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www.vnmath.com NGUYEN vAN TRAO PHAM ｎ ｇ ｕ ｙ ｾ ｎ THU TRANG HAM "' "'" lEN r _ t-ofUC www.vnmath.com ｎ ｇ ｕ ｙ ｾ ｎ VAN TRAO - PHAM ｎ ｇ ｕ ｙ ｾ ｎ THU TRA NG e BAI TAP:> HAM BIEti PH(JC NHA XUAT BAN DAI HQC S U _ www.vnmath.com MA06, 01.01.1 2/18 _ DH 2009 Lei n6i dAu Mil d5u ｶ ｾ ham ｢ ｾ ｮ phuc I-Hun ch in h hlnh va li ｴ ｨ Chu6i Lauren t, Ii thuy€t ｴ ｹ ｴ Cau chy ｾ ｮ ｧ du' va 43 ap d\mg d7 H \t(Jng dan g ia i va d a p 56 123 Ta i ｉ ｩ ｾ ｵ tham khao 136 www.vnmath.com Loi n6i dliu ｾ ｉ ｯ ｮ hoc II H am ｢ ｩ ｾ ｮ phuc" dlfl:,lC giang ､ ｾ ｜ ｹ lJ hqc kt !lam tlll'{ hai khon Toim - Tin, trttong ｄ ｾ ｩ hQc Stf I'hl,\lTI 1Ii\ NQ! Trollg dnttlng trlllh dElO ｴ ｾ ｯ theo lin chi , thiJi \Hong hoc -1 :::1':, - -2 -:: , nen 'lInn J \YT = e-'- 1t+21.:lf R=e-'- (k = 0, 1, 2) hay (k=O, I ,2,3) hay = {-.12 +;-.12 .12 +i-.l2 .12 _i-.12 -.12 22'22'22'2 c) -2 + 2i , hay O tOn ｴ ｾ ｩ 6(£") > 0, thoa mall vdi mOl z'.;" nu\ 1-_'1 < l' 1-"1 < " 71 1:'-: Ｑ ｾ ｬ ｮ II Nhuug: 1-' - "1 < {; , ta c61-'- _1 -::." '_I < e: 1-::.' \'8 1+) ｎ ｬ ｉ < lal < thl vdi tnQi ;: lB 00 f ez) ｾ t(l.i O feu :) ｾ Do lim a n % = nell f(;;) n-CJ;J f(aoz ) ｾ = n_oo lim J(a - f(·"')· n ;;) = /(0) V(\y +) Ni'u I Cz) ｾ www.vnmath.com I Ｚ ｾ ｦ laJ > l(aD thi trollg IAn c{i n V(\y f ; +:,) - ｾ I e;) Do " lim ｾ = nt ll '>0 u" eua O co Ie,) ｾ Ｎ 1(;;) - ,,lim -.:> 11 - I(-=-) = -i(: - 'Z) o BAi 1.23 11m pMn thvc va ph&n nia cae ham sau ,:2 + ::; + 0) Ｎ Ｚ Ｎ Ｎ Ｎ Ｌ Ｎ Ｚ Ｍ Ｍ ］ Ｍ Ｚ ｾ 0) JCz) -" + 2z', i::; +.::- I'(.1',y) = -2-' ｾ I BAi 1.24, 1'1 ""n ,' lU un 11",,0 ", L vi , glal a kfnh hOi ) Izl < Y t lJ, ) ｾ 1.(\ Ii\ r c6 n Vｉ ｾ i lｉ [ 1f = - T, ' oi;; = ""2 - '12 = : ｾ = tl Hen 2(x_y)\L'.