oloklirotikos_logismos_2

BAffEAH 'NANAA|OY

oAoK HPOT|KO>AOt-t>MO>2

oEoPtA - MEOOAOAOflA1600 AYMENE> A>KH:EI>

Etr)r.ernrrrd * Arn)rd * TprnloOIoKInp6poro

l-ro 1pr1or1 rtov gor4rciv rov A.E.l. r<ot T.E.l.

4N EKAO:H

(AlOPA,,BtBhro6f 14 Avciteptov MoOqgottrdrv

AnoyopE0erol n dvqnqpoyoy{ pdpouq i dlou rou p0xou duro0 [e o_noro6inore p6oo i rp6no (prlxovlK6, nêKrpovtKd, qororunLKd. (In.)odug(J)vo lre rouq N. 2387 / 1920, N. €01 / 1929, ro N.A. 3565 rou 1956.

12:!11? Kgr r9!U. 1 q0 / 7s. xoo(c v.lrlr'm ^6"

np6trl 6K6oon: 1988Ae!€prldK6oon: 1990Tptq 6K6oon: 1995TCropq dK6oon: 2000

ISBN SET 960 - 7007 - 00 - xl sBN T2 960 -7007-02 -6

A(epo 040novEntqrnpLoKd pLp^(o

tuyypoedoq: Bqyy6^nq :rovddyoqTirôq Prp iou: OloKiqponKdq noyto!6q 2Copy ght tuyypog6dq - EK66oerS ,A(Opo'EtlrlflEtd 6K6oqqS: Ntrq((n Tpdulo!Eknin0roq-OAps-Mowd(: X. Zoxop6flouôq, A. :ttdpoq -

:. Zqxqponoiôu O.E.AEUK66oq I (Mooxd,ro), rnl. 48.17.981

OY'IKOMAOHMATIKE: EKAO:EI:- BIEAIONAAEIO (AIOPADlrE:040ffl0Y1 * 106 81 AOHNA * TnL.:33.01.269 _ 33.02.622rnl€oUororuno: 33.02.622 * n)€Krp. 6/vort: [email protected]

np6Aoyoq

O r6Foq ourdq (6eftepoq rnq oerpdq "OIoKInporLKdq noYtou6d), o-neueiveror orouq eoLTnrdq r(])v Avotorov Kqr Avor6p@v :Xokbv Kol Yevt-Kd oe 6oouq ooxoloriwor ue ro [o€r]!o Kd.

tro [p6ro Kegdtrolo (nou onorele{ ouln^f]popo rou A r6l.tou) Y(verotpLcl [opouoiqon oplopivov ovordpov lo8d6orv oloKlipcrons (oôd.n-poon pe rn Boilee(o o€poloFdr(l)v, oôKNlpoon Fe m poieELo r(l)v ou-vopriqeov Biro Kol fdpq, K.d.). fivetoL oK6po qro Keqdlolo our6 pto

nclpouo(don lov eMErmxdrv olo(lqpop6rorv pe noÎdq nuu6veq AoKn-oeLq.

:To 6enrepo KeedôLo nepLdxsrot rl eeoplo rov ttnl6v oloxlnpo]pd-rqrv, or egoplloydq rouq Kor 6rdgopeq poeo6elserq. :ro i6ro KeQo^qton€pdxowor 200 nepinou ovolurtKd nu!6veq AoKloetq KoL [lqpo6e(YForo.

To rpiro KEqdldto ovoqCpcroL oro tptnld oloxllp6porc Kol drtq e-qoppoydq rouq. nepLdxer pego6oloYid Kot rlepinou 130 qvq].unKd NJpd-vEq AoKioeLq Kor nqpo6e(Ypotd.

nrq€lo drr Kor o r6uoq ourdq eo onor€Moet dvo onpowlK6 FoionFoyro rouq OolTr]T6q Kd rouq :nou6oqrCq. An6 q eioq ourn 6Eop6 uno_

Xldooi gou vo errxoptornoo rouq Fo8rlpottKo0q Mopiq Mlovol6v (T. li-Kropo rou flov€nrornuiou rnq TEpYdornq), fr6pYo no6dpo (QpowlornAvordrnq EKnoi6euonq, r. Bonod tou E.[,4-n.) KoL E dvn :ouArovonoi"Aou (Koenyirp(o AuKeiou - ouyypogio) yro nq notuit+Eq uno6ei€€LqTOUq.

A6nvo OKr6Fproq 2000BoyydAqc :rov66yoq

S IBAIOIPA'IA

o) Ellnvlril B]Bl|'oypAlo

l- TMANH EY>.: 'AoKioElq Avurx6pi,rv l4oonuorLKirv" Aoivo 19672- TE0PTAN0n0YA0Y f.: ":roLXELq ihensorLKiq Avol0o€oc", Tdtoq 8'

(E(6doeLc Aooa KupLoktdn) eeooal\ov,Kn 19853. AA:KAA0n0YAoY A.: "Avdr€go uoo4lorlKd III" Aoiw 1987

{E(66oeLs tu!Edlv){- $f lnoY A,: "AnELpoor!(6q Aov!o!dq", Tetxoq T'5. MI>oYAH 8.r " l {oonlorLKi Avdluon 8 "6. nPIKq [ , : " l lqoi lqto TEvrxl iv l {o0n!orr({ i lv I I l "7. nANATI0T0n0YA0Y A. : ">roLX€ Lo AnELpoorLXo0 AoyLo!00 2"

(EKOdoE !q Kopounepdnoutrou )8, >EPgtTA E.: " l lq0n!orL(d" A0'1w i9779. T:ATm f.: "Avdx€po l,{a€nuorrKd ll" o€ooolovaKn 1986

f0. T:IT>A A.: "l'lqgisoro T.vLKiiv llo€nuotL(liv" rdu, I l-Aoi!,u 1985ll- oAYTZA H H.r "l.bongorLxdq AovLogdc llo/ouvopxioer( nolldry !€-

roo;\ni!:v" (university Stydio Press) 8€ooalova|(n 198512. XAINH I.: "l{oeiuqro Avrrragr,rv lbenuorL(ri,v" rou.0'-Aeivo 198413. XPY:0oAKH A.: ":xoLx.ad 0loKlnooruod AoyLouod" Aoivo 197514. XPY:oYAA A.: "oloklnpfiLk6c loyLo!6c' I I Aeirr 'o 1965

9) =ev6yl&on Bt,8llovpqad

I . A S T L U N D E . :

2. AYRES F. :

3. BALI [ . P.:el{ York 1963

el{ o€lhi 1985

er Delhi 1986Analysi s" l ' loscor 1976

6. BENGT KLEFSJi i : " lentamest natef ial t i l l ANALYS II I" Lund 19707. BER|IAN G.: "A Problem book in } lathenEtical l4oscow 1980

AnalYsi s" ( l4IR)

8. BRAND L.: " l4sentorLKi Avdluon" MErdqpqon E. i l .E. Aedvo 19849. BURKITL C.: "A First Course in Mathenat ical Analysis,

Canbridge lJni versi ty Press 1970

Aeiw 1967A0r1w 1986Aoivo 1950Aeivo 1986

"A First Course in lhtegrat ion" l {ew York 1966

"Dif ferent ial and Int€gral Calcul usr l(Schaum-s ol t l ine ser ies )" Integral Ca I cul us"(Larmi Publ icat ions )

4- EALI N. P.: "Engineering f lathenat ics"5- BARAIIENIOV G. Kln.:"Problems in l ihthematical

( M I R )

10. DEMIDoVICH B,: "Problens in l ,4athenat ical Analysis" i4oscol{ 1976(i1lR Publ isheB)

l l . EFIMoV A,: " i lathenrat ica] Analysis" l . loscow 1980( lR Publ ishers)

12- ELLIS R.: "Calculus" New York 1978(Harcourt grace Jovanovich Inc. )

13. XIAF A.: "Calculus Refresher" l {ew York 1950(Dover Publ icat ions)

l{ . l . lALlC S.: "} lathematical Analysis" Eombay 1995(t{ i ley Eastem Lmt. )

15. PARZYI{SKI l { . : " lntroduct ion to MathelBt ical Analysis" London 1982( Int .rnat ional Studeht Edit i on)

16. TRANDAFIR R.: " l ,4atematic i superioarer '(Editura Facla) 8uclrett i 1976

17. I IEISBECKER H.: "The Calculus Problem Solver" N€w York 1985

18. l { lLLIAl lSol l J. : "Lebesgoe Int€grat ion" l ieYr Yorl 195219. ZAANTN C.: ' r lntegrat ion" Ansterdam 1967

20. DEvl l lATZ A.: "Advanced Calculus" Nel lJ York 1968

{Holt Rinehart and } l i nston)

2l- GI8SoI{ 6. A,: "Advanced Calculus" ew York 1958(St. l lart in 's Press)

22. HARTMAN S. andl"The Theory of Lebesgue l , leasure and lntegrat ion"I , i IKUSINSKI J. : l (Perganon Press) oxford - London - Neu York 1961

23. HAHKINS T. | "Lebesgue Theory of Integfat ion" Madison 1970(university of l , l isconsin Press)" l n t rodL rc t i on toCa lcu lus " Massachusse ts1969(Perganon Press)" A n a l y s i s " I , I I ,(Addi son l , lesley Publ ishjng ConDany)

2{. KURATOIISKI K.

25. LANG S.:

25. IANG S.:

27. l.lcSHA E t. J.

28. sll,tiroNs G. F.

"A Second course ln Calculus" Readina 1969(Addison l{esley Publ ishing Conpany)

Reading 1968

London 1961"Integrat ion"(oxford Universi ty Press)

"Introdyct ion to Topology and Modern Analysis" 1963(tic6raw-Hill Eook Company Inc, )

29. l \ATAl iSoN I. P.:"Theory of Funct ions of Real Variables" Vol. 1,(Frederic Unger Publ ishing Company) Nen York 1955

30. NIKoLSKY S. l '1. :"A Course of l ' latherut ical Analysis" I , I I t loscovi 1977(l ' t IR pubt ishers)

31. PHIILIPS t . G.: "A Course of Analysis" Canrbr idge 1960(Cambridge Unive|si ty Press)

?- PISKU 0V . : "oi f fer€nt ial and Integral Calculus" l loscow 1969(f l lR Publ ishers )

3. PRoTTER lt. H. &l'l,iodern llathg||atical An.iysis" Reading 1972l. toRREY C. 8.: l (Addison lesley Publ ishing Company)

31. RoYoEI{ H. L.: "Real Analysis" (znd Edit ion) l {ew York 1968(The ftacmi llan Conpany)

35. RUDI I . : "Pr inciples of athematlcal Analysis" l {ev/ York 1954(l4ccravl-Hi l I Book Company Inc.)

5. RUDIN . : "Real and Coflplex Analysis" (znd Edit ion)(l,|ccl"al'/-Hlll Sook Company Inc.) Nelil York 1974

37. JEFFERY L.: "The Theory of Funct ions of tv{oReal Variables" Toronto 1953

38. HAYES A.: "Problens in l4athematical Analysis" N. Jersey 1952

39. ol tUt)oNNE J.: "Foundat ions of Modern Analysisl Nex York 1960

{Academic Press)

{0. HARDY G.: "A Colrse of Pure l lathematics"(Cnnbridge l jn lversi ty Press) 1952

NEPIEXOI' IENA

XEgdloro I: Ynor\oyLo|ro( olod4pt,'ldr.lv - EMeLftLKd otroKlrp6scra

YrcloyLou6c opLou6vou oloKlnpduqroq ue rn Boig8LqnE[epqoudvou oopo (ouoroc. . . . , . , . . . . . . , . . . .oel16q 9

0 i o K l i p o o n ! e o v d r r u v l q o e o € 1 p d , , , . . . . . . , . . , . . . . . . . , . . , . . . . . . . . ' 1 0

Yno.\ovLsldq o,\oKtrnprrrudrov uE rn BoiE€ro r0rv suvoprioeov Biro KsL

T d u u o , , . . . , . . . . . . . . . . , . . . . , . . . , . . . . , . , . . . . ' 1 4

A Y M E N E : A : K H : E I : . . . , , . . . . . , . . . , . . . . . . . . . . . ' 1 5

Yflolov[ot6s oioxtrnpot&r,rv uE rn Boi€ELo uLoc nqpo!6rpou. , . . . , . . . ' 27

EtrtreLnrLKd otrol(lnpdlorq. . . . . . . . . , . . . . . . . . . . , . . . . , . . . , . . . . . , . . . . . " 31

A Y | , | E N E > A > K H : E I : . . , , . . . . . . , . . . . . . . . . . . . . . . ' 3 5

KEpdlslo II:

ALdeopec unolvio€rc qn6 io

[w^d olorl4pdFto

xd)po R'., ' 39

"45" 5 t"53,54

"58'66,77' 88n 1a2. 223" 232

' 251" 254' 259' 260" 264, 270" 274" 276

lLf i t rd 0loKtrnpriuqrd.. .

I6rdtntEC tou 6Ln,\00 otrox,\qpdrUor o(. .ALd6oXLkd oloxlnpduqlq. , , , , . . . . . . . . , . . , . . . .YnolovLo!6q rou 6l1110! o,\oKlnpLl!fi0q. . . . . . . . . . . . . . . . . . . . . . . . . . . .

6LdqopE( l le0obe0o€L(, .l 4 c r o o x n / d r L o ! 6 C l o u 6 L n , \ 0 0 o l o x t r 4 p d , U o r o q . . . . . . . . . . . . . . . . . . . . . . . .

E0dpuoytC r0lv 6LnI6v otro x InpoUdrl,v , .Auolnp6r€p4 0Edrpnon rou 6Lr,\ot oloKtrnpri)loroc..A Y M E N E : A : K H : E I ' . . . . .TEvL(€uuCvq 6Lnld otroKtrtpd)udro. . . .A Y T 1 E I I E > A : K H : E I : . . , . . .

l:€dldro III: TpLnld otodt0t[rqYnoloyLoudq ruv rpLrld)v oloKlnpoldrolv. . . . . . . .npoo6ropLo!6q tu)v opllrv rnc otroKl p0ronc rpLntrot oô(trnpdruoroc...Yno,\oyLo!6q r0lv rpLf l Idlv oioKlnpoUdrov U€ lerooxnlo!1otd,, . , . . , , .M€rsoxrl t /drLol.r6C o€ KutrLv6pLKdq ouvT€TqvUdv€q.. . . . , . .MerooxnpdTLolrdq oe cpoLpLKdC ouvreToyUdveq.. . . . . . ,Eqoouoyde ruv !p!r^0:v otroKlnpdudrrNT€v LKEUldvo rpLntrd oIoKtr4pdUoro.. ,

KEOAAAIO I

YNOAOTI:MOI OAOKAHP(lMATrrNME AIAOOPE> ANOTEPE> MEOOAOY>

EAAernrrrd OAoxAqp6poro

- I. YTIOAOTIDIOI OAOXAHPOI.IATON

o) Yr.fovqac rd ogrluGru, otort|l*rtoc g€ q 0o.10€ra rcir€rl|d-voo dgorqEroc

noptro€ lyucrc :

l. io urcloylooca ro ororMg$a lr'zdx-

lOoI:E (vou r= ! l ; 6=1 = 6y=1 1 ) .

I x ' ?dx= l i ! ' h [ 1+ ( 1+h ) ' ?+ ( 1+2h) ' z+ . . . + l 1+ (v -1 )h ] ' ? ]=I n to _' = t i ' t ' 1 t *1 t * z t ' *1 r . )+ ( 1+2 .2h+22h , )+ . . .+ [1+2 (v -1 )h r ( v -1 ) ,h , ] l =

= l imh [ ( 1+1+ . . . +1 )+2h [1+2+3+ . . .+ ( v - l ) ] +h , L l , * zz r . . . r ( v - t ) , l l =

- l imh fv *2h ( v ; l ) v +h '? ( v -1 )v . (2v -1 ) l -h ' o I l

' r T r l= l im

Lhv , hv (hv -h )+ T (hv -h )hv (2hv .h ) l= l imL l+1 ( 1 -h ) . + ( I - h )1 (2 -h ) l =

To opLo!6vo otroktrnpli,Ioro unoJ\oyLcovtoL (oL ov ro 06ooioolu€ ocv ro60Lo ev6c nercooou6vou copoaotrcroq.

'Eror ov n ouvdprnon f €avo[ oovExiq oro KI€Lor6 6ldornto [o,8] r-oXt€L o €n6u€voc rlnoq (nou rpo(inrE! cn6 rov op!o!6 rou opLotravou o-IoKtrnpd!sroq):

onou r,= Au9F

I f ( x ) d x n i ! h l f ( o ) * f ( o + h ) + f ( o + 2 h ) + . . . + f l o + ( v - 1 ) h ] 1 ,J h.o

t0 XEOA-I\AIO I

t+1++2=2++=+.

r | . r r z I 1 1EElrlS€uon: tavol lr 'zdx=l i ' l-=i -.3- =:.I L J I T

Ilapcrilpnor:

0 nponvod!€voq nlmq no(.pv€L (qr rn lopei:

Ilaodb s ryua :

tta urolovroo€! to olroxlilru: Jx3ax.

M('|:ECvqL r!=2 =r

"=h. 'rxout,.,

I x3dx= ] i n ( r -1 ) ( 1+ r ' r 3 + r ' ? rs * , . . * ru - r r ' u - ' )= l i r n ( r -1 ) ( 1+ r r+16+ . . . + r " v - ' )=J P 1

= t rn ( r -1 ) g : I l =11p -1 11 'u -11 . r im d 12n- r1-r ' t re_l ts1 re_l F1 r{_1

=tsr i r 1 =f .r+1 r3+r2+r+1

B) oforMF&n Ie ordnluyF oe oerpd

l.lLo EuvduooeLpd uIIog€L vo llopqyoyLooe{' i vo otroKlnqreEa 6po flpoq 6-po, oe onoLobdflore 6lddr4uo,nou repL6x€rq0 oto dLdotnuo nou quri ouv-klav€1. 'EroL ,\oL[6v unopo0ge vc unoloya(ouge otro(trnoduqro, oood IIOon-yo!!6v{rq ovond€ouue rnv und !o oJ\oKl\iO{dud ouvdOrnon, oe oeLodl.4ac laurin* (i o€ o€Lpd (ord Tov r0no rou 6Lrdvuuou rou Newton+,), Av

* o r\tros tou uac L6uin .i!dL:

r rx)-r(o), i r ' (o) ' t ' r " (o) . i i , - " r0, . . . . - l f r 'u ' ro)* . . . {u.n*)** o rJros to! 6ldviuou ro! Nerton eildL:

(1+x)d=1+ + x+ d(; i ) * 'a+ o(el)(h2) x3+.. . a€n

I

J f ( x )dx= r im(F I

dnou r"=*

+qrf (or)+or '?f (qr 'z )+.. ,*or ' - ' f 1cru- ' ) ])fof(

ls\?

o )

>0

OIOK HIATIXOI IOTIXMO' 2 Xtulodyou B. 1 l

n o€Lpd (duvqloo€Lpd) ouvKILveL, T6Te ro otroKtripr,luo eLvql aoo ue to d-

opoLouq rov oloKtrrpoudrov rov 6pov rrtq.

IlapdoE rylrq:

l. l{o unoloyrooel rc otor}lmn: I e-t"ar,r,l o

^dol:E tvs l : e ' -1+ n . h . n . . . - f t - . . . (1 ) . H oeLpd our i ouv ' ( r . -

veL yLq Kdes t€lR. H {1) yLo t=-r, iz, 6Lv€L:

"-'"1-s. s s - t2)

EneL6i n o€Lpd (2) €lvqL ouyxl lvouoo yla KdgE oen, 6xouuei

L-'"=lt'- * - + - S -...1*=l*- * i,,,.. j,. i"'*-....d 6 r ' . r i . r o - . r o

, . " 1 f r , r r l ' t f r , r 5 1 "=t,l;- + . t 1.. n _ + l,-...=* r.l-' "'. "t -'- # "'.'.

2. ilo moloyrooE( ro otortirqrc: JS

*,(olorMruro mu Dirichlet)

ftton:Elvor: 1px- + - $'$ - i i -.. . 'r- 'r" Tff i *... , ono,",

Y=,- j--+"+. (1).H oeLpd rou 6€urdpou !6louc rnc (1) ouv(I(vEL vLa Kdoe x€R. Enog6voq

ecuo., Jg *=l(- S. $ - f . ..;*==Jd,- J* d'. I+ d"- Jf .". ..='-#*"-++,'++,.'....

3. l{a unoroyLo0€a ro olodigqE: i t*-'' *.

ldott:

elvou: e"=t*o* $ - $. S. $ +..., -@<rr<&. Tr.o (,,=-x2 dxouu€

12 KEOMAIO I

5. llo uroloyld€l ro oloxMgan:

r.dq:

0 , 3

[ -"t dr.) /140

ELvoL: ouvx=r- $'S -. . . - t- ' r af i - . . .

f -tr-"r-r=r. t ".;+ "'. H+ ''.....l:f|s.fi;l],'....H npi[n o€Lpd ouyklaveL yLo Kdo€ xe]R kqL n 6€dte9n vLq -1<x<1. 'Exou!a:

" -ur={1-*-* - . . . ) l r -+-* : - } ' \ y I ^ ,

r 1 - x - s - i ' +

T 5 . x ' - " '

H oeupq ourx lL !€L yLo - t<x<1.

OIOK,1HPOT.IXO' AOflLttOt 2 Lrcrodrou B. 13

j' H"=j' (,-+ +".. +", )*=[,-{_ii.;i ...]'''=0,",i J " ' - ^ 6

6. ilq unotroylooe! ro otodipqr: J#.Mor:

geroirtrE lnx{ + x=e', tcvql dx=eodo. 'Apo:

ldx re"d' r ' . t .zr-S- * rr f . t , , , t ' , , .J t* =J =,i =J ---t- d(.r: J T-di,,+Jdt'+ Jt du+ Jtr d',r+

r . . .=tn o l*o*2| ; r , r r . "+.r r ' - . . . | . - tn, tnxr* tnx+* ( tnxt ' . r f , t r* t ' . . . .

* H d€Lpd * +. * - $ - . . . ' " r" i ' "" rLd xdeE o€rR.

l 4 KIOAÂIO I

y) YnoloyLoudq olorMq,rfsrv !s Ir| gorloEld Blv ouvogrilded/ Bilta

KoL ftlulo

Ilapcrignon:

Tn o| lyKtrLon rou yevLx€uu6vou otroKlnpd)uqloq f(n) 6€|: toue orov np6ro

r6uo (o€trLba 365). 8€lxvoule r{ i )po xsl ro endurvo:

f ,To fcvlxEugevo ofoxMpqq I x' (l-x)_-'dx (nou opae€L rrt duvilgqon

B(r,n)), ouyxtravsl egdoov n)0, nd).

A1166€Lf[:

o) 'Eorr r

1:m, 1<n, r6rE n duvdprnot f (x)=xd-r(1-x)" I eLvor ouvExiq

oro [0,1], dpo otroK^npdoLln oro [0,1] (ro otroxi \r ipous ornv neplnroo! ourr i

6ev ELvsr yevLKEUudvo).

0) ' tor0r 0<m<1, 0<n<1, 16r€ ro otrox;\ iprruq €lvoL y€vLK€uuavo. ECvoL:

| " ' - - t - * t " - a " - | " ' 1 t -x1 ' - o " ' l r ' - r { t - r )n - rax ( l ) .

t o ) tt '

lo otror(I ipuuq ixd- t f -r l "- a* ouyktrLver, bL6rL ov oeupioouuE rn ouvdp-J

r n o n g t r t - - ! e t v o u t i n * j j { - } = t i m * r ' - " t x ' - ( 1 - x ) " - - 1 . T o v € v L x € r , -

u€vo 6loJq o,\oKlipl,us:

:rov lTpd[o r6uo (oEll6o 370) €i6oue rouq opLotouq rov suvoprio€I,]vB{rq xqL Tduuq, xcgdrq KqL r:Lq r6L6rnr6q rouq.

:ro €n6lEvo nopc06Touu€ ouunltpolorlKd 8erclo, r(ou ovsqapovroL ocLqouvcprio€Lq qureq, nopqedTouue oK6uq KqL oioKinpduorq nou unoloyt<owqL

!€ tn BoiSeLo r:0jv ouvoprioeov BiTq KoL Tduuo,

YneveluCeou!€ 6rL o! ouvoprio€Lq Biro KoL Tduuo opa<ovroL ovrCoroL-

Xo qn6 rouq rUf louq:

g f * .n t= i ̂ ' - r f t - ^ t ' - 'ax . n .n 'o)

f ( n ) = I e - ' t " d t . n ' 0

o o(ÎiPOTII(oX l9lI[ttOE 2 rrav6dro!, B.

t l

I g{xldx-

| G,- dx, erEL6i l - r<1 ouyKtrCveL (EivoL t-m<t € m>0, nou) ! \ '

roXU€L on6 Tnv un60eOnj.'AOo ojueovo Ue yv0jar6 (gttipLo Kot ro yEvlt(Eu -t

lavo otroKtripoto Ixt-111-r1"-to" ouyKltvEL. €o 6eLEolu€ &jpo 6r[ to I.)

u. Jx- (1-x)" dx ouv(ILv€l. eedpouu€ rn ovvdprnon p(x)= I

I ( 1 - x ) ' {

ELvoL : r im- l l l =1 im 11 -x )1 -nx ' -1 (1 -x )n -1 - r .x *1 - a l x ) i ; i - '

En€Ldi ro t,0. | -0, ouyrtav€L (€avoL t-n<t € n>0), ar€roLi ( r -x) '

d r r (oL ro T .O . J " ' - t t - * ) . - t a , ouyK tav€1 . [ t o r t roLndv ond rnv ( l ),

r po ( tmE! 6 rL ro f . 0 . l x ' - ' ( t - x )n - ' dx ouv ( , \ ( v€1 .

'Aoo ro T .o . l x ' - ' 11 - *1 ' - r0 , , , y ro mro , n>0 ouyKravEr .

AI'IIIENE' AXF{:EI'

t. rk 6€.r@€. 6t. B(.,n)= +!l;i.l! , r,n€r*.

m- l I - - t- i : | " - - ( l - r ) " d " + 8 ( m , n ) = j : : : l S ( m - 1 , n + 1 ) ( t ) j +

+ B(m-t,n+t )= Tl sh_2,"*z).'EroL

n {1) ypdq€roL:

' 'orou ! r 8dlouue ro ! -1 roc drou n to n+r .

(XOAÂIO I

B(m,n)= +H? 8(m-2,n.2) (2t.

tnoylJivLKd EXou!€:

B (m,n )= (n -1 ) (n -2 ) . : . [m- (m- ] ) l B im- ( ,n_1 ) ,n+ (m_ t ) l =n (n+1) . . . Ln+ (m-2 ) l

= #*Lli$ffi err,n*,-r1 (3)

' E r o L n ( 3 ) 6 L v E L :

'(',n)= ;rn#l].{{*};ffir . nnin*+:6-r1" =

= r*-#*#.-trirh;r*,r' = ('il*!i;l)' .

2. tb 6E!)Oca 6t. B(l,n)=r, n€|l . .

l0on:/ n - r \ r l . - r \ l

H lo6rqro 8{n,n)= ]]:;=::]+T;r: , yLo n=l 6LVEL:

, r , - r - ( 1 - 1 ) l ( n - 1 ) l - 0 ! ( n - 1 ) l - ( n - l ) l - t", ,,",- -TI,n-:Dl-- - ----fi..r- - nln:TIT - -i-

3. th 5r!roe! dr! 8(r,n)= HP tt t, .,n."'.

Mo't:/ - - , \ r / - - r \ r _ r (m) r (n )Elvor. : B(m,n)=f f i= i i f i i i j '

( 6L6 r r f ( v+1 ) - v | + f ( v )= ( v -1 ) l ) ,

nopcri9||on:H (1) Loxiol qK6ud (oL ov m>0, n>0. H on66eLln daverol oov ddKnon

oro 6Lnld otroxlqpdluoto (d€lf6o 164).

{. lh 6€rx0ea drr yr.a q qndorlq f(ln ox{e! o rrlrcc:

raltl**)= 4- rto,t (!'o)\ . ,

2 2 n -(rtrcq 6L @rduort ylc tn ouvorndn r@uq).A[on:

o OXÎPOTIKO' ̂oII'!D[ 2 fta

Et'vdr: B(n,n)=+HP-9e1.,"t-ff i (r).

E[w! wtFt6 oK6gd (Btr&s At tdro, oe^[6o 372)' 6rr:. l *

g1m.n1=2 [ auz.- 1xo,rvz.- 1 xdx g, B(rn,m).z Inu2-1xouv&-1xdx=- ' - " - J - JO0

; tz=zi 1

znut** ;'"-' a -" i fu t nuzrrtu-1a'=7]1 f tnr2x)2"-1 zdx.

000=(o!rouu€ zx.o + 2dx.dtd"otaN t'0' tFo l(cl 6tov x=tl/2' tt=[)'

I r /2

. j=i I tnu"tt'-'a" # 4 (n!0,)2r-1do"

t ' o ' o

,

'Ato:

A116 (1) to! (2)

2

- fr zJu--',o,""t'** # tfr'*)= #B(,,').7kffixf (,).

rlm) r 1iTfrt 'v,=' + r(i ')r(r '-+)-Fr(rm)'

5. h uDroyrdca ro orclUr r.,,. I ru =!- ' v.r.L fi--if

ir'|lol fro x'=o eavdr zxdx=dt. To dpto tnc olorltlooonc noooravouv tq (-

6 !o . 'Apo l

r , - [ * * - [ xu 'x(r-x ' ) - lax. ] J {" ) ' t t - " ' ) - tz 'ax=i /r-x- i, o

I r(r)&+ | r(r)d:.I t

tlo0Kunt6Ll

A! t (2dz)= t ( t ) , {d re

18 KIOAMIO I

'Apo:

A v v d p l L o ( , a o t 0 l v = 2 ( , ( x € ) , t d t € n ( i )

- / v | \, 6 ' \z 'z l^v- -Z-

;fv--::\-' \z ' ' l { r) .

6Cv€r :

12) .',"=+ $i#.+And rnv monyodtevn doKnoi eavo!:

'G'+)- ++i# -,(.+)- 6 f-:'Y'EroL n (z) y(v€ror: r2k- + + #F*i - r".* -!- fffi r:r

Av v |leolttdC dfio v=2k+l (r€I), epya(dldor€ ovdloyo.

6. th 6e!)€ca d.t' '| (t-+) r:-rdrdiB(t,n+t ) (t>0, n>-tt.

fttdn: o

odloute + ={,J + dx-ndo. 'ordv x=0, r,Fo, 6rov x-n, o=1, 'Apol

n 1 1

j (t-+)"'"- '*=/ (t-r,r)"(no)t-1nar,r-ntl r,rt-rf i -r,,)ndo=niB(t,n+1 ).

l

7. rd 6e!x0ct drr J

x'1r-x'1n6'-{ r(# , .l (D or sdqrv0 1

va urolovrdea ro olor{ga. Ix3(t-r3)tdr

(n,.,p,0).

Mct: 1 -1-,o) €lroule x!-r,r + x-tjd €avoL ax- *r,rn aor. ,orqv x=0, o=0. ,O-

oôMH?aTIKo' ôIIXMo! 2 ttav6dyou B.

TOV X=l , O=1. EXOUUe:n 1 . t - - - '

I - ' ( r - "" )ed"-J 'n( t - " )p*"o ao-f J r , ' (1-o)edo= FB(+,e*1).

8) :uyKpLvovrqc ue ro nporyoiu€vo oloKAiputro, axoule nF5, n=3, p=3''Apq ond Inv (1) npoKtnr€l:

i , . , , iJ xs(1- , ,3) rdx=+B( + , 3+r-+B(r ,4)=+J x ' (1-x) 'dx=(ncpovovrrx i o-

. r f ( 1 - x ) r l r 1 f , ( 1 - x ) ! 1 ' rroKriruon)=+ * iiij i l.-+ 11 iGij- dx]= rz J {t*t"a"=- '6 ' ,

1 | { r - x ) 5 l ' I 1= 12 L 5t:Ti- I,= 5u '

8. frd rqrc oerlrcrtc oprolodc r, n vd 6e!x0e[ drr:

r -e-1B(! 'n)=l -- ' -- dx

i ( t+xr_--ftton:

odroule t}xl "d .t x= * * ar= -!- do, 'orov x=0, tJ=o. 'orov(1-( , l " )

x + +-, rdre o=]ll fi =1. 'Aeq:

J # d,=i(ri;)'-'('-)'.' fr *=('"" 1*=1. * = *)=. \ r + x /

= J i ! ' - ' ( 1-r,r)"- 'dr,r=B(m,n).

6€Lxoc. 6E! ,(*)=*.

urn,nr=1H# ra:r- B(+,+)= lq4+'\T 'T )

( r (1)=r ) . 'Apo ' ( * , - ! ) * ( * ) ' (1 ) .

t (* '*)=l l - ' t ' *11- '6*=J x-r1t-x1-rax'

19

9. llo

tulol:

- \TJ ,

E i.vd L:

'(+)'= --iTrt- =

20 KEOA/ItrIO I

eelouU€ x=nUao + dx-znUoouvode. 'orqv x=0, 0=0. 'otov x=l, 0-n/2 .

lzt.n / 2 a/2

'Ae., B(+,+)-l ofu J* rn,ro,"ra.=rJ ae+1e]ii'?=nO O

An6 (1 ) (o r (2 ) rpoKdnr€L o ru r ( { )=n .\ . /

r - _ r / . \10. rb 6erx0Ea drr J e-" dx=r(+J.

îlor:oarouue x'={, + x-or/!. tavoL ax={er- ' l lao. ' f tov x=0, r,r=0. 'orcv

x + F , ( ! + s . A p o l

I - ! | . . t ' 4 , t I - 1 | / r \ / , \

J e- dr1 e-+0, 'db"+ J e-u! , ' do=+r(+]-r (+) te.vo! T(n) =

=(n- l ) r (n -1 ) ) .

r /2 tt /2

It, tk 6€!)€ca Otr [ "6ae. [ + -.' " i@,tdq:t /2 n/2 1/2 \ /2

I nrrae. [ --!9- = [ arlse,rv,sae I nu-leouvoece='" ,o "i-'E

'. to

' l ' r t r- , 2+-1 ' l2 r !- , . z l - t= lnu 'eouv '8d0 . lnu" 6ouv z ode=

) t

=+'(+,+)'(+,+)=+

(fr ),11/4) __trf,14, "'

_1

=+r/2

IB ( o , D ) = 2 . 1 n u

0

3 \ . / 1 \ - / 1 \ - / / 1r(-i)r(+/r(+)r(+

2n-1 ?d- 1

0 OK APmIl(Ol lolttlot 2 tralodyou B. 21

(ono to"

/ / 1 \ - / r \ r / r \\€rvoL r(+,1=r'r rcu r(-fl =rir(ril, 6L6rL: I'{n+l).nr(n).

/ . \ r / r \ \or""' r\-f-l=*r(ri//.

12. k onofoylddv Eo otoxl||rqlE:

fr,o n=*

, /2

" n/2 a/2

at Ir. ,| r1l.ro,vu!fr, p1 I"= J ru'xar, v) I,= J

/A d000

ldat:n /2 , ! - . . , " ' . 1 7 \

o) EtvcL l,= J nuz'z-:,ouu' E -'16r'{ "k, il

- + ji$1 =o ' f .T /

, r(z)r(+) , 'rr(+) , r(+) , l.(+)=-z --;7T5\-'7 -7rF\- "z -TI-iz-ifT =T -Tt--1=r-7\'i|' \T/ , \Ti T, \ r - l T T' \Ti

(€(vd! r(n+r)=nr(n) !:!4 f(+).+r(+)=+ +'(+).t/2

8) Elvou t"= J nu2'u-rrouutl-trar'*s(+,*).0

, r(4)r(+)' '(0.+)

3fi

'+ #4 - t#l =.+9=r =.---3+?-,r-'\?/ z,\2) zT'\z) -r -r z ' \-r)

=-7-rfl-7Tf-t - , -r ' r t t , I

" 48 / i = l9 .

105fr

,r rr*" r,J| 1/W or=J'.rr.*"-t.0.= i'nr'i-'*"'i-'ror=0 0 0

. /3 \ . / t \.+'(+,+)=+ +H.#-+&+= +" riho rou 6!-

rtrosldduoo 2a-1r(n)r(.{)=n rf z.f , ym n-f, rpoxdrter.:

rtr(i)r(f,)=n r(*) * (*),(*)=rnn *,(f)r(f)=a ")

XIOA/\.AIO I

13. io E€rxoea 0tr I e*x'-1ax- I(nl'n o"

iilor:e6rouuE ox=01 + dx= ff . '0tov x=0, o-0

'Apa: I e-*x" -1dx- l

" * (g)" '9

- - ! [q _ ! ' ,' o

(o,n'o).

K O L o I q V X + 6 ! ( ! + *

e*r,r"-ldo- l(!} .

3z=t + z=* .

rG. io rrlroroyrd€( ro olorMqrn [,,lf;H e.

fit(r|:06rouue lnl-=2 + x=e-z. ELvsL dx=-e-zdz. '0!ov

x'f tqv x=1, z=0. 'Apo:

I

| ,.fy'"(*) 6a-l e-2"t2 (-e-z)dz=l e-szzldz-(earoure' t v ; i

+ 0, r6t€ z+-r-

E(vor dz= + ro dpro mpo!6yo0v - coot=ie+ I +*=+-i"..,0.=I ll " 3/r t,

- _l_ [ e-!t i- 'dt. 1 ,/A\= | , . .- 3'a l '

' "* 3,o '\-tl= - r(+.1/=# +t\+)-#n=]i

t7. lid 66!)€ea dr! yr.! r't rq! q>o caw! I.t""*'"'or=-t=rt'q!)| 6P ' ' t ' 1

orfirrv vo 6e!,€s( *. L.*""*" + .

ittq:- o) ,6rour" atxrEz + r.+. ,1*1 or=f or. 'Otov x-0, r.(). '0-

rovx++a ,2+ { - . 'Apo :

f b -r 'l' EL f !r1-,r= | + i" i- a.- -L, | "-"2

2 az. -{1 | e-,zz a,-i o - - - 2 t 'o 2d , ' i

| - / n + t \- ; ;+r ' \ - r l '

8) To npolyoduovo olortrtpolo vLo n.0 dav.!:

J "**"*=*'(*) * J e*"""dx. * (r).0 0

Ecvo'. 6sos I "'"""4'+ J "-'""6192 S . $ t"c*l ytod!6 6rL

J t1tlar=eJt1r1a,,' ov I f elvor dprlo ouvdpqor).

O oX]\llPmIXo[ IOIIEIIO' 2 trai6dyou B,

18. Av p or q eawr osr!.oa o0!0||oc y! !€!r@a 6E!:B(p,q+t) = B(p+t.q' _ B(p,q)

qpP lq

gir., g(p,-q.r) =1 j ,,p-, f r -rlqd"= * J

( r -x)qxts1dr=(noocyovtr.rlfttq:

ol

24 KEOAIAIO r

or'oxr,4pr,rn1= 1 f [11-x1c d l'- [ 011*yc-r 1-r ) + dx]=+ ; J xe(1-x)q-1d"=9 L L P I O J

= B ( p . _ 1 , q ) . , A ! s B ( p , q + 1 ) _ B ( p + t , q ) l i )p - q p

B l B ( P + l ' q ) = a [ , n 1 1 - , 1 1 - t r , - 1 t - - 'p p J p j x " x ( l - x ) a d x =

= 1 J ' r - ' J r -11- ,11 11-" )q- 'a , - * J xn- t11-*y , r - t6" - -L

J xp-1( t -x)qdx-

| - , . 1 - ,= l- 6rp,q,- -E- 6( p,q+r ).

'Apo. B(p+J ,q) =1e(p ,q) -Je(p ,q*1 (D B(prq+1) _=1-s(o .o t_c P P q P

- " ' ' '

- l s (p ,q * r ) + B (p ' -q *1 ) +0 (p ' , q+ l )=g& l3 ) + (p *q )B (p ,q * l )=B(p ,q ) -Pqpppqp

+ B(p ,q+1) - B (p ,q )qp+q -

19. t5 urolov.dodv E otalflrA[E: o] [ t' dr, !] f f.rt-Ttdt

/tdotf: tt rzn {

o) €droule x.zt. 'Apo ostsl. ' txoL dxoule:

[ * *= | S- zat ,en I t .1r- t ; ' la t - r ,a Jt ' - '1r- t1 l - '6s=io4 iEE i

, f r , . . "z ' \T)

=+-9=. = Yt -t5-* =E+- .T z ' \T) TT z ' \T)

B) e,tou!€ {'?-o'zt .7 69= --9- 6g Ko[ Osts1. Enolavnc Elvo!:2lE

J,r'c-['a,r= $ J ,irr-'r!a,= $ Jt]-'rr-l +-'at = +'(+.+)=llg6-3a'

o or']\sPoTll(oX tlolrlltoE 2

m.

Mq:

[(vor,:

tk umtovrdea Eo olorMgs [ !!d3-"' -'--t

1, (t;;iif- dx (''n>o)

1 1 t: l f

) i r* l-( (r-xJ' )n(r-x)-.dx-1r' t+x' r \ l+x, , /

- t [ ( l *x ) ' l r f ( r - " ) , tn_ t t__ :_ ;_ I t_ | dx ,j rL r+x ' . l L t+x , J ( t+x) ( t_x)

odtoule !!4: =21 a. ! l1+xl2 -,f+12

' - - Z 1-7 =t ( l ) ' E(va! d(dud:

r_t_t_+ (t+x)z * I l1gl1 =,-,- l+xz " 1*x,

H (l) 6av€r: -b1 6^.j1 .+ dx- { lrxr). dt( :+x r y . t - x .

-1 , a,= -J!:11- dt= !!4)(t.x") ,. | 1t - r r 11-,z j l f l " ' - ( , * , ' ( | ;not-zTA]:Udt.

'mo: Jz\9"1raf .=l- dt-2!+D-z i t!-r ( t-t)!-1at=,'h-?s(n,n )_; z , t ( t_ t ) ;

".."* ,ltl${i) .

21. llo uofoyldatv E, olorl|I1*rro:

al r= j-3-r,*'dt.

,ldq:

o, E(vd! :

! ) I = [ I dx .'. 4lni v, r=J x-{tnrl.tx r,n>o

j' +-t=J {eln3)-rx"ax=J

"{*"1o3d", earouue 4x.ln3=t + x=

=m-t' + dx= -:_ __l_ dt. 'Apo:

'= J "* # t-rdt- -+- J .r-'".a,- ;# r(+)_ #.8l odtouls -lnx=t + x-e{ => dx=-e-tdt. ,Apc:

26 KEOAAAIO I

I = | e - " t - t 1 n 1 - " - t 1 6 t - ( - t ) ' I e - r b + 1 ' L t b d t !( -1 ) " r (n+ t ) - ( - 1 ) n . n I(m+1 )n+r (m+l )n+1

* Bratr . { ioknon 13, d. l i6d 22.

oAoKIHPATIKoE lorlluol 2 tra!6d-rou B. 27

6) Ynolovlo!6c oloxl'|qUdEv Ua m oodo€ld tltdc moq|etoou

0pLoUdv€c uop0dC oloktrn&iudlov uno,\oyC(ovrct !€ ELoov0rytl uLoc rlopq-p6rpou ornv un6 ro olodiqdlo ouvdprnon xo! Eeoptovi l(ordnLv rou (ov6-vo rou Leibni tz*.

nspa6€ [ytora..:

l, l{c urotoyr.o0e( ro oloxtiF|! I # *.

ldon:[Lodyoulr€ uLo nopdlerpo q qtnv un6 ro olo(lipota ouvdornon. 'Ero[ Ea-

voL : F (q ) - | + ;+ dx (1 ) .I I n x

EavoL np@ov6q F{l)= J i# di(. EnELEi ouvrpaxouv oL xotdl\ntrec npounooC-

o€rq, ![opodue vo nopoyrJyLdouue tnv (1) oC npoc o, on6r€ dxouu€:

dF (q ) a I " " - t - I a / x ' - r \ , f a / ' o r \ .- nq =ad J ] - * " J i o \T6F / " , " J 5s \ i " i - r ; i / d , -

t / x " t nx " \ - f . , f x ' - ' l r I= J \ _rn" _,i *1 * "x=1;;r J,= fi .

( t? dF (d )= d ;T dq ? F (o )=Jd ; l da + F (o )= ln (o+ l )+c (2 )

0o urtoloycooutrE rn qroeEpd c.

I " o - r IH ( 1 ) v L a q = 0 6 t v e ! : F ( 0 ) = | i = - a r + F ( 0 ) = l o d x + F ( 0 ) = 0 . H ( 2 ) v L o

I r , r ^ J

o - 0 6 L v € L : 0 = l n 1 + c + c = 0 . H ( 2 ) y r o s = l ( E a v d l c = 0 ) 6 a v € L F ( 1 ) = 1 n 2 .

* Bl i r . xoL rpr i to r , i !o del i6d 228.r* l tc err juevc rapo6er j ruqro uox, jouv r5res o!" rpouroodd€Ls yLd rnv .odp -

r,ovri ro,.t xcv,jvc tou leibbitz,

KEOA]\AIO I

t lvor. 6ruc F(t)= | {;l a*. 'rpo, [ {;} u"=rnz.

J " ' ^ I

z. h urotoyr.dca ro olorl4,ru: I1--'*"r*.,trtor:

| --z' rotor F(o)=J e ̂ (tuvoxdx (l). Eavs! npoeov,c 6tL F(t)=J e-x-ouvxdx

0 0 j $ {e-*'ouvox)an * # =J - [e-*' (nuo, ) r] ar.

t r t l=*J nuox (-z"e-* ' 16.= | .J (nuox){e-* ' ) 'ax:} l r i ' l rnuo,r; . ' l ' -- Ls*t L

-i "r*"o"u-'al -] [-i-",**,,t-'ol -- gi1o,,uo"y"-*"a,=- g r101.

'Aea dF#) -fr1oy = !F{*}-+" * JSorr"r=J--ic" *

+ tnF(o)=- | +c (z ) .

0o unolovCooutlE tn ordoepd c.

H (f ) yLo o=0, 6(vEL: F(o)=J "-"d*.

EauoL drrc I "-*"a,= T..

,0.

F@= + .H (2 ) v rd o=0, 6cver : tnF(o)=c * n $ =" . .Eror n (2 ) 6b.", ' . G

v€r: rnF(a)= *""* >r(o)=J ** 7 + r(o)=+gT g

+ rlt)- ri "-t. ooo, J "-*'ouu"a*=r{r)= $ "-i.

3. lh umloyldc( to otortllo|!

itttol:

[ ( v o L :

* rcooyovturri olorr'ipoor.** Brar€ A! r6uo, o€l l j6a 3?2.

I e_n l | r .I --r- *'

O OXMrtnI(OI ltotlttlot 2 E8\,6d1ou B, 29

rtot=[ellu dx (1] (q>o),onfte: F(r)=J "Trr dx. H fi) 6[vor!

arlp..f + /e-Tnux)o*=[ :I{!ul6x.-[ g**nu,6,0c J ac \ x l - l x I

* s. -J e-*rurdx (2).0

rtwr Je*nrrxax- J e{(ouvx ) 'dx.-e {tuvx+ Jouvx(e*)

'ax,-e-axouvx-

-o J e-orouvxox=-e**bu**

[J "-*tnrrr t 'ar] .-e -"ouvr-o

["-"*nur*oJnuxe**]==-.-uorvr-dequx-o.

J nlle*ax.t l'Apa: J e-"*ntxdx.-e-*ur..-*nr*-o' 1 e{xrtuxdx + (li{!)Je'axtuxdx=

=-e{rouvr-qe{ttux + L-o*nrrxar.:95 (ouvx+crtux).J l.dr

'€ror n (2) 6rveu: dF#) =f e* (*ur.onrrr *

+ dFSt) = -l +L l*o. J" - t+or

(et"o., Il ff r-""-'l''t'# U* p (ouvr,r+oqett)'o)

+ dF(q)-- .-! ao * r(o),- f-J- 6o*" *1+o. I 1+a2

+ F(o)=_roq€oa+c (3).eo ulolovaoouuE tn orooepd c.

'orov o + r-, t&e n (3) 6t'ver 'llm F(o).-lin roE€0o+c .+CH

+ l|| | F(al=-++c (4).

H (l) 6av€!: lm-r(o)=lim_ [ eTlx ar.O, 'Eror i (4) 61"e1 g--{+c +

a, g.{. 'npo:

F(d).-roEE0o++ l:1- F1)=-rot€.01++ .r F{l):+++ + F1).+ .

I "l'u'*-+'Eno!6vric:

30 (IEA]\AIO I

itor:

'h(l+,C)'z

tld uDloyrd€a to olorlrbtn:

oEr,rpouuo rn

H ( 1 ) 6 a v e L I

ouvdpqon s(q)=[ -9 (t). rcuot,Jo Cl+X'

s ror=f toreo( | ) Q).l d \ l d /

s ' (o )=- [ - -g- (3) .,o (0+x- J-

I

H (2) 6rv€L: s ' (o)* zd*T-;*reo(f )

(4 ) .

(3 ) 6xoure: s ' r l - [ - i - 'ro ( t+x. ) .

(4) lxouuc: s'( l)-+-+ '

I d' =+.+.Jo ( l+x! ) '?

An6 lnv

An6 lllv

'Apa r

O]\O( !PATIKo' AolIEMol 2 trq!6dyou B, 31

e) Ell€Lmrrd olodmqrtq

l. Ellcudl €MELmlrd oloxlmdtaEo (il olorhpd ard rou Jacobi)

0votd<ovror EII€Lf l i e l leLnrLxd oloKlqpdUqrq d' , B KsL y'eLbouc,

ovrLoroLXor ro oloKlnprllqro:

l ) F , ( u . x ) = [ d u' Jo /(1-l,l5(it:rrT

0 . ! . 1

0 .u .1 l2 )0

0 .u .1 (3 )3) F3 (u,x )= [ ----gJ ( 1 + K d ' ) / ( 1 - u ' ) ( 1 - u ' o ' )

r =[--------!tr' / ( l+xr,r ' ) / (1*: ' l ) ( 1-u'zo'z)

! € 0 < ! < 1 .

napariplon:Tq etrieLnd €itr€LnrLKd otrokllp6uoro

lorLou6 o=nu0** noLpvouv rLC €nduaveq

t l . ( r : , 0 ) = [ d P' I /I:trr|u-0

0 .u .1 (1 )

0

x oloeep6c qpL0u6c 6Ld0oOoc rou pn66v*.

2. n \ilfi €ll€Lmlxd olorlll9ttoro

A! x=1 , r6re rq nponyodlevd otroKtrrtp(i)uqlo ovoudaovlor nlilpq €ll€L-rnlrd olor lngdgcm q' , 9 ' KqL v' el6ouq, ovraoroLxq. ' [ roL Tq f l \ i lpnetrtr€[nILKd olo( lnpduqlo eLvd! xo:

( 1 )

( l ) , ( 2 ) K o L { 3 ) I e r o ! € r o o x n -

uopqdc (uopqdq Legendre):

'1 -V262

Av r=0 , r6 re ro o roxrdpoud d ldyeraL

f r a 0 : 0 + O . C r a , y u ( o - r + a = ]

drn ! rpor f lopr r t .

( ;d t r l . x i ! f t1 dxdud d !=du lqdo.

KEOAAAIO I

I

l ,(!,S)= | r '1-u'4u'e de0

L t , ( r , , O ) = [ : d e ; IJ ( t+Knu'q) r '1-! 'nU'A

0<tr<l ( 5 )

0.u.t (6)

(K oxo0ep6c 6L@opoq xou Un6av).

1u 6={ , t6re rq nponvodgsvo oloKjupdluoE ocvouv ro nlipn €}kulrr-

Kd oôktrno6lord d', 9' KoL y' ea6ouq, ovraororxd,

Yrcloyr46c udv €ll€lrrrdy oloxl'lHdq|I

To EII€Llr!Lkd oloKlnpdruoro urotrovaaovror npoo€vylor!(d 6Ldrl rq o-vi(orolxo c6proro olo(lnpduoro t(av €IlErnrl(d]v o]o|(Juooudl|rv ,6ev €CvdL6uvordv vo El(ppoomtv !e r€rcpoolavo aploud oroLx€!0r6d,v orrvoprioelrv.

TLo vo ufloloy{.oou!a avq elle!ft1(6 oloKl|lp{,rtrq xpnoLuorcLo0!E roviuno:

_J_*-11_,1- l-1- 1 *- lJ, ' . HJ " ' - 1+9J "". . . . -.nnI .3 .5 .7 . . . ( 2v - t ) v' z-.t-i:r:::ftit- r "" '

H deLpd roiJ EEillCAou lttrouc olvxâveL dnilq ecvd! yvrlord oucJ\d Vro-1sx<l. 'H XpnoLloroLoul6 rov rdfio:

r-x)r=r--f '-,h,,- z5+ "'- l;*fu *-... 12)

oerpd rou d€urtoou udlouq ooy(I(v€! ogoJ\d ylo -1:Xr1.

'EioL €avoL ouvotdv va oloKlnp(l€€t KdOE 6poq rnc oerpdc (l) |l (2)6vo dLdotnuo [o,9].

( t )

Ildpcb€ly|lotd: I2

l. ila moloyld€a to ordr{3rp: ,,(r,+)=l .;!- .\ - , t t1- l r .W.l

0' dL{ivuuo 10! Ner.ton.

oIoKIIAOTIr(OX ]|oIDUOt 2 Erc!66you B,

/tdq:

E(var: := -[t-rznurrl-l-1* I 1'nu'oi'* l:i [r,'nu"o]'.' / l - ! .n ! .Q

.l*8 fu"nr"t".... g6vs1 6nf4g2e<1.

H oElod ro! 6€rrr6goj !6lo{rc ouyxlave! o|dtrd' o|l6te |Inopo{l!€ w oô-Ktrnpiioourc (doc 6po rnc d|rd 0 ,oc [/2. 'Erq! 6xourc:

rt2 xt2t - - - . - lI. | $ -l lt'-i tr,'nu'ol- iii Dhr"ol"...lao=to ll-tznt2q to | -

r /2 a/2 N/2| | . - - I t . t

",J 1de+,| +u'nu'ade+ '| ' la-i s'n!'ode+'...

0 0 0t/2 r /2

(1 ) .

r12oo umloyaooute ro oôrl{prolq I ngzvp@, vetl* (mu e(wl n vE-

)0

vlnl uopo|l n$ olorlrfl{rd$,v aro o€rtteoo ualoc qc (l}). Ea6oue otov |utbto tduo oelaoo 372 (ouvoprioglc Birq ro! fdnro), dt!:

r/2

B(m,n)=2 [ n!2"-tgouv2n-r0d0 (2) (n,n'o))0

(o! qx6ud 6rL: o(m,nt. {ffi 131

H {2) yro 2n-l=2v, 2n-l=0 (or6tE nr= eF ror, n"+,1 6CVEL:

. rt2e($,f)+J nu'?"oao.

0

voq alvdL:

Eavdr 6!0c

3

z I- )

0t t l tz )=f r ,

nuzuodo=. / zv+ l \ . / t' t --z-/ ' \z

T(v+ l )=1 '2 .3 . . . v . v1 ko t :

( 4 ) .

. Eno!6-

34 K'OMAIO I

n / 2

i*0

,( aF)"(**)=("-+X"-+X-+) +'(+)'Apo q (4) 6t 'veL:

- (2v-1) (2v-3) . . .1 f r f r ,zu* ' t ' z . : . , .1v- t 1v

n / 2

1-T

'Apo:

0' E r o L I o L f l 6 v n ( 1 ) 6 ( v € L :

. n 1 , 1 n 1 . 3 " 1 . 3 r'=z* zP zz* 24 u T4z'Ero[ t rorn6v eLvqL i

'..(+ii)".-(;+:;)"'- l

,l./,. I )= f"/'z----!e.- =' t z t ta t i_ iw \

= +1,.(+)','.(;;i)','.(#*)','- l (T r )

' =l t,.(t)'

r / 2

"1"n/=J

ouolKd ovdtroyq u€ rnv npooayyLonrEpouq 6pouc rou oepolouqroq,

2. ih uroloyrd€a ro olo(Mpq!:

^rrol:

nou e6loule noapvouuE KqL [EpLood

4:I7iIT ae, o.u.t.

, _ _ _ . , . l 1 . . 1 1 . 1 ,LxouuE ( l -u ' ru 'a ) '= l - iu 'nv 'a - T ; t u -nu- (0 - 2* u"q ! 'o - . . .

(en€L6i 0<! 'znu20<1, n o.Lpd ouyK^lveL ouotrd, on6rE unopodue vq oloKlnp6-oouu€ Kdoe 6po rne on6 0 Cuq n/21, 'Apal

n/2

i 4:u-nt'e dr= I lr - ] u"nu'o- 2f, u"nu"- ffi ,'*'r-21ffi u'nr,'o-..]ao=0 0

+ r i v c e n o o t , j 6 t L : f ( x + 1 ) + f ( x ) , - . . ' ^ * , . ( ? r i ) = r ( - i ) =

=,[(" +).']{*+).(. +)=(" +)'[(' +).']=(' +)(' +)'(''+)-*

oloKlHPoJI(ot IoIItl.loE 2 xro!6dyou B. 35

r / 2 n / 2=J ac-]u"J nu'oao-

0 0

1 , .,1T u- | n!-qde-

J0

1 . 3 . .Z;Z:K U- l qU-QdA-.. .=

I0=+-+,'++- l*,"*'i *- 1i+" H+ +- ..=

=+['-(+)", ' ( r ! ) ' t ' (#1) ' r ' - . . ] (dn,rq ecoolre oro npon-t / 2

yodlevo nopdo€Lyro, eavoL: J nu'uoao= Llffi* +). o*,0

naparipnon:

noÎd oloKlnpduqro ELvdL duvardv vo unolovLo8odv u€ rn Boi0€Lo el -

IeLn!!K6v otro(ltpoldlorv.

l'l€ rn Boie€Lc €lA€lnrlK(iv oôKtrnpoldtl,N unoJ\ovL(ovtdr (oL Io olo(ln-

ai lcro rr t( lopaiq JR(x,/0{x)

)dx,6nou R.avor p41i ouvdplnon rov

6GT Kor a(x) ECvoL nolud)vutto rplrou r l r€tdprou ooe!od*.

nupo6€ly|rora:

a / 2

olortr|gar: [ {qrv")'ar,)0

0o trErooxnloxlooute ro olo(]ipllld oE €,\tr€IIITLK6. otTou!€:ouvx=ouv'?z (1) + -quxdx=zouvz(-nuz)dz + quxdx=2ouvzntzdz .+

, ,4rdi\ dx-2ouvzr\ p262 4 "4-l!!rf ax-zouvznuzdz

> tl---or;v,z "IidiTi o"=2ouvznuzoz -+ ax- -&E- oz.

l lE ro t/crsoxnuorLou6 (1)ra vto 6p!o lnq otoxtipoonc elvou 0, f . Eror.

dxouu€:

t A ! Bo iua(x ) :s , t6 re ro o lo r l ipd ld ! f to roy i ,qerou ue r f Eo igeLo ro ! le -

yo !d lov u reoer r€Lr t r ,kdv o lo rhooudro ! .

l. No moloyldec ro

A'td[:

,.(,'+)'J0

I / r \ ) " / r \ , / 1 . 3 \ z - . l/ 1 -!. nu. e do=.i L1 - \i /,.- \z.z/ 3u" -(Ztrl su" -...]

( T , )

GOA]\AIO I

a/2 n/2 r /2 r /2

J touu'tlo'=zf #a,lf ffi1d,=4,/tra;li dz-'o ,o , l+ouv'z t o

n, o

r/2 r /2 r /2 n/2- t d z - l _-21 ;: j* =21 /z+louv,z-1j dz-zt =:L =z[ O--62 ar_,o r ' f *ouv .z ,o Jo @ Jo - ' ' - ' -

a / 2 t /2

az- -2- |E]

0

) ,/Z:ffi f

'Ero[ ro opx!(6 oloKlipouo !€rooxnuotao[nK€ oa 6lo0opd 6uo ElIaLm[-K(iv otroKlnp0rudrojv nou unolovliovtoL t€ In goi8eLo r0lv tdnov (Tr)l(or (Tr)owLotoLxc.

dt

,68

a/2

2. tlo lnoloyads.' ro olorftlqtr

r /2 r/2

dz

a / 2

r - ll0

Mon:oatoule ouvx=ouv2r,r (l). 'tro! dxoula: douvr=dolrv.o + -n!xdx-

=-2ouvunuodr,i + ntxdx=nu2odoj + dx= !49 6,1 1, 6r= ___!EQ_ 6" J!-/F6tt:;

+ cr= $ + dx= nti2odt, + 6,.= Znlr,rouvtrdu,/i:nrrn"o- ,f:dnrd /Fo,rnzd

* nurl-,.*"t,

+ 6'= 2ouvt'tdl'tt1+auv.tn

TLo x=0, n (1) 6lve0 ouvo=ouv2t) + ouvao=l .+ 0-0.Tto x=+, n (1) 6lver oov { =s,oyrr,r + O=ouv.o + o=*.

'EtoL dxo!t l€:

r . [ dx - [J 6ini J

r/2

lT+innizd - l0

2o0vudol/,l;di7--o'6intd

/{+^,,rTo rel€utqao oloxA{pous €{voL €J\i€Lrrtlk6

eELo rou rdrou (Tr ) .Kor unolovd<€tdL tl€ rt goi-

^ l

I0o

n/2

-Jo

d(.)d(!n /2

OJ0

1-\V+ )\uzz

3. [o u.royrd.t -o olo.Mm: '1"

*o *.,o

ttdct: r/2 r/2 rt2Elvdr l-l 4+4n!'x dx= | 4+4-4ouvrx dx. | r'5-4ouv'x dx=

t ! t0 0 0

t12

=1'/sfrJ-Iif a'.J l / \ q ,0

o€loutE x=t-t'r + dx--dt. fLd x.0 eXoUU€ rl.'.-t xdL VLo x=+ E[voL o=0

Etvor o'(6uo orrvx=ouvf+ -!,) + ouvx'nuo,, 'EroL:\ . I

0

r"l

OIOXI\HIOTIKO' lOrIlHOt 2 trov5dyo! B,

t / 2 a/2 -(f )i,", *To teleulo(o oloKlfftpo eavor a^lErrrl[d ror ufloloyl(€tor dn6 tov tilno:(T , ) .

r/24. th urolovrrs€( E oforft|.glr t-l

,ltot:eerouye f .{-r,r + x{-20 +

x.+ eavoL r,r={. 'noo:

1 / 2

0

dx.-2do.fLo x.O e(vot o.| , vuc

| -2dor, l"t

t/2=t

I0" /""*"W

^ f d o 2 |*r/,m=,.5lt,

2 | * r _2r J /.-l w-\z " .fi

" ,/r(uQ)"*" Et" 1/,-(ff) ' ,, '"2 lt)

dtt

To 6uo t€tr€urct'o olox^nq;,Uoto ecvor eltr€LlEttd (dL ulolova<ovxot, Ko-rd ro w(,ldrd.

38 K'OMAIO I

5. tlo uttoloyrdea !o oforMAd!: I= [ tl*a

i{tol:ecrouue x= ;fr5 e 4x= -!.!i9- dq. 'EroL eavdL:

,-J

I nuodo r nuodo f do

J {f:ouvtetlfE; J nrrnj]Sorl"tr , ,i.-]diF,

r d | ! r d o l fJ /1;5h_-i:nTr,iT ) /4_::jnj''6 L )

To rEleutqLo o,\oKlnpolq €Lvo[ €,U\eLrcLX6 odproro oloxlipq]q. Epyq(dloorE

ovdloyo !e to f lood6eLyuq (1) qc o€lC6oc 32, on6rE dxoupo:

, = i ll *. I I l(,f+)'*.,] *. J i* l(G )'*",1 *... | =l t r / 1 \ f ^ 1 . 1 / ? \ r f l=z Lr.z \-il J nu'oda' i:i \-il 1 nu'oo,p. ..., .

Td olo(lnpdDoro qC oyK0JUc unol\oy((ovro! lte q 9oieero rou ovoyG,yL{oo

r0nou in( doKlonc 7 (oetr06q 66) rou Al !d!ou.

,(fi)'n",

KEOANAIO II

AINAA OAOKAHPOMATA

ALaloooes uno[v{oelc an6 to x6po n:

Al/atapov!e !.puti6 loroloil,rda 6\,\,0uea lou otopoit! to €r(!.60 0x0

(6nrc6{ ro! xdpo R' )

o) l orcdn 6i to dnu.r:0, \ , . 'Eor. ! o(xl '01) ' 8(xr,0r) 6Uo on|). | :a rdu

.r',id6ou 0x0. H ordorcon d(o,0) ft,v d'u.tuv ourd! opCBatc|, 6rd tnv cx6

on a 1 n, 3 1 =VQ'-{Jtii[:f IITO(tr,t,

B) 'Eoro o(x0,00) dw onu€fo tou erLra6ou 0x0.

AvoLrrds r t osr|le ,,vrpo ro o rd! drt'ilo t

to onleLood\Jolo T.{ (x ,o) €R2!(x-xo)r+({ - to) 'z < r 'z}

lqErords rll,tlos*!€ x;lrpo ro a |lc', atttdla r

ro onleLodd\ro lo T={(x,0) €R' : (x -xo) '+({ -+o) '?s 12}

r) A\)oLrrd op!oyd\,ro. To dnlEooodvolo

T = { ( x , o ) € R l r c < x < L y < + < 6 } o v o l d r e r d ! d -

vourr'3 oooordvuo. To avoLrl,3 oployrivr,o surFoli(E-

roL ro l 1 : (c ,6)x(1 '5) (610u (u,B) , (y '6) qwL' Ia

6) rq€roid opiold\,!o. ro onu€uoottvolo

r : { ( x , + ) € R ' z : d 3 x S O , r 3 t 3 6 } a v o l d s e r a ' ,

xleudrd opooydl,r,o. To nl.Lfid opOol.ivLo du!!ord-

( e l d u d L r : t d , 6 l x l 1 , 6 l .

t-,WororttdC rdrdog

ffiqQb/ilaror6f d|(}o(

' f i xux lLr l3s 6 icxos.

40 !€OAAAIO II

e) Xurr ' , r f r6rLox6.o loud(Eral rux lLr i tEploxr i €v6s

ro! R?, o l lo lxads x i , los !E xdl rpo ro a rqL ar t [s r . H

dulBorCl€ra! ue 6(o, . ) .

(rt) n€!!ox{ Gvits on|l.(ou a. Xdoe ureportwlo

A, (A c R2) lLas xui l l r {s repLoxis d(o, r ) o!o-

!d l . rdr , r .p lor f iou onl l .do! q. I r . rLoxi .vds on-

! . to ! o ourrBol [<. roL !€ d(o) .

() Eobteclr6r .ect€plrd, ourjoplaltd, !.uolr(,-

'Edro T 3!o on! . looJvorLo (T c R'z)) raL a(ro,$o)

avo or le | :o rou enLrd6o! . Tdre ro or !€ i io a ldy. rdL:

Eoorerld onuEao too I, d@v ui.ipr€L reploxi

or . 6(0) cT. ( ro o 'J loro ro! .o i r r r€pr , rd! onuEio!

To o .(!d! eddrepLx6on!6to ro! dnu.Loou-

To o E(vor E€di .pLr6on!E(o rou onreLoou-

To d ' l lo lo T edloLuUd neplox{ tou orr

e(o) tou d rd toLo d-

ro ' i T lay€rd ! €d@rc!L-

ro onleCo o .dwLdulopLdr6 on!€6 1ou

onreLoouldlou T

Eidi.poril sn!€do tor, t, dra! urdpx€L iEpLoxi 6(a) rou d rdrolc rou

d(c) c R'z-T. (To 061010 rov eaorcplx i jv onu€tov iou T l ry€taL .eor€pLrd

tu\,otlcrd ot|u€r:o rou T, dtc! oc rdo€ r.pLoxf e(o) tou o d!{rou! on-! . i ! tou ovvdrou T rdL roD ouvdlou R'?-T.

IqbvottAto dnr.r:o rou T, drqv urdpr.! nEpuoxrl 6(o) tdtouc rour n ! ( a ) : { o } .

oloMHPoTIKot AoI I tMot 2. t ' rd !6dyou B. 4 1

n) Ed\lopo on!€loduvdlou. To oJloro tov ou-

lopLardv onuei6! . !ds dfueLoould lou T ldy€raL

dnlo lo Iou T rdL oul8or i4eraL ue 9T.

o) xl.Lord onlruori\)olo, 'E!o on!eLooiloro

T , ( r c a ? ) , r d r e r d r " r r € L d r d d ! r € p L , i x . L x d L

o n v o p d r o v ( 6 r r d 6 i d r d v s r c T ) .

L) A lo l r rd oDeuod6vo lo . ' ! s

o ' r ! .Loo i lo ro

T , ( T c R ' z ) , I ; 1 e r d ! d l o L x t d d ! 6 . ! n e p u d x e L r o

A! a ld onue loo i lo lo c {vdL dvoLr rd i6 r . d lc rd on ! . to ro ! . i voL e-

!o !d t€ IdL du !no tds .

!6 ) tu lEx t l r6 on leLod ivo lo . 'Ew

on! .Loo i lo lo r , ( r c R 'z ) , o !o !d t€ tdL

o! ! .x rL rd d td ! rLo r r io€ q€ iyos dn !e io ! rou d ,8 undoxeL ldd rcn ldo !a ln

' rpo ! !6 ! . dxoa ts o fue id o rdL I iou d ! { ,eL o ld r lnpn o to T .

oa) Aldu.rpos on!€Loould lou. 0vo!da.ro!

ouvdrou T,(T c R'z) , o apLluds 6(T)=suFd(o,B)( o , ! ) € 1 i T

!8) toat|l{lo onyeLodd\,olor. 'E!o

onu.!-

oo ivo lo ld1 . rq ! {pdy ldvo d ! ! rope i !o e1-

x l L - o 0 c j o i u o r i r r o L c l E l E p a o r d v n d r t i l d r

L t ) tu !Byds onr , .Loo$!oro . 'Evd r l ' Lo td

I !v . r lL rd on !eLodr i lo lo .

' : A , i d o r - , o o i v o l o . : w . @ p d y - ' v o r d - 'dpLn!dE rdL d ! r fo rpoea.

6 L d u . r p o < 6 ( T ) e v d s o n ! c L o -

T

6!d !e1 !o< d l rd lou

( 6 n l d 6 i ! < 1 - ) .

rd r -ood1! tvo onu€Loo l lo Io o -

Ih ou ! . r rL rdonUeLos! !o lo

| 6 L d ! e t p d < r o ! . ( ! a u r p d y u d l L r d s

- t

IGOMAIO I I

Kaur i rq { ypdu!{ i rdgo. 'Eolo i=o ' ( t ) , tee, A c R lLd ou!sx '1s ou-

vdprnor l . 8 .DpoJ!e ro 6Ldvuolo o i= i t t l a-u o n opx, i ro! dqdsv, -o lov n

rdp( iu€rpos t l l€rqBdl€ io! oro or jvoro A rdr . ro 6Ldvuor jd ; ( t ) !e loBdr€ -

roL raL to dr l jE[o r i rL ! .J taL dro erL. re6o. To oJ lo lo to! gao€dv rou roro-

lauBd!€L ro onue' :o I drq! o t u6ta8dl le tq! oto A o loudq€iaL ra l l r {h i rd-

€o i lpo! ! oro e l i i€6o. 'EroL lo ! i6v ro 06!0^0 c=tu lo ' i l= i ( t ) , t€A) . i -

wL r r id xal rJ ln oro er i r .6o.

aLd ar loroCnon rou oplo lo i ovold4ou! . xo!r f in c tnv ouuiptnon r ' . i ( t )t € A .

Xr.€uot i ro l rdh. r ' l id xo!rd^n ;= i ( t ) , t € to,Bl ovoud! . idL ' i l€r ,or i

"o, i<ol= i te) . rd of !Eta A( i (d)) , s( i (E)) o lo! , tqol rar dpx{ roL idpds

r(oroulo lns raur i lns, 'EroL t r ,o Nol i i l f l jyeraL r le lot i ar r n opx{

ro rdods rn< rdur ico!rdL, .

Atftt )ireroftl xalro\n. MLo xl.Lotd rour']ln rd-rElaL a!l{ d! 6.! *rE!on!€da ro! f iE pe rov eourd tns.

Ilrooa,/alolldua\,n mlt$rin Mld rduniln ldy€roL rpoocwrorL4dvn drav

ax. ! rdoopLogei n good 6Laypo0is rns, ! in ! reptr ldon rns r lELdt is drr is

xqln i lns oElpEi toL SerLr i eood n 6Lotpded rns ra l r i lns xotr i tn ! d! tdO.-

r r oopd xoL " iuno"

1"e 6"""16" rou ooroyLoJ.

l . r rc ro l r ro ln, MLd xdlnJ^n ;= i ( t ) , t€ Io,Sl o!o!d<.rdL rc(o dra! u-

rdpxEL I rdp( iy@yos i ' t t ) lo , ' , " €dw! dulEr i ls yLd xdoe t€ [a,g] .

T| ln l l r r . | td l€r :o xcurd ln. MLd rd! t rUln layEtdu t ln ld lLrd re ld q! dro-

ter€r : rd! drd n€r€pdo!a lo i ) , i ro< 6Lq6oXL)t . i ! 16Co! r .do! rd!nJtov.

Le) x l . ldrds rdro i , i lerdrd i€ ,ox i r l r tELdtd xopao.

6 -

'E !q u roo i lo ro T to ! R 'z (

i xÊLo l i r€poox i d@v d l rd

opJaeraL d ld uLd or l { x l€L-

oflj rdur n i roL ord dU.s

o t l f s x leLd lds ra ! rd les* i€ -

rEpao! ;v .s ro r ldsos rou BpL:

Eddr€pLrd rns

rp6rns rq l d lo raon le r :d rov

io ! eU\EL ou !Er rLrd) l ;1erqL r IELotds td ros

' r !n i r .6n !dur i ln* t _ Y r o r ' . . c , q L 6 r " o . c r r j s , . c , o - d E r o u r i , r c s r o . o p 1 r , o , , r o r c t o r e r o i v t c -a t r 6 r c r c o o o L , ' ! d , o r l i l s o s 6 ! a 6 o x L r 6 r 6 E o r , . u c . o . i o . . s r r _ o p r i s , = r t x ) ir , rE !op ' r i ts x=e(( , ) rou ou ouvqpr iqeus r xcu o e ivar nopc-r ,y iour , .s oe rdga o,r -

oIo(^HP0TIXo, loIItttOE 2. trdr6dyou B. 43

ml i i rov ( ro o i loô r {v duvopLldv o4! . idv) av ixou! oto r .

napat ipnon: t lc e id lEla dra! ra le rdf to i reolox i 9a Ewool ler lec

Arl.f duv.rrlr6. rdros, A! dvas rdios (xl6Ldt6s rl cloLrrds) dxeL flv

L6Ldrnro drL oroLd6{ io lE o, r l ' i r lEoot i ro l rdh ldoo o 'aurd! l ropeu votrou-

otdr€drrouv.Xols (do< dro! Er9!rLdr€[ d ' j r )d or le jo) Xopjs !a €.eJyeL otrd

tov rdro, tdte o rdros layetdL d i ld ouverro l id<. [ inv q l rdoeln rEpl r roon

lEyerau .oUarld onvexrr,rdc.

Ar ld lu \ ,ExtLrd. ldros I lo l lqr ld duvrx l r rds rdros

nopatilpnon: xrnv r€pdr&,qn rou 6edr.oo! d)(ilaros to eooiepLtd !n1pd! !od ' iLaou l \ )o xopJo o !o !dq€rdL io r r j t r .

llcpc6€[ygctc:

1. To orJlolo

2. To odvolo

, { ) € R , : (

-1IT--

( > 0l .ds0 6w qlo l , r rd

- - - ' r '=0--- - - - - -a

r: oI .d !d! E\)a ) i r .Lold

laL ro odwro { (x , { ) €R'z:

44 (XOAr\AIO II

3 . T o o J r o r o T = l ( x , ! , ) € i { ' : I : i } S 3 ,

2 S { : i s } e ivou d la x ldLord raL epdyu. lo

onreLoo i lo lo . Aud!€roos ro ! on !€Loour6 lou

eUlaL n 6ua- r ivLos tou opSo lo ldou A! fd nou

€ilaL ion !€ 6T)-li'+t1=G.

4 . To on lo ro

E i lou 6 !d r leLord

l o . T o d o l o p o i o !

{ ( x , , 1 , ) € R ' ,/

To orueLjo Mr\

vopL,cx6 o f l r€ io . To

w L € w . o o r e p L r d

l , l3 ( 2 ,2 ) € iwL dw

5 . T o o J l o l o T = { ( r , i l , ) € R ' ? r 1 s r , + { , , < 4 , x : 0 l l E j l d L d l d r ^ e r -

drd rdu gpqy!d!o of !euooi lo lo.

To oUvopo rou T EtwL ro o0!o lo l (x , , r )€R'z, x 'z+9'?=t , 1 '?+,1 '?=+, 1=,1, ,

T={(x,+) €R'z ' x 'z+r] , '1 s 1}xoL epa l !a !o on !euoo l ro -

T . i !dL io o ! !o lo :

r 1 \-= ' - - - )v 2 v 2 i 1 1 \of le io r4, \ ; , - - / e i -

on!ELjo. TdloE ro on!€ jo

. € o r e p L r d o r t r s U o ,

nepuoxt js . [ ro T 6ev d l i rou! d losra

on! . ia tns 6Lrorduou o^ (6udrL x lo)H 6Ldu€rpos ro! o! !dro! e j ldL

6 . T o o i l o r o T = { ( x , q , ) € R r : x r + { , ,

u .Loo ivo lo qpd l !d lo d l l . i 6 .v e i laL

o i l re o lour16 onr . x leLord .To T ie

pu la !8d ! .1 ra o r ! . ia ro ! f r rLxux l i '

ou AfB, rd d f !€ ,o 1 ] )s 6La la rpo ! AB

rdu rd on ! . id r rs ypdr ruooxLdd!d lns

< 4 , 0 Z 0 , r l ( , ] . E [ !01 { i !d on-

,:/

II

i.1

- -rtr'

l l

!, _L' )

oIOy,^HPnTIKot I0 I l tMot 2 . tno !6dyou B 4 5

A Lrdd oôxlno{ius ra

0€0rpoir !E !Lo (npoyuqrLKit) ouv6prqon f(x,U) r tou €lvo! op!o!6vn xoLqpqyu6vt oe 6vo xi€LoT6 KsL qpoy!6vo Xrrrplo T Tou EnLf ledoLr 0Xt! . YnooaTou-

!E 6rL ro e!9q66v rou xoplou elvsL A. Yf lo6roLpoir lE (6Louepl<ouue) ro

x o p L o T o E v r o n i i 8 o c X o p i o T ; , K = 1 , 2 , 3 , . . . v u € E u B q 6 6 v A K , K - 1 , 2 . . . v

: e K d € E x { i p t o n o t p v o u u € t v o r u x o L o o q ! € L o ( q K , n K ) , ( = 1 , 2 , 3 . . . v ( e v 6 L d ! € -

qo onleio). :xnusrlColuE ro depoLolq

S , = f f ( € K , r l K ) A k = A , f ( q , , n r ) + A , f { E , , ! , ) + . . . + A ! f { l ! , n ! )

(€v6ldueoo depololo). To depoLot io S, yLo uro boouavn ouvdprnon xor yLo

lLq Dosuevn nepLoxi €{opdrqL on6 rqv urc6ldlpEorl rnq n€ploxiq Ko! cnd

'Eoru 6rr vrc Kdge 6ro!6oron roJ x0rplou _

!E narA< - 0*r (ct Yto xd-

0 r v L 6 6 a . v 6 r o u L c n r v o 4 u € l 0 r v ( E - , 4 - ) , ( k = 1 , 2 , . . . v ) r o 6 e p o L o l q

t f ( € " , 1 _ ) A _ r € l v E L o t v s k q l r o v o u r 6 n p o v u o r t . d o o r e ! 6 L ( 6 r l o 6 i € { . -

v o ! l i m L f ( 1 . , n - ) A - = L ) . T 6 r € i € J € d r r r o u v d p r r o t f ( x . { ) € t v o t o t r o ( , \ 1 -! + 6 K : 1

p6oLun Kqrd Riemann ( i of l l6q otroKlnpr,)oruf) ornv n€pLoxi T.

, l Td xoodd TK. ivoL n .n .pdo!a ld ro n r igos , r i i !e xop io TK e ivdL urod i lo -r o r o ! T , r d o e o r u € i o ( x , { ) 1 0 u T d l i r e L o ' d l d r o u r d r L o r o ! x u p j o , r d L 6 i orop l :o Tp ' Tp i l i 6e ! i xow koL ld d f ! . id i a ! ;Xou! roL ! . i dn leJo durd 6e !

. t !dL eootepLr . i o r !e io oJr . rou Tp o , i rE rou Tp+1. H 6ua(oeon ro ! Xop io !T o € x o p i d l K ( K = 1 , 2 . . . ! ) r o ! d x o u ! r L s n p o n l o , j ! e ! . s L 6 L d i ' r r e < o v o ! d E e -rdL 6LauEpudn ro ! ld ! io ! T .0 !o l rd<erdL l . t r rd rnro i ( !do l r ) ins 6 !duEoLdns: ' L . y c r n r . ! n . o , d - " r o o s r ! v o o L r . i , ) - r \ b L a L i o r o , t . . H \ , r . d , n - d o u B or ' . . _ r d L . P ' l .

' 0 , o v r o . r i q o . r b , ^ L o i , v d L c d l c r d r o r , o L d p L o r o o r l o -

a i d rd ! ( ! - +@ ) rd r . ! r o .

To ) (op ,a lnop.d !d . i ldL xdL op$oyr iv la o r f ! i€prnroo f dur i r rEnrdrn-rc rns 6Ldt rdpLdfs e [ !dL n 6udyr ivLos to ! l reyd t r j l€pou oogo- ro l l ;ou ., : , r A l id duuBdL ' leu d ld ! ro n r r igo< to ! lnoxdp id ! dug. j le rdL dnepLdpLord 6 l l -rs6 i d ra ! ! J +@ .

IGOMAIO II

0 npavlarLK6q opL€p6s L mu e€oprdrqL p6vo ofl6 Inv ouvdprnon f,voud<€rqL 61116 opLotcvo olorliiHnl Ine ouv6pqonq f ornv fi€pLoxh T

our rBo t r tE€coL uE l l f ( x ,U )dA i l l f ( x ,0 )dxd ' ! .T T

'EroL ; \ot [6v €Lvo L JJtr,,,r,ra*auT

=l im i r ( tK ,n . )AK f i )

napotilonon 1

0 r(,noq l l f(x,J)dxdo-l lm E_f(EK,a()AK 6LvE! u6vo oErdpnrrxd tov

opLoU6 ror, 6L(l\oir oloxtrnodUoroc.H €oqptoytl rou rtlflou ouroir etwL npqKiLKd oxE66v o6tvorn.>qv neplnroon nou n f l€pLoxh T eLwr op8oydlvro (oL n 6[ou6pLon P d-

nor€l€LroL dr6 opeoydrvLo KoL oxdgo o r0noq rqe suvdprnonq f(x,U) 6€v €L-vol notrln,\oKoq unopot!€ vd ono.\ovloouus avq opLotr6vo oloKlilpouo u€ lovop[o!6 6[oq eoLv€TqL orq sn6!Erq,

Ilaporiipno! 2

:rnv nEpLnrbsrt nou o r6noq T alvoL tva opooydlvLo KqL n ELqu6pLonrou

T aflor€lElTql dn6 v oogoyd)vLo, nou !q opl(ouv tc 6Lqo[iumq

t x j _ 1 , x j l , i = 1 , 2 . . . K K o L i ' r i - i , 0 i l , i - 1 , 2 . . . 1

!E Kl=v, Xo=o, XK=B, oo-y, 0l=0 (qL €VOLdU€oo onlalc eLvoL ro:

* i rLd or rd 6L t r Id o ror r ' jpd ld .

I oIoKrli{PmIXOt AOtItl{Ot 2. tro!6dyou B

(l j , l i )€ txj_1,xj l , t0i_1,0i1 t6x€ n Lo6rnio (1) notpveL rnv uopqi:

47

l l f { x ,U )dxd0" l im _ t , : _ f ( t i , q i )6x i 6 { iti *6 j=1 i=1 J ' r

6nou 6X.=X.-X.-1, 6(r i r l , i - ' ! i_1.

To vLv6uevo 6xi 6Ui=(xi-X. i_1)(Ui-Ui_1) €xqp6(€! ro eu0o66v* Aj iTou opooyovlou t j r , '

' '

l 2 l

:xnv n€plntuon nounsplf i&]on nou K=l=v). H

napa6Ely laaq:

l. llq unotroyLoo€|,

io nli8oe v opoovovlr,lv elvaL v'z {6nlo6i qlnv

Lo6rniq (2) nslpveL KoL rnv lopqi:

l l t tx , , l taxau = l im t : f ( t . ,a , )6x. tu ., i ! + - i : 1 i : 1 - . -( 3 )

T= t0 ' l l r t 0 ' l l .

A(En:

H Lo6rnra {2) 6[v6L:

l ltrE rov oolou6 ro oloKlioluo llcdxd0. '0fi0u

T

[fcaxarr rinr i ]rte',,n,tox,o,r,,=! \+- j=1 i=r r - J -

6€uBc6d! A ro! opsoror iou r : ta ,Bl ih , ,6 l 6 i lEroL urd tn! ox;onA:( B_d ) (6_y)

0

t-.

I

I

I

I :T

WI ii

48 KIOMAIO II

K l ( t- l im t L c { x . - x . _ . ) ( 0 , - 0 ,_ . ) r=c l im t ( x . - x . , ) f ( { r - 0 r_ , t -

! + o i = r i - t \ 4 6 i : r | | r ' j l

=c l im_ l ( x r - xo )+ ( x , - x r )+ . . .+ ( xK-x ( -1 ) l t ( 0 r -Uo )+ (0 , -0 , )+ . . .+ ( r l r -0 r_1 ) l

- c l i ' n ( x . - x0 ) (+ , -00 )=c l im ( l - 0 ) l l - 0 ) - c .

'APo lJcdxd,!=c.T

2. Ilo umloyloo.t ss rov oproud to oloKrtq,;n lJetdxd!

6rovEIo,1l,I0, 1/2). T

A0oq:

llLdtr€pt<ouu€ rov r6no T oE v, ro nliooq opeoy6vro. Aur6 ylv€rqLoq E€iq,o€oJpotru€ U!o 6lq!6pLsn rou 6Loori l loroc [0,1], toxo lnv

o.xo < xr < x, < ,. ,< xK-1 < xK < ... < xu=l r6tor.o f lou &-xr_l= +r((=1,2,.. .v).oeopo0u€ {oL !Lo 6Loutprsn mu 6loori loioq lO, 1/21 toru

rnv o=Uo <Ut <0 , < . , , < 0 "_ r . U* . . . . . + "= + i 6 loLo nou

o--o*r - * *** (x- t ,2 , . . .v ) .

An6 ro o4u€to tn( no6rnq dloutp[onq ptovouue .u6el€c mp6tlnieq npocrov dlovd 0 |lJ KoL orld To onU€ld r4q 6e0T€pnc bLotlaplon( {06pvouu6 €u-8et€c nop6ltr t , \€c npoq lov 6tovo X'x,

'Erot 6xoulE n€l jxe! u[q 6!o!EpLonP rou opgoyovLou T o€ vz To [tri8oq op€oy{itvLo nou 6xouv (o€tva Bdon

x * - x " - , = - | , t * = t , 2 . . . v ) , f i t i o c { < - 0 < _ r - } , f * " r , 2 , . . . " t x q L e ! B o 6 6 v

,| . eer, :oo0ue oqv ev6ldlEoo on!.Lo rd o4!E!q ( tk,r lK) !€

. _ x*-r*X. -"1 . - f , - - , - 0, - r4. -#-7; *- , - . . ,

\= ----- = ' - --Z- =

Zi- Kor 4,= =---- -$

* ! i w - t : o , 8 , . t d l L I , r d L E L d . : ! I ol - l L = 1

_ ^ _ I L : 1

- ) : t L = . " u : t "

* * ' E t o L . [ ! o L 1 , = 0 , r , = ! , r . = f , . . . , r " = ] = r .

