CMBUk6 GMgeRkl 6- GMgeRkl ( The Rem Iegrl ) MeB¼eø¼TT&l J = [, ], egrevgé Jdy J. MeB¼MEk P =,..., é[, ], eø¼rg I k = [ k-, k ], k,,..., ehafeø¼rgti ké MENk P. ym&y ei P g P' CMENké [,] ekyyf P' C «KMr» é P ei P' P. yk P CMENké [, ] ehiy f CGuKm_T&leAelI [,]. MeB¼eø¼rgTI k émeke¼ek gekleli geklerkmé f edy M m edyerilkçn e¼ ekkm g U( P,f ) = L (P,f ) = k k m M (f ) sup f () (f ) k (f ) I k k (f ) I k f f () : : I I k, k,,...,. CplUk Rem eli g pluk Rem erkm erógkñé f efowgmek P. vckr,s f m k (f ) M k (f) MeB¼ k,,..., dueñ¼ L(P, f) U(P,f ). RTwsþIT MeB¼GuKm_T&l eaeli [,] eká ( ). U(P ', f ) U ( P, f ) g L( P',f ) L( P, f ) MeB¼RK P' P (). L(P, f ) U (P, f) MeB¼ RK MEk P g P - 69 -
y M k = M k M " k k k ( ru 6. ) ( ). CkRKŔK eyigg P' = P {}, I k Edl I k Ceø¼rgTI k é P. eiekg M' k = supf(): [ k-, ] g M' k = supf(): [, k ] e¼ M' k M k g M" k M k M' k ( k ) + M k ( k ) M k ( k k ). dueñ¼ U(P', f ) U(P,f ) + dukñmeb¼vsmpb épluk Rem erkm. (). g P = P P e¼ P P g P P ehiym ( ) eká L(P, f ) L(P,f ) U( P, f ) U(P, f). kur UElr MeB¼ GuKm_T&l f eaeli [,] ei kur UElre¼gHjf smum U(P, f ) : MeB¼RK MEk P é [, ] g L( P, f ): MeB¼RK MEk P é [, ]CsMuMT&l. GuKm_T&l f eaeli[, ]ekkm _ M = supf():[, ] g m = f supf(): [, ] e¼ m( ) L(P,f ) U ( P,f ) M( ) MeB¼MEk P g P é [,]. f f U( P,f ) : MeB¼RK MEk P é [,] sup L( P,f ) : MeB¼RK MEk P é [,] - 7 -
CGMgeRkl Rem eli g GMgeRkl Rem erkmerogkñé f. edy L(P, f) U(P, f) MeB¼MEk P g P ektjá f _ f ym&y GuKm_ f eaeli [,] CGuKm_mGMgeRkl Rem ei ekgmélryme¼edy f ehiyekehavfc «GMgeRkl Rem» é f eaeli [,]. ekk¾g gvedy f eiekg sg; F étaeli eø¼nmyyedl f km Méledy eaeblekǵrmµn_kñúgkrp TMk MgrvgGMgeRkl g edriev. «ThrN_ skßlkçn GMgeRkléGuKm_dUgeRkm dmen¼rsy ( ). f eaeli [.] Edl kmédy f() = ². yk P CMENké [o, ] EdlekIeLIgedykrEk [o, ] C EpñkesµIrKñ. edy f ekikñúgeø¼ [o, ] e¼ M U k f K k ehiy mk (f ),f k k k k,f (k ) (k k k P g L P ) edy k U P,f 6 6 f ()d() k,,...,. eká kln + 6 ehiy LP,f kln + _ f f. - 7 -
dueñ¼ f g f f f eksñdæf f mgmgerkl Rem ehiy ( ). GuKm_ f eaeli [,] kmédy, (), f ()d. f yk P CMEkmYyé [,]. edymyst ggstmlkçn ñéakñ úg eká M k (f) = g m k (f) = MeB¼ k,,...,. dueñ¼ f U( P, f ) =. I k g L( P, f ) =. g k lkç&nðrkŕk «mf f epþógpþĺkçnðe¼kw mmekedl P, U(P, f) L(P, f ). lkç&nðmá L(P f) + eká f U(P,f ) f f f. k I k ehiyeyigsñ dæf f KµGMgeRkl Rem et. lkç&nð Rem : GuKm_T&l f eaeli [, ] mgmgerkl Rem lu¼re m MEk P é [, ] Edl U(P, f) L(P,f). f dueñ¼ f f m&yf f mgmgerkl Rem. ei f mgmgerkle¼ / f f eká - 7 -
