Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 48
Текст из файла (страница 48)
IlOCKOJibKY lzl = 1, TO 11CXO,[\HbIH 11HTerpaJI 3KBl1BaJieHTeH CJie.llYIOI.UeMy HHTerpaJiy:izPEWEHI151 TI1fil14HblX 3AJJ,A q315if:"I e;8 -1 Ie;8 dB= if:" .J2-2cosB e;8 dB=f. BBdB -J,r~2 sm-sm. B . BdB =i/ -! •=i ~2 sm-cos122020TaK KaK 2sin ce I 2) cos e =sin (3 e I 2) - since I 2), a 2sin ce I 2) sine=cos ce I 2) - cos (38 I 2), TO 11HTerparibl /1 11 Ii paBHbl COOTBeTCTBeHHO11 = (cos(3B I3/22) + cos(B /2))12n = 2/3- 2 + 2/3- 2 = -8/31/2I?= ( si n(B I-0,22) _sin(3B I 2)Jl3/21/2vCJie.[(OBaTeJibHO, 11CXO.[(Hbll1 11HTerpari paseH"=O.08.--1.36.27. ITycTb z 1 11 z2 E D. Tor.[(a B c1rny BhmyKJJOCT11 o6JJacrn D OYpe3oK,coe.[(HHH10ll(11i1 z 1 11 z2 , ueJJ11KOM JJe)!(J.fT s 06JJacT11 D. A B c11Jiy aHari11Tw1HOCT11j{z) 11HTerpaJJf' f(z)dzHe 3aBHCHT OT TiyYH, coe.[(HIDl!Oll(ero TOl.JKl1-1z 1 11 z2 11 JJe)!(all(ero B D.
ITo:noMy Mb! MO)!(eM npe.[(nOJJO)!(l1Tb, t.JTO 11HTerp11posaH11e Be.[(eTCH BJlOJlb 0Tpe3KaZ1 Z2 .B TaKOM CJJy1-rne JlAA .f{z) = u + ivHMeeMIt f (z)dzl =It + iv)d(x+ iy)l It udx-vdy + iJ,'.' vdx+ udyl =(tudxr (tvc1xf (f,'.'vdyr+ (tuc1yr ~ (tudx/ +(tuc1y/~22+M22=(u+r(t dxr +(t dyr] =M2jz1-zl.TaKHM o6pa30M, Jl1lH aHarIHTHt.JecKoi1 s 06nacT11 D ¢YHKU11H 11cxonHoettepaseHcrno JlOKa3aHo. Ecm1 Tenepb npennoJJO)!(HTb JIHlllb ttenpepbIBHOCTbHCXOJlHOM ¢YHKU1111, TO MO)!(HO TIOCYpOHTb npHMep (nOCYpOH:Te ero caMH),noKa3bJBa!Oll(HH, t.JTO ttepasettcTBO nepecrneT 6b!Tb septtb!M.6.34. 4) .ll:JJH JlOKa3aTeJlbCTBa attaJil1Tl1l.JHOCTH cpyHKU.HH f(z) B o6nacrnRe z > 0 .[(OCTaTOt.JHO nposep11Th, t.JTO s KroK,!J,OM TOt.JKe z.o, Re Zo > 0, Cyll(ecrnyeTrrpomBO.[(HMf' (z 0 ). ITycTh h E C 11 h-::/; 0, mr.[(a(j{zo + h) - fiz 0 ))/h=PEWEHI151 Tl10114Hb!X 3A)J,A4316J' (z0 )CymecrnosaH11e rrpoHJBO.!lHOH6y.ueT .UOKaJaHo, ecm1 Mhl .UOKa-)f(eM, 'HO MO)f(HO rrepeXO.UHTh K rrpe.ueny rrp11h-+0rro.u 3HaKOM HHTerpana ..[(IDI :noro .!lOCTaTQqHo .UOKaJaTb, qTQ rroCJie.UHHH HHTerpan CXO.!ll1TC5! paB-h, 0 < lhl < o, r.ue o .uocrnrnqHo Mano.
