Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 45

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(T- e )-µ,.II'0dr + c" ' I-2e J--1,iT-er:dt. 4) -cr:(l+rs)21s dr·Jo (l -l" )2111'r; - Jo (l - 15 )415 'r; -~t 14 dt + C";5) r; = crJl6) ecm1 ypaBttett11e rrapa6oJibI p=2I +cosqJ, TO r; = i cos " Fz.2I'Raea 14: avJ dyav.14.3. /(z)= _ -dx--dy+C+iv(x,y),r.L1ez=x+1y,zo=xo+y0-<>dX-rrpo113BOJibHa5i TO'-!Ka 06nacn1 D, C - ,L1eikTBHTeJibHa5i KOHCTattTa.14.8.I) -(x 2 -y2)/2+C;2) 2xy-(x 2 -y2)!2+C; 3) (x4) arg z + C;5) rq> sin q> - r log cos q> ;7) y sin y ch x - x ch x cos y + C;6) -2xyl(x22y+ y2)+C;+ y2)+C;8) 2 arctg (xly) + 2xy + C;9) x sin x shy - y cos x ch y + C.14.9.I) z 2e'eia, a - L1ei1cTB11TeJibHOe;2) A exp ( z 2 I 2), A > 0;23) exp ( z + ia), a - ,L1ei1cTB11TeJibHoe;4) Aze' , A> 0.14.11. A, B - ,L1ei1crn11TeJibHb1e KOHCTaHTbI:1) u =Ax+ B; 2) u =A arctg (Y/x) + B; 3) u =A log (x2 + /) + B;4) 11 = A(x 2 - / ) + B; 5) u = Axl(x2 + /) + B; 6) u = Axy + B.14.12.

A, B - L1ei1cTBHTeJibHb1e KOHCTaHTbI:I) Ax+ B; 2) A arctg (vlx) + B; 3) Ax/(_/+/)+ B; 4) Ayl(J + /) + B.14.13. [flz)J - HeT, arg.ftz) 11 log [flz)J - ,Lia.rOTBEThl K 3A.ll,A 4AM2953) B sepXHIOIQ nonyrnIOCKOCTb.13.91.1) Bo sc10 nnocKOCT& c pa3pe3aMH no ,nei1cTBITTen&Hoi1 OCH a.non& .II)"lei1(- oo, -1], (1, oo);2) B npaBYJO nonynnOCKOCTb c pa3pe30M no .neiiCTBHTeJ1bHOH OCH B1l0J1b.II)"la [ 1, oo).13.92.1) B nonynonocy lul < 1t/2, v > O;2) B nonocy lul < 1t/2;3) B nonynonocy 0 < u < ]r./2, v > O;4) B nonocy -1t/2 < u < 0.13.93.l)w=~z-b,w(a+b)=i;2z-a-1)( z+z-2) w =21+ tg 2 -a, w(x + Oi) > 0, npw x > l;23) CHa<rnna c; = ez, ,nanee KaK B 13.93, 2);4) CHaY:ana c;; = ez, nanee KaK B 13.93, l);5) cHaY:ana c; = ch z;6) cHaY:ana c; = z- 1 orn6pa)!(aeT Ha nonocy 0 <Rec;< T 1 c pa3pe30M1[T ,T 1];7) CHaY:ana c; = T 1(z + z- 1) OT06pa)!(aeT Ha nonynonocy 4-l <Rec;< T 1,Imc;>O;8);= ( z- a;9) cHaY:ana c;[T (a+ a-1), l];1b) b=a; w =; + V+i, w(oo) = 0;= z- 1 orn6paJKaeT1Ha nnocKOCT& c pa3pe3oM no OTPe3KY110) cHaY:ana c; = T (z + z- ) oro6pa)!(aeT Ha BHelllHOCTb O"IpeJKa [T 1(a+ a- 1),T (b + b- 1)];111)w=~r 1 (z+z- 1 )+1.13.94.

