Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 20
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Ern11 p - uenoe, TO LAp L(JI5! Bcex A, 3a HCKJIIOqeHHeM, 6bITb MO:>R:eT, OL\HOfO3HaqeHH5! A.10.27. HailTH HCKJIIOqHTeJI1Hoe 3HaqeHHe L(Jl5! cpyHKU:HH=f(z) =sin z ·e ~1(cM. rrpe.L\bIL\YlilYIO 3a.L(aqy).10.28. )J;oKa3aTb cJJe.L\yIOlllYIO meopeMy eouHcmeemwcmu OJ/R 14eIlyCTb ueJJa5! cpyHKUH5! f(z) HMeeT rrop5!.L\OK He6oJJbllle p. EcJJH cpyHKUH5! f(z) o6palllaeTC5! B HYJJb Ha rrorneL(OBaTeJJ1HocT11 { Z11} c IlOKa3aTeJleM CXOL(HMOCTH L > p, TO f(z) = 0.10.29. IIycTb ueJJa5! <PYHKUH5! f(z) HMeeT rrop5!.L\OK He Bbillle p,HJIH rrpH I10p5!L(Ke p THIT He Bblllle CT.
)J;oKa3aTb, 'ITO eCJlH cpyHKUH5!f(z) o6palllaeTC5! B HYJJb Ha rrocJJeL10BaTeJI1HocTH { z"}, 11 rrpH 3TOMJ1bZX rPYHKlfUU.-nJim--> crep,"-+""'I z,, IpTO f(z) =:0.10.30. IlycTb 0 <H rryCTb ueJJbie0 < r < oo,P11lz 11 1 = r11 •lzil~lz2I~...~lim I z11 I= +lz"I <TaKOBbl, 'ITO I::=! (;,r·00,11-700<00rrpH mo6oM r,)J;oKa3aTb, 'ITO rrpoH3BeL(eHHerr:=IE(: ,p"..... ,,-1)= rr:=l(1-~)x"viixexp(~+~(-;-)zll"112+ ...+~(-;-)p,, -IlP"....II5!BJl5leTC5! ueJJOH cpyHKUHeti, KOTOpa5! HMeeT HYJJH TOJlbKO B TO'IKaX{Z11}.131UEJib!E ct>YHKUHVI VI P51Ll,bl Ll,l1Pl1:XJIE10.31. .[(oKa3aTb TeopeMy o pa3JIO)Kemrn: ueJIOH cpyttKU1111 s 6ecKOHe'IHOe npoH3senemte.10.32 .
.[(oKa3aTb TeopeMy Eopem1 o pa3JIO)Kett1111 ueJioi1: cpyttKUHHs 6ecKoHe'IHoe npo113senem1e."'\'~10.33 . .[(oKa3aTb, 'ITO cpyHKU11.5I L..11=0z"s,(n +a) n!a > 0, np11HaL{Jie-)KHT KJiaccy [ 1, 00]' T. e. ueJia.5! cPYHKUl1.5I 3KCilOHeHUHaJibHOro rnna,- mo6oe semecrneHHoe 'IHCJIO.s10.34. IIycTb cPYHKUHR g(t) E C[a, b ] . .[(oKa3aTb, 'ITO cpyHKUHRf(z)= f be; : , g(t)dtnpHHaL{Jie)lnfT KJiaccy [1, oo).(/10.35 .
.[(oKa3aTb, 'ITO cpyHKUl1.5I f(z) 113 npenh1nymei1: 3ana•m npHz--+ oo no semecrneHHOH oc11 crpeMHTCH K ttymo. BhrnecT11 0Tc10na,'ITO cpyHKU11H /(z) 11MeeT 6eCKOHe'IHO MHOro HyJieH.10.36. IlycTb cpyHKUHH f(z) E [p, oo), cp(z) E [p, oo), Z1, Z2, ... HYJIH f(z) KparnocTH mi. m2, .... <l>yHKUHH cp(z) HMeeT Te )Ke HYJIH(MoryT 6bITb H L{pyr11e), KpaTHOCTeH He MeHbllle m1, 1112, ••.. .[(oKa3aTb, 'ITO cpyHKUHH F(z)= cp(z) I f(z)E [p, oo).10.37 . .[(oKa3aTb, 'ITO ecJIH f(z) = A / 1(z) + B f 2(z), TO cpyHKUHH,accouHHposattttaR no EopeJIIO c f(z), y(t)A y 1(t) + B y2(t), rney 1(t), Y2(t) - cpyHKUHH, accou1111posaHH01e no EopeJIIO c cpyttKUHHMHf1(z),J2(z).=10.38. HaH.rn y(t) LlJIH cJieny10m11x cpyttKu11i1::1) Ae "';2) sin z;3) cos z;4) sh z;5) ch z.10.39.
