J

I. N. Bronstejn, K . A . Sze rne ugyajev

D. Musiol, H . Milhll g

MATEj\IATIKAI . .. KEZIKONYV

Hurlnpcst. 200G

v

I V..rlag HarryDeuueh. FrlUlkfllrt am~I",i".

Tbull 1999.

fyr-u-Illi tankOflYV

E kemyv brt..-lik kiAcl...... ,.;,; ilkt...... kllrllowri'un oemt...... alapjau ll.Z Oktat8lkony__rkc-U'tifd"l'iu..,;" .. miv"," tipogriJia ..u..nhP kkikollyllVi> J~ltak... lllalRltlAtdlll f..JluvozruU.....t~! minlt'gy lUikai wrj...ldmiikJx.n i. . t""'L1'111t kijllvi~ tlUlunk. Ehben ...mUllki,b"n kwm..lkP lr""I...1i hO"lz{ljlUull"", nNkiil ,",'m II tdj"" unt. ",'m IlIIHi1k Ii',,,,, .... mllliMe ("rnl/,h"" (fotoko!,ia. mikrofilm V>tgy IlL.... ",lllthuulm,o) 1l"]I1 .'I(}k.~7.C,,"l nl1 .... k ,

u , ,

1.1.4.

'1I, I..'>.

1.1.4.1 1.1.4.2

,9II II II II II

, ,

U.4.3. 1.1.4.-1.AJ~"bra.i

S~

u10 10

Ioti:aritlll'''''*

J

kif..~k 1.1-,}.1Do1inkiOk. 1.1.::1.2. A L brAi kif.jM'A....k.-u.lynMM.

I

1.(;.

R.\< tfo"Y',.okrl' 1.1.6.3. Sp'"..iAliM k"pl"\"k1.1.6.-1. Rin< l iAIL >fl "

"'i':..

,"".-l

raI,~

1.1.7.

l.l.it1.2.

1.16.:>. Ki'l pohlJ"''' l"KIl"K.vohh k&,;;" "..;G!. 10

\'i'~''''''''rok

1.2. 1. 1.2.2 , 1.2.:\ 1.2.4 1.2.:>

" " " "IS Pl 19

Viii

1.2.:>.1. 12.;:;,2

S;uUlltl\lUknu-p ~lirt.\ni k;qj>p .

wW'.. .... . ... . . .

1.2.5.31.2,5,4

HI\l/nnnikuII k07Rp:-;"'lO'1.1""" ko-..-p .

I.;; .:J.:J. 1.;;.3041.;;.35. 1.5.3,0.

(),.,;INoCg ...... eu t.

1.3.3

Tor~ti-u,mitNo

1.3.3.11.3.3.2

Tor~I""

,

1.3.3.3.1.3.4.

Egyrn16tOc.......tlirt'.rd.. t",-..k In'" trig, 1101111 t..."., f i~"...k

ca

61

Trl\U_.......,..lI-~

a2 2 I1.221 2.2.2 J 2 2.2 I 2:2 2;, 2 ! 2 1.>,

", u "62 62

A pmhliIUlHn.~~,-"""" """"IM'lllOZkli,..... ""'t" i_.lioto-],p 2.7.3.3. Jto.zr.r-k .....kt....d~.-..mjll 2.' .3A . ~k ...mapil""'" Ciltlolllp,rikw- o1Kgvinn!F. !N-),; IUu-U. hil">lll IH"!ty...lI!..V"lu .24.32,4.4 2.4,:;,

"' 67

Hipt.1 2.17 .H. Mh hvAlto:r fiiggv"ny..k ..... 2.1~.1. I:kliukii>;.,. ..lMllit';"" 2.18.1.1. T i>h hv;'lt,lIZiio- fUqvpuyt'k .. lo'iNlita..., 2.1S.t.2. T ijh hv N l.< .ro. filuvPlI.y..k Il"',mctriai ahr&zol.lt.AA. 2. IS.2. K iJlOnfpl.. htclnK'7i'lgl,.j" f' .... ""10'7,13.1.4 .3. Romhn",".3 .1.4.~ . 'I'rap':-,.. 3.1.4. 5. AltJllAnO'l ni'I\Y~""ig Sikhdi .."h l.iig..k . Sikh di kiinuaku t o k 3.1.6.1. Krht'>k rlMlOiJdA,;.'u'1I AlluUu.... lrh..tk 4.2.1. DofinkiOk 4.2.1.1. Do,lA.rtnu......... >k 4.2.1.2. AJdt'l.o:>t..tok > 3.::'.1.4, y"kto"'R)'ult'\.k >. Euklidtotozi Ytkl.ortkOaa>.-tIl.kop""bb~~';;A.l.;;.Fi~

3W 32032 \ 321 321 322 322 322 323

'" '" '"aae3J>j 3J>j

:m329 329 331

324 32' 326

".4.2.

Lintoarb Diopbant.2,2. M,1.....kou-k h.o.hlW'.... klo.lZi .2,3. ::;.24. 5.2.::;.

283 21'S3 2RJ2~

2117

2"1$7

RrI"':iOk .... "'k~k Ekvr.....Joop',rtok k01J)n.i I..k""",,"""'k '::'.3.3.4. Lll~"''') I>orUlk '::'.l.3.5. Lie-Illg.-h" ' k, ':'.l.:U,i. FH..gy"unl Lie-nIOpo rtok ]e krVC'd.,... C""I>ortnk >f>I.4.3. Kwwucncii>.k .... IlIAI"~LiJ)'Ok ".4,4. F.Zi,2. Titk"";tN.5.4. TitkOllitAo04.1 . .-\klaI-ik,,,,kri~~ :;.5.4.2. ~ WopPtl t,t~lAo> :-'.::;.4.3. A Vig............kod :>.Zi.4A. Matrix llf"l'o'f'tl",i~k Zi.5.Zi. A kJ-iklllO krip~ lIl6dArrei 5.5.5.1. StAtiR:7.t.ikll"",u..liz;" .'>.5.5.2. A K....i..ki Fn...lmMpr-t.A Zi ..'>.6. On.... Time-T&j>C 5..'>.7. :'\'yilviwuo; kukr.8.3.

396

5.9.

:>.8,4. ".11.". ".116. :>.11.7. fUDY :>.9, I.

SlkgriLfnk Pj.JY'k irAriyilnU srMokban

saIlita.i bAIOsawlt

J.Xl 3:.9 360 361 361 J6J364 J6.I J6.Io 3W J&l JOO 369 370

6.2

ui'iIo6Prto'krk ~tArosM& 6.1.5.5. Implicit alakbM adnu fiigrkty ~h'inck~h&tamziIsa. TOObv&ItoIZ6Io ~uvn: ditf........riilliIAa 6.2, I. P...-ciAfu; dmv.ltAk

6.1.::;.4.

Ab8z0hi t

(~)

kib

:>.9.2.

:>.9.3

3.9.4. ".9.". 5.9.6.

A fllay kib &lapja :'.9.1.1. AnlUYhalm.uk~ 5.9,1.2, T~ fiin;vmY""k ".9,1.3, fllzry halmuok fUZlly baimAzmd~k 5.9.2.1. .-\ltaliDoB funy halmunul.....letf'k :'.9.2.2. A &yAkod"tloau bAKlu&lt fuzzy 5.9.2.3. A~iQi ""ItY lwwpntAl6 ~ ::'.9.2.4. KitM'jf'lozu"loi I17AbiUy 3.9.2.5. FUDy koml'lrnoenNuggvkJy Fuzzy ,..liciOk ".9.3 I. FIUZY I'f'Iilri6k fW1llA ::'.9.l.2 Fuuy conatrelkii> R 0 S Fuzzy kiivMln:lt't6l"-"u~k Kik-t#b-\oIooi (.MnuyfikMilllol ~....kThd~i

6.2.l.1. 6.2.1.2. 6.2. L3. 6.2.1.4. 6.2.1.5. 6.2.2.

Fiiggvo'ny J'l'll'ialiukt'iviIt;a Gf'QlllO'triai jeIl"Ilta. kk vilt..... _klA dill..""I>ci&I ~ma A dilI.......nciAI f&t, tllllljljOd!OZt'l'Alkaln~ pfldAk

378 379379

"" -oe "" -cesee407

",..

.OJ

JSO3I'l1

.,7

T u.lAMlal'u intnpou..:-i.2 74.

A.iml>U>tikUOl~.

AlCI.imrJt>tik, IwvAnY""""Ok

f~-lIUruk 7..l.1. TriRon,xnetukw>~k F'ollri~NlOr.

425 426 427-127

;.4,2.

;.23 9.1.3. 1 A pmbkItnA~maziI.M ;>23 9.1 .3.2. A ~Jr.,.k", jit.t~o"o~k f6bb lulajtJowr.Mgai 524 9.1.3.3. A fiiuvf!nyt'"k rinti ....-fr.-ju", ,j24 ;>25 9.2. Pl\l("iaWo ..wr..r.-oc~nlru-k 9.2.1. E1Io6rmk_ ,,2:j 9.2.1 .1. E1fo&nondlllinMriA pArriiliA diff,,",~k>uok ;;2:) 9.2. 1.2. EWl'f"udQ nem.1i.no:t\rill povciII.Iis diif~niMf'k ,j27 9.2.2. MNoOO.,""ltt liJlf'irioo Jl"l"ciiliR d ~k 5;J) 92.2.1. Kf-t l'Ugn"'n VliJt0z6ju Ill&.oodn-D(hl diffl'Tf'~k ~ lyoeil.M", tllla~ ,j;lJ 9.2. 2.2. TObb. lllUllkk ~tl"ll v&lt0z6t \&t.almaW mModI"f'Tl(hldiff..renmAk i 10.3. 1 A variAci6ot k ..

572';'71';':"3

~.".,-"",talymM

Ma..odCajuFrrobolm -f"J..int"'Cl"~1.-wk

,jJ2

9.2.2.3. 9.2.3.

M.w.ocUrn(hlliu"an., Jl"l"ci&li,;~

diflt~nlM1;2 ,";2 ;>1;2

"'gy ,nl..t.,k10, Var iltci6l1dl.lllitb 10.1. A fdlld"t kitllz.... 10.2. Kl""",,,ik,,,, fd ",t"tnk 10.2.1. IWI"'rimptrik'lIl prnhl {>m" 10.2.2 . A hr""hiM1.torhr"n-prol>li>m"

11.2.1. ElfajulO lI>agl'1 inl"lP"~nl.-t..>k 11.2. 2. A "'-'fOZAtoo Inf"'&k_Iith< (AuJr., ~ .. "I>l'ruxirnAciil)~. r-.:..nTl\Alln-foOf 11.2.3. FT1"gyf-nlrt..k mrgold&.w.. 11.2A .l. AzinlPgralApJ-lnoWu.ilo:-iO)a 11.2A .2 . M. -AJlprolllmkto 11.2.4.3. Knlk:ticii>oo ~. 11.3. Fmibolm- f" lr " hoiif" jli int~n1.-wk 1l.3.1. E lfa ju lO m." inlf'gr&1o>gyf-nk-tr.k 11.3. 2. F"l(>l1maJt. analm..Jwli ~PdroozkOzOlt 11.3 .3. Az intep;r~1rt viloozA_ ..... liIlf'J.rior. orgyt'rUrtl""Tiltornk .... fllnkrinMlnk . 12.1.::>. 1 u- k0'p007Jow>1F: 12.1.;;.2. HnmomorfiZU lll>" ""OOrnorfum"". 12.1.;;.3 bomorf ..... ktoW"I1"k . 12. I .6. V.uo......ktrU1"O'k knml'l"fldomk .. Hilt-1 tJor....n 652 12..;;. Kompakt on...-ljllugAltOJl1l 12.9.3.2 . A:t- illt "l~rAl ui'hli.u y lul nj, I"Il"",iI;" 6.:>9 12.9.3.3. KouVIZ&lWyai . 13.1.104. \".. ktorfiigvffiy.. k ""'ylor-sorfpjte..c 13.1.2. SkNUInPZUk 13.1.2.1. SkAlAnnnI5 VRY.kaJAr JK>lIlffiggvCu~' 13.1.2.2 FOllt........bb WlArIIll'Z6k . IJ.I.2.3. Sluo1&cUte'liik wqr;adAsa .. lroonliniUak tiiggv"'nykpnt 13.1.2.4 ~tf.liilftl,..,, . 13.3.1.2_ A yonalint"PioItnrduwikAijr~I1lh . 13.3.1.3. .-\ voualiu~ lulajdnnMpi t3.3.1.4. A ........Jin~. wwl mAooodfaju AltalAnno.tLp""'-1 vonaJinte,;ral 13.3_1.;). \,..kt.JnnOtt\"~~U--S 13.3.2. F'e-J~li ilW1P"Al 13.3.2.1. Slklx-li ....lUlrtdaraLwktono 13.3.2.2. F..hilM.i il1t~n.I kio;Amiu-. 13.3.2.3. F..h-llrti inlql:rloI e. me-z6k BUlI'- . 13.3.2.4 II. tip'''''' friiilrti inlo'ral Jus&wjla... dPr+k,GM;;~7

6t4

--m..n.

