Đánh giá phép các phép biến hình á bảo giác lên mặt phẳng bị cắt theo các cung tròn đồng tâm

B=BovoiBo lam~tph~ngphucmarQng w bi celtdQc p clingtroll d6ngHimt(;li0 saocho Bo bie'nthanhchinhno bai phepquay;=/~w va g(w)=awlwr-1voi lal=1. Chungminh: XetPBHKABG f E F mi~nA lenmi~nB, theo(2.24)taco: P\ <s,1- S'(0,f), f EF :rRK D1!avao(1.11)va(2.20)tadu'Qc 2 ~Sl <s,1- M* (0,gfK ,g E G , :rRK(g) (3.3) trangdod~ngthucxayrakhivachikhi w=f(z)=g-l(z)=bzlzr~-lvoi Ibl=1,tuc B =f (A) =Bo. 33 Tif (3.3),taco: * ( )-1: PSIM O,g ~1-~,gEG, 7rRK hay M*(O,g}~~ 1 ~1,gEG. 1- PSI2 TrRK Nhu'v~y taco (3.1)voi tru'onghQpd~ngthuexciy rakhiva emkhi (3.3)xciy ra d~ngthue,tueIa B=Bova z=g(w)=I-I (w)=awlwr-Ivoi lal=1. M~tkhae,dl!avaoeongthue(2.32),taco: ' ( ) c(R,/)m 0,I ~ -1- 1-' I E F 4 PRK 2 =>m'(0,/) ~4"c(R,/) d(R,/) ,/EF, ke"thQpvoi (2.20),tasuyra M* (O,grX~4*~,gEG.d V~ytaco (3.2). 3.2Danhgia Ig(w)1 DinhIy 3.2:Du'oicaekyhi~uvagicithie"t(jphffn1.2,VgE G,WEB ta co: 4-~lwIK~lg(w)I~4~M*(O,g)lwIK. (3.4) Chungminh: Theo(2.31), V/EF,zEA,taeo: 4-~m'(O,f)lzIX ~I/(z)1~4~Izlx. Thay z=g(w) va f(z)=w , tadu'Qe 1- 1 1- I 4-pm'(O,/)lg( W)IK~Iwl~4pIg( W)IK, 34 tlido .... Ig(w)l~ 4pIWlKK va Ig(w)I~4-~lwIK, m'(O,j) ke'th<jpvoi (2.20)tadu'<jc K I K 4p IwK K I ( )1< ) -1' 4-plwl ~ g w - M*(O,g suyra (3.4)8 3.3Danh gia ban klnh R(g) Djnhly 3.3:Du'oicaekyhi~uvagiathie'tdphftn1.2,VgE G tacocaedanhgia: K K 4-PdK ~R(g)~4P M*(O,g)CK, (3.5) K R(g» [~(1-M*(0,g)f) ] -2 voiSI>0. (3.6) PSI Chung minh: Tli (2.32),chungminhtu'dngtv'(3.4)taco Rt ~ ~(R,j) va Rt ~4-;d(R,j),j E F, 4-"m'(O,j) hay K K K 4pc 4-PdK~R~ K,jEF. m'(O,j) Ke'th<jpvoi (2.20),tadu'<jc(3.5). Tli congthuc(3.3)tad~dangsuyradu'0, R-i (g)~~(l-M* (O,gri), VgE G,PSI tuctaco(3.6)8 35 Danhgia(3.1)coth€ lamchos~channho H~qua3.1: D~t E= PSIc!(;~~O),ta co: 7rC2p M*(O,g)~(l+E)t ,VgEG. (3.7) D~ngthucxiiyrakhivachIkhi B=Bova g(w)=awlwIK-1voi lal=1. Chungminh: Ke'th<;5p(3.3)va(3.5)tasuyra PSI 7r4;C2M* (O,g)t ~1- M* (O,g)-t hay M* (O,g)t ~1+ P~I . 7r4PC2 Tli dotaco(3.7). D~ngthuc(3.7)xiiy ra khi va chIkhi d~ngthuc(3.3)xiiy ra hIedo SI=0 keD theoE=O,tuc1aB=Bova z=g(W)=f-I(W)=awlwIK-Ivoi lal=1.. H~qua3.2: Trangtru'ongh<;5pK =1,M*(O,g)=Ig'(O)1nen(3.7)trdthanh Ig' (0)1~../1+E, Vg E G . (3.8) D~ngthucxiiy rakhi va chIkhi B=Bo vag(w)=awvoi lal=1,ba'"td~ngthuc nay s~chanba'"td~ngthucc6 di€n Ig'(O)1~1,Vg EG voi K =1(xem[10],IT.350). 