, MEk P ' Edl U(P, f) f MeB¼RK P P' gmmek P" Edl L(P, f) f MeB¼RK P P". dueñ¼meb¼ P = P' P" eká (P,f) L(P, f) «ThrN_ ekgerirtwsþite¼ CmYy«ThrN_ ( ) geliedim,itjb k f GuKm_ f mgmg erkl edymmáḱngmgerklet. ekm U(P, f) L(P, f) = k k (k ) k k = dueñ¼ mlkç&nð Rem eyigá f CGuKm_mGMgeRkl. 6- lkçn égukm_mgmgerkl ( ) MeB¼MEk P é [,] eká kln + RTwsþIT ei f g g CGuKm_mGMgeRkleAelI [,] e¼ f + g g kf Edl k' CGuKm_ mgmgerkleaeli [,] ehiy ( ). f g f (). kf k f. M k ( f + g) M k (f) + M k (g) g m k ( f + g) m k (f) + m k (g) MeB k,,..., dueñ¼ U( P, f + g) U( P,f ) + U( P, g ) g L( P, f + g ) L( P,f ) + L( P, g ). edy f g g mgmgerkl e¼mlkç&nð Rem eká U( P, f ) L( P, f ) g U(P, g) L (P, g) L(p, f ) + L(P, g) L (P, f + g ) U( P, f + g ) U ( P, f ) + U ( P, f ) L ( P, f ) + L ( P, g ) + U( P,f+g ) L( P, f + g ) mlkç&nð Rem ektjáf f + g mgmgerkleaeli [, ] ehiy L(P, f ) +L(P, g) g g f L (P, f ) + L (P, g) +. - 7 -
E L (P, f) + L(P, g) f g f g dueñ¼ f g (). smrydukñ. f MeB¼MEk P é [,] eká M k (f) M k (g). k,,...,. g f g L( P, f ) + L ( P, g ) + RTwsþIT ei f g g CGuKm_mGMgeRkleAelI [,] ehiy f() g() MeB¼ [,] e¼ dueñ¼ U(P, f) U (P,g) ehiyedy f g g mgmgerkleli [,] e¼ f g. MeB¼MEk PNmYy é [,] eká M k (f) m k (f) = Spu f() f(y) :, yi k, k,,...,. EMeB¼,yI f () f (y) f () f (y) M k f m f Mk (f ) mk (f ) g U(P, f ) L(P, f ) U (P, f) L( P, f ). edy f mgmgerkl e¼mlkç&nð Rem eká f mgmgerkleaeli [,]. mü geto f f e¼mrtswsþitgeli f f. CTUeTA Mrs értwsþite¼mbet. «ThrN_ BüemIlGuKm_ f eaeli [,] kmédy f() = eyigá f () = MeB¼ [,] f g m&yf ()d f g()d. RTwsþIT ei f CGuKm_mGMgeRkleAelI [,] e¼ f mgmgerkl ehiy f f m&yf ()d f f () d.,,. - 7 -
dueñ¼ f mgmgerkl E f KµGMgeRkleAelI [,] et. RTwsþIT ei f CGuKm_mGMgeRkleAelI [,] e¼ f ² mgmgerkleaeli [,]. MeB¼MEk P é [,] eká M k (f²) = [M ( f )] g m k ( f² ) = [ m( f )]², k,,..., M k ( f² ) m k ( f² ) = [ M k ( f ) + m k ( f ) ]. [ M k ( f ) m k ( f )] M [M k ( f ) m k ( f ), k {,,..., } Edl M = sup f () : [,]. U ( P, f² ) L( P, f² ) M[ U( P, f ) L( P, f ) ]. mrtwsþitgeli eká f mgmgerkl ehiymlkç&nð Rem ektjáf f² m GMgeRkl. RTwsþIT ei f g g CGuKm_mGMgeRkleAelI [,] e¼ fg mgmgerkleli [,]. msmpb fg = (f + g )² f ² g² gmrtwsþitgeli ektjá fg mgmgerkleaeli [,]. eyigbüemilsmumégukm_mgmgerkleli [, ]. CBessMeB¼eø¼ [,] EdleKg smume¼ edy R[,]. smum R[,] mtmrg BICKN EdleKGTjáBIRTwsþITgelIe¼. ei f g g CGuKm_mGMgeRklelI [,] e¼ f +g g kf Edl k k¾cgukm_m GMgeRkleAelI [, ] ehiy R[, ] ClMhlIeEG rbmym&yéplukbirgukm_ gplkunégukm_ myyedymyb. dukñe¼edr fg CGuKm_mGMgeRkleAelI [,] ehiy R[,] mtmrg BICKNmlkçN Rl mym&yéplkunébirgukm_. RTwsþIT ei f CGuKm_T&leAelI [,] g ehiyei f mgmgerkleaeli [,] geaeli [,] e¼ f mgmgerkleli [,] ehiy f f f. - 75 -