' ! * * ' E r o L € j s u { r o : 0 , 0 , = ! , , . = * , . . . , r , = * = i

* i * * A r ^ o o i . o t r , n < c [ , o o f i o - o r \ o o L i o d r . " o L o o - r L d r - . t r . . , r _ l[ + . _ r , r . l . e o o - o o - L r ( o ( E ( . 4 < , , i \ - r o r , ' \

" . - " r o o e a , : ; ' o " 1 .

oloxlHP0Tll{ot lorrD{ot

2<-L

EtwL f(x,0)=ex{, - f (Er,nK)-e6. 1+=+ . +-#. *o rnv Lo6rnrs (3)

211-L

l l e q q x o q F r r m L 4 e . - z ; '/ r \#6 r=1 KE1

KoL 6xK6{r = ( xK-&-1) ('IK-UK-1) =

6xoo!e:

=y_#A Fi +=( +.+.+.....i+ )=

2r-t: _-7; z(-l

2 ! -1 . *

H-#(

n n - L

1 3 5

t -zv'

1-t*#FI

l ' i a - tE l{n E+ e-1 ----:-- v'-

r *t"\e '_ " , / - 1 . 1 . t 1+3+S+. . "+ (2v_1 ) l * =

F-r1

tt ' i ( - i : . - -5J-) l lnr e" -

e 1 l v - 1 ' t r + -

I

= * r . - t t 11. - -J- r rm eF. * te- r t ." vlr6 el/v-l

\Fr€--r1

(rL*'" rt' "",'1 .r) 'mo

[[ "'u,aro,r,- -]- t"-rt

, !..r_ _r/= -i

'AgpoLold ! dpo! "reo!.tpLris rpo66olJ.'AgooLouc ! ripdv cpLguntLads nood6ori'

Iz

KEOMAIO 1I

np6rootl

e€0Jpot!€ rnv ouvdol0on f(x,U) nou eLvoL op!o!6vn (o[ qpdy!6vn orox{, iP{.o T-to,Bl " ty,61.

Av n f(x,0) €lw! oloKtrnpdrown oro T r6rE un6pxoL ro dplo

y-l&+g-!I ,1, ',r=, t(* (r-1[B-o) , v+ (J-rI6-v)

) xo. er,,oL

Loo !E J.Jf(x, ' !)dxd0t

nop66€ r ytto:

lito urlotov|,oo€l lo olorlip.rro [["*arul o*u T-to,1]rt0,|.t lnton: T

H ouvdpqon f(x,U)=e" eLwr. suv€xic orov t6m T.'Apd urdpx€L* ro 6L-nld otrodipotto koL eLvol Loo !€ ro 6pLo

, , j l u t- l . u i- ll i n - r t t e "= l l n - - i : z t ' l Ee " " l i n r - ! v Eeu=\46 i : r j - r \ *6 '

i= r i=1 r * - '

t=1

L 2 ! - 1 t 2 ! - 1= l im -L v leo*eu*e ' * . . " * " t } r r . ' / - : - ' - - \

i *- -i- (r*"

' -e " *"'.e - J=

/+t r= l r ,n J k ) -J . t im +

g ! =1s-1 ; 1 i1 , - -J - = (e-1) .1 -e-1 .e1^_1 Fr6 '

J /v_1 \F6 J/ !_1'Aeo

jJe^dxdil,--€- | .T

-feauerglr{ conve0o tou 6rdod oloK}Ir{Eroc'Eoro

llo ouvexilq ouvdpqon f(x,o) oro Xoplo T rou filna6ou oxl,,y!ornv onolo €LvoL f(X,0) :0 v (X,0) €T, r6te rc 6Ld6 olodipr, l ld

l l f (x.U)dxd' l , 6[v€0 rov 6yxo rou orepeot rcu nepLxIeLerdL on6 rnv €nL-

qdv€Lo S uE e€Lo,ro1 2-p1x,q11,.0 t{dol.o T (oL rnv xuAlv6plri €n@dveLdrcu axet o6ny6 (o!n(,ln m oivopo rou x{dpLou T (oL yev6x€Lpaq mpdllnte(npos rov dtovq z z. (BAdrE ef iduevo oxi lo).

r X. l i6a s1, f lpdrdon,

OAOMHPOTIIOE AOI'IEMOX

Ihpctrlpnon:

A v € L v o r f ( X , U ) S 0 v ( x , U ) € T r o r e o

q n o r 0 0 l r i o o l o k i i p o l o - l l f ( x , { ) d x d 0 .

T

0vl(0q rou OTEpEOir 6Lvero0

B) Av €tw[ f(x,0)=1 yro xdgE (x,0) €T l6re ro 6rn/\6 olo(J\igouo €(_ropd(EL ro Etj0o66v A tou Xoplou T sro €ni.ne6o X00.

r--;i----r' [ ro r €Lvor lA = l ldxd i ! . 1l r l

no6rdd0:

Av l lq ouvdprnon f elvqL ouv€xiq oe 6vq (A€Lor6 xoL qpoy!6vo XoploT r6re n f el.var otroKlnp{ioLu4 oro xogto T.

IIaOaripr|ort:

: !nv oEl(60 99 ovqetpouu€ Kldtr lEq npordoelq oXerLK6 [e Tnv oôK^n_pr,ro!!6r i ro U[oq ouv6pino4( f(x,qr) o tvo xopLo T.

- I6!6qr€c rou 6u od oloxtn qEros

l l Av o€R kot n ouvdprnon f €l ,vo[ olortnp6oLln oro xr,plo T 16r€ Ko!4 ouvdprno4 af E[,vqL olo( lnpdol ln f io T t(oL LotueL:

JJortx,utaxaoT

= oJJrtr,+rara,rT

4- i-f -r - r--++liri-il.l

52 KEOMAIO II

2) Av oL n€r€pooutveq ro nl ieoq ou,, ,opr ioELq fr , f2, . . . , f . € i-vgL otro-

Klnpd,oLu€q oro xoplo T rcr ol ,o2,. . . ,cr EtwL oro€eool npoylorLt(oL oplo-

!oL, T6T€ Ko[ n oovdplnon g:orfr+orfr+.. .+oKfK €LvoL o]\oxlno6own oroT |(qL Loxir€l .

JJr", r, r,T

,0)+q, r , (x ,0 )+ . . .+o<f . (x ,0 ) ldxd+=o,JJr r (x ,0 )dxd++T

'q,llr, (x,q,)dxdo-...ro*JJr"rx,rr:r axarr,T

3) Av n ouvdprnonf ( x ,o ) t 0 v ( x ,Q ) €T

5) Av or ouvoprio€rq

o x ( j E r f { x , 0 ) s 9 ( x , 0 ) y r o

f elvoL o,\oK,\ng!oLun olo xr,Jplo T xsL 6l,vs!

f l(oL g el!o! oloxlnpdrcLtr€q oro xoplo I xol L-

xdoe (x,S)€ T 16r€ elvo!

JJrtx,oraxa,u > o

{) Av lrd ouvdptnon f . lvol otroxtrnp6ot ln oto XrrpLo T (oLt! f ()( ,0)< M

vLo K6g€ (x,U)€T (6nou m KoL l l € lvoL owl,orolxo Io xoid[apo roL ro ovt i , -

reoo n6poq rnq f) m[ A €LvqL ro qr9o66v rou xr,,plou T r6Te elvoL

rnn s JJttx,,utaxa,l s rr

6) Av Tt,Tr, . . . ,T" etvoL nen€pqoutvq to f t \ iooc X(,JpLq ro! XUpLou Tr a r o L o n o u T r U T , u . . . u T r = I x q u o x 6 u o T o X o p L s T t , T r , . . . , T ( 6 e v t -

Xor.,v roLvd €orrrr€pL(d on!€[o KqL n ouvdprnon f sLwL ouvaxiq oro (l€Lor6

KoL qpoylevo Xr,,pLo T tbre LoX0€L

t t t t t ll l f ( x , { )dxd t = l l r ( x ,U , )dxd0 + l l f ( x , t )dxd ' !-.. . -JJr(x,{,)dxd,r,

Tr7) Av ro xripLo T €LvoL Ki€Lor6 (o[ rppoyltvo KoL oK6!q elwr ouv€(rF

(6 Ko[ n ouvdprnon f tlvol suv€xiq oro xroLo T 16r€ udpx€L avo rould-

xlorov eo!,rep[(6 on!€{.o ll()G,00) tor, xr,rplou T rcftLo nou

JJitx,,r,raxa,r < JJo tx,,r taxa,r1 1

oôKJlHloTIKoX AofMot 2. t{a!6d1o! B.

JJrr,.,r,ra,au = rrx,,u. r JJaxa,rT T

8) Av I ouvdptnon f €Lvar o,\oxlnodbLun oro xoplo T 16r€ (oL n ouvdp-

rnon lf ElvoL olor(InpdJoLurl oro Xop{.o T xoi. LoXirEL

JJrr*,,r rarar, t .T

JJ r(x,,r)ldxd,t,T

9) Av 6(10 suvoorioEL! €Lvo! oloKl4pdrL!€c o'6vo xogl.o T 16r€ (dr royl!6uev6 rouq €lvol oJ\orlqdiDLti4 ouvdgli]on oro T.

Ildpdrionon:. , t tTo 6LnId olo{I ioouo

l ,Jf(x,U)dxd! eLvqr Ut6dv orLc Er6uEv€q nEornLri -

o € L q : To) H ouvdptrton f(x,0) €lvor nepLrri t,rq npoq X Kor, ro Xr,ipto T AxE

6lovq ouuterplac rov dEovq U'0/.9) H ouvdpxlon f(x,U) €CvoL n€prrr i {dc nooc 0 (oL ro xr,rpto T 6x€[

6€ow ouu!€tpLa( rov d[ovo x'x.y) H olv6prno0 f(x,0) €lwL nepurut oq npoq x Kor 0 (o[ ro )(uJplo T

txer xtvrpo oullerplcq xnv opxil tl,rv o{6wv.

-Etuvdll|ltrlrd (i 6u6oxrrd) olorlnpdl(fro

0!o!6(ErqL endvo.lnrrlL(d i 6La6oXLk6 oloKlip{,rlo !LoC ouvdprnonc f()(,0)l(d8e oloK,\iprruo rnq lopaiq:

f , r ' " ' ' ) 1 / t f

r o ' ? ( ' l ' ' r \I IJ r(x,u)drldx (1) (i t4c uooo'ts J I J r{x,,Udx.]do (2))

9 1 1 ) ( / Y d r t { , )'onou qr,q2 ( i or,or) ELvqL suvopdoeLq rou X ( i rou Lr) i orog€pac.

To 6Lo6oxLK6 oloklipr,rtro ( 1 ) urc,\ovl<€rqrr(,rc)e€iq:

YroloyL(ou!€ np6ro ro o;\oKl iooryq F(xl=J f(x,+)d{ (6Lar4puvroq ro

x omo€pd) ,.or (or6nLv unoiova<ouu€ . t"-li l; [P,rtor. ro*ôro ,no-mlovt(etdL xo[ ro 6Lo6oXLX6 olodiptdxo (2)).

iH rL! i f ( r0, {0) ovour i lErdL !don tu! i rns dovdorno,rE f dro xupjo r .

Ilaparipnoq:

[Lc ofilonolnoi tllv ouugolLo!6v floltrtq 0op6q oL qvKl,i€q o[q [Lo6ox!-

xd oloKtrapd,uora nopo,\3l.novrsL,

To 6ro6oXLK6 oloKl i tpt i ta (1)vpd(o€rol r(ol t /€ tr tv uopei:

. B r Q z ( x )I dxl f (x,0)d0, €vdr ro 6Lq6oxLK6 oroKl ip0r lo (2) Ypdq,€roL r(oL !E rn !op-

qi j drl j f(x,0)dx.Y o r ( { )

IIopcdE Iv[qrs I

t, ild urrotoylooet rc 6.dox!d oroxrioouo l-[ [[,,+otlou.

AlJont, t r ' r " r ' r j oEl.vqL I xudx=Ul xdx"0 l+l^ =+ +.

'ao" J'[[ ',+a,lau,-['] 'rd,!=+ l+l' -+ k-+\- + '

J r L J d ) J r .

, " , , 4x 12) tito motoylooet ro 6La6oxlx6 olor,,tieqra J, ljo x'ldtldx

Aton:

,tuo," Jr,r.o,r,=rJ ;o*-, l+]-' * ($) +"'m" ['f [ -,r,a,r,la,=[",'a,=e[i'cx+[$]' ={# - +)"''l r L J .

KEAMAIO ]J

-Ynolov.ou6q tou 6L od olorlnpqdtoq

0 unoloyLou6q evdq 6Lnlot olo(Inp6uqroc !€ rov opLouo €LvdL epYo_

oto notrtnloxn eLOL{d ornv n€pllr[oon noo n ouv6prnsn f KoL n fi€oLoxi T

6ev etvqL onl6q. To €n6u€vo €€opltlorq 6L€uKol0vouv nolt rov unoloYLolo

ev6c bLdoo o,\oKltpduotoq.

* I ro [6Lo aro t ; leo ]1a e td louue dv e lou . iqouu€ Io lE pd lous ro ! u . tdE ln-r0 ! x ,q , .

' r rdL e | ; !d r

'ilf**P'=t''*,, 34

- 1 l

0A0l (1H!nTIK0t A0I I IMOI 2 . I rdv6 . i ]o ! B 55

S€{pnuo 19 (O€dpntra Fubini )

Av q ouv6prnoq f(x,L!) €LvoL oioKinp6orun* f io opgoy6vLo T=tq,Bl ' ty,6l6

K c L r o o , \ o K l i p u t o h { x ) = l f { x . { l d 0'l

u[ ipx€L vLo K6€€ onu€Lo x, tou

[o,9]r6!E ro oloKif tpouoB 6

| | | f { r ,+)d{ ldx un6ox€L Kot eL-

[[rrx.,r,raxa,r,= [FI f trx,+ra,r,loxi ,, '"}

naodriipnol:

Avd;\oyo 6Lorondv€lqL

trouq rov uErqBlnr6v x KoL

nponyotuevo SEdpntq

T6Te 0uoIKd €LvqL

ov evotrtrdqou!€ Touc p6-

0fltpnla 29

Av n ouv6prnsn f(x,0) €LvqL o,\oKlnpdoLln

T { ( x , 0 ) : d < x < 0 q ( x ) s U s o ( x ) } r * x o L r o

oTo Xl,rO i,o

16rE To otro K,\ ilpr,rtr o

! s h (x )= l f ( x , ' l ) d ' !

B o ( r ) - I r ( x J

[ [[ rix,,r,ra,r,loxundpx€L vlo Kdo€ onu€io x rou Io,BlUf iOPXEL KqI €LVqL

I d ( r lf f l

f ( x . { )d ) (d ' != | l l f (x ,L ! )d0 ldx' i ' " "0 (x )

Avdioyq 6LoruldJv€rsL !onpanyo(rlevo 0€6pfuo o! Evotrld-

Eoou8 rouq p6;\ouq r{,iv u€roB,\q-rorv X kdl (r , ' t roL ov

* . 0L d ! *p . r ioeu<

"^",^l*u. JJti*,+;uxdo undpx.L.

.1,=r1r1, ,1 ,="1"r1 EiwL d! !er€ds oro td,Bl .

t r & r r ?I l f (x ,U)dxd!= l | | f (x .0 )dx ld !T I "

KEOA]IAIO II

T = { ( x , 0 ) : t ( u ) : x < s ( , ! ) , v s i l 5 6 } , e l v o L

f t r 6 t r 8 ( { ) |l l f ( x . ' ! )dxd != | l l . f {x , { )dx jdo?

'Y "tr,r,)

naporipnon:

To en6g€vq rcplouqro 6tvouv rnv n|o ouyn0iqr6vn nporEtxi |rc6o6o u-no,\ovLoto0 rou d!nlo! o,\oxlnp6uoroq.

n6plotd 1,

o€upo t ] u€ rov t 6no T= { (X ,+ ) r qsXsB , h {X ) SU<q(X ) }(6nou h (oL q €tvoL ouv€XeLC ouvopri loEtc Kq! h(x) s q(X) vLq Kde€X€ to,8l). Eo!,r 6!L 4 ouvdprnon f etvor. ouvexiq orov 16110 T r6rE E[voL

mpLolio 2.

eeopouue rov I 6no T - ( ( x , q r ) r y s0s6 , S (0 ) sxso (0 ) )(0n00 g, s €LvoL ouvex€[C ouvopil ]oELq xor g(0) : o(0) vto xd6e 0€ly,6l)'Eoro 6rL ol ldprnon f €LvoL o0v€Xi( otov r6no T 16r€ slvoL:

6

l l f ( x ,Q)dxd0= l l lT Y

lf ( x ,0 )dx ld {s { { )

nopo6€[vtrara:

l. [lv€f,or- q ouvdpxnon f(X,o)"X{, oprolevn mov r6noT={(x ,o) : 0 ;x s 2 , o s{ <2x}

lb unolovld€[ rc ororriijso J.l)(sdrd'].A0on: T

, . . 2 . . ? X . . 2 . . - - 2 XEtvq! llxodxdu = | ll xua+ lax-l l4-il or=l 2x3dx-zlx,dx='; '"

=z f-$ l'+.t r l 0

2. lb urEtov.oo€|. E 6!rt6 oloxliiqds JJ nu(x*Ulax+, OtovT

I a ( r )

J lJ f(x,'!)du.jdxJJrrx,oraxao =T

oIoMHPoTIK0I r l0r l t t {O[ 2. I ia !6d-ro! B.

T{(x,0): 0sxs+, oa05 n i2 ' I

o) oq oroxl\qpdooule np6lc oq npoq

"to

Soit\f\ \ i"\St r\ I

*." -,, ,r*, o. z i;r*\ ' ! i i , l --{i i \ '" i r- " . ' - .1 Lr" ' . " . , t . ."- t , t . \ ' " .1;, .-1, : . . , . \ . , , ,

=[**#];.+.,=+F) 0o o iox, \qp6oou! . np6rc l l rq npoq l l j , To 6pLo rurv X (oL U €LvoL ro

i 6 ! o . ' E x o u ! E l

jjr,'.,r,r a,a,r,=J' lJ' r,.+rau]a,=

I l^'*- I l 'a*-[ 'r2,.-zra*=

='z [+ '];=, (+.,)=+

0A0K HrQTIkot ôrllMot 2. Xro!6d1ou 3. 5l

l0ot: \ /2 a/2

eLvo. f[nurx+raxau=f I' ; 1 1

T=((x,0) : 0sxsf , o )

au(x+U)d(x+0)]dx-

.sq ,s

a / 2- t I

r/2

nu(x*U)d0dx

IlT

.a/2

(nrx,n/ ' , f /2 tn/2= l l -duv(x+ ' ! ) l - dx- l

| 1 r / 2=l-ouvx+npxlo =2.

3. lrd urotroyro€eL !€ 600 rp6noug m 6lnl6 oio(tr/toorlo JJf,,'+la*af,6 rc ! T=t0 ,11 , t0 ,21 . r

itol:

o) eq olox;\np6ooute npdro oc ngoc x. To 6pLo tou x €[voL 0, 1, rq6ouq rou q r ELvoL 0 , 2 ( eLwL 03XS 1 , 03 ' ! s 2 ) ,

, , , " i , , " - " t , r ' t , \'Apo l l (x ' : * r )dxd0=l l l (x 'z* , r )dx ld{ ,= l l -+. tx l d, r= l t - i .uF0=' i \ , ' "I I r . z 1 2 ) ^ I- l+0-+r.-+.2=+.

p) 0a oôK^npd,oou!€ np6ro {,rc lrpoc {. To 6p!o r0rv x (oL U elvol, ro

i .6Lq. Exouue:

If r"-,no,a,r=[ [[ r"+ra,r,lo'=i 1

= l lx 'u .# | dx=l {2x '*2)dx=

1 t / . \ a=2 l -+ *x l =2 ( i - -1 l= -+ .

IGOA]\AIO II

IL@O€C tl€€o6€toElq-flaporr|pioerc orov unoloylo!6 tou 6!r o0 olo(^npcld-mq

It0d6€uon l:To 6LnI6 oloKtnpduaro unolovc<ovroL dnoq a(bolE

I s ( x )

JJ f ( x , , r )dxd i ! -J IJ f (x ,q i )duJdx1 _

h ( x )

6 d ( { )l l I f t l

J i r {x 'o)dxd, l1 | i r ( r .o)dxldoT Y g ( { )

0116 Tov irJno

( 1 )

i an6 rov r(rno

To o,g orov npdrro t i ,no €{.voL !o 6pLq rou X €vir ro h()() , g(X) srov l -610 rrno elvor ro 6pLd rou U. Irov 6e0repo r0no y,6 elw[ ro 6p[o rou' ! KaL o(| l ) , 9(U) €tvoL ro 6pLd rou x.

t2)

Tq 6oLd lou x i Tou U orouqtov rp6rc nou opl(€tor io Xtipl,oo) Av ro ollvopo rorr xrdplorr Trluv€Ior oe 6to ldvo on!€Lo o-116 KdSa Euealo noodllnln npocrov dEovs X X xor qII6 kdBE €u-oELo nopdltrn,\n rpoq rov dEovq(r ' | ] r* ,r6re o r0noq (1) io r0-f ioq (2) 6lvouv Inv tLt j i rou6Lnloir oloKlnpi'uomq {6nlo6i,

nponYouu€vouq ruflouqT. 0o vvooL(ou!€ 6!Ll

xo€opl.aovroL

r r

on6

urcoot!€ vq oloxlnp6ooulE rpdro 0r( rpoq X i npldro nq npoq 0).Iqv flepLmosq ouril vLd vq 6pot!€ ro 6pLo r(,lv X xoL (r 0o Epvda.6!u-

oTe |ljq EqilC:

I) Eos.6ra ottou!€ l,! oloxlnpaoqrs tr'{rE oq rOoC I (6nlo6i vo xonor-lono[noouu€ rov r0no ( l )) .

otpvoupe !Lo ruxoio euo€lo noOdi lnin npoC rov 6tovd U'0 KoL Bpl.-oxorrtr€ {otrvOptlto€L rou X),rlc r€rovutv€q rlJv Kolv(iv onu€Lov Tnq !E roolvopo roir t6nou T,6ToL npoK0niouv !o 6plo rou 0. TLo w gpotu€ ro 6p[orou X,Bpl,o(ou!€ 600 onUELo rou r6mu T,nou 6xouv rnv !LXpdT€pn (cL UE-voi(rr€on x€rlnuAvn. €ToL npoxtnrolv Io 6pls rou x,

III 'Eau.6t! SGlout € w olortnpbuu€ np&ra |{ npoq X (6qlo6it vo xonol-uonornooule rov r( lm (2)).

t r n r r e p i r r o s n o r i r o r o o a o o , o u , l q . . o " n o , o u " n d .

I or,ar+[\ ' \ "

O Ol(^HPdrI1<ot /tolllllo[ 2. trclodyou B 59

06pmuus ud ruxd[o eoo€to mtdllnln nooc rov d&r4 x'x xor Bp[-o(ouue (ouwprioeL tou 0),rls r€tunllv8c rdv koLld,v otrELov rnq €uoel.oq

!e ro otvopo rou r6[ou T. 'Elo! 6xouu€ ro 6pLo rou x, flo vo Bpo0u€ ro6p!o rou 0,0o Bpotu€ 600 ontol.o tnc tregroxilc T,mu w 6xouv tnv rrtp6r€-pn xol pevoltrspn totov!6rn. 'Ero! lpor0rroov ro 6p!o rou 0.

Idp66€ryto:

It 0mlovLogel, !G 600 rp6nouc ro olor,\ig,to JJnUrO,l

O-"

T.( (x, '1, ) : o sx s 2, 0 <{s f r } 'Iton:

ol rCDc: 0a olo$npdoourE ntdno oc ttpoq 0

,, ,^,. : l |Etvo! llx'ldxdo.l l l x'ldu ldx (1). ELvo'-

T " "

,f ,fr r u.r rVl r r.^rI M'r,.xl {dr.xl -i- | .xl ..fi- - +l - +.

'tror n (l) 6[vEL:J n J o t . J o t 1 1 J

JJ*,a,arJ #a,. + t + l;. + .8l rp6rbc: 0o olodnOdJoolge rpdm l,8 nooC X.

0a Epoou€ rd v6d 6p!d tlnv x rcL U.llcgloEdvouue ytooud tov 16It0 T,

06pout/8 lLd €uo€Lo mo6ltrnln npoq lov

dEolo x'x (or gplo(ouue rlc reunu€vecrrlv l(oLvdJv olueLov tnc €uoetoq !a to lt6-pl,ytoud rou r6flou. ECvo! xil,2 rc!x-2.

'EroL ro 6gLd rou x ehqL U2 ror zH ull(pdrepn ieloyg6vn tll|v on!e!L,v muT slvol 0 xol n leyoJloraOn elwL VZ.'Ero! m 6plo rou U eLvdl 0 xor VZ, Eno!6voq ELwr

,t, zllxodxdx =l I I x'ldr ldl, 12)'{ 'o "t,

xax+ f# l'-+[z- 9'l -or- # . 'r-'L - r t . I

Elvo L n (2) 6l,ve L

60 ICEOMAIO II

v't -

JJ,,a,', =J (zu- g )o,u=z l+ l: + t#l:=+B) Av o r6lloc lnc olox/\iftjonq

vq ralv€rqL !6vo oa 6iro ortu€Lo qnd

{ Ur,€v6 €ueelo flqpdtrltin npo( rovdaovo x X T6!v€L ro otrvopo rour6nou o€ nepLoodr€po 016 6to on-

!eLo*, 16r€ yLo rov unoioyLo!6 rou6rnlo! otroxlnp(i t rotoq XpnoL!onoL -

oi te rov r i rno (1) (6nlobi oloklq-p6vouU€ flpdro l,J( rpoc {). :tnvn€plntl,jort ouri ro 6ord rov X (oL

V) Av ro otrvopo rou r6nou T!6!vEroL !6vo oe 6i jo on!€to on6K68€ €uo€Lo nop6A,\4ln npog lov d-

[ovo X'I Eui euo€lo nopdl,\nAnnpoq lov dEovo 0 | ] r r t lveL ro o0-vopo oe n€pLoo6r€pd udv 6!0 on-y€Lc** 16r€ ylq xov uroloyLo!6rou 6rnj\00 o/\od\npd,lqroq Xpnorlo-noto!!€ !6vo rov rr)no (2) (6nld6i

oloxlnp vouu€ npd] lo l ,E npos l) .:Tnv n€ol,ftoon oulh To 6pLo rolv xornv n€pl l l I l , lo4 q ( I l ) .

txEL rtroLo lopei dpr€ ro o0vop6 rouk6e€ rupd,\lnln 6Lrg€[o npoq tov dqovo

0 BpLorcwol 6r0rq n€pLypdoouu€ ornv n€plnroon o ( l ) .

tqL 0 8o unoioyLs8oiv 6noc llEp[yodoou!€

6) Av ro oi,vopo rou x0rplou T rtlverol on6 xdge 6ug€[q nopdllnJ\n n@(rouq d€ov€q oE naploo6rEpo ond 600 on!€Lq*r* 16r€ XopL<oule Io xoplo TKoidl , \qlq oE unoxoplo Tr,Tr,T3.. . . To o0vopo kdee unox(,rplou np6n€L volatvarol o€ 6to oiu€to on6 Kdge aue€Lo nopdtr,\n;\rl npoc €iq rooldxlorov d-[ovd, T6r€ ylo K60€ unoxirolo xptoLuomloirle i rov drno (1) i rov rtno

* 0 to ' ros dutds ovo] ]dqetdL !n rd lov l ius* Tdros un ra lo l l rds' t 1 6 r o < ! n r o v o v u 1 6 s .

,!n

oôy'^HP0TIKoX T1OIIXMo[ 2, trc!6dyo! B

(2). TLo vo unoloytooutr€ ro dlntr6 or\oK,\ttpouo 0'610 ro XnpLo T XpqoLlo -

noLoi,ue rnv 6! 16rdrnro (oela6o 521 rov 6Ln;\drv oloxlnpt.|]dtov. To 6plotrdv x r(ol ! BpLo(owor 6mdq n€plydqouU€ ornv n€plnr0rn ( l ) / l ( I I ) .

nop66€Lynd:

0€opo0!€ To X0rplo T ro{r ox/ l loroq nou txouls XtJpLo€L oro Xoplq TIT2,T 3 .

eivoL JJrtx,utaxao=fJr(x,0)dxd,!'Jjr(x,,1,)dxdu-JIr(x,'!)dxd,l=T 3T ,

t r t e 2 ( t ) r Y r s q ( ) )

1 6 / s 6 ( \ )

= l I f ( x , 0 ) d J d x + l I f { x , ' ! ) d 0 d x * l I f ( x , { ) d 0 d x .t r s r { r ) 4 / e r ( x ) l , g r t x l

a) Av ro xurplo T €(voL 6vo opoovrivro nopo^Ju]6vpoutio ue fiA.up6c no-pdiinj\€q npoq rouq d€ovEq 16r€ €pyce6Uoor€ 6nu< n€pLvp6qouu6 sriv o nE-

olmtrrn !€ Tnv noporipns4 6rL 16T€ ov XpnotuonoLiooute rov x( lm (1) €a3i.vot h(x)=cr 9(x)=€r, (cr,c, oldeepaq) Evo ov xpr lo l t rof lo Lioo u! E rov r t-n o ( 2 ) g o € t v o ! g ( ' ! ) = c r , o ( 0 ) = c , ( c r , c , o l o e € p d q ) .

tlnv neplflroon nou ro xoplo T el,vor 6w op8oy6vLo noooAtrnA6vpoutlo u€n^€upaq nolal intr€q npoq Iouq dEovaq xoL n ouvdprno4 f(X,Ul etwrrnS!og-'rne f(x,{)=h(x)u(0), (x,0) €T r6rE Loxi.L KqL o r inoq

B 6

I l f { x , ' ! )dxd ' l r= l h (x )dx l w( ' ! )duT '

( t )

62 IGOMAIO 1I

Ilopd6€ Ly|ro:

Atsn:

E l v o L

lh unoloyloo€l to 6Lnl6 oloxliprruo llx'!dd'! 6lov T=to,i ]rto,l lT

l t | |x0dxd!= i I x'!dxd!= | xdxl Ud!=il

T

=t+ l; t+l =+ + =+Ilpoooxi: fro vq olo(Anp{pouuE npdjro oq npoc $ npbnEL qa Evdq uev

xd€€ EUo€Lo nop6tr lnln npoq lov dqova U'0 vo r6Uv€L ro oovopo rou Xopt-ou o€ 6tJo on!€Lq Kor o0 erdpou vo r6lver ndwo rlq [6LeC roundlec rou

Av 6€v ouugolvouv xoL o! 600 dur6c nepL[r6oE[q rdre [p6n€! To x(,r-plo !q X(,DLoo€L o€ urcx0jpLo. To cvi lsroLxd [p6nEL vq oulBolvouv ov 06-

,\ou!e vd oAoKlnp6roulr€ flpdjro llc noo( x.

ndpd6€LyUo i

I'h urcloyLoo€l !€ 660

T nou rcpLXl€LemL crl6 rrc

A(,on:

rdnouc ro 6Lnl6 olorlig,rlo

(qlntlec 0=X, Xz+0'?=4 !€ X,0

llxdxd! oro xop !oT> 0 .

H ypqqLki flqpdoloon ftu xopaouT olv€roL on6 ro noporitr€0poe oxiuo.

d) rp6mq: 0o Xpno l lonoLf toou lE Tov

6 o( ( , )t t I r t rl l f ( x , ' l r ) d x d 0 = l i l f ( x , ' ! ) d x l d Ui 1 " e r + l

6nlo6i 0o oloK,\nqiloou!€ np6ro 0cnpoc x.Aurd €LvoL 6uvor6v 6L6rL Kd-

8e €uo€lo nopdl,\qln npoq rov d€ovooe 6iro orlusl.o KoL 6K6!o rtlveL ndvrs

0a BpoirlE ra 6pLo !(,JV X l(oL 0.d4ovq X X. 0L rErun!6vEq Tov KoLvdv

x x r t tva! ro f leplypcuuc rou xoplouTrq L6LEq xotrntr ieq.

06pvouu€ uLs euoElq flopdl,utrn npocrov

onuELl,Jv rnq euo€Loc lie ro nEpLypcuuq

0A0K,, \HPnTI l0E IOf IEl lo t 2. t t rd !6drou B

rcu xoplou €tvo! 0 KoL Va-{r . Apo ro 6pLq rou x €lvqr q] KqL \r i :F.H t !-Kpdrepq reroyutvt rov onu€Ll, jv rou T €Cvq! 0 KqL n u€yo,\ tT€pn reroyuevn

€LvqL VZ. 'EroL ixou!€

,l; \tr= ,1. 'F- 'tz O| /A- t '1 !7 \

l l f (x . r )dxdu=l l l xdx ldr - l I i I dJ- l ( -z- i )d ' r .' i n " l , 1 . , !

\l; -r 1!) i1 tr,/) ?,h 6,/Z_Z,JZ a{?= l (2-0) )d0= l 2 . ! l - l " r l =zn- :+=" ' : ' -= :

J , t l . L - t .

B) rp6floc. TLo vo oloKlnpiooul8 np6!s uq rpoq U nptfi€L va X(,rpLooupe ro

xopl.o T oE unox0rpLq 6!6rL Kdo€ EUo€Lq nqpdi. \4ln rpoc rov dlovq U 0 6ev

6 3

rtuveL ndvrq r !c L6!Eq ypou!6q rou f leplypdt lqroq.

'Ersr .LvqL [[rt x,,1ta xo,l= [lrtx,,1Ioxa,l*) ) ) )T T r

,, Ft ,, ,.ru'?+ i l f ( x ,Q)dxd0= l I xdodx+ l I xdu rdx=- . v )ii \E=12= l - x t0 l "dx l

x t0 lo dx=

t A z , J Z t- l x ,dx. l xv4-x,dx=l+ l -+ l (4-x, )1 rl, vz

4,tZT

Ilapariigto{;

Av ro X{,JpLo T onor€,\elrdr ond 600 KovovlKd X0rold !e tvo xo!!6 on-

!eLo 16r€ 0o €pvdsroirue ndlL 6ftlc rnv flponyoLruevn r(€pinr0ron.ouoLKd €6dr6ev qnoLTeLToL XopLoUOq rou xoplou 6L6rr €LvoL f ipoQovnc.

I IapdO€!y!a:

I\b umloyrc,oe[ ro olo Kîptitlo

dsl€rd! 0|16 rrC (d!r0lec {Ex,0=X3

Alton'.

I = l l ( x - l ) dxdT

r:d(4-x, ) =

6rot Io xop[o T[€9r--

64 KEOAAAIO II

EtvqL r=JJ(x-1)dxdu+JJrr-, roro,r=

7 .( f tx r .0 r . r - l

= l l l ( x -1 )d+ ldx+ l l l ( x -1 )d ' ! ldx=, o L i x , I J _ r L r r J

= l r tx- t t+: i ,ax. l r (x- r )u l : dx =

= l (x -1 ) (x -x ' )dx+ | (x - l ) (x3-x )dx .' o l r

l

lleod6euon 2:Av 86lou!s w umiovloouu€ 6w 6Lo6oxL(6 oloxîq,lo Kor n npdr -

tn o,\oxlltprdon 6€v €lvoL €ux€pitc, 16r€ Kdvoute q/\trovi ornv oELpd rnq o-IoKlipr,ronq (Bpl,okovroc auoLKd ro v6o 6pro rnc oloKlipoonc).

fTi"'+J0,.',& v1+x,+0, J

lldgd6€ ry|lo :

It umloyro€€l ro olorlipolo

i0ot:

E[e!6tl n oAo(lipnEn t,)q rpoc 0Od rnc o,\oxAipdonc. nponyouritwq6ut')c 0o Bpo0u€ ro vtc 6pLd r(,rv XKoL 0. 0 r6[oq I rn( olok]ipr,Fnq

T=(x,o) 0 < x < 2, 0 501-42-)

H yeolr€ip (it nopdoToon Tou T €[-voL n nopdrl€upn.

TLo vo gpo0uE ro 6pLo rou Xeapvouu€ trlo ruxoLo EU8€Lo no-pdli t ln npoq rov 6lovq X'x xo!BploKouli€ rLq r€runltveq l{,lv onu€Lov rotic rnc !€ ro nepLvpotilo rou r6nou T.

Boloxouue 610[ l=VF rau x-2. Aurd €Lvot ro 6plo rou x. 'Apo \20<x<2.eo Bpotu6 rri,pa ro 6oLo rou 0. fLo rc oKo[6 our6 ovc<nroole 6r]0 onueld rouT6nou T nou ixouv rnv lL(p6TEpn (oL rnv u€yoltrEpn r€rovutvn.

(0, - r )

6Ev Elvor. cuxepiC 0o ol;\d[ouua Inv oEL-

oôMrPmllot lOrlDO[ 2. tro!6dyou B 55

Soto(oul€ ||,.0, 0.2. 'Apo 0 S '! 3.2. 'Etdr o r6roq T otl,<elcl xdr

oe €€lic:1= { ( ) ( , 0 ) : 0 s0s 2 , @3xs2 }

' ixouue lor,n6v

[T"", o,t,=[T[' , oJ*J,& /r.rE J "$''

vno;6 .1(1 )

( t+x,+u,) -1' d ( 1 +x,+u, ).[ ( r +xr+9, ) rrl' =y5.q.- 1'pr 1.

1 r 0r".,, [f * "-1-l- ac[r. ['r "rq-- 1,1., I 101,.Jo

[o vl+x'4' I 'o

f rns.

EtwL I x dr=+ [' or' , = I f' atr*x'*,1't" -ii;[ /FFfrr ' Jfi, (t+xr*{r)r ' ,,66 tt,y,,9,1.

-tlvi

'EroL n

It0068uor 3l3. 'E,! anld opLol6vo olod\igqro nou o unoloytop6( rou 6€v alvcl

€uxEpilc u€ rL( ouvnelouav€c !e066ouc €l,vqL 6uvar6 vq onotroylo€Gl, uslnv 9oiloera ev6q 6!nl\00 olox^npduotoc.

0 rp6m( apyooLdc 0olveroL oro er6levo flood6eLyto,

f f t f ,*n"o<!s!.

.l J l'. r:-'"I Inx lo Inx

(x,0) € I0' 1l xtd,Bl=T. Ero!

Ilapd6aLyla,

io urolovlooGl to

l0ol3r d ,

elwr Jxqal. fr *c.

l l ouvdoqon f(x,0)"x{ el,wuaxoule

lorl

Apo r

uv€xi

olo

'Apr

tit'I,

' " '

kdee

t :

tllo.!o

. l;L!r!t

J

tc vro

!',!

nc

Jj",,,o=[ [f *,-].,r,=t t+t: *=f * *=1++.nn{urr)rl.B*ld+T

. ln(B+l )- ln(a+l )=ln

]G]OAAAIO II

ELvoL 6ur,,c Kor JJ--r,o=f [1j,..,r,]r,.=1" t# ]:--l'+f -

A116 (l) xar (2) npoxtnr€r 6rL ax=1n €il .

lletooxdrqE LqD I

0L !n r ioptEdLav{s du! teraf !d!€s o!o!d(ovraL, rq! rUi j lpoU!eS ouvlEtcy-

ldves (droL i .X. oL io lLrds oLrvr€tdy l ld !€s (p,o) e lds on! . to ! M ro! erL id-

60u E[BL rd! iu ldypdy!cs owrErdyrdves) .

0L ra! ru l6ypquu€s ouvr .By!d! .s (uru) ou!6dolrdL !e r !s , ropreoLqrdE

r l€ 6w letdoxnla lLold ! l tns lop9is ) ( :h(u,u) , ( , :g(u,u)n. O letooln ldrr ,o ldsd!r6s arELxo! [qeL ro , I€Lord xcL epayldvo xopio T .ou ErLra6ou 0x ' !

r lE lo ld rqL gpcyldvo ropio Tr rou 6! ! r30ou oruu. Av or6yq Loxi€L o a! r r : -

o lpoeo< !erqo)( fuarLolds M-r rou rp6nyoirsvo! ! . tdoXnrdlLo!o0 M, 6 i rd66 o

u e r n o X r ! o r t o u d < u : h r ( y . + ) , u = c r ( x , S ) i 6 r c 6 L 6 i o o o i r ! i v e s r t o c o x d s T

,o! Tr or . ! ro! {4o!roL ! . ioqf rous ue ! !c o)(aon , '1-1, , .

-N tqkrSLavil o9l(ouo

tt066€ud4 4: lb[oxnuorloudc ror, 6!ll]ot olodrrrdlatoc

0 unolovrou6q €v6s 6rnio0 oi\o(^rtpduotoq !e dLoaoxL(i or\oxtriplo4, l,lqnpoq t[q KoDraoLqvtq ou vt€tov! a vEq, ovqe6p ero L oe lLq 6Lo!6pLon xou r6nouT,rou yLvETor !€ tnv Boio€Lq rou ouorhuoloc irrv KoprEorovrirv ouvt€tov!5vovlbAltq qoptq o uno;\ovr-ou6q ev6c 6Lr o0 oAoxJ\npd'!@oq,elvor nLo elxepic cv

n 6lou6plon rcu r6nou T yLveL !€ Kqunll€c un (cpt€or.ovov ouvi€Ioytrav(,jv( koUnul6yoolli,rv o0vreroyutvov).

l 2 l

f ' xB-x "J,

-Tnr

€.opo0!e rLs ouvqpi{d.L< x=f(u,u) roL ' t :s(u,u) ue (u, ! ) €Tr , (Tr ef -

laL du r ) ,€uord rdL opdyudro ropjo! ro! erLrd6ou ouu). Yro9;rourr . dru or ,

ouldpt ioeLg f rcu g € i !a ! rcpc. toy ior , les os rpos u raL u oro xopio T! . o-

slid<ou!. IarogLcli opd(ouoa 6.urdods rdl€os rd! d\lvaprio€@! f rd! s (!Je

1 lod td r re ro{6 xdpr€d la ldv rdL no l ! , . i ! duu€royrd ldv

x=o du !g :h (p ,9)| ! : p n ! 0 : g ( p , S ) ,

n 'EroL r . r . d tus . iwL

Ol0klHPnTIh0E A0f I tMoX 2. t rd16dtou B 67

, o o s u , i ) v a - o u ! 6 o r i c o ! 1 . f f i

, - ' " " 1 . " *

ax axau au

44, a0au a\)

-lleloq(n|Erroldc rou 6!11100 oloKlngdgaroc

Ocupot!€ ro 6lnid otro&\lsruo f l t tr, , t , t tra, l . oEupoUte xqr rov u€ro -) )T

oxnldrLo!6 X=h(u,u), t ! -S(u,u). Av Ldxriouv oL En6UeveC npounoe6o€Lq:

o ) To xtr€to$ XoJpLo T Tou EnLnadou oxqr qn€rKovl(eroL (u€ Tov npon-yotu8vo uErooxnusrLot6) qlqLlovooiuqwq oro xleLsr6 X0JpLo Tl tou €nLnt -

6ou ouu.B ' )01 ouvoor i to€Lq h( ! ,u ) , 9 (u ,u ) €LvoL ouvexELq !a ouvex€Lq lEpL-

KtC flopoy{iryouc oro Xr,JpLo It.

v') H lq(0rBLovl f , f f i EtvaL dLdooon rou un66v oE <doE onu€Lo roLXoplou Tr. (EKr6s Looq on6 lvd nEr.pdo!6vo ntr i8oc onu€lov rnq [EpLoxicT r ) .

T6Ts Lox0€L o en6!€voq rorloc u€rsoxntorrouol Tov 6L[tr6v oloK,\ng,Jud-

JJtrr,urarau.=JJ Frh(u,u), s(u,u)r I ff i l*0,T T r

(6nou F €LvoL n ouvdpqsq f rKQpaouAvn olLq vteq ouvr€tov!6veq).

. Av n oôK,\ipoon yLv€! npdro Lrq npoq u r6rE o nponyoo!€voe rrlnoq yLv€-

(6nou o,0 elvoL rq 6pLo rnq !€rdBtrnuls u Kqr ur{u), u,(u) Elvsr ro 6pLo

rnc uEroBtrnriq u.

(1 )

l 2 )Frh(u,u), s(u,u)r lB{fg|.o,r r I B r u 2 ( u )l l r (x , , ! )dxd0 = l I' f

" 'u ' (u)

*Ano6€rrleraL d." oru,r,= i#t$ lu"o,.

68 IGOMAIO II

(6nou v,6 €{.wL ro 6pLo rnq !€ra8lnrfu u xol ur(!), ur(u) etvol ro 6010rnq lErogtrniiq u.

. Av f(x,0)=1 yro Kdoe (X,i!) €T r6!a ro EuBo66v A inc neploxlK T 6{,vErqLdno rov lillo

,-Jfaxa,,.JJ | $f,itil**.,TI

Ihpd6€!ylo:

Itr umlovloo€l !€ lswxqgotlq|6.o€ xo|r|lul6ymrt €c drwsEoytrav€( to

6r|t16 olorli!|F ll)(odxda 6rov o ril|oq T r€prrlel€tql ard rLc rdurtl€c

a Av n otro(Iio{,,on ylvErq[ I|p{irro t,Jq npoq u rdr€ txoutr€

Tf,=x! a,.a, r,{, r,-af.

iton:

Tp6rcC xo! o€Lpd Epyaoloc

o') xopdoouu€ xo y€qrerpLKd

oxild rnc ll€ploxiq T u€ qv Boi-e€Ld rov stLodJo€r,rv rilv lponyou -

lt\n,tv (o!n6ôv. EroL 6xouu€ romponj\eoftrq oxilo.

B ) An6 ro sxiUo ooLvEro!6rL oL ouweftyt tveq (x,{ ,) rou

ruxo{,ou ontEtou l,l rou T €LvoL

6 u r ( u )

11t,*,*,*r*={ L,,,,Frh ( u, u), e(u,u ) r l$l,i*i l**

SerL(6q, (oL oKduo 6rL LKovorcLoav rLq oxto€Lc U s x '?S Zl i ( l )

x< , ! ' ? s2x (2 ) . 01 (1 ) ( oL (21 6Lvouv r r #s2 xo ' , r s$sz .

OL rE,\€oroL€c ox6o€Lq loq odnyotv orov uetoornyoruoud u . -# xor

u= i ! e1_<u<2 , 1sus2 . E roL ! € l ov uE rooxnuo tLou6 ou r6 o

r6noq T rou ElLnd6ou XoUJ onELKovC<€toL surrLuovoqiuovrq orov tdfloTrrou€IILne6ou 0ruunou eavo! op0oy6vLo,

(3 )

O OKIIHmTII@' loIIXrcE 2. tr!6d.ro!, B 69

0 r6noq Tr 6(,verdr 016 ro €n6u€vo oxlitd.

y') gpl.o(ouuE tnv lor0r8Ldvi opl,<ouoo

+if#l-=--eI-au

euau

Erwr B#*+. -lGlnrECi;UI au

au-50-

"r_r_-i_. .Aro 1fi . f f i1,+.

6') EkQpd<ouIe rnv utr6 !o oJ\odip0,!o ouvdpqon oa xcuruidypotrU€CouvrElcv!avec.

'ExougE uu= i ._fi =xt. .Apo f{x,,1,)=x,lj=uu=F(u,u).

e') E0qg!6(oule rov drm (l) tnc o6^[6os 67, ordts exou!.

T r= -! o-ur+-l= -f" = -f .

4-0-

-x'z-uz

-02-F

2tx

=* J*'10,'; t+1, t+1,

IGQMAIO II

l,lerooxnuorlou6c rco ELntrot otroKlnpd, oioq de nolLkaq ouvt€Toyltv€C

. tlvo[ vv0or6 6rL ot Kopreoldvaq

66v!q! !a r lq notrLKtq ouvtErdyutvec(p,e) rou [6Lou on!€lou I tE iLg oxa-oeLq: x:oouvo=h (p,€) |

I (1 )' l=on!e =9(p,8) I (p i 0)

(o p e6di eKqp6<el lnv nol[Ki a(TL -

v d ) .

nolld 6rnld oloKlnp6uqrq urlo-iovtaovrol €u(ol6Tepo oe noAr(6q ouweroylev6( rcpd oe ropr€olovlc.

:rnv n€plrftoDrl quri o r0noc uetooxnuortotlo(J rou 6rnô0 olox/\npdltoro(E Lvo [ :

JJrrx,,r,raxa,r,T

=JJrto,etoaoaeT I

( 1 )

'onou F(p,8) el lo! n ouvdpf i ton f , Ex0ooouav4 oE nolLKCq ouvtaroy!6v€c.

. H laKo8tovi oplaouoq ornv neplftujo4 usrooxntoxtolot oE no]\LKac ouwe -

ouwerovu€veq (x,0) cv6c onlEtou ouv-

royuavec elvd! Lon !€ p (p : 0).

a lrnv n€pLnr0En lslooxnuotLouoi o€ [o/\txtq ouvr€tovltvEq ro €u0oadvrnq nepLoxltq T bLv€TqL on6 mv rlno

rJJaxar = JJoaoaeT T r

fhpot ' |o i lo€ Ls srnv rrEptnr{rsr uetcouucrr"osorj o€ no}.Lxtq ouvr€-1C1/U av eq

Itrdpatilpnol| I

o ' ) Av n repLoxi l Tr n€p!d\ELero! snd rLq notrLXaC y(,rvLes e=Or O=0,(6r < er) KqL on6 r[q xo!n() l .c u€ etLodlo€Lc p-pr(8), p=pr{O) !€pr{8) s p,(€) v0 e tor,0zl 6ntro6lt ov €lvoL

T , = t ( p , o ) € R , / 0 r s 0 s 0 z , p r ( g ) l p s p , ( e ) l

r6re o r(,rcq (l) ypda€rol:

t 2 )

0I0K H!0TIKO' Iotltt4ot 2. Xrqrodyou ! 71

(3 )

npoooxi: : rov nponyotu€vo r l ro np6n€L ro p1(0), pr(8) vq etvor !nopvnlLXd oro dLdortuo lErsBoi iq rou 0 6nlo6i l ro 6L6sqra [€r,er] .

0 ')Av n n€pLoxi Tr n€pLKtrEt€rq! qn6 rouq xirx iouq !€ eqLo6o€Le p=pr

p = p , ( p r < p r ) K o r r L C K o u n i r , \ € q ! € 6 q L o 6 o € L q 8 = e r ( p ) , 0 = 0 r ( p ) , ! Ee r { p ) s 0 , ( p ) v p € I p r , p , l 6 n l a 6 i q v e L v a L

T r = t ( p , 0 ) € R ' z / p t s e 3 p z , e 1 ( p ) < e 3 8 , ( p ) ]

T6re LoX0eL o r0noc:

( 4 )

napdrllproq 2

:Tnv fl€plnroon rn( ovrLKordoroon( x-poovg, !r=pn!6 elvor ndvro

D (x ,+ ) -^D6,3T ''

naparipnot 3

0 lETooxnuoxLo!6q o€ flotrLKtq ouvr€Toy!6vEq vLveror srnv rIEoLtrr0Jon

nou ro o0vopo inq nEpLoxiq T etvoL K(Jxloc, i l tpoc KiKIou,q yrvlxd xo!-

n(] in nou n €Elooolt rnc nEpLaXEL x2,{rz.

o 'Eror ov ro oivopo ror.r rdflou rnc oioK,\ip{,ronq €LvoL KUK,\oq (i lapos([KIou) ue xtvrpo 0(0,0) xqL oKrlvo s 86rou!E x-psuv8, o=pnue. ]oxr loov

ro en6lEvo oX€!LK6 !€ ro dPLo rov P, e:

d ' ) K l K , \ o q r 6 r t : 0 S B S 2 r I , 0 s p 3 q

B ' ) n u L K i x l l o : 0 5 0 < n , 0 ! p S q

Y r r c t o p t 0 y u l ( ^ t o : U J s J ? i r U S p S o

. Av ro o0vopo rou r6nou rnc oloKl iooons ELvq! KUKioq !€ Kdwpo K{Bty)*

r6rE x6vou!€ ro !€rooxnuorLo!6 x:pouvo, 0=pn!0. TLo vq Bpo0!€ rc 6pLo toup, €6roulr€ orqv EELo0jon rou KiKiou x=pouv0, o=pnue.

'EroL npox0r(r€L !LotreurepoBdguLa €{Looon oq npoq p, Av n €t looon ouri ax€L pt<q ro lnbdv,t6-

I l l no,erocolae, e r L ' p r ( ! )

F ( p , e ) p @ d e =JJtr,,,1*ar,JJT T r

p 2 ! 2 ( p )

f f r tx. . r , raxar,- [ [ rro,eroaooe=[ [ [ rr" ,er"ae]ao' i + "

rp r l r n r rp )

r i v o L o r a . 6 u , o c i , r c u t r = o i r = 0 .

72 KEOA]\AIO II

re ro 6pLo rou p €LvoL ro !n6av (oL n dliln pt(o av etva! €erlki h n on6 -luin rrt i rnq ov a(voL cpvnrL(i. Av n el(o{.on 6€v lX€L pa(o !o !n6dv r6re!o 6pLc lou p opLo(owoL l l€ oxn!o!Ld!6 KmdllntnC qvroorLxiq oxddnc. f[ovo gpotge ro 6p[c tou 0 el€td<oule rrq n€plnrd,o€ Lc:

,-i-v"

o-6 )

. Av ro oovopo rou t6nou E[vo! (('I(Aoq !€ kavrpo k(B,y) t6r€ gmoo0le !a(ord^l4ln owr(ordoIoon vq lrrooxnudriooutla o€ K(lxtro !e l(6wpo O(0,0).ouoL(d e6d, €t\ro! 6uw6v vo owLlaronlooet lpdBlnuo orov urcIoylou6 TouoloKlnpdi|roroc,vLdu!6 €v6€[KvuroL n rponvo0uEvn Epvool,o.

' Av (o o0vopo rou rdnoo ELvoL n al l€rqa + * $=r, eetour. j-=u,

' n t "

i ' l-i- =u. rorE €[voL ;it:ii =oB r(o! q tM€rqrn uElooxnlort(EroL'otov (tKJ\o

u2iu2=l . O6rouue orn ouvaxeLo u:pouv€, u=pnpo. tOte "tvor ffi

.o "cro 3 p 3 l .

* r4ropoiuL " r

r , ivo,uc ,or ' ,s ' iov """

to r ,c , "orryo ' "oy, j

r :apoue9. v :drn!0L e o : o : 1 , o r s . 2 r . ' t , o .

n d r , r , " + n r , e t c o r n u o r i r , r d r o r o o o o o y d , v L o [ o , r ] , L o , r r ] .

l(irrloql

nU!|(0Klro:

xETOpmKiKtr!O:

-F. e <10<o< +0<e< +

KlrKloc: *. e .. S

nq[K0(J\Lo: *. e. n

r€ roo lokaK lLo : * . g .9

x0K loq : nS0SZn

n ! ! (0 t ko : n . e . *

lE top tox rJx lLo : nS I<+

,-f.., ,,,.- kh(t\oci ososn

u, ( | ) t6 te ( -nu tx t *Lo : oses*

---if- \ teraPtox0xtruo: osg<*

Y) tote

16re

oloMHmTIKOt rlollDtot 2. ha!6dyou B t3

f[o ro 6p!o rou

a ) Av ro olvopo

B ) Av to olvopoy') Av ro o0vooo

napa6€[y|r(rto:

1- llo unoloyr,og€l

oloxlilqonc og((€ror

,ldott:

I Loxtouv ro €n6!Evo:rou !6nou €LvoL o^6xlnpl n 6^tre[0t,t6r8 0 3 0 3 2r€Lvqr t!r[ l l€L0n,1618 0 s I S n.eLWl teIoPrc eÂ€lo4qtTorE u S u ! ?r

6lnt6 olodipqro Jl" -u axaO. 0 rdmq T tn(

Ttn otaol x 'z+t '?Ja'z, o>0.

ro

ond

0 r6noq T elvqL xuxtrLxdc 6loxoq !e (6vt9o 0(0,0) xdL dKrtvo q.06rou_

!€ x=pouvo, 0=pnuo. ECvoL dxdu=pdpd8 (ot f(x,r!)=ex'+{'=eo'=r(p,€). To 6-pLo* rou 0 ELIqL 0=0,0=2n rq 6€ 6pLo ro0 p elvaL p=0, p=o. Ero! 6xo0tl€

T 0 02 \ 2 1

r 1 l ^ r t d I= l + e ' d 8 = lJ ' L l o )0 0

0 0

] r"" '-r rae-{ t""-r r ;e;l"=

-1) ( lb rov nponYotlEvo !ero_

T leEooxn|rmL4Ers! orov t6no T1).

to 61 6 olorlilllar JJrdrrf, 6Ev I €(vc! to qri-

oO€oyrrvCo d[dwv rou'trepLexstca ueroA, IIU a0xl.w

* r [ w ! o 3 x ' z + { , ' ? 3 q ' z - o 3 ( p o u v g ) ' ? + ( p n u s ) 2 3 o t r o : p 2 3 d ' ? - 0 5 ] < oEtrELoi o rdt ros T er j \ ldL rurrLxds 6Coro< E! : !d! 9€ l0 ,2r l .'EroL o rdros T ! . rqdxnlat i !€ .d! o l rpr , !o tooiual td oto ooooY{ i lLot o , d l r t o , 2 r l .

' * g ;* , y2+1,2=,r - (pdu!6) ,+(pnuol t=2?- p '=2 '1 - o=2. c jwu or t jL ,c r ' :+r ! 'z- . t- ( p o u v g ) ' l + ( o n l g ) t = r - p ' = l + o = 1 . ' E x o u l r ' t r i o n E : ( x : o t a L l , z o ) '( p d u l g > o r o L p n l g : o ) - ( d ! v o Z o x o ! n u s > o ) - n € [ o , t / 2 ] .

, , 2 i d 2 r oJJ"*'.,,'axa,r,=J [l u'**'=l [J ]*''1,-

=+ (e" -1)zn=n(e*

oxnto[rold o r6lloc

2. 15 umlovlo€€[

|[ qq Inc v(,l/adc uat,x'z*z=4, x'z{'1-l./vbr:ll€rqoxnuorC(ouu€ o€ nol!Kaq ou-vteroytrtv€c. Eavqu x=Oouve, 0=pnlo (p:0). Tq 6pLo*r lou e €r:-vqL e=0, 0=n/2. To 6p!o lou p

74 KEoAAro rr\ / 2 , t / 2

troL Exouu€: f[ro-a,u-[ [fr'","ecrloe-f (ouvs) |'+l'd0=+ [ ;;"eds =' i

" . "

J r o FJ , ' 1

=lnvetl/'?'1l t , ,

3. lto unolovlog€l ro olor,\lir{rr/o {r"(+

- F }xdt 6nou n neoroxtrT n€old€l€ror drd ynv tlt€do .+;.

# -t, (o.g , 0,.

n0on:

06rouu. x.ou, 0.8u. Elvor $f,l*i -ou *0, r(x,u)= + - # ",'.,'== F ( t i , u ) .

0 r6noq T !6 Tov lerooxnuorLo!6 out6 lErooxntotL(aroL orov t6no T,I Iou nEprx;\ ,L€rdL qn6 Iov x0K,\o !2+u?=1.

'EroL Ervor r=j'J (+ - jr )axa,r,.llr,,.,, rooa,o,.Kdvouu€ rov !€rooxnuotlo!6 u=9ouve, u=pnu8. EroL €LvqL

0 3 uz+u , 3 j - 0 s p rouv re+prn l l g< l - 0s p s I

€Lvo! ox6!o 0 s e < 2r xoL $fot#i =o *ot F(u,u)=u,+r,?=p,. ,Aes

tt / v'1 i '2 \ t t rrp,pogapdo l[ [ l?"p,oeaelao =t=JJ\+- .d>Jdxd, !=J j (u ' -u ' )o0dudu=' ;1 . r0Lro lT T ' T '

*o Jo'a,fl,*e l# ]l ,,,;"- 2[oB- --r '

ie rnv Bo40€!a rov 6LrMv oloK^npoldvdv lropotge vo onoloyt.Coula 6!6_ooto Vevlxawlr/s olo&\ngdDfio, rou o u'loloylol,6q touC €l,w! 6uoxar/tc.0 t06[oC sgydollc qtlvgro! oto ar6lrw lrop66€!vuo.

ndpd6aLygc:*-"d'r'#i*'.ot.t to o.lo.riooxo fJ-"'e d x

oEopoo!€ Io xoplo Tr, Tr, T3 nou o9l,<ovro[ ovrl,molxo (K €[lK:'o rdros r" €iwL ro opsolr i l i ,o to,11,to,2r l

oô(^HPiTIKO' l0fItlrol 2. trdv6ciYo! B

T != t ( x ,0 ) : 0 : x So 0 S0 so (o>0 ) ] ( r e rodvovo )

T.=t(x,,1,): O ! X, 0 S U, X'+U'1-q'z] (r6ropxo Kirktrou)

T3.{(x, ' !) : 0 .: x, 0 s0, X'?+{'z=zo'z} {rtropro x0xiou)

8€doouu€ rnv ouvdornor f(x,0)=e ^ * t0,#opLou6vn oro 11,plo out6. To o,\oKiip{rlo rnq (o,o)ouvdprnonc quric (y!q rnv onoLq ELvor

f ( x , q r ) > 0 v ( x , 0 ) € T r ( i T , , f t T : ) ) o u -

[6v€rql xoeo( ro xop[o efiE(tellowoL. ErsL

jJ e- ' r 'u rdxd0 =JJ " - ' " * 'dxdr s l le- 'x

' " 'dxd0 (1)T 2 T r T l

8q unoloylooutr€ rq o|(polo o.\oKlnqi,uoro rnq ovLodrqroc (1). Kdvouu€rov U€rqoxnlqx[oU6 X-pouvot o=pnt€.

fLq rnv nEpLoxi T, txouu€:

0 : x ' .S ' s o ' - 0 s o ' suv 'e .p 'nu '?8 3 o ' z . 0 s p ' l s o ' z - 0 s p aq .ECvoL x : 0 , U i0 doo pouvez0 , o4 !€10 -0345 { . H l oxoo lov iopl<ouoq,6nuK €L6q!€ ornv II€pLml,jon rov nolLKdv ouvrarovu6vov,€Lvq! p.

1 / 2 d t / 2 d' t !o ! t xou i i€ [ [e - t

i " " ' )axa l - [ | " -o 'oaoae- l l Ie -o 'a t -o ' toe =

+"

r'1€ tov L6Lo ip6m BpLoKouue 6rL [J"-lt ' t 'u']xau,= l tr-"-^'t

(e l .vor 0 : p 3or2 xoc 0s es t '+ ) . E tvoL cKduo

JJ"-<x'-+'raxa,r,=T r

-x e'q dxdu= I le-x e-'r dxd0 :

J , Jo

: +i"i*' ] 'or.- ; {' l*'-,,0e- } re-"- r r r er"/'= f, r r -"-"' r.

t lJ J"

0 S x : o0 _ s { , 3 d

i o r d n o s r n s o r o r l i p u o n s . j w L o p i o . r d v L o .

KE{AiA]O II

12 ) .

-f "-*ixf ;+'a,r, : f e1'dxf rxix=lf cr"dx]'.

'Eror n ( i ) rLveu f , { r -e-o ' ) = [J " - - 'ax ] ' ,s ] { r -e-2"1

rLvau r im f , { r -e-o 'z)=} , r i rn f , t r -e- '?" ' t -X .

'Apo dn6 rnv (2) nooKorieL o" r* [1""-t"a*]'=+ -

[it '

5. ih umloyrdEL lo V€vrk€uutvo oloKlilpl,llo

dx-+.

(o,B > 0)

E[.vo L

J.=_l-ox

> 0 ) .

Aton:

o€opoare ro otroxlipow I e-!xdx, (uJo

I e- lxoxarm Ie-r 'xax= r im [-* "*f-'o y+- 'o y+6 '

'Aoo I e- !x ax. -L (1) .

H (l) oloxlnpd)vowqq ro 600 !6In on6 s

8 + o I E + o

I f[ " '" ' ]0,=[ -ir - I tf e-lxdx]dun

J d L J 0 r t - t . t

=rn 9 - J*-1"*t-"- t ' ' t

o"=rr 4," J . \ X /

"-ur.a'u

t-"- !x tBl= - ldx '

l=-Ll !tr-t.+

6(,rq B 6[veL

n +-- |

B 6

l f ( x ) d x = l f ( d ) d d .- l LwL Y ldo ro o rL

olor lmTIKot IOrI$rot 2. trov6dlou B 77

EOAPI4OIE: TON AINAON OAOXAHPOII|ATON

- l. YlloloyrotdC EIBd66v enln66(,,v xtpaov

T0 €ugo66v A €v6q .IILna6ou Xrrptou T 6Lv€ro! cll6 rov !0no:

a-JJa'au,T

(o€ (opr€o[qvaq ouvr€royutvEq)

i on6 rov rtno:

Ihpatilgnoq:

A v e L v d L T = { ( x , U } € R z : o S x S B , h r ( x ) s r ! s h " ( x ) } r 6 r e r o € u B q -66v io! xl,)ptou e Lvc!

B h r ( r ) IA - l l dxd0= l l l d0 ldx - l Ih , {x ) -h , {x ) ldx

r - h r (x )'EldL Korol iyouuE olov r tno nou 6Lv€L ro e!Bo66v enrl lCdou Xoplou ! . onl l t

I Iopa6€tV!ura

l. tlo gpE0€1. ro €!Fo66v A rou Xrplou T rou entn66ou 0x0 rmu opl.(etoiort6 rlq rotn0lEq 'lF3-2l'?, 'J.Xr.

I It l .voL A=l ldxdo. €o gooir ! . ro 6010

1'rorv )( KqL L!. En€L6h Kd€€ .09€l,o nopdt-,\nin n9oq tov d€ow 0 i! suvovrd ro ne-paypoluo rou xr,Jplou o€ 6to onlElo (oL

ox6!c ouvovrd Kq[ rrq 6uo Kdl lnt i€q,uqqoul0CpeL vo oloKlnpiiroouu€ nodlro 0Jq nqoq

0. To (or.vd o4u€[q !!o( .uo€tos nopdt-i4, \nq npoq rcv 6Eovo 0'0 KaL rou r€-pLypdllqroc rou Xoplou 6xouv tEToylt -

A = l l p d p d 0 (o€ ro,\LK6q ouweroyutv€ql

v€s o=x'q, '!E3-2x'z. H ![(p6repn xo! n uEyoJ\0r€pn r€runrtvn €lvol ovrlororxo

XEOT]\IIO ]I

-1 |(or l . [ro[ 6xouti€:

, , , t , , " - " ' , ; , , 3 - 2 x 2 r to=f o^*=1,1J,,, d'fx=1,[{J"1x-i,

xlF2,

A0ort:

r :+x.*"rax-[ :x- ]x,- ]x ' ] ' , = -tir'u'

2. fh Bpeoel to elEod6v A loo x,DLou T llou oo(Eeto[ aflo rlc xqlniilec

rr*r e=JJa,,a,r,. Tq 6pLo lou x

€LvoL x=f , x.2W. Tq 6plo rou 0

elvqL 1, 4 (ro 6pLo Splo(owqL 6ft,1(n€oLypi@ulr€ orrtv oelLdo 58).

r'fr,W l ir r'6'Apo A=l l l dso- l lx l du =' t (2 / t ,

) ' tL 12 /4 ,

- [ (z to-#) a6-$ -2 1n4 - . rou6o. .

-2. YnoloYrdtdt 6Yxt|N'Eo!,, f(X,U) uLo ouv€Xic duvdptnon oplo!6w omv x,\€Lor6 r6no T xou

€nLn66orJ oxu u€ f(x,U) :. 0 v (x,U) € T.o€rlooale rnv (ulLv6p[(i enLqdveLo !e o6nv6 ro orjvooo rou r6nou T

xoL y€vAr€lpEc nqpdlln,\aq rpoq rov dlovs z'2. H KUILv6pLKi EnLpdv€Lq, nerl ledv€Lo S nou opL(€roL ond rnv Ellooon z=f(x,0), xqr o r6noc T opL(ouv6vo orEpE6 nou odyKoc ror j ElvEtoL on6 rovrtno:

v =Jjrrx,,uaxarT

( 1 )

IlapofllpioE!q

l) Av v|.o rdoe ()(,0)€T €l.r,o!f(x,0) s 0 16r€ o 6yKoq rou orEpeo(r 6tvE-rol ord Iov fl:rno

v=-JJttr,,rrara,rT

t2 ]

oIOMHPnTIKoE r\ollrMox 2. trdlodyou B 79

2 ) Av f r ( x ,0 ) , f , ( x ,0 ) ELvq rouvEXeLq oovoprilo€!q opLoutv€c otovno T xoL ylc Kdoe (X,'r) €T €LvoL

rd-

fr(x,0) < fr{x,$) r6re o 6yKoc rou or€-p€ot nou opl<ergl sn6 rLC elLodvErsqr2=fr(x,0), z=f,(x,{) KoL orrd rnv KUIL-

v69r(i enlqdv€Lq u€ odny6 rc ol,vopo rour6nou T {oL y€vareLpec nopdllnêq npoc

rov dEow z'z btvETqL q' l6 rov i0no

(3 )

(dnou | €LvoL n ouv6pr4o4 f oe f lo lrxaq ouvr€rqvtr6v€q).

napsripton I0 t0roS (1) oe nolL(6q ouvx€rovutvec vlv€mr

v =JJorto,etoooae

rYror i9erdL drL rdne n !L€uS. io Xzdu!d . ) rd r { igE l id € rLed lELo d 'd \x l

I ) 0 16110( ] Elvdr n opei npoBotrn oro €nL[€60 oXU rou or€pEo0 KqLEL!o! Euvqrdv !o 6[v€ro! (nopd66rvla 1) h 6X[ (nopdbeLvlo 2).

l l ) Av 6€v blvEror o dnoc T dr€ f lpoobtopL<€roL oq €Eic:

o')Av ro orepE6 nEpLKl€L€lor on6 !LC enLodv€L€q z=fr(x, |J]) ,2=f,(x,0)!6!6 ondlolqoule rov z 0116 rLq €{Lodio€Lq !,Jv 6io €nlqov€Ldlv. Ero! npox0-nrEL uLd Kolnti4 c,nou €lvol q opgi npoBoli oro EnL[€6o 0x0 rts Kourd,\nqcr,xord T4v ol lolo reuvovm[ (oro ldpo ) oL 600 €nledv€L€q. H Kounr] ln c sL-\6r ro otvopo rou r6rcu (nopddELyuo 3).

ndpqrilplon 2

T[c vo unoloylsoutr€ Ue 6Lnl i o, \ox,\hpoon rov 6yxo €v6q oxEp€o| l ( f lou

opL<€roL 6[0rq r lapLVOdUouE nponyouufvr,{) €o vvr,rpLaouu€ 6r[ :

rdpr i l ln ln rpos rov . i€ovq z '2, ( lK] €T)

v=JI t f , (x ,q , ) - f 1(x ,0) tdxd ' !T

K'O!]IAIO II

B ) Av to or€p€6 n€pLKI€LeloL qn6 rnv EnLpdv€Lq z=f(x,U) xot rnv

xul lv6plxi enredveld q(x,U)=o, t6ra ro otvopo rou rdnou T eLwL n (dt i -

ni ia 0(x,U)=0 (r Io enLn€6o 0x0). (nop66eryuo 4)

>Inv neplfituon ouri €LvoL duvotdv ro otvopo rou r6llou T vo KoeopL -

<610[ (oL on6 rLq (o!nr i , \Eq f(x,{}=0 tot 9{X,U):0.

y ) Av ro oi€p66 neprKlEl€xor. on6 rnv E[Lodv€Lo z=f(x,0) (1) r6ta

ro oovopo rou r6mu elvo! r l Kolrnoln f(x)r !)=o nou npoxanreL on6 rnv ( l )

Ylo z=0. (nopd6€LyUo 5).

6') Av Io or€pe6 n€oLxtr€L€ro! on6 rLq ef l [edveLec z-f(x,{ ,) er{x,{)=0qr(x,{)=o 16!€ ro ol ivopo rou r6nou dnor€tro(rv or Kd!n!}Ec ar(x, ' ! ) - 0,

9,(x,0)=0 (oro enl.n€60 oxu), ( 'Aoxnon 128 o.IL6o 198).

Ihoa6€tyudra:

f. ih urDlovldcl. o 6vmq iou orep€oo mu 6xc! Bddn tctoty$/o rt€ugG

I xar to OlDq rou os rdo€ oru€lo 6[vaEr ord qv enr0dvero 0=x2+22+1.

Abn:

0 <nroo!€voq dyxoq e{,vo L I

v= l l f t x ,z )dxdz . l | {x ' *z ' * t 1616,i 1 - o

=l 1",. # - ' l 'dx=l (+*1,)ax

4 , , t r ! l ' 4 r 5= - - r r x t ; * l - - - | - - f + - r - -T

K.trovd6a(.

2. lh urDlovldEl, o 6yrcC tou r€rpoa6pou nou neplrleldrol dtt6 to

x+f+z.l rar !d dr[.rt€6o rr',v o[6vov.en[-

rl€60

A{ron:

H eEloooq rnq 660oc ABf el.-wr z=l-x-{. [ror n ouvdpqonz=f{x,{ i } El .wr n f(x,0)=1-x-| ] , , 0rdnoq T eLvoL ro rpl.y{,tvo A0B (dto

€nlne6o IP0r) nou opLa€Ior on6 iLCe u € € [ 6 q x 4 = 1 , x = 0 , 0 = 0 .

oloMHmTl(ot ôtM€l 2. trdlodyou B

' t xo ' ruE v 1J f (x ' t )dxd , l , -

JJ (1-x -U)dxd, r . To 6oLo tou 0 ELWL 0-0 , 0=1-x .T T

Ts 6pLo mu X ELvdL 0, 1. 'Ero[ €Lvor

n= [ ' [ ["*. , ]0,=f ' [ [ '

*, ' -*-*,orlor=[ ' I f , ,-rr+- -Jl ] ' ' lo, =

I r o t j o I J o t t , ) o J

=[ frr ' t t r- ,r- Ll+ ldx=l (1;x)? or.- 1 | " ; ," ] l=* ". .**.r o l . . ) ) o

3. lb utoloyld€|. o 6v(oc rou or€peo0 rEo retld€leloa o ro rdpo -

9olo€!66q !€ e€tc[,x'n x?+0'z+z=4 xo[ ord to en{,ft6o z=0.

A0o4:

H f lpoBotri rnq erlLpdv€Lqq X'z+t'z+z=4 (1) om Entrl€6o oxrl opL(€LroX0l-pto T !€ o(,vopo rov KtKlo x'z+U'?=4(npoxtm€L qrr6 rnv (1) yLo z=0).' [ ror 6[vc[ T=t(X,U) € Rr:Xr+q,r34)

Etvo[ ok6!d x '?+U'?]z=4 - z=4-x'?-o'?- f (x,0)"4-xzr! 'z. €rooxt lor l<ou-

!E oE nolLKtq ouvr€Ev!6v€q. 06-rolpe X=pouv8, ql=plU8, ELvqr

0 3 X'?+U'z s 4. 0 s p'1s 4. 0sps2

xo!0s0sh, ; i6 ;6 i=e.

f( x,{) = 4-x 'z-{ 'z=4-p'z.

2 \ 2 2 \' apo v - l lp r (p ,8 )dpdo= l I p (4-e ' )ded€= l | 3 - + lds=' i "

2 1 2 \

- l l ? p ' - - ) - o ' I d € - l ( 8 - 4 ) d s = 4 J 0 1 : n = 8 n K . u o v d b c s .

4, I{d ufloloylooet o dyxoq rou dtEO€o0 rou n€gLx]€aelor dd rlc enLod-v€L6q X2+{,2=1 xor {Fz=2.

N €nLqdveLo U+z=2 elvqr €nlnEdo nopdl lnio npoq rov dlovq X X evdl n E-

a2 KEOAÂIO II

[Wdv€ro x2+{2=l el.val KUlLv6p!(i tiE odtv6 Kouri^4 rov r0KAo X2a02=l {rou€nLn66ou 0X0) xoL yEvtrerpEq naOfilln/\€C npoq rcv dEovo z'2.

' A O o € { . v o L T = { ( x , 0 ) € R ' ? : X ' z + U ' z s 1 }KoL z=2-U - f (X,0)=2-0.

r VT-x?'Aoo V = l l f {x , t )dxdu- l l l (2_0)d , r ldx

" t ' '-6=7t

| | . 1 , l- l lz t - i - | ax.-vr - I .

= | | zr'i:Ir- !-l;I:_L +2r'FF+ t r;x- , ldx=l r l J

f -=41 vl-X.dX= (0trou!€ x=n!|Jr, of l6r€

dx=ouwdoj, orov x=It ur+ (oL olav

,= - r ,6=- |1 "

\ /2=,Iv1-n!'zo ouvi.,dl,l"4l ouvzodor=)-r /2

5. llo unotroyLo€el o 6y(oq rnq oFl.po( X,+U,+zr=t.Aoon:

tl oaaLpo X'z+o'?+z'z=1 (1) 6xet (tvrpo r4v cpxit I|,)v o{6vov (oL oxrlw1. A6yo oullErpl,oc go unolovlsouuE Tov 6v(o xnq o@tpoc oro np{iEo 6y6oorov o€6wv. H op8i nOoBoIi rnc €nlodveLoq q( oOaLpdC o[o EntnE6o 0X0 .t-vqL o xik loc x '?+o'z=l ( f ipoKolrt . ! on6 rnv (1J y[q z=0). 'Ero! xo xoplo Te l , v q r r o o o v o l o { ( x , U ) € R , : X , + U , S 1 , X t 0 , 0 > 0 ] .

=d lou 'eauu * $ l ' -

= l { -zn r .uov66ec.- r _ i / 2 -

' E iwL v l -n !?u=ouw 6 rd rL ,€ l -+ , + I' * Br i r€ rp0j ro rdpo "csrU]L{o j - r '1nou. ' o i r jo" 0. , .

oIoMJlmTI(ot IOIIT|'|O' 2. trdv6dyou B 83

H ( l ) 6 t v € ! z - 1 v l - ( x z + 0 2 ) .

'Apo z =f (x ,0 )=!4:n:;02J.

Ileroox4lqrL(ou!€ o€ miLx6q

!6vec, t tvaL x-por. ,ve, 0-pn!8 p€

0 < o < 1 . ; : ] + l : + = o x o lu t D , o l

r(x,rlt =r'lilz+rtrz t -Vi:iz

'roo -$=JJrrx,,r,raxu,r, =T

, a / 2 . . t _ . r / 2 , l-J, ll, o"i;:t-o'oolae :1, aeJ, or:a-o =

=rr'/ '?\,- + "r+,ar r -p" t =- ) (a -o)J r r-o,rr or r-0, r=- f $[rr-o.r;] '- i r - r : - i 'Apo v= * - S * . uoutoec.

0 6ytoq rou or€p€od nou oXnlGL<€roL ond rlv nep[olpoqi €vdc eflLna-6ou xrlpLou T (xou en[na6ou X00),v0p0r drld rov deovq x'X 6cveroL on6 to!rrJno:

v =JJz"uara,rT

v = JJznxaxa,r,

( 1 )

t2)

o . e .+

oa9,Eo(t 'Dtt roort|Il€! d6 tnv r€Olorpotli rouI, X' ! V I V-xl y0qr 0116 rov 6low x'x.

-3. 'oyrol or€9€6v (m6 negrsrpogll

'oTov n f l€pLorpoqi v[vEL nepl rov 6lovo { 0 o ovTlorolxoq r inoq Elvdr

IIap66€Lyra:

I5 8g€0e1. o 6troc ouIrpl.oo T={(Llr: 0sXS

0J----- _

' n o p " - i p n o n 1 n s o e l i 6 d s 6 1 .

KEOAAAIO II

jt0on I

To Xoplo T ncpLoidvaroL on6 ro €n6u€vo oxiuo, E[voL:

rr rtrvt lv=ll2n'ldxd'! - 2nl l l , '!d'qdx =

f ' ou r " r

="1"1-F lj.*,,l*x.rax ==-!f x. uord6ec.

-l. Etrgoad Enloovsqlv

ee{,rpo0u€ uLo €nLedveLo no! opt<eroL ord lnv eqLooon z=f(x,0)}. 0s0Fpo0ue 6vo luila G rns €nledveLoq ouriq,nou nepLd\e[eE! on6 lro (l€Lorixoln.ltrn C. 'Eoro 6rL n op8i npo8oli io!, rui[oroq G rnq elrr06v€Lo(, oroenLn€60 oXV elvo! to Xoplo T. To etgo66v S rou rUiuoroq c rnc enLodveL-oC z=f(x,0), 6Lvero! ond rov r0noi

,, - r---:-------:-:s ll /t.( ji)'.(jfr)'axo,t,

T

nogadlprton:

Av n noogoli ytve! orq dltroenlnebo ouwetoylavov r(oL n eflL-qdv€[o opl(eto[ an6 rlq €[LodFaq

x= f (q r , z ) , 0= f ( x , z )

r6re txoupe ovrlorouxo rouq r0-

, =IIlqif1l#*.T

, -JJT

dxdz'.(#)'.(#)'

* t oru ipt r lon f (x , { , ) € i lcL ouvexr is rcu r ixer lepurr is topardyou< hs tda€os

0I0K HP'TIKOE lolllMot 2. Xrd!6ci u B .

fioporipr|oq:

A v 6 € v 6 l v e r o L T o x o p L o T T 6 T e o u r O n p o 0 6 L o p l , C e r o L d n { , l c

n e p L y p d L l o ! s o r n v o E i [ 6 s 7 9 .

Ilapd0e Lyua

ib BpEe€l ro €uBo66v rou r l iuoroq rnc €nLqdv€Loq z= ! t^ 'L, , l t ' / ' ITou onotou n npo8oli oro entnEdo xou opi<€TqL sII6 rls oxtoeLq 0 < x ! 1,

o a 0 : 1 .

Aton:

E C v o L I = i ( X , L ! ) € R ' z : 0 s X s 1 ,

0 s ' r : l ] K q L

r(x,ur)= + (x "+u " )

'Aeo s=lJT'Exouu€:/ii;ffi)0,0,,,,

+=+ (+;r. ,4.0)=x,, "s,

#=+ (t-+t ' )=u* . ' r 'o, n (1) 6LveL:

s. [ [rt-,+ a,ar,= [ | t r-,-,r,rro,r,a,-[ $ | r r.r.,r'" I'or-

- i I l , r - r r ' r - , ' . r r ' , lor= * lo*r1 ' r , -y,*r , ' ' " l '= ' . . , . uoursec, J , | I r I l .

5. YmloyLo!6c gdaoc, (6wpou gdcoc rol podrv odpdv€mq ltoq mtn€dnq u-

lLKic €nredveroq.

8 5

Av unoe6ooutr€ dT! 6vq xoplo T ELva[ 6vs ulLK6 ai , \ ; \o KoL 6rr n nu_

xv6rnro rnq !d<aq oE x60€ on!€Lo rou (x,U) 6lvero[ qn6 rtv ouv€xit ouvdp-

rton 6=6(x,{)*}r6Te n ouvolLxi u6eq m nou KorqvaueroL oro x0jplo 6lvEtc!

( 1 )'=[ futr, ,urara,r

*A! 6 (x , i l , )=c (o rd lepd) id r€ ro xop io l ; y€ tdL oroy . !ds .

KEOlrlAIO II

. 0L ouvrEroyttveq (i,0) tou xCvrpou ud<oq ( rou Xilplou blvovrdl ord rouct(Jnouc:

. 0L opLolnlaq c[ou( r0nouq (2) noproEdvouv rLc (ororLKac) pontc ( i npdr-!€C ponaS) r, lc nooq rouq dlov€q 0'U, X'X ovTlo[otXo. Ero! elvoL:

rr=JJxotx,urxauT

Ka! r)(=JIo6{x,0)dxd.!1

l2)

(3 )

( 4 )

Ilapdo € Lypq :

Ib 8ps0€[ ro

{,,=t, '!=X3 6tov nvdgrool ltuxv6tfioq

i0oI:

ffurtx,,rlaxaunl lxb(x , { )dxd ' ,

x - T l - - '

l .J 6(x ,0)dxd0T

6(x, ' !)dxd'!

. 0r nponyoolevo! uinor (21 ypdqovior l= * , f * .

. 0L pontq o6pdv6[oc (i 6e0rEp€c pon6q) Ir, I!, oq lrpoc louc dEov€c x x,(r '0 , kor n notrl( i poni oopdv€Loc Io oq f lpoc lnv opxi 0 ELvovrqL ovrl-

, orotxo ond mu( t0trouq:

Ixll ' l iz6(x,,!)dxd0, I+=Jjx'z6(x,U)dxdoTT

Io=IJ(xz+'!z)6 (x,{,)dxdi! ( i I0 = Ix+ I!, )

(avt9o !6(os too x,plou I mu oOa(aror ord tLc rqlnolecratovolt r4q !6(oq ornv rcploxi T dtverot sni r4v ou-6(x,0)rI.

H y€r,uetpLri nopdordon rou

x(,jpLou T dlverdl on6 ro nqpo-n^€0pLrq oxilo. ELvoL

;-S,o"$. n*,'{Jo rr,u ru*ar JJ,r,ora,r,.oo llorlnqjJoouuer flpd]ro oc npoq 0.

oloK[BPmIrOt lOIItl|Ot 2- Erd\,6d1ou B 87

To 6old rou t elwr 0=xr KaL o.rfi ro 6pLo tou x E[vq! x=0, x=1,

'Erdr 6xoorE [[ or r,,r,l*ou= [' lw,rnurr- | r 'r"rW r i' " lsJi ,oJx! ,, [+j,,'ot=zJ

(x-x')dx'ii-'

ervoL re.JJx6(x,u)dx*=1"J|,**o*J'- t+ ]:r'.+ J,t*rr*= o! .

, , , , , f r , , , -vtElvor 0(610 i|x.'|,|06(x,o)dxdl,-J Jx,u"Mx..iJ t''lrr ax-

-t j, tvlt-x'tax. -ih . Etor. txouue:

i'+'+ =$ , t'i"''qi =-l+

88 KEgAAAIO II

IUTTMIIEId oE|dE Vra u.d orEr||$r€pn o€fu|ot EU Eurlot oloxl|I'agoEC

'!oto f lLa oo!o!aw ra! gpoyt levn ouxiprnd ' t oro op9oyoi lLo 1: lo , l l r t ' r ,61'Eoro axdlq uLq 6 la lapLon P tou opootovfou T d€ m qtoLx€! .66n opgoyi -

Mi ,d r€ror ,c 6ualapLor xqtddrevd!€to! d! edpoulE t r rs €u$6(€s x: l . , ra !

{=, } j ( i=1,2. . .D, i=1,2, . , .h) lcodtr l4 l€s rpos to, . ,s dto! .s { , '0 rdL r ' r q ! -

Yroodrou! . drL a:Xo < xr <. . ' < xn:B rdL y={0 < { r < 0, < . . . < { ,n=6

o € o p o U l e t c o r o r , x € l d o n o p o o - r d ! L q T 1 1 = [ x i _ r , x i J , l 0 1 _ 1 , { j ] ( 1 = 1 , 2 , . . . n ,

j=1,2, . . .h) !E e!8cod! A. . . To e! !a6d! A rou opooyov(ou T e ivoL ido !e

A=(8-d)(6-y) raL ord ld . fEL i :oo !E ro dspoLoro ro! erBa66v Ai j r@v oroL-

x . L o 6 6 ! o p e o y o v i o ! T , ; . ' r r o u e i v c u A = t : A r r .

! iwL axdla Ai j=(Xl -x i - l ) ( { j - , ! j -1) .6x i6SJ.

oOrcrtC l. toro lij ,o4rj ro dv6reoo ro! rardreoo rdods tns f oto

o p i o y . S v L o T . . ( 6 n r d 6 { M . . : s u p f ( x , + ) , u r r = I n f f ( x , + ) )^ ' ' ' ( l , o ) € r l l " ( x , , 1 , ) € T . .

