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

12 CHUaNG 2 CAC CONG CT) Trangchuangnay,chungtoi lieUmQts6dinhIy, b6 d~va cach~quadn thi€t chovi~cdanhgiacacd<,tiluQnghlnhhQcd6ivdicaclOphamF va G. 2.1nilt diingthucCarleman,caeh~quavamdr{)ng n6 d~2.1:(niltdiingthucCarleman) Gia stt w=fez) la mQt PBHBG don di~p hlnh vanh khan A={zl(O<)r<lzl<R«oo)}leu mQtmi~nnhi lien D khongchuadi€m 00 vdi bien trang C1va bien ngoai C2 sancho Izl =R tuonglingvdi C2.GQi S la di~n tich(trong)cuat~pmddo C2banbQc, s la di~ntich(ngoai)cuat~pdongdo C1 banbQc.Khi do,taco: S2(~JS. (2.1) D~ngthucxayfakhivachikhif(z) =az+bvdi a,b la hlings6va a:;t:O. w=f(z) ~ A Or R wz Hlnh2.1 Chungminh: Xem [4,tr. 212]. 13 H~qua2.1:(Dinhnghiamodunmi~nnhi lien) Gia sami~nnhi lien D quacacPBHBG / va 1;l~nhiQtbie'nleuhaihlnh vanhkhanH={wlr<lwl<R}va H]={w]h<IW11<R1}thl R =RI r 1J (2.2) Ti' s6 nay duQcgQila m6duncua mi~nnhi lien D va duQcky hi~ula mod(D). Chungminh: / ~ 1; ~ D HI O~ z WI R] R 0 ffinh2.2 XetPBHBG j:j;-I mi~nHI leumi~nH, rheab6d~2.1,taco ffR' ~[ ~'J ffr' hay R RI-2- r 1J (2.2a) TucJngrtf,taxetPBHBG 1;0/-]mi~nH tenmi~nH], rheab6d~2.1,taco [ ) 2 2 R 2 1rRl 2 -; 1r1J hay R] R-2-. 1J r (2.2b) Tli (2.2a)va(2.2b),suyfa(2.2). 14 H~qua2.2:(Tinh ba'tbie'ncuamodunmi~nnhflien) Ne'umi~nnhi lien A eo caethanhphffnbienkh6ngthociiboathanhmQt di6mdu<;1ebi 'nbaagiaedondi~plenmi~nnhilien B thl mod(A)=mod(B). (2.3) Chungminh: g ~ f~ h~ ~ HI O~ Rj 0 w Rz Hinh2.3 GQi f la PBHBG dondi~pmi~nA lenmi~nB . X6t g la PBHBG dondi~p mi~n A !en hlnh vanhkhan HI ={sh<Isl<RI}va h la PBHBG dondi~pmi~n Bien hlnhvanhkhan Hz ={tlrz<ItI<Rz}. Theah~qua2.1,taeo: mod(A)=RI va mod(B)=Rz. ~ rz D~tcp=hf thl cpla PBHBG dondi~pmi~nA len hlnhvanhkhanHz. Theoh~qua2.1,taeo: mod(A)= Rz va Rj - Rz r --z lj rz V~ytaeo mod(A)=mod(B)We(2.3). 15 H~qua2.3:(Tinh dondi~ucuamodunmi~nnhi lien) Trangm~tph&ngz chahaimi~nnhi lien A va B vdi moduntu'dngung la R va R], coHnhcha'tAc B va A ngancachhaithanhphilobiencuaB. r lj Khi do,taco: R Rl-~-. r 1j (2.4) D&ngthucxayrakhivachikhi A=B. Chungminh: w=f(z) ~ R Hinh2.4 VI mod(B)=RI Dent6nt!;liPBHBG ddndi~pf mi~nBIen hlnhvanh lj khan 11={wi'i<!wi<R1}.Khi do, quaphep bie-nhlnh f mi~n A trd thanhmi~n nhi lien A vdi mod(A)=R cobientrongla C] va bienngaaila Cz saDcha C]r baaquanhha~ctrungvdi !wi=1j vaIwl =Rl baaquanhha~ctrungvdi Cz. GQi S la di~nrich(trong)cuat~pmdda Cz baabQc,s la di~nrich(ngaai)cuat~p dongda C1baabQc. 