Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 16
Текст из файла (страница 16)
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IlycTh)J{op.n;attosa crrp}!MJI}leMa}l Kp11Ba}lo.n;ttoCB}l3HOH orpattw:1ettttoii 06nacT11tta Hay . .Il,oKa3aTh, qTQ .UJI5I paseHcTBa,n:11Mo 11 .n:ocrnToqtto, qT06b18.29.<Ji rp(t) dtyt-fr rp(t)dt= 0, z E i5,t- Zzcp(t)= 0, z Etterrpephrntta HaD,D,I1oKa3aTh, qTQ ycnos11ey . .Il,oKa3aTh,0ED/1yecTb rpaH11uarrp11Ha,n:Jie)Kl1TqTQ .n:m paBeHcTBa= -1, -2, ....HeJih35l 0T6poc11Th.8.30. IlycTh cpyHKUl15l f(z) aHaJil1TJ1qHaCTl1 C 11 y.n:osnernop5IeT ttepasettcrnyC, a -ToqKa ztteo6xo.n:11Mo 11 .n:ocrnTOqHo, qT06b1t/' rp(t)dt =0,r.n:etteo6xo-=0I1ycTh )J{op,n:attosa crrp5IMJrneMa5I Kp11Ba5I11 cpyttKUl15ly ecTh rpam1:uacp(t) Herrpephrn-11 cpyHKUl1}!p/' rp(t)dt =0, n = 0, 1, 2, ....O.UHOCB5I3HOH orpatt11qeHHOH 06nacT11DDHa KOMIIJieKCHOH IIJIOCKO-KOHCTaHThI, He 3as11c5ImHe OTMHOroqJieH CTerreHH 11 :::: [z..Il,oKa3aTh, qTof(z) -a ].8.31.
.Il,oKa3aTb TeopeMy JIHyBHJIJI}J.8.32 . .Il,oKa3aTh,qTQ ecJIH cpyHKUH}J aHaJIHTHqHa B pacurnpeHHOHKOMIIJieKCHOH IIJIOCKOCTH8.33 . .Il,oKa3aTbc' TO Otta eCTb KOHCTaHTa.rrpHHUHIIMaKCHMyMaMO.UYJI}lattaJIHTJ1qecKOHcpyHKUl111.8.34 . .Il,oKa3aTbTeopeMy: eCJIH aHaJIHTHqecKa5I cpyHKUH}J orn11qHaOT IIOCT05IHHOH B 06JiaCT11D,TO sepXH5!5! rpaHb MO.UYJI}l cpyHKUl1H HeMO)KeT .uocn1raThC5! HH B o;:i:Hoi1 ToqKe 06nacn1D.1088.35 .
.D:oKa3aTb TeopeMy: MO.UYJJb aHam1rnqecKOM cpyHKUMM B 06nacn1 D 11 HerrpepbIBHOM B D .uocT11raeT csoero Ha116onbrnero 3HaqeHH5:! B rpaHHqHOM TOqKe '.HOM o6JJaCTH.8.36. IlycTb cpyHKUH5:! f(z) = I~= 0 a,.z" aHam1rnqHa B Kpyre{ z : JzJ::; r} . .D:oKa3aTb, qTQ cpyHKUH5:! lf/(z)= L~=o a,,, z"n.Ul15:!, aHaJJHTHqecKa5:! BO BceM: KOHeqHOM ITJIOCKOCTH_ r -k e1:112 ,IIf/<k>( z)j<Mr.ueecTb cpyHK-cHk-012- , , ,... ,M-KOHCTaHTa, He 3as11c5:111.i;a5:1 OT z.8.37 . .D:oKa3aTb, qTQ ecJJH aHaJJHTHqecKa5:! cpyHKUH5:!j(Z) B o6JJaCTHD orn11qHa OT KOHCTaHTbI 11f(z)=F0, z ED, TO inf Jf (z) J He .uocT11:eoraeTC5:! BHYTPH o6nacTH D.8.38. IlycTb cpyHKUH5:! f(z) aHaJJHT11qHa B 06nacT11 D 11 rrycTbinf Jf (z) J = µ > 0. .D:oKa3aTb, qTo HJJH lf{z)J > µ .UJI5:! Ka)f{.uoH: BHYT:eopettHeH TOqKH o6naCTH D, HJlHj{Z)= µe ;a, a -;::i:ei1:CTBHTeJlbHOe qHCJIO.8.39.
