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

CHu'ONG 4 "" "'" CAC DANH GIA LOP HAM G Trongchuangnay,chungWi danhgia caed'.lilu<;1ngd~ctru'ngchomien chu§nclingnhumoduncua caeham g E G. Vi~cdanhgia banldnh R(g), g E G theocaed'.lilu<;1ngdfibie'tgill mQtvai troquantr<;mgtrongvi~cdanhgia caed'.lilu<;1ngkhac.Do do,tab~tdffub~ngdanhgia R(g) vacaed'.lilu<;1nglien quan. 4.1 DanhgiabankinhR(g)va Ig(w)1 DfnhIy 4.1 Vdicdcgiathie'tvakYhifucuam~c2.3,vdimQig E G, WEB (w*O), taco: K M*(O,g)2(;)2 (~1), (4.1) 4K K M*(O,g» 2-p( ~) , (4.2) K R(g» [ ps 2 ] 2 ne'us>O, ps+S-1! M*(O,gtK (4.3) K K- - 4 P dK <R(p,d,Ot ::;R(g)::::;4P M*(O,g)CK, (4.4) K K 4-Plw1K<R(p,lwl,ot ::;lg(w~::;4PM*(O,g)HK. (4.5) Ddngthact{Ii(4.1)xayrakhivachikhiB =Bovag(w}=a»i»iK-lvdilal=1. Chungminh 22 GQi 1 =g-],gE G,theoc6ngthuc(2.3)trong[18,tr.l046],taco 8'(0,f)~8 .1r M~tkhac,tudinhnghlacackyhi~ud ffi1,1C2.3taco m'(0,f)2~8'(0,f). V~y m'(0,f)2~81r (4.1.a) Ke'th<;1pvdi (3.30),suyfa: M*(O,gt~ ~8 ~ (4.1).1r f)~ngthuc d (4.1.a)xay fa khi va chi khi I(z) =hzlzli-l, Ihl=1, tuc B=f(A) =Bo, do do d~ngthuc d (4.1) xay fa khi va chi khi B=Bo va z=g(w)=f-1(w)=a~wIK-],lal=1. M~tkhac,tuc6ngthuc(2.13)trong[18,tr.l046],taco d i ( ) -1 i * ( ) ~ ( )- <2Pm' 0,f =2PM 0,g K ~ 4.2 .c Tuc6ngthuc(2.4)trong[18,tr.l046]vab6de3.5taco: 2 ps =[8]-lr 8'(0,f)]iK KK ] - - 2 ps 2 ps => R(g)<=(S,-.-s'(o,f)] <=[s,-.-m'(O,f)' ~ K r ] 2 R(g)2 ps 2 8] -lr M*(O,ftK ke'th<;1pvdi 81=8+ps tasuyfa (4.3). M~tkhac,tuc6ngthuc(2.16)trong[18,tr.1O46], 23 tac6 d,,;~p,Rk,m)";T(P'Rk.O)=t. TheodinhnghlavaHnhdondi~ucuacachamT(p,r,s)vaR(p,t,s)suyra I _! RK =R(p,t,O)~R(p,d,O»4Pd. Titd6tanh~ndu'<Jcc~ndu'dicuaR=R(g)trong(4.4). Mi,Hkhac,titcongthuc(2.16)trong[18,tr.l046]tanh~ndu'<Jc _! I 4 P m'(O,f)RK~c ~ I I RK ~c4Pm'(O,ft1 Tit d6tanh~ndu'<Jcc~ntrencuaR =R(g), Vg E Gtrong(4.4). Tit congthuc(2.15)trong[18,tr.l046],tadu'<Jc m=4-;m'(O,f~zl~:5lf(z~:5T(p,lzl~,m)<4;lzl~ Ke'th<Jpvdi(3.30)vachungminhtu'ongtl!nhu'(4.4)tac6(4.5). H~ qua4.1 VgE G, fa (4.3)suyra K (1» R(g)>[p::sr lacntu (tr » ps ;:::canst>0 va S ~ 0 thiR(g) ~ 1. HelnmIa,ktth(lp(4.4)v(ji(3.17)tacokhi d ~ 1 Ktr2 I-R(g) <l-RK (p,d,O) ~K[I-R(p,d,O)] ~ 8 2pIn- p (1- d) nghfaiakhid ~ 1thiR(g) ~ 1nhltngratchqm. 