3.4Dauh gia g6cmd 2~(g) Nhu'tadffbie't0<~(g)<7r,Vg EG. Bay gio, ta timcaedanhgia co th€ s~c P hancho ~(g)trongmQts6tru'ongh<;5pnaGdo. 36 3.4.1C:}ndumcua r3(g):(DungphuongphapdQdaiqie tri) Dinh Iy 3.4:Voi caeky hi~uva gia thie't rongml,le1.2, gia sa c<d, khi do VgEGtaeo: f3(g) '2 IT - P n ITK21n4' dM*(O,g)t c d 2p f- dr c rO(r) 1£ '2 IT _ / nO.oK2In4 PdM* (O,g)t p ~ c2pln~ . (3.9)c Chungminh: ~ ..R...... O """"::::::::"':§~::"':::::~ o """'" '/ '" \ \.(Tj (T . 2 . : 0 : : \..::::':.:::::::=::::<::/ A ..(""""""""""""""""",~ ( t >""~:::::::::4.::::""" j "'~) / 0 """"":~L: """'" ..., '. I (.. " ~:::::::::::::::::./ i \ z ), Hinh3.1 Ap dl,lngb6d~2.10vaobailoandangxetvoi Bo la tugiaeeongcohai e~nhn~mlIen haiduangtronIwl=c va Iwl=d; haieanhconl~ila caeeungcua (TIva (T2va dQdo p(z) =1~I'zEAo, Ao=g(Bo),taco: Ip( Cr ) =fp(z)Idz1=f~,- - 1 z I co' c, voi Cr={zllzl=r}nBo,c~r~d,Cr=g(CJ. B<)tz=re'<P,taeo: Idz I =Ie'<Pdr + ire'<PdqJ 1 = 1 dr + irdqJ I '2 1 irdqJ I =1dqJ I. IzI Ir I r r 37 (Ba'td~ngthuctrenco duQcVI c<;lnhhuy€n cuatamghicvuongkhongnhohon c<;lnhgocvuong). VI v~y,taco: lp(c,);, fld~;'2a~t -P). c, M~t khac, do tinh d6i xung quay (1.4) va d~t B' =BnH voi H={wlc<lwl<d},A'=g(B'), ta tha'y m(c,g)~lzl~M(d,g),gEG,taco: A' n~m trong hlnh vanh khan Sp (Ao) = Jf p2 (Z )dxdy =! H p2 (Z )dxdy Au P A' =! Ifdxt =! If ~dY2=! Ifrdr~qJ P A' Izl P A' X +y P A' r ~! 2]dqJ1dr =21l:1rlM(d,g). P 0 m r P m(c,g) Tli dotheob6d€ 2.10tasuyra 21l:InM(d,g) ~.l(2a)2 J dr ~.l(2a)2~ Jdr , P m(c,g) K c rQ(r) K Qoc r tuc a~ M(d,g) 1l:Klnm(c,g)~ d dr 2pfrQ(r)c ~Kln M(d,g) m(c,g) 2pInd c Ngoaifa,theo(3.4)k€t hQpvoi(1.8)va(1.9), taco: -K K m(c,g)=4PcKvaM(d,g)=4PM*(O,g)dK,gEG. 38 Suy fa 2K n-Kln4"M*(O,g)dK a::::: I cKd < 2p f dr - c rO(r) 2K ~Kln 4" M*(O,g)dKcK d 2pln- c VI f3=TC-a tac6(3.9). p Nhanxet: Ne'uc=const,d=constvacho Do~O ma M*(O,g):::::M~=constthl a~O ~ f3 TC We ~ - . p Vi dV3*1: ;=h(w) ~ - ' - I K-l ( - )z=zz =k z ~AB /"""' CS ~ 2J \\ /. ~ ;.. y """"""""\ /""""""""'~""" ) '...... r (:_~)red \R ( t""); -,) Ii (( (-'; R \ \ :; / \ : ~::: / \ :.::::~ j ------ gEG ~ Hinh 3.2 Gia sa Bo=Bn{wlr<IwI<R} e6d<;lngnhuhinh3.2(p=2),c6dinh c, d va eho Do~ 0 ta chungrninh M* (0,g), g EG kh6ngdffnde'n 00 khi Do~ 0 tucla M* (O,g):::::Mo =const. 