Yq11Tb1Ba5! Mao 11 TOT cpaKT, qrn Re z0 > 0, rro11yq11M c11e.uy10my10 oueHKY csepxyHOMepHO J\1151 scexJIOCTh.UIDI rrocne.uHero 11HTerpa11a:-J(2tdrlzol+ c5) o(It 2+Zo2 I- 2 lzol c5 - c5 2) Cl f 2 +z2 j) ~2 I z01 +c5 -20ultJo(t +Zo)22 2 ''ITO 11 OJHaqaeT paBHOMepHyro cxo.u11MOCTh HY)f(HOro 11HTerpana. Cne.uosaTeJihHO, rrp0113BO.!lHa5!/, (zo) CYlllecrnyeT 11 pas Ha!6.40.
3) B'()=I·z0Im f(z 0 +h)-f(z0 )=_2 z0-J 7tsint7 7 d t .,,_. 0ho (r+ z0t6.39,cmry TeopeMhI Mopepa 11 peJyJihTaTa perneHH5! 3a.naq11.UJI5! .uoKaJaTeJihCTBa Toro, qrn .uaHHa5! cpyHKUH5! j(z) He HMeeT rrepsoo6pa3Hoi1 B YKaJaHHOH 0611aCTH, .!lOCTaTQqHo ITOCTpOHTh JaMKHyTy!O Kp11ByIO, ne)!{alllyIO B .uaHHOH o6JiaCTH, .!lllil KOTOpoi1 11HTerpan He o6pa.I.UaeTC5! B ttyJih.PaccMoTpHM KpHBYIO yz: jzj io} .
Tor.[(a={=l2/Te;e d(}~-dz= iE J'B .1r sin z)0 sin(EeJferno BH,!leTb, qTo rrpH E -++OITOCJie,!lHHH HHTerpan CTpeM11TC5! K2ni.3TO 03HaqaeT, qTO He BhlIIOJIHeHbl ycJIOBl15!, ccpopMyn11:posaHHbJe B Ja.uaqe6.39,T. e. rrepsoo6pa3HOH s 3TOM cnyqae HeT.661• •>0IIycTh a11: rrycThk > 0 . IIocKOJihKY j( x)=xsmaxx2 +k25!BIDieTC5!qernoH. cpyHKUHeH, TO HCXO.UHblH HHTerpan paBeHl-v.p.2J- xsinax dx,10- x- +k-KOTOpblH CXO.UHTC5! rro rrpH3HaKy .[(11p11xne-A6eIDI. .[(anee, paccMOTpHMzeiaz¢YHKUI1!0{ z:j(z)= - 2- - 2KOMIIJieKCHOro rrepeMeHHOro z B 0611aCTHz +klzl < R, Im z 2: 0} \ {z: jz - ikl < E}.HOi1 0611acrn HMeeM, qTo 1''Yaofo:.£fRxeiaxdx-RX2+k2+~zeiazdz-2--1lo-ikl=< Z+k-+(z) dz=0.11HTerpan rro rpaH11uezeia~dz.
