Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 10
Текст из файла (страница 10)
.IJ:oKa3aTb,'-!TO j(z)= const B D.5.6. IIycTb cpyHKUirnf(z) = u +iv ammHm'fHa B o6Jiacm DC CHBC!O.UY B D au + bv + c = 0, r.ue .ueHCTBHTeJibHbie KOHCTaHTbI a, b, cHe Bee paBHbI HYJIIO . .IJ:oKa3aTb, 'ITO j(z)5.7.= const B D.IIycTb cpyttKUH5!f(z) $. const aHaJIHmlfHa B o6JiacmEy.uyT JIH aHaJIHmlfecK11MH cpyHKUHH5.8. IIycTbcpyHKUH5! j(z) = uf(z)+ iv,v(x, y) .nttcpcpepettuttpyeMbI B o6JiaCTH DHDcC.j(z)?nptt'IeM cpyttKUHH u(x, y),c C..IJ:oKa3aTb, 'ITO ecJIHBCIO.UY B D BbinOJIHeHO O)J.HO 113 ycJIOBHH:a) cymecrnyeTlim Re 6.f ;6.zLl.z - o6) cymecrnyeT6.f.1im Im-,Ll. z- of).zTO cpyHKUH5!j{z) aHaJIHTH'IHa B D.5.9.
IlycTb f(z) = u -r iv, rrptt'iCM cpyttKU.HH u(x, y), v(x, y) .umpcpepeHUHpyeMbI B o6Jiacm D C C . .IJ:oKa3aTb, 'ITO ecJIH B Ka)l(.uoifD cymecrnyeT limTO'IKe 06JiacT11JI1160 cpyHKUH5!5.10.f(z)IIycTb f(z)u11pyeMbI B o6JiacTHDcymecrnyeTattanttmlfHa B o6Jiacm=uD+ iv,cTO JI1160 cpyHKUH5!f(z),D.cpyHKUHH u(x, y), v(x, y) .u11cpcpepeH-C . .IJ:oKa3aTb, qTO ecJI11 B Ka)l()J.OH TO'IKe H3lim arg 6.f , TO cpyttKUl15!f{z) aHaJIHT!1lfHa B o6Jiacm D.Ll.
z-o5.11.16.f I·ll.= - o 6.zIIycTh z6.z= re iq>'f(z) = u(r, <p) + iv(r, <p) ..IJ:oKa3aTb, 'ITOycJIOBH5! (CR) B nepeMeHHbIX (r, <p) HMeIOT BH.Uou 1 ov -ov= -1-ouor r oip ' orr oip-=--5.12.IIycTb cpyHKQH5!f(z)a BeKTOpbIOTss= u + iv aHaJIHTH'IHa B o6JiaCTH D CC,H n Ha nJIOCKOCTH nepneH,UHKym1pHbl, np11qeM noBopoTK n COBeplllaeTC5! npOTHB qacOBOH CTpeJIKH . .IJ:oKa3aTb, '-!TO eCJIHIs I= In I= 1, TODzaea 560auavOUavon ' 8nOS--8s5.13. ITycTh cpyttKUJurf(z) = u +iv amUIHmqtta B o6nacm DC C..D:oKa3aTh, qTO B Ka)l(.ll,OH TOqKe z E D crrpaBe.ll,JIHBhI COOTHOilleHHH:= u~ + iv:, ; 2) J'(z) = v~ - iu~;3) J'(z) = u:, - iu;; 4) J'(z) = v; +iv: ,5.14. IlycTh cpyttKUHH f(z) = u + iv H cpyttKUHH1)J'(z).n;HcpcpepeHI.~HpyeMhI B o6nacmdf0=D C C .
.D:oKa3aTh, qTo0f dz + f dz, dzozu(x, y), v(x, y)ozdx + idy, dz==dx - idy,af = ~(af _ i af ), a~= ~(of+ i of).oz5.15. ITycTh f(z)2 oxoy= u + iv,oz2 ox8yrrpHqeM cpyttKUHH u(x, y), v(x, y) .ll.Hcp-cpepeHUHpyeMhI B o6nacm DC C. .D:oKa3aTh, qrn cpyHKUHH f(z)aHaJIHTHqHa B o6naCTH D Tor.ua H TOJihKO Tor.ua, Kor.uaofa-z= 0(yc;zoeue /{GJZaM6epa-3Wr.epa).5.16.