24 H~qua4.2 Khi K=I, M*(O,g)=lg'(O~nenta (4.1)taco Ig'(O~~~(~1) (g E G) (4.6) Ddngthacxiiyrakhivachikhi B =Bovag(w)=aw WYila!=1. H~qua4.3 4 Dl tlmm(JtddnhgidcothlsiichCln(4.1),d(itC =2-; P: (~O),Vg E G,tacoc M'(O, g) ,{';,c)f (4.7) Ddngthacxiiyrakhivachikhi B =Bovag(w)=a~wIK-l wYi lal =1. Chungminh Theo(2.1)trong[18,tr.l046],Vf =g-', g EG, taco: 2 2 -- 2-- s, (f) ~1rS'(0,f) +psR K ~1rm'(0,f) +psR K. 2 -~ 2 Tu (4.4): R-i( ~4 PM* (O,gfK c-2k€t h<Jpvdi (3.30) Suy fa 2 -~ 2 8, ~1rM* (O,gfK +4 PM* (O,gfK c-2ps =>(4.7). H~qua4.4 Trangtn{(Jngh(lpK=1,khi do M* (0,g) =Ig'(0)1,(4.7)triJthanh: Ig' (0)1 :?o~1r+ C (:?o')8, (g E G) (4.8) DdngthacxiiyrakhivachikhiB =Bovag(w)=awwYilal=1. 25 Ddnhgid naycungv{Jiddnhgid (4.6)sachl1nddnhgid cddiln Ig'(0)1~1, g E G v{Ji K = 1 (xem[10, tr.352]). 4.2 Daubgiag6cmdP(g) R5rangta1uonco 0 <fJ(g) < 2" ,Vg E G. Tuynhien,tamu6ndanhgiat6t p hdn trong nhung tntong h<Jpnao do. Djnh Iy 4.2(c~ndumcuaP(g) V{Jinhilnggill thiefnhutrongm1:tc2.3,taco: p>(d~Crz; K 7r ntu (0 <)M*(O,g)cK ~ 4 PeP (4.9.a) ( ) K K 7r K 13> d-c 21n!i. 1 ntu4-Pe-P<M*(0,g)cK<4-P(4.9.b) 4 4PM* (0,g) cK Chungminh: - TheoThao[13],tr.1O9t6nt~iduynha"tPHBBGu mi€n BIen mi€n E 1ahlnh tron lul<1 trir P nMt dt theo ban kinh l j ={u 0 <Co,; lul,; do,a:rgu~ 2:} sao cho u(0) =0, bien C thanh lul=1 va cae CYj tu'dngling voi cae f j (j =O,l,...,p-1) (hlnh4.1). - D~tk =gu-1. V~yk 1aPBHKABGmi€n E 1enmi€n A saocholul=1thanh Izi=1,caenhatdittheobankinhf j thanhcaenhatdit theoclingtrolld6ngtam Lj (j =0,1,...,P - 1)vak(0)=0. f)~ t A ~ A n {z IIzl >R'}, Ao ~ A n {z-; <arg z <;}, ]3= g -,(A), Eo=g-I(Ao), C'={wllgcw)1=R2} (hlnh4.1). 26 GQi C" =u(C'), E =u(B), Eo=u(Bo)(hlnh4.1). R5rangAo =k(Eo)(hlnh4.1). - GQi Z =h(z) =Inz Ia PBHBGmi6nAo Ienmi6nmi6nnhilienAl cobien ngoaiIa hlnhchli'nh~tcokichthu'oc.