7'

660

13.J.3.

Int.-grAl~1Pk

ees671 671671 671 (ill

13.2

131.3.:>.~ Tlorlwli dilf~itM.

13.3.3.1. G_ int.-grAlti1...w, So intforAlfonntllaja. 13.3.3.2. Stokn iu~lktdP. 13.3.3_3. G........ iulrl{JiJlkl-lci 13.-1. ~1l"Z6s-.WnitNr.>k UU. Ti... ta forra..m..w . 13.4.2. Tixrta V"(O' furrlw,.IDj... 14.1.1.2. KUlUpl...x v,;,ltuz,{", fiiggYl-ny !".tAn'rt"k.. 14.1.1.3. Kom pl..x vAltl"b" ~ fooju"oP 1-1.3.1. Kuoll.... x ~'I _ok knnwt'gf'nriaja . 14.3.1.1 Kntllplf'X t"8'l !ld V!\lo ,~...r,'iAACo; . 14,6.2. J", ..,hifd p fiilUtv" uy>"k 14.6.3. Th'-t.. m~"'lLy; 7J>; 7J>;

142 145 716

"",4(;;~7

'"749 749 750 7:;1

,::;1

75'75J 7:>3

716 716 71,71~71~

i54706 759760 700

719 719

tm762 763 763 763764

119,19

720 720721 721 721

7tiS

m m m

766767 767

7""769 770770

,24

7:l4 1:./::; 1:./,7:.1t)

no

711

7:10731

m

7:.1:l 73:1735

772 772i73 773 773 773

15. lntegraltrlln."f"rmllri ok 15. 1. A~ int.gril!tran-.:fvl lu;,,'i{, fO~l\Jm ... 15.1. 1. Al intPj(rAIt,rll,"'d orJ"...-iokAlt..J,uIdil!i,. ill t.'~rilltn .. l,.'l.f"rtllr\(,iok 1;,,1.. . In.....n ll'lk 161.2 Komhinilciok 16.U. Vari ilciiik

XXXII

TIlrl."lo"'J~g~z'k

XXX III

XV

17.3.2.3. Glol,.uu. homokllmkus hifllrkMiOk 17.3.2A, A torun r..lhomlilM '

l'ti.'> ,.7S1l 1171 "!71 1171

18 . 0pt imal izAlu 18.1. l.i""AriI< ol'liulI.lid"""i f..iJ.Ku.tok IK3 .2, P "ldak di.~zkri't dOllt"'i moc.ldl.. kr.. 11".3.2, I. Bcv........ rl4.~i M... lat Ill.3.2.2. Haliz...ak-feJt.dat 111.3.3. IkllmAnn-f.k fw,! 9!:K> 9!:l.S

926 927

'.I'J'J9!:1!.1

XXX IV

I

19.4.1 .. . '1'0],1.1'1"""'" ..ljAri\lll",&,k riWi.1 jtlltlll7.f,. 20.1.2. Bn...".,to vi'k1.8.k 1\ k-gfollt........hh A1kA1lI11\ZA.~.9!. ~)

20.3.4. Algdm,i kif..jcz.....k20.3 .::;.Soro~atok Nt

Ii"ti.k

9!. !J 1

XXXVI

Ta,talo",j~gy:iI;

XXXVII

W.3.li. T Ahl........t./'" tlIIh'< uk t uni.k. wktorok .... matrixok h W.3,li. l. T Ahlw t - ';" tiimh"tn,kHlrl"k 20.3,6.2, ElO' lilllrokr" n. mAtrixokr.. vonatkoz""tMa 21.:>_2. IrrarionAiill fiiU pny.. k int,w41jlli 21.5.2. 1. :\.;; a' ,; r kif..jn......k,of. tarta.lJIlll>'.o inltw4lok 21.5.2. 2. Egys b .;; kif"j..,.p,..t tarwnli\i/i> inloluuw> i1,hWAI"k 21." .2.4 . A .jar ... b ho .j'J'i'+'9 kif,-jn;io>.rko... t.vtalm"".o intt>gr41ok 21.:;.2..; . A ~kif"jP7: Ilartaimazo.'>inlgrlUok 21.5.2.6. A ~ kif~J '2J.,.t tarta!rna30 intqi:riLk>Jr. 21.5.2 s. A ~ kif..~t W-WIlIa7.6illt.-.,;rl\lok 21.'-",2.1", A,fa?- + L ,. kif..~t w--ta.lmad> illtgrlUok 21.5,2.9. E.,;y;'h irr,....ionAli.~ kif..~b1. lartaltM7.o inl.-gT4lok

10.J6

10:>6 10561051

1051m~1

1058

1059l OW

11161 1116310;;

106 7 106 7

21.5.210. Rrktlnlv formu.lak .. bUlOlniA!iI< tliJI.,rtu.iA! i.lJt.twAijanaklr.i:yt ~ w"-'lttaJok 21.;;.3.5. .-'r.. kol-anfirnl;.fiiAA"Pnyt t.v1alrtlNli> in~4Iok 21.5.4, E,;:,'l-b tr~ .... nJ.AAVPuY"'k inl~A1jai 21.".4.1. A hiJM'rbolik,~~ n'AA"f>nyoolt illt"tiIJjai 21.::'A. 2. Exp0!lIw",,fii~i IlY 21.9. Dt-.-l- fiiwpny" k (h"ngerfiiW'plIy.1. Sta nda rd norma li. d"""IAA. "h,,1 0,00:5 r :5 1,9' ) 21.15.2. Sta nd llrd normali. "1 lr,,,. IIh,,1 2.00 :5 r :5 :UIO ""7. a.is. , 2.,'lo,d""21. 11 F i>vM1a7.O jPlOIo'.w-kM. haMuMjllk (tN.rI Hal llU\U'l mfl.-.t . 2'),. old.J; "~f'U'S-"mok IuolIlllWl: ti _ to. L :.!. J . .}.""pgi->czllMlDOk~nP'~

TargyID utal6

1125lUll

Z_ I

-2- 1.0 1.2,_

"racionAli.. ....iLmnk h.oaIlJIAU.' .

Q _ {.r J"

A ten~ !oi,dt...... hatt-. 1Ml'f'_ A ~rl ... !fIIii.. pan t. .-\Z ' PR. u....t ~uim'"9!1"'nl',.n.k np\,{"LZ 'i k. Mivr l " r""lonilli. "".Nllolr. h"huaz" ru;u< mind..u pontjliboz ,.,.Arnot tudjunk (l'u'''']'';. Az MAlizi.~ tAnkolllVf'i a:z irrRcionl\lj,onlull'.kr" ,...tal.>al.t:ol ddinicibt ..In;Ur..I''''ldA,,] int ''fvall""",''''t,,lyA za-,J. A lrt ........ rao--iuw\.ll>llldmol aJ.Iit e10. ~I.",rordit va. wiud"n .... I'ftcion&Ii>IIlZArn IIl"g"dMti> v~. ~-...nllAnrti>rtkPnl(ponu.an ki1.C& mOkra van !lZiik.~) ~t kal.juk,

,

"' ="''''+''1' ..1 = ......I ..3

A ,..lOO -'mokaAk .. ko..",k.,z6 alapt,l1Ajc~aiV>UlI1Ilk: A ,.,.J60.!IZAmok halmAU v;'K1lO""-..... nh!a... tif'n. mrt hillly",bIl)liw.dli I.i1.llllyia". mint"", iiss;r.,,,, wti,h ajo vo p;rfU.

1.1. 2 ,2. Ind lrekt [ell ent.mondas...a l tort.t no ) b lzo ny ltasA q " Hita.. himll yllru.... ,I,ljft.h"'l" q t"'J,,,h,hol ill, I" lu" k ki. i""ut6bhih61"ID' h "Hli~ r illht"" l'I'.kiiw,k, -". tet iink, v"l'Si~ q =;. r (1"",11ll"'~ :lIS;'. "l d .). Ekkor ""." llha" ij is h",,,i.~ kpil hogy I'K""", m..rt implikki(, ,~tj"u CNlk harn i~ f" It""""l>llljlltl",tuuk l"'mu; Mlitftsra (1:",,1"'" illlpli"'":i,, i"a """l't"hliV .at{",ak I. .. ",'IO' 1 '' ''' "lO'

6

J.

ilnt",~hh

I, I . E'EN'" ., z,im"M.CZIlIl'llIa k "ftVt'IOO I-t.6I n-ig" ingy nO . ) .

3. A k;ilOnOO7.o ajap"logaritmlll.ok I\fAnyl'l'k gym",""". ltl{Vb"lP1o:f. a alal't'" vntlAt kozi>logacit nu~""" iR)' tl!(..;

' 1 > 0, "!o(i",~)

1. A 10a!"I"""Amra vonhl.kow logari tmu""k>tt Il':,.,. al"I'oJ v"lO" (lR IGGsFlt 1t'Y'mh""""Aw.k nevezxuk.J"l"!;";;ik: logll, x :lgx, ':",f" lllu'in Ig{rIO") =Cl+lg.r. 2. A~, e "j hl", ...A lIlrll vOllatkod, l""ari tm"""k>tt " 'n ",;,' ::d.,'. "'40' zii k. J d ii!h\ v,.,;y ~ ta.. k "'~l"'k " IllCI'\k";IWyi~ l.or" tI7.....,.]"P...loK...-itm,l'lkat hasLn.ilt3lr. em' .. l'02.ata IuoI~A~. 10" ",,"1i:B r -y...og"D&Y f/'i'Do'vk lr.il....OhPn. ~till""jel alan vagy ~i'fl)..k Mgtlmrnt"mal....:nt _ff'J...In..k.

If'frt...,.

1 .1.6. Racio n a lis cgesz kift'jf'zioM'k 1. 1.6. 1. EI6allitas polinomalakban~lil>do>n r""ionAJi,;.-g. f'g),ul......g:i 1u.1"",olat.ot ka ;u.,"Wt """"'"""" li!rlno-lc n....'f'7OZlik, A tcl.-! mindrn ~ ~'lwrnplcxa", b""Ji.mra.. to....,.... mindPn n = I. 2.... fi1iokn- ~_ ...\ rovidk hino miAli" l'gyiitthat6j .., " lIlk m"ll.hatarozA.'.,,, ..2 2a.r 3' 2lIilo;.....