(;Y+Y -.J:2+ y'1 _ T_ 1) o = y - 1: vex, y) = 2xy v'5, a) + cos x + cos2x + + cosnx; 21: L,_ I,}W' z- l - i Btli 1.26, TIm cae tdng: + ｮ ｾ Ｑｉ z -l-i C huoi t ro l hanh L00 1J)", Chu61 21- ｹ ｢ Ie Ｇ ｴ p ｨ an ｾ thue t'a phiJ.n no eua oj u (x , y ) = x + y ; v (x , y ) -X-y _ 6) u(x , y )=:r'1_ y 'J_ 00 ella chu6i Ii\ I 2, _ 1< hay Iz _ i - II < 12i - 11= vg, Iii hlnh t roll rna tAm i + ban kfnh ｺ Ｉ Ｌ ［ ［ ］ ｸ Ｋ 11 n - hlji t\l ella chuoi IA hlnh lron c6 ba n kIn'll hoi tv r = ?vlien hQi tv kl ' b) ｄ ｾ ｴ w (2 , 2(x-y) xY+y+x -y2+X+ l } I( 1- i -,,-z" ; ">0 L a) ICz) - ,Cx - ;y) + 2(x + ;y)' = 'ix + y + 2(.r _ y2) +i4xy = 2(x - y 2) + y + i:r(1 !.I} Ｇｭ ｲ ｴ o Bni 1.25 Tim ban A,;inh hpi t" va mil n h (ii t!J, cua cdc chu61Cf,i.y thita sau ReI(z) _ l -i + -2-'E· V"y 1(') - u(x, y) + w(x, y ) - (1 + ')0, b) Ttfdng llJ ta c6 f (z) = :;2 + 2i.::- - I U'i gi.ii a) TI\ c6 = :r:(1 + l +i u(x,y) = -2-::'+ - 2-=; l +i COT1Sf iy: z = x - iy m' n r Tu: I -i fI" f(-"-) - ICO) all V(iy Kef( z ) _ 2(.' ,.) - 11 + y ; h nf(z) b) Gia i tlMng tI,l' t6 c6 + Ll1i gia L a) Ta ('6 :; = x til b) sin x + sin 2x + + sin nx + Lui giiii Til ('() ('; ｾ ('oS.1" (.lnr + (,\,r + f"'lr + www.vnmath.com l ｾ ｮ ＮｲＮ r(m:." _ VI E lR 00 d6 (1 + eo.'1 C + ('os 2:r+··· + LCli giai i S1ll2xy)1 ｃ ｏ ｓ Ｑ ｾ VI! tnli c-iia (Uing t lni(' t rell lil tbllg ci'm n + ] sO hang clia lito citp so nhull co ('onp; b(,1i c,.z ·il s6 ｨ ｾ ｬ ｬ ｧ dUll ｴ ｩ ｾ ｮ lil I N('11 c'Or + (iI.r + ('11 : + _ + eon.r = _ (I - cos(/1 + 1).c) + i:sin(n (I cosx)+isin.r = to - cos(n 1- (.1" elr Ttl c6 W gia ｴ ｨ ｾ ｌ = hay t- 0) y2 - le'''\ = = O suy rA ｜ ｦ Ｇ Ｇ Ｇ Ｇ ｾ 1I'{c-o - Q) (ei'l'a- l)(e-;'Po I) = 1, tt"tc 18 XCi = eifJ • V;:ty nuh xa phAi tim co ､ ｾ ｮ ｧ Ｚ z-a o OZ -1 Bai 1.32 Tim dnh X(l phiin ｴ ｵ ｹ ｾ ｮ tink bt€n nt{a mCit pilling lreR 1m:: > kinh tron ddn Vl \wl < tla dle'm Q htln thanA tam III = cua hinh tnJn LCii giaL Ta tim anI! Blli 1.31 Tim anh X(l phd t " ' thi1 h hf I ' • n uyen tmh bien hinh trim n c n t n cltu dtem ::: = a bie th' h _ , = 00 n • n =1 = Xii_ -, w=e,1J ') T l1c ' W e' a \ ' = 1'\1>1,·e''PO' +i + = + z -2 + i + k2rr ［ V3) + k27r ｾ = I'a ( ｉ lem Ncu z =- e 'lP ta co \1\ t:ua cos.;: ､ ｾ ｵ a) cos.;: phall ｴ ｵ ｹ ｾ ｮ tinh duol (tEmp;: XI;\ X{I- phan ｴ ｵ ｹ ｾ ｵ tinh d udi dl;,.ng: lV=).Z-::o z - ::\ Do (t ｮ ｾ ｮ Ｎ ｴ www.vnmath.com a ＬＮＮ trollg d o 11\ sO tlnJ'c va lmo M$t khflc, vlo, vA d6i XUllg que tn,.Ic tllIle n E!1l w(o) = vA 1('(0) doi xltng qua dU'bng tron Iwl = 1, nghia Iii weal = 00 Do = = - -" = > z-a Do (·ftc dil!m ｴ ｲ ｾ ｮ tr\lC thvc bien vao dttbng tron dOn VI ｮ ｾ Ｑ ｬ j ｾｴＮ = 1).1 x-al, I -= Iwl = Do ｾ ｉ Ｇ 1,\1 = hay ,\ I W 02:-0' = 0; dn ］ ｸ ｾ argw'(i) Ｇ ｾ = O I Tu La co () o Vij.y allh XI,\ phiii tim la w = - - E III +l Z cAn tim o =｟ ｾ ;;-0" lhanh hznh tnln dOn Ltfi gjai Anh XI;l tIllwl < clw w(4)=O; phai tim c6 ､ ｾ ｮ ｧ Ｚ ｷ ｾ Ｎ XIiL bie • • I nUa m6.t phAng treu phAu tmng VI, du«;Jc X8c dinh theo cong thuc: ｷ ir._ Bai 1.34 Tim ]J hep bien d6i phii,n 11Lyen linh biEn hinh trim!::I O ｖｩｴｵＨｾＩ ］ ｯ ｾ ｡ ］ ｾ Ｎ w=e ］ ･ ﾷ ｾ Ｘ Ｍ ｾ ｡ - az Do d6 ;0::-1./12.-::-1 Ｍ Ｍ Ｍ ｾ ･ -ｬ Ｍ ＺＺＺ 2-.: ｡ｲｧｷＧＨｾＩ］ｏＮ www.vnmath.com eho nen z - 4i hoy Bai• 1.35 Tim ham J an.h ｸ ｾ hinh tron Iz - 4il pIllIng v > 11 cho f(4l) = -4 va f(2i) = O < len nUa m",t A z - 4i -2- Dod6 (z - 'i)(w Lai gim Ta thAy anh Xl,\ Z\ = :: ｾ 4i bi€n hlnh lenhlnhtrollddnviJ=II It thAnh Olin ｭ ｾ ｴ phing tren Ta c6 111 1U Bai 1.38_ Tim anh cua cae Ｍ ｮ ｮ ｾ ｮ Ｚ a) ffinh trim oJ' + iV X(l thl (1) tro { x> o·, oS + !J ' > X; 11 I +-y2> 4;Ytrabo h( ZJ, t2,':3 E cho J (z,.) 1; J( z:;1 = va f( Z4) = 00 Ch Ung thoo \a +' dl > < '2, h ay = w ong tOmg Chung nUllh ｲ ｡ ｬ ｬ ｾ mil1h rang J (z) = (z, Z2, Z3, z,d· nell a + d kh6ng IA s6 t hvc thl f IS loxodromic t = B lti 60 C hang minh rAng vdi bAt ky phep bi€1l d 5i tuycn tillh \0-8 b6n ､ ｾ Ｑ ｕ =t , '::4 eua t: thl I (TiLl h bAt ｢ ｾ ｮ ella ti s6 cross) " 6I ChoI () az + b • , B tlI1 Z = d khongdong llh»t vOi z,ad-bc= cz+ ' In mot phep bien d6i ｴ ｕ ｙ Ｑ ｬ Hnh Chlmg minh rAng, fU? U a+d = ｨ ｯ ｣ - thl I co mot di€m co dinh =(diem th6a mall J(z) = z ); {'on cac truCrng hQp khiic t hi f cO hai dii!m e6 dtoh B Ai 1.62 Gia Sl't 0: vA j3 la nh COIg d iem c6 djnh n o i I rong bAi Ｑ Ｎ ｾ ｬ cUs I · Chung minh rAng, l1(:u (J =I (3 thl tIl = J (z) d UQC cho ｢ 0' ｾ w-(3 = K