Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 27
Текст из файла (страница 27)
, a,,, TaK KaKt h(;)d;+ Al'-2 tte113secTtthIX rrapaMeTPOB: A > 0I1 TaK KaK arg h(x) = const .um1 x E (ak, ak+i), TO011Ak+1-Ak+2I= Afa," lh(t)ldt, k=l,2, .... ,n;3,UeCh cq1naeM, qTo a,,+ 1 = ai, A,,+ 1 = A 1 H qTo O.UHH H3 HHTepBMOB(ak, ak+l ), k =1, 2, ... , co.uep)KHT BHYTPH ce6H ToqKy z = oo.3TH COOTHOIUeHHH II03BOJIHlOT orrpep;eJIHTb BCC rrapaMeTpb!.13.1. ,lJ,oKa3aTh, qTocyrrepII03HU:H~OL(HOJ1HCTHh!X cpyHKUHH HB-J1HeTCH O,UHOJIHCTHOH cpyttKU:HeH:.13.2.
,lJ,oKa3aTh, qrn eCJ1H attaJIHTHqecKaH cpyttKUHH f(z) O.UHOJ1Hcrna B OKpeCTHOCTH TQqKH Zo, TO j' (z)*0 B 3TOH OKpecTHOCTH.13.3. ITycThft.z) = L~= 0 a,, (z - z0 )", iz - Zoi < p, p > 0, a1=/(Zo)f:-0.,lJ,oKa3aTh, qTo cymecrnyeT PiC: iz - zolS> 0, p 1 < p TaKOe, qrn .UJ1H scex z EPi ciJyttKU:HH.f(_z) O.UHOJIHcrna. I1p11secrn rrpuMep, rroKa-3hIBaIOUlHtt, qTo o6paTttoe yrnep)K,UeH11e ttesepHo.13.4. ITycTh y - 3aMKHYTaH Kycoqtto-rna.uKaH Kp11BaH )Kop.uatta.ITycTh ciJyttKWrn ft.z) aHM11rnqHa Ha int y H rrpttHHMaeT Hay Ka)K.uoeCBOe 3Ha'iemre TOJJhKO 0)1,HH pa3. ,lJ,OKa3aTh, 'iTO.f(_z) O,UHOJIHCTHa B int y.13.5. ,lJ,oKa3aTb, qrn rrpene11 paBHOMeptto cxonmueH'cH rrornenosaTeJihHOCTH aHaJIHTJ1qHb!X 0L{HOJ1HCTHh!X cpyHKUHH HBJ1HeTCH 111160ciJyttKU:Hei1 o.utto1111CTHOH, n1160 rrocrn~HHOH.
ITp1rnecT11 np11Mep rrocJienosaTeJJhHOCTH 0L{HOJ1HCTHhlX cpyHKU:HH, paBHOMeptto cxon~rueH:cH K cpyttKU:HH, TO)K.uecrnetttto paBttoH: HYJIIO.KOH<t>OPMHblE OTOEPA)!{EHI15117713.5.1. Beptto 1u1 yrnep)K.n:emre rrpe.n:b1.n:ymei1 3a.n:aq11, ec1rn y6paTb ycJIOBHe attaJIHTHqHOCTH rrocne.n:oBaTeJibHOCTH cpyHKUHH?13.6. Orrpe.n:enttTb o6nacrn o.n:ttonttcrnocrn cpyttKUtti1:l)z",nEN;2)z+z- 1;3)ez;4) sin z;5) cos z;6) tg z; 7) th z.13.7. )].oKa3aTb, qTQ perymrpHhie BeTBH cpyHKUHH O,ll,HOJIHCThI:1)ef;.,11EN;2)lnz.13.8.