R xsinaxJ-2--,=1J-Rx- +k-,dx+l,+/R.r,+koZ=IIo TeopeMe Koum .um1 MHOroCB5f3-rrpe.[(CTaBl1M B BM.Ile CYMMhI IpeX 11HTerpanos:JDR,E8DR,EPEWEHID! nm11qHbIX 3A,QNI317IR ----> 0 np11 R ----> +oo no neMMe )l{op.uatta (cM. 6.48). 11HTe!£ no 0Kpy)l{HOCT11 lz - ikl = £ paseH11HTerpanrpan(t:e;e + ik) exp ( ia(t:e;e + ik)) e;o?8edfJ.JoEe'- + 2ike'. 2rrIE= -1 rJlerno s11.ueTb, 'ITO np11 £ ---->+O11HTerpanI,cTpeMl1TCH K senwrntte-ine-ak .TaK11M o6pa:mM, 11cxo.utth1i1 l1HTerpan pasett=xsinax ix_,, -I()Ho np11 a > 01f=xsinaxd _ 1 -ak-v.p.,, x--tre .x- + k2- x- + k211 k = 0 11cxO.UHbIH 11HTerpan ecTb 11HTerpan )J,11p11xne:r0sin ax dx = ?!.....x2Cne.uosaTeJJhHO, cpopMyna s 3a.uaqe6.66.Tipeo6pa30BaHMeM <llypbe6.61 crrpase,UJI11Ba np11scexa>011 k 2: 0.f (~)¢YHKUIIBj(x) Ha3bIBaeTCH 11HTerpanJci;)=-1-je-i.xq f(x)d.x ..fiii_Jicrr<O BH.lWTb, 'ITO npco6pa3oBaHHC <PypbCMO)!(eT 6b!Tb npe,UCTaBJieHO B m1.ue:llIIHj(~)= ~e-q2 !4af-v2tr-j(.!;)HCXOJJ:HOH cpyHKUHHe-a(x+iqf2a)2 dx.Bbl'll1CJJeHl1H IIOCJie.Uttero 11HTerpana paCCMOTPHM cpyHKUHI-0 KOM-nneKcoro nepeMeHttoro j(z)=· e- a-2- , a >0, B npHMoyronbHOH 06nacT11 ttaKOMnneKCHOH nnocKOCTH: Dn,= [-R, R]x[O, l;/2a] .UIIH ~ > 0 11 DR,c,= [-R, R]x [l;/2a,.UIIH ~ < 0.
PaccMorpHM TOIIbKO nepBbIH cnyqaii - BTopoi1 pac-OJCMaTp11saeTcH attanor11qtto.Tio TeopeMe Kourn HHTerpan rro rpam1ue 06nacT11 pasett ttymo:1<Ji e-azaoR.:c .upyroi1dz= 0.CTOpOHbl, 3TOT )Ke HHTerpan MO)KeT 6hITb npe,UCTaBJieH B B11.UecyMMbI qeTbipex 11ttTerpanos:fR-Rf.; I?e-ax- d.x +i2a_,f-Re-a(R+ir)- dr +o,,e-a(x+iq 12a)- d.x ++Rif0.,e-a(-R+ir)- dr=.;na=l1 + h +h + [4.JicrKo BI1.UCTh, •no np11R----> + oo 11ttTcrpanb1 12 11 14 crpeMHTCH K ttymo.TI03TOMY MO)!(HO nonyqHTb paseHCTBOPEWEHI151 Tl1Dl1qHhIX 3A)J.A4318I1HTerpan B rrpaBOH 'laCH'.I IlOCJie,[(Hero paseHCTBa Bhl411CJilleTCll CJie-Je-"'' dx = /. Tor.ua, np11MeHllll TeopeMy <1>y611-.UYlOll(l1M o6pa30M.
ITycThHl1, nOJJyYl1M paseHCTBO.,.,oo.,2TrooTC0Oa12 = f e- 0 <x-+y-)dxdy= fe-a,.-,.dr f de=1lfe-a 11 dry=-.R'0ITo:nbMy 11MeeM I=cpyHKUIUIJ1l I a.Cne.uosaTeJihHO, npeo6pa30BaH11e <I>yphe1f (x) = e-"'_, paBHO f• (;) = --e-J2ii;214la -a=l1'2/4- - e-'='J2;.a, 4TO11 Tpe6osanoch .UOKa3aTh. CpaBHl1Th c 3a.[(a4ei1 6.49.Fnaaa 77.8. Hai1,D.eM Hyn11 3HaMeHaTeJill cpyHKUHHfiz) = l I (l + z4 ):z =~=cos( 1l+:k1l)+isin( JL+:kJL } k =l, 2,3,4.1l + 21l4.