"V--""" gyi>k"i "lllIllAk,(I,.m

D: r+,;'i - (r .. -

'Y>(r1 -

r -x~+W.r -.'!i +

W)

=-

r-z~ .. W o_U III 1.1.6.3. ~I""'''''''' 1.'. ---,~,. "",r _~ "'-"1",1~+

P(z) o,,:r"' .. -alr ... ,,:: ~ ... ~ _ ..... ~ Q(r)" (r-o!l(r-02)(r-o.. ) r-n, z-n2 .1'-0..

Irt..kM r..lJu..tJiAa; = a; +l - a; = d=mn~t ,i,"",~..... l\ndh d

i;7= i;='''''''r,:

a1

a.

k iilo llhooPg" konstans, v"ID'is

t..J,f,t

nrt(1.6Jh)_ p . 100 p =-~ k. Lr,;vrn

ftlI:

'ij

K htoOltt- v.....ri tj"k. un ..nyoij tu

pnr!mfony(1.69)

=

Ja

2 ;

/12 ,

100-piUA

A

'''''9J~rt...

l:Ou'purk" thUl k""tijk I~"'hh ko't kiilOnbOW.( l. 64b)

2. a=b ......W1:1

KL

1.3. Penziigyi matcmatikaA pi'nziigyi 1ll"tPrial~-...k.I"'KY ....k pgy llagyozAmll ~vam.atot lW'tIll'k.2. Kamatos kamat

100

( L1U)/I

K tiiki'Tf' p'K

~liv,,1 a tok,; a kametot lila, ",illlkn ily"n luunlllp.'riOclu~ vi-lt~n hoz1i1lrj4k lIZ IIktuAli k.....-!"th"n K ,

(!.ti5 )

toki'l\(~_ Ekkor a kamatp"ri6nti kaJ""t K.........- P-.100m

N< n,',v uti;'n. alLlI"lY"k unudegyike m

1. Ant",d lk"

-" "" f ....k (kama'pPriM'lso k) "7111na II;< ",li-,'.~ ,",'~l,'ll;'''' tiirlrti>k,.., no,...k""ik (.1. . :oo . n t,;k.. ouudy i,'i ; .2%-kal k..mal,r~ik. ti ,',v aJall al i'V'f'll1I",k:

S

,,) Zn kamat az100

n~ed ik kamatP"'riM tl ~ra ,

_ ,S [ Z. = 1 - -".

I (" -,- I)) . (I.,u..l m, _2m

b) Z ~z ....il ett kamat az S ad""'sag mN mll.. tben. S ka m a l pt'r ic'xltls .... p% kamu l lAb ",.. lIett tort ~no tlt 6 la!':os l orl....zt .....

....npl:

no.u

z- t.z~_~~[-" ;l _m2r:,I]

.-,

0.7bh)

tc =t.- ('1'-1) 'I' '1'-1

(l.7:.laJ

K _ E 'l"' ~ 1

'1'-1 .

(1.731

3. t" ~lll'l gyakurilob bf'fiz..tfool foil . ' e ll kl' ut i tok iI.... A z ~t. ill. A kalw"nlil bu.o; "l'l:

h.",[..

_(m-11PJ'l"'- 1 ::000 '1'- 1

(I;~)

Ko - E[m.

(m -l)p]'1~- 12110 q-I

11.';jh)

uorhP""'" ItZ pg""z t iir]p"zt ki

~b..n AlWiJO.

EllS I"",-t ....P = 5.2 ..,.....ka.u...r.1Ah 1ll..ll: IIImuitAprii.ol"""'ltk"llt m 1.U/Io">uld,;,...... ,ilj "I..,1 (1.i 9)-'" " "'!'luld tjll k "". A. S. q. .... mennyi-egek h" hflrmpjyikp,..,. ha .. tobhi..k ....Ipk. i' llll'J'l . A : Egy tiO 000.- F t -oo l o rlf':'lz t.p,..." 1I,1i......ag "v i K..lllllt.. !I%. ,"':. i'v ""lit. kd l UirtP>ozf.t>ni . M~kkOfa ""

1. Ant""'I"",,"'TN A. tiir1"'7ot''''~ Il hllvi" tnrl....ztbPn.. po'ru:llgyi(,v foIYlUltiul AZ .........ji mdtd1OMZ."tj mrgludljAh61 ,. lljO l"'u. hOlO' .. 11lIUtI"ulu'riilldlllOi>rti>kT'" Vlu,tkwl ... 1011 ..w.ali>k"" IrIr.... '' '11l.'IIl..Jlu.~l.ik_

lndulo-

LPira.~

co ooo411JOO

..00irtHI: .......i'n AZ 0 = 1_2.. 10 ....It'a) linellrlll:"J\noo kiT.... ,",I" ~.._Vrt,,"i oor"''''1 "'....rint Nli'>k k,,"o l.. irMrOlllli",;;',..;"r" Attrrni .

.. 0",='/

)

1

l"' IIY Il"KY lIi1' IW'V"'l'M'.,.i "-r... A = ,',0 (1)). Ft. Ezt a g"l'd;; ev al"u JigitAli""" Il.Z U, - 10000, ft, "llliI.T",lvilJlyhtikn kdllb- rop;A1ma lIi"n. "rtI'l~.pl. kOlUl'h ~kra""10' ...-ktorokra... Ill jfollPl bel)"'1h ...it!wtji,]lt. Ha ....1Il iUl1tjuk kLl.,,jc=ttoon a7. ..llr,lkt-I6jk.... "zpl'f1>lli ~k vaIOsak. 2. A ;ro llOOl . "'1O'PZo irimy(l...11f>ntl>t" iranyil k ekvivajens "'lO"' nI5twn..wg"k 1. AUlllO'" ('IJ'.. nl6t"'ruJ.gek -.Ie. amdyrk .. b...miik lIlJIl"T'1'fllO bMl1,aiwb6lUI1>Ok mind.-u {Oni'1u't-r u-l;""Ulnek. 2. Egy..w ir /ronyu "O'"fIl6tk-u.~kr6l t-.'IUnk. ha ki.-t "lO"nllOtlrn""g mimio'gyik.. Ill; I. tipu boo. vagy mindlwlt6" II. tlpu..lJA tArto'tik 3. Elk-ntH..... iranyu egyt"nll"itlf>n..Pg..krlil "-J.hink. ba a:t ~ik t"IO""nl6t~..,.I. tlp'........ " ma..ik a II. tipuaba tNtauk. .t. Ekvh.,.l.. n~ ..g:J....n~tk-~..kk ..1 vaa ~mk. ba krt. ugyallA>:l>D i. Ill""'" hPt:i1iYf'1 jf'II:IIl'll'ad; ba ..iN:,t a -Un 1101:'I"1/... akkot- "'" pgyl'lll6t],o~1 irauyat IIlI"'gboll fnrd iuni. 1. Reciprok ht;.ktokre~'onatkow.-gy.. nlt'll",n.~

Ha

""y

g vAllo zasa Ha a>b. Akkor b b ,b..prt> vonat koz6 egyf'nlo t le lls~al~"2-"''''''~ i....--,tt illl~,ijokr.. Vl',nalkOl'i>C Al:CHY SCHWA'l1~....,16tJ .. ll...r,.; a:t. (l.lOtih) kt'plrt awUogOllja:

.u::"

.,-"7

l"' . . ~ :......... '5.l'l~

_41

2:

-- -d l..

2

(1102)~v '-"K.V'-'ulO .llLin\

(t,.....)' (t,.. ') (t,..,)ktl'k ,,"-,h,t ;'rt kcire vonatkozo ('g)'('nI611('n~g('k K;'t II ffi II rmltiv Yftlh, ~ll. II < b .,z.Nuuuti. rno"rtIuti. lw-rnonik,,,, i-.. u"l:r,.1>..... b~ 1L:! + .

v.. gyi.~

(alb , + a,l>. + .. , + .... b,.j2 $ (Ill~ -.- .. / +. u,.~)(b,2 + ~2 t + b~l), (l.llJl.ih) KM vPg'''' ""'''m''' ''07.Atrll.lIlIl('!yt'kn ..k lIlingj..1"SIlk IIkkor .. rv... ny..... h" "I : hi = ", ' b-J = . . = (In bn (ll a va[" mdy i_no (I, (v"ltV b, j '" 0, akkor b, (II.) is n.] Ha " = . . "" II," {" I. a 2, IIJ}, {b I>J' I.j } "~,AllIhAfln"~()k,,t vt'ktorok o."" ,,,INl" jc>l ,h "1tI....m ~ k"t !IOC kmlW'rp'nrilI.}AhOl k.w..tmik a b&l oIolalon A.ll6l1Of kon~jA_ 2. Holder~ ..nl o t k-n...Pg inl.-gra lokra Ha f Ir ) i'S 9'(z ) ko't mi>rlK'tofiiggvPnY A7. I X. ...t1') mho ti-1r.t.c'rl"1.l (tud fAA'!. oIn.). akkor

mM>dfohl f"l(W'lllOtk~k ~oI 0 1llf'gO!.-IA.~ - .;;;; < r < ~ .;;;; ( 1" < ..r.;i) .

( L1 2Oa)

)' lin~ki-,,,CYf'nlOtl ..n.I>g

l!IOrokra H.. p 2: I. tovibba cr y., E C ki'l szAln'.ortlllAt ...kkor( 1.I 13B)

(1.l2Ob)

1A ,3.4 . A m a..';odfok'l pg)'pllleit IPIlMojl; a ltalanos csctca.?~b1""'c>O1"~ +p.E .,.q. trivuw.. Vl'IlO' rnm' n0'gy7Rtti> ",.)0 k~il"-"IMOtl,k(1.113h )1" -+( I 12111~v""I 0 , (3" -;- 3)2 :r

2+2'

> -G.

A I. P!\'y, 'u!,i t k'lJ""'l'; IIWIl"""" ("lj....;i!.

IIl"kra . II mA....-.dfokl",k" 1"j.\"I\}>

m

( LI t:'!...)

v~

(I

11,1,) ,

c. -2:r~ +

].I x -

20 < 0, (1" _of

~ )1 > ,. , .,3 ~' _ ~ < _~ 7 .", 2 2

"lakr... IlZ altlllill1"" ......tt ...n I_lip; IlZ

A 11lt'1\"1,I",,i l.art."mAuy"k 1" > 5 ,..

34

1. A.nlm,trill

I

~fogl"JvlI. 3 m"!d,.]..l/llll....."lfnk,1 ''RV''n~ gyiik.. i altai. 1;IlPghaui.rozutt ..,'gelgt"wn nyflt iutNv...lJUlIllluiU..tinki6 ""A'Tint ko't. kompl..x ,'!O'('ul6,1ut. ""M. rlHMk ill. i"pul" r',z.~k j,. "R,\']," .......k~... 'l,l>.. f~"h1t" k')j"I'!o-X,,"'-t":iooi-R.-t rrrtu...o:( 1 , 1 Z:~1

....&mot a : kornpI..x ~ oi,!:"" z kOlIlI'k-x.......n .. \1: "'IZimot : "Gl6.< r"~ :"lI.. k." bllllAlnnt : h?:,.t ..~ n'.:;.1lO"k Df"WZZIa:

'h )lU'llJ:-

p=~_

( 1.12",,)

ilLw.b~O.