IIycTh cpyHKUH5!w = z + a2 z 2 + a3 z 3 + ...z: lzlattan11rnqtta Ha BHyTpettttocrn e.n:11tt11qttoro Kpyra {< 1}, D KOHcpOpMHbIH o6pa3 BHyrpeHHOCTH e,D,HHHqHoro Kpyrn rrptt OT06pa)KeHHH w w(z). )].om1:mTh, qTQ rrnoma.n:h D paBHa=.7Z'+.7Z'_L:~=2nla,, (3To yrBep)K.n:emie Ha3hIBaeTC5! eHympeHHeii meopeMoii nJZOU{aoeii.13.9. )J;oKa3aTh, qTo B rrpe.n:rrono)Kett115!X 3a.n:aq11 13.8, ecnttL~=1 n Ia,,Is:; 1,TO cpyHKUH5!./{Z) O,ll,HOJIHCTHa BO BHyrpeHHOCTl1 e,D,HHHqHQfO KpYTa.=13.10. )].oKa3aTh, qTQ ,ll,JI5! Bcex cpyHKUHH w w(z), y.n:oBJieTBOp5IIOIUHX rrpe.n:rrono)KemrnM 3a.n:aq11 13.8, MHHHMYM rrnorua.n:tt D .n:ocTttraeTC5! Tor.n:a, Kor.n:a D COBrra.n:aeT c BHyrpeHHOCThlO e,D,HHifqHoroKpyra, T.
e. w(z) z.=13.11. IIycTh cpyHKUH5!a, a,w= w(z) = z +-+~+ ...zz-z: lzl0,ll,HOJIHCTHa BO BHelllHOCTH e.n:HHHqHoro Kpyra {> 1 } H aHaJIHTHqHa BCIOJJ:y, KpOMe 6eCKOHeqHO y.n:aneHHOH TQqKH, r.n:e OHa HMeeT2rron10c rrepBoro rrop5!,n:Ka. )].oKa3aTh, qTQ L~= 1 11 a,, :<S 1. 3TO yrnep-I l)K,ll,eHHe Ha3bIBaeTC5! 6HeUtHeil meopeMOLl llllOUfadeil.13.12. )].oKa3aTb, qTQ ecntt cpyHKU115! f(z) 0,ll,HOJIHCTHa B OKpeCTHOCHI TO'-IKH Zo, KOTopa5! 5!BJl5!eTC5! IIOJlIOCOM, TO 3TOT ITOJlIOC HMeeTrrepBhIH rrop5!,ll,OK.13.13. ITycTb cpyttKUH5!178rwea 13w=f2(z) = z + a2 z + ...O.UHOJUICTHa BO BHyrpeHHOCTH emrnwrnoro Kpyra {z: lzl <1} .
.[loKa-3aTh, qTQ cpyHKUI15I13=;:+-a,z + ...F(z)=2 -TaK)!{e 0,UHOJIHCTHa BO BHyTpeHHOCTl1 e.u11H11qHoro Kpyra.13.14. IIycThcpyHKUH5Iw=z + a2z- + ...1aHaJIHTJ1qHa 11 O.UHOJIHCTHa BO BHyrpeHHOCTI1 e,UHHJ1qHoro Kpyra{z: lzl < 1} . .[(oKa3aTb, qrn la 21:S 2 . .[(oKa3aTh, qro :no HepaBeHcrnoHeyJiyqwaeMo.13.15. IIycTb¢YHKU115Iw = w(z) =z + a 2z2 + ...aHaJil1Tl1qHa11 O,UHOJil1CTHa BO BHyrpeHHOCTl1 e.u11H11qHoro Kpyra{z: lzl < 1 } ' D -o6pa3 BHyrpeHHOCTH e,UHHHqHQro Kpyra npH OT0-6pa)l{eHHH w = w(z)crpattm~ei18D . .ll:oKa:mTh, qTo mo6aH ToqKaDD y.uanetta OT HaqaJia KOOp.UHHaT Ha paccT05lmre He MetthHa 114.
3Ty nocT05IHHYIQ Ha3bIBaIQT KoHcmaHmoil Kii6e.rpattHUbIwe, qeM13.16. IIycThcpyHKUH5Iw= f(z) =z + a2z 2 + ...aHaJIHTHqtta 11 O,UHOJil1CTHa BO BHyrpeHHOCTH e,UHHHYHOro Kpyra{z: lzl < 1 }. .[(oKa3aTb, YTO:1) .[(JI5I JIIQ6oro z E C: lzl < 1 cnpase;::i;m.rnb1 ttepasettcrna1-lzl <,< l+lzlc1+ I z DJ-1fCz)1- c1-1:: ll.3TH HepaBeHCTBa Ha3bIBaIQT 111eope.11ou UCKQ:J!CeHLm.2).[(JI5! mo6b1x;: 1, z2EC :J::,J::; r, I :: 21::; r, 0<r<1,HepaBeHCTBa( ~) : :; IJ'C::,)I ::;(~)4i + ,.3).ll:JI5! mo6oroIJ' <::2) I41- ,.z: Jzl < 1 cnpaBeLIJIHBO ttepaBeHcrnoIargj'(;:) I:::; 21n ~:: ~:.cnpaBe.UJIHBbIKOH<DOPMHbIE OTOEPA)l{EHl1511793To HepaBeHCTBO Ha3bIBaIOT meopeMOU epaU{eliUR.13.17. IIycTb cpyttKr.urnw =f (z) = z + a 2 z + ...
