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

26 CHU'ONG3 cAc DANH GIA CHO LOP HAM G Trongchuongnay,chungWi danhgiacacd(;liluqngd~ctrungchomi~n chufinda:neuclingnhumoduncuacachamthw?clOpG. 3.1Danhgia m*(oo,g),bankinh R](g)va Ig(w~ D!nhly3.1 V6i cacgillthietvakyhi~utrongm\lc2.2,v6imig E G, WEB (w-:f.00),ta co: K (05)m*(00,g)5(:)2(51) (3.1) 4K ( ) K m*(00,g)<2P ; (3.2) ~ K 4P m*(oo,g)dK5R](g)5R(p,c-],OtK <4PcK (3.3) K R](g)> r PSI -2 1 2vo-is]>0 mn*(oo,g)K-s (3.4) ~ K 4P m*(oo,g~wIK<lg(w~<R(p,lwl-],orK<4plwlK D&ngthucxayraa(3.1)khivachikhi B=Bovag(w)=awIwlK-]vo-ilal=1. (3.5) ChUngminh: Theocongthuc(2.18)tacov6imif =g-I,g E G, tucVf E F, S'(oo,f)~~+ PSI >~2 - . Ji - Ji JiRK] M~tkhactir(1.11)taco: M'(00,f)2~S'(oo,f),Vf E F, nen M'(oo,fY~~. Ji (3.6) Kethqpv6i(2.28),taco 27 -2 S m*(00,f)K ~- , Ji tuc CO(3.1). Dogillthi~tv6bientrongC cua B, dangthuca (3.6)xayrakhivachikhi I w=f(Z)=g-I(Z)=bzlzIK-1 v6i Ibl=l, tuc la B=f(A)=Bo, dododangthuca (3.1) xayrakhi vachikhi B =Bo va z =g(w)=f-I (w)=awlwlK-1, lal=1. Theocongthuc(2.27)va(2.28)taco: Vf =g-I,g E G, 4 4 d - - -I - <2P M'(oo,f)= 2P m*(oo,g)K c tudoco (3.2). Tu (2.26),theodinhnghlacuahaihamphl;1T(p,r,s) va R(p,t,s) vatfnhdan di~u(1.14),(1.17)cuachung,taco: ( -1 J -I c-I s,T p,RIK,O =tl ~ R{ =R(p,tpO)~R(p,C-I,O). Dodotacodanhgiach~trenchoRI(g)trong(3.3). Tu(2.26)va(2.28)taco: I I -I I-K- -1- - I - - d 4 P m*(00,g)K d 4 P m*(oo,g)dK <RI nentaco danhgiachiftndu6ichoRI(g)trong(3.3). Do(1.11)va(2.28)congthuc(2.18)coth~bi~udi~nduaid~mgsau: * ( )-2 S ps I / .m 00, g K >- + z VOl Sl > 0,Ji - JiRIK tudosuyra(3.4). ! I ! ~ I Tu(2.25)taco:If(z~<4PM'(oo,f~zIK nen!wi<4P m*(oo,g)Klg(w~K -K =>Ig(w~> 4p m*(oo,g~wIK, tuc co danhgia ch~fmduai cho Ig(w~trong(3.5). 28 Thea(2.25)taco: If(zfl $;T(p,lzl~,O)=t2~ Izl~ =R(P,t2'0)~R~,lf(zfl,o) nenIg(w~$;R(p,lwl-I,OtK , tilecodanhgiachifmtrencha Ig(w~trang(3.5). D~lamsitebondanhgia(3.1)tasechUngminh: H~qua3.1:D~t -4 D=psIR(p,c-l,or ~2P P~J20c (3.7) Ta co Vg EG K K m*(00,g)$;(S:D)2 $;(:)2($;1) (3.8) Dltngthilcxayrakhi vachikhi B =Bo va g(w)=awlwlK-1v6'ilal=1. Chicngminh: Thea(1.11)va(2.18),Vf =g-I,g E G, tile Vf E F, taco M'(oo,fY ~S'(oo,f)~~+ PSI1i 2 1iRKI Tu dothea(2.28)va(3.3)taco: m'(oo,g)~~~+ ps~~~+ PSI -2 =~+PSIR(p,~~,or 1i - 1i 1iR(p C-] 