1.5.2 . Ceomeu-lai slt>IIl!i-!tt>lk 1.5.2.1. F.li'ial!ftas vektoralakbanA VRJ6IIlo'Il '" ~,'kfw.n f"!Q;,k ,'.,. a koordi""t",..ndw...,. k"",j"'I'.mtj"t"'l "z ill,otlll""'ll", lUll!.'" ( 1.6 . ahra ) . .-\ kompl"" ,.,..Am"",,"t "'1".1 !,on!.okW iIO.

hAbO.

; =

ru.p '" 0(1.12:>01)

...-dg-

! 1\I"1g-

i -,

To

bao

> O.

!lao=O.b>O.

0"0=0," kivonjuk (",'.!I ..) A B '" (II_) (b....) = (II.... b....l

(I 37) (3 -50) (' -21)2 -I'; 2 I" .; 08kommutativiti. ~i

.-+-

m~4.1. abrL

n[Jtl n

3 -5 0

2 1

B

Am

-12 26 11 -3 14.2, iI.hc...

Irn A B

A miltrixok .:....z.,,,,laM koll!llllltaUV'" -.no-iAtiv mnv..\fot:

nabaly: A ... B = B

A

(421b)(4.21")

... u>c ia t i v itbi !lUlbAly: ( A + 8 ) - C '" A + (8 ~ C). ~

uon.an uammal Egy(m. n) tipl...u A nloi.trixolll2.-.7i-l;II,k. H...AZOrIhM az II (n. I) tip....! n ..b ( I. m) tip'uni. akkor a IoOI"W"ktofllAk~' n ..J..mil 11 = (I>,.b:z.. .!>.)T oszlol.....ktorral kepzctt ...kaLirlrtOr7Atan az

alibbi.

~

i-rtjUk

,. a,,1>~ =

= (oB}A:a( A B)=aA/l B

(oI,2'lrl

di sztributivit bi uaba ly : (tl 3) A = n A ::l: J A.

(4.:l2d)

.-\ ......n.fLro komm".....tivitA... l'ZII.halya iI.l.talAl"," nrtll ",, !T "" !! 8OITl"lldji>t pontoAAn he kdl IAlUn i. " ........-nd M"".~.I\7A7. loaT '?Y diNIik.... ..,.,..,.Atvt ....... Im"2.... , a,,)T oo.. ) """,,,kIOlTa! vrtt daodokw .zor..atiul az alAhhi In m) tipUM" ma.lrixot rrtj,ik

L ..../I.. -' erv"uyo- Itt

.

(4.24)

:>. !o.latrixok u o n ll511I. Az AB lIzonat u A ~ B miouixok or.wr....ta. .J:aldrio mtitru:lwrmfn.. k i~ rw.....,.ik. noak .. kltor kfpc:thontlu< mAn aM.nr 't.-I\t l'lO'...... f>g, II.m..lyl". 1L a t/\lt"k vIIIlh"tri

(4.2,,,)

AO - O .... OA =O (-13.: ru",k .. lI'l.ab&Iynalr. .. mcgfonlita.... altaW>N1 urw ~h1Y""'. IQ.IIZ, az A B = 0. h6l TIf'rrt lIz'-1k~k."P"' kiWrtkk ~1dILooA

A _1 ( , -b) -. ad-oc _=-~ loll ,..,;ymIJlIJoon~ kiPlo'1Pllk l':ft"lIi.oal kor"pot""~'''''k llt ""uk. H.. I. ~ik ~fv,iIIUWd"lI A wAt~ 'W l!. IW'mn.. lI..... lnor ugy. bogy Il;I; ~wo..'l: m'l.l)...... llkkor~

A " ....uruYlnn41

4l

.! .....l1~ tiblll ......Lilt m6trixnormA.olIk

D:".ktrAlnonnll: IIAU -ff Alb. b ) ....ru....zeKno fm" , v'A-.-( A TAJ

(4.;,0\

40,402

Q~

--1.2.2. Detej- minansok szamitll i s7.ahftlyai ..A !.ArLAc E-ff l.. kif..jli.. i tfl,llrWlU" kimon k.. 1LZ Ir '" !: 1 Ln"S",lj..."", 3 v.. kl " , I~t

""">1'. iw.ufii~ """h6l '''' a kiftjti.,.i IM... l.o ""l'jHk !

-,

(4.56}

A odj A

AA adJ = (dct A) F-.

(4.l' W ",.vh,)'." l " (4.:-'2) ", ,.,. " (4 A!l). L"< ' v"l"lllitlt " (/.,>'1)r.. ,... (.1.:;0 ).,... u,.

f\h,,11lZ A tnatrix A odj adl' ''' !J,ilt IIlli/nr';'t "., A dc,tn,'i,,,, k f\~ ,.j,; j,.j,"" " ld, I." ' Hlill'm 't ,..". LTnf'lY"k lIZ"11. an ... a.. ...tli> alau YallllU. null6Vl1l ~l.6k A .-lMrd inil.Ai. k;.t kiil,jnl.0z6 fl . t ..... ji: koordilllltl\Ct"lt.-l"""rn' von"t,k,u.o..u. 2. Einst .. in_fele kOllvenci6 A (Hi4IhPly..rtvagy

!

= A~

i, '"

lIUrl

+ 0:2"'2 + ""..13

i" = LtljW"'~i~.

_ .

(I' =

t . 2 . :l)

2. H a rm lldrt'lI\hl d..t t'r lll illlUl>l "rtekoA SARRUS-"zohtil. ",weilll,. "Tn...ly ~ h"nllAr7..-m:l

n'l'll"l...ntidja . E zt npv7.ziik iLl< A. .... l:! wktmok d""J,~". umZ(lIAn"k. 4. Redukcio Tegyunk " gy ","lr",,Iruuodfok'll'JQ"'nl.-t gynk,i. A D tr>UllOT.fomLiloriOlllitrixd, . ., ;"' d, .-.zlovwktorait a ""jli.t.... rt;;k..kl,..."

l

I

Sp(T) = I" - l:.r::z'" 1]3 '" ill + i; '''l, IJ.l)i', ....kt"rral br-vl'n ..... " ...k ..Ikotj./lk1\

D", _ & 0"

1

"'I

gl . akkor A rj, "' a if'

tf'kintht'lvk kcit)t ;olkalJUO>:l-. II kLriAu.~. illnvt' .. k"ntr"v;uiAw; l...w.-W'kt.... k Irfrho.tOk o tn8 frvi'nY'"' Il KROSECKER....imhOlulluwtJ Il ,

To

, - au. (Jx. r ......

1

Dr... 0'"

if,.fj'",r ~tozatllm mar...-!. "tt>.

4.3.4.4 . Szamftasi see balyc k

A 264. okWon mAr lU''glMI"U """,I,.ijyok lUdlrtl prvlonr-k A ko~...tkW............"v...i.......t:wiJyok: 1. ~zeadu, ki vonu A~ ll~MlIJ"'IIrPnrt..lmpb..n II t.prinwr~i6nill" kowl.kf"ZO t'I!"jrl"nAAgu: 11_ = -6.... rIA>'''''too. (4, n,,) Prt,lm{>!>r" ..gy m.\>lodrpnd'l ".."mrt r rNlmcnY"".

ti"".

=

=

27 0

aJIlf'lym'k" kompon..n....i t" r ",II) (i,).= 1. 2, 3) . ~lilldr..ndfi tenzor ..IMllitlll.tl, ' 11;)' IIJ.imnllrlimllill"a.-i.. ''f/..\lf'llltotr..ndszer..k..t Po. lillf'Afi..o ptimalizAliu.i f..ll\datoka t oMha tunk meg. :\.1. ..l..mcser....!..p''''t a

- aA l

(i,j=1.2.3).

(-t.lOO)

A K-rd';ll "",wu, ,,trikllll",,,,, (4. 100) i-ppt'll lV. (.!! x !V v..ktor kompoU..ll'. it rn.lmrny.-Ii llZ ~ ti-nyt"ZO ki .

" 0" " 0" ", 0. ." 0"

r

'. 0,. 0,~

W1.l"IhPI. Ugy bogy" So: - (! x !v lUtiliJi,.; wktor" 'I. '2.~ kompo~kel, ..kinthud'l t-,udutl) s: = (c t- Cl.l'))T f..rdm I17.imJrwtrik,," mNoodfrndlI C tMlZOl. IUIH'lyno~:

c ='" = ( -~ -ci ~"J ) ~ 0 '2

(4.102)

o, oMnAkkaI Vl'".-.ctjirk be. ahoIlI. hAlolooj

2.

0.. u ... _ _

S. FH"drendu pl8u-uootlP'nz.orok "J.-MK\oo.kal&rNo" Jlf'Z"\KJo....k tor fog~ IlZ alWArxa.it4""'-n~ n-f"d~d ~llht"i>l ko..w,J..uiillcol,....w..t.al

alaI..... a,a:laI

a'"

(4.lro..\

'1 x,

Fl

1 r~ 0 1

It I2 0 0

y.., =

a...,r J

a - Ia

'" B".

Itt lIZ:

1 -100 : 1 s 2 0 11, 0 2 0 1

H"",m ..l..n"........ utAn (I"..]d ftlll ll, - I . . !h - '" 111 - 'J) Ju..l>j"k:

c:e"'lc="'"'1'C~'ir-i1 :r~.1'1 1 1 0 0 - 2 Y3 - 3 (ill 2 0 10 r, 0 - 2 0 1 0

A=

(::.

:~, :~J

..m (:J m z -.

A~ 1lJ2 = 0 IW"U nino; lo""hhi d ,-n"""''''l'' ld,"'~'g Po. .... YI "" 2YI - 3!fl ... 10 fiig;g(,,;{>gi r rt-nd.""",ret1."III,'" (J,ZZ, ... . . . ... GZ1I , ... GZlZ'''' .-

r] -+- 2J 2 - .cJ + x, _ I 2.1, - 'c2 + 2.1J 1" 2x. = 2 :iT, --- 'c2' -+- 3.r. - 3 x, -Jr, -3.r .r.O

x,

,.s..z ''lO'''uktek f\:1.Aml\ 1 Ut1!:.IO'...... ik ItZ ism ..r..tlt-n ..k szillnAval, ItWllhM" rll.llp;( A ) = 2 . r"II~:( A . i) - 3 miatt Il ....nd""..,[IWIn m~uIJhato.

a,.I. =G200S"

III Q2

=

2, A hOlnugi-n re nds z..r t r i via li. mn..k ho zzaren< hl'-' A (4. 1000) ",r " old "",ihoz ",", A~ '" '" lin rfuis "lI.. "lrtrrllc!",,,rl,,-1. hnzzru;"l< fiigg""'"Y"k rrndl'Zm, t. amrl yrr a1kn1rr"",.znk az l kiol VAAh lltjllk. ..

,\1.

2, Linear-is e g..vE'nl..t rendsz erek " " 'goldh" t liltAg" A (4. 110a) li" "Ari" l'wn],t",nd"z"[ 1"",1,0""" Itkkr ol,lhl\to n"'g, Ita" lill,'flrl>< fiwi'ny"k hmzitrf>n .. lt. inrlll!H.tll"''' vi.... Wik. . I lUIId:v..k.ll~I;u m....adrral "R"iitt . k .. qs: (r kon.~tf'4 ri>o&u. fllAA"tlrn.l!. ""jilt klor. rti>k -l'rubl"IlWlI (4.1"12). ",m"lyh.-n

A uu\trix

val,,,, sziltlm,trik",. m"trix. a kii""t -

ke7.6 tM"]"k i-rvi'nyt_k:I.

n) van. = 1.2. Aln..lyLlu.. or.....j&t..... tono YNl.. k Gram-Schlllidt-r..le o rtogona llza la..i E'jaras ~ \'~ 2-4

,Wi ( A -,I. [

.4. ....jil.ti-.-v-k..k,l., =0,1.,:>:: 1.,1., \Pt...-nd~hoIIutIAmr.zl\k """l .\, :0; 2r, -3r, _ r U 71'" Jr, U -5Il 2r2 - 4r, U

31 -,I. 3 _ - X _,1.1 ~ 2), - O. -?> 2--1-,1. -2 A""j..tw-lttornb.I"hodornuioiO a AAjAIPTt"k..kM (,; .. sximm..tf'i,it ulf"lI:l>u"tj... A (4,117) ""'I jd'llti. IlORY 111"'''11~ trawWocrnilIlIIli7.lilhAtkpk A, = Al '"

-l",~ = 2,k.