w =~5 + .J z13.95. w =iJ2 ch13.96.13.106.\112+9 .(;a= ( Z II /2 +.J Zarchz),?/1111-1) -0 <a< Tl I 2.297OTBETbl K 3A)J,A qAM14.14.fit) =at+ b, a, b = const, TOflbKO ecm1 u2du1 du=const.2du14.16. !J.u=-, +--+--, =0, u=Alogr+B.r dr (!qra,.-14.20.1. He cne,11,yeT.14.20.3. Yrnep)K,11,em1e tteseptto. ,[(ocrnTO'lHO paccMoTpeTb cj>yHKl.\11!0u(x, y) =Re(e') B o6nacrn D = {z = x + iy, x :S 0, 0 :S y :S 1}. Ecn11 o6nacTbD orpatt11'1eHa, TO yrnep)K,11,ett11e septto.14.20.4. He cne,11,yeT.14.22.

Ymep)f(,ll,eH11e Heseptto. ,[(OCTaTO'lHO paCCMcnpeTh clJyttKllillO u(x, y) =.xy.14.26.2. ConpIDKeHHM cp)'IIKUIIBv ( r, (J)=2rR sin(fJ - rp),R 2 + ,.--2Rrcos(fJ- rp)•14.29.1) u(x,y) = 1;2) u(x,y) =x;3) u(x,y)= -,-,....::Y_ __x-+y-+4x+44) u(x,y) = 2.xy; 5) u(x,y) = x 3 - 3x2 -3xy2 + 3/ +12x - 1.14.30.1) u(x,y) = l;ty+l,, ,,x-+ (y + 1)-2) u(x,y) =3) 11(x,y)= 1/2 -5) u(x,y)2x= -arctg-.(lhr) arctg ((1 - x)ly);4) u(x,y)x,;x-+ (y + 1)-yJr22_ l f2" (r - R )(z 0 +Rei"' dI 4 .31 . u(z) - -J,0,,rp,2ffR-+r--2Rrcos(rp-fJ)14.33.

u(x,y) == ,z = Zu +re iO , r > R.x2 - y2,, -1.x-+ y-Ilog I z -c;I tp(fJ) I de; I+ Cff 1,1=114.43. u(z) = -- Ji== __!_Jilog I z-e;o I tp(fJ)dfJ+C, I z I< l.ff 1,1=114.45. <DyHKUH51 H ee nepBbie np0113BO.IlHbJe orpaHH'leHbl B Ja,naHHOHHeorpaHJ1'1CHHOH o6naCTl1.14.54. Yrnep)f(,ll,ett11e Heseptto - .nocraTO'lHO paccMOTPeTu u,,(x, y) = xy, ao6nacTb D = { z: Im z > 0}.298OTBETbl K 3A,IJ;A YAM14.55.p, =x, q, =y, P2 =x -y, q1=2xy,43p 3 =x - 3x/, q 3 = 3x2-y3, p4 =x -6.x2/+y4, q4 =4x3y-4x/,2P11 =i1' cos n<p,2q11 =/'sin mp, n = 1, 2, ....14.56.a) YTBep)!()J;em1e a) HesepHo - ;:i;ocraTO'-IHO paccMO'IpeTb cpyttKUH!Ov(z) =Re (exp(-i/z)), a o6nacTb D = { z: Im z > 0};6) eCJil1 cyruecTByeT cpyttK.l.lilll V(t), HenpepbIBH3H Ha BCeH )leHCTBHTeJibHOHOCH 11 orpaHH1-1eHHM, TO yrnep)!()J;eH11e BepHoe.Diana 1515.6.1) O; 2) a; 3) 0; 4) 0; 5) jim wj; 6) jlm wj; 7) jRe A.j; 8) jRe /...

