Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 47
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IIp11 lzl > 1309PEWEHl151 Tl1ITWlHhIX 3A.[{A4pH.U paCXO.U11TCH - He BblflOJJHHeTCH He06XO.U11MOe ycnos11e CXO.U11MOCTl1pH.ua. ITycTb lzl = IA, rnr.ua1~1= (n+lf(l+l/4"+ )·4 = 4n +8n+4(l+l/4"+a +I (4n 2 +2n)(l+l/4")411 2 +211l+l/4"121J=11=(l+64~++ 2n)(1--~-J=l+~·~+Q1 ·-/-,e>O,IQ,,l<C(const).4"+ +42n +c411-11IIo np113HaKy raycca pH.U CXO.U11TCH a6comorno rrp11 lzl = 1A.11rnK, pH.U cxo.u11TCH a6comorno np11 lzl '.S 14, np11 izl > 14 pH.u pacxo.u11TCH.2.75. 2) ITp11 z = 11 , nE N, 6ecKoHe•rnoe rrpmnse.ueH11e pacxo.u11TCH.IIycTb z =t n, rnr.ua2( 1-..£)e=in= (1-..£)[1+..£+ z , +o(-I,)] =nnn2n-n-0 (-I,).1-~+21in-J13 cxo.u11MOCT11 pH.ua L~= 1 l/ n cne.uyeT a6comoTHaH cxo.u11MOCTb .uaHttoro6eCKOHe1moro rrpOH3Be.ueHHH.OrseT: rrp11 z = /1 6ecKOHeqHoe npol13se.uem1e pacxo,UHTCH, rrp11 z =t nE N6ecKoHeqHoe rrpo113senem1e cxon11TCH a6comoTHO.2I'lla6a 33.26.
15) .[{oKa)l(eM CHa<rnna yrnep)l(.uemrn. s 3a.uaqe 3.26 9). Ilo orrpe.uenemno cos z 11MeeMei:+e-i:e·''+e-·"e-Y-e-"+ i sin x - - 2222Cne.uosaTeJJhHO, .UJJH lcos zl cnpase.un11sb1 paseHcTsa:cos z == cos x22(eY+e-·") + cos(2x)-1 = ch1 y - sm. 2xIcos z1 =22,2Icos z 1 =(er-22e-r ) + cos(2x) + l = sh'-y+cos-x.,2AHanor11qHo, .UJJH sin z rronyq11M paseHcTsa. zI'- = sh1y + sm-x,. 'Ism. zI'- = ch1y - cos-x.'IsmIIo:noMy 113 nepsoro paseHcTsa .UJJH lcos zl 2 BbITeKaeT HepasettcTsolcos zl 2 ~ ch 2y, a 113 sToporo - ttepaseHcrno lcos zl 2 ~ sh 2y. AttanornqHhIMo6pa30M .UJIH lsin zl nonyq11M ttepaseHCTBa jsin zl 2 ~ sh 2y, lsin zj 2 ~ ch 2y.0TMeTl1M, qro paseHCTBa B 3Tl1X HepaBeHCTBaX B03M0)1(Hhl TOJJhKO.UJJH z = nk + iy 11n11 z = n/2 + nk +iy, k E Z.3.28.
12) .[{oKa)l(eM rrpe)J(.ue scero yrnep)J(,UeHHl! B 3a.uaqe 3.28. 8). Iloorrpe.ueneH1110 ch z 11 sh z 11MeeMPEWEHl151 Tl1CTH4Hb!X 3ALl,AY310ch z =sh z =e-+e-ex-e-xex+ e-x=2e--e - z2ex-e-x=22cosy+ icosy+siny,2. ex+ e-xl2sin y.IlonoMy 11MeeM cnenyiouu1e pasettcTsa:2222222lch zl = (e ' + e - ')/4 +(cos 2y)/2 =(ch x) - sin y =(sh x) + cos y;222222lsh zl = (e ' + e - ')/4 - (cos 2y)/2 =(ch xi- cos y =(sh x) + sin y.Ifa 3THX paseHCTB cpa3y )!(e BhITeKaeT, 'HO2(sh x/ $ lch zl $(chx/;2(sh x) $ lsh zf $(chx)2.A no 11 ecTh Tpe6yeMbie HepaseHCTBa.