HaH:rn y(t) L{JIH KBa3HnOJIHHOMaP( Z)r.ue d;, i= 1, 2,"'\'k= L..;=J A;e,, __·- , A;=;t: 0,... , k , - seplIIHHhI BhmyKJioro MHoroyroJibHHKa D .10.40. HaH:rn onopHoie cpyHKU1111 .UJIH Kpyra { z:pe3Ka [- cri, cri], cr > 0.lzl < cr}11 L{JIH OT-10.41. .[(oKa3aTh, 'ITO ecJIH K(cp) - onoptta.5! cpyHKUHH MHO)Kecrna--G, ro onoptta5.I cjJyHKUH5.I GE ecTb K(cp) + £ , me GE - £-pacw11pett11eG.I'Itaea 1013210.42. Jl,oKa3aTh,qTO onopHa5I cpyHKUH5I K( <p) - HenpephrnHa5IcpyHKUH5I .10.43. Jl,oKa3aTb,qTQ eCJIH onopHbie cpyHKUHH BhinYKJihIX MHO-)KCCTB paBHhl, TO H caMH MHO)KCCTBa paBHhl.10.44.
HaM:rnconp5I)KCHHhie ..QHarpaMMhI ..QJI5I cne.uy10rmrx cpyHK-UHH:1) Ae0' ;6) ch z;2) sin z; 3) cos z; 4) cos z · cos (iz) ; 5) sh z;7) j(z) - KBa3HnOJIHHOM.10.45. Jl,OKa3aTh,HOro THna,qTO ecmrf(z) - uena5I cpyHKUH5I 3KCnOHCHUHaJib-y(t) - cpyttKUIDI, aCCOUHHpOBaHHa5I no EopeJIIO cj(z), TOf(z)= - -. ~12m cy(t) e:.1 dt ,c - )KOp,LlaHOB cnp5IMJI5IeMbIH 3aMKHYThIH KOHTyp,r.ueIUHH conp5I)KCHHYIO ..QHarpaMMY10.46.
IlycTh j(z)DOXBaTbIBaIO-cpyHKUHH j(z).- cpyHKU:HH 3KcnoHeHUHaJihHOro THna,D -co-np5I)KCHHa5I ..QHarpaMMa, H K( cp) - ee onopIW5I cpyHKUH5I. Jl,oKa3aThou:ettKy: .L\JI5I n106oro e > 0 cnpaBe.z:i;JIHBO ttepaseHCTBOIf (re;rp) I< A(E)e[K(-rp)+EJr.10.47. IlycTh y(t) -cpyHKU:HH, accouHHposaHHaH rro Eopemo ccpyHKUHCH j(z) 3KCIIOHCHUllaJihHOro Tttna.
Jl,OKa3aTb, qTQ B nonynJIOCKOCTllRe (te;qi0 ) > h( (/Jo),r.ue h( <p) - ttH.L\HKaTpttca poem j(z),y(t) eCTh aHaJIHTYJqecKa5I cpyHKU:H5I H crrpase.z:i;JIHBO npe.ucmsJieHttey(t)= J f(z)e-:'dz,ttHTerpan 6epeTC5I no nyqy arg z= <p0.10.48. Jl,oKa3aTh meope.111y llo!lua:nycTh cpyHKUH5If(z) -uenaH,3KcnoHeHuHaJihHOro Tttna c ttH.UHKaTpttcoM: poem h(<p), D - conp5I)KCHHa5I .nttarpaMMa,h(cp)K( cp) - onoptta5I cpyHKUH5I MHO)Kecrna D ; TOr..Qa=K(- cp).10.49. Jl,oKa3aThCBOHCTBa HH.UttKaTpHChI poem1) HerrpepbIBHOCTh;2) maxh(q;i) =CJ (CJ= Jim~);k -+ O<I3) h(cp)?. - cr.h(cp):UEJibIE ct>YHKUHH H P5I)J.bl )J.HPHXJ1E13310.50 .