0 !pd1!dr(16a doLStds

repo d0po!dr6 olorlilpFns (rn

0 npaylarr , r6s ap!ouds

u ( P , f ) = : t" i j

^r i

! f lLq in ! 6Ld ldoLon P) .

L ( p , f ) = E ! r , r r A ; r

o lor l {posns ( rns f rLa t r ! 6LaldpLon P).

8.opoJ! . rnv 6Ldldplo1 P xcr ro onueic ( t ; ,n ; ) t "v oroux.r -

T . . ( i : 1 , 2 . . . , n , i = 1 , 2 , . . . m ) . T o o i v o r o r u ! o n r e l u v c u r , j vq xdL oloudqeEL oit\,olo to\, e\,6Ldrdoo! onu€C.v rou opooyo -

I ! .d tn! 6 lq l ipLon P raL ro o,Svoro t , tov evououioov onueiov,

d n o o L o u d s ( P , f , t ) : : : f ( t r , n r ) A r r ! o u o ! o ! d ! . r d L d g p o L -

dopoLdrd ololllilpocns i €!6duedo depoLora) rns ou!(ipqons f

00LoU6q 2.

oprqrdc 3.

oloMIrPnTIKo! lotltMot 2. lravodyou B 89

ooLoldc 4. 'Eoto A to 061010 dl@! ror 6Ldlep|joed! rou ooOoyoliou T.

o!oudqo!!. ouir€po 6lrrd olorldpouq rns f oro T rcr oulBord<ou!€ fT f io!

rpal ld1Lrd dpLsud inf u(P, f ) , To! rpar ldrr , rd sup L(P, f ) o loudAou!€ ldro j -P€A P6A

iepo 6Lrrd orox l {pota 1n< f oro r rar owgor i<ouDE [ [ f .

ilf : i n f U ( l , f )P€A

l ll l f : s u p ! ( P , f )

- T ' "

0pLCry6C 5. H ooqiptnon

ororr ' tpr ioLrr r ) oro opsol6vro

0 rpd t ld rLx6s o ! rds

y6vuo T rqL d ' i l9o l [ ! . r ! r ,

f rdyeraL rard Risann o lorrno. io i l i ( i o t r r . is

( ! € R )

6Lr ld o lo r l ' i pd ld tn< f o ro opoo -

t t l l!e . lJ f d l . i

T T

f{Xi4)dxdqr.

nspsrnpioeLq:

1. lua rd ie 6Ld lapLdn P ro l r T LoXJ.L !A S L(P, f ) S U{P, f ) S l {A t .

2 . r L o N d $ . 6 1 o ! ; o L o r F x c u v u o r r i s e € x r o y i 1 0 ! o u ! 6 r o u q r u v € ! 6 L d -

! i o , r o n r , e i , v , e i . r " u L ( P , f ) l s ( P , f , q ) : U ( p , r ) . _

3 . r ' Ld xdoe 6Lo ldo lon p roL r , o r iEL : r rp , r ) s i j r s JJ r . u r " , r r ._TT

np6ldqn I . H duwiprnon f rou . jwL opLo! ;vn roL epcy ldvn oro opgoy; -

v lo T e isL o lo r lnpdo l ln o to T drav raL ldvo d td ! tLd dge E > O urdpre !

! L q 6 L q u a p L o n r , ( P € A ) r o u T , r d r o L o n o ! U ( p , f ) - L ( p , f ) < . .

Opt o t t6C. ' ro t . t . . ro r j { ros tns 6Ld. ro ldou 10 ! d toLxeLd6ous opgo16! i -

o ! r . . ( i : 1 ! 2 , . . . D , j = 1 , 2 , . . . n ) . B ) € r r d r n t o l l e l l r n s 6 r c y a p u o n s r o p { a e -

t o L a r d l n o x a d n l l p l l f t a x ' . - ( i : 1 , 2 , . . . n , j = r , 2 , . . , n ) .

tb6roon I I . A! n ourdprndn f € i \ .uL epd-y ldvr sro opooydl lo T rdr€ e j -

* p xdL 14 e ivou "v . lo rouxa

ro rd i i rEpo xoL 1o d !6 repo t rEpoE rns f o ro

ooooydDLo r . A .a laL 10 e !Bo6d! rou r .

lG=ff, , t) ) J )T

- 1

r i ' " u ( P , f ) = [ [ f x q r u . r ( p , r ) - [ [ fl lP l l - 0 ' i l lPl l -o !1

90 KEOAÂIO II

Il06toon III. To 6Lrrd orLoxliporrd rns f oro opso.rdlLo T urdpxeL rdL

e[w' , | :oo ue to! rpayldrLrd dpLl l ld ! rdre rq! ] rd!o tdr€ a! yLd rdg€ e>o

undpxeL 6 > o r6toLos ro! s(P, f ,E)- l . l < e y la xdge dopoLdua RieEnn

s ( ! , f , € ) ! e l l P l l < 6 .

0p!o l r6q. uLa ouuiprnon op!o!a!n raL epoyrdln oro opooldrLo T ra l re d-

tL E[aL rdrd Rimann o lorrnpddl ln ( i o lor lnpt ioLu) oro T,dro! xaL !d!odra! urdpxEL npq1]rdrLxds qpLgtrds I rdroLos r , ldre yLa ) idg. E > o !d u idp -

xeL 6:6( . ) > 0, ddr€ yrd rdg€ 6 laudpLon P to! T !e l lP l l < 6(E) rcL yLa

x d e e o d v o l o E ! 6 ! d ! d o q J v o n u e J o v ( 6 i , n j ) , i = 1 , 2 . . . n , j = 1 , 2 , . . m , s . t ! s L

| .E, : . r{x i ,oj)At j - t < e

lhpo6 e a.yuqta :

1. Ar:!et6L n oudprnon f(X,{,)=X+$ ooud}d\rn dro opoordv',o r:to,3lrtO,3llh yr:veL !|,d 6!d!€pLdn P ro'r I crd 9 oployrirrLd r(d|" r6i6rr,v \ia 8c€0o6! oLcpLeloc U(P,f)r L(P,f). Io GBhleo.er: sq ou!4(eLd n axedn:

! A : t r ( P , f ) < U ( P , f ) 3 x A ( 1 ) .

€€opo\ i le lLa 6Ld ldpLon tou 6Lddr { ld ros I0 ,31 o€ ro ld 6Ladt { la ta rou

1t le lcL ord rd o roLXe id lov owdrou {0 ,1 ,2 ,3 } . 'E roL

6Ld lep tqo ! !e ro T

d . 9 o p o o y u l l L o . T . . ( i = 1 , 2 , 3 , j = 1 , 2 , 3 ) u E € l , g d O d v A i j .

0 5

0 5

1 :

lLo ro op9olr iu ,o Trr dxou!€ dt ! ( r

x S 1 , 0 3 ' I S 1 - 0 < X + q J < 2 -f ( x , + ) < 2 . ' A p a

! r r : 0 M r r : 2 , "

ILo ro opgordvLo Tzr dxouuer

x s 2, 0 ! Q < 1 - 1 < x+q, 5 3 2

: : f (x , ! , ) 3 3. 'Apo t r2r :1,M2r:3 I

T

T* 1,,

T i , T , .

T . r

0 1 2 3

i = 1 , 2 , 3 , i = 1 , 2 , 3 . ' A p d

Au! loyd Bpddaou l l€ d rL t i3 r :2 , t43 l :4 ,

! 1 2 = 1 M r 2 : 3 r ' t 2 2 : 2 M 2 2 = 4 ' r 3 2 = 3

l ! 3 2 : 5 r ! r 3 : 2 U r 3 : 4 , r r z 3 = 3 M 2 3 = 5 ,

u 3 r : 4 [ 1 3 3 : 6 . l ( w L A . . = 1 Y L d x d o E

3 3 3 3 3u r P , r ) - I I t , , . r , . . I : H . . = I ( M i r + r ' r i r r M i r ) -

:Mr r+M12+Mr 3+M2r+M22+U23+l '13r+U!2+M33:2+3+4+3+4+5+4+5+6 : 36.

IolOKlIHPtrlIlOt ]|olItl{OI 2. Irv6fiou B 91

3 3E':wL cadud L(P,f) : E E urrAr{: !1r+!r2+!rr+u2r+u?2+!2!+!rr+lr !r+!! !=

:0+1+2+1.i2+3+2+3r4 : 18.

E ' : B r , 0 S r S 3 , 0 S { , < 3 - 0 S x r $ 3 6 - 0 S f ( x + 4 ) 3 5 . ' A o a

!:0, |:6r, Er:\to ,ror, A:(3-0)(3-0):9, troo n (1) r|:\,€tar,0 . 9 < 1 8 < 3 6 < 6 . 9

2. il,v|,'d/. n duldpqd! flr,o)+-2x op|,dnaq oto op!o1dl,!o r=t1'2l't3,51

? ? nd') lh ' r roroyldoo'rv o! apLo}or i u(p,f) , L(p,f) , l l f , l l r (ro u-) ) t t

6|.c|ldp!d{ ror, f drd rrr opCo1t6rnc i&oua rou rdc€

o"us f , -uf , t .B') Ia oroloytsl€( io s(P,fr;).

1') Ita urolorrooEC to Jj(+-2x)dxdf ue 6'Jo

rl0ol: T

c') 8.0,p00!E uoo 5Lolapr,on iou 6uocr{lctos [1,2] crd v 6',cc?fuara

t16-'rs 3:-L .!, €dtu tn! {1=xo < tr . X, . r.r < xv=2}**.

oEopolltrE rcr ut! 6ud!6p!dn tou 6rcorihqrcs [3,5] ard ! 6l,cdrrluaB

'ld-"s +.+, ,oro trr\, {3={0 < 0r < $z < .., < 0!:5}fir.U€ tLs 6Lc!.pr:dELi ourda ro opooth,r,o 6Lduepr:tet6L oe v2 opoopivcc.

Tkk l rr_t,x|<l r [{x-1,{r.1 , k:1,2...!,

^xr=(xk-r"-r) (+r-0K-1)= +. + = +

. !=iDf f(X,t) , n=sup f(X,$)(x'o)e! (x,0)€l

.. Et$r, r0:r, xr=l++, r,jr++,...,r*=r+f, ,...,1u=1*-lt-=..

..r Ed\dL {,0:s, sr+++ , a,r=3++ , t.=o+ 3!,...,0v=3a + =s.

.-'! TrE ,X€r 6ldord-

tpdrorrs.

92 KEOANAIO II

Iro opool0iv.o r<K drouue )k_1 s r s x< I l-2r<

s -a( s -2x.-r l

+k-L a u s {( l '

I o*-r s +so. I- 0 " - r - 2 x . < { - 2 x S 0 . - 2 x K _ 1 - 0 K _ 1 - 2 x K g f ( x , 0 ) S 0 K - 2 ) ( K - 1

^". """=ili j:{;:,

=,fK-2xr_l,,d1 UKK=if;,,ts'!,)aK_1-,xl<. Ero!;vos:

["t0, '+,- ' '*."

ucp,rr= i j ,"""0"-=+ A j , (ok-2xK-1)=

[A'"t *-, i'"i.,'",] =_ 2

2 I)-2! ( xo+)(1+" ' +X. ,_, ) l :

= + tt(,- + ) (* *)-...'(* aJl,[*('- + )-(* i- )-...-(" *J]l= * [[,*f r '-,".-...*r]-,[* ]r '-.-...*,*',,] ='Yu

Erolavue U(?,f) : 2Yu eC*" axdrrd L(p,f) :

= + j, .i (+(-r-2xK)' * -i ' j, ,*-,

= * [ r r . * , * , * " ' . {u - r ) - : t x ,+x ,+x ,+ . .+x , ) ]=

= * [[*(3. *) {* * ).....(, 3lir)]-[(* ;;4,- +)*...(* g]l= f [ . - f , t t - r- . ' .*<* ' r r-r [* !<t*r*. ' .*r ] ]=

= * l.*i ,'i," -,1** ,r#.r l] =+ [.'*-.-,*r*'r]== f- r*zl. 4:I . '1oo 11p,11= 2t4 .

Eiw! l l(q,-2x)dxd,l,= r in r(p,f)= 11' 41 =2

llr+-rr,lora{= lit u(P,r): ri ' 411 =2.'i '*- \+@

*?, A '*^""=-rx")=*u i , r . - r - r r" ,=

OIO(ÎIPmIKO' AoII$lOt 2. XBv6dyo! B

9') 8.opotrr. oa! o{.voro erJ6LolEsov onuedov ,rd onu€rd

K:1i2,...! roo e(wL id xe\)tpd $rr op0oyo!':o! TKK. EirJcu

,. 5l .". -!, =

^K -1 ^K I

- !

_ ! _ 2 \ , * 2K -1 *YK-1 YK

-K 2 2 2r - ''1K= --'a- =

_ 6!+4r(-2 _ 3v+2k-1- - - "2- - - -T-

( CK,nK ) ,

3+ 31511 +3++

s(P , f ' g ) : E

! / t"+zr-t. t-_-;-

t f (€K,nk)AKK:

"'\3*')=+E' r= l-" i , i t=

'r ' ) c) pdros _t t l t

EreL6i . iw,. JJf lJf :r ,

. tetdu drLT T

B) tpdr.s

Ed!aL l l ( {-2x)dxd, l ,= t tn s(P,f ,6) ='f \>x-

9. o€lportu€ fltv ouldprtdn f rou e(\E! op!o!6\,n ist gpoyxa\)n oto op-

to$vro T: [o, r ]x l0,1 l l tdL rou op(rEtsL uJs sEis:

I t ov x,(! Pnrot oPLolod ro! Tf ( x , { ) = I

L o a\, x,t|, drpntou cpi,ollor: rou T,rb e(eEot.C Bv a f ef\Jat, olorlnpolrLun orc oplo.rdlLo l.

Tdro P lLd oroLd6i t rore 6LautpLdn rou T.

Ed\aL ocqds dtL xdae drouxeL66€s opoo.rdvLo 1. . reprd leu cncpodt l to

dnlecd yLd rc oioia n f ndrjpver tnv rLuri 1 xaods xal. onueJd 1t"d ra o-

l lI l (0-2x)dxd0=2T

lin 2=2 ,

!Coo! tou 6uoodlaros tx._r, l" l

laoov rou 6loodldtos I0K_1r {,,< l

94 KEOMAIO II

rodq n f rqi;pleL rn! r!!i 0.' A p a

: . : 1 , d L r " : . = 0 v i : I , 2 . . . D x d l v j : 1 , 2 . , . r i ' f r d L . d w L

U(p , f )= E E u . . a , .= E E re . .=e=r

(ro e!8a6d! A rou op0or6! lo ! r . r : \a ! 1 6!drL A=(1-0) . ( r -0)=1) iqL

L ( P , f ) = t E u , , A , . = o ( € o ' d o o v u . . = o )i : r r : 1 _ l r l r l

'Ersu loLidv o i loo\)d !e rnv rpdtadn I n f 6€v ECvaL o lor t rnp.SoLln oro T.(&pd ro drL ed\ lcL {payudvn), 6LdrL n ox6on U(Prf ) - l (P, f ) <. 6ev L-

rr. 6dr,.roo ro oploydvLo T:to,llrlo,tl rr! ltd dutdpr4c|l f !.

f l d v x = 1r(x,*)= Il o a ! 0 s # S l r r

tk Ep€ooitv oL dr lolod r , , r , u(P,f) , L(P,f) J j f , JJf .

rrn dudx.6 ET T

6.rX!.4 !.6do rrdroua dr! n ouvdpnsn f ed!s! olorhpdduln dro T.

ldor:

q ' ) ! i j w ! 1 s f ( x , 0 ) < 0

'A!o U: sup f(x,+): l( x '0)€T

v (x , { , ) €T .

!= inf f (r ' { )=o(x '0)€r

E') 8.opoi!€ !Lo t l ]xdio 6La!CpLon ?,aid rus rdpdt lnles rpos rou! d-Eo\Jes EUoeCEs X=xt, q,={, j , !e O=xo . x. . x2 < . . . < xn- l < xn=1

o=40 < {r < { , < " ' < {n-1 < {D.1.

E. xd5€ srolxELqi6€s opool6vLo

r r ! = t x r _ 1 , x i I x t + r _ I , , 1 , . J , e t w u r , . = i n i : f ( x , { , ) = o' r ( l ' o ) € T i i

' ^ p o L ( ! , f ) : E I u , , , 1 , . = I E o e , . = ol = 1 i : 1

- " - J i = l j = 1

Irc opgoy.3lLa tx,_1, x. l ' to j-1, +j l=trD_1, r lxt , ! j_1,Uj l , j=1,2.. .8 'EpLd-

xo l raL or le [q yLd ra o ro id n ou ldprnon @ip !€ ! ?n ! rL ! i ! 1 .

Il@x lotIEoI 2.

'Erdl 1|.c ro opootdvLa dud E,:wL !

r.,a ft lrdroLra opooydvLa rns r*"t:j"l;. "r*,l : r , 2 . . , n . 'E roL exoq re

u(P'r)= .E ilr trr^ri=

,:=' (r=t*trtJ=

U i . 3 0 , i = 1 ! 2 r 3 . . , n - 1

'rI=' turrrr,,*u.,q.,'. '.*un-r;An-1.;*uo5no., )=j to'o*....o*u..0o.,-

' :!i ""1^':= ,!,

*"= ,t

(r'-xo-1)(qj-0,-1)=

=rrn-.n-r) r!,

(0J-oj-1).(x,-xn_l) I(0r-0! )+(9r-0r )r.,.+(+n-0n_1)l :

:(xn-xn-l) r. 'Apc u(P,!)=xn-xn_1.

r') rcrlc,, l l t= rrtr u(p,f)= u'' i lrPll 'o ltP lr .:\ '-\-r )=o (1)

x*" Jfr= sup p,r):o (2) (n ,d,. iir. l, ",",r,=0,.:i p€A ,1 ltr rr- o6') Ard (1) ,ta', (2) rcoi0ne,, drr n f cdvql olorhpdsr,ln sro T ral,

ud^'.6ra JJf( r, 't, )d)(d0=0.I

Itjnopo0le w 6e[qovu. dri, I f er:\,cr, otorr,npdor,ln dro T ro|, os €eis:'Eoro e > o .rl€you!. in! 6la!!p!or p arcL .!dr€ \E .,:vcr h_L_l < e rd-te .';wr, u(p,t)-!(p,f)=h-xn-r-o=xn-xn_1 < e (yLo iid0€ .'> ii)l ,rpo ,rou\jdotnon f .r:\Et olognpdorsn oro T.

5. 64.\,fl1 n dutdcflFr f(L0)=xr{ r.! rda. (r,t) € t1,2lrt3,4t:t.a ' ) rb 6€!xo€a dtt L(p,f) <sSU(r, f l .

8') x! u|oloyrolec ro oroddp4a Jj trtlaxal.

lrJon: T

a') toto Pr:{r=xo < xr < l , < . . . < xr=2} ,sr

pr=t3:00 < 0r < { , < . , . . 0n=4},660 6la!€pdo.us $! 6lodrnldrur, t1,2lrc,, I3,al !\,tdorooxd.

ue tLg 6 ldu .p i ldELs dur6s ro opooYdvLo T 6 !d !epd<€tqL o lc oP9oYdv ld

T i i = t x i - r , x i l , t , l r _ r , { . L i = 1 , 2 . . . n , j = 1 , 2 , . . , n u e € r r 9 0 6 d \ )

Ai j=( x i-x i_1) ( ,1, i - ,1, j -1). t ro opgoy6!Lo T. . Exouue

x . _ , s l sx . I. l l . " . . ! . I

- r ,_ . , *n i_ , l r+ { i x .+ ! . -+ r r_1+( i_ r < r ( \ , ! ) < / .++ .

l - r - - I ,

' A o o M . ; s u p f ( x ' " ) - ' . r d L ! , . j i n f f ( x ' , 1 , ) = r . - t + , 1 , - r" 1 1 , v ) € T ; .

' ' { 1 , 1 t 6 r ' ,

r i uL x , - , s * s r , I' i - r r_, . ' -_,s f r r r_r*1.r *

{.- ' * '1'r Iq )_ \ s -? s+ j )

* | <or-r++, I s xi+!, j + (xi-ri{ j-1)axia+ j < + ( xi-1+ri)AxrA,, j+

++ (,l l-1+,!j )AxiA,l, j s (xinrj)axiail j (r)

(6r@ Ax.Ai l , .=(xt -x t -1)( , l , j - , ) j_1)=Ai j to euBd6d! lou op9oYovdou).

udr ipvoutre ro 6Lr ld dgpoLoua tov 6 io uEld! t ! ! (1) . 01dt€ axourEr

t t ( r , _ ,+ , r , . _ , )A r ,a " , s f E I ( r ] - x , z - r )av ' i +i : I i : r

r r r r - l = 1 j = 1

ti = 1

L1r 2 t ({i-{, j-, )Ar. : tI r - - i : l

( Xi*{j )axia,l,J

- !(!,r) s | ! r*ir1-rr .i r+,-r,r-rr** i t*r-*r-r) jr,t]*;-r,

s u(?,r) - L(P,r) sltr;-r; lro,-,r,,1* |( \-xo )($;rl,;) s u(P,r)

* L (p , r ) < | t , ' - r l t+ - r l * | r : - r l (16-e) s u (P, r )

+ L ( P , f ) S i s < U ( r , f ) .

B ' ) ErEL6r{ yLd rd lE 6LdudpLor l P aou T,o oplo lds s € i lcL c le6dptnaos tns

P , d r d L ; o t c L ( P , f ' < 5 s t ( P , f ) , c l w l [ [ r r r , , ! r o , d " - s .1

OIOMHPnTI(oI lOlItMOt 2. tiqv6dyou B 9 7

Terooyrrv [oLtro-H avvoto rou

urootvola ro0 R'z-q!8o6ot €t6C 0pc$dFu uroo0tdtou rou n"

ddrpo opooyovfou. To! ; ipo (u{o)) Evds oployol rou O,(O q R'z) , !e 6La-

drdo€Ls qr ,q2 opdC.raL ord ro! rnro ! (0) :or .o, (owpeodr:sote o€ yvr loLo

6 erourro!avo op€oydvLo).

To larpo Elds opooyovdou (dros roL ro u i tpo xdo.6L6Lddrotou 6Loor i

l d r o s ) E [ v c L r o . ! B d 6 d ! t o ! .

IroL)(erd6eg onlElooJlolo. 'Ew eoo-ruflo onuEloodlolo A, (A g R'z)

rdre ld! oro!xeL66.s dtav. iEL r tudn rerepoqr4\Du i l foo ls opooydviov (xd-

p{s roLui eoorrp! rd onu€id ) .'Eoro AoL'dv dtL ro A e[ !dL n d lddn r0r ! opoolov io! or ,o2, , .oK

( A d r u 0 , u . . ' u o N ) .

To larpo ! (A) . !ds oroLl .Ld6ols duld lou A 6 i i ! . ru1 ard rov rdro

u ( A ) = ! ( 0 r ) + ! ( 0 , ) + . . . + l ] ( 0 K ) .

TEip61l,!ftr,ld uroo0\,ora rou R'z

-Eoro T dw epdyydw urooiwlo

'Edro Q. ro o l lo lo t6! oroL -

X€uo6d! onleLoouvdlor rou R? rou

i€p!€Xou! ro T rd! r . io o 'horo rov

oloLX.Lo6d! on!€uoould lov tou R2

ro! neOrdxovrdL oro T.

Eaw! rpoeqvds d iL . l lgo6d!

P l 5 . ! 8 d 6 d ! Q i , i : 1 , 2 , 3 . . . Y r o n e -

! * ( r ) : { s u p ( € ! 8 . P . ) , P . c T }

Etw! rpoesvas oxdud drL ! r (T) 3 u*(T) . A! r r* (T)=!*(T) tdrE n ro lv i rL! '1

tov h(T) , I l * (T) r6rerdL e!Bd6dv rou dnle lodlvdlou T. t r r ! r€p[r t@tr ou-

t i '6n lq6i otn! t repl t r iddn rou !* (T)=!n(1) to oJ lo lo T o!o!d<. tdr , ie tpdtd -

l idLuo ( i u€tpuoLuo rdrd Jordan). o apLi rds !* (T)=!*(T)=S ld l . rd! 6u6ui -

atoro larpor ,iotd,Iodan! tou T.

napqr iptoq:

I d X J o u v o L e r d u e v E s i p o r d o € L s :

1) At Ew onleLoduvolo . iwL retpapl ioLlo rdr . ro o, i lopd rou €re!

h d L ! * ( T ) : { i n f ( . u 6 . Q i ) , O i > T }

98 !€OMAIO II

!6 tpo un6d! ( !16€v lxd euga6d) xa ! a l rLo tpoec .

2) A ! rd oJ \ )o lo Tr ,T2 , , . , ,TK E i \ rcL t€ reayo ldoLr ]a rd re r d !6o i to , )s

xd&is xcl n rolri rolE eJw!" lerpc.rolioLrrd olwtd.

3) Av ra o ' r lo rd Tr ,T2 , . . . ,TK e lvdL.Erpoyo l ioL la xc ! 6e ! d rouv ro ru i

€oor .pLxd onu€ ic rd re € !B(TrU T , U , , , U TK)=€u!Tr+e l1BT2+. . .+€ !gTK.

-Ynoo0vold ro0 Rz !E n€p|,€xd!€vo gn6!v il udrpo ut66v

0pto!6c 1. toro c d lo uroonvolo lou Rz.

8d ld !€ dt ! ro c 6XeL aEp!€xdlE\E un66!* , drd! y la rdse e > O, ro crdriri.rc! ard r€r€pcorew iOos xl€Lstd opooy{ilLd rou R2, rou dxouv d-gpold lo e!8a66! u l rpdlepo 10! €, (6f la6f G g or u o, U . , , U oK rqL

! ( 0 r ) + l l ( o r ) + ' , . + u ( 0 K ) < e ) .

09ro!6C 2. '!drd c ds unooilolo tou R'z.

8a leuE dr ! ro G dx€L l i rpo un66v*nj dtd! yLd xdoe € > o ro c xdlJ-nrEtoL drd ! !o d) to loustd r leLor6! opgoyovuo! rou R, rou 6xou! dooor,d ld . ! -

F d O d ! l l x r d r € p o r o u e . ( 6 n r a 6 i G E O r U O , U . . . r a ! , l l ( O r ) + ! ( O r ) + , , . < e )

I lapdr ipnon 1. Av a\E urodJvolo c tou Rz dx.L repLexdrEvo !n6dv rd.€axei , Nd! !6rpo !n6dv ro a l rddtpoeo 6ev Ldxie! .

ndoafllprron 2. Av c €Lj!a! fw rl.lord xdL qpqyudvo uroddlolo rou R2tdre o! dv loLes r r ro c ar .L rep!exduElo !n6dvt ' ror , " ro G dxeL !arpo un6dv"

IIqpa6€ [Vtara:

1) To oLd.rpdl r rd rns ouv€roJs o lv lprndns f : la ,Bl + R dr€L repL€rdle-!o l rn6av rd! ldrpo !n6e! .

2) H eog.dd 0=ox+8 dx.L ldrpo !n6av evd 6e! aXeL t rEprexdle lo !n6a! .3) To o\ i lopo e lds xreLotoJ rqL qpdyy;vou xopto! rou R2 dr .L r .pLe-

XdlEvo !n6; ! rdL u; rpo !n6d! .

no6rdon: ' !w epal ldvo d i lo to T,(T c R,) dx€L ErBc6dv d! xaL !d!o d!

to d i lopo rou T dxeL l rn6ElLrd euBd6o'v ( ldrpo xard , rordan !n6, iv) .

* B !€rpo xdrd , t@dan ur66v.** H udrpo rdd lebesque un6iv.

otOKABmTIrct lollXlloI 2. Xtrdv6drou B

Stqac EU 6r o0 olorln,4lctoc rtro o'6vo e0ovltro uDotroto lou R'z.

Ei5ou. otn! qeli6c 89 ro! opr,dtd rou 6',rlo!t otorlnod|lcros rd\,o d'

6!0 'rl.r,ord opoolt|vuo. Itn! r€pdrtoon rou !r,o rovdptnor f(x,{) ed\EU op',-

o!a!n lai, qpa1,lldvn o'aw r..Ldrd raL rerpd.rov{oLlo XuJpdo T,(dt(u trdr'svdy-

xn opooiiv$), LdxrrEr o oplolds rou 6L od oloihpdudroa dtus 6d0ue drnv

s6l i60 89.

Ar\Jorrle ardya orn\, r€p(r dn svii rid! ro! .rd!e!o oor,dud:

op!o|r6c, 'Eol,r c dlJa gpdluAro ontr6Looflolo (e = r'z) ,o,, f lrd qpay -

!6!n duldptnsn oro c. !9 'doov ro c edvcr 9po1!6!o,ano6€txv€rsL dtL lJndp -

x€|, eld xr€Ldrd opooy0lvLo T tou R2, nol) r€pr,€x.u ro c (6nro6i c g T). Oe-

dpod!€ ltc du\)dptnon g oproldrn o1o T !E

99

. ( r ,s )= {

f (oo) av ( x ,0 ) € c

av ( x ,0 ) g c (6l l l ,q6 i o! (x ,0) € T-c) .

8a l6!e dru n ouldptnon f €Cva', tiard

Rft,llnn otrolt^npddl|tn dto G dtd\, n g

etw! olotilnpdol,lrt dro op9oyotvlo T.'ErdL Io!nd! op (aoul r .

JJr(x'o)axa+1 Jst x'+lara+r

0

np6roon l, A\, !!a cul,dprndn f e{!s! duveris o'a!a ,irELord rdL r.tpc-

lou:oulo od\iolo T,(T e R'),idr€ et!o|, olorhrddlln rlrd it€oann dro T.

np6tool 2. Av lta dr,vdplnrn f €(\,aL r'd ra1,n o'ava rrelcrd |icL te-

rpapu:oLlo o6sro T,(T c R'z) rau axEU r€rEpoouavo rrioos onrt.r:d! dou\,3 -

x€Las'rdr. Eij\)o! oloxlnpdolln xcrd Rt€narD oro T.

-npot&Elc oE|v olorl4q'lludtf|s l|'oc dry{qEutc

t l* Aro6e(xleidl dr! ro olorlip@lc

JJs(x,{)dxd0 .i\la|" alE{dprnro rou opoo-

T

lov(ou T, 6nLa6il yLs rdoe dllo rl:ldrd oplorilr,o T'lDu iEpdxEL ro c .CwL

J Js{ x,+)ax+11gtx,4,1at64.T ' T

100 KEEAAAIO II

nO6loon 3. A! lLd su\xirntdn f .t!ar gpoyldu o'dw rreLdld risL te-

tpqyovior ,uo oJ loro T,(T c R2) NaL ard ld .dwL ower is dro Tt extds drd

to onl r . lo e lds ouldrou U,(U c T) , t rou dx€L ldrpo rard JordaD r1n63v tdr . If E iwL o) 'o{rnpdour xord RieErn oro T.

np6tdor t 4. A! l l ld oudprnon f . iEL qpd d ln o ' !s x l .Lord xqL r€-

rpo-ro l ioLrro onvoro T,(T c Rz) roL € isr oxeOdv rovtof ou!€x6s* d ' durd

rdr . . iwL oroxrnpi iq lun xold RieEDn oro T.

noO66€LyUo l. A(!€rcu r ouldprrFn f opLo!3vn dto! idro

1 1 r+q , ov ( x ,+ ) l ( 1 ,2 )T2Ux,+, / x'++' 391 | lE f(x,U): l

t o dv ( r , { , ) : ( 1 ,2 )

Ns et€rsdt€C a! n f .dwo olorihpdolln orov rdro T.

E d l d o l f ( x , , ! ) l = l ) ( 1 + + ' : 1 = x ' 1 . r , r ' s e y L d r d o € ( x , 0 ) € T . ' A p a n f e C w L

H f . jwL duv.r {s de rdoe ou€io rou T € i rde 1ov (1,2) ro oro lo id-pouordC€L qouval .La.

EU!o! I ih f (x ,0) : s l f (1 ,2)

( x ' 0 ) + ( 1 ' 2 )

'Apo d iuqow !e lnv npdtdon 2 n f .JsL rd id Riemann olor lnpddLln drov

Ilop66€Lyuo2. Na . tErodrer : "a! n/duldotndn f (x ,o)=d' r ! i - e( r rL o-

"p6o i i *o r= \0 , i1 ' ( \0 , i l

f "-+ ", (',, '.)€(0,+],(0,+]duldprnon s( x,{)= I

rv

I 0 o! x:+=0

H 8 E I E L o p o y l d " n o r o o p o o . y , r l L o t , = [ . , + ] , 1 " , + ]

6 L d r L o < e ( x , + ) < l' r !q rdn€ (r , t )€tr .

H s . i s ! d u l e x i s - " t { . , + ] ' ( t , + i * " d d w 6 x i < o r o d u l o ^ o

'r Anla6rS e ivdL ou!exr1< o lo T Errds drd ta onueic evds d l ld lou U!(U < T) , !eldtpo rdtd l ,6besque !n6Ev.

0 o(lUp0TIt<Ot AOIIU{o' 2. Erav6d.You B 101

c={(x, ,1,) €R2/(x, ,1,) o 66a

Er.r,6i ro airoro c aXEr, !6,100

tov rleupul! OB, oA]

J@daD un6;v,n g e|:!dr, olorilnrdoLln

Tl ( rpdrdon 3 oe)L i '6d rco) .

"t*" JJrt,,,lu,+{Js(x,{,)dxdo rT l T

l t t t t l+J J s(r,+)dxd,r1.Js(x,+ )dxd++o1.J f (x,+ )dxd{

102 KEOAÂIO II

AYI.{ENE' AXXII'E I:

- o) YnoloyLotdc 6Lnl6v oloxlmqdto\,

l. ilo urotroyloo€t se rov ooLo!6 ro 6!nl6 olorliitr uo [l (zx*t)ddl,,drou T= I0,l I r10,2l. ' ;

A6on:

e€opo0u€ trLq 6Lo!6plon rou T o€ vz ro [I ieoc op8ovdrvLq. Auri n 610-ucpLon yLv€io! oq €{iCl

eadpo0u€ uLo dLuutplon rou Dlodritaroq 10,11 aorr,l rnv

o=xo ' x , 'X . < " ' < xx - t < xK < " ' < xv - l t c ro lo nou x " - x ;a -1 =

= | ' , t . = r , 2 , . . . , u ) , ' t r oL e ( vo r

x.=0, x,=f , x.. f , x,=$,.. . ,x"=f =r

e€opo0le (oL ur-o oLoucpron rou 6loorit/qrcC [0,2] eotu rnv

0= ' !0 < ' r r < 0 . . . ' . . 0 "_ , < q rK < . . . < 0v=2

x t ro ro nou { ,a0 ( -1 - f l . f *= r ,2 , . . , , u f . 'Apo €LvoL

(ro=0, o,-+, u,'f , +,=$,...,,r,,= -!t +An6 rd onu€lo rnq npdrrnc 6La!6p!onq qtpvoule eu€at€C flopdtrlnl€q rgoc rov6[ovo 0 0 KoL on6 to o4!€to rnq o€in€ptc 6ro!6plonc OcOvouu€ euoeleg m-pdlintrEq l lpoq rov 6[ovq x X. [ro! axoute net{,x€L uLo 6Loutplon rou op-oovovlou T oE v'? ro nlieoc op€oyd,vLo nou axolv KogAvo paon x"-x*-1= |,(K=1,2, . , , ,v) 60oc Ux-+K-1=+ , (x=1,2, . . , ,v) Kor euBo66v ^" . -+.++o€opo(lue oov ev6Ld!€oo onlei,o,ro (tvrpo K60€ op8oytlvlou nou 6xouv t€run -

X--r +X-l ivn {K" - :+-^ xo! rErqvUAvn

r ; bLoldplon tL 'v . rdL r jc v oucor ' i " " t " " rUtous f

= {, r H 6LaldoLon r ;vcta l l l . ! Ouot , i r ,or r , r t i rous 2:9 = lL

0loMHmTIrc[ t0rID{0[

. 2 r - 2 + 2 K

n.=v<-i.u. +=y# +E{.voL f(x,g)=zx+{ - f6",n")=2 2t;1 * 3IJ - r1;",1"1= +. +

= rlq*,a*1= qx;z

ELvqL oK6uo 6x.60.=(x.-x,(-1)(0.-q,._,)=(+- o.-!" ) i+

- 4f1I ]

=

=+ +=+ 'EToL o r0noq

f ( x ,0 ldxd0= l im I I l f ( q ._ ,n_ )bx_60_\ f f ( = t k=1

ocveL JJtzxrslayas.T

TT

l im : t : i - - # = l im l . L l L ( 2 r - t )=\4@ <=1 <=1 ' '

\H v - K -1 ( r l

= l r,T_ + \,t2( 1+2+. "+v)-vl.

- $-$ rut* ' r-ul =

=*-s "'2. l{o uroloyLo8el !e tov opLold ro 6Lnl6 oloxlipola

6 n o u t : t 0 , 2 t t t 0 , | .

A(jon:

tpyo(6loor€ ovdlovo ge rc nponyo(J!€vo rop66sLygo, (6nto6i 6@!€pL -(oute !o 6rdo[nuq [0,2] o€ v 6Looripdro nf6rolc 4 (oL ro 6Ldornlo[0,1]oE v 6roorhlcro nrdrouq l*

) . ' t tor gxnuoit(owoL v, ro n;\ i€oq

op€ov6vlo uE EuBobdv A( . -bx<60N- +.+-+e&JpotIE oov €vdldtl€oo onu€lo, rq x6vrpo (doE opgovovLou TKK, 6nô-

6i l rq onlsLo:

. 2 x - 2 2 K K - l xv*-r tv. - I - * i zr- ls^=#__#_ =.!',.".

JJr*'.ara,ar,T

2

-:- =xp'lp | | 110 ',t30x139 ql 'g

,"_,". p _,- * -,"r,= fr-,ft l_oo(,* _F) Jr{r fr -,n +z) )=='r'n lr"t. *z I J='1'nlxnr"*o'1 lj J=mtrrr*,t'tt J J ror ront:

. , - , * *

=1 ,a-na l - f -1 ,a -na12.1=

- [x1,"- ,"1. * ; ]+{,- ,*xz) J=x{,a-xf - ,a-xs) J=

r - r 1=xe,[,r4, -*vl.J-xe[4'P{ro*ax),J

J,J-xmrtne*4,x),J,i'on1]tr tr

"t rr :tlD0v

.'t9xp ( 4a+4'x) z J rJ.xpoPt+s.'ix ),J, J

:1r9 130xr 39 4t' t

f 'lh,fl'o r . 1 t r r o r r t r o r t t

=4'{24'+ i),J =4e,lxzf. -{-J,J= 4!e[xP(24-,x),J l,J=nrxn{,n-,x),J,.1

. "r o, t*u

t.MxP(.fr+.r) .J .J 't1t tt€xrt9 !t '€

''= i+- diir'+ - #r n"it"rfil =:l^- ( l+^)^+e+( t+^ )9- ( 1* i 111rn1i1 $- f f i

{n-1n'....i*1 1i* Elffirr )-:ri;il-t;ffi ] + -ffi

-('-,,+r-,r".,J'^?' 'i' +-il =;,'-",';6nzrJi' '"i

+-n- + t (-",t -,ffi ,J'i' ?

.fil =r0,0,r,,,,,,1'1 ooo.

II OIYWODI f 0 l

O o(^HPnTIKoX lotltllot 2. tro!6dyou B 1 0 5

i(roo I

1. q;"*'L" [{l'*J*=l , ''i:',,,=J'-. ,,..*,0,={#,,- 4 ll,{",.*) {},.*) =i

6. llo urlolovioo.l. to otorlfF,ro: J-f . f f *{,.aaUr,

Aton:

f f .r,'.wroo,{'If ,r,'-:+.,ror]o*=J'[,',r' $ .,,r,f .0,==1"(-,. + x,,zx-x"- S-".),,=l ("- # ", * ),,==[-'- +.+- + ]: =,-+.+ -+=+

' vrr?7. ib urblovrooel. to otorlhdE: | |) "n

Aton:r vIT'.t

ftt .-J -* ar- 1t""e.t',ti * ttntr*vi;al :'= f, ttnl rrf t- tn r :=

=+ tno*f).

8. tl ur.toyrdet to oroxlitu! J f "olt4ar.ltol:. t . x ' . r . . x ' - - r - ' / x - x ' z

,J j ""*,-J ll ",/xd{,ldx=l [fi"], o=J ,ra-la,.-(clox-lirpoon (ord nop6yoweq) =rx{et-xtl l-J r f"x-xtO*-r - [ex- {] '

=

="-'-G'*-')=*'

4drt+r'{'

1 0 6 KEOAAAIO II

f ,o, ,!64=lv/o-4x-4I:dxd0l, t'j" {a-GFJTdx ldo=

+ {o'-,1,') rornu ==Ll@ o,r,-. /o,_e, )o=f d# +*_+1",* + l:_+G,- g = +(er*' Jr"-*,ar=

tf--r' . $ -rnr, -f)

10. tb uroloyro€al. ro 61 6 oloxltp.Ird:

I-JJfr+faral 6lav r.{(x,0}: x"*!" s p', { a o}.

i0on: T

To xrdolo l ax€L odvopo ro nulrdd\lo roo oxiluqtoq,eo olo(lnpdFouue flO{,no (lJC trpoC 0.

eo Bpo0la ro 69Lo rov x (or (r.

06pvou!e ruxolo EUoelo nopdllnlnnooc rov dtovo U'U KdL ooloKo!!€ r lqr€rovucv€q rov rao LvOv on!a{.ov rnq uEro n€gLvpouuo Iou xrdplou T.

gg1,q1 U-s, {,=Vbt:tz. H yLrp6-

reot r€qrnllvt I|,,v onletov too xt!9Louel,vo! X--p mL n lEYoJ\tx€9n X=P.

'ExoL El,voL T=t(x,0): -p .< x < p, 0 :0 s VEa:f4,

,, p r1@t=f;, 1 pf ,1,21,1d2=7r'Apo l l (x++)dxd0= l l l ( x+ ' ! )d ' ! ldx= | , Lx ' !+ + lo dx"

',i ,o LJo J rr

{i [,,"r4" ;,""," r ]*: ]f , r*-* r! a1o"-x' 1*'f Joo to'-x'tax=

[*' $ ]",=o.f o.=f c.

oIoMHPtITI l io t iOl l t l lo ! 2 . t rdv6dyou B 101

11. llo uml\ovlo8el ro 6ml6 o]\oxttip.dtionouc, EhsL T={(x,0): {, ! 4X, X ! 4}.

il,on:

oxquo.

(x'-xtr)dxd'! !e 6to tp6 -ilT

0 r6noq T flogLsrdv€Tsl oro srdp€vo

o ) rp6rce, 0q oloKlnpdjoouuE npbrqlljc rrpoq x. otpvouuE ur,o ruxqLq EU0€Lo

nopdi. \nln npoq rov dlova X'X. 0L r€rtn-

!6vec &rv Ko Lvrirv dn!€Lov rrs eue€Loq !€ro neplypotuq rnq rrepLoxic (6nlq6h ro6pLo rou x) €lvsL X= + , X=4. H ! [-KpaTepn xoL n !€yoi0Tepn reroyutvn (6n-

lq6i l ro 6pLo rou U) €LvqL 0=-4, U=4,'EroL txouU€i

" z t rI l(x'-x,!)dxdu,=l | - tx'-xUlaxo,r=l l-{- - -1- u I -ar =

4 4

=l f- #".J: -a,r* 9\,, - toz+J_"\ 192 32 - ' 3 )" ' -

--T '

B' l rq6noc. 0q oloKlnpdjoouu€ npiurq oq npoq 0. 06pvouuE uLq xuxols €u-88l.o nopdllnln rpoq ro! dEovo U'0. 0r. rEroyu6v€s rov Ko L\,riv ot!€Lov rnq€uoElcq u€ ro n€plypduuo rou Xr, iplou (6nld6i ro 6pLc iou U) ELw! U=-2Vf ,0=2{. H u[kp6rEpn kdL n levol(rrepn r.run!6vn (6nlo6i rc 6p[o iou x) E[voL

x=0, x=4. EroL 6xou!E:

l l (xz-x{)dxd+=l I tx ' -x+ta,r tx=l lx '+-x i l dx=l 4x 'v:xdx= # .'i '" '-t,fr 'ztE 'o

12- lla Ulroloyrdoel. ro 6lnl6 oloxliF,ruo JJx'Vax+

O.ou o tO-c T

rE r.del.erdl dli6 rlc xqmtl€c x=0, x=r, ,lFo, iz.

I0o[:

To v€ri!€rpLKd sxiuo rou r6nou €LvoL ro €n6lEvo.

a') rp6ro(. 0q oioxinpdooule rpirrd oe npos x.06pou!€ l lq ruxsLo sugeld iroo6ltrq^n npoq rov dgovo x'x. 0L rertnufvec

r08 (EOAÂIO II

rtl,v Ko Lvriv on[Elrrv rnq €uo€[oq !E roorrvopo u rdnou elvql X:0, x=1. H

uLm6repl (or n leyoi0lEpn r€roy!6-vn elwr ovrLololxo 0=0, { i=2. t toL

txou!€:

l lx' idxd'!=l I x'{dxd'!=l 0l-i- l d0=' i n1

=[ 9a,r=llgl '-+.) , . { L . i o

B') tp6roc. 0o oloKlngiooutic npdrro r,Jc npoc 0,o€ovout/a trlo ruxolo €u-e€lo nqpdl,\4trn npo( rov d{ova U 0, 0L reIovlavaq urv xoLuilv onuELov rncEU-o€loc !e ro olvopo r{ru r6nou eLvdr U=0, 0=2. H lLrp6repn l(oL n u€vor\0repntetu4utvn €[w! ovttororxo x=0, x=j. t tol 6xouge:

I lx '&xa{,=[ [x ' rauax-f r ' [ { l 'or ' f z ' "1 ', i Jo L . ro , , t 'd ' -?L- i l , "? '

V ) rpdroC. go €@pu6oouti€ rov r0no rnc o€At6oc 61, Elvor

[[,'ua,ar,.f "*.[**- [#l'.[#f =+. + =+'i )o

13. lh uDlovldo0v ro 6! d otoxtno{Eto

o'l l.Jjx''ldxdl. 0 r6m( T [€o..rt€lsrd. drd rlc euoel,€c x=-1, I-2,H, a=2. r

B') 11J(4-o)dxdl 6tav r-t(x,S)/{=l-x, -2 s x s l}.T

v') I.llrurx'dxd,l,. 0 r6roq T n€prxl€aeror od r|'( mudrtcc U.x3 x=3,T

| - | , " ! 1 2 t <=l zx.dx=l :+l=ri + +_ =6.

B ) eo otroKlnpdooutr€ np6ro 0)C npocU, l4ld euoeLo ndpdlln,\n flpos rov dlova

010K^!PnTI I@t A0f I tMot 2 , tnd !6d lou B

0 {r ouvovtd ro oirvopo rou xtdplou oro o4uEl.o (X, 1-x), (x,0}.' A p o r o 6 p L o r o u { r € L v c L 0 K o L 1 - X . T q 6 p L q r o u x E L v c L - 2 , 1 . ' E r o r

1 0 9

axouu€:

_ l l 7-t \-r

Y ) 0q oloxlnpdoou!€ llpdrrq oq npo( 0.0o gpol le rq 6pLo rou U. 06pvou!€ uLo eu-8€La nopd,\,\nin npoq rcv dlovc U U nou 16-lveL ro olvopo rou xrJpLou oTq onteLo{x,c, ,(x,x 'z). 0L reroy!6v€c l | rv ontelov ourdvElvot ro 6pLo rou U, Iq snueto !e rnv !r-Kp6Tepn KoL t t€yql l rEpn Teqlnl tvn eLvqL xq0 KqL A qvi l ,of trxo. E!d! ra 6pLo rou€LvoL 0 koL 3. Apo 6xouu€:

, ' , ,x,I . l I j "nurx 'a+lax=l l+nr^x, l ar" l x ,nunr,ax=[ * r ,nx,c.rx,=

I t - l != :r | **"1;= * j -","22"-,"01= j ; rr .rr= S .

14. lilo moloyr.d€€|, ro olordrlipolq JJ tarol O*u T €[vcl m xoplo nouT

negLxl€l,€td! ard rLq mpc0ol6c

l ton:

U'?=4x lTo stornuq - I 6lv€L (X=0,

\ '=4tt )

0=0) ' (X=4, L!=4). To onuelo 0(o,o) KoLA(4,4) €LvqL ra Korvd onu€Ls rurv 6iroaLcypouldTov. 0a oioxlnpdoou!€ nqlrd 0Jqnpoe 0. oEpvouUE ULo cu€Etq nop6,\;\n-in npoq ro! dqovo U 0 nou rtuv€L Tootrvooo rou r6nou..T ord orllElo I KoL fu€ r€rdv!6v.q f , zVf ovrloror.xo.'Apo

+s U< zy'i. To onu€Ls !E rnv

uLKp6r€pn (oL leyol(lr€pn reunt6vn €LvoL ro 0 xor A !€ X"0, X-4- .fuo.0 r x 14 . 'E rsL eavoL :tt t \ e,,/ i t ' r . , . ,26llu,a*a,r-l l-' ,r,a,r,ar=l l#l'-^ r,=t(r,- .'{)r,-[,'- # l'='u- * -g' i

" ' " ' t ' .

J a / L r u v r o

15. rb U|lotov.ooet to 6.dr6 otoKlioq|a JJ(t-a,rxllnrdxdU 6,Du1=((x ,0) : I sx sz , o f Us t )

n0on:

EtvoLf f t r -z , r , , r t rn"ara l= [ ' rn* [ ' r r -z . l1" I1r'; ', ",

-x)du'ldx-J' (lnx)[{r-u'+x{J0dx =

.[ ' , r 'a'.[$ ]*l '-[ '++ dx=z lnz- I

16. ih unotroyrooet ro 6LrtO olorl\iipouu JJtUOrO,l 6.ou T GlvcL to

x0,p(o rou mrn66ou Oxl uErota rov o€6wr U.U, Tx'x

roL rnc xc[lnolncvf,.rrFt.

Aton:

eo oloxlnp6ooup€ npdEo nC ngoC 0.To 6pLo rou 0 El.voL 0 < U 3 ( l - , , f )r .Too P L d x o u X € L v o L 0 : X 3 1 . A p q

r r r ' r r ( l - v f ) ' z rl lx,!dxd,l,=l xl | oduldx=f I xfi-Vf)rdx- i n '1

l-"'= 26r .

t7. rto molovloo€[ ro 611116 otortiqflo II fqii- ** "*

r=t(x,u): l .xsz, t=U<*ll0on:

To x{rolo T noplotdv€ror yeote-rpr,(d on6 xo mpor €0poc oxiuo,

,t""' jj ffia,,ar,'r,vl ". r /r i lx

1, tl r-'Tia.l-' d0Jdx-J, l'oreoxtl" ax'

o-i-rf )

oôMHP0TIKoE lor l tMot 2. r ta\)6dyou B 111

.[ 'rote,rxax= ['x roecorcr-[r.otrrrl '-[ '* ;x, dx=2roqEez-ro{€e1 -

L J r J r

+ i !1s+ =2ro[eo2-ro{eo1- -l- ltntt*x't 1'=rror.rr- n ' ' .tn o

18. ilq moloylooe! ro otoKlituro t=lJvrl-r.Or* 6mu o r6noq T n€-

olrl€trror f i6 ro rolyrrvo 0(0,0), e1l,ri, 11to,t1.

A0oI:

0L a[Lqjioerq rov nlEupdrv rou rpLvdivou€[voL X.U, X=10U, 0=1. eq oioxlnodroou!€n9d,ro oq npoc x. otpvoule lro €rj0€lo no-pd;Un/\n lpoq rov dlovo x x rlou rauv€u rlqnl€upcq rou rpLyd)vou oro sryieLo T |(dL tr tr€x=U, x-10'!, Apo 0 < X 3 10(i. To on!.Lo u€tnv u!(96rapn KdL lEyclG€on rErovuav4 aL-w! ro 0 KoL 8 ue 0=0, {,=1.

'Apo 0 S|], S 1

, = j jr++ -, or,= J' J"h=u,., *= f I rl$+-1r1 ]"

-f t ts , l '1 ' r ' -0,0,

=-? | + 27r3du=181 a 'd,=r8 l+ |

19. ilo onor\ovr.o€€|, ro olokiipoF€oy6vLo t3 ,4 l r t l , 2 l .

' txou!€ .Jl(x+'!)

- zdxd! '

T

= l l l (x-0)- ' :dx ld!= l l - rx- { ) - r l dr=

, ' , . r \ t i , . r 1 z=l t ;i;h - r; ld!=l rn ;# | =rn i;

20. tlr uroloyroset. to oloxlfitqro

Aoon:

d0-+ l

fiT

ilT

(x*,!)- 'dxdl 6noo T etwl ro op-

,S or* o.* T=to,l I'to,l l.

112 (EOAÂIO II

t r - 2 t t t . r 1El.wr ll rfur dxd'l.l | | rfur dx ldJ.

-l *'lSl.*"* l.S '-| trorewti=|roreol-{ro6eqo .

- * ' * - * ' ' = * .

21. ila 6elx0cl, 6rL [*{wn,.o*-f *f1,.,r'*.

al J+ft x.,u ax

Aoon:

0eototl6 tnv €ue€to 0=x rdtnv ltogoBo^li rl,=t/X nou exoov Korvdon ! . [ a (0 ,0 ) xou ( i ' 1 ) ,

geopot!€ (or ro Xoplo rooreprrl€l8rd! ord ric rponyoouE -

vac xolndi\oc, Eavot:

r r r ' r r f i ll lf(x,t)dxdr-l ll f(x,U)dt ldx.T ' ^

(2 ) . An6 (1 ) rd r

(2) nmxorE€. n cno6sln6c.

22. rb motovlo€.t ro 61 6 oAoKAtp.r6 JJrWar+ o"ouT

r.{(r,0t: 0 > x'1, x > o, x'zr{'z < 6}.

^0oI:

0 r6rcq T nopLor6votol on6ro ndoclrtr€0orG oxito. (To rccv6oruelo A r{,,v xdlr0r\ov 0-X2,x'*U'l.6, (x I ol EploremL on6rnv l0on rou ouoxiucroc:

(v'{t)

. l dx l f (x ,0)dU (1)

rrwr. oxouo JJr,r,r,**-l fnr,r,**-[

oôx^HloTl(Ot lorl'tlo[ 2. tncv6dYo! B t 13

=-? JtLru-,',"' -*'1 d,= ! l- | rc-*t' ^ , l,li- rx ' l = " '

l4dx11,*^rlrrt

23. ih uEloyrooel m olodlF4aI - t0 ' l l r t 0 ' l l .

A0oI:

, ' r , t t t t t trL*, r"Jll *#,)%l,*l ffi -#),"_[1n x*{4I l'"rnJ:E _ rn_L .t y*@12 tt 1+tti lZ

'24. flo urroloyro€el ro olorligotro

T={(x ,0) : 0 > x ' ,noort:

.' r\^!ELvor, I-l | {x'z+rr)dxdu-

J o ) t ^

-l l-+ +'!x | -dr"

'l (+"'"-{u,'+')a,r,=="'- T{0

25. ih urotroyroge[ to d\oxlaptuo

ILJ,I'Ccf'oxal, o€Rr 6rlouT

T={(x,U): x,+0, < q,}.l tm:

rLvou r=l I (,t,, vo, - x. d,! ) dx=)< L\Ez=i,

8

r=JJtx'*utax* oto"Tx > 0'z),

tt-rz, ^z+r!,=6) )

wJ[T

1 1 4 IOOAÂIO II

26. Ilo umloylo€al ro oloxlfur{,poT= t1 ,31 t10 ,21 .

A0on:

tL*' r-J J lx-z ruuuxaulJ lrz axJ nuua,r=(J r1-2 6x)1-ouw:'=

=(1-ouvzJ l I I x -2 iax* l l x -2 dx l= f i -ouv2) l - l (x -2 )dx+ l (x -2 )dx l=I L ) r J , )

= (1-ouv2J l - l * -Zx l - 14 -2x l l=(1-ouv2) .1=1-ouv2.

27. flo urotoyrooe( ro oôrîpqld r-JJ Ufu cxal 6rcu n reploxi rT

repLd\el..roL on6 rLq €uo€lec {:0, x=d, x=U, (d > 0).

A0on:

eo otroKlnpdooutE [p6xq (,rq npoq 0.0o Bpoipe to 6pLq lou 0. 06pvouue lLoeueElq nqpdllntrr npoe rov dqovo (r'UnoulEuveL tu oirvopo rnc n€pLoxiq T o[oon !€ l . q T (X , x ) , A (x ,0 ) . 'AOo 030aX .Tq onlElo !E Tnv lLKo6r€on xqi, ueyol0-TEpn rElun!,vn elvoL ro 0, A ue x=0 ,x=d . 'Apo 0 <x 3o . E ro [ €LvdL

rct f'( , I f I | ,h tx font=J, ll t'ir d'ridx1, x[Tto{Eai], dx=j+

28. Ito urcloyrosel. ro or'oxr.ior,uo JJ tx'zT

on6 ro rptyr,rvo 0AB u€ Kopuqas ro onleld {0,0)

I= I I lx-2 l4u0drd! 6nouT

-X$)dxd{. 0 rdnos T opl<srdL

l1 /2 ,1 t21 , ( 1 ,O ) .

ax=f,rxri=f .

+ B l d I . l d p a t i p n o f o € r i 6 0 s 6 1 .

Or!OK-{HPaTIKO' IOt I tUOt 2. t ra!6dyo! I 115

Ation:

0 r6noq T sLvcL To loooKe,\tq

Elooon rnq ni€up6g AB Ioo tpL-yr i)vou EtvoL x+0=1. H ntr€up6 0A

axeL E{Lo(,ron U=x. EroL rq 6-pLq rou x ELvoL \ ! , 1-0 To dpLq

r o o 0 e l v o L 0 , 1 / 2 .

'Exouu€: JJ

(x'-x'!)dxd', =T

r\r rr-a= l l l ( x , -x { i )dx ldu=r o L J o J

lplyovo 0AB xoL !o EouJtEpLK6 rou. H €-

=t* t+' - 4,r, l t-oor= - [1r-r,r ' .rzr '-s,r*z)ou=.'.= +J o L r ' l + " J ,

a. ib unoloyloo€l ro oroaiprr. r=JJ"2r*3'1' axao 6mu T elw! toT

xoplo mu nEpLd\€[EroL crd tlq €uoel€c x=0, {o0 xot x+*1.

ntoll

Ei,kotro eolverqL qn6 To sxilo 6rL

0 < x s 1 , 0 s U s l - x . ' A p o

r I-x

r= [[",r*3+a*a,],= [ [ ezx*3{au,cx =i

' a ' o

_ l l r e r I+3{ ' l - " dx=

= * j,r",*-"*,0, =* ;*'-'-* ""]l=-+[""+ "'-"'-+] ==-* [*'t"-rr--! c'-r)]= + l"'- # ]=*,"-,,,r""*-,, -

=+ (e-r ) (e-1) (2e*1)= + (e-1) ' (2e*1) .

30. tb unoloyloe€l u€ 600 ro6mus ro oloxllipqr JJ tt"+"totnl t.ou

o r6noq T neplrtrEl.fior. 0116 rlq m|In0l€s x=2, 'lFl, o=x'z.

a(i +)

1 t6 KEOMAIO 11

iton:

o') rp6mq. 0o oloKlnpdoouUe

nplj)Iq oq flpog $. ELvdL:

l l ( x ' + ' ! ' ) d x d ' ! = l l l ( x ' + t ! ' ) d ' l r l d x =

T . .

=l I x''t+ q- | dx=

= | ( x,r -i x,-x.- -,i. )ax= ri? .

8 ) rpdmq. 0o oloKltpboou!€lpdlro 0C npoq x, ELvor:

If r'+,r'ro'o-[ |[ r".*'rorlo,r=[ | -$ *'rl' a,r=[ ($*u'-dq'z' -*'r' 10,i

" ' iq ' \q 1"

3t. atv€roL ro 6ln^d oôxriirurc r=t't'['%kx.

a') llo tr@o6|'oplor€l o r6noc T qq olorlip0Enc cao erlm6o 0x1,.

8') th unoloyloo€l ro oloxlipatrc !e olloyli rnq oelpdq rnc oloxltolprq.

n0oI:

o') An6 rov ouuBollou6 rou oloKtrnodldroq oolv€toL 6xr 0 < x s 1,2x <' I 3 2\q.. 0 rdloc T [EpLKI€LeroLon6 rLq xolniJleq X=0' X-1, U=2X KqL

0=2Vi - U'z=4x. ErsL eLvor

T={ {x ,$ ) : 0 .s x s 1 , 2x .s U s 2 f }

B') 0a Bpo0u€ rq v6o 6p[o Tl,]v

x (qL 0. 06pvoutr€ €uo€Lo flopd]Un^rl

npoq rov dEovo x'x .

An6 ro sxhuq eoLv€TqL 6TL

0sUs2 KUL += xs+.

B(# {') r(S vl

QAoMHPnTII@t I0r t tMoE 2. tno!6dyou B. 117

32. No

toL cno tLc

n0on:

unoIoYLoE[L ro olorlipGrtro ll odxd, 6rav o t6noq T n€plrlEle-

1€uoeles 0-X+1,' lF-x+3, [-1-X, 0=x-1,

To xool,o T nqp[ordverol on6

En€L6i E!8elq rcpd)\in,\n npoC6aovq x X i rpoq rov d{ovo U'4r6Uva[ fidvro rLC Ldr€C xoun0lec

Xripld)uU€ ro Xrrplo T o€ 600 XrrplqT2. 'EroL 6xouu€:

l l l l t lj J0dxd{,=J J0dxd{,1.J'rdxd{

=

ro €n6levo sxilis.

6€v0o

an. lld unolovLooe[ ro

roL ('|!6 trC rqrlr0lEc 0=0,

Aoort:

.l,i9l,l.'-= i,trr+r: - glr]ax. J'I t-xi:r' - txtr' ]ax=...oro.rfsq,o

JJxf--x.dxdl,. 0 r6noc T n€eLxte[e-

x=0, l-r, x=r]rcc*.

0 T6noC T 6Lv€rq[ on6 Io endlevo oxiuo.

En€L6i .ue€Lo nap6ll\nl4 npoq roucd{ovec 6ev t t lvsl ndvro rLq [61€q xo!-nilee,x0Dl(ou!€ rov r6no T o€ 600 16-I louq Tr,T2. EroL € LvoL

( l - t t -

J J xvl -izdxdil,l J xvl-x,dxd'lt J xvl -x,dxdo=

T , T r I " t t Z "

=[.,'n=" [ [lu,lr'-f 'rvi:"f

I a,r,la, -u .1 , . !

-f xt-x" t\ax. I J'rrt -r' rtar=- * g -r' f 14'

=+ +.+ +*l lxt r -x ' t t

T , . t '

" T ' -

=[ [[--, io+]o*.[ [[ '- j*h'= [ ' [{ l ' ' '" ' i - r

'

" " r - r

LoL ' r r -

- ro€nrx] '=

fi8 IGOIAAIO II

34. th umtovloo€L ro olorliqdro j,J (I+l)dxdir. 0 rdrcc T regld€|,e-T

Io! fli rlv r(t|lnafn 'F||rX (oL rLC €uo€t€C X--n, I{, rFo, |lF2f , 'lE3-X.

A0on:

0 r6noc alvql n Evoon Bw Unoouv6Aov Tr,T, t]€

T r= t (X ,o ) : -nsxs0 nuxs0s0)

T ,= { (x , { } : 0sUs2 r!-T 3Xs3 -0 )' [ror ixouU€i

f Irx-rrcxa,r,=[[rx. loxo,r-[ [rr-rra,a,r,=[ ' [ [ ' rx-rraq,lax-[ ' [ [ lx-rrax]a+-' i ^ i ; "

^ ' n " n , * r ) a L ) + / 2

a I to r_ | . , , t3-{ ,- l lx$+'!l dx+l l- i .xl d0-l (-xnux-nux)dx+,n!x , ! . _ ,+12 )_a

.l tt+t.r:+,-$- * l,-.i (+ +. $.r+--$: $) ",r,"-

| (xn!x)dx-

| nlxdx=x,r.I .

35, ilo uflolovroo€|, lo otoKliqdrc JJ,f"raO

O*" T €l.vcl ro xdplo ro!T

lodrou redgmroplou r.v oE6udv rou reprxtelerol *O u" ale*n $.-$i:f

OIOK^HPoTIKOI IOt I tMOt 2 . XnavSdyou B, 119

Iton:An6 To oxhus eolvEroL 6rL:

0 3 X s q , 0 s { 5 ; v o z - x , .

L ,r"z:;at t P e

' "'Aeo: Jj llidxd'!1-

'J" ud{.dx =

! Ya2 - t zr d r 1 ^ t d o z t d -= l l - i , ! ' l dx= r - | (o ' -x ' )dx=

..!Jlu*d:=# [",_+]=+35. I{o uno,\oyLoo€l ro oloxtioolo

JJ*'U'oroO O-o"

I-{ (x,qj) € R,: rx,r{ , ,

< r iA0on:

An6 ro oxi l id eol lErqL dir

- /T -0 tsxs /1 - , !2 , -1 : , ! s 1 ., , , r , \11-+"'Apo llx'?o'zdxdL!=i | -_-r. ' ,1'oxau =T

- ' - _ Y

t' t,tT=67=1, l, zx'zo'zdxdu=

I ' f - ' l l r f f t l z=J, 12", '+lo d' !=

=[ ' { a.11-r,, ' / . or=l 4,r,rr-,r , ,r

06touue qr= nue, on6r€ d0=quv8d8. orov |])=0, €:0 6rsv l l j=l, e=n/2.

ll ̂ 'vaâu.J" /'

4 rrq2e ouv3B.ouvgdo= I Jnl2 nr'r.ouon66e.' i "4 1 . 3 ' 1=T '6 ; { ;2 ' -T = - r r .

37. Iilo unoloylo€o0v rd olorlnd,|r.rEd o'fr,=JJ axaU,

o 'r I ,=JJ1ra1'6r* 6rov r={(x,{) €Rz: 4. '# tr t .T

o,o)i,t,r

120 KEOAAAIO II

g) g€rrpo0le oov i6rc oôkAh@Jonq oJ\dKlnpo rov Eltr€!fiLkd 6[0(0. troL

6xouue: - -fi VF.{ . r: + vEr-TT -B s qr s B. Aoa! \Gvj-

8 8 - ' +vE'--u'

A0ort:

0 r6noq rns o;\oddruonc EtvoL ella[rftLx6c 6LoKoc flou El,w! s!!!stpL-(6c oq llpoc rooc dlovec. 'Apq eo olo(trnpd,ooupe orcv r6no T1 nou €LvoL lor6ropro rou €l\tralrrrKoo dtoxou to! to oloKtriptuo nou 0o n9ox(lu€! eo no^-iqn;\ooldoouu€ €III 4,

0o otroxlnpdiooule npdlto oq npoc X. 06pvou!€ uLo €ugelo nopdl,\nln npocrov 6tovo x'r flou ralv€L ro ol,vopo rnc [€pLox/t( Tr o$ onleLo A, I !€

x=0, x= * vF,=oT. Apo O < x < * vEilI?. to onuelo s€ 14 ulKpdr€pn

KoL lnv uEvotrttEpn lErovutvn €l loL ro onu€Lo 0' f !e U-0, 0.9. Apa 0s0s9B + \F:V'rrou txouue Ir-J lJ dxldu.J + vf,:u-.d{,=

' * r B ^= -fr [;,t'.ftu'. + B,rotnrJ -F], -fie"ornut=A ' Apa

i _ 4 (o0r) -_o-| l - - l - - !P " '

r.-[[rx+r'axat-[ ['" 3'+'*rroto,r*- [1 f""-g'*',**-t^r '_a )_+ VB2_t2 !B!;vE:it

A'+ti':F " s ',;o:tz'i-,!' r r-P**r=Su-,f

"'.l l^ ___3'!dxdu= | | 2(x,+0,)dxd0+0.'-B t_=vgr_+z t _A\

(€c6oiv (x,*rt") olvol lto 6pr[o ouv6prnol rououv6prnor rou x xdl zxt E[wr llo nEpltrllouvdprnon rou x).

0IC.VJ\H!nTII(0I lotIIMo[ 2. tiav6d.rou B. 1?1

B-21 |, - 8 .

B

-lvB,-u,ld'! =

oiltuo.

F.l'***'l'l '* =

xQn-xfiT

-rJ 13fi1 re'+'r'4. f o' wt']au,.

ottoqre r1r=9 n!8, d(r=! ouvgd€. 0rov 0=0, €=0, 6rov {,=8, g.r/2.i / 2

'rpo I,"qI l# e'o*'e. * B'?qu'ze BouvelBouveae=J. LJp" p l

, ^ / 2 - .* l j$ ouv"eroo 'nu ,e ouu,e lae- $ .8 .

= |o3on+ |cs'n= f os16a*sa y.

38. lld rrrloloylog€t to otrodigr d Jlxdxd0,

oro urlooovolo roo f,2 ron

rm0d€l€tor 0[6 rrtv e$e[c x=l xcr f,nv (rynofn Xr+X,-'l,r-0.

l0on:

0 T6noq T nopLordv6toL on6 to €fi6!€voXopl<olu€ rov T acouc r6nouc Tr,T,

ELvoL Xr+x'?-0?=0 r t l=tvlzTx+-lT :

t-rx{;T. To r6qo frA0 opL(€roL oc .t i ! :

U= 1fxTT, 0 < x s 1. To r6to ̂ f ,O ogt.(€-tor os €tiq: {-X1,FT, -1 s x < 0.To 16{0 lT30 opl (€ lcL 0116 r [C oxtoeLq :

' !=-x4;T, -1 3 X s 0. Tetroc ro rdlof{20 opl<€ro! on6 r[q oxtoELq:0= -xFT , 0sX s1 . E t voL :

l l l t l ll l xdxdo= l l xdxd0* l l xdxdu {1 ) . 0o uno -

lovLoouue ro npdjro oloxtrltpr,Juo. ElloL

voL 0r=0 KoL ylo X:0 et loL rd=l. tavqL ox6!o X+l{ , , - Xr=(or-1), Kar dx=zrr4r)

-zf fr.-r t 'rz"a"-of (o'-2o,*t )o.dor=.. ,= # . nop6!o!o unotovL<ouue KoL

-1

t. rr , rrf i :T , ,ol lxdxdr-l ( l _xdr)dx=l -2x)f iT dx=(€troulE {;I=u on6re ylq x=-l E(-+J 1 , \ J l v '+ r / r r

122 KEOI]\AIO II

to d€0repo olodipolo 6roL eLwL:I xvlTr

llxaxau-l (l xdr)dx=...= T.li+ -;* . .Apo Ervar+; t v"vFr

jJ,a,au=- ft - T# - *? = f# -,$tT

39. lb uroloyrd€l ro otoxllg,ttro rJJxrf.araf,6mu T €avor ro rogto

nou ttegLKlsleto! 0[6 rov x0xlo x'*{"1 -rTrrc eue€[€c x=0, {=0.A0oI:

E ( i ko lo qoLve ioL 6 ru 03x3 f1 l [ , ,

rr r' tv1-d,2l- l lx'?U':,Jxd0-l I x,O.axcO-

. .JT:,-,i: .r= l lu ' + I ' - ' ao-l {u11-{ iz ir I , ,

oarouUE 0-n!8, + d{"ouv0d6.'orov 0-0, 6.0, 6tou ,1,.1,6-|. ,&o

[ [r 'r 'oro,l- ["" * q!'o.ouv!6.o'v "or=

I l" ' nu.e.ouv,0d€=i ' ' a - ' r ,

=+ . ;i+ .+=+.40. I{d rrroloyrooe[ ro otrorJ\tplyo [ ["#- **.

Atoq: t'i+l<r

H ouvdgr4on f(x, 'r)= F:ffr+T elvoL neglrr i oq npoc x,npdyucrl 6[vdl

11-x,, l , l= --- iLI): =- --{- =-f(x,o) v(x,+) €T. To xoelo T E[vdL {u-{ -x) '+'1, '* l x '+0r+1

KtrLK6q 6l,o(oq Ue Ktvrpo ro onu€|,o 0(0,0) KoL qxtlvq l ,on u€ 2. Eneroi nouvdprqon f(X,0) slvq0 flepLlrll l,rc rrpoq x {o! ro xoplo eLwL ouulerpLKd ocnpoc rov 6qovo {,'Q,ro oj\oxîpr,Uc atvsl Lo|o !e Un66v.

j, nu'e'o'u'eoe=

olOMlfnTIXOt loaI tMot 2. !&!6d1ou B. 123

41. !l,ve!o! to olodllpqlo

l-3t t I l t x - . 1l l ' tdxdo-l l l *d'!ldxT "

llo npoo6loolot€1, o r6no( T xol w unoloylooel ro oloxlltpoga !€ ollovi tleoelptrnq otroxlip(r0c,

l0on:An6 rov ouugotrLold rou o,\oKlnpiuoroc qolveroL dTr !q 6pLo rou 0 €L-

var U=1! J= i i . Ta 6prd rou x €{.vol x:ot )(=r.'EToL To f l€plypol]uo roo

tdnou T KseopLc€rqL on6 rLc(ounL'Irc { i : l t 0= i i , x=0

f!o vo ul loôyloouu€ ro

olo{litp0uq XopL<outr€ rov 16-

no T oE blo rdnouq Tl, Tr.

(BI6n€ sxi l !o). tLvo! |

t t l t l lI l0dxdu= I l{dxd++ I l+dxd'! =

T l

=i,'f ;-'p-;;1*,1,'{"i.*[(,.'. 6! )a+=. .= ] ].,^zqr-r

(Tq dpLo rolv X,{r oro o}oKtripouq l ludxdr! goloxovror oc €[tc: 06ovou!€

T ,ULd €u8e[o nopd]/Uln npoq rov d[ovo x X. 0L r€r!n!6v€C rorv Korvd,v on-

letov rn( euo€toq !€ ro ncplypoluo rou r6lou €Lvor r= f f i . 1npo"1, -

f l r€r of i6 r tv ' != i j f ivovroq urq rooq x) (qL x=I. Aurd €Lvor ro 6prorou X. Tq onlElo rou rdl lou rcu txouv lnv lLKp6repn KoL lEycl( l r€pn rE-rovu6v4 €LvoL cq A |(or f uE rerovutvrc { ,=} . 0=2. Au16 €LvoL ro 60rc

r o u i l j .

42, No unoloyLo€€L ro oloKl ipir lo I .JJI r '+ ' fara,f , , 6nou q f l€groxlr

n€prxlEleio! dn6 r lq xou r lec x=2, 0.f , ixt .

124 I€OAÂIO 1I

nton: ELvo! T{ (X,0) € R'z:

0o o,\oktrIpdroouuE npdltq orq fipoq

U u€ 6pLd 1 xol xt , rou Kqt6nLv oqnpoc X !€ 6plo 1 xol 2 6noq eolvE -

rqL KqL 016 ro sxiuo sl,\d KoL ordq v ( 1 ) .

E { .vqr I= l l | (x ,+0 , )dUldx-1 ' 1

=l l , 'u.$l ' '0,=

=[(,".4 ' '-*)a, " f+'+ --$ +l'= t]!er l r r u 3

43. ih rrnolovlooEa ro olo(,\lirruo: I{JV;or:*:U-,oxu,l,

r=(o s x s 6,-,r,?l o <11 so)A0or:

| | twz-:- ' ;11,td'=ll- t **Etvo! I10 l jo vo, -x, -0,dx1d010

l,J0 v(q,-l lJ,)-xzdxldu :-

-J [xv(o,-0,]-x, , (o.-r,) roEnu Ci= ]1" '

, '* =

=l f v",{,. 4;-u--",r{/ * el91 16qap @ lo* =

4 r t z - - f ; , _ ! " , l -

=l ("'rr',"r*,)o*=1" n# + d,!=+ J (o.-,!.)du

* 0 T u n o p e f ! d y p d q E i : T : { ( x , { , ) € R t , 1 < d , S 4 , { F a x 3 2 \

'=[ ' f [ ' -r".c 'ra*]o- f ff - "Ji::io* ci*" I

vaz_\,dr= î:__-4:, _i rcr,L + .

1sxs2 , 1s0 jx ' l * (1 )

0 OKiltlPmIloI /IOIIIUoI 2. tE\,6daou B. 125

44. Ik 6erx0rl 6rlr:

JJxl-t"ar+' d"zo"r +ft${tt:I

o r6mc T rctrdEteu! ord tov r0rlo x'z+0z-cr', ue I >0' a >0 (l'i > 0)

A0on:

Elvar T= { (x ,0) : osxsGt:ot , os03c.}

r,[[xr.' ,r,.-' axa,r-[ [ ,zr-r rz"-ro**-[ [*a'-t) i r o l

Eno!6vlr

.2r tEz4r-ttr l, d{'

( i ).f #,,""-r",r-ora,I. # f0h-1 r."+,rra,r

fLo rov unolovlo!6 rou rEiEuloLou olodnp6rc$q 06rouue

J . . - LUr.czo - {,=ctr' - d!= -;- o ' dr,r. 'otav U.0 eLvo! t!.0 Kor 6tov 0.c

ol,vor n l. 'Eror txouts

(t-ur)l ar,r.!

[ ,*!. =!- ['.t-t c2r c or-1 .-"), d* g'(t*), , l o -

- 2 - " * ' - * T [ -

-2(l+n) -2(l+m).cfrl-s1n,'.1y =.";" +H##.45. flq urolovloooov m otrordneduoro

"'f r,.JJrt'arar,

r r 7B'l lr.llxtldxdl,. 'o.ov o t6rDc T n.ordelctol ot6 tov rotlo I'+Ut-|.

/tlon: T

d') Elvot xr+0'< | + x'?s 1rl ' . . -y'F0t s x s VT:Vz. ELt/oL ok6uo

1- { , ' : 0 - l > . r l r ' .U " s i . - l SU s l .'trol 6xou!e:

. lrerr,ri re tus ouwprioer.s trBitan llc|. rldluan ropsriurouue oto! a'idllodelr:6a 3?0.

126 KEO}]\AIO II

r \4-i$z I v,FUt' ,=[' | [",.u,,a,1.,r : [ 'z[ '- ' , ' ,r, 'axoo-['* ' l4l**-: r - : f f i l r l " l o

= [,5 r -,t'" t" " or,= | Iu'? ft,],' ; h d'lt I $ J +' r r -,r,' r' /' 4,r,.

etrou!6 U=n!8 .du=o!v8d8, ftov U,"0 - O-0, 6xov 'lr=1 - O=+ 'Apo

n/2 a/2

I ,=$ [ nu'e.ouv'e.orueos=] [ ns'e.ouuueae=$ #+ + =

12rr r" E:tz - -2T ,

B') Etvo! x'?d! '?< t - -r ' i :xzs0 < VT:J:, xqL -l < x s 1.

,t ,,{r'-it'EtoL txouue: t:-J lJ x'0'duldx=0'. (n duv6prnon f(0) 'x' :o' €l.var

neprft i) . - '-vf-F

46, tla ubloylao€l to oAoxltoqro

rl€plrl€t.€roL ard rLc euo€t.eq x-0, 0.0,

A0o4:l l . - . 1 t r

EtvoL JJe'" "" dxd'r=

T

_f [[ "zx'3oaxla+=

dlvdpr idn f (x ) e r :wL

ouq;prno f f (x ) . ' : !aL

JIe4+30 dxdr 6rav o rdEG TT

ao"- .a-. I rtxrax=r II

ree! t l i rdre j -df (x)dt=o

oôK,l\filoTIKot lorltMox 2. trav6dyou 8. 127

17. l5 umlovlo€et. Eo olorlltt o llrfaraU O-",-ia') T el,!u! ro x.p[o nou tr€plrlel€ro! d|i6 rLc |rodor€c 0=2x'?-4x,

l'2x-x'. .B') T el.vo! to xop[o rDU |l€or.xl€asrct an6 rlc tlopo8ol6c U'?={x, U'z-s-x,

!c Fo.l0at:

a') tLvoL JJx'!dxdu=T

r'r ix-x' l= l l l xod ' ! ldx =

=[[,9]"- 'o,-

=[ ( - * r ' * ' " -0 " )a ,= . . .= - *

8 ) tLvor J.lx0dxd0=T

ttr ts-q2 t r t r " . t5- , ! t- l l l "x tdx ld0= | l0 * i - | ^d ' !='0 "-l:

- '-t:4 4

=+ t(- l l . l ' -ro+i.zsu)a,r=

=. . .=53 .

48. l{o urctrovro€€L ro oloxlioqro Jj(rA)dxdr! 6nou T el,vor ro xoplo

rnu neprxl€Lgmr on6 rov rtdo 1+,1'-p'lrn" er,\Er{,n + * $ =t ,or, .oua

oerrxobc nlr6lov€q 0r, 00 (a,B > p).

l0o0:(0 ,B)

1o ,p )B'

A(x , + (o '? - r '? )L ) , s (x ,h2 -x2 r \ )

f ( r , + (d 'z -x '? )1 , ^ (1 , o )

2x-xz l

( q , 0 ) x

128 KEO1iAIO II

Xnp{,aou!€ ro Kdpto o€ 6l,0 X{,rplo Tr,T, 6ft,)q @Lv€roL oro npoqyo0trevooxituo. ElvoL

,, 1, i to"-r'\" ,JJ

(x-2+)dxd{1.J (x-2{)dxd'll j (x-2'! )dxd01" lJ - -(x-2'!)diijdx.T ^ T r T 2 ^ " ( D ' - x ' ) ' ^ ,,d. .o3

(o'z-r 'z) ! ,9

(o '-r ' , ) ! ,o , i , . ' - r ' ,1.

1l l , tx-z,rtaulax1, Lxu-u'1,",_*., i - l l '*-0'J, d)( =

=f [ +r,*-r ' , t- $ ro'-r ' )-rto'-r ' ) !- tp'-x")]ax-

-l[ *,"'-, ',0- $ r"'*'r]a,-

=l- * r"'-*"r'a- #("'* $ )-Jyro'"'r"'*"- $ ]' -

t A a2 / - " l

\ ' tq+[- t (a '-x ' ) ' - i r io 'x- t ) j "= - . ' -^.

{9. ilo urEloyro€el ro oloxlioqla t=ljot* o*u o r6rcs T r.9!rl€a-

€tor ord rov (0rto x'1r0?=t6 (o! ord tnv i^"ufn $ - $ .t.(rco' lden lhths series" Bali P.)

qJiton:

l6Yo ouuu€rolog rou r6nou !,qnpos rouq dlovEq €d oôxlnfllEouleoro l/4 rou rdnou T (Tou npdEourercprntopLou) xor ro 6Eqv6levo0o no^/ooutE !e 4. Etvor

| 5 -r I r f i vro_si

-i- =l | | dx ld0=' 4.(dFI"

/r6-0,,-\I

- \ \'16- ' ! ' . ' ! ) \ \-1'

' . \o o

)/u '

q,,

=l {+ v16-U,-Vt6-q,, ld' tr =+ | v16-U,dU=(Btrouue 0=4nut on6r€ d.u-4ouvtdt4 \ { / 4 J o

6rov ' l=0, t=0, 6rov 0=4, t= n/2) =*

- Eis l ! -16-16n! 4:af t - q l t=qoul t 6LdrL tqto, r /21.

=+a/2 r,/2I qouvt.qouvtat=4[ ouu'tdt I 4 [ ouv!4ut +J o ) o l .

* =n - t=on.

50. Ib urDloyLd€[ ro olodal'qro llf(x,l)drdt drw

o') f(x,{}.xn!(x+0, KoL T-to,nlr l0, n/21

E') f(x,0)-x'zU ex{ l(o! T=t0,l l t l0,2l

v') f(r, t)=x'{,ouv(r{,1 .or T=lO, r/21,10,21

-i- i-ca *-n.

Aton:

o ' ) ELvaL [ [ r r r ,urarau,=[ [ [i

' t " oxn!(xa)d0]dx-l

I l

l-xouv(x+0) I dx.

{ [.,*("*) -r"'"r]or.J (xrux+xo,vx)dx=l xnuxax*J xouvxax.

-:*[-xoru1*41r1*xr1rx*ouvxl" -n-t-t=n-2.) r t r t r 2

o ) Etvq! r-JIr(x,{)dxd{,.I [l x'+c"d,!]dx (1)7 2

e Luor, Ji.+*{a'l+J,r,{er+ )'d,r=x t,l,exo l,-xl 0' exod,!=

=x(2e2r)-J ex{d(x'!)=2ye2x-1gx{;l=2)(.zx-.zx*1. Apq

r.{ rzr""-Jr'lar=J'rr"',0*1,i ' or1'*=f"'r-+"'. + J,.rj,=*"-+ ""-+ *a-l tl ,2.

ou" ta t= " " | "n ! ' * * * " .

Jrt"-11'a"=-ro,"*.1 oovx<tx:-xdu!x+nlxlc rd!

J x( nrx l'a1=1,1.,1f r,1ax:x.!x+ou!x+c,

- "r*" J

Jxo,"xax=

130 KEOMAIO I]

n /2 2

v'r rivou JJ r{x,u)dxdu=J IJ x',touvtx,r,'ta,r]ax.' ; " " ,

11 " . \ . / 2 \ / 2- l l lJi,",,*",0,,n,]ox-J + I4!(x,r, ')r;dx=J 1,nla',*-

n/2 t /2

- - r I n ( -ouvqx t 'ax - I t -xo ,uax l " /2* * [ ouv+xotqx)=

=f t-xouv+xrl/2*j2 rnu+xrl/ '=f ( +) =-+

51. Ito 6elx0r(. *, JJr*t 0n-1 tt-x-j1P-16x6,1,= +HS#!IT

(n > 1, n > 1' p > 1 ). 0 rdrcC T opl,(srqL on6 rLq €u€€leq x"0, 0=0,x+0=1.n(rort:

Ervou [ [) )T

Kdvou!€ Toi!= ( 1-u )u

r= | '

l -u

x'-1,l,n-1 ( 1 -x-,1)P-1dxd,!=Jx'-1,tn-1t r-x41n-r 6,1]4,l r_xrl

v !erqoxnuqrLoud

E IvoL

(1 )

0

1 -u= l - u

6xou!€ o=0, u=0 lt

E l . v d L x + U - 1 = 0 - u + ( 1 - u ) u - 1 = 0 -

u + u - u ! - 1 = 0 - ( u - 1 ) + u ( 1 - u ) = 0

{ u - 1 ) - u ( u - 1 ) = 0 - ( u - 1 ) ( u - 1 ) = 0 + u = 1 i u = 1 . ' E r o ! U € r o v U o r o o x n U q r L o t 6

(1) o l6noq T qn€LKovl<erd! ouel l lovooiUqvro orov r6no Tr nou opl(Ero00 1 1 6 T L q € u 8 6 1 € q u = 0 ! u = 0 , u = l , u = 1 .

'Exou!E Ii x'-1 ,tn-] (1-x-{,)!-ldxd,!=

=1, J , u ' -1 l r - , tn -1u" -1{ t -u )P-1{1-u )p -1(1-u) dudu=

. [ ' f J ' - ] ( r -u ) ' *e -1u ' - r ( , -u 1n-16 ,o ,= f 'u ' - 1 r -u ,n rn - .16 , | , r - r 11- r ;o - r6u=)o Jo

=B( r .n+D)8{n .o) - rJT) f (n+P) . rJ lJ f (? ) - r ( I ) r (n ) f !p )r t m { n r p l t t n + p l t t m r n r p )

x , l - x )

OloKlHmTIlOt loIIEllOt 2. tn\,6dyou B. r31

52. Av oL ouwpriloel( fx{, foxpta€rol on6 rlc oxloerc o 3 X s B,

E[wr ouv€xetc oqv reproxi T rlou o-

v3 {sd r6 te E [ vo r :

/t0on:JJIrrrr',ur-r*,.r',ur ]oror+

I t l l t t t l *'Exorru€ jJL fx+(x,0)-fvx(x,0)ldxd'!1Jfrt (x,'!)dxd'!-JJf{x{x,U)dxdo=T T T

8 6 6 8 6l l t t I. l

I f ) .o (x ,0 )d , !dx- l I fu r (x , { r )dxd0= l t f r (x ,6 ) - fx (x ,y ) ldx -11 ^ - a r . ^

6- [ t t , O,+ l - r , t o , , l t : a , l= I r {8 ,6 ) - r {B ,y ) - r (o ,o )+ t (o ,v ) r -

- t f ( 9 ' 6 ) - f ( 0 ,6 ) - f ( 0 , y )+ f (o , y ) l =0 .