39 GQi ;=h(w) la PBHBG dondi<%pmi€n BIen mi€n A la m~tph~ngmd rQngbi ca:t dQcp cungtroll tam 0 thai h(0)=0, h(00)=00 vakhaitri€n Laurent cuah(w) trongIanc~nw=00 c6 dc,lllg h( ) a1 az w =w+ao+-+2+'"w w (3.9a) tucla anhcuaduongtrOllIwl =R voi R ra'tIOnbdi h g~ntrungvoi duongtroll 1;1=R . N6i cachkhac m*(oo,h)=Ih'(00)1=HmIh(w)l- w-+ooIwl - 1. TheoThao[ll, tr. 109],ham h(w) Ia PBHBG dondi<%pmi€n B !enmi€n A A c6 tfnh d6i xung quay ca'p p. GQi z=k(;)=;VIK-lla PBHKABG mi€n A lenmi€n A trongd6m6iduongtroll1;1=R duQcbie'nthanh va do d6 mi€n duongtroll Izi =RK . VI argz=arg;nenk(;) clingc6tfnhd6ixungquayca'pp. Khi d6 z=g(w)=ko(h(w)) la PBHKABG mi€n BIen mi€n chufinA c6tinhd6i / ,.(' H - * ( ) I. M(R,g) I. RK 1 V " Gxl1ngquaycapp. onnua m oo,g= 1m K = 1m~=. ~ygE .R-+oo R R-+ooR GQi C, la anhcua C,={wllwl=r} voi r ra'tbebdi h va C~Ia anhcua C, bdi k; ZlEC~ saochoIzll=M(r,g), ;1EC, saochok(~)=z,va w,EC, saocho h(w,)=~.Tac6: M. (0 g) =lim M (r ,g) =lim1:J =lim I k (~)1=lim I ~I K, ,-+0 rK ,-+0rK ,-+0 rK ,-+0rK =lim Ih(WI)IK=lim h(Wl) I K = l h'(O)I K -:I:-0 (VI h IaPBHBG) H-+o Iw,IK H-+o W, 40 Mi;itkhac,nSu r~O thl ta colh(w)I~lh'(O)llwl=lh'(O)1rtuc Cr g~ntrung du'ongtron1;1=;, voi ;=Ih'(O)fr. NSun6ihaicungcuanhatcfittrongmi€n anhAd€ du'Qcdu'ongtroll 1;1=R) thltrongmi€n B clingsecohaicungn6itu'dngling.Nhu'da:neutren,anhcac du'ongtroll Iwl=R, Iwl =r voi R d't IOnva r ra'tbebdi h g~ntrlingvoicac~ du'ongtroll 1;1=R va1;1 =Ih'(O)lr. Khi cho Qo~ 0 do tinhba'tbiSncuamodunhai mi€n nh!lien quaPBRBG R R R) ~ .:. ;=h(w)taco R) ~ d' Ih'(0)1r r V A,.,!' ,.,!' 1/ , ,.,!' b / h' - d Ih ( )1 R) d Ch '<;lyneu R rat on va r rat e t 1 R) ~ , '0 ~ - ~ -. 0 r ~ 0 vac c R ~ 00 taco I h'(0)1=d .c T6m I~i,tac6 M' (O,g)=(~r<00. V~ydanhgia(3.9)lacoynghiavatic$mc~ndungtrongtru'onghQpnay. 3.4.2 C~ntren cua ~(g): D~u lien, ta chiami€n B lam p ph~nb~ngnhaub~ngp du'ongcong JordanYj(J=1,2,...,p) n6i 0 va 00,du'ongnQchuy€n thanhdu'ongIda bdi phep quaymOtgoc 2nj. Cac du'ongcong Yj nay chiami€n B thanhp mi€n nh!lien p B~,(J=1,2,...,p)voibientronglamOthanhph~nbienG"jcuaB. Ki hic$uC(a,r)chidu'ongtrOlltamt<;lidi€m avabankinhla r. Tren B] ( baadongcuami€n B] =B; ) ta co th€ ve-themhaidu'ongtrOllphg: Du'ongtrollthunha'tla C(W)'1)) gioi h<;lnmOt hlnh troll dong chua thanh ph~n 41 bien (}j;Duong troll thu hai la C(w2'r2) chuatrong Bl va baabQc C(WI'1j). GQi B2la mi€n nhiliengioih~nbdi C(w],1j)va C(W2'r2). z=g(w) ~ B A ......... c:::::>/. .. ~ . . '. . Q . .;.. ..f />3fI\;\ B, ) B, / 0 ...