0TMeT11M TaK)!(e, 11To 3HaK pasettcTsa B 3Tl1X HepaseHCTBax B03MO)!(eH TOJlbKO rrpH z = x + ikn HJIHrrp11 z = x + i(n/2 +k n), rne k E Z.3.34. IlocKOJihKY s YKaJaHHOH o6nacrn cjJyHKUIUI f (z)= e- 11 'HBmieTCHcyrrep110311U11eH Herrpepb!BHblX cPYHKU11H, TO .ll.JUI JII06oro a HCXO)lHaHcP)'HKUHH HerrpepbIBHa . .ll:nH 11ccnenosatt11H paBHOMepHOH tterrpepb!BHOCTHyKaJattttoii cjJyttKU11H cnenaeM 3aMeHy rrepeMettttoii: (, = -1/z. B JTOM cny11ae I CI 1 I lzl H arg C n - arg z. Ilo3TOMY 11cxonttaS1 JanatJ:a cson«Tcsi K113y11eHHIO paBHOMepHOH tterrpepbIBHOCTl1 cPYHKUHH g(/;,) = eC B o6JiaCTHDR < I /;, I < 00 , larg Cl > n - a} .
.ll:nH ycTaHosneH11H pasttoMepttoi1 tterrpepbIBHOCT11 cPYHKU1111 g(I;,) = ec )J,OCTaTO'IHO rrposep11Tb HaJil11111e KOHe'l-=== {(,:HOro rrpe,JJ,eJia Jim e~ rrp11 /;,ED. IlycTb~ -t~s= ~+ ill Tor,JJ,a ec = e~(cos (arg s) +i sin (arg /;,)), larg Cl> n - a. Han111111e KOHe11Horo rrpe,JJ,ena cjJyHKI.(HH e ~ Ha6eCKOHe11HOCTl1 6y!leT HMeTb MeCTO TOJibKO TOr,JJ,a, KOr,JJ,a ~ ~ -oo. Ho 3TOB03MO)!(HO TOJihKO !lIIH cnyqaH, KOr,JJ,a larg SI <:: n/2 + £, £ > 0. Cne,JJ,osaTeJihHO, 11CXO)J,H3}! cjJyHKUHH e - llz 6y,JJ,eT paBHOMepHO HerrpepbIBHa B yKa3aHHOHo6nacrn ,JJ,JijJ: a$ n/2 - £, £ > 0.3.39.
IlycTh 0 < p < n/2. 06o3tta1111M Dp = C \00LJI{z: z - mi;I< p}.n =- N=.D:oKa)!(eM rrepsoe 113 3THX HepasettcTB. IlycTh z x + iy H z E DP' 3rn223Ha11m, 'ITO !lIIH nio6oro n E Z lz - n1! <:: p, T. e. (x - n rr.) + y2 <:: p .IlycTb i = x - 11n. Tor,JJ,a O!lHO 113 pasettcrn Ja,JJ,a1111 3.26 9) rrepe1111weTcHCJie,JJ,ylOW:11M o6pa30M:22222Jsin zJsh y + sin i, y2 + i <:: p .I=tv111tt11Manhttoe 3Ha11eH11e rrpasoii 11acT11 rrocnenHero paseHcTsa 6y!leT!lOCTHraTbCH B cnyqae, KOr,JJ,a y 0, ax HaXO!ll1TC}! B rrpe,JJ,enax (11 - 1) 1t + p$ x $ 11 n - p !lJlH tteKornporo 11 E Z.