53. Ae oL ou\r@rto6lc f xoL g €twa uaoc let(8lntlK mt oloxlngd,oLlacrto 6r0'tnuo [d,B] ror to vrv6p€!6 muc €twl or\oduo{Ealn olv6rtnon aroT.to,ol ' td,! l 16r€ w 6erX0D[ 6rr:

- rB rB rBo t * l l l r lx )g l , t , l - r (0)s(x) l '?dxd ' I . l l r {x) ' :dx l t l s (x} 'dx l - t l r (x)e( r )dx1 '

"itB

8 ) + JJtr(x)-r({)r Is{x)-s({ ,) ldxdl, ' (9-a)L r(x)s(x)dx-T

rg tF- t l r (x)dxl t l gtx laxt

A0ort:

- , l l - -o ' ) ELvor + lJ t f

(x)e( . l i ) - f ( ' r )s(x) I 'dxd0- j JJr ' (x lg ' t , l toxo, t , ,T T

+ z jJf? ('!)s" {x )dxd0-jl r(x ) e{'!)f(0) s(x)dxd'!=T T

, r B r E , t B t B r B t B= i I t' txtaxl s' t+ta0+ +L r'z('!)d'll e'2(x)dx-L r(x)e(x)dxl r('!)s('!)dU=

' ! .9* : | : l :

r ! ! .owao,6d'uv fxo, ,or f , !x Ftooodl i (EL tnv \ t&odn rdv 6L-

132 KEOMAIO IT

+ . t g 1 8 , r 8 1 8 t B t B= i l f l tx taxl s ' (x lax** l r ' (x)dxl s ' (x)dx- l r (x ls(x)dxl r (x)s{x)dx =

tB t8 rB rF= l t ' (x )ax l s ' (x ldx- l f (x )s(x)dx l r (x)s(x)dx=

t . 3 l r r g l t t q=l I r ' (x)axl l I s '(x)dxl- l I r(x)s(x)dxl .

, t t l ] t 1B ) + JJIr(x)-r(0)] [s(x)-s( ' ! ) . idxd' !=

Tt l l , l l I l l-+jJ f {x)s(x)dxd,!- +JJf ('!)s(x)dxd'I- +J.l f(x)s({r}dxd0+

T T T, t t t rB rB r rB tB++ l l f ( i ! )s ( { )dxdu,=+ ld ' ! l f (x )s (x)dx. + f , f (0)du l s (x)dx-- ' i - b 1 r

r rB rB t rB t9- i l t { x }dx I f ( , ! ) d ! *+ ldx l f { ' ! ) s { ' ! ) d ' ! .

r r B , r B t B i t B t B= + (B-oJL f(x rs (x)dx- + L f(x)dxl e(x )dx- 7 |

t {x)axl stxtax*

t t B r B r r B t r r 8 l++(B-o) l f ( x )s ( x ) dx= ( 0-o ) | f (x )s(x)dx- l I f (x )dx l l I s (x)dx l .

54. lilo 6elxe8[ 6xL:

q ) 2a s l j( l+2x'+'1,')dxdu s 432 6rcuT

B') -27nl/2 < ll{r-d,)drdr < 271V2 6n,ou- - l lT

T= t -2 '21 ' t -3 '31 .

T-{ (x,i l ,)/x,+0, 3 9}.

n0o4:

E[vo[ vwcr6 (LbLdrnro 4 oetrLbq 52), 6rL:

< l l f (x, t)dxd! .< l , iA (1), 6tuu A ELvoL ro Eu0o6dv rou xr iJotou T KoL

Tll etwl owLorcLxo ro KordJTEpo xol ro ov6rEpo n6poe rnc f.

rB rBls(,1)ds= | ' (x)dx

olo(lflmTlxot lOIIruoX 2. ,rd!6dror, B. 133

o') To egFo66v rol, op0oyov[ou t €[vdL Loo Ie (2-(-2))(3-(-31)-4.6 ==24 r. trov66€C.

H ourdotnon f e[vo! ouvExic oro T doo moouo]6<El r6yroln tor eld-xlqEn rui. Et vcr

I s t+ZXr+0, s iB v(x,0) € T'Apo t l t . ] , t | .18. 'EroL n (1) 6lveLl

24 s. l l ( l+zxt+0r)dxdo s 432.z

t r0') To e!Fq66v lou T el.vu! A'lldxd{. tu uErooxngdrlooule oe m]L(Cc'i

orrwaloyllvec. 0ttou!6 x.p6(iv€, 0.pn!0. 'Ero! el.vo!

0 3 x ' ?+ r l ' ? 3 3 ' r 0spz s 3 ' 1 .03 p3 3

E{ , ' / o ! a r6 t ro 03es2 r . 'Apo

tt pr t, tzrr ^r.trA.lldxdt.l I dsde. I l+l ds"i tot;'=etl r.rDrt6€q' i ' o ' o i ' - , 0

H olvdprion f €[wL ouvexic olo I dpo mpouord<er rlvrrrn |(or atrdxt-oq rlu6. Eltor

x-vgiz 3 x-0 < vgio-'zrl l2).

eo 8po0!e rq q(o6rato fic ouvdpqon g (0lrvgi02-2-.'1,. ELloL

- -,illg t{,). -:: - l . g'(t). -0-W-'!t2wil,r W{l,,

uE g[(a u'- -!- . 16 qy rllil oudl elwr s" (- + ) < 0, 'Aoo n g ex€Lv2 \vz t

layLoro ro g (- *) +8. 'ouold 6.Lxvouu. 6rr n h(x).x-v!:It axer E-\ v t t

ldxloEo ro -3€. lgtor q (z) Etverr -3Esf(x,0) s 3v?. 'Apo m.-3v2,

x=3f, H (l) 6LveL -27nl. s l l(x-{)dxdl S 2hA.'i

s5. b uEloyrdc( u 6rrld oLruh|! r=fJk +rltC., c,r, r "c

vsr o op&ravlo t0,rl'[0,2]. r

* 1o [r+0+! dl lgor iEeL ro ar3paLo ! ipo. rou r+0+1.

134

A0or:ElvoL 0rx<1, 0<U<2 + 0:x+{:3.

o) 'Eor(,r 0<X{r<1 (1), rdr€ o-pLe€rqL ro xopao:

T r= t ( x ,U )e nz : x>0 , l | l >0 , x+0<1)H (1 ) 6 l veL 1 :x+U+1<2 + [X+0+1 ]=1 .

0) 'Eour 1<x+rl<2 (2), 1616 o-pL(erqL ro xopLo:

T,={(X,{) €lR'z: 0sX31, i3x+{<2}.H (2) 6aveL 25x+'!+1<3 + lX+U+11=2.

Vl Eoro 25X+U<3 (3), i6te o-pL(€ToL To xDpaoi

N ( 1 , 1 )

T 3 = t ( X , U ) € n ' z r 0 5 0 3 2 , 0 i x i 1 , 2 i X + 0 < 3 ) .

56. llo umloylooe( Io

Xrp[o tto! neplrlelEraL 0tr6

H (3) 6LvsL 35x+{+1<4 + [x+U+l]=3.

6) 'Eor0r X+0=3 (4), rdre ople€rqL ro onlEao M=(1,2). H (4) 6lv€L:

x+0+1 .4 + [ x+U+1 ] -4 . 'Apo .

II = II r II r IJ r II = ilo,ot, JJ *ot-, JJ o*00+o=( €lrBd6dv T, ) +2 ( eloobdvT T r T 2 T s 0 T r 7 z 7 z

T, )+3(e!Bo66v r , t3{ r . r .z . r * } r . r } .2- }+.

t lbfilit oloxMftt[ .].1 (X-rlr)dxda, drdl T E(yoL ro

Trlc rc|Ilr|tlec f'z=4x, x'z=4{, x+F3,0=3.

( " C a l c u l u s , , E L L I S R . )

oapvout/€ rqv eueelq +=2. 'EroL

To Xr,JpLo T X0pL<€TcL orc Xopao:Tr=t (x,qr)€ lR, | 1s0s2, 3-U3xs2r'U) &

T,={ (x,U)€ rR'z: kt)R, +l)'z<x<zt|t

t lJ.J (x-u)dxd(J=T

To erBq66! ro ! naDor r r t lo -ypd l r l lou rxFK.L ldL i ,do !E (KN) . (KP)=1.1=1.

€! tsd66v rou tpLy;vo! NM! " i ' " ' ] tunlrurt=]r .r=f .

l l l l r " r '6 , l , lGJl rx-0)dxd.!- JJ (x-, lJrdxd!-l I (x-t! 'dx ol l ] .1 l lT , 1 : ' r L ' ] - {

(x-,!)d{d.l ,=

1 , 2 1 /

0IO( HPoIIKOX ]lollt!,!ot 2. trd!6d1ou B. r35

57. t& molovrdEa m 6ufd ofoxfrs;n JJl(ld4, dnou I €lwl ro xe-

glo rou ngd$i, rer@rnrcoco0 rlou treplrl€C€lul aII6 rlc rqrn0lec x+'?, x=

tulol;To xlJjpao T 6{,vol T=((x,0)€R'zl

| | t6r P-{2 1JJ xudxdu-l ll " x'!dx ld'r-

-0(3-0)]d0.

(novfi !or{rLo oeosolovarlc)

0' l$(i4-q, 'z, 0i01, 'Exour€:r,l2,/zl

1 3 6 KEOMAIO II

-9) Ynolovlo!6q 6!nMv olorlnpoltrov !€ u€rodxnuorFld

58. Av ro xrdpl.o T ne9lxleLrro! on6 muc (0llouC p=l xct 9.2 (oL tr.c

€uo€[€s 0=o xoL €=B !e 0 s 9-o < ?n, ro fiotroylootl ro oloKlipqrq

A0on:

I'JJ(3x*sa')dxd,l, 6Iav:

O=| , O'tto=0, Bd, v', o.o, 9.2n.

l4elqoxntrorL(outE 06 nolIxtq ouviE-rcvuaveq. garcu!e X-qor.,ve, 0=pn!8.tLvoL

a ) 13932 , 0 . O . +

f ( x, ' !) =3X+8{r'z=3psuv€+80'zn! 'z8=F(p,8) xdt

ffi =0. 'Aoc e LvcL,

I=f /'IJ'

t:oouue.so'r'eroao]ae-f /'J'

r :o,ouve+Bp3n!,0)dpd€ =

=f /2to'ouve*zo' nu'el'ae=f

/2 {zouve*:onr.e)ae-

=f /2 { zonve* t s- t souvzs ) d0=[ 7n!€+ 1 ss- Jl n ze)',/' -:.

] n.

B ' ) E t v q L 1 s p S 2 K o L 0 s O s n .

'roo I=1,"If t:oouue'eo'nu'o )oao]ae =

{z"e*rsr- f *ze]-=rs".

y ' ) ELvq r 1<ps2 , 0sg<2n .

'Aed I=f 'f

(3eouvo*8e.nu.e)papdo=

=[zrue.rse-f nr26]'"=,0n.

e=ito

O OX HmTIKTI jlorltllox 2. trdv6dyoo B. t3?

59. tb unolovlo€eL ro 6! 6 olodtiprdlo:

_ f f . " . " - . . .11l(x'+'!')dxdo 6rov ro xr,plo T o9[(etorT

dr6 rrq oxaoeuC p-0, p-20 U€ 0 3 0 3 3n. 0

Aoon:

To xopl.o T 6Lwror an6 rologonletftJC q(iuo.

llercoxn!ft [<outre oe nolr-Ktq ouvreravuevEc. gcrouue :

Xrpouvo, X'pn!e. ELvoL e3ts20

0<0S3f l , f f f i=o ror

f tx,0).x'z+U'z.p'z.F(o,e).

'Aed r.f"ifre,edede=f"t+l:" -" + I'i"*= + t+l:' " + "'.

50. io rnEtoyl&ov rd olorlfltqEto:

c'r I"*lJrtr'o"r'a *o o'r I",JJ,nrx"*'r'ao'o

x>00 > 0

nton:

o') 0o letooxnuottoouu€ o€ notrlxaq ouweroYlrcv€q. o€rouus X'gouvo'U-pnp8.

E tvo t Osx '?+U ' :s l -0sp2ouv2O+p ' !np 'z0s | -0sp 'zs | -

osos1 . E tvor oK6us 0s0s2n ,

f(x,{)=(x.*0.)'/, -(p, )'t, =p, *, B{i# ,.,t 2 l t r r 2 r r - e r r . t 2 n' roo I r - l le?ededo=l l+ l de=+l d0-+ te l ;^=+.

* ,old! lerodxnlotJqouu€ d€ rotlaa. duvtetdl|rd!.s n rcroBucvit opiaouoc

. is ! idn u€ o (3 l ; re der i6o ?0) ,

2S

€=3n

Sx

1 3 8 KE4AAAIO TI

B') €o letdoxnuarLoou!€ oe mtr[ktq ouvr€rcvu6v€q. oCTouu€ x=pouvo,( r = p r u € 6 n 0 J q n p o q v o i r u e v o B p l o K o u ! € 0 < p S 1 . E n E L O i l € [ v q L x o L x z 0 ,

0 Z 0 ro X{,ip{.o T r€pLK;\€lerol ond r6Eo r€roproKuKllou KoL rouq Esrr-

Koirq nuLd€ovEc. Apo 0 <e < +.ELvo L oK6!q f (x, ,1,)=xL!(X'*, ! ' ) ' / " =p'z( nuoouve)p3=psn!oouve=F(p,e) xoL

D(x ,0 ) _ -D_6,ET -''

'noo t,=["/ ' [ 'o'*eo uueapde=|"/2 | Io't]nueo,veae=] i"/ 4!0ouved0 =! t J r

, ta / 2= ?ir I auzsd2s= "i

I-ouv2el"'"= ?,8- (-ouvn*ouvo)= 7i' (1{1)= -2i- =

_ l

5t. d') ilo unoloylooeL rE 600 rpdnouc ro oj\odipous I=JJfx+taxOl

d rov ]= t (X ,U) €R1 : X > O , U > 0 , X '+0 , S 1 ) , r

B') ih unoiov[oo€L xo oloKAiqJuo tr€ u€xooxnucrLotr6 o€ notr[rtc ouv-reYoy!6veq.

I ton:

o') 19C tpdnoc. To XuJpLo T eLvd[our6 rou oxitloroq. 0q unoloylooute ro6Lnl6 oloxtrhprrlq o;\oK,\nptirvovrqc npdrsorq npoq U. 06pvou!€ uro euoElo nqpdl-;\ntrn npog rov dqovo $ |]r nou r6!v€Lro otvopo rou r6nou orq on!€Lq A KqrB lra rarcyltveq dvrloloLxo U=0,{,=fFx,. Apo .{ .voL 0 30 : Vl-x, .Et-

wL xor 0 <x < 1 (Bl6nE oxtua).

,' r ,vT-Ia r ,t, ,,,rrVT-"'Exouue : I= l I | (x .J)d ' ! ldx- l lxu- +1" dx=

=f [ 'vi;, .-f,rr*.r1ax=f- | 11-r,1'r ' , f r-] , , ' I = ? .

29c rp6lroe. 0q otroK^npluooutr€ np6rc oq fipoq x. Tc 6pLo rou x elvdi,xoL Vl-r l , . . to 6pLq rou 0 €LvoL 0 KdL 1. 'Apo:

oAoMHPnTIÔI l \o f l tMot 2. t t rd!6d]ou B.

i r t "En r r , r " . 1y ' r :F r ' ; 1 - ,12 lI - l l i rx* ' ! )dxldr- l l+ .0xl d"- l l+ .uvr-u, ldqi=

=[4+ S-]r '+'r" ' l l 4.

B ) K6vouu€ rov lerooxnuorro!6 x=psuvo, l ] r=pn!6. Ta 6pLa rou I €Lvo!€=0, e=f l /Z. Tq 6pro rou p EavsL p=0, p=1, (0 < X'?+U'z < 1 +

0 s ( p o u v e ) ' z + ( p n l e ) ' z < 1 - 0 < p ' z < 1 - 0 s p : 1 ) . E l v s ! q K 6 ! o

ffi =o *o,- f(x,0)=x+L!=p (ouve+n!e),

'no" JJr,-,r,rarar=JJr"ou"e,pnueleae6s.l,.'f f

/'touve'npeta6]60=' i

i .1 r / 2=f o'l nr:e-ou"e I do=l ze'1do= +

62. tb unotroyLoo€L ro 6Ln^6 oôkîpr,rJa l-f f JctT'l-r,-,r"a,l]0, r..ol.

noon:

And ro 6pLo xnq olokiilpoone eqlvEroL 6rL 0 s x s o xoL 0 < rls 6?:xr'EroL o t6[oC tnq oiokl ipr, ionq Etvdt T=i(X,0) €R,: X,+0, S o'z, x>0, Ur>o]

0d lelooxn!f iLoou!€ oe m,\!x6q ouvrETqyut!€q. oeTouu€ X=pouv8 ,

o=pn!€ . E lvoL 0aea + , Ospsa "or f f i

=p . ELvo! o l (d lq

f ( x ,U) =U, -x ,4 .= ( p .nu .O ) p .

t i / r t p 1 I / ) t a , t t / 2 I ^ \ ' t a'Aea I='l. t lp'n!r8de]d810 a!,6dsj" e"do. i , l , (1-oJv2€)d0

[-i-1. =

n 5 | t 1 / 2 1 t 1 / 2 I ^ J I= - ra [J, d-z J. ouv20d28j= fu I t8] ; " - i tnuzel " ] = i f .

53. tb umloylooEl ro ol\odrliq,ud: IJe-(x'+'l')

dxdl

' f tdv T={(x,U) €R'z: x'+{ ' <o', x -t O,t , l t O} (o > 0).

Aton:

0o unoioyloou!€ ro olo(liplJuq !e Xpfton no,\LKdv ouvrsrovlavov.

139

140 KEOMAIO II

8€|,JOo0pe rcv u€roolnuorLo!6X=pouvo, 0=pn!9 (1). To 6pLdrou € €LvqL 9"0, e=+, ro 6proIoU P ELvoL P.0, p:c*- t lvo[ o-

(drqf(x,U)=e-(x 'z+{ ' 'z)-e-P'z xqL

D ( x . r lr-o.6i -p.

APo l .Je " ' ' dxdo=

T

/ - d

65. llo umlovLooet ro oloxtripogo

T={(x,U) € R,: x,+0,

r=iJ \ iffit dxdu 6nouT

<1, x>0 ,0 :0 )

: ! E d w L o s x 2 + + 2 < a 2 ' o :

o : o . * 6 L d r L d x o u ! .

p2ouv2o+p2u20 S d2 o o S p < a. r jwL ar6!ctar lp lo r r r r lou ue xdvrpo O.

=1"[fu,',0,1o'=y"[(- +)"-'1"*.- + 1,"/'r"*'-lo,= X r r-"*'r.

64. l{o urcloylo€EL ro 6r|116 olo$ipqro I.flpapao 6nou o t6noc t

nEpLKie{,6ror. On6 rr( kqtn0lec O.2ouVo, O=4ouVo. T

{ S . a l i k a t . A n a t y s i s )A(,on:

0L l(ot,ln0i€C nqpLordlouv al-, \ rLqrELc (o€ rclLx€q oovt. ioylav€q)

L r v o L - 7 . u s * ,

2 o u v 0 : p < 4 o u v 8 . ' A p q

rr /2r / rdulo 1I= l lJ pap lae.. x / 2 \ 1 2 d u r a r

= + J:i; t', l;,:::,'= | !',',',,û"*,-u!',','*u.eae+[efue . I ]"/'=:n.

CIoMHP0TIK0t 10r1t l to [ 2. Xno!6dro! ! .

l0on:

l ' l€rooxnufi [<ouus os noi!KCq ouvr.royltv€q. 06touu€ X:pouv0, t!=pnu€

E i l qL 0< 05 ] { exouu . r t r qp ro x0Kôu ) ,0<o< 1 , f f i =o rou

f ( x ,0 ) =

tt,f-'++, o*,=J'" J' ( r. ;;)'- oo*u=

=; l/ 'o,i"r, 't ' ff i" 0,=; rer;z'{ (l;d)" rr' 'r 'ao=

=+ tl lfr# rr.o'rl'-J' rr-o'r[ (];fr* I i'l==* f f t r -o"r5J' . i to

* lt r, J, !tj;i

=* [ r r -p ' t ' ' * -oenrp ' l '= *

a" ]=dp i=+ l l ( r -p" ) ' l - l :J - L L l o J i V t - p a

I n . ll ,

- 1 1 .

66- No unoloylog€l

n neplo)( i T nepLXl. l .erd[

A0on:

0q !erooxn!orIoou!€ o€ noiLXaq ouvr€roy!avcq. 06rouuE x=pouv0!

0=pnu8 (1). ELvsr x '?+Ur'?=x - ozouv'0+p2r!2o=pouve - p 'z=pouve - p=ouvo.'Apo 0 3 p s ouv0. KovovL(d €o 6np€n€ vs rdoouus e€ I0,t l En€L6i 6uos

oro 6ldornuo qur6 ro ou!€ vLv€To! r(ol opvnrLK6 nolpvoulE 8el- f i /2, rr /21

l l - - lro 6rn l6 ot ro( I ipu lo I= l l (1-x ' - l | l ' ) 'dxd+ 6rov

Tord rnv (o!nt , \n x 'z+{ 'z=X (x0i ioc) .

( B l d r e o e l ( 6 0 7 1 ) .1.4' rov uclooxnuor-o!6 ( l ) l rcproxf

T onErrovt<erar queLuovooitucvLo orqv 1€-pLoxi Tr nou opL(€ToL on6 rov Krr( lo

p=ouve, Exouue 6ffi =p kor

f (x , { r )= ( 1 -x ' -L ! ' ) t '= ( 1 -p" ) != r (p ,e l .

142 ICOMAIO II

t=[['no.ucoo" ["'[[""'i,r-p, ooo 1o.=- t

["'l [""'lr-o, rtrr-0, r lae -' r ' , ' ' t t ^ ) - r a / 2 " 4

, \ /2- -ouue -n/2= +l l r r-p,1'2,1 ds=-+ I L/ i ' j : i i i ,o),-r tae--

! r / 2 1 ' o - ! n / 2

r n / t , t t / , . t \ / 2 , r . / 2

I t1 - (v1-ouv,o) !1d0. i I t1 - tquor r ldo=+l d€-+ l Inuor 'de-! \ /2 . t /2 ' -n /2 " -n /2

, t n / 2

Io f ' . - - I I ( -nus ) rdo- "1 | (nue) 'd0 -

=*.-f [-*"Y*' i **1_,,;* [- nr'e"uve -!*,e]'"/ '=

3n-4=" '= --r '

67. l{d unoôvuo8eL rc otro(liootro

_ l.T

1=T

rerxovL(€ioL quq!povo-

oiuovro orov r6no T' =

= { ( p , € ) , 0 s p s l ,

-+.s.+]

'Apo ll r.:+-r dxd'! =

T

ri}luu- a*a'!s1

0

0

ro,- i t

.IIAtort:

H ouvdpnon ttx, ' t , l= ff i : 6xEL 6L6ypduuq ouu!€rpLx6 oq npoc rov

6tovo 0 '1.

l t l l'Ae" -'J I 1;i#.[7 dxdqr=, ^J I 1;+{) dxd,! (1)

r '+ / ' s r x ' . . ' 3 -l zo

t loo uno,\oyloolu€ ro olo(îpoua t= Ji ]tiot dxdl! 6nou o r6noc T opL<E-

rol qn6 Trc oxdo€Lc x'z+0'z< 1, x >_0. oo leroox4ldl loouu€ To oloK]ipl, luq

u€ IIotrLk6q ouvr€royuAvEc.

EtvoL x=pouv€, 0=pnue (2). l4s rov trErooxnudrLol6 (2) o r6nos T o-

( 1 , 0 )

oIoKAHmTIXO' lo l lXMot 2. l t ro !6d1o! ! .

=11. +iP *",'1 r# ll,',,',^* *1* = i,z( r - fr ")ao =z r o-,o r, * ti = z- * .'EroL n r) 0,"",

,,j*l.rrr#h, oao=z(z-i) +*

b8. Ito rrroôyrogEt ro 6lnlo otro(J\ttftlo f[ '-o€qglj(:cl axa,l'l Fw6rov T={ (x ,U) €R, : 1sX, ' t r<4 rd l x> to t ) .

Aoon:

0 r6noq T Etvol our6q rou lpdJrou oxilcroq.

KovouU€ rov !ErosxnUoTLo!6

x = p o u v S , ( r - p n ! S ( 1 ) ,

E L v o L 1 s X ' z + U ' ? < 4 -

1 s (pouvo)1+(pnuo) 'z 5 z 'z -

1 s g ' z s Z ' z - l s p s 2 . E l v o L

xz l t l - l u l < x - - xsQ :x

- -pouv8 < pn!€ < pouvo -

-ouve < n!0 s ouve +

t4u6 t < ouve - e€ l - + ,+ |

ffi =0. 'Etou 6louPe:

[[ xro€€o(x'z+0'? ] dxddr -tl '/(;(

=JJoouveroteoo'aeao -,r. ,/,

=l proE€ep'zl I ouvede idp=- , , : r / q

0

(0,+ )

143

/2,,2)

r l -' /1 ' Lr

0 ,0 ) ( 2 , 0 )

l

l.l€ Tov lerooxtuotLo!6 (1) o r6noc T onelxovl(€rsl ot/eLtroyooit/ovro o[ov Td-ro Tr nou €LvqL ro opeovdv lo r , , , t , [ - i , *J .

tLvor ax6uo f(x,rr)= xrotEa(x"+'?) = !9! ! !19$90' =sgyg16{ceo ) =F ( 0. o ) (oL\,@ lF

( 0 ,

t44 KEOAAAIO iI

(n*"r"ro"oo=, !i$) roE€apzdp=vz [+: -"*']'-vz lrr$,r-

-2\tz $E a4- )'.r"rr-, [* t"rt*p-r]'rvz roeeoc- $ - ) nn-

- 9n ,"r.

59. lb unoôvloeel rc 6rnl6 olorlatq,,lq

T={ (x ,U)€R, : x ,+U,<4 , xz l , 0>0 .

n(|on:

r=JJ -r"o $ axar o.*T

eq !Etqoxnlortoouu€ oe nolLXaq ouvt€rqvu6v€q.

06tou!e X=pouvo, qr=pn!8 ( i )

ELvoL x,+| l jz-4 - ptouv,0+p2ny'z0.4 -p -2 . H x=1 6LveL :

pou le= l - p= *Jq6 .

' eoo * [ . 0a ,

E[voL p=2 (oL r= aJW - t- arl"g -

. ouue=* -8 .+. .Apo

'EXou!E cK6uo

;-+A-++ =0. Aoo

f(x,0)=ro{€0 + =ro!e0 3#%' =ro€€0(€qB)=8-F(p,o) xqL

,=f"IJ' .,*],,-; i'[*']',"# ' f i is

=f " rror- ;f' " --e- ar=lr']l/'- *= + +[e"oe*tntouuet]" /3=

*=; 1"31,- -g y.=

j *ito do =

a , E t , - I- -E- -2 " 'T '

;3* =J tt*ur=u*u-1- "r*" J e05d5:0 eo0+1n I o!!9 | + c.

OIoKlHPllTIKot l0IItMOt 2. lrd!6dyou B. 1 4 5

70. ilo unolovrdel. to durl6 oloxlllFt o

o I6mc {(x,l) €R': (r-{)t ' s c'z, x > 0,

Aooq:

71. l{o unoJ\oyr.ooel ro oloxlipqo:

opl<etor dn6 rrq oxdoelq T={(X,{) €R' l :

Aoon:

M€rooxnuorL(o!!e qE nolLK6c ouvr€rqylrtvac. el iouu€ x=pouv6, 0=pnug.

l=JJtx'+'laxc onou r rtvorT' r>ol .

ffi*., n nepLoxi r+ { , ' z .<1 , x>0 , 0>0 } .

Etvot (x-o)'+J'? < o' . Xt-2ox*ot*, l , t s o'. x'1+rl'z s 2ox . O s p'z 5 zopouve0sos2qouvo . t l vo r o (6uo o<es f ,

fiffi =o xou r(x'0)-x--0-=

,. , \ /2 ,2door ''roo I'JJo o'touv'e-nu':o)ded€=J" J, o'(ouv'e-'u'e)dpds.

.a /2 .2dou)$ . \ /2 - - . ,2ddv \6 ,a /2=l I, o'ouv2sdpds-J, ryt [/J, do=+ Jo (ouv28)(zoduvo)'d6-

. tn/2 t rn l2=* J" tzouu'e-rtreo'ouv'odo-+ J, tlzo'oru'e-teo'o,"'e)ae-

-so' f/3ru'eae+o'f/'oru'sae=[ao'r,

-no"r. ]"" onou :,=J oru"ete.

ttvou 6uorc !"=] ouvu-1onue* # tu-r. ,ro, gpto(ouue t:. l l /2= + ,

r : .r : / '=+ ft" i l2-+. 'Aoo r=sc"-j !-ao"-i*-. - i- .

TT

eo u€TooxnlctLoouue o€ nollktq olvt€toyg6v€c. 06rou!e x=pouv8, o=on!€

E t v a L 0 S x ' z + 0 ' : s 1 . 0 s p ' : s 1 - 0 S p 3 l . E t v o r . c x 6 u c 0 3 0 5 {(rErcproxirK^Lo ue Kevrpo 0).

14€ ro u€TdoxntorLdu6 out6 o t6noq T snELKovL<ttdL dtov l6flo

10

r45 XEOAAAIO II

iil;tr'

r , +o, r: ' to, f L

'EXoutr€ KoI f(x,{)=

.{ lr.;i *,0..,.

'o*'.JJ ffio,or,-T

-',"'l'orE ooor=["f'ott-o' t 0", (r).,o ,o t ''p_ ,, J, vF6i

E1uo, [ 'e ! : :d ]60- l ' ( o - o ' ' \ ao-* [ ',. \4:pr ro \vTii VF;{ / ' ,o

.+ | i+-+ ao- { | rr,'r'r 0,,-o', =} |:{otoo!e oro lfx;rro oAo(îB$o 92{r) =

=] r.otnur:l*f 1o-21-| lroqn!t -toqnpol- |ra/2r -'EroL n (r) or*, r-f '-(* -+ )*-(+-+)

Lo** l '4o,v1-p. a ro v l -p l

to ' * tzYt=, ; ' t ' -VT-,i: '

=+ +-+-+-+, ^ , \ / 2 n 2 n

72. ilo woAoyLooe! ro oloxliiprdro ll -#L axa{, Otc"J) X''{J'T

T=t ( r , i )€R ' : x '? 'o '?s9 ' : , x ' . .0 ' . :o ' , 0z0r g >o}

lGon:

earougE x=pouvo, {,:pn!6.Etw! o' s X'z+U'? s 9'? -d'? s p,ouv2€+o2nu'zo < o'z -q5p<0 . ELvd r oK6uo' ! 20 -pn !820 -n !0 :0Ene!6i el,voL (oL x > 0 t xso€Lvo! 9ouvg u 0 6 pouvo s 0 +ouvez0 t t ouvg<0 .'A9o 0 < 0 < r. tl.vor rou fffi

=p. >uvErdrc

0A0MH!0TIK0! lo t l tMot 2. t rq!6dyou B. 147

JJ"t+ -*=l tJ"lt 'e*er,0,]*=l lf r**'.*r*]or==f louve+nuet rolBde={9-a)f (ouv8+nus)de=(B-o) tnre-ouvol;=2(9-o).

Il.

n!!xodLo

^6onr

tft rrDlovloo€|. ro oloxliq|ro I | 6-itl ,-,dA 6Edv T €[vor !o'i

x,{,=dx rou rE&rou rar@t|tlopl,ou.

06!0uu€ X=pouv8, U=pntg.ELvoL xz+0,-oX - p2ouv10+prn!r8=-doouvo. p2=opolvg - p=oouve.'Apo 0 O - 0 S e s |. re rcvtr€rcoxnlorr-ou6 our6 o xdnoc Tolr€L(ov[C€roL otov 16Lo r6no. E[voL

f ( x,0 ) -6I;-, -U,,. l [ , - p, =f (p,€ )

xoL f i f f i "0. EloL Axou!€:

J lvq,-x,- ' ! , dxd'!='f

, f l /2 .douvs .1 /2 .dd \ )$ . f /2 - -oouvo=1, ,j, o\-o,-o,aoao=1, ,/, o (o,-e')rdode=J. IJ,

--]r",-o,r"or",r,rjae

t , / ) t , . , . r c d u v o , t a / 2

1 , f -+ +ta"-e ' ) ' " l , ot - -+ J. I (o 'z-o 'zo ' rv?e) ' / ' : -ot ld6-

I t , / 2 ' n l ? " ' - o ' u " 1 ' r ' - t l o e ==-+ jo Id r (1 -ouv ' )8 ) " -o ' :ae=- tq ' l

=-T a, 'J, (n!ro-l)dr= j ' l+ +l .

1 4 8 !€OAAAIO II

71. llo umrtoyro€€l ro otodlpqlo .jJm€[!(x'+tl,')dxd0,

6mu T ehrol oT

rdnoc mu n€plrlslsror dfl6 tnv xqfiarlt (x"rot)t=$t, 0 u 0-

A0on:

Kdvoute rov lerooxquorLou6 x=pouvo, 0:pn!€ (1).

E { , v o L : 0 s (x'r0'?)i 3 U'?- 0 s (p'?o0v'z€+p'nu"0)'s p'nu'e -

o s p ' sp"u t€ - os pn < nu t0- o0doo 0<esn .'EToL

u€ rov uerooxnuorlotd o !6noq T p€tooxn!f iL(8rol olqLlovooiuov-

! o o l o v ! 6 1 0 l 1 - t ( p , 0 ) : 0 < p s V i u O , C < 9 3 n ) n o u n q p L o r d v € L x 0 { I o .

ELvoL ox6!q BIl:8] +. r".r,t*c

JJ.ornutx'+' laxau-lJ(ro!nup'? )pdodo=

=l{i*"'u"Y*\,6;6

=* f(J ."r.,"0")or:

, r r r ^ " _ l v i ! - g r r r= * | lp'.ornup'.ui:r;"1 ae=j ltnr:etotnr: (4ue) *y'-t -nu'+ t lae.

r r r 1 ta/2=+ | tau0roEnu(aue) - touvEl -11d6- ' ! l In !€ ro lnu(4u0)+ouve-1 ]de+

r rn , t i /2++ l I n l s ro€nu(nuo) -ouv8 -11d8= + | (0n !0+ouv0-1 )d8 .

r/2

r l nr j I I {n -e)nue-ouvo-11d0- . . . -+ (4-n) .- 'n tz

i ndpdyo!rLx i o lorr ipoon

oAoMHPIITIKO' Aot I tMoI 2. tna!6d1ou B.

75- Iilo unoloylogel ro oloKlipl,xla f = [[axa'l O*" T e[wL ro )cnpl.o mu

n€pLd\e!€rol on6 rnv xount^n (+. # il + + (1) (o,B,h,K o-oLouoL 0€ILxo[).

n0on:

Kdvou!€ rov lerooxnlorLolb x=opouv€, | ]J=gp4ue (p > 0). EroL n €q[ -

ooon rnq xqunuinq yLvetou (p'zor:v'?g+p'znrzurz= o'p'zourvzg - 8'zp12g'10

1-

"'' ({+#-{5t )u5 o={ $ ",",e-

-!ul ',e. Er.vo!

o<Ps V# oru 'e-

1 4 9

n€! jz ouv'o- B"

roqeq lF

EneLdit ro Xop[o T €LvoL o0l lerpLkd oc npoq tuuc dtovec €o epyooro0!€ dronpdto tEtcprqudpLo (on6re 610L unotroyL(ou!€ !o 1/4 rou o,\oK,\nsiuoroc).

. 0 < e < r o E € o ; g . E L v o L o x 6 u on lzj oLVEL 8nolEviJq EAU _< .gfi-

Diffi -oeP'

0q Bpo0!€ rq dpLo rou €. Er€r6i p€R np6-

-l : nu'o < +- ouv'8 - €a'8 <;;- (2)- E ' n '

-c., {}(+ ",",u- + nu:6)d8

r_+- fT ; ;r=]]e^ - dxd, ordv

Tx :0 , t ! >0 ){

0 , 2 )

B ,

'npo *=f rl=" = +t (+ - + ) .r',ff;.ff;1 .

75. No unotroyro€€L yo 6[n]\6 oloKtr/tp0uq:

T= t ( x ' 0 ) €R '? : 13x+ {s2 ,

nton:

l(dvouuE Tov u€rcoxrtuqrLo!6x4=u , x - ' ! =u ( l ) .

ou (1 ) 6 l vouv x= $ ! ,

{ ,=# (2).

#nu'e.nu 'e>o-

(0 ,1 )

( 2 ,0 )

150 XEOMAIO II

H n€ploxi T o9[<etoL on6 rrc .uoEl,€q x+U=2, X+U=l, X=0, U=0.f ld x+0=2t X+U.l n f ig;rrn and rlc (1) 6[vEr u=2, u=|.Tlo x=0, U=0 or (2) 6tlouv u.-u, u.u.

'Etor uE rov lerooxnticrlo!6 (1) n n€p[oxt T onerxovt<€mt otroluor/ooftUqwo otnv n€pLoxi Tr UE Tr={(u,u)€R,/ I S u S 2, -u < u 3 u}

ax

a'l-t

I:u -f_ttvoL 0l<6!0 f(x,U)=ex+{.e u =

=F (u , u ) . 'Apo

JFl i ,=+t u(e-e-r)du- "-:- ' | " l ' -

L L u . ) , . L . J l

- , . . | - : -a-r r . _ . u l x r { r , _ l d u"'"' nl!:l,-t -l ^,

l d u

* lro

] ' * *

t 1TT11

T-T

(

l " * l l l e " au lau-. r. l) I

. . r . .u.L=*J, ["]+ of,' * t " - " - ' 1 '

s l { s2 , ' l s0<2 )

77. ih motovloo€! ro oôxribqD 11 ##F **.

"*L1 = ( ( x , 0 ) € R , : I

i0oq:

To lreplypollc rou r6nouko€ool(€xoL ofl6 rLC Kopntlsq

x' !-1(- 0=T ), x l | j = 2

t- U-i ) {,-1 (oL {,-2. 0r6noq T nopLordv€tor on6 ronopontre(lpi,rs oxilo.

oeopoitrE rov uErooxnuo-rLoud s-{,2, u=Xt (1). ELvoL( l sX0<2 xo l 15Us2 ) -

0"2/xf i ,1 ) i

( 2 ,1 )

040(^HP0TIK0' loIIllrOX 2. trd!6riyou B. 1 3 f

- ( 1 <xos2 (oL 1<g2s4 ) l g ( 1 su<2 KoL l su<4 ) .

0 r6[oc T u€ Tov !€Igoxn-

uf iLou6 ( l ) oneL{ov{.<€rqr o!-qLuovooiltqwo qrov r6no Tr tou

rrdporl€(rpoq ox|llarcq.

Elvor f(x,{,J= 9"fi = =(0 '+x0-1) '

= uW : =F(u ,u) Kor(u+u-1) '

D(u ,u ) _D-|r;TI 'tu1rAu-5x

auTTA U-5r

02u

0x=-202 =-2u. 'Apo

Bif*i-+ - ID(x,u) lfir-.nTl =t-+ t-+

'Aro [[ -il4udrr!"[[ ,w , a"a,{J! (u+xu-1) '? +1 2u(u+u-1) ' 1,

-t' + tl G**," 1,,= 1+ t .+- 11,-l' l* -*# 1,,-

W(3+u) r,1

T {w -*,{g l'-'['.(\ID"]=(,or", y'$-,og", # ) =(.""'VT-+ ).

78. Ito urtotroylooEl lo olorlifttro JJ tn:O; ax*. 0 r6troq T rEpr -

rletgEar 0|16 lnv Uu.uFt x'z+x{,raz=l itl.

lton:

0 r6noq T €[voL ro slvotro rov onu€l(,]v/ | \ 2 t \ l i \ 2

{ (x ,0 ) €R 'z : x 'z ,x !+{ , ' z< 1}= t (x ,u )€R 'z : (x++{ ) . ( ? ,1 ) = t l .

EneLEi o 0noloyLou6q rou oô(trnpbuoroc etwL nolfuIloxoc e€{,igo0!€ @v !€xq_

(u > o ) ,

--14- 6u6,12(u+u-1) 'z

du- ; l

152KEOAÂIO II

oxnuorLou6 x- +0 -', I

- I l 2 ) .

Yr=" I0r {2) !E t0oon oro r€Tpdvr,rvo xo! Kot6nlv !€,ro6o0€on dlvouv

(r-+r)"C ,r)"-,-,. - x,+xr!+rr,=u,+u, € u2+u2 s 1. ,EroL o r6nos

T OnalKovl(eto[ olgLUovooiuovro Olo T6[0 Tr={(u,U), ur+u, S l} .

au au

eLvou or6uo ffi =

' ̂ ^ - D(x .0 ) 2 ^ry " Dru ; I l - - - . u ronoc

Tr noprordv€Tot on6 m nqpo-rli€(rpos oxiljd. Eto! txoule

I l -

llY4-3,r. dxd{= | | -! 2\nizdudu='; '.' ,t3

, ,!4-r,= j ^ uT=,, tJ dL, ldu=

o r' -.rrf f i t ^ rt=; - | vr-u: |1 duldu= + | '4 : t r zr ] - - r a"= 4 [ ' { t -u,)du, E4vs '_r .1vI_;" , v3 1r ,t1 )_t

79- ih unoloylooea ro olodllrqrd [[ ,tdt o dmc T rcor-' i (x+0)'z(x-0)'z

d[[Ero! oM xo rpLtnE tr€ (opuotc (.1,0), (3,0), (Z,l).

Ator:

8a,roolue rov lt€Toox4uor[o!6 u=x+0, u.x-U (l). e xov letooxntqrloldourb o r6noq T sneLKov[<eTo! ourtLl lovooiUovto orov tdno Tr no! nepLKiELeToL

1+_6- -T

au au

O]|OKI|BH IlOt lotl'lpt 2. tB!6d1ou B, t59

0116 to rp[yovo !€ Kopuotc ( '1,1), (3,3)

ELvar

lBir+t l. Ittw! 016!0 f(x,t) '

( 3 , 1 ) .

auau lTF Tr l t t r l

l ' l | *2. 'Amt t l

au au I l t -1 |ax auI t l-T I -T ,

ffil;;'r!'rt"'"r' 'APo

{ G#*-F'l[ *-'.a"o'JJ'l-'"*"{ [l,'+ *]"'

.1;"ti + -1,"+,# (+-+)-l (# - # !*[- #. + ]:"+ar. a utDlovldet Io oloxNloi{o [[ {iE laril, 0 r6rDc T mpr-,i

tleleldr old rouc r0xlou( x'zr{r.4' x'l+0'.9,

l0on:

0d u€rooxruortoouuE ro oloxlttflrlro o€ nolartc ouw€tcvrEveq.'Exow€ x.po!vo, U=9ru0.

'Apo 4 3x 'z+1 l ' l s9 .4<p2ouv2o+p2tp2e<9.4 <p2 <9 +2 sp s3 ,

El.w! 0<es2r id! f ifr i i =p.

154 XXOMAIO 1I

t t . . . . . . . . . . . . _ t r - r? ' r r t r -'roo JJ4tE? dxd!1J!6'oufi6-nrrnut0 ededolo [J, eede]d0 =

T T *

r 2 r f ^ ! 1 3 r o 1 2 '=l l-+l d0= -+ I d0= + r.

81. lb uDlovloo€a ro 6rnl6 olodalr|n

nsdrrlel€m0 ar|6 r|,c itfi(,lec $=4, ,=2,

JJ rr'*'taror. o.o-. rTx,-a,E9, x,{,=t.

Kdvoule rov uerooxnlorroud X'?-o'z=u, xo=u ( l ) .

Elvor 1 < x 2-U '? 3 9 - 1 s u s 9 .

[ L v q L o x 6 u q 2 s x U 3 4 - 2 3 u ! 4 .'ETo[ o r6noq T onEL(ovl<€rq! duoLlo\ooilt/ovro orov r6fio

T t = { ( u , u ) € R ' z : 1 < ! 3 9 , 2 t r 3 4 } .0q €xepdoouu€ rnv ouvdpxnon f(X,U) ouvoprioeL rLJv v6ov u€rqotrnrdrv.

Alron:

oIoK^HPnTIKoI IoIID,Jo[ 2. ttrd!6d1ou B.

ElvqL x'z-U2=u + Xr-2x'z0'z+0r=u'zI roods.I - x ' -2x' , ! ' * , rn=r '1 +4u' - lx l+&'1 ) '1 =

2\\b = 2u - 4\zA2= 4u2 )

=u2 +Au'z - x 'z +,! 'z =l iz i t iz. Apo f (x,U).x,+Q,=Viz;Tu' t=F(u,u). ELvqroK0uo

lo=B{+*J-=l-DTu,nIDIx,[t l-':'l

ztx'+q')exAu

-[

A U-5tr

z.lizitriz lzfiriTnzl z\trzidiz

'EToL lo Ln6v txoulel l l l - | 1 l ll l ( x ,+0 , )dxd0 - l l vu ,+4u2 - - - - -dudL=+ l l dudu=r) rr 2\G7;dG . ))T T , T I

I l , t 'du=+l sdu=4tu l l=8.I | . J

duldu"+_ r t t l-T) t l

82- l{d

TOI Cno tLC

A(,on:

unolovLo€€L ro oloKtrlprr/o JJf'f,*fara,f,. 0 r6noc T neplxlet€-

eu8eL6q U=x+i, 0=x-3, ,1,- llx- +, q,=. +x+s.

0L eqrodo€rq nou opt(ouv rov rdro ypdqovroL | ] / -x=1,,r,-x=-3, u.+x=+u.{x=s.

EhoL -3 s 0-x s 1 , +3 0++x 5 5

06 touPe g -x=x , ' ! ++x=u (2 ) . 'E ioL o !

! =u.s.

ffi.rtuo'ffi=r 31"&Y+ i=+

EtvoL f(x,U)=0-x=u-F(u,u). 0o unoloyLoouu€ rnv IqxoBLqvfl opL<ouoq

au Au-x -atrau au-x -tr

( 1 ) .

( 1 ) K o L ( 2 ) b l v o u v - 3 ! u < 1 ,

, 1 4= - L - T = - 5

-1 1

+,

155 I<EOI]\AIO II

3=- ?-

'Apa [[t,r*ra,ar,=] [' [['*,h"=* t' t# l'*=* [' (+. +),"' i 1 / r " - r ' 1 t ! ' , t r '

[ ' oo*-r[ ' du=-3lul ' .-8,I r t l r " 7 t '

c r r d(1+u)

83. ilo 6euo€. c.. [ [[ rrr,urolaol I r (#, Hf )fih,**lion:

An6 ro oAoxîF{rd rou rgdnou !6louc Oc.'vetot 611 0 S X S a (1)

osu3x ( z ) . geo0or i s6 rov l e taoxn tqELo !6 x=# ,v . i ; $ . 0 ! )

Ko t ( 2 ) 6 [ vouv 0 s1 f t 3o -03uso ( l+u ) ko r

Os0ix -o s f f < x- 0 suu sx(1+u) . osuus # (1+u) .osusl .

l ax a l l , | | u u IlrE- ru I l- T1;Or TT-uI' IEtwr cx6yc fl.1{r{f. I r. r I rl a r a ' ! | | | u Il - f i - TF| |T; r 1 ; r |

ax a0T' T'

'- n+lr'- nh"'-'{+'-ti'=- Tth" - lBifi$il' l - n*!-r, l' n+'T'a l , o( 1+u )'Ero! lxour/€ J [J'r,,orar]a,{ ' [J' (#, i# )rThF *]0"

0 0

' , , t-81. tlo urotoyloo€t to oloxlrfriouo

'|'| C-tdxdf !E

TT ' { ( x ' 0 )€R ' : x>0 ' f >0 , x+ ' } s l }

Aarott:

lGvoull8 lov uercoxntor Lo!6

x+rlFu (l) tfu -u tzl

H (2) 6[v€r 'lFu(x'{t'!L {=uu (3)

0r ( l) rcr (3) ue oeol.peon lCvouvx.u(j -u)

o OnHmTIKOt lOIrEilOI 2. ,nd!6d1ou B. t57

E lvdL 0s x+U < 1 -0sus 1 . A [6 rnv { ,= l l u nooxo rE€r 611 0su .E { , vo ! x > 0 -u f i - u ) > 0 -u i t .

..1: .o a

" a 1.'Ero! !a tov rr€rcq(nuc[Lolo o roroq T oneliovl(€rol orJoL-l rovoo i l . r owd d ro ! l 6no T r : { (u ,u )€Rr : 03u3 l , 0sus 1 ) .EIvor f (X,0)=d(+0=eu=F(u,u).

'o^ | *[,_&*i | =r-ur.u. Enom,arc

JJ"fto,o,r,.["',**.1'U'+*,Jr".f ""[-J' a".]r*rl. ]. r._r r.

" i,i;=::*:,:J/.tioor:

( )('{'' < Brl x'+o'l > t'l

l-3! aoEl,vor,or<6uoff i .

l : ' : :tif #

€6rouse 1191 ., *dL qq: .u 1)'Exo!!.: x,+U,-Bro s o - rff. .6, xoL

x'z+u'?-8,o 2 o r {i€ te" . .Apo s('vdr

Ii4€ 6poLo o(p!g6q rpdno rorolivoula ot4v dvro6tnro

lsx0€! 6rL: $ - j- xou ,= #i, U= ##, .ELvoL l,rJ -ff i, t,r,r,=rh .*i-,,,{11 *r",,,.

o r > o r > 0

B r > ! z > 0

(u'z+uz ) '? t l

8 r<usBr .

d r : i u S o r

;f ouo, .To oiodiq4Jo Vlverq!:=[,+i,+-'"t+l'1.(+).I=II+*;]:

158 KE9AMIO II

86. Eo[tr f ro xopl.o nou opdooEroi ond rrc euo€[ec x+tlFl,x=o, 'F0.

ttd 6etE€r€ 6rL: JJ"* ffi ara,r" $.T

Aoon:

otroUtre: (X+0=u r(oL X-(r=u) (1) r

TLox>0 -u+u :0 -ua -u .

Tr.o ' ! : 0 - u-u Z 0 - t 3 u.

EnConC axoule: X+U3 1 - u S 1,0 !6no( !€ rov uerooxnucl-

qr6 {l) ofleLxova(Erdr, dtioLuovo -

oilovrd orov rdno

f 03u311Tr= | (u ,u)€n ' : Ir - usu !u ,

lxntoxL<outrc rnv loKdLovi oO( -

aouoa J- ;.iffi ELvo!

{2x=u+u, 20=u-!) - (x= 9i , ,r= f)tzl

. l " ' 1/2

'- l-,,, +1/2t t

" -T , - -4 -= -T . tuverdK lJl= +-. 'rxouu"

l r / " r , \ , t r r r " t r t l r t u / , , \ / , , \ ' l'-lJ"* ti#) dxd,,=J, l+ J_"*" (-i) *lo* - .J" ["J-,","(-5-)o(#)Jo,, , \ ] u=+ | l " ( nu{) l a ,=+ lu tnu( t t ) -a! ( -1} tau= j - [2rLnurdu=

=[ {nur)udu=nu1 t+l ' = +

rr dxd0 , To xr,,pto T op!(erol drd87. dlvsrol ro 6LrI6 olodft9rdlo JJ f*tr+t.T

ro rpLynvo 0AB gE Kopuete (0,0), (1,0), (1,1).o ) lilo uroloyLooe{. to orrokliqrld.

B') ilo fioloylo€eL to olo(lfipolo !erooxn!ar!(6w6c to (cr6lldo.

O OK[HPmIrcE lOrID{ot 2. t[c!6d.rou E. t59

Aoq:

o') 0o gpolrlE ro 6plo ro0 (r(0o oloKlnpdEoule np6to l||q |l9oc

U). oeogooue uLo ruxsLq €ueelonopdllnAn tr9oq rov dtow U"!.

To Ko!v6 onte{.o rns Euo€[ocI€ to nEp[v9o!t/o rou x0pl,ou 6-Xouv terovu6vEq 0.0, 0=X (ourd

eLvo! ro 6p!o ror, {,), }l urxo6t€-pn |(qr n UeYcl(,repn r€rgrucvn rov ongs[ov rou T €t\ol X=0, X=1. (6pLd ror.,

t ' r r xEtvor. l l -J- 6;a,l- | l l 1 o,lldx-l troE€0(u-x)lldx.,,i r*(u-xt. Jo rro 1+(r!-x), r ro

.I t-ro*ot-xttox.Ii.toteoxdx-[ x'$t€oxdx.txroteexli-J x(ro€eex)'dx.i , . ro ,

t t --roEEot- '|- 1+r dx=roleel-

[+ ln(l.x')lo.tot€el- + 162-f-lnv?.

F') oarpotue @v uetooxnlorlou6 u.U-x, u.x (l). l.l€ rov getdoxntrotl-oyd cut6 ro rptvovo dlre0(ovt.(eroL olrellovooilowo oto rg[yovo 0A'B'!a rc-pWaC (0,0), (-1,1), (0,1) (rpoxtrEouv dt6 rrq (1) 9d<owoc 6r,o6oxrKd d-rou X KqL 0 rlc olvr€rovucvEc rltv kopuodrv tou rpLvd,vou 0Ag).

rtvou ffi.

t

t;,k d L f ( x , U ) = 1 - 1 .

1+(U-X) '? 1+(u+u-u) '2

iTmrUG;U'

I .-1au au-FEau a!-r -50-

: = = = = ' = _ '

= 1l;, =,ru,,;. "" IJ "nh.lJ ir,

**=f[J" i* *],,=

160 KEOAÂ]O II

=l lro€equlo du=l [ro[eo0-roq€q(-u)]du.l ro(esudri.(t6Lo €pyools u€ rnvnpo-

4YOUU€V4 naprlr&,'O4 t: ?- - In/2.

88. l{d onolovLogel ro oj\oKlflpolrd dxd! 6rdv o

tdrcq I o9[ ( . ror o116 r rc oxtoerc +-# t t , x>0, 0>0.

A0on:

o€rdpo0rJ€ rov lirqoxntorlo!6 * *, *.u - x=ou, |l,=gu. ELvoL

(+) ' . (+) ' - ' * ' . # '$* '*" -u '?iu25r. Ervor o.ouo

* ao , * : o -u20 , u : 0 . 'E toL o x6noc T ue tooxnuq r [ ( € rdL d r r -

q f . l ovoo i t i ov ro o ro rdno T r= { (u ,u )€Rz t ! 2+ ! r s1 , u >0 , ! z 0 } .

rlvou ffi= -oB.a0A U

TLo vq unoloyloouuE ro

o 0

09a{-u

x€J\€uroLo oloKlipr,rt/o xdvoule rov !€TooxnucLou6 d=9ouvo, u=pn!€ ondre

0 : p < 1 x o L o s O < * . ' E r o L 6 x o ! t / s :

tr t Ell l .: r,/2rt ^ l l ln ,.tzr' -f ,^,T= I lo0 \/ i# dudu'l laBe t/fi;z dede=qg | | p \ .h;7 dpde= (ro oroK^t-

) ) | ' -T r '

qru; ourd urotroylornr<€ crrv ooKnon 6rFaB (+ - +) .

ololoHlnTMt I\OIIAOX 2. trd!6dyou B. l6l

A U

a0Tn-

r. rsr89. lh r€rEmrd.roEel to 6!doxrr6 oroxla!|l|q I-l dll f(x,a)dl !c,o ox

tnv goiioelo tnc awLrar6otlrrlc u=xr$, uu={ (l) (0 < o < F, c > 0, u/l)

160||:

0r (l) 6[vouv X=u-{,. X=u-uu - X:u(t-u) (2) mt 0.uu (3).

_!! -!!Ecw! B[ut*] .

l1-r, -u

tu u

au

a0-5F

€o 0oo0lr€ 6plo r!,,v u (or, u.'91qy 1.9 -{.?.1i, u 1 1 -g I .g L9191 'r.9

'orqv x=c rdre n (z) 6lv€r u. #. 'Aoc os u 3 1$

0o Fgotrre rq 6pLo ro0 u.'0tov t1.q1 16at Exouuo old {2) Ko0 (3) uu"ou(l-u) - u.c-ou. u(l+o).d

. * +. . 'Orav U=Bx !e tov (,ELo rpOro Aolorou!€ 6rt 0. S , 'eror

3a'i:i i-

e('vou I. l I f[u(t -u),uu]ududu.

JF

90. lb urolovlooet to olodifi|la r.JJtx+*"laxal onou

l-((x,t) € Rr: r'z+a0'z-zr-01+1 s 0)

^lion:' "' LU:el .t (rlre.,r,n).ELvoL X'z+20'z-2x-8r!+5.0 - #, ,n r"

'Ero! o r6roc T aGprxl€l,etoL o|t6 tnv oll€L{,n ge (6wpo (1,2) (cL lirrlruLoE6lr,,v 2 (aL fe

* E i \aL 0 S X S c - X Z o . E( !aL dx S { S BX - 0 > o . '4oo u=Xr4 , : O.'ElsL E':BL lu l:u.

11

162 IGOMAIO II

l(dvouu6 rov !€rooxn1orrotd + =u,

g:3 =u. g)(6uus ur+gr 3 1v2

(eq.6oov eLvor , (x ; l ) ' . (o -2)z <1)I \EJ '

_ /'EroL o r6floq T onerKovt(€roL q!-

0Luovooiuovto qrov T6no

T r = { ( u , u ) e n ' z : u l + u ? s 1 } .

20

OErLvou ff i= =2'17

(oL f {X,0)=X4+0'?=zu+ 1+f2u+2+2u2++4+ 4'12!-2u+5\'/Zu+ 2!z +7 .

'A0o I=JJ (x*U'o.)dxdu =T

=ll | zur sItu + 2D z +7 ) z\,Zdud u.T l

l.lExqoxnUor[(ouu€ rd]pc o3 rcl!ktqouvT€rcylaveq. ELvqL u-pouvg,u=pnue.E L v q L 0 s p s l k o r 0 < e < 2 n .

LrvoL qKouq ffi;fi-i

=o xor

F(p,B )=2pouv0+5tr2pn!0+20'znu'18+7

' noa rzfz!!otzoouve+5't1p\vo+2p2\u,e+7 )dpdr=ra ( [itzr""*e*rrro,n ",

+zoi\v'zs+T defde=2,lz i l+ rn,"r- $ o,nue, S nu"e* + r.]ir.

2n 2t=r, I,(+ouve*!€nuo* lnv'a,!)ae=za [*J *"*'.

. uf [".,r0r, | ['-nu'eae. ] l'-aaf-...=rnn.' J n

U

91. tld uroloyrorEa to olorlllHr tou polsson [ " * " r t

A0ol:

eEopoose to oro*iiorgo JJ"-tx"*+") a*u,I, dno,T

T"{ (x,U) r x"*0" < R'}

x-pquvo rt t to ' r "

t .onr , l ' , ' * "0 . . o3p:R, €€to,2r l .

I lo&r9rovi 6l,wr: lJl 'p > 0. 'Apo

JJc-<x"+'raxo{J"-c' r,:rooceJJ ;o'ooo*.lj orl'".'ro.TnTrRTr

""l r '"(-*) ar+"r.z"J(-|)". it+' l"zn(-l)r"-. '-r r=nrr-"-^'r.

'mo JJ c(x"*lxom1r-"-R'1.T

To rErpdy{rvo Tr'{(x,0):

-R<x<Rrc r -Rs0SR l ! t € -p!6xe! tov dd\o x"*0".R' rounEPoaxeroL on6 rov (0Klo

x'z+0'.4R2. Ero!6u!c

J Jf ""'t axaus J J;(x'*o&a,r,:

x'+{'3' -Rs(lR-Rs038

. J !<x'+'ruxorx2+{,t s 'rn:

anô6i 6xou!. r'fi-e-R'?) s I lJ e-R -R

R R

s J lJ ;rt'*+"a,1]ax s n{r-e-aR') - n(1-e-R'R

) 5

R

s f"f"-'"."0*+s n1r-6-aR"; - n11-s-R'; = J_"*' orft-"i1 .

lf,'."3 n1l-e-aR'1 . n(t-e-R") < )dxl s n(t-"aR")

164 lco&tAro II

- l i m ("rR ,

r ) . r i ' I f "* 'dxl s r i ( in-e-4R']) -' R.€ r.,R , n*. )

- " = (1"--i4'= " - ,'. L;--"* = r- |l-."*=,

Abon,, t r /2

Et.vou* B(m, n)=l xn-1( t -x )n-1ax rl a(n,n)-zl ouvh-1ery2n-leae

tlvor dxduq rt ')=[ t .-Ie-tat I r lry=2[ ,z'-r"-* 'dx lz,t) r )

'rxouue r(m)r(n)-4[' xzm-r"-x'?or['-*zn-r"-+"o6, o ! "

- 11_+ Jxh-\*'ax[ *zn-r"-,r,'69. ri, lJJx u'-fu2'-r"-t x".,r'r uxo,t,

0&,po0u€ ro xup {.0 |

T r= t ( x ,0 )€R , :0sx<

T r= ( x ,0 ) €R2 : o : x ,

T ! - ( { x ,U / ) €R ' z :03x ,t ( vo r :

lJJxr.-rur"-t.-rr'*,r'ro*o* =T2

s 4JIx h -1,1,'?n-1"- ( x'.,,') dxd1,

.. 4JIxr'-1orn-1"-(x'*o' )dxd(,T3

92. lb 6€ix0el. !e tnv lolio€Ld 6rnlo0

t(',")= rl?ll*i)

osU<

'!

lo,sE )( ( ) , o )

olo rltp{Uamc 6tL

(D>0 ,n>0 )

t1 )

(3 )

os0,

os0 '

o (o > 0)] (rftOdyrvo)

s otl (r6roplo (tKtou )

< 2ozl {rtropro l(0klou )

Io,0l GA,oj )(

Bi rd o ro ! A ' ld lo o . l i6d 3?0.

( 4 )

*Bl,are oXerlxd u€ rLs ou\jcprioeLs Jdyua ic!

oIoMHmTII@t IOI IEMOT 2. t ra!6dro! B. 165

M€ rov lerqoxnlqrLoud x=pouvo, U:pnuo axouu€r / 2

4[[xr'-1,!rn-1e-(x'+,r, ')cxa{,=+[ ouuh-terrrn-]sdsI p2(n+n)-r e-o'zdp (-L)

, J o

zB(m,n)[ p2(.*n)-r "-o 'dp

*o. ' a[ [x?n-102n-re-(x"{ ' )dxd, l , -

\/2 ila=tI ouuh-1enu2n-reasI g2(n+D)-r e-o'zdp:]

,lh2B(m,n)l p2(n+n)-r e-e'dp. EroL q (4) vpdeero!:

B{m,n)[-zp2t '+nt-r "-o '60.* f [ r : ' - t*zn-re-(r 'z++'z)dxd,! <

" T r

< g(m,n) [ 2p2(n+n)-1 e-ezdo.

llclpvout]€ to 6pLo d,rv uEAdrv rnc 6L iq ovLo6rnroc 6rov o - a- ondra 6-

xouu€ ue rnv ooioeLo f l , ,v (2) Ko[ (3).

8 {n ,n r l ' ( i n+n ) r T (n ) r (n ) s B (m,n ) r ( ' n+n } . a t r , n t= 1 * *1 .

fetuo. rr ' [ozo2('*nl-r "-o'60=[

zp2(h*n)-r e-o'?dp!Dr(m*n)

*o, t i ,n [" iozt '* ' l - t "-o'60=[

2p2(h+n)-r e-e'zdp=r(o+n).)

"*- J, )o

l l t93. lls unolovrd€a ro oloxliorlc J1 df orO, drou T Eavs! to xrpao.

nou n€pLKI€(Gttt 0116 touq xrtxr\ouc (cr)r x'z+o'z=zx, lcrlI xz.{'z=x, lcalx'z+{'z=4{ xol (cq): x'zrt'z=3{.

ftlon:

H e€ao{, ioi ro! KUKrou (c1) ypda€ror {x-1)'z4r'z=1 ((6woo (1,0} drt lvo

1). H €€LooJon rou xu*trou (c,) yed,leror (x-+)'"! '={ (t*. (- l , o)

qKrfvo +). N eEaooron rou (c3) vpdeErqr x'1+l -2) '1=4 (Kdwoo (0,2) o-

t 66 KEOAAAIO II

KtLvo 2). H €Eaooon rou (cr)^ / 2 \ z o

Yedq€rqL x'+l ' l- +,) "+. To

Xoplo T flopLordveroL ond to6rrlovd oxitro.

An6 [o sxiuo eqCv€rol 6-

r [ : X'z+0'z-zxS, X?+0'z-xi0,

x'+u'z-4u{, x'?*o'-30r0.

Kdvou!€ !o !erooxnudt!0!6:

, -o i , u=t (1)=

g x= -!91 , 0. u'zuurau2 u2+u2

E(vor: x 'z+{?-2Xs0, X'z+U'?-x tO'Apo:

Elvqr sKdlq: Xz+U?-4{10, x 'z+o'z

'Apo I

:uvendc:

"> t{sz, , ' i t ' . t = r. x ' io ' .2.

-r,t.o * t'to .+, :, *'i0' .,. x'il. sl.

35us4.'EroL

trE to u€tooxnlotlo!6 {1) to X(,roto T on€LKovl(eroL qloLgovooiudvroqro X0rplo Tr, f lou €[vqL to opgovdNLo l lu:2, 3guj4. 'Exoule xoL:

l u2 r t t " ) '

zuu1(u'?*u 'z) '1

!x9i&!l = J 'u l U , U I I a 0

l -n

ax lau l

aur Iau l

2 u 1 u(u t .u '?) '?

1u'z+u')2

Ii*1

dxd!=[f .l!.-ui: l- ,",' lo*,=['l'i ' , , ' r ' | {u?+'r 'z) ' I rr"

-1r",1 ] grnul ]=rnf r"$-r"2.:"$.

lu, nJ, ) ,

/ , , , * , , ? \+ u=x\ ni / + x.n'n-

uu d'rd'r=J u dul rdu=

E:"oL: Xu:x?r+? kcL ' r=X)-{ , , - } xu=""

x ,= l l ! !1 6;"., ,=*.ff = "-G.')

oloXlHPmIKot llolltllol 2. [ad!6d1ou B. 167

94. l(YeEot n dnitpnlort f, yro qv droad f'(x):2xf(X), f(0)=1. tlo

6ELr0€a 6rr: t= JJr1ryt191aan1t-"*"1,T

dru, T eavq! to N+Co rd, rcoldcaero! on6 rov x0do xt*{"=q,. (E.I.n.)

A[or:Advoou€ rn oLoeopLk{ e€looon f ' (X)=-2Xf(X) (X!,9r<ou6v{rv lero8trnrdjv)

f ' , " \ f . , - \ | | .E(vdL: t i j .:A * J

'r i i j dx:-2lxdx*c + .JflxT df(x)=-x,+c +

+ ln l f (x ) l - -x ,+c 0)

fro x.0 n {l} 6cve[ ]nf(o).c + ln1.c + c=0. 'EroL eCvqLi

tn l f (x ) l - -x . ' r 1 r1111=s-x" + f (x )= te-x" A 11x1-s -x ' . 'Apo ocvoL:l l t l . 2 . , 2 t t t . 2 . , 7 \

I= JJ r(x)r(u)dxdu= JJ e-'( e-v dxd0" JJ e1' a 'dxd0'TTT

MetdoxquorL(oule oe rclLkcq ouvrarov!6v€q. Stroute x-pouv€, 0.pnu0. EC-

voL ()loio, ororzn *or ffi =0. 'apot

r" I J "-"'papae=['"[i-c'06p46-f "as["pe-e' dp-[o]21['-+e-e'?d(-p'?).

i i Joro Jo Jo - -oJo '

=2"(- {) f"' i '=-"f"-o'-t1 =n1 1 -"*',.

* EavoL f (o)=1, r ipc crorre ierau r (x)=-e-x" .

K!4AAAIO IIt6a

=ii-f %rP=f i','i**-l'i***[ 3"'3"' ( * " ) " | . . * " .

-y) E4opusydC rl,,v 6mlAv olodnqNtdEw

95, lh uoloYrooet to qr9o66v tou )op[ou I rcu n€pldteletot oil6 lnv

eueeto x- * Kcr cn6 tnv nopdol'i tJ'z=4x.

Aton: '!

t'?=4x

(x,?rx )

0 , T

(x, -2li ')

x

96. rb unoloYroo€l to qr8d66v tot, xstou ] nou rtEplrl€l€ro! dllo rlc

sooELeq x=0,U-0, 5{,=3 xoL lov rtx,\o x'1+0'?=t,

Aton:

*'4"=)/-/ 3 /5 t)=3 /5

( 0 , t )t rrt=Ot,,r,)

0 (r-o x

X=0

To (nroiruevo €uBobdv 6lv6roL

A" I ldxd',T

Afl6 ro oxito oolveloL 6tL: i

o . r . f , xcu -2W<0s2W.

!/4 26'roo r-[[axau,=[ [[ a,rla* -

'i 'o "-2\E -

An6 io oxirus QoLvercL 6rL:

Os0s i xo ! 0sxsv1 - r l ' .

To (nloiu€vo €!Bo66v 6Lvero!

016 lov rturc A=Jldxd{.

,, , t t i r{t-F'Apo A-l ldxd0=l I dxdu, =

T

=["l"rviT*= ["h=o'* ilo

-o Jo

-no paro rt r, r 11 o ro r I {podn.

8-T

O OK^nmTII<ot AollEuot 2, trd!6dlou B. 169

,3/5 -- [ E]s- ] -m*1" ' - - * -*-err* r . !ov66ec.

97. tlo unoloyLoo€L ro d!Ba66v tou Xoplou mu repLxtr€letor ond lLCKqnr0tr€c: 0-0, x'+{'.1 kor t:l/z.

Aiion:

To <nrc(l!€vo eIBo66v 6{.v€roL!E ro €!Bq66v rou xuiplou oro ntdro

A xo oxiuo qotvero! 6tr"

0s xs v l_ ' r . , O<0s j_ ,'ExouuE enoUtvoc:

tL/2tEF,+.lldxd'I,=l I axa0 .- ' ;

rr/2 lT.:if tr/2 _=Jo. txlo d'l=.lo v1-'!,d'!=

r,hJ?:it I l/2=l=1:-itornu'r1" '

=;ff,l,otnut-

loon:

El.w! A=lldxd't=l | | d'! ldx.f

' , " 1 8 '

1, (e"-lnx)dx1, e"dx1, I nxdx=

.te"l'?-tx tnx-xti=-e'?-e-2 ln2+l r. uovd6ec.

on6 rov r0no n=JJaxa,l. e" uflotrovLoou-i€loptnlooto. T

l l n ^ ^ l / ! n \--F- t -rt + A-.\ --F- + -rt I r . !ovdoec

98. l5 urDlovrd€l o fl8d66v ou trpl.ou

T - t ( x , 01€R ' ! : I s x 32 , l nxs { , 3ex }

1 7 0 (EOMA]O II

99. llo unoloyldgel to e!rBo66v tou (tKlou x2+02=r2 uE rnv EoiSELd 6r-nloo olodnpA[to(.

A0on:

go unoloylooule ronp{inou tEtopt6Ku*l Lou,

A1.ldxdoT

' o n o u T = { ( x , 0 ) € R ' z I x :

;'?+r1'? 3 r'?l,

ed !ErdoxnuorLoo0t€ o€ nol/K6c duvreroytr6v€c. €cxouue x=pouv€,0-pn!s,

El,vot 0 sX'z+0'?s 12 + 0 < p'1ouvz€+p'.1p' l83 r 'z + 0 s p < f . ECvoL o-

'!

x,'/FqF)€gBo66v A rou

0 ,U: '0 ,{ r ,0 }

K6rc 0€ [0, n/2].

,a/2. F , .1 /2. ._- , r /2'm" r=J'-ffoao]ae=J "111" *1"-'i ae= $ rcl/'?- j:'Apo ro euBq66v rou K0Klou etvoL 4 #.nr' (r. uovdOec).

mrololec 0'z=r ror lErr.

A0oI:

0 r lnoq mu 6[ve! ro (nroir-

urvo Eu9o66v € Lvq L:

Al JdxdoT

And m oxiuq eolv€xol 6r!

0 j X s 1 K o L X ' < U < ! & .

, ' , , d ' ''Apo A=l I I do ldx =

T- -1-

100. ilo uroloyldel, to €!lo66v rou xrlou mu n€olxletDlo! ctd rrc

=l tul":dx=J (V:x-x'?)dx=-i- (r, uovdoEc).ro l ' ro

oloK m0TIIO[ lof lEr lol 2. Ero!6drou B. 171

10'1. lld l9e0€t Io etBo66v ro! )@l,ou ftru n€plrlsteIa! dr|6 tnv mpoBo-li 0=x, xsL rqv eooeto x+{=2.

Aoot:

And to oxiuo qrolv€rot 6rL-2sx<1 (oL X rs0<2 -x .

'Apo €fvot AlJdxd', -T

at I r2-x 1 rr t--= l l l d ' ! l dx - l t 0 t - "dx =!z " xz

' ' - z \ '

1 (2-x-x , )dx ' . . . = + r .uordoec.

102. l& unol\oy|,o0el tomdrro 00lô tnc rc! ,lnc

iton:

qr9o66v rou Xr,plou T no!p=4r30.

lr€P L tI€ LrEor olld to

EraLii ovo0ap6losr€ o€ no-tr[(tq ouw€rovltvEc ro emo66v6LveroL dt6 tov r0no:

A1Jeded8.f

To 6OLo rou p eLwl 0 KaL nu30.To 6pLo tou 6 €l,vor rq onp€td

YLo ro orolo p-o,6nro6tt 0 (or+.

rr rr l3r rn!30'Eror txoule: A.l lededo=l | |-,i n .'.

I r i r r r l 3 --+ | +B- +r nu6e | = _i; r.

A0on:To (nrc0levo

100. 15 umlovlde t. to €!9o66v lou )(orl.ou

ooo]o'.f /';41.""

!0v66.c.

T-t (x' f) € R': 4 s r ' l+l ' < 4x, 0:0)

qrFaddv 6tvEror an6 lov ttno A1Jdxd0.T

x '?-xl

0=0

de=+.,o i ! 'z38d8=

172 KEOAAAIO II

0o !€rooxnuorloootr€ oE noll|(ecouw€rdYuevrq.

ELvoL x=Pouv€, 0=Pnu0,0L EELdtoeLq rov x0K\0v x2+{r2=4 ,x'z+01=4x ypt,pvtou p=2, p=4ouv0 ov_tCsrolxo, 'Exouu€i 4.x'z+{'z + zsP

Kor X2+{,2 s 4x - p'ouv"Ot

+p'?n!'?o < 4pouvo - p2 < 4pouvo

t rn/3 r t r l3.i J.

(tsouv'e-l1ae=1,

- o s 4 ouvo. To onuelo louiq A rov K0Ktrt v o=2, p'4ouvg 6x€L ouw€_

rovlev€c (q,o)=(z, [ /3), 6rot el.voL 0 s 0 < * To EuBd66v 6tv€tdL ond.. . ' /3..qouvo . , i l r . . ,qo.,"0

rov ttno A-llpdpd€. 'Aoo A=l ll oaolae=l l+l dp.t t )o tJ2

(8ouv'z8-2)do=.'.= $ +rG r. uovdoec.

104. ilo utElovLooeL to 48o66v lou xstou T m0 mglrloleftL dtd rl(x4r0lec 0'z.2ox, U'z=2gx, )([-v, nl"6 (0 <o <8, 0<y<6)

(Prcblems jn l ' lathenatical Analysls B. Denldovich).l0on:

To xr,Dlo T EtvoL our6 nou6€txveL ro oxipd.

l(dvouu€ rov !e$oxnlotlou6

xu=u, fr=', (1)An6 rLc (1) npo((,nrel :

/tx=V:ft., t,.,lffi

l i l€ rov lerooxnuor[o!6 (1)

ro xdplo T uetooxnlorl,<€rqr oxo XoDto * T r= { (u ,u ) €R ' ! : y sus6 , o< t i sB l

* r lwu - f 3 rs j - t<q , rs6 - rsus6

2 a x 3 \ , 2 < 2 B x . c 3 j i S B . q o )I

t t t t I r , r 8 , 1 1 1 6 r ,'Aoo E1.Jdxd01'l + dudu-l+ J + *'|L.J duj=+iln,r:.r',r;=T T r q l

=+(6-y ) ( lnB-1no)=+ (6 -y ) ln + r . lovd6€q.

105, ih uroloyLo€sl !o 6uBo66v rou x{lou nou rcprrleleto! ord lo 6-dfteglx6 rff (ap6Lo€!6otc g.o(l+ruo) ao! ols ro eAdreplrd lou .6rlou p.o.

lto4:

to (nroou€vo eugo6dv6[varoL cn6 rov rino:

t lA" I lodPdo

T€o 0po0!€ ro 6plo rrrv p(oL 0 . ELva [ 03€3 [ .

tLvoL o(duo o < p < q(l+n!e)

'EroL 6xouuEi

rr .1r .a( t+nuo) 1 r r r - ,1a( t+. ]un)A=l ledpde=l l l odpldo=l I+ l d0.+ l td , (1+nre) , -o , lde=,d , r l r r oL . ra

-2 l \= + j , (2au€.r! '0)de=...= + (8+n) r. !ovd6ec.

t06. lb moloyLooEa ro €t8s66v rou [Daou rFo n€rrrle{.srqr srd 600 e-d,,rEol|.d €Tmdl€vouc |(0rlouc !e oxilveq Rr, R: (Rr < Rz].

Aaron:

eo xpnoLlomliooule fiotrLxAq ouvreroyltvEq. A6yn oullerploc 0o Spotuero e!9466v rou lloot Xoptou.

A116 ro nopqKdrlli oxitq txoltEr pr=zRrouvo, p2=2R2ouv0.

174 KEOMAIO II

'Apc 2Rrouvo < O s 2Rro0v0

E lvoLoKduo0<s<+ .0

r0roc A=llpdpde 6Lver:T

i rn l2.P 2=2R2ou!0

+=l I Pdodo =' P r =2Rrou!$

. t r / 2 - - . 2 R , o u v 0=l l-9: I do=

Jo t z l2Rro ! !g

-f"1tltu:oiltu:e ]ae=f/'zrnl-nlro*.e*-r,ra*1,[ *. # 1,"=- . . . -+ (R?-Ri ) . 'Aoo A=n(Ri -R i ) r . uovd6€c.

to etBo66v rou xr,Jp(ou tlou oot(€rol dr6 qv moo-

x".l'=36 oro np&ro r*orrtr6oro.107. l|o 0norrovloo€ [

Seli 0'1=5r m! toe r0rlo

Iton:

An6 to oxilq qolvEror,

0<932V5 xoL

. x < vTG0r

6r!

rt:p,J\t tl\6=iz 1

E{.voL A=l I I dx ldr!=

.f*,,,{{o*=

=l*(*=r'-+),,=,2,18 ,ztl\ .z

=J /t-rl,'d!- J" * o*= (xdvouu€ roiron lou

Jr'-,.-a"= g va=', 4 -oqnu * ){ * -'+- * .", * - * ]l*==. . . -18 ,119 r . uovd6ec .

O oXAHmTITO' ̂ OII'MOX 2. trsv6dyou B.

108. ih BoEee[ tl) e!8d66v EU xqlou rou opae€tot qd xLS nupdot€Sx'?=stl' 0?=5x ro! ord mv r0rr\o x,+,=35-

Aoon:

ELvqL A(2y'5, 4), B(4,26) ([poK0mouv 0116 xLq tr(roeLc xov olornldflrvrlllv €Elorioe(ltv rolv t(olr fulv),

EneL6i Eu8€l.ec nqpdllnlec npoq rolq dEov€C 6€v rtuvouv 6l€c rLq |(o!_n0trEc, Xrdpl(ouu€ ro Xopl,o oE 6io XoOLo eapvovrdc xnv BTl/00..

=;{-" + l";', ,2',/E-r l x"ax= *. t . r .

',r"*. o= ["f frlulor-['*f [*-fr]or=f",*rT,or-['*rrr*-,er-n ' i r / s t \ , ' 1 2 1 5 ' ' a x . / 5 ) \ \ . / 5

(v56ix:- + )ax'v5 J-r'ax-fj x'dx*J r'36trax-

109. ih 8pe0€1. ro €|lgo66v rou d\ru Uloot rnc Kopdro€rdotq p=2o(l-ouv€)

I0ort:

0o €pyo(r[o0I€ !e nollK6q ouvreroy!6veq. To 6pLo rou 6 elvoL 0 j(qL n.To 6pLq rou p €tvoL 0 KoL 2o(l-ouv8).

To (nro0levo €r6q66v 6lv€xoL on6 ,ou ,gno e=JJoaoae.T

, ,76 (xorlllAlo Ii

. { .?d( 1-ouvs) .n. , ,20( l -ouvo )'*" o=J,'l, pdeds=j, l+1, do=zq,jo (l-ouvo),do=

=zozf 1r-zouve+ouv,o)do-2s"fe-2nt)s+ + - $-9 ],+"{*o.i-o)=t".!. uovd6€c.

ll0. lb Bo€oet to ql8o66v roo nptoup=@u€ xdl dr6 rttv ropbr"o8Ldli g.d(l+ouv€)

(Probl€ms in l ' lathematlcal Analysls B,

|EU ogl<clca od rov rtdo

Denidovlch).0iaro|t:

A6y(,J rnq oulu€To{,oq rou X{r)-plou Il,C npoq tuv 6lovd X'X go

epohre ro €uBo66v rou xoplou nougplo(eftL ndvo qll6 rcv 6€ovo X'X.To xopl,o our6 ro XnpL(ou!€ orq

XorOLo Tr,Tr.

r l v o r f r : t { o t u i / u 3 o 3 i ,

o o u v o s p < o ( l + o u v 8 ) 1

r1 l2 r ra ( l+ou! ' ! ) 1 r1 l2 r ^ ,14( r+ou!0)'Apo Ar = l l l eaplae- l l - t : l d8.l , 0 . do ,us l / ! L - l ddu !$

. , r / 2 , l / z=-lJ, I rot r-o*er r'-(oouve)'] do= t L tr*zouua*ou""e-ouv'e)d0.

-r/2

" # J- ' - t ' , roruttor- $ rot l / t .o.1nus1""= S *o,.

l2

{ /

0I0K HmTIrc[ Aorltuot 2. trov6d.ro! B. 177

ELvoL ox6uo T ,= t (e ,e ) /+<Osn, 0sp<q( t+suvo) )

r d( l+oorn) r o( 1+ou\a5)'rpo r,^-f [[ orl.t=[ f4] ao= -9' [ (r*ouve).ae =- . ) r / 2 1 ' r t t r t 2 L . r o . t r / 2

= f' I rr*zorue*ouv,€)do. + r€)1,"+o,tnuet"rr+ $ tnueouve.e]'r, =' t a / 2

-...= + -"'. Etwr \,+\,= -F *.- li1 *,- liC .

Eno!6wrq A=2 +{ = +C r. !ovd6€s.

lll. trd unolovloosl ro e!0o66v rou K,paou mu neolxl€tsto! ord Io d-[o tnc ru(losl6o0q x=o(0-t!0), U.o(l-ouvg) mu Eo|'oreto! oro nparEo retoo -rfl6pio, kor daa tov 6tovo x x, (o > 0t.

Aoon:

To (nro0!€vo ElBoddv 6LV€ELon6 Tov r0no:

A=l ldxd{,T

ELvqL x-o(g-4!e) - dx=o(1-ouv0)de

u € 0 < 8 < 2 n , T o 6 p L o r o u 0 E L -vor. 0 xoL d( 1-ouve).

2 i o( l -ouvo) ,2r'rroL txouue A=[ [ o(1-ouvg)dude=[ c{r-ouvetf+1]"-""i1=

r2a .2\= | o2( 1 -ouv0) '1d8.o, | ( l -2ouv€*ouv20)d€=.. .=d2(zn+n)=3o2n t . lovdb€q,

ll2. lh uDloyla'o€|, ro ql0d66v rou x'ptou |Eu ragt rlel.etoL ord rLC|o||||tlec o't o={$30, l') o=d 2e, (o > 0).

n0ort:

q') A6vrr o0!uerp{,os eq unoloylooute ro etBo66v lou Ev6q g(,lôu.

E [vo ! 0Ses + xo r 0sp<4n !30 .0c eecpu6oou lE mv r0no :

EITi?-c'tdrro o.li6o 3e4 'Aoxnon 2.

1 2

178 XEOAÂIO II

r /3 4n!30

A=l lededo. Etvdl + =l | | eap lae=f

' o " o

=* [ " " ,o . ,unutu oe=I [ " / t ronr , reae=

4/3 i /3=8,Jo nu'z(30)do=4j, zn!'z(30)do =

-o[n"t ,-ouu{ur)to.=olr- nulser l " /3 = +1.J o t o J o

B') A6yo r4c oulu€rploq 0q unolovl.oorru.

t Lvo l os 65 * , o .o .on r t t .

eo eqopUdooule tov r0no

r{J'a'ae.T

. rr/2 Pr'v2o'rpo * =l I edpdo -

. r / t= r I

- to , fn r2eo_ | J -o ,nu?t2€ tds . . . ._

t ) o

r , !ov66€q.

'Apo A.4n r. !ovd63q

ro €pgo66v rou ev6q eotrloo.

nd2 - tto2lr - -?-

ll3. ib umloyaoo€[ ro q8o66v lo0 X|ol,ou9!x6 qC rqlr0tnq (X,+0,),-8(IZ-{,)-O' xdr atdx'z+(t-z) 1.,1, (x > 0).

^0qn:

rcu opt(ema @6 ro €t{,rE€-m eafiaplx6 rDu xorr\ou

(A .A .K la f , Ca lcu lus )

es u€Tooxnlct[oo!!E o€ noJ\L(6C ouvtetovltvaq.etloiJlrg X.gouvo, 0=pnI0. H €{Lqlon rou lnlvloKou ypde€roL

Ph-8P'z(ouv'zo_nlz8) '0 - O'z=8ouv20.

H e€[qron roo Ko|(trou ypdq€roL x'z+0'?-4r! - p'=4pn!0 - o=4n!e. To<nroiru€vo qr9o66v etwl too !e ro e!9o66v iou xoolou 0ABT0 {doro Tl),u€Lov ro EuFq66v xou Xr,paou 0ABA0 (6or0i T2).

* In! ! fdros rou U! .pvor l t rL

OÔX EloTllot lotlt OI 2. t8\,6dlou B. 179

t t t l t l' E ro ! €LvoL A= l l pdpdo . l Jpdpd8- l l pdOdo (1 ) . go opo tu€ ro 6OLo rou e . t [ -T T r T 2

vor p2=8ouv2e, p.4n!0 6110 t6nure.8ouv2e - 2nr'?o=ouv2g - 2\u2e4-2m29 -

rue= 1+ 'Ero! 8r=+ E[wL xoL gr=+, deo + s e s +. Td 6prd

rou p vro ro xopl,o Tr €[vo[ 0 < p 3 4ntre. Td 6oLo rou o YLo ro xrlpl,o T2

e lvsLosp<v6 ; ; i26 .

