Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 53
Текст из файла (страница 53)
ITmrnMyJci+xrae;;.xdx=±+;+0(;2}4TO 11 Tpe6oBMOCb 1lOKa3aTb.16.55. BocnoJJb30Bas11111cb qiopMynoi1 ITyaccoHa (cM. 16.53) ,11,nsi <PYHKu1111.f(x) np11 x > 0, ITOJl)"111M paBeHCTBOf=L00(x)n=--oo')e-1.-x= L J e-1211m.e-1-xdt =000011=-00--00.'PEWEHI15I Tl1fll1l-lHbIX3443A~A 4s KOTopoM nocne.n:m1i1 HHTerpan s npasoii '-!aCTH ecTb rrpeo6pa3osatt11ec!>yphe (CM.paseH (CM.6.66)6.66tPYHKU:HH e-t11n112x'B3HOe B TO'-!Ke6.49) exp[- n~i2} J'.h:~ = 2nn.3TOT HHTerpaJITaKHM o6pa3oM, HMeeMnp11 x-++O:f(x)={1i f exp[-7[V~11=-=2112x2 2n)= v~f1i(1+2fexp[-n"='xJ]-IIPIIJI O)KEHIDI K MEXAHIIKE II <I>II31IKEAnnapaT Teop1111 attan11Tw1ecK11x cpyHKUl1H Mmf<eT 6bITh 11cnonh30BaH.z:1n51 113Y'-leHl151 noTeHu11anbHO-coneHOl1,llaJibHblX seKTOpHbIX nonew, ecn11 HXon11cb1BaTh c noMO!llhIO KOMnneKcHoro n0Tem.111ana.
PaccMOTJJHM KOMnneKCHhIH noTeHI.{HaJI B rn,llpO,llHHaMHKe, :meKTPOCTaTHKe H TepMO,lll1HaMHKe.KoMnJieKCHhIH noTeuu:uaJIl{upKyJTJLLfueu eeKmopa nom1 A(A" Ay) B)J,Onh 3aMKHyroro KOHTypa CHa3bJBaeTCSI KpHBOnHHeHHblH HHTerpanfc = fcA.(x,y)dx+A,(x,y)dy(Hanpasnettue o6xona no KOHTYPY - nonO)f(HTenhHoe).BeKTOpttoe noneo6naCTHAHa3hlsaeTCSI 6e3euxpe6blA1 11n11 nome1-14uaJZbHbl.M sD ecnu s 3TOH 06nacn1 cymecrnyeT cpyHKl..\HSI u(x, y) TaKaSI, 1iTOA(x, y) =grad u(x, y).Ecn11 seKTOpHoe noneA HenpephrnHo s o6naCTH D, TO u11pKyn5!Ul1S1 noD, paBHa Hymo TOf,lla Hmo6oMy 33MKHYTOMY KOHTypy, np11Hanne)f(aiueMyTOnhKO" TOr)J,a, _Kor.z:1a seKTopttoe noneA -noTeHu11anhHOe. Ecn11 np11 3TOMD OlJ.HOCB513Ha 11 '-13CTHbie rrpOH3BOlJ.HbJe aA,Jay 11 aA_Jax HenpepbIBHhI s D, TO ycnos11e noTeHuHanhHOcT11 non51 A 3KBHsaneHTHO ycnosmoo6naCTboA\"oy=oA,(1)OXEcn11 ycnos11e (1) BbmonHS1eTcS1 so scex T01iKax paccMarp11saeMoro no.TUI, 3a 11CKJII01ieHHeM KOHe'-IHOfO '-111Cna T01ieK Zk>k= 1,...