0) 1i 1i1iRK , ,I =>m*(oo,g)f $; ( s +psIR(p,C-1,or ] -1= ( s+D ) -I =~, 1i 1i s+D tudotaco(3.8). H~qua3.2: TrangtruanghQ'pK =1, v6'i m*(00,g) =limm(r,g) = lim Ig(zl~ = I g' (00~ r-4oo r Z,-400 IzII ~ thl (3.8)trathanh: Ig'(00~$;~ 1is+D (3.9) 29 v6iD chobai (3.7),Vg E G, ding thucxayrakhi vachikhi B =Bova g(w)=aw v6i lal=1. Vi s ~TC,D ~0, batding thucnaysitebanbatding thucc6di~nIg'(00~ ~1v6i g E G vaK =1(xem[91tr.217). 3.2Daubgiagocma2fJ(g) D~ti~nchovi~ctrlnhbaydinhly taseIanluqtduavaocaekyhi~umaiva chUngminhdinhly truac,saud6ph<itbi~udinhly sau. Buac1: Theo[6]t6n t<;tiham WI=~(w)v6i~(oo)=00 bienbaagiacclandi~pmienB lenmienE, ={w,hl;, I}, chffit p nM! ci\! /, ={WI0<Co,;;IwII,;;do,argw,=2:} saochobienC thanhIwll=1vacae6~tuangUngvaicaetAi =O..p-l). G .' { - Tc TC }<;nBI =BI n p <argWI<P . G9i B' =~-1 (B;). G9i A' =g(B') Tathvchi~nphepbienhinh;=If/(z)=1- bienmien A thanhAl namtrong Z 1;1< 1. Lj cua A tra thanh 1'j cua AI, Do arg; =-argz,zEA neng6cmacuacling L~bangg6cmacuacung Lj (bang213). G9i A; =If/(A'). G9i s =s(;}am<)tPBHBG mienA;I len mien A2 v6i s(O)=0 c6bienngoai laduangtroll Isl=1vabientrongIanh<iteM r =~O~Isl ~t',args=O}(hinhve). 30 G9i u=u(s)la mQtPBHBG mi~nA2lenmi~nAJ saocho {sllsl=I} thanh {ullul=I}vanMI oit r Ifdthiinh{uliul~;<I} (hinh ve), Ham rJ=InWI thvc hi~n PBHBG mi~n nh! lien B; len mi~n B,={I/R(I/) >0,-; <1(1/) <;}\ HI/o ,; R(I/ ),; 1/" 1(1/) ~ 0) (hlnh vo) v6i 1/0;=In Co' rJI =Indo. Ham s =iprJ 2 thuc hi~n PBHBG mi~n B2 len mi~n B, ={.91-2'"O)}\ {.9IR(.9)=0,.90,; I(.9)';.9,} (hJnh ve)vm: S - iprJo - iP I S - iPrJI - iP I d0 - - nco, 1- - n O'2 2 2 2 Ham (; =sinS=-ishiS thvchi~nPBHBG mi~nBJ len mi~nB4lall11am~t phltngtrencochuanhatciltthltng[(;0'(;] (hlnhve)vai: (;0=sinSo=sinC~Inco)=iSh(~InCo} (;1=sinSI=sinC~Indo)=iSh(~Indo). Ham t =K((;)=~:~:thvchi~nPBHBG mi~nB4 len mi~nBs v6'i ~-1 (;1- (;0- (;0t = - 1 (;1+(;0 ~+1 (;0 K(0)=0 cobienngoailaduOngtrollItI=1vabientronglanhMc:1t r =~os It I s tpargr =o} vai Sh(flndot - Sh(f Inc,) - Sh(~lndo)+1 Sh( ~ In Co) (O<tl<1) 31 D h' h b' K' h ( ) x-I -,' 2 0 -" -' ( ) - 0 amt vc tent vc tl X =- co tl = ( ) 2 > Vffix >1tuc tl x tangtrongx+l x+l khoang(1,+00)vahamthvcbienthvcshxtangtrongkhoangxacdinh. (do)f- (dof:- [(co)f- (cof:] (do)f- (co)f+(cof:- (dof:t - - 1 - p -p