Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 51
Текст из файла (страница 51)
.LJ:eHCTBl1TeJibHO,2JT ei(lnR+iB)IId:S I.R2 e1W + l01141:::RdB ::;R2n e-edeRJ-2-::;-2o R -12nl ei(lne+i8) I£-->+O~O;R -1 R-)+oodBfo E 2e;w+ l cdB ::;E 2nJ-: : ; - - ~o.1- E 2 l - E 2 e-)+oe-eE0y crpeMJillll R-->+<XJ H £-->+O B OCTaJibHh!X HHTerpanax, nony4HM paBeHCTBOcoeilnxco ei(lnx+i212')--dxdx = 7r(e-n12_en12).o x2+ 1o x2+ IJfl1TaK, HMeeM:coilnxe- dxox2+lf12'1r( -n/2( -n/2= l-e-2ne-e n/2) = Jree-en/2) =en-e-n12=12.29.
3)C.nenas 3aMeHy nepeMeHHOHt= x!,11cxoAHhIH HHTerpan nepe-muIIeM B BH,ll,e-lcoJta-Ip0Bo3bMeM cpyHKUH!O f(z)cos tdt1cop0J.1= - Re ta-I e11 dt, a=-.p= za-I eiz KOMIIJieKCHOro rrepeMeHHOro z B 06nacT11DR.• = { z: lzl < R, Re z ~ 0, Im z ~ 0} \ {z: lzl < £, Re z ~ 0 Im z ~ 0}.B yKa3aHHOH 06nacT11 MHOr03Ha4Hal! cpyHKUHllIIO cpOpMyJieZaza orrpe,!1,eJieHa OAH03Ha4HO= e a In z = e a [Inlzi +i arg zJ, a cpyHKUHll j(z) l!BJil!eTCll aHaJil1Tl1-4eCKOH. Ilo:noMy rro TeopeMe Ko1m1,i;,'foDR .Ef(z)dz = 0.rrpe.ncrns11M B B11Ae CYMMbI LieTbipex HHTerpanos3ToT HHTerpanPEWEHl15! Tl1fUf4HbIX 3A1lA q332fcRf ta-I eir dt +za-lei:dz + (iy )a-1 ei(ir) dy +fI:[= RERl m:<'.O.
Re:<'.0f+I :I=<Im :2'. 0,Re :2'.00 <ITocKOJJbKY< l,aTO 11p11MeH11B neMMY )l{op.L\aHa .L\JJH 11HTerpanaIilim / 7 = 0 , Jim / 4 = 0.HeTPYAHO 11on y411Tb , 4TOR-7+ooE-->+O-YcTPeMJJHH11/4,oueHl1B He11ocpe.L\CTBeHHO 110):\bJHTerpaJJbHYl-O cjlyHKU111-0 B 11HTerpaneR-Hoo11E-H-0B .L\pyr11x 11HTerpanax, 110JJY411M paBeHCTBO·a-If y00f000t a- I eirdt=z;.'.E.ca- 1)a -I e - ·"dt=e 2rea ),0f}:\e f(a) - raMMa-cpyHKU11H 3Hnepa. lfraK, 11CXO}:\HbIH 11HTerpan paBeH1f cos(xP )dx =-Refta- 1e=oop0.1101;.'.:.(a-1)dt =-Ree 2r(a) =p;p.= ~ r(a)cos(~ a-~)=~ r(~ }inDw6a 1313.5.IT ycT& 110CJJe.L\OBaTeJJbHOCTb {f,i(z)}, cocTOHll!aH 113 aHaJJ11Tw1e-cK11x , OAHOJ111CTHblX cpyHKU11H CXO.L\11TCH B 06JJaCTl1DK cpyHI<U1111 .f(z).
