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DI,CIi vito Huh chit hlnh binh ha nh , tfnh chat dttong thAng ~~g s:'ng " ta de th~y dttong thAng nay song song voi dttbng I}hiin glaC e uo.. man yeu bill toon.[r]

www.vnmath.com NGUYEN vAN TRAO PHAM NGUY~N THU TRANG HAM "' "'" lEN r _ t-ofUC www.vnmath.com NGUY~N VAN TRAO - PHAM NGUY~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 v~ ham bi~ n phuc I-Hun ch in h hlnh va li thuy~t Cau chy Chu6i Lauren t, Ii thuy€t th~ng du' va 43 ap d\mg d7 H \t(Jng dan g ia i va d a p 56 123 Ta i Ii~u tham khao 136 www.vnmath.com Loi n6i dliu ~Ion hoc II H am bi~n phuc" dlfl:,lC giang d ~\y lJ hqc kt !lam tlll'{ hai khon Toim - Tin, trttong D~ i hQc Stf I'hl,\lTI 1Ii\ NQ! Trollg dnttlng trlllh dElO t~o theo lin chi , thiJi \Hong hoc <:il8 mOn h(){ IIAY hicn Imy In tin chi , voi s5 gib bal t~.p chi COli 11,\1 ti(·t (50 phtlt) rho mQt luAn ~I~t klu\c, VI day la mi)t mon hQc tltong doi kh6 doi vCli sinh vien, lien d~ cho sinh ViCIl nA.1l1 bdt ChfQC cBe IlQi dung dH1 y~u ella mOn hoc thl vi¢(; cau trllC l~i phAn bit! ti.\.p vii Illfong diin sinh vien lam bai tij.p In rlit efin thi0t Do v~y, thlmg toi bien SOI,lIl clIon "Dal t~p haJll bif!n phuc" val Illl,)(' dlCh giup eho sinh viall de dang hon vii;<: tiep lhu man hQlo: N¢i dung ella cuon sach gom ba chucfllg : Chu(mg 1: l\11I etAu ve hAm bien phuc Cilltdng 2: Ham chinh hll1h va H thuyet Caud1\' Chucmg 3: Chuoi Laurent, Ii thuy8t tM.l1s dtt \., ap d ug NQi dung cac chuong tlfdng lIng: vdj giao trinh Ii tbuyfl"Ham hien ph(fC" hai Uie gia Nguyen Vi\n I,hui', Le Mf.u biID soon vo xultt ban tai Nhs xufit bell Dl,\i hlJC QuOc gill H1 HOi "Ai n~m 2006 Trong ph51l dAu moi chltdng cua culm Sticb, cbUD& www.vnmath.com mQt sO hili t(tp mAu nli Wi giiil chi tiet giup cho sinh Vil'il l~ull quell n1i \"il,'<; gifi i cAe btl! tmin t11110(' ph~Il Philll (,lIoi moi dlltdllg: Ib IIlQt sO bAI qlp 11./ giii.i (vOi gOi y 1I0(\.<, dap s6 cJ ('IIi)) S/;~ch) nhAIll pllli.t huy nllllg h.rc t~t hQc va khii nii.ng tlt duj' d<)c /{l,p eua sinh "iell Chung tai khOng dlfa WIO CllOn sneh lIay mOt ~ hlQng qua Idn bai lap illS c1ni )' nhi(1u han d{in f inh chuJ.n ml,t c vii s\r d<l d~llg ('us cae bAi t~p Den qUlh ('uOn "Bid t~p ham bi~ll phlie" b(ln dQc co th~ tham khilO bai tli\p ciie stich da dllOt: xuM ban trlIClc nInt CI1011 ItBoi t.\ip hilill biE-n pink" ella cae tile gi~1 Le M{iu Hili , Sui Dde Tile; ('liOn IIlntrodllctioll to complex AnalYsis" cus tAc gia J Noguchi - Chung tai xi'1 chan cam all GS TSK H Nguyen VAn K~u~; GS TSKH Le r>.liiu Hai ; COS TSKH Do Dtic Thai eta climh Il~H1U c6ng suc eM doc ki ban t hao va citra nhi~u y kien e uybau I a I Ian dau " tien duoc xuat ban nell khong trunh • C'" uon SCI khOl nhl1ngll·~ - Ct-, llIong nil/Till dUric St( gop', lleu sot IUllg to! b1,\n dQc v ) ., CliA Cae tac gia Chlidng Md d§.u v~ ham bi~n phuc Bai 1.1 Vih cac so phl1c sou dudi d(lny l1t\1ng gu1c d(mg mti: bJ l+iJ3,' aJ -l-i; cJ -1 ~ 't Loi giai fi) Tn c6 I- 1- il - ,f2, dJ (J3+i)'(I-i), 3rr arg( - I - ', ) - M( -4' Do 3rr 3rr)) "'2 -,'!!! - i = v2 005(-4) +tsm(-4 = VL.e ~ b) Tn co II + iJ31 - 2, arg(1 11" +"iJ3 = 2(cOS"3 (I + iJ3)J; '" - 3' + iv3) + I Sill 3) = Do d6 ,~ • ;r 2.e J www.vnmath.com Xet On (~ + ,V3)" 7IT hl2(W:-i7 +lsiIl I -i 7n T )= In ,Ll!" Iv2.