.D:oKa3aTb, qrn .rum cpyHKUHH f(z) = sin z crrpaBe.UJIHBhI coOTHorneHmr:1) h(cp) =!sin cpl;2) !sin zl < e h(<p) r;zl > A(E) e h(<p),. BHe OKpeCTHOCTeHlz - knl < E, k =0, ±1, ±2, ....3) !sin10.51. HattTH HH.UHKaTJ)HCbI pocn CJie.uyl{)m,Hx cpyHKUHH:1) e z; 2) sinz; 3) cos z; 4)shz; 5) ch z; 6) e:" ; 7) e z + z 2.10.52. IIycTb ueJia5! cpyHKUHR f(z) HMeeT HH.UHKaTJ)Hcy pocTah(cp). Orrpe.ueJIHTh HH.UHKaTpHcy pocTa cpyHKUHH F(z) = P(z) + f(z),r.ue P(z) - MHoroqJieH.10.53. HaiiTH HH.UHKaTpHcy pocTa h( cp) ,UJI5! KBa3HIIOJIHHOMa.10.54 .
.D:oKa3aTb, qrn 6ecKoHeqHoe rrpomBe.ueHHeh(z)=n-•=' l(1-~J..1z , O<A.,ioolim~=cr<oo, ,__,_-1,1) eCTb cPYHKUH.sJ: 3KCIIOHeHUHaJibHOro THrra;2) HH.LU1KaTpHca pocra: h(cp) = 1w !sin cpl, 0 ~ cp ~ 2n.10.55. IIycTb fl.z) - ueJia.sr cpyHKUH.sr 3KcrroHeHUHaJihHOro THrra 11y(t) - cpyHKUH5!, accouHHpoBaHHa5! rro Eopemo c fl.z) . .D:oKa3aTb, qTo.u11.sr cpyHKUHHHMeeT BH.Uf111l(z) accouHHpoBaHHa5! c He!{) rro EopeJII{) cpyHKUH5!t111y(t)-t111-1f (O)-tm-2 j'(O)- ...
- fc111-1J (0).10.56. IIycTb D- corrp5!:>KeHHa5! .u11arpaMMa cpyHKUHH f(z) . .D:oKa3aTb, qTQ HH,lJHKaTpHCbI pOCTa cpyHKUHH f(z) H ee rrpOH3BO.lIHOH J'(z)COBIIa,Ual{)T, eCJIH HaqaJio KOOp.UHHaT eCTb TOqKa HJIH BHyTpeHH5!.sJ:.UJI5! D, 111111 BHernH5i5J: .LIJI5! D, HJIH rpaHHqHa5! .um D, He 5!BJI5J:l{)m,a5!C5!BHYTpeHHeif ToqKoif orpe3Ka, BXO.lJ5!Ill,ero B cocTaB rpaHHUhI D .10.57.
IIoKa3aTb, qTQ ycJIOBHe Ha corrp5!:>KeHHYI{) .UHarpaMMY D Brrpe.uhr.uym,eii 3a.uaqe 5!BJI5!eTc5! cymecrneHHbIM.10.58. IIycTb F(z) - ue11a51 cpyHKUH5J: 3KcrroHeHUHaJibHoro THrra,D- ee corrp5!)KeHtta.sr .uttarpaMMa . .D:oKa3aTb, qro ec1111 D c G, 06-I'.rzaea 10134= { leInaCTh Gz:z - l < 1}' TO cpyHKUIHI F(z) p3311araeTC5! B HHTepIlOJUIU110HHhltt pR;:i: HhIOTOHa_ t:! F(O)F(z)=,L, . 0k!z(z-1) ...