3rn 03tta11aeT, 11To p :S Ii I :S 7f. - p.TaKHM o6pa30M, HMeeM uerro11Ky ttepasettcrn !lIIH z E Dp=PEWEHI15l TI1f1WIHhlX 3A.LJ.A lJ311jsin zj 2 ~ sin 2 x ~ sin 2p,'ITO 11 Tpe60BaJIOCh !lOKa3aTb.3.45. IlOCKOJihKY nm1 !al < 111lzl '.S 1 BeJil1'll1Ha 1-a. z* 0'TO 11CXO)l-Ha51 paUHOHaJibHa51 cjlyHKU1151 5!BJI51eTC51 HerrpephIBHOH, KaK cyrreprr0311UH51HerrpepbIBHhIX cpyHKUHH . .[(anee, H3 paBeHCTBaKaeT paBeHCTBOz=a- w- '1- a· ww=a) /(a. z -1)(z -BbITe-'ITO 03Ha'-!aeT, '-!TO 11CXO)lHa51 cjlyHKUH51 ocy-mecrnm1eT B3aHMHO-O!lH03Ha'-!Hoe coorneTcrnwe.
Ocrnnoch noKa3aTh, '-!TOlwl ::; 1,T. e. l{TO 11CXOJJ,HOe OT06pa)!(eHHe He BhIBOJJ,HT H3 eJJ,11HH'-IHOroz = x + iy, a= a 1 + i a 2. BKpyra. IlycThTaKOM cnyqae ttepaBeHcrnolwl::; 13KBHBaJieHTHOHepaBeHCTBY<1(x-aY+ (y-a2)22(a,x+a1 y-l) +(a,y-a2 x)2 -11JIH HepaBeHCTBYx2 + / + jaj2::; jaj2cx2 + /) + 1,KOTOpoe JJ,JI51jaj < 1 3KB11BaneHTHOttepaBeHcrnyx 2 + /::; 1, '-!TO11 TPe6oBa-noch JJ.OKa3aTh.I'Jiaoa44.11.
1). Ilp0Bep11MHe06XO)ll1MOe ycnoB11e CXO)ll1MOCTI1 p51na: 06m11H:jsin zl =~sh 2 y+sin 2 x'lJieH p51na !lOJI)!(eH CTPeMI1ThC51 K ttymo. TaK KaK(CM. 3a)la'-!y3.26),z=x +lirn IC sin nz) I nl = 0 TOJlhKOiy,TOB cnyqae2nzl = ~sh ny + sin nx 'sinI n--y = 0.2nIlprry = 0rrpH 3TOM06m11H: '-IJieH pJina//~<><>rrMeeT BH!lI~=' sin nxsin(nx)/n.Ilp11MeH5!5! rrp113HaK .[(11p11xne, rronyqrrM, '-!TO p51!lcxonHTCJI JJ.JIJI mo6oro xEnJJ,HMOCTH S!BmreTCS!4.18.Im zR.IfraK, orneT: MHO)f(eCTBOM cxo-=0flyCThf,,(z) = a (a+ l) ...
[a + (n- l)]b(b + l) ... [b+ (n-1)] z"n!c (c+l) ... [c+(n-1)]Tornaf,,+ 1l(z)I ='(a +n)(b+ n)f,,(z)(c+n)._z_,(n+l)= hl.,Cb+n)(a + n)I ·n+lc+nPEWEH!15! Tl1fll14Hb!X 3A)J,A 4312I f11(z)11o3TOMY Jim J, . +i (:::)I= z. Cne.uosaTeJihHO, 11cxo.uHhii1 pi!.ll cxo.u11TCH a6co,,_.~JIIDTHO np11 jzj < l; np11 jzj > l pH.ll pacxo.u11TCH.PaccMOTpHM e.u11H11 1rny1D oKpy)l(HOCTh jzj = 1. Ha :noM MHO)l(ecrnef,, +1(z) I ( 1)1c+nI= n+ (b+n)(a+n) .l f,,(z)I1peo6pa3yeM Bh1pa)l(eH11e(n + l)(c + n)(b+n)(a+n)-(1(11+ l)(c+n)(b+n)(a+n):+ ~J(1 +~J(1 + !:.J-1nnn(1+ ~J-1 =n( -l J] [ 1---+ob +a( -l J] = l + c + 1- a - b +o ( -l J .c+l= [ I+--+onnnnnnI1TaK,_[J!l_I =Ir+ c+ 1-(a+b) +o(~)I =nnl111+1(z)= \( 1+Re (c+ 1)~ (a +b) + ilm (c+ 1) ~(a +b) JI=1=[(t+Re (e+l)~(a+b))'+(rm (e+l)~(a+b))'J" +a(~)==l+Re (c+l)-(a+b)11+o(~J.11110 11p113HaKy faycca .llJI.fl 3HaKOIIepeMeHHhIX pi!.llOB CJie.UyCT, 'ITO a6COJIIDTHaH cxo.u11MOCTh pH.ua I~=I !,, (z) Ha MHO)l(ecrne jzj = l 6y.ueT TOJihKO np11Re( (c + 1)-(a + b)) > l 11J111 Re(a + b - c) < 0-TeM caMhIM 3a.uacm peweHa.4.25.