,. ' \ "l,''

' A 9 o n ( l ) 6 L v € L :

{ l2fnvh , I lzrr 'M r rr l?r ^z tLrnlo , I lzr ^rr@ ,A.l I ouoae- 1; I oapael.l l:rl do-ll l-+l d€l=tr /6 ' r t ' r t6ro | ' r t6L - Jo 'Jr l51 - ro

t r /2 , t r l2 , r /2 , , , /2. l 8nu'€d0-lJ 4ouv28d8l.4l (l-ouvze)d€-2ll ouvzod(20)1"' n / 6 ' r r / 6 ' ' \ / 6 ' t a / 6

=qf'I ,' rc-zln/ ,' *uzearzerzlf'/.2 ouvzaatze\*( + - + ) -unuzeti/,!-' a t 6 ' n / 6 ' ' r / 6

-ztttv;oti;t= * "*.1- $l . $.v::-'r= $ r. !ovd6€c.

ll4. I5 urElovadoel E q&66v @u ltlou |DU neolrlalctou or{ rnv rqr-ir|| (x'+'}'z)'z{'?(x'l{'1).

180 KEOAAAIO II

lton:

H {otn0trn ELw[ o Anlvlo(oc rou gernou]l i .

:s rc]\L(tq ouvr€royuev€c (earoou€ x:9ouvo, U:pryo) n Eltftlon tnq rou-rlqq ypdgeror p!.o2p'z(ouv'?O-nu29) 6 p'z-o2ouvzO.

eo unoloyl.ooule ro €!9c66v noun€pLxlELeL ro npd,ro qtllo (l6vo tnqouuu€rploq (lr npoq rov dlovo U'0 ),

t lvqt A=JJpdpd€. ELvoL yvrror6 (o'T

r6uoc o€l[6c 396) 6rL -+ < 6 <I

r<or. 0 s p s o6it26.'Eto! 6xoure:

rrla rd!6J!-ZE 1 . tr/\ . ta/4A"l l l ocolae=-1 ro, t:vouv'zod8= + | a,ouv2sdo"

. l l t r ' t o J - . r , / q " - r - r l \

- 11" o*"0"= -f;1 rnuze:l{)"" $ . 'ooo o.^=^=, # -o'r.soudoec'

ll5. ih rolovloO€l, ro €u8o6dv tnc ar"tO'* $ * $.r.lton:

eo KdvouuE €rrop[oylr rou rtnor.,, l.JJaxa{,,6nou T elvoL ro xnplo nou ne-

prxtraleroL ond rnv altr€L(rn. es Opo0le 1o 6OLd rov X KqL 0.0q otrod\npriroourr€ rpdrrq oq npoq x. elvor $: r- $ t xr."'(r- $)

["]: *,0=f,

f, ,,r-+',u=lBl6

= rfif 'ar-e-q,ar= + tqry., $ '"rn, g ]:= + (+ *€,r,,)==2oP(*) =noe

".uouoo"..

OIOKIUPnTIKoT ,{ofltl4ot 2. Ira!6dyou B. 181

116. No unoôvLogEL ro euBd66v Iou X(,jpLou T nou [€pLKl€L€rqr on6 rnvypqll i arx+gr'!+yr +lo,x+g,' !ry, l=u (t) u > 0 (6[v€rcL 6rL or8:-ozBrl0).

(R.Trandafir llatsnatici Superioafe)Aton:

lo <nro0uFvo FlBobbv 6tv€ror qrd rov r ino r=JJOxaU,. rtuoru. , .u o"r,-T

Kordo .oon o rX+9r0+v r=u , s rx +Bru+y ,=u , E !o ! n (1 ) 6 t veL u l+ lu l=u (2 )

t l

1l--ll1 = q'jr'?-:Fro'z

'Aeo liffi | = 1;;tr;_6*,. Enoudvoc

l t t lt = l l dxd0= l l dudu" '

r i i r l s ro , -B ,o , I l o r0 , -B ro ,

An6 rqv (2) npoK0nrEL 6iL ro xr!p{.o Tr(oro onolo u€ rov !€rooxnuqxLo!6 nou Kdvo-u€ ol leLKov((erqL quqLuovooilovio io X0JploT) €LvoL tErpdy0rvo !€ e{Lodlo€Lq n,\Eupiiv

u+U-u, -u-U:tt, -u+U=U, u-U=u. H ntr€updTOU r€rpqvuvou €tvol y'2-2!. Aoo To Eugo6dvrou rerpoyuvoL €Lvq! j ldudu=(v2u)?. Eror n

I T r(3) 6Lv€L: E= ft; ! ; ! , o;I 2u'z.

ll7. I{a motroyloo€L ro €!Bo66v rnc enLqdv€[oq rou opl<€rol on6 rLqox6oeuq z=chvilitT, 0: x,+il, < oz (d > O).

(fl.lleisbecker: .The calculus probleo solver.)Aoon:

To <4roir !€vo EuBq66v Elversl qn6 r, - f f -rv runo r=jJvr+2.+:; dxdrir , onou

T = { ( X , U J € R ' z : 0 < x ' + $ ' z s q ' z } . r

rtvou z. = (sh{,irr") -L , =1.6y6;:1 2{zvxt.'F ' ,rFra,

txouu€:

l l dudu (3 )

T r

o , ! )

142 KEOAÂIO II

'Aeo f-l-tl;Vt-(=r',r'aO' ffi -

rnoutvr,re e=JJ il*"{';[' dxdo (1).M€rosxnlor[(oule oe no^[xaC ouv!E-T

royu€veq .o t rouu€ x=pouvo , {Fpnue . E tvo r 0<pso , 03€<2 l l xo r .

nG;E' -. ^""

,21 ,2n t2a- - ' rsho-cho+1 ) I d6-2n(osho-cho+1)=Jo t9snp-cnp J!ou-Jo losno-cno+cnulou=(r l0

:nuea@n:H nponyot!.vn ddxnon dnd nopoopoti 0pa0n*€ orn gdon quri. 0 ovoyvrir-

ornc vo eEopioe! 6!r ovixeL qlnv o!d6o rov doxidEl,Jv 136 doc 150 (s€Ia6q

205 6|,)c o.trC6o 217).

t18. h uDtoyrdcl ro qdo!6v roo x!p[orr:I

I - l {x ,U) n ' : , r*2, osu { l .I x ' j

Addn:To (nro0!€vo euBo66v ELv€toL on6 xov ttlo E=JJaxau,.

'r.o, exouu.,T

= l ' dx= lx 'dx= l -' r x -

=-+,1-i r. uov66eq-

'=1, llf *]*=1' r'rY"=1, ( ; ";*=r,

l t l* t i l d ! | o ( c h p ) d p = l p ( s h p ) ' d p : p ( s h p ) - l p ' ( s h p ) c l p = p ( s h o

t J )-p(shp)-cbp.

r \ c _ . { d u L r L s u r € o B o r L r i < o u v c p r i o e u s r r s r c p o r u r i o e - s

odrouods roos, roporalnor l€ otor npr i Io r ,5 l lo ,orus oer i6es

t 1 l 'L x I ,

,-l

1+sh rfzl0z,

0llo( llmT]KOX IOIIDiO! 2. ,rdv6d1ou B.

l^], ,1":::": . : : f i) 6ev.ouvovr6 ro en(n€Do 0x0. H Enlqdvelo (2) €t_

;:j"::^j::i'::.,:-::lyd (ormi^. rov _,r_r" x,_,1,.*,'i"l uo;;;;;,';" ":_

nlrl€do 0x0) Ko! y€v6r€[p€q rop6lltJr€qnPoq mv dlovc z'2. A6y(| ouuerpLocToiJ o[€oaot 0o unoioyacroupa tov 6vrctou step€oo |Iou Bp{,o(eto[ oro np6!o0y6oo rrrv oE6wv. Et vorT-{(x,0) €R'?: 0 s x s o, 0 s 0 sv6i_x-4M€Tqoxntq!L(orrtlE o€ noiLx6q ouvratoy_tev€c. 06xou!e xlpouvo, 0rpi!0,ELvoL osos

| * o . o5 ; . * , 1 , ao .

-(xt* ,1, t ) .de-p, (oL

>

T

,t}:F

tx ,0

l , " i , i r '= a ' i 0 < p s o . E ivo l ox6uq f (x ,0 ) .oe

fiffi =o ' o.'r"

+ ={ r(x,0 )dxdo=J'l'?lf op" -" ao]ae! ",

/ " [_ i f "- ir,. r Jae=-t J"" rco"qor--t l"tt"-""-,tor=-1. (e-o,_r) rerl/z= $ rr_"*.1.'Apo

v=n(1-e-o')q K. !ovd6€q.

120. Io x[616ykAr1 o 6yroc tou crpdotosr66c z-4-x2+, (tl -. .o "nln"oo*ff.

*u [c,!xl€t8ut! oti to nr-

183

tl9. lb urolovrd€a o dyxoc rou ornra6y64q 2-g"-(x'iqi;-- i,iil **ltt-o

mu nr'lxl€lerdl ord tLG €-

i(to.l; (21 (o > 0t io! ors ro e|rhreoo Fo'

l0ol,

Etvot f (x,U)=4-xr-g,. H rpogo,\ irou ropogoloElbolrc oEo €rll,ne6o XoVtl.voL o K0doq Xr+02=2, (npox.nrer. o_n o r n v ( l ) y L o z = 0 ) . 0 ( t x i o s o p l , < E Lro xopLo T,

0 <qrotpevoq 6VKoc 6lv€roL on6rov rl,no V-JJ f (x,0 )dxd,l.

T

!x,/i:Vr)

-2

,/"x,-Ji:xzl

t 84 KEOMAIO II

,, i ,,*-:xr'npo v=l l{q-x"-,! ')axa{=l | |- f ' : ' :v* i '(4-xz-0,)d{,ldx.l

J 2 ,4Ir

z

-x')U-

0 ,0 ,4 )-vr:i.

d l

-zl, Io-x'rnn-x- tql ' t ' '

Jax=

'ETo[ 6xouu€:

(4-x'z lr l ' : dx=(Sarouri€ x=24u8)

= * | ' j*"'*t.ro^6n€ ox€rLXd mt-

ro rdro J ouvuxdx )=

= $ ||*" 'e"e.$*"e'e-f e]" ' ,

l2l. llo urclovloo€L o 6y(oc ror.,vttelg€c rou xull.v6oou g=orr€, ndlu

to! xdu', on6 ro erlrEoo )Ol.

I tdn:

H E|IL0dv€rq S opl(rroL ord rnvouwerqvUtveq tz-2, X.pouvo, 0=04Ue)YpdqetoL z.f (x,{ j )=16-p.

H npoBotr i rnc xulrvbpIxiq En[-gdveuog p=6gv6 qTo e|rLn€6o 0x0 elvoro K(,xl\oc g=o0v8 nou opl,a€L To XtJpLoT. 8d 8po(lue ro 6pLo tr ,Jv p Kor o.El-v q L 0 < p s o u v g .

(ovovL(& 0o Lnp€fle w fldpouu€0 S I < n. EneL6lt dloc ro ouve otodLdo[nuo oud nol,pv€r, xoL opvnr[l(ter L l e e t u 0 n o o o u u E - i s o s t

Yn€vouul<ou[E 6rL o[ov r0no

=8n KUgL(6c lovd6€c.

orepeoo nou It€oLKI€{,€rot ord ttc ye-ar6 rnv rl,rvl(li eraodvera z=|5-vlt;Dt

€[touon z-16-q]{-t nou oE xlr,\Lv6pLK6q

dP , (e )V= l I p f (p ,0 )cpde ro p r (0 ) , o , (0 ) € l voL !n

i ' p r ( s )t t ta /2 toorg i /2 ,o ' rX

v .JJf(x, ' ! )dxd{-J! / , I , 'u-o,oooor=i_,r rJ,

( tso-o ' tcoae=

oIOKIHmTII<O' lo l l t 0 t 2. Erav6dyou B. 1 8 5

122. llo woloyrooe[ o 6vrcq tou or€peot ro0 rcplrtr€tem! ord rLc €na-06veleg yr4yz=t ( l), 2.4 (2).

Aton:

H ElLedveLo Xr+0r-z e{,voL tdOoBotroEtd6q, n EnLqtdv€ro z=4 E[vc[ ellne6o[op6,\lnlo npoq Io €ntne6o 0x{,. 0o Bpo0u€ rnv opoi llpololi rnc rouitc ftiv6!0 enL@v€tdlv oro En{,fle60 oxu.

t f l2 f ^ ! touln t r /2 / ^ , . , ,16- l lep ' - -+ l d0= l {8ouv ,s - Yq--n/2 - - \12

, n l 2 ,

| (++louvze- ouvu + ouvgn!'u

)ds=

I - , , o - , , ! o 1 1 / 2=l4e+2\v2e- ri: * "t" | = 4n- -#' ) - \ 1 2

K. !ov6dec.

fLo io oKon6 o!i6 kdvo0ue drotroLei lounpoxoft€L n e[Lo0jon x,+0,=4 'rou roplordv€Lrc L on6 @v ttno :

v- | l l fr (x, ' !) -f , ( x.0) ldxd0 6nou

Tf r ( x , ! )=z -4 , f , (X ,0 )=2"x ,+0 , . 'APo

v= l l l 4 - ( x , . r l , ) l dxd ' ! ( t ) .

TMErooxntorl<outrE oe notrL(6q

06rouu€ X=pouvo. 0=pn!0.Et-va !0<x ,+0 ,34 -0Sps2 .

z o n 6 x [ C ( 1 ) K d L ( 2 ) , o n d r .

Kir( io.0 <nro0!€voc 6yxoc 6lve-

n ( 1 ) 6 t v E L :E L v q L q x d u d 0 < e

v= | l(a-e, )edode= |;+'

'o

=i'(ryY-+

. Zn, ffi =p KoL 4-(x'z+0'z)-4-p'?.

r+p+,rap fie+"I'rp -0"tao=nlY -

)=znte-+1-an K. !ovd6€e.

'EroL

o r l

1 8 6 GOAÂIO II

123. l{a woloyrogel o 6yrce rou sllel0oel6o0e -t - $. $ =l |l

( o , 9 , Y > 0 ) .

A0on:

0o kdvouu€ €' toployi r0u r tnou V=l l f (x, . ! )dXd' l .

Tp€00 Ba BpoitE rov 6yKo oro npdjTo dy6oo fldv aE6v0rvouvo,\LKori dyKou).

l-------:----;--'

N (r) 6lver '=tvf /t- [4 - -.1!")r

En€L6i z Z 0 €LvqL z-f()( ,0)=

p€00 tE To €nlnebo z=0 elvor 4 6l-ÊrLln - tz + -6? =r tnpoKunr€L dno

rnv ( 1) yro z=0). H 61tr€LLl in I I€pr.k. \EL-

€t ro xorpLo T. 0q Bpo0le io dp[o ! ,Jv

An6 xo b€irrEpo oxftuo oolv€rqlo r t 0 s r s d , 0 s 0 < + v o ? - x ,

o t l o u ! € ( = : v c 2 - x 2 .

," +=f [{,'f l{ t+l' utz:oz a,r,]ax( 2 )

X

Lil

( 0 , 8 ) l x , ; " q ' - x ' )

A6yo ouulerpi,qq rou o[€-

{rcu el ,vql ro 1/8 Tou

A-8.

(o ,o )

0o unoioylooutE to otroKr\ipqlq

l -

I vK,- ' !rdL!. 0610uu€ Knut=0 + d0=xouvtdtpe te l__1 ,_ l l l vk t : f , dUr =

=J**;*r.t"*tat=J*'""".tdt=*'J 1+ou't2t dr # l"|*"]=

' q 2 9 2

'o* *=f, f l#[,".,*.*l+|:,=+ I # +,,=

oÔKIHPmIIOI loltXllot 2. xDv6d'rou B. 18,J

=+ f S *-+ f,fli ,"'*',*= ii+ f"'*- Hg f,'*==tP["'"+], =i$ t"' +] =+ H=+ noey' Emucvoq

V" + nogv (. Uovd6eq.

l2l. lb umlovroo€t o dyioc tou otepco0 mu morlle[eto! ot$ ro rDo-

Boloe!66C z=x'z+{2, ro rpln.lro 0AB tou etrlr!6ou )Of,E €ELdoGrc nl€u0&

x.0, 'lFix' tsx+t ror o|s to tplo et[|te6o |bU nep\oov on6 rLc |rlcrDtc tourpryd,vou rol €tvo! r@eu mo( rc EtttrE6o )Ol.

ltoli

0 r6[oc rnc oio(Itorronq T optaetqt ujc elic:

( ( x , ' ! ) €R ' l : 0 <xsz * y aq r3x+ l )

H erlodve[d 5 qc onoldc n roo9olli oro erLreoo )Ol etwL o rdrcc T

6xEL e[Lo0,o[ z.x'z+02. 'EroL o <nrol,uEvoq 6yrcq e[wL

u=[[rr'-u'roro+=[ [ ['.tt,'*,r'to,rlor=['Ir'+- 9l**'or=' i " " r * t ,

' ' 3 ' /2

=[ (- -llr'.zx'.r--1-)ar=..'= -if '.. lovdoeq.

l1tiiiiiiJiilllr

188

. l ( . -0 Cnrodu€voq 6vKoq E(vo!: v- J' | t /x,. ' t , -(x,*{rr)]dxdu. M€roox4uorL(ou!€1

oe noiLK,C ouwErovleveC. oarouttE X=pouv8, 0=pn!8. ELvoLl

osp51, or€r2n, Blf;g+=p

'npo, v= [ '? " I tp typ- r

-p , tap lce= l?"a6 [ r1oz-or16o=1s1z" f o ' - a l t -, O L O . l J O J O

' - o L J t l o

/ 1 i \ , - -=ft(t--r,)= ii =+ ( lovdteq'

l25.io u|ofoyrdea o dyro( Y rou dr€pcarl |rq, ncplxl€aelol dw a[6 m m-

pdotoerddc ,=+(t, f'), rdE drd ro enanedo o)d xar €|rplxit olrd! \ o ' z 8 ' z l(8, phi t ippoy: .probl&es d'

qv rulrvbplli srl|dv€rq L + !: =1.- ---'--- c. B. Aratyse lhtidGtique')

,\don:

H npoBol i T tou or€p€ou oro enan€bo oXU €(voL o el I€LnrLX6c 6(oKoC

fl * f . r . 0 (nro0levoq 6yKoc 6lv€roL ond lov rdno:

v- [[.axa+=-f I[ t"' . *' )*o*' i " i t s z gz1

KdvoutrE ro trErooxntqtLotd X=opouvo,0=qpn!0. €avdL 0spr1, ()seszrl KqL

ffi *oo. rr,

v- )7ll ro, wu,a-o,ny,B)oBpdodo= + ["f]ogp,oo66= $ ['"oeIto,ao-' i i ' Joro ' )a )o

125.h uroloyld0e.' o 6yaoq u, oE€r€od rm, rcglll€aeBr drd ro e( |r€9! -

orgq{C mpdolo€rbec z=x'+{'z (l) xol rov rgvo z'=,2|',, lzl.

,trton:

An6 t lq {1) rcL (2} npoKdnreL 22=z +€ z=oj z:1. [ roL ro oEep€6 gpao(€TcL ge-

tqtJ rov enLr€6ov z-0 Kqr z=1. H ioui rounqpdBoloeLooiq ue ro EnLfledo z=1 €LvoL oxdrloc xz{rz=l. H npoBoli tou oteOdoU oro €-naf l€do 0X{, eavo[ o KUXALK6C 6aoKoc X2r02J1.

0l0Kl \HPOTI(o[ AOII IMoI 2. Lrov6dyou B. 189

= :i- K. Uovd6€c,

127. tlr ryoo6loplorel ro dEcp€6 mu o dyroq !o! 6[vclcr ard to otro -

=+ t'r;"1+l'

i .\,S ,rffeqta: J J{r(2-I-f)dxdil, mL oEn ouvdx.Lo vq utDloyr.og€l !o olortligqrq.

Aton:

0 o6nyd( rn( Klr\LvdplKis €nr06veloq rou opl<er ro orep€d olvor ro)opLo nou tl€plrêl,eldl on6 rrc |(altro]€C (mpq8oldc) (x='i' ,X=i/U+0E{,0.X' )H en!06vero S (rcu el,vcL ro Enl,ns6o n) 6xE! €ELodon 2-x{=0. 'Erol roorepa6 dlvetor dno to efidue\o ollita. (To ol!6to A €lwr ro[vd onualo louXt,plou T mu opl,(Ero! drd rLq ndpoFolec (crrou€nlnadou t+0-2{) |(o|' 6[vdrA { r , 1 ,0 ) } .

rtr rW .r rv0''rxoru" v=f [J*, (2-x-{,)dxld'!-J [2x. + -ttx]+dt=

l ' - l

1 (2w- +0-nw-a)1+t'-0" )dl-.... i0- x. lot66eq.

128. lb urDlovloo€|, o 6vEq @o oEeoEoo DU 6x€L 860I oro filrcao oxlto x'pl,o T.t(r,{) €nr, x z 0, { > 0' xt+Ut 3 l} ldr netarootdr drir om€trl,re6o z'40, r>0.

Jl'-r-fr,

190 KEOAAAIO II

. - r l

4

l6on:

o C4ro0pevoq 6vxoq dtv€ral ond rov !(,no v=JJttx,OtalaO. rtuo|'T

i=f(x,0)--40. Apq v1Jzdxdo1J40dxdtT 1

€o opoijue ro 6pLo rov x,qr. Elvor 0<Us 1 xou 0 s x s y'T-I".

i,,,r-v'EroL 6xouu€ v-l l l '- '

c,r,ax la'r,=l 4q x lf:E'a!= [ 4qf-i!,du=, r l r o J J o ' J o

1r- ,y ,1!r 611-,1.1--2 [ (1ay;) '4 ] ' -

{ - t )= } x , r : ov66ec .

!-\,r'2ox-x' I

IIII

tx,o0 ( d , o ) 2ox

129. l\b EOeo€L o 6yioc rou orso€o0 mu mp!x.l\€|,€ruL 0It6 rnv an!q{-v8to Xzrt2-2ol KcL to ertreIo 2.oX, z=BX (d > 8).

AtoI:

H €nrqdv€ro X'1+0'?=zox €[voL (ulLv6pL(lt tia o6nvd (olnoln rov K0Klox'z+Ur-zqx nou op{,oKEro! oio €nl,re6o OxU Kqr yEvtleLpEc nopdtrtntraq[9os rov dtovo z'2. EtvaL x?+U,=zox i x,+0,-2ax+ar=ol i (x-o)r+0r=o2.

A6yt, outru€rplcq rou ctep€o(l!K npoq ro €nl.r€60 0zx,0o Bpoiuerov 6vxo rou m€p€00 rDU BpLo(€-TqL olo npdJro 6v6oo rov dl6vuv.t l . vo t 0s0sQE;TZ xo r05 Xs 20 , f ( x , { r ) "dx , 9 (x ,0 )==Bx. 0 <nro0u€voq 6yxoq 6{.v€rolorc rcv rinoi

?- 1 J I f (x,0)-e(x'0) ldxd'!"

T

W,lllJ

I,,"Il*", *,*]0,-f"t*errrurfF'"ox-(o-B)l'?ox,zo'x--x.dx -

oÔKIHPnTIKOI IOII'MO' 2. Xrav6dyou B. 1 9 1

! o ' t q -g l * . r pq v=q3 (o-9)n <. !ov66ee.

130. lb uroloyldel, o 6yxoc rou or€ocot rao Bplorelor oro rgdno 6y-600 urv c[6$v xoL mu lt€prd€l€ml drd rnv €nl96v€Lo 22fl!2.1 rcl to e-rtttedo x.0, {,=x.

Alro0:

H €nlqtvero z'z+Ur.1 €Lwr xuA[v-6pL(i u€ o6nv6 l(oyn0ln rov x(/(,\o z'zrt2=lnou BpLoKeiqL oto €nlneoo ozu xoL ye-ltr€[geq flopdllnl€C fipoc rov d{ovo x'x.

H npoooli rng enlqdv€mq to! or€-qeo0 oro eIICrIeoo z=0 ElvcL ro rpLyovo0AB nou n€pLKIELE! ro xojpaoi1 -111 , , 1 ; en r r 0<Us 1 ,0<xsU )

* tLd ro! , )roloyLold tou orox)Lnpdloro< ,=f"*C.**r, odtoule x=d+dn!oordr. <rx:dou\r ido !€ .-+ (d- ' r=o),

' "=f (did ' x:zd). 'EtoL E!w!

1= [n/' t o*nu. I v6t--J,ri'.roou* la"=", f "/2 r r *n," lou","a.=

t \ 1 2 . ^ . 1 / 2 . -

^'[ [''' *"' **1"' "*."*-.1-, | ] *,", +"- ""X'" l"'. z l / 2 . a 1 2 ' - | | 2

192 KEOAAAIO II

'Exou!€ z'z+rr2=1 - f(x,0)=VT:ot, 'Apq

v= | I f (x,0)dxd0= | ( I -1-0,dx ]d|l-i 1 " o

=l ,Il7rx ttdo= | {ir'j:[td{=

=-l I rr-,r,r*orr_r']"-+ tLr*]_' I '

=-* [ r t - , r , '1 'a] '=-- f r - ' r=- ! ( . lovdoeq.

l3l. ib umlovloo€a o 6vxoc rou oregeoo [D0 rcplde{'gtdl drd tov rt-lrv6@ x1+rr'=4 (l) xoL rc un.p0olo€L6as x'!+tlr'?-z'z.| l2).

A0o[:

H npogotri iou ufi€pBoJ\oeL6otq orc Entne6o x0{ €tvqL o rixltoq x2+r1'?=1(npokorEaL on6 rnv (2) vLo 2"0).

H rouh rou €nLn66ou xou !€ rov xttrrvdpo eLvdL o kurdoq x'+V'z=2 . H[€pLoxi T opl(erqL ord rouq Ktxlouq x'?+ri, '?=1, x'?+$'z-z'1.

H (2) bLv€L z=x'$ft1. A6yo ouuu€rploq Od unolovl@Wc rov 6vxo

vLq z < 0. 'ErsL Etvar + "Jlvx'*U'-t dxd!. 0q lEcaox4lqrlooulE oE no-T

ILKaq suvleroylaveq. Odtou!€ x=pouve, 0!pn!8, ' [ IoL €lvoL

z=f ir++4=\/p,.1|- . Oo 0po0!e ro 6pLo ro0 p. ELvoL

1 3 X ' z + o ' < z ' z - 1 3 p z s 2 2 - 1 s p ! 2

ELvoL o{6!q 0 < e: 2n, Aoo

, t 2 r r t ) , , r 2 t , , 2 , . r r t , . , . L 1 2- lv = l - l lpvp; : raplae-* l - l l1o: -11*610,-11lae=] l lu f i l l - lo6-

I t2 r_ t2 r=l J

/zds=j r3do=!5tel;"=2nv3. 'Apo v=hv3 (. uovddEc.

oloKlHFlTI(oE ̂ or lDlot 2. [ndv6dyou B. 1 9 3

132. It fimlovlo€et o 6yKoc tou or€p€oi, rou n€grxlElerdl on6 to €nl-nebq z:0, X+{,-z+2=0 kc! rnv Ko^lvdplKi €nLqdvELo Xz+qr2=16 orny n€otmiionl l o u x > 0 , 0 > 0 , 2 : 0 .

l0on:

H npoBol i tnq (ui tvbpLxiq s[106v€[oq sto enln€60 X00, {z=0) €LvaLo KirK,\oC X'z+0'?=4'z. En€16/t €LvoL X:0, 0:0 I o6ny6c {o|rn| : l , \n €LvqL ro ra_Topro rou Kirxlou nou Bptox€rol oro [piirro reropiqgdpLo, Kor nEpld€t€! roXoplo T. H €nlqdv€Lo S €tvoL Exelvn f lou op[<€!oL qf ld tnv EELqls4 x+0-z+2==0 - f (x,0)"2=x+0+2. 80 (dvouue eqoployi rou i0nou:

l l t lv 'J i f

(x'{)dxd,!1 J (x.,1,+2)dxdu. 0o Boo(,!€ ro

T T

6pLo rov X xqL (r, orGJ( 0olveroL ondoxi lo etvdL 0 : X S 4, 0 s U l r4O-X,-.

r r rV16- !2'E roL txouqr : v= l I J (xn +2)dr ldx =

{xviz-F-rs,"qnu }x- } r re*, r'l, *r $]'

= L!9 +an x. uovd6ec. {r-z+2=0

'!

'*o'= 16

133. lh npooDloplorea o 6yroc Iou orsg€oo oro rlltm 6v6oo u.v o!6r,jv,mo negLxlEl,eru! on6 rnv (Ul!v5p!(tl €ltqrdveLo {,=ex (or an6 m err(,n€6o X=0,

loon:

fl KU,\Lv6p[Kil eltqrdv€Lo 6XEL o6nv6 Ksuriln oro €nln€ao 0X0 rnv Kqunt]n{r=e" xor v€v6t€Lp€q nopdllnl€q npoc rov dqovo z,z.

H npoBoli r4q (0tlv6p[xie €nLedvsLdq o=ex KoL rov €r!n66{rv q[o entna-60 X00 6[vEr lov rd lo:

194 KEOAAAIO 1I

T= t ( x ,0 )€R ' : 0sxs l

0 6yKoq 6Lv€rol sr6 rov r0ro VlJf(x, ' !)dxd0T t

'Aoo txoutie: v=f[,r,a'a,r,=[ l[" **1,r.[ [4]', ' ;

=* | " '?tzx=l r"2*r:-+(e.-1).

0<0s ex l

6nou f{X,0)=z=0.

dx_l l:r_l dx_

l3{. lelfr8 6tL o 6yxoC ro0 c[ep€ot rcu op{.{Erdr- dr6 tlc €rlodveLec

x2+{,2=o, (o! z lo(x)+o( ' t i )FBo{x)+yo(0) (6nou o,g,y > 0 ror o(x) > 0) et-wL dve{dprfioc rnc d(x).

Aton:

r(vo!osz. !:{ i ] j*fo9

E{.vdL sx6uo VlJz(x,0}dxdo=T

_ [ [ Bo (x ) *yo (0 ) , r , , { , , , =lJ 0{ x l+o{u} -^--T

"IJ:{ilgffi *II3{*]*ifr .€aroule 'r=)( (oL X=Y on6r€ lJl=lr.'Apo to tlMrc oloKliptruo vtvEior.:

l l ^ t - \ ( l - t v \ l l - t t r , \I'1j "niieiIT

axar=jJ ogji6iyy dxdvlJ arn-=itrh- dxd0-I,, 6sdq €tvaL:T T T

{0 , r ,0 )

! . J o i u s o \ ' ( o ! p . r r ' . o r - c r c v i o o i s o " o o .

oIoKIHPoTIKoI I0I I tuot 2, tna!6dyou B. 1 9 5

v=JJz{x,+laxao=Br,*vr ,=(0+y) I , . (B+y)T

=] ro.',rJJarau 1 + (o*vlno'.T

0q unotrovtoouuE rov 6vKo tou or€p6oi, yro X > 0.1/2 rou <nrol lEvou 6yKou. H npogoli tnc KUiLvdp[Kicxo(r elvoL o KlK,\oq x'z+0'?=2'z. Eror o

135- l[vero! n xullv69rki er!0dvelo X2+{,24. l{o unoloyLoo€l o 6yxocrou olepeot rct ogLc€rcL crl6 rnv (ulrvbpLKi onlqtv€Lc (oL cn6 ro snlnsooa=0, z+4t-4.

nton:

Irr,-r.r=|re-urJJT

o(x )+o (0 )oril+-ffI dxd'|)

0 6yKoc oux6e elvqr roEnredvEroc olo €nln€60

(0 , 2 )T6noq T elvoL:

11x , rP ) en ' : x : 0 , x ' z+o '?<4 )

H suvdprnon rnv onolq eo o,\oxln-prioouU€ €Lvqr 1 z"f(X,rl)-4-0.

To 6plo tlJrv X,U elvor 0:x s\A:[t-2 < t ! < 2 . 'E ro ! bxoo t tE i

i t tw-+'-V = | l(4-r!)dxdr!- I l l (4-r!)dx ldrr.

= l (+ -1111x1" -v dx= l (4 -0 )v4 -0 ,d { r =

-. ' '=8n KUB, !ovd6€c.'Apd V=16n K. Uovd6€C.

t t -* r i w r l l d l d + = t r d 'T

xrou x 'z+r1,2=a2 ro!

(/4-, l t ,U)

( 0 ,

136. Nlo 9p€0el o 6vrcc rou ol€p€ot rou ltpdnou oy66ou niv ol6urv nou opt-

<€roL on6 Tnv Kutrrvdpr.dl en[gdveLo x2+{,2=oz (oL ond to s|l(,n€6o z=xfl!.

6Ldru ro 6Ldd orox l , ipo la napLdrdve! ao e !Bc6d! rou 16-

€ J u L r d , , ( r = { ( x , , 1 , ) € R , : x , + , 1 , , < o , } ) .

1 9 6 KEOAAAIO I I

A0on:

0 <nrolur loq 6yKog 6[v€roL qII6 rov

r i r n o : V = l l f ( x , l ! ) d x d ' ! .

TELvq! f (X,0)=z=X+0. odroule

x = p o ! v € , U = p t ! € 0 1 1 6 r € € l v q ! 0 S p s o ,

ose:*,Bti#*.'Aeo V=J I

p (ouve+aue)pdpdo=

$ t n / 2=J. o'zdp Jo (ouve*n!e)de. v=[t]"

x. trovd6€c.

v l l u2 - t r ]ELvqL +1J tdxd ' ! .T

eo uarosxnlorldoutr€ o€ notrLKaq

ouvrexoy!6v€q. 06Tou!e x=pouv8,l l j=pnlo. EroL ,XouUe:

f ( x ,0 )==L=+=F(e ,8 ) ,

f,iffi =0. eo BooirlE ro 6oto r,,iv

p K q L e .

ELvqL 0 s x 'z+92 < 2ox - 0 s p'z <

0s0s i . Enou.vus + v= l lT

137. d unolovio€eL o 6v(oc rou oE€oao0 mu op[(etc! €rldc rou nopc_

9oloel6o0c x2+|lJ'z=oz, aw6c rou (ulLv6pou x'z+02=2dx (cr ndvo cnd to erL-ne6o x0r,

n(ron:

H npoBotril rou Kullvdpou oro €nlnebo xou ELvqL o xtKôc x2+$z.2ox.A6-y(,J rnq ouutr€Tploq,eq unoloyLoouue rov 6yko xou o[epeot oro flpdTo 6y6oo

r6v oqdwv. 0 r(rnoq T op[(EroL rdc E€ic:

T = t ( x , 0 ) € l i ' ? : X > 0 , U > 0 x ' z + U ' ? s 2 q x )

tnuo-ouuolX/2= $ r t . t t - v=

zopouv8 + 0 s p < 2qouve. ELvoL oKduo

,rc.,*=["[j '"0 $ oo]ou =

2ot-3_

\ i = - - / -

o oK\HPATIKo' loll'lirot

138.06vE!€c

Atul:

l& urFlovaooel, o 6vrcq tou orepeo0 mu rE ldelfio! cr6 rlc cnt_

H €nledveLo :-X"U'? (1) elvoL ropq8olo€166q n 6€ fiLqdvero :.1 (2)elvoL Enlne6o nopdlUnlo rcoc ro €n[n€60 OX|l,,

H mogoli inq rogic tov 600 e[!-Qover6,v oio Enlra6o 0x{, €l,vd! o K0-(loc x'z+0'?.| (n €[touon lgo(onler on6xLq (l) xol (2) u€ onotoLei rcu z).Oo(dvoule €0optovii tou t0nou:

v 1J I f, (x,'! )- f r (x ,0 ) ldxdit,

1

ElvoL f, (x,0).1, fr (x,0)=xz+{,r. 'Apo

v=JJrr-x"-u"taxa,r, r)A6yo tuluerploc rou ol€peot eo unotovLoouus rov 6vio oro md'to 6v6oo

ELvo ! T r= { (X , { ) €Rr : X r+02s1 , x :0 , r l , >O} ,O tu rq oo [vs td r 0 l t 6

. ,a /2 ] /2 a/2=*J to ' : ; ' "* 'ae.f j r rso.ouv.e)d0ao'J ouv.eae=.. .=fno,

{. uovd6ec. ' l9o v-}no' (. pv66€q.

ro oxtllo .[voL 0 s X s 1, 0 s U sy'FIt '!

" r 'r rVFx,'nPo : i_ - l l l ( r -x ' - { '?)du ldx =

=J'[,r,-,r,.-g]@u,= (x, /j:irr

=f [*r'*,nt-* lt=rin:r ]dx=_ [' 3vi17-3xzvF7-rrr:it+x,fi-x-, ,__ 1 .l' p6n _2r,rr1:p1ar=J,

- -_---T_-- ox= I r.

198 ('OMAIO I]

=+l V]:?-dx- + I x'zvl-x,dx=

(crro 6i,o oloKtr4p6uors xdvoou€ rqv ovrrKotdotoon x=ntt on6E dx=suvtdt6rov x=0! t=0, 6rov x=1, t=+) =

=!["'^""o,-!l/'nr'touu'tdt: $[ oulnut - t 1'/'?-

-et-W.i',"-+'+-+'.eoo v= $- v-+ K. !ov66oc.

+=:*- + =-jf r<. rovddec.

139. tb moloyrooel, o 6vxoq rou sr€p€o0 rou npdnou oy66ou nrv oE6-wv rcu n€pl(leLEtoL dn6 ric e[!odv€!€c z=x'n1,'?, o-x'z, '!=1, 2.0,

Aoon:

H erLodv€lc z=x'?+U2 al.vo! ndgoBoloar6,q, I 6n!0dv€Lo 0=xz €!vd! ru-lrv6pl(i !e obnv6 xolroln nd\tu oro enlnEdo 0x{, rnv (alntilrt (noo@oln}

$=x2 xor vavcr€ro€c mg6.[Inl€c nooc rov 6qovo z'2. 0LrEnLc6vEL€c {=1,z-0 El,vdr enlne6c. 0 6vxoc V 6lverol 016 rov rom V=Jjf(x,o)dxd'r.

t l voL f= { (X ,q r ) €R '1 : 0 3 X S 1 , X ' s 0 S l } xo r 2 - f (X ,U) -X ' z+0 ' z ,

* s;*" Jouuz6l=

oultnut - -j- *" J", t-,,tdt=l( n1, touvt ) ,dt=

= * J-'r.u'= * f nu"tr.rorr.r. * (- *^t-* - *, )

A(x ,x 'z ) , s (x , l )

oloKiHPoTlKOt Aotlxl{ol 2. ttrd\,6dyou B. 1 9 9

.|40. tld Bp€081. o 6vKoe roo or€pao0 nou nepLxr\El€rol ott6 ro elrlmooz=0, m noptrBoloel6€q z=X2+4,{2 xor rlq (ullvfptKtC flrq6v€Le( o.X'z,X-0'?.

Aion:

0L (u,\Lv6oLKEc en!o6vEL€q 0=x'zrLn€6o 0x! rLq nopoBol6q 0=x'z, x=o'zovrloroLxo (oL y€v6!€Lp€q nopd,Un -

iec nooq rov 6[ovo z z. 0L nopoBo -

r6c 0=x: X=U'z opl(ouv ro xopto T !.

I = t ( x , 0 )€R '? :osxs 1 , x ' ? < 0 < r& l

0 Ctroo!€voq 6yxoq bLveroL a|r6rov r0ro i

, X=U'? txouv obnvo0q xqlnt i€C oro e-

'i

v-JJf(x,0)dxd(rT

6rou f (x,0)-z-x'1+40z. ttoL txouu€:

. . . f r \ h

v1J{x'ao')dxd{-J I'J. (x'z+4u'z)d{,Jdx

=l lx'0.4 -tl ^ dx=

=[ [r 'w.* (n) ' -x '-4xular .

=r4.+4---TT

=....$ r. yovdoee.

l4l. ilo unoloyLo€€t. o 6y(oc tou oregaoo roo neoLdelato0 016 tlq t0-

l\LvEprK6C €[u96vE!sc x2+{2'o2 ' V2+z2co2, 22+x2-(l2t (o > 0),(A.A.Klaf.Ca' lculus)

i t xu lLv6pLr{ ereodrew z2+x2=d2 axeL o6nyd squr i rn oto €r i iE6o o2x tovx jx lo z '?rx2=a2 roL yevdr€lpes Epdl ln l€E ipos to! d{ow + '0. Aui lo lo oul ] -repdoyata dxoulE rd! 1L! iLs d l l .s 6 io ru lL!6pLres €r !9d! . ! .s .

-I- -+ +t'.

2 0 0 KEOAAAIO II

A0on:

0q Bpo(lu€ rnv roli r(,JV KUlLvopLXdvl4E cnoloL{oit rou z rpoK0rrl€L o2-tI2=a2-x2

enlr (Ebo 0=)( , 0=-x. H npoBoAi rou orep€o| j

Ef lLqqveLd)v z2=o2-U2, z2=q2-X 2.

i 0=+x. 'Eror n roui €Lvqr rqo[o eItlII€60 oXU bLv€ToL 0116

TO npd[o oxl]uo.Abyo oulruETplqc rou m8p€00

npoc To enlne6d Jv dE6vov KdL

nPoc ro enlne6q U.tx opkeL vd u-

f ioloylooule ro 1/16 rou 6yKoo rougtep€00,

rto ro r t ipo rou or€pao0 nou

n6pIKle[EioL on6 to €nln€6o oxz

oxqr, U=x €lvqL z=\6i:I?. 'Apq

. v l l -€rvoL t3- ElJvd.-x.dxd{ ' l€

T" t (x ,U) €R'zros+s T ,

U < x s /A-[t ).

l4exooxnuoxlCouuE o€ no^LK6qouv!€rdy!aveq. 06!0uue x=poov8,qr:pnuu onolE ELvoL u 5b < f ,

0spsq, i f f i -0 .

f ( x, r! ) =\-s, - X r= v6T -ptitnv,6.

'EroL axouue: X

" i h r P-Tfr - l I I ovqr-p?ouv'?8do ld€. 0o uno,\oyLoouuE ro €o0rrepLK6 oio(rno0rus.

r c _ l 4 , . c

) e ' ts '?-prou\ 'e dp. l o(o,-p.ouv,8) rdp:- 2 .J

(o,-p,ouv,e):d(-0,) r

t e.- , .Uv]3 I

(o,-p,oLv,o)rd(o,-p,oLv,o)=- f6[ izd l {o,-p,ouv,8) / , ]u=

o r - o r n U 3 0 _=- 3ohz6 t(o,-q,ouv,ot'/ '-o,t=- :; luze to'(1-ouv,s)3/, -o3t= !

= + He,e = + Hi,8=+('.itffi'*)=$ (,-h"'*)-ouvtf

-

(o ,o )

A(' l ,0 )B(,62-rlrt,$)

0M4!!,01:IK![ l0rlxMot 2. tna!6d.rou B.

v . r r r l q r r r n , l ' / \ , t ^ 1 t ' t 'Enoutruc +-+i lF,trE.n!elde=+i - iu"*-#J, nuror-

- , t t / 4 , ^ , : - - t= + J, #o. ae- g r-weri : + [-, "*l-]:'"

- $ r-,*rl/'

= $ [r- --:--- ]- 4 [' -f- I r. r:ovooec.r*.0 *

,

142. th umlovloo€l o 6vxoq tou ot€p€o0 nou ool(ero! on6 ro €nttr€6oXE0' z=0, 0E2-x, rov napoooll|(6 x0ltv6po 'lFX, .oa rnv €nrodvelc z=X2+2f2(qro ndno 6y6oo lllrv oE6w,,vt.

A0or:

201

To €nlrabo !=0 eLvol ro €nlf lE6o oXU, ro60 ouz. H npo8d,\i !r,Jv elLnt6dv X"O, 0=2-X olo

€nlnE6o

eftLnE6o

{

X.0 €LVqL ro €Ir{ ,nE-0X{ el,voL oL eueal-

eq 0:0, u:2-x.H npo8oAi rou nopogolL(ot kul\Lv-

6pou oto (,610 entnedo slvoL n rcpo8o-lh 0=x'?. 'EroL op{.(eroL xo xtjpl.o T rouoXluqroc,

0 <nrol lavoc 6vxoc 6Lv€ta[ on6

I tv -JJf(x,'!)dxd'!

16rcu z=f(x,0)=x'z+^r' ondxe

'-11r".r' r"*r=1, [',' r,,.r,,or]*=f [,tu, +,r,, ]';'d,==f[, 'ra*r-$rz*r'-*,-ir]or-.... ff i *. uouoi.q.r odlou!€ ee+ :0 o'dr€ de: ffi, *" *o= 1$

2do

",*" J rfu *{ .% =, J ;r, o,=,Jt *.)-,d( r+o)=-2( 1ro)-, =

:-2(1+€eg)-r =-, _

l --3_ (Brdr€ xdr, rp6ro tduo oetr6a 123).lree

--

t0,2

(2 ,0 )

KEOAAAIO II

143. oa,r@0ue rlv KullvdpLri erlfdvem no! txEL obny6 mv reol,le -

rpo rou tgry6ritu O g !€ ElldqerS r €ug6v X=0, tso, ++{El xd! v€ve-telo€C nopdl^r eq rooq rov d€olro z z, tlo Boeoel o 6y(oc rou oreoeot nouopl(€lor dn6 tnv xullvdprxi erllodvalo, ofi6 to eltl,ne6o l+0]z=3 ro! dldro un€90olri6 rcpdolo€166c z=Xr.

(Integral calculus LBal l )Aodi:

0 (nroiuavoq 6yrcC 8oBp€e€l on6 rov rano:

v=J JIrnx'+)-t ' tx ' t l Joxol.T

ELvqr f , (X ,U) "2=3- ( x+U) ,

f r ( x 'U )=2=x0 '

KqL 0 < ) ( <2 . O . ' t , . t - - l !'Apo

" r -X

v= ['[ [- t t: - tr.,r t- r,l tal la, =

Jo t J6 I

= l l ro-x,r- # -x #l- 2 cx

: / " x \ ,=[ [:( '-4)- '( '-])- \

-;z/

I ' - - t i r- x l 2z l dx - .

. . = t x .uovd6ec

144. Nq unoloyLooeL o 6yxoq roo ctepEoo nou op((€Iql on6 tnv Enlodveldz=oe-(x'r''') (o > 0) on6.o €n(,|I€6o )d, (dl cn6 rov (otlvb@ xr+or=rr.

Aton:H rpoBoli rnc €n[edveLoq rou or€p€00 oro 6rlnE6o 0x0 elvcL o xtxloc

I3 ,0 ,0 )

I 'r-+)

i B l d r e o E r i 6 d 7 9 .

0I0KnHPnTI iot A0r l t r4OI 2, Xnov6dro! a. 203

x'z+u'z=r'znou opl<€r rcL ft xopLo T.

Etvol z=f(x,t)=qe-(r t '1" ) 'o vuoKdee {x,$) € I.

A6yoJ outrtetplqq rou arepEo0 0ounoiovlsouue rov 6yxo oro nod'ro 6y6oo (

I{lrv O€6vov. 0o lErooxnuolLoou!€ o€ flo-lLKtq ouv!€myutv€q.

t tvoL X=psuve, 0=ott i€. txou!€ :

0sp3",0<o<+ f f i=o .o," t (x . , ! )=oe-o ' nps

t t / 2 r t t . 1 / ^ . . r- f =11o" ' ' ' ' , 'dxd,! ' l l l oe-" 'oaolae=- + I - l I 'e-0 'd(-e ' ) lds=

' , i , , . , 0 r r o t

=- f [ ' " ie-e'r ioe=- I ["" t"* ' - lae=- ] r"-"-rrre: ; / '?=

= ii tt-e-' ) - v=no(1-e-' ) |(, uovd6E(.

145, llo unoloyroe€L o 6vroq rou oi€geo0 nou epdo€to[ n6vn on6 rovnqpoBolLx6 (illv6po z=4-{r, (1) KoL (drd on6 xo el;\€rrrL(6 nopoBoloe! -6Ac u =x'z+30'z (2).

A(,on:

0[ 6to €n[edvel€q rtuvowor oe uLo xolnutrn, !€ npoooli oro €nh€-60 0X{ qv n 6i l .r-Un X'?+402=4 (npox(rlr[€r 0116 rLS (1) (oL (2} !e qrlo -

IoLai rou z) t ' + - + -1. 'Ero! loln6v o rdroq T oo[(€rq! qr6 rnv

i^Êrun -t- + - i- =1.

Adyo oultr€rpLoc rou o[€p€o! 0o unotroviooot]€ rov 6yKo oro npdro 6v6oo

O (nroi,u€vos 6y(oq 6LvEroL 0116 rov r imo v=ll f f .(x,,1)-f,(x,0)ldxd,t,

6nou fr(X,U)=z=4-0'z, fr (x,U)=z=Xz+30'z. r

'Eco r eLvoL + 1J (4 -40 ' - x ' )dxd , ! (11 .

1

Kdvouu€ rov s€loox iorlou6 X=2pouve, 0=p4!e. 0n6r€

'!

)

- f , 0 ) 0 ( r , 0 )

o= + .+ < I -0sx ,+40, :4-0<4o2ouv,0+4p,nu,B. 4 :

20t+ KEOAMIO II

€ 0 s 4 p ' z 3 4 e 0 5 p < 1 . E l . v o Loct lon !e J-2p. ELvo[ ax6!o

f (x,U)=4-4U'z-x 'z=4-4p'1qt/20-4o2ouv'z0=

-4-4p,=4 (1-p, ) ,^ / 2 |

' n p q - ; - - | 1 4 f l - o ' ) z o d o d e =

(sp-8p:)dp]do =

(oL O s I s +. H lox.i0Lavi opl<ou-

rlI"1,

l{6. l{o urclovlogel o dy|(oc tou oE€p6ot nounledveL€q z:X+rr! Xr,.l, rt}=2, 0-X, '1.2X, 2.0 (X

A0on:

molll€l€roL ond llc €-> 0 , 0 > 0 ) .

(Golden lLths Serles)

0L EnLqdve0€q U.X' U.zX €lvoL €nln€60 xd8erd oro €nln€60 OX0. H enL-0dveLo 2.0 €[vor ro Enl,re6o 0x0, H erlgdve|o z=t+d, €[vo! tvo €n[n€6o nou[epvd on6 tnv opxi rov ot6vov. T6loq oL enLedv. lEq XU.l , XUJ-2 elvot l (u-i[v6Orx6c EIILOdv€leC U€ o6nyo0c clto €fl1,n€6o 0X0 rtC un€pBotraq X0-1, X0=2,xqL yav6r€Lp€c napdllntrec npoc rov dEovo z'2,

H npoFoli rou orep€o(J o[o €n[- ,,[E6o oxu a{,val ro Xrdplo T rou npdr -

xou teroprnuopl,ou nou nepLKleL€xoLon6 rrq (qunt,\€c xo:l! Xrrrz, {,:X,0=2x, 6rruc qalv€ro! oro oxius.

' txouue f (X,0).2=X+(r.

'Apo v=l l f (x, ' !)dxdQ=l | (x+,!Jdxd,!.TT

t[€L6i m xurpto T 6€v elvo[ Kdvovt(6,u€ rnv .ue€{.d x=l ro XtJoL<ou!€ oroXrDLd Tr,Tz.

E{.voL Tr ={ (x,U) € R? |

8 (

' ' * =

'12-T

( l , 2 )

sxs ' !<2x) ,

OIOKIHPoTIKO! AOI. ttOt 2. trov6dyo! B. 205

Tz={(x ,U) €R'z : 1 :x s ' t | , x s , l s f I

' rxouue v=JJttx,,rt oxau-JJ r'-,r, r*a,r=JJr x-,r,raxaa*JJ t x.rr;taxa,rr=l 2

tt f t2x t t"/7r rz/x-l*,,l l j),,"',*10'-1"'["",'*'*]0,=.. .= v#. + x. uovdo€c'

147, ltd unoloyloe€t to e!Bo66v rou rliluaxos rnq "n.q6u",oq

,=fxtl'drav n npoBotri rou oro m[n€6o oxl, e{vda ro opooydrvro T-t0,3]!t0,21.

nton:

0o €popu6ooule mv rtno:

*o'. -1fr.0. Aod o r0noc (1) x

,=;1,w;i+r,=4",mirx =[g rx- r r' /, ]i= I -. **0.,.

._t t" - l )

T

t l .voL

6 L v e !

t48,

(o > 0 ) .

lton:

H npoBoli oro €nln€60 0X'!lou tlilorcq owol), too Kultv-6pou, etvoL ro xrrpLo T IIou o-pla€Iol on6 rrq oxtoarc:0Sxso , 0s ' ! 3x .

0o Kdvouu€ eeoptroyil rourono!:

ifr locf€l !o qr@6y tou qrauatDc qC 6rLt6v€Loc rou (utr[vopou

mu og[(e@! c|r6 ro €n1rc6o 'I.r, 0=0 610 ltld}Io 6y6oo lt|v o[6vov

/(fi.)'- (-ffi)'.r axau rl

,-!y'/, . ' too &=in^ =

:-_-l-:=

s"JJJ(#)".( jfi )'.1d-d{.

206 KXOAAAIO II

tlvor. z-GzlF. ExoutrE

'*" "[!'1-p,*u.

=1"( [' o or\0,-o f-1-,r'*0,=r, rr, Vtr-f- / )6,6-ii - 'a

* [ + ax=-] [ " {o ' -x '1-"c1o,-r ,1-

=-* [ t" ' l ' r t-To'z ! . uovd6€c,

a z - tai -

6z:F '

]d=-o1 1oz-ozlr- 1o.-g" y";=

149. No gpas€l lo €!9d6dv rnq €nLetvE[oq oqotpoq oft('wq R,

A0on:

H €{ louon rnq oOoLpoC €LvcL X,+0,+2,=R, ( t)

0q Bpotu€ ro EtBo66v rou ntLorpoLptounou 9ploKeroL n6vo on6 ro €nlnedo 0X0.H npo9oli lou rluLogoLpLou oro En[nE6o ou-rd €Lvqr o K|: l l ( t roC X,+0,=R, (nooxtnrEL q-

n6 rnv (1) vLo t-0) tuu opl.(€L ro xr,Jplo

T{(x, ' l ) : x?+{ 'z 3 R'z}. 0q Kdvoup€eQoplroYn tou runou:

( -R,o) (R ,0 )

' ro" fr=]rn,-1'-,1,1-"

*fi= f rn.'"-,r,,1-"

rLvor- z=VF7-xzlor

-?x ): - ----i-L't] xr;1,71;nz

-z,l). -tf-Ti.:;6iT-+[:

Enou6v.,rc 1+ (#I'(*fiI= - R '- Fz:-IF+|FT

'=Jhi'.(.F)'.(.*r )'**.

'EroL €LvoL +=tf + dxd{, 0q u€rooxnuortoouur o€ rotlxtq ouv-. J J VF:TF;[,T

oIor '^H! f lTIKo' lo f I tMOl 2. t ro lodyou B. 201

r . I q Y u e v € q . o a m u U E X = p o u v 8 , U = p n U g - 0 5 X r + 0 , < R , { 0 s p : R . E L v o Lq K d u q 0 s g S 2 n .

B{f#. V[r(x,+rtr, ) !tr,-p,R 'EXoute

* =[ '^[ 'o r 6p6s-i I" [ôto'-* ' t 0. .- + f" [ lo !-(r :s ' ) lae' ro r VF:iT ' J, Jo vi,-o, t )o rD \Fliz )

" r2n. at - - rF ^ ,2n= +1, [ l ! i#-l,ro=- ] J, r-znrae+'retl '=znn, -] =znn,.

'Aps S=4nR'z r . !ovd6€C.

150. l{a unoloyLoo€L ro egBa66v rou Wfuoroq lnc enLodv€Loc rou (utlv-

6pou x'z+r!'z=o'z nou i6!vero! €EonepLxd on6 rov KttLvdpo x'*tt=ot (o r 0).

A0on:

eo unoloYi,ooulr€ ro e!go66v lou 1/8 rou rlhtoroq rnc0oli rnq EnLodveLqq Xrlur=q, - r!=!6tllt oro Enlfi€60 OXz

eIILedvEroq.H npo-

eIvor o K0Kloe

Ltr'Eror m XopLo T opl<eroL on6 Ttc oxaoeLq

(x ' ?+z ' z=s ' z , x>0 ,220 ) .Ene!6ft n f lpoBotr i t y lvExqr om €nlnebo oxz

8s XpnoLtolo Li)oouu€ rov r()no

,= [ [ r / ' - ( - .1^ . . f - ( - l ! / - t , 'n ro . (1), iv \d^ / \d t /

t lvqL .u=,,6):.X-t. tror txouuE

ay .= :L l 9 -0 . . rpo 4 (1 ) 6 rvEL :. ̂ re-iaj t- [ f " o.o*-fJ1-r-! o,;*=" )i \6i-V rd Lro 'te * |

P* 1,,- - Lzrf ' -"ax-o | "S ar-" l"cr="', o V o , - x . l n v a . - x .

'Apo S=8q'z r . !ovdo€q.

208 KEOMAIO II

151. Nlo umloyroo€|. ro qt !d6v tou qritroloc Inq €nlqdv€Loc tnq ('tol.-

poc x2+r$2+z?-4 (l) z 2 0 nou BpLoxEla! eqdreglx6 tr|c rultv6ptxtK €nL-

qdvelac X,+(',-l )?-1.

nton:

H nOoBoIi oro EnLneoo 0X0 xou rthloroqnou 6xer olrvopo iov xlKlo x'?+($-1)'z=1 (2).

0o {6voutiE €gopuoYi rou r0nou:

._ff" l )

6tv€L z=v!ltT. tLvoL

-2x ,z - -ztt

,t{:iti" ' -aU-

,t!:ir[r

| ^ 2 \ 2 / ^ z \ 2 4\ ax / \Tu/

+'= {-7,:u'

s= [[ _L dro,r (3J xT

tiErooxntlqrL(ouUe oe nolIxtq ouvrerovlEv€C. 0610ute X=pouvg, (r=pnU€.

ElvoL 0 s € s n. An6 rnv (2) Axouue p'zouvz0+(9nu0-l) 'z=1 I p'z-2ongg=o '

o=zn !e . 'Apoo39s24ue .

ELvoL oduo ff i =0, ' t tou n (3) 6LVEL

t= | [ ' "" [4 oolau= [ ' f " 'u-ro-o,r-+ 61a-o'1=f -zt 1r-o' tLt i" 'uae=Jo J! LV4-O2 I ro r.

=-2Jo (v4-4q!,0-2)d0--aj. {vr-t lu'8-1 )d8=-aJo ( louvo l-1 )d€=

--+lf/'o*eae-J'rrou"eoe ]-+fae-+ lr *e r l/2- tnuo) nrr'l< rell-+{ t*t ),rn=

=4n-8 r. trovd6eq,

152. l\h gpeeel ro elEa66v rou Illiumoq rnq Knvlric err06vtroc lz"'x't4t'l1

mu gp('ox€rdr rtw ar6 ro €nl.ne6o z=0 xor €w6c tou rul[v6oou 4lEx'z+{'?.

xnc oqotpoq €LvqL to xoplo T

1

H (1 )

aX

0n6r6

'Apo

oioK^gtnTlKot lotltMot 2. Xnov6dyou B. 209

Aoon:

H npogoli rou ruiudroe ouroir oloTo <nrol,!€vo €uBq66v 0Lv€roL

on6 rov rirno:

€ncn€60 0x0 elvar o k(Jl( loq xr+$r=4v.

'LIT

r' $i)" . $fi)'a,a,r,.'Exouug z,= r'j i l" -Zrrr=4

- 1=+ -;i=+E(vq! oxbuo .2. x2+{2 - 2r2r=

-#"*=+'tfi-+/ r , \ , / r , \ , v , n , 2 ! 2 - n , 2 t 1 \ 7 ' 2 1 t'E loL ELvoL 1. $ i ) - \ ; f r / -1. #. # =r+ 4e;5- ts ' - # ' .1-+=+

'0o Bpolrue rq 6p!o rov x |(oL U,

tavor x,+0,.4$ - ,z=4,1,-,1,u r x-1v@-qz. 'Apo -VE[]iEr s x 3 vtrfi{z.

ECvqL qK6!o 0 .< U, s 4. 'EroL 6xout/€

lt t,lql-Lr........_ " 1 r r' c r' - -s.l | | -+ dx ldu= -r- | txl '"q-' l-d{- +- | {v4{-0)+v4u-0?)d{,Ee Lr\4v-,1,, v3 I v3 x -C,r-t. v3 Jo

, t \ - a \ EE +_ | v40_0rd,!:. . .r y+ r r. uov66€c,v3x

153. llo unolovlooel. to €uBo66 rou rut|lcroq rnc @tL.E.c x'z+4'z+22-2oz-=o (l) nou Bp('o(EroL oro s€r,n€p!r6 too drtou z={58 (2).

A0ort:

H (1) vpdeeroL x2+ 2+22-2s2+o2-o'-0 - x'z+{'z+{z-o)'z=o'z (3).'Apo n ooo[90 6xa! Kavrpo K(0,0,0) Kol aKrLvo o.eo gpotue rq rotv6 oru€Lq xnq oodlpo( xdl rou K{dvou, To oio[4!o r(,ry

( 1 ) K q L ( 2 ) 6 l v € L :

x2+02+22-2oz=O I| - 2 2 ' z - 2 o z = 0 . ( r ' 0 , 7 = o l .

22=i,+1, I

' t 1 0

rro z=0 n (2) 6iveL x'+U'z=o +X=0=0.

lLo z=o q (1) 6[veu x'z+rl 'z=q'z. Apo ro KoLvd ltpoq u€ro€0 tns oeo{.-pse xqL rou Eqon€prxoir rou kduvoue l v o L r o o n u € L o 0 ( o , o , o ) x q ! oK0x,\oe x 'z+o'z=q'z rou BpLoxersL mo€flLn€bo z=q. 0q 0po[,U6 rqv €t(-ftJOn rou uepouq Tnc oqqLpqq flou

BpLox€tcL eKr6C rou xdvou.H ( 3 ) 6 l . v € L :

(z-o) '=o'- (x '?+{ 'z) - z-o =

=VAt:iI4[T - z=q+ !67:] x:;0T (4)

H (4) nopLordv€L rnv €nLedv€Ldtnq ono{.dq (nro0U€ ro €t0q66v.

H npoBo,\it lnc €nLqdv€Loc quiiq

r l o u o p L ( e ! r o v r d n o T = { ( X , U ) € R , l! lnou:

oro enlne6o oXU eLvor oX,+0, < q,). ed Kdvou!€

azax

,.f "lJ:

To rel€uro[o oloKll]pouq lerooxnuq! | (ovro C se flotrLxtq* ouvrEldy!Avec 6L-

r :_o aolaed[ ' ie)* [a d(0,) =2n+ f o lo,r -G "

r \Jo / z )o \6r:p, z k ,/o-d>

f to. -0. t -1u

to. -0. t = - zn t zlto" -p" tLl" =z16" -=-r"t

* Sdto l tos x :pou lg ,o s x'+0' <

s=JJ V1.( ji)',( -3fi )'dxdoT

xou {fr=-'!

,te-F:V' E x o L o r 0 n o C ( 5 ) 6 L v e L :

Eqcptoyi rorr

a 2 - o s o 2 s a ' - o s p s c , a L o s $ s 2 r .

oAoMHP0TIKO' lo f IXMoX 2, t rovodyou B.

tlvoL z=2vx 6pa "i= fi

, + "

E t v o L o K 6 ! o 0 s ) ( s 1 ( o L 0 s 0 < 2 f ,

=f"y'F ror?wr,=l"zr,*rar=rl,' t,*' tlotr* tt = { lrr.r"'" l',= ! e

=! re tT- t l . ' npo s= * (2 f -1 ) r . !o !d6€c .

211

154. ih ip€OE[ ro .g8o66v tou tttatoc r[( efitedv€lce 22={t nou x6-

8eto! arr6 rqv €Irr96v€Lo {r=4x Kdr to erllnedo x=1.

loon:

H enrqdveLo zz=4r €Lvor KutrLv6orKh u€ o64v6 xauni,ln fldvo oto rnlnE-

60 ozx rnv rqpoBoli z'=4x KoL y€vtreLpEq nopdltr4lEq npoc rov d{ovo + 0.

H E[LedveLo U'?=4x €LvoL {ulLv6oLxl] !E obny6 Kotnir l t r&w oto €nL[€_

60 0x0 lnv ropoBoli 02=4x xo! vevtreLoeq ltopdlintrec npoq rov d{orq z z.

H npoBoli Tou ruilroro( uC Enredv€Lqq

zz.4x oro erl1fl€60 0X{ €LvoL ro xojplo T nou

nepLKtr€l€rqL qnd rnv ndooBol i l q] 'z=4x xoL

r t v € u 0 € L o x = 1 .€o Epoir!€ ro epBq66v tou rliuoroq rqq

e[Lgdv€Loq sro npiito 6y6oo 1,,v oldvov, To

'ld'1,ldx=

155. nh gp€O€t ro €!Bd66v rou lgfufioc lnq €nlr0dv€Lcq r[q @otldq

x'zj07+zz-25 (l) leroto rov enlnadDv z'2 rcL z=4.(schat. 's out l ine ser ies]

eo npoBdtrouuE oro €nLnedo 020. H ngogoli tou rlilgorcq tnq EnLq6vELoq

rnc saolpas oro eninE6o 020 6lver ro Xl,,pLo T nou [epLKl€L€loL on6 rLq eu-

aEi, .c z=2, z=4 Kor rov x0(Io ' ! '?+z'?-s '1 ( [poKi,mel 0116 rnv (1) YLo x=0).

€o Kdvouue €eopuoyir '", '*", t=11y,4g;13:p".

e!Bo66v olv€roL otr6 rov l tno:

+ =JJ/ r.(*)'.(*fi )'dxd,!

212 KEOAJ1AIO II

n6yr,r ouUtetplaq eo epvoolotlso[o nqiJto 6v6oo ttlv qldv{dv.

ELvoL x=\E+:V - += =L.v vz5_t, _2,

ax_ 'Apo,lrs-G:7-

I t- r...,,,_rK-- rr E " ,JJ''-ZT=flGffi--., *o..lJ.ft,..,-'u,"sJ [l ab *],' -

troVl.ooule rox-,tv$.

l l i l v z r _ z . I t / - \-51 l roqau + | dz -s l (& tn !1 - ro€4!0)dz=51 I , j - -0 )dz-v 2 5 - z ' J o

" I tzl,' + 2=5n. Aeo S=2tu r.uovd6Eq.

155. ilo unolovLooet. to e!Fo66v tou qr4lcroq I||c €nrqdveldc x2.0'?+22 (l)

nou Bplokaror u6oo ornv enrodvEro x'l+{'1.ot (2) to > 0, x > 0).

A(/on:

H €nlqdvcLo x2=r!2+22 elvcr, kr,rvLki ue koouei 0 xd! dlolo orJl letpl .o( rov

dEova x'x.H €nledv€Lo x'?+o'z-o2 E{.va[ KUlLv6pL(i !e o6ny6 Kounaln rov K0|(tr0

x'?*r!'?=ot (oro €ntn€6o 0x0) (q[ y€vbrELp€q lop6llnl€q npoc rov dtovo z'2,0o ngo06ôotr€ o[o €ntns6o 0z{. I npoBoli rou ruigoroq rnq €nL06velqc

(l) oro €n[ll€6o 0z{,6tvEL xo xrrplo T nou ro o0voo6 rou npoK0rta! on6 r[q(l) Kor (2) u€ onot\oLqi lou x,

'EroL npo{0lrr€! q Kolntr\q z'z+zr!'?.o'? i

+ -

"++

=l nou Etvdr ttrlEL(rn. t lvoL x'z=0':+2': - x=!6?iF. 6a uno-

Vr)atrFo66v oro npdrro 6v6oo rov ol6vr,lv (l6yo oulu€rpLoc). ELvoL

0d (dvouu€

*+. u.'t @r7

Eeopuoyi xou **, r=JJ/'- S){gy dudz. Er\,,or

# =#' Etor €[wL:

< l l / t r ,2 - . t

+ =lJV'zii:;t.l Bl/E:V Jl

0 oMHP0TIKoI lo f I tMoI 2. I ra!6d.ro! B. 213

q t f I t l _. i- =ll1/1. r;,, , ,r= d,rdz=l1,12 d,tdzT I T

KdvouuE rov lerooxnuorrsu6 0= -q pouvo,t,

+ 0 xt lorL<ou!e rnv IoKUBLovi opL<ouoo.

a0 au-at -iF

az AzTp Tfr

( 3 )

z=qp4u8. EtvoL 0 3; + fhFlw

€ < +.'Exouu€:

- I pnrev2

opouvg

I oruev2

ortuB

-6

" i / 2 , , t , . t / 2Ero,n(3) 6LvEr +=1, Ll f foaolae=1" dso, ' loedo=i +ro,r;== __ . ^po 5=ro- r . uovoo€q.

entgdveLoc rou nqpdotroal-€Lvor o unvl,okoc rou 8er-

(Schaunrs outl ine s€ries)

Ady{,r ouu!€rploq 0d Bpotue ro €uBq66v xou rpi trdroq r4C e[LedvEtqq oronp6ro 6ydoo rov dq6v0v,

157. ilo Bpeo€[ ro eugd66v rou tutlloroq qq6olrc x'z+!z--z nou n npoBoli rou oro erllnEdo z=0noul l i p'z= i-ouv20,

n0on:

To <nroir!€vo €uBq66v 6lv€roL qn6

dXdL!

'Apa f

=Jjv1+4( r,+0,) dxd' lr.T

MErdoXnlcrL<ouU€ oe nolLKEC

f f i$ i =0 . r i voL 0 1p '3 = : :

5 r * . e ' " U u o o e l a 6 d 3 9 6 .

ouvrercvuCv€q. EXouue X=pouv0, Ur=pn!0,

- o=o=@P xou 0305f , "

rov rono:

+=JI\,(#)"(ifi)',Er*,(#)'.( j f i ) '= tz,+ear,

214 IGOA,IAIO II

\FNzr !6!!-2fr< ta/a | |

-- 2-- - .r , r./a r-"-f'A*

+ =J, ['J, e/ir-4ede]d0= + 'J, J, (r+4p'z)'d(r,40') .

. V6$-ZF

- + '+ f ' .[(1-4e'r ' l ' fr ae= {- f '"tr,*o*rut'" - ' :or=

rl\ t/4 r/4

- i[, reour'"1"" -r:at. -.! J zv:zow'eae- ] J ae-

nh r/\=3J o*"o1rrv-{rer i /n=$J rr-qv'e1a1aue1-;=

= $ rn,er;/u- $ rnr,'e:;/u- + - ++ - {- 4 - * =

1 1 n s l l= -F -TF - -4F ' - 3F - -4F

'Aoo S=41+ - iL l= -i- - +-\ J o 1 6 | t

l5g. I{o uflol\ovldoel ro €!Bo66v rou qrtucroc tnq efllq\tveloq qq o0o(-

w x2+{2+22-d2 too onoaou I npo0ol/i oro drh€6o 0x, eLvor to r9a'lo gillo

tnc rqntlnc p.douvze (o > 0). AtvataL 6rL r > 0.

r. !ovd6€c.

A(ron:

En€!6i n npo8oli rou rliucroc tncenLedveloq BploK€rdl oro En(Il€6o 0x{ro <nrottr€vo €!Bq66v eo 6o0€L on6 rovrho:

T €LvoL o r6noq nou nEpLrl€ieroL on6loflpdr.o qrillo rnq xounttrn( p.oouv2o. A6-y{,J ouutr€roLoc 0o Boo0!€ ro €uBo66v rouruitrcroc oro npdEo 6y6oo rov oEdvoiv.

'-Jh/'.(#)1(*fi)'o,.oo onou

Flvor. ji = a(6:Ir:[z) = -x "o,

jl- -odx dx ,ta _Ffr dv t3=i1:v

oIOXnHP0TIKoX l0I Ix l40! 2. t rd!6dyou B. 215

0o U€Tooxnlclldouue oE nolL(eqE l vqL yv l l o r6 d r ! 0SeS+a xo r

8tftti,. Apa

ouvraroylav6c. 06rou!€ X.pouvo, U=pnU60 s o < oouvzo. E(.vqL sx6uq

s rr-T = l l

Tr * -r1-q'-x '-qr '

. [ [+o,o*=[""IJT'Vqr_x ' _0 '

"

L

a"--$q. oto,r,=

Igoo]or=

.-qJ 11oa-ezyllad!!1066=

,r/4 ,: /+=-ol I (o,-ororrv,20)L-old6--o, I' od

=- # i,'' n rrorr-o, ('Ir: +{-ouvzerr/a*o,rerr/q--'noo s= *

-0, r. !ov66€c.

( ( l-ouv,20)!r-1 lde--o, (n!20-1)d0=

--z- '

I

)

-2-

159. tld lp€os[ ro g!!c!6v rou rlflaroc utc e'r46v.rdc 0r+2,=x, fi)nou r66€to! dr6 rry efla9dvslo X"+2-n2 l2l lc! Eo €nan€6o i.t, F-t ,( 9>0 ,a>0 ) .

A0on:

H €IlLedv€lo r1'z+2,-;, elvou K(,JvLri !€ Koguqi 0 koL 6Eovo ouuu€Tplqqrov dEovo X X. H €nLedv€Lo X.-r!.=de[vcu xul!v6p[(i !e o6ny6 t(oun0lnrnv xour(r ln x,-dr,=o, ( i -4 - -{ -to- LoooKeittq unepgoli) nou BoLo(._Tqr oro enLnE6o oxUr.

' r r c L 6 i . o d u l r . l c o 1 { s X r o u O d r n v r a l r , j t r p : o o u v . g e i w L d p t L o E ( k = 2 )r r ro l ru^n d(or . lE i rqL d6 2 t (=2 ,2-4 oJ l la . To dp ld rou o E jvoL 5=o raL

"= + =

+ (d lqeEpd lao tE s .o uLod tpqy loo , t laorE lo oJ^ lo ) .

216 KIOAAAIO I I

H npogolh rou ruituo[oq rnq enrQdvELqq

T nou ro o0vop6 Tou n€pLKI€L€ioL 0116

Ksunir ln (EUe€LEq) z=rq (n EEtooo4

ouri rpoKtm€L qn6 rrq (1) Kqt {2)

!€ onoioLqn rou xl .0o unoloyLooutr€ I6y(,) ouuu€!pL-

oq ro €lBqdbv rou TliltuoToe rnc €-

fl redveIqq oro npdJTo 6y6oo, (dvowsq

e9optroYn Tou TUrroui

(1) o[o rnlneto 020 6LveL ro Xoplorrq EUe€L€q U=8, {r=-B KoL on6 rnv

Ervou if = ',9*, = #

ax a(6Tttt z- - - - - - = - : .

Unz.,2tr w* r v]#uo, =

'^," +"llT

=J, l1 " "1*=1t*,':iar*rJuau,*o'4. Apa s-8o0y'2 t. uovd6Ec.

160. [h Bp€eal ro e!Bd66v too tuilatoc rqq Efiupdveloq 2=x0 nou nEpL-

€xalol laoo ctov K0lLv6po x'z+{,2=o'?.

nton:

H erL06v€[o z=x0 €LvoL un€p0otrLK6 noodBo]\oaL66q.

H npoool i rou ruiudroe rnq ElLedv€Lqq z=xu oro enlnsbo 0x0 elvqL xo

xrlpLo T !e oivopo rov x0K,\o X2+o'z=d'1.To <nroiu€vo €uBq66v dlveroL qrl6 rov rr]no:

'=IIT

rL"o' ff=,r,, jfr=x.

'no" s=JJVf.f+q-a1du. Hcrooxnuqrlaouu€ o€ no.\LK€q ouvr€royltv€q. o6rou!€T

x=pouve, ( r=pnue. ELvsL 0<032r (K tk loq) , 0sp<o l<oL # l+J i =p .

r.(j; )".(-ffi)i'au,.

oAoKIhPllTIKoI AOIIII4oX 2. tnov6d-rou B,

Ae" s=f '[fe,4;rd,lou=f iu f t';"rr dt1.e,)- + t"g.]: f, =

= + t (1+q , ) , , _ t l r . ! ovd6€q .

161- No BpEo€l ro e!9o66v rou rutucroq r4q enuodveroc z=ro[eo ]rou llpdnou oy66ou fl,Jv cl6v&N IIou n€pl6xeroL €w6q rl,v |(utrl,v6pr,rv x'z+{]z=o2,x'1+{'z=g'z (o < B).

A0on:

H flpoBoli rou r!i!o!oc rnq anlqdve[oq nou flepL6xero! ew6c ttlv (0-

llv6oov,o[o eflLn66o oXr],ELvoL io r6ropro rou KUKltKot 6oKlutrLo0 noo o-pl(EroL on6 to0q K0K,\oue,X2+U'z=o', X'1+U':0'z.

To enlo0lEvo €trBo66v 6lverou ord

dxd0 "

=[frf . --r- - ---:-'- o,o*=tl , l (x , *u , ) , (x ,+0, ) ,

= [ [ . , [ ' . J - oro, ! .,i \/ x'*0'M€rsoxnuqlL(ouue o€ notr!K6c ouvx€rqyutves, ELvdL x=pouv0, | lr=pnue u€

oses * , ospsB (eLvoL q ' z< x ' *U '?< B '? ) . 'EXo0 !€ qK6uq

t\ /2t tB.0. Aed s=1" I I e ( 1 ) .

0o unoloyLoouri t ro lvp' . !dp.

ELvot I=lvp'z+1 dpJp vpz+1 dp=

' ' ; '". i f -t."c*j-r*- -. r f r,=, . . ' ,

H .(*r..; r" = _r r*,"x

,:q-

- 7i6'

,=JjV,.(f;-)' .GfiI

2 t8XEOAAAIO I I

=etf,tl- lpplFit' ao=o\6zu-lp -4_ dD=ov6"n- [ --{ ,.=' ) 2'Fn ),li"if

-evb4r I gl ao=pt6-t-[ $_JJ rr-f J_ ao-p4;.r-:.[ -! co=, vo, t1 J ,JF,t J 641

- J ,157n ""-

=p$zit -I+tn 1p+rf i i i ) . Apq 2t=p/6?-+1+tn(p*v6+T), ,rrou n 0) 6{.v€r:

=f [o'Oua-t"r*V6'arr[-+ [uug4T-",6r-"," ffi ],. ,ouoo...

162. liq Fp€ooav o[ ouet€rovlAv€c rou k6w@u Bdoouc rou )(opl,oo rou tle-pLxlEletol on6 rtc xounolec Ur=2X rcr X.Z, {F0, 6rdv ornv enledv€td tou rd_rovctreto! ldld ruKvdrntoq 6{X,0)=t. Unton:

0o (6vou!E €0ootloyi Tl,Jv rtrftnvrnq o€iLooc 85.

z . y ' : xt t ,o, ' - l l l aolcx- l ru,rrxox-

=l a i dx-ql t '4 + | - . . .=gJ , r . l o

ELvoL ok6!o r,=l [Jr\o*]or=...* *,

.Aoc ;.Je=+, .l=+=+

163. ilo 9p€0€a to xawpo Edpouq rpr,yilvou 0AB U€ E(!(6o€[C nt€Updv 0=0,.{FX, X.l 6rdv or0v elrgdv€rd rou (mdvatErot !d<o nuxv6rtltcc 6(X,0)=x+,}.Aton:

Kdvout€ €qoptoyi rov r0Rdv 2 rnc oeiL6oc 86.'Exou!6 m=ll(xn!)dxau,=l I I " r x.u raq, lor= J lr,r. # lâr"

r f , ' r r-[ (," 4) cx=f' !# ax=* I-ll] l'--lro ' , l ) , z - ^ 2 l 3 l r ' T r ( c r

0l0Ml{P0TI(0t norr4tor 2. rrc\,6

r*=JJ,rr r-,60*o+, [' [ [t,r(x.,tr )d0ldx=T ' 0

'i" t*"#J1,,-r t t - a

1t+-+)ax=f, Jx'ux-=+ t+ t:=+E(vol ok6ro o,.jj, r*,r,roro,r.J' I j]x r x.,r, rau]ax=J' I x.,r,.x $ ]xan

[(,'. + )r'=+ [,,*=* [-l- l '. 3z t - . 2 t o . ' - ^ z t f Jo -T .'o*r.*.*."+ , o"*.*."*

154. llr fiolovlo€otv or po|rac obpoveloc oc [!oc touc daovsc x.x, 0.frou oloyevo0c ullxo0 Xl'9lou

T . t (x ,o )€R '? : ( ) s0<26 i , 0s ) (sd , o>0)I0.D:

0o (dvou!€ Eoopt/ov/t rov t0rft)v:Ix"JJ0'?6(r,0)dxd0 , r+,llx,6(x,0)dxdu.Ene6i ro Xopl.o etvo! ouovevd( dXo'ute 6(x,r!)=c 6nou c orooepd. ,Ero! E(.voL

, l t . , p11z6 i - r - ' otr-11,r,.ccxau,.cJ [J ",r,,d,1,]dx=

* f rr,r3wr *= +r fr*.rF-or.= * f to',, r",, or- $ .o'a ['r'a ar- -g, *'n [*I . + *,.j, ' t"ELvdL a(6uo l*.ff '-^x,.au**f

x,t,r,rl6i. ar..J'2r.6;ox.z"fo tx'/, dx==2"\[$ l'=4 *:'z ' / 6

220XEOAAAIO II

165. lta unoioylo€e( ro K6w9o t/6(nc ro! [ ponli cDoavetaq I0 rns €r!-odveuoe nou lleoLde('€ro! on6 r[q kounfilEs U.xt, ,1.+X (x,,1> O), Oto"6 (x , ' ! )=1 .

A0on:

t (

o) rcvor i=* JJxorx,uraxa,lT

O= f. [[,r,0 r x,,r, r*0,r, *o." '=[forr,,r,ror,r T

r r r 4 x 1'Aeo m1! lir3dojdx='l (ax-xt)dx=

'["'- 4 ]i ".- r I f lexx=?-J tJl,xd,rJdx= +'J x(4x-x!)dx=

=*[rq, '* ' ror=* [o -r f - J l '=e J o

o .*l I ' l,,*,],,"*/ r,r,,:lrox=]-+

A(x ,x3 )B ( x ,4x )

i6x'z-x5)dx=+ [ 'u4-+

0) Etvar Io=lJ(x'.,l i ':Jo{x,rr,raxau=l | | "

T ' o . ' \ '

"J 0^'- +,,'''- + ),'=.""= # .166. io utrolovld€( n .bri dgov€toc t_ utalo0 x,ptou uE ouvdornol nu_

rv6Emsc 6(x,ll= f tlor ot tO r.a|'xl€tEtda d6 rlc rqlntlEc ,f=r, ,*i.

A6on:

. e Lvcr. r,-JJu.o(x,,!r**=jj A r,*o*=l ffj,j+ "_p,

_=l"ll;;i-,dlj, t+)i,i J,= {,,[ rs-,r,r.(f)"]a,r,-

1',-l . -

r,",r,' r+]a,=J'[,.'u- $ ]'\,=

oIOK\llPmIKO' AOIIXiOI 2. trcv66yot B. 221

- J5 J (zs-ro,r,+'- fi)ar, =

=*l (rouor+,r'- -|fl)o,r =

.* Wv- gu,* $ *o r"u]'=

167. No urclovloo€l ro €lr9o66v xo! n gorii a6pdvrLoC r,E nooc rov dEovaU'0 tou xuptou T.l(x,g) €n,: - |ouv:x < | < ouvx, -i = r s $|

(R.Trandlf!r: (Irtrratlcl syperloarc'li0on:

tt ta/2 | f''r\ 1 tn/2o ) El,voL Eljdxdo=j t.l , d'Ildx1 (ouvx+ +ouv3x )dx.

T -r l2 - +o!e3r -r/2

f/2 | tr/2.J ouvxdx+f I ouv3xd3x= t uxJ l{:/2+ +InBxt::'/2=h +t-1-1).-nt2 -1/2

.2-a= + .

etvcr J x"ouvxax=J x"(rux)'dx=

.x' nrx-J zxnrxax=x"nrx-zJxnrrxax=

.n'z

_ - 1 . - - - -

r/2 st)rx rl2t t " ? f I I Ig') ExouuE ly='lJx'dxd01 t'| , x'd0jdx1 x'z(ouvx*+ouv3x)dx=

. " -r l2 -

icu!3x -a/2

r/2r - ' !.21 (x'orrvx++x.ouv3x)dx=

'/2 \ /2.4" x'ouvxdx*$J x'ouv:xdx.

=x'?4!x+2J x (ouu( ) 'dx=x'nux+2xouvx-2J ouvxdx=xz nux+zxouvx-znrx.

222

ELvo! aK6!o J x'ouulxax- { fx't nu3x)'dx= + x'nu3x- + I zxn!3xdx =

=]r'*lr- -|J xnusxax= *x"nuex* i I x(ouv3x)'dx=+ x'za!3x+

.f x"*lx- f J suv3xax= | x'nv3x+ $ xouv3x- f nu3x.

'Apd r+=2[x'1n!x+2xouvx -zn rtX/'*? f ]r"*r*.f, xoru:x- f ry:x]"/2=

"(*").+ (+ +. +) =+.4-#.# .

KEO&'\-AIO II

o, \oK l l lPaTIXoL{OTI IMOX 2 . Inq !6{ i ro ! B 223

fEvlreweytr (tri yy,pro) [mld oloxlryrqram

:rq nponyo(|u€vo oploqu€ ro 6Lnl\6 oioKiilpouq yLo mqyttv€q ouvcprio€Lqs€ KieLoTtq Ko! apoy!6v€q n€p!ox6q. eq yevLKeLjoouu€ r ipo rnv tvvoLqoLntrot oloK,\npduoroq otLq n€p!f l r6o€rq nou o )ro nedlo rnq oloxl ip(,ronq (nf l€pLoXi I) el .vo! 6vo Ki€Lor6 xoL I i qpcygElo (onepL6pLqro) unoot lvotro rou RzB ) |] un6 rc 6lni6 oioKtripouq ouv6prnon f(x,o) 6€v €[wL epoyu€vn sro fiE-6Lo rns o,\oK,\ifl,ronq. iv ) 1600 n nepLoxi T doo xoL n ouvdprnon f(X,0) 6€v €LvqL qpqyttvec.

q') To rcD[o T rnS oloiliqroK €twr xl€Lor6 mL Un epqyUt\ro.

Yno€6rou!€ 6r! c! (v=1,2,3.. . ) €LvoL uLs ruxolq sKoioooLq Ki\€Lor6vKo! QpovUtvov uf loxopl{dv rou Xoplou T. Ynoocrou!€ 6TL n qKo,\ou€lo G! !x€Llnv LOL6rnrs 6!r tnop€l vd "6LoorotrEl" Kqtd T610Lo rp6no dlore vq xqi t lo€L610 Io Xoplo T. Tnv L6L6rnlq duri rnv ropLordvoole !e G! r T i t in6Gv=T.

Ym86routrE ok6!s 6rr n oovdprnoq f(X,U) €lvo! olo( lnpdjoLun sE Kd8€ xl \€lordxol QpqYUtvo unox0jplo tou XrrJplou T. A! v[o xt€€ o]<Olouglo c! !€ G! - Tx o L G v c G ! + 1 ( v = 1 , 2 . . . ) u n d p x € L r o t i m - ] j f ( X , 0 ) d x d r ! K c L € L v q L 6 v q q

rpqyuqrLKdq dpLS!6q t 16r€ n suv6orlon f(X,qr) i6yEroL o,\oK,\np6ol!n ( !€ ye-

vLk€uuavn 6vvoLo) oTn n€ploxf l T xsl o opLeUdq ! ovold<erol y€Vtx€uUevo o-loKruplr tro o e(6ouc rnq f orn n€pLoxi T KoL oulootr l (€rq!:

>rtv n€plnl l lon aurl t , \6!€ drL ro y€vL(eu!6vo oôKi/ tpl l l lo rnC f ortv rr€-pLoXi T Lrrdpx€r i t 6rr ouyKiLvEL. Av ro l im [ [ t f r , , f , lOra, l 6€v undpx€r i l**- a;€LvoL ro r- r6rE l t !€ 6r! ro y€vtK€uutvo o,\ox,\ i tpujuo rnq f ornv r€ploxi T6Ev LrndpXEr i 6rL 6€v ouyK,\Lver, i 6rL qno(lLve!.