floTeopeMe BeH:epwTpacca cjlyHI<l.IHH .f(z) aHaJJl1Tl14Ha B 06nacT11D.Tipeti:110-JJO)!(l1M, 4TO OHa He HBJJHeTCH TOJK}:\eCTBeHHO KOHCTaHTOH. ,[(oKa)l(eM, 4TOOHa HBJJHeTCH OAHOJ111CTHOH. Tipe}:\110JJ0)!(11M 11pOTl1BHOe, T. e. 4TO cymecTBYl-OT, 110 Kpai1Hei1 Mepe, ,[\Be TaK11e T04Kl1 z" z 2E D,z1-:fz 2, 4TO .f(z 1)=.f(z2). TaK KaK cpyHKUl1H.f(z) He eCTb TO)!(JleCTBeHHaH KOHCTaHTa, TO cywecTsyeT KOHTYP.f(z)*.f(z,)r,TaKOH, 4TO ZJ, Z2=.fCz2).BHYTPl1 o6JJaCTl1Er11Dr CD ,11p11 )TOM Ha KOHType0TCl-Oti:a CJJe,[\yeT, 4TO <PYHKUl1H g(z)Dr, 110=.f(z) rKpaHHeH Mepe, }:\Ba HYJJH, Ha rpaH11uer.f(z,) 11MeeTo6JJaCTl1Dry <PYHKU1111 g(z) HeT ttynei1:. Tor,[\a 110 TeopeMe fyps11ua (cM ., Ha11p11Mep, rn.12) Ha411HaH c HCKOTOporo HOMepa no= no([) MH scex n >no([) <PYHKU1111= .f,,(z) - .f(z 1) TaKJKe 11Me1-0T, 110 Kpai1Hei1 Mepe, }:\Ba HYJJH.
3TO 11poT11-g 11 (z)sopeq11T ycnos111-0 O.L\HOJJl1CTHOCTl1 11ocne.L\OBaTeJJbHOCT11 {f,,(z)}. Cne.LlOBaTeJJbHO , <PYHKUHH.f(Z) 0}:\HOJ111CTHa.Tip11MepoM 110CJle}:\OBaTeJJbHOCTl1 OtlHOJJHCTHbIX c!>YHKU11tt, paBHOMepHOCXO}:\Hll!ettCH K cjJyHKU1111 , TO)!(,[\eCTBeHHO paBHOH HYJJl-0, MO)!(eT CJJY)l(l1Tb110CJJe}:\OBaTeJJbHOCTb {f,,(z)= zln}: lim f,, ( z)fl -700= 0,zEc.PEWEHH.51 THITH4Hb!X 3A.[(A 413.11.333Ha1sam1e "TeopeMa rnrnwa.ueH'' npo11cxo.u11T OT rnro, 'ITO :naTeopeMa 51sm1eTC5! aHaJI11T11l!eCKHM Bb1pa)!(eH11eM cJie.uy10wero reoMeTpHl!ecKoro cpaKTa: np11 orn6pa1KeH1111 o6JJaCTHjzj > r, r > 1, cpyHKu11ei1 roOCTaeTC51 HenoKpblTOH 43CTh TIJJOCKOCTH(J)6oJI&we HYJJ51.lfaw(riP)W:3.Uh(J)= w(z) = ll= w(cp), 0 :S+ ivcp :S 2rr,jzj= r,r > 1, nepeXO.JlHT np11 OT0-B 3aMKHYTYIO attaJIHTHlJeCKYIO Kp1rny10(J)=orpattw-rnsa10w:y10 tteK0Topy10 o6JI3CTh, m10-2[ m((fJ)+W(q}).
m'((fJ)-.~ d(fJ =022" [ re;rp+ re - irp0=J2ox [ re;9 +2re-irpIlp11 JJI06oM r'l3eM, 'ITO21~ a e-11irp + ~e"irp+L:IIII~ na,,e-i11rp + na"e;"rpl-L,, - - - - -- - 1111=12r> 1 rrrroma.llb S > 0.IIlX2r11= 1d (/J - 1rr2_~ I 12:rr L,, n a 11 .11=1IlepeXO.ll51 K npe,D,eJiy np11 r--->l,noJiy-2:L~. 1 nIa,,1 ~ 1.13.39.1.IlycTh T04K3 z 0 nepexo.u11T B T04KY w0.JlHT B TO'-IKY, CHMMCTj)H'lHYIO c TOLJKOHw~TIJ!Oll.!3.llh KOTopoi1s KOTOpOH BbJLll1CJI51CTC5! no cpopMyJJe (CM.