, N, B KOTOpblXycnos11e (1) ttapywaeTCSI 11n11 TepSleT CMbicn 113-3a 06paiuett11S1 s 6ecK0He11HOCTh XOTSIKOHTYPYC,6b1Oll.HOH 113 np0113BOll.HhIX, TO u11pKynS1uHS1 no 3aMKHYTOMYo6xonSIIUeMy KaKy10-n11po 113 3T11x TO'-leK 11n11 rpyrrny TaKuxT01ieK, MO)!(eT 6hITh OTJIH1iHOH OT HynS1. Ecn11 u11pKyJIS1Ul1S1 no KOHTYPYC,orpatt111111sa10meMy o6nacTh, BHYTPH KOTopoif HaXO.lll1TCSI n11wh O)J,Ha 113TO'-leK Zk, OTnH1iHall OT Hynll, TO TO'-IKa Zk Ha3bJBaeTCSI euxpe60U ln04KOU.fcH33bJBaeTCSI u1-1meHCU6HOCmbJO 6UXpfl B TO'-IKe Zk (11MeeTCSI B 811.llY Oll.HOKpaTHbIH o6xon no KOHTYPYC).IIomoKoM eeKmopa nonS1 A 11epe3 Kp11syro C Ha3blsaeTCSI Kp11son11ttei1HhIH 11HTerpanNe= fcA ds,11r.neA 11-npoeKul151 seKTopaA(x, y)Ha nonO)!(l1TenhHOe ttanpasnett11e Hop-Man11 K Kp11soi1 Cs TO'-IKe (x, y).
Ecn11KOHTypC3aMKHyr, TONe= fcA.,(x,y)dy-A/x,y)dx(ttanpasnettue o6xona no KOHTYPY - nono)f(11Tenhttoe).346nPYIJI0)f{EHH5I K MEXAHYIKE YI <PYl3YIKEBeKTOpHoe noneAD,Ha3bJBaeTcH coJ1eHouow1bHbl.M s 06nacn1ec-ntt cymecrnyeT cjlyHKUHH v(x, y) TaKa.ll, qTo nonHhIH ee .uttcjlcjlepeHUHandv =Ax dy - A_,. dx.<DyHKUH.llEcntt seKTOpHoe nonenoTOK seKrnpaEcnttAseKTopttoeOAJoxoAJoyHAv(x, y)Ha3bJBaeTC.ll ¢YHKi1ueii moKa..l!Bn.lleTCH coneHOHJJ.an hHhIM s o6nacTHqepe3 mo6oii JaMKHYThIH KOHTYPAnoneoA,D, TO He06XO.llHMbIMA .sisn.sieTc.si ycnos11eTOH .llOCTa-oA,oyOX(2)D,paseH ttymo.HMeeT ttenpepblBHhie qacTHbie npott3BO.llHhieB O.UHOCB.s!3HOH o6nacrnTO'-IHhIM ycnostteM conettott.uanhHOCTH non.siEcn11 ycnos11eC c D(2)BhmonH.sieTc.si so scex rnqKax paccMarpttsaeMoro no-nH, 3a HCKJJIOqeHHeM KOHeqHoro q11cna ToqeK zb B KOTOpblX OHO HapyrnaeTC.ll HJIH TepHeT CMbJCJI, TO nOTOK BeKTopa IIOJJ.llAqepe3 3aMKHYThIH KOHTyp,BHYTPH KOTOporo co.uep)f(HTC.s! TOqKa Zb MO)f(eT 6bJTb OTJIH'-IHblM OT HYJilI.To'-IKa zk HaJbJBaeTc.si ucmO'lfluKOM, ecn11r.uec-Ne> 0,H cmoKOM, ecnttNe< 0,3aMKHYTbJH KOHTyp, orpaHHqHsaIOmHH o6naCTb, BHYTPH KOTOpOHco.uep)!{HTC.ll nHIIIh o.uHa :na TO'-!Ka zk.