mmek P é [,] g P [,] Edl U ( P, f ) L( P, f) P, f ) L ( P, f). g U( eyigbüemilmeké [,], P = P U eká L(P,f) = L( P, f) + L( U(P,f) L (P, f) dueñ¼ f mgmgerkleaeli [, ]. E L(P, f ) P. P, f) g U(P, f) =U( P, f ) + U( P, f ). f f L(P, f) + ehiy L(P, f) krsµ f f g dueñ¼ f L ( P, f ) + f f f. ei f CGuKm_mGMgeRkleAelI [,] eksmñsresr f f g f. 5 «ThrN_ smum {,},{,,}, {,,,}, {,,,,, 8 } sut ECMEkéeø¼ [,]. «ThrN_ GuKm_ f() = ² CélI [,]. MEk {,,,}Ek [,] Ceø¼rgI. [, ] = [, ], [, ] = [,, [, ] = [,] EdlmREvg = =, = =, = = erogkñ. MélGrmé f ykeaelieø¼tmge¼kw MélG,rmKW M = =, M f 6, M f (). - 76 -
dueñ¼ m = f() =, m = f =, 6 m = f =6 7 U f (P) = M + M + M = () 6 6 9 L f (P) = m + m + m = ( ) 6 6. «ThrN_ ykgukm_ f() = eaelieø¼t [, ]. MEk P =,, Ek [, ] CBIreø¼KW [, ] g [, ] eká U f (P) = () «ThrN_ ghjf ei f() = MeB¼ [, ] e¼ dmen¼rsy 6 f ()d ( ). g L f (P) =. yk P =,,..., CMEkmYyé [, ]. edy f ykmélefresµiwg eaeli [,] e¼vefr esµiwg eaelieø¼rgimyy@ [ -, ]. dueñ¼ M g m mmélesµikñkw M = m =,{,,...,} ektjá U f (P) = + +... +... = ( + + + ) = ( ) eyigexijy g,s f L f (P) ( ) U f (P). edyvsmpbe¼epþógpþćmeb¼rk MEk P é [,] e¼eksñ dæf f ()d. lt plugerkye¼gsresr d duc d ( ) 8 / d 6. ei EdEdleAeRkmRkhVe¼CuekNEkgmkMBs EdlsgéLIgeAelIeø¼ [,] ( emilru ). GMgeRklgelI pþlńuvrklépþéueknekge¼. - 77 -
- 78 - «ThrN_5 ghj d. dmen¼rsy yk P =,,..., CMENkNmYyé[, ]. eaelieø¼rgimyy@ [ -, ] GuKm_ f() = mmélgrm M = gmélg,rm m =. eká U f (P) = + +... +. g L f (P) = + +... +.,,...,, MélmFümeAeø¼BIrMY edykunvsmpbe¼wg = eká edyukmélbi = dl = eká L f (P) ) (P U f E ) (P U P L f f. edyerciserisyk P ersþ e¼eyiggsñdæf vsmpbe¼epþógpþ MeB¼RK MEkPé [,]. dueñ¼ d. eieø¼ [,] sseagsþmémukl O e¼ededlsseaerkmrkhvé f() =, [,] Cu eknbñy ( emilrugerkm ). GMgeRkl d pþlńuvrklépþéueknbñye¼ KW f() = y
- 79 - A =. «ThrN_6 ghjf d. dmen¼rsy yk P =,,..., CMEkmYyé [,]. eaelieø¼rgimyy@ [, ] GuKm_ f() = ² mmélgrm M = gmélg,rm m =. ektjá U f (P) = g L f (P) = MeB¼,,..., ekm edykunvsmpbe¼wg = e¼yknþletac ) (P U (P) L f f f() = ( ru 6. ) y y f() = ² ( ry 5.. )
L f (P) Uf (P) Lf (P) Uf (P) Lf (P) Uf (P) edyeyigerciserisyk P ersþ e¼eyigsñ dæf vsmpbmeb¼rk MEk P é [,]. dueñ¼ d. 6- MENkfñkḿYyMYéGuKm_mGMgeRkl RTwsþIT ei f CGuKm_m UNUUT&leAelI [,] e¼ f mgmgerkleaeli [, ]. ekrkebüemilgukm_ f ekieaeli [,]. yk P CMEké [,] EdlekIeLIgedykrEk [,] C Epñkesñ I@Kñ. eká M k (f) = f( k ) g m k (f) = f( k ), k,,...,. dueñ¼ U( P, f ) = k f ( k ) g L (P,f ) f (k) k U(P,f ) L(P,f ) f () f () kln +. dyeñ¼ f mgmgerkleaeli [,]. «ThrN_ GuKm_ f eaeli [,]kmédy f () f () ei CGuKm_ekI dueñ¼vmgmgerkleaeli [,]. MeB¼* RTwsþIT ei f CGuKm_CéAelI [,] e¼ f mgmgerkleli [,]. - 8 -
