Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 40
Текст из файла (страница 40)
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pa,n:11yca r2'.0 Ha3bI-BaeTcJI MHO)!(eCTBO TO'IeK { z: IPk(z)i = r, Pk(z) = (z- z1) ... (z- zk)}. ITycThE - MHO)!(eCTBO TO'IeK, rpaH11uei1 KOTOporo JIBJIJieTCJI CBJ13HaJI JieMHYICKaTa. I1oCTpOYITb MHOfOlJJieHbI <Da6epa .D:JIJI MHO)!(eCTBa E.17 .69. IloCTpo11Tb MHOro'IJieHbI <Da6epa MJI MHO)l(eCTBa E = [-1, 1].17.70. Ll:oKa3aTb, 'ITO BO BHYTPeHHHX TO'IKax MHO)!(eCTBa E c .n:ocTaTO'IHO rna,n:KOH rpaH11ueJ:i:8£ HMeeT MeCTO paBeHCTBOr.n:e Fk(z) - MHOrotJJieHbI <Da6epa ,n:JIJI E, onpe,n:eJieH11e cpyttKUHYI <D(0,n:aHo B 3a,n:aqe 17.66.17.71.
.[(oKa3aTJ>, qTQ nmcpopMyJie (CM. BBe,n:eHYie)QeJIOM cpyHKQHH f(z) BJ>lqHCJI.sieTC.sI noa= lim ln If(z) 1.Iz I1:1--+~17. 72. Onpe,n:eJIYITh THn uenbIX cpyttKu11i1:1) P(z) e0 z;2)ea: -l P(z);z3)P(z)cos(crz+a);4) P(z)cos(a.Jz 2 +a)r.n:e P(z) - anre6pa11qecK11i1 non11HOM.17.73.
ITycTh uenaJI cpyHKUHJI.flz) TaKOBa, 'ITO.flz) =Co+ C1Z + ....1) Ll:oKa3aTb cne.n:y10rrme HepaBeHcTBa:- .-ln I f(z)1lffi1 : 1 --+~Iz II<_ (J",-n,,11:1 lim -\II c,, I= lim" n!j c,, I~ a1n1-.~3KBH:BaJieHTHbI.en--+~2602) .IJ:oKa:mTh, qTo THII cpyttKUHHj(z) EE paBeHCJTor.ua H TOJihKOTor.ua,Kor.ualim~n!I c,, I= CJ.n-?oo17.74.
ITycTh j(z) E Ea H sup If (x) I< 00 • )loKa3aTh, qTo rrp11 mo.rER6oM HaTypa.rrhHOM ksup I !<kl (x) I~ cl sup I f (x) I.rER.rER3To ttepaBeHCTBO tta3hIBaeTc5I 11.epa6e11.cm60J11 Eepmumeu11.a.17.75. ITycTh cpyHKU115I j(x), x E R, 6ecKotteqtto .z:i:11cpcpepeHUHpyeMa 11Isup f<k>(x) l~M d, k= 0,1,2, .....rER)loKa3aTh, qrn cyruecrnyeT ¢YHKUH5I F(z) E Ea. KOTOpa5I coBrra.z:i:aeT cj(x) rrpH z = x ER.17.76 . .IJ:oKa3aTh, qTO eCJil1 ¢YHKU115I <D(z) HMeeT rrepHO)J; l > 0 HcD(z) E ET, TO<D(z)= """.t...,k=-,,cke 2:rik:/I , r.z:i:en:::; [Tl/(2n)].17.77 .