qttcnoHCTO'-IHHKa(Ne> 0)HnH CTOKaNeHaJhrnaeTC.ll 06iLJ1bHocmb10(Ne< 0).IIpeJinOJIO)f(HM, '-!TO Henpephrnttoe seKTOpttoe noneHO H coneHOH.UaJihHO s o6nacrn:D,A(x, y)noTeHuttaJih-T. e. cymecrnyJOT cjlyHKUHH u(x,y) -noTeHuttanbHa.ll cjlyHKUH.ll, H v(x, y) - cjlyHKL(H.ll TOKa, onpe.ueneHHhie c TQqHOCTbIO .llO nOCTO.l!HHbIX cnaraeMbIX, TaKJi!e, '-!TOdudv=A_,.dx + A-'.dy,= A_,.dy - A,d.x.PacCMOTPHM cpyHKL(HIO KOMnJieKCHOro rrepeMeHHOrof(z)= u(x, y) + v(x, y), z =x + iy,Ha3bJBaeMyIO KOAtn!leKCHblM nomeHtfUGJlOM. KOM!lJieKCHblH llOTeHL(HaJI .l!BJI.sieTC.s! a11amm1qecKOH ¢YHKI.(tteif s o6nacrnA= A_r(x, y) + iAy(x, y) = f'(z),JlttHHH u(x,y)D.IAI = lf'(z)I, arg A= -argf'(z).= const Ha3bJBaJOTC.ll 3K6UnomeHtfUGJlbHblMU JlUHUflMU HJIHJlUHUflMU ypo6Hfl.
JlHHHH v(x,y) =const Ha3hJBaIOTC.ll JlUHUfl.A·IU moKa. JlHHHHypOBH.ll H nHHHH TOKa o6pa3yIOT opTOroHaJibHYIO ceTb. JlHID!H ypOBH.s!cjlyttKUHHnon.llAu(x, y) opTOroHaJihHhI seKTopy noJI.ll A. Cne.n:osaTeJihHO, seKTopv(x, y) = const, T. e .ttanpasnett no KacaTenhHhIM K nttHH.l!M ypoBH.llJIHHHH TOKa .l!BJI.l!IOTCll BeKTOpHblMH JIHHH.l!MH paccMarp11saeMoro BeKTOpHOro nOJill.
3aMeTHM, '-!TO347CTPl1JI0)l(EHH5J K MEXAHl1KE 11 <l>I1311KEEcm1 np0113BO,n:HaHj'(z) BHyrp11 KOHTypa[c +iNcTOqeK Zk, TOc HMeeT KOHe'rnoe ql1CJIO oco6b1x=2n(f,::,;:=.:,resj'(z).KoMnJJeKCHbIH noTem~uaJI B rHJlPOJlHHaMuKea-ECJIHnomoc <PYHKl..\1111 f'(z), TO s TOqKeacPYHKU11H fi:z)HMeeT pa3-JIO)!(eH11e BH,n:a!( z) =fOBOpHT, qTOHUK 0611JibHOCTl1C_,,(z-a)"p l+ ...
+ - - - +2n z-a[+iN+ - - . ln(z-a)+-C0 +C,(z-a)+ ....2mf+iNqJieH - - . - ln(z - a) Onpe,n:eJIHeT B TOqKe a 6UXpeucmO<J2nzN H 11HTeHCHBHOCTl1 r, o603HaqaeMbIH (a, N, [), qJieHp l2n z-a-- - - - owwflb c MoMeHTOM p, o6o3HaqaeMbm (a; p) ( p - KOMnJieKcHoe q11cJio), pa.n:rryc-seKTop.D:H!llee qepe3 TOqKyaponpe,n:eJIHeT HanpasJiem1e OT .n:11noJIH, npoxo-B HanpasJTeHJ.Hi JTUHUUTO!Ca.Ecmf Ha 6ecKOHeqHOCTHn ) c ,pr + iNc c_,.1,z= ,,z + ...
+-z+--.- 1nz+ 0 +-+ ... ,2m2nf+iN ITO qJieH - - n2niHOCTl11.Nzzonpe.n:em1eT Ha 6ecKOHeqHOCTH BHxpe11CTOqHHK o6HJib-11 HHTeHCHBHOCTl1f,qJieH_.E_ Z2n- .D:HTIOJib C MOMeHTOMp.HaiiT11 noJie cKopocTeii .D:BH)!(eHHH HeC)!(l1MaeMoii )!(11.D:KOCTH, KOM-nJieKCHbiii noTeHuHaJI KOToporo paseHPeiue1-tue. f'(z)= a,az, a > 0.TI03TOMY BeKTOp CKOpOCTlf napaJIJieJieH .n:eiiCTBH-TeJibHOH OCH, HanpasJieH B noJIO)!(HTeJibHYIO CTopotty 11 no .D:Jil1He paseH aBO scex TOqKax TIJIOCKOCTl1.2 . .[(Bl1)!(eH11e )!(H.D:KOCTH 3a,n:aeTCHKOMilJieKCHblM noTeHuHaJIOMj{Z)= z 2.HaHTl1 noTeHUlfaJI CKOpOCTeii, <i:iYHKUHIO TOKa, JIHHHH ypOBIDI, Jil1HHlf TOKa,seJIHqHHY H HanpasJieH11e seKTopa cKopocTHVaxCTH11VayHa OCH Koop,n:HHaTPeiue1-tue.