mlkçn kmuá k égukm_c f T&leAelI [,] ehiymrtwsþit Hee, f CésµIeAelI [,] m &yf, Edl f () f (y) MeB¼RK y eieyigerciserisyk * Edl Gg; [,] C EpñkesµIKñ e¼eyigá f () f (y), z,yi g k,,...,. E M k (f) m k (f), k,,..., U (P,f ) L(P,f ) ) k. ehiyyk P CMEké [,] EdlekIeLIgedysrkrEk M kf () mkf (y. dueñ¼mlkç&nð Rem eyiggukm_ f mgmgerkleaeli [,]. RTwsþIT f CGuKm_T&leAelI [,]. ei smumémud é f sseakñúgeø¼mmyr Gs EdlmrgVsśruUCg e¼ f mgmgerkleaeli [, ]. yk P CMEké [,] Edleø¼rgéMEke¼EdlmMudé f mrevgsruu Cg. edy f T&leAelI [,] e¼ M, f () M, [,]. due¼ Beø¼rgEdlm Mud é f MEkEdlpþl U(P,f) L(U,f) M. f CéAelIeø¼rgImUy@ f due¼ f mgmgerkleaelieø¼rgtmge¼. ekgrkkmr P' é P Edl M k (f) m k (f) eaeø¼rgimyy@ é P' EdleAelIeø¼rgTMge¼ / f mgmgerkl. MeB¼ f eaeli [, ] eká U(P',f) L(P',f) M + ( ) dueñ¼mlkç&nð Rem GuKm_ f mgmgerkleaeli [, ]. «ThrN_ ekegygukm_ f eaeli [, ] kmédy f (), { : s }, { : s }, - 8 -
Mud é f elikelgemyymyrǵs sseakñ úgeø¼ [, ] ehiymurǵsédleasl GCFuésMuMrgmMYrǴsÉdlmREvgsruUCg. dueñ¼ f mgmgerkl. kur UElr eigukm_t&l f eaeli [,] mmudŕǵs e¼ f mgmgerkleli[,]. kur UElr ei f CGuKm_T&lmGMgeRkleAelI [,] ehiy g CGuKm_T&leAelI [,] Edl f() = g() MeB¼RK [,] elikelgemumyymyr GséAkñúg [,] e¼ g mgmgerkleaeli [,] ehiy ()d f () g d. BüemIlGuKm_ g f eaeli [,]. eká ( g f )() =, [,] elikelgemumyy MYr Gs eakñ úg [,]. yk P CMEké [,] ehiyrkćeø¼rgtmggsédlmmurǵs Edl( g f )() mrevgsruucg. eká g f CGuKm_T&l KWf M, ( g f ) () M, [, ]. P,g f U M g L (P,g f ) M. dueñ¼ g f mgmgerkleaeli [, ] ehiy g f ()d. E g = (g f) + f mgmgerkleaeli [, ] erb¼vcplukégukm_mgmgerkl dueñ¼ g mgmgerkl ehiy ()d (g f )() f ()d g f ()d «ThrN_ ekegy f CGuKm_eAelI [, ] ( ) kmédy f (),, MeB¼GuKm_ g eaeli [,] EdlkMédy g() = + eká g() = f() elikelgerg =. GuKm_ g CéAelI [,] G mgmgerkleaeli [,]. dueñ¼ f mgmgerkleaeli [, ] ehiy ()d f g()d. - 8 -
«ThrN_ ekegy f CGuKm_eAelI [, ] Edl f () q ei = q p ( p, q zmrvgkñ ) ei ei = MeB¼MélNmYy * MurUPBEdlFMCg mmyrǵs. yk P CMEkmYyé [, ]. RK eø¼rgémeke¼ EdlmMuEdkM Ckĺk gmmurupbfmcg, mrevgsru UCg BIeø¼rgTMge¼MEkEdlpþl U( P, f ) mrgvs YCg U( P, f ) k¾mrgvs UCg Edr. due¼ U(P,f). edymyst gmygstmlkçn ñéakñ úg e¼ L(P, f) =. ehiyeø¼rgeaslé¼mekedlpþl dueñ¼ f mgmgerkl ehiy f ()d. 6- RTwsþITsM @sþigmbivfikn RTwsþIT ei f CGuKm_mGMgeRkleAelI [,] e¼gukm_ F: [,] kmédy edy f mgmgerkleaeli [,] e¼ f mgmgerkleaeli [,] MeB¼ [,]. eká F () F () f ()d CéAelI [,]. F(y) y = M f ()d f ()d y f ()d y Edl M = sup f () : [,]. BIvsmPBe¼ eyigtjáf F CésµIeAelI [, ].. - 8 -
RTwsþITe¼gHjf et¼i f mcŕgŕk MueAkñ úg [, ] k¾edy k¾gukm_ F ( EdleK ehafgmgerklmkm é f ) CéAelI [,] C. «ThrN_ GuKm_ f eaeli [,] kmédy dŕg = E F: [,] Edlmrg ehiy F CéAelI [, ]. MeB¼ eká F() F() f () F () f (),, f ()d f (),, = f () f () d M () RTWsþIT ei f CGuKm_mGMgeRkleAelI [,] gcŕg ],[ e¼gukm_ F :[,] kmédy Edl M () = sup f (y) f () : y,. edy f CŔg e¼ (f,) = KWf M () kln. dueñ¼ F mdiepr g EsülRg ehiy F'() = f(). «ThrN_ ekegygukm_ f ealei [,] kmédy f() = F:[,] mrg F() = erb¼ f mdćŕg =. F () f ()d mdiepr gésülrg ehiy F'() = f().,, ehiy F mdiepr gésülrg = E F'() = f() = RTwsþITRKw¼ ei f CGuKm_mGMgeRkleAelI [, ] ehiy F CéAelI [, ] gmdiepr g EsülelI ],[ ehiy F'() = f() MeB¼ ],[ e¼ f ()d F() F(). - 8 -