(' _ (L~: 1+ I~:I \ (1·.1 + 1"1l)' A C~:I + 1::1) C~:I + I~:I) (I,d + 1·,1)' (1 + 1+ I":I~;'I + 1:'1~;d) (1.,1 + 1·,Il' c61(V3 + i)'(1 - "1- 2')2 = 16 12 va 1i rr 511"" Atg((V3+0'(I -i) ) = •• ,g(V3+<)+a,g(l-i) = 46"-:1 = 12' d) T V(iy o 5IT 511" ~" (V3+,)'{I-i) = 16)2,(005 -12 +isin -12) = 16)2,.'" B a; 1.2 Rut gon,' a) z •JAJI "' gl a L ) 1- i = ; J+l b) = (! + i/3)3, 1- i = (l - i)' = _-2 = 2 Z= - - +-i M IT 2(\,.1' + 1,,1' +2,zdlz,1 + Re (~~:II"I) + Re ( ~~~i IZ21) , -1 + 2Re(z\zz)) 11" b) Ta co + 1v3 = 2(cos"3 - isin"3) :::::> ;;; z'" )(Izd' + Iz"I' + 2I"llz,1l "'lIzd 2(1 + Rc "~I"~ ,Hlzd' + ",I' + 2Izdl.,1) !;;;2 IZl (2 + 2Re = (1 + i/3)3 = 8(cos1I" + isin1l") B = (21', + "I)' = -8 o = 4(z, + ,,)(z, +"') = 4(lzd' + Iz,I' +2Rc(" ,,)) Ta 5C cill:Ing minh A S B Thij.t vij.y: A B i 1.3 Chflr!9 minh rli.ng: Vl _ 2".11',1 + Re (~~>d) + Re (~~~IIZ21) :S 1'''' + Iz,I'+2Rc(z",), (I) Re(z,,,,):s Iz,=,1 = Iz,lIz,1 n.n Re(",,):S l,dlz,l Dodo :S " (lz,I-lz,I)'Re("z,) :S (Izd -1·,I)'lz"I',I· UJi '" CUll Tit chU'ng lUinh bat dAng tht1'c tUdng dUdng • I1',1" + 1.,1" I(lzd + I',I):S 2(1" + ',Il (.) Suy www.vnmath.com + :::2)(:::1 + :::2) + (.0:\ - Z2)(!';"""" =;) (::\ + z-.d(z\ + ::2) + (:::\ - ::2)(::1 Z2) = (:::, _ ;,:, + "::1=2 + '::2:\ + Z2~2+ :::\=, - :::1=2 :21'\ +- l o 2{/=, /' + /=,1')· hoy (1) dU(jc ch(rng minh Va" A :$ B hay (*) dlr(jc dnrng minh Do Bai 1.6 nm c(in cac 56 phuc sau: ,,) -YI, o ",.=,/' - /=, - "~I' ~ (1 - I', /')(1 - n) b) (I - '5\.=,)(1 z"") - (=, - ,,)(z, _ ") (I - ,.",,)(1 - z,.',) - (" _ ,,)(z, _ z,) - ::] Z2 - Z I Z2 + -1~1-2-2 ~ -:; ~ -:; /'01 /"/' -/=01' -/=,/' ( I -/=01')(1 - /",,/') - R - + ',', + ~ :: - -'>-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 <I B~i _,v'22} = -/8e'4" Do .:; + 2i = o Bili 1.5 ChUng mink 11'01 /"/') LCli giai Ta co )I - ::].z21 - IZI - <:21 1+ c) :;r'2·+~2' Loi giai Bili 1.4 ChUng mtuh /1 - -YI; b) + 2i = L1 Chting minh fLghia hill/! h(>c 3"+111.;,, V2e;-'-'- (k = 0, 1, 2) " he'lT "" } { J2e'"'"iT; /2e;'J; II~I - 1\ ~ [arg zl.""" z I- 0", o UYi giai \'id :: r( cos\>, www.vnmath.com + I sin ",), d6 l' = 1::1 va ~ = Arg.::: Khi '!i'::3 1-=-11 1=1 ICOSIP- l + i sin <r'l= )2(1 - cos 1') 211'/21= 1<;>1 Y nghia hinh !(cos "-1)2+::.in V ,,., ~ = ) 4sin' ",/2 = 21 sin ,/21 h()C: N~ u ve duang t rim d On vi a gOi A la dit.m I ~ - 11 1"8 d' , d"61 eua • d B lei I-I &y A Co n I a rg =[ Iii d o dai cua cung All If; thi1e ch(tng minh n6 i len di) diii ella d Ay AB phai nho hay bang dq dai cung AB ~lr6 hOl~ Ba i 1.8 ChUng minh riI.ng n ~u ': + :2 + z·~ = va )- ) ~ )- ) , -1 -,, ) _ thO h ' )~3 t n ttng d t PrTt Z j , ::'1, Z3 iiI ba dinh cua m,'t t I' ' _~ v amgl(J,r(t'lL not ttep htnh iron don ut "6 Li1i ghii Vi , Xet hi¢u: ), )-1-) • 'h2~ - "3 = nen can e Inlg mmh I" - I - IZ2 _ 21.1:'2, Z3 Z2 )=) Z2 qj2 - IZI - Z31 -1::2 - 2312 -;:;'~:::l - ~ Z2( -=, ¢ ':;2::2 - !\:::t - '::1-:\ = '::-323 + zJ,'::3(d o ;::1 +'::2 + Za = 0) = )=,)'(d(mg )=,) = \:,1 = I:,) = I) V~\y IZL - :::2!2 - \=2 - =31 = \.::, - <:3\2 -1'::2- Z3\2 Stl y \'<:1- "';11 \=1 =:.1, Tlr(Jng l1.,1 ta co 1.::\ - Z'l! = \":::2 - Z3\ Do d6 ta.m gial Z\:::2':::.1 lit lạm giắ d~u va llQi t i@p t rong hinh trim deln vi · Bhi 1.9 Tim (lieu kicn cdn va d"tl de" ba diem Z I ,Z2 >Z ;J Wng dOl mot khac nhnu eiLng ndm Lren m(j ' duCJng lhiing Lei giai Dit-u kicn cdn: Do =1, :::2 =3 cling na m lrcu mQt duang thling nell ;:;1 - ~:I = k( =, - '::2) v{Ji k la rnQI, 56 tll\fe V~y ~ = ::- , =2 :1.1 - ZJ XI - Z2 Im- - - = O fa thA)' ro rung d i~u ki~n tren di ng 1ft (Heu kic:n Iltt, Ta cbl!ng minh • h" • t = '!i:-.1 + '::2'::3 - + ZiZ2 oiAt hai dinh lien ll ep ':: Tim din h :;::3 ke viti Z2(Z : of ;:; d· Bili 1.10 ('It o , ::\::2 Z\Z2 - ::2z3 - ZzZ3 = -z'::3 - z,ZJ - 3i':1 :"1.1 ':::3) + =i( -Zl - '::-3) = Z"3(-::j" - '::1) + ~(-.1 - ;1) t huQC dUdng trim - z31 = IZ3 - zd, z:d2 121 - ¢:} zzZ1 - z,-Za = V-lilY d leu " k·· ' d ~' b a d"lem '::-1 , L""l , '::-3 ", II t ' l 111 !" D (0 z, - Z2 I ffng doi mot khac va c ling llA.m f ren mQt chtang thA.ng 1& ),.1 = ch" + ::;2':::' - <> I:,)' = I argzl bi~u dien eho s6 va B III u i~m bieu d ien cho s6 ~ thl don ",' v{l.y: vc P {U