p -p - p p -p -p (dO)2-(do)T +(CO)2-(co)T (dO)2+(CO)2-(co)T -(do)T G9i Ji =Ji(t) Iam(>tPBHBGmi~nBslenmi~nB6saocho~lltl=I} thanh JiIIJiI =1 vanh<itctttr trbthanh~IIJiI=rl <I}. (3.10) Do dophepbienhlnhmi~nB6lenmi~nA3co th~xemlahqpcuam(>ts6 - 1 PBHBG vOiPBHKABG z =g(w)nenlam(>tPBHKABG. Theo(1.2)thl r ~r? - G9i S Iadi~ntichcuaA;,dotfnhbaaloantfnhd6ixUngquaydip p quaPBHBG lenmi~nchu~n(xem[171tr.l09)tacoS=1'( . P a) V6i p'?2: Ap dungb6d~2,7,v6i D' =lJI(D),taco f3 ,D' , RI rs , [ RI t~ln1- t,2 J=arCSIn =-1~arCSIn~ ~ arCSIn - - 2RI 1'( 2 P 1'(2 -In(1- tr2 ) R ~-ln(l-t'2) f3~arcsin 1 2# Theo(3.3)taco: RI ~R(p,C~I,OtK~R(p,C-I,OtK Tu batdlmgthuccuaThao [171tr.110d6dangsuyfa: -I 1 ( -I ) -1 1 1 4P R?Z<T p,Rt,O ~co~c~d~do ~4PM'(oo,f)Rf,Vf=g-l,gEG Theo(2,28)taco: 32 B A' ~B'W A 1 ;-) \ ; z0 R-I BI/ B' ( ))l0 I- I Z Co do WI 17=In W I I Azo---1-1 0 iT[- u=U(S))p170 171 -iT[ 17 B- 2 @) I) =ip17 P I I 1)1 2 B3 I 1)1 (;=sinI) -T[ I 1)0 IT[ 8- 2 - J1 1 (;1 2 pp) )B4 I (; t=K((;) (;0 G -1 0 1 1 I1mh3.1 t 33 I I I I do:::;4Pm'(oo,g)~R?(:::;4Pm'(oo,g)~R(p,C-1,at <4Pm'(oo,g)~c =do 1 Co;, T(P,R,.i',O r>4;R,* >4;(4-;m' (00,g}d Kr=4-: m'(oo,g)* d ~Co sh(flnd,) 1 sh(flna;t Sh(~Inco) Sh(~In~) Vi v~y(I = ( ) < ( )sh~Indo +1 sh~Ind; +1 Sh(;Inco) Sh(;Inco) (do)f- (co)f+(cof:- (dof: (do)f- (cop +(coff - (dor: - ( = < - - ( 1 (do)f+(co)f-(raft -(daft (do)f+(cop-(coff -(dor: I r, ~ R(I,t"O) t ~ T(1,r,~,O) <T(l,r,~,o)~ I' D d' . R(p,c-l,atK~-In(I-?2) 130 0: 13<arCSIn C = 1 2 ;p (3.11) (3.12) v6i (' xacd!nha(3.11). b) V6ip=I: Co th~xayfa 1T:::;213<21Tnen duOngkfnhD cuanhatd.t Li coth~kh6ng phaiIakhoangcachcuahaimutcuaLi .D~kh~cph\lcdi~unaytalamnhusau: Gqi w=h(w)v6ih(00)=00 bienbaagiacdondi~pmi~nB leumi~n B ={wllwl~I},chuanhatc~tl ={wla<c:::;!WI:::;d,argw=a}saochobienC thanh !w!=1. Ta c~tmi~nB bairiaargw=1Tthltrongcacmi~nB, A secocacduOng congtuonglingnoi Cv6iw=00ho~cIzl=1v6iz=00. 34 I~ ~, B \,~ " W ". ~ z =g(w) A z'~Fz J~h(W) 1 r~ 0;0 d B Hinh3.2 Dunghamph1;}z"=x(z)=...Jz(ch(;mnhcinhJ1 =1)th1Jchi~nphepbi€n mnh mi~nA l~nmi~nA" (hinhve).Khi do:R"=x(Rj)=..jR;,L" =x(Lj )G<?igocroocua cungL" la 2fJ", thi 2fJ"=fJ <Jr n~nduOngkinhD" cuanhatcAtL" chinhIa khmingcachhaimutcuaL" . Hamz"=x[g(w)]th1Jchi~nPBHKABGmi~nB bi cAt l~nmi~nA". Tuongt1;tnhutr~ntaco: D'" =If/(D"),R'" =If/(R")=~Rj-j . (7r . ~-ln(1-t'2)D'" . V2 <arCSill fiR: fJ"- 2arcsin-:S;arcsill ~ 2 2R-- 2R'" Jr j2R'" -In(l- t,2) . R(l,c-j,O)~~-ln(1-t'2). :S;arCSill 2..