MeB¼MEk P é [,] F() F() = F ( ) F( k k k ) = '( ) F mrtwsþitmélmfümedl k ] k-, k [, k,,..., k k k k f ( k ) k k k f ( k ) k k U( P,f ) L(P,f) F() F() U( P,f ). k = E L( P,f ) edy f mgmgerkleli [,] e¼ mmek P Edl U( P, f ) ()d F() dueñ¼ f g L( P,f ) f ()d F() f ()d f ()d F() F(). RTwsþITe¼gHjf eieképþimcmyygukm_ F CéAelI [,] gmdiepr gésüleaeli ],[ ehiyekegygukm_ f eaeli [,] EdlesµIwgGuKm_edrIeveAelI ], [ e¼ ei f mgmgerkl ekgrk F eaeli [,] BI f edykrkngmgerkl. «ThrN_ ekegygukm_ f eaeli [,] kmédy f() =, s, eká f mdiepr gésüleli ],[ ehiy f() = s, os., E f ' mt&l dueñ¼ vkµgmgerkleaelieø¼rgté ],[ Edlm et. «ThrN_ KN d - 85 -
- 86 - dmen¼rsy eyigm f() = e¼rbimitivé f KW F() = mrtwsþitrkw¼ eká 5 F() F() d «ThrN_ KN d 8. dmen¼rsy RBImITIvé f() = 8 KW F() = ². due¼ d 8 = F() F() = () () = mmµsbøe¼ eká ½ + 5 d + 8d eyigm d d, * dueñ¼ duc + d + d keßm F() F() GsresrC () F () F. d
RTwsþT ( RTwsþITMélmFümTImYyMeB¼GMgeRkl ) ei f CélI [,] e¼mmyb ],[ Edl MY f() e¼ehaf MélmFümé f eaeli [,]. f ()d ( )f (). «ThrN_ MélmFümé eli [,] KW d + d dueñ¼ MélmFümé ² eaeli [,] KW. edy f CéAelI [,] e¼ f ykmélg,rm m g MélGrm M eli [,]. edy g eaeli[,] e¼ mg () f()g() Mg(), [,]. RTwsþIT (RwsþITMélmFümTI MeB¼GMgeRkl ) ei f g g CéAelI [,] ehiy g e¼m MYB ],[ Edl mg ()d f ()g()dk m g()d f ()g()d M g()d Mg()d Eedy g(), [,] e¼ g ()d. eká + ei g ()d e¼ f ()g()d ehiyrtwsþite¼bmeb¼rk ], [. f ()g()d f () g()d. MY f() e¼ehamélmfümmdk TMg g ( g weghed verge ) é f eaeli [, ]. - 87 -
+ ei g ()d e¼ m M f ()g()d g()d ehiymrtwsþitmélmfüm mmyb ],[ Edl f ()g()d f () ()g()d f () g()d f g()d 6-5 GMgeRklGMeRá ( Improper Iergrls ) GuKm_ F eaelieø¼tmt&l ekegy f CGuKm_kMéAelIeø¼TmT&l [, + [ Edl f mgmgerklelirkćeø¼t T&l [, ],. ekgkm GuKm_ F eaeli [, [ edy F () f ()d ei F() mlimikln + ekyyf f m «GMgeRkl Improper» rymeaeli [, + [ ehiyekkm GMgeRkl Improper é f edy f KWf f ()d lm f ()d. ()d kln F() + (erogkñ ) kln + ekyyf GMgeRkl Improper rikvc m ( erogkñgvc m ). dukñe¼edrekskßgukm_ f kmélieø¼tmt&l ]-, ] EdlmGMgeRkleAelIRK eø¼tt&l [, ],. ekkmǵmgerkl Improper é f edy f ()d lm f ()d. «ThrN_ skßpbrymégmgerkl Improper égukm_gerkm - 88 -
( ). f eaeli [, + [ kmédy f() = (). f eaeli [,+[ km edy f() =,p. p s, dmen¼rsy ( ). f C due¼ vmgmgerkleaeli [, ],. MeB¼ p, ei p =, dueñ¼ d p d d p p p p p p kln + ei p + kln + ei p = + kln + rymkln p ehiyrikkln p., (). f C due¼vmgmerkleaeli [, ],. MeB¼, * Edl = + y, y y. eká f ()d f ()d E sbørym. y k ()d k k k f ()d f ()d y f ()d f () d y f. eieyigg k = k dueñ¼, k. y d d k kln k + ehiy f ()d. kln + k f ()d e¼ k qøsśbø k k k CplUkedyEpñk ées riqøs - 89 -