eua hai dAng thuc tren bAng Th{i.t c(mlt t,d '::-2 clia gu.ic diu n Uti gi:li V it,! Vi j'., -:II Z3 - (='1 ;:.1 -':11 www.vnmath.com + {;3 - z;,} 11! ::3 - ':2' (':2 .::,)c n Do d6 "::3 = ::2 + (Z2 - nil" 11 + 56 pln1c ,-Jt1i lm (z.zk) ;:> 0, :S k Bhl 1.12 Gho I' 211'" va gee gilrA ZJ -;;2 Vdj':2 -::1 n -;-' nen mnq ::dc n Dai 1.11 Chf"lg rnmh rdng cd hai gin try j;;2 - ndm t:rfn dtt(mg lilting dl qUfl 9& to(1 d9 va song song t/oi duong phiin giae eua gOc etia tam gi6c vOi dinh t(li cdc dilm - 1, 1, ;; va dTlilng phdn giuc dl qua dllm z Lai giai Ciaau :;2 - = r(eosr,o+isi n 'P) vai { ;T(eos v:::.! - Iii T > 0.0 $ t.p 1)(z+ 1), nen a rg Z I = Nhu vi.\y, nell lin:: > + i s in 0) tbi ° ! = ! (cos9 l~in91 1<1 z thl 1m!z < O Do Iin(:: :,.) > O.'Qk = 1",· , n nen :::.'" # O,'tIk - 1,' , n Thea nh(lll xct tren ta co 1m (t ,1,.) < + sin 't") , VQ.y "u (L.~_, Z/.) t- O Suy 1m (E~= I :,.) -!- , va d6 l:"10.: = !c :: '" O thuQc ~ ( arg( z-l )+arg( ::+l)} DI,CIi vito Huh chit hlnh binh nh , tfnh chat dttong thAng ~~g s:'ng " ta de th~y dttong thAng song song voi dttbng I}hiin glaC e uo goe t~i dl'nh ctlH t am giac c6 ba dinh Is 2, - 1, ~ ~~ do, 11IU i)cduullgthangdiqua g6c t OI,\ dO t hoa man yeu bill toon Tu d6 tjJ.p gia tri J 22 - thoa ma n yell c~u hili toall B a i 1 Cht1ng minh rdng, n €u trinh :::3 - = thi =) , 22 , ZJ la n9hi~m cua phlJdng Lo~ giai D€ t hAy ;; = la nghi~m clla plncdng lrinh :3 - = O Nh lf vi,\y ta co th~ gia s t! ':::1 = Khi dAng tlu1c cAn chlkng m in h tU'dn g dU:Clng voi 1+.;::; +Z; = z; + z~ + =;',.:::~ <:::::>1 T hoo dinh If Viet ta co o 14 1';::((;05 L~ ! z, #- o Bling vit;'C w h1l1h va su d \;l1lg If lu~n tren ta auy z] nAm tren du:ong I hl1ng chua phan giac t rong t~ drnh ella tam giae eo ba dinli lit O,Z - l ,z + C;U = < 211" ~ + sin ~); ;r(eos(~ + 11") + Sin(~ + 11"))} z~ = ( z - tid: z" Hay rhttnq minA k= Do d6 , to chi cAn chung lJiinh z, = y'r(cos;e2 • duong thllng Ta eo ~ 11 o Li1i giai Nt3-U z Khi d6 t ~p g-iii lrj ella =,': I Z2.'··, = 21 '::2.2J = hay Z2·ZJ = Do d6 :;·4 www.vnmath.com Bni 1.14 (:111£119 mtnh rdng tJdi 91a 11'1 k tntll! > O k of: l)luUlng , t _ -.0 L en h~l £~ va n., BM 15 ChUng mmh rang ncu cJlUO' z-b ~I~k hi plu/d1l9 lrinh littiJng iron Tim tiim va ban I.;inh duiJng tron (16 Lui giili Xct pho'dng lrlllh I, -al -o-b (I) "'ang duang vd' Do Llji giai DM arge n = ~k (I ) 1c '" Re C n, COSCk ~ VI chui)i (l-k')' lal' - k'lbI' 1- k2 en hQi t1,l nen chuoi L He en hi;li tv, d6 chuoi ,,=1 f: 10,.1 hOi W (theo (1», n=1 (2) Ta <6 la - k'b!' - (I - k')(lal' - k'IW) ~ lal' - 2Re(ak'b) + "'1&1' -Ial' + k'lbI' + k' lal' - k'lbI' = k2jal2 - 2Re(a,l.;:.l1j) + k21a\2 ~ k'ia-OJ' Do (2) suy ( I) ~ E n:=1 aI -k' V~y chuoi o ~ L e" hQi t\,1 tuy¢t dcH n=1 e h(h t\/- Chang Bai 1.16 Cia S"tl cae chll.oi L':=l en va 2:': mi1th ning nitt Re en 2': tJdi mQi n till ehtt6i 2::::'".1 Ic PI di.ng h()i t"\' Ta kla-bl l o-~I_ - k' - II - k'i LCfi gi.ii Dtit Cn=xn+iYn Do chu6i (3) Til (3) suy fa (I) la phUong trinh dltbng troll (k a - k2b tiim t{l,1 ::0 = -1 _ /,-2 va ban k'III I1 R =:-k-!.lu' :b,,1 II - k'l' 16 Izl'-2Re(oa)+lal' ~ k'(lzl'-2Re(ob)+IW) I:- k'bl' ~ In - k'bI' ICn!·e',p" Khi dO en = Ta co Re C n = Icnl-cos\Pn 2-.lc nl-C050 Suy fa :1' _ 2Re(a - k'b)z lal' - k'lbI' _ 1 _ k'l + I _ k2 - nen !.pn f n=1 c.,., hOi tv nen tv· T I) voi o f % hQi Thco giil thi€t Xn > nen v6"i n du ldo thl I < 1" K.bOD& • I ~'" Ova% <Iva_ mlit Hnh tong quat ta c6 t ie gu\ so' In -~ <x ~:c,.bOittJ n >1 _ D d 0' O<x'n · n· suyra chlloi L., , , , www.vnmath.com tI,l nell " , L (.t'~ -!J,~) ,,;-1 = E(ak+up+l) <X> L Ic n j2 = V(iy OQ 00 n_1 """,I E (.c~ + y~) = E f len 1'2 hOi n.' x~ - 00 E (.c~ - =t y~) hi)i ll.1 Xe' o B 1.17 Ch1mg minh: = + u21'l = Vt phai vdi 11 = p 2: 2, tuc lit (lad'l + 2Re8kap+d + p/8p+d'l , = (%Uk) (Eak)-(t,Uk) (ktl n.) ~ (t Uk + a,+I) (t iik +ap+l) -( t Uk).