Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 24
Текст из файла (страница 24)
I1po1lOJI:>Kl1Tb11.89. I1po110JI:>Kl1TbaHan11rnqecK11 BCcpyHKU11IOsin (z) IaHaJIHTI1'IeCKl1 cpyHKU11IO j{z)= 2:.:z.0.::.IIB711=0 ......C\{l }.-XI11.90.TifiecK11 B11.91.~ c(l +F(x)<DyHKU11IOC\(-oo,= r x _e_dt, x > 0,Jo l +tnp01l0Jl:>Kl1Tb aHaJIH-0).CTycTb cpyHKU115!fit)HenpepbIBHa np11t > 0 11 jf(t)I ~tra, 0 ~ t <+co, a> 0. .[(oKa3aTb, 'ITO HHTerpan rnna Kow11/(~)=fox ~~:(5!BJI5!eTc5! aHan11rnqecKoi1 cpyHKu11ei1 B11.92. DoK33aTb,C\[O,co).'-ITO raMMa-cpym<.u1110 3i1Jlepaf(~)> 0 11 (-' = e<:-I> 1111 , MO:>KHO aHaJI11T11'IeCKHC\{O, -1, -2, ... } CJie1ly10rn11M o6pa3oM:rne Re zf(~) =I:"°11=011.93 .
.[(oKa3aTb,(-1)"n!(~+ n)+ f "° (=fa" t :-ie- dt1npo11011:>KHTb B-l e-I df.1'ITO cnpaBe1lJI11BbI cJie1ly10rn11e yrnep:>K1leHmr:1) r(z + 1) = z f(z);2) f(z) f(l - z) = n I sin nz;,..,) f( ~ ) -4) f(-).-. -- e; 2 .,:1_ 1 ·, e -~.c:-1.,, c./Cc,,,.J5)I-rhueJia51 '-!'YHKU1151;f(~)- '=-1-.J C:e~d~.2m ·JAHAJU1TI1lJECKOE DPO)J.OJDKEHHE.
MHOrG3HA lJHblE <PYHKUHH155r.ne y - KOHTyp, 11306pa)!(eHHbIH Ha rrp11BelleHHOM HIDKe p11cyHKe.y11.94. IIpo.nomKHTbaHaJI11rnqecK11 B C\[-i,1rF(z) = 27ri J1[1=2d~ce +I)(~ - z)'i] cpyHKUHIOI z I< 2.11.95. IIycTb cpyHKL~1151 flz) peryJI5lpHa B rroJiyKOJibUe {z: p << R, Im z > 0}, HerrpepbrnHa B ero 3aMbIKam-rn: 11 Im flz) = 0 llJI5lneikrn11TeJibHbIX z, -R < z <-p, w Refi.z) = 0 .llilll lleikrn11TeJibHbTX z,p < z < R . .LJ:oKa3aTb, qTo flz) MO)KeT 6hITb aHaJIHTwqecKH npOllOJI)KeHa B KOJibUO {z: p < izl < R}.11.96.
IIycTb cpyHKUIDI./(z) peryJI5lpHa B ITOJIYK01ThUe {z: p < izl < R,Im z > 0}, HenpepbIBHa B ero 3aMhIKaH1111 11 rrp11H11MaeT .nei1crn11TeJibHbie 3HaqeH115l Ha MHTepBMax (-R, -p) 11 (p, R) .nei1crn11TeJibHOHizlOCH . .D:oKa3aTb, 'ITO cpyHKU115l flz) MO)KeT 6hITb aHaJil1Tl1'!eCKl1 rrpollOJDKeHa Ha BCe KOJibUO { z: p11.97. HattTHcpyHKU11110)< lzl < R}.06JiacT11, Ha KOTOpbre oTo6pa)KaIOTC5l c noMOLUbIO= e':1) np5lMOyrOJibHl1K 0 < x <a, 0 < y < b;2) rronyrroJioca 0 < x <a, y > O;3) rronoca 0 < x <a.11.98. HattTl1 o6JiaCTl1, Ha KOTOpbre OT06pa)KaIOTC5lcpyHKU11111)0)c ITOMOLUbIO= COS z:rroJioca -n I 2 < x <TC I 2;2)rroJioca0 < x < 2 n.11.99.