f ()d lm m. dueñ¼eyigsñdæf s d rym. f CGuKm_kMélI EdlmGMgeRklelIRKćeø¼TT&lkñ úg. MeB¼ ei f () d f ()d mrbmkñ e¼ekyyf f mgmgerkl Improper eaeli ehiekkm vedy f ()d f ()d f () d eakñúggmgerklimproper gelie¼ krerciserisyk msmét. «ThrN_ skßpbrymégmgerkl Impropre égukm_ f eaeli kmédy f() = e dmen¼rsy GuKm_ f C gmgmgerkleaelieø¼tt&lkñ úg. k k e f ()d e d k f ()d k k k g,k, kln + k edy f CGuKm_KU e¼ kln. dueñ¼ ()d. f dukñwges riedr ekgeg;ilkçn vä&ysmr skßpbrymégmgerkl Improper. lkçn vä&yedlekerictuetampusmrǵukm_vc mkwdukñwg «lkçn vä&yeróefo» MeB¼es rimyvc m lkçn vä&yeróefo GuKm_ f g g kméaeli [,+[ mgmgerklelieø¼tt&leakñúgedkmé¼ ehiy f() g(),. ( ). ei g ()d rym e¼ f ()d rym. (). ei f ()d rik e¼ g ()d rik. ekgtjált plgelie¼y ggyedyeri vsmpb f ()d g()d,. ekttylált pldukñ MeB¼GuKm_kMéAelI ]-,]. - 9 -
«ThrN_ skßpbrymégmgerkl Improper égukm_ f eaeli kmédy f dmen¼rsy GuKm_ f C gmgmgerklelirkćeø¼tt&leakñúg. eká, e eróerfoeyigá e e ehiyedy d e d rym. d m ehiyedy f CGuKm_KY eaeli e¼ dueñ¼ e d rym. e. e rym (m«thrn_geli ) e¼mlkçn vä&y GuKm_eAelIeø¼T&lmT ( Fuos o Bouded No-Closed Ierls ) ekegy f CGuKm_kMéAelI [,[ gmgmgerkleli RKćeø¼T [,] ],[. ekggukm_ F eaeli [, [ edy F () f ()d. ei F() mlimi kln e¼ekyyf f mgmgerkl Improper rymetaeli [,[ehiyekkmǵmerkl Improper égukm_ f ClImIe¼ggvedy f ()d lm f ()d. ei F() + (erogkñ ) kln e¼ekyyf GMgeRkl Improper e¼ rikvc m (erogkñgvc m ). dukñe¼edr ekskßgukm_ f kmeaeli ],[ Edl f mgmgerklelirkćeø¼ [, ],], [. ek kmǵmgerkl Impropre é f kñ úgkrnie¼edy f ()d lm f ()d. e d rum. «ThrN_ skßpbrymégmgerkl Improper égukm_eaeli ],] kmédy edy f Cé¼vmGMgeRklelIRKćeø¼TT&leAkñúg ].]. MeB¼ p, MeB¼ p =, d p p p p kln ei p + kln ei p d + kln. f (), p. p - 9 -
d dueñ¼ rymkln p ehiyrikkln p. p ekegy f CGuKm_eAelI ],[ Edl f mgmgerklelirkćeø¼tt&leakñ úg ],[. MeB¼ ],[ ei f ()d g f () d sresr f ()d m e¼ekyyf f mgmgerkl Improper eaeli ],[ ehiyek f ()d lm f ()d f ()d lm y y f ()d dubimuedr krerciserisykmél msmét. ekk¾mlkçn vä&yeróefodukñ MeB¼PBrYmé GMgeRkl Improper RePTe¼Edr. lkçn vä&yeróefo GuKm_ f g g kméaeli [,[ mgmgerkleaelirkćeø¼ttleakñ úgede¼ ehiy f() g(),[,[. ( ). ei g ()d rym e¼ f ()d rym. (). ei f ()d rik e¼ g ()d rik. ekttylált pldukñmeb¼gukm_kméaeli ], ]. «ThrN_ skßpbrym égmgerkl Improper égukm_ f eaeli [,[ kmédy f (). dmen¼rsy GuKm_ f C due¼vmgmgerkleaeli [,[. eyigm f ( )( ) E ],[ eyigá d dueñ¼ d kln, [,[. rym ( mlkçn vä&yeróefo ). PBrYmd g rymmlkç&nð (Asolue d Codol Covergee) - 9 -