.D:oKa3aTh, qTo ecJIH ¢yttKUirnf(x), x ER, - tterrpephrnHa5I ¢YHKUH5I c rrep110.l(OM l > 0, 11 eCJIH cyruecrnyeT ¢YHKUH5I'l'(z) 3KcrrotteHuHaJihHoro T11rra, MeHhlllero qeM T, TaKa5I, qTosup If (x)-lf/(x) I~ CJ, TO cyruecrnyeT TPiffOHOMeTpHqecK11i1 nonH.rERHOM':rikxll<D(x)TaKOH, YTO sup.rER=L~=-ll c"e-If (x)-<I>(x) I~ CJ ., n :::; Tl/(2n),OTBEThl K 3A)J;A. qAMDiana I1.2.1) Re w =0, Im w =1;2) rrp11n=2k+l : Rew= 0, Im w = (-1)\rrp11 n = 2k: Rew= (-1/, Im w = O;21 21113)Rew= 2°"r"'C/1 (-l) 111 ' Imw=O·'~m =O4) rrp11 n = 2k + 1: Rew= 0, Im w = (-l)k+I;rrp11 n = 2k: Rew= (-1/, Im w = O;5) Rew= 114, Im w=O; 6) Rew= -413, Im w=O;7) Re w = ""'"'21 (-1)"' C2"'x"-2"' Yi,,, '~1n=On11-IIm w = ~ 1 2 1 c2,,,+1 ( - l)mx"-2111-1L..Jm=O112,,,+1 .y'8) Re w = cos (2n a), Im w =-sin (2n a);9) Re w = 2" ( cos rp )" cos nrp Im w = 2" (cos rp )" sin nrp ;22 ,2210) Rew= 2" (sin rp)" (-1)" cos nrp,22.
rp )" (- l)"+I sm-.. llfP_ ( smI mw= ')"221.4.1) lwl = J2i5, arg w = arctg 3;3) lwl = 2 12 , arg w = O;0,n=4k,5) argw={-n/2,n=4k+l,n,11=4k + 2,n/2,n=4k+3,6) lwl = 1, arg w=5rr./6;2) lwl = 8, arg w = n;4) lwl = 14, arg w = 0;kE Z ;7) lwl = 1, arg w = -n I 4;8) lwl = 2lcos a!2J, arg w = a/2;9) lwl = 1, arg iv= a+ 2nn E (-rr., n],10) lwl = 2 cos cp/2, r.D,e cp3rp I 2,arg w = - 2JZ" + 3rp I 2,{2Jr + 3rp I 2,= arg z EI rp I:.,:;11E Z;[-n, n],2JZ" I 3;27r I 3 < rp :.,:; Jr;-Jr < rp < - 2Jr I 3.262OTBETbl K 3A)lA 4AM1.5.1) z = i~, cp= n/4 + nn,11E Z;13) z = x +iv x = ±--y x < o·.,.J3 , - ,2) z = i(l - '12);4)z=3/2-2i;5) ecnH n = 2, TO z =x, x E R; ecnH n f. 2, TO z = e;~,= 0, ±1, , ±[n/2], npH mo6oM /1 EN, z =0;6) ecnH /1 = 1, TO z =x, x ER; ecnH /1 f.
1, TO z = e;~,r.n:e cp = 2krrl(11+ 1), k =0,±1, , ±[(n+ 1)/2), npH mo6oM n EN, z =0.r.n:e cp = 2knln, k1.11.2r"+ cos(x+na)-r"+ 1cos (x+(n+l)a)-rcos (x-a) +cos x!)~~~~~~~~~~~~~~~~~~r 2 -2rcosa+l2r"+ sin(x+ 11a)- r"+ 1sin(x+ (n + l)a)- rsin(x-a) +sin x2)~~~~~~~~'------'---'----'-~~-'---'~~2r -2rcosa+l) cosx-rcos(x-a)) sinx-rsin(x-a) .34r 1 -2rcosa+I'r 2 -2rcosa+I5)2:7= 1 cos(ia)6) 2:~ = 1 sin(ia)1sin( (n+l/2)a)-sin(a/2)2sin(a/2).a-::e2'ITk,kEZ:''cos(a/2)-cos((n+l/2)a)2sin(a/2),a-::e2T1k,kEZ.1.13.1) IlpH r1 = 0 - OTKpblTbIM map c l..(eHTPOM B T04Ke Zo 6e3 T04Kl1 zo;npH r1 > 0 - KOHI..(eHTpH4eCKOe KOJlbl..(0 c l..(eHTPOM B T04Ke Zo 11 pa.n:11ycaM11r1 11 r1;<r2 - <r1B T04Ke3) npH = 0 - MHl1Mal! OCb c Bb!KOJIOTOM T04KOM y = O; np11OKPY)!{HOCTb c l..(eIHpOM B T04Ke (l/(2C), 0) H pa.n:11ycoM l/(2ICI);c j:.