.J2 + l.J2-Jr + 2JrITyCTh Z1= cos - - - +I Sin - - - = - -5JL4J2 .J2. . 51lz,- =cos -+1sm=---1 4422 ,9JL22 ,71l4. . 71l4J2 .J21li-1l(l - i)=cos-+1sm-=--1-7~3•22 ,J2 .J2. . 91lz4 =cos-+1sm-=-+1-4422BHYTPl1 KOHTypa y ne)!{11T TOJihKO TO'-!Ka z 2, rro3TOMY<ji~=<jir l+z_dz_r (z- Z1)(z- <-2)(<.
- Z3)(z- Z4)=2Jri--------=(-iJ2)(--J2)(-J2 - di.) - i(l+i)../27.20. ITycTh T04Ka2..J2z ~ i5. Bo3hMeM oKpyJKHOCTb .uocTaTOlfHO 6onhworopa.u11yca R c ueHTpoM s T04Ke z, TaK, 4T06h1 .uaHHb1i1 KOHTYP y 11 T04Ka zne1Kan11 BHYTPl1 noi1 OKPYIKHOCTl1. Tor.ua rro TeopeMe Korrm 6y.ueM 11MeTb1f(z)=-.1<ji J(r;) dr;-.<jif(r;) dr;.2m l>-=l=R r; - z2m r r; - zPEWEHY!.51nmWIHblX3A.[lAY319lth f (r;)PaccMOTpl1M 11HTerpan - . 'f --dr;. C.uerraeM 3aMetty rrepeMeHHoro21l! l>-:i=R r; - Zr;- z= Re;rp.
Tornadr;= Rie;rpdq;11_I_.f(~) di;= _l_. 2[ f(z + ~e;rp)iRe;rp dq;2m 1,_=l=R ~- z2m 0Rep= _l 2[ f2rc(z+Re;rp)dq;.0B cm1y HerrpepbIBHocrn cpyttKu1111fiz) 11 ycrroB11H lim f (z) =A 11MeeM;:-+ooI '"·If()Jim - f f (z+ Re''P) dq; =A, ll03TOMY - . cp-r;-di; =A - fiz).R-->~ 2rc 02rci r i; - z1B CJI)'t.Iae, Korna TO'!Ka zED, no TeopeMe Kow11 11MeeM - -. cp f(~) d~ =2m r i;-zp1- -.f (r;) dr;. IfoTerparr crresa He 3as11c11T OT R, rro3TOMY27lf l>-:l=R r; - Z1- -.pf (r;) dr; = Jim - 1- . p f (r;) dr; =A,R-->~ 277:1 lq-:I=« t;- Z277:1 r ( ; - Z'-!TO 11 Tpe60BaJIOCb JlOKa3aTb.7.24.