dukñwges riedr ekgmgmgerklrymd grymmlkç&nð. «mf ekgśkßgmgerkl Improper é f eaeliedceø¼nmyy. + eigmgerkl Improper é f rymekyyfgmgerkl Improper é f rymd. + eigmgerkl Improper é f rym EGMgeRkl Improper é f rik ekyyfgmg erkl Improper é f rymmlkç&nð. «ThrN_ skßpbrymd égmgerkl Improper égukm_ f eaeli ],] kmédy f () s. dmen¼rsy GuKm_ f C due¼ vmgmgerkleaelirkćeø¼tt&leakñ úg ],]. eyigm dueñ¼ s s d, ],] ehiy d = (- ) rymd ( mlkçn vä&yeróefo ). kln «ThrN_ ghjf GMgeRkl Impropre égukm_ f eaeli [, + [ km edy s,, rymmlkç&nð. dmen¼rsy GuKm_ f C due¼ vmgmgerkleaelirkćeø¼ TeAkñúg [,+ [. 6 ei (k ) + k. 5 6 e¼ s 5 k (k) 6 f () d f () d (k ) ( k) k k 6 f () d k k f () d rik. kln + k edy s rym e¼ eyigsñdæf f ()drymmlkç&nð., k*. eká - 9 -
lmhémeroti6 79. rk L f (P) g U f (P) kñ úgkrniimyy@dygerkm ½ (). f() =, [, ], P =,,, (). f() =, [, ], P =,,,, (). f() = ², [, ], P =,,, (). f() = ², [, ], P =,,, (5). f() = + ³, [, ], P =,, (6). f() =, [, ], 9 6 P =,,,,, 5 5 5 5 (7). f() =, [, ], P =,,,, (8). f() =, [, ], P =,,,, (9). f() = ², [, ], P =,,,, (). f() = ², [,], P =,,,,, (). f() = s, [,], P =,,, 6 (). f() = os, [,], P =,,, 8. Bül ehugviácgm GMgImYy@geRkme¼ mb. yk P CMEké [, ]. (). L f (P) = g U f (P) =. (). L f (P) =, U f (P) = 6 g (). L f (P) =, U f (P) = 6 g f ()d. f ()d. 8. (). ekegy P =,,..., CMEkmYyé [,], rk L f (P) g U f (P) MeB¼ f() = +. (). edyerismyr (). KN 8. smyrdulmh (8) MeB¼ ( ). f() =. (). f() = +. d. - 9 -
8. KN d edyeriplykekleli g plukeklerkm. ( ENM = ( )( + + + ) ) 8. ekegygukm_ Drhle f eaeli [, ] kmédy, f ()., (). ghjf L f (P) Uf (P) MeB¼RK MEk P é [, ]. (). BüléhuplEdleKmGsñdæf f ()d. 85. «mf f CGuKm_CéAelIeø¼ [, ]. ei g kméaeli [, ] ehiyusbi f ERgćMumYyMY r Gs e¼ g mgmgerkleli [,] ehiy duc GuKm_ g () g ()d f ()d., 7,,, usbigukm_ f() =, [, ] ERg =. Ckr,sŃs f ghjf f ()d 8 edyghjf 8 CMYEmYyK I EdlepÞógpÞ vsmpb L g (P) I U f (P) MeB¼RK MEk P é [, ]. k f ()d 8. 86. edyeri k ghjf GuKm_ f eaeli [.], ( ) kmédy f() = ³ f ()d. s s k s CGuKm_mGMgeRkl ehiy 87. edyeri s Edl m MeB¼ m, ghjf GuKm_ k siuus RgÜmMeB¼eø¼[,], CGuKm_mGMgeRkl ehiy s d os. 88. ghjf GuKm_Gs,Ú NgÉsülEdlRgÜmMeB¼ [,], ( ) CGuKm_mGMgeRkl ehiy ep( )d ep(). 89. ghjf ei f CGuKm_CeNþIreAelIeø¼ [, ] e¼ f mgmgerkleli [, ]. rk f ()d. - 95 -
9. ei f CGuKm_mGMgeRkleAelI [,] ghjfgukm_ g eaeli [ +, + ] Edl kmédy g() = f( ) CGuKm_mGMgeRkl ehiy f ()d g()d. ( ENM yk P =,,..., CMEké [, ] yk P' = +,..., + CMEk é [ +, + ]. ) 9. ekegy f ()d 6, f ()d, f ()d KNGMgeRklgeRkm (). 5 f ()d (). f ()d (). 5 f ()d (d). f ()d (e). f ()d (f). f ()d 5 9. ekegy f ()d 5, f ()d 7 KNGMgeRklgeRkm, f ()d (). 8 f ()d (). f ()d (). f ()d (d). 8 f ()d (e). f ()d (d). 8 f ()d 9. edyeriplukekleli gplukeklerkm / ghjf ( ).,5 d d, ().,6 5 8. 