~, 2In ~ vabientrongla nh:Hdt theo do£.lnth~ngr n6iZI=InR - i ~ voi Z2=InR +i ~ tiler codQdai j3 (hlnh 4.1). Tit mi6nAl cohaitru'C1ngh<;1pxayraho~ctaco thetImdu'<;1CmQtmi6nnhilien A2, co bienngoaila du'C1ngtrODHim J (InR,0) (trungdiemcua r), banklnh b ~ min {In ~,;}vabien !rang la r (mnh 4.1) ho~ckhong thl! urn d1f(1cmi;;n A2 nhu'tren.Do do, ta xet2 tru'C1ngh<;1psau: .Tru<tnghfjp1:Giasat6nt£.limi6nA2 thoacaetlnhchffttren. Khido: mod(AI)=mod(Ao) (4.10) Theotlnhddndi~ucuamodun(3.3),taco mod(A2)~mod(AI) (4.11) - PBHBG h =i(Z -In R) bienmi6nA2 tenmi6nnhilien A3 cobienngoaila dliC1ngtroDtam0 banklnh1vabientrongIa nhatdt th~ngr' =[ - ~,~ ]2b 2b (hlnh4.1). - GQi p la PBHBG mi6n A3 tenhlnh vanhkhan A4 ={ph<Ipl <I} (hlnh 4.1) saocho Ihl=I tu'dngilngvoi Ipl=I vaP( ~ ) =ro thl ro=R( 2,~,0 ) dodinh 2b 2b' 27 nghlahamphl;1R(p,t,s). Khi d6: 1 mod(A2)=mod(A3)=mod(A4)= ( p ) . R 2,2b'0 (4.12) - GQi EI Ia mi~nnhi lien chuami~nEo c6 bienngoaila du'Ctngtroll lul=1va bientronglanhatdt £0=[co,dJ. Khi d6theotinhddndi~ucuamodun mod(Eo)<mod(E1). (4.13) - D6dangthtlydingt = u - Co laPBHBG mienEI lenmiennhilien E2c6bienl-uc 0 ngoai la du'Ctngtroll It1=1 va bientrongIa nhatdt th~ngn6i di~m0 vdi tJ = do- Co (hinh4.1). I-d C0 0 - GQis la PBHBG mi~nE2lenhinhvanhkhanEJ ={sII(<Isl<I} saochoIt1=1 tu'dngungvdiIsl=1 va s(tJ)=s ( do-Co J =1( thil( =R ( I, do-co ,o J dodinh I-dc I-dc0 0 0 0 nghlahamphl;1R(p,t,s) (hinh4.1). Khi d6: 1 mod(EJ)=mod(E2)=mod(EJ)= ( d - - ) . R I, 0 co, 0 1- doco (4.14) Vi phepbie'nhinhk =gu-1tumi~nEo lenmi~nAo nhu'dffneula mQtPBHKA BGnen mod(Ao)~[mod(Eo)T. Ke'th<Jpvdi (4.10),(4.11),(4.12),(4.13)va(4.14),tac6 28 K1 l 1 R(2,L,0 )~ R(1 do - CO OJ2b 'l-dc'0 0 ~ R ( l, do-co ,O J K ~R ( 2,L,0 )<L.1- doco 2b 2b Dodo: (do(3.24» fJ >2bR ( I, do - CO,O J K >2b ( 4-I do - CO J K I-dc I-dc 0 0 0 0 ( J K 2b d - c ~ fJ>- 0 0 =A. 4K 1- d c I0 0 M~tkMc, tit (4.4), b =mill {In ~, :} tasnyfa: (do(3.24» (4.15) b2::min { ln ~ 1 ,Jr } =a 4PM* (O,g)cK P (4.16) Tli ba"td£ng thuc cua Thao[13], tr.ll0 0 <Co~C <d ~ do<1 (ba"td£ng thuc d ~dola tru'dngh<Jpd~cbi~tcuachu'dng5voiK =1)ke"th<Jp(4.15)va(4.16)ta co ( d - C ) K P >PI >2a 4 =P2. . Truitngh«jp2:Gia sa khongtant;;Limi€n A2 thoacaetinhcha"ttren,khido ( Jr J P 1 1 1 - > - 2::-In- P2::21n- P 2 2 R2 R ke"th<Jpvoi(4.4)va(4.16),suyra P >21n ~ 1 =2a>2a( d - C ) K 4P M* (O,g)cK 4 V~y,tlihaitru'dngh<Jptrentaluonluonco 29 ~~ ;----- " I ,' I " " I , ' I ,,/ 00 ' C' \! ~ 11I ,I ,, ,\ , - , '- B , Bo ", I , ", ,,/' ' Kabg z =gM I I ' I K9)I 1 /~ /kmnm lZ=ln' i 1£,, E ,/ ~O~UI - :E , 0,,, £1 £0 0 Co 0 t1 s o~ 1 1 Hinh 4.1 30 AI {: p 0 ------------------------------- 1£-i- PIn R2 -i TC P A, r z, p -2]; Oro t- O 1 b+InR P 2b 1 p ( d ~c )K vdi a =mill fIn '£ . 1 K '; }/3>2a 4 l 4PM (0,g) c K Jr - Ntu (0<)M*(O,g)cK~4-Pe-Pthla=ff ,khidotaco(4.9.a). p ~ -~ -~ 1 ff -Ntu 4 PeP <M*(O,g)cK <4 P ~ln K <- 4PM* (0,g) cK P thla =In K 1 lientaco(4.9.b). 4PM* (0,g) cK Chu y4.1:Theochungminhtatha'yntu la'yc=co'd =dothltakhongdn thay /3, bdi /32« /31)dodotrongdanhgiad dinhly 4.2coth~thay(d ~cr bdi ( J K d -c ,:? "., ~ 4-K 0 0 de du<;lccacdanhgla sacbon. 1-d c 0 0 ? , Bode4.1(Kiihnau) Trongm(itphdngzchomQtmiinnhtlienA gicJih(lnbiJi dui1ngtrim Izi=1va nhatcdtL(t) ={ziO~Izi ~t(<l),argz=O}.GQiFl LaleJptatcacachamw=fez) co tinhchat:m6ihamf E Fl bienbaagiacdelndi~pmiinA LenmQtmiin Bf co bienngoaiC(f) va bientrongc(f) saGchoL(t) thanhc(f) vcJiD(f) =1, Vf EFJ trangdoD(f) LaduiJngkinhcuac(f). GQiS(f) Ladi~ntichtrangcua miindobienngoaiC (f) baabQc.Khi do,mQthamIv EF1thoamiin: S(f)~ S(fJ= So' Vf EFl' , ( ) 1n(1-tz) , -ff cod(lng fa z = ( 2)vaSo= ( 2 )1n1-t In 1-t 31 1 ~ C(/) w ~~/OC)l, c(7J 10 Hinh 4.2 Chungminh:(xem[12,tr.288]). Chu y4.1:Voi nhG'nggiiithie'tcuab6de4.1nhu'ngD(/) (>0)bfftkythinho phepcodan,taco 8(/) D'(J) ?S. hayD'(J),; S(J)80 voi 80=80(1)xacdinhnhu'trongb6de4.1. DfnhIy 4.3(c~ntrencuaP(g) Viti cackYhi~unhl1trangm1:lc2.3,diJ,t: d - ( .!.. ~ J fo = ~ 1 ' do=T p,4PM (0,g)K c,O , 4PM' (0,g)K 1; ~ [(d.)~-~][l +c~(d.)