-'L- , ,Q - 3 ! "'L "" ,r,2.~,

(4.134 1d~UAlll"'lrixa. U

..takot kapjuk. amdyb..n

A7.

L IIlC A matrix

tiJrtko6 fOrgAw..al, atlll"iyhPn a di"4onAlf'\PJI>Pkd, =

(_~) M.!2 '" C, (_:)I ,.

.......tnrnbl.&hoI C I (,;

Cllt~ kow-I"'

J..

....r.,jn1 (Id_J>i'kW,\ 27~. '*l.).

4 .5.2.3. Kvadratikus alakok ((jlengelylransz(ormiLci6ja1. Va u... h adratik us alak. d .. fin ldo ."u. ,r1..... S. "Al~'Jk f-~a ~ algoritmu..1 dolgoztak lti. KM. l r, ukor l\Z A TA ",,,trix >." ....jAt.. ,ti-k,'in..k .. l"r~itiv u(-gy",'1gyOk,.it ... d~ '" v'r.' (v = I. 2.... r) """"'okat az A matrix "Z"'9rzOI\9"J,iru ",.j,t"""mM ll....~1.'ik...,~ A AT mAtrixnak az r darah I11l111i.t0l k,ilonhiO'o A. Mjilli-".'lco' "all. mint az A TA matrixnak:

2.

Va\68 poLitiv d .. finit k""dratiku llllak. tulajdonsagok

1 . Egy Q viii"" J..,zitf".ldinit h.,.,lr.. tikIL,,.Jald~"l" h",""'" szimmetriku A llllitrix llIi",I"1l fOAtU>lmll ,,110 d ..m, pm.itiv. 1l7.ot:lO1..

"g}'''""".

",,1

>U

(i= 1.2. . . n).

(4.131)

A pO/.itiv ddi"it.....I(l1- '" hal ol.-lati ~zing"IAri" v..kl.orok k01.iiui OAA1.,'f'iAA'''' A!!.,. = d. '!.." A T.Y.,. = d~!!.,

(-1131a)

(4.1371

! =-Ci

(4\;12)

Ervpn}',,,,:E Rr r I'ar,~{> (TIl. n) Up""'1 A wlltrix".. k Vi\l >r p01.iU" ,l. IV'" I. 2, . ,,) ~"ill~"lIyan vll.ltcm'>k.;un('ly"r&tort ..lru.gyjuk. P"I AM

F =T.

A v11 .. A8 vC .3. IgauilgfiiggvenyeK Tulajdonibunk II kifr,;..zn. minden itiolrt v4.ltnzOj&UU~ i!:a:I~krt_ FA

A = .-t...kd"~

(5.16)

'W'ik lrotl"l-

~ b.1..~.1.5. ) _

E'.zrko>t .. _hilynkal f..lha.'

.-tB 'I=

e ",

B V C ~ az

..w>biak -.-illt iguolha&O: A V 11 =-

AB V C =

r rr

r

F

r r

T T T T

T

T T

r

T

T T T T

r

T T T

MindPD~tl'I.t iti-M lIymodou "0 PIv.ilto:zrt"k,l iF:",,_~fiiflJ.,'Vt:'. A:~] kovtkf" ik.B l "10' 1M r u egmut eth etjnk , hoI'S 11-Ml koV.2Oa)

V>\gy

~4=B =Al1=BA =B -A1l' ~A_

(:>.201:(:;.200::)

5 .1.2. A peedikatumkalkulus kirejeztsciA matnnalikai logiJu. m~..J~lU: it;'!fotWk"hlllll&l uagyobb kifejIZlW>ok K h~ fo,ti"k ........ .-\Z q()'Cdi ek-u....Ir. t1lla~t Ij"'ld" l1l prilllawkat (l'i' ldiml win ki.....-bh mint n"l pr....tik4tumokDAk n 'l'r Q(r )) , '7 x (P(r) _ Q(r )) =:> ('7'1" P(: '" 'I r Q(r . d31"

Z.

3 r P Ix ) fonu&hoon ifbatO..,. .". kvautifikaci6utjAn;u: ~ fl"'dllu\'tu mokbOl i~"'took ~ . ~ r.; .. ~t""lIZ&mok lWmaza e. p az ~n prin-.Am~ "&Y""l"'*" pR'dikilum. 6JwI' A 'f n P (n) bawi>o l< pnodikatnlll kijcli>l".,{'1)luut i. tlO' .-gy w i forllluill. in""TJ'l"'q'rct",,",6 ......t(>n igaz, ha az prt~lm'l""'i tRrt.,.rnimy "leru..it .. "Z"..... l vAl101l,klm tl,tI'l' JlI,'R'''' m P(x ""l~'"'' VrEX Plr ) o!f, er i6t"r (xE XIIP(x )) Iid"""t 3r XP(x )

5.2. Halmazelmelet5. 2. 1. A halmaz fogalma. kflldnleges halmezok.0\ halm,.,....lmi>lrt alapjllit GWRG CANTOR II!.'-I'; HilS ) fogll.1nUl.Zta Dll"K. Az Alt&la alkalmezott fo. ~omalkolAA j ..lt !i M""r"IM/w"l. p/'lrl"111 M = {'tll k h"ltu,"". n

Q :

{ ~ I p, q E Z 1\ 'I f

O}

ra"io"ilIi. Mr.mo k 1 ",11ll1V.>I,val..... "za.n",k hll1m"" k .... komplex I07.,u-nnk hll1m"7 .....

!i."

nC

288

.UI. Halmllu Jmild

289

2 . A halmazok kiz a rola KOOI..a gana k ~Ive Az A PI; B helmesok po n\.ollan II.kkoreg.vpnlOk. h.. 1110'''11"""kat az t lrm.. k"t tarLo.lulAZl'.4Ir.. ..:f.ro:: .4 = B .. 'v'r (.r~A .. :rED ) . (:: ti) d A (a. r. 3. 1, 2} .... {I. 2. J. 7} 1..~llI-.ok wl: A Hn;6 B ) jeliilt helmae, mpl)'Tf' 1AUB ""{L II EA VIEB} (5.39 ) Anwnnvibf'fl.4 Cs B 1l7. E, iIl..lvr"" 1 tnlajd,,~kW lll.-ghatArozolt.lI.kkor a.z .-tUB ..lellleire legeIAhh "" ".-gyik tlllajoJon.~ igI\z. wogy;... a ko'!. !JAlm1L7. kmiillrgalllbb II.:< '1O'iknek .. lOOf'" ba cgymA."lIAk rP..31:1) Ut .. tionyt 1D'>Wan f..~juk Ie"t halma:< ~ bizonyitasMA. 4. H a t viloyh a lm u E,;y .11 hAlID10Z Yl'I1ArnPnnyi rP.!Izha.IrJlAU, aJkotja .. halm.v. MtvUtyhalm.vA.t . J..IOIj: ... iun~""...k ""lO' w .liwolit.A, AAnak n.."""z,ik, "" ranl.1I V"IIY n"'ha IM I irAAm6ddai jlOljiik. A kar.:;'71(:> :;OJ)(:;.61)(j.63)

.-1uI = A.AU.\I =.\I

An',.,I.

Tilbb h almaz Descart u o rzat. .o\,.>m'f>c! ~I~I .... Jk

'''{'''lIoM veY ""'mm",tnk\l>O k f'C"'I'lf'lI halmiU ..1..-"..--1. ""tlY k'ilOul.00z6!"wmlUlnk ""'''''II,,>Wtli kaj.. -....I.aU>ka, irnak .... Ax . .-1, .-1. hAllI\.ll7.Ok kiiKOtti rl'"ltlkllloo i.~ !I:i'..okNotltl. t P,;ldak,;I'J>"u"" A - {I, 2. 3. ~I halma" ,,,,,,,tl,,,~ ,.,),.,.ii>il tArg~nujllk. T '" {(a.b) I a, b E A A a OfIO. I\ZOk .D=(A\lJ)u(lJ\A). (~.ti4\rtow r..I...i6oo.k i'rvcnyt'!W'k, 1 .(RU S ) o T =( Ro T)U( S o T ),

Vil:"" bin"",, r..I>\ci6k ,..;cto'-I..." {"'u-lu,t .. lulAjJUnMgoMt ti>bt,lIyir.. " lIyill kiolV>l.~h..to, 1KIlO''' T uoILllll1iom....i6;ilt. anwly .. ,...,k~ onrru.gil>ll vilozi At. H.. d jdiiLi "",*",~ 2".; .. ~I ,'aiOdforxat&wruogAvai vetl k1fZt'JrO& Oz..

D~

= Vd..P..... r-', ", do.. . r'" I

Az ~'D""'"t&o a1bolnuoz.A,. tr ..uxdonnAciOj';vAl D. nlOllOrtut. m..I}...t a~portn.U 'WVf":/;,lnk, EJ>. Ix-n .. od,Uu "'~. (J,t,od :1.4.3.3 .). Am bAlm"""'. H...\I 'l'J...tl..n O'lrmhOl;\\1. ..kkor < M > riklikll" 3. Csoportok re nd]e, hal.. los job b oldall m ..Upkos ;tt a lyo k A CllOportPlu,,,lcotbc-u ,ogy v"ogc~ C ClOOIM'l't "[''l.lI,ill..k ~l.at onlG jrlOli, A11l.tlllyi],..u .. ,"""port '1'J-' II ,.1.I1K' altai gtll..raIt < a > ciklikwc r""""''M'IKort v"'I:"'" ..Jtkol 'm,,k uncljiot lIZ II ,.1.", r..ndjinek mend juk, vatO'i.~ ord< a > o,da. uogy,n (V. )" (C, .j '''''llOft l. ill..tv.. t....1 oltlali mpJl~k""'7,tAlyfll1"k >01,011"" "Mim'l. llf'Vl"tf'I.{'>;('n oraV e!..mf' Vll.1I. Ennek fulyUn "johh oldali lllrhPn" \Pupduribilis rfm lIZrir .. ",,10 rP'"zkfoi I . Az S M sz im metrikn" csopor t AT. S.w ""imlllrtrikWl.....-.port 11'u,.,\f >'I~>'l?: Fr.l ~') A f""o"ztA.~ grafllr."" ~tAciOjadobozkilkbJ tilrtklbl1.. aaudy.~tainkal .

(:;'130)

EfDEE~~r,k.....