I lcpqripnq:

Av n ouvdprnon f(X,0) €lvdL un opvnrLKrl*or lv repLoxft T dn^qbn av el-

* A ! e i l d L f ( X , + ) <t n E f r n ! l f l r d L

0 v (X,+) € T i I f a l ldE. ! t rpddn lo rd re i€dpo, j le d l ta€p- rded lodre nr i lL l le !ud dwrexp! !d rn d ro lous id .

l l f{x, ' l r)dxd'!G

JJr(x,,!)dxd,r,T

224 KEOMAIO II

vot f(X,U) > 0 V(X,O €T tdt€ n aklovi ! !oe xoL Udvo ouyKExpLltvnc oxo-louoLoc G!, opK€L vlo vo uoc eeqoqolloel tnv orrvKllon tou yevLKEuucvou o-troK^npii)troroq, (€e'6oov qus!xd eLvoL l im l l f(x,0)dxdo.r , (a€R)).

* -6

nqparip||o||:

M|-o nLo onli nopouolool rou 6lnloi yevu€uudvou oloKllpdloroq q' eL-6ouc e(vor n €ndlEvn.

'Eorl,t 6rL rt nepLoxi T 6€v eLvor Opovlavn (oL 6r! n ouvdprrton f €tvoLoioxtrnglorun oe KdoE d€Lord KoL epov!6vo uroo(,volo c tuu T.

i1e llo ruxolo Kopllttrn Cv (!n orqeepi)* onoKdmou!€ qn6 m otvotro T6w xlelor6 xcr epoyutvo unoo0votr6 rcu, G!, npo@!,tirc rc 6[[16 oio(J\ipouo

JJf(x,0)dxd0 undpxeL.G!

'Eorui 6r[ d €lvd! I dn6oroon rnq opxiq r{iv oq6wv on6 rnv Kql]n(,trn C!

Av ro . t im l l f (x,Ur)dxd0 undpxELr* VLo r6€e o(olouol,o (owl, l rdv Cv (v=1,2..)

xol €LvoL npovlorLxdq opr€!6q,t6re 4 f J\ ty€ror oloxlnd,oLtn ( !e yevL(euUa-

vi 6vvolo) oro T.

Av .{ .vdL f(X, ' r) > 0 V(X,o)€T dp(€L vo n6pouge l lo xo! U6vo orryx€ -

K p L ! 6 v t o K o l o u o l q K a U r t t r o v C ! , ( v . 1 , 2 , . . . ) , ! € d - + - .

nspo6€ tyllora :

tg) ilo €terErct oyu6w 8') ilo rDodropLorEl,

narol:

to trpto T={(r,l) €R2: x2$' ,221 €avoL rpoy-Urq oKoloootc G.. looytrA\ro! urDx'ptr"v rou T.

o ) To Xrrplo T Elvol Ur.o ! Igpoytravn n€ploxi nou [epL6x€L roortu€Lo rou xuxtrou xz+i|''z=2z Kdrrou €!,JrapL|(oi rou.

H[a qxoloueLq G! qpoyltvov

UfiOXI,JPLOjv ToU T €Lvo! n

{ npdr .L loL yLo lLqn * , o r d ! d + + 6 r d . e

drolou9to xoui'lro!.

O oKAHP0TIIO' AotIXl{Ot 2. trdr6dyou B. 225

Gu=( (x ,q , ) en . , 2 , s x ,+U , : v , , v=3 ,4 ,5 . . . 1 .TLo v=3 npoKfuEEL ro epoyucvo x0rplo Gs={(X,0) €R'z: 2, s x,+U, s 3,}

fra v=4 mox0rr€L to o9oyri€vo xopao GI=((X,O) eR,: 22 sx,+{, s4,I

ELvqL c3 c cr c .. ' Tevrxd e{.vo! c! c Gv+1 VLo v-3,4,5.. .

ILva[ oK6!a ] irr G\tT.

29) ilo urctroyLoeet rc oloxlipr,ruo: f[ "-rt-{'axa!

o.ou,iT= t ( x ,0 ) €R '? : - 6<x<+e - -< ' , < + - ) )

Aoon I

lo XnpLo T nopLordv€L to €|rlfieoo fl,Jv qE6wv. oeojpotue xlv oKolou8lcrGu={(x,0) en': x'+rlr 's v' , v=1,2,., .} (6nô6i1 o€ooo(lue !Lo or(olou0Lo o-loKdvrpr,Jv KUKirx6v 6LoK!, !e K,vrpo ro onueao 0(0,0) |(qL oKrlv€q v=1,?,3 . . . . ) .

e t vou t= [ [e -x ' - "axa l - r i ' [ [ " - t ' -+ ' a ro l I ) .; * . ; ,

MEiqoxnlort<00!a oe noluxtC ouwerqvucveq. Otrou!€ x=pouve! 0-prl!8.

ELvqL 0 s X ' z+o? s v2 - 0 < p ' zg v ' 1 .0 3 o3 v . ELvqL qKd lo 0 <8 3 2 r ,

Blf;*i =0. ',.o., n {t) 6LveL:

I. ti.n [["-r"-'r 'arau,-

- 1 1 ' [ [ p " - c " 6 p 6 6 =

- tr ' +[ [ [ r-"-o'a(-p.) las=

, 12\= l in - i l I -e-P l l do=- r im +u * - - , '

'Apo .J.Je-x

-'!dxdu=r.

Ti-E r e L 6 i € [ s L f ( x , q ) = e " ' > 0 y l cqro loug(d< cox. i yLo w .qdoeul io . r ,

1 5

-u" - r )de=- r i r ] t " -u ' - , ) rn .n .

rdg. (x '0) €T n ex loy i ! !cs ouyrerpuldvnstn! ' i rdpqr '1 dxL ro, ) yEvlxeuldvoo oroxr . r -

ri"

226 KEAMAIO II

-I6rftF€C uv V€yrrEuucvuv 6!116, olorltr &Ev a' €tDouc

l lIr) Av ro vEvLxeL!avo o,\oKIiporlo l l l f(x,Ulldxdt* quyxtrLvEL 16rE ouy-

T

dLv€L rc[ ro v€vl |(euutvo oloxl iprarc l l f (x,0)dd], { | (o[ qwtorpoqo).

T

Ir) Av f(x,U) €{.vo! u!o ouvdprnon opLo!6vn orrt un qpoy!6vn n€pLoxh TxoL 9(X,0) €Lvor lrLo !n dpvrtr lxit ouvdprnort opLoUtvn ornv L6ro n€pLoxi ](oLyro Kdee (x,0) €T €lvqr, l f{x,{) l S 9(x,0) 16r€ qv ro y€vr(€uUtvo oJ\ox,\ igo-

l ( t (!q JJ9(X,0)dxd0 ouyxlLv€r, KoL ro y€vrl(€ultvo olox,\ i lp'r!o

,JJf(x,U)dxd0T T

t t t touyxA tvEL xo r Lox i€ ! l l J f ( x , ' ! ) dxd { l s i i 9 ( x , { )dxd0 ,

opLo!6v6c or4v l x ,U ) €T (oL

Ij) Av f Ko[ 9 elvoL 6(10 ur l opvnlLx6q ouvqprho€LcqpqYpavn r l€pLoXh T,yLo rn! onoLo loxtEL f(X,0) S 91X,U),

x6ud ro or\oxî f tpo l l f (x, t)dxdrlr 6Ev ouyxl\ tvel , r6rE 6avT

oloxl ipuuo I lq(x,0)dxd|]r .T

lj4

f |apc6EIylord:

l . l lo 6.rx861, 6rt to

dl,vcr. Et\roL T'{()(.0)

A0oo:

ycvrr€uutvo oloxlir,trra I.JlT

€R 'z : x€ t l r+ - ) r ' r€ to r l l ) .

nux{,x" ( l+U ' )

dtd'l ouy-

H n€ploxi T 6€v alvoL rppoy-uavn. €eopo0le uLo orclouoLo xLr-pl|,lv 6! taro|-o nou

G!= { ( x ,0 ) € R ' z : x€ t1 ,v l , U€ t0 , l lv=2,3...), TLo rnv qxo;\oueLo G!

y€lLreu!avo o loer ipora j J f (x ,0)dxdi , ouyrr i ! . ! aro^ i ros drdv

r r Ioroaripold

J.J f(x,{) dxd4.T

010Kr1HP0TIK0' IoIMot 2. Etrcv6d,rou B. 221

roxt€L 6rr l im-G!=T KoL drL Gv c G!+1 v=2,3,.. . . oArouu€*

f ( x , ! ) - . - ! ! I l I - , o (x ,q ) - -x ,0+02)

- x , ( l +L r , )

Etvou l--! !-IL | . t ELvoL oK6uoi x ' ( 1 * , 1 , ' ) | x , ( 1 *0 , )

r ' - [ [s(x, , r , )dxdJ,=[ f I d] , ldx=l 9x [ ' ' or : '

" i j r , r ro xz ( l - , ! ' / ) I r r ^ ' r o ( ' - . ' l )

=[ ]] 'r ."+.nrr;=-(] -r)(toqeor-roseoo)= + (+ ')=+('-+)

Etvq! H-t;= g-+ (r-

"1-)=+ 'ooo Jlrr,,para,r=*.:,rr*uo p

rnv l, L6L6rnro !o 6oottvo otroxi\ iporuo ouyKilv€!.

2. l|o eletoorel. oq moq Inv otyKlLon to ofoxlipnlo l"llouv{x'z+U'z)dxd'1,

q)6 l o v T - { { x , ' } ) € R ' z : x > 0 , 0 : 0 }

To XoOLo T 6€v €LvoL qpoy!6vo.

o€opoirte rtv oKotrou8Lo r{,iv Xr,rpLovGv r6roLq f lou

c!={ (x,0)€ IR'z: X2+(r2:v '?, x:0, U:0v = 1 , 2 , 3 . . . 1 .

E L v q r G ! - T K q r c ! c c ! + 1 ( v = 1 , 2 . . )

M€rooxntorLcoLru€ o€ nol!(6q ouvraroyutv€q. 6trou!€ x=pouv8, q=p4!€.' I \ o r J€ 0 < x ' + r - 1v ' ? -0 < psv koL c r6uo 6cL 0=e= * , B fo l * i ,

1 / )r r - l' L ro r L t vc r I -

l oLv {x 'U ' )dxd . ! - l l l poJvp 'do ld0 -" l '* r rEL6 i n ! t rd rd 6 rn ld o lo r l ipd ld duv ip rno f 6e ! 6Ldrnoed drdsepd fpdonr ro

o to le6 io o ror r ipodrs , ro o lox l ipb ld I 6e ! ] ]nope i !d u ro loyudg€ l rd r€u-5 ' : d \ r c . o d k o ' o u ' d a c ,

r * Yn€!$u ! [4o ! !e d tL d ! o rd ros rns o lo r l fpodn< € i ldL opgoyu i lLo rdL n u t rdr o o , . 1 d o ) o r ^ i o d l ] d d . d p - r o a . i r L r - < l , o p e i q g ( l \ h L v \ r d , : \ d L(Bt tb 1 tB t6

| | i er I )h(! , )d{ ld): I s(} )dr I hr ! , )di t ,r , . y

228 KEOIÂIO 1I

I=|*u'rrtl"=X nuv". 'our,rc ro

oioK;\ftpnuo I 6€v oovKl{,veL*.

r /2 r / 2

tnuo'llae= | tnuv'-nrotJ

l i m

d0-

I qgv'z6sv undpx€! dpa

8') H ourd0rnor f(x,l) tuv ELwi ggoytCw or[v n€p.oxll T rnc oloxlll-

eeopo0!€ !Lo ouvdornon f(x,qr),nou etvoL opLou6vn o€ 6vd xtraror6**xo-pLo T eKr6c on6 6vo ontr€[o l ,{(Xo,0o) rou Xoplou . Ynoe6rouuE 6r[ n ouvdp-tqo4 6€v elvo! qpov!6vn o[nv nepLoxil rou onuelou ll, oeopotue yLo oKotrou -

8Lo Pv ovorKrdJv nEptoxd,v rou onueloul l r 6 & L o f l o u P u * , c P u ( v = 1 , 2 , . . . ) .

YnoStrou!€ 6tL ev ELvo! n oKolou€Ld

!,rv 6Loltro0jv ToJv 6pov rn( Pv xot o-K6!d 6ru l in €.,=0 (r6re €[voLP , ' t ] ) : * -

-

8e0ipol,!E ! Lo oKolou€Lo KlelofljrvxoL rrpovlavov X{iplov G! rSrolo nou

G,=T-P! u€ l lm Gv'T-{ l i ) Ko! c!ccv+1Av yr.o (dgE oKohou€to Gv (dnoq rnv

oploc{r.) un6px.r to*** l ln' lJv + 6 ; ,

f (X,0)dxd{ KqL elvoL avoc npqytor[ -

x6q opL0u6c 4 $r€ l t ! . 6rL n ouldplnon f €LvoL oJ\o{trnodroLtn {u€ yevL-

x€ultvn tvvold) oro xr,Dlo T. 0 npdylor[K6q op[€!6c ! ovoud<€rcr, v€vrrcu-g6vo ol*MFtr I' ea6ouc rnq f oto XtipLo T.

lu!9o^[<ou!€ !e:

* t ro ropd6eLy ld d l td n u rd ro o lo r l6po!d du ldprnon 6 . ! 6udrnp€d dadSepdf tpdon lo . A ! ro ! rd ! qerL lo iodyE rn ! 6Ld6Lrdd ia u ro loyLduo$ rou o lo r l rpd ld -ros ! -q l ,Lo ouyr .xp l lCvr a rorouo lo ldp iov Gv rqL Eo io 'o l -c d rL

. l i l l l f ( x ,+ )dxd '1 ,=r , ( l€R) durd 6Ev 9d egdoodrL ! . rn ! o iy r lLon rou o lo -v+.?;G x l no.iuo ro <.

* * * Yro t i re rdL r j ru n ouvr ip r ron t (x , { , ) e i tqL o lox lnp idLuf de rdse d ' i voro

l l f ( x ,U)dxdoG

Jjtr,,uro'0,r,T

oIoXAl tPo] I rcE lor l tMo! 2. Xnorodyou B. 229

Ildpar/ipnor 1.

Av n ouldprqon f elvo! un opvnrrK/t r6rE n EKioyi ULqq (c! ldvoouyx€KpLutvnq qKolou€Loq €Coo{ulL<6! rnv ouyK,\Lort Tou ysvLK€u!6!ou o,\o-

t lK ^ n p d l o ( 0 q € ( ' 6 0 0 v ( , u o r < 6 . t v o L l i m _ J j f ( X , 0 ) d x d r = r ,

( e € q ) .

cIlap(lrionon 2.

Av n uf l6 ro 6lni6 oloK,\ i lp{ruo ouvdprnon 6€v Etvcl opov!6vn ornv r(€p!-oxi n€pLooorlp{rv 0116 6vo onlelov,r6rE 0s0rpo0!e ro ot]vo;\o Q rov onUEL{Jvsur6v, oplcouu€ !e rov LbLo Tp6no rnv oKotrou8lo Pv,on6r€ ylo Tnv qxo,\ou -

8Lq GV=T-P! axoute l im.G!=T-n. Kord Iq dl , \o loxLrouv ro LbLd dnoc ornv n€-pLmoon rou €v6q onUELou.

nooqriptot 5.

: ro y€vLK€ut6!o oôKlnpdJUoTq B

!E qLrr6q nou Loxt loov oro vevLKEUuavo

eL6ouq !ox0ouv ovdloyes L6L6rnrec,oloK,\npdlqro o' a L60 uq.

tlqparlipnon I

MLo nro dII ; \ i nqpouoloon rou 6Ll i \00 y€vLKEUtAvo! oloxlnpdJuqroq B, ,a_douq €Lvor n Eq{c:

Ynooarouue 6TL n ouvdprqon f 6€v €lvoL {rpqvutvn olnv n€proxi €v6qsn!€Lou H rnq Ktr€Loric KqL apaylcvnc n€pLoxitc T. 'Eoro 6rL n f €lvqL o.\o-xtrnpdol ln oE x68€ Ki€lot6 xol Qpoyutvo unooLvo)\o G rou T nou 6€v n€pL6XELt0 11. I ' l€ t lLd ruxqLo xl€lorr t Kounl) in C! (un oroo€pi l )r , nep!Kt€Lou!€ rco4!€lo I \4. 'Eror opl<eto! lLs r€ploxi G rou f i . npopows ro 6Ln,\d o,\oKt i -

t t 'Pqra l l f (x,{)dxd0 undpx€0.

T-Gu'Eoro drr d Elvor n ordorqon rou on!€Lou t4 on6 rnv KqunLtn C Av ro

C ) k q L € L v o Ll im l l f (x 'U)dxdo uf ldpxeL** (yLo xdo. oKotrougLo xolnutrov

rpovUorLKdq opL0U6C, 16r€ n f i tvEraL oloK,\npdol ln (UE y€vLXeuU6vn 6vvoLa)drov r6no T.

' np6r€L.aL yLd oro lo !9 [a* ' ' O t o ! d ' O 1 d t € G r { M } .

230 KEOMAIO II

Av €Lvor f(x,U) z 0 yLo Kd8€ (X,U) €T opKel vq ndpoulE uLd koL !6voouYr(€xpllAvn oKolouolo (olnriturv Cv (g€ d - 0).

nqparipnon 5,

H evvoLq roLr yEvlKeu!6vou 6lnlo0 oioxlnpduqro( €nEKr€LveroL xoL se !nqpqytaveC ouvqprioELc, nou opl<ovtdL os un 0poy!6vo KtreLord oivoio. :qvn€pLflroon ouri ovoydlooas orLC 6uo npotyodtev€g n€pLrTdoet(.

l lqpo6€ [yBdta:

l. 8.r,,po0u€ ro oloxlipi4,o JJ ,AiU, OrO0 6nou T-((x,U) €R,:x,+{,,s t}T

o') Nd eqetoqrel ov npdrcLro! ylo y€vLKeugtvo oloKiipouo.

B ) I{o npooolooLot€l ulo oKolouelo Pv xo! n ovrtfiorxlt inq Gv

A0on:

o I H ouvogr4ot f(x.{)- i4ur €Lvo! oprot iavn oinv nEgLoI i T eKt6q q-

nO ro on!€Lo 0(o,o). H f 6€v €LvoL epoyu€vn or lv napLoxi rnc ooxiq rov o-

{6vr^rv 0. 'Apo ro olo( l i iourua €Lvo[ y€vt xEuuavo 8 €L6ouc. ElvoL f(X,{ ,) .=

x t - l t t O v t o x 6 e € { x , U / l € T - { ( o , o ) } .

B ) 6Eopo0!€ osv oKoloueto Pv (oKotrou8ld ovoLXr6v nepLoxdlv tou onlaL-o u 0 ) r t v o x o A o u o l q P u - { ( x , { ) € R ' : , ' - { ' . + , v - l , 2 , 3 , , . . } . t L v o L

Pr+1 c P!. H 616!€rpoc xnc P!

dou xr+q,r= $ ) *cr r,oxte,.

t im E! - l lm * =0 . 'Exoru t

qx6!o 6iL l im P!={0}.

H o(oôu8[o c! etvoL

n oKotrou8Lo:

G!:T-PV=t (x, | ! ) €R'z '

o .$ sx1{ '< 1 ,v=2,3 . . . }

elvq! n oxo/\oueLo eu=f (n cxrlvo rou xrl-

ue G! c Gerl xor !€ l im G! = T-(oI

0A0K( f i lnTl l (ot , \0 I I [M0[ 2. I rov6dyou B. 231

l l t r ,2a-a2. ltvers! ro olorripouo I=lj !+ 6nou T=t0,11't0,11.

fxq') To olo iprrro €[voL y€v!(€rru6vo;

B') I{o moo6r.opLor€[ uLo dxo]ou8(,o ovorftdrv Xriplo,v Pv (o! n owlsrol-xn rne orclougld Gv.

y') No El€Taor€[ ov ouyKltvsl ro oloxliprduo.

A(,on:

o ) H ouvdor4on f(x,{)= -+- ol l€Lpl(€ro! Kord uiRoc rnq nl€uo6c x=0

(6po 6€v €CvoL gpoygcvn qrnv n€gLoxi rlrv onuet{rv tnc nl€updq x=0). EroLIoolo(,\ iprrruo eLvor v€vr.x€uucvo 0' et6ouc. tLvoL f(x,t) > 0 vro x60e(x ,0 ) €T - { ( x ,U ) € R ' z : x=0 } .

9 ) o€opotuE oqv oKolouglo qvoLxrdv nEpLoxdv xnc T, rnv o(olouolq

P \= { ( x , { ) €R? :0 l x< I - , 0s0S1 v .1 ,2 . . . } . E t voL

l i n P != i ( x ,$ ) €R 'z rx=0 , 0 3 0 S l l

H qKoôuglo q opl<etoL o tnvox6on:

G,=T-PV={ (x ,0 ) €R '? r+ 3x s l ,

0 < 0 s 1J ELvoL G, . Gu*r. Et-

I im_G!=T-( (x,U) € R'z I x=0,0 s0 < 1)

v ) tl urd ro oloKlipolo ouvdprnon €LvoL trn dpvrlrlKit. 'Apo opK€l. vo n6-pou!e uLo ouvx€(pLu6vn oxoiou0Lo. Exou!ei

l l -v -oxar - l u ,d , ! l -+ ax- l . r , ,a , r , J x - -adr= 5 [ {3 ] ' tx ' ' , 1 ' l v .l. t x." )o ,), . . x."_ r / ! I / !

=+ [,(+)'^] eneLoir : im ] l,-(+)'^ l=+.npo(rtnr€L 6rL ro yevlx.utrCvo oloKllp(|lo I ouvKJ\[v€L Kc[ €[voL loo !€

5

KEOAAAIO II

AYUENE: A:IXI:EII

l. llo uroloyrd€l ro 6Ld6 oloxll|0qlo l=ll(x'z+'t')dxd0 6!ovT

T={ (x,'r) € R'z: x € i 1,31, {, € t0,+ -)}

Aoon:

H mpLoxi T 6€v el,voL 0poy!6vn.cu={ ( x ,0 ) €R ' : x€11 ,31 , 0€ lo , v l ,v=1 ,2 ,3 , , . l e lw l G ! - T | (q !Gu 'Gu* r . Exou le f ( x ' u )>o KoL

Iv-,J'J (x':+'! '?)dxd{i.,|. IJ"

(x+r')d0ldxG ! ' -

- l lx ' , r-+ lax=l (x '** )ax-

Earrpoiue rnv oKolouo{.q xoo{,t0v G! u€

"" (+ -+) . { z ' } tze*zu ' t E lvoL 6!dq l in } (zsv*zv ' t . * - ''Apo Io o,\ot l ipoluo 6€v ouydLveL.

2. b etet!'eL |,C rpc nW 0r4f4 ro ycvlxet||trc 61 6 olorlf|r|ro

I.ll Fi:rr drd .

A|.r:OuoLXd ro R7 6Ev elvdr Opdvuavo. ELvoL f(X,qr)= ]; fr6r r o v,,o *te.

{x,V) € R'?. Oel,ipoi,!€ rnv o|(olou€Lo ftrv Ktr€lordrv l(oL epdvuavov flepLoxdrvG!= t0 ,v l t t 0 , v l v= t ,2 ,3 . . . t l .w r

'"=l{'*.tlt *m t-=l+tt" d#=J-f ."r"orwra+=

]a'r=J - f r-r,ox'r': ;a'r=

OIok fiPoIIKot ôIIXMOX 2. t&v6dyou B.

(Otrouue W{ 6rov {=0 - r,r=0 6rov 0-v. o=v'? Elvor vd{,=do - d{-31 ) =

=l froeeo,a" | -ot;l*a,' + | + +'Apo I . > + l nv . E re !6 i l i n + l nv=+- ,

'Apo ro yevLX€uutvo oloKlipouo 6€v ouy(lLv€1.

3- lla 6elx0€[ 6rr m yevLrcularo 6!n\6 olodipqo I=[[ tri xtfi:'tiFft

ul6pxEr, 6tr00 I={(X,t) €R'?: x I l , { 2 l}.

A(ron:

l l neploxl l T 6€v ELvo! epoyutvn.e€ooo0ue rnv okotroue{,o* I(rv xtreLoxd'vKor qpqyutvrrv n€pLoxdrv

| , l r r 1Gv=l r++,v11r+ v,v l , v=2,3. . .

etvqL r,,= [[ dxdo -- Er x{,vl F:1TiI-l-t

"l ox I d'! =

i+ !x f -T l i1{ ,WrI

l r n i l ; =+ lnv r= i lnv

eLvqL KoL l ln Iv-+ - .

- [z ,oq, r f r :T l " Iz ,o t .o4]1" =a(rot 'ow- i - ro{ .0 ! ) ' . , tuo,r r r r r , \ y ' i l

r r n 1 r ; .ri,n rv= r,n_a(to{Eaw:r-"{* .} )' "(1-o)'-'.'Apc to v€v!x€ultvo oloKl ipor ld I un6pxa! KoL eivor, { ,oo ue n2.

4. Ito urctroyrooEl. ro yevu<eurAvo 6rnl6 oloxlnpoud r-JJ*-o(x*1)axa*

6*u 1=111,111 en,: O s x, 0 S '!). 7

x+r'd-r x,FlIV ( x , ' 1 , ) € T .

234 KEOMAIO II

ioon I

H reploxi T 6€v €l,vol egoy!6vn.€at po6!e qv orolouolo ftJv nepLoxri]v

Gu= t (X , { ) en ' , 0 < x < v , 0 < { , 3 vv= l ,2 ,3 . . . ' . t l . vo r .

r =[[,r '"-{(x+rt r"r,, ,-' " JJ - -

=f ll-e-,!(x+1)dr-|li{x.1 ) ld0-f -e-,,(x+l) t,du--ljre-,!{url)-e-,r ro,r-

={"-+ ar-,r,t- $ fi-*r'*trar-,r,tu*1 ) t=-re-ot!+ #=-t"-'-1'# ,"-"-u-,1--"-u"1* *;""-

"* =

=- +.'- # J-

- #-r==" o+r+o-o=r.'Apol l m l v - 1 - I = 1 .

ELvor f(x,0)=0e-s(x+1) ' o v{x,,1,)e r

te-{( \ , r1) l ! -

5. l{d utlolovro€el to

'onou T={(x,{,) €R,: I 3

A0on:

To x(,ipl.o T 6€v €LvoLepov!6vo. E Lvorf(x,U)=e-x" > o v(x, ' tr)€re€opoiu€ r4v o(oJ\ou8to tLiv

G! - { ( x ,o ) € R ' z r 1<x < v ,

x<U<x ' zv=2 ,3 . . . ]

yevLx€ulavo 6rnl6 oloxlioqro t.[le-xaxO.t )T

x , xs0<x 'z } .

rc"cu r =[f"-raxau,-f [[ "*a,i,1ar.[ "-rtr]r'ax=[ e-x(x,-x)dx=

! J J J , t i x I I X i ,

0/\ol(^HPOTIlO[ loIIXl{Ot 2. tro!6dyou B. ?35

=l x'e-xax-l xe-ldx= (rogoyovrLKi otoxAiaron) --l xr(e-x),dx+[ x(e-x),dx-

- rv ru r !E-tx,e-xlv+l 2xe-xdx+txe-xlv-l e-xdr--lx,e-xlv=Zl x(e-x),dx+lxe-xly+

l" rt+l e-xd(-x)=-tx'?e-xlv-2t;e-rlv+21 e-xdx+lxe-xl!+te-xll=[-x,e-x-xe-x-e-)(]y.

=-(v2e-v+ve-!+e-!-e-r-€_r-e-r )-3e_r-e_1,(vr+v+1).3e-r- v2+v+l *- >3e-r-O=

e!

=3e-r. 'Apo l im l !"3e-r=J,

6. tlr urotoyldEt ro olortttr.{ro lj ;i*' *" 6mu

TT = { ( x , f ) € R ' ? : 0 3 x < + 6 , 0 s { , < t }

Aton:

tt r€gloxi T 6ev El,vo!epov!6vn. ELvol

f(x,0). .:+-q-? z o v(x,U)€T

oEgvorru€ rnv eugelo X . v(v.l,2.3... ) r@6ll n rgoctov dEow {,'0.

eeopo|Jle rlv oKolouo{,o Tl,iv d\oLordjv Kqr 0pdvtl6vrjv xtdplov

G ! { ( x ,0 )€R ' : 0s xs v , . 0< t s I v . i , 2 , 3 . . . )

*,", r"-J"[f ro ;fo; ao]r.J"14';t,,"r+'r,n,r,1ar.

-*fi ,J';,"*',0,-* -,r r]ax--! i4"-,-'-",1'0,=- | i, ["-("'l "-'*""]0,=-] ir"-,,,.+,-1i*;*in=*

| "-,',".,or-,,-il-f i"-,i r-,"r=-| ["-,".',];+ 1"''1'=

236 KEOAAAIO II

_ l-T ("-("-1)e-t)-+("-'-"'). eruo,

,=* ri ' 'J"-r"-'r- I *-"".i].+(1-+). m" r-f (r-|).

7. I{o e[erooreL uc flpoc rtv 0[oo(n to oôKtipr4ro [[ 4xdJ ,r onouri (x,{,).t={(x,0} €R,: x,+,t, > t} (|(€Ri)

Aoon:

0 r6noc T 6€v elvol 0pov!6voc.

ELvoL f(x,0)= --.. :-!- > 0 vLa (dgE(x'+l lr ')^

(x,U) € T. 0e0po0u€ rnv dxolouoLo

tov xrdpt0v G!.{ (X,||,) €R'?:

1 s X ' * J " < v ' z , v=2 ,3 ,4 . . . )

ll€rooxnlorl<ou!€ oe nolLk6q ouwelov!6vEq, E[wr 1 s p s v,

o s e s 2r, *[o!*] =o "0..

t1x,,11. J,. rooo '

l lI . - l l

Gv

- 2rr- z:n

6mu

iaEr:

! 2 ndxd+ , =[[ - ! . - 6o66=[ o-2. ' : .oo[- d.- I

^-2(+2 1! ^--* l l F **-1,, "l ""-l - 'r-=r j,t€r;"=

, -t.rr, , i . -2.r2 ., ( -- ov -2x+bo - x<lL P l r = t : t L v - r , + {* - t

+ av -2{+2<o-D1! 2 [

16re e t "cL t . - [ I ap f ae=t rnp t l te :? r .2nrnv - + - .

dootrt to otrord\ i lpold ouyKILv€L 6rdv K > 1. T6r€ e(vor l im Iv= * =1.

tb urDlovldel ro v€vr.€wlvo oroxrbr o f=f[ t[4

l={(r,a} €R'z: r: o, t e *e ," "

-, - r? (l+xtt(lr ' l

N r€pLoxi T 6€v .LvoL apoy!6vn. ElvdL f(x,U) > 0 yro Kdo€ (x, | ] r) €T.

O oMHmTII(o' lotltttot 2. tiav6dyou B. 237

Kdvoule rov uErooxnldrL(ry6 0=02

ELvoL d|(6!o T'={ (u,u) € R'?lu : 0,

dudu

laron:

o€r,rpoige rnv q(oiouolo rnv Xtiplov

Gr- t (u ,u)en ' : osu sv , i < u s u ,

v - 2 , 3 , 4 . . . ) ' E r o L € L v d L

\=[[ ;-;1- o"a"["r]- a,['.=[ o," / r ( 1 + u ' z ) ( t + u ' z ) r o ' r u ' J , r +

=2ltoEeQul;troEooull/v=2(roEeav-roEe00)(IoEeov-roleof ) !1-3

,(+-,X+-')=$.,n,*$.

f. ilq U|totrovldea lo vevrrcutrcvo 6!nl6 olollf|'qra

r=J{"- t+ t-x' or*.

+0

x= ll. Eauot

2u

1 .

H nepLoxn R'z 6ev savqL qpoytrEvr. E[voL f(x,U)=e-l0l-x'1> 0 ylo Kd€e(x,U) € R'?, o€r,Jpotits rnv o(oloue[o tov x(oplov

cv= { ( x ,U )e n " : - v3x3v , - vs { i s v , v -1 ,2 , . . . }

r tvou r"- l le- l { l - r 'axaU=l I e- t+t- r 'axau I l e-x 'axl g- t+ta{ , I -" t " t l v J_v J_ ! J_v

* H teOLoXf i rns o lox l ipoons . | :w! opoot jvLo." tuldprJtdn dptLq.

238 I(IOMAIO II

=41 e-x'dx l e-1,1,ldl lJ=4le-x-dxl e-ad'U.J . J ,

- -+1"'apo l i r r v=4 r i r _ l e - {

dx l j r _ l * ' ' J ,

-

=ri "-t 'o,

: $- a-zy' ir. rnoutvuc I. t im I!=2

4 i "

^ d X l i m t - e - ! + 1 1 =

JJe-t-\ ax*T

10. ilo unoloytog€t to Y€vtx€u!6vo 6tdd oroxl,lq o

6 r ! o u I = { t x , U ) € R ' z : 0 s X < + o , 1 s u 3 2 ] .

^0o[:

H nEpLoxl l T 6Ev €LvoL QpoYtlovn.

Elvou e-xr l-k > o YLq xdee (x,$) €T.

o€opo()p€ rnv oKolouoLo l(,lv xtlpLulv

cv= t (x ,U) €R 'z r 0sXsv , 130 :2 ,v -1 ,2 ,3 . . . ]

Etvdr r.=[[e-rJ-L dxd,l=[ e-idx[ , l-L al=-1"-t d(-x)l 0- 'r d, l-t-e-r] '2t ' ! '1' l=" l J

=211-e-" I l \E-1)=2t,/2-1 ) (r- -. . | -> 2('r2-l).

'Apo 1 im l !=2 (V2-1 )=1 .

l!. ila unolovLoo€[ ro YevLx€ulavo 61fltr6 orodiq,,uo t= [[ olot,,

6nou T e(vcl ro €rl( 'ne6o 0x! (k€n)' ] i( l tx ' .{, ,) '

A0onl

ELvoL f ( x , { r ) > 0 v to x6g€ ( x ,L ) eT .

0 t6[oq T 6ev elvo[ QpqY!6voq. Oe0Jpo(l!€ rnv oKotrouolo l(,lv x(,]pl0iv

Gu= t ( x , , 1 , ) €R ' : x ' , , ! ' s v ' . v -1 ,2 .3 . . . 1 .ferooxnuott<ou!E oc nolrxaq oLvr€toYutvEq' otTouuE x=pouv8, 0'on!0'

FLvoL 0<es?n , 0spsu - t f f i =p . 'Apo

Y tq K l l Ecvo t :

* t i w L r w o t d d ' L | . " d x = ' "

o ol(AHmTnOt ]lOIIttO[ 2. tBv6drou A,

2 a | 2 i v 2 r

t , ,=[ [ [ d?,1 .o-+ [ [ (1*p'?]-rd(r.p')-+ [*rr ' r ' r '*r ' ." r o t J o ( t + p , ) K f , ) o ) o z ) r t - a '

2 rr l= f l r I t t r -u ' t ' - * - r tae- zr l i r t { r*v, )1-(- r11e1lr . { - l t r*v. t1-"- r1

o ) 'Eoru ( < l - l-x > O, T6r€ El.vol l lm fv. l ln+ t( l+v')r-k-l l=*-

'AOo ro olodilr,I/d I6Ev ouvrAl,vaL.

B') 'Eoto r< > | - l- l( < 0. T6l€ elvar

l l l_1,. lT_* t( i rv ' )r-r- . ; ! . ' roo r"$.\ , + - - ; '

' 2 t v

v ') 'Eoro K.r.T6re eLvou I, , . I l f -*] .*4t l Io( l+l) lor.J o L r o r r r v - / J . \ L ) o t + e ' !

- . .2r= i I t tnr t .e '11oas"+ nnt+e') l ; l de=n tnn+v,) .

'Agu l lm J.,. l lmln ln(l+vz)1.+q 'Eto! ro olo(I ipolo I 6ev ouyKttve!.

12. b 4stotcL dv dryrl0vr! !o otorl,lryro I-[[ I o-,'i u.'T=t(r,0) €R'z: x,+1, s lI

Alior:

H ouvdptnon f on€LOL<exoL VLo (X,0).(-j,0). 'Apq 66v eLvdL qpavutvnolnv l l€9loxi t tou oQu€lou l , l ( -1,0). El .vo!

Xorp[rJv G\)r

Gv= { ( x ,0 )€R ' r - 1+ v Sx s 1 ,

-VFFs0sfFP, v= l ,2 . . . )'[Xoull€:

,, ..,/r-f"l lg=l l l 4la'=) ! v l+x t . r l -Y l+x Ju! _11: -v r -x -

f (x , , ! ) . + > o v(x ,0) €T-{H}Vl+x

e€opoou€ rnv o|(olouo[c tov

,-,.r/..,

'!1 .E-T

. . . .'\',i$\.

, l ,

,'4r'4(x , - / l -x ' )

244 X'OMAIO I1

, t r , rF r r r= l l+ , ! l dx=l -=' ' L v l * x l , / ! : i : . '

, v 1 * x

--* lr'-*r'a | =+ (,-+) '

rcuo, r i ,_* ( , * ) " '=* * . ' r "

(2fFF)dx=?l Vl-x dx=). . 1

r= l im I !=+ v8 .

13. l{o uroôvlooel. to olo(trlDtuo l=[[ !f!9 onou,,i f{T={ (x ,0 ) €R 'z :0sxso , 0<0sx)

ioon:

H ouvdprno4 f 6€v op{.(Erol YLo d0€

o|rELpt(Eror (Enolavoq 6€v ELvoL {ppoy-

u6vn qrqv nepLoxi rr,rv onlefuv rou 0B)

otpvouue eu8€Lo nopdtrtrnln npocrnv eu8€Ld x -d i , rnv x -0=+ (v€N*) .' [roL opL(ou!€ rnv oxotrouglor

c != ( ( x , t ) € R ' z : | . t . o ,

o<0sx - | v - t , 2 , . . . 1

d 1

=- l l2(x- ' ! ) "1 'dx=-21

Il r U\

Id x - : .

EL*. [[ q-cq =f tt " is- ld'"-[ti n*o 'r, rri vx-{ I j/,

( x , 0 ) € T 6 L 6 r l

o > 0

(016 !iKo( rou 08

0

Ir / I - nI I {r-,1,)-, d(x-o) ldr=t J D !

[(,"'*)* -'t ].,=-zf rrr a,.zf w a,=

=-z 'tv' rxti,"'! rx"/" ti," --z 'llli to-ru'* * l"'," -( +In ].

t tvou t i rn [ -z f l i (o- 1 /v)++ ( ; ' -

# ) ] =* . ' . ' roo I - ]q '4 .

t rEL6 i n ou ldp lndn f ed laL !n dp ln tL r i rd ip !ou ! . !Ld ouyrExpLr ld \Jn dro lo lg to

o oK HmTIrc! lolltuot

t4. ito unotroyroe€l to ororrrroorc r,JJ # 6rout w-x

T- { ( r ,0 )€R '? : 0 3x 10 , 0s0s B} , B > o

Aoon:

H ouv6prnon t(x,t)= *W-x

onelpL(€rdr Kord Itkoq roo 0B6po 6ev €{,voL epaygevn.

g.or9o0!€ qv eueeCo 0-x" *nopdllnln rpo( rnv €uoEla 0.X.0pl,CouuE aror rnv o(olou0Lcr,

cv- { (x ,o) e n '? : } =u=0, or*

'EroL txouua:. I

tfg-t [[ '410u.....-,

1' W-x '1 ,.. "o W-x I

\'.1-> +F!/,.

,,rm r.$0.4.

l0onl

H ouvdptnon f(x,U). J- 6evELvoL qpovutvn ornv nepLoxi rnqapxic rov o[6vorv, Ei,voL

;fu' o " tx,,t,t e u to,ott.g€opoole rnv oKolougto iov

deLoEiirv KoL apov!6v0v xbp(.rrv

t5, Ild unoloyLo8e! €o'6oov undoxer ro oAorlioogo [[* a*aU O*u

T. t0 , I I , to , t l . r

v .1 ,2 ,3 . . . )

g.',./T+-+B'/,-+ -!

'H f c iwL un opvnr l r i ou ldp tnon y lqurc

16

io{pEqrE llLa oqr€tp!!6!n arotrou-

242 I€OI]\AIO II

P != { ( x ,U ) €R ' z : } . * . , , } s0s t , v=2 ,3 , . . . } . E l vo r

l l . , ' , r 'r"JJfr*ar{ [J ,fu*].'-l tJ i}fl]o*=J t'"r''1rPr L/r L/e I l t I ln

cx=

?in+2

, --ilT

tJ.. ,I = l l

G

dxd'IEF

Ilr I /e

"z rnz-2- (+.i)rn (+.'I (+.,)-('.'+)"(,.+) -('.+).2n W 21n2-2+1+1=2 in2. 'Apd l jn I!=2 tn2=L

16. d u|loloylogeC to vrvr.€ulCyo 6url6 olorligi|ld

6mu T. t (x ,a) € R ' : x 's ls tL

iton:

E [voL T " { ( x ,$ )€R , r x , <U3 t ,

H oovdprnon f{x,r!}. * 6ev eLvq!v0-x,

opov!6vn oxnv nepLoXi x6e€ on!€Lournc (oun(ltrnq ||,=X2 (dL €Lvo[ un opvn-

oeopotpe tnv oKoloueLo rov xn-

Gu= t { x , { ) € R ' : x ' + l s r l t s 1 ,

.*.y'If u=r,,,...,

E t v o L

'.{--Tlt rdxd0 -f I tVtr - "z ) L l -. " _VT:T/ii x.+ l

-+l dx=l-\n:-V;

t,t-x" t-1at,l+'t]ax

. J'"*,=J .r***'--Jr'*r'*,-'--J*ddd:orno-o.

* rt*" r-_f r. f, =0.

- l s X < i ] .

OIO(]\HP0TIKo! A0IItMot 2 tra

-ft+n. ,, .t{-rt= l t2vdr -x / | dx .z i (v l -x , -v1 lv )dx=

, L ) ) r -,/i*1-1lv

J:--:--| v . _ - / . . . r _ 1 v r t / e

=21 ,ll-rax-2,!wlxl * =lrotnux-xr'1-x l - -' -

-6t;

-rl-v;

'APo l lm I !-n 6nlo6i I=n.

-'L-v'

1E (/ +. f" t-) -"e^,,f1.,/] /T+,."r*(-f -r) r/{ Uqq -',fr /;Y =

t7. tto unotovloo€a ro yeylx€urravo 6.d6 olorlliplj|[ I=fJlnrr+raro

6nou T.{(x,r!) €n'z; lxl+l{l s l}, or elwe vwtlt 0t. t yJ(t"".

Aoon:

H ouvdprton f(x,Ul.1n lx+016€v eLvor ooovuavn ornv [€oLo-

xi rov onu€Lov rnq €uoELoc x1{,=0To xoplo T opt<€ra! on6

r!C .u0€LEq X+qr=1, -X-U-1'-X+'l=1

x-qr=1.

oeopoiti€ rnv oxolouolo I{iv xtrpfu,rv

Gv= { {)(, ' l l € R'r -: i s x+U S l- . - 1 s x -U < 1 , v= j , 2 . . . 1

- J'r-^,"- "=r+Jvi-*n "-*,=J"-..",=+ (,, +2t)=+(,,^,;"*.) =.f <roenux+r{-f).

244 IGO]1j1AIO II

n6yo rnq ouutr€rptoq tou oxiloroq ge0rpo0!€ rnv oKolouolo rrdv Xopl0rv

c ;= { (x ,u ) €R 'z :+<x+U!1 , -1 <x -0<1 , v=2 ,3 ,4 . . . }

Kdvouuc rov uErooxnumLoU6 x-!-u, X-{=1.' E r o L e l v q r r , = { { u , u ) , } < u s l ,

- 1 < u < 1 v = 2 , 3 , . . . }

- ' B1t++= +, r(x,0)=rnrx+0r=rnrur=rnu.t . , l l , l l l I t f'oro

z" = i ) ) rnududu=i J I J rnuduldu=+l turnu-ur; / !du=E! - r 1 /v

='J: t' + ,"+.+ ),'=+ (-'-+ '"+ .*),:=:-' -'.I'npo l im # - -1 - l im I "= -2 - I= -2 .

18. No unoloyro€€a ro y€vtxeu!6vo ofoxfrlgao I- [[1nfT+Tzaxa!, ov

eavo! vvrdotd 6NL wr,^Lyar. To xoglo t tregLrla[etaL ard rov Ko(lo x'zro'z-|.

n||on:

H ouvdprnsn t(x,0)=tn#i[Z Oev eLvou rppqvuevn ortv f lEpLoxr] rou rrn-66v .

o€(,jpo(l!€ rnv oro,\ou6lo rov Xoptov

T != { (x ,0 ) €R , : { - s x , *0 , s t , , t =2 ,3 , . . . J

M€TooxnuarL<ouu€ oe notrLK,e suvr€Toyudveq.o t rou lE x=pouve , {Ep4 !6 . ELvoL 0<B < 2n , } sos r , f f i =o

. 2 1 . r , 2 a . , 1 ^ . , - . 2 4 - -'noo I ,=l I rprnpdprds=l l l ( f ) r"oaolao=l l l#rr l' t , 1n 4 t ' . r , t ' ' I J . l l ' J r l \

r 2

1,,"(+' l).,].'-J, [[+'no],,, * ro'r',,lae=

{#'"} j,#)',,3"-^(-,i'"}-* #)'*, ;'Aoo l im I ,= -+ 6n lo6 i r I= -+

O oKAlfnTItOX Aoll'lrot 2. Xrd!6d1o! B. 245

19. ild 6€LX0€1. 6rL ylo 0 < x < | ro Vavlreulcvo olorliFua

f f - #*= 6nou r={(x,0) €R,: x,+{z s r} ur6pxEL.f (I-+{,-)

Aoon:

H ouvdptno4 f(x,0)= ; l ; ; 6€v eLvqL qpoyuavn ornv n€ploxi ror i on-t x ' + 0 ' ) "

!eLo! 0(o,o). (€v6 €LvoL 0poy!6vn de Kdee unostvolo rou T nou 6Ev repLdxelro on!e{,o 0} Ko! eLvol un opvnrLki .

oeoootjte lnv oKolougLq r0JV xtdp(0Jv

Du={ (x , , r ) en ' : $ - < x ' * , t ' s 1 , v .2 ,3 , , . . }

efvcu I"=f[ -{I!$-. . tterooxnu,,ra(ouuE o€ rotLXtc ouvr€rdvuav€c. o6roui/e6' (x-+U- )

x=oouve , 0=pnu6 , ELvoL 0s0szn , $ sx ' *0 ,< t . f = r . t .'Eror n okoAou8lq ruv Xr,rpa0jv D!, u€looxntor L E€xo L olnv oKolouolq

D i= t (p ,e t l | . o t t ,0<os2n , v=2 ,3 ' . . ] . ELvoL o r6uo f f i=ol i' '

u4;7-- 'r 2 t r ' . _ f p - 2 K + r , r . l ' r | / t \ 2 ( 1 _ K ) l'Aeo rv-JJ + dedo=Jodoj

. -?i; de=znl L:z..'"-l;rlt/, = # l]f "J lD; . " r/',) "

[ 1f t ou o '* ' tELvoL i in I != 1

[ + - c v K > ]

' A p o y l o 0 < x < 1 e [ v q L I . *

20, t{a unotoyloset ro yevlxcuu6vo 6LnI6 otroxtrttpouo t-[[ l#HA{,on:

i)

H n€pLoxi R2 6€v €LvoL qpoy!6vn KdL n ouvdprnon 6sv E[voL 0poy!6vn ornvn€pLoxi rnq opxiq 0(o,o),

KgOA]\AIO II

El-voL f(x,t)= *"1q, 'O r"o xdoe (r,{)€R,-t(o,o,}. O€tlpot!€ rnv

cxo loue [o r i dv x r rpL r , v Gv={ (x ,0 )€R, ' $ <x=u , }+=u , v=2 ,3 ,4 , . . . ]

r tvor r , .J ' [ ' l ** ] r* ( i ) . . rxour,E1/r t / \

J"sla" a*=[ f *r"-{- ]" . f -+of - f -"r.0( } .f ) rt=-1/r Llr

f .+ "=ft

,roo ',L" +*=+ J" f ao={rr,or;,,==+{ lnv- ln - r , )=# tn - i - =+ iv , ) Le>. - .

t'Apo io oloKlip@q 6ev ouyKtLvEL.

21. a moloylo€€l, ro vevrx€uIovo 6uil6 olorliift,uoT=10 ,11 r10 ,11 .

A0on:

H ouvdpEnon 6€v €avol

E t vo r . f { x , { ) - * >0VX

V l o x d 8 € ( 0 , 1 )

{x,0) rou Tnou n f opL(erqL.o€r,ipotU€ Tnv oKoloueao tuv Xt,pa-

lf4 oroo ono,'i nx

opLouavn Kdrd !i(oq tnc nl€opdc OA rou x€ipoydr-t l . r l *

x-'rdxl ,!d{,=ztxtl:, + IU.t:-

22. llq motoyroo€( ro yevr.€wavo 6urt6 olorliqdlo l.[[ axaO

6nou r . t (x , ' l ) €R. , t 3x , 0<{sxr } . r i l *x l '1

nton:H reploxi T 6.v elvor opqyutvn (o! n ouvdprno4 f{x,o)- rul)z 66v el-

OIol( tHPnTI loI io l l tMot 2. t rav6drou A. 247

voL Qpov!6vn o[nv nEproxit xdo€ onlslou rnc

f ( x ,U ) > 0 v (x ,L ! ) €T - { ( x ,0 ) €R , : Q=x } . t 1€nepLoxlt T ' ILc n€pLox6q

T1={ ( x ,0 ) €R ' : x : 1 , xsUsx , }

T ,= { ( x ,0 ) €R ' z : x > l , 0 s { < x }'EroL e LvoL:

eu0eldq U=X. ELvaL

rnv euesao Q=x xopt<ouu€ rnv

0 s x 'z , v=2 ,3 . . , )

v=2 ,3 . . . ]

d,td)( . fufr(,r-xr

'J, J,

r= [f -q!,! - f I dxd'! +{ ' to -xr '41 1* -* , '

+ l I dxduir (u-x) '

(1 )

8€upo(r!€ t!C qKol\ouea€q lov XopftJvi

G - { ( x , . ! ) €q2 : t . x . v , x * l .

Ko t d = t ( x , 0 ) €Rz :1<Ysv , 0s0s

'Eror txoulE I =lI dxdLl ' +[[ dxd'!' l r . { { , -x ) ' ) l / . (u-x) ' :

=ri.J, tt I0o ulloloyloouuE rs 600

I

r r r ! IJ= l l l d+ ldx= - l! r r Lro ( l l j_x) ' I I

'-f ,

it. *!"_!g!L(U-x)'z

r€i€urotc otroKtrnpdJuorqi E Ivqr1

Ir,r,-,r-'l*- o*=-{

l0-+ 4-'-,--,'1*=

T " r T t T u r r , t= l ; . , dx - l vdx t i r l ( - " ' . r ) dx+v (v -1 ) - t im t t nx - t n ( x - l l t ' - v { v - r )=

a- l

244 KEOAÂIO II

= I im l l nv - l n {v -1 ) - l no+ ]n (o -1 J l+v (v -1 )= ln J - ' r r r ' b9 : l - l * l u - r t .qr.t v- |

crl

' A p o l i m J l = l l m l n $ + r r m l n $ + l i m v ( v - 1 ) - 0 + ( + - ) + ( + 6 ) = + 6 .v+q ' F re " q f l

'EroL €[voL 'lin l!.(+- )+(+- )-+-. 'AOo ro opx[(6 ol\ori\ilol,,Uo 6€v ouYKlL-

23. o 6Erx0eL 6rL xo ytwxeu!6vo 6LnI6 otro(trio0,lq'

I = f l f l u ( xa { , . )dxd , ! 6ev ouvx l l . vEL . E tvo | , l = { ( x ,U )€Rr : x20 ,0 :O} .! l J

TA(,on:

H nepLoxf T 6Ev eLvoL qpoyutvn.

o€opo0ue rnv oxo,\ou8lo &rv xopl0vl

G != { (x ,U)€R ' : : 0 sx sv , 0 :0 3v '

v "1 ,2 ,3 . . . ) .

etvou l,=JJnu(x+U,)axarJJ(nrx'ouvrl'*ouvx'?iuu'?)dxd{,=JlnuxeouW'?dxdu}Gu Gu

.f J"*r,nu,r,axa,r+ JJnux,ouw,dxd0=2f nrx?dxf ouw,du."rr@,tLvoL rjn_r!-2 ri'_ [Jirr,arf*w.a+]=zj nux.axJ 'ouvold{, =

2 +V+ +V+ =T ' '

Ard !o etoy6uEvo 6uoq our6 6€v !nopo0!€ vo oulnepdvoutl€ 6rL To o,\oxlil-

pr,ruc ouy(^LvE[ 6[drL n ouvdptnon f(x,0)=n!{x 'zru'z) 6€v 6La[npEL arogepd np6-

onuo yLo l(dOe (x,U)€T, €roL n ouyK€(pLgdvn oKolouoLo xtrptov nou n{pau€ 6€v

etooeoll.<e! rnv oovdlon roo otrod\npd,lotoq.

*oloxlipola rov Cay]ey.**IIpdr€LrdL yLd rd olorhpdlara rou Flesnel .Cwr I or.rvx'zox=l ryx'2ax=

)n J t

=+Ut B lErEq ' rd r ,ooer i6q 36e.

cloMHPoTIrc[ lollII{ot 2, tra!6iiyou B. ?49

Aq 6e0jpiooule d)pq uLo dAlt q(otrololo Xrrplov rnvc ! = t l x , 0 ) € R r : X " + { " 3 v " , x 2 0 , 0 > 0 v = l , 2 , 3 . . . } .

METooxntqxl,(ovlqq o€ no,\LXC( ouvr€rcy!6v€c 6xouue :

0sp<v, 0sest . i i f f i -p .

t t t r /2'Aoo I , - l ln ! (xz*u ' )dxd ' != l de l {n !0 , )ede=totX/ '+ l n le ,de '=. ' " ' .

=f t-ou"o't|.f {1-ouvv'z). To tim +(t-ouw'z) 6ev undpxer,\*@

'A9o ro oloxl ioolo 6Ev ouyxlLveL,

24. Itd unotoyr(oEt ro otorltipoto l.[ [ "-tx'*+'lor*

Aoon:

Ar ro dpLo rnq o lox l l l poonq ooLveBL d r ! 0 gx s+ - ,0s0 <+ - .'Apo

n n8oLo)(it T rnq oloKiipoonc Etvo[ ro flpdrro raropro r!,v oq6v{rv.H oloKlnrurto olvdpqon el,vd! Serr(i, oei,ipotle rnv o(ol\ou€Lo !,rv

xoocov rv={{x,{) €R'z x'*0' s v' pe x :.0, 0 z 0}l eraoxnld{.(ouueoe rollKtc ouvx€royltvcc. OtxoutrE l=pouvo, {=pnuO. Etvor O 3 O S + ,

0 sx ' : * r ! ' : s v ' ? -0 < p ' zs v r -0 s p s v . ELvdL o*Auo f f i =0 .

'EroL n sKolou8lo flJv XopLov larooxrlpcrl(Ero! orn! qxotro!8(o

T i= { (0 ,0 ) €R '? : 0spsv , 0 .ea+ } ' . ' E rou axou rE\/2 \ n /2 \ n /2

' . , - f [ ["-" ior lor=l t +"-o'01-o.1l6e=-][ 1e-" ' r ]ce=

=- { u"i; t :/2=- +r"-"-"'r *= +(' j") r=.+'Apo l im I != + = I .

n-r

KEOAÂ|O ill

TPINAA OAOKAHPrrMATA

To rpLnl6 otroktrnpduarq opL(ovroL ue ovdloyo rp6no !. !o 6Lntrd ((ol

Loxtouv dr(oL86c qvdlovo eerrpiuara oloKlnpooru6rrFoq xo! ovdloveq [bL6 -

rnr€q). >rnv nEpLnroon rorv rpLntrdv oioklnpqldfl,rv n ouvdprnon f eLvc!.lLqouvdprnon rp[dlv u€xoglnld)v Kor, o r6noc rnq oloxlifldonc tlvoL uLo deL -

srl t xoL qpov!6vn neproxi E rou R3, (olEp€d n€p[oxi). 'EtoL IoLt6v:

o ' )Av E . ( i "1 ,2 , . . . v ) €LvoL uLo x lELo [ i xqL gpov lavn uno r l€oLox irnq xlELoriq Kor epoyucvnq neoLoxic E, (E c R3, E, c E yrq kde€ i"1,2..v)U i ' ( i " 1 ,2 , . . v ) ,€ l vq t o 6yKoc tnq E . Kd t ( l t , n i ,< i ) eavqL 6vo ruxoLo on -

uELo lnc E , , ELVqL

JIJ','E

,U,z )dxdudz= l im t f (q i , n i , ( i ) u i

B ) Av €tvoL E=Iq,0lr ty,6l t te,( l (opgovdlvro nqpoi ln. \e[Lne6o) r6tE

| | l f ( x , 0 , 2 ) d x d ' l d z = l i m t I r ( t t , 4 r ,<L ) 6x . j 60 r6<vi= l t= l

L

6nou v-p.tr.u ELvo[ ro n]i€oq rov o[oLxsL0rbriv lropd]trn]Enlll6bov nou xse6-

vd* ax€L 6yKo Vj ik=6xj60i6<k.

v ') Av EavoL E=to,Blr ty,6lr tE,<l Ko[ ro nl iooq rov op8oYlJvLtdv nq-pal lnlel lLn6bov elvqL vr r6re eavoL

* t tn ! r€pi r loon aur i n l€r rd lnrd!Los to! !e-rd l ! t€pou opgoyowou

l lP l l rns 6!a!EpLonE P e{uL n 6La-r . i -rdpdl l r I . t rLna6ou.

O OX HPnTII@X ]loIIDOt 2. trar6dyou B. 251

JjJr r , ,,r,, . ra xa,r,a' =!

r r - t \ _ t . r ! " z \ x " 5 r x -\ * o i = 1 i = t i = 1

_ ' r r I r

Y'loôyloE6c rov rpt ral6v ol'orl.lpoldrov

- o') EttdrrolrFtrro tlurld olod||](|cro

Tq elovolnmLxd rpLntrd oJ\oklnpd,lcro oplaowor. kor unoloyf<ovroL oxpL-

Bdc u€ qvdtroyo rp6no !a ro €novol lnm0xd 6Lrld,6nlo66 oL oloKtrnpdro€Lc yL-

vovrqL 6 Lo6oX [ l(d, I el,pdvroc 6l€C rtC tiaroo,\nttq sto0ep6(,el(T6q qn6 €xeLvnoC npoC inv ono{,o ytv€roL l(d8€ eopd i otroKlitpr,jot.

napo6cl,vpata:

l. llo unor\ovLo€sL ro enovclnritrKo olo&\fiodUo:

- l t r ' r r ' . . r . r - .r"t | | I (x+,tr lz )dz tdx]dt,rD r ro LJd I

Aton:

,r*,. ,=J [J [,,.,r,* = ],*],r=l[l 0*.] ),,1*=-1" [- ** -i,],*'l ( ]"*, ),t'1,,,',,r,r4,r,=[ru- $ ]''r.

2. l{o uttolovlooe[ ro oloxliiqrlo:

t' r t"r t'*"'1 tj i'1 (x.u-z)d{,]dxldzE l ' ' 6 . . x - z

Aton:

tL"c, r-J,IJ, Ix+- # ' '*]"_,*l a'=J.IJ, xrx, 'r. $L 41y,21-

-\t\-z) - (x-zr' -'r*,r]orlo'=J,IJ G\2*2zz)dxfdz=1.:zzx,.2z, xtiaz=

= l 4z :dz=o ,J - l

KEOAÂIO III

- 8'l Ymloyro!6q urv rplrliN olorlrp+6u.v.

Ioxi,ouv ro €r6ti€w e€opiloro:

0E6o[!s l.t t l

Av ro (pLnl6 oiodieot/a JjJf(x, ' ! ,2)dxd{dz undox€1. 6nou e eLvor roE

nopollnl€nLne6o to,Bl ' ty,6l ' t€,11 KoI ro opLoutvo oloKiiprruo

tch (x ,0 )=J f { x ,u , z )dz undpx€L y ro (60E onu€ [o ( x ,0 ) t ou Io ,B ] r I y ,6 ]=T ,

t t l t - 1r6!E ro oloKi ioolq

Jl I j f (X, ' ! ,2)dz]dxdqr un6pxEL xoL E[vqL

l l t t t l l - lI I l f ( x , ' ! ,2 )dxdudz. l l l I f (x ,+ , r )dz ldxd0I ] T '

{ 1 )

jjJ,r,,,r,,, ro,ol,*.f If lJ'rr,,,,r,,, r*]o,rlo,

e&onn 2.

Av ro 10!rl \6 oloki l lpolo JJJf(x,U,z)dxd0dz

un6ox€L,vr.o uLo ouvdornonE

f optolavn o€ 6va Kleror6 rdno E,tou oodooeroL ndvo roL xdud ord TLq enL-odv€Les z=g(x,{,), z=o(x,{,), (o(x,U) > 0(x, ' !)) ' (dL ro otroxtrtpoua

I(x,U)-l f(x,{,Z}dz (oL urdoxcl ylo l(608 onti€to {x,qr) rou €nln66ou' q ( x , 0 )

Xrrplou T (nou elvoL n opoi nooBotri* rou E oro ellrlE6o 0xU), r6re ro otro--iii-irdlew drc! ).,iue npoBoli oa €wooile 1I(l!@ opsi rpoBori.

oro*r1pra,o 1611x,,!16,1 (lnou t(x,,1).fr(x,,l,z)az)

r6i€ on6 rov tl,no (l) 6xo!!€:

jjn,,,r,**"Ju[J'r.,,,,rra,r,]a*J'[[ [J'rr,,,r,,,ra']a,r,]ax

IlqpdrilOrpn r

Av undpxcL to

vLa (d0€ x € [o,B]

jJ j f (x ,u ,z)dxdudz.E

'Apo elvoL

q49!.lHPoTIIot ̂ orlxuot 2. trq!6dyolr 8.

"r*," IJ 1J,,,,:i,,,1,zyaz]axorp, erLonq undpx€L xqL eLvcr

l l t r r r ro (x ,+ )I l l f ( x , ' ! ,2 )dxd ' ldz= l l l I f (x ,0 , r )dz ldxdQ' ; -

; "e r x ,+ )

I]aporiptol :

A v € l v o l T = t ( x , U ) € R ' i o < x < B , S ( x ) < , l ] < t ( X ) ) ( q L r o o i o k l i p | , r -

rt(")to I I(x, ' !)d' l undpx€! yLo xd€€ X€to,Bl r6!e on6 rnv (o) KqL 0116 rov'e ( x )

rtno (2) txoup€:

t t , 8 , . t ( x ) , . 8 . 1 ( x ] . d ( x , @ )

lJ j r(x, ,r ,z)dxd{dz- l l I (x, , ! )dxd+=l l l r {x, , ! } c,r l rx=l l l l l r rx, ,r ,zrazlaulux' ; ' r

' o "g ( ' ) ' ' o ' 6 (x ) ' o ( r ,+ )

* ^ f rd6i n r .p loxr i E rou Rr repLrre ie lqL ard rd € i r : i .6o x=q, x=8, ord r !sru lL!6pLx;s . rLa. i !e !€s ro! ; louv o6nyois xo! idreE U=g(x) , 0=h(X) xcL y. !d -t€Lp€s ropdlh l€s rpoE to! d[os z 'z \ * dn6.Ls e i lod le les z=r(x,0) t z :s(x! { )H opsi t rpo8ol i tn< r€pLol is r oro . r i t r .6o oX4, e| :wr n r .oooxi ( Iou R' ) , T !eT = t ( x , { ) € R ' : o < x < 6 , h ( r ) S , ! S s ( x ) . 1

l 2 )

nqparipnort:

Av6ôyo o€opiuoro ue rq nponyoo!€vo 6lorund,vovroL qv EvoiidEoulE rou(poiouq r( l iv UeroBtrnrdv X,0,2.

lldparionoq:

To €n6!€vo n6pLolo 6Lv€L rnv nLo ouvneto!6vn xqL npoKrLKi u6€o6o uno-ioyro!00 ro0 rprnlol oô K,\ npdrloToc .

lltgLoUo:

' E o r o 6 r r E = { ( x , U , z ) € R r : q s x S B , 9 ( X ) s 0 : h ( x ) , r ( x , U ) S z s s ( x , 0 ) } *

o ( x , 0 ) I lf(x, '!,2 )dz ldoldx

e { } , Q )111',,.,,,,,**"=1'1Jj',11

254 KEOAAAIO III

6I Iou oL ouvogrio€rq g,h,r ,s etvql oov€xe{.c,

Av f(x,{ ,z)Elvol lLa suv€xhc oto E ouvtprnon r6te e[w[

jJJr,,.r,.,**'{u11""11E q g ( x ,

s( X,,! ) I .lf (x,U,z )dz ldu ldx

! ( r , q )

Ilapal{prlon:

:rov nponvoi,u€vo rtflo or

onJ\6C oio(Inpdjo8rc yLvowqL U€

Tnv o€!p6 f lou Eelxvouv oL qYxt l -

leq. 'EroL no(iiio yl.v€rcr n oAo-

KAiooon |,rq n9oq z (ra x,0 0€{,,-po0wqL oro8€p6) tot6nLv YLVEToL

n olodipr,rs4 r,lg flpoq 0 {ro x e€-

opeLroL ota0€p6 ) l(o! r6;\oq y{.vE-

roL n oioKillooon (,J( npoc x.

ouolxd avdtroyo !e rnv lopel)rqq fl€proxiq E €tvo[ 6uvor6v ot

rp€lq on,\tq ol\o(i\npdrc€Ls vo ylvotlv xoL u€ 6ltrn oelpd.

z=s (X ,0 )

rM!og.o!6c dv op[rrv fiK olodtrom €v6c rpLnloo olorlofuroc

- t(!lov!(i r€ptoxi I rou Rr

l lLq roLooLdoldin xtrEtorh Kor opoY!5vn nepLoxi t , (E c R3) tr tY€ldl . kd-

vovrxir rr( rgoc 6vo dlow !,rv ouvr€rcYg6$lv,ov Kdo€ eu8alo mpdllnl4 npoc

rov dtovo our6,no0 [ep!d on6 tvo Eo0[€pLt6 on!€to il rnq E,rtt]v€L Tnv €n!-

qdv€Lo (ro o0vopo rnc f leploxiq E),ro nol0 oE 600 on!€{,o.

-lhod6logrry6c u.v opttN tnq oloxllFotlc

Av n f i€proxh t €{,voL KcvovLKl] ,rdTE 96pvou!e !Lq €uBELq noodl ln^n npoc

tov dEovo z'z KsL yLq ouoolp€ro x (o[ 0 nooodLopl<oulJe ro onp€[o z=zr(x,U)

z=zz(x,0) nou n.u€el-o t6!ve[ ro oavopo xnq E. Td ol leto ourd eLwL rq 6-

pLo rou 2, npoBdloutE |(qr6rlLv inv n€ploxi E src €nln€6o 0x{,. EroL opl<a-

ro! ![q neploxl] T ro! anrnabou 0X0. To 6plo I(,Jv x,U npoo6lopl4ovrqL on6

inv n€pLoxlt T xcrd ro Yvt,o[6.Av n nEgloxi t 6ev €i.vor- rcvovLKi rdle rnv xtpL(ouuc (6 rc (oL ornv

n€o('mdon rou dLnôt oloKlnpdjuoioc) oe (qvovL(ec unon€pLoxcc, gdpvovroq xo-

ldllnlo enLne6o.

CIo( IHPnTIIO' lo l I t t4ot 2. E@!6dtou B.

'EroL lorrdv o[nv fl€plrrtrrsq uloq KovovlKhq nrpLoxiq E,ro rpLn,\d otro-Ktr ipolo !LoC ouvdprnone* f(X,U,z), €(epd<erqL Ue rnv 6Lo6oXLXi otr0,. \ ip{, lon:

255

'EToL xdvovrqq npdro

oioxiitpr,rtro rqe rlepLoxiq T

rnv oioxiilp{rort ('Je

on6Te epyo<6!qore

Kqrol iyou!€ s ' tvo 6Lnl6IIPoq z

kotq

z

napatigqon

Av ro rpLo6ldoxoro xopao elvoL tw opgovdvLo nopotrlntr€nln€doE=lq,Bl ' lY,6lr l€,nl,uE EEp€q tsp6trtrn,\€c npoq ro ErILnEbd r0rv ouvrerqyutvovKqL n url6 ro otroKtriEduo ouvdptnon €LvoL t4C Uopqh(f { x , ' ! , 2 )= f r ( x ) f , ( { ) f 3 (z ) yLq xdee ( x ,U ,z )€ E , !6 r€ xq r ' ovq iovLq rE ro 6L -riO oloKlilporuo** 6Xou!€:

x L,n

"c ,ouur i oreped nepuox' i

Ea doo! qudLrd undpx.u.

I ln repi r rdon ro! f (x ,0)=f i (X) f r (+) raL r r€pLoxr i T ef roL opeol i i lLo

dxdr.I t t t z r ( r , + )

I I l f ( x ,0 . z )dxd ldz= | l l I f {x ,0 , r )dz l' L ' J i t J z l ( } ' . | , ]

t6 llf , ( x ) d x l f , ( 0 ) d ' ! l f 3 ( z ) d z

t t t 1 8

' l iJr(x 'o ' ' taxa+a'=lE q

KEEAAAIO III

Ibpa6 € ly|rarq :

l. fio Fp€€o0v ro 60!(t urv X,0 mr z qro r0!d6 olodiq o

l l l f(x,t,z)drd' ldz 6nou I elwl n f leproxh rou npdlou ovddoo lov oq6kv trouE

neglrl€leta. 016 tlc snLq,dv€LEC X2+4V2-16-2, z.O,

loon:

H €nL06veLo xz+4t!2=16-z ( i ) eLyor nopoFoloeLotq evdr n €nrqdvelo z=08{.voL ro €n[n€60 0x0. H llpoBoli lou ndoo8olo€Ldoiq dio Enlr€60 oxr! €LvoL

n t i l€Lun x'+4t ' .16 - - i , - + =1 (npox(rm€L ond rnv ( t) vLq z=0).

ll 6;\i€r.0n Kor ou oerL(oL nlrd[ov€q opl.<ouv ro xoo{.o

T=t (x ,o) €Rz: 0sxs4, 0rOr f t i r t06pvoLrtrE lLo €u6€Lo nopdllnln npoq mv dlovo z'z nou n€pvd on6 tvo

eqJxcpLx6 onu€Lo !4c or€pE6c neplox/lq. H €uoeld ouri r6uv€r ro o0vopornc n€pLoxic oro onuElo A KoL B !€ z=0 ror, z-15-x'?-4r! 'zowlo[otXo.

' A p a 0 s z < 1 6 - X ' ? - 4 , 1 t . ' E x o [ t r o L n d v € L v q L

, @!P ro-1'-,,,1,'l l l I t t " 1 ^ t t I IJJJ f (x , ' ! ,2 )dxdr rdz- j l l ' l l f (x , ! ,2 )d , ld , l jdx' ; '

2. l{o molovloo€L ro p!nl6 olo ipolldl

j l l (x+U.ztdxdtdz 6mu E={(x,{,,z) €R': x:0, 0:0, z z O, x++}z s t}.E

0 ,0 ,16 )

oAOK,\H?0TIKO' IOrItMot 2. tM!6d-ro! B, 257

I0on:

H neploxi E ep6oo€rdL on6 rq €nln€6d rov qE6vtJv

x+'!+z=1.H flpoSoli rou or€peot oro EnLr€60 0X0 6LVEL ro

o6oeLe ntr .uPtv X=0! U=0 KoL X+U=l.

Kqr qtl6 ro €nLII€60

rplyovo 0AB u€ € l r -

And ro 6€ t r r€po ox iuo0o lvEro r d rL 0 sxs 1 KoL 0sU< 1 -x .To 6pLo mu z BpLo(ovidL oq €l iq: 06pvouue uLo eue€lo nopdllnln f lpoq

rov dEovo z'z nou rauveL to o!€pE6 € oro onu€lo l4l,M, UE xqtnYutvec z=0,z=1-x-U ovrLo[oLXq.

'Apo 0sz < 1 -x -0 . E roL €Lvd r

r r r 1 - ) r 1 - l - { r l r / 1 - x r r l - \ - + I l' . j I I (x.,!-,)dzd'ldx=l lj ll

(x*.!-,)dzlduJdx=

r t r r r - ) f . , r 1 - \ - ' l t , t ' , 1 - ' / l - v - , r , \ . r r= l I l l tx ' . r )z ' ? I doldx-J l j I (x+ur)(r -x-ur)+ -- ig l d ' ! fx=

_ i - 1 x . t , t - x- [

| t r . l -+r ( r ' , : , . ' .# )o**=[ [ ; {1-x- r ) (1*x . , ! )d tdx=

=f l,'-*+ ,, ,**,',d{,d-=1, + l',- l}#r],-' *=+ I,(, , *'-- ).'==+ l"(+-*+!,=+ t+, +.+1,=+

(0,0 ,1 )

0 ,1 ,0 )

KEOAÂIO III

3. ih unlovld€|, ro tp! 6 ortorttgqlE= ,2t,t3,41xt5,5t.

Aardn:

H o[Ep€d napLoxi E Etw[ op€oy6vLo nqpalAnl€nlfi€Eo.

t l lrr*" JJJ'+a,or,a'=J ll [J '{,d']d!]0,=l 'o'[,ror[ a'-' ; '

= t+ll t+1, r,ri.(z- |)(e- |) re-sr= fl

{. No unotroyLooel. ro olortipous JJJrOraraU,

6tlou E €LvoL Io no-E

potrl0AEr(nE6o !e to9u06c ra onueto (0,0,0), (3.0,0), (0,2,0) rc! (3,?,t).

A0on:

H npoBoli rnq erLodv€Locxou or€peoo oro enln€do 0X{,€[vo! ro op0oy6vLo ue xopuetq{0 ,0 ) , ( 3 ,0 ) , ( 0 ,2 ) , ( 3 ,2 } .E l voL 0<xs3 , 0< { i <2 .(o l 0<2s1 . Apo 6xou !e ;

JJjxuzdxd{dz=!

. l l l l l xuzdz ld i i ldx "

l l l 1 l= l l l -z -x td0 ldx=

.l -i x j- dx=l xdx=

=-z-=-t

JJJrr araa. o-uE

2."%/%'ffiltltl i ,2 ,1

lill,j,ilillililiiiliiriiriii iiiii

OIoKIHFnTIKO' lolltMot 2. Inc!6.iyo! B.

flsparipnort a'Evo roLnl6 olo(I ip{ iuo l l l f (x,qi ,z)dxdudz EtvqL aoo IE ro !n6ay drsv:

I

I1ap66 € LyIa :-o olo{ l ipulro | | lx( .-{ , )dxd, ldz (dnou

E

n or€ped nEproxi l E elvoL duri rou oxi lqroc)ELva! ,n6tv 616rL 1 oLv6prnol f (x,+,2) ==X(4r1'z) €Lvqr n€pLrl i oC f lpoq X xoL n orE-ped n€pLoxit E €Lvq! oulUeiprki l r lC f ipoc To€nL[E6o 0(rz.

259

o') H ouvdprnon f €LvoL n€plrr i l l jq f ipoq lLq lEroBlqr i kor r t or€pdne-ploxn E €Lvq[ ou!!€rplxi oS f lpoq ro €nl l l€bo nou 6€v ovcq6pErqL n !€rsB,\n -

B ) H ouvdprnon f ELvaL fi€pllrlt orq npoq Cvo <eriyoc u€roglnxdlv KqL nor€p€d nEpLoXi €Lvo! ouuU€rp[ki oq f lpoq rov 6qovo nou 6€v ovqetpovtoL o!

!€IagItrtq** fty ) H ouvdplnon f s lvoL n€pLrt i oC npoq rLC ipeLq terqglnrtC KoL r l

or€ped nEproxlt €LvdL ouUtlExpLKi uC npo( rfv opxit Tov qtdvrjv.

$titfx

T

1

Yrclovlo!6e udv rOLlMv olodnq#drov !€ latdoxnlcrLou6.

fevlxd: 'Evo rpLf i l6 oloKtrhpolq elvoL 6uvqt6v vo unotrovLoe€L nlo e0Ko]q o!xdvoutr€ o qurd ol,\ovil u€rooinrd,v, tlEroBclvovrdc 6roL on6 xqprEoLovAq ouv-r€rcYU6v€q o'6vq dnlo oirornuo ouvt€royUdvov (u,u,w). o€l l lpolrue Iotn6v rovu E t u o x n l o r L o l l 6 { x = X { u , u , $ ) , 0 = ' ! ( u , u , ' { ) , z = z ( u , u , w ) } ( o r ) .

l ' lE rov u€rooxrt lorLotr6 our6 n otroklnpurto ouvdprnon F(X,0,2) lErdoxn -

u o r L < e r q L m n v o u l d p r n s n F l x ( u , u , w ) ) U ( u , u , \ { ) , z ( u , u , y r ) 1 , n o r E p e 6 E r e -pLoXn qn€LkovL<€rqL qleLpovooit lqvrq rrnv f i€ped f lepLoXi Er KoL €l ,voL

dr l lo6 i t r .1 . d ! n f E i laL t r .pL t t r i oE rpo< { , xdL r r d r€ped nep lox i E e i lo lou t j le rpLr r i ds npos ro e r i1 .60 0x2 .

* * An lqo i f t .X . d ! n f € (sL r .pLr r r i os npos X xaL 1 , xdL n d rep . . i t r .puox i EeCloL ou ! !€ rpLxr i os rpos to ! t t tae z z .

KEOAMIO III

dxdlrdz=|f{f*#l dududw*. 'Eio! loxoeL o rtnoq:

Td 6pLo rnc otrokl ipoonc {qeopt<ovrct cn6 rrq Lo6rrtr€c (o,)xol on6Tnv ocEPEd nepLoxlt E.

l.{€tooxtgdarou6g o€ rol,Lv6prxCc ouvrEToy!Cv€q

l. 0plq6q xullv6pl|6v ouweroyutwv

H 06on €vd( onU€lou l.j=-(X,U,z) ro! )(d,Oou R!,(o0opt(erqL (oL on6 tou(npovlorL(ooc optoloic p,0,2 dnou p.(014 ), (M' €( 'voL n l lpogol i rou onu€L-ou 14 clo EnLne6o 0X0),6-A0l ' l ' , roL z (6nou z etvdr n xf tnypavn lou l t l ) .{olarle €n6lEvo oxito). An6 !o ox/llo rpol(omer 6rL

{1 )

(X-poov8, {,-puo! z-z}. (1)**ovotrd(oou€ kulrv6oLt(6q ouvrE@V!6-v€C €voq onu€Lou tt=(X,U,z) xou Xdr-9ou Rr) rou( rgoylarL(o(lC opleuo(lcp,€, KoL z,nou €notr48e0ouv tLCoX6o€LC (l). I lo vo bntui,oouue 6rLn Saon rou on!€lou ll (ooopl<€roLo-116 KU]1v69[(aC ouviEroyg6veq, vpd-qou!€ l l- (O,e,z ),

:rrq €qoproy6q rovof i€ vo € lvqL p 2 0,

r Y iE !0u !dEou! . d tL r op i louoa .ou

,' =;8,u,qfi,"(*" '.n r,"

A z-;

-qLaua$

(Idu8!a !6 op i (o uoa )-!rat

_3Lat

az

0(u,u,w), z(u,u,w) ; | ff[#$lo,ou*jJJrc,u,.ro,ara..JJJFrx(u,u,,.),E E r

/./\\ji.-/.^---l!-----f

(!iLv6pLd,v ouvt€tdy!6vov ro p,e,z €xltvovroL 6roLO € t 0 , z f l 1 .

-!Lau

-3Lau

A z-5;

o-"e * {i* Ard t r "s (1) rpor '3r teo p2=1'z+4,2,

OioKIHPnTIKoX Iot I tMOt 2. tndv6dyo! B.

naporipnon l.

0L opL0uol p KoL g €Lvq! o! f io l lK6q o0vrerqy!6veq rou onu€Lou l i l '

Ilcporllpnon 2.

0 yeoterpLK6q r6noq rorv sqlElov 11=(p)0,2) rou Xdrpou Rr u. p=o (q >0

m s € € p 6 q ) , 0 € 1 0 , 2 n 1 , K o L z € R , € L v q L o o p 0 6 S K u x i L K 6 e x l l L v 6 p o q , n o u 6 X € L

€l lo0Jon X'z+qr '=o'z (AToL arxoLoioy€Lro! o 6poq "KUiLvbp!K6c ouvr€Tqyu6 -

2. laraoxruarloy6q oe xulLv6pLx6q ouw€rdylEv€c

:rnv n€olnrLron nou etloule vs unotrovLoou!€ 6vs rpLnl6 o,\oK,\ipo!o xoL

o r6noq rfq oloxtr i lprronq n elvsL avq s0voio ouutr€rpLK6 oq npoq €u8€Lqj loq

o!€uxohlv€L vo xdvoule oliqyi !e&B,\ntdr! oto o,\oK.\ilgnuo xoL vo uEro9o0leqnd xoprEoLov6q o€ xulLvbpLX6C ouvT€rqy!6vec. 0€0jpotpE , \oLn6v !o !€!o-oXnUoILo!6 {X-pouve, ' ! "pnUe, z"z} (1).

l '4€ rov u€rqoxntorLoud qur6 n oloxlnporto o0v6plnon F(x,U,z) rdlp -

v€L rnv loppi l Flpouv8, pnU6, zl , n or€p€d n€prol f t E on€LKovL(€rqr q!91-

lovooi l rovro o[hv or€p€d nEpLoxh Er KqL ox6uq €LvqL

axooaz=l f, ifff i ldpdedz - dxdudz=pdodsdz. '[roL LoxrrEL o r(lroc:

I I lF(x,{ i ,z)dxd' ldr=l I lFtpouvo, p4!e, z lpdpd8dzJ J ) ) ) )

( 2 )

To 6p[o lnq oloxl ipl , ronc Kq€opl4ovroL on6 rrc Lo6rtreq (1) xsl

rnv srEp€d f l€pLoxi E.

. "r*" Bto-}ig:i =p. yrE!ou!rrou,r. d.L

261

ax

3{-

az

aXasa+a 9

-9

.?L

a{azaza;

p( X, ' l , , z) _D ( o , 9 , z )

262 KEEMAIO IiI

fldpdriplol '1.

To 6pLq rov p xcL 0 unoloyt(ovtqL !e dv6Aoyo rp6no u our6v nou €L6o-

!e otov ufiolovLo!6 Tou dLnlot oô(trnqjJUoroq U€ !6rooxnucrro!6 o€ noIL(6C

tloporipnon 2.

YnoS6rouuE 6 rL E l € l , voL n o rep€6 nepLox l l

E ,= { (p ,e , z ) : os0sB* , h r (0 ) spsh r (0 ) , 9 r {p ,0 ) szss r (p ,0 } } , ( 6 -

vovioq iov !€Iosx4uorrogd x=pouvo, $=pnue, z=2 Loxtrel o rirnoq:

JjJitr,,r,'rara,r,a.JB h r ( 0 )I IT Jh r ( s )

r r c r ( o ,o )

U eF(e ,e ,z )dz lde ]d0

8 r ( P , 5 )

llcpq6 € [ygctc:

t. lito uroloyrooe[ to otoxtftporo r.[[[xaxou, onout l t

E=t (x ,U,z ) €R3: Xr+ t , Sdr , X 2 0 , 0 :0 , 0Sz:h l ( r t rdpro op€o{ , ru -xllKot xul{,v6pou).

Aton:

0q unolovlooule to otroxtr ipt ipo !e Ietooxn!f iLou6 oe KUlLv6pLKeq oowe-

royuav.q. 0trouuE x=pouve, {=pnuO, z=2. 'EXouuE 0 S e S + (o rdnoc

lnc oloxAitpr,Jonq ElvoL rtTqpto op8ot K0xtrr .xo(J xulCvdpou)0 s x ' * 0 ' ? < o ' z - 0 s ( p o u v o ) ' z + ( p n u € ) ' ? s o ' ? - 0 < p 2 s o ' ? - 0 s p s d K o L0 s z s h. Etvor ox6!o fiii;:;f -p.

( a L f ( x , 0 , 2 ) = X = p o 0 v o = F ( p , 0 , 2 ) .

lle rov lerooxnuor[otr6 our6 norep€d n€ploxit E on€!|(ovla€rcL ou-

eLuovooi[iovro ornv orep€t nEploxi

E = { { 0 . 0 , 2 ) € R : : 0 < g < +

0 S p 5 o , 0 s z s h l n o u e l v s [op8ovdvLo ropo,\lnl€nLrIEbo.

(3 )

' E t l d u 0 3 8 - a S 2 r .

0l0K HP9TIKoI lorltMot 2. ttra!6d1ou B. 263

t t t tn t t ra I e' Ero! .xou!€ : '| .| ,l xdxd'ldz=,J J 'j e(oowo)dededzlo t 'Jo I J e,suvedelde ld,=E E '

l h l ' 1 2 - . r c , r / r r ^ ) t o h o t=l dzl ouvodol p'dp=hlnlo]:" l- i- | . +J o o L " l r

2. llo urotoylooEt ro olorllrq,o r{JJr tr'+'faxaCor.x,+{,s 1

Aoon: 2 sz s3

l ' lErooxn!o!L<ouue o€ {ulLV6pLKt( ouvrsroy!6veq. 06TouuE X=pouv8,' r-pn!8, z=2.

H n e p r o x i E = { ( x , 0 , 2 ) € R ' : x ' z + r l ' z < 1 , 2 3 z 3 3 } ! € r q o x t l o l l < e i o Lq l q L t o v o o i u o v r o o t n v f i E p t o x i E ' = { 1 9 , 0 , 2 ) € R 3 : O s O < 1 , O s O 3 2 n * .

2 s z s 3) . EtvoL cx6!o $fffi "0. 'aoo

t 1 P r t t , t 1 z r 1 - - r 1 r t t p , , r r r - r 2 'I - l I lz .p 'z.pdod0dz=l I l+ l d€dFl | + zaeaz=l l { - e l az=

-[ *a'=[ l ; .1l--9l-"=+ .J r . L r 1 2 q

3. tfo urolovlooe[ to roid6 oloxllpq,o flllaxAU, 6nou E ,lvo! rlE

ot€ped rcgLoxi rou np/rrou ov66o! ndv oE6vov rcu ot[(€tc! ord tnv lUtrLv -

6oari 6nltdv€ra x2+02-1 xor, a 6 to rnarc6a z-0, xa{+z-2,(C.Llone "Corsodl nnalisi DteDtlca")

n0on:

ll€tooxnuarLcouU€ oe xul!v6pLx6C oovrarovlav€q.e6rou!€ X=pouvo, 0=pnU0, z=2.

Eavo l 0s0 . - | , O<o<1 KoL Osz < 2 - (X+0 ) - 052<? --p(ouvoinro). Etvor. o"iuo fl{ffi -0.

r Blate rdpd. f ipr ron oel i6ds 25s.**To xopio T.{ (x , { , ) €R'z: x 'z+, I , '1 s r } € l \aL x l r lLrdE 6{dros.

KE0,1 AIO III

a / 2, t-J"

a / 2

[o'- $ r*"e-ret]'oe-

-J Ir-]r*"e.nrer]ae-

=[e- d r ,ue-o,uer]" i '=*-4

,/2'Aeo

ll ldxd'rdz.l

lil€raoxtgar|.o||6€ o€ ecrorxtc ouvr€rcy|roveq

l. ooiqdc Colpldv d!vrEtov!6!ov.

