=O n!0,'tz 0,t <0.15.21.
<l>yHKLJueu Eeccem1 J,,(z) nepeozo pooarrop~}.J,Ka3hrnaeTrn pernem1e m1cpcpepeHu11aJibHOro ypaBHemrnn E Z Ha-OCTEPA~HOHHOEMC4HCflEHME2172 d1wdw2zz --, +z-+(z -n )w=O,dz-dzKOTOpoe rrpe,nCTaBJI5!eTOI CTerreHHb!M p5!,nOMJ (z)11=(z 12)" L=k=O(-1/ (z I 2)2'k !f(ll + k + 1),IJ:oKa3aTh, qrn:1) J 0 (at) +3) J (t)+Ja2+1 p2 ;(~ -II~ p2 +l15.22. Ilycn, F(p)2) J 0 (2Ji) + p- 1e_P-1;p)"= A(p)B- 1(p)-pauttoHaJihHa5!cpyHKUH5!, rrpttqeM cTerretth arrre6pa11qecKoro MttoroqJietta A(p) Metthrne cTerrettttaJire6pa11qecKoro MHoroqJietta B(p), ttMe10mero rrpocThie Kopttttp,, p2, ..• , P11 E C.
)loKa3aTh, LJTO opttrnHaJIOM F(p) 5!Bm1eTC5! cpyHKUtt5!f(t) = """.L..k=115.23. IIycTh f(t) -;-F(p).A(p,) eP•'B'(pk))loKa3aTh, qTO lim pF(p) = f(+O),ap->~ecJitt cymecrnyeT limt-t+oof (t)=f (00 ) ,TO lim pF(p)p~O=f (00 ).15.24. IIycTh F(p) - pauttoHarrbHa5! cpyHKUH5!, aHaJIHTttqecKa5! B> ao, ,nJI5! KOTOpoii: BbIIIOJIHeHbl CJie,ny10mtteIIOJIYIIJIOCKOCTM Re pYCJIOBM.SI:[J F(a + ib) Idb cxo,nttTC5! ,nJIH mo6oro a> a0;Jim max IF(p) J= 0, r,ne r R - .nyra oKpy)J(HOCTH JpJ = R,R->+= pEr"1) ttHTerpa112))J(ama5! s rroJiyrr11ocKOCTM Re pJie-> a 0 . )loKa3aTh, qTo opttrttHarroM ,nJIHcpyHKUHH F(p) 5!BJI5!eTC5! cpyHKUH5! f(t)=l:kres[F(p)e"'], rnep,HYJIM 3HaMeHaTeJI5! cpyHKUMM F(p).15.25. IlycTb F(p) = A 11 (p) I B111 (p),r,ne A 11 (p) H B111 (p) - aJire6pattqecKtte MHOroqJieHhl CTerreHM 11 H Ill COOTBCTCTBeHHO, rrp11qeMn<11111 A"(p) HB111(P) He ttMeIOT 06m11x Hy11ei1.
)loKa3aTb, qTo ecJIHDwea 15218pi, p2, ... , Pt - Hy1m B111(p) KpaTHOCTM mi, m2, ... , m1 cooTBeTCTBeHHO, TO)-fL,, (F( )-'-f(p .t -l1) Id'11k-1Ill• c/ k171k -k=I-I[(pp-pk)111'F( ) pt]p e p=p,•15.26. Hai1:T11 mo6pa)l{eH11e F(p) .ZJ:JUI cpyHKu1111f(t):l)f(t)= tp, ~ >-1;2)f(t) = t"e i', n EN, ), EC;= t" sin mt, 11 E N, m EC;4) f(t) = e i.r cos mt, 'A, m E C;S)f(t) = (n + l)h, t E (n r, (n + 1) -r], n = 0, 1, ... , r, h > 0.3)f(t)15.27. HaH:nr H306pa)l{ett11e Jiarrnaca F1i(p) cpyHKUHH1f,(t)={h -,0,IO<t<h,t?:. h.,,_,+0 F;, (p) . 3TOT rrpe.z:i:en Ha3bIBaIOT npeo6pa3o6aHueM JlancpyHKUHH o(t) .