16 Khi do,taco: s~m/ (2.4a) va S 5,1rR)2. (2.4b) Vi mod(A)=R nent6nt(;liPBHBG dondi~pg mi~nA leuhlnhvanhr khan D ={tIr 5, It15, R}. Ap dl,mgb6d~2.1choPBHBG w=g-) (t), taco: s~(:Js, (2.4c) trongdod~ngthucxayrakhivachIkhi g-) (t)=at+b vOia,b la hangs6, a;t:o. Tuc A lahlnhvanhkhan. Ti'icack€t quatren,taco: (~J <~(:J. Ti'ido suyra (2.4). f)~ngthuc(j (2.4)xay ra khi va chI khi cac d~ngthuc(j (2.4a),(2.4b)va (2.4c) cungxayra,tucA=B hayA=B- B6 d~2.2: (Md r{)ngbilt diingthucCarlemanbdiThao[12,tr. 521]) Gia su w=fez) la mQt PBHKABG hlnh vanh khan A ={zl(0<)r <Izi<R(<oo)}leu mQtmi~nnhi lien D khong chua di€m 00 vdi bien trongc) va bienngoai C2saocho Izi =R tu'onglingvdi C2.GQiS la di~n tich (trong)cuami~ndo C2baabQc, s la di~ntich (ngoai)cua t~pdongdo C) baabQc. Khi do, taco: 2 s~(~)K s. (2.5) 17 D~ngthucxayrakhivachIkhi fez)=alzr~.-I+b vdi a,b la h~ngs6va a~o. Chungminh: Xem [12,tr.521],[17,tr. 13-14]. 2.2Md r{)ngcaebittdiingthucGrotzschvaKiihnau B6 d~2.3:(BittdiingthucGrotzsch1) Giasaw=fez) la PBHBG ddndi~phlnhvanhkhanH ={zl(O~)r<lzl<1} leu mi~nnhi lien D vdi bienngoai Iwl=1vabientrongc, saochomi~ngidih~n boi c luauchuahlnhtroll Iwl<s,(s<r).Ki hi~uM={MaxlwllwEC}va gia sa , ? 2bi rangtren C cop diem Wk =MeP ,( k=O,1,...,p-1). Khi do,taco: MsT(p,r,s), (2.6) trongdod~ngthucxayrakhi va chikhi f =fo lamQtPBHBG ddndi~phlnh vanh khan H={zl(O<)r<lzl<1} leu mi~n nhi lien D ={wis <Iwl < 1}, (0 s s <r <1)bi c~t boi p do~n 2 thang Lj ={w s:>!wI:>/,argw~2;j },(o:>s </ <1),j ~1,2,.--,p. Chungminh:Xem[6,tr.372]hay[19,tr.18-20]. B6 d~2.4:(Md r{)ngbittdiingthucGrotzsch1bdiThao[13,tr. 63]) Gia sa A la hlnhvanhkhanR<lzl<1vdi pn,(p=1,2,...;n=O,1,2,...)nhat - hi c~tcungtrolld6ngtam0 saochoA trungvdi chinhnoboiphepquayz=zeP . GQi f la PBHKABG mi~nA leumi~nB n~mtrong0<Iwl<1 saochodliong troll Iz1=R tlidngling vdi bien trongC giOih~nmQtt~pdongchuag6ctQadQ, du'ongtroll Iz1=1tu'dngling bien ngoai C cua B . Hdn nii'agia thi€t B trling 2~i vdi chinhno boi phepquay;:;:'=weP . 18 Khi do,taco: M~T(p,R*,m), (2.7) vdi M =max{lwl,WEc}, m=min{lwl,WEc}, 0~m~M <1. Ding thucxay ra khi va ChIkhi w=f(z)=ah(u),lal=l,u=bzlzlt-I,lbl=l, h la PBHBG don di<$phlnhvanhkhan R*<lul<1 ten miennhi lie~saocho lul=ltu'onglingvdi bienngoaiC={wllw\=I}, conlul =R* tu'ongling vdi bien trong c ~{~I+m}u{~m,;H,;M,argw~2;j,j ~1,...,+ Chungminh:Xem[13,tr.63]hay[19,tr.33- 35]. Nhophepbiend6i z =Q va W=ml , b6de2.4trdthanhz w H~qua2.4: GiasaA lahlnhvanhkhanQ<lzl<Rbic~tbdipn,(p=1,2,...;n=0,1,2,...) 