fi:z)=x2-V,npoeKu1111 seKTopa CKopo-Ox H Oy.=z 2 =(x2- /) + i2xy, n03TOM)' nOTeHUHaJI CKopocreii u(x, y)y2 , cPYHKI.llfHTOKa v(x,y) =-2.xy. JlHHlflf ypOBHH u(x, y)const - rnnep6oJibI, JIHHl111 TOKa v(x,y)= 2.xy = const -q11Ha BeKTOpa CKOpOCTl1 H ero apryMeHT eCTb=x 2 - /=rnnep60JibI. BeJIH-348f1Pl1J10)KEHJ15! K MEXAHl1KE 11 <f>l1311KE!VI= lf'(z)I = l2zl = 2Jx1+y1 , arg V =- argf(z) = - arctg(ylx).IIpoeKUHll seKTOpa cKopOCTH Ha KOOp.llHHaTHhie OCH Ox, Oy: Vax = 2x,Vay = - 2y. Ha'-lano KOOp,llHHaT l!BllileTCll TO'-IKOH rIOKOll lKH.llKOCTH.3.
,L(s1rnceH11e lKH,llKOCTH 3a,llaeTCJI KOMnJieKCHhlM rIOTeHUHaJIOM .f(z) =r c qepe3 OKPYlKHOCThC = { 2lzl = 3 }.In sh 1!Z. HattTH BeJIH'-IHHY nOTOKa, J..!.HPKYJilll{H!OOnwem.r c =0, Ne= 6rr2 .4. HattTH KOMnneKCHhIH n0Tettu11an j(z) Te'-leHHll lKHJlKOCTH, ecm1 H3secTHhI ypaBHeHHll 3KBHrIOTeHl{HaJlhHhlX JIHHHHch z sin=0.y + 2xy = const,f(O)2Omeem. j(z) = - i(z + sh z).5. Te'-!ett11e lKH.llKOCTH onpe,llellileTcl! KOMnneKCHhIM noTettu11anoM.HaiiTH n0Tettu11an CKopocTeii, cpyttKu1110 TOKa, n11tt1111 ypoBHll, JIHHHH TOKa,BeJIH411HY 11 Hanpasnett11e BeKTOpa CKOpOCTH, npoeKl{HH CKOpOCTH Ha OCHKOOp,llHHaT:2a)f(z) = z + 2z +2;6)j(_z) = z 22;s)./tz) = ln(z - 1).J0 meembz. a) u(x, y) = x - y- + 2x + 2, v(x, y) = 2(x + l)y, JIHHHH ypoBHllx 2 - y2 + 2x = C 1 - rnnep6onhI, JIHHHH TOKa xy + y = C2 - rnnep6onh1; !VI =2Jcx+l)1+y1, ArcrV=-arctcr _Y_ +nm,m=-l,O, l; Vox=2(x+ 1),oo x+lV0 _" =-2y;6)u(x, y)x 2- y 2=0ex-+y2t0,v(x, y) =C 1(x-' + y' )'-, JIHHHll TOKa xy =-2xy,(x-c2(x2 ++ y-)02,JIHHHH yposm122x - y =' ' IVJ = (x + 2y2 )3' , Argy-t,22v3arctg(ylx)+(3m - l)n, m=-l,O, 1,Vox= 2x(3/-x 1 )(x2+/)3V' oy= 2y(y2-3x 2 ) .(x2+y2)3'12'ya) u(x, y) = -ln[(x-1) + y], v(x, y) = arctg--, JIHHHH ypOBHll2x-12(x - 1) +y2 = C 1 - OKpYJKHOCTH, JIHHHH TOKa y = C2 (x - 1) - rrpl!Mhie,IVJ=1Jcx-1)2+/,ArgV= arctg-y-+nm, m=-1,0,l;x-l=349T1Pl1Jl0)!(EHI151 K MEXAHI1KE 11 <Dl1311KE=Vox6.x-IV(x-!)2+/' o_,.=y.(x-!)2+/flocTp011Tb KOMTIJieKCHbltt IIOTeHI.UiaJI Te4eHl151 )!(11nKOCTl1, eCJil1 113-BeCTHbl ypasHeHl151 Jll1Hl1tt ypoBH51x2+ 2xy + x = const,f(O) = 0.- /2Onrnem.
