Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 50
Текст из файла (страница 50)
l) Pa.nHyc cxo.nHMOCTH R p51.na L~=o z paseH l. BttyTp11 Kpyralzl < l cyMMa j{z) :noro p51.na ecTb aHaJIHTw-recKa51 cpyHKUH5i. Tipe.nrrono)l(HM , 'ITO TOlfKa z = 1 ecTb rrpaBHJibHa51 TOlfKa. Tor.na cyw,ecrnyeT cpyHKUH5iF(z), aHaJIHTHlfeCKa51 s £-OKpeCTHOCTH Ur.Cl) TOlfKH z = 1, TaKa51, 'ITO F(z) =f(z), z E {izl < ln Uc( 1)}. TaK KaK rrp11 0 < r <TO f(r)---4-+oo rrpHHarpam:1uerp =1117r2= L~=o r 2",r--4 l - 0. Cne.nosaTeJibHO, TOlfKa z = 1 ecTb oco6a51 TOlfKaKpyraCXO.UHMOCTH.TioKa)l(eM,'ITO=zTOlfKak0 < m < 2 , m,k EN, eCTb oco6a51 TOlfKa Ha rpaHHue IqJyrak - i,TiycTbl cpyttKL\1151: j(r)ei<p,izl = l.z = re;,.,, 0 < r < 1.
I1MeeM00im2 11'\'f(reY')= L..r eir 11 111Jr 12 k·- I00'\'=L..r11=01·Ref ( re rp)2 112 11 m7r. '\'2k-l11=011=000=Lr21111=0- ~ 2"- L.. r11=021111 mcos k=I2k=L r2112" 1117r22111117rcos ------;:=t +211=0cos~+002 11 1171[2 11cos--+iL..r cos--,+ooL2k-lr2112 11 1177rcos----;::::(211=k+I=~ 2"L.. r .11=k+I00LTip11 cpHKCHposaHHOM k H r--41 - 0 cpyHKUH5i g(r) =211r--4 +oo.11=k+I0Tc1D.na cne.nyeT, 'ITO TOlfKa z= e;'P,cp= (nm)/2k-i,KaK MHO)l(eCTBO TOlfeK z = ei'P, cp = (nm)/2k-I, 05iBJU1eTC5i oco6oi1.
TaK< m < 2\ 111,k E N, eCTblzl = 1, TO rpaHHUa211.naHHOro p51Aa L...11=0 z HMHO)l(eCTBO BCID.llY ITJIOTHoe Ha e.nHHHlfHOH OKPY)l(HOCTHKpyra CXO.llHMOCTH COCTOHT H3 OC06bIX TOlfeK'\'00p51.ll He MOJKeT 6bJTb rrpo.nOJIJKeH 3a rpaHHUY Kpyra CXO.llHMOCTH.=lz - ll11.12. .[{aHHbIH PM cxo.nHTC5i B Kpyre pa.neyca R 1, T.
e. B Kpyre<1cjJyHKUH51: j{z) ecTb aHaJIHTHlfecKa51 cpyHKUHH. Tip11 zx, Ix - I < I, crrpa-=00se.nnHBOpa3JIOJKeH11ef (z) = L (-1) 11 - 1(x -1)" I 11,11=lIx - III< I. C.npyroi1PEWEH115I Tl1Dl14HbIX 3A.[lA4327~cTOpOHhI, lnx= In [l+ (x-1)]= IC-l)"(x-1)" In, lx-lj<l. Cne.nosa11=0TenbHO, fi.z) eCTb aHanHrn'!ecKoe npo.nom1<eH11e cpyHKU.HH In x c .nei1crn11TeJ1bttoro 11HTepsana (0, 2) a Kpyr lz -1 I < 1.~ z 2-11.57. JJ:aHHa51 cpyHKU1151l 51BJJ5leTCll MHOrD3Ha'!HOH, HO OHa no-nycKaeT Bh1neneH11e O.llH03Ha'!HhIX serneH: B nsyx o6nacrnx, Hanp11Mep B D 1= { z: lzl < l } , D1 = { z: lzl > l } . Tio3TOMy, ecm1 x > 1, TO fl.x) =z=x,x<-1, TOfl.x)~x 2-l .