33,!l,3'lY 6.7. l))S = Yu·v'd(/J =TO+ iv,:noro reoMerp114ecKoro no5!cHeH115! s&neKaeT 11 cnoco6.llOKa3aTeJI&CTB3. IlMeHHO, OKPYIKHOCTh6pa)!(emm=u= ro(z)=oo .wo;= 0, TOf.ll3 z0nepexo-Wo = 0 - ueHTp Kpyra,w(z) = /..(z - z0 )/(z - 0 ).TaK K3KzTor,D,3 npeo6p330B3HHe HMeeT BH.llBocnOJJb3yeMrn TeM, 'ITO rp3tt11ua 06nacT11 np11 KOHcPOPMHOM npeo6pa30satt1111 nepei1.ueT B rpatt11uy o6JI3CTH, T. e.
np11 zR=IAI ·1~-:ox- ~I= IAI,A= Re;r:p , w( z)= Re;r:p=x, jw(x)j = R. IlMeeMz-:_o -:no ecT& 06m11i1 BH.llz-~.llp06tto-mrnei1ttoro npeo6pa30B3HH51, r.ue napaMeTpcp -yron nosopornKpyra, npH 3TOM Kpyr nepeXO.ZWT caM B ce651.13.62. ,[(oKa3aTeJihCTBO .IlyCTh cymecrnyeT KOHcPOPMHOe OT06p3)!(eHHe,rrepeso.u51ru:ee KOJibUO r 1 < JzJ < r2 Ha KOJI&UO R 1 < lwl < R, 11 Kornpoe 33.llaeTC51 cj.>yHKUHCH j(z), 3H3JIHTH'!eCKOH B KOJl&Ue r1 < lzl < r2 H 0.llHOJIHCTHOH.IlcnOJib3Y51 npHHUHil CHMMeTpHH P11MaHa-lllsapua, noJiyqHM 3H3JIHTHl!eCKOe npO,llOJJ/KeHHe cpyHKUHHj(z)Ha BCIO KOMIIJJeKctty1-0 TIJIOCKOCTbHCKJll-O'lCllHeM, 6b!Tb MO)!(CT, TO'-IKl1 z= 0,HOe OT06pa)!(eH11e p3CWHpeHHOH KOMIIJieKCHOH IIJJOCKOCTHBmMO)!(Hbl .llB3 CJIY'l351.c33rrp11 :HOM j(z) 33.ll3CT KOHcpopM-c Hace651.PEWEHI151 TI1I1WIHb!X 3ALJ:A LI334IlepBblH CJIY'IaH, Kor.ua TO'!Ka z = 0 nepeXO,U11T B TO'iKY 111 =f(O)= 0,= z g(z),KaK OKPYIKHOCTb lzl = r16ecKOHe'iHO y.uaneHHal! TO'!Ka nepexo.u11T s ce6ll.
11MeeM f(z)r.ue g(z) - uenal! cpyHKU11ll, g(z)11n11 lzlf. 0, zEC.TaK= r2 nepexo)J.11T B oKpyJKHOCTh, TO g(z) = const.BcaMOM JJ.ene,.uocTaTO'iHO paccMoTpeTh cpyHKU11ID In [g(z)], Kornpal! aHan11T11'!Ha BKpyre lzl ::::: r 1 11 Ha rpatt11ue lzl = r 1 .uei1crn11TeJihHall 'iaCTb Re In [g(z)]= In lg(z)I eCTb KOHCTaHTa. 0TCIO):la cne):lyeT, 'ITO rapMOHl14eCKal!cpyHKU11ll Re In [g(z)] eCTb KOHCTaHTa BO BCeM Kpyre lzl::::: r1, no:noMY 11 In [g(z)] ecTb KOHCTaHTa.
0KOH'iaTenhHO, f(z) = a z 11 r 1/r 2 =R 1/R 2 •Brnpoi1 cnyqai1. ToYKaz = 0 nepexo.u11T sKy, a 3Ta norne)J.Hllll - B TO'iKY 111l/f(z),6ecKoHe4HO yJJ.aneHHYIO T04-= 0. Ecn11 paccMoTpeTb cpyttKU11IOg(z) =1TO np11.ueM K nepsoMy cnyqa10, a Tor.ua fi.z)= llaz = bz- • 11 B 3TOMcnyYae r 1/r2 = R i/R2.13.94.1.