+ a211z" + ...attaJIHTHqHa H OJJ:HOJIHCTHa BO BHyrpeHHOCTH e;::i:ttHHqHoro Kpyra.1) ,l.l;oKa3aTb, qrn la11I <en .z:i:mr mo6oro n 2::2.2) _aoKa3aTb, qTo erntt Bee K03cpqrnutteHTbI a,, - ;::i:ei1:cTBHTeJibHbieJa,,J :::;qwcna, TO ;::i:m1 mo6oro /1 2:: 2 crrpaBe.UJIHBO HepaBettcrnon.2IIpoBepttTb, qrn ¢YHKUH51I (1 < 1, y;::i:oB11ernop5IeT,BCeM ycJIOBH51M 3a.z:i:aqw H JJ:JI51 ee K03cpcpHUHeHTOB HMeIOT MeCTO paBeHCTBa a,, 11..zz) Jzl=13.18. L(eJibIM JIHHeMHbIM rrpeo6pa30BaHHeM Ha3bIBaIOT rrpeo6pa30BaHtte w= w(z) = az + b,a E C\{O}, b Ec.HaHTH 061.UHH BH.Uuenoro JIHHeifHoro rrpeo6pa3oBaHtt51, nepeBOJJ:51I.Uero:1) BepxHroIO r1011yrr11ocKocTb { z E C: Im z > 0} Ha BepxHIOIO rro11yrr11ocKocTb { w E C: Im w > 0} ;2) BepxHroro rro11yn11ocKocTb { z E C: Im z > 0} Ha HH)KHIOIO no11yrrJIOCKOCTb { w E C: Im w < 0};3) BepxHIDIO rro11yrr11ocKocTb { z E C: Im z > 0} Ha rrpaByro rro11yrr11ocKocTb { w E C: Re w > 0};4) rrpaByro rro11yrr11ocKocTb {z E C: Re z > 0} Ha rrpasyro noJiynnocKocTb {w EC: Rew> 0}.,l.l;oKa3aTb, qTo BO Bcex c11yqa51x rrpeo6pa30BaHwe OJJ:H03Haqttoonpe;::i:eJrneTC51 3a;::i:atttteM O.UHOM rrapbl COOTBeTCTBeHHbIX BHYTpeHHHXToqeK (z 0 , w 0 w(z0 )) HJIH ;::i:Byx nap (zk, wkw(zk)), k 1, 2, rpaHHqHbIX ToqeK.===13.19.
HattTH uenoe JittHeHHoe rrpeo6pa30Bam1e w6pa)!{a10mee IIOJiocy, 3aKJIIOqeHHYIO Me)!{.uy rrp51MbIMH= w(z),OT0-)' =kx + bi' y =kx + b~' z =x + iy'Ha rronocy { w E C: 0 < Re w < 1}, H y;::i:osneTBopmomee ycnoBHIOw(b1) =0.=13.20.
Hai1TH uenoe JIHHei1Hoe rrpeo6pa3oBatttte ww(z), oT06pa)!{aIOI.Uee BHyTpeHHOCTb e;::i:HHHqttoro Kpyra {< 1} Ha BHYT-z: lzlI'Jwea 13180peHHOCTh Kpyra { w EC: lw - w01 < R}TaK, qrnfa1w(O)= w0 11 rop11-30HTaJihHhii1 n11aMerp nepexomrn B n11aMeTp, o6pa3yIOw,Hi1 c noJio)KHTeJihHhIM HanpaBJieH11eM nei1CTBl1TeJibHOH OCH yroJI a:13.21. .lJ:JI5!1z- , zcj:>yHKUHH w =0 :'.S a :'.S 11:.= x + i y, HaHTH o6pa3bl CJie.llyIO-ll{l1X JIHHHH:21) oKpy)KHOCTei1 x +/=ax, a ER;2) oKpy)KHOCTeii x2+/= by, b E R;3) np5!MhIX y = x + b, b ER;4) np5!MhIX y= kx, k E R;5) np5!MhIX y - y06) napa6oJI y=k(x - x0 ), k E R;= ax2, a E R.13.22. y CTaHOBHTbo6pa3bI nptt OT06pa)f(eHHHw = (z - z0) - + h, h EC, z =x + y:11) np5!MoyroJibttoi1: ceTKH x=e, y = e, e ER;2)R, arg (z - Zo) =a, R > 0, 0 ~nompttoi1: CeTKH lz - Zol =13.23 .
