Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 52
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f112m l~f=R,f2m l~f=R,;J(;); 111 d;, 1112'.0,C=const.lTaK K3K BTOpoM. HHTerpaJI paBeH Hymo, HMeeMaf f(R eirp) [ _e-i11rp__ + C R111+1ei(111+l)rp dm..IRllI't'=- l2n2II1r 0I±1IIonO)l(HM C = - - -1 , m + lR;'+m+B ncpaoM cn)"-iae a 11= n.HTaK,l2n2;rR;'0= a;. + ia;, = - - J(u + i z) 2 cos mp dcp .B o BTopoM cnyqae a11. nrp )d rp.= a, + ia.
,, =-r- 2Jllc u + iz. )C - 2·r sm11112trR;' oIT03TOMYll=--Ju(R 1e''P)cosnrpdrp= --J v(R 1e''P)sinnrpdrp,trR;';rR;'2na;,.2:r0012n1 2"a;,'= - - f v(R 1 eirp)cosnrpdrp = - - -trR;'.;rR;'0Ju(R1 eirp)sinnrpdrp.0I'Jiaaa 1515.6. 3) Qqestt,nHo, 'ITO noKa3aTem cTeneHH poem cpyHKUHH tv paseHHymo, IIOCKOflbKY Ha 6eCKOHelJHOCTl1Ilo3TOMY 113o6pa)l(eHHe Jlannacat" pacTeT He 6bICTPee MHOro•rnetta.fF(p) = t"e-P1 dt, p = p 1+ip2 ,0MO)l(HO paccMaTp11BaTb .nm1 p: Rep > 0 . .[{JUI Bbl'IHcneHIDI 3Toro HHTerpaJiapaCCMOTpHM cPYHKUHIO f{z) = z"e-z KOMrrneKCHOro rrepeMeHHOro z B o6nacTl1 DR, rpatt11ua KOTOpOH npe.ncTaBmleT 06be,[{11HeH11ePEWEHM51TMnlill.!HhIX3A,UAY338aDR =[0,R]u{z: lzl=R,O::o;argz::;a,tga= P?. ,p1 >O}uP1U{ z: z = ( p 1+ ip!. )t, t E [0, R]} .=<DyHKUl151 .f(z) aHaJI11Tl1'-!Ha B yKa:iaHHOH o6nacrn, a B TOlJKe z0 HMeeTHHTerp11pyeMyJO oco6eHHOCTh ( v > -1 ), !103TOMY !10 TeopeMe KowH 11ttTe-pf (z)dz = 0.
Ho HHTerpan flO rpatt11ue npe.ucrnB11M BrpaJI no rpaHHUeaoRs11.ue cyMMhl Tpex HHTerpa.rros:RRf t''e- dt +f10z'·e-:dz+ fCpt)"e-P 1 d(pt)=I1+h+f3.l:l=R0RO~arg :~aIlp11 R -++oo HHTerpan /?. crpeMHTC51 K HYJIJO, nocKOJihKY .UJI51 f!O.!lhIHTerpaJihHOH cPYHKu,1111 cnpase,LlJil1Ba oueHKaIz"·e-:j ~I z j"e-1~ R''e-R. YcT-peMHB R -++oo B HHTerpanax / 1 11 ! 3 , 113 flOCJie.[lttero paseHCTBa nonyq11M:Jt"e- dt- p"+I Jt"e-P dt =0.1100Cne.uosaTeJihHO, F(p) f'(v + I )/p"+ 1, '-ITO 11 -rpe6oaanoci, noKa3aTb.KaK <iacTHhTH cnyqaw :noro np11Mepa paccMOTPHM cPYHKUHJO Xes11caw.L1aH(t). l13 :noro np11Mepa rne,LlyeT, 1no 11306pa)f(eH11e Jiamiaca cpyHKUH11Xes11cai1.L1a paBHO lip.5) PaccMOTPHM cPYHKUMJO sin wt 11p11 HeKornpoM KOM11JieKCHOM w. Tor.ua=.
wt 1-IsineIiwt_ -iwt I leicvt1+1e-i{t)tl-lmw1lm(J)fe<-- e+e<__ e11mwl1 .2i22lb 3TOH oueHKH Bb!TeKaeT, '-ITO 110Ka3aTeJih CTerreHl1 pocTa cPYHKU,1111 sin wtpaaeH IIm wj. I1o3TOMY 11306pa)f(ett11e Jiannaca F(p) MO)f(HO paccMaTpHBaTh.!lJI51 p: Rep> !Im wl. l1cfloJihJY51 np11Mep 3), rronyqHMF(p) = J sin(wt)e-P 1 dt2t0l f(l)2i p - iw=~J e-<p-iw) 1 dt-~ J e-(p+iw) 1 dt =l21 00f(l)(1)2i p + iw15.21.