Ecn11=-~x 2 -l.11.83. TaK KaK o6nacTb lzl > l He conep)f{HT TO'!eK sernnett115! z = 0, z = lcpyHKU1111 fl.z) 11 nonycKaeT Bh1nenett11e onH03Ha'!HhIX serneH:, TO no TeopeMe JiopaHa mo6y10 0,[(H03Ha'!HYIO BeTBb cPYHKU11111z = - - 11sp51nJiopatta.11MeeM:1-z1- 11z1lz = ei1ll3 (1-~)- = ei1l/3113V~~= ei1ll3 (l+~11=11(1+3) ... (311-2)],11 1111 13 z~ 1-z z MOIKHO pa3JIOIKHTb~ =~ 1-!lz-l\J!=Z11f (-.!.)ll =OZ. TaKKaKj(l/2)>0,To~(-.3!. )(-.!.=1 ) .
. (-.!.-n+l)33II .lzl >I.Fna6a 1212.S. 1) TiyCTb cPYHKU.H51 fl.z) aHMl1Tl1'!Ha B TO'!Ke z = oo. Tio onpe,[(eJieHHIO :no 03Ha•rneT, '!TO cPYHKUH51 g(~) =fl l/ ~) attanl1TH'!Ha B TO'!Ke ~ = 0 H,cnenosaTeJibHO, B oKpecTHOCTH TO'!KH ~ = 0 npencrnBHMa a s11ne p51na Tei1nopa g([;)= I;=oati;tCTHOCTH TO'!KH z. 3TO pasHOCHJJbHO TOMy, '!TO cjJytt~1151Jlz) B OKpe-= 00 npeACTaBHMa B BHAe p51Aa Jiopattaf(z)=l.,;=oatz-*.TiocKOJJhKY res f (z) paseH KOJcjJcjJ111J,11eHTY a 1 c o6paTHhIM 3HaKOM, a;:::;;ooK03cpcjJmJ,11eHT a 1 MOJKeT 6brTh nonyqett no cjJopMyne a1= lim(f (z)-a0 )z,r,[(eGo= Jim f(z);:---700z---+oo=f(oo), TO OKOH'!aTeJibHO HMeeMresf(z)= lirn(j( 00 ) - f(z))z,-=oo;:---700'!TO 11 rpe6osanocb noKa3aTb.2) Jierno BHAeTb, '!TO Jim z(f (z)- f:->~(00 ) )= Jim g(;)- g(O)~->0~=g'(O), r.nePEWEHf.15l Tf.111I14HblX 3A)J,A4328g(s) = f( lls). lloc11e.uH11i1 npe.ue11 cymecrnyeT B C11ny aml.JI11TwrnocrncpyHKU1111/(z) B TO'-!Ke z 00.=12.10.
6) Ilpe.uno11o:>Kl1M, '-!TO a I- b. IlycTb CHa'-laJia z = 0. llOCKOJlhKYLn~-a= ln1~-a1+iarg[~-a]+i2k1l,~-b~-b~-bTO, 3ac}l11Kc11posaB k E Z, nony<rnM O.llH03HaYHYKl BeTBh 11cxo.11Hol1 MHOro3Ha'-IHOtt cPYHKU1111. Ey.ueMBeTBbz11B.UaJihHettweM Cl1MBOJIOM fk(z) o6o3Ha'-laTh(Ln ~-a).,-b k1) TiycTh n 2: 0. Tor.uaf;.(z) 6y.ueT aHa1111rn'-!Hares fk (z) = 0.B=0.
Il03TOMYTO'-!Ke z:=0=2) TiycTh n -1. B :.noM cnyyae TOYKa zPH.llKa cpyHKum1fk(z). llo:.noMyres fk (z):=o= lim( zfk (z)) = lim [in Iz : -+o=az-b:->o= 0 CCTh nonKJc nepsoro no-I+ i arg .:z --ba + i2kn] =In 1::1b + i arg::b + i2k1l = (Ln ::).bk3) IlycTh n :S -2. Torna TOYKa z = 0 eCTh nomoc nopll.UKa (-11) <PYHKU1111fi(z).