TaK KaK 11CXO,UHal! o6JiaCTb Cl1MMeTpH'·!Ha OTHOCl1TenbHO .uei1CTBl1TeJibHOH OCl1 11 rpaH11ua o6naCTl1 .UOJIJKHa nepei1Tl1 Ha .uei1CTBl1TenhHYIOoch, TO np11MeH11M np11ttu11n c11MMeTp1111 P11MaHa-Wsapua. PaccMOTpl1Mo6JiaCTb D1, rpaH11uei1 KOTOpoi1 l!BJil!eTCll .uei1cTBl1TeJibHal! OCb 11 OTpe30K HaMHl1MOM OCH: y E (0, 1),HO npco6pa30BaH11J1: 111 1x = 0.=z2llp11MeHl1M K 3TOH o6JiaCTl1 nocne.uosaTeJib-, 1112= 111 1 + 1 = z 2 + 1, w3 =Fi=~z 2 +1, np11:noM 6epeM TY BeTBh KBaJJ.paTttoro KOpHll, KOTopal! nepeso.u11T D 1 Ha sepxHIOIO nonyrrnocKOCTb.
TaK KaK w 3= ~ z2+ lnepeso.u11T o6nacThsepXHIOIO nonynnOCKOCTh, TO 11CXO,UHal! o6JiaCTb -D1HaBHelIIHOCTb Kpecrn -nepei1.ueT Ha BCIO IlJIOCKOCTb c BbI6poweHHblM 0Tpe3KOM [-.J2, .J2]no.uei1crn11TeJihHOH oc11. Tenepb np11MeH11M .upo6Ho-n11Hei1Hoe npeo6pa3osaHl1e- .fi.-llj .fi.-J: 2+1--,-;:--- =nepeBO.Ullluee 11CXO.UHYIO o6naCTb (BHeWv2+11j .fi.+~ :2+! 'W4 -HOCTb Kpecrn) Ha BCIO KOMIIJieKCHYIO nnOCKOCTb c Bb16poweHHb!M Il0JIO/Kl1Ten&Hb!M nyYoM.11nocne.uHee npeo6pa30BaH11er. =iv="\/ w4.fi.-0i~.fi.+vz2+1nepeBO.U11T 11CXO.UHYJQ o6JiaCTb Ha BepXHJQJQ nonynnoCKOCTb (6epeTCll TaBeTBb MHOf03Ha'iHOH cpyHKU.1111, KOTopal! nepeBO.U11T Ha sepXHIOIO nonyIlJIOCKOCTb).11rnK, w(z) =.fi.-Mr.fi.+Jz2+! .PEWEH1151 Tl1fU14Hh!X 3A,llA4335Dwaa1414.8.
6) Tiep6ineM K nom1pHhIM KOOp)llrnarnM x = r cos cp, y = r sin cp.y CJIOBl1e conp1DKeHl1H rapMOHl1'-leCKl1X cjiyHKUl1H u(r, cp) 11 v(r, cp) 11MeeT Bl1)l2?? . ?, 1 , ,I , T() r cos-rp-r-sm-rp cos2rpu,.= -vqi, v,. = --uqi.r,u,.aK KaKr=-2cos 2rpr3,0TCIO)la v(r,ip),ltqir,rpII2sin21p=,.2sin 2qJ=,, vffJr=-42cos2qJ,.2sin 21p= --,.- -+ f(r) =--,.- -+ g(ip),1c (const). lfraK, v(r,ip)1r-0TBeT: v(r, ip)=-2xy1,,v,.=2sin2qJ?1(x-+y-tr3noJTOMy_/{r)sin 2qJ2rcosqJ·rsinqJ= ---+c = +c =41TO?,.-r= g(cp) =-2xy011(x-+y-t+c.+ c.14.17. TiyCTb u(x, y) - rapMOHH'-leCKaH cjiyHKUl1H B 3aMKHYTOM Kpyre pan11yca R 11 c ueHTPOM s TOLJKe (x0 , y 0 ), 11 TOLJKa (x, y) JieJK11T BHYTPH JTOroKpyra.