<DyttKu11ei1 EecceJI51 J,lz) nepsoro po.Lia nop51.!lKa n E Z HaJhJBaeTC51 peweHHe .!l11cjlcjiepeHU11aJihHOro ypaBHeHl151., d 2 wz- - .,dz-dw.,.,+ z-+ (z-- n-)w =dzKOTOpoe npe,LlCTaBJI51eTC51 CTeneHHhlM p51.!l0M0.PEWEHllil nmvrlfHblX 3A.[(AlfJ,,(z)(!..)" f:=2339(-l)'°(z/2)2' .+ k + 1)k ! f(nk=Olbo6paJKem1e Jlarrnaca HMeeT su.n:i=J1 (at)e-p dt =-JJ (t)e-prladt.=10oao0=,Ulli! .noKa3aTeJibcrna 1) .nocrnT04HO paccMOTpeTb CJiytiai1 a1.
EcJIHrrpe.nrroJIOJKHTb, 4TO Re p ~ s0 > 0, TO p.si.n, rrpe.ncrnBlli!IOll.(HH cPYHKU.HIOEeccelli! J0 (t), MOJKHO rr04JieHHO HHTerpuposaTb. CJie.nosaTeJihHO,=fl()-prd_~C-d1·t et- L . - - · o ok!k=O22k1=f2k-prdt=· t ef'(k + 1) o-f (-1/ 1 f(2k+l) 1 - 1 f (-1/ l (2k)! 1 k!"2_2k" f(k+l) .k!" 22k "k!"=_!_ f (-1/ (k+l)(k+2) ... (2k) 1 = _!_ f (-l)k 1·2·3· ... ·2k l- k=Op2k+l --;; k=Ok!p k=O1=22kp2kI?k-.l=(-1/.zkkI•lk!p k=O(-l)k 1·3· ... ·(2k-l) 2-4- ...
·2k= -P k=O2:-k·p2k -k!22kp2kl·----v:=pll1=- 2:--·-·(l+-)· ... ·(k--)·----v:=p k=O k ! 222 p .f=_!_(-112)(-1/2-1) ... (-1/2-k+l) ·-1-=p k=Ok!p2k=_!_(1+-1 )-112=1pp2~l+ p2.15.27. 1) Jlerno BbI4HCJIHTb, YTO Fj, ( p) = ( e -pil -1) I( - ph) . 0Tc10.naCJie.nyeT, '-ITOlim Fj,(p)'1-HO= l. IlpH3TOM Mb! rrpH.neplKHBaeMC.s! CJie.ny10m11xorrpe.neJieH11.si H o6o3Ha4eH11.si. Pacrrpe.neJieHHe 8-<J:>yHKU.HH ,UHpaKa MOJKeT6b!Tb BBe.neHO CJie.nyIOll.(HM o6p330M:< 5,rp >I "= rpCO) = Iim -Jrp(t)dt,DJ'1->+ohOr.ne cjJyHKLl,H.s! cp(t) .s!BJI.s!eTC.s! 6eCKOHe'-IHO ):IHcjJcjJepeHLJ,HpyeMOH H CKOJibyro.ntto 6brCTpo CTpeMHTC5! K HYJIIO Ha 6ecKoHe'-IHOCTH.
JI106a51 rrpoH3BO.!lHa5! 8-cpyHKUHH MOlKeT 6b!Tb Bbl'-IHCJieHa no npasuny< 5<"), (/J >= c-1) 11DJf< 5, (/J(ll) > = c-1)" Jim -I " (/J(ll) (t)dr = c-1)" (/J(ll) co).'1->+0 /JOPEWEHH5! nmttlJHb!X 3A)J,AlJ3402) l'.bo6pa)!(eH11e Jiannaca F(p) 8-rpyHKUHH onpeaem1M no rpopMyne-fF(p) =< 8,e- p1 >= Jim <f,,,e-p1 >= Jim 1 " e- pr dt = 1.h-70h-7+0 h 0A 11306pa)!(em1e Jlarnrnca F 11 (p) n-11 npo113BOAHOH cpyHKU1111Jil1M cnet1y10m11M o6pa3oM:o (n)onpet1e-F,1 (p)=<8< 11 l ,e- P1 >=(-l) 11 <8,(e- P1 / 11 l >=<8,p 11 e-P 1 >=1"p= lim-Jh-7+0h0111e- P dt=p1"11Iim-fe- 111 dt=p'1-7+011,h0'ITO 11 rpe6oBaJJOCb AOKa3aTb.15.39. L(JUI peweHm1 3TOH 3aaaq11 npeanoJI0)!(11M, 'ITO npHMbie 11 06paTHb1e npeo6pa30BaH11H Jlannaca cpyHKU11H, sxonHm11x B ypasHeH11e, cymecTBYIOT.1) lb pe3yJJbTaTa peweH11H 3anaq11 15.13 MO)!(HO yTBep)!(t(aTb, 'ITOF(p)= cD(p)L(p) ,rt1e F, cD 11 L - 11306pa)!{eH11H Jiannaca cpyHKUHH f, <p 11 K cooTBeTcTBeHHO.IlycTb L(p) I 0 .