11yCThz2f,,(z)= (l+/)",Tor.uaI z 12'"~IJ,,(z)j=ll+z 2 IJim~ I /,, ( z) I = - 1- 2 • 11ono)l(l1M z = re;~ rro yrnoBHID jcpj~n/4II->~11 + z I44H Il +z2 12 =4(l + r2 cos2cp) 2 + r sin 2cp = l + r + 2r2 cos 2cp;::: 1 + r > 1, ecJIH z '# 0.11o3TOMY p.l!J( I ~= 1 !,, (z) cxo.UHTCH a6comorno Ha MHO)l(ecrne {z: /arg zj ~ :rr.14 }.11oKa)l(eM, 'ITO Ha )TOM MHO)l(eCTBe HeT paBHOMepHOH CXO.llHMOCTH. PaCCMOTPHMPEWEHI15J TMITML!Hb!X 3AJJ:AY~~7~~313llS( z ) -S11 ( z ) = ~ k=u+lfk( z) =z- ~ k=u+I (l+ z2)k-2 -k ,(1 + z)z * 0, S(O)-Si0)=011 ccpopMyJI11pyeM KOHTp-ycJIOBHe K Kp11Tepmo KowH paBHOMepHOH CXOllHMOCTH cpyHKU110HaJibHOro pHna::le:> 0 VN:Jn~N, ::lz E Z : jS(z) - S (z)I11~ £.B3HB B Ka4ecrne £ =112e, z11 = 11 J;i,, 6yneM HMeTbIs( ~)-s1 ( ~)I=-vn-vn4.33.
6) ITycTb f,,(z) =113aMKHYTYIO o6JiacTb Re z <L~=' n11e-112011e11 ' :,>_!_, Vn~n1(l+l/n)TOr)l,a112eI J,, (z) I= n-8, 8 > 0. B JTOHcxo):(11TCH, cJie):(oBaTeJibHO, pH)],11e11 '0.Re:.o6JiaCTH Jf,,(z)j ~L~=' J,, (z)PaccMOTPl1M1111e-1120•PH)],cxo)l,HTCH paBHO-MepHo B 3aMKHYTOH 06JiacT11 Re z :S -8 rro rrp113HaKy Bei1epwTPacca. A TaKKaK KmK):\bIH 4JieH f,,(z) pHna HBJIHeTCH HerrpepbIBHOH cpyHKU11eH, TO H cyMMa pH.L\a L ~= 1J,, (z)- tterrpepbrnHaH cpyHKUl1H B o6Jiacrn Re z <-8)l,JIHJII06oro 8 > 0. 0TCIOlla CJienyeT, 'ITO cyMMa pH):(a HBJIHeTCH HerrpcpbIBHOWcpyttKu11ei1 B o6nacrn Re z < 0.(-1) 11 + 14.45. ITycThj,,(z) =Re z > l PHllL~= 1 lle·;c·(lnll )Re·e-l= ~. B o6Jiacrn:n -CXO)l,11TCH, ITOJTOMY cxomncH pH.L\-(-1)11+1rrpOH3Be)l,eH11e n ~=I l + e:ln11(JiaCTb Re zl-----=-i;;-;;-, rnrna IJ,,(z)I =L::J f,,(z) I 11J CXO):\HTCH a6coJIIOTHO.