0 -2) 4aCTb nJIOCKOCTH, orpam14eHHal! yrnoM paCTsopaz = 0, KOTOp&1i1 o6pa3osaH ny4al\!H arg z = cp 2 11 arg z = cp 1;c==4) np11 C 0 - nei1crn11TeJibHal! ocb c BbJKOJIOToi1 T04Koi1 x O; np11OKpy)!{HOCTb c ueHTpOM B T04Ke (0, -l/(2C)) 11 pan11ycoM 1/(2/CI);c j:. 0 -5) ceMe11cTBO nyr OKpy)!{HOCTefi, npoXOlllllllHX 4epe3 T04KH(BKJII04al! npl!MOJJHHei1Hble 0Tpe3KH, COelll1HlllO!lll1e T04KH z, 11 Z2);Z1HZ26) 3JIJIHTIC c cjloKycaMl1 B T04Kax z, H Z2 c 6oJJblllOM nonyocb!O, paBHOM a;7) rnnep6ona c cjloKycal\rn B T04Kax z 1 11 z 2 11 nei1crn11TeJibHOi1 nonyOCblO, paBHOM a;263OTBETbl K 3A)J.A 4AM8) KOMrnreKCHall nnocKOCTb 6e3 napa60Jibl x'=-1 + y-14H 'laCTH KOM-nneKCHOH nJIOCKOCTH, Jie)f(all.(eH BbIWe 3TOH napa60JibI;9) 06oe.rurneH11e OTKpbIThIX Kpyros { z: lz-il<-V2}, lz+il<-V2} 6e3 HXo6meH: qacrn;10) OrKpbITal! o6nacTb, orpaHH'leHHM Kp!IBOH lz-zdlz-z2l=R2, r.ne z1 11 z2 Kopm1 ypasHemrn z-' + az + b = 0.
3rn Kp11saJI npe.ncrnsnJieT fMT, npo113se,neH11e paccTOJIHHH KOTOpbIX OT z 1 11 z 2 nocroJIHHO 11 paBHO1.14. ITp11 k i- 1 - OKp~HOCTh c ueHrpoM s TO'IKez0np11k= eiz, z1lk--11 - lk--11-?--=1--?- -11 pam:1ycoMR2.kr = - 2- - I z 1lk -11z2I;npJIMaJI, npoxo,nJimaJI qepe3 cepe,n11Hy orpe3Ka [z" z 2] 11 nep-neH.nHKYJIJipHall npJIMoH:, npoxo,nJimeH: qepe3 TO'IKH z" z 2.1.15.a)l) BttyrpeHHOCTb e,nHHH'IHOro Kpyra c ueHrpoM s TO'IKe z = 0;2) e,nHHH'IHaJI OKp)"'A<HOCTb c ueHrpoM s TO'IKe z =O;3) BHeWHOCTb e,nHHH"IHOfO Kpyra c uempoM B TO'IKe z = 0.6) He Jie)f(HT.1.16.
