Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 35
Текст из файла (страница 35)
e. S'(x0) = 0, S"(x0)A- H oo cnpasenn11sa cpopMyna"7'0, TO npttMETO,UhI f10JIYYEHI151 ACI1Mf1TOTI1YECKI1X OUEHOKF(J..)=ei[,\S(r-o0 nS'(r )][)+~s4 "'-o2""'"l229f(x )+O(J..- 1) .J..jS"(xo)IoMeTOll nepeBana. PaccM0Tp11M 11HTerpa.rr s1u1.aF(A)=fr f(z)e,isc:ldz,r.ue y - Kyco'IHo-rna.uKa.H Kp1rna.H )Kop.uaHa, a cpyHKu1111 /(z) 11 S(z)amurn:m'IHhI B HeKoTopoM: 06nacT11 D E C, co.uep)l(aw:ei1: y.
IIocKOJihKY cpyHKu1111f(z) 11 S(z) aHa.rr11m'IHhI B D, TO MO)l(HO .uecpopM11posaThKpHBYfO y B D, OCTaBJimI HenO.UBH)l(HhIM11 KOHUhl y H He Memrn 3Ha'IeHH51 HHTerpa.rra F(/....) . Ilpe.unoJIO)l(I1M, 'ITO np11 .uecpopM11posaH11ttMO)l(HO IlOJIY'IMTh TaKyfO KpI1BYfO y, 'ITO Bh!IlOJIHeHhl ycJIOBH.H:1) max Res ( z) .UOCTttraeTCH TOJihKO B O.UHOH TO'!Ke Zo E y' KOTO:erpa.H 51BJI.HeTC51 BHyTpeHHeH .UJI.H y;2) Im S(z)= const nptt z E y B OKpecTHOCTM TO'IKI1 z0 .IIycTh, KpoMe Toro, S'(Zo) = 0, S"(Zo) ~ 0, M B OKpecrnocTH TO'IKI1 z0KpHBa.H y npoxo.um 'Iepe3 o6a ceKTopa, B KOTOphIX Re S(z) <Re S(ZQ).Tor.ua np11 /....--+oo cnpase.un11so ac11MnTOT11'!eCKoe npe.11.crnsneHHeF(A)r.ueek == e;.s<::o> [2trJ..jS"(zo)If(z 0 )e;o, + O(J..-3n))'(n - arg S"(zo))/2 + kn, k = 0 MJIH 1 I1 BhJ6op 3aBI1Cl1T OT Ha-npaBJieHH.H HHTerp11poBaHI15! B.UOJih y.16.1.
IlycTh cpyttKUI151f(z) MMeeT aCI1MnTOTI1'IeCKOe pa3JIO)l(eH!1ef(-::.) - L~=oa,,z-",z ~ =.n ::::: 0 cnpase.unusa cpopMyna11-J-k•a,,= Iimz II [J (-: . ) - .....L,,,= 0 a,z .)J,oKa3aTh, 'ITO .UJIH Ka)l(.uoro: ~ oo16.2. )J,oKa3aTh, 'ITO eCJIM cpyHKUI15I j(x) !1MeeT CTeneHHOe ac11Mf1TOTW-leCKOe pa3JIO)l(eHI1e npH x --++oo, TO cpyHKUI151 f(x)+ e-xMMeeTTO )J(e CTeneHHOe aCl1MCTTOTI1'IeCKOe pa3JIO)J(eH11e.16.3. <l>yHKUI1H j(x) =e-xsin (ex) np11 x --++oo 11MeeT cne.11.yFOw:eeCTeneHHOe ac11MnTOTH'IeCKOe pa3JIO)l(eHFre:I'Jwea 16230e -x sin (e·') - 0, x _,+oo.Mmirno 1111 JJ.aHHoe ac11MITTOT11qeCKoe pa3JIO)KeH11e JJ.11cj:Jcj:JepeHu11poBaTb?16.4.