( t = C'~l U.)Up+l = t k,.l k""l +(t::, alc) It_I Up+I + \a)l+11 V~ ,Ai k~1 + lap+'\!!: = L CiA:) p (aka,,+1 +akUp+l) 2Reak6p+1 +la,,+1/ • k=1 , d 101 +) C ia sir d Ang t hu-c dung "'" k"" IEak\2-1 f Uk !'l LC1i g ii i T n chUng minh b.1tllg qui n~p theo n +) Neu n = 2: V~ t rlii + Up+l) ,1;. \ ",1 tl,l (0"" k ' hrii 1t,1 Tit fa co ", , 2:' (a.\: + UpTI)(i't k +<l, ) ~ 00 TA l'O ~ ,.~ -y~+2i.r"YI" Inil ~ c~ hOi ( I' )-V ph" (I) ta co: (I) Ta se cluIng minh dAng t hii'c dUlIg vdi (p - 1) p+l p+t ~'-I k_ 11 E la I' + II:a.I' ~ E = P l:$k<":5:P+1 +1: la, + a.I' (1') = t (\Uk\2 + 2Reakap+d +p\a +d p k_l • 18 = V€ plllli (1')- Ve pha i (1) Do d ( 1') d ttc;lc chlmg minh o Ai 1.18 Clw (lflh ,T(llll = z2 a) An" rua cae dllimg.r b) T(lO W = ;;2 = c'2 - 1;;1 Nl,u c > ta c6 = R; y2 C2 _ + 2i.cg nen u = x - y"l, V v- O tn,le < ta co CllH n Anh cue dUClng 1.; = Ie dubn mnh ci18 duong trbll u2 + v = R.4 g lhAng Y = ±r _y2 1\ = 1; = -; u x2 + y2 .r-ty = ~+ x Y ';I' V{l.y y = , x'l + y2 =C ta xet hsi l.ntbng hOp: +) N~1l c=O: Tn c6 u=O vo v= · Do -00 < y < +00, y y "f: nell -00 < v < +00, V:f: o V~y imh In true Ou \ {o} j u +) N~II C f 0: Ta co U +V2 = -' 1' Nlnrng: -z + = C x +y r y Iw _ '2 ' r - R Day la phuclng nell u + v:.! = Z;' Tu ta suy (1/'-1ZC)2 + v =C vao plntang '2_ Y - Ill plntdng tr'inh cua hai duong anh Ie chtong troll tam A , (2C)2" Vo.y = (2C; 0): ban kinh R = 2\e)' Ta tim aoh ellS Jz - I I = nhu sau: Ttl \z - 11 -= = 1, hay ;r2+y2 = 2x, Thay vitobi.!uthiltc6a*L (x_l)2+y2 c61t = 20 I:: - H(ly hrn: b) TQ.o anll f""1ia duang u = C D~ tim finh dla duong x b) TIm t{lo iUll! ella <luou · _ trinh u = r2 _ y"l tA dl1 C 1~ - C : Thay u Qc =~,:;"# O a) Anh cua (lltiJng x = C; ~I 06 Iii l1Iia tren Cua truc Ov C = thi x2 _ X(lW 11 {::: ~x' N~u :c2 (v'C)' LCti gh~"i fl.) Ta c6 w = - = - - ;;; x + '1/ x dUClng :, = y Ie dltong !-x y'l (v'C)' - - - - - - - = I Do li hypC'rhol vuong Day la 11\1a hen trai tn,lC Ou (We Ie (-00; OJ c JR.) Anh = D6 H\ (htdng hyperlxI\ Dy Dhl 1.19 (,"ho tinh = 2ey {U: -"' P ' - y2 :f: ta co y Nell C = till Nt;\I C = , ,v' = -v \'8 U C - - - j,ay v' '2C 4C' u) Day lit phuong trinh purabo) co tn.lc OIl ; Neu C " vuong co tn,l.c 1ft Ox co = { V - .l· = y = C' ta dltc;1C ll 4C2{C2 = C, atilt eva (/JliJrlg {/ "'" C Lai giai a) TR co 2TY Tho), r www.vnmath.com (Jc (Jc)2 JUiy tim: ~ Vlj,y snh Iii dttCJng thAng Reu, = ~ t Int(" .1' www.vnmath.com =C b) TIm t{'O nuh ('lUI du()ng u ("us U l/l d\/Qc Tha.y 11 = C vile bi~ u - - c .r t- y2 I Hitm 71iiy co hen t-uc MIL tron9 \.::\ Nl>u C = lhi :r = 0, y :f: O T(l.o 8nh cua dUOllg lit tn,lc 80 g6c tQa d~ = trlr di N~u C "# thi x + y2 = ~ 2~)2 + y2 = Tuc Is ta c6 {x - (2C)2' Do d6 tao {mh cua dU"Ctng 11 C Is dUCtng troll t~m = (2~;O),bAnkinh2r~\ ' oj J(z) ~ ~; tro 121 z" o UJi giitL Ta c6 th~ gall duoc neu tOn tl;li lim f(z) a) Ta c6 z" "'" ! n _ kl11 n 00 nell .: : = n- , n _ 00 n(!n I,', Re lim '1-00 1-_'1 < l' 1-"1 < ." / i - O DOd6d~tf(O)~O l , a d <lu 1-' - "1 < {; , ta c61-'- _1 -::." '_I < e: 1-::.' Idn va d(lt ;;' = - -;;;" II , - - - thi n +1 1t 1 n(n+ 1) <0 1 1-(1 ) n _ _'' ,, - \ 1- ( - -,,-+-,) ~ , > e o Bai 1.22 Chl'ing minh ning vb, a :f: 0, luI ~ till phtldng tnnh Mm f(::.) = f(u.:;.) kh6ng c6 nght~1n Iii ham kh6.c hfJng 1'6 lien tuc t(li;;=O 2., z." Tir tao suy khong ton t~l hm Re z z_o z b) Ta c6 < I~I _ zRe z Izl - IRe zi :S 1.::1· V~y z _ o 71 \'8 1:'-: 1~ln+ -nl~ = A Re -n ' -z,,- = 1; n ~o n_gg t1,lC 'I\IY nhll~n hilln khong lien t\,ll"' deu vi nhl no li{'t\ tU( df.u thl vai E > O tOn t~i 6(£") > 0, thoa mall vdi mOl z'.;" nu\ Nhuug: <-0 