I1ycTh cj:>yHKUHH f(z)=u.umpcj:>epeHUHpyeMhI B o6nacm D+civ H cpyHKUHH u(x, y), v(x, y)C. .D:oKa3aTh, qTo HK06HaH npe-06pa3osaH1rn OT rrepeMeHHhIX (x, y) K rrepeMeHHhIM (u, v) BhJqHcJIHeTCH no cpopMyne18fl 18f 1D(x, y) = 8z - 8z22D(u, v)5.17. (I1po.ll,OJI:>KeHHe 3a)l.aqH 5.16.) .D:oKa3aTb, qTQ eCJIH cj:>yHKUHHf(z) aHaJIHmqHa B o6nacm DC C, TO\!'(z)\ = 1:~1.5.18. ITycTh cpyttKUHH f(z) aHanHmqHa B o6nacm Do6nacTh GcD.
IlycTh, KpoMe Toro, G1-cC Ho6nacTh, HBnmoruaHcH61)]J1<l><l>EPEHUI1PYEMOCTb <l>YHKU11Mo6pa30M o6nacnr G np11 oT06pa)KeHHHfEcn11 f(z) O.llHOJIHCTHa Bo6nacTH G, TO nnoma.llh S(G') MO)KeT farTh Bhrq11cneHa no cpopMynef fc:IJ'(z)=u5.19. ITycTh cpyHKUH5I f(z)21dx dy.+ iv 11 cpyHKUHH u(x, y), v(x, y).[(Ba)K.[(hI .L\HcpcpepeHu11pyeMhI.
):(oKa3aTh, qTo<ff<ff82fl:::,.f=.-+-=4--.8x 2 8y 28z8z5.20. ITycTh cpyttI<l(H5Ij(z) aHaJIHTttqHa B 06nacT11 D C C 11f(z) I- 0BCIO.llY B D. ):(oKa3aTh, qTo BCIO.llY B D cnpase.[(JIHBO paBeHcrno~log I f(z) I=!_ J'(z).oz2 f(z)5.21. ITycTh cpyHKQH5I f(z) aHaJIHTHqHa B 06nacT11 Dc C.):(oKa3aTh, qTo BCIO.llY B D1) fl lf(z)j2= 4 lf'(z)i2;=2) .6.log(l+IJ(z)l2)4 if'(z)i2..;2(l+lf(z)l )23) fl log lf(z)I = 0, ecn11f(z) j 0.5.22. HattTH1) f(z)ofoz=lzl;4) f(z) =,[(Jlj{cne.ny10ru11x cpyHKQHtt:2) f(z)~I z - a 126) f(z) = jz -al I lz -=lzl P, p ER;+ Iz- b 12 ;bl;5.23. IlycTh cpyHKQttj!f(z)3) f(z) =ePlzl, p ER;5) f(z) =log jzj;7) f(z)i+ I z I= arctg--,I z I< 1.1- I zI= u +iv aHaJIHTHqHa B o6nacTH D cC.):(oKa3aTb CJie.[(yIOlll,He cpopMyJihI:1)~I J(z) I=!_ I J(z) I J'(z);8z{)3)-v8z21I= --! (z);2f(z)08z{)1 I2) -u=-f(z);4) -cp(j f(z) I)=2{)z1 /-<p2(I1f(z) I) I f(z)f (z)1-,f(z)I'lta8a 562r.ue cp(t) - .umpcl>epeHu11pyeMa}l cPYHKUH}l .ueHCTBHTeJihHOro nepeMeHHOro t.5.24.