~')(a).w.p.~,w

(5.127)

A ItolljugAit i YOl''''G-Ia-rrI..k ..lav.iM ,,1IOmk "" .-.IlOI,,' rl .. ll.." trAlis ,,['Illl-l...k a f"\ram''!.r I."rl-...n II {O... OJ n'.. fd ..l m..g. v3g}'i.~ dOl" ... o-~ } "" e(O., .. Il). AT. R.. ti' rl>Oct A B L-c.o po rt . lIllytoll"". Yllgyn. 1lZ .. . b Iuindm n' ki">ruy..-oto. 'oo ~ k "'C U .. kiJrIl)"'7K", tl"'i ..klwr folytol_ Vlloi6oo fiiggWoIlyt>i ""' ..... II() .... J p.&clLlllrtell.'iIwk,r.~y tlllajo.ioI~ i .lUnrtnt LIE-,"""p"rtUM IIf'Vf"ZUnI. ha iR.......k A:I: a1ahbill.lr.:

D,= E_ ,;,X,_Oi,:Jl ( i -=o 1. 2. 3), ab... E a baromdim..w:i6ti ~.,..-gmAt rixot j ,,!i,]j . lIZ 1 19y"'" alabbi matrixok:

mAtrix ,,\:2 ) ordOjW\k.

x, ~ (~ ~1 -~) X2 = (~I ~ ~) ' Xl "" (~ -~ ~) 0 0 lIO 000~k II.Z S0{3) Lrg-c-oport intillitl-~illWi. J:.- ,,",A~onuMk ..,;y 1.1E -aooport..J.-mci --.k A

(5. IJSJ

:\.,;.

.........,

1,= .,:.(a!>. . O ~ . 3, ... ,,1..) (I- I. .. , n). tt.. I32) G....li !ll'.hrUl tot IllCl':bat/u'm;lAk ........1>r.... r..j d M'to.-k A "",,,u4Woel: fiiggvkiy ....j~nAk (lNod (a 133)) . , fiKY"l.-m.... v+uoVOWI. c(,o) = 0(0 ) . fl(d) """"",Uol.-m ho ",,,ii, in.......... ..l.-m " , J>IlTNl.eWck-.-(i == I . ..) .. kOv..tkrWlwt bpj1lk:('; ,134)

.b a l. a-,. $J ~l\lIl""",,k h.-I\lMl a ' furgatN!i ,... ""' a knordiwOtareudszPr kn,Mpontj&n It.t rrwOO. ho /I ~eDVlyiriIn)...... mUlAtO Ii ~ ktor at. "2. OJ knm l>Ollt:ll"""il ill mrgNihA.uuk voIna II. fotgatMi nooport ",km",inl'k meghataro.. wd-ljat,..,.:

O(R )2.

= .,(. ,.y,.... x._.y.jgg.. I. A z ElLERf{>"" oziig IIlkalmll1.NU.1I 1 kOvt-t I-zoir.i>ppm k;'p""_hrt6 "": D(R) '" e~x'eu' e"~ ' . (3.H 2)5.9.

a 1.11i:-Y f NAAV{>ny f>1gg lIZ i ~ (r" rJ _ III 1",IY""ktortOl, f .. f{X), &kk"t lIZ i krdinii.ta t''''I\t.,,"ny &< X I"'nl.Mll "II:YlUlllZt J'(/, /ortik.t vt.... i r.-[mint /lZ f f;iggv{>ny" 0;' f hdyl'll. ~ &kkm "z infilliw7.imii.li_ v"luY,AIl" iJ/ih. ,liff""'Il:L1",:.r2- - 1 3 - .

a

a

iJ.l 1

iJ:.eJ

L 2 "' r .1-

iJ f):.el

- < ~",rt"k k"'l""lik fllktol"N\r>:rlllAl.....t.i>kra: A : H \ { O } .q \ { O } " ~ 1I111v ..\I'l.ivda. (,\(O}~por-tjAi. B : ." pA......rgVf>tkoWl rn.H>lOponjai V>Ulnak: E = {E}. C I - [e.p.]. V 2 - {t.I'l}. L'J = {t.,Pl}. L't = (tPt PsI. SJ. A nem l.n\-Wi~ "'-"""pnrtok V,. U2. l, ,-.,. Uj ..ikhln1>oak. minlhogy "" ""'lI"'k lW\Ul" luil" k'll """l_ ben ~m UlIYlUUlkknr 5J hl"m riklik" A triviaiio nom.aLo.oztoko" ltivw .... 5:l oooportll"k..-k Al: l'. ~oort ttnnniUo-.t.'>JIl. AltaWli\ll ~ G r.;opor1llak miutko ulY"J' U r""zow'port AITlt"I)r p jlil = IGI/2 oo~t1ljll. ." !OZimIDf'lnk,,,, S.. ""'I,rt.!>t'-'" ~>Ortjai""t pPfmutAci6 '"""f".rtuk DAIr. 1IP_,lk. D: Az 'I " " lui'r'1.,l r"luJAri~ GL(n) tru p pzeSl:'k1. 1. Homomorlizmu!lOk .... izomo rflzm u""k C90porthomnmornzlTlull AlllphTlt.i ~trllkt,'uit,k kim'tl. ",,,,,iib,,. _,Irllkl',ral.arto"k-k{>p"''''''''k., f"K" llk '~ni: Uogyf'n.. k (G,. oj ..... (Gl.o) """1",rtok. l::gy h: G, - G~ kk"'p'",i1P7i'l< ,,'G 2 Uf'''lTaJi-~ c J,.mro.. VIA AI .. h k-ko.'J'f'Z"'- tnA&JAI a1kotj&. A h m"ftia G, norlllAbztoja. 2. Csoportiromornz m us ."mnIu"'lhm /I h OO''i,,>rtbnmnmorfiun'''' m;upurt ...... D 3 ilii-twmooport "IlV~ \ZI.>UlOff..k.l"1Idj;ik 6';'; &Z eg)'f"nlOokWu Mn)ll~..u.: [ ~)O t irjAk ko.

s..

2.

C",y le )'

~ SL(f1) DI'"~t.-,jA" GL(f1) . lI1i~ aa SO(n) ~ j A a:l'. O(,,} o;nportMk. E : A Iwmpk-x "It'llArioo u";'luxok l"[[IIlltN-lo""'1>r1Ia1. F~' G. " ,....'>UI"t....,..zaI" P perm ,TiIriOr. UlPU..lros7t.AhAill'" h..J.mlU'-lI lIlZ

1. Dd inid ok bo

{A. 0 A) .... (8, 0B ) n;nparrok . A~ A. x B DESCARTES-

'V ' milvcl.;II't. un-lvre

aSobS = rwkn. ord A "A ~,j ' : =

'"'2.Vi

B = "1OITIliI.k-wn.. YOMlknzi> fakl.or uooJl'll"l" jAllak ",vpziink. jPlol....... GIS. A7. ..J4hhi Ihpl o..v.('f'-'I'\I':'-"'I ad m'l!: l t (I,ol11 ",n",.f) kkl'I"''7'"'''' '' (val"" Vouport ok Eg:. ll&OIt'>l komlwolli..ue mond"llk, li>i R 1_;unl"tll\l1 va.!ltnlr!'''.j". E~' f,;I"lQ'",,'nl LIE-lilli"'" rAllalr "in"""n AIlELfr,l, illV>lTiJtlls rt'",~a1I-:,'hli .ja. A kOlllpakt. LJE-''''''!'Oltok '1~y"zf>nkk t. Cashllir-op.. rfllor :\ C CAsIMrn-OI'f>I',;,t","Iran "IIt'rMof, ItJu"ly ruind"n gen ..rat.olTlt] ,..,. i~' a LII-;-"Ig"hra mind"11 ,.]nni. vel kommllt"L F.:? "g'lIWJi!.tnkh"n ..!tY kVI"ll'alik,,~ aJa k. C=

C

~

Sp/2n.Q

A TJA

j r...j

(_~. ~. )=

C(G) = ~p(2n.C)

milldplI D

E q/(2n.C},aJ,uI D Tj

JD

':'IV~n\'rt ..I"", \(Ll:a!l:t..", '1tY D( G) 1,'k(.I-""'""""r t'Wli'n lUI I PrtJ'kct ..-i rpl. H" .. = / (_}-lJf"n." hd yp t...(9.900.J

.... /\hol U Il2 .r = !. Y = 1/ pontban "" 1 hti-kM ........a fd , lOVAbb4"" U loA" V li.l~y rnf\....,.jreodil l}'ladat o k mE'goldis.inak Oreen-tele mooszl're Tl'gy'lk fE'1. bogy a

z(z.t) = i !f (r - ml) + I(r + ml)1

I

2

'J-[,-~

g(&)

1m .("lm"'- ('-'I') _ 1(" )'dI,("/m'". (. ,.,..m

;/,.,..2t2

t&

'1') ] ~ ~.....

( !:l.~)

XJ2

+eu =1 {9.92a) fh t1ifl.rt,,>oox!"'",1, linpAriK [>Ard hlillt1i!l""'ndAlpgy('nl..tl,,,k .. l ar t "mA y b"'tArfln nW!\"'lntt ('rlA',k.... 1 n

4~

II II~s

ds -

4~

kct f..lVf>Vli m''golda-, Akk"r ,IM1 '" I,Mtmul!.nyG{r, .or parl\mPt..r], A GREES-fliltJl;Vl-ny 1, 1l1"jtlo,,~ai:

v.e. 'I) CREEN-/ ugg,":" yPi: k"""",iik m"l\ (! (.,. 'I

1v ,Grdd 11 d Yz,

(9 92e)

A RI I:: MASN i>fla G R EEsJIlUtlK7~'lb"llaz It kiY hogy p]&rWr a ditf"r..nd4legy.."h,t ''flY sl"'k-ti I'ruhl"'mAlt mpgo1;t 6Iwt ~Olii ~ kOrk 6-6 m..,. .. kiinM'k .. u....rLlMuli'gPvl'l1\Z adolt probIkna

,;)M("')"'"

(9.93d)

POiSSON int.rKrilt bpjl.lk. A ;.1Ohrlt uylUlJIU)k. mint frot. "(T) jl'kti ..oak "kar f>o IrihaV>LIlMrhati> kiiZPg.k. ]>"kl';'lIl, I , t1 A hHI1Am..Ito''rlo-~ (1a..I ;:"lA. okl) '''10' A !lm."".f'\:""i ",!o-. ,u.1.(li.. 1.-..at.. 01,1.). A ...." kta. Ik L" ti~ikM-wo.n iro. miul 1'1. "kO:to'l':"k,;" lli,wk U('lU (01.'0" . tonoo j,,,k f..lisnl n NalllllO k\"lll'lnul"...Iw,ilW""" Iwtnj.-ijn. dilf,nII(~nl.--t..kk...1 tro.kil. lto>:unk. A kvautUlllllll"t"h.aniluo. ......U-hrn """" SaJ' 'lIY m,,~lda- I ..hpt. 2. A lIll'goldiu;nak roly wu"",w k,'ll (;;AA,'ni" "1. ",lnt"kt./'I ("t"hililk1)