tmnqll<l tll1fdn~ cila hal hRIIl lien h.IC oj J(z) ~ zRe z z < J.:holl()v • I I < I vi no I" '- vaz lii'n -:; khtic 1::1 < La, giai R6 l"A.ng w = ChQt1 B,Ai l:~O C6 thi 9?n gi6 try t(1! Z = di cae ham sau h.am hen t(lC t(lt diem, dl1C1C kh(Jng? , ltOc m¢t ha.m Ii~n tl,lC thl Lui giai Cia S11 f(:;.) lien l\lC l~i :: = vo f(::.) - /(az) Vl a =/:- 0, lal =/:- nell < lal < ho(l.c lal > 1+) N~ lI < lal < thl vdi tnQi ;: lB 00 f ez) ~ feu :) ~ f(aoz ) ~ - f(·"')· t(l.i O Do lim a n % = nell f(;;) n-CJ;J = n_oo lim J(a n ;;) = /(0) V(\y www.vnmath.com I :~f +) Ni'u laJ > thi trollg IAn c{i n I Cz) ~ l(aD ~ I e;) Do " lim ~ = nt ll '>0 u" V(\y f ; 1(;;) - eua O +:,) - co Ie,) ~ ,,lim -.:> 11 - I(-=-) = -i(: - 'Z) o BAi 1.23 11m pMn thvc va ph&n nia cae ham sau ,:2 + ::; + 0) : ,.: =-:~ 0) JCz) -" + 2z', i::; +.::- I'(.1',y) = -2-' I BAi 1.24, 1'1 ""n ,' lU un 11",,0 ", L vi , glal a kfnh hOi ) Izl < Y t lJ, ) ~ c6 1.(\ Ii\ r n oi;; VI ~i I = ""2 - '12 [ 1f = - T, ' l = : ~ = tl Hen 2(x_y)\L'.(;Y+Y -.J:2+ y'1 _ T_ 1) o = v'5, a) + cos x + cos2x + + cosnx; 21: L,_ I,}W' z- l - i Btli 1.26, TIm cae tdng: + n~1I ella chu6i Ii\ I 2, _ 1< hay Iz _ i - II < 12i - 11= vg, Iii hlnh t roll rna tAm i + ban kfnh y - 1: vex, y) = 2xy 00 z -l-i C huoi t ro l hanh L00 1J)", Chu61 21- z),;;=x+iy b Ie '~t p h an ~ 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_ 11 n - hlji t\l ella chuoi IA hlnh lron c6 ba n kIn'll hoi tv r = ?vlien hQi tv kl ' b) D~t 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} ' mrt ~ 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 ('() ('; ~ ('oS.1" (.lnr + (,\,r + f"'lr + www.vnmath.com l ~j n r VI r(m:." _ E lR 00 d6 (1 + eo.'1 C + ('os 2:r+··· + COS1~ + ,(sinx+sin2r+'·· +!-i1Jl 11.'1:) 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 h~llg dUll ti~n lil I N('11 c'Or + (iI.r + ('11 : + _ + eon.r = _ (I - cos(/1 + 1).c) + i:sin(n (I cosx)+isin.r = - e,( +I),r to - cos(n 1- (.1" elr t- 0) LCli giai i S1ll2xy)1 Ttl c6 W gia thi~L = hay y2 - le'''\ = = O suy \f''''~ 1I'{c-o<l2.rll t" rA Tuc lis ta c6 y b) Ta ('6 Rc - = -, , = a X z; +y N~u a "'" ta co x = 0, y g6c lQI\ dQ x hOM: 11 d\1bng tron tam (2a; np nghi(!m Ia + y'l = (2~P , 0); ban kmh tn,lC : 'IlP (1 Bai 1.28 Tim phan th1jc va phdn 0.0 cua : 21,;05.1.+lsin (n + 1).1; - sin x -siu nx = n) cos( l - 2i); C08.1- '2si n'J.<!+ siu ('l'HI).r £ Sill :2 S IIl Z + sm(1l+ I ).l slJl;r-si u n.c 1-~-'-';i -;;~::::-.:::'::'::= "" 2-2co&_r LCli g iai a) V(Ji :: cosz = b) sil).:r + sin 2:r + + sin flX = sin en + l )x ~nh· 1.27 1!m , m(lt l,itdng pht1c - i sin x) sin x - sin HX oos.1 tdp nghi~m c11a phUd n9 ,I = a , a E JR b) Rf' - + i sin x ) + ell (cos x 2 a) +cosx+cos '2x + _ +oos n x = 2sin ~ + sill ~ sin ~ si n r b) sin i = x + iy t o c6 p-lI(COS X V~y = cos x chy - isinz shy Do d6 cos(l _ 2i) no tnf di nghi~m U\ o 2\a\· - cos.r) - i sin rJ cosI)2+::;in :r = - ('os.£" + cosnx - cos(n + l }x J Il~n : # 0: a =F ta co (x - ; a }'l + l ).z;) + isin(n + 1).1."11(1 .r2 V~y t(l.p nghicm Iii i d\1CJng phAn gi6.c ('iu m l,\t l)h Ang phu{ L N~u + 1).r a) = cos eh( -2) + i Si ll sh e-2)_ b) TUdng tl,1' ta co s inz = siu r ehy +icosx shy !len sini ~i Bbl www.vnmath.com Biti 1.29 Glu/Url mmh ning sin :: till ham hi ch(1n tn~n C Uti giaL Thco bili 1.28 fa I sin.;:1 = j'sin r d/" (:05:: kMng IJht'il iii nhU1l9 Loi giaL Tn tim anh phall tuy~n tinh duol (tEmp;: XI;\ ::: - Zo w=). , Do e6 r oos2 :r sh 2y_ JSin2' x + sh2y~ Ishyl leos.;:1 "'" Jcos2';c + sh 2y 2: [shyl· 0- Ie -1 N' Dodo 0 ncn =0 = Qua dU'Cfng tron don vi, diem «oi xunp; Q • d 01 S.