.[J:OKa3aTb, qrn:HaH:nr liml/QCQ1) o(t)-:- l ;2) 3<"l(t)-:- p", 11 EN.15.28. Hai1:T11 op11rnHanj(t) mm cpyHKUI111 F(p):1) F(p)=p-a-i, -1<a<O;= p - 1e-a.fP, a> O;F(p) = p-" - 1e-"- 1 , 11=0, 1, 2, ... ;2) F(p)3)4)F(p)=(p 2 +l)-Y2.15.29. 8bl'HICJ111Tb 11HTerpanbr:. ,;rc1) f" +r= _e_ d{2) f" +i= _t_· d{a-iao ~11 +l{1-ioo ~n +l'e,;r.4) f" +.i= -,-d;; 5)11-1= ;-+ 1'ft1+1·-- -'=';:e,;'l ;:.,- (if::,,11-1=;-+ 115.30 . .[J:oKa3aTb, qrn ecmif(z)---+ 0 rrpw Im z---+ oo, a 1 <Re z < a 2,Hj(z) attan11rnqHa B IIOJIOCe a,< Re Z < a2, TOCHT OT a, a,< a< a2.r:;: f(;)d;He 3aBl1-OCTEPAUWOHHOEHCYHCflEHHE21915.31.
PaccMOTpHM 3aOal/y Koiuu{a0y (11) ( t ) +a 1y (11 -I) ( t ) + ... +a11 y ( t ) =0,11y(O) =Yo,)/'> (0) = Y1, ... , i -o (0) = Y11-Pr,ne ao, a,, ... , all, Yo. Yi. ... , Y11-1 - 3a,naHHhie rrocTO~HHbie.EcJm { \jh(t)}, k = 0, 1, ... , /1 - l - cpyH,naMeHTaJibHa~ CHCTeMa peweHHH, T. e.a0 \jli">(t)11/j) (0)'t' k+ a= ()k;.
=11 11\j/i - >(t) + ••• +a,,\jlk (t)l'{ 0,k••= 0,= }"'k-:1=1;k,1=0,l, ... ,ll-l,TO, Kai< H3BecTHO, peweHHe 3a,Uacm KoWH HMeeT BH.n y(t) = L:~:~ y, \j/, (t).Dyen\jlk(t)-T\jlk(p),P,, (p)= a0p 11 +a1p"-'P.k ( P ) --aop 11-(k+l) +a,p 11-(k+2)+ ... +a 11 ,+ ... +a11 -(k+ IJ'.D:oKa3aTh, qTo:_Pi,(p)-.1) \jl.(p)---,p-0, 1, ... ,11-l,P,,(p)2) \j/,, (t)="IllLi i~o res ( e /JI1'1rne pj,j= 1, ..
., m, -Pi,(p)) ,P,,(p)HYJIH rron11H0Ma P11(p).15.32. PernHTh 3anaqy KournylV + 2y" + y = 0,y(O)=y'(O) =y"(O) = 0,y"'(O)= 1.15.33. PewHTh 3anaqy KowH2x"(t) + f... x(t)x(O)= a,=0,x'(O)f... f. 0,= ~·15.34. PaccMoTpHM 3a,r:i:ai1y KowHa 0y<"l(t) +a,/'- 1>(t) + ...
+a"y(t)y(O)rne a 0, a 1,•• .,=f(t),=y'(O) = ... =/'- 'l(O) = 0,a 11 - 3anaHf{b1enocTO~HHhie.220I'Jzaea15ITycTb y(t) -:--Y(p),f(t) -:--F(p), a \jf,,_ 1(t) ecTb (n - 1)-.si cpyHKUH.51 H3cpyH.UaMeHTaJibHOH CHCTeMbl perneHHH B 3a.uaqe 15.31 - :ny cpyHKUHIO Ha3bJBaJOT rPYHKLfUeu eOUT-lll'IT-1020 mo'le'IT-1020 UC/110l/T-lUKa.,UOKa3aTb, qrn:= F(p)1) Y(p)P,, (p)'2) y(t) = 1- f' \jf 11 _ 1(t-r)f(r)dr,ao101111-Ir.ue P ( p ) =a0 p +a 1 p + ... +a,,.1115.35. PernHTb 3a,Ua'-ly KornHy"(t) + y(t) =sin t, y(O) = y 1 (0) = 0.15.36. PernHTb 3a,Ua'-ly Koum:1) x" (t) - 3x 1 (t) + 2x(t) = 2e 3', x(O) = 0, x' (0) = O;2) x" (t) + 4x' (t) + 4x(t)13) x v(t) + 2x,,11=t3e-2',x(O)(t) + x(t) =sin t, x(O),4)x(t)+w-x(t)== 1,x' (0)=2;= x'(O) = x''(O) = x'"(O) = O;{sint,O<t<n,,0,t > n, x(O) = x (0) = 0.15.37.