Hawrn o6JiaCTb, Ha KOTopy10 c ITOMOLUbIO cpyHKU1111oTo6pa)KaeTc5I nonoca 0 < x < n/2.11.100. IIocTpottTbU115I ro= e:_,Ci)=tg zp11MaH0By rroBepxHOCTb, Ha KOTOpyro cpyHK-oTo6pa)KaeT z-rrJIOCKOCTbC.Towea 1115611.101. ,[(oKa3aTb, '-!TO ecm1 a - ;::i:ei1crn11TeJibHOe 11ppau110HaJibHOe '-IHCJIO, TO p5!LI:'\'"'1_ e2.:11ai)L--11 = 1 2 ,, (zlzllzlnpe;::i:crnsnReT co6oi1: B o6nacrnx> 1 11< 1 attan11T11'fecK11ecpyHKu1111, ;::i:m1 Ka)K;::i:oH: 113 KOTOphIX oKpy)KHOCTb {1 } 5!Bn5!eTC5! ecTecrneHHOH rpaH11uei1. Ecn11 )Ke a - ;::i:eikTBinenhHoe 11ppau110ttaJihHOe q11cno, TO yKa3aHHbIH pR;::i: npe;::i:crnsnReT co6oi1 pau11ottanbtty10 cpyHKU11IO.z: lzl =11.102. ,[(oKa3aTb, 'fTO p5!.U'\' "°_l_In:=e:"",0 11=In:+n'cxo.n11TC5! np11 Re z > 1 11 ero cyMMa - attan11T11'-lecKa5! cpyHKUH5! cecTecrneHHoi1 rpaH11uei1 Re z 1.=11.103.
D:oKa3aTb, '-!TO cPYHKUl15!aHaJIHTH'-IHa np11 Re z > 0 11 HMeeT np5!MYIO Re zseHHOtt rpaHHUett.=0csoei1 ecTecT-11.104. ,[(oKa33Tb, '-!TO cpyHKUHIOr.nea,,= (-1)"+1, \ k - i= 2k, \k= 2k+e- 2k, k =1, 2, ... ,MO)KHO aHa-n11T11'-!ecK11 npo;::i:oJJ)KHTh B nonynnoCKOCTh Re z > -1.11.105. ,UoKa3aTb, '-!TO cpyHKU11IOf(-;,)=fa"'- e< e' sine'dt,1onpe;::i:eneHHYIO B nonynnocKocT11 Re z > 0, MO)KHO aHan11T11'-!eCKHnpo;::i:on)Kl1Tb s nonynnocKOCTb Re z > -1.11.106. B KaKHX o6nacrnx C cnpase;::i:nHBbI cpopMyJJI>I:711) f "" t : -i costdt = f(-;,)cos 7IZ; 2) f "° t: - i sintdt = f(z)sin z.Jo2Jo2I'llaBa 12BhlqEThl. BhlqHCJIEHHE HHTErPAJIOBC IIOMOIIJ;hlO BhlqETOBBw1emoJ11 o;::i:tto3tta'IHOH attamrT11'IeCKOH cpyttKU:I111 f(z) B 11301111= oo) Ha3bIBaeTC51 3Ha-poBaHHOH oco6oi1: TO'IKe Zo (B TOM q11cne I1 ZoqeHHe HHTerpa.rrar;::i:e HHTerp11poBaH11e Be;::i:eTC51 rro 3aMKHYTOMY Kycoqtto-rna;::i:KoMyKOHTYPY )l{op;::i:atta, co;::i:ep)KameMy BHYTPH ce65l TO'IKY;::i:ep)KameMy ;::i:pyrnx oco6b1x TO'IeK cpyttKllJHff(z).z011 He co-Ilp11 :noM HH-Terp11:poBaHHe Be,n:eTC51 B IIOJIO)l(lfTeJibHOM HarrpaBJieHHH OTHOCHTeJibHO o6rracTH, co;::i:ep)l(ameH: ToqKy z 0 .
3HaqeHHe 3ToroHHTerparra B CHJIY TeopeMbI Konm He 3aBHClfT OT KOHTypa HHTerpHpOBaHH51, o6rra;::i:aIOll!ero YKa3aHHhIMH CBOHCTBaMH. ITo3TOMY rrpH BbJqlfcJieHHH BhilfeTa B TOqKeTaTh, qrny-oKpy)l(HOCTh { z:pa;::i:11:yca, a B rnqKezo= oo -Is - z 01= 8}Zo f.00 MO)l(HO cqH-;::i:ocrnrnlfHo MarrorooKpy)l(HOCTh {s: Isl = R};::i:ocrn-ToqHo 6orrhllloro pa;::i:11:yca.3Haqemre BbIIIJeyKa3aHHoro 1rnTerpana B cnyqae z0K03cpcp11u:11ettTya-I -cc 1np11 (z - z0)K03cpqmu:HeHT rrp11z -I=/. oopastto=1- ,as cnyqae z0 oo pastto - a_i, r;::i:eB nopaHOBCKOM pa3JJO)KeHI111 cpyHKU:I111f(z) B OKpeCTHOCTM TOqKH Zo.Ecntt TOqKa z 0# oo ecTb rromoc1res=:=:orropH;::i:Ka cpyHKUimj(z), TOl'"-1!i~~[( z - z0 )"1 f(z)].(111 -1) ! ---o dzEcmi q)yHKUI151 fiz) amlJIHTH'IHa B orpaH11qeHHOH 06nacT11 D(6b!Tb MO)KeT, 11 MHOrOCB513HOM), KpOMe KOHetrnoro 'IMCJia TO'IeKZ1 , z2, ..