- xtrng Vvl A' dli5m W I W ~ = I'a ( I"lem z-o = ). :;-li W = 00 n • n =1 vdl = ,.;;I Z-Q = Xii_ -, crz-l Do d6 :: n)i xa true thl,.rc, ci:lIlg voi y tang, modun eua si n z \1\ t:ua cos.;: d~u tang vo h{\l1 Ncu z =- e 'lP ta co j Bai 1.30 Giai phudng trinh: a) cos.;: = 2; b) cos 2::: " • '," < L dl gla a)Tuglathict 1a c6 = sin{:: + i) Do e'· - " e1: +e-'~ 2 hay (e U )2-4e'%+1 = Til ell = 2:r v'3 ho~c e'l = - v3 Do i:: = Ln(2 + v:1) = In(2 + ,,/3) + ,k2rr ho~c ;z = Ln(2 - ,,/3) = In(2 - ,,/3) + ,k2rr, YOI k Vay::: = -i In(2 + 13) E Z + k27r ho~e z = - i In(2 - V3) + k27r ~_iTlk211" - - 6" - + ho~ :: = 7r -2 + i + k2rr (vai k E Z) o lI~n - \),0'\ (e iIP - a)(e-;<'<> - Q) (ei'l'a- l)(e-;'Po I) = 1, tt"tc 18 XCi n 28 an 1=1 < • tam hmh tmn = eifJ • V;:ty nuh xa phAi tim co d~ng: w=e,1J z-a o OZ -1 Bai 1.32 Tim dnh X(l phiin tuy~n 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 _ , \ b) Ta1t co: Tu gUl thiet suy r cos2- - ",,(rr ') T l1c ' , ~-~2-:;-1 la 2.:: = - - ~ _ l + k2 I l<~ iT 2'" 7r IOo;u :: = - - + z + i + k2n hay _ \ c'Y'rt I' e' a \ ' = 1'\1>1,·e''PO' X{I- phan tuy~u tinh d udi dl;,.ng: lV=).Z-::o z - ::\ Do (t n~n.to , www.vnmath.com a trollg d o 11\ sO tlnJ'c va lmo M$t khflc, vlo, vA d6i XUllg que tn,.Ic tllIle n E!1l w(o) = vA 1('(0) doi xltng qua dU'bng tron Iwl = 1, nghia Iii weal = 00 Do = = - -" = > z-a Do (·ftc dil!m tr~n tr\lC thvc bien vao dttbng tron dOn VI n~1l j = 1).1 x-al, I -= x- a Iwl = Do ~I' = x_a 1,\1 = hay ,\ I W argw'(i) giAi Anh hinh tron d dn ir ir._ e - 2' = - 2" I Tu La co () = O z- , :r; Vij.y allh XI,\ phiii tim la w = - - E III Z +l o Bai 1.34 Tim ]J hep bien d6i phii,n 11Lyen linh biEn hinh trim!::I<l I = e ~ A snh x~ = 0; - 2" argw'(i) = () - arg 21 = = eiO w(i) ,,- v(}i IUQi x E JR cAn tim o B~ 1.33 Jim phep biin ddi philn tuytn tinh bi~n mat phdng tren Imz > thdnh hinh iron dan vi Iwl < cho ;;-0" lhanh hznh tnln dOn Ltfi gjai Anh XI;l tIllwl < clw w(4)=O; phai tim c6 d~ng: ~-a w=e·8~ = _~ XIiL bie • • I nUa m6.t phAng treu phAu tmng VI, du«;Jc X8c dinh theo cong thuc: w =e'~~ III 2i' (x - <>)(x - ol _ (x o}{x _ 0) - , voi mQi .02:-0' = f' •• (z+i)-(z-i) (z + i)' 2· =e'o '_ (z+i)'· fit Do d6 V.·yt It co l.c)j Suy , w'(i) = e'O_' khAc I ':+ til til M ~t w=e·/I~; o Vtiy > O - az Vitu(~) = o~a=~ w=e Do d6 ;0::-1./12.-::-1 -~e - l-~ ::: 2-.: argw'(~)=O www.vnmath.com eho nen z - 4i hoy Bai• 1.35 Tim ham J an.h x~ hinh tron Iz - 4il pIllIng v > 11 cho f(4l) = -4 va f(2i) = O < len nUa m",t A z - 4i -2- Dod6 (z - 'i)(w Lai gim Ta thAy anh Xl,\ Z\ = :: ~ 4i bi€n hlnh troll Jz - 4iJ ~ zw <2 ¢:> lenhlnhtrollddnviJ=II<lvaanhxow -e-'~ W b·· , ml,l.t • 1len nua phang v > It thAnh Olin m~t phing tren Ta c6 <:1 (2t) = -i; 2] (4i) = va Wl( -4) Do d6 anh X(L call V ij.y w = - 4e-I~ Ta can 11m Ani; x~ biel] 11\.16 m(i,t phing t ren len hinh tr ' Oil don vi vbi g( - 4e-' 4} = 0; g(O) = -i tim c6 da.ng: + 4;) = 2(w + 4) l6 = 2w + + 4zi - 4iw + w(z - 4i - 2) = 8+4zi!" a an h + 4't - z = -8 - Xl;\ ' tim can Slt anh X$ cAn lim co dl;lng w = a z + b Anh XI.l lis hQP CUA phep quay vectd mot g6c blmg arga, phep bien d6i dong dl;lng hi: sfi k-Ial va phep tinh ti~n tbco vecW Vi -i~ -i = g(O) -_ e 4e ~ .4e'4 " = 1,suyra9 = O nene b = e,g e -' _ is ( - e -i) Do an h XI;l bien D len cbinh no nen erg a = hotc arg a "'" Do d6 w = kz + b hoi;ic W = - k :: + b 11' Trong ca hai tnto:ng h~p ta df u co k = VI n~u k " tbl dai dli eho c6 luth IA diii co rong khnc wi dQ rOng dii d6 dIo Do ho~c w =:: +b hol;iC W +b = -;; Nell w 32 zi Bai 1.36 Tim dQ.ng lang quat ella M.m I.T'1Jen tinh nguyen, biPu D = {D< x < 1} len chinhn6 Lbi giai Cia Suy w+4 = W+ = z + b thi b = ih, hER :\t It u·- www.vnmath.com +/)!hlb V~y anh l +lh,hER 11\ gM phAll t.1l' thl( tu cElt bOlllllj\ \nnh troll 111' Q x~ Jukovski b) Ta co iinh BAi 1.37 0) Ti", 011" Clia fO U' < R(':; < 1.1111 :; > O} quo (l1Jh xjl I =- I I _(., + -) [hI (z, - z,)(1 - - ) 2:'1 b) Chling mlllh allh X(l JukolJsA:t dOn try 1-1 trcmq TlIea m(l.t "hdng ttin ~(.: +~) tv = I ZIZ-:!: • Nr-II '2~("1 ~I -+- I ) =0 V6.y iLnh Xf.