Ilo3TOMY1res fk (z) = lim:=O(::. -" fk (z)I(z-a)(-ll-t)= - - - Ln-(-n-1)!z- b k)(-11-tJ:-+0(-11-l)!:=O=(-n-1)!1(-n- 2 )!(-1)-11-2[1_1]=(-a)_ 11 _(-b) _1=-n1n:11+1+ 1 ( b11+1- a11+1 )=a11_1b11+I1=Tipe.unonmK11M Terrepb, '-!TO z oo. BMeCTO ¢YHKUl111 fk(z) paccMOTp11McpyttKlVHO gk(z) =fk(llz) (CM.
12.5), KOTOpaH 6y.ueT HMeTb 811.llgk(z)=z -11(L za-1)n--zb- 1.kEy.ueM 113yqaTh cpyttKu111-0 gk(z) B TO'-!Ke z = 0 (CM. 12.5).1) llycTh n :S-2. B3TOM cnyqae gkCO) = limgk(::.) = 0 11:->0Z-gk'(0) -- 1·1m:->011(Ln :a-l):b-l k --z0.C11e.11osaTenhHO (cM. 12.5. 2)), B 3TOM cnyqae ;~;, fk (z)= 0.PEWEHM51 TV!DI1YHbIX 3AL(A Y2) Dyen, n329=-1. B 3TOM CJI)"fae no-npe)f{tteMy gk(O) =0, HOz" (Ln za-1)zb-1 k, (O) _ .gk- 1Im.= I1m:->O[iz:->0IJza-11:a-l +i2kn =1"?k 7r.n - +iarg-bz -1zb-1Cne.noBaTeJibHO, (cM. 12.5.
2)), B 3TOM CJIY'fae res fk (z)= -i2kn.3) ITycTb n = 0. B 3TOM CJIY'fae g;(O) = (Ln l)k = i2kn, no3TOMY( Ln za-l) - i2kn'(O) _ 1.gkzb-1 k- 1mz:->0.- 11mIn I za-l 1+ i aro :a-l_"'zb-1 _ bzb-1-z:->0-a.Cne.noBaTeJibHO, res fk (z) =a -b.=114) Tenepb nycTb n > 0. B 3TOM CJI)"fae TO'-!Ka z 0 ecTb non10c nopll.UKa.nm1 n1060J;i sern11: gk(z), 3a HCKJIJO'-!ett11:eM k = 0. PaccMOTPHM pll!l JiopattacpyHKl.(1111 gk(z) B OKpeCTHOCTl1 TO'-IKl1 z= 0.I10CKOJlbKY ~~~ fk (z) eCTb K0-3cpcpm.(11eHT c o6parnbIM 3HaKOM np11 z B pll.ne Jiopatta cpyttKu1111 gJz), TOHeo6xo.11.HMO nony"lHTb p!I.11.
TeH:nopa a oKpecTHOCTH TO'-lKH z = 0 ¢YHKUHH( Lnza - l) .zb-1 kITp11 3TOM ttac 11ttTepecyeT TOJibKO K03cpcpm..111ettT 3Toro pll.na1np11 z"+ . 3TOT K03cp<j:J11u11eHT, KaK 113BeCTHO, paBeH_l_(Ln za-1)(11+1) (0) = _l_[n!(-l)"a"+1 - n!(-l)"b"+1(n+l)!zb-I k(n+l)! (za-Ir+ 1(zb-1)"+ 1=l:=0n+la"+l_b11+II1o3TOMY res fk (z) =:=~n+ 1.
3a.ua•rn nonttoCTbJO pellletta.12.23. 11) Ctta'-!ana nycTb a> 0. PaccMOTPHM <j:JyttKUHJO KOMrnieKcttoro= (1- eia:) I z2 B 06nacT11:nepeMettttoro .f(z)= {z:lzl < R, Im z 2: 0) \ {z: lzl <£,Im z 2: 0).B yKa3aHHOH 06nacT11 cpyHKUHll .flz) attan11T11'-!tta, no3TOMY no TeopeMeDR,sKolll11 1.,'fuDR.c= 0. 3TOT 11ttTerpan no rpatt11ue 06nacT11 npe.ucrnBHM sf(z)dzs11.ue cyMMbl-E_ eiaxRl-eiax1I --2-dx+ f--2-dx + I-RXEXl:l=Rlm:;,o1 _ eiaz- -2- dz +zIl:l=Elm:;,o1 _ eiaz- -2-dz =11+/i+h +hzPEWEHl151 nmW-IHb!X 3A,ll,A4330I1HTerpa11 / 3 ouemrnaeTCH cne,D,y!OlllHM o6pa3oM:l+e-a-' Rd(J< 3-Jde=2Jr,Rz- RoRff:l=Rlm :~Or,D,ey =Im0.z 11 a>Cne,D,oBaTeJlbHO,f,=/4lr->O1- eia: d = fo 1- exp(iac: e;e)l=I=<zz2rrp11R-->+oo.h:PaccMOTpttM HHTerpan~";w.