fiz) = (1 - i)z + z.7.flocTp011Tb KOMIIJieKCHbltt IIOTettu11aJI TeYeHl151 )!(11nKOCTl1, eCJil1 113-BeCTHbl ypaBtteHl151 Jll1HHi1 TOKa: cosx shy= const,/{0)= 0.Omeem. f(z) =sin z.8).Hai1:T11 u11pKyJI51u1110 rro OKpy)!(HOCrnMlz ± al = a,ecm1 113secTeHKOMIIJieKCHbltt IIOTeHUHaJI TeYeHH51 )!(11nKOCTl1f(z)Omeem.9.= 5i In (z2 -a 2).r c =-I On IIO 06e11M OKPY)!(HOCT51M.flo 3anaHHOMY KOMIIJieKCHOMY IIOTeHU11aJIY orrpet1eJil1Tb o6HJihHOCTb11 11HTeHCHBHOCTb B11Xpe11CTO'-IHl1KOB, MOMeHTbl nHIIOJiei1:, 11 11CCJienosaTbTIOBCp;CHHe Te'ieHH>I Ha 6eCKOHC'iHOCTH:a)fiz)=Cz(C=cx+i/3); 6)fiz)=z"; s)j(z)=z- 1;r) fiz) = ln(z 2 - a 2 ), a > 0; A) fiz) = z + R2z-'; e) fiz) = z - R2z- 1.Omeem.a)ex -r = 0, N = 0, Ha 6eCKOHe'-!HOCTl1 nHIIOJlb cMOMeHTOMp= 2n C, v =i~;r = 0, N = 0, Ha 6eCKOHe'IHOCTl1 MYJibTl1IIOJlb nop51nKa 211;2B) r = 0, N = 0, B TO'-!Ke z = 0 n11rrOJib c MOMeHTOM p = 2n, v = -r-l / q>,z = re;qi; V= = O;r) B TO'IKax ±a 11CT04Hl1Kl1 o6HJibHOCTl1 2n, Ha 6eCKOHe'IHOCTl1 HC6)TO'IHHK o6HJibHOCTl1-2n '22v= ~;z -an) B TO'IKe 0 n11IIOJlb C MOMeHTOM-2nR 2 ,Ha 6eCKOHe'IHOCTl1-27t,1V= 1- -Rr-, e;2 91 , z = re ;qi,V00= 1;e) B TO'IKe 0 n11IIOJlb c MOMeHTOMMOMeHTOM2n, V=1+R1·2-, e'r-91, z.91= re'2nR 2,,Ha 6eCKOHe'IHOCTl1 n11rrOJib cV 00 = 1.350DPvU10)1{EHl15l K MEXAHI1KE YI cDl1311KE10.Hattn1 KOMrIJieKCHblH JlOTeHUHaJl Te'-leHm! BO BCeH JlJIOCKOCTH, o6-pa30BaHHb!H BHXpeHCTO'-IHHKaMH {(ab6ecKOHe'-!HOCTH 3a.a;aHHYIO CKOpOCTbOmeem.,[(11110JlbN_=I11.TO'-!Ke=f(z)c.MOMeHTOM:= Nk-.;;:' Nve-·a z + L.k : If k2nve-ia1111HTeHcHsHocTbl{)1= 1, ...
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