(z-k+I).10.59. ,UoKa3aTh erue ontty meopeMy IloJZua: nycTh uenaR cpyttKumr F(z) ecTh cpyttKUlrn 3KcnoHettu11anhHoro THna, 11 conpmKettttaRDc;:i:11arpaMMaG (G= {z: le z - l I <1} ). Ecn11 F(k) - uenh1e q11cna,k = 0, 1, 2, .. ., rn F(z) - MHoroqnett.10.60. PaccMoTp11M cpyHKUHIO f(z) = e z 1" 2• He npoT11Bopeq11T n113TOT np11Mep TeopeMe IIon11a, TaK KaKflk) = 2 k?10.61. IIoKa3aTh, qrn ecn11 pR;:i: ,U11p11xne L~=I a,,e-"":, 0 < A.,,ioo,CXO.IJ:l1TCR B TOqKe Zo, TO:1) pRJl cxon11TCR B nonynnocKOCTH Re z >Re z0 ;2) B Ka-/K}lOM ceKTOpe larg(z-zo)l~8<n:/2 CXO}ll1TCR paBHOMepHO.10.62. ,UoKa3aTh, qTo ecn11 pR.ll ,U11p11xne cxo;:i:11TC5I a6conIOTHO BTO'iKCz=zo,TO OH CXO.lll1TC5I a6coJIIOTHO 11 paBHOMeptto Bno11y-IlJ10CKOCTl1 Re z :'.".: Re z0 •10.63 . .IJ:oKa3aTh e,n11HCTBeHHOCTh pa3JIO)J(eH115I B pH.LI .IJ:11p11xne, a11MeHtto: ecn11f(z) = L~= 1 a,,e..1,,~ 11 flz) = L~= 1 b11 e-..1"~, TO a,,= b,,,n = 1, 2, ....10.64.
IIp11secrn np11Mep p5Ina ,U11p11xne, KOTOphIH:1) CXO,Lll1TCR Ha scei1 IlJ10CKOCT11;2) BCIO.IJ:Y pacXO,L111TC5I;3) nnR Hero cymecrnyeT KOHeqttaR a6cu11cca cxon11MocT11.10.65. IIp11secrn np11Mep pR,Lla .IJ:11p11xne, KOTOphri1:1) CXO}ll1TC5I a6COJ1IOTHO Ha BCeH IlJ10CKOCTl1;2) a6conIOTHO pacxo;:i:11TCR BCIO.lly;3) }lJ15I Hero cymecrnyeT KOHeqttaR a6cmi:cca a6conIOTHOi1 cxoJll1MOCT11.10.66. Hai1T11 MHO)J(ecrno ToqeK cxo.u11MOCT11 cne.uyJQm11x pR.uos:(-1) 1111.:1) ""'~ e-"-e"-J;;:.
2) ""'~ _e- · 3) ""'~ e- 11'(e 11 ':+e- 11' : ) .~11=0'~11=1II2'~11=0'135L(EJlblE <bYHKL(vU111 P5!)lbl )lV!Pl1XJJE7)'""00L..tu;;;:lei":.112 '10.67. HaihH o6nacTH cxo,n:HMOCTH cne.uy10uu1x pMOB:1) L~=Jn2 +n3)(e": +e-i":);3)'L..11=1""00 (e;,,': -e-;"'=)·'2)'""00T"(ei11: +e-i"=)·~11=!,10.68. IloKa3aTb, qTo .umr a6CUHCC npOCTOM, a6coJIIOTHOM H paBHOMepHOM CXO,UHMOCTH (c, a H T COOTBeTCTBeHHO, CM. crp.
126)cnpaBe,nJIHBbI HepaBeHcrnac :=:; -r :=:; a.10.69. IloKa3aTb, qrn ecntt limn/ A. = h, TO a - c :=:; h (a H c - CM.11ll-'tOO10.68).10.70. 11puBeCTHCM. 10.68).npHMep p.sr.n;a,)l;JISJKOTOporo a - c= h (aHc -=10.71. ,UoKa3aTb, qTQ eCJJH h 0, TO a6cu:Hcca CXO,UHMOCTH c MO)KeT 6hITb Bb1q11cnetta no cpopMynec = lim ln Ia11u-tooII A,,.10.72. Hai1TH a6cuttcch1 cxo,n11Mocrn, a6conIOTHOH 11 paBHOMepHoi1 CXO,UHMOCTH p5I,Ua ,U11p11x11e:1)'""00 _1 e-:11.L...11=1 1!22)4) '""00 e_") e-11':. 5)L..u=l'""00_1 e-:11.L...11=1;;;,''''""oo(-1)" e-:1n11..L...11=1''"" 00 _!_e-:ln11.
8) '"" 00 (-1)"7) L..u=I1'L..u=lI!6) '"" 00 (-1)" e-:ln11.L..u:;:;:ITlTle:-lnln11.'''"" 00 e-"' e-11':.9) .L...11=110.73. ,UoKa3aTb attanor ttepaBettcrna Koum .UJJ5I K03cpcp11utteHTOBCTeneHHoro p5I,na. IlycTb fi.z)= L~=I a11e-;." =, 0 < A. joo, a6cuttcca a6-conIOTHOH CXO.UHMOCTH a < +oo.
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