PenmTh cHcTeMbI .UHcpcpepemwanhHbTX ypasHeHHH:x'(t) + x(t)- y(t) = e',l) {y'(t) + 3x(t)- 2y(t) = 2e',x(O) = 1, y(O) =l ;X8(t) - x(t) + y(t) + z(t) =0,y" (t) + x(t) - y(t) + z(t) = 0,2){= 0,y(O) = y'(O) = z(O) = z'(O) = O;zu(t) + x(t) + y(t)- z(t)x(O) = 1, x'(O)= 0,x(t)+y(t)=I{1'0,0<t<1,y(t)+x(t)={1'0,O<t < 2,3)ft > 1,t > 2,OCTEPAUHOHHOEMCYHCflEHME221x(O) = y(O) = 0;x'(t) = y(t) + z(t),y'(t)=z(t)+x(t),4){=x(t) + y(t),z'(t)x(O) = -1, y(O) = 1, z(O) = 0.S) {x'(t)-ax(t)-by(t)=be"',y'(t) + bx(t)-ay(t) =0,x(O) = 0, y(O) = 1.15.38.
Penrn:Tbl1HTerpanbHhie ypaBHeHl15I OTHOCHTeJihHO Herrpe-c:p(t):pbIBHOH cpyHKU:l1l11) rp(t)= f~U-4')rp(4')d4'+sint;2) t = f~e 1 -~rp(~)d{15.39. PernttThttHTerpanhHhreTeJibHO HerrpepbIBHOH cpyHKU:l1l11) f(t)=BoJihTeppaOTHoc11-f~K(t-4')rp(i;)di;;2) rp(t) = f (t)3nrypaBHeH115I<p(t):+ f~ K (t -i;) rp(i;) di;.ypaBtteHH5I Ha3hIBaIOTC5Iypa61ieHU5/Mu Bo.nbmeppa nep6020 uemopo20 pooa COOTBeTCTBeHHO.15.40. PernttTb ypaBHemrn B <1acTHhIX rrpott3BO)J,Hhrx:l){u= uu, 0<x<1, t > 0,11ui{4)ulr=O= 0,u .u + uII -- 0 '{ u \ = sinxt t=O'll ..{D+ull= 1,2u\..= = sin wt;1-= < x < oo '-= <x<oot'> 0 ' ul 1=0 = x 't > 0'u\f:;::Q=0'-Xu 1 \ 1=0 =x e .15.41.Hmr1u11 =ur,=u1 \_=0,.\.
O<x<l, t.>0, ui_I -01- 02)3)uL=o= uL= = 0,= sinnx ' u1 \t=O = o·'t=OHaiirn orpaHH<IeHHhie rrpHx 2: 0perneHH5Iu(x, t)ypaBHe-222I'Jzaea 15u1 =2a u~.t'x> 0, t > 0, u1 li=O= 0, ulx=O=15.42. PelllHTh ypaBHeHmIa4i1 a~ u,1) - 4 - - 4 = 3xr, 0 :S x :S 1, t > 0,axat3u(x, 0) = x , ur(x, 0) = Uu(x, 0)4u(O,t)=taa,f(t).= Um(X, 0) = 0,u(l,t)=l;aa' +3--u-+3~=x 2 e-'2) ~+-L_t333axu(x, 0)3at at= ui(x, 0)3axat 2'at 3= uu(x, 0) = 0, u(O, t) = 0, x > 0, t > 0.15.43. PelllHTh ypastteHmr:?1) Liu = a-u.rn a > 0, x > 0, t > 0,ul1=0 = 0, urlt=o = 0, ux(t, 0) = Vo(t - T), V #- 0, T > O;22) Liu= a Uxx + QU 1, a > 0, X > 0, t > 0,ulc=O = urlt=O = 0, ux(t, 0) = VO(t - T), V #- 0, T > 0.15.44.
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