IlycTb cpyHKU:H5:!j(z) aHaJJHTHqHa B o6nacTH DH HerrpepbIBHa Ha D. .D:oKa3aTb, qTO ecn11 lf{z)J ecTb KOHCTaHTa Ha 8D, TO cpyHKU:115:! j(z) 11J111 eCTb KOHCTaHTa B o6naCTH D, HJlH OHa o6pamaeTC5:! BHYJib xorn 6b1 B O.UHOM ToqKe 06nacT11 D.8.40 . .D:oKa3aTb JieMMY 111Bapu:a.8.41. IlyCTb B Kpyre { z: JzJ < 1} cpyHKUH5:! j(z) aHaJJHTHqHa 11lf{z)J::; M, JzJ < 1. .D:oKa3aTb, qTo ecm1 Zi. z2, ... , z,, - HYJJH cpyHKU:HHj(z), JzjJ < l,j = 1, 2, ... , n, TOJf(z)J:$Mz-_:,.z-_:2 ·... ·1-z-_:,. I·z 1-1-1ZZ2 ZZ,, ZrrpHqeM paBeHCTBO MO)f{eT BbIIIOJ1H5:!TbC5:! JlH6o .UJI5:! BCeX ToqeK Kpyra{ z: JzJ < 1}' JlH6o HH .UJ15:! O.UHOH.= z" +a, z"-' + ...
+a,, . .D:oKa3aTb, qTo xorn 6b1 BO.UHOM TQqKe OKpy)f{HOCTH { z : Jzl = 1 } HMeeT MeCTO HepaBeHCTBOlf{z)J > 1 HJJH paBeHCTBO j(z) = z ".8.42. IlycTb j(z)8.43. IlycTb P(z) - MHoroqneH CTerreHH n H Mr= maxi P(z) j.l:l=r.D:oKa3aTb, qTQ rrpH0 < r, <r2 crrpaBe,UJJHBO HepaBeHCTBOCTETIEHHb!E P5f,D,bl. P5f,D,bl 113 AHAJII1TVI4ECKI1X <DYHKUI1H109-11M 'ifj -11 >M,1 r2'T. e.
M,. r" - HeBo3pacrnIOmaH ¢YHKUirn Ha (0, +oo).8.44. IIycn, P(z) - MHOroqJieH CTerrem1 n H ,[\JIH z E [- 1, 1] HMeeM jP(z)j :::; M. ~OKa3aTh, qrn ,[(JIH JII06oi1: TOqKH ztf. [- 1,l] crrpa-Be,[(JIHBO HepaBeHCTBOjP(z)j:::; M (a+ b)",a H b - rronyocH 3JIJIHIICa c ¢oKycaMH B TOqKax 1, - 1, 3JIJIHIIC rrpoXO,[(HT qepe3 TQqKy z.8.45. ~OKa3aTh, qrn TOqKa Zo eCTh HYJih rropH,n:Ka m aHaJIHT11qecKoi1 ¢YHKUHH f(z) rnr,n:a H TOJihKO rnr,n:a, KOr,[\a B HeKOTOpoH: OKpeCTHOCTH TOqKH Zo cVYHKUirn f(z) = (z - Zo) Ill cp(z), cp(z) - aHaJIHTHqeCKaH ¢YHKUHH B OKpeCTHOCTH TQqKH Zo, <p(Zo) :f.