ITycTb cj:Jymmm1 f(t) 11 g(t) HerrpephIBHhI np11 t:::=:a , cj:JyHKU1151g(t) > 0 np11 t 2: to> a 11 J~ g(t)dt = +00 • )loKa'.3aTh, qTO 113 ycJIOBH51(/f(t) - g(t) np11 t --++oo cneJJ.yeT, qrn np11 x --++oof J (t)dt - f' g(t)dt.xaa16.5. ITycTh BhIIlOJIHeHbI yc11os1151 npeJJ.hIJJ.ynieH: 3aJ.(aq11. )loKa3aTb,= o(g(t))qTO 113 COOTHOIIIeHJ151 f(t)x -7+00ro([ g(t)dt),= O(g(t))a 113 COOTHOIIIeHH51 f(t)x---->+=f(t)dt =nptt t --++oo CJieJ.(yeT, qTQ np11rf(t)dt =np11 t -7+00 CJieJ.(yeT,qTQrrp11oc[ g(t)dt).16.6.
ITycTb cpyttKUHM f(t) 11 g(t) tterrpephIBHhI rrp11 t 2: a , g(t) > 0np11 t 2: t0 > a 11f' g (t) dt <oo.)loKa3aTh, qTo TO ma crrpaseJ.(JIMBhI{/CJieJ.(yIOIIIMe yrnep)KJJ.eHl151:1) ec1111f(t) - g(t) np11 t --++oo, TO np11 x --++oorf(t)dt -rg(t)dt;2) ec1111f(t) =O(g(t)) rrp11 t --++oo, TO np11 x _,+oorf(t)dt =ocr g(t)dt);3) ecmrf(t) =O(g(t)) rrp11 t _,+oo, TO rrptt x _,+oorf(t)dt =ocrg(t)dt).16.7. ITycTh cj:JyHKU1151 f(x) MOHOTOHHa rrp11 xnpH11:::=:0. )loKa3aTb, qrn-7+00L~= 0 f(k) = J~' f(x)dx+ O(f(n)) + 0(1).METO.[lhl nOJIYLIEHI151 ACI1MnTOTMLIECKI1X OL(EHOK23116.8. ,ll;oKa3aTh, qTo nptt n --++oo cnpase)l.JJHBhI cJJe.D.yIOmtte acHMIlTOTHqecKHe paBeHCTBa:1)L"ka -na+ik=I2)a=-1,Inn,{/(a+ 1), a> -1;Lile(lnki' -{ (lnn)P+I/(/J+l),k=2na+I(lnn)P/(a+l),a= -1, fJ > -1,a >-1, /J-mo6oe.16.9.
,ll;oKa3aTh, qrn nptt n --++oo cnpaBe)J.JJHBhI ouettKH:1) L~= 1 sinkxlnk = O(lnn);2) L~=l ka sin kx = O(na ), a~ 0,KOTOphie paBHOMepttbl noBaHo, lrt (rr l -xc, 1!l + c), r)l,e c > 0 H cpHKCHpO-=0, ± 1, ± 2, ...16.10. IIycTh F(x) = f~' .J1 + t 2 dt. ,ll;oKa3aTh, qrn nptt x --++oo2F(x) =x I 2 + (lnx) / 2 + 0(1).16.11. IIycTh F(x) =f~' ~v(t) + k2 dt,r)l,e cpyttKUIDI v(t)~0npttt ~ 0 ttenpepbIBHa H fo~ ~ dt < oo. ,ll;oKa3aTh, qTo nptt x --++ooF(x)=lklx+r~ar)l.ek-v(t)dt+o(l)v(t) +k2 +I k InpOH3BOJJbHaH )l,ettCTBHTeJJbHaH KOHCTaHTa.16.12.
IIycThpyeMoii nptt t EcpyttKUIDIf(t)HBJJHeTCH ttenpepbIBHO .D.ttcpcpepettutt-[O, ] . ,ll;OKa3aTb, qTo nptt s --++Of t+J (t) dt =- j(O)ln£ + 0(1).£016.13. IIycTh BhIOOJJHeHhI ycJJOBHHqTO nptt s --++0npe.D.hI.D.)'I.Qeii 3a)l.aqtt. ,ll;oKa3aTh,r' (CO, dt = + f (0) tr+ O(ln £).Jor+E-16.14. PaccMorpttM HHTerpM2EI'.w6a 16232r.ne E > 0, a H ~ - .ueHCTBHTeJlbHbie qHCJla H ~ > 0 . .IJ:oKa3aTb, qTQ nptt€--t+O cnpaBe,UJlHBbl CJJe.uyromHe aCHMCTOTHqecKHe paBeHCTBa:l1) F(c)- -/J Inc2) F(E)--~r(a;l)c-ca+IJl/JnpH a= -1;npHa>-1.16.15.