2/T' cungtrolld6ngtam0 saochoA trungvdichinhnobdiphepquay; =zeP . Gia saf la PBHKABG mienA tenmienB namtrong0<Iwl<00 sao cho Iz I =Q tu'ongling vdi bien trong C] baag6ctQadQ,du'ongtroll Iz1=R tu'ongling bien ngoai C2 cua B. Hon nua gia thiet B trungvdi chinhno bdi phepquay 2/T' P w=we . Khi do,taco: m2 2:: [ ( ;1 ) *,!!!L J ' T p, R M2 (2.8) vdi M2 =max{lwl,WE C2}, mj =min{lwl,WE Cj},j =1,2. 19 B~ngthucxayfa khi va chikhi w=f(z) =ah(u),Ial=I,u=bzlzlt-),Ibl=1, hla PBHBG hlnhvanhkhan Qt <lul<Rt len mi~nnhi lien E saDcho lul=Qttu'ong ling voi c] ={wllwl=m)} va lul=Rt tu'ong ling voi c,~ {wI1wi ~ M,}u{wllwl ~ m, ,; w,; M"argw ~ 2;j ,j ~1,...,P}. Chungminh:Xem[13,tr.64]hay[19,IT.35- 36]. Be}d~2.5:(Ba'tdiingthucGrotzsch2) Giasa w=fez) la PBHBGdondi~phlnhvanhkhanA={zl(O<)r <Izi<I} lenmi~nnhilienB n~mtronghlnhtroll Iwl =1,c6bienngoaiC2Ia du'ongtroll Iwl =1 vabientrongc) saDchoIzl =1tu'ongling voi C2. Khi d6, du'ongkinhD cuac) thoa D::; Do =2T(2,r,O), (2.9) trongd6 D=Dokhivachikhi c) la do~nth~ngnh~nw=0 lamtrungdi~m. Chungminh:Xem [8,tr. 220]. Be}d~2.6:(Ba'tdiing thuc Grotzsch2 md rQng) Gia sa w=f(z) la PBHKABG hlnhvanhkhan A={zl(O<)r<lzl<R}len mi~nnhi lien B c6 bienngoai C2va bientrongc) saDcho Izl =R tu'onglingvoi C2. B~t M =max{IwllWE C2}.Khi d6, du'ongkinhD cuac) thoa D';Do~2MTH~rol (2.10) trongd6 D=Dokhiva chikhi w=fo(;) voi;=azlzl-t-],Ial=1 vafo laPBHBG dondi~phlnhvanhkhan A=FIr-t<1;1<R-t} len hlnh troll Iwl<M bi ciit dQc do~nth~ngnh~nw=0 lamtrungdi~msaDcho 1;1=R-t tu'ongling voi Iwl=M . 20 Chungminh: .:. Tru'onghQp1: K =1, C2 trung voi du'ongtron Iwl=M w=j(z) ~ R B (] Bo M 0 M w w ------ w=fo(z) ~ Hinh2.5 Chi dn thljchi~ncacphepco dan ; =~ va; =; , d~dangdu'atru'ong h<,Jpnayve tru'onghQpcuab6 de2.5voi mien A thaybdi A={;I~ <1;1<I} va B thaybdi l3 nQitie'ptronghihtron1;1<1.Trd vecacbie'nz va w tathudu'Qc (2.10)vdi K =1clingke'tlu?nchoD =Do' .:. Tru'onghQp2 : K =1,C2Ia bienngoaiba'tkl cuaB GQil3lamiennhilienchuaB cobienngoaila !wi =M, bientrongHi C]. Do tinhdondi~ucuam6dunmiennhilien (xemh~qua2.3), taco: mod(B)~mod(B). Theoh~qua2.2, taco: mod(B)=R.r M~tkhac,giasa l3comodun mod(l3)=~ .r 21 V~y r r-<- l?