9. ei f CGuKm_mGMgeRkleAelI [, ], ghjf f mgmgerkleli [, ], [,]. 95. ( ). f CGuKm_mGMgeRklelI [,]. ghjf mgukm_ceniþr g g g eaeli[,] Edl g () f() g () MeB¼RK [,] ehiy g () f () g f () g () d d (). MeB¼GuKm_ f eaeli [,], mgukm_cenþir g g g eaeli [,] Edl g d. g () f() g (), [,] ehiy () g () ghjf f CGuKm_mGMgeRkl. 96. edyerilmh 95 ( ), ghjf eigukm_ f mgmgerkleaeli [, ] e¼ m GuKm_ C g eaeli [,] Edlf() g() MeB¼[,] ehiy g() f () d. 97. ghjf MeB¼RKǴuKm_C f eaeli [, ] m [, ] Edl ()d f f (). - 96 -
( ENM erilkçn kmuá k égukm_c ). 98. ghjf ei f CGuKm_vC meaeli [, ] e¼ f ()d. egy«thrn_edlghjf MeB¼GuKm_mGMgeRkleAelI [,] Edl f ()d, f mmáćgukm_vc met. 99. MeB¼GuKm_vC m f CéAelI [, ] m [, ] Edl f(). ghjf f ()d. ryghjf ei f CGuKm_CéAelI [,] Edl f () d. MeB¼GuKm_vC mekid f CéAelI [, ] Edl, ghjf f f () f () f f () f (). ryknmélgmgerklgerkm ½ ( )., (). Ar s d d. km f eigukm_imyygerkmmgmgerkleli [,] et ( ). (). f () f (),,, {, * }. MeB¼ ekg F() = d. (). rk F() (). rk F'() (). rk F'().. (d). sresr F() CGMgeRklé (e). sresr F() CGMgeRkl. MeB¼ ekyk F() =, MeB¼Mél epßgeto d (). rk F() (). rk F'() (). rk F'() (d). sresr F() CGMgeRklé (e). sresr F() CGMgeRklé. MeB¼GuKm_ F myy@gerkm KN. (). F'( ) (). F'() F'( ) (d). F"().. e¼ f() =, [,] - 97 -
d (). F() = 9 (). F() d (). F() d (). F () s (5). () os d F (6). F () 5. Bül f eiehugviác GM GMgImYy@geRkme¼mB d d (). U f ( P ) = MeB¼MEk P =,,, g U f (P ) = 5 MeB¼MEk P =,,,, 6. MeB¼GuKm_ F kméaeli kñ úgkrniimyy@gerkm ghjf F mdiepr gésüleaeli ehiykmǵukm_edriev F' eaeli. ( ). F () d (). F () s d (). F () s d (v). F () d 7. MeB¼GuKm_ f eaeli [,] ekkmǵukm_ F:[,] edy F() = kmŕk Mu [,] Edl F mdiepr g Esül ehiy F'() = f(). ( ). (). (). f () f (),, (v). f CGuKm_Þ q,,,,, p q f ()d 8. ( ). ekegy g CGuKm_mDIepr gésüleaeli. ekkmǵukm_ F eaeli edy g() F () d. ( p,q zmrvgkñ ) ghjf F mdiepr gésüleaeli ehiy F'() = g'()g () MeB¼. - 98 -
(). ei h CGuKm_mDIepr gésüleaeli ehiyekkmǵukm_ G eaeli edy G () g() h() d ghjf G mdiepr gésüleaeli rykmǵukm_edriev G' eaeli. 9. GuKm_ F : [, ] kmédy F() = s(s ) d. ghjf F' m ehiymdiepr gésüleaeli F( ],[ ) ryknmél ( F )().. ( ). km GuKm_ f eaeli gmyefr Edl ()d os (). kmǵukm_c f eaeli gmyefr Edl 6 8 f ()d f ()d,. 8 9 f,.. ei f CGuKm_mfñk C eaelighjfekgg f edypldkébirgukm_vc mcéki eaeli. ( ENM sresr f ' CpldkéBIGuKm_vC m. ). skßpbrymégmgerkl Impropre gerkm ½ ( ). d d (). (). e d (v). d. skßpbrymd égmgerkl Improper gerkm ½ ). s d ). s d ). e s d d).. ekegy f CGuKm_eAelI ],] kmédy f() = e s d f ()d rym EGMgeRkl Improper f () d rik. 5. skßpbrymégmgerkl Improper gerkm ½ ( ). d d (). os (v). d (). (v). e. ghjf GMgeRkl Improper (v). e d d 6 d - 99 -
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