H e6oq ev6c oruel.ou ll.(X,0,2) rou R3 (o6opt(eror Ko! on6 touc rlpoyt/o-tr.(o(,q ooleuoic p, 0, rol e,'onou p=(0M), 0=A0li KoL o.T0N.(Btrtre oxi!o), An6 ro oxfuo*rpo(inraL 6t r: {X=pouv8n!e,0-pn!8ru0, z-pouvq)** (1).

ovoud<ouue qpaLplKac ouv-reravlavEc ev6c di!€aou ll rouRr, rouc npoylqrLxooq opLglotcp,0 (oL e nou enotrno€0ouv rr-CLodrqr€q ( I ).

fLo vo 6nldroouu€ 6iL n ee-on rou onle(ou l{ koooPl(erqL o-n6 oeoLpLK6C ouwerov!6v€C, vpdtooule =(p,0,0). ttLC eeopuoy6c u,lv oeoLOr(drvouvraroyltwv m p,8,e edeyowo! aroL 6ot€ vo el.voL 0 < p, 0€t0,2[1,0€ [ 0 ,n ] .

U' . iwL n rpoBorf rou or leco! U orotns ond ro M oro! dtow z '2.

Ard t !s (1) rpor '3r i€r , p 'z=x2+{, '?+22,

€ru ie6o oxq raL r to i lws tns rd!e-

s=toqeq f i , e:rotouv +^ vxz r!,2 +27

0 oKlHPoTIl(oI IorlEUoX 2. tra!6dyou B.

Ibparilptq:

0 y€otrEtpL(6c t6noc rov ontElov l{ '(p,e,a) 1ou x6Oou ue p-o (o > Oo [o0€p6q) 0s0<2n , 0ses r . t va [ n oeo lpd !€ €E tq ro4 x r+ ( r2+z2=o '?(610L dL(oLoloy€LtoL o 6poc "oeo[pLKaC orJwEtov!6v€q".

2. teromrarrqac oc Otplrcg drvtetlvutyec

:rnv nepLftrron flo! o5,\outrE vc unolovloolle tvo rpLntr6 olo(Ihpouo xoLo r6noc Iltq oloKîftlonc E elvo[ avo d(,voAo oulUatol(d tq flpoq on!e{,o, loqdLeu(ol0v€L vo k6voupe olloyi u€toB^ntdv orc oloKlipotto t(o! vo !€roBo0ue0116 xopr€oLsvtc se oooLpLKeC ouvt€toy!6veq. oeopo0u€ trolrdv rov uElooxn -

uo!!oU6 {x=pouvonlg, {r=gnuongo, z=pouw) (1). He rov usrooxnuoiLou6 our6n oJ\o(iuporto ouvdprnon F(x,U,z) [oLpv€L tnv lopqi Flpouv€nle, pn!on!e,pouvql, n orepe6 rrEpLoxi E on€!xov{,(€toL ouoLlJovooilqvld olnv oreo€d nE-PLoxit Er r(or. oK6!o elvoL:

dxdudz=l fif$5il dpd8dq - dxdudz-p'znladpdodq.' [ tol Loxo€L o rtnroq:

t ( l| | lF(x,0,2)dxd0dz=l | | Ftoouvonuo, pnri€nue,pouwl, ' 'znlodpdodq( 4 )

lo 6p[d rnc oloxlipoonc (q0opl,(ovro! dn6 r[q lo6xnreq (1) Kor an6orEp€d neploxi [.

Ihporipnoq l.

Av n (ft€p€6 nEploxi E rEgLd\el.e&Lo ) An6 rn oedLpo t€ .€(ooon x'+0' jz2=0'? t6r€ eLvoL 0 a p < q

"; - . , . - ,i Ersu ;i i ff i =-p.nue. EiwL ,iq, #e-&k6i =e.nua.

a lT6'-!!-aoaz

-!Lap

#azap

_.?.x.30

J9A 2ao

YnElolui4oq,e 0"" B{i*# =

266 ]GOAÂIO III

S € [ 0 , 2 n ] ,g ) A n 6

( Q : 0 ) r d r e

0o x0z, z0|])6 ) A r d

a € I 0 ,n l .ro 4ULoqoLpLo UE cqlooon X2+0'z+22=o2, KoL to en[nE6o zoxe l vq r 0 s p s o , 0€ t o ,n l , q€ [ o ,n l .

X0z, z00t X00a€ ro, +l

To rEropTo Tnq oAoLpaC !a €glooon X?+02+2,=a2 xoL To €nLnE-( x . : 0 , U > 0 ) r 6 . E € L v o L O s p < o , 0 € t 0 , + 1 , q € l 0 , n l .

To 6y6oo rnq oaol,pqq ue e€(dion X2+ 2+22.02 KoL ro €nlnaoo( x 2 0 , $ > 0 , z > 0 ) r 6 r e E [ v o L 0 s p < o , e € t 0 , + ] ,

e ') An6 to ntLoeqlpLo !€ €lLo0ron xz+{r 'z+zz=cz KoL €nlne6o X00 (z > O)rd re E tvc l 0spso , €€ t0 ,2n l , {0€ tO,+ I ( . r . I .

Ilaporilpnon 2.

Av n o[€p€d n€ploxh n€pLKtrEl€ioL on6 lnv o(DoLpo (x-o),+(U-0)r+(z-y)r=-r '? ( !s xtvrpo ro onu€{.o (o,B,v) xoL qKrLvo r) . Tdre Ocroutr€:

x=o+pouv€n!9' 0=B+pn!enuq, z-y+psuv9.:rnv nepLrrloon quri rq 6pLo r(,iv e,e, |(oL p €Lvo! iq l6Lo 6n0rq o[nv

[€pLm0Jon rn( ooqLpoc !e k6vrpo ro onuEl.o (0,0,0). E[vo0 qK6uq

I D.ii;tjil=e'n!0.

Ilspatllplst 3.

Av n orep€d n€proxi l lepLKIeLeroL on6 To €1,\€LooEr6aq

-&- - - i t - ?

=1 rdre 06rou!e x=ou, 0=Bu! z=vor, or6rE npox(rf l re! n

e[ ior,Jon u2+u2+r,12=1. : tn ouv6x€Lo lErooxnldrL(ouu€ oe oqoLpLkSe suvraroyut-vee Koro To Yvrdord.

. n / v ' l ' r \ l>r4v n€oimu04 ouri €rvor lf i f i f,ail "dBvp.nla.(Tc 6pLo r0JV p,8,q €LvoL ouid nou ovo{ptpovToL o[nv nqparipnon 1).

Ildpariipnon 4.

Yf lo8arout i€ 6rL Er elvdL n or€p€d n€pLoxl lE r = { ( p , 0 , a ) : a s e 3 B * , h r ( 6 ) s a 3 h , ( 6 ) , K o L g r ( q , 0 ) s O 3 S , ( q , 0 ) }

Kovovtoq rov U€rooXnlqTLoU6 X=pouvonUQ, (r=pn!ente, z=poulQ LoXil€L o T0noq:

* t j w L o s 8 - a < 2 n .

0 o|.^HrtTlt<ot AOTMO' 2. !trd

{ 5 )

Ilapatiipnot S.

nol laC 0opaS cvr l rou U€tosxnUotLoUo0 {X=pouv8ntQ, 0=n!6nUq,z=poove}XpqoLlonoL6[roL KoL o trErqoxnlot10!6q (X=psuveouvQ, ll,-pntl8ouvA, z=ptle]ornv nEpLnTOon rou !eroo)(n!oi !qUo0 oUtO0 €lvoL ]Jl-prouvq.

ncp06€ [y!qra:

l. rlq urcôvro€€t ro orodiporro r=JJJtx..,t'*,,taxar,a, OnouI

E={ (x'0,2) €R': xrro'z+Z'?s tA0o[:

M€rooxntot t ao qj€ o€ o0oLpLKtq ouvieroylavec. otroutrE x=pouvonto,U=Pn!8nu0, z=pouva.

Etvcr. 0s X'z+r], '?+z'z s 1 - 0 s p2 s j - 0s p s j .

E tvoL qxd lo 0sBs 2n, 0s0<r (oL l$ f f f i l r . . r . xorf (x, ' ! ,2)-a'z +gz +2, =92.

. . r 2 t r r ' , r ' . , ' n , n , ^ " , , , ' n r , t . ,rpo I=l [ ] l l o ' 'o 'nr:oao ja,r lae=l I l* n, ,pl 'opre=l l l *a, ,a, |a,r . J o t J ! J I

21 2a= -q1- [ ro*,p:]ae= { [ ae- -f 1613.- -on' ) o , . J r 5 ' _ ' , 0 r '

2. t{a unoloylo€€l ro oloKtipqo t=[[[ otOU. - - 6noo

E=t(x,U,z) €Rl i o ' S X,,+"+2" s 8,1 a,B , O.

Airon:

11€TqoXnlqtLCouu€ oe oqc!pLK6q duvr€toyu6v€q. e6To0lJ€

{x=pouven!0, {-pruon!o, z=pouvq} (1}. ECvoL lffiffil *,*, ,.

e . . h I ( S ) _ . g , ( L 0 )

l l l ' j rF(e,€,0)re,nualdeldq']doh r ( 9 ) s r ( a , g )

JJjr rr ,,r, , . r ara,r,a' =J

268 KEOA]\AIO III

0 s€s2n , 0sqsn KoL o :p3g r . E l vo t ok6us f ( x , q r , z ) -

= - l- = -L =F(p,€,q)' l4e rov lcrooxnuqrlou6 ou!6 nlx2 +{z +2, )" t I h")-h o'

ol€p€d n€ploxi E an€L|(ov[(EtdL oueLtrovooiuovto ornv o[epef lteploxfl

E ' = t ( p , o , q ) € R 3 : o s p < 8 , 0 s e < 2 t l ' 0 < 0 3 n ] n o u € [ w L o p e o v d r v L o

flopollnA€nln€bo. Apar 2 r B,1I

E

= | iuodel ds

dXdrdz =lll + (,"n!e)dedodg-l [ [(**')"]*]."

-

dp- l -ouve l ; lo l?n t lnp l l -2 '2n( l nB-1no ) -4n ' ln +

(x'+0'*z

2r

43L+ (9'.+ l=+

I

[ 1

3. tlq molovlooet to rpld6 otro(liloqrd JJJataftt o-" t €[t/o! n

E

oretad |rsgtoxif r|ou Bploxerc! ltod 'ln oFlpo x242+22'z rot dw tou d-

vou z,=x,{ 'z, (z : 0).

A0ort:

l.1€rooxnuorL(ou!a oe oooLotKAc ouweidYueveq. oerou!e x.pd!v€nue,{,-pn!8i!e, z-pouvq. H et[ooon tnq ogolpoq Ypdo€toL p2=oouvQ - p"o!w.

H EItooon rou xdvou vpdeelor p2ouv2rr.p2ouv2en!2rp+p'nut0nu'o -

ouv , , p=nu ,q - , p= * . ' E r sL e l voL 0s0s2n , os ,ps * Kd l

0 s p s ouve. tlvqL oK6ua I Uii## l"o'rr.

.2\ . .a /4, , !uw t2nt t r l4 t ; rouvo rm"l l laxa+a.=l l l l l o'nloaolcoJoe=J [J" [$nuoJ, aope="E

I i r t t r / ' l r , t2r ta/ \ I .^.2 ' l ouv"ol ' /a.+ | l l ouvroauodo ldo. + | d6l -ouv'odouw= f t€];'[- =i-Y],

* 'EXou! . -d '?^ l a2+4,2+22 3e? ^

l+ d ' : s o 'dw'?enu'zorp2ruztnr-2o+p2ouv2o s 82- c z < p ' r r u ' o ( o v v ' $ + r l ' 6 ) - p ' d ! v ' e S g ' - d ' < p ' ( . r U ' o + o u v ' e ) < 8 '- a 2 s p 2 o)

A(,on:

€ t rou l re -L=u , * * , i * ,

'E toL Dr .p lox i E ue rooxnudrL -

<EioL ornv neploxi t ' ((u, i .r, t{): uz+uz+w2 31}. ELvor,

0 (x ,0 ,2 ) -trlu,u,wI -

-!! _oL,u aua'! e{au au3L .9Lau au

-!L

a'!

-5r

q00

090

00Y

fro vo unotroyloouue ro retreuroLo oloxtrip0ilo lerooxn!fil<ouue o€ oeaLplK6couvrerovu6veq, El,vdL u.pnu0ouvo, u-pnu0nu6, \,{-pouvq, ue 0 s € < 2n,0so<n , 0 su2+r2+u2 < 1 - 0 .< p , s I - 0 s p s 1 . E l , vq l ax6uq

, R*rS* | =0.*r KoL VT:it--ir-i"=fi:GtTt+w-u=VT-rt.

l r r - , 2n , ' r ' -'Apo I =oBv | | lVl-i-t. p, nlqdpdsdo-ogv I ilsl auol r'i:Dro,do=))J Jo Jo Jo

. . 1 . - " , * ^ ^ ^ r _ o B V r ?-qFVtur;- t-ouver;Jd vr-9'9 'op = opyzn.. Tt = --r t- .

r To Jo

v1_o"p.dp unoro.r [4E1a! uL 1rv av lLrardd,ddl o:n!u. .ErdL L isL

i _ - t t / , _ I / 2 1 t a / ,

Jovr-e"o.do10 vr- lu ,u q! ,uouud! : jo n ' l ,dou"dd!= { J i

n" ,2odu:

f 1 1 1 n / 2 -= [ ; r ; r u q d ] , = ; '

270 KEOMAIO III

EFpgof 6g tov rplnJ\dr{, ol'orl'too|ralto}v

Il Ynoloyrq6c 6yr,ry

v -JJJotar,a.E

0 6yxoq v €vdc ot€peo0 E oLvEloL qn6 rov rtnoi

To orap€d lou onolou <nrc0pa rov 6vxo,npo(irnreLxo eoot€pL(d lou (ul l ,v6pou,xo rdooBolo€Ldtc.

ndpa6E[Vtara:

1. lk t,|lolovLd€el, o 6yroc rou oteoeoten{,rc6o 0x{, ko! opt(eEl o116 rov ialrv6po

n0on:

sout/€ oe KUlt vdprl(6( ouvt€rov!6v€q,ELvq! x=por.,vo, {,=pn!0, z=2.

H npogoli rou oT€pEo0 oro snLne-60 0x{, €lvor o x0&\oq x'z+{,z.o2.

E, nou oplor€to! [tvD 0tl6 rox'?+0'.o2 xor, io nopolotroer66q

ov oeoLp6oouue ondz

go utrolovLooule ro 1/4 rou 6y(ouxou orEpeo0,,\6yo tnc oullerplcc rouoq tpoq rouq dlov€q. [lvoL

i- =jjJaxaU,ar (l), so u.rooxnlort -E z=X, r !2

E{.vqL 0s 9'?ouv,o+p,n!.8 3 o. - o < ps o.

o5z:p2* rqL B{f ;F j } , .a/2 n p2 r /2 d

*{ [["']"1*{ [J','ri'2

l+ ],-= + reri/'= -$ l-.* .

E L v d L o k d l o o . e . l .

' lpo v=ou * K. uov66€c.

a3ztp7

_t-),

de ldo. l l l erde ldo=

* t i vq ! o :z lx2+4,2 : t o jz3p2(o ! !29+n!20) +

OITOKIHPOTI(o' IOlItl,lOt 2. t8v6.iyou B. 271

2, lb unoloyr.d€l o 6yxoc rn( gqtpac x'?+l'+zr=o'.

Afunr

0 a4roilavoq 6vKoc 6aveio[ on6 rov r0no:

l t Iv - l l . Jdxdudz . onou E=( (x ,0 ,2 )€R3 : x ' 7+ { ' z+2 ' z <o ' : l .

E

|.l€lqoxntorL<outiE oE @oLpLx6g ouv!6rcy!6v€q,

06rou!e l-pnuqouvo, U=pnlqnue, z=pouve,

ELvoL0soso 0seszr, 0s{Dsr ' , 1f f i f f i l -0,*r .- 2 \ . \ ̂ - 2 r , r - -'

I lg]l lq laoae"Apo V-ll ldxd{dz=l | | p'auodpdeds.lt t t r o J o L J j o

.{ 'tf +,'1,,=f'l-I "*'1, *=1"9 *= $ r0,2"=+no,.|(. !ovd6€c.

II. Ymtoya4dc gd{oc, rawpou ldGC ro! todv afgdysadc tltdc areo€dqreproxic.

c'l lt(o. Ynogtroule 6r[ oe uLo orep€6 n€pLoxi E elvor (orovEunutvn

!d(o' l ia oirvdornon (ouv€xi) nuKv6rnrqq,6-6 (X,0,2). T6xE n ouvo^LKi !d(dnou kstov6u€Tol olnv rlap r. oxi, 6 tveror dnO tov tono:

' =JJJo rx,,r, ' r axa,ra.E

{ 1 )

B') >uvr€xqvUtveq xawpou ld<dq0L ouvrerov!6v€q rou K6vrpou u6<oq rnq orEp€dq n€pLoxiq t,6l.vowqL o-

r0 rouq t0fiouc:

I=+JJIE

x6(x,{,2)dxd0dz F+JJIE

{6 (x,qr, z)dxdldz

2 =*fJf",,,'r,.ro,oro.E

( 2 )

272 KE{I]\AIO III

(m 6{.voL n !6<0 rnc n€pLoxnc rlou 6{.v€rol an6 rov r0no {1)).

V') Ponac o6pdv€|'oc

0L porAq obpdvetoq uLdc olepedc n€proxiq E 0,q nOoC rcuq dEovEC0 0, z z ELvowaL on6 rooc €[6!evouq r0nooc ovrLoroLxo:

I,=JJ/r,r,.-', rurr,,r,,, ruxaua,E

Ir=JJJtx'rz' to tx,,r,z taxau,a.E

r"=JJJrx.-,r,, ra rx,,r, ' raxa,r,a.E

(3 )

=1"" [1," [i,u*,0,10,1*=l'' l' " l4*,1'r*r= if '[f ii*,1*=

=-*f"touurrl'1r=* f "rt= +.

naod6€ ryua:to lo€eoov o0 ouwercvlcyEq rou xewpou ld(oq tou or€9€o0 n|otou E

nou €tvr! to rpdno 6y6oo qo(poc ort{,voC Fl 6tov n xatqvouil tnc u6aoc o.ow6 6tv€ror, an6 tnv ourdptnot ruxv6tnroc 6.1.

/loon:

0o !erqoxnlo!Looute q€ oqorpL(6c ouvr€xcy!6v€C,Etvor, X=94y9qur6, o=pnlenue, z=pouvq

!E osos + , osss * , osps r .

*' I tip"f,-Aj.l =o,n!Q. E{voL

'=J J ,l loxa+a,=J

J J P'n,'PaPaeao =E !

i "x1%'- t - - - - -;:'--T)i-'

O OKAHPIITIKOI Ôtltllot 2. ttu!6d1ou !.

Etvol oK6uo JJf-.raraut'JfJrr*,u60"*raeuqae=JJfo'nu"qouvodpdqdo l

E E ' E '

,N/2 ,n/2 , t /2=J

-","emf -nr'ra*fo;ao= ..= *.

' [(oL €LvqL i=+ + "+.

Ervor o(6!o JJJ,r,.r*aua.JJJonwn16.o'*6oaoae-JJfo'nu'enuodpdedo=E '

.a/2 ,a/2=f

-reoeJ -*"a"fo'ao={. ee" o=**=*.

'Exouu€ x6ôc JJJr.raraoa-JJJr*"ro'nwaoaoae-JJfo'otvonuoaoaoae-! '

. r /2 ,a/2 ) .1 /2 . t /2=fief'-*",*'a'J o'ao{ ief $aoJ o'a"-...-{.

'Aeo t=++=+. 'Aoo r{(i,0,i).r,t(+,+,+ )

213

* r l :w l l , r : { ( x ,+ , z ) €R t / x2+q ,2+22 <1 , x : 0 ,0>0 , z>0 } .'Apa E ' : { ( p ,o ,e )€R3 / o= "= + , ose : * , oaos l } 6n ra6 i o

rpLo6Lr idtaros rdros r ' € l :wL opaoyd!Lo ndpdl l r l€r inE6o.

18

IGOMAIO I I I

-ev|'rEt 6vo rplnld olorlnodlcro

:o y€vlK€uUtvo rptnld oloxlnpdluolo opl(ovtdt- o{pLBd}S U€ ovdlovo !p6no,. :J 6lrld KaL Loxtouv oKprflic ovdlov€C [6L6rnr€C.

r- Ito €{.Erootet cv ouvxltl,v€! ro y€vl(€ultvo orodiq{ro l=flfq4dz))) Xz 402 +22

6nou E={(x,0,2) €R3/ x2+{ '7+22 > 11.

Aton:

E t v o L f ( x , l l j ' z ) > 0 v L o x d e . ( x , 0 , 2 ) € E .N n€pLoxi E 6€v elvoL qpovlAvn. 8€l,rpor i le rnv oKolouotq f l rv nepLox6v

E ! = { ( X , U , z ) € R 3 : 1 s x ' z + r r r + 2 ' ? < v z , v = 2 , 3 , . . }

1t"" , I . - [ [ [ gmlot, r . leroornldrLouo s€ orDolpL(r( ouvrErovut-" ) ) r x 2 + a 2 + 2 2

v € q . ' E x o u t i € : X = p n ! O o u v 8 , 0 = O n y 0 n ! o , z = p o u v e ! € 0 3 6 < a l , O 3 p 3 n

x c r 1 s x ' ? + + ? + z ' z < v t - I s p ' S v ' ? - t 3 p < v . E l v o ! o ( 6 ! 0

ll]4.{4.} I,o nuo RoL r(x.0,r)= . r. , - 'X, +, I , +2, 92

^* r -l[[ ",d]q =lff o:!!e!!!p!, -[[hwo*0,=["or['4uode[ dp =" ) ) ) \ / - I z . r , ) | , p 2 ) J J ) - r . J .

E;

= 2 n [ - o u v a ] o ( v - l ) = 4 r ( v - 1 ) . E n € L 6 i l i m I ! = l i n 4 n ( v - 1 ) = @ r o o l o x l h p o u q

I 6 e v s u y x l L v € ! .

2. llo €€araor€l oy ouyxl(v€! ro y€vlreu!€lro o]oxlfUun:

r=[[ [ --qd{dt 6rou r-((r ,0,2]€R!: x2*4t2+2, sr] .)r , v xr+82+22

A0on:

H ouvdprnon f(x,{,,2)= + 6€v ELvoL opovp6vn ornv neploxi rouvX2+,1)2+zz

on !eLou 0 (o ,o ,o ) Ko r €LvoL f ( x ,U ,z ) > 0 v (x ,U ,z ) € tR t (o ,o ,o ) ] .o€uJpo{rle rnv qxolou8tq r0rv or€pEdv xorplorv:

O O(^tq0Tllot ]torlttot 2. tEv6dyou B,

E != t {x , ' ! ,2 )€Rt / -d r s x "+"*2 ,3 t , v .2 ,3 . . , }

et*, r,,=[[[ .$ . ilerooxruota(ou!€ oo oqolprxtc ouw€ruyltveq.

ELvdL x=pruOoovo, {Fpnggnte, z"gouw tl€ 0se<2n, 0Sosn xc!

$ sx ' . , t ' * . "< I - $sp"s r - * <p .s t . 'Exou !6 o r6 ro d r r

r{x,o,,)-#.+ *o' | ffif} l.o.nuo.

'm r"-f[[ g .tlt+?Pcoaear.['noe['nrrar[' oao-'L. vx.+ttt.+z' ,i. vo. 'rt"

=^.,l+1' . =2"(r- -!-\ *.- ' 2n. Apo rrn1 rv.zn. 6dd6ri r.2r.L . t l l r

gno!6wc ro olorlilpoto olvrlly€r.

215

?75 I<EOAAAIO III

AYMENE: AXH:EI:

- o) YnolovLoldc rp$klv olroxlnoqdrov

l. lla utolovLo8oov io oloKlmdDfio

o') | I l(x+Ulr)dzdxd. B ' ) | | l e " ' -dxddz .

= t ( e ' ? - 2 e * 1 ) . e z l ; = ( e - 1 ) , . e - { e - 1 ) , . ( e - I } r .

2. tlo unolovro€€a m otoxltlulo l=JJJf.arUr 6rrou E €tv!. o opscE

ruil.K6c r{ rvbpoc !a gdot rov kutlLx6 6[oxo lr-21,+lt-21, s I xor oloc h

iton:

H opgi npoooli* lou xullvdpou gto €nlnaoo oxu €lvot ro xopio

T - { ( x ,$ ) €R ' r l x - z i ' z+ l ! -2 )z s 1 } ,

* [ to €rr i le ld drsv I i ! . | rpo8or i ' t 5a ewoo6!€ op96 rpo8or{ .

040( HPnTI]<oX IoIItMot 2. tno!6{iyo! B. 271

E l v q l o K 6 ! o 0 : z < h .

' ̂ '" JIJ-u',-,u,'JJf J",o,a, ]a*ao=

-;- dxd+= -;- llx0dxd0=T

t z+ttG(i=il,h z I l l l= - r I l l 0x dx ld0-

, 2-y1- ( \ ) -2 ) '

{ : 2 + n ! o=4 -- l i | 0v1-(U-2)?dU -

-2h'zf 12+nlr,r)ouy,r.rdt,l=2flh,L \ / 2

{J,-T

o16

!o

Elvor A(2-r'fi lFa7,o)Bt2+/1:6aF,,t))

3. lih urc,\oyro0El ro otokliqolo r=JJJzxaut1g1aya,1az 6rouE

E=t l16 , 1 /31 " t 0 ,n l ' t 0 ,21

lton:t ! / ) - . \ t t ? r ! / t r i t _ r " r , rrLuo, t- l / l l l tzxnux.rtozlaulax=1 l l | ! r nux,r , I a+lax=' r / 6 " 0 " n

, ' / ' r r ' 1 t l / ) " t ) / :=l l l ?x4ux'!d't ldx= | i-2ouvxul;dx=l lz-zouvnxtd^=' ) / 6 " 4 ' ! / 6 ' ! / 6

= l2x-f nurr | =+* " t -r ' t ) .

4. 'Eoto T to rplyr,lvL(6 Xr4r[o rou eltln66ou OFl, nou rcpr|(l€lero! leEaE0e!€e|,d,v G0, {FX Vro 0 < x < 1 ror E n o[dpEt n€ploxf nou nsplxlElrro!rlc e[!06v€!ec z--!'1, z=X2 y0o (X,{, €T. tb unoloy.roe( ro olo(^liqr-

I.Jj j(x+l )dxd0dz,t

Ir lr i

KEOA]I]IIO III

Aaron:

ELvoL 0sX3 l , 0s .U jX Ko ! - 02s z <x2 .

'o* I- [' [ ['I f ' rr.lo.lu,rla,- [' [ | r,., r r,r*]a,r,lar-

1 "o ' : { , - , ! ' J

r l r r x r . l r r t=J If rx.l r',-,r,,rau]*=J If rx,++x,r,'-,r,.rc,r,]ax-

=1,[,'*,'t. + - + ]:,'-+ l" r,".,'ro,=] t+ . + ],- +

5. l{o umAoyLo€.t ro oloxlipr,Ic I"JJJUVf=f'taxaUr 6nou E €tvcl nE

otepEd tr€Or,oxi nou treordeterd! cr6 rlq erwdv€rcc Xr+rt=|, X2{lr+zr.l (l)

/\0on:

To oieoed E n€plkleLaftL fi6 ro enL[€6o 0=l rov |(il!v690 X2+22E1 Ko!q oooLoct: xr+v2+22=1 (2).

N opei trooooli rcu orepeoo oio €n[n660 oxz elvoL o KUKtr[Kdc 6loroc

x'z'2, < I (n eE[doq rou (tKlo0 xr+zr=lIrpokorr€L on6 inv ( j) yLo {=0 KqL €[vcLn noo9oli rm err0dveLoc rou Kullvdpouoro EnLnEdo oxz). A|16 ro 6E0rEpo oxiuoQ,oLv€roL 6rL -1 s x s 1, -114s?< {1-7

An6 to nAjiro oxilo oolv€roL 6r[

_Vi t i - ' 'sUsr.'Etot

€ { ,voL

r r'u.z ,t=l | | '1y'i--1-' u 66261 =

t t t_' _\6:F _6_r,_z

r t-r- ! tt t / \= i I (z ' r -x 'o ' l )dzdx- '

oloKrlHPtlTIXoX r\olltMot 2. tnc\rodyou B. 279

=J' [,.rr*,r-{rr-,.r,].,=J' ( i,..*,.-*),,=

{-+,,.+,,.+,)l',={+

, fi:t=t l=l | + v1_x,tx,+zz)dzdx=l +: ' : , , t=

5 . 'EouaEno r€ps6z=3-xr<lr, z=-5rxr+u, te

Atoq:

.11=;z

, ' , -* . ' ) l -^

dx=- .t-!5:F

negLoxi ltou lr€pLr]te('erol otr6 ro nop@loe!6fX Z 0, U 2 0. l{a unoloylo€el. ro oloKtriipttro

l l lr l 'dxdtdzE

i\4e ono.\olqi rou z lerql0 Tov EqLodJoeov r(,JV nqpoBo,\o€!6dv nooxtnlELn npoBotri rnc €fll06vELqq rnq oreo€dS rlepLoxitc E olo en{,ne6o OX0.

' EXouus 3-x'z-Uz=-5+X'z+o'z - x '?+U'?=4'EroL opi,<atdL ro X(,lp [o

T = { ( x , ' ! ) € R ' ? : 1 " 0 ' s 4 , x : 0 , U > 0 1 .

Elvq! 0 s g S 2, 0 < x < Vf-rl-z (oL

-9+X'z+ttr '1 s z.S 3-X.- ' ! , .*

r - r , w ' l . r r l - r ' - i / - r 1'Aeo I=l I I I I Lldz ldx ld'! =.15* , z * r :

a f rW ,1, ' r '13- t_- ! ,_ l= l l l | rJ l , l gx ldu=L J_5+ r -+ i ! . I

| | r \u-q'= l l l u(8-2xz-2u, )dx ldu=

= | | | l{8-2u'z)x- +ux, I d0 =

t Etiii-s+x'z+4'z < 3-x2-{,2 6LdrL

z=-5+X, At,

r d " : o b o _ d u r i . ( w . . 0 0 6 J s - r u c t n v 1 ' ? + , g ' 3 .

280 ](EO!ÂIO III

=l [,r,r*zu' rva-+- -iuo+' r'a

';fIro+'r"']"= tff '

U (4r! ') 1n-,1' 1' l ' 614-r',=

I=JJJxtarar o-"E

o 'z ,xzo ,0ao ,z>0 i

oregeoi E oro en{.r€6o 0x$ €l,vol lo raropro

( 0,qr ) (r';zT,'!)

1o. -rz y' t , 4'1 i

. ...- +q".

h'=*l ' ," aU=_ 1[

7. ilo urEloyioo€l ro oloxllb o

t=t (x,a,z) € R3: r,+{z+z, s

nton:

H npoBoli rnq anlQdv€roc tou

rou K(,KIou x'z+U'z.o'z.'EroL

opt(EroL m €ILI I€60 xoplo

T - { ( x ,0 ) € R ' z : x ' ?+U 'z s oz ,x :0 ,Q :0 )

eEopotle rnv €ueelo l€) nqpdllntrnrtpoc rov 6lovq z z,nou nepvd on66vo €s{depLKd on!€Lo rou o!€pEof,Ks[ nou rtuv€L tnv €n[9dvsLd rouE oto on!€lo A, B !€ 2=0,

z=tror=F -@- ovrLoro LXo.'Apq oszsVEZ:F :G .

0o unoioyaooutr€ ro 6LnI6 oloxtrlt -

pqrs.EtlqL (0,\6ne dEtr€po oxl l lo),0SUsoKoLO<x<@- r l * .

'rp" I=l I I xGzT::I;ax lau,=

, /rr r \6 'z: i i t , I r f- + l | (o?-x , -0 , )1 d(o7-x ' - , ! . ) ld0=t j

1 / 21 | ^ r | , " 1 ' /2- i" ' ' j ouvrudo. + l30i* +nu2Lr.n!20rouv't!1,

* s d o a ! . 4 , : q n ! u o r d r e 0 S u s t

IIII

- lYz:

0A0MHPoTII(ot I0trtuo t

8. flo urotoylooEl. ro olodis{ro t-[[[z'aya]dz 6mutlt

r={ (x , r ; l€F: f+042, s . t }A0on:

H orep€d rcploxit E neplx^€Lero! dn6 tnv oqol9o f+$'?+2,=1. H npo0oli

rnq oeolpoq dro €ntn€6o oxu elvol o |(0(,\oq X,+{,,=l {npoK0m€L ofid rnv €-

lao|,on rnq o0otpoc ylq F0).0o Bpot!6 !a 6pLo ir,rv x,U. H nooBoli rnc oxep€dq neploxic E oro €n[-

n€6o oxqr,opi'<.r !o xr,plo T= t(x,0) €R'?: xr+U,, s l].An6 !o oxilo 0alveroL 6rL -1 sx s 1

-tl-xTS0 < v--X,.8q Bpol, le io dpLo ro! z.

'Exouu€ x 4(J2+2'?=1 : 2-1!/i--x rq 1

lllo euo€la flopd,\Inln npoc rov 6[ovo

z z nou l€pvd cn6 6vo €c,,repLk6 onlelo

rnq oqoLpoc rblver rnv €n!edve!6 rnc otoonlel,o A,B !e z=-\4---xr-'!1 z.Vi;tItdvrl,qroLxo. Apo-I{--!I.:IF: z < \--x,i!2. 'Exouu.

[[L.c,orc.=[' ItF f tqt{.,lola, =[' lw [-u'\i,u,uo,"r' ar Lr _v,FF rr _vJ;T:t7 r r t_, )_,/L:7

"- t' tut=F I #]ffi oror= [' fff { 1 r -r, -,1,,v',, 6,r,6,-.r J -tj:t; - ro r_t ) _\87 "

f_ f v1 - r , 4 | . - " . , 1 ' ' ,- l . , J , T L ( r - x - ' - { r - J

d r l dx

etxoule 0=lF1-x huo - d0=li:;-l -x,ouv€do. orov 0=0, e=0, 6rov

' re" [ [ [,.araor.= [' [ "" -1 f r, -r, 1ou",r1 ", ti:-x,ouvsd'dx=

'; '

) , r / 2 ..1, | +,'-*', '","'sd0dx=1, +(r-x')'z # .i *-{ j,

q1=Vt -!?, e- n

( 1 -x " ) 'dx .

2A2 I(EOAÂIO III

=z.i J rr-x'r'ux=i J rr-zr'*r"tor=.| I,,-+-+ ],=

9. llo unoloyloo€t ro oloxlipqro JJjaratf,ar 0""" E €Lvol n r€p.oxii iou

n=T

R! nou ool.<€ror on6 rov xtllv6po

n0o4:

0 KtriLvbpoc xt*z'=o' 6xer ye-valeLp€q nspdltrnlec npoq rov dlolo(r '(r.

To enlnebo 0-0 €LvoL ro €nL-r€60 oxz. To €nane6o x+U=o elvoLmpdl.\nio rpoq rov 6eovo z'2. Hnpogoli roo orepEo(, E cro enln€6ooxz €tvoL n (€pLoxh x

T= { { x ,0 ) en ' : - c szso ,- v6 " - "gx3 \6 : : y ' 1

'tro! , \oLndv to 6oLo r(,Jv z xoLx axouv (o8opLoeal,. eq opo0us rq 6-pLo rou 0. l ,{Lo €ugelo nop6llnln npoqlov dlovo 0 0 nou nepvd ond avo e- (

d,tEOr,x6 oryi€Lo lou E r6lvEL m o0-vopo rnq nepLoxiq orq stuEl,o A xolB ge {,=0 XoL 0=o-X cwlorolxc.

'ApoeLvo l0s0sq -x .

Eno!tvl,JC o LvoL

/ . 2 1 \ n /B \ 4nt r--r ' r , = t t_rFJ = _rr-

E

r ' .z t=ot xcL rd €nl | le6o { ,=0, r+0=o, q>0.

lo 'z -zz ,z) / \ ( /0 'z-2 'z ,z

i C

'-/x

r t t r r r t - t t , , P r r@l - l l ldxd, !dz=l l l l o+ laxaz=l l l , r r i "dxdz" l l (o-x)dxdz=l l l (q-x)dx ldz ="'' ';\1a '; '; '"" -'6':7

oIoXlHPl TIKo! A0I t lMot 2. Erc!6dyo! B.

(O6rouu€ z=qnuo orrdre dz=souvodor. '0rov z.-q, .-] , O.o" r=o, "-tt=

rI2 \/2=zo[ 6t-q,n,o oo,r

"urdr,r=2o' I ouv,ua+2o' [wryu*'1'/ '-.. '=""'.t - t / 2 ) - r / 2 L . r - t / 2

f0. lfo unoloylo€e[ ro otoxlltgrJo t[JJr'axdilr. '0rou E €[w. n or€-E

9e6 raproxi rou replKl\aletcL 016 tnv690 x'?}0'?.ox (o > 0).

Aton:

o{x!(oa x'.+0':+z'z=o'1 ro! ord tov r0trLv-

H noogotri rou fi€peo(J E qro enlnEdo

0x0 e(!oL o x(Jxloq X,+U,=oX nou opL(EL

r o X U p L o T = { ( x , u ) € R } : x ? + { 2 s o x } . o a p -

vouue rnv EUoeLo AB rqp6tr l4i \4 npoq lov

dlovo z z nou tl€pvd qn6 avo €orfieprKo on-

leto rou E (o! nou rt lveL tnv enlqdveLo

Tou or€oaoir oro onueLc A,8 !e

,=-13-7-V, z-\Gt -F=V ovrlorolxo.' A o o - V o r - x ) - U , < z < v o z - X ? - { , r . ' E t o L

r=i[[,,oaoa'-[[ [[6'i":L.].lxa*= l[ 3 u,'-x'-t)'r3/'1dxd+. €q unorovt-rr Lr _\I-;----;--- J r)

E T d - - x - - Q - T

ooute ro 6Lnl6 oioK;\ ipolq !e u€lqoxnloiLou6 o€ no^[xAq ouvr€xoYutv€c. e6-

r o u l e X = p o u v 8 , ( r = p n u 8 . E L v o [ 0 s x ' * 0 ' s o x - 0 s p t s o p o u v o -

0sp<oouve KqL 0s6:n, & l**J- . , ,o ' - r ' - r " r ' / ' =ro ' -0" , ' / " .

' no" r. [ [ | ro'-,'-+' r "' o*o+= $ f 'otf ""i o' -p' t' t' poo= * "'f ,'

-*t,*=' i '

11. o unoloylo€€L ro oi\ortr i tqro I-[ [ [ tr '+' , . laxaUt 6nou E eavol nt t t

orepEd ll€oloxi rou nep[K^€leto! dn6 to fldgo8oôer66c x2+t2-2, lov K0lLv6po

.. . . - . | - --

fffilt

2U IGOAÂIO III

x2fllz=t xoL ro ma,re6o Edv d{6vrrv.

ntal:

H npogotri rou m€peot o[o €rLIIeoo 0X0T={ (x,U) € R'a: x'*{ ' . : 1, x z 0, Q : 0}Lq eue€Lq nopdtrin,\n fipoc roy 6qovo

z z nou n€pvd on6 6vo €o0[€pLK6 on!€LoToLr E,ouvovrd ro owopo Tnq nepuoxheo[o onueLo A,B l]6 xornylevec dvrlo[oL-xo z=0, z-y'z+1tr '?. 'Apo eLvoL0 < z sX ' z+0 ' . 'E roL 6xouu€ :

I=.J J J

(x'+{, '+z)dxd' ldz=E

, , , , t 2 + . : . 2= l l l l tx ' .+ ' .2 raz laxc,: ,= j - | l {x '*0 'r ' , rxdu.f ' 0

' " i

polo unotroyL(eroL

(x ,+u , ) ,=p \ .

I ta /2,npo r= i l

Ue U€rooxnuqrroUd oE notrLKaqE lvqL o<os | , , oso

To r€.\€urqlo 6LnI6 olo(I i -

o!vi€roy!6veq. 06Touu€

sr, ffi+,

I o ' l ' n[ - r ] " -E'

t2. t{a unotroyld€t ro olodiFua t= [f[t,1", +.,x'*x ,9r laxaur.'L''orcu

n n€proxii E |r€pld€a€rdr dni6 mv klllvdpo x2+r2=2d, xot dr6 tov r6vo

z,-r2 (x,+{, | (a,x > 01.

Atsn:

H moBolh rnq EnLodv€Loq rou olep€oi l oTo €nlnebo oxqr ELvqL o K(lx ioq

X'+t2=2oX nou opl<Er ro Xurplo T={(X,$) €Rr: Xr+{rr S 2aX}. OEpvou!€ ULo eu-e€Lo AB,mpdtrtrn,\t npoq rov 6Eovo z z nou n€pvd qII6 6vd EoorEpLK6 onuELo rouE KoL nou r6lveL rrv €rLadv€Lo rou orep€o0 o[o onuElq A,B tE z=-Kqle,

do l p 'pdp=; +

opL<EL ro xoplo

' r r i w L o s o < f o " ; . " r o x o p [ o T e i w u r € r a o r o r i x r L o ,

OIOXIHPOTIKOT AOII'MO' 2. Ina!6dyo! r. 285

z=xvx '+{r ' 'Apo -Kvx '+t ) ) ' t z S rvx '+g ' .

t t l l I I lr= l l l l | ( ' r ,+xz)2 ,+xz ' ! , idz ldxd ' ! :'rl'1*n-;01'

=21 | l+(xa0,),K,+x,0, IKvx,+u, ldxdoT

'lo dLn;\6 otroxtr/lprluq unoloyl<Eror !e

tErooxntqxLold oe flolLK6c olw€fiy-

!6v€q. 06ioutE X-pouv8, U=pn!8.

L L V O L U S C . < i , U S p l Z o O U V b

_ 512- -2r

r/2 26ou!0'=+l I*o, f'f*.orr,rr,ouu,6ny.s166= ffi Kq?(K,+8/33).

( K'z+3ouv' lonu' l0)P6dP]ds-

t t l13, lb U|loloylooel. ro oloxlilgqrd JJldx,iltaz

0nou

E.{(x, ' ! ,2) €Rr: Zy2+412 3z .-4-I2lnoon:

H npogoli rnc or€pedc nepLoxil E oro enln66o 0x0 ELvqL n €nln€6n [€pLo-

Xi T rcu opl<Eror on6 rnv dx6ot 2x'z+r)t2 s4-0' . ty '1+rl '32.'Apd T=t(x,U) €R'z: x 'z+q'1 32\,

r r r t , r t " - l ' lELvo ' . I= l l ldxd0dz= l l l I dz ldxdu= l j (4 -2x ' : -2 { ' z )dxd{ .

)J L ) t " 2 +1 ,2 )I l T - ' - T

0s unoloyl,ooutr€ To r€l€urolo oioKl/tpouo u€ Xpion nolLxdrv ouvr€Toylevnv.ELvoL X=pouve, | l ]=pnuo !E 0 ! 0 S 2n Kq[ 0 sxt*r l t s 2 + 0 sp <uE.

ELvoL qx6!o B{rS} =0. ooo

jjt+-zr"-z,r'rara{,=lll 4-zp' )pdede=zllep-p3,**"1 ";1i*p"11 "T T ' T '

r T - II I e',de lde

' E r o L € L v q L

K'OMAIO III

"(' t-J| *4, f+l%e=+ ter l" z r e t l"=e"-+n=+n'

14. I{o umloyr.ogcl ro rplnJ\o otrod\fp(ryo f=[[[tx*lr,rrrt-taxaUa.' !6nou E={ (x , { , , z ) €R ! : X :0 , 0 > 0 , z > 0 , X+{ ,+z s l } .

(Colden l,{aths series )A0on:

E lvoL 0s Xs 1 , 0s0s 1 -X , 0S zs 1 -x -U (g l6nE nopd6€Lvuo 2 oe -60 256). 'Apo

'-{L:-'l{ * 'rx.*',-l-,o,]au]ax-ft1'; u+ir1"'-' ]or]0,=

_i[f 1 (x.q,.1-x-,!.1 ):;-{x-,r,-1 )-, 1,,r10,=f ;f-'r:grurl_,rp,.

=-if[l-*',-'-,*r-, r-lou]*: J5J' Ir',r,- r"9ir r-'1'-i,=

=- Jaf , r- ' , r* ,- , , r+r-x+r )-r-(xrr )-r ) ldx : ! ! ' r r ' , r -*ur- ' - (x+1 )- ! ldx=

=-* lr'"r-r-" { -z-'rrntx*rt]1: | [z-"- | rz-'-t^r]=

= - * - * ' ' .

15. lld orbtovloo€t ro rernl6 olorlliporo r= [[[,f,araf,a. 6rDU t €lwrE

n negLoxli rbr, neprdelato! ord m €[tnE6o 0=x+2, z=x+3, qv ruIrv6o|. l €nl-qdvElq {=x'z rd! to tEpo8olo€166q 4z=x'z{t2.

(colden ihths se.ies)Aoon:

H [poBoî rns neploxr1s E oro ellLnEbo 0x0 ELvqL ro xldploT= { ( x ,U ) €R , : - 1 <x<2 , xz <03x+2 )

oIOKr\HI ' ITI (oI lOI I lMot 2. tno!6dyou B, 247

::1'-- j

- | tx .zr 'x ' - { tx-z t "

16. llo urcloyto€e{. Io otrox.l\f|polo

nou napldE{,€rdL qfi6 T[g €nwtveree

z=x+3

x,..-':.'---+ttr=xi.

I=l I Ixdxdtdz 6rou t€LvqL n nEoloxE

z'z=x0, \/x +W=1 KoL ro €nlnebo oxr!.

dx=

v d€ovo z z r{t ^snuEfo rou \,r=x, I I A'l:,";:,"1; \ IHr

3 ,€',1

= r/'l' 2

'[J*."rft-, +-, +r,),-]*=

*1,,=J- 19, ,y'-;1,-, d-+,1','

.t!

l4lo eueeLq rqpdtr;\nln npoq ro\,noo flEpvd qfi6 6vo eoor€p[kd o

E rauveL xo obopo lnc nepLoxu€lo o (qL H ," .= | tx'*,1. )ov!Loro tXo.

'eoo f (x.4") s z: x ,

' E r o ! € L v d L :

2 x+2 x+iI= l l l " l l , ^ " odz ldu ldx=

. r r r x . r J j ( r z + ! , 2 )

2 x+2=[ f[ ̂ ur.r" ' ô+lo,=[! | L t ̂ . ! / 4 ( x . + 4 , . \ r ! \

2 X+2=[ l f . (,r,*-:+-*+*,-*u,)a1 r r J x /

\

-x ' - :x ' . f x.- { .x"]ax=*..r .= - r i t t x + z / x + r ( x + z t -

(Ibv LoriuLo Aov6l.vou)

288 KIOMAIO III

A(|on:

Kdvouu€ rcv u€rqoxnlqrLo!6 4=u, Vu'r . EroL ELvoL u'z=x, u 'z=l l j . 'Exou-

rplvovo 0AB xoL ro €oo!€pLx6 rou. Ei jkolo

o€r,rpo(l!€ !!o €uo€Ld nopdllntrn npoq

rov d{ovo z'z nou IIep!6 on6 5vq eoor€ -

pLX6 onuelo rou E Kq! nou tauv€L ro olr-

lopo rnq f l€pLoxiq orq on!€Lo H, 0 l ]€

z=0, z=uu ovi loloLXq.' A p o 0 a z S u u . E r o L a x o u u €

. . - D(x , ' r , z ) - i . . . . *" tG'U;t -* '

H nepLoxlt E ue rov u6ro-

oxquqrLo!6,onELKovL<etoL o!-qLuovooitusvro olqv n€pLoxi E-

no! n€prKi€LerqL oII0 Tnv Ent-

rpdver-o z2=r,r2u2 . z-uu (unep-

Bo,\ !k6 f iqpoooio€L6ac) kqL rc

€n[nE6o u+u=1**, z-0.

0q BporllE rq 6pLs rov !o-roBlnrdJv u,u xoL z.

H npo9o.\i lnc aEpLoxic

E om €nln86o 0X{r elvqL !o

2 u 0

0 0

0

f

p c l v € r o L 6 r L 0 s u S 1 , 0 s u : l - u

,=11J,',*0,,,,=f [1,' "[',"',"1*1,,=1

Jf "ou',r,ri"o,]ou=

=l | -",,',,*,,*,1"

ll'u,',,1r"1",'[*1]',,= + | u,'-,,'0"=

=-ll" ,'r,-,*r'-,'ra,={1,t,'+,'.:,'-,',*=+ [+ -, ii " 9 ti ],=

6 +v[ =1

AOII IUOX

= r(x)r ( . ) r (n)r(p) rf (l{+n+p )'olrou

n nepLoxf E oDl.Cerol on6 rc snl.ns6o X=0, U=0, z-0, X+o+z.l,(1'n'n'p z l)

(nov€rrodplo Aorrotwu)A0on t

17. lb oErxe€t 6rL r=JJJxl-t1'-t'"-t, t -r.l-z )p-laxdtdz=E

Iiue{rva ue !o nopdoeLylo Z rnc oa^Ldoc 256 6CvoL0 S X ! 1, 00sz3 l - x -0 .

€trouliE x+o+z=u, x+U:uu, x:uun.,Eto! elvor x=uuw, r1-uu11_*;,Ener6i ro x,0,2 €tvor OerLKd i un6Av (o! x+{j+z s I go €tvo! 0 <

= l - u ' 1 u l . u 2 u . A 9 o

sus l -x

0 < u s l , 0 s r s

ErwrlS+fit#

I= | | | ( uuv , ) ' - r f uu ( i - u )J ' - r [ u ( r -u1 ] n - I1 1 -1 , )p - I u ,ud r :dudw=

= l u r - r . h - r+D-1+2 { 1_u )p_ rdu I uc_1+n-1+r f i - u )n_ ]du f *1_1 ( r_ * t , - ra r=) ^ J

I , ' * t " , {1 -u)p- ldu l u r 'n -1 1 r_ ,1n- r6u [ * { - r1 r_ , ;n -16 ,=a l o

=B(x+n+n ,p ) 8 ( [ ,+m, n ) B ( r ,h )= r (1+m+n) r (p ) r ( r+m) r (n ) r ( x )T (m) -r({+m+n+p )r{r+ft.n )f (r+l||)

- r (P)r (n ) r ( c ) r ( 'n )f( l .+n+n+p )

18. llo urcloyloo€|, !€ /oa|ol rulr\r6pLdlv ouwetov|tei&tv ro olodtpoiro

, ' ,F i l , tr-l I I x'aza0ax:, . 'vF_rz

ir,+{, ),

l 9

Br ; r€ orEaLr. i !€ o lwpi io .Ls r r iwa xoL Bira oto! rpdro td lo o. r i6a 3?0.

290 KEOMAIO III

Abo{ i

An6 rnv lopoi rou oôdnp6loro( 0qlveroL 6rL -2:X:2, - ' . /a-f!{r3!ry

( x ' *+ " ) " : z 31 (1 ) .e.rouu€ X=pouvg, =pr g, z=2.An6 To 6plq fllv X KoL 0 {polv€rqL 6rL q or€p€6 nepLoxi E ornv onolo

YLveloL n oloKtriprlon npogd,\Ero! oIo €nLn€60 oxu Kord rov KUKttx6 6Loxox"* { " 34 tz) .

An6 rnv (2) npoK(]nr€L 611 0 3 p'z 34 - 0 Sp < 2.ELWL oK6Uo 0<e32n . An6 xnv (1 )npok l . rn re r 6 rL p rszs 1 . 'Apo

E L v d ! :2 \lF-:iz 1 2a 2 7 2a 2

I I I x 'a 'aoax=[ [ [ tpouuer.oazaoae=[ [ r (o,o,u,er . ] tapae='-t '-,6nt irz-"zY

2 a 2 2 \=[ [ tp ' -o ' to,u'eaooe-t i (+- - l l )

J . L \ + o /

| / . | / . \ l-28 l, \ + + + ow20 )ds"-ll4(8+ + nu2e)l

2 t

ouvzel ae=-2gl ouv'od6-

2 1

=-28n.

19, llo un loylore{. !o tprd6 otoxt,ip.4ro r{JJO,axOa. o*"E

E={(x,0,2} €n!: I 'z+0':+z'? s r '1, t u 0}

Aoon:

M€rdoxnpor[<oule oe o{roIpLK6q ouvr'!ovtrEv€q.oarouUE X=pnuqouv€, U=pn!6nUe, z=pouW.E tvd l0S8Sn , 0 jAS f i * l ( o ! 0<p< r .

t lvor ox6!o | ; i"^-f.# | =pznua. Apo

I-l I I p'nuzsnu':a.p'?q! ' !dpded',=l 4!!edql n! 'z0d81 p-dp=

=ioud' _",u,11 I el' I_ft lr .... znr5L r - " .1 " [ 2 41 . 15 l "

' -T5

* 'Exouu€ n!LoeoipLo, BIdn€ napat iprdn l oel i6d 265.

oIoKnliPmI(OI IOIIXMOI 2. trd!6{iyou B.

20. ilo unoroyrosct ro olodilpq! ,JJJaraUt o-"E

E={ (x ,U ,z )€R! : x '+o '? : l , 0sz 3x '?+o ' z }

A6on:

l4€TooxnuorL<ouu€ o€ KUiLVbprx6e ouvr€Toyufv€q.t Lvo l X=ps ( r vg ) U=pq IO , z=z ue 0 <€ :2n ,0 sX " *U 'S1+05p '?3

- 0sps1 xq l 0<zsp '1 . E t vo l oK6uo $ f f ;&# =0 . ooo

p r 1 [ 1 1 o 2 1 1 1 2 t 1 | ^ 2 1 i a t i ] r 2 1 r ̂ c t l .r= l l l l ledz ldelds ' l l lp tz l :dp ld8" l l lpsdelds=l l+ l d8=

- -T ' " - -T '

21. ib uroloyrooo&v !o oôdndFfl:

291

r'.jJJxaxar,a', r.{JJxr,axar,a',E E

Is=ll lxtJzdrd' ldz 6rou

E={ (X, ' l , z ) €R! : x ' * ,1 ' so ' , x 20 , { >0 , 0 sz <h}

n6on:

€o uelooxnlorl,oouu€ oe xulIvbprxdq ouvlerdyu6veg.ELvqr x-poovo, U-on!e, z=z ,. BlolrS#

=p. ElvoL 0 < p s q,

u 3 v s - T x q l u s 2 s n . A p o

a' ) Ir =J j jxoxa+dz=l.J.Jp'ouvedpdodz=E ! '

t \ r r / 2 / &= l l l { l p ' z o u v o d p ) d 0 l d z =

i t \ t t / 2 r . , r h= + l l l ' r u v s d o l d z - - ; - i d z = + o ' h .' 1 " r ,

t tu ta/? / 4B ) I , - l I l xudxd, ldz= l I lpsouven!odpd8dz= l | | ( | o rouvenueao)ae laz=' - L '

^ \ t \ t r /2 r ^q rh= f- | l l ouveaueag ldz= + I dz=+ oqh.

liiiitr,Nie ; l p i

=+ltl'"

22- b uDtoyro€el. E oloxlii&tt

E={(x, l 'z} €R!: r ' l r l '

= | l , **"a," - ; f ouvtr*z '1a1r*2.1= { r ryz, : ' - } rn!( r+2, ) l :=

h r I2v'l t ' =JJJx,r'axa,r,a'=JJJo'ou"enue.apaea'=

[ [[r ' i ' (J o'o,venlezao)aelaz-

ouvoalezdo]dz= # J ro,= { ".n,.

t - I J l2zrr lx'..l' +2. I dxdtdzE

S l r o ! 0 s 2 S l I .

A0rn:

eroo)(nuorL(ou!e o€ Kult v6pLk6C ouvt€toy!6vec.E t v o L X = p o u v e , 0 " p t u 6 , z " z ! e . O S € S 2 n , 0 < X r + 0 r S 1 i 0 S p S

Kcr 0 s z s r. Etwr o(6!d ff[rt*;] =0. .ooo

12\ t t I t lI . l IJ I j 2eznu(e, .2,)de' ]dzJd6 (1) .

Ef w! J 2p2n!(p,az,)dp=loaqp, +2, 1dto, +2, 1 =7lony1e, +2" p1p".z, 1=

=-z touv{p '?+2 'z ) l : - -z louv(1+2, ) -ouvz , l , ,Eror n (1 ) 6 { .veL :i r r r '

IJo I ' / I zowz,-zouv(1*2, ) ]dr ' ]do t2).

EIvoL J! Izouvz'z-zouv( 1'2, ) laz-["zou,vz" az-l,zou,t 11 +22 )dz.

-+ (nul-r!2+4!1 )=nsl- +n!2.'Apo 0 (2) 6r.v€L r.f "(nut-

+nu2)do=(,r v1- +nu2) t[t?,=zn(nut- fnuz).

oIoK,\HP0TIKo' lotltMot 2, tra!&!Yo'r B.

23. ttd umloyloo€l. .o oloKtiqq/o [J[ztx'+'nxa!lrn

6muE

E={ (x , { , z ) €Rr : x ' . { ' ?< l , 2 3z 331 ,

n0s4:

It letqsxnIotLcouu€ o€ KUILv6pLK6q ouw€ftVuav€e.ELvdL x.pouvo, 0=pnue, z=2.ELvqL 0<es2n xoL 0 s x ' z+Q2 < 1 - 0 3 p ' z < 1 - 0 < p 3 1 ,

ELvsL qKdlq ;too## * xo! f(x,U,z)=z(x'z+U'z)=zp'z.

H n€pLoxi E usiadxnlotL(€roL dlqLlovooilqwq ornv n€pLoxft

E ' " { ( p ,0 ,2 )€R3 : 0 gp3 1 , 03e< 2n , 2 sz s3 l ,

) , 2 i ] , 1 , 2 n )'Apo l l l z (xz*0 '? )dxd0dz= l | | zo 'aooeaz= l zdr l de l psdo=' 'E '

/ r \ 5n= [-z-1, tet;"1+1, 1+-+] (2r-0) (+) = + .

2t. ttd urotoylo€El. ro olodioq! lJJJaxawr 6rDU E n orEped nEoL-E

oxi nou |regLde[eror 0116 tLq en[odvslec z-4-lt2, z=2x2+{z xol rd enl.rcdo

0=0, x=0.

A0on:

l l snLedvrro z=4-0'z (1) tLvaL xulLv6pLKi !e o6nv6 Koumltrq rnv f iopqBo-

I i z=4-0'z xdL yEv6r€Lp€c nqpdl lnlEc f lpoq rov 6qow X x. H €[LedveLo

z=zx'z+Oz 12) Elvor nopoBoioeLbaq nou 6xeL dEovq ouulerplqq ro! dEovo z z

xqL ouvqvrd To enLn€60 oxu olo sn!€Lo 0(o,o,o), l4E ondloLai rou z u€rqE0

ftrv (1) KoL (2) npoK0nr€L n elLol ior l x '+{r '=2 (Kl,( Ioq) rnq rol ie l ( , lv 6!0 €-

nLqoveLdv. H flpoBotrrl eno!6wq rou orepeo0 oro EII{,ne6o 0x{ ELvaL rc xoplo

T = { ( X , { ) € R ' z : x " . ' 1 " s 2 x > 0 , 0 > L r .

A116 ro 6el j rEpo oxiuo odtvEtoL 611 0: x < rQ, o <t ' < ' lT*.

e€opoir!€ lrLo EU0€Lo nou l€pvd on6 6vo 6oorEpLK6 onu€Lo rou or€p€oir nq_pdllnin npoe rov 6Eovq z z.

293

KEOMAIO IlI

z-2xz atr'z

itiliiIll4ililH eue€to ouri orJvowd xo o0vopo tou E oro ouEto A,B !E 2=2x'?+0'z,

z=4-{,2 owlolorxo. 'EtoL etvoL zx'?+t'? s z g 4-0'?. 'Apo

tt|t ttl--izt ,q-t'z 1 1 1,,/\ lttt--'' lr= l l l l l - , ,dz ld0ldx" l l l t {a-0 ' ) - (2x '+u ' ) }d ' ! ldx-

"[*[(,*"'r- $Jl *= +1', 2'a' 1' /' 4,=n.

25, l{d urcloylo€el. m olor}ipr,ruo I=JJJr,f,.axa,fu., 6rtu n orEped ne-E

ptoli n€9txr\sa€rol ard rnv €nl!6vero x'*,!'=o" ror tc enl,|r€6o I.0, {r-02"0, z=9. (d,B > 0).

Iton:

H EnLedv€Lo x1+02=q" €LwL KUILv6pLKl l(rou enln€6ou 0x() x'z+U'z-o'? xoL vEv6r€Lp€cTo orro€6 t Bg(oxeror olo nqiro 6y6oo rl,,v

ll€rooxnlor[<o!!a oe kulLv6pLKCq ouv-

i6toyutveq. 0610uu€ X=pouvo, t !=pn!8, z=z' E r o L t [ \ o l 0 < o < o . 0 . g . * .

o s z s B, iifrifi =e, f(x,'r,z)"x'lz ==Pi{ouvsnrs)2.

'Apo:

!€ obny6 Koln0lrt

napdj\lnj\€c npoqrov dEovo z z.oq6v6,v.

iir, I iri

oAoK,]\HPnTlKot lolltllot 2, tnd!6d1o! B. 295

\ / 2 d I \ / 2

t=l I lor(ouveae6116,6p66=1

a/2= * [ nuttort

=*"'u'.t+], t+ ]u=f r-*,'r;" + +=+ ++=

ouvenrreoeJ o3ooJ

26, tlo unotroyrooEl ro olorltDr4rd r= [[[znutx'+')axa{az dnou E E(w.t t t

ro oqqlplr6 ryilo !e !t-q 86o[,nou rcolKlea.eis! ord qv owLpa x2.{2+2242(t) xo! qi6 Io erlr€oo z+ (2), 0 < I < o,

A0o[:

H tolfl rnq ooolpoc !E rc €nlrle6o z-9 s{,voL o ktKtroc xz+o'z"qz-o'z(npo-Kirr ier on6 x4 €lLodio€Lq (1) xoL (2) u€ onoloLqi rou z). l i l6tqoxnpqta<ouueoe KUlLv6oLK6c ouvtardvtr6v€q. €6toti !€ X-pouv8, U:pnug, z=r.

Eltor 0 3 X'?+0'z< o'?-B'?- 0 tp" 3o'-B' - 0 5 o 3 6E:Ez koL 0 s 0 < 2n.MLo eu€elo nou nepvd on6 6vq €d,r-

tepLx6 on!€Lo rou E,nopdtr lnln npo(rov dEovq z'z rduveL to o0vopo rouo[€peo! ord onu€Lq A KqL B, !€ z==/62:X-t:ilr xou z=g ovr(o[oLxo,,Apd

'EroL 6xou!€:

tzrr r67:Ezr .6.t=dz - I II= l l l l l pznup 'dz ldp ldo=

-f' lY+ pnup, r z, rflp 1p]ds= tr (" lJ"* ^*' "'

-,

. pt t\Fa| a, - | *814i t\81=62 J=+j" duj" ttupz ta'z -p2-s' )da'1 - + lJ

(q'z-B'z)rue'de'z-j p?rup'zdpl '

l- -B- )de ld€ =

=-T[-to"-a"trouvo.tfry- 1p.6t)vp.a()ye"16z=E' ]....-+ ta'z-o'z-n!{q'z-o'z) l

296 KEOMAIO IlI

27. Iita umloyLooel lo oloKlnsrso I=jJJV-Xr+{Fdxd0dz 6rou n oE€psd ne-

ploxi E ll€pLKl€lerar on6 rlc flltdvel€e X2+t!,'?=ax(6,roc).

A0or:

(xtllvbpoc) xar( lldvar Lodh, !o Agnrrdv)

H spoBoli rnc netLoxltq E oro€[Lne6o 0xUr €LvqL o krJktrLK6q 6L-dkoct X2+U, < qX.

MerqoxntftLaouu€ oe K!J\ Lv6PL_

Ktq o!wetoY!av€q,8610!!e x=pouv0, U=Pnue, z-z

E L v o L 0 < x ' + + z < o x .

0 S Orouvro+pan!rB < qpouve -

u S o s o o o l u u E - i 3 v s - z -

H eELqron rou xdJvou 6lve[

x2+q2=22 . p '=z' . Apq -p3zs p,

ELvoL o(6uq Uio"ffi =0. Eror6xouuei

n/2 aou lgl t t I l l

I"j jJeedodedz=J tjE ' - n / 2 '

. r /2 .4ou lq= l l l p ' l ' lo -cp lae=

) _r / 2L)n

IJ ,,0']oo]*.

:i H t rpoBol i rns er lodleccs ro!x'irlos x2+{,2=ax.

' t * BI6ne dxerLrd dto! A ' td lo

II-lII

=,J_1i,[J:"";t*]*= * J_1,,,,"':"*' *. | !'.'.',";*;*û t" /2-+ I ouv-ode =

q r I I J n l-z l -Tr- -T- l

-f ouv'enle+3f/2

{ ouvonlg+€) | =s l2 L

- 3cnn- -Tr

xurJ!6porr x2+i1, '?=ox oro er i r€6o ox+,e iwL o

(awyo1LNoi r , l rc ! ) .

CIOMHI I IT IvO[ ÔI I t r '4o t 2 . x rdv6dYou B. 297

28. llo unolovrog€|. to tp[n^6 oloKlho{4ro t4e ouvdprnols f(p,o,2)=p'z

6nou n g[ep€6 n€ploxi E n€pldEt€ioL crl6 ro nopoBolo€Ld6e z= -p'z (l) rol

ro €IILnEbo 0x0.

Oo €pyqoroitr€ o€ KLrtrLVbpLK6c ouvr€ToYu6v€q. H npo0oli Tou nspoBoloeL-

6o0c o[o €r lr€60 0x0 sLvdL o xix, \oc p'z=4 (npoKtnr€L on6 rnv (1) Yrq 2=0).

t 2 a ( 2 f - er=l i l p'zpdzdpds=

psd negloxi nou raprrl€tetol dr6 rLc EltLtdveLeC U+z=4t z=0, x'z+{'z=16.

A0on:

t L v o L 0 s p ! 2 K o L 0 < e < z f l ,

0e0po0ue uLo €ue€Co f iou nEpvdqn6 avo Eo0T€pLK6 onuel,o lou

orep€o0 E rcp6,\,\ttr4 nPoC xov d-

{ovq z'2, noo rdlv€r ro o0vopo

lou E o[q ont€lo A, B, !e z=0,

z = 4 - p ' ? q v r L o r o L X q . ' A p s 0 l z < 4 - p ' ? ,'Ero! 6xouuEi

H orep€d I I€pLoxi E n€pLK;\€L€!cL

on6 Tq enin€bo r!+2.4, z=0 kol rov

xlnLvdPo X'+U'z=16.H fipoBo,\i rnq E oro €nln€6o xou

ELvoL o xuKlLK6c 6l .skoq x'z+{r 'zs 16 .

eo €pyqo€otu8 o€ xulLvdpLKeq ouw€ -

rovttv€e. S6rouue X=pouvo! {r=pnu6 ,z = 2 . E L v o L 0 s X ' z + r ! ' z < 4 ' z - 0 3 p 1 4

E L v o L o K 6 u o 0 s € s 2 n .

-1"' !' o' vf,- o" ooo'--['"f e' (4-e. ) deds=f "[e' - + ]i€"f "( 1 6- + ) d0=

16 .^,2r 32n- l - r o r o - l - .

29. tto unotroyloo€l ro oloxtlp,[d [[[Vf .;0-*ranr, 6rou g €[wL n oE€-I

! \ ' i\-i'.'t - '

')rffilfnllll.L!il

x'z+t 'z= 16

298 KEOMAIO III

E0xolq qolvEtqL on6 To nponvotuevo oxiuq 6TL Tq onuelo* Ar g Exouvs xorly-

Uav6q qvr lcEoLxo z=0. z=4-l) . Apo 0 S z : 4-0 - 0 S z : 4-pn!9. Enoldvoe

12( r l . r r -on l l | 2 r r r \

l l l {4[ 'dxdrd.- l I I p.odzdpde=l l l ro 'zr i -e- !"colae=t t t

_ 512 -- - l 5 "

f l ^ r n t 2 n r ^ \ 1 l / t r / a' 'J ''o'J nuoouvuar{ ae= [+'] (,J nuornu,r)rerj','o i nz,+

. l+]1{"," nuoanu'r)terl"=f=*'?'*( i)"

| " , ,2^

1a /4 t I " , ,2^ t r

l_f | 2n,+ lT | 2^=_

3n/ \

.d0r .pLxd drr ! . io rou o lep€oi .

=f"Ifr*",,*,r*]*=f"l+',- g n,']'*=f"(S *on,')o'=I t . . r 2 n=[ : fe*t+o*e1,

30. llq urotoyLd€[ ro olorlibqrc r-JJJrarOa, o*"E

t=t(xr0,z) €Xr: 7,+{,+2, 31, x.,{" 32,)(colden Ltl|s serles)

Iton:

KdvouuE Xpion oeo!pLK6v ouvr €roy! 6vov: x =pllrqouve, U=Dnuenuo, z:pouWeo 0po0!e ro 6pLq r(,rv 8,rp kdL p. ELvoL 0 S e S zfi (oL x,r{z <22 -p'zn! 'zeouv2e+p'1nu'zqnp28 <p'?ouv'?o - ntr '?rp s ouv'zrp - e0'z0 31 (€neLdh ortco(rorpLK6q ouw€roy!6veq elvqL e€ [0,n]), axoute

I n I l r - 1o€ 10 ,+ l I ,oe

l i ; ! , n l . E tvoL 0 5 x '? . r l ' ? *z ' s 1 - 0 s p I 1 (qL

lffijfiff l=o'nuo. roo

t1 i /a r2n

l l lzdxd0dz=ll le'znuo pouwdpded'l=l ordpl nuoouwdol de'' L '

'\ A8// zz' taL n A0 rEpvi ond dld

oÔy\^HP')TIKoI IoIIXMO' 2. tra!6dyou B. 299

l t t - - -31. Irh utoloyrd€l ro olodfur4lo

Jjj(x'+{'+z')n zdxdldz 6nou

E=t t x ,0 ,2 ) €R ' : x " r ! ' , 2 " t . " j -

A0or:

eo Kdvouu€ xphon o0oLplKd)v ouvreroyg6unv.06mu!€ X=pnu{pouvo, U=pnUenUOj z=pouw.t l voL 038s2n , Os93 r I xou 0 l x2+ r ! ' z+22< rz .

0 ! p'zn!2eo!v'z0+p'zn! 'zrp1y'z8+p'zouv'?q 5 rt - 0 I 0'n! '0(ouvteln! 'o)*.ozouv2q s r2 - 0 < p'n!"orptouv20 s r '?- 0 3 p'z< 12 - 0 < p ! r.

ELvoL 0(6!0 | f ioif f i l+'nuo KqL f{x,0,2)-(xz+{i ' :+zz)x0z=

=p 'zp3nu '?qouvgn tOouW. 'Apo

I= [[[o'nuoo'o'nu'ooruenueouvodpdedq= [2nor.rvenreae Innu',rouvoao l 'o'ao='l'. )a

.f2nnueanuefnnu'dnuo["0'= [$1" t+1" f+f =,) o ) o ) o t . J o L ! J o L s r !

32. tlo 6e!xo€1. 6rr yLc D > 0 Etwr I= IJJ (x"+r"*z'tnay4az - ,$

A&on: x'z+q2+z2s;L

M€rqoxnumL(ou!e oe seo!pL(6q ouweiov!6v€q. ElvoL x-pnleouve,( r=pnuqnue ! z=pouva . 0 f l 6 r€ 0sg3z r , 0sasn , 0Sp<1 .

l$fffi lr"*', 1x'z+4'z+2'?;m=ph. Apo

r :f"*f*o,1,'0"60=1e1f"1-ouvo:| lffi, =^', "#

= #i.

33. llo umloyLoos[ Io olorlis|lo lll Vl?i axOaz 0muE

E={(x,0,2): x"r{".2" s l , x z 0, '} t 0}

* H r . p L o x r i E ' = { ( p , s , e ) € R : : o 3 0 3 2 r , o S Q < r , o < p < 1 } . i ! d L o p -9016!Lo t rapol ln ler i r€6o.

XEOA4AIO III

i6on:

l.lErooxnlot[<ouu€ 06 orroLpIxaq ouw€Toy!6vEq.tLvq l x=pn loouvo , o=pn !an !e , z=pouve ue Os05 |

* , 0sqsn

KoL 0 s x?ru?+Z, s | - 0 s p s 1. ELvoL ox6!o l$[ff i [ -0,*, -","

vlz l=vplouW l. 'Aoo

r L t / 2 , " 1 1

- - l l - -r - l vo p_op l du l v l ouw l r lUodg=++ l v louvo l nUede=+ l V louw lnuodo=1 ) r r t ' . J o ' t o

a | l ' / 2 , , : " - ( . - . 1 a I l n /2 x- l l vouvQn!9dc+l v-o'rwnuedo l= + l- l ouv'odouvo+) 1 / 2 I / L J D

* [ ' t -o,wtr cr-ouuorl=] 1-1 ouu'n r1^/ ' -1 [r-o,wr ' / ' l ' =. . .=j l) \ / 2 J / L J l o r t J t 2 2 1

34. lto unotoyrooet Io rp! d oloxliei4ro l.JJJtox'-ol+v.'taxar,a, ono,E

r=i(X,U,z) €R.: yz+r!r+2, 3 l )

A0dn: ( l lathneibt lc ' l Analysi s s ' l la l ik)

go epyqolo(rgE !e qa[p!(€c ouvt€ioy!6vec.o t r o u l r € X = p n u e o u v 8 , U . p 4 ! O I u e , z - p o u v e . E L v o [ 0 s p < 1 , 0 S O S 2 [ ,

o<o<n, l&fff i l=o,nro. 'epo

I1 J J

(oe'n! 'zqouv'z8-Bo?n!'?q4!20*vp'orrv,q)D,r!edpdBdo.

E l

Xr,,pL<ouuE ro lpLr 6 oloKliplJtto o€ depo[olo rpldrv oJ\ox^r]pr,rtdr{rv. ELvoL

t t lI=J J Joe'a!'co'rv'?edededqt J '|Beqrqro4u,€dedBdo+J J Jvo.nuoouv,oaoaoao.l r E l E l

'Eollr Il, 12, 13 ovil,oTolxo ro oloxlnp6lois rou oopotolcloq. ELvoL

* ' e x o u u e t d t o p t o o o c i p c s r i p o d v r r : o 5 o s 2 r t u d p l o u l E o 3 o = ! .

oôlqHPllllKo: AoflIllot 2. trdv6dyorr B, 301

. L . r i r r ^ 5 r l rl . .ol o-ooJ nurqdrpl ouv'?eco=q | $- | | -ouw- !-ql | . louv;4r'u + u

l- -

4no

!l€ ovdôyn Epyqols BpLsKou!€ 6rL t.= 19 , t , . T .

'rps I= +i ' iT. Ti =ii (o-e+y).

t l l -35. Iih unoloyrooel. lo otrodifl,uo lilz'dxd{rdz.

'onou E ELvor n or€p€dE

neploxi nou n€prKlel,EroL otr6 rLc oqotpec p-l, p=2.

Ator:

0o €pyoolotue !e opoLpL(6c ouvrerdyutvEq.E { , v q L 0 S 0 < 2 l r , 0 s q 3 n , 1 S p s 2 ( q L z = p o u v q . ' A p q

t 2 t t 1 t 2 t z i t a t z

l l lz'?dxd{/dz=l | | (o':ouv:a)p'znuedodeds=l I I prouv'zanuadpdade='L' )o )o Jr

-lj"fF ""*,,*r1'

o**=f "l {l *"',nrqd*do=f - +{ + (-"!"',r)]"ds=

={i l rae= . lS n.

36. rb motrorloo.l ro oroxlturolo r=[fl --!f!E:-- . 'o*",))) v1-x2-l)2-22

E = t ( X , { , f z ) € R 3 : x 2 + ' l t 2 * 2 2 s 1 , X > 0 , V z 0 , z : 0 }

l6on:

(lhvfirlorihto /lovdlvou)

MErqoxntor l<ouu€ oE ooolpLKtq ouw€lqyuavec, EroL €ivoL

E '={ (p ,€ ,0) : 03ps1, 0 . t . } , 0 .<0< + } .'aoo rl '/ '1""lt 9lr!9 6p666, ! fl '?|/ '?fl '? nr"ll lfuut dtd'd,r=

r! )o Jr \n:f,t Jo r. Jd

-€*o!e p=n!t .

TEOAAAIO II]

Aoon:

=i'i,'iji,,',,*.,r, *l)'' l" + in*,*ae-f /'[- ]*",1"/1e==f"+ *-+ rui"- #

37. Na ono,\ovLo€€[ ro oAodioolo ]J.ltx.+.*tz-tt ' ]axa,!a' 6mtE

E - t (X , ' l , , z l €R3: x r+Ur+(z - t ) , s t

K6vouti€ lov uErqox4usrLou6 xEr, U={, u=z-1. rtvoL fl{ff i -r.

nepLoxi E u.rooxntorL(eto! ornv n€proxi t ={ (x,U,o) € R!: X' l+0'z+o'z <

'noo r-JJJrx'-,r'-trr r'taxa,ra'-JJJtx.*,t'*,'taxooa".

llerooxnuort<ouue oe opo!pIrac ouw€rovuavec,ELvoL x=pouvonue, $"pn!6n!0, r,r"poirvQ on6r€ O s x'z+0'?+r,r'1 < I

0spsr e lvor xor 0s€s2r , 0sqs" , 1f ; { f f i | =0. .0.

'ooo 1- [ ' [" ["o,o, nrraror3o : [to.6o ["ce [ 'nur6q= | zn. z= * .J , J ,

1 )

38. tb uroloyrde[ .o or'orrrto,.l,. I=JJJ

6mo E={(xro,z) €R! : #-#.#. t t(6olden lhths sc|.les)

llron:

K6voup€ rov u€rooxnlmlou6 x=ou, {=0u, z=w (1).

1.1€ Tov trErcoxnlotLo!6 out6 n otep€6 neploxi E o[€!Kov[<€roL ougluovo-ohtrowq otnv or€p€d neproxq E'"((u,u,vr)€Rr: u'z+uz+nz s l ] .

dxddz

* I rEpLo1t l in< o lor l f ipoors e i*L opoo-yduo idpal ln leni t r .6o.

oô(lltlmIKot lotlt!{ot 2. ltov6dyou B. 303

Ervo! B{ft#=

fit.-.--l-:--zt[Lvo( c(6sc f(x,,! ,2)=e/ F ' -E? ' - |r

=

- e u t * r t + " ' = F ( u , u , t )

'Ersl ro 6oogavo otrokj\iouuo Y{,v€rdL

"ffi(oBr)ordro*

fLo rov unoJ\oyr-o!6 tou releutqtouoô|(lnp6laroq €o !erooxnu6t(oou!eo€ ogotpLKaq ouvraroyutv€c. e6-routE u.9n!0ouv0, u=pll nte,

w=oouvo. E(voL l9J-g'!,4 l=o.nur.l u ( p ,u , rpJ

M€ lov latooxnloELo!6 our6 n ot€-pEd n€pLoxi E' o[aLKovC(€roL o!-ellovooiliovrd ornv otEp€d n€ptoxltE " = { (p ,e ,o )€R ' : p ' z s 1 u€0 :ps1 ,0 j 0sn , 0seS2n .

EtvoL 0(690 F(u,u,r)=ep=h(9,O,0)

'Apq I.oBv.l i I e"o'nuodpd,rde.E -

En€L6i n n€p[oxi rnc olo(liooonceLvo! opeoyirv!o nopoll €nLlrEbo (u€

reroylavl,|v) 6xoou€:

_aL -!'L _!Lau au ata0 au a0au au aw

1n -5!- -5i

t:-1"

l0

0 0

8o0 y

"IJJE '

2:-----,rt--1---:)--,-;*,--l--;, -.::\$.t.')S

C6pEs nopdtrtrn,\€c rpoq ro enlrlebo tolv duv-

304 KEOAÂIO IlI

2 r r 1 1

I=ogyI dol nrqopl epp'op=ogvzn.zI

=qnaBytePpe-l zpepuplr-

=enoovieeo'?-zJ (eP ) 'pdp I l=

=4nqgy1sop,-zeop*zf epool l=

-4nogytePp'z-2ePo+2eP l ' -

=4nogy(e -2e+2e-2)=4no0y (e-2).

'Apo I=4rqgy(e-2 ),

x'z .a2 2 ' .. E. _ iz

ax ax _!Lau liJ atia'! a0 aQau au aw

az Az azau au aw

< lI (o,8,v > or.

q 0 0

080oov

=oOy

ILvoL f (X,0,2)=uz+u2+ri 'z.

l l l'EroL ro aoolavo olo{lhpouo ypdpetol I lJjdy{u'z.u'z+|{ '?)dududw, 0q unotroYt-

sou!€ ro r€Iewoto oloKlip0ruo ue !erosxn!qx!o!6 o€ oQoLpLK6q ouvr€Toyltveq.

(eP ) 'p '?dp-

3e. rto urotov.oo€! ro rolnt6 otorriiouro "llJ

(+ - # . #It*r

6nou E-{(x, l ,z} €Rt:

ioon:

iluo, $fffi=

Kdvouus rov u€rooxqucrlo!6 r=ou, {i=ou, z=vw (l ).11€ tov u€tooxnlotrou6 our6 n gt€p€d nepLoxi E oleLKovt(EroL ctlQLuovo-

oiuovro oEnv or€pe6 napLoxi E'=i(u,u,Nl) €R3r u'z+u'?+flz < 1l .

Aot l [Mot 2 . hd !6d1o\ )

E L v o L 0 l p s 1 , 0 s { o < n , 0 s e 3 2 r .

18{#tH$} ' '" Ko! u2+u2+w2=p'?'

'*" ,=f f f ",*rr'r'nue)dedodp-qoYfo'dpf *rorf

'or=

=*' [4]' t-oJv,tl"" t6l: '=oBy {z'zn= $ "o'.L J I .

40. lto ufloloyloo€t ro rgrnlo olo.li!4ro [f[ 4 ornNt onou) ) ' vx24z+2,

n or€pe6 rep|,oxi E Bp(oxerol oto mdtto 6Y6oo tt v o[6vu, xol nepldr€letol

ofid rry otolgo ;'z+r1'z+2'z-6 rot touq drvouc x'.02=2t' 9-io€€92r

nton:Melooxn!fi [aou!e o€ oqdLPLKeq

o=pntanu8, z=pouve. H e€Looon rnc

oqo{.poc ylvetou 9'z=6 - g-\8. l1

e[iooon rou npdJlou x(i,vol., vlv€roL

0" f ; - - . eLuor ospsG' ,

0303 + xau fsoaro leo2

Ervor oK6uo B{f*6} =e'znle,

t tx ,+ ,z t=4-*Vx'z+{ '+z 'z P

'EtoL axou!3:

l l l1^))) q.lT;7-

a/2 rcEee2 Vi

,a,r,a,-J [J,,_ t{ f,o'nuoao]ao]ae-

E E i o r o n r i l o u o c o o o u p u r i s d u v t . r a Y U f ! . s .

E(ret a2+t '=p2 - p2u2oo, . rv2s+p2ru2q19zo=p26uv29 - n!29=ouv2o * ouv2o-n! 'z ' l

:O € ouv2e:o - ou!2q:oJ! ;

ouv!€roy!avEq €Lvor x=PnuqouvS,

\ iq,

2a

KEOAÂIO III

n /2 rcEee2 .E 1/2 roa.q2l v t i l t l=l I I l-+ nlq I de ldo=31 dol

r r t ' 1 / r ' t ' t t "

. t | t r \ 7 E | 1 |-3 + I -suvro€cez+duv + l - + l -+ - -z \ ' 4l . \!-2 t5

n!ado=3 rsl;/'?r-ouvet:i;"e2.

) .

{1. its uroloyLd€t ro E rnl6 olodtba& I=JJJfx'.f+z'faxafaz OnouE

E €('wL n dt€prt ncploxi rou rcpl(lleaftoi an6 tnv ao[pc (x-o)'z+(Fo)'z++(z-o)'z-o'? (d>01, (lbvenronluro o€@lovlncl

ltdn:

(6voute iov tr€lqoxn!filo!6 x-o"u, U-a'u, z-0.r,{, |,4e rov uatqoxnuo-!Lo!6 our6 n or€ped l€oLoxl] (x-o)2+(U-o)'+(z-o)'? j o'z Ct| l€!(ovL<eto! o!-rpLuovooltgovro ornv sr€pEd fieoloxi Er: u2+!2iir2 s 02.

Ervo! B{ft#=\xu) ( , 100

010

001

'[!oL 6xoule jJ J (x'+'! !+rr )dxd{dz1 JJ t

(u+o)r+(u+o) !+(lf tq) 'zldududr=

-JJJ,,a,o"a*JJJ',a"a,a*JJJ-,a,a,a*r".JJJ,a,aua*t,JJJ,u,a,a,-

*".JJJ"a,oa''*", JJJuuauaw*ro JJJ {uz*uI.wz touauaw.

l(dg€vo on6 ia nd,Io EI! olo(lnpdrtoro e[voL !n66vi,To ag6ouo (ar6 oerpd oô(trhpr,uo 6(veL rov 6vto itq oootpoq d&Lvoq o.

'Aoo etvor 3orl l ldududyr:3d! +ro!=4no.. 0o unoloytooug€ ro r€trEurcco olo -l t )

(]\io{,,!o. Er

+ ncpatr ipnor oc i i6as 2s9.

oAOr\^HPOTIKO! IotItllol 2, Eid!6drou B.

M€TooxnuorL<ouue oe @oLpLx6q ouvrerqyuevEq.€arouu€ rr=pnti0ouv€, u=pn!pn!€, v=pcruve, Etval 0 3 p S o,

o: e s 2r, 0 s a: n, I 3[#t$i I =pzn!o. Ero!6voc €LvqL

r 2 n d a z r , a

30[[[(u,+u,.],, )dududw+" [ [[ [[o,o.nroao iae lao+oI nuuaoI aeI o"ao =' ; "

=3ot-ouvel; Ief; ' LiJ" = ri:- . Eroutvoc ElvoL l=4no5+.i1!- .

307

-B) Eoogloytc ttrv t9wl6v oôd\ipqlfrov

42. llc unoloylooa[ o 6yxoc opoot xuxl.xoo xularEpou UG okr[vo l6orKo xa|, ifo< h,

l0o[:

0 K(rtrLvopoq opl(er !o onlerooovolo E rou R!:

E " { t x ,0 , z )€R! : x rau , 3o r , 0 : z s h }

E{.voL v"j. jJdxdudz.E

ll€tooxngora<ouu€ oe xul!v6O!KtC ouvrexdvuavec.

E [ voL X .po r r vo ,0 -pnu0 , z " z , 0<0 !2 r l , 0SpSo , 0S .z th ,

iti;tfi -0. roorh.2rp tz r f i .h , . I ^ r . tov " l l ldxd, !d , - l I I oapaea'= | d€ l pdpl dz=rg l " l# l r ' : l =zn * h .

t t t ) o ) , J o o L . t o

.no2h (. !ovd66c.

43- th unoloyLd€l o 6vro< tou or€p€oo tuu r€plxt€lero! o16 r|'c €ir|,|6-velEc X2r02=1, 14*2.2 xoL z=0.

Aoon:

H €n!0dvElo X2+U2-l elvot Kullvopl(f U€ o6ny6 (o!rd,ln (oro entn€do OX!)tov KirKlo X'z+o2=l (oL VEv6r€Lpeq ropdtrlnl€q npoc rov d€ovd z'2, H elLqr6v€Lox+0+z=2 elvqr en(,r l€60 nou rcpveL rouc dEovEs x'X, 0 {r, z'z om onueidA(2 ,0 ,0 ) , B (0 ,2 ,0 ) , t ( 0 ,0 ,2 ) owLo toLXo .