j.15.26.1)r(/J + 1)p3) n.1fJ+1,/J > -1, Rep > 0;2)n!, Rep > Rd.;(p-A.r+ 1Im(p + iw)"+ 1,, 1 , Rep> I Im w j;(p-+w-r4)P -,),, , , Rep> (Re A.+ I Im w j);(p-A.)-+w-5)h"'[.~;o n+ 1 e-nrp(l-e-pr), Rep> 0.p1- e-P'115.27. - - ph15.28.1)2)ta.rca + 1),1-<l>(z~} r)le <l>(z) = };r Jo' e-3) t" 12 1,, (2.Ji);11'dr;- cpyHKUHH ourn6oK;4) J0(t) , r;:i;e J,,(t) - cpyHKUHH EeccenH nepsoro ponanopH;:i;Ka n (cM . 3a;:i;aqy 15.21).15.29.l)(n!f 1t"; 2)(n!f 1en"t,t> l;O , t< l;O,t= l,n> l;T 1,t= l;n= l;3) (n!f 1e"'t"; 5) sin t; 6) cost.1 "+;~dp1,15.32.

y(t)=- ·. eP1 4=-(sint-tcosf).2;ri x-i~p + 2 p- + 1 2f299OTBETbl K 3A)J,A4AM15.33. x(t) =a cos f...t + CPI 'A.) sin At.= (sin t -15.35. y(t)t cos t)/2.15.36.1) x(t)= e' -2e 2' + e3';2) x(t)= e-21 (1+4t + t5!20);3) x(t) =_!_(3-t 2 )sinr-ltcost;84) x(t) =8r'(sint-tcost),O<t<n,-2-'ncost,t>n.{15.37.1) x(t)= /,y(t)=e1;2) x(t)= 2ch (ht) I 3 + cos(t) I 3,y(t)=-ch (ht) 13 + cos(t) I 3;-r1-e '0<t<1,l-e-1 -sh(t-l),1<t<2,-e-1 - sh (t -1) +ch (t- 2)t> 2,3)x(t) ={y(t)-e-1 ~~~-;;_l),1<t<2,-e- -ch (t-1) +sh (t-2)t > 2;={0<t<1,1=e-r, z(t) =0;4) x(t) = - e -t, y(t)5) x(t) =eat sinPt; y(t) = ea (sin Pt 115.38.

1) <p(t) = (e11).1e- )14 +(sin t)/2;-2) <p(t)=1 -t.1 fx+m15.44.1) u(x,t)=-1 ( q;i(x-at)+q;i(x+at)+. lf/(z)dz ) ;22a ·'-"'2) u(x, t) = -1it2a odrfx+a(t-r)x-a(t-r)F(l], r) c/17, r.n;eF(~,t) = ~ f~ e-i{'f(x,t)dx;-v2n3) 11(.x,t) =4) u(x,t) =Ic-: f~2a-vmIc2a-vn/-~-~q::i(17)exp(~i°drL/(17, r)(x-77)2, d17;4a-texp(x-77)'),( 4a-(t- r)JNt-rd17.= z(t) =300OTBETbl K 3A,UA qAM15.45.ex p ( 4a2 x21) u(x t) = _x_ f' µ(r),2a./n Jo (t - T) 3122) u(x,t) =-~v1ir0v(T).Jt-T)(t - T) dr;exp()drX24a 2 (t-T)'3) u(x, t) =2 a~ f~ _!~:):[exp(=(x-,;)24a2(t-r))-exp(x+,;)2 )]4a2(t-T) f(,;,r)df(15.46.1) u(x, y, t) ==J- f-- exp ( (x-,;)24a2t+(y-17)2)rp(,;, 1J) di; d17;1(2aJ!ii )2 -2) u(x, y, t) =( x-~")'-+(y-q)-,X= (2a./n)2r°C-lt-T)2 s--s-_e ,dT4f(;,17,r)d;d17.'r<r-rJ15.47.1) u(x, y, t) =X_f(2aJ!ii)2 -2) (x, y, t )=-[d,;r e(.r-~)'+~_,.-iJ)'4a·rJof- eYCJ,f' -dr-,(2avn) 2 0 (t-T)" --(x-c)'+e(x-~)'+(y+IJ)'}4a'r;:c~,1J)d1J;r'.~4a·(l-t) f(;: T)d;:.~'~'3) u(x, y, t) ==(aj,;,)' CdqJ;[/'-"'.;~;-"'' + e'•-<>;;~;' '"'' ]N .n) dn215.48.