.,z,, E D , 11 ttenpepbrntta BD, TOI'wea 12158r.ue CJD ecTb 3aMKHYTbIH Kycoqtto-nrn,UKHH KOHTYP )Kop.uatta B cny1.fae, Kor.ua o6nacn D 5!BIDieTC5! O.UHOCB5!3HOH, 11 06ne,U11Hett11e KOHelfHOro lfHCJia TaKHX KOHTypoB, Kor.ua o6nacTb D 5!BJI5!eTC5! MHOroCB5!3HOH.Ec1111 cPYHKUH5! j{z) attanHTHlfHa sc10.uy Ha C, KpoMe KOHelfttoroq11c11a TOlfeK z0 = oo, z 1, .•• , z,,, TO crrpase.u1111sa cpopMyna'LA""'" =o res f (z) = 0.:=:~ITp11 BbilfHC11ett1111 tteco6crneHHbIX HHTerpanos cpyHKUHH .ueii:cTBHTeJibHoro rrepeMettttoro cymecrneHHYIO ponb 11rpaerJleMMa JKop.11:aua.1.
ITycTb cpyHKUH5!j{z) 5!BJrneTC5! tterrpepbIBHOH B o6nacTHD = {z: !zl 2: Ro, Im z 2: a} ( { z: !zl 2: Ro, Im z11.f{z) -0 rrp11 z -::=:; a})oo, z ED. Ecn11 m > 0 (m < 0), TOJ~J,. e; : f(z)dz = 0,111r.ue yR - .uyra oKp)0KHOCTl1 { z: lzl=R}, Jie)!<alllM B 3TOtt :>Ke 0611acT11 D.2) ITycTb cpyttKU115!j{Z) 5!BJ15!eTC5! HerrpepbIBHOH B 06nacT1111.f{z) -D = {z: !zl 2: Ro, Re z 2: a} ( { z: !zl 2: Ro, Re z0 rrp11 z - oo. Ec1111 t > 0 (t < 0), TO::=:; a})J~~lr. e-':f(z)dz=O,r.ue YR - .uyra OKp)0KHOCTH {z: !zl=R}, ne:>Kama5l B 3TOH :>Ke o6nacm D.Crrpase.u1111s npw-1i1un ap2y,11enma aHaJIHT11qecKoi1 cpyttKUHH: ecJIH cpyHKUH5! j{z) attan11rnqHa B 3aMbIKaHHH D O.UHOCB5!3Hoi1 orpaH111:1ettttoii: 06nacT11 D (T.
e. j{z) aHaJJHTHlfHa B o6nacTH G H D C G)sc10.uy, KpoMe KOHelfHOro lfHCJia rro1110cos ~k E D, kHMeeT KOHelfHOe lfHCJIO Hyneii:Olk= 1, 2,... ,11,11ED, rrp11qeMj{z) :/= 0 BCIO.UY Ha CJD,TO rrp11pamett11e apryMeHTa cpyHKUHHj{z) B,UOJ1b KOHTypa 1 =CJD rrpHO.UHOKpaTHOM o6xo.ue 0611aCTl1 B I10J10)!{11TeJibHOM HarrpaBJieHHHpaBHO 2TI(N - P), r.ue N - rronttoe lf11c110 Hyneii: (c yLieTOM KparnoCT11), a p - I10J1HOe 411CJ10 ITOJIIOCOB ( c ylfeTOM KpaTHOCTH) cPYHKUHHj{z) B o6nacTH D, T.
e. Var[argf(::)Jj:e= 211(N - P).BblYETbl. Bb!Yl1CJ1EHI1E l1HTEfPAJ108 C OOMOW,blO Bb!YETOB159ITp11 JTOM )lOKa3brnaeTC5I, qTo127r-Var[ar 0f(z)JI127rif 1(z)y f (z)=-j--dz.o:Ey1IHTerpa.rr B ITOCJie)lHeM paBeHCTBe paBeH cyMMe BbJqeTOB cpyHKu1111 J' (z) I j(z) B o6nacrn D.B11qeT cpyHKUHH f1(z) I f(z) Ha31rnaeTc5IJI02apucjJ;iiui1ecKuM 6bl -LtemoA1 cpyHKUHH f(z), ITOCKOJibKYJ'(z) =!!__(Lnf(z)).f(z)dzHerrocpe.ucrneHHhIM cne.ucrnHeM rrpHHUHrra apryMeHra 5IBm1eTC5ITeopeMa Pyrne. EcnH cpyHKUHH j(z) H cp(z) aHa.JIHTHqHbI B 3aMbIKaHHH D orpaHHqeHHOH OLJ:HOCB5I3HOH o6JiaCTH DBCIO.llY HMeeT MeCTO HepaBeHCTBOIf (z) il:EDD > i 'P(Z) il:EiJD'D cpyHKUHH f, f + cp, f - cpTO BHYTPH o6nacTHqHCJIO HyneH: (c yqeTOM KpaTHOCTH).I1Ha rpaHHUeav11MeIOT O)lHHaKoBoeTeopeMa rypsuua.