PaCCMOTPHM 06-> 112. IToKa)!(eM, 4TO B JTOH o6rracTH rrpo1nse):(ett11e cxo)l,11TCH.(- !)11+111crronb3YH Janaqy 4.31, 11MceML::=, ~ = L:=, (-1) +1e-:ln" . ITpw z =11e'x > 112 JTOT pH)], CXO)l,11TCH, cJie)l,oBaTeJlbHO, OH cxo)l,HTCH 11 rrp11 Re z > 112.11rnK, pH)].L::=,J,, (z)cxo)l,HTCH, cxo):(HTCH 11 pH)], l: ~J!,, (z) I". IToJTOMYno Janaqe 2.66 6eCKOHe4HOe rrpOi-BBe)l,eHHe TaK)!(e CX0lll1TCH.I'Jzaaa55.1. 17) 11MeeM u(x,y) = e"', v(x,y) = e''.
ITposep11M ycrros11H Kow11P11Matta u'x = v'.1., u'_r = -v'x. Torna u'_r= ex= v'.r = e'', u'_r = 0 = v'x· IToJTOMYx = y + 2kn:i, kEZ.314YfTaK, OTBeT: TO'-IKM ,llJHpcpepeHUHpyeMOCTH cpyHKUHH j{z)1+ie-'° pac-= y + 2klfi, kEZ.nOJIO)f(eHbl Ha C'-!eTHOM '-IHCJie np.s!MbIX x==e=5.4. 1). IlycT&j{z) u(x, y) + iv(x, y). 11MeeM u(x, y) const. Tor.na 111 ycnos11i1 Kow11-P11MaHa v'x(x.y) 0, v'_,.(x, y) 0, T.
e. v(x, y) const, noJTOMYj{z) - KOHCrnHrn B D.f5.22. 6) 3an11weM=(z)=lz-ajz-a=- =I g(z) j, g(z) =- =u +iv.z-b I1<l>yHK-z-bU11Jl g(z) attaJil1Tl1'-!Ha BCIO..Qy, KpOMe TO'-IKl1 z= b. MMeeM f (z) =~.ofi (af .af)OZ = l OX - Oy , TO0TCIOLJ.a, TaK KaKlof =-·lll/·-(2uux +2vv..\ -iuu!' -1vv!.) .2 '-Jll-+~ 2v--'.:\ozIIIcnOJI&3yeM ycnosmr Kourn-P11MattaD03TOMYdf=I I z - b I[2 Iz - a II•I.,.nn.si.I•/•IcpyHKUl111 g(z): u'1]1 Iz - b I2 Iz- aI= v'n u'_,. = -v'.n.I- = - - - u(u . +tv )-tv(u +tv .) =---(u-tv)g_ =oz·'·''·'_!_lz-bl(z-a)~-a-b_lz-bl_2lz-alz-blz-bl22-lz-aj·2__1__lz-al (z-b)jz-bl 2 (z-a)-=_!__lz-bl_ (a-b)2 lz-aj (z-a)(z-b)5.25. 1) Ilo orrpe.z:1eJiemuo cos z =Tor.na 11MeeM ypasHeH11e t~w 2 - l22tw-eiz+ e-iz IT2+ 1 = 0,.yCT&izerro3TOMY t=t,eiz+ e-iz- - - =OJ .2= w + ~w 2 - l, meHMeeT ,[\Ba 3Ha'-leHl1Jl KaK MHOro3Ha'-!HaJI cpyHKUHJI; Tor.naln(w + ~w2- l), r.z:1e0KOH'-!aTeJI&Ho:ln(w + ~w2- l)- MHOro3Ha'-!HIDI cPYHKUIDI,~z = ~1ln(w + 'iw--1)==o6parn~ Ke;' .arccos w - MHOro3tta'-!ttaJl cpyttKUIDI,lo6paTHaJl K cpyHKUl111 cos z.I'!laaa 66.6.
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