e,9 =-:: !! :-(3 ' l( > 0, B E JR ｐ ｉ ･ ｰ Ｎ bien dOi ｴ ｵ ｾ ｮ t fnlt I d UQC gQi lit hyperbolic ｮ ｾ ｵ e ,9 = elliptiC ｮ ｾ ｵ K = 1, va loxodromic !'rong nhung truung hQp khAc: B ili 63 Gia su va n nhu 70 t , e" , C I f' v u:tng minh rAng, neu () = /3ur = f( z) dl1Qc cho bOi , 1 :w=-o = ;=a +, (0 = (3 =I 00), w = z+, (o ={3=oo) Trong trubng hqp n ay , f dltGC gQi In parabolic 40 B ni 1.65 'TIm ､ ｾ ｬ ｬ ｧ tGng q ua.t ella phep bil-n d bi plum ｴ ｵ ｹ ｾ ｮ tinh biw tan A(R) , < R < 00 Bili 1.66 a.) Tim phe p ｢ ｩ ｾ ｮ dbi 11hl\.l\ ｴ ｵ ｹ ｮ Hnh hit n O 1, 00 thu.nh i, + i, + i l U'Ol\g {m g b ) 'Tim phep ｢ ｩ Ｑ Ｑ di\i phiin ｴ ｵ ｹ ｮ t{nh bien - 1, i t hanh -2, i, t uong I1ng www.vnmath.com HUONG DAN GIAI vA DAp s6 Chttdng 1.39 ｖ ｩ ｾ ｴ Ｌ t.:hAng ｨ ｾ ｮ Ｌ J7 = r + iy Khi r'l - y'l = va 2xy = Tfr dang thuc thil nhfit suy co x = ±y, ttt daug thuc thu hl:li suy x = y Vi;ty x = y = ±1/v'2 1.40 u = ± JVx'/.V2+ y2 +.r ; 1.41 Si't dN! lP ､ ｾ ｮ ｧ (= 1) II (z - v= ± V2y J";x'.l + y'l +£ nhat thuc p - j) = z· -1 ｾ ( - 1)(.'-' + + 1) j=l 42 Sli d\,mg Hnh hQi h,l ｴ ｵ ｹ ｾ ｴ d3i 1.43 11m z" = aZo j-a %1 - 1.44 Hay danh gia ｨ ｩ ｾ ｵ n 125 www.vnmath.com n EJe> = '" • {IoO E c ,.""r U (C \ E.) thl 1.45 LAy tilY Y ao E I N'eu ncr • I ''''' n E < i < I ('ho E C uj=I(C\ Eo,)· Do d6 tOil tI}l H 1(\ ,' no , n, _OEo • :: 51 Gia 511 ｴ ｾ ｮ to-i day {f ｽ ｾ ｟ ｬ hQi tl,l t!;Li UlQi ､ ｩ ｾ ｭ thuQC ｛ ｾ ［ Ｒ ｾ Ｑ Ｎ ｄ ｾ ｴ fez) lim /",,(x) Tllf >Q djnh If ｌ ･ ｢ ･ ｳ ｾ ･ hQi tu bi = j I eMn t8 suy nt f(t)dt = lim] ! ,,(t)rU, Tfl co J ve fn,,(t)dt - O , 0 Do d6 ta c6 f(t)dt t- , O Suy f(x) = hau khap ndi va VI , n,::l: (n d) "r3" t l)(log(n + 1) - t Gv 0'_0 va S = ｾ Ｎ Ａ Ｎ Ｎ ｾ 'In f == hOi t.v bi ｣ ｨ ｾ ｮ ta co :1" =j " Oon· Vtli Z E 6.(1), ta (;6 / (:;) == 1- z s = (1- ,) *J 2m IIta-1 khac J Isinntldt= o Mau thuan xiy "p = O lf(t)ldt + I)p ｾ n == zn n""O f:(," _,)," ｾ f ｏ V8i : E D{l, COOT), tOn tl;li hAng st, dVClllg K (= _2_) "'" £, n ｾ 51·1'1" + L: Izln) n '''0+ I ｬ｡