~] (di+£1)[1- cf (do)fJ ,i' =R(I,~,O),T =T(I,i'~,O} UA' ~-ln(l-t) 1lVeu <l' 0 d ? ? - 1 2JPR(P,d,ot - vac> ubede 4PM' (O,g)Kc <1thi: p<2arcsin)-10(1-/) - 2JPR(p,d,Ot =PI vitiP ~2, (4.17.a) 32 ,8<4arcsin )-In(1-"f) - 2-J2pR(p,d,O)~=,82voi P =I. (4.17.b) Chungminh: - GQi WI=If/(W) yoi If/{0)=° la PBHBGmienBIen mienB giOih~nbdi duilng trim Iwd=I va p nMt dt l j ={w, 0 <Co,; Iw,l,; do'argw, ~ 2;j}sao chobienC thanhIwll=1 yacae (J"jtu'dngungyoi cae f!/j =0,1,...,p - 1).D~ thffyding: - Hams=InWI th\ic hien PBHBG Drien nhi li~n Ii, = Ii n {-;<arg w,<;} len mienB2(hinh4.3) yoi So= In do'S)= In Co' - Hamv =-i ~S thgchi~nPBHBG mienB21enmienB3(hinh4.3)yoi: .p .P I d .p ,P Iv =-1 - S =-1- n v =-1 - S =-1- nc . 0 20 20') 21 20 - Hamu =sinv thgchi~nPBHBG mienB3lenmienB4(hinh4.3)yoi: Uo=sinVo=sin(-i ~Indo), UI=sinVI=sin( -i ~InCo). - Ham r =k(u)=u-uo thgchi~nPBHBG mienB41enmienB5(hinh4.3)yoi u+uo k(uo)=0cobienngoailadu'ongtrOllIt I =1yabientrongla nhatdit y={tl°::S;lrl::s;tl<1,argr=O} ,. Ul -uVOl t - 01- ul +uo - sin (-i fm c} sin(-i fIn d.) - ish(fIn t) - ish(fIn ;j-J - sin(-; ~Inc} sin(-; ~Ind.) - irh(~In:} irh(~In~J 33 - Sh(flnt)-Sh(fln*)- (t)f -(trf _[(*)f-(*fJ - s{~ill c~)+Sh(~In:J - (:j - (:.t+uj- u. r~ 1 E 1 E --C 2 --+d 2 P P P P ( P P )E 0 E 0 do2 -Co2 +co2do2 do2 -Co2= Co2 do2 = 1 pIp p p p P ( P P )E -Co2 +E-do2 do2 +Co2-co2do2 do2 +Co2 Co2 do2 (dof - Cof)(1+Cofdof) ,=>tl= ( E. E. )( E. E. ) tucO<tl<l. d2+C21-C2d20 0 0 0 - GQi r;; =r;;(t)lamQtPBHBG mi€n Bstenmi€n B6(hlnh4.3)saocho{tlltl=I} (4.18) thanh {r;;IIr;;1=I} vanhatdt r trdthanh{r;;IIr;;1=r <I}. I -Ham';=h(r;;)=r;;K lamQtPBHKABGmi€nB61enmi€nB7(hlnh4.3). - GQi J1=~(,;)lamQtPBHBG mi€n B71enmi€n Bs(hlnh4.3)saochokll,;1=I} tMob tullpl=I}vii {4' 14'1=r~} thiinb y'~ tulo ,; Ipl ,; t,argp =O}.Then dinb ngliiacuahai hamphoT(p,r,s)va R(p,t,s)thl t=r(l,p,O) =r( 1.rk,0) vdi r =R(I,tl'O). - GQicPlaPBHBG mi€n Bslenmi€n AI(c A) vdi Al =g-l [1fI-1 (HI)]vagQis* la di~ntfchcuami€n Al dotfnhdolxungquayp IgnnenS*::;7r. P GQiD Ia du'ongklnhcuanhatdt L j (j =0,1,...,p -I). BeHmquah~gifi'aD va 34 Bz B4 1r 2 S( B3 Uo I I I I : WI I : f!j 0'-:Co I I I :13B I I I I I do ,I So v VI Vo 0 1r 2 u / / ~I// )if / )( / / 0 to r t tl WI =ljI'(w) ~ p .V =--S.I 