1) UeH'I]J OKPY2!<HOCTH s TO'IKe -bla. pamryc paseH J(b 2 -ac)/ a 23) ypasHeHHe rrpJIMOH:z- e'"z-e'"Z1 Z2 -Z1 Z2[ z1 - z2;l= 0, tp = 2arg(z1- z2 );5) IIJIOll.(a,nb TpeyrOJibHHKaS=lz3-z1I· lz2- z,I2sin ltpl,ltpl=larg(z3- z1 ) - arg(z2- z1)I;1 l2 2- 2 1llz,-z3llzJ-z2I6) pa,n11yc R=-.2 IIm(z1z3 +z3z2+z2Zi)I 'i lzl(z3 -z2 ) +lzl(z 1 -z 3 )+1z3'2(z2 -z 1 ~ueHTp z 0 =- - - - - - - - - - - - - - -21.19. IIp11Im(z1z3 + z3 z2 +z2 zJnosopoTe ccpepbI Ha180° BOKpyr ,n1rnMeTpa, napannenbHoro.neikTBHTeJibHOH OCH KOMIIJieKCHOH nJIOCKOCTH.1.20. CeMeHcrno oKpy)!{Hocrei1, Kaca10m11xc}l .upyr .upyra B no1110ceP(O, 0, l); rrpllMOH , npoxo,nllll.(eH qepe3 Ha'laJJO Koop,n11HaT, COOTBeTcrnyeT60JibWall OKPY)f(HOCTb, a IIpllMOH, napanneJlbHOH ei1 H OTCTOllll.(eH OT Ha'la-264OTBEThI K 3AllA 4AMna Koop;:u-rnaT Ha pacCTO}IHJ-!e p > 0, - oKpy)!(HOCTh, Jie)!(all.la}l s nnocKOCT11,HaKJIOHeHHOH non yrnoM arctg p K Mep11n11aHaJibHOH nJIOCKOCTl1.1.24.1) BHyrpeHHOCTh Kpyra c ueHTµOM s T04Ke z = 0 11 pan11ycoMR2!-./l-R 2;22) sHewHoCTh Kpyra c ueHTµOM s T04Ke z = 0 11 pan11ycoM -./1 - R IR;3) OTKpbITaR Hl1)!(HRR nonynnocKOCTh nJIOCKOCTl1 C;4) rrpaBaR OTKpbITaR 110JIYI1JIOCKOCTb, 3a 11CKJll04eH11eM 3aMKHyroroKpyra pan11yca '13 c ueHTµOM s T04Ke (2, 0).1.25.1) ('13 + i)/2, (-'13 + i)/2, i;2) e -i(x!S+krrl2J.
k = 0, 1, 2, 3;3)±efi. (cos n/8 - i sin n/8);4)if5 e iCaJ5+2nrrl5J, n = 0, 1, 2, 3, 4, a= n:- arc sin 3/5;5)3e-irrJ ;6) ±'12,±i'16;7)~2cos~e;"' 14 ;128) ±'12 Jcos(<p/2) + sin(<p/2)J e ia, r,[(e a= ..!..arctg + tg(rp/ ) ,1-tg(rp/2)29)±'15e;'l'12,r.ue<p=-arctg2;11) e -iCrrll 4 +2"''17 >, n = 0, 1, ... , 6.1.37.1) .LJ:ei1crn11TeJihHbIM11;10)2' 1116 ei<rrJ32+inrrl4J,n=O, 1, ... , 7;2) q11crn MHl1MhIM11.I'Jla6a 22.3. 113 ycnos11R 3a,[(aq11 cne.uyeT TOJihKO, 4TO Jim lz,,I = lim lz~,J.11-40011-too2.5. Yrnep)!(.ueH11e HesepHo.2.8. Ilp11 ycnos1111, 4TO Jim z,, = 0, mm Jim z,, = = , 11n11 Kor.ua z1111--+oon~0,11-too~no.2.18.
He cne.D:yer.2.21. Jim z,, = (z 1 + 2z 2 ) I 3.11--+oo2.22 . .D:nR scex z, He RBRR!Ollll1XCR 'll1CTO MHl1Mb!Ml1 q11cnaM11. Ilpe.uenpaseH 1, ecn11 Re z 1 > 0, 11 paseH -1, ecn11 Re z 1 < 0. I1p11 z 1 =Ci, r.ue CE R,npe.uena He CYlllecrnyeT.2.23.l)lzl<l,z=l;2)lzl<l;3) ecn11 k > 0, TO lzJ < l; ecn11k<0, TO lzl4) JzJ < oo ;5) JzJ:S 1.:S l;265OTBETbI K 3AL{A 4AM2.26. <p = 2kn, k = 0, ±1, ....2.29.= i", =1) Ecn11 z<p nQ, r,L\e Q - 11ppau:11ottarrbttoe q11cno, MHOJKecrnorrpe.L1eJibHhIX ToqeK ecTb OKPYlKHOCTh lzl = 1; ecm1 <p = rrQ, Q - pau:110HaJJhHoe q11cno, TO q11cJIO rrpe,L\eJibHbIX ToqeK KOHeqHo.