\\jJ) )., ~~ ~ ~ / AI I::i'l ~A, . \ /// ..~ --./ w z Hinh3.3:PBHKABG z=g(w)bie'nmi€n A leDmi€n A voi p =4 Thea h~qua2.3,taco: mod(B2)~mod(B]). D<)t R. =min{lwllwE C(w2,r2)}' ~=max{lwllwEC(w2,r2)}' tuc RI=lw21-r2 (O<)~ =lw21+r2 (3.10) Saudotatie'pt\lCvehaiduongtroll C(0,RI) va C(0,~), tucC(w2'r2)n~mtrong phftngiaacuaBJ vahlnhvanhkhanB3={wiR,<Iwl<~}. Ta tinhtie'nvaquaymi€n B2 r6i apd\lngb6d€ 2.11,mi€n B2coth€ bie'nbaa giac dondi~plen hlnhvanhkhan B4={slr <Isl<I} saGrho C (W2'r2) tuongung voi Isl=1, 42 2 h2 2 I( 2 h2 2) 2 4h2 2, r2 - +lj -\I r2 - -lj - lj ~. I Iva r =r(lj,r2,h)= , VOl h=W2 - W] . 2ljr2 (3.11) f)~t Al =g(B1XcA), A2 =g(B2XcA]), gEG. M~t khact6n t(,liPBHBG don di<%p; =; (z) mi~nA2 ten hinh vanhkhan Bs={;lr'<I;j<l}. Vi phepbi€n hinhhQp;ogos-]mi~nA2ten Bsla mQtPBHKABG nentaco: r'< 7c_r . Theatinhdondi<%u(1.17)cuahamphvT(p,r,s), taco: T(2,r',0)~T(2,r7c,0)vdi r xacdinhnhu'(3.11). GQi D la du'ongkinhcuamQtnhatcatclingtrOllLi' D' la du'ongkinhcuaanh du'ongtroll C(WI'r]) bdi z=g(w),g EG tilc du'ongkinh bien trongcua A2.R6 rangtaco D ~D'. f)~tm=m(Rpg) , M=M(R2,g), gEG. Thea (3.4),taco: m 2::4-: R] K =m ,M ~4~M* (0,g)R; =M . Theab6d~2.6,taco - N€u p =1, d€ coquailh<%D=2Rsin~ c~nthi€t chavi<%ctimc~ntren cuaP(g) tac~nthemgiathi€t 2p~n vi n€u 2p>nthi D=2R. Giathi€t naydu'Qcthain€u (D~D) =2T(2,rt,0)M (R2,g) <2.41r7c4~M* (O,g)R; <2.4-~dK« 2R), tilc 1>:1-1. * -1>: 4P+'rKM (O,g)R; <4 pdK. (3.12) Luc naytamdico th€ apdvngdu'Qcquailh<%D =2Rsin~. 43 - Ne'up ~2 thidu'dngnhien 213s 2n S 1[dodo taluauco D =2Rsinp. p / ? , Ap dlJngbade2.6,tadu'Qc: D's 2MF(2,r-K,0). Mi,Hkhac,taco: DsD' va M=M(Rz,g)sM. Suyfa DSD's2T(2,r-K,0)M. Tildo Ds2T(2,r-K,0)4~M*(O,g)R~ s2.41r-K4~M*(O,g)R~=4~+lr-KM*(0,g)R~. M~tkhactaco D=2Rsinp Suyfa sin13=D < 4~+lr1-M* (O,g)RK 2R - 2R 2 0 -K Vi R chu'abie't,tathay R b~ngc~ndu'oi,nghlala R ~E=4p dK . Suyfa 4~+1r1-M* (0 g)RK 4~+1r -M* (0 g)RK0 13< ' 2< ' 2sIn - - K . 