ｊｾＬＮｲｬｬｨｩ･ｮ Khi d6If( z)-.'l1 ::511-zICE 18n - "I.Q ｉ ｾ < 110 00 01=0 11 - zl (L 18n - 'I + L 1'1") n=O n ",O = lI - 11 _ "I no 'I n::::O L ISn - 81 +< -1 < 11 -.1 L I'" - sl +.K - Iz =0 nIl 2,,- n , I 1'( I I ) - B{m kfnh hei tu 11\ I n=G f: s",", va kh; I(z) - 00 I;m jlf".(t)ldt j 1og "+1 1.53 Do phtp dOi bi@;n z (z, tit c6 tha gia su (== D&t cho ISn -81 < O 2" 126 = nｾ j=1 L: logj < 'l2 cho 11- :1 < f{(1-lzl) Voi £:> tily y, tOn tl;li Lli'oi theo dinh Ii Lebesgue ｶ ｾ 1.52 a) ( lIn 11 = 00 n_O o " v;.y j lf(t)ldt = Iqln - I va mf Ifl so: O fl, = c) log(n')fi'7 = 2" 1.49 5\'1 dVIIS khai trliin Taylor t(l i :: = O 1.50 sup 1/1= ;;/Iql"' 0) L ISin tldt= JI J If sin tldt =2 sin tdt =4 LAy z cho (1( +1) ｾ 11 - L: ISn - sl < " •- a, 1.54, DM fez) = e " -z+ l 1ft ,) - f(z') 11(,) - 'I < < I,.a 1'1 , $ r ta c6 lal < I, 1.1 :5 :5 (1 -Ial') :: =- ｾ Ａ :51: =- ;;! T www.vnmath.com 1.55 00 sm(, Ｋ ｾ Ｉ - sill; V8 cos( z Ｋ ｾ Ｉ ｾ - cos z ｮ ｾ ｮ ｴ ｾ ｮ Ｎ + ｾ Ｉ _ tal' z Gi!.su tim ｴ ｾ ｩ < ｾ Ｇ < cho tan(; ｾ "') = ta" z KIll d6 t c6 Ｘ ｭ ｾ Ｇ =O Til d6 ta suy sin(z +h ') = sin z MAu thuAn xAy ,a vi sin; LIIAn hoAn ehu ky Ｒ Ｎ n 1.58 Bang quy ｮ ｾ ｰ Ｌ ta c6 2n+l_ l Il (1 +,'") = L v=o z" Cho n 00 ｖ ｾ ｏ i' - ｏ ｾ ｴ II,) ｾ Z-Z4Ｇ Ｓ ｾ ｩ Z2-Za Ｂ - ［ Ｇ Ｎ Day la mi) l phep ｢ ｾ ｮ tuyen tinlI va liz,) =1; /(Z3) '" 0; J( z.) =00 1.59 dili Ap ､ ｾ ｮ ｧ djnh If sau, "ClIo b(i ba ､ ｾ ｭ ('" '" oJ) va (w"w"w,) I hai bi) ba diem phiin bi¢t t Khi d6 ｉ ｾ ｮ lai nh!t phep bi!n dOi tuyln Huh / cho / (z,) = Wj, i ｾ 1,2, 3.", tac6 J bday I.duy nhat vA do J(z) =(t,z""" ) 1.60 Dat g(,) = (J(z),j(z,),j(',) ,J(")) ' Khi g(z) fA phep ｢ ｩ ｾ ｮ dili tuyen tfnh va 9(',) = 1; g(,,) = 0; g(t,} ｾ 00 Til d6 ta c6 g(z,) = (z" z" '" ,.) 1.65 Oat g(z) = R'e,e 2-a -az+R2' a E ll.(R) - BE IR I 1.66 a) J(z ) = (2 + i)z +i ,+1 b) J(z) = 6izt z +3, ... ｾ ｎ VAN TRAO - PHAM ｎ ｇ ｕ ｙ ｾ ｎ THU TRA NG e BAI TAP: > HAM BIEti PH(JC NHA XUAT BAN DAI HQC S U _ www.vnmath.com MA06, 01.01.1 2/18 _ DH 2009 Lei n6i dAu Mil d5u ｶ ｾ ham ｢ ｾ ｮ phuc I-Hun ch in... viall de dang hon vii;