\ na.y ddn trl 1- t mi~n nao (\0 vi\ ("hi D khong chua =1, Z'l ma 21';:2 = l I d i 4; • no ra.ng =1 \fa ~ lfQC sAp xep nhU' sau' Nell mljt diem thuQ( z, 11\11\ Ill~t philllg trcn thl di~m thuQc mea ml;l.t phing dUdi ' ~"' a )Do w = ;-I ne n LAJlI giAi Rcz = V~y anh x(' don tri 1-1 1lI1a mtl,t phing tr(\n .;+~ 1.+.1 !ti !!.: = : =~ = _,_ _ Rc - JIll U' /w12 w'iiJ - W T uang tl,llm z = - - - Do d6 t~p {O < R ~ Iwlll ' C _ < l:Im ::> O} • h I , q 1I3 I\n xe 1lJ = :; bien -1m {o <Rel'wl- < 1'lwl '_ > 111 1U Bai 1.38_ Tim anh cua cae -nn~n: a) ffinh trim <)l z l>1 qua anh X(l {Re w > ; Iwl'l > Re w; Ian w < O} (I) = oJ' + iV thl (1) tro { x> o·, oS + !J ' > X; 11 I +-y2> 4;Y<O} = ~(z + ~) D~t z = re'''', ta co 1 1 Z r r = -(r + _) cos <,0 + i-(r - -) Sill (;? Do d6 Jukovski w = u+iv= !.(rei "'+-'-) = ~ (r(cos lP+i sin r.p)+ ! {cos ({,-iSln rei"" r < O} bay {.r>O;(.l· _ l), R < 1; &)1.1 < 1; Lui giaL Xct w 11' 1.;1 < } hay Viet I , u=-(T+-)l"05~ r ip)) \'/1 www.vnmath.com I " I B AI T P -2(1 - -)siu'r" TV C IAI r h) H.) jf; ,) JI v"1"'+f; Bai 1.4.0 TUlI phim tllllc va phim 6.0 a) m nh 1=1 < R < 1 hic:n thhuh mi~n ngoAi ellip (,'0 I Clla H() phuC' .Ji -&- iy (t lu , hai hil'l\ r, V) ('H(' han tn,lca=-(R+ );b=-(R ) R R ' b) Hillh 1.:1 < bibu thilllh mM phAllg Chi'! di dOtUl ( I ; 1] c) lfinh 1.::1 > bi~u thallI! C \ \-1: 1}, o 21rJ Bni 1.41 Cho n dl~ll\ P J = cosH , 171") -,n E N,n ~ J +tMIl /I n tn'n hlllh tron dOli vi Chung minh dU1~ 116 poP} lu khnR.lIg cach gift8 Po va PJ" Bili 1.42 Cho < r chubi < L va On E JR,11 ,."{<:os6" + < n;-: AlP, -= n () = 0, 1,2, • C!lI rug minh isin 0,,) (1 BAi 1.43 C'ho diiy {.:: l~=o uoi ;;,,+1 - z" = 0(""" - t" 0< lnl < TIlII gidi iU,UI lim z" qUf\ '::0 \'8 ~\ Bru 1.44 Gilt stt :::;., _ .:: # IX) va (n db _ ( Ch(rng minh rAng: ;::\.(" + -'2·(,,-1 + \- '::,,(1 " d, tJ _ ( Bat 1.45 C'ho Eo C C,ll' Er in nhfOig t~Jl compact tbO& En, n n Eo~ I- \-di tnQ i M h Ull h(tll E o \,'" • E, ~ millil nflf-rEo • I- www.vnmath.com = {.: B ro 1.4 Hay t!nnlg miuh VAllh khan R (Tl,r2) < r2}(O :S" T, < "2) Itt llJot mit-n I.' Bili 1.48 Cho E lit t(lp D\ E cflll g Ie r C11 r{l C E C;r l < !Io ll;i\ khlu , v(Ji m i~n D C h{Olg minh ng mi)t ll1i ~ n 1/(=)1 t,en 6(1 ) z" - ] =-, z- J = ~ c} ~(n' )I/" zn., (p > 0); TIm supremum t 1,l va b) f: q'" z" OQ L d) + ",,0 Bal 1.53 (hfien Stolz va d ' h (Iql < z" 'I-I n "",,0 C« G«· G« ,.) a z n 19 chuo\ lUy thila hOi tlJ tr n ~ (1) B tli 1.5 C ln tn g m inh ng Aut (.6 ( l» If! m(lt hQ uhung h im' dbn~ lien tlJC tr['n cac ti,i p compact coa 6.(1) Sill ::: D ili 1.55 Ch ung minh r il.n g t an.:: c \ { 211 + 11" 2-; It E Z } • v(1i chu k y la m Ot ham (~+ )({3 ) ~ v +v + v)b + v) lr~n 71" D ni 1.56 C h un g millh ng, n€u 8.) CluIng minh rAng: r) n 6( ' ( ' cos r) c { zE 6(1) ; n «(;cosr) 3B L ~ 1); (E c(o: 1) Ki hi(!u r) dl~ ( ~') v~ ti~~l H: 1l tl,lC Abel) C ho t hAng di q ua ( t {l.O g6c ~ 7r Ia mien n Am giu a h a i dltOng To G( < r <"2 VOl dltbng thing di qua va ( a gQI ( ; r) III mien (g6c) St o lz vdj dinh ( oo;r} < }\.} n «; 2w;.;r) r:: f hthl ( inFimum (,"l'a cua cae chulli liiy thua sail: •• /z-(/ 1-/.:-/ < (O: n-/'2) H M f: C!u·n J!; ll1inh rAn g nt-u L:~=o a (n h Oi t\l thl 11 """" 2~{ n=O = Bai 1.5: Ti m ban kinh hOi ,~, n'z" 11=-=-1;\ E 6.