.lJ:OKa3aTb,a :'.S2n.qTQ .npo6Ho-mrneH:Ha5! cpyHKUH5!w = (az + b)l(ez + d), ad i: be, w(- die)= oo, w(oo) =ale, a, b, e, dE CKOHcpOpMHO OT06pa)KaeT pacw11peHHYIO KOMOJieKcttyIO flJIOCKOCThcHa pacw11peHHYIO KOMITJieKCHYIO flJIOCKOCTb13.24 . .lJ:oKa3aTh,c.qrn cynepno3HUH5! npo6Ho-mrnettHhIX orn6pa-)Kett11if eCTb .npo6HO-JIHHeHHOe OT06pa)KeHHe.13.25 . .IJ:oKa3aTb,'ITO o6parnoe K .npo6HO-JlHHeHHOMY OT06pa)Ke-HHIO 5!BJI5!eTC5! .npo6HO-JlHHeHHh!M OT06pa)KeHHeM.13.26 . .D:oKa3aTb,qTQ np11 npo6HO-JII1Hei1HOM OT06pa)!(eHHH o6-pa30M OKPY)!(HOCTH 11JI11 np5!Moi1 TaK)Ke 5!BJI5!eTC5! OKpy)!(HOCTh HJ111np5!Ma5!.13.26.1.
.D:oKa3aTh,CqTo ypasHeH11e s11.naonpe.nrn5!eT np5!MyIO, ecJJ11A= 0,A.::z + Bz + BZ + C = 0B i: 0, C E R.EcJJ11 )!(eAsi= 0,KOH<l>OPMHbIE OTOEP NKEHM51A E R, C E RAC > 0,11 !Bf -181TO ypasHeH11e orrpe.uemieT oKpy)K-HOCTh.13.27.ToqK11M11M'''R0Kpy)KHOCT11 pa.u11ycaHa3bIBa!OT c11MMerp11qHbIMH oTHoc11TeJibHO0,c ueHTpoM B ToqKeO.UHOM nyqe, BhrxowimeM 113 roqK110ec;rn 0H11 Jie)!(aT Ha=R2• .D:oKa3aTh,11 !OM] !OM*IlffO rrp11 .upo6HO-Jil1HeHHOM orn6pa)!(emm TQqKJ1, CHMMeTpWIHhieOTHOC11TeJihHO 0Kpy)KHOCT11 11n11 rrp5IMOH, rrepexOJJ.5IT B roqK11, c11MMerp11qHhre OTHOCl1TeJibHO o6pa3a 3TOH rrp5IMOH HJil1 COOTBeTCTBeHHO OKPY)KHOCTH.13.28 .
.IJ:oKa3aTb,qrncymecrnyer11en11HcrneHHO.upo6Ho-mrnei1Hoe OT06pa)!(eH11e, KOTOpoe Tp11 pa3Jil1qHbie roqK11 Z1i Z2, Z3rrepeso.u11r B rp11 pa3JIHqHhie roqK11wi, w2, w3.3To OT06pm1<em1eorrpe.ueJI5IeTC5I cpopMyJioi1W - W1 • W 3 - W 1 _ Z - ::: 1 • Z3 - Z2W -13.29 . .D:oKa3aTb,13.28,JIOH B 3a)J.aqew2w3-W1Z -qrn cjJyHKum1 wZ2.::::3=f(z),-:::1onpe.ueneHHa5! cpopMy-r.ue Z1i Z2, Z3 - pa3JIH'-lHbie TO'-IKl1, H W1, W2, W3 -TaK)!(e pa3JIH'-lHbie roqKH, KOHcPOPMHO OT06pa)KaeT Kpyr, rpaHHUaKOToporo npOXO.U11T 4epe3z,,Z2, Z3, Ha Kpyr, rpaH11ua KOTOporonpoXO)J.HT '-Iepe3 TOqKH Wi, W2, W3.
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