ITycTb cpyHKumrf(x) y.uoBnernopReT ycJJOBHIO feJJh.Uepa Ha0Tpe3Ke [O, 1] H nycTb qHCJlO ~ < 0 . .IJ:oKa3aTb, qTQ npH€--t+OJoo f (x) e-e./I dx = - f /J(0) In E + 0(1).XI16.16. ITycTb cpyttKI..I,H}lj(z) aHanttrnqHa rrpH z E { z: IIm zl < n 12},a qttCJia f...., k = 1, 2, ... , oo, y.uoBJJeTBOp}{IOT ycJIOBHIO 0 < /... 1 < /...2 < ... H/...k --t+oo. ITycTb TaIOKef(z) HMeeT acHMITTOTHqecKoe pa3JJO)KeHHef(z)- L~= 0 a 11 e-J.,,:, Rez.IJ:oKa3aTb, qTof'°'--7+00 ,IImz j<Jr/2 .'(z) - - L.J11::= 0 a II AII e-J.,,:, Re z --7 +oo, IIm z I< 1r 12.0016.17 .
.IJ:oKa3aTb, qrn npH z -too, Re z > 0, cnpaBe;::i:mrno aCHMnTOTHqecKoe pa3JJO)KeHHeooe-IIo --dtz +t00(-l)kk!Lk=Ok+Iz.16.18 . .IJ:oKa3aTb, qTQ npH x --t+oof00-I .r-1.\. t efGt -"oo (- 1) k k' ·IL..,k=Ox k+I.16.19. ITycTb (a, b) - KOHeqHbIH HJJH 6ecKoHeqHhIH HHTepBaJI,cpyttKUHR S(x) MR JJI06oro x E (a, b) y.uoBJJernopReT ycJJOBHIO S(x) :S c 0H npH HeKoTopoM /... =Ao > 0cxo.UHTCR HHTerpanF(A) =J" f (x)e;.scx> dx.(/.IJ:oKa3aTb, qrn npH JII060M /..., y.uoBJJernopRIOIUeM ycJJOBHIO Re /... :=::crrpaBe.UJJHBa oueHKa/...o,METOLl,bl TIOJ1YL!EHI15! ACI1MTITOTl1L!ECKI1X OUEHOK23316.20.
IIycTb cpyHKUllilf(x) 5IBJI.HeTCH KycoqHo-ttenpephIBHOH np110 11 6ecKOHeqHo .zu1cpcpepettu11pyeMa B OKpeCTHOCTH TOqK11 x = 0.)J,oKa3aTb, qrn np11 "A _, oo, larg "Al :S n/2-E, E > 0, cnpaBe,[(JIMBO cne-x~.nyIOw:ee ac11MnToT11qecKoe pa3JIO)Kett11e .UJIH npeo6pa3oBamrn JianJiaca cpyHKU1111f(x):J; f(x)e-J.'"dx-I~=of<">(O)A_,,_ ,1r.ne cpyttKUM5If(x) y.nosJiernopReT rrp11 x16.21. IIycTb> 0 ycnos1110Jfi:x)I :s M ec\ c ~ 0.2 r' ,<I>(x) = Ji J~ e-r dt - MHTerpan sepo5ITHOCTH.
)J,oKa-3aTb, qTo npM x -T+OO crrpaBe,[(JIMBO acttMCTTOT11qecKoe pa3JIO)KeHMe22<I>(x)-I- e-·'_e-'.Jiix .Jiix16.22. HeMAta BamcoH.a. IIycTb a,npH t Et= 0.[O, a]L~(-1/(2k-l)!k=I~zk X2k> 0, cpyttKUMRj(t) tterrpephIBHaH 6ecKOHeqno .[{H<pcpepeHUHpyeMa B OKpeCTHOCTH)J,oKa3aTh, qTo rrptt A. -T+oo cnpaBe.[(JIHBO acttMIITOTttqecKoepa3JIO)Kett11eratp-re-;_,a f(t)dt-_!_I= A.-<11+/J)tar((n+fJ)la) f11J(O).Joa n=On!16.23. IIycTb cpyHKUM5I f(x) 6ecKotteqtto .n11cpcpepeHuttpyeMa B OKpecTttocT11 x = 0, 11 nycTb 11HTerpaJIF(A)=J; f(x)e-J.xdxa6COJIIOTHO CXO,[(HTCll nptt HeKOTOpOM ~ > 0. )J,OKa3aTh, qTQ ,[(JI5IcpyttKUHH F("A) np11 "A _, oo, largA.I :S n/2-c,, E > 0, cnpase.un11Bo ac11M-nToT11qecKoe pa3JIO)Ketttte tt3 3a.uaqw 16.20.16.24.