0.8.46. IIycTh rnqKa Zo ecTh HYJih rropH,n:Ka n ):(JIH ¢YHKUHH f(z) HHYJih rropH,L:\Ka m .L\JIH ¢YHKUHH g(z). qeM HBJIHeTcH TOqKa Zo .r:\JIH cne.r:\YIOIUHX cVYHKUHH:1) j(z) g(z);2) fi.z) :t: g(z);3) fi.z) I g(z)?8.47. Hai1TH rropH.r:\OK HYJIH z = 0 ,n:JIH ¢YHKUHi1:l)zsinz; 2)z2(e:'-l); 3) esin:_erg:; 4) 6sinz 3-z3(6-z6).8.48. Orrpe,L:\eJIHTh rropH,n:oK Bcex eyneH: MH cne,n:yIOIUHX ¢YHKUtti1:l)sinz; 2)cosz; 3)shz; 4)chz; S)e'-I;7) (1- ctg z) I z;11)8.49.sin(~J;sm z~OKa3aTb28) (z + 9) I z49) sin (z-;6)sinz-2;1);!O) cos (z -1);112) sin(--_1 J.coszTeopeMy e,n:rrncTBeHHOCTH.8.50. CymecrnyeT JIH ¢YHKUHH, aHaJIHTHqecKaH B oKpecTHOCTHTOqKHHHH:Z =0 H IIpHHHMaIOIUaH B TOqKaX1) 0, 1, 0, 1, 0, 1, ...
;Zn= 11-1CJie,L:\yIOIUHe 3Haqe-2) 0, 1/2, 0, 114, 0, 1/6, ... ;3) 1/2, 1/2, 1/4, 1/4, 1/6, 1/6, ... ; 4) 112, 2/3, 3/4, 415, ... ;S)j(n -1)7)j(n- 1)=e-";=j(-11- 1) = 11- 2 ;6)j(_11- 1) = (2n + 1)- 1;8)j(11- 1) = f(- 11 - 1) = 11 3 ?110T.wea 88.51. .IJ:oKa3aTb nepsy10 TeopeMy Bei1epwTpacca.8.52 . .IJ:oKa3aTb srnpy10 TeopeMy Bei1epwTpacca.8.53 . .IJ:oKa3aTb, qTo ecm-r pH,n: L~J f,, (z) I cxop:i1TCH paBHOMepHO sHyTp11 06JiacT11 D, r.ne cpyHKu1111 f,,(z) - aHan11mqecK11e B 06nacrn D, TO pH.n L~=II.t,;(z) I cxo.n11Tc.srTaK)!(e pasttoMepHo BHYTPHo6Jiacn-r D.8.54.
Ha 06JiacT11 cxo.n11MocT11 pH.nos 11ccJie,n:osaTb cyMMhI pH,n:osHa aHaJII1Tl1qHoCTb:1)L=n=l1Iz(n-1 + z-)·'8.55. IloKa3aTb, qrn cyMMa p.sr.naL~=I (nze- 110 -(n - l)ze-< 11 - 1>0 )eCTb aHaJII1TJ1qecKaH cpyHKU11H Ha o6JiaCTl1 CXO.LJ:l1MOCTl1 pH,n:a, HO pH,n:Ha 3TOH o6JiaCTl1 CX0,[(11TCH HepaBHOMepHO.8.56. HccJienosaTb T3rn-cpyHKU11IO 8(z) = L~=-e- 1"'' 0 Ha attan11T11qHoCTh.8.57. OnpeneJIHTh o6JiaCTb aHan11T11qHocT11 cyMMbJ pH.Ila"~ sin nzL..11=!2" .8.58 . .IJ:oKa3aTb aHaJIJITl1qffOCTh CYMMbI pH.[(a s YKa3aHHbIX o6Jiacrnx:1) L~= 1 1·c'(n+zt', z:;t:-1,-2, ...
;2) I~= 1 (-l) (z-nr 1 , z.cl, 2, ... ;11113) L~= 1 1C (n+z) , lzl<l;115) L~ shn.::•n! '11=17) L~=' e110',zECsinnz 7 E C4 ) "~~11=1I '"'11.'n/4 < arg z < 3n I 4, -3nl4< arg z <- n I 4.8.59 . .IJ:oKa3aTb, qTo eCJil1 ceMeHCTBO aHaJIHT11qeCKl1X cpyHKUHH{f,,(z)} orpaHHqeHo BHYTPH o6Jiacrn D, TO ceMettcrno {f,,'(z) }TaK)!(eorpaHHqeHo BHYTPH D.CTEOEHHbfE P51L(bl. P51)..(bl 113 AHAJll1Tl14ECKl1X <DYHKU11H1118.60 .