2R 2A-1idK V~y ( 2.42:r1-M* (O,g)RK J f3(g)sarcsin dK 2 =f31(g), 44 2K ydi di€uki~n 2.4pr1-M*(O,g)R; dK ::;10 M~tkhac,apdl:mgb6d€ 2.8,taco: D'::; IS In (1- (2 ) -7r ydi (= T (1,rt ,0) , 7r [ 2 ( ) 2 ( )] 7r [ K * ( ) K _K K] - S::;SI =P M R2,g -m Rl'g ::;P 4pM O,g R2 -4 PRJ =S. (3.13) V~y D ::;D '::;,I SIn(1- (2) -7r 2Rsinp~~SIn~~t2) . Surra 13(g)::;arcsin SIn(1- (2 ) -7r 2R K Thay R =R =4-PdK ta duQc ~sIn(J-t' ) fJ(g)::;arcsinI _K~7r 1=132(g),4 p+2 dK '0 dO;:; ki' lln~~t')YOI leu <:fn K I ::;1. 4-P+2dK 45 Nhu'vi;tytadatlmdu'Qci;tntrencua fJ(g) du'oid~ng: Dinhly 3.5: Du'oicacki hi~uvagiathie'trongm\,lc1.2,vanhu'moilieU(j tren, Vg E G taco Ne'u p =1va thoa (3.12) hoi;lcp'22 thl fJ(g):::;mill{fJI(g), fJ2(g)}, trongd6 ~ (g)=arcsin [ 2.41fr+M* (0,g) R: J dK ' (3.14) fJ2(g) =arcsin S In(1- (2 ) -7( 4-K+1 dK (3.15) voi r,~ va S xacdinhnhu'd (3.10),(3.11)va(3.13). Vi d1}3.2: gEG ~ A ( ~ ~ Y4 w z Hinh3.4 46 Gia sami€n B c6p =4 thanhph~nbien O"j' j =1"",4 la cac du'ongtroll c (aJ'&) voi aj =euta; & du'dng,du be, Trang d6 thanhph~nbien 0"1la du'ong troll C(ap&)voi al =a,&=fj >0(ffinh 3.4), Ta ve du'ongcongJordanrl la du'ongphangiaccuag6cph~ntu'thilnha't. Saud6dungphepquaytaxacdinhdu'Qcr2'r3va r4la 3du'ongphangiaccua3 g6cph~ntu'conI~i,Cacdu'ongphangiacnaychiaB thanh4ph~nb~ngnhau, Ta ve themdu'ongtroll C(a,r2)saDchobankinh r2 Ia khoangcachtu a d€n du'ongphan giac cua g6cph~ntu'thil nha't.Khi d6 mi€n B2 chinhla mi€n nhi lien gioi h~nbdi haidu'ongtroll C(a,&) va C(a,r2), Mi€n B2 c6 th€ bi€n baa giac ddndi~pleu hinh vanhkhan r <Isl<1, Theo h~ ? 22 / h b"'" b' "'" ? ~ d ,;:, h ' I' ~ / 1 r2 / &qua. , tIll at len cuamo unmIenn ~len, taco - =-, tilc r =-, r & r2 M~tkhac, tac6: ff a r2=asin4"=12' V~y r =&12 a Ap dvng(3,14),tac6: 2.4':'2'EKM' (O,gJ(a+;JK PI(g)=arcsinI aK (a+&t Cho a c6 dinh kha lOn, &~ 0 , M* (0,g) sconstthi PI (g) ~ 0, 47 Tu'ongtv, apdvng(3,15) Sln(1-t2) V -Jr) , I I' /32(g =arcsm4K+!(a +&t voi t =+(<:)',0).s=: [M2(R"g)-rn' (R"g)]~: [4'M' (O,g)R;'-4-' R,'r a a va RJ =a- .[2,R2=a+.[2 . Theo (1.24),taco: [ I ) I &.[2K &.[2K . t ~ T {--;;- J '0 ~ {--;;- J -> 0 kh1& -> 0. V~ykhicho a c6dinhkhalOn, &~O, M*(O,g)~constthlln(1-t2)~O,tuc ta clingco /32(g) ~ o. Nhu'v~ytrongtru'ongh<;1pnay caecongthuc(3.14),(3.15)Ia khonghi€n nhien vati~mc~ndung. ._.