ITycTb 3a)laHa ITOCJieL(OBaTeJibHOCTb cpyHKUHH {f,,(z)}, orpaHHqeHHhIX B 06nacT11 D. ITycTh 3Ta rrocne)loBaTeJihHOCTb paBHOMepHO CXO)lITTC5I BttyrpH o6JiaCTH K cpyttKUHHj(z) ;j:. 0,z E D. Tor)la )lJI5I mo6oro KOHTyparcruero qepe3 ttyn11 j(z), cyruecrnyeT q11cnoD, Dr11 0cDI1He rrpOX0.[(5I-= n0(f), TaKOe, qTo .[(JI5Imo6oro 11 ~ n 0 (f) q11cno ttyneH: Nr,, = Nf B o6nacTH Dr.12.1. Pa3JIO)!(eHHe B p5I.n; JiopaHa cpyHKUHH j(z) B OKpeCTHOCTHTOqKJ1 Zo = 00 HMeeT BH.[(f(z)=c 0 +c1z- 1 +c2 z-" + ....HaifTH res/(z) B ToqKe z 0= oo.12.2.
,lJ,oKa3aTb, qrn eCJIH Zo # 00 - rrpocrnH: ITOJIIOC cpyHKUHHcp(z)/\j/(Z), T. e. cp(z0) i- 0, \j/(z0) = 0 11 \j/ 1(z0) i- 0, TO1res cp(z)/9(z) = cp(-;, 0 )!cp (z 0 ).:::=:-oI'.Tiaea 12160**12.3. IIycn, cpyHKUHH <p(z) 11 \j/(Z) amUIHTHqHbI B TOl!Ke Zo 00 'rrpttl!eM <p(zo)0, a \j/(Z) HMeeT B TOl!Ke Zo HYJib BTOporo rrop5l,I(Ka.~'fJ(Z)H attTH res-- B TOqKe Zo.'lj;(z)12.4. HaH:rn res (<p(z) f(z)) B TOqKe z0 , ecn11 cpyttKUH5l <p(z) aHaJittTHl!Ha B TOl!Ke Zo, a cpyHKU115lf(z) HMeeT B 3TOH TOl!Ke IIOJIIOC rrop5l,UKa k 11 nrnBHyio qacTb nopattoBcKoro pa:mo)l(eHI15l cne.uy10Il(ero B11.ua:~+ •• • +z-z 0a_k k(z-z 0 )12.5.
JJ:oKa3aTb, l!TO eCJIH cpyHKUI15l f(z) aHaJIHTHl!Ha B TOl!KeIz = oo, TO crrpaBe,UJII1BbI cpopMyJibI:1) res f (z):=oo= :--+oolim z (f (oo)- f(z));2) resf(z)=-g'(O), r.ueg(z)=fil/z).:=oo12.6. HaH:rn res f' (z) I f(z)) B TOl!Ke z 0 -::/=- oo, ecntt:1) z0 - HYJih rrop51,UKa n aHaJIHT11qecKoH: cpyttKI.J;IiMfiz);2) Zo- IIOJIIOC IIOp5l,UKap cpyHKUHHf(z).12.7. IIycn, cpyHKUH5l <p(z) aHan11rnqHa B TOl!Ke z0oo. HaH:rn-::/=-res ( '{J(z) · f '(z) If (z)) B TOl!Ke zo, ecn11:1) z0 - HYJib rrop5l,UKa n aHaJIHTHl!eCKOH cpyHKu1111f(z);2) z0 - nomoc rrop51.UKa p cpyHKu1111f(z).12.8. Hawrn res f(<p(z))) B TOqKe z0 -::/=- oo, ecn11 cpyttKUH5l <p(z) aHa<p(z0 )*0, a cpyHKUI15Ifiw) HMeeT B TOl!Kenon10c rrepBoro rrop51.UKa c Bbil!eTOM, paBHbIM R.JIHTHl!Ha B TOl!Ke Zo 11 <p'(zo)Wo=12.9.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