ITycTb fiz)ff= u(x, y) + i v(x, y), Tornafff(z)dxdy=rSl:ISRCncrraeM 3aMeHy nepeMeHHbIX: xffv(x,y)dxdy.rSl:ISR= p cos cp, y =p sin cp. Toma2RRf(z)dxdyffu(x,y)dxdy+irSl:ISR= fdp f u(pcosq;,psinq;)pdq; +rS/:ISROR2tr+if d pf v(pcosq;, psin q;)pdq;.0izlTaK KaK cpyHKU1111 u(x, y) 11 v(x, y) - rapMOHH'!ecK11e cpyHKU1111 B Kpyre< R, TO no TeopeMe o cpentteM JlJIH rapMOHH'!eCKHX cpyHKUl1H 11MeeM~~00Ju(pcosq;,psin qJ)dq; = 2nu(0,0); Jv(pcosq;,psinq;)dq; = 2nv(O,O),TeM caMb!MfJRRJ (z)dxdy = 2nu(O, 0) Jpdp +2niv(0,0) Jpd p =fi0)n(R 2 - r2).r5l:l5R7.53. 2) 11peo6paJyeM HHTerparrrPEWEHl151 T11n11qHhIX 3A)J,A4320dtI I - I dt- = JIln(l - t )dt- - - JIInt--.Jl nt- ·-t-:t-zt- z0080nepsoM HHTerpane cn enaeM 3aMeHy nepeMeHHhIXdt = -dv.1111l· - t =y,TOfllat= 1-y,1r 1n(l-t)·__:{:._=J ln y dy =-r lnydy . IIycTb 11HTerpan0 I- y - zt- zJo y - (l - z)Jodtf In r-= F(z) , TOrna HCKOMbIH HHTerpan ecTb:t- zI01-tldt-F(l- z)-F( -;, ) .f() I nt - ·- =zf-F(z) KaK HHTerpan rnna Kow11 eCTh aHan11TwrncKa.sI cpyHKUH.sI BHe[O, 1).
Bb1'111cn11M 11HTerpan np11 nocrnTO'IHO 6onhw11x no Mon ymo z,Cf>yHKUH.sIOTPe3KaTOr,I~a no TeopeMe en11HCTBeHHOCTl1 3H3Jil1TJ1qecKl1X cpyHKUl1H 11HTerpan6yneTBh1q11cneHLlJI.sIz ~scex[0, 1).Pa:3JIO)l(l1MldtfoInt-,t -zT. e. - - , B p.s!Ll no CTeneH.sIMHa OTPC3Ke[O, 1): -I- = -Itlz,t-zt -zl:(1- t/ z).sIAPOttHTerpanapaBHOMCpHo CXOLl.sI!UHHC.sI= --l L~-o ( !__ )" . Tor.u.az-zl dt ""'~_ I JI (lnt)t"dt.flnt--=-L...t- z,,_ z"+00t"-1 I'JI (In t)t"dt =--IntTaK KaKn +1o~dt10-1 0ll---ft"dt = - - - , ,+l(n + 1)0/1l,=L _= F (z )fInt-t- :,,_0 (n + 1)- z"+1TO nOJI)"-laeM- p.s!,!1, cxo.u.11Tc.sI, TaK KaK paccMaTptt-0saeM TOqKJ1 z, Monynh KOTOpblX LlOCTaTOlJHO 6onhWOi1. I1MeeM :FI- =- L ~(_ )l11=0(n + l) z"+2= - -l I:~z11=01l=-In(11 + I) z"+'z (1- 1/ z.),,l( 1-l -) .MTaK,F.(1z)=---ln1- zl-z(f In 1-1t _.!!!.._)'t- z=_ l_In ( :-1).
JIn 1-t . .!!:!__ =_!_(111-z-)200:(:-1)z0tt-z2z- lPEWEH1151 Tl1CTYf4HbIX 3A):\A4I'Raaa8l8.2. 10) Pa.m1yc cxom1Mocrn R =321lim ~=Jl-tOOllim ~I cos( in) I. TaK KaKJl-'tOOcos(in) = (e-"+ e" )12, rn lirn~ Icos(in) I= e, R = l/ e./l-'tOO8.7. 3) TaK KaK paimyc cxo.u11Mocrn pH.ua R =,,f"l:J = 1, TO cpyttKlirn '.!j I a1111-tooUl1ll f(z) = I :=o (2n+ 1)- 1 2 211 +1 amUI11TwrnaIB Kpyre /zl < 1, a ee npo113so.u-Hal! pasHa!'(::,) = I:=o z " = 11(1- ~2TeM caMb!M1lf (z.) = -·I+27In--~ +1-zc'1),I z I< 1.HO f(O)=O, U03TOMY OKOWJaTeJibHOl+z/(z)=-ln-, lzl < l.2 1-z8.18.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