1) u(x, y)= __!_Ji;' f(x-,;) sina,;d,;;a?)(. ) = A e -Jr· cos2x--xsmx.B.- 11.x,y215.49.301OTBEThl K 3All.AlIAM(2nl+x) 2Jnl)u(x,t)= 2a 1l t312X2)u(x,t)=2a/;i t312L~-,, __ (2nl+x)e-~e-x'2 /4n'2t;x-'3) u(x, t) = _x_fµ(r) e- 4a1(r-r) dr.2a/;i 0 (t - r) 312I'naaa 1616.3. HeT.1 ~ (-l)kk!16.31. F(t)- - Lk=D~' ecmr a< l;ttF(t) = crJt,r,[(eCo=r~ ~dy' eCJIH a= 1, I1Jo 1+ yF(t)- (a-l)t-a logt, ecna a> 1.16.40.

.)21l IA cos(A-1l I 4) + 0(}.- 1).16.49.fat~ 3 .(t) -a16.50.f(t)7'~sin(wt + n-/ 4).(I)116.54.f(x) - - .2x16.55.f(x) - .J1l Ix· (l + 2e-'h'), x~ +oo.I'naaa 1717.2. JII06oeOTKpbITOe MHOJKeCTBOFB HOpMHpOBaHHOM rrpocTpaHCTBeHe .SIBJU!eTCll MHOJKeCTBOM cymecTBOBaHHll.17.5.1) x= (x" ... , x,,)E R"11n11 C" ,11 x 11~ =maxI xk I -He cTporo BhmyKJiall Hop Ma,Ji pII x II"= ( "L~J xk I" ) , p > 1, cTporocTporo BhmyKJiall;BbmyKJiall, a rrp11p=lHe302OTBEThl K 3A~A 4AM2) f(x)E LP[O,l], p2'.l, llfny10iall, a npH p3) f(x)=lllP=(f~lf(x)JP dxf P, p>l,crporoBbl-He crporo BbmyKnall;= C[O,l]II f II= sup I f(x) I 11n11ttnH L~[O,l],OS.r'SlII f II= essup I f(x) I He crporo BbmyKnb1e HOpMbI.OS:.rS:l17.9.

TiycTb X= R 1, F = (-1, l], x1 = 1/ 2, x2 = 1,£(1,(-1, l])=0,l]) =0,E(l +Yi, (-1, l]) = Y2, £(2·1, (-1,1]) = 1.217.10. IIycTb X = R , x =(xi.Fror.aa £(1/2, (-1,X2)ER2,II.xii= Clxd 2 + lxi) v..= {x =(x,, X2) ER, X2 =0}, x, =(0, 1), X2 =(0, -1), E(xi. F) =E(x2, F) = 1, HOE(x 1 + x 2, F) = 0.17.23. HattnyqumM rp11roH0MeTp111-recKHM noJIHHOMOM .11Bn.11erc.11nonHHOM a cos nx + b sin nx. Tip11 mo6oM p ~ l 3TOT Hattnyqurnif noJIHHOMe,[(HHCTBeHeH.17.24. Toqtta.11 HH)f(Hllll rpattb paBHaq+~ + rr:=llq:~~:J1(a-../a 2 -l)"17.45. HaHn)"fwee np116nmKeH11e paBHO -'----,----'a- -117.46. TiycTb 0 < k < 1, y = sn(x, k) - cPYHKUHll .5IK0611, KOTOpa.11 onpe.z:1eJU!eTC.ll H3 COOTHOWeHH.llf'dt0~(l-t2)(1-k2t2)x- J,-Tor.z:1a HaHJIY'flllee np116n11)!(ett11e paBHO ( 1 -r,[(e1/L= k2u+I IT"'1-1 sn 4 (-2/ -1 K , k) , K2n+l= J,fl0J1 - \2)/ (1+~l+\ 2 ),df.~(l-t 1 )(1-k2r 2 )17.63.

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