- R' Theatinhchfftdondi~u(1.17)cuahamph\)T(p,r,s), taco: T[2,~,O)~T(2,~,O} ~ (2.lOa) A ,,' B ,/ O. "MI .. I' I I I : C Cz,'I I \ 1 I \ ,\ ,\ ,' ,' -' -' -'-------- R Or CIG Hinh 2.6 Ap d\)ngtfu'onghQp1,taco: DQMT(2, ~,oJ Ke'thQpvoi (2.lOa), suyfa D:; Do=2MT(2,~,0). Tuc (2.10)vOiK =1. w M B 22 .:. Tru'ongh<;lp3 : K ~1, C2 la bienngoaiba'tld cua B ~ A R ~, " B '-' \,/ G \MI II I ~ r I : C c2,'\ , '- J " \ , """' ~~---- Or u~g(w)I " '\ l»I O~ u Hinh2.7 Mi~nnhi lien B co th~bie"nbaagiacddndi~pbdi u=g(w)lenhlnhvanh khan BI={ulo<fJ <lul<Rj} saocho C2tu'dngungvoi lul=RI' Ap dl;mgtru'ongh<;lp2 cho PBHBO .w=g-I (u) hlnhvanhkhanB( leumi~n B , taco: DSlMT( 2,~,0). (2.lOb) M~tkhac,hlnhvanhkhanBj coth~xemla anhcuahlnhvanhkhanA qua phepbie"nhlnhh<;lpcuaPBHKABO f voi PBHBO g, tucquaPBHKABO gof. Do do,theo(1.2),taco: ~ ( r ) * -< - RJ - R . Hdnnii'a,theoHnhcha'tddndi~u(1.17)cuahamphl,lT(p,r,s), taco: 24 B6 d~2.8: (Ba'tdiingthucKiihnaumdrQng) Trangm~t ph&ngz chomQthlnhv~lllhkhanA={zl(0i w=f( z) Ia PBHKABG bie'nmi€n A Ien mi€n nhi lien B co bienngoaiC va bientrangcsaocho Izl =R tlidnglingvdi C. GiS Ia di~nrich(trong)cuami€n dobienngoaiC baobcva D la Quangkinhcuabientrongc. Khi do,taco: SIn(1- (2 ) , . ( ( r ) t )D s;1/ -i( , VOl (=T 1, R ,0. (2.13) D&ng thuc xay ra khi va chi khi f(z)=fo(~)=bln(1-(~)+c,lbl=1vdi In(l- ( ) 11 -1.. ~=a~zK-' RR ,lal=1. Chungminh: w=f(z) ----------. z R deAOr w s=g(w) Q'i 1 B, 1 s ~ Hlnh2.9 25 D§u lien,bi€n baagiacddndi~pmi€n B boi s=g(w) leuhlnhvanhkhan BJ ={sl0 < fj < Isl <I} . Sando,th\lchi~nPBHBG u=h(s) hlnhvanhkhanBJ leumi€n nhilien B2 gioi h~nboi du'ongtroll lul=1va nh£itc~tL(t)={uIO<lul~t,argu=O},<t<l, saDcho Isl=1 tu'dngung voi /u/=1. Theo dinhnghlahamph1,1ta co fj =R"(I,t,O) hay t =T(l,fj,O). Ap d1,1ngb6 d€ 2.7choph6pbi€n hlnhhQpg-Joh-Jmi€n B2leumi€n B, taco: Sln(1-t2) ". D ~~I , VOl t=T(1,fj,0).-1( (2.13a) M~t khac, BJ co th6xemla anhcua A quaPBHKABG f.g la hQpcua PBHKABG f vaPBHBG g. Dodo,taco: ~~(~r. Theo (1.17)v€ tinhddndi~ucuahamph1,1T(p,r,s), ta suyra T(l,~,O)';TH~r ,0). (2.13b) K€t hQp(2.13a)va(2.13b)taco(2.13)voiphatbi6uv€ tru'onghQpd£ngthuc8 2.3Ba'tdiingthuctheoIy thuye'tde)daictfctri Ly thuy€tdQdai c\lctrib~tngu6ntumQts6cacba"td£ngthuclien h~ giii'amoduncuamQttugiachaymi€n nhilien,di~ntichmi€n dovadQdaing~n nha"tcuadu'ongcongthuQcmQthQdu'ongtraitrongmi€n