[i 35 ~ /3=2/3"::;2arcsinR(l,c-],o)f~-ln(l- ('2) 2-/2 Lamm9tcachtuongtVnhuphana tac6 . R(l,c-],o)f~-ln(1-(12)=/32 /3<2arCSIn 2-/2 (3.13) v6'i(' xacdinhb (3.11). Chli y: Trong(3.12)va(3.13),neumi~nB bandauc6d,:mgcuami~nB] (hinh3.1),codinh d =do vacho c=Codandendo,khi d6 /3]~ 0,/32~ 0 v6'iVK nendanhgia(3.12)va(3.13)kh6nghi~n hien. BuO'c2: D~chiami~nBthanhpmi~n hilienbtmgnhautathvchi~n husau Ve m9tduemgcongJordanYon&mtrongB noi bien Ccua B v6'idi~m W=OO. Giyj Ia cac duemgc6 duqcdo Yo quaynhU'ngg6c 21lj(j =l..p-1). Cacp duemgyj naychiami~nB thanhp mi~nnhi lienv6'icacbientrongla p thanh phanbiengj cuaB. GiB] Iam9ttrongpmi~n hiliend6v6'ibientrongg]. Ky hi~uC(a,r) chiduemgtrolltiimt';lia,bankinhr. GiC(wprJ Iam9tduemgtrollgi6'ih~ m9tt~pd6ngchuagl' GiC(w2,r2) Iam9tduemgtrolln&mtrongB] (baod6ngcuaB]) baobcduemgtroll C(W]'r]) . GiB2lami~nnhi lien gi6'ih';lnbbi c(WI'r]) vac(w2'r2)'Theotinhch~tdon di~ucuam6dunmi~nnhi lientac6: mod(B2)::;mod(B]) (3.14) Dilt ;;] =mill ~wl,WE C( w2'r2)}tuc ;;] =IW21-r2';;2 =IW21+r2. ;2 =max~wl,WE C(W2,rJ} (3.15) 36 vagiasu ;1 ;::::M tuchlnhvanhkhanB3 = {wi;}<Iwl<;z} nmntrongmi~nB. T at!nhti€n vaquaymi~nBzr6iapd1;lllgb6d~2.9thl mi~nBzc6th~bi€n baogiacdO'Ildi~pb<ris=s(w)leumi~nB4=~r<Isl<I} v6i r =r(r r h)- rzz-hz +rIz-J~zz -hz -r I z )Z-4hz r z I' z, - I 21jrz . (3.1-6) trongd6h=lwz-WII. ~ z=g(w) >=Inz) 27r P ~ A3 M" in--;-. m Hinh 3.3 p =4 37 D~t Al =g(BIXc A), Az=g(BzXc AI) vaig EG, M~t khac t6n t(;li phep bi€n hlnh bao giac don di~p ~=~(w) mien Az=g(Bz),gE Glen hlnh v~mhkhan Bs: r' <I~I <1. Vi phep bi€n hlnh hqp ] ~ogos-ImienB4 leu mien Bsla mQtPBHKABG nen theo(3.13) taco: r' ~rK Ap dl;1ngtinh don di~u (1.14) cua ham phl;1 T(p,r,s), ta to: I t=T(1,r',O)~T(1,rK,O)yair duqcxacdinhbbi(3.16). D~t m"=m(rpg),M" =M(r2,g) v6i gEG, !