2 }"'ill' \T-~ J U+Uo 1 ~ bg Hinh4.3 35 I , I " I , JI! : W \. BU/~O./ ' ,I ,: B( ", ", "---------- z =g(w) ,I , , A ( -.. ) z I I 0: f {O I '-,AI, I, 010 rp }=In".1r s}:-1- p f.l r' t 1 fjJ bg BOp 1 B6 h K abg , Or 1 f3 tac§nxethaitntongh<jp: a)Vdi p ;:::2 thl f3<2tr<tr dod6 D chfnhlakhoangeachhaifiut cuaL.. Khi p J d6 D =Rsinf3vanhochuy4.1,suyra:2 2 D . # ...r; f3 =2arcsin- ~2arCSIn .JS: ~2arCSIn cc.s2R 2R o 2R" plJo tr vdi So= (1 2)-In -{ dod6 (4.19) ~-In(l- (2) . ~-In(l- (2) f3~2arcsin .JP <2arCSIn C ( d O) K 2R 2vpRp, , vdi I ~ Tl1,r ~,0). r =R(I,I, ,0) trongdo I, dtt<!cxac djnh ohtt trong (4.18). M~tkhac,theocongthuc(2.16)trong[18,tr.1O46]va(3.7),tac6 m'~1f) :>co:>c:>d:>d.:>T(P,Rk,oJ keth<jpvdi(4.4),(3.16),(3.26)va(4.3),suyra: do., T(P,4J; M" (O,g)~c,O)=do<4~M" (O,g)L, m'(0,f) R ( d 0) m'(0,f) d - d - Co;::: I p" > 2 - 2 -fo- - - 1 4P 4P 4P M* (O,g)K do d6 , =(d,~-cJ)( I+c,k~) <[(d,)f-£!][I +c~(d,)~]=TI ( p P )( P P ) ( 1'- !'. )[ 1'- - 1'- ] 1 dol +Col 1-coldol d2 +f~ I-C2 (do)2 suyra: 36 r =R(1,1,,0) 1=T(1,r~,O)<T(V~,o)=T- V~ytaco(4.17.a). b)Vdi P =1coth~xayra 1r<P <21rDendu'ongkfnhD cuanhatcfitLj coth~ kh6ngphaila khoangcachhaimutcuaLj' £)~khficph1;1Cdi~udo tadungham ph1;1Z = If/ (Z) = ..rz (chQnnhanh.Jf=1)tht!chi~nPBHBG mi~nA leDmi~nA (hlnh4.4). Khi do: R=If/(R)=.JR.,L =If/(Lj)' jJ =~ vahamZ=If/og(w) tht!chi~nPBHKABG mi~nB leDmi~nA. z =g(w) ~~~ ~ ",~~~ ---'-, C , ,, , I ' I ' I ' I ' J j , I \ I ', . I '. 0 :1 , I, ', ' , I, I, ' "-- """~~~- Z=If/ (z) ~ ~~~====~-==:::10 R 1 ,~/o""""" -" Z"""II Z=If/og(w) Hlnh4.4 GQi jj Iadu'ongkfnhcuanhatdt L, tu'dngtt!nhu'trentaco: p=2arcsin ~,; 2arcsin -1s. 2R 2R pSo Den Fr P :::;; 4arcsin2-J2pRSo 37 vdi Soduqcxacdinhbdi(4.19). Dodo: ~-ln(l-t2) . ~-ln(l-t2) . <4arCSIn !5.- p::;;4arCSIn 2.J2pR 2.J2pR(p,d,0)2 Chungminhtu'dngtvnhutrongph~na)taco(4.17.b). ChtiY 4.2:Trong(4.17.a)va(4.17.b),ne'umi€nBcod~ngB(hlnh4.3),c6dinh C=Cova cho d =dod~nde'nco'khi do l1.~ 0,712 ~ 0 vdimQiK nentaco p<l1. <21rvap<712 <21rtucdanhgia(4.17.a)va (4.17.b)khongphiiihi€n p p nhien.