308 KEOMAIO III

H EnLptv€to z:0 €LvsL ro €nl,n€6o 0x0.I,4ErooxnuotL(olll€ oE KUILv6pL(tq ouvt€Toyuevec,€ a t o u l e X = O o u v o t 0 = p n ! 0 , z = 2 , E l v o L 0 < X 2 + q , 2 3 j - 0 3 p , < 1 l

0 < p < 1. ELvoL oK6po 0 s 0 s +

r(or x+0+z 5 2 - z < 2-{'l*X) +

0 s z s 2-p(n!0+ouve).'Exoule (o! ffi$ =0. 'ooo

v{JJorarr*E

1 + 2-p(n!o+ou\ i9).t tt ' lt po.laela,=

J . l

1 /2 r 1/2^ , 1 l-l llplz-ptnue*ouvet]oplae.l I o'- $ (nuo*ouve) l ae=

n / 2 'l l I 1 r r f / 2= l l1-+(n ls+ouvo) ldo=|6-+ ( -o ' rv€*n!€) l . ' . .

44. llo unolovlor€l o 6vxoc rou dr€peob E nou 0p6oeft! dr6 rnv drro6-v€la F4-X2-{,rt xol 0'16 ro €nar€6o oxl.

ioot:

r(vo! v=JJJdxdodz.E

H lpoBol i rnq €nl0dvei-dq 2.4-X,-U, (1)o[o €[Lne6o 0X0 €[voL n nepLo-

xi T ue o0vopo tov r(id\o x2+{,2.4 (nOox(,nrEl cm6 tqv (1} yro z=0).'EroL opl<eroL ro EnLn€bo Xtdol.o T=t(X,t) €Rrr xr+S, i 4] . Elvor

0 s z s 4 - X ' z - { i ' ? . ' A p o :

\ -x2 -[2

V=l l ldxd' !dz- l l l l dzldxd' !=l l {4-x ' : -{ ' : )dxdl l ] . ApK€L vc unol\oyLdouu€ ro

6Lnl6 oloKlipr,rlo. M€rooxrtudrL(ouU€ oe rto.\LK6q ouvrExoy!6v€C. EtvoL

X=pouvg, ' t pnt€ g€ 0 < g s 2n (oL 0 < x2+02 3 4 r 0 3 o, < 4 + 0 S O 3 2.

OIO( HPOTIKO' Aot l t l l o t 2 . tna !6d-yo ! B .

ELvsL aK6!o fiff i + *, f(x,,!)-4-x'z-0'z=4-(x':*0':)=4-p,.

2 a 2 2 n -'Apo

l l (4-x?-0?)dxd0=l l l p(a-p ' )dp ld€=l l+ - + I d€='; '" "o

2 a=[ ta-r lae=+tet]*=en K. lovd6Eq.

{5. ih fiorroyloo€|, o 6vKoq rou gte9eoo nou neplKl€Lero! on6 tlc etr!-9dv€L€s (X-1) 'z+U' l=z ( l ) ( 0o3oro€!6€c) ro! 2x+z=2 (2) ( f l tn€60).

A0on:

H rlpoooli rou o!€Oeo0 oro €nlne6o 0X0 opL(EL dvo Xr,plo T nou ro o0 -

vop6 rou ELvq[ o K|1Ktroc x2+0'z=l (n eqlo{,Jon ouli npoK(lnreL ue cnotroLqi rouz dn6 rLq €€Lo6oeLC !0rv 600 €nupov€Ldlv). l,{ErdoxnuorLcoute oa xu;\LV60L(ACouwerovtrtvec. 06rou!e X=pouvo, U"pnuo, z=2.

E l vo t 0<X ' z+U ' zS1 -Osp '? f 1 -0 jOs1 . ELvsL oK6 !q 0 jOSAr .tou (x-l) '?+u'z sz s2-2x a pz-2pow8+1 < z s 2-2pouv8.

ErvoL vvr,ror6 6rr fieij;f =0. 'Aeo

2f, I 2-2pou$ 2i 7I l l l l I 1 | l l - 1v - l l l l l odz lde ldo=l l l e t2-200uvB-p ' *zpouv8 - 1 ) de Jd8=' o

" o ' ; , - 2 e d u ! o ; 1 ' / o L r o

2 a 7 2 1

=f Iforr-o'raoloe=[

= + =+ k. uovd6rc.

46. 'Eoro E n orep€6 ll€plox/t rou ndrrou ov66ou d,rv oE6w,v nou nep!d€|.-atcr cft6 TLq enLedv€l€s z-X'z+zu+l KcL z={+2. l{o urolovro€€t o 6vxoq Iouo[€p€ot E.

Aoon:

ME onoioLqil rou z tr€TqEt rov eELodjoEov rrdv 6io €nlecv€Ldlv npoKtm€L nop8i npoBolil rnc €nLedveLqq rou or€p€ot oro Erlln€do ox{r.

309

3 1 0 KEOMAIO I I I

Etva! x '1+20.1.U+2 - l l j=1-x'?.'Erot opi,(€Tor, To X(,Jpto

T = { ( x , 0 ) € R ' z : U s 1 - x ' , x > 0 , | ! 2 0 } .

ELvoL o(6!d x 'z+z!+1 szsV+2.

(elvoL X'z+2$+1 < U+2 yro xdoe (X,0) €T)

'Aeo v =JJjdxd0dz=E

7 7-x2 $+? 1 1-x2=it1 ' ti.,-'11.'1*=l ll-' ,,''i',*-,*].'.1 ti

'

47. lio unolovloo€l, o dvroc rou cr€pEoa, nou n€pLrtrEl€ror. dr6 tnv ru-trlv6pLxat enlqdv€Lo x2+0'l.2x ro! rq er[n€6o Fx, Fzx.

A(jon:

H enLqdvELo X2+0'z=2x €LvoL KulLv6pLxit tr€ oOnV6 Koun()trn (oro €ll(ne6o

- l( 1 -x ' -0 )d0 ldx =I

0x0) rov (6K,\o x 'z+U'z=zx h (X-1) '?+U'z=l

KoL y€var€Lpeq flo96ll4laq npoc rov d-

€ovq z'2. Adyo ootrterploc rou olepeot0]s npoq ro €rln€6o 0zx,0o 8po0!e rov6y(o rou olep€oi, nou opLo(€rol oro npdr

To 6y6oo rov ol6vrJv.

ttvo! (BI6ne npdlro oxitio)0 3 X s 2 , 0 < , ! s V Z X - X . .

0o Bpo0u€ rc 6pLo lou z.

oeopo(ltE ur,o Euoelq nou n€pvd q-

n6 6vd €d,lrEpLx6 on!€l,o lou st€o€oi

nop6l^nln npoq rov dtovo z'2. H €u-€ELo quri r6uv€L ro obopo rou ore-

+ (1-x ' ) 'dx=

x,/Ti-TI

T

1 ,0 )

x ,1 -x ' )

( x ,0 )

O O(^HPoTIKOT lor l t l lo ! 2 . hov6dYou B. 3 l l

peoo orq on!€Ls A,B pe z=21, z=yovrlorolxo. Apo €tvol X s2< 2x'EroL 6xolue:

t r | | | , 2 , , 6F7 ,px , ,-rJJlaxa,ua'=j, lj, lJ d'Jduld'=

,4167 l=l ll (zx-x)d'! ldx=

. ,l;:==[f ,l " " dq,ldx=f xr{,lfwdx=J o L r o I J r

T'Apq

V=rI K. liovd6€c.

48. lb unoloyroo€l o 6yKoq toupofr 9.3rro.

Aton:

0o €pvoolootiE o€ or0oLpLXtC ouvx€rqy-u t v€q , t Lvo ! 0s0s2n , 0sos r Kc .0sps3nue . 'Apo

v= l l l dxdudz=E

2 r r 3 n ! o 2 r r ^ 2 r r=t t t p'nuqapaqds-l [ 14 *rl"""oroe=[ [gnu'oarae=

=J, t L-* nu'oouw- f npeouve* f o l,*J, # ae. f x. u0u66.,.

ctrgeo! mu r€9rxlslerot otr6 rov 16-

r ' B l ; t r . urodr le ioon del i6cs 191.

** T6pos oloudqelal n eilgd!.La rou icpoyero! ue

(x c) 'z+,r ,2=r2 repr l rov r ieovc , ' ' .

( x - 1 ) ' z 4 , ' z = l

t r .pLorpo,o ' i ro ! Ni r lou

312 (EOA]\AIO III

49. lld unol\oyro€el' o 6vxoc tou sr€peo0

enr0dveLeq X2+{,2+2'z=oz (t) xoL z'l-x'?+$z

nton:

H erLodvaLo (1) vD&r€ror

!€ xtvrgo ro onueto (0,0,+)

|,4€rqoxruorlaouue o€(rEpnpBtUe, z-pouvrr.

fi €lc@on rtq oooLpoq yp6e€to!p'1ouv'1€n!2q|p'?n!2€n!'?0+p'?oui'z0 =

-opouvQ . p,n!,0+p,oUv2e.opouv0 -pz=qgouvq . p=qouvq (3 ) .

H eq(q.|on rou |(djvou ypd0€roLp'?ouv?q-p'?ouv'?en!2e+ptnutenuto -

p'?ouv?rp=p'11ytrp - 6gv'?9=qy'?(p .

o=+ (4), An6 rnv (3) npoK0nr€L0 S p S oouvo, An6 rnv (4) npo(l-nx€L osos f . t t vou oxd lo

ogoLpLXaC ouvr€royutv€q. etrouuE x.pouvonuo,

E nou [€o!d\€[€Tor ond rlc

tzt.

= -$ . 'or" etvoL oootoo

. H €nLedveLs (2) €(vqL

'!

o s I s 2n xoL iiplffi l.o'nuo.

d8= i- K. uovd6Ec." -2\- -r/4

o ' l I s u v ' o l- -T l l - - - l

"*'. ('-+)'ror orr(vo f

d8" ++ J,

50. lo u|rolovr.o€a(' o 6y(oc rou o[€oeo0 rou n€pL.l€(EcL dr6 rrc mlgt-yEf,€S z-4-{,'?, z+2+2 xcL rd e|rllrEba X.-1, X=2.

itor:

0L €n!06v€L€( z=4-tt2, z=$z+2 €LvoL xulLVdpLKtC !€ obnvo0c t lq (oW(JI€(ovlloroLxo z=4-qrz, 2=02+2 ([ou Bplol(ovro[ oro €nlre6o ouz) NoL y€v6reLpEc|lopollnlEq rgoc rov dEovc x'x. H op8i llpogoAi rou orepeoi, sro fiLne6o oxuE{.voL ro Xop{,o T nou n€OLKtrELEToL on6 TLc Eugel€q x--1, X=2 KoL to <€riyoq i0jv

7i?.:

Itrv euosr.dtv {,2=l € {Ftl {flpoKtflt€L

lou z), lnlooi ro xuPl'o T €[vol

ro oP€ovd,vLo ABf!. 'EXouUel

v-JJla*alo. (1)

'Ofl(,rc 0olv€tol ott6 to fiofto oxn-

uo . [ vo l - 1 3x32 , - 1 303 - ]

eo gPooua ro 69[0 to0 z' oeooou_

!e avo Edit6olK6 onueto to! otE-

peoo iol uLo euoEto ltou rePvd o_

n6 to onleto out6 ndPttrlntrn ngoq

rov 6Eovo z'2. H Euoolo outi i ' -

lveL ro sovo9o tou or6Peot oto

onuetq K raL I ue z=t!'?+z' z=4-0'

dwLotorto. 'Aoo '1"+2 I z34-U'

. ' ? . . r . A -o r r tEnou.v,,rq v,J

,lJ ,tj{._2d,lduldx "

'1,[,,"*"''"'*1'- "=1,[,,"*'"r*],'=

=1,[-'+.^]','-==1, t--e'-+.4ax=1,+

51.t{ounotrovrooelo6Yrcclo|,qt lolotttoult?9!t^![?to!qI6@|tooo9olo.

.16' ,.'l$ 11t rq! ot6 ro lrtnrlo '.+iLj:I*, osoooôvrrne)

016 tlq z=4-U2, z'02+2 ue ofloôlen

ax=$'a = e K. uov66€q'

3 1 4 KIOA]\AIO III

l0on:

H npoBoJ\i qC rouiq rovKil(IoC (X-l )"'{,'?=5 (npo-( l ,nr€! 0t l6 rLc (1) Ko[ (2) pe

orqloLQi rou z).(dvouuE rov terooxn!fi1 -

0!6 X=1+po0v0, { ,=pn!8, z=2.

E t v o r 0 S 0 S 2 n , K q L

0 s ( x - 1 ) ' ? + 0 ' z < ( V i ) ' ? -

0 5 o z < ( v 5 ) r + 0 s p s v 5 .

E[voL oK6!o ui65fi = o.oE(llpo(lle ![o eugeLd nou fl€pvdsrl6 6vo €oor€pr.kd onlel-o touqraoeo0 lopdilnln npoq lov d-

€ovo z z, H €uoelo rclveL looivopo rou or€peo0 oto onuelo

A , B ! E z - r i l , t = * - t

qvrLoroLxo. 'apo e[vou (j{1< z s

6uo sr€pedlv o[o enl'n€60 oxz etvoL o

,=+.1

---r

( x - 1 ) ' ? + 0 ' z = 5

u++1 -

l+2pouv8+p, <. .!E!!9,t.

r s1; '=

,t ,znr ry'i r l:!rf]9 *1,' rpo v=l l laxaua'=i l l l l ' o ir i lao]ae=t t^ t ' , t ' , ' ' r+zoouua.et l

zr y'E 2r ,la=[ [[ r ['.i* *1- r*2pouve*o'

hrl*=J *l Io(2+zoouv€+4-r-2oo've-0,)ao]

2r tE

rso-o,rao]co=|J [$ - #j* o,=;

= -+ r(. uovd6ec.

| 2s 25 1L?-- l - l

1 f f f=TJ LJ

_ 1 " _ 2 5

52. ilo uroloyr.d€l o 6vroc tou aw,eot rou {t6tou ov66ot rw dE6wv muopL<.ro. ffi rLC 4|.&ve.ec x'z4,.a, lll, a2+22-o2 (21.^(tott,

0L 600 €nLOdvELss €LvoL (ulLV6pLXtc. H EnLOdveLo (l)6xar odny6 (ountln

oIoX, lElnTIKoE lo l l lMox 2. t rav6dyou B. 315

oro €nhe6o oxu,rov |(oKlo x2+02=o2 xoL yavtrelp€c mpdllntr€q nOoc rov dto-

H €nlqdveLo (2) exE[ o6ny6xoln[trn rov K(jKlo X2+22=o2, orto€nh€60 oxz KoL yevaieLp€q nq-p6llntr6c npoc rov d€ovo $ U. HnpoBoli rnc iouic ft,v 600 orE -pEd,v oro enln€60 0x0 €Lvor o |(0d\oq X,+02-o,. An6 ro 6E0r€poox iuo eoLvEroL d rL 0Sxso ,03 ' t r< !6 t : x f .

eE(.,po0ue ulo €ue€Lo rcu II€p-vd 0116 6vo eoorapLK6 onlelo rouox€peoo flopdlln,\n flpoc rov dEovqz z, l l €uoelo r6lver ro oovoporou ore9€00 qro ongeLo A, B !ez=0, z=6r? ovrLoror.xo. 'Eror.

e t vaL 032S\67 : I? .

. Ae" v =r I lr" T J-** 1. ] d,, ]d, "

tot tvd2- r2 - I ao- t r - : - - - - - lq - - aa . .= l l l vo , -x ,d t ldx : l vo , -x , t ' !14 ' - r 'dx= l vq , -x ,vo , -x ,dx= l (q ' -x ' )dx=Jo LJo I r ,

=1"',- $ ]i -' + = + K. !ovd6ec.

o 6yKoq rou or€p€00 E nou neqLKl€[EtoL olld tLc €nlqdvE[€q4z=x l2l.

("C!lculus" R.El l{ot)

53. o Bpe0€1.4z=xrrl, (l ) xoL

l6srt:

H ngirln eflL06v€Lo €i.vol ndpo&Ioe[66c |(o[ n 6€0r€pn €Lvo! €nLn€6o. Ano-

Iolqouuc ro z u€roEt dv {l) Kor (2) o[6re [poxtrEEL n €llo0ron x"x'z+o'z ((t-

K;\oq) nou €LvoL ro olivopo rtq (pogoîc T rnq enLqdv€Loq tou ot€peot oto €nL-

ne6o oxu. 0q !po0t€ rq 6p[o rou z. 0e0jpo0I€ uLd €u6€Lq nou 6LtpxErqL 0116

3 1 6 KEOAÂIO III

6vq €o,Jr€orx6 onuato rou ot€-pEoir nopdl,\ntrn npoq rov d€o-vo z z. H EUoeLo r6UvEr ro oir-vopo rou E oro onuEto A,B tr€

z= i- (x'+' lr ' ) , z= - i-x ovrt-

o[oLxq. Ero[ e LvoL

i (x1 ' ! ' )szs i -x . 'Aoo

7I l t l l t n x lvlJJdxd'ldz1'lLJ, !zldxds=E r i ( x '+ { ' )

-* JJtr-r ' -u' tarou. r 'o uo

,- -}Joor*, o:pn!e. ELvor

+pouv6 -p2 t * - oaoa

unotrovLoouue ro 6L[i\6 olox;\ipouq oatowe

x'*r!' s x - i*porrve*p'or.,v'e*o'nu'es-f *

-i Etvo! q(6uo 0 s I < 2n, 6i6iif =p.

' = + f' f p( **po'ue- * -po'ue-o'o'u'e-o'nu'e)aooe-

_ 1- 16r

=+lj'f t+-"!,'* +l:"1+ + +):*=+r"6-+)*=tetfn= fl x. uovroec.

54. lio unolovr-o€€{, o 6yKoc rou o[spEo0 € nou n€pLd\€t€Tol on6 rnv K0-lLv6p[Ki enLpdvelo {,-X'z roL rd entrE6d 'lr=x, z-d, z=A a,B > 0, B > o.

Aton:

H fipoBotri rou or€p€00 oro €IItnEbo 0x0 elvqL To x0Jplo T nou n€pLKtr€l€-

rdL ond rnv KqUIIt ln 0=X2 ko! rnv eueei,q 0=X.An6 ro 6€irr€po oxftuo eolverq! 6rL 0 < x < 1, x 'z30SX.E t v q l o | ( 6 u o s < z S B . A p o :

4z=x" "!"

x'+1t"=2x

oIOKIIPOTIKO! lolltMot 2. tta!6dYou B. 3 t 7

u=JJJa'a,r,a'=E

r l r r l r rB r r-j, lJ,,IJ.d']doJd'=

=1,U".t ' t ;au'1a' =

. (B-o) l l l ,d ' ! ldx =

=(B-o)J I { l } ,dx =

=(9 -o ) | ( x - x ' : ) dx -

- {B-o) l+- - i - l .

=ro-r (|-{)- fx. !ov66eq.

55. 'Eoro f(X) ![o ouvsxic ouvdptQon orc 6t6qE4!o to,8] !E f(X) > 0vx€ [u,!] Ko! T ro xr,pto rou €nlr66ou 0x{, rou opL(erdl qn6 ro 6l6ypqrgarnq ouv{ptnonq,tov 6Eo!q x'x roL rrq euoeteq x=d, x=8. 'Eolr,r E ro ore-gEd nou npor0rrlel on6 tnv n€plorpoqi tou T nep[ rov 6€ovo x'x. llo 6e!x0al.6rL o 6v(oc ro0 oiep€ot E 6lv€ro! qn6 rov rhrc

v-n [Err(x) ] , ax.) d

0 (nro0u€voq 6y(oq dtvetol dr6 tov r0no v=JJJarau,a'.I

A(ron:

3 1 8 KIOANAIO III

'Eoao Tr To x0jplo rou fiLn66ou oxu flou nepLKl€L€TqL cn6 ft bmypdulo-

rs rrrv -f KoL f a[o [o,B], E[vdLosXsB , - f (X ) sq rs f ( x ) . Tocr€p€6 E nEpL(t rEl€rqL qn6 Tlq e-

nuedveuec z=-,/i?dttiF KoL

z=' / Lr(X) l ' - r ! ' . L loL €XouUE

ttt ,6 ,f ( r ) ry'[??ITT'lI7

v= l l l dxdodz= l | | l dzd {dx';' ,d, _r( x)lvl?Gll]-{,

rs rr(") - _ u{?dtlr:[r .=l I z o,yo; =' o ' - f ( x ) - V t f ( x ) 1 ' ? - { ' z

-B , f( x)=l I z'(?(xTr-lFa,uox =' o ' - f ( x )

=(oerou!€ 0=f(x)nu0) =

2

t 2f (x )ouv0lf (X)ouvededx/ 2

2

tf (x) l 'z( 1+ouv2e)dedx=/ 2

- - t / 2

x ) l ' l €+ i nu20 | dx' - a / 2

f- r l t f (x ) l ' zdx '

=fJ".f r.:=Ju,,,

56. l{o unotroytd€t o 6Yroq tou ol€peot E nou llEp|,deletol cn6 tlc mL-

qdv€L€q x'z+{,2=2, X'z+U'z'2x=o xo[ sn6 ro €n{'n€6o 0x!t1,.(llovsnlctiuLo iov6tvou)

nto4:

H €nLedv€Lq x'z+U'z=z elvsl llqpoBoôelb6c, evdl n erlQdv€Lo x1+$'z'2x =0

elvql K[trLvdpoc us o6nY6 Kolni,ln rov KirKIo x'z+U2=zx KoL Y€v6reLp€q nopdl -

ln iee poc rov d€ovo z'2. H npoBol i i rou or€p€oir oro enln€60 0x0 elvoL o KU-

(x , f (x ) )

oÔT, IHPOTIKO: lo l l tMot 2 . hd !6oYo! B . 3 1 9

KlLKdq 6loKoq X'z+U'? s 2x (KAv-

r p o K { 1 , 0 ) x a L o K r l v q r = 1 ) . 0 o

Bpo(l le ro 6pLq rou z. €eopotu€

uLq €ugelq l lqpoi in^n npoe rov

6govo z'z rcu nepvd on6 6vo s-qrr€pL|(6 o4!e{,o rou a[€p€ol. H

€u8elq ouri ouvovrd ro otvopo

rou orepeo0 orq on!€Lo A KqL B

nou exouv z=0, 2=y'z+rl 'z ovr l-

mo[Xc. 'Apq ELvoL 0 szs X'1+U'0 dyxoq lou olep€oi 6Lv€YqL qn6

Tov r0ro i

x r q

v=l I ldxc,rdz-l l l I dz ldxd{,=l I (x'.,! ')dxd,, (1 ). ApK€t vo unoroytoouu. roI T

61ntr6 otroKlhp0rtro. 0610uu€ x=1+oo!v8, | ]r=pn!8. tLvoL

x"*' !2 s2x- (1+poove)'z+(on!8)'s 2(1+pouve) - 1+2po0vo+p2 32+29auve -

p ' : s1 -0sps l . ELvo ! q (6uo 0seszn xo r f f i =0 ,

f (X,U)=X'?+Q'z=( l lpouvB ) r+(pnuo ) 'z=1+2pouve+p'?

57. l{d unoloyroo€L o 6yxoc ror, orep€o0 E nou rleprdEl€toL on6 rnv €rIL-

tt/Jrtlo X2+{2+22=o2, (z > 0) xoL on6 rnv €1rtQ6v€Lo z'z=lx'z+{'zlop'zlJ !e

0 < | , < n .

i(|on:

Oo unoloyLooulE rov 6yKo rou orEpEo0 o[o npdlro 6Y6oo ruJV qEdvLN.H npdj-

rq €nLqdvE[o ELvoL oQqLpLKi Ks[ n 6€lrepn KovLKi. M5moxnuarlcouue oe oao[-

pLKdq ouvrErdYu6v€q.

oEroute x=psuv€nlo, U=pnu04!q, z=pouvq. ELvqL:' ' < o L 0 a d .

u I p : o , u a v t 2

2 \ I 2 \' rpo v- [ [1x '* ,J '1a1c,r=l [ [ prr .zpouue-o' lar jo.=[ [4 -z 4 ouue- 4 l 'ae=t l J o L J o ) J o t . r c r o

2 a 2 a=l (i.i**.*)',=1 ouu66s= f zn+ | tnueJ?-= $ *.ro,,.2 a

i*.i I"

x",i '=2x

320 KEOA.ÂIO III

Ervol oK6!o I B{H$#l ..'r.6 a / 2 u

'Aeav= l l ldxd{dz" l | | p 'a lododedp=

"E-

=J *roJ aeJ nrrao=

= [4], rtri/'r-o*rr"=

=.$ lrr-o"u.r=$*r,*'Aoo Vo^-4V= $o:n.rp'?f x. uovdOeq.

58. Ilo unotroy[oeEl o 6y(oq V rorr dr€p€ot E nou nep!(I€|,€roL sfid ro €-rl,n€6o z-0, rtv rulrvdglxd Gn|'dy€rd xr+||J2=o, xo! rnv nopdolL(li xu^lv -6gl(i en[0dv€Lo o'?qnz (n > 0, : : 0).

i tdn;

( t . i l .n. l

0 4nroir lcvoc 6yxoq 6[vEroL ond rov l tno U=[[ [OrO,tOr. A6Vd ouu!€rpto(t t"t

eo ulloloyLoou!€ rov 6y(o rou qt€peo0 oro nd,to 6v6oo r{,,v ot6wJv, rtou €Lvo[ro l /4 rou olL(otl 6yxou.

E l vo r o52g -9 : ,

'ro" ,i =JJJaxa,ra'=E

=lIIJ i t l

a']axa,r"JJ r' rf '/'axar,=

=lJ#**'T

To XopLo T elvoL o KUX.lLxdc 6Co(oq x2+02 < o:llerooxnUor[<ouIe oe nol!x6C ouvreroyldv€C. etrouUE X=pouv0, 0=pn!€.

OIOKig!0TIKOI AoIl[[lOt 2, trov6dyou 8. 321

t lvoL ospsq, os€. + , 3{*$} . .a/2 d r /2

' t " { ' [ f -s1 a*o+=* [ [ oo '* 'eaoae-] [ nu 'eae[ o 'ao=', , i-

"* [].-]n,*1""t+], =+ + +-#. Aeo v=#.

59. ih unotrovloo€[ o 6Yxoc tou oteosoo 0 mu oo[(eB! ard rLq oxa-

oe rq z2 - {20 , x ' 1 -220 , { ' - x >0 , z2 s2 { , x2 322 , V2 s2x( llove[lodu ro oeoaElovtxnc )

A0on:

To oraoed neoLKleL€To! df l6 T[q KulLv6p[Kdc €f l lqdv€L€c 0=22, 2=x2,

x='!'z , 2'!.22, 22"x2, zx=lt' . Kdvoup€ rov lerqoxnudlLou6

-i- * , * =u,

+ *. An6 !L( 6oo!6v€c oxdoELq rpo'(( ln!€L 6tr .

- j_ ' r- -L > l. - lL > 1 xou -j- s 2, + s2, + s 2. Aeo| 0 - ' z - ) ( r v

1 3 u < 2 , 1 < u s 2 , l s L < 2 . ' E t o L n o r e p € d n € p L o x i Q o r € L K o v L ( € -

rol q4Llovooiuovro ornv otep€d nepLoxi nr: t l ,2 l , t1,2l ' t l '21 (r l rBoq) '

n - 2 2 2 2" -iF -r

rrucu $fffi = lu,, un ",1 -l+ o g

0 -z 2')2z 0 - x l =7

- izxo

D ( x ' { , 2 ) - 1rtvou 6uuc ii"iffi ;i]Jlffi -1 - ot_.ffi =z . Aoo

v=JJJa,a,r,a-JJJla"a*,=*J *J aJ a'=f r,r; r'rri rrrl=+.a a r

60. l{q moltoylo€sl o 6Yrcc tou otEgao! € $u D6deror ott6 trc e[!odv€!_

'c z=x2+{2' z'2x' (rhv€n.oiirr.o ,toybtyqr)

2 l

2tb "x

l

322 KEOAAAIO III

Aoon:

M€ qrlcloLqi tou z Epto(ou!€ invoro enltrE6o ox{r.

ELvoL X'1+0'z=z, z=2x - x2+tt2-2x. xz+(r 'z-2x+1-1=0 - (x 'z -2x+1) +o' l=l- ( x - 1 ) ' ? + U ' ? = 1 .'Apo

n npoBolij rou o.Epeol, eLvo! oKoxtrrK6q 6Lo(oC

T=t lx,U) € R'z: (x-1)'z{r'z < 1}

ELrrcr. x'?+0'z< z s 2x. 'Apo

2\t t t t t l t l

v - l l l dxd0dz= l l l l - -dz ldxd{ "I f " '

l l ^ -- i .1t zt -t '-+')dxd+ (1) Kdvouu€ tdr-T

po rov trEiooxntorLo!6 x=l+pouv6, U-pqle on6r€ eLvqL0 s ( x - l ) ' z f l r ' : < I - 0 g p z g 1 - O s p s 1 . E L v o r 0 ( 6 1 0 0 s e < A r K o !

f (x,U)=2x-x'z-0'!-2+zpouve-j-2pouvo-p2o0v'zo- o2\v2e=t-p2 *or ff i =p.

2 r 1 2 n' rpo [[rzx-x'-u'rcxa+=[ [ f rr-o'r 'aolae=[ f4 -f- l ' ,", i , , . , , J ) o t . ' r o

=(t-+/ r0l ; '=2n+=+. A* v-+ ( . lovdo€q.

61. th uloAovro€€(' o 6ymC tou.oreg€oo nou nsoLrtct€ro! ord r.q Enqd-v€LeC X2r02.2, X,+r!,=4, z=0.

A0s4:

H €[LedvELo X2+U42 €tvor nopoBo,\oer66c, €vd n snledleLd Xz+92-4 eL-vaL ope6c KUKII(6C l(tiLvdpoc,rttro( n afiLodv€Lq z=0 €tvo! ro .Illneoo OX0.

A6vo oullatolds 0o Bpoiiue rov 6VKo lou otepEoi sro nd,,ro 6v6oo &Jv o-€6urv.

'[toL ELvor -i- =JJldxamr. H npoBoli rou or€peot oro erlneoo oxu EtwLE

npoBoli r4r enLedve[o( rou drepeoi,

( x -1 )z { ) ' z

oIO(^HP0TIKo' lor l t t4ot 2. t la !6dyou B. 323

o KUKtrLK6q 6Lokoe X'z+U'z :4,I Iou oPi,<€L Ue roue e€TLXoirq

nuL6€oveq f,o €nlne6o xoploT{1x,0) € R'z: x 'z+U'z < 4,x:0

0 : o ]Tq 6pLq l(rv x KqL 0 6tvov -

TqL Ond rLq oxloaLc 0sx52

o s 4 s V?:tt. o6pvouus uLaeu8€Lo fiqpdllntrn npoq lov

dlovo z z nou nepvd qll6 6-vo eoot€plxo orlleLo tou or€p€oi. H €uee[o surh ouvovr6

m oovopo rnq oT€p€oc n€pL-

oxiq oro onleLo A xoL B !€2-n, 2=y2 t t12 svrLoro Lxo.' A p o 0 s z s x ' z + 0 ' z . ' E r o L

axou!€:

r r 2 r r w - t ' r r X ' + { ' I l_F = l l l l l dz ld , l ldx=

={ll*'"*"*1*=

=11,,r.+11*==fi,"'t=,. rtgr'

'Apo V=8n x. uov6669.

62. lib umloyloosl, o 6yf,og too or€p€ot E tDu n6plxleltEoL dI6 rlq EnL-

edv€leq {Fl-X'?, 20=z xol or$ Io enl.treEo onl,.

' I i

" ' - / v2 t \ 12

ldx=faro lnu i .v4-x , x \+ -T

) ) " =2t1.

n0on:

H f ipbrn erL06v€rd sLvoL KUlLvdpLKi , !€ o6ny6

Ko! yEv€r€Lp€q f lqpdi in^€q npoq rov dEovq z z. H

(colden l4aths ser i es)

Kounlin rnv lTopoBoli 0=1-x 'z

6eirrepn €nLadv€Lo €LwL €n[-

(x , /4-x ' )

324 KEOAAAIO III

ne6o. eo 0po0!e rd 6OLc rou z,€€olpo(,tiE llto eu8€to nopdtrln,Unpoq lov dlovo z 'z no! 6[ tpx€-ToL dn6 6vo eoJTEpLK6 onleaorou E. H €uo€lo ouvovtd lo o0-v0p0 tou oTEp€oLr oTq on!€Lq A,B UE z=0, z-2$ ovr lotolxd,' A p o 0 < 2 s 2 0 .

H llpogoll] rou orep€oo cro €IIL-n€6o 0X0 €l,voL ro Xrroto T rou6eir6pou oxhuotoq. And ro oxi-uo gotverot 6r[ - l S X S | (oL

os0s l -x ' .

'ErsL €Lvor vl. l jdxd{dz=I

f _ r r_ " r f _ " l=l l l I I dz ld'rdx =

I l - r 2 r - ,=t tt z,r,arla*-z[ f4l'-'1 r . ' o r l l L l r o

=['-z{.+]'"...=--lgdx- | -x.),dx=l ( t -zx,+xr )dx=

K. Uovd6€q.

63. tla unoloy[d€l o 6ytoc roo sr€p€oo E noo lp6o6ror ar6 lng (F(ooxz+{'z+22=4 11) rdl on6 ro nqpoBolo€rbcq xr+or-3z (2}.

(Calcul0s F.Feudor)Aton;

t{vot v' lJJdxdodz. H rooBoli lnq enlo6vclcq rou orep€ot E oro €n[re-I

60 oXU elvor o KtKloq X2i! ,=3 (616i! on6 rLq (1) xo! (2) npoktrr leLz2+32-4=O ) z=1, z"-4. To enlne6o z=1 onor6nrer, on6 To nopdBolo€166c rovKirKIo X'z+(r '?-3"1 - j ( 'z+0,=3).

0 ( tdoq opl(€r ro xrdpto T={(x,01 €R,: x,*{ , < 3}, Otpvourjs uLq €u-oEla AB rcpdllqln flpoe rov dEovq 22 nou [€pvd ord 6vo €otdteplxd on!€Lo

;,xfl,flfilritx,1-x ' z )

OIoKAHP0TI (o ! Io I I t l4oE 2 . I rd !6 . i yo ! B . 325

rou E, H €ugeLq tauv€L rnv a(Lqd-v€Lo tou or€pEol t oro o4!€(o A,8

!e Kolty!6veq z= + (xr*,1, . ) ,

z=!att:[r. ,Apo

-f tx, ', l ,ts z s t/ i.F-i ' /-. Etot

6xouu€:

'=lJtJ,' : : azlaxau-Jjlvs;10z-r 5 ( r '+ t ' ) T

- -| r,.-,r,"1]d,.r.

TLq vq ufioloytooutr€ ro 6rni6 oio(tripouq uetooxnuqrlcoule oE no,\LK6( ouvt€_xoYti6v€C. El.vor X=pouv€, {=pqqe, !€ 0 < O S Zn,

0 s x ,+0? < 3-0so,s3 -0sp<V' , f f i=o *o .

v4_x, - { , ,_+(x4| l ]z )=v4_e,_+p, . .Apo

,=1;'i,1,' 1,a;- + !,,=f i,[/'*;',0-J* 1i* ]=

=2"[-][ o-o,r'a ]6- -[ ro"rf]= * x. uovroec.

i60[:

0 6vKoC rou €lI€LVo€L6o0C 6lvExo! sn6 rov r(lrci

E-{(x,{,,2)€R!, #,}1 .+1 ."

=rrrl ' [- ] Jrro,, rrrro-0,,- I Jri,o"]=

lt rou EJ\trar,{.'oEL600q oro

(rpoKtuft€r ond qv (1)

64. lilo unoloyr.o€€[ o 6yKoq tou elt€r-0oel6otc -.t( ' $ - $ =t trf

v=J/Ja*a,ra. ono,E

Enln€do oXU 6Lv€L rnv 6lêuln

YLo z -0 ) .

l lIIL

.- - - - t -'-\:t4t :i ---- r.

H npoBotr

326 KIOMAIO IlI

'- -n"r - -Fr , 1- 17 - -tt

SnIeL{in opl<er ro €ntn€60 xrdpLo rdfx,Ot: $ -# a ,t.

( i ) l0vovroc oq l lpoe z 6tvEL

H

< z : c 'EroL €awl

"lf,-E--gt t t t t l t - r - d 2 8 2 1v= l l ldxd0dz= l l I dz dxaU =zc [ ['L' 'iL'_sllig_E ' 'i

To rei€uro{,o 6td6 olokliporuo unolov{-<em! !e rov t/€looxnuftLou6

- f -oouve, *=pnlo, 'e .o. etwr f f i =ogp, 0so<2n,

- + dxd0.

1-+- - i t=v l -p ' z . 'Apo

rou opa<Elor 016 ro €r\l\€LrrrlK6z='l (2).

z, f lpoKi ir l [€L n t \ t r€r{n x44q],=1

o= +-+; s.r-o<psi KCI

2 a

T

={nogc x. uo vd6 ec.

65. l{o !raôvLoo€|, o 6yxoc rou or€p@.,rlrpo8oloe!6cc z=Xr+40, (t ) roL ro €rdn€6o

Aton:

| i le qruloLei te[o{0 rov ( l) KoL (2) roumu €[wL n rcoBo.\i rnc €nLedv6Loq louor€p€ot 6[0 rnanE6o oxq]. H etao0ron rlqtilEl'lnq vpdetrol t -

fi-y =1.

\-T IELvoL -1 < r<1 . - i . l ; [ < r < ] l + :

A6Vo ouljrsrploq lne ElI€LUns oc lposroue dEov€g rE { .pvouue 0<X<1 ,

'l1:i7

1-+-J' '- -e - -8,

oôKrlHPnTlxot lolltMot 2. xra!6dyou !. 327

E(vou xou x'?+4rlr '?s z s 1. 'Apo

, vi:F ,* =[[[.**,.=[ f [?[[ o.lotlo'=

.,E ,2 x2+4+z) '

, Ji:i"

= l l l 2 rr -x ' -q,r ' tcu lax-.)

1 ft]:z

" l l , rrr-x 'r-9 u' l 2 ax=

=1,' *,,*', "' *= * | [rr r-r"r'i' - ]rrr-r'i/'- ]."arx]'. * .'Apq v.+ (. !ovd6Ec.

55.o6vero

l(,on:

E (vo r v= | l l dxd0dz ( l ) .

!

H npoooli rnc enLodveLqc rou oi@€00 oroEn(rc60 z=0 eavdL to rpayoivo 0AB.Ard ro d€fteoo oxiuo kooue 0 s l, j 3 xoL0 s x < {, oApvourr€ ulo au€E(a mpdlJ\nAnrpoc rov d[ovq z'z rou novd ord 6vo €o0J-rapl(6 onu€Lo rou E KoL nou rl lv€L rnvEne6vELo rou orEpm0 oro onu€to r xoL dlta z=o, z= UitT owlclolxo.'Aoo o<2 . -A , E rou eawu-

ri- q' 'r I

v - l l l l l d z l dx l d ' ! = l l l f , ; dx l d ' ! -Jr LJo t ro I Lo LJd v'- ' .)

Ilo urDlovlo€€a o 6vroc too orap60 |DU r€orrl€asror ori mv €1tr-4 .

z. U:;T rdl old td e|lane6d 0EX! {F3r I-0 (o1 2E0.

. l # i r d0-21 i l ;+ d (u ,+1)=2t ln (0 ,+ l ) l ; -2 (J. v_- |

o,(r

I n l 0 - l n 1 ) = 2 1 n 1 0 ( . p o v d 6 E q .

67. llo utotroYLo€sa o 6Yroq tou qr€peoo rDU fleplxletemr dr6 rlq ertr-

frdve!€g x'z+{z-z'z (r6voq) ror x'?+{,4(z-z)'?=4 (@ts'a) '(scha|''s outlllne serles)

orpoLpL(h ouvrErovu6v€q. E(vo[ x=p4loouve,

lton:

Irl stooxnufia<ou!a o€

Gpntqnue, z=F'drrw.H €qtooon rnq oeoaooq Y[vetol

H €lLooon rou (0vou Ylvetol

,F+

AOY(,I ouuu€rploq eo onoloY[oou-

!e rov 6v(o tou orar€oo oro lp6jro

dv6oo tov oq6virv.E l vo r 0 s e s ; i ' o sas i i

l n r " i , , l I "o s p s 4o0w Kor lfiili,roi l=e'nuo. r 1 2 . , \ / \ . , \ o v e o . ^ t n l 2 t t r l { I'roo * -[[[aroua*[ ll ll o'nu'ao la'r lae- g1 ll o'rvrenlode ldo' 4 ) J ) J o L J o r r o

t

-+ t ' ;o ' + . ' rpo v= I x . uov66€q.J J

58, tto ufloloYrooel o 6Yr(K to(, orop€o0 ttol' tr€pltlglEtol ott6 tlv o9otoo

x4,lt4z2-o, (o > 0) rar tov rdtro z2nu'td.(x?+{')ouv1, 6tDu tt €(w! lio

r r o8@dr t l o ro |nU0<o<n .

Aoot:

Oo unotroYioouuE rov oYxo tou or€p€oti dro rpdtro 6Y6oo fl,)v ql6lov (l6Ylt

ou!!Erpaac rou orEpeoi).oo u atoox nlor Loouu e oE opol,pLXh ouvterov!tv€q' O|Iou!€ x=onuQouv€ '

i!=Pn!en!o, z=PouW.H €(Cooon qc oqqtooq YpdQctol p2=o2' 'EroL Elvsr 09<q'

t, y;jas xr+62+(z- ,=4 e X2+l)2+zz-42=A - p2-4pdu!9:0 e o:4o!!9

r .* Ejw! r r+r l , r=2, - o2r tu2es! !20+p2n!29ru2o-p2d! !29=o - nt ,2q(ouv 'zo+r! 's)_

-ouv2p=o a ouv2o-nszq=o F ow29:o - z9=l - q=L -

OloK]lgmTIKOt lofltllot 2. tra!&i1ov B.

H e[aolxrn tou xd]vou ypdqEroL p2ouv2onlr'zr,r-pznu'?eor:v'zr,r -

ouvz{rnu'zl l -q! 'zeouv2(,F0 e ouv2qnu2r, l -(1-ouv2q)(1-nu'?r,r) - ouv2en!2r,r-1+

+ouv2qln!2o-ouv2 qn!'?r,r=O - ouv'?p+nu'?r,r-l - O+r,

'EroL € .vd l 0s asu. t [vo l ox6uo 0ss<+ * ' lB{**# l=e 'nu,

u r ' / 2 r t ' r r o - l r r n l? r ru r ^ , ra l+=J LJ LJ e'auedeldalde-l lJ l fnrvl uvlae=

329

=i'''Apo

69. tlo unoôylooel' o 6vroq

90vcr€Cr p-oouvo, X2.$2+22-d2.

AlroI;

H orep€d r€ploxi opi(€Iolon6 Tov krltrLvbpo p=oouv8 (oL

tqv dPoapa x2+0'z+z'?=c'?.eo €pyooroiu€ o€ xullvdpl-

xaq ouvrEroyqtv€c.ll npoBoli rnq €nlqdveLos

lou orepeoi oxo €nlne6o 0x0 ei-w[ o Kul(loc p=oouvo. Eavq[

u < P S q o u v u , - - - i S d S t

E[voL dK6uo z'?=o'- (x'?]{,'?) F

22_a2-p2 _ 2_1'628 . ,Apo

=l "li + *,o,Pu=1"[ S [ *,*]0.=f" g r-,*ri*-

-3- -t r /2 - t_[-ouvorl]d€= f (1-ouvr,r]{0J, = i I (1-ouv.,J.

v- + $ (t-ouwt= $ 1r-ouw) (. !ovd6€c.

tou otepeor, tlou repL|(leasrol cr6 rlq enl-

rH rpo l tn e ruedreoc edwL Nv lLv6p l r i xd ! n 6 .d tepn oedLpLr i .

330 rcOAIAIO III

,a / 2 dotvn ,\E z--o r-vo2-pz 1z < va,-pr. E(voL V=l l ldxd0dz=l | | pdzdpd8 =

E '-a/2-o --\82--62

t\/2 potro t{Az=6,*J, J, Jo

pdzdpde=

*J" .J, pvo,-p,dpdo=

lq!- (or-qrouvr€) 3/,

lde-

t t l2r t i , tcdulo= 4 t t - ; - { o , - o r ) 4 l d 0 =

4 l n / 2= - T l

^ t n / 2 t r \ / 2- +o3Jo ( l -au'0)de= +orJ,

" f,o, t:n-+l (. lovd6e(.

t 1- (1-ouv,e)n!e1de- +0,Ie.*re- ]*",r]"/1

70. [o molovaooa( o 6vroc rou ot€p€oo nou rcprxt€a€flr c|l6 rLC mr06-ue.e< o4z2.d2 {t) (ofo(pot !!! ztoqto (2) {dvoc),6nou I etvol n wvkmu oxnpdrl.Cer n YevGcipo tou (6vou U€ lov 0€rtr6 nUldlovo 02.

Aoot:

o ' ) Eo ru 0< 9s + , r6 re ro o rEp€6DLvetoL dn6 xo npdrro qxilro. l4E cflcloLeri rouz on6 l lc (1) KoL (2) npoK0nr€[ n nOoBoif llnq Enlqdv€Loq mu or€psor, oro anlrle6o0X0. 'Ero[ txolu€:p,+(poq0),=or - p,( l+00,0)-q, - p-onue.H Teleuro('c €l looon ndpLoxdv€L Kixlo.ed €pyooro0pe oe KUI\[vEp[(tq ouvrerovU6v€CE iwr 0<p<a4uq , 0<692n , z -2 .H (1) 6(ver zz-o2-p2 . z: ldr:dz. E(w[ ok6uo ooao 3 z 3 6Ziz. Ero! 6xouu€

| | t t2a anpg tliEdz t2r ,ottve l\1847v=l l ldxdodz' l | | pdzdpde.l I pzl dpds=

t t " t tn Jo Jpooo Jo Jo lpoeo

OIo( lHPnTlKO! IOf I tMOt 2 . Erd !6dYo! B . 3 3 1

= jj i"n'n,o*u=Eu-o'"*, ooo.=

i- I rc''o't' n -$ "*

I """ae=

i/, _ o3!! ie *r_ {ae=

- , 2 n. f (-ouv.o-nu,rrouw+l)J. 1de=

= : i : [ - (ouv?0+n!) ' r)ouvo.1]= : j : (1-ouw) x.

B') nv f < o < n 16r€ ro orEp€6 blv€rdl an6 io 6eL]t€po oxi lo o-

ndr€ o (nrol lEvos dvxoq v €twL Looq !€ lov 6yKo vr rnc opoipoc, u€Lovrov 6yxo V, rou o!€peo0,nou n€pLxlei€lo! on6 rLq 6nLqdveLaq in( oqolpoc(o[ Tou xtivou. 0 dvKoq ow6q {Eovo<6!oor€ ovdloyo ti€ rnv nponyoi,trEvn rI€-pinroon) €LwL looq u€:

v , = l t L t 1 - o u v ( r - e ) 1 , ' A p s v = V r - v , =

= -: (1-ouv0) ( . uovd6€q.

l--| ro'-o'* '*r

2a

{

2 n

I,

!ovdoec.

qd - ic t1-suv(n-e)r=

oteg€ot E mu nEolkl€lera! 0116 rlq €-( o > 0 ) .

71. I{o unoloyLood[ o 6yxoq rou|l[06v€r€C x2{t'?+2'z.3o2, x2+'4t7-2a2,

^0on:

H or8ped r€pLoxi E r€pLxlei .-

erqL qn6 rnv oeolpq x 'a+' ! 'z+z'z=3o'?(1)rcL ro nopqBoloeld6c x'+q'=zoz 12)0o Bpo[le rtv npoBolil T rou dr€p€oi

oro enlne6o 0x0.tlp6nEl enottvi,rc w Bpoiu€ rnv

€{iolrrn rnq Ko!flt,\nc nou nEpLK}Ea-

€L To Xr,JpLo T. N €lioosn suTll [po-

xrimeL t€ onqlolei Tou z ti€To€rl Tov( 1 ) K o i - ( 2 ) .

KXOMAIO III

An6 rnv (2) 6xouue r S( . or6re n (l) ogu* ,+a+(f{3)'=:c,

e {X'z+' ! , ) ,+4o,(Xr+{,)-120.=0 - (X4t lzq, i x4{!-602 (or loppl l l t€roL))

'Apa n npoBoii rou alep€oi oro enLnE6o

0x0 elvoL ro xopto I={(xt{) € R,.x?+0, <2o2j- A116 ro ox/t t lo qn[v€61 6-I L - o Y Z S X < d \ 2 ,

-lEt:F <V s r.Zriz:Iz. eo Bpo0u€ rq6pLo rori z. oE(,jporiu€ ur,o €ugEao noodl-trnln npoq tov dlovq z'z rlou nepld cn66vc €orrT€plkd ontELo rou t . H eu€elor6lvEL To oirvopo,rnq nEpLoI i( E orc on-ue(o A,B !€ z= !ig: , z=V5dzT,:iE qv-

r lo roLxo. Apo i iL <z<V3o ' -x ' - { , '

l.l€rooxnlarL ou!e o€ xulLvfp!|(6C ouvr€rov!6v€(. Oarolg€ X=pouv8, X:9n!0, z=z'Ero[ €( ' \o! 03 x':+0, s 2o2 -O 392 s ?or-O < O slzo. Elvot oK6Uo

o < I < 2r, ${rt$,i} * ror j} s z <v3ot:ot. 'EroL 6xou!€

ttt 1?q 1l-zor 1liit-ir 1 1 .2r t.'.2d ,E;t:;zvlJJdxdudzl. l.)" | 0",^

edzlde jd81o lo Io o,,ro aoae=

-["!tuotne:o-fttaoae(1) . Eavo | ,

It^ 4t*- - * )or= ; Jr" -via, - o, 3o, -0, I - ; Jm'', ao=

t r,,ho f2a=- i l " t3o , -e fh d (3o , -c ) - -B ; rp " l !

-*lr*-zef n -(3,p)'a ]-+"'-+ ro,-vDo,r-]*=

=| e ra7-rr-]o=o'tq.l -+]

' rViz"=- t[ tio'-o' t"a 1" - f r'.

'Eror n (r) 0r"". v=J""o;{=r -+ ]d€+*{q-l

-}) *. ,ouou.,.

oIoKiHroTIKO[ Aot l lMot 2, t rq!6dYo! B. 333

72. 0 utrtKoc=l+I02. tlo 90e€€[

Aiot:

rtBoq rou otiua@c ax€L olvdplnor ruxvdrlT[oc 6(x'U'z) =

n ouvollri u6<o D tou oE€oeorl.

l,t-J"-l l"t'.+*;*,'=1,1-+-' ],-

d'!dx -

(r.fx)a'{'+" 1,=+73. la unoloYld€.' n ourctrrrll !6<d rou dteo€di E rcu |r€olxleletol att6

rrc ertodvercc z2=x1r{2, F:O, z-l 6oY I outtottprt tlun'6rnroc €|'6t

6(x,f,z)-xz (rotloepd).

loon:

H en[odv€Lo z'?=x1+U' €Lvor xrirvoc ue xopuoi to onlelo 0(o'o,o)' H €-

[rodvelo z=O €Lvor to Ell('n€do oXU e\iir n €nr0dvelo z=1 El,vol €nLm6o IIo_

gdllntro npoq ro €nLreoo Ox0. I't€ onoloLoh rcu z 0116 ilc eELodlo€Lq z2=x'z+!"

z=l npol(im€[ n nooBotri tou oleoeoi oro enLfia6o 0xt! nou €lwl o l(uxtrLxoq

6LoKoq x'?r0'1 < 1. ELvo! /PlW < z : 1.

' l oo m=l l l xzaxa0az-x l l l l zdz ldxd0=tLt J., Lr \t2tu?

- F[ [ r . ' : ' oxar=*[ [ l r -x ' -o ' toxat t r l' )i fr2+,1-t ' 'i

Oq unoloYioouu€ ro 6Ln;\6 oloKliptlgo ue

u€rdoxnuqr!0!6 oe nolLK6c ouwetoYu6ve(.

E [vo ! X=pouv8 , U :pn !6 U€ 0Sp5 l , 0<eS2n ' E ro [ axouU6 :

1 2 r 2 t I

m=f [[ orr-0'z)aea'=+[ *[ (p-p')dp=+te]3'1fr-+1 =

* - / 1 1 \=7zny7- t ) - -v - .

l t lEtvoL m=l I I 6 (x,{, z)dxdldz=l I I ( 1 +x{,2 )dxdUdz=

) ) )

334 ( 'OAAAIO I I I

74. lto lp€{el I ouvollxi !d<o rou orsp€oo E,om ('ro(o xaEv€I€Er,

u6cd u€ ouv6orr|('r ruxv6rnroc 6(x,'l,z)=2,6tdv duE6 tr€9rxle[6tol otr6 ttv€ntt6v€ro z=l-x2 mL to enCns6c x.0, 4.0, z=0, F2.

Aoon:

€nledv€Lo,!e obny6 rnv Ko!ll!]ndtovo {, 0.

,= t _x .

H €f iLedv€Lo z=1-X'z el .vdr xulrv6pLxi

2=l-x2 xqL v€v€rerp€q nopdtrlntr€q llpoq lov'0noq EiiKoio eoLveror 0116 ro oxl]lrq

E L v o L 0 S x s 1 , 0 < 0 s 2 , 0 s z 9 1 - x '

'Aoo mlJJ2dxd'ldz=

a.]a,r]ax=

I

7 2 1-x2=,tlttt

75. tls 9g€ootv oL ouvteroyu6vcc tou rcvtpou B69ouq rou or6p€oo E noun€prxl€('sror. m6 rrq €n4dvel€q z-xt+{t, xt+0"=2x rdL to elltns6o 2-06rcv o'oord Kcrovtueto[ !d(o uE ouvdprnoq nuKv6rrFoc 6(x,{,2)=1.

1 . 2 I 1

" l [ j ' ' ' ,ru]. ' ' [(r-x'rt+r"'?ax+Jir-x'tu-=,1,- * ] '=.( '-])=*.

Aton:

A6yl,l sulu€rplqq rou ftapeot 8o

epvdsloius oro nAl'lo 6y6oo rr,rv dqd-

vov. 0q €pvqoao(llE oE (utrLvbplxic

ouvr€roy!6vec, 06rour€ x=pouve,'r=0n!0j z-2. EroL I cnlgdveLo

2=xz+U'? (nopdBolo€L6eq) ypdp€ror. z=p'zKo[ n €nLedvELo x'z* ' ! 'z=zx (xi l lv6goc)

ypdeeroL p2=2gouv0 - 9=2ouv0.H flpoBotri rnq |(u,\Lvdpr.Khq EnL-

qfv€roq p=2ouv8 oro €nln€6o 0x0 et-

vqL o (lKloq g=2ouvo. Apo osDszouvo

E[€L6i 6t]oe epyq<6uqore o[o npdrro 6v6oo t(,rv oE6v0iv,n npooo.\i ELvo[ nul(!(,\Lo,dpo 0 < 0 s n/2. 8€opotue uLo €ueeLd ndo6trlnln npoq rov dEovq z z nou nepvd

EtvoL 0 - + JJJ,ro r',,r,,' rorr+a'= f JJJ+o*a+a':o.

^ , t /2 ,1au\ t .p ,i =f JJJx0rx,,t,,ztaxauaz= fr .z), l, .lo oouur.oo'ooo.=

^ ta/2 t2ovr i .^^ ,a/2=+1, l , o'ouv0dpd6-i!*J. *" 'eae="!p=i

_ , t t t . t n / 2 t 2 a u \ q p 2 ^ t a l 2 t 2 o w oz = liJJJZb (x,0,2)dxdq,a.= *"2J, .J, J, ,oeodr= f"'j, J, p,code-

, r r r ' / 2 -o=rfr +J. ouv'od€'*- +=+'

OIOr'ÎIP0TIKOI IOII'MO' 2. Xna!6dyou B.

ono avo Eorr€pLX6 onleto rou E KoL !6tlveL ro oivop6 Tou oro onuel,o B, AuE z=0. z=p2 ovr lo[olxo. 'Apo 0 s z 5 p2. 'Eroc 6xoupe

r i l2 t t2oor l t p2 . . .a /2 , ,2outem1JJ6(x,0,2 )dxdodz=zl l l l l oazlaoloe=zl l l p,aplae=- i 1 " o " o

= * f-" ro' r1" ""3r=. [*"ouuu ro.= an.' ) o " l ,

76. lld moloylo€o0v oL ouwelcylcvsc rou (€wgot gdcoc rou olEp€ot rcuneprrlal.€ror, a|16 rov r0lLv6po rot oxflatoc 6rdv d'our6 (fioveuerar !d<o !eouv6prnon nuw6rnroc 6lx,l,,z) =20-22 .

A0on:

0q rpyooro0u€ !€ |(ulLvdpLKdq ouvrErdvpAvaq.ELvqL 0Se<2n , 0sps2 KoL 0<2s4 .

* H oor )dp t r tdn f (x , { ,z ) : { e iwL r .p l rd i os rpos 4 rdL "o

o t€ped eUEL duuuetpLxdos rpos ro .n i r€6o OzX, dpa to rp l r ld o lox l ipoua . iwL un66v.

KEOAÂIO III

'Aeo m=JJ16(x,'!,2)dxd{dz=

2r .2 A=J" J. J, "o-"

'oo'ooot=

t2r t2 / . r t ,4

J, J,\''' f)1, oaooe

i ' r2 , r . ' , r ^ r2 t n ""J" J, f edoot- i l " -T

704 _1d0.t 2I de-

EtvoL a(6uo [[ [{or x,+, ' laxa,ldt=[[ [u{20-2'? )dxdtd': 0.' t ' ' t '

[[[tort,,r,. loro,rr '-[[[x{20-zz )dxdrldzl' o''L', ' t-

f f [.tr,,r,,, r*o,ro.'JJJ,,.*"*o*,.f 'f'f

Qoz'z' tadzdpdo''L' ' t '

,2a ,2 , r2a r2 ,2n t2 t21

"J, J,(roz,o- -P )l,oror=J, J,seouooe-1, 48p':lode=Jo 1e2d0=384r.

'Apo 1=0, rlr0, t= *+ =+ (o€lt6o 271 rinoL (2))'

77. gs{,olwe rov KUr.lLx6 nAvo o- _ft z u€ drtlw g6onc r xo 0{oq h)'

orov onolo Kdrov!!€to! g6(0 !e ouvdpt|F l noxvdErlioc 6(x,0'z)'1'

o') ilo Boegoov oL ouwetcvlevrc to! rewpou !6(nc.

B', flo 9o.0o0v oL ootrac o6o6votdc lz, Is.

* Aurd rpox ' f i r .1 e i te crr i tov uroroyuolq tou oror lnpot latos,e i t€ ord to Ye-

rouds i " r , n ouuiptndn f (x , , l ,z)={(20_22) €{sL !€pLrr i os rpor + rdLnolFp€{ i r .pr ,ox i ! E iwL d lur le tp l r i os rpos io e l i r '6o oxz '

t r * H o l ldptndn t<x, ,Lz)=xeo'22) . | j \sL rEpLat i o< ipos X i rd! n oreped nEpLo-

x i E eJs! duu! . tp l t , i d . rpos to . rdr '6o 0+z '

tillri

oAoI,]\IIP0TIKOI lorIXtlot 2. Ito]]llpqL 337

Aooq I

o') A6Y!, oultretploq tou

rov ol6wv. 00 €pYoototlua oe

m1lJ6(x,0,2 )dxd'ldz-I

t \ /2r t t t rn-41 l l I l . 1 .pdz ldp lds=' ! " 0 " ! ^

oteoeo0 eaipo0!€ ro tliuo rou nodllou oY66ouxultv6p!(6q ouvrEroy!6v€c. E(vo!

(rc on!€Co A ttoL B !xou!

,= *- o *r z:h drtddto,,-

(r'o-{")aoat-" [ [

= -'rn""

- l,( =i-

n/2

J ae= ]nn"'. rtuor

JJJro r r,,r,,' Nra,ra,=fi ot'E

[ -+ Jl,|,!6(x,t,z)dxdudz= ,r 0-0,r . E(voL orduo!

a / 2 ? h

[[[,0rr,,r,,. roro,rn.. [[['o*o,r,o,.o [ [ [. .oa'coae'' L ' " E ' ' o ' o ' !

o

r/2 t - r l2

=rl [(*'- { o')o'oe=} n'"'J ae"}nr,'"".

'Apo i -i- Jj jzdxd{dr=+ h. 'Eror trorndv ft (tv4o ud(oq K ax€L ouvr€-

tovuau"c (o,o,f r') .

2 2

H duqiprnofi f(r,{,,z)=x edw! reclttr5 @s rpos xwu suuurtpui os rpos to €ttr66o 0{,2.

-Apc io

Ardloyn lr. rn! rponyort!.ln .punlgda.

n! n drepEd r.oLoxi ei_olo,llrlpd|lo .':$L !n6av.

t t t { = p n u iB ) 'E rou !E l . - l l | ( x ' * t ' )6 (x ,U,z )dxdudz :

!

"J"tJ.tJ"oo"a.]ao]ae".,.= f nr'".. E{.vdL oK6uo

a / 2 t h

I, j ,=ll l(x'.2')dxd{dral | | (pzouv,o*2,)pdzdpdo='E- 1 1 ' , ! ^

"l | [("'- ] o')"*'e.](,"- $'")],'*=]*"('' '.; '. ')

78. io unoloyro€el, n !d(o tnc otrped( rcpr,ox6( E nou opl(d!o. on6rrc en4dv€lec z-O, E-2, F.Z, z=1-x2 xc! Cxet nuxldlrEd 6(x,U,z).xr.

iturt:

ll (nrotlevn ud<o bavErot on6 tov rono:

E0o Bpoi,qe xo 6prq r(lry

x , U | ( q L z . 0 L e n L q d v € L € q

z=0, (r !-2, ' !=2 eLvor EnL-n€6q. H enLqdveLd z = l -X2ElvoL (u,\Lvdp!xi !e 06ny6xqlni,,\n (oro enlne6o oxz)lnv rlopoBo;\i z=1-x'z (oryEv6!eLp€C lopdllni€q ftpocrov 6qovo 0 U.

H npogoAi rou o[€p€ol,oro enlnedo 0X{, €avoL roopeov6vLo !€ €qLCliJo€Lq nl€updlv x.-1, X=l,-2 a 0 s 2 kor 0 < z : 1 -x 'z . Apo n (1 )

t l I t 2 | , 7 - \ 2 . t . . 2*J ll I l" *,a,la,rla'=f If , 'rr-,'ra,r,]'-7 '42 ' 'a J ) r_y_2

=r[1{x,-x.)ax=+ t+ - + l ' =+.--1

l,=2. 'Ero! 6xouu€ -1 s x 3 1

2

l (x ' -x")duldx=

+--2,6 L v € L

1dx- l I'-l-

',=JJJorr,,r,,.raxa'ra'=JJJr'.r*" (1)

fiffi

oIoMHPaTIKOt AOtI tMor 2, t icr6dyou B, 339

79. tla Bpege( n owolr.Ki !6(0 tou or€peo0 rcu opa(Ero! dr6 rle ox6-oeai X2+{2+22 s zdz 11) , z2 Z X2+02. H owdprllon ttutvdrlTloq e(vol6(x,0,2)=2'z.

I60r:

H enrqdveLq x2 +['z+22=2oz -

xo onleLo K(0,0t0) *oL qxrcvo o.

H €nLQdveLo 2'?=12+r1r'? eLvou xrir-

vo!. ed Fpo0le rnv rotrit ro,v 6to

€nleoveLdrv nou npoKdnr€[ qn6

rnv loon rou ouorilctoqy2+t1'.+7'?=2s2 1 ^

| - Zz'=2@ -xz *'!"=22 |

z ( z - o ) - 0 - ( 2 . 0 i z " o ) .

f[o z=o n npdEn €{ao(,Jon 6a-

ve! X2]o'?.o'?. 'AOo

n toUi lov 6tro

€nl0oveL6v axeL e€[o(b€[q

{X'z+rlr 'z-o'z, z=o}.

And ro dx/tuo eolv€roL 6rL "of<On= S "1=f .,p -f . Xavouue rov

uErqox4uorlou6 x=ptur0ouv8, Gqn!8n!e, z=pouve uE O l8 s 2r, 0 S e s +0 s p s 2oouve (npox0nrEL 0116 rnv {1)). Etvor oxouo | ffiffi|

=o"nuo.H u6(q rou or€pEot davEroL drd tov (tno:

'=JJJo r x,,r, ' r axa,r'=J fi," arava:..[[[o" *"", p'nroapaeap=

x'z+0'z+(z-o)'z=a'z EavoL oqoapo !€ Kdvrpo

=lj"[^[I:**'u*"'onuade]dq]do=f"If/a (zoouvo)u o'u'*nu'aq]ae=

t^s t2\ t r l ta _ 1 b - t2r t r . \= -- . l" iJ" ouv'e4!odoJd0=- + 0'j0 dojo ouv'qd(ouvo)=

64no5 | ouv'o l ' lq r+no' , , (#)', ,rr"=- -]- L

-- 'B-- l"

= --5- L3-'--E- j =--.

340 KEOAAAIO I I I

80. tlo Bp€oe( n ponli o6gdv€@c oq rpos tov 6[ovo xou sr€pooti IIou ne_pLeX€roL uEroti d0o ouoolovrr6v KUJ\(.v6pov,Ue sxrLvee o,B (o,B > 0 tlE o<B)ovrlotoLxo Ko! orlioq h. d(vetor 611 6(X,'!,2).y, (y > 0).

Itfi;

To orepE6 nepLKle(€ToL qn6 TouCxu^Lv6pouc x'z+o'z=q'z, X2+U,-8, KoL o-no ro EnLnedq z=0 Ko[ z=h. l,lErooxn-lcra<ou!€ o€ xu,\LvdpLt6e ouvteroy!e-v€C. etxoutr€ X:pouv8, o=pn!€r z=2.ELvoL 0s0 < 2n , o < ps B , 0szsh

xoL ii!4,!f;# =o' H (nrouriEvn pontl

o6pdv€loq 6lv€tor m6 rov x0no (d-tovqc z z).

Iz=.J Jl (x'+0'z )6(x,0,2)dxd0dz"

!

-y) fevlreuuCw rpLnld oloxlnpdlaro

81. tlo e[€ro'le( cv ouyKl(veL to olo(liiptuo

6 n o u E = t ( x , 0 , 2 ) € R 3 : X , + U , + z ! < l , )

A(Jon:

H ouv6ptno4 f(x,0, z )=

=v l l jo 'ooooea.=vf dzfD'dpI de=vh (9 ' ;o ' ) 2n.- ; '

1rz*qz*rz 11 t t

lou on!eIou (0,0,0 ). l . t€rooxnlart(ou!rELvoL X=pouv€n!0, qr=pnuanlo, z=pouvq,

0sX '?+ t l r ' z+z ' S 1 - 0 <p , 3 1 + 0 5 p

f(x,qr,z)= Pouv8?lQ = suv9!!0 *0| ' ]t o , f t , u . I

------J- drd{dzlx, +V,+2211 ',

6€v €Lvo! epoyuavn qtnv n€pLoxlt

ndiro o€ oqoLpL(eC ouvrerdy!6vEq.E l v o r 0 S € 3 2 r , 0 s q s n .

S 1. EtvoL oK6uo

${ffi/*,*0.

,=il1E

.--r---ji---- i - -- . i \I oL---)"--)9:---/L ,

o] \OMHPI IT IKOL IOI I tMoT 2 . I tdvodroJ B. 341

'Apo l=[{[o'ue']uq p,np,pdpd0d,p=[[ lounenr,.adedode. oa,rporiu€ rnv oKoôuoaq) ) ) o ' t t t

U ,Jv xu rp iuv rou R l , E , - { (p ,€ ,e )€Rr : uL ao a t , 0 i e < 2n , 0sa<n l

EneL6i n ouvdprnon f(p,€,q)=orrvgn!2q 6ev 6x€[ oroeEpd np6onuo, €Eerdcouu €

oq rpoc rnv o(lyKILon ro oloKl ' jpolq JJJlouv8ln! '?adpd8da=Er;

- I '2a= t im l l l rouverqu,oapd8da= t i 'n I ao l rouv6rael nu,oo I

e # ' I / ) ' o

/ , \ -= l in [1 -T J a+=2n.

'EroL ,\orn6v ro otro(i ipouo .JJl ouv8 alr '1adeded@ elvoL loo u€ 2n,E i

6ntro6i duvxl[v€1. Enou6v0rq KoL ro olo(Aiouuo JJJ

or"enr'.aoaeat

ouyKlLv€L (o0trer,)vo !e yv0fih L6L6rnrq), E

82, ilo unoloyroo€t ro y€vlx€uucvo r9Lnl6 oloxlluoqo I=JJJ"-"araOa,6nou E={(Xtt,z) €R!: X2+{2 Sz,, z ZOI E

nton:

H napLoxi E 66v ELvoL epovu6vn. ElvqL f(X,0,2)=e-z > 0 yLq xd€€(X, ' l ,z l € E, l lErooxnuf i L<ouu€ oE KUlLv6pLKEq ouvr€royu6v€q. 86tou!6

X - p o ! v 8 , ( F 0 n U 0 , z = 2 . E L , , t d r 0 3 8 S 2 n , 0 < X 2 + r ! ' z < 2 2 -

0 s p ,onv)6*p ,nu ,e <22 -o sp2 s z ) -0s p s z . E(vq ! B{ i iF , i } - ,

KoL f(x,0,2)=e-2. 'EloL ro oloxl ipouo !€rqoxnuo[L(€rqL oro

I . l l l e - " o d p d g d z 6 r o u L = { { 0 , 0 , 2 ) c R r t 0 s p 3 z , 0 < 6 s z n L .) J J

,2n t t ) t , t31/2 ,2n* EiwL I Idulo loe- l oulede- l ouv$d€- l oJvod$+l ouv$ds-

) \ / 2 ' t ' z r / z

. [ - ! o ] t r / 2 - t l u e l ( . - - l r L e l l n / 2 + [ n u € 1 2 t r . - a .

- , t - I* * " * "

I n ! 2 9 d 9 : - r q - f p 9 d u v q r .

342

8€rdpo0u€ Tnv ol(otrou€Lq rov napLox6v

E e = { ( p , 0 , 2 ) € R 3 : 0 s p : z S v , 0 g 0 s Z n , v € N * } = { ( p , O , z } € R : :

0 < p s v , O S z s v , 0 S g s 2 n , v € * ) .

v ! 2 n'ExouuE I " - l l le - 'pdpdoaz. l l l l l oe- 'ue laz lao=l l l lznoe-" taz lao='; '

=l znot-e-"t-ao-l 2ro(-e-u*e-oJdp=-[ zroe-u dp*[ znpe-pdp.' a P J ,

=-zne-vf ouo.znI pe-odp : -zne -'[41" -znro.-o."-o]]--n"-u(u,)-L . l a

-2n(ve-"'e-v-r )=-n ! -rn - -zn -L .zn f13 zn.e" et

' A P q I i n I ! = 2 n - 1 .

83. l{o uroAoyroo€|. ro yevlx.uptvo rput6 olo|(tipolo I.f[[ dxdodz

6nou E={(x , { , ' )€Rt : . ,<xs t , 0s0sx, , 0 . .=r r , ' i ' ' / f f i

n0o[:

H ouvdprnon 6ev €LvoL qpqyltvn ornv l l€ptol i xov onuelov inc €n[ed-

E lvo r f (X ,0 ,2 )=vlr, -x, I

otoJpolt le lnv qxolouolo r0rv r€proxd)v Gv.{(X,U,z) € R!: 0 3 X < l ,uSzSr , 0S ' r sX . -? , v€ r ) .

r r ' z -a rErvau r . f [ [ dxd{dz - l I f ' l I u 'v rar Vt0;rT ro Lro tjo V|TFFT

' -r-1 , 2 1=t-t^ tf' I ],,r,,=f'tt'

- -'--' or1o, =,, ro Uo !4i:u r " Jo LJd vi::o I .

l,rl*-

Jo.-ooo=-J or.-o yao=,"-e*J .-e

O oMHP'TIKo' IOIITMOX 2. ttra!6dyo! A- 343

-(06rou, le

I

r f - r ' r ! - v ,-*=,,= IIl," rx'-o ' 'at]ax=-zJ

Irx'+r ']" dx=

I x'e " dx.

eLooue orr J.i-x'ox-Vn-.

xd{dz-

To r€lau&Lo oloKlipopo €LvoL

R3 6ev €LvdL rppqvutvn). Elvou

€ R 3 .

aefnloaofo'"-oio=

=-,1({+*)d,- ,y'T., l+l' =i-,y'T '*-' 1.'APo l im Iv-1 on6re I=1.

8{. No unotrovLo€sl ro olo(lip{l|la

A0on:

>rnv oetrt6q 163 ( 'Aoxnon 91)

'Aoo I e-^ dxl e-' d,!l e-" dz=fififi -J - ) - ) ^

- - t l | - t \ z + , j . 2 + 2 2 \- nvn - I= l l l e ' ^ Y - ' dxdudz= nvn (1 ) ., : :

vevL(eu!avo rprn^6 o;\oxlhprjlo (n n€Oroxh

f ( x ,U ,z )=e - ' x vLo Kde€ (x ,U ,z )

od,,Oo!!E tnv oKotrou€Ld llljv Xtrplr,Jvl

t . . = { ( x , { , z ) €R3 :0sxs * , os , l , s * , oszs * , ve r * )" v3v3v3

M€TooXnuorL(ouu€ oe oeoLpLXtg ouvr€rqyu6vEC.eExouU€ X .pn !Oouv8 , U=p tU{ rn !o , z=pouvq !€ 0S0S 2 r , 0SA<n ,

ELvoL 0< X2+0 'z+2 ' z < v ' ? - 0 < p ' z s v , -0<ps v . E tvoL q (6 !o

111,4,,y="-(x'*+'+,"r ="-o' "o, I ${ffi | =e"*,r.

'roo r"=JJJe-rx1+"*'")axa,r,a,=jjjo"nuo"-o'doded, :l:I E '

=2nt-ouvol: J, o'"-o ao=nnJ, o'"-o-oo.

* H t repLox, i o lor l iouons e iwL ro opsoy.3!Lo nopsl ln l€ni t re6o t o , 2 r I ' t o , r I ' t o , v l

KEOAAAIO III

E[voL Etor6wc t= li'n I,,=4n lin fe'?"=o'jo

- 1-anf-*'"-x"6, (2). And

(1) Ko! {2) nooKomel 6rL l"f-x';t 'ax=nvi - lT'"- '"ax.

f i r:t.

: [--x."- ' ix.[*^'e-x'dx{-3) fr - +' +.

8s. k 6elx0e(' 6rL ro otoxlilltrrd ,.!hffi, 6mu r !n ovur!-E

r6C O!q6C ror E={(x,l,z)€R!: X'z+{2+22 Sl\ ourx}[vor y.d 0 S r . 3.

i0o[1H ouvdprnon ttx,0,z).;ff i iF 6Ev elvor epoy!6vn otnv reoloxll

iou onteLou 0(o,o,o), eLvoL 6€ f(x,U,z) > 0 yLo K60€ (x,0,2) xolJ E (€K-r6q rou (o,o,o ) ) .

€Eopottre rnv o(o,\ouolo r0jv XopLruv

Gv={ {x ,U ,z )€R! : } <x r * ,1 r *2 , 3 I , v=2 ,3 , . . . } .

ll€rooxnlarL(oulE oe wolpL(eq ouvr€roygdvec: x=Oouvonue, 0=pnu€nup,z -gouvg , E (vo r - i r so ' 3 l - - - sos 1 . t t vo r o (6po 0<6s2n ,050sn . 'Apo .

t..-[ [ [ g!0d' , , -[ [ [ e'?luq 60666r= ["ae["nr,[ ' o- '* 'oo.! Jt) l \ l f l f +v)K ))^), pK J, J, ) t /"

=t0l: ' t-ouwtl 11 o-r+z6p=ln[] p-"*2ap {tJ.- " ) u t t t t u- e**r. a-, x=-o - <lt=-dd drq! x:or d:o drd! x J -6 r 0 r +-.

r* f -x,;'i"=J*<-,),t(-')"d(-)-f -.,.-"-=f -,";"a*.

(crrdyi ! . tdBhrr is),

oIoKlHPnTlXOt IOIIIUO' 2. trov6dyou a, 345

['-(+I.."]f

t lnelr / ,=rnv, ov (=3

[ ["5u-"],,,=*I!=4II]nv r +6 ov r=3

r,=4"* [,-(+I*3]

II

EavoL Iav r l3.

- i ! s v - r+3>0 -0<x

nopo[noo0u€ 6iL o )

9 ' , )

y ) I " = 4 n * [ ' - ( * ) - - - ' ] - * - ( + 3 < 0 - x > 3

'Apd ro oloxl ipqio ouyrtrLvE! yLo 0 < x < 3,

85. ia urolovad€l ro y€yLtsw€vo rp! 6 olorlatgtld

I-I | | ---s------r- '- dx4dz 6tro'r E.i(x,a,zl€R!: r 'z+l2st xor osz).' ; ' t I-+r-, '

i0ol:

H n€ploxi E 6ev e(vdr Opoytlavn xoL n ouv6prmq f(X,0,2) 6€v €Lvo[ Opo-y lavn o rnv n€p [ox i ro r i 0 (o ,o ,o ) . ELvo [ f { x ,U ,z ) > 0 vLo K6€€ (x ,0 ,2 )€E -- { 0 ,0 ,0 } .

Seoporile rnv d(oloueLq iojv reproX6,

E != t ( x , { , 2 )€Rr : l z sx40z s 1 , 0 < zsv , v -2 ,3 ,4 , . . . }

ll€rooxolar{'Cou!r oE KUlLv6pL(aq ouvt€rdvgtveq,€trouge X=pouve, {=pnu8, z=2.

E l . vo ! - i t sp ' 11 . - i - sp<1 , 0<0<Zr , 0sz<v

Kot f(x,{,z)- -7- , * i**=i.p.

'EroL n qKotrouetd rr,rv xrrpL0D Ev tlErsoxnlorl<€roL ornv o(o,\ouolo

E"= { (p ,0 ,2 }en ' : T sps 1 , 0<0s2n , 0 t z t v )

- ! 1 2 n _ v l

tcvou I..-[[[ e""dxd'&dt, -l tt It C* *lorlo'-[ [[ o'4 "-.1e1z"aelcz=) t l J . o ,+q , l h t oL ! l l r o ' h J

- r J0 l J r l v -

- ' ' ' 0 - - J -

346 KEOMAIO III

v 1=,4,11,.o' t' "-"ao]a,=zn[,.-'a,l ,,,

! 1

p " dp - -z t l e - "d ( - z ) l p ' r dp =' o ' I / )

" t . / I \ ' 4 1=-zr l [e-".r i j - Ip " l i l ,=2n(eo-e-") . i - t '1.r , ) l=

=*( ' - -1 )( ' - I ) - $rr-orrr-or= jF. .Apa r in r . .=4 =r.e v / \ i 1 7 t / .

87. Nd urolovld€l ro yevlxeulavo rgllrM olorlllqdtd

I=j j ]dxdldz 6mu E-{(x,0,2) €Ra: xz{ar 'zs l , 0 s r}E

Aoon:

ECvoL f (X ,U ,z ) "1 > 0 YLd *deE (X , { ,2 )€ E .H nepLoxi E 6ev ELvdL 0oqvu6vn.g€opo!!e inv oxolouoLo rllll/ nep[oxrirv

E ! - ( ( x ,U , z )€R3 : x ' +0 " 31 , 0<zsv , v= j , 2 ,3 . , , ]

M€tooxnudrt<ouu€ oe ku,\Lv6pLK6q ouvt€Tovu4vsq.o t rouUe X -ps r j yo , U=pnuo , z -2 . ELvo [ 0S0 :2 f i , 0<OS] xoL

0 < z s v. 'EroL q oKoloueLq Ev lerqoxnlorL<etoL o[nv oxotrouolq

E ; - { ( x , 0 ,2 )€R ' : osp< 1 , oso<a r , o3z 3v } .2 r 1 ! 2 a ! 2 \ L 2 r 7

I l l t t I I I t t I I r t 1 IE t vsL I v= l l l l i pdz ldp ldE= l l l p r z l ; dp ld8 " l l l vpdp lds=v l de l edp=

=vl€f; ' l -+l = 4| v - *-. 'Apo !o otro{tr ipnlq 6€v ouv(trLvEl.

88. lla unotrovldoE[ ro y€vrxeugtvo roln^6 oôxlit&Ird

,_f*- f* f*_ dxd,l,o,'-J, J. J, (x.,{r,l,lrn.lon:

A116 ro 6pLo rnq oôxtri lo{r)olq npoK0mEt 6rL 0 s X < +6, 0 3 U < +-,

OIOKTUPnTIKOI IOI I 'MoI 2 . tnc !6 ! i1o ! B .

0 3 z 3 +-. 'Apo o rdnoq rn( otrox,\ipoonc elvoL !o fipdJro 6v6oo r(,rv dE6v0rv(n€proxi !n opoyutvn).

H otroK,\tpr,Eta ouvdprnon ELvoL 8er!xi. 8€ooottrE rnv oKo;\ou8[o rov n€-p roxdv t v= ( ( x ,4 ,2 )€R3 : x ' z+ r ! ' z+z ' zsv ' z , x20 , U :0 , zz0 , v= l , 2 ,3 . . . )

llciooxntior[<ouuE sE oadLpL(t( ouwetoyucv€C.oarorrtr€ X.pnlqouvS, o=pn!en!€, z=pouw.

Er"o. 13.{ i## l=0,n, ' , osesv, 0sss +, 0ses +(6xou!e 6y6oo oeolpog Bi6rc oEil6o 266)

i / 2 t n / ) d t t l z i / 2' A p d I ! - i | | p ' n ! , r ( p ' * l ) - ' d p d 8 d ' r = l d 9 l 4 u ' r d q l o ? ( p ' ? r 1 ) - ' z d p =

. " b l ! ^

^ - . f v " ,

* n I ^ l !=+ t-ouval: , ' I p.(or+1)-,dp-+ | ______L do - + l - : ,+ +rotE@o i) o ' \ b 2 + 1 l z r o

n t - v - l w - n n . - .=*[ vi;r *oe.r"] t=='+ + (ttvot lln rot€ev- + ).

'Aoo lim I!= + =1.

89. ilo omloylo€el, ro y€vlkaultvo !91116 olo(Illpat o

I=JIJ"^ * -" dxd{dz 'onou E €tvdL r qr€9ed reproxlt nou op[Cerdr dr6 ro

E

En0re6o xO{, KoL rnv (ullvdoLxli €nwdvero x'?+02-o?, 0 s z.

Aton:

tluo, f {x,U,t )-""*'t" - ' ' 0 vuo Kog€ (x,{,2}€E.

ll o[€p€d rl€pLoxit E 6€v eLvqr 0poylavn.oeopo0!€ rnv oKolouoto ft,v nepLoxrl!,

E ! = { ( X , r ! , z ) € R ! : x ' r U ' : q ' , 0 s z s v , v = 1 , 2 , 3 . . , }

llerooxnumIaouue o€ ko;\Lv6pIKtc o0vr€ioyu6v€q. 06aoulrE x=pouvo, |l]=pn!8,

z=2 . ELvo ! 0 !psq , 0sBs2n , 0szsv xo l B { i# j } . .

347

'r Brdn. A,rduo o.î6a 81 {unororLdlds rou oôx^np. iuqros I ; t ' ,*

* ,

KEOMAIO III

'EroL 6xo!!.r r"=fJJ"""'r'-*oro,=J" f f *t"roara-

=J'ief If*'-"a,]ao=znf r-"c'-1ioao=znf r-"0'-u."0',ooo-

=1f 1 -"0'-u*"c' 160,=n1-eo"-u*"r' 1o.n,-"o"-u*eo'*e-r-t ).

'epo r. 1im_ru- g_(*" * *"1

* -r) =n(-e"'0*e"'*o-r)=n(e"'-r)

9it. ilo uroloyide! ro vlyLxeqrlvo rplnM otoxttoqro l.ttt=4t ))v1-x2-{.-2t

6nou E.{(x,{,2) €R'r x'1+{, 'z+zr s 4}.

A0oI:

H ouvdprnon 6av e[vd! oocvu6vn ornv neploxi uw ou€|'ov roc snlqdv€roqxnc oeolpoc Xr+!r+2.-4.

Elval f(x,| l , ,z)= O ylo |(d€€ (X,0,2) €R' !e X!+{,r+Z, <4.v4_xz_l)2_2,

e€opoi!€ rnv oxolouolo rtrv or€pedJv reprox{rv

Ev - { ( x ,U ,z ) €R ! : o s x?+ ' t ,+2 , r ( z - } ) ' , "= r ,2 . . . rll€rooxnudr{,(ou!E oe @oLpLk6C ouwerovgcv€C.

86rouu€ X.pnuqouve, 0=nwn!8; z-pouve. E{.voL Os O s 2n, 0<q<n,

o. o. r-* , I 8{*{-:6}l -e!n!o. EroL exou!€

r -ttt4 =t"tt ' t f- i -r:r- t2'1 " '2- ! ^'v ,r v4-x,-02-2, J! LrD Lro r'{=or--

doldtldel, dojo nledejo !fr?e-

-rer?ir-ouuerx [z(,"a,]-9 /,- 4;f i =

=2r.2.2lrolnu --+- - --"LL '

'APo l ln l ! '4n'z=I-l*-' 4;-.1=*'.

oÔK^HP0TIKo' loIItI,lOt 2. Irav6dvou B. 349

91. Nd unoioy[de[ ro y€vlxeu!6vo rpln^6 olorlipqrq

I1jJvxz+'1,,+z,dxd{dz. 0nou E.{(x,0,2) €R!: x,+I,+2, z 1)-

E

loon:

H or.p€d neploxh E 6Ev eavo! qpoyuevn. ECvoL f(x,0,2) > 0 vLo rdo€(x , { r , z )€ E .

0€0jpo(,!€ rnv o(oiouo(o rov crr€9€d,v napLoxrilv

E != f (X ,0 ,2 ) €R3 : 1 s X ' z+q , ,+2 , S v r , v .2 ,3 . . , 1

l,{erooxnufi{.<ouu€ o€ oeotpLKCc ouvreroyuavec.06rou!e X=pntqouv8, U=Onuen!6, z=psuve.

EavqL 0sgs2n , 0sqS l I , 1S !Sv , l , J l =p rnuQ.

.2ar . \ r . r r24 , l'Apo l"- | | lfr'-U'.7" dxd,rdz-l | | | | pp'nuqdo lao lae=l dol nuodel'o'dp.- 'L ' Jo L io r r1 t r )o lo Jr

= te j j " t -owol1 | 1- l =n.z ( - - f \ * - , . -\ { 4 I

'Apo ro oloKl ipqto 6ev ouyKl lvaL.

!f2. o urolovroo€l ro yevL(EUUEvo rpLnt\6 olorllbJ|lo

ftt -=!tSEa - 6noo t et.voL n orEped regroxii rcu rcp.xtre(ztor 0116' i ' v l -x , - ' r , (1+z ' )

rov orrtdv€Lo: {X'z+0'z=l, 0 S z}.

Aoon:

H ouvdpr4on b€v E[voL qpoygevn o[nv n€plox,1 n v otlsardv {X,U,z)€nt ueX'z+u'?=l xcu 4 o[ep€d nsploxi 6ev eLvo! apovlavn. Elvo! f(x,0,2),0,vLq xdSE(x ,U ,z ) € E - { ( x ,0 ,2 ) €R3 : x ' ?+U '? -1 } .

o€r,tpotuE rnv cKolouoto tov or€O€tiv nEploxdv

t v - , ( x , r l , z )eR3 : 0sx ,+ { , . ( ' - * ) ' , 0 < zsv v=2 ,3 ,4 . . . }

350 KEOMAIO III

ilerdoxnuar!<oute oe xutlvdplrac olwerovulvec.

06 touge; .p61y6, 0 .pU0. E [vor ( )SOS2| r , OSp.3 t -+

0<zsv , l , r l=p .

'EroL ervdL 1=[[[ ----qgrr-J" l- + r' p , . .-""' "'"' \]jlffi il.,,)1 J, I ffu.. *',"''r2r .\) . .1-+=['-l oh "{

-oft "',r,?',,0.*,rlr-vm rl-*-

znf .n'.

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