£- K Theo (3.5) ta co: M" <4p r 2 =M" , (3,17) ~ _K m">4 Pm*(oo,g)Yl =m" gEG, (3,18) G9i D la duemgkinh cuabientrongcua Az ' D la duemgkinhcuanhatcat Lj(J=O..p-1), R6rangD>D, * Dungbdde'2,7: - N€u p =1, d~co quailh~D =2R]sin13c~nthi€t saunaytac~nthemgia thi€t phl;1213<TC, vi n€u khangthl D =2R],Mu6nv~yd~tR]=4-Km*(00,g)dK(~RI) Do D ~D thldieuki~ndud~213<TC IaD <2R] 2T(2,r~,0)M" <2.4-Km'(oo,g)dK T(2,r+,0)M Q 4Kd-K <m*(oo,g) I - K Q m*(oo,g» 2.4ZKrKd-K rz (3.19) - N€u Vp ~2thl 213<2TC<TCdo do ta 1uanco D =2RI sin13' p 38 V~yneuvp~2 ho~cneup =1thoa(3.19)thlapd\1ngb6d~2.7taco: D =2R,sin fJ ,; D' ,; 2T( 2,r* ,0JM' ( I ) [ I~_K } M"T 2,rK,O 2r* M" . 2rK4P 12 . =arCSIn -K . <arCSIn -. K ~ Ji'; arcsInI R, - (R') 4 ' m (oo,g}d { 2K I K } ,8~arcsin2.4P rK;2 d-K[m.«X),g)jl =,83 v6'ir xacdinha(3.16). * Dungb6de'2.8. Ham r;=Inz (laynhanhdontri In1=0) bien A2 thanhA3 la mi~nnhi lien (3.20) v6'ibientrongIa m(>tduangcongkin co duangkinh D" baob9Cnhatciit th~ng A =[InRI-i,8,lnRI +i,8],tucd(>dai cua A la 2,8vabienngoaila m(>tduangcong kin Damtrongtugiaccongv6'ihaic~nhla cacdo~nth~ngclingd(>dai 2ff Damtren p Rer;=In!Ji va Rer;=InM" vahaic~nhconl~ilahaicungco thech6ngkhitleD nhaubOipheptinhtieDq=r;+i 2ff , tucdi~ntichgi6'ih~nbOibienngoaicuaA3Ia p M" m" " , 2ff M"m S ~-ln- pm'" Ap d\1ngb6d~2.8taco: ~ 1/2fflnM" I ( " M" lfJ ,; D" ,; !" = ~ m" In(l- t') ,,- lln 1- t')In~. In(1-t2) p -ff V p -In(l- t2)InM,:' ,8~11 m 2p tilc 39 !rong d6 rxac djnh II (3,16), t=T(l,r* ,0). M' xac dinh II (3,17), m' xac dint II (3.18). Mifltkhactaco: ( =T ( I r* O J <4r* =t, , , dodoneu( <1thi ( 2) 1 1 1 -In 1- ( =In~ <In ( - r =In 2 >0 1-( 1 ( -- 1-16r K ~ _K 4PY2 >0 va In -K - K 4P m*(oo,g)Yl ~ _K 1 4P Y2 V~y,8$;./1n 2In =!!... _K 1-16rK 4 P m*(oo,g)Yl $; /In 1 2 In {4'; [m'(oo,gW(r:- r; J } =,84 1-16r K 1 { 2K 1 ( - K -K J} ,84=./In 2In4p[m*(oo,g)[ Y2-Yl 1-16r K (3.21). Djnh IS'3.2:C~ntren cua,8(g) V6'icaegiathietvaky hi~unhutrongffit;lC2.2: - Neup =1 thi,8 $;,82. - Neup;::::2hoiflcp =1va thoa(3.19)thi ,8$;min{,8p,83,,84}' trongdo a . R(p,c-I,Or~-In(~-?2) ~ -, 1 ,R(P,C-l,Or~-In(~-?2)1-'1=arcsm C ' neu « va C $;1. 2vp 2vp 40 fJ ° R(I,c-l,o)f~ ~ -, , R(I,c-l,0)fv'=!n(I-?2)2=2arcsm J,.. , nt::uI <1va J,.. ~12",2 2",2 { 2K 1 - } 2; i - . P K -K * 1 4 r Y2 1 fJ3 - arcsm 2.4 r Y2d [m (oo,g)j , nfu K *( ) ~-d m oo,g 2 2K ( K K J j 4 P Y2 - Yl fJ4 =JIn 1 ~In {4': [mO(oo,g)t'(;;: -;;~)} "IOu mo(oo,g) >1 1- 16rK ~ 16rK <1 V6i I' xacdinho(3.11), r xacdinh0(3.17) YpY2xacdinho(3.15). Vi dlJ 3.1: Chop =4 B2 Hinh3.4 Mi~n B ban<1ftuco bi~ntrong0 Iii duangtrOllC(O,I)vii P =4 thanhphAn boA , 1 ° 1, 5: C' ) /. . i~ Fz Th' h h' bo leD con c;u a Vi = ,ai's VFz - an p an l~n2-1 00chinhIii duangtrOllC(ao,s)v6i ao=a,r1=s, v6i S du<1I1gdube. Ta IffyduangcongJordanYoIii duangphfulgiaccuagocphAntUthatU. 