Toraa no cpopMyne Men11Harp(t)l x+i==- .21ltrt1e xJ eP,-x-100F(p)L(p)dp,> s0 , a s0 - noKa3aTeJJb cTeneH11 poem QJYHKU1111 <p(t).2) AHanOrn'lHO rrpellblllYll(eMy cnyqa10, pewem1e <p(t) ypaBHeH11H BTOporo pona MO')KeT 6b1Tb nonyqeHo rro cpopMynerp(t) = _I_ x+fi= ePI21li-x-1 00F(p )1-L(p)dp.15.43.
L(nH rrpOCTOTbl H3JlO)!{eHH5! rrpearrOJIO')KHM, 'ITO u(t, 0) = 0 H HaIIOMHl1M, 'ITO npeo6pa3oBaHHe Jiarrnaca 8-rpyHKUHH paBHO 1, a ee npeo6pa30BaHHe c 3ana3.Ub1Baa1-0ll(HM aprYMeHTOM pasHo e-pT, rae T > 0 - 3ana3AhIsaHHe. L(anee, npHMeHHM npeo6pa3osaH11e Jlannaca no nepeMeHHOH t H,Y'lHTbIBaH 3aaaqy 15 .11, rrony'lHM B 11306pa')KeH11Hx Jlarrnaca 3aaaqyp 2U(p,x)=a 2v:r<p,x),V~(p,0)= ve-pT, U(p,0) =0,r.ue U(p, x) - H306pa')KeH11e Jiarrnaca u(t,x) . 06mee peweHHe ypaBHeHHHAlli! U(p, x) HMeeT BHAU(p,x) = A(p)ch px + B(p)sh px.aaPEWEHl151Tl1011YHbIX3A,ll,A4341Y'-IHT&IBaH rpaHH'-IHbie yc11os11H np11x=0,11erno no11y'-IHTb, '-!TOB(p)=av e-Tp.U(p,x)=av e-Tpchpx.PaTaKHM 06pa10M,PA(p)= 0, aITo cpopMy11eMe1111HaVu(t,x)=~p 1+i=pt-pTJ _e_ _ ·ch px dp =p2m ,,, -i=aaV P1+i= ep(t - T +alx)+ ep(t-T-alx)=-.dp=2m p,-i=2pfav= -{ H(t-T +al x)+ H(t-T-al x)},2r.neH1a,na'-111eCTb cpyHKUHH Xes11cai1,na, 11 Mb! socrron&30BaJJHCb pe1yn&TaTaMl115.6 1).I'llaaa 1616.7.PacCMOTPl1M CHa'-!aJJa CJJYlfaH, KOr.na cpyHKUliHy6&rnaeT rrp11x?.