IIycTb cpyttKUllil g(x) attaJI11T11qHa 11 orpattwqetta B 06JiacT11,BKJI10qa10rueH: .neH:crn11TeJibHYIO ocb. )J,oKa3aTh, qrn nptt "A _, oocnpase.uJI11Bo ac11MnTOT11qecKoe pa3JIO)Kett11eJ g c) e=-oot-h212c211-1)!! a1 - J?-:rr:I=A 11=0An2n'~tr.ne K03cpcp11u11eHThI a 2,, 6epyTc5IB OKpeCTHOCTl1 ToqK11 z = 0:m pR.ua TeH:Jiopa .UJIR cpyttKUtttt g(z)I'!la6a 16234g(z) = I~= 0 a,,z".16.25. <PyHKlfU5l MaKOOHGJlbOa K0(x) onpe.uenetta paBeHCTBOM\" r ~K 0 (x) = e-· J,0 ~.[(OKa.3aTb, qTO npH x-+e-rxt(t + 2)dt.00 cnpaBe,UJIHBO aCHMTITOTHqecKoe paBeHCTBOK 0 (x) =e·_ ,Jf(1 + O(~)).16.26. .[(oKa.JaTb, qrn cpyttKI.JJU! MaK,UOHaJib.Ua K0(x) MO)KeT 6bJTbnpe.ucrnsneHa B sw.uelir.r2I+rK0 (x)=-[.~16.27.
IIycTb r(x + 1) =J; t.re-1dt.dt, x > - 1, ecTb cpyttKI..1)151 3ifne-pa . .[(oKa3aTb, qTo npH x-+ +oor(x+l)=x'e-'.J2rrx(1+0(~)).IIonyqHTb OTCIO.Ua rjJopA1YllY Cmup!luHcan! = n"e-".J2nn(I+o(~)).16.28. I1cnOJib3Y5f paBeHCTBOnj e-"«I+x)"dx0=2+ IO-l/n)· ...·(l-(k-1)/n)k=2H cpopMyny CTttpJIHHTa, noKa.JaTb, qrn npHn-+ooF (n) = .Jm112 ( 1+ 0 ( ~)) .16.29. I1oKa3aTb, qTo npH n-+ooJ;' sin" tdt =.Jn /2n( I+ o(~)).216.30. I1oKa3aTb, qTo npw n1-+J (l-x-I2oo)"dx-.Jnln.16.31.
HaifTH acHMCTTOTHqecKoe nose.uettHe npHt -+ oo cpyttKUHHMETO,[(hl I10JIY4EHI151 ACI1MI1TOTI14ECKI1X OUEHOKe-tx00F(t)=J1 +ta XO235dx,r.ue a - npOH3BOJibHOe )J.eikTBHTeJibHOe '-IHCJIO.16.32. .D:oKa3aTb, '-!TO eCJIH > 0,00.Io x-a.•x dx1TO npHt- t 00~J[ ___!___q_ 111at 2aeeea16.33. .[(oKmaTb JZeJ1wy Pu.11taHa-Jle6e2a:rpaJIf:I f (x) Idx (anycTb cymecrnyeT HHTe-H b Moryr o6paruaThCH Boo); TOr)J.a rrpH /...
--+ oocnpaBe)J.JIHBO aCHMilTOTH'-!eCKOe paBeHCTBOf f (x)e;;._,.dx = a(l).b(/16.34. I1ycTb a> 0. .D:oKa3aTh, '-!TO npH /... --+s;(1 + xra eiJ.xdx = iA_-loo+ 0(.,1,-2).16.35. I1ycTb cpyHK.I.J:MHj(x) N pa3 HenpepbIBHO )J.HcpcpepeHQHpyeMa Ha [O, 2n:] M /kl(O) /kl(2n:) .a.irn k = 0, 1, ... , N - 1. .[J;oKa3aTb,='-!TO npHn- t 00J:,,. f (x)ei"xdx =o(n-N).16.36.
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