41 Ve duemgtrollC(a,r2)saochobankinhr2lakhmmgeach1611nhattua den duemgphangiaccuagocphflntuthunhat. Khi domi~nB2 lami~nnhi liengiOih';U1bOiC(a,&)vaC(a,r2)' M' . n: a A a r2 =asm"4=.J2' rl =& nen 2 2 f( 2 2)2 f,...r2 +& -"\j r2 - & & &"1/2r= =-=- 2&r2 r2 a . Theo(3.20): { 2K I } /33 =arcsin 2.44 rK r2 d-K [m'(00,g)jl Taco p =4,d=1 { I } K f,...- - &"1/2 K - I f3 = arcsin 2.4 2( -;;-) r2[m'(00,g)j K I /3~arcsin2.42(.J2)KrJm' (00,g)jl ~. I -&KI-- a K (3.22) choa (>tJ c6 djnh, kha Ian. Ta (hay khi 8 -> 0 n€u m' (00,g) 0 thi /3~ 0 , Vi v~ytrong truemghqp nay danh gia /3~/331akh6ng hi~n nhien (0<jJ <;) vati~m cijn dung. . Theo(3.21): Taco: /3~_/In 1 ~ In {4':[m'(oo,g)t'(r:- r~J}1-16r K 42 2 ~2 J 2a 2 a 2, r22+&2_~h2_&2Y 2+& - 2-& 2&2 _.J2&Mar- - -- - - 2&r2 - .J2a& - .J2a& - a . - a.J2 - a.J2 /" 1, / h ~r2 =a+r2=a+-=c2a, rl =a-r2 =a--=cla, VO'!CpC2a cac ang2 2 soduong. fJ'; Iln 1 2 +';[m'(oo,g)t'aK(e;-en} 1-1{4q 1 { 2K 1 ~/35, Iin 2 fIn4p[m*(oo,g)jlaK(c;-CIK)r 1-16(.J2)K~ aK ( 2 ) 2 K 2K Ii,; .I-In 1-16(,/2)< :f +'[m"(oo,gWaK(e;-en} 2 1 I { 2K 1 1 fJ'; j16(.J2r a~t 4 P [m'(oo,g)raK(c; -cnr'" (3.23) V& ffiQia(>fJ c6djnh,khi "->0 n6um'(""g)~o>OthlfJ->O. VI v~ytrongtfuanghqp nay ta da chi fa danhgia /35,/34Iakh6nghi~nnhien (0<P<;) valati~mcijndung. 43 lJinh Ii 3.3(C~nduOicua fJ(g) ) VOi caegicithi€t nhu trongb6 de 2.10 taco: l 2K °oK7rln 4 P d K P~;_./ m'(oo,g)I~1] 2pln d c (3.24) " Chfrngminh: z=g(w) ~ Eo w z/~>\ r\1~2~ 'i.0 J ,"'-..' ' "~:::] ",", ,,-' ct':: , \"""-'-:"/"" " 0,1 T :', """'", """" ': L "-1 ,'., ".., ",:: 0 " , ", ",' ", , , , , "' " ' Hinh3.5p =2 Apd~ngb6de2.10,xettugiaccong: Eo =(w=re'"le~r ~d,r(r) ~e ~r(r)+n(r)}vffiO d:J(r),; no(,; ~)vap(z)= ~ taco: Ip(Cr) =fp(z)ldzl=fldzl vOiCr=g(CJtrongdoCr={wllwl=r}n{wlwEBo}tuc~- Iz I Cr Cr ~ Cr lam(}tclingnfuntrongA noiLovOi L]. 