06o3HaLJ11M 3HaLJeH11e cpyttKu1111 u(x, y) Ha rpaH11ue Kpyra LJepe3u(R, cp), a 3tta'ic1rne cjiymn.urn u(x, y) BHC rpaHHUbl - •1epe3 u(r, 8). Ilo ~aHHOH rapMOHHLJecKoi1 cjiy1-1Ku1111 u(x, y) nocTpo11M aHan11rnLJecKy10 cjiyHKu1110j(z) s 3aMKHYTOM Kpyre - TaK, LJTO_/{z) u(x, y) + i v(x, y),j(z 0+Re;"') u(R,cp) +iv(R. cp). no 11HTerpaJibHOH cjiopMyJie Kou11111MeeM=J f(¢)d¢1f(zo)=-.;:=., Zo =xo+tYo·2m 1;-;il=R s - ZoIIepexo,11,H K rrapaMeTP113au1111di;= Rieiqi drp,f(z 0 )·2n2m0= - 1- . J f¢ - z0 = Re;qi' qJ E [0, 2n], nonyLJaeM1 2n(z. 0 + Re;qi) dqJ = - J f (z 0 + Re;qi)dqJ 2n 0cjiopMyJia cpe,11,1-1ero 3Ha4emrn )lJIH aHaJil1THLJeCKOH cjJYHKU1111.
Terrepb OT)leJIHM nei1CTBl1TeJihHYIO 11 MH11MYIO LJaCTl1 B JieBOH 11 rrpaBOH LJacrnx 3TOHcjiopMyJihl H noJiyLJHMu(x0 , y 0 )=-l2n2nJ u(R, qJ) dqJ,.r.ue u(R,rp)= u(x0 + Rcosrp, y 0 + Rsm rp).014.29. 2) 3arr11weM pewe1-111e u(r, 8) B s11,11,e HHTerpana IIyaccotta:1 2" (l-r 2 )cosrpdrpu(r,8)=-f, Q:-=;r<l,8E[0,27Z"].127' 0 l+r--2rcos(ip-El)C.ueJiaeM 3aMeHy qJ - 8 = a - n/2 , rnr.uaPEWEHI151Tl1I1114HbIX3A.lJ:AL!336(l-r 2 ) 2"+" 12 - 0 cos(B+a-n/2)daf=- -u(r,B)27r0l+r--2rCOS(a-1lf2)n/2-B2212 0-(l-r "u- "f2n)12 022"+" 12 - 0f"12 _ 0"+" 12 - 02TaK KaK)[f2sinB·cosada+cos B.2l+r -2rsmacos a da2.-l+ r 2 -2rsina -_(l-r-_ 2nsin(B+a) da=f"+" 12 - 0= 0,]•,l+r--2rsmasina da2.fasinada?JT, 2 _ 02l+r -2rsma"u' 2- 0r2n2l+r -2rsma 1-r(cM.
12.13. 10)). OKOH'laTeJihHO rro11y4aeM u(r, 8) = rcos 8111111 u(x,y) =x,x = r cos e, y = r sin e.JT/2-0OrneT: u(x, y)n/2-0= x.14.48. IlycTb r < R 1 < R, rnr.na cjJyHKU:HHfiz) aHaJIHTHlJHa B 3aMKHYTOMKpyre { lzl :S R 1 } 11 ee MO)KHO pa3JIO)KHTb B pli:,n Teifnopa rro CTeneHll:M z, pH.LI6y,neT paBHOMeptto cxo,nHThCll: B Kpyre { izl :S R 1 } :f(z)='i.~= 0 a11 z , lzl~R,.11Ec1111fiz)= u(x, y) + i v(x, y),•()(ux,y ) +ivx,ya11 = a;,+ ia;,, .11 z = re;o, TO'"'~= L.11 =0• ") n i11B•a11t +ia, z=x+1y.11 re(u(re;0 )= u(x, y) = L~= 0 (a;,rv( re iB)'"' ~ ( a r= v ( x, y ) = L..11 0 11I=11II11cos nB-a;r sin nB),.smnB +a11"r II cos nB) .Ilo TeopeMe Tei1nopa 3arr.11weM K03cp¢.11u:11eHTbI B BH,nelf(~)d~a,,= - .,g11+12m l~l=R,f.(3aMeHa,;,.,,= a11 + 1a11 = -2l2"f(R/'P)Rie;rpd(fJ· oR11+1 i(11+I)rp1lll2"2n0= Re'rp , (/JE [0,2n]) = -fef f(R 1eirp) e- . rpd(fJ.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