dotinhtheomQtdQdo ba"tkydu'QcAhlforsvaBeurling[l]d€ xu'ongnam1950dfftrothanhcongC1;1huu hi~ud6giainhi€u bailoant6iu'utrongIy thuy€thlnhhQchambi€n phuc. 26 Trangm~tph~ngz=x+iy,chotugiaccongQ cocacdinhlfinIu'Qtla A, B, C va D. QuaPBHBG dondit%pw=f(z)=u+iv, Q du'Qcbi€n tenhlnhchunh~t Q'={w=u+ivIO~u~a,O~v~b}co dinhtu'dngling lfin Iu'QtIa A', B', C', D' sao choA'B'=a;B'C'=b. GQir Ia hQcacdu'ongcongr n6ihaicanhd6idit%nAB vaCD cuatugiac cong Q, do p=p(Z)~O,ZEQ saochodit%ntichcua tu giac - congQ theode>dop lahuuh~n,nghiaIa Sp(Q)=Hp2(z}iS<+oo. Q (2.14) De>dai cuacacdu'ongcong r theode>dop du'Qctinhb~ngcongthuc lp(r)= Jp(z)ldzl(~+oo),rEr,pE<D. y (2.15) B6 d~2.9: Vdi cackyhit%unhu'tren,taco: Sp(Q)~al~vdi lp=inflp(r),b yer (2.16) d~ngthucxayrakhivachikhi p(z) =kif (z)l,zEQ,k=canst. Chungminh: Taco Sp(Q)=Hp2(z}iS=Hp2(z)dxdy Q Q 2 dudv atb p2 (Z) 1;/ =[fp (Z)jf'(zt =II !If'(z)12dvJu ~ b p2(Z) b J du 1 ~ b p2(Z) b }= J 2dvJdv ~=- J 2dvJdv u0 olf'(z)1 0 Jdv bo olf'(z)1 0 0 27 <: 7; A J I;'~~Idv J' du (Do apd\lngBDT tichphiinSchwarztchohaiham p(z) va 1trendOc;ln[O,b]) If'(z)1 1oJ J 2 =bdlJp(z)ldzl du (ruIa nghichanhcuadoc;lnth~ngu=canst,~ 0~u ~a,0~v~b ). 1 a =b JI~(ru)du~a120 b p' Ding !hac a(2.15)xaY fa khi va chi khi II I~gifta hai ham I;'~;)IvaI lahang sf),d6ngthai Ip(ru)=lpvoimQiru,(O~u~a)p(z)=kl/(z)l,k=const,zEQ, vi khid6 Ip(ru)=Jp(z)ldzl=kJI/(z)lldzl=kb=lpvoimQiuE[O,a]. Yu Yu B6 d~2.10: Trangm~tph~ngw chomQttu giaccongBoc6 haicc;lnhn~mtrenhai duangtroll Iwl=cvaIwl=d,O<c<d.B~t O<O(r)=Jldlpl~Oo(~2Jl"),trongd6 c, <p=argw,Cr=Bon{wllwl=r}vagiasa O(r)khatichtrendoc;ln[c,d]. Giasa z=g(w)la PBHKABG mi6nBo lenmi6n.40cuam~tph~ngz . Ta d~tCr =g(Cr) , 0<c ~r ~d <+00. Hon mIa, gia sa p =p(z)~0 duQc xac dinh trong .40 saD cho 1p (Cr) =fp(z)Idz I~ 00, c ~ r ~ d c, va Sp(.40)=Hp2(z)dxdy<oo,Z=x+iy t6n tc;li A theonghlaLebesgue.Ngoaira Ip(Cr) ~I~,c~r ~d. 28 Khi do, taco: 1 2d J dr Sp(Ao)~K(l~) crO(r)" (2.17) Chungminh: GQi dS la vi phancua Sp(Ao) tu'dngling voi [r,r +dr]c [c,d] , tuc dS xa'p " Xl dt theodQdo p(z) cuaanhmi€n D =Bon{wlr <Iwl<r +dr}bdi z=g(w).Do dr(>0) ra'tbe va O(r) khatichtren [c,d]coth€ thayD bdi 15=Bon{wlr<Iwl<r+dr,a<arg(w)<a+O(r)}voi a la argumencuamQtdlnhtu giaccong15n~mtrenIwl =r Hamt =Inw bie"nmi€n 15lenhlnhchii'nh~tvoicacq.nh I r +dr I ( dr ) dr 'n ( ) V ," d ? / ./ D - I 'n-= n 1+- ~- va,!,,!;r. lv(;J.ymouncuatuglaccong ar r r dr dr mod(15)=O(r) =r.O(r). Theo [ 3, tr. 19], taco: 1 dr 2 1 dr ( 0 ) 2 d/ ( ~ ) / . dS~ ( )Ip ~ ( ) Ip , trong 0 Ip =Ip Cx VOl r <x <r +drK rO r K rO r La'ytichphanhaive"tren[c,d]tadu'cjc(2.17). 