11cnOJib3Y}l cPOPMYJIY .LI.JI}! np0113BO.UHOH o6parnoi1 cPYHKUHH, .UOKa3aTb, qTO1)IefZ(efi) = - , nnz=2) (ln z)'z--HaTypaJibHOe;1;3) (arc sin z)'14) (arctg z)' = - - .,;1 + z-=1~2;15) (arccos z)' = - --l- z 2•5.25. Hai1TH 3as11c11MOCTh Me)l(.uy cJie.uyIOm11M11 cPYHKUH}lMH:1) arc cos z, ln z;2) arc sin z, ln z;3) arctg z, ln z.5.26. llycTh cPYHKUH}lj(z) = ln (z - c) =In lz - cl+ i Arg (z - c),r.ue c - HeKOTOpoe cp11Kc11poBaHHOe KOMnJieKCHOe q11cJIO. llposepHThBhmOJIHHMOCTh ycJIOBHH Koum-P11MaHa .LI.JI}! 3TOH cpyHKUHH.5.27. Hai1TH MHO)l(eCTBO ToqeK Z, B KOTOpb!X K03cpcp11u11eHT JIHHeHHOro pacrn)l(eHHH np11 cJie.uyI-Om11x 0To6pa)l(ett11Hx paseH HYJIIO:l)j(z) = z4).f(z)2az+b;2).f(z) =sin z;= z", n -3)/(z) = - - , c:;cO, ad :;c be;cz+dHaTypanhHoe; 5).f(z)11= -(z+-).2z5.28.
HaHTH MHO)l(eCTBO rnqeK Z, B KOTOpb!X yroJI nosoporn np11cJie.uyIOm11x oT06pa)l(eH11}1X paseH HYJIIO:l)flz) = z4)flz)2;= z", n -2)flz) =sin z;HaTypanbttoe;az+b3)flz) = - - , c;eQ, ad;ebc;cz+d15)flz) = - (z21+ -) .z5.29. llycTh cpyHKUH}l j(z) aHaJI11rnqHa B TOqKe z0 11 f '(zo) i=- 0 ..lJ:oKa3aTb, qTQ yroJI nosoporn a KpHBOM y B TOqKe Zo 11 K03cpcp11u11eHTJI11Hei1Horo pacrn)l(eHHH R npH oTo6pa)l(eHHH f onpe.ueJI}leTrn cJie.Ll.YI-OlllHM o6pa30M:ll)1<J:J<J:JEPEHQY!PYEMOCTb <J:JYHKUHH63a= argf(z0 ) +2kTI, k E Z, R = lf(zo) I-5.30.
ITycTh cpyHKU115! f(z) amummqHa B TOqKe z0, f(z 0 ) i= 0, 11rna;:i:K11e Kp11Bhie y 1 11 y 2, npoxo;:i:m1u1e qepe3 ToqKy z0 , o6na;:i:aIOTcBowcrnoM Re f(z)= Re f(zo),z E Y1, Im f(z)= Im f(zo),z E Y2-.D:oKa3aTb, qTo B 3TOM cnyqae Kp11Bb1e y 1 11 y 2 nepeceKaIOTC5! B ToqKeZo IlO.ll np5!MbIM yrJIOM.5.31. ITp11 BbrnOJIHeH11H ycnoBHH 3a;:i:aqH 5.30 npe;:i:nonm1rnM nonontt11TeJibHO, qTO lf(z)I= lf(zo)J, z E Yi. a arg f(z) = arg f(zo), z E y2..D:oKa3aTb, qTo KpHBbie y 1 H y 2 nepeceKaioTc5! B ToqKe z0 no;:i: np5!MbIMyrJIOM.5.32.
ITycTb f(z)= u + iv,cpyHKUHH u(x, y), v(x, y) HenpepbIBHO.llHcpcpepeHUHpyeMbl B o6JiaCTH Dc cH J(JI5! JII06bIX TOqeK Z1 HZ2 113 D= lz1 -lf(z,) - fiz2)I.D:oKa3aTb, qToJ(z)=11e; z+a,nneKCHbIH napaMeTp, HJIHf(z)z2I .rne 8 - n·eH:crnHTeJibHbIH, a a - KOM-=e;11 z +a.5.33. ITycTb cpyHKUH5! f(z) attaJIHTHqHa B TOqKe z0 H d(zi. z2) paCCT05!HHe Me)J(ny CTepeorpacpHqecKHMH npoeKUH5!Ml1 ToqeK Z1 11z2 EC . .D:oKa3aTb,qTO cymecrnyeT.