( 1); h) C'ho fez) = Hili 1.51 eho fn{ x ) sin fi X, n J , 2, " lil day cae ham bi~n t~l,l'c x E [0; 211'J 1<111 d6 {In{£) } ~=1 Is bi ch.tm dell C luIng mi nh 'lang bat ky day uao eu a (J,, (.C)}~1 dell khong hoi tl,l t rE!n 0; 2./ a) {= l ; r = «:Ol' I u«;'2CORr) B Ai 1.49 Tinh cae gidi h ~ 1l snu: " sin Z J" c"" - I J' log( l + :) J,m - - ; lin - - ; 1111 ,-0;: *-0 :: z_o :: _ Bat 1.50 Cho f (;; ) lIl inh > I( > l hi Q n ( - -=;) hQi lu "" I fl dt~u t1f n CaC t~ p compact eua C n (1 ~ Bni 1.5 C min.h r;;'ng .:;n) hQi 1.1.,l t u yi,!t (lol ·a d@u n= t r{'> n chc t ~p compact ella A {l ) Bni 1.58 Chltng minh rling D Ui 1.59 Ti bei sO cross cu a n ~( + ,,=0 b bn di~1ll " )= =Mn = 1" 1 _ ;; .,':; C\18 C dur;tc xac diDb www.vnmath.com () tiily, nUl ZJ nan hang 00, t ill v( p h"i hi~u theo nghia Is B oo 64 Gilt s \t f{;;) dU'Qc ch o nln t t rong M i 1.6\ C1 l1nl ~ mmh h(\11 ::1 -+ • C ho / E Atlt(C) 19 Vh~p bi ~1l dd i tUY<'-'11 tfnh rilng, n ~u a+d la sa t h l,tc thl f la hyp('rbohc, eilipti('" hl\y p>trabo 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 rall~ 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 di~ 1U =t , '::4 eua t: thl I (TiLl h bAt bi~ n 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 tUY~1l Hnh Chlmg minh rAng, fU? U a+d = ho~c - 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 1.~l cUs I · Chung minh rAng, l1(:u (J =I (3 thl tIl = J (z) d UQC cho b0' ~ w-(3 = K e,9 =-:: !! :-(3 ' l( > 0, B E JR PI~ep bien dOi tuy~n t fnlt I d UQC gQi lit hyperbolic n~u e ,9 = elliptiC n~u 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 d~llg tGng q ua.t ella phep bil-n d bi plum tuy~n tinh biw tan A(R) , < R < 00 Bili 1.66 a.) Tim phe p b i~n dbi 11hl\.l\ tuy~n Hnh hit n O 1, 00 thu.nh i, + i, + i l U'Ol\g {m g b ) 'Tim phep bi~ 11 di\i phiin t uy~n t{nh bien - 1, i t hanh -2, i, t uong I1ng www.vnmath.com HUONG DAN GIAI vA DAp s6 Chttdng 1.39 Vi~t, t.:hAng h~n, 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 ; v= ± V2y J";x'.l + y'l +£ 1.41 Si't dN! lP d~ng nhat thuc (= 1) II (z - p - j) = z· -1 ~ ( - 1)(.'-' + + 1) j=l 42 Sli d\,mg Hnh hQi h,l tuy~t d3i 1.43 11m z" = aZo j-a %1 - 1.44 Hay danh gia hi~u 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 t~n to-i day {f }~_l hQi tl,l t!;Li UlQi di~m thuQC [~; 2~1 D~t fez) lim /",,(x) Tllf >Q djnh If Lebes~e 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 = ~.! ~ 'In == f :1" =j " Oon· Vtli Z E 6.(1), ta (;6 / (:;) == 1- z s = (1- ,) n == zn n""O f:(," _,)," ~O V8i : E D{l, COOT), tOn tl;li hAng st, dVClllg K (= _2_) "'" £, n 51·1'1" + L: Izln) n '''0+ I 110 laJ~,.rllhien ~ "I.Q Khi d6If( z)-.'l1 ::511-zICE 18n < I~ 00 n=O n ",O 01=0 11 - zl (L 18n - 'I + L 1'1") = lI - 11 _ "I no 'I n::::O L ISn - 81 +< -1 < 11 -.1 L I'" - sl +.K - Iz =0 ~ *J 2m IIta-1 khac J Isinntldt= o Mau thuan xiy "p = O lf(t)ldt + I)p f 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~2 j=1 L: logj < 'l2 cho 11- :1 < f{(1-lzl) Voi £:> tily y, tOn tl;li Lli'oi theo dinh Ii Lebesgue v~ hOi t.v bi ch~n ta co 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) 11 - L: ISn - sl < " •- a, 1.54, DM fez) = e " -z+ l 1ft ,) - f(z') 11(,) - 'I < < I,.a 1'1 , $ r ta c6 .0 lal < I, 1.1 :5 :5 (1 -Ial') :: =- ~! :51: =- ;;! T www.vnmath.com 1.55 00 sm(, +~) - sill; V8 cos( z +~) ~ - cos z n~n t~n( + ~) _ tal' z Gi!.su tim t~i < ~' < cho tan(; ~ "') = ta" z KIll d6 t c6 8m~' =O Til d6 ta suy sin(z +h ') = sin z MAu thuAn xAy ,a vi sin; LIIAn hoAn ehu ky 2~ n 1.58 Bang quy n~p, ta c6 2n+l_ l Il (1 +,'") = L v=o z" Cho n 00 V~ O i' - O~t II,) ~ Z-Z4'3~i"Z2-Za -;'~ Day la mi) l phep bi~n dili tuyen tinlI va liz,) =1; /(Z3) '" 0; J( z.) =00 1.59 Ap d~ng djnh If sau, "ClIo b(i ba di~m ('" '" oJ) va (w"w"w,) I hai bi) ba diem phiin bi¢t t Khi d6 I~n lai nh!t phep bi!n dOi tuyln Huh / cho / (z,) = Wj, i ~ 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 bi~n dili tuyen tfnh va 9(',) = 1; g(,,) = 0; g(t,} ~ 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,