"KY

LAPLAC&--lU dilf,....nriAlpgy{'nl...t1 . kik..r""hd.6. v" lO' ILZ illVt' l7. io.. f"clUlll" ajl\l,jAn klll'l li1wm,'g. 6. K ozellt~ m 6 moo"~,,,r ..kf'\: . Kiilou l....!':d kdl t..nni fllUllit ik"" I'll n",nerikll~ modszer ek ko--.rott .'"riVl\.ltj""':nllk ~l"U'k" ... I"t~. ElIil,tikll" n lIZj,.IJl('rrtlf'fl fiiggWnyt lulu Il"m wtaJll1IW.OliIll.II.l-I taghan k'-donbi'rzik.aronlvw IIlNo ho'U1"lt,;n fl'il'''t.... lekk..\. EhbPz ..legpndO. hAM iIolJ\f"l"f'tlrn fiiUvfuyt belyt>ttn.itjiik Mjat lIlf1gll. ~ uyt, bogy Mtott inhOOOtl:"'n k..nyt c "" JUI - r d" + {u d.-filliitlj.&):;I'3)"~ at.ler ivitJtat tartalm....o WltoJu-V< dilfr ....oili'O jf'1I.. mzol< A kV/llillunD1Pdla.nikibAn m indffi nWr~6 IDPfIny~k.-,;y "P"1*tor f..""1 JnPg_ A lP1 At .....wi (9.1W&)-hAn " (9.I03b)--hAn f,.\IPpO ff (~ HlJmdlmua,t) IIUIILTON -oJ",rll.u t (1Noo:1:'211. ,*L) jPk-nu'or 11. wnrJ ~ja. am t .+.a lDO"q,>isi'" a l>OIl1('1>wl "",t lut. ju k. IIIJl(.V 1"" ..... p iff _ ii/- D.. .J.T .= Qw {r.l ) . (9 1UOtll2. A hAromdi m.. nzi 6oldi ffrlzi 6ll t>gy.. n l.. t A hO....,OZlml...... 11I..~m l6M II C knnN>ntnviot,uci>.lo11 !dv.-t1 n~ fugg It.Z ..dot 1 pruhl"m" krzWttiloOlcJ.onhatO ..r6t01. i>o. AltalAl,.llI,uUooJ,,uo.l ~16._~ mmd e harom ~ k,iIOn-k,oollll" Al~. 1 Ebl..... ..z """,hoon .. pIVl:iAlis tliff"""rK'~1M hNnm ~ difl'c""lK'~nIM.,.,. ik ,..""t. .....LrP'iJ, .... ..:. rP'iJ ... ...!.. rP'iJ.01'. d.r'iJ. dy'(z,.r2.zJ,f) '"" i" (z [ . l"t .lJ)r-ift(9.104h)A m:u"Cl'k.- All>t.j>OlAt !wul _" 2..,;.g.,. ho,;,v II. ~ .. d" lJorf,'ltAt' iMt&. _ C' '" V(r,. rt. L3)' &klmr JIZ iUlIr.potok JpIrft.~.~>d6A:1 ;,ootOl fiiggootk-t, SC HR60l"GBJH~llM .. t k~t6 "'(r,. rl. Xl} lrullamlllgg\'fny. A lM'm>otAo:-ioD4ri""SCHROOI NGEIl~h61 (!1 10M) 11.(9. 1().I1o~~ _bPt6 ""j,"' + ~';' ( - F}'''''= O .IIf'lI1rrWivUoztlku.~I! UOCwoI_1.",", (9. loc.h)9.1w.r.1 " . =A. , sinll:. x. !/I = A. "m k. lI !/I.=A .....u k,:. . - Oha.x "" 0., = O~ &hnI A. A,. A. al1and6k. IR.' 'I kidl,dti ... r.....tb-~ ~dU-to>k-k.-t:'I z = O. H"lY 'I = 0 x = G. II = b ....: = c ,...-tio II' t..tjrlb,'u a ""intionl.= - '-, l;= - -. 11:.=-" "'.... nI,..hoi n...... ~ n, pg""" .,.-VnoIi:. Ek knr AZ oWo7l'J1L e>of'tkn 1\ ~Iu> All"llOtAt k """"d.ul f"lln.lJA~ n.) +n,.' + n,' i'wo:7..-g kijlOflhOzOAllapotokbnn uK,\'lUUl.Zt ... cnklu1. \rt-i f.1. Eklwr dftJpJl di/npt>tokr61 ,-"lfmk; M I .11l.f1lh A:tOllOll rnergiaj" a1lapotwl.k V>I.lI ..klwr I-n.. ",,_ dfO:7Ji linrAn-n ffilOt"t]"1I Mjat(ii,uv('nyrk 'rjit.k I." lWICIY "k""k 11gyllJl>\l. "'" ellcr~jllk_Et//.Julgllnk.n, k-PtuffilllZAlnok null" kiv"U'li-vPI mindrn tf/,ist. ~t f"h",bctm'k. H. a 1Iulla ....u,ktot ;,. fr-1Yf'btni'k. Akkor Az IW jl>lPntmlr>U', .. ... VIr) potl'OCW MI.AM.JAt l 1Jln t( .; ).lb. (9.111 .. H lwlwry.,ttnin}" o...:J .A 1nf"gIf>1r16 iLl1llnd6t ...l......,.,l m~ "npk vh1Nrztani. JJ. ha thtii.~I Ilf'm dlto:t.t..t,ia ..16.icl"t. ill. pppeb drllniAljuk 6kfo(; (-1)1Ij"" d''''-(~''-lt f;(co;d)=2ijt( 1_0000 (atn;dy' . . (9. 1130:")~=1 2 ../f;,.(9. 1Ha). ' '"If.(9,1l7b)E... =~k~ =.; r2 .(9 .117.,)_~zl = 0 .. m = ... ('011 = r~I"..,.wj"tw1.I>f'flriMOr"jli LEGENDRE-pol inomot h -WI 19.:.6 iranya ki """ tiilltrtw. pl. ha llh ;:; n lIZ ltlUlYlw.n m... u........ ti'"''' MJ ..,J M. ..,.;'rt II. 4' '" 0"'- ~ 8",(:,; = J!!!"""I"'" {til '" 0, 1, 2.".).A" til k""ntum~7 .AIll"t n"i!J"~u~ h'",otl",.. ., i"' nll-k uevezzu k. 6. A s zogt61 fUgg6 fTH'KokiAsok (!J,Ill,') ht,lmPl",,, a polar- "" az a:lirnntali'l ''l\Y''nl''l('k,'' n....;.~. kdl !I7.on,.l.m:.,.,pO/t = p--i'.12 + (V' - I)"] d y2 d y2 dy H.. l>{:,;~SI"Pi(mlI mcgfddo klaAAzik"!l hArmmlikl... ar.ocillAtor __,,"'""':r~.w =lit.a =J:::~=~.(9.122e):>.agy .. kV>UlI'Im......amok""",Uon .. lrwwt.lImmfY-hallikai~Iot.~ 1I kIN-.ikWl kiYu-p&t.Ckhcz JIDu..9.2..1.. Xemlinearis parcialis diffcrcncialegyenletek. szolitonok 9,2..1..1 . A problema elmCleti fizikai megkozcluescI . A HzolitoD fogalma .-\uohtotolr: - VlIgy......lith hullawok fizikar.1as f'&"t nmllinoWls kl.rK WIC nlO'Z6 ~ mOdon 1obIiz.BJ.1 u.\'lY&i; II lDf"'gftolr16 ..~ valame-Iy tartomAn)Tl'I Y>lll koDcrotrilV>l.. Ez aje1etWg ..k>ford,li: .wliud teeekben. pl. anbarmimilruu-=-:.kbAo. JOSEPHSO:'lr. if; cltolnjrDrk. u-l.i-llu"' .....U a.._l _ O-nak. a.. = 2". a.._ 1 = 0 ftOg::: utE. t} uhol :r 1\ h..tykoordin!l.ta.... t a:7. idokoordim't... Az I"gyt"nl..I.kPl ; utAlI (II:)' vi"dkcdn"k. minth e lithatoltllk volua cgy m,,,,,," , 11:'''''' mintll'gyik Mmliton fnrlllij" {~,,'I..-->x' ""zimpl.otiku> n"1~" ':mmi"it ..nck. Ekkur cla.'ztikll>l kiiklliillhlltMr61IK""' ....llink. Az ut.6bhi ekvival..n...gy J\'-s'l.Olitonn\l'goloI,io;nnh"l""l r"lylUTlAt utAu I - .,. OO- rt" 1Il''g111t """.lmptot, k'L"\I1 k"'t uI!YlI.J\f}lvau "'-'1-:1':'1 neru knJoonru..to ",,~,ht"llh m""11I'k 4t.. A ..roolliew,fr~gr;ny ..kn..k i>o ll" ......I.11i." ', ~ 6( .... }2 + U. a'"0(9. IJOA)k1 tacWID>lUlI" ki>vt-tk..w 1,,mliJlArUol.&jtnW,A Lin""",, rvol,j, i6f, rgyt'llkt.kkPl ..lI"IlIi't!>f'Tl" (9 .123 ). (9 .12-1)&. a (9 .12:') rtf'flllill.Ari!satok. uwgoldasa i..-kt [tf'diK "&y k"t --HitolllDt'gokl&M. A 2..-, = II brlyrt~~ (9. 13t.lIl;~htIl. ~ (9.126 ) ~~ k..pjllk. A (9.129) o16 ka!, '~ p"ltw.: nPI1lh~""'"~L H.. (9 .126l-hAn II. +6uu. ~It ~611" . ""," brI)'tf'f eo. t". A ( 9.:n, Ab rAn) to = I. EY til';' kUfi lIemli~ rif..kt..-aJ ,uhm),; """'lnl...n. &mikor .. t h w l : . ,",,--,,-.~.I. .. ~ npti1cibal:i. lWIikuI" lIZ " ~utath ffi~ "" [ clrkt runlOK ":.r,,~~. :wn I . II. KERR....flc!t.two.ahoIn( E ) = ... + '>lIEI2, ..... "2 '" A1landhkkr. ." "'" oogtllvitalO __ k>tck hidrndiMmikAj..ho.n.llffii" pWttik""5pir4lbrok k-lJ-f\>,AtlP.gOldhaiIf (I.f ) ='blrh!21J(I +~t..u U I'\'OIIiciOs fiiggvi-nyrc \'OWIot&uzO D f..l"", dtlhhi 1 1 2'Jf..."" n . = 2'.,!ii;(I - I'"t ~ ;;ft) (n = 1, 2) jPliil"""kkcll'l kiivctk..-,Jj alak,t:nplkl11i JlIITlUllMl'r jdl" lIl~i; .,.{,., ... \ A tl1l11"'m ......m"l'l tmrkolo ja " - 4{ ...,l*"""gg,'l m,,,wK." hllr, kolo hllJlAm fA:7.iAAcl..---'-gc 2('" A KdV' H "ulit Ollllai (9. 127) ,n.. H hf'" ax (1/) "U1p- -b nt lit'Igi>b61 allO mha.n.ikai ",~l IPllf"Vl"I.ik. M..gforditva. be. IlZ......,jUcillll f'-~y.v. iUJauJO c htfhOI a nuo..."rtfkTf' " Nt, alU .. K;;.... t k..,il iilwr...f1 1AA"'" &II renn;~ = 4K(k)k~. k = I .-tion. 1l7.ll7.. w.), _(~nnb)a ~ II. kOvrt~ llWtu:u(I.l) = l arct g e, ( - otl .(9.111c)pld...... ""-10-59 .2 4 . Abr&.510(9. lol::')(9. 136)Kink-ant lkink_u oliton 11'" 0 -U-p (9 .134. 9. 136)- hnl egy lgek moodll. lnill! (10.3) 8/"..,jA!i.