EcJIH izl < 1, TO rrpe,L\eJibH351 TOqKa z0 0. Ecn11 jzj > 1, TO rrpe,L\eJihttaa TOqKa z0 oo;==2) 33MKHyr&IH Kpyrizl ::; l;=3) ecn11 const 0, TO MHO)l{ecrno rrpe,L1eJI&HhIX TOqeK Re z = O; ecn11const t- 0, TO MHO)l{eCTBO rrpe.neJibHbIX ToqeK - OKpy)l{HOCTb (x - c/2)2 + y22l/(4c );4) ecn11 const = 0, TO MHO)l{eCTBO rrpe,L1eJI&H&IX TOt.JeK Im z = O; ecn112const t 0, TO MHO)l{eCTBO rrpe,neJibHbIX TOqeK - OKpy)l{HOCTb x + (y +22l/(2c))1/(4c) ;==5) 33MKHyrbIH Kpyr (x - 1/2) 2 +y2 ::;6) MHO)l{eCTBO (x - 1/2) 2 + y2114;:'.:':27) 33MKHyrhIH Kpyr x + (y + 1/2)8) MHO)l{eCTBO x2+ (y + 112)2:'.:':2::;1/4;1/4;114;9) MHOlKeCTBO a ::; arg z ::; (3.2.35.
Boo6me roBopa, HesepHo.2.42. He cne.nyeT.2.55. Cxo,n11Tca rrp11 a> 0, ~ > 1. A6comorno cxo,n11TCH rrp11 min( a,~)> 1.2.56.l)a<O;2)a<0;3) a<O;4)a>l.e111nQi2.57. Harrp11Mep, z11 =ln(n+l), Q - 11ppau110HaJibHOe q11cno.2.58.z =0, z = -1, Re z < -1;2) cxo,L111Tca rrp11 izl < 1, a - mo6oe; rrp11 jzj > la< O; rrp11 lzl = 1, ecn11 z = eicp , TO rrp11 <pi- 2kn1) Cxo,n11Tca rrp116oMrrpH <p=2knCXO.UHTC51 rrp11 a< -1, k3) CXO.UHTCH rrpH izl < e;pacxo,L111Tca rrp11 mocxo,n11Tca rrp11 a< O;=0, ± 1, ... ;4) CXO.U11TCH rrp11lzl ::; 114.2.59. 1) z-::/- n, nEN; 2) z = -n, nEN; 3) z-::/- nn:, nEN; 4) z-::/- ±nn:, nEN.2.67. He cne,L\yeT.2.68. He cne,L1yeT.2.69. l) Cxo,L\HTca; 2) MO)l{eT KaK CXO.L\HThCH, TaK 11 pacXO.L\11ThCH;3) CXO,L\HTC51.266OTBETbl K 3A.[(A lJAM2.74.
1)2) cxon11Tci:i a6comorno; 3) a6comorno4) a6COJIIOTHO paCXO,llHTCH;5) a6COJIIOTHO cxo.UttTCR;Cxo;:iJnci:i a6comorno;paCXO.llHTCH;6) a6COJIIOTHO CXO.llHTCH.2.75. 1) ,[{JIR lzl < 1 CXO,llHTCH nptt JII060M a, npttJII06oM a; np11 zcpf. 2kn ,= e;'P cxon11TCRTO CXO,llHTCR np11 anp11cp= 2kn,< -112;2)lzl > 1 paCXO,llHTCR npttk E Z,ecJitt aCXO.llHTCR nptt z< -1;ern11f. n, n E N;3) cxonttTCR np11 z f.
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