44 - - - - D~t z =; e1rpthl Idz I= Ie1rpd r+~r e1rpdipI= Idr+~r dipI~ Ii r ~ipI=1dipI Iz I Ir I r r n~n~lp( C}t -P J G9i Ao=g(Bo) thl ( - ) 2ff ~ -n - rp r +- Sp(Ao)= ffp2(Z)dxdy = H~d; =Hrd~2dip = Hdr_dip ~Mfd_r fdip= 2TrIn M(d) Ao Ao I I Ao r Ao r m(c) r rp(;) p m(c) trongdoA c {z:m(c,g)~1z I~M(d,g)}tucla A c {z:m(c)~1z I~M(d)} Do dotheo(2.17): ~ [ 2 ( Tr - 13 J] 2 ~ Jdr ~2TrInM(d)K P °0 c r P m(c) d~ta =(;- jJ J / 4 a2 d 2Tr M(d)taco --In-~-ln- K °0 c P m(c) llOoKlnM(d) Q a ~ I m(c) d 2pln- c K -K tathay M(d)~ 4P Id IK, m(c)~4P m*(oo,g)1c IK vaotaco: K °oKTrln 4P IdlKK 4 P m*(oo,g)1 c IK d - 2pln- - c 2K - K 4 P Id I °oKTrln m*(oo,g)1c IK d 2 In- p c a~ 45 2K vi 4' IdlK d Q"KJTln 4' IdlK m'(oo,g)lcIK>0,-;:->0nen m'(oo,g)lcIKd2pln-c 2K ~O 2K - K 4 P Idl noKJrln m*(oo,g)lcIK d 2pln- c A fJ Jr Jr nen =--a ~-- p p tiletaeo(3.24). Nhfulxet: N€u c=const,d=const,m*(oo,g)~canst>0,khi no ~ 0 tm a ~ 0, tile fJ ~ Jr , P eonghiadauhgia(3.24)ti~me~ndung.Di~unaysedu<JeminhhQabm: Vi du3.2: Cho.P=2 ~ ~ '~~ \r r:~Qo WGd (~.Rl ~~1 R. \;=h(w) Z=k(;) Hinh3.6 46 GQi B Ia mi~n Iwl >1 bi khoet hai tIT giac cong dong Vj ={we<; Iwl <; d,(j -I}n- +~o<; arg w<; j1T - ~o}o<no <1T, j ~ 0,1 lac cae Ihanh phanbien (;1'(;2IacacbiencuaVb Vz (hlnh3.6). Cho c,d co dtnhtITCIa c=const,d=canst.Khi °0 ~ o thl m*(oo,g)thay ~ d6i.TuynhientasechUngminhm*(00,g)kh6ngclanden0, khi °0 ~ o tilc chUng minh m*(oo,g);:::mo=canst>0. Theo[17]t6n t1bt e~tdQcp clingtrolltam0 saDehoJwl =1 thanh1;1=1,h(00)=00 .Hannuakhaitri~n Laurentcuah(w)quanhw=00 cod<:tilg: ( ) al a2 a3h w =aw+-+2+3+'"w w w (3.25) TITC Ia Ih'(oo~ =lim l h(w) 1 =la! * ° z->oo W (3.26) Vi Z =k(;)=;I;IK-J Ia m<)tPBHKABG nen z =g(w)=k[h(w)]dingIam<)t PBHKABG. VeduemgtrollIwl =r v6ir ratIOn. Vi r IOndo(3.25)Ih(w~ ~ lal\wl tITeimh cua \wi=r baih galltrimgv6i duemg troll 1;1=; vai ;=lair. V~ylmhcua \wi=r bbi g =kohgantrungv6i duemgtroll _K Izl =R vaiR =r =lalKrK. Tudotaeo: m*(oo,g)=limm(r;g)=lim ~=lalK*°. r->oo r r->oor 47 Neutab6 sung haiclingcuanhateMtrongmi~nA d~duqcduemgtroll 1;1=R] thltrongmi~nB clingsecohaiclingn6i6]va62, Khi cho Qo-+0 vae6dtnhe,d,do tinhbatbieneuam6duncae mi~nnht lienquaPBHBG h thlR]-+c ,r r r r h - c ( .;:: ? h' h h' h ' h kh )va: - -+- =>- -+- ay r :::;r - mIentrO't an III van an R d c d d] Lucdotheo(3.26)co: ;:::; rlh'(ool c Dodo: Ih'(oo~=lim Ih l (W I ~= lim~=limr d =~=const>O ~ 1wI--'" W 1wI--'"r r--'" r d V~y theo (3.24) trong truemghqp nay ta co f3-+; khi Qo -+0, trong khi 0<f3<; , tiledanhgia(3.24)ti~me~ndung.