d(J(z). J(z 0 ))1llTI ---'----'---'--'z ~ zud( Z. z0 )H HaHTH ero.5.34. ITycTb cpyHKUH5! f(z) paUHOHaJibHa H d(zi. z2) - paccT05!HHeMe)J(ny CTepeorpacpHqecKHMH npoeKUH5!MH TQqeK Z1 H Z2 .. ,[(oKa3aTb,qTO ecJIH d(j(z 1),f(z2)) = d(zi. z2) .llJI5! JII06b1x TOqeK z1 H z2113 C, TOf(z)= az + b.CZ+d5.35. ITycTb cpyHKUH5! f(z) paUHOHaJihHa H attamuwqtta B TOqKez 0 . .D:oKa3aTb, qTQ eCJil1 B JII06oi1 TOqKe z aHaJIHTJ1qHOCTH f(z)cnpaBe.nmrno paBeHCTBO=I'llaBa 564IJ'(z) I1 +lf(z)l2i1 + lzl2'TO f(z) 5IBJU!eTC5I .upo6Ho-mrneHHOH cpyHKUHei1:.5.35.1.zIITycn, cpyHKUH5Ij(z) pauttoHaJibHa 11 attaJIHTHqHa B ToqKe= 0 .
.[(oKa3aTb, qTO ecJm B HeKOTOpoH: OKpeCTHOCTI1 TOqKH z = 0crrpaBe,ll.JII1BO paBeHCTBOIJ'(z)I1+ lf(z)l211+ lzl2'TO f(z) 5IBJU!eTC5I .upo6HO-JlHHeHHOH cpyHKUHei1:.5.35.2. ITycTbHOCTH TOqKH zcpyttKUH5I .f(z) aHaJIHTHqHa B HeKoTopoH: oKpecT-=0 I1 B :now OKpeCTHOCTI1 crrpaBe,ll.JII1BO paBeHCTBOI J'( z ) I1+ IJ(z)l2121 + lzl.[{oKa3aTb, qTO .f(z) 5IBJU!eTC5I .upo6HO-JII1Hei1:HOH cpyHKUHei1:.5.36.
ITycTb cpyHKUI15I .f(z) pauttoHaJibHa H aHaJIHTHqHa B ToqKe=0 . .[{oKa3aTb, qTo ecntt .UJI5I mo6oro z, I z I< 1, BbIIIOnHeHbIzycnoBI15Ilf(z)I< 1, I J'( z) 121- lf(z)I1l-lzl2'TO cpyHKUI15If(z) 5IBJI5IeTC5I .upo6HO-JIHHeHHOH.5.37. ITycn, f(z) = P(x, y) + i Q(x, y), r.ue P I1 Q - MHoroqneHbIx I1 y c .uei1CTBHTeJibHbIMH K03cpcpI1UI1eHTaMH, H rryCTb f(z) aHaJIHTHqecKa5I cpyHKUH5I Ha C.
Cne.uyeT JIH OTCIO.Ua, qTO .f(z) -OTMHOroqneH OT z?5.38.BepHo nH yrsep)l(,ll.eHHe, qTo .UJI5I n106oro MHoroqneHaP(x, y) cymecrnyeT aHaJIHT11qecKa5I cl>YHKUI15I .f(z) Ha C, rnKa5I, qToRef(z) = P(x, y)?5.39. ITycn, cl>YHKUH5Ij(z) aHaJIHmqHa Ha Kpyre Iz I :S 1 I1 rrpHHHMaeT Ha rpamrue 1<pyra Cl z I= l) neifCTBHTeJI1>H1>Ie 3HaqeHllil . .[(oKa3an,qTO f(z) = const.I,O:Hct>ct>EPEHLJ,HPYEMOCTb ct>YHKU,HM655.40.
Ilycn, f(z) i: 0 11 aHaJI11T11qHa Ha Kpyre I z I :'S 1. .UoKa3aTb,'ITO f(z) = const, ecm1 jf(z)I = 1 Ha I z I = 1.5.41. IIycTb cpyHKU1151f(z) aHaJIHTifqHa Ha orpaH11'leHHOtt 06nacT11D 11 rrp11tt11MaeT ,n:eifcrn11TeJibHbie 3HaqeHH5I Ha ee rpatt11ue. Cne.nyeTJIM OTCIO)J;a, qTo f(z) =canst?5.42. IIycTb z= r e;o + a,r < R. LJ:oKa3aTb, qTQ cpyHKUH51Re;" + (z - a)J(z) = R e ;,, - (z -a )rrpH cp11Kc11poBaHHbIX R 11 cp ecTb aHaJIHT11qecKa51 cpyttKUl151 OT z BKpyre lz < R, rrp11 3TOMalR 2 - r2Re f( z) = - 2 - -2- - - - - R + r - 2Rrcos(cp - B)5.43 .