.[(oKa3aTh, lffO eCJU! ceMei1crno aHan11rnqeCKl1X cpyHKUHHorpaH11qeHo BHYTPH D, TO OHO paBHOCTerreHHO HerrpephIBHO Ha mo6oM KOMrraKTe H3 D.8.61. IIycTh {f,,(z)} - rrorne.uosaTeJihHOCTh aHa1111rnqecK11x B 0611acrn D cpyHKUHi1, 06pa1y10m11x orpaH11qeHHOe BHYTPH D ceMei1CTso . .[(oKa3aTh, qTo ec1111 noc11e.uosaTeJihHOCTh {f,,(z)} cxo.u11TcH HaITJIOTHOM B D MHO)Kecrne, TO oHa cxo.u11TcH paBHOMeptto BHYTPH D.8.62 . .[(OKa3aTh, qTo 113 nocJie)lOBaTeJihHOCTH aHaJIHTHqecKHX B0611acrn D cpyHKUHH {f,,(z)}, 06pa1yIOUJ.JiIX orpaH11qeHHoe BHYTPH DCOICHCTBO, MO)KHO BhI6paTh no.unocJie)lOBaTeJihHOCTh {fk,.}' paBHOMepHO cxo.uHUJ.YIOCH BHYTPM D.8.63 .
.[(oKa3aTh TeopeMy B11Tan11.8.64. IIycTh q11c11osoH: pH.ll I~J a I cxo.u11TcH 11 cpyHKIJJrn .f(z)atta1111T11qtta B 0611acT11 D . .[(oKa3aTh, qTo 6ecKoHeqHoe npomse.ue11H11e TI~=' (1 + a 11 f (z)) ecTh atta1111T11qecKaH cpyttKUHH B 0611acT11 D.8.65 . .D:oKa3aTb, '-ITO raMMa-cpyHKUHH~TI 11=1.: + n-1r(z) =Tl11+l(:-l)ln-eIIHBJIHeTCH aHaJIHTHqecKOH cpyHKUHeH BO BCeH KOMnJieKCHOH OJIOCKOCTH, KpOMe ToqeK z 0, - 1, - 2, ....=8.66. IIycTh {f,,(z)} - rrocne.uoBaTeJihHOCThTJil D ¢YHKUHH 11 fncxo.u11TCHi- -1,paBHOMepHozaHaJIITT~ecKHX B o6Jiac-ED .
.[(oKa3aTh, qrn ec1111 PH.ll I~J f,,(z)iB D,TO6ecKotteqHoenpoH3Be.uetttteTI~=' (1 + f,, (z)) - atta1111rnqecKaH cpyttKUHH B 0611acrn D.8.67 . .[(oKa3aTh, qrn rne.uyIOm11e 6ecKoHeqtthie rrpomse.ueHHH rrpe.uCTaBJIHIOT co6oi1 attaJIHmqecKHe ¢YHKU1111 B YKa3aHHhIX o6Jiacrnx:1) TI~= 1 (l+z"),izl<l;2)TI~=' o+ ~)e-''", I z I<3)z I z I<n ~ 1cos-,11n11=0000,u'= -11;7[,z:;t:n(2k+l),kEZ;24) TI~=I (1 + (-1)" ..::.), I.: I< oo,.:11:;C1(-1)"+ n;1128Tnaea5)TI ~, -II-sin(z)/ 11z I 11,I Z I< oo , Z ~ k Jr, k E6) n~=l ch (r'' z), I z I< 00 ,Z·,z ~ 2" (2k + l)Jri,7) n~=lcos(r" z), I z I< 00 ,z ~ 2"- (2k1k E Z;+ l)7r,kE Z;8) TI~= 1 0 + r" z"), I z I< 2.8.68.
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