;ede = ·f" exp (ia£e;e-1) dBEzezoE elm :;o,O;e.Ee£-->+O ttMeeT£-->+O. Y cTpeMJ15151 R-->+oo 11IlO.Llb1HTerpa11bHa5! cj:>yHKUHH B nocne.[(HeM ttHTerpane np11ia .rrpe.[(en, paBHbIH£-->+OIIo:noMy / 4-->-narrpttB OCTaJlbHhIX HHTerparrax, rronyqHM paBeHCTBO~v.p.l-eiaxf --x -dx = 1ia.2Cne.[(yeT OTMeTHTh, qTo 3TOT )Ke pe3yJihTaT no11yq11TC5!, ec1111 rrpHMeHHThpe3y11hTaT perneHmr 33,D,aqttBCrry'Iae a<0,12.22.C!l,eJiaB 3aMetty nepeMettttoi1x->-x, nOJIY'fttM paBeHCTBO1- eiax~ 1- ei(-a)xv.p.-dx = v.p.dx22xx~f --IIOCKOJibKYf= Jr(-a),(-a) 6y.[(eT Berr11qHHa IIOJ10)!(HTeJibH351 H MO)!(HO rrpHMeHHTbpe3yrrhrnT rrpe,D,hI,D,yWero c11yqa51.
TaKHM o6pa3oM, OKOH'laTeJlhHO 11MeeM~v.p.12.27. 6)l-eiaxf --x -dx =Jr Ia j, aE R.2IIocKOJihKY HCXO.LlHhIH HHTerpan paBeH ReCMOTpHM cj:>yHKUHIOf~eilnx0x +lf2-dx,TO pac-eiln~(z)= -,-z- + 1KOMIIJieKCHOro rrepeMeHHOro z B KOM-rrneKCHOH IIJ10CKOCTH c pa3pe30M ITO IIOJ10)!(HTeJ1bHOH qaCTH ,D,eHCTBHTeJlhHOH oc11 11 c BhIKorroTOH TOqKoi1 zcj:>yHKUHHInz= 0.IIp11 3TOM B yKa3aHHOH o6rracTHO,D,H03HaqHo orrpe.[(enetta 11 paBHaInz= In jzj + i arg z.
Pac-CMOTpHM 33MKHYThIH KOHTYP HHTerp11poBaHH5!fR,£={z: jzj= R} U{z: jzj= £} U [£, R] U [R, £],r.[(e YK333HHhie OTpe3KH pacIIOJIO)l{eHhl Ha BepxHeM H HH)l{HeM 6eperax pa3pe3a. IIo cj:>opMyrre, rrp11Be,D,eHHOH BO BBe.n;ettHH K .[(3HHOH rrraBe (cM. TaK)!(e12.26),IIOJI)"1HMPEWEHM5J TMDW.!Hb!X 3A.[(Al.Jleiln:·{eilnzeiln:J. ~z=2mr e s - - + res-222/ R.£ z +1z=i z +l:=-i z +ll331·{eilnieiln(-i)}= 2m - - - - - =2i2iei(iargi)= 21lil1HTerpan no KOHTYPYeiln:2ir R.ER eilnxMO)f(eT 6bJTh npe,!1,CTaBJieH TaIOKe B 811,[\e cyMMbl£f -2-dz+ f-2-dx+ fl:l=R Z + 1+1EXexp(iJn(xei2JT))2RXeiln:f - 2-dz=I, +!2+!3+/4.dx++11:1=< Z+1.LJ:oKa)f(eM, 4TO 11HTerpaJibI / 1 11 / 4 crpeMl!TCll K Hymo np11 R-+t<XJ 11COOTBeTCTBeHHO .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