2.4Caeb6d~khae B6 d~2.11:(Bie'nhaidu'ongtroDl~chtamthanhhaidu'ongtroDd6ngtam) Ne"uA lami€n nhiliengioih(;J.nbdihaidu'ongtroll Izi=1va Iz- hi=lj voi 0<h<1, 0<lj <(1-h) du'cjcbie"nbaa giacddndi~plen hlnhvanhkhan r <Iwl<1 thl r =r(r),h)= 1- h2+r)2-~(1- h2- r/ Y -4h2r)2 2r .) (2.18) 29 Truong hQp A la mien nhi lien gioi h~n bdi Izl=r2 va Iz- hi=r) voi 0<h<r2,0<rl <r2-h thi r =r(rl'r2,h)=r22-h2 +r/ -~(r22-h2 -r/Y -4h2r12 2r)r2 (2.19) Chungminh:Xem [18,tr.20-22]. B6 d~2.12:("D~oham" cuahamngtiqcchoPBHKABG) Voi caeki hi~ud phfin1.2,giiisa W =f (z)laPBHKABG cuamienchua z=Ovoi f(O)=Ova m'(O,f»O. f)~tg=I-I , taco: I m'(0,f) =M*(o,gfX, I M'(O,f) =m*(o,gfX. (2.20) (2.21) Chungminh: Lfty R>O du be , d~t CR={zllzl=R}va C~=/(CR), r6 rang t6n t~i WIE C~va z) E CRsaD cho m(R,f)=lw)I=lf(z))I=r, r>O. f)~tLr ={wllwl=r}va Lr=g(Lr) Vi Lrn~mtrongIzl~R,taco M(r,g)=lg(w))I=lzII=R. Dodo I ' (0 1) =1" m(R,f) =1" r =1 . [ M(r,g) ] -X =M*(O ) -t m, 1m I 1m ) 1m K ,g.r->O - r->O ( ) - r->O r RK M r,g K Tudngtlf, lfty R>O du be , d~tCR={zllzl=R}vaC~=f(CR), r6 rangt6n t~i W2E C~va Z2E CR saD cho M(R,f)=lw21=lf(z2)I=r, r>O. 30 B~t L, ={wllwl=r}va I, =g (L,). VI Izl=R nflm trong t~pdong gioi h(;lnbdi I" ta co: m(r,g) =lg(wz)1=lz21=R. I Dodo M'(O,f)=limM(R;f)=lim r -'- =lim [ m(r;g) ] -K =m*(O,gft. ,~O RK ,~Om(r,g)K ,~O r H~qua2.5: Cho K =1, ta co m'(O,f)=If'(o)1 va M*(O,g)=lg'(O)I. Luc do m'(O,f)=M* (O,gft trdthanhcongthilcquellthuQcIf'(O)1=lg'(OfI. (2.22) 2.5Cae daub gia eholop ham F BS xay d1!ngcac danhgia cho lOpham G ta c~ncac danhgia duoi day cholOpham F , tilc lOphamnguQccualOpham G . DfnhIy 2.1: Duoi cac ky hi~uva giii thie"td ph~n1.2,voi mQi f EF, zEA,z *-0,z *-00 , 0<R <00 , taco: S'(O,f)~l, PSI ~(l-S'(O,f))1Z"Rt, 2 2 S'(O,f)1Z"RK~S(R,f) ~1Z"RK, (2.23) (2.24) 1 m(R,f)~RK , M(R,f) ~Rt~S'(O,f), (2.25) (2.26) (2.27) m(R,f) ~4-;m'(O,f)Rt , (2.28) 1 -'- M(R,f)~4P RK, (2.29) .l -'- D(R,f) ~2APRK, (2.30) 31 4-im'(O,J)lzlt~IJ(z)I~4ilzlt, (2.31) 4-im'(O,J)Rt~c(R,J)~d(R,J)~4i Rt .(2.32) M6i d£ngthuctu(2.23)d€n (2.21)xayfakhivachikhi J(z) =azlzlt-lvoi lal=1. Chungminh:Xem[19,tr.54- 56].