0. B TaKOMIIfa ffix)MOHOTOHHO Hecny'-!aek+l11-l11-lL: L f (x)d.x $ L: f (k + 1),(x)dx =k=OJ~' f (x)dx =k=O11-I11-Ik+lL: L f (x)d.x?. L: f (k).k=Ok=O113 3Tl1X .nayx HepaaeHCTB BbITeKaeT, '-!TOL f(k)- f(n) $ f f(x)d.x $ L: f (k)- J (0)."ll,,k=OOk=OIleperr11cas 3TO ,nBOHHOe HepaseHCTBO B Bl1,nef (0) $IIOJ1Ylfl1M, '-!TOltaIIllk=OOL f (k)- ff (x) d.x $f (k)-1 ff (n),(x)dxl $1 f (n) l+I f (0) 1. .[(JJHB03pacrnl0w:eH: cpyHKUl111 ,nOKa3aTeJJbCTBO aHaJJOrH'-IHO.16.13.11cxo,nHb!H l1HTerpaJJ MO)l(HO rreperrHCaTb B Bl1,ne:f (t)d: = r' f (t)- f (0) dt + f (O)ft ---5!!, =Jot2+£o t2+£2ot2+..,II'][ lt=Jo J'(;(t))-tdt+f(O)-arctg=/ +/t+££co12212.MOHOTOHHO He-PEWEHl15J THTIWIHbIX 3A)J,AlJ342TipH noJJy'-leHHH HHTerpa.na I, Mbl BOCnOJJb30BaJIHCb TeopeMOH JiarpaH)Ka.ITOCKOJlbKY cpym<I.(H51 fix) HenpepbIBHO .11.HcpcpepeHUHpyeMa Ha 0TPe3Ke(0, 1], nerno nony'-IHTb oueHKY .ll.Jl5! HHTerpa.na / 1:J +£11 1I::; max I J'(x) I /dt.rE[O,IJ2Of= max IJ'(x) I·(Jc.in (1+& 2 )-lnc) = O(j ln c I) ..rE[O.IJ12ffKpoMe Toro, ITOCKOJibKY arctg- ~ - , HMeeM, '-!TO npH £~ +0 cnpasennH-2c-Ho £Bo I 2 -f~O) -~.TaKHM o6pa.30M, OKOH'-!aTeJibHO ttMeeM, '-!TO rrpHe~+o= f(O) .!:.+o(ln £).f0' ~(t)d;r+c2E16.17.
PaccMOTPHM cpyHKUH!O F(z) Btt.11.aoo e-'1oo e-'F(z)=fz+t- d t =-f--dtz l+t/z00.ll.Jl5! KOMITJJeKCHOro ITepeMeHHOro z, Re z > 0, B OKpeCTHOCTH TO'-IKH z3aMeHHB z Ha s = l/z, paCCMOTPHM cpyHKUH!O g(s) BH.11.ac;) =gOOJ0= 00.e-f dtI+;tB OKpecTHOCTH TO'-IKH s = 0 nptt ycJJOBirn, '-!TO Re s > 0. Tenepb paccMoTpHM cpyttKUHlO <D(s) .11.ei1CTBHTeJ1bHOro nepeMeHHoro s Ha OTPeJKe [0, l]BM.II.a <D(s) = g(ss). <DyHKUH51 <D(s) HMeeT Ha 0Tpe3Ke (0, l] ITpOH3BO.ll.Hbiemo6oro rrop51.11.Ka, ITOCKOJlbKY-Ioo<D(k J(s)=(- 1/k!;kJ0e tk(1 +st;/+'dtH HHTerpa.n B rrpaBOH qacrn paBeHCTBa CX0.11.HTC51 paBHOMepHO ITO s E [O, I]H s, Re s > 0.
0Tcmna BbITeKaeT, '-!TO00f<l>Ckl (0) = (-l)k k !;k e- 1 tk dt= (- l)k k !;k['(k+I) = (-1/ ;k (k !) 2 .0Pa3JIO)KHM cpyHKUJUO <D(s) ITO cpopMyne MaKJiopeHa np11HbIM '-!JieHOM B cpopMe JiarpaH)Ka:<D(l)s= 1 c ocTaT0'-1-= <D(O) +<I>' (0) + <I>" (0) + ... + <l> c11J (0) + <l>c11+1J (B) , 0 < (] < 1.2!ilOCKOJibKY <l>(l)CTaBJieHHe= g@,a <l>(k)(O)n!= (-1/ (k!)(n+l)!2s\ TO cnpaBe.ll.JIHBO rrpe.n-PEWEHllil Tl1Dl1YHblX 3A,UAY= l/z,C.nenas 06parny10 3aMetty nepeMeHHOH ~(-ll k'11L411M paseHcrno F(z) =zk=Ok+l•+ (-1)11Ml! cPYHKU1111 F(z) nony-e-1!11+1=+343f1z + Bt) 110 (+2 dt, r.ne Re ze-1t11+Idtf (z+Btt+' = o(l/z-I+=z-oo, Re z > 0, (-1)' 1)J:anee, nocKOJJhKY np11> 0.n+l).0B TaKOM CJIYtJae .ll.Jil! <PYHKUI111 F(z) np11 z-oo, Re z > 0, nony411M paseHcrnoFCz)=112:1(-llk!+(-1)"+ (n+l)!k+lk=O11+2Z11ITocKOJJhKY rrp11z-oo,Re411M ac11MnTOT114ecKoe rrp11=z > 0, oz-oo,(-1) n!Jf-dt -Zz11+ 1(Ree-rO t+•Z1=(-1)"+ (n+ l)!z"+ 2, TO nony-z > 0, pa3JJOlKeH11e= (- l)k k !2:k+l,Zk=O4TO 11 Tpe60BM0Cb 1l0Ka3aTb.16.34.
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