.LJ:oKa3aTh rrp11se.ueHHbie HmKe CBoifcrna aopa IlyaccoHaR 2 - r22R +rrrpH 0 < r < R, cpIf2-2Rr cos(cp - B)R - cpHKCHpoBaHbI.1) 5!.upo IlyaccoHa - rapMOHH'lecKa51 cpyHKUH51 OT z B Kpyrelz -al< R, r.ne z =a + re ;a.21) 2n3)f"(R22r )dcp2Rrcos(B - cp)-= l.Ecn11f(z) attaJIHTH'lHa B Kpyre jz - al :'SR, TO°R2+ r2 -221 f 2" J(a+Re ;")(R -r )dcp;e!()z =-Jn .,, z =a+ re .22n " R- + r - 2Rr cos(B - cp)4) Ecn11 u(z) - rapMOHifqecKa51 cpyttKUH5I B Kpyre lz - al::; R, TO221 f h u(R,cp)(R - r )dcp;ou(z )=u(r. e) =-J,z=a+re .'2n ° R 2+r 2 -2Rrcos(B-cp)'5) Ecn11 h(R, cp) E C([O, 211]), h(R, 0)= h(R, 211), TO cpyHKUl151Diaea 566u(r. B) = ~'rapMOHttqHa B27rf"0R2h(R, tp)(R2 - rz)dtp+ r 2 - 2Rr cos(B - tp)Kpyre lz - al< R, lim u(r, e) = h(R, cp).r--+Re~,;I'llaBa 6HHTErPHPOBAHHE <l>YHKQHHKOMIIJIEKCHOro IIEPEMEHHOro.HHTErPAJibHAJI TEOPEMA KOUJHIlycTb cpyHKUIUI f(z) KOMIIJieKCHOro nepeMeHHOro z onpe.uenettaYi.
y 2, ... , y11c IlOMOmhIO ToqeK Zi, Z2, ... ' Zn, 3aHyMepoBaHHhIX B nop5I,UKe HXHa cnpHMmreMOH KpHBOH y. Pa3o6beM KpHBYIO y Ha .uyr11cne.uoBamrn no KpttBoH:y.HaqanoM .uyrH Yk 5IBJI5IeTc5I ToqKa zk-t.KOHUOM - ToqKa zk. Bhr6epeM Ha .uyre Yk TOqKy ~ H paccMOTPHMHHTerpaJibHbie cyMMhIEcJIH cymecrnyIOT npe,UeJibl HHTerpaJibHhIX CYMM(JnJi['(Jn'Yk> k = 1, ... , n, cTpeMHTC5I K HYJIIO 11 nptty Ha .uyrtt11 OT BhI6opa ToqeK I;; k Ha .uyrax Yh TO 3TH rrpe.uenhI Ha3hIBaIOTC5IKor.ua MaKCHMYM .UJIHH .uyr)TOM npe.ueJibl He 3aBHC5IT OT cnoco6a pa36HeHmI KpHBOHYkKpU60JlUHeUHblMUwm1e2pa.1Ul.MUnepBOroHBTOporopo.uaOTcpyHKUM5If(z) B,UOJih KpHBOH y H o603HaqaIOTC5I COOTBeTCTBeHHOJ, f(z) I dz I= ,~~~a", J,f(z)dz = ,~~a;..ECJIH cpyHKUH5I f(z) MMeeT BH,U f(z)= u(x,y)+ i v(x, y),TO cnpa-Be,UJIHBhI paBeHcTBa:J,f(z) I dz I= J,u(x, y)dl +if, v(x, y)dl,r.ueHHTerpaJiblf u(x, y)dl, f v(x, y)dl1-KpHBOJIHHeHHbieHH-ITerpaJihI nepBoro po.ua OT .ueH:cTBHTeJibHhIX cpyHKUMH u(x, y), v(x, y), 11J, J(z)dz = J, (u(x, y)dx -v(x, y)dy)+if (v(x, y)dx + u(x, y)dy,Ir.ue HHTerpanhIDzaea 668J,(u(x, y)dx - v(x, y)dy), L(v(x, y)dx + u(x, y)dyeCTb Kp11BOmrnei1:ttbJe 11HTerpa.Jibl BTOporo po.ua OT .ueikTBHTeJibHbIXcpyttKU11H u(x, y), v(x, y).
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