- ......tM. &hoi F llfKl.mPntumaikawtt I ,. v>l]tozO" koVabtM. IWw:w: I .. ( , nem "IIll"IJrn . lIW I""t"I eti'....,u .r(o).y(n ", lLl j.c(P), y( d )} pontok TIlilru"hmk:y(x)oA(a, 0)B(b, o) x...10.2. "l>r3.y{J") = Yo(r ) + ''1(z ) (10.13) aIlol '/(: hdyeu ... iti'olot....1>\.7. f lY] h..lyd",1t:7. (-wi fiiAAO10 .1. "hr...f(f ) :fF(:r,Yo + f'l./i, + fI/)dJ" .(10.14)fii~vi-ny 1 1': azt a kiiwlf'lm"nyt ]>..-lig. Tlli~lPrint I\:'; f ly] fll11k,.jnll;"u y(r)-nll """ j,.o;.rti>kct vcgy..n fd, ... .. kiivf'tkl'l.6 fdtCtl"l vAltja fd:~. ,\.al'anW1.rP Jo.q "" .. f...tmn .. kbt-tt'llPgkisrhh? F - F(y. .v) =2~~_ti-n (10_Zld-h6l llZt Mpjuk. holo' Y oE~y~y"l '"Ez .. .ti lf,," ...ciak..~\:t"lI"' t, = ..ltt'ln.ik:., '-"I >illa.mlO az ,(Q) -=-- A .... Il:I: ,(r) _ B IW'I?Illfplti-tt'1 ~~i-vrI h"t.Voz!la.tolI>ja ".( 1O.2Ih)~G(r, y(.e). y(r)) ,(l O.2tJ}I"'fIl.IlK.f>r ..... olj d.r a j F_ fir {)y' l~y fJy d.r!'tV1\_''''l'',1 ,-a!tmi>\("nt. 'l"cl< int.~iik 1\ ki;w,tk,,.,.liu..."M'fii~ ..... t.:( 10.22,,) JH (r . y{r),!I'(.e).~) = h.. l), u, ""U....."',1 k"l'juk Ill.. g. ,ul A (10 .2.1.) izoperim..t rikll" prol,u. llll\(DR)'""'h,1\:1.1. kal,juk. h01tY1[,,] = [.,II + Jfl,lr "" mill.(1O.2:J1\)H { r, y (.c ) , y' (x ) .~ )= y+ ~.jl-I-,( 1 _( IO.2'J a)566MiV/'.!r ( F,.) "" -2,,// . b ( F,..) -JI.F.hb61 "" kOVltk:ik.11OKY F, ~ - 211.\1 . F,. '"4 ) ...... "".-2ook par"'IlJi1.ert~M1l1",m fiigg. Ahboz . hogy" (lO .;r.1a)bdi inlfyAlPrti'k"I...u.. &Z (a. ,4,) '"" (b. B ) J>Ontokat iWlzckotO gor!,..t61 {'lggjOn. rJ,-, .. v,w",ztoU ]lIU"IUII{ ..,J6A.I lftil1 rWIll. F mk pmitiv (>bo6rPlldii 110'Ragen f..9'J'l't'r.~D"k k..lIl"nni, .. 3...... a 4. ~tm.{,j ban. ILZI\Z u.1,it"'ii!l1i.!u-U...... ( IO,4U) F(r.Y.l'i./-IfJ ) = I,F(I.yi".fJ ) (I' > 0) Mti-t.lu"k. ~tivel a (10,39"') vari4cio.. l'whl';ma ( 10.34) ~1,,~ i.olJ i" .....t...lIrk t,'killlh"to. P7 '" ml";{,.],,lo EULER.-fel" ,Prt difJ"rpndAI"gyI'lllctek a kov('tk....li ...J1lk"t ;;ltik' .211EULER-fel,' diffprrllciaJ"8..\,pul..t " k&""tk"J.dy1+ oyrJy =U:;00.Ez l'gy MltUldll ..gyiittb"t" jlj 1"'gy[I' ,lifJl'l'bH . ~ H (r . ~. i, ~) : Jl + ), J j-l + II T ~V("7~~ ....1. J...u..... . q ,' H r .. tcl;....ill .. ( 10.40 ) f..!tftr!. AZal JJ pw.itiv .. IM5mntI'l 11< w V "lly. l ovAbhA.. lluVPII ,f' Ji +2(t)r (l)dl _ I ( lO.-lJh)fOllDll1f\ ""git>Iag. akkor lIZt kapjuk. bov;y' 1""kjtj,-lk .... rl..&rl"lIcl'l~/1(u~_,~.: )dr d,.(lO.4~)(10.4$)m .....( l O.II"""Il')''Kf'n.lo.u ""k",,....! ,liff........nf'iaIhat-i>.' 1....1igl'arall\'1.,r. A" .. _ Il(r y) filltll,. veny grfLfja '''?'IV. ,,,,(.r . ~) p;rAfja ..Ik"u.. [..!ii[,ot1-f..jkoozl,H lMoi allAM md k tt a varii\ci6A I'roh lfotwi.k lIltti olyan o....z".riiggfloo.,klwl f.a.(I)I3",(,h-illl"'1'(1I6A.)0= f(z)-oj-Af K(z.,M,)d,.(11.2&)"ir) = I(r) ... AJ .,\ ft.-lli-1'6 lllU'g ralok marliink:~ fUwtrkz d1t.euOtOI. t.-bi.t-~ ",""uk. aD'Pl)oPkPt A.va1 ~(11.Gb)K{r , , ).. y) d", (l LIb) iA.=l a. (, )I.: y) dll ,k= I. .... n.o =-f(z) + >. / K(r." :-(y)dy.(I Ue),:>(r)=/{x) +), / K(r,')o?(,11a,. (112g"t '-""""",rhILlL ",,, IlodJl'l. bogy l17.Am,,", ,IJIf"tI"ncW..,;ypnlrt. bd,...rt"" .. wYaA tll.lt ()W p( J') mf'gnl'A .. ih(x)a.(x) dr .(11.78.)IJ:./( (.y(())d{(1 1.;1)Ennek "'" '-gyo.l\gllt """ i~mPrtn('k tekiruve all.a1AbMi.. pgy hoIIlOl':l'll illtegrlo.Wgytonwt lIlf'g"I,l.Uhalmaz4t mil)"'" {{'It'wl..k m..n"tt van nll'!tolda...a. >I.Z iuhornogl'n integrAl"tO""llrt''''k PW7A'1.jiik" (11.4..) inttyrril~nl~j tmn.upontiltjdt: (.r) "" J(z) -'- AfK(y.,r}v{Y)dll.(lUlol.),, {11.4&) inhomog"n integr~~Il"tllf'k"!Y ~d.AM- Ekkor Il>f'g I..h..t mlltntni, bogy),,, trenszpon.iJ.t ~tIW'k j,. !'ajil.thtCk(Z)Iiz ) .,..i,)d,( I L9h)Ai. utOO.OinugrNhAnll:T; !... bjx ". bj ha.ln~ (tI.I3M)-t minr\kM. ~mOrionAlkaJmaz,,,,~~n'ltiik azkApjukNlin~ ~. IJA "" J ( I ) zavarofii,l',gvi-ny ortogmuIb.. A tTAI1lI>:I>n4lt bouiosffi in"'Y~t ugy&DlIZOII ~-hal; tAnO&i> '-""" ~ A . A u-rofiiw+nYn- VODAtku-Ui ortosonAlit4Ai ff'lti'k-I ~kii {I I .7d) n~IJA~...J, de anuB! kouuyebben P1woorid>M.6, tovwbi n'-r... MIK"lyO' 0(>.) =II->.-'II'-).~ I = I _ >.1 _ 21'"4If ..EnnckA ~ket riw.ll'tcz"" clOezOr j,." 0 . Ekkor,;', (z ) = I (r ) ~>. / K (.I",,)J(II)dy .(l l.l lh)toohM11 >'.. .... >.- -2 11'(1_ .\ ) F 2 ' ). eem ....jAtkt..koo u m~&kgyPnk1.IH'k bl A l = ~ . Al = ---:?, " k ilrra 1 _).1_ 1_>.2_'"aA ~t i~ ..~.... .....nnt m.. kif,,~ (11.IO) jnhh oltJaJi"n hfo.lwll tw;1Y'-"t~~ ,i ! ) "; lyPhf'. T~ ,.jbo.,'iJnoc cflj4bOl .. ( I I, tt b) ~l>PD /lZr.ft -V"am, llItfV"Ci6II\;jJtoai>~ ~ra v.'Ito:r.w.juk. K"fljuk:A-krauint~~nlf'trtW80ld4M ... z) = s + ~ [ (At -2)..wz -;- I (I i1_ ), 1_- ),) CUoZ] A"in-.. .,(r) = 1(.1")'" >. .i~..-nkot ........t i'nl'u;)'1 =E:.nnclfogva(k = 1.2 ) ..... .e) = i! . .\2"" -! .~l"sin( r .... r};"i,) dr hom~ in~I.1IJPkVMnAk",,~(z) '"'=1( %) - AfK (z. , )[f(,).. >.i>.' /K (" 'l )J('l ) dr,] d,K (z ,,)!{, )dl/ +..f( ll.lle)K {r. , )K {' .'l)/ ('l }d,dr, .( I U ld)Be-t_ .. K 1 (%. y) "" K (r . y) m KJ{z. , ) - [ban fck-rfolv.....zK (z ,i )K,(i ., )d{ ;rJObt. tovibM .. lk u ,-t>. / K ,(% , )/ {y ) dll + >.' .,;>t(z ) .. kOvetb-zOkPppcn irbaW:A... ,( z) "" I(z) AllAlOg mOOoll,&'1"fK,{z .l/ }! (y) d,(ll. th')1I~r!AMik iA ll~.>., _1ion 1~ x (lii-n x COtiEUliatI >.,~tCn 1~ r( siIu + oou)drGO0f.oltncl, Ami l!. vAlto7AJ'\f .K,(J:II)/(lIl dll +.. + x-fK. (.f.l/ )!(II)dy(1l. lIg)A kiT;'On....g' .. ditf"r'Il!d 1l"'801,1"";;'u..k i"'I'" tiv m..ghet...r07.AAA.r"'. E ....lh"l, kiilldllJV'" '" .1(7) +t,>."fK~(x.y)/(y)dy ,R_I (11.12};;(x ) = I(x) + >. / 1\"(" , y)",(y ) dy ,hatvAnysor, ""( 11.10)sn'm. ;..l Et:MANN-l'Or kOllv"rg'nd"jAt. II.... K(l' , y) ho a 1(" ) fiiggvi'lly korlAto, ",",,,", fenu(a $:.1" $: II,IK( x, y)1 c ua S y Sb ) NoI/( z)1 < '" (a S x S b) ,(11.13a )..kkoraz,\, L >.~ IM(b -o)r...,..II ( i 1.121h..h M~ m.. jor""...... Ell II!V'01lK"lIiai!lOr konVf"l"~. ha-11. 2.3 . Fredholm-fe te m egoldAsi m6dszer, Fred ho lm terclci 11.2.3 .1. Fredholm-fele megoldesi medszc r(11.I 3b) ). A Kinellto JIIego ldas d iszkret iz.iIA. -...1,. (x)=/(x ) ... >, / K (z , ').,;>(II)dyl I>.J < M (bp l il(I Ll:0)(11.130') FIlOHOU.lA& m&-tf",ji, intql;faJegyrnlPlfot ki'>zAriIl~tnn.w,..ITf'I. Joohf>t mcgoIrlIUli. Tegyiilr. fel. bogy '" K (x . lI) a. az / (x ) l1,",,",ya S r S /l. 0 S r S "....,ten folytollU'o. ..... (11.15) ~ben ~ int'"l(JNt a h&l oI. - 'I\. (x .,} _.+ K (q r..);"(J-l] . -(11.1&)(11.1-101.)EIr..hK_ ).; . '"" / ... A reedseer pgyl ,tthat6ibOl kep7.Mt ,ktcmllllAi1lo (1 - >.hKll ) ->.hK I2 ->.hK 2, (1- >'hl\-n)(11.17c). A ( 11. 13,:) korLitra ......10t.. kintl."lt.1 < I . lllO'IUlUo II\(z .11)1 S M '" I A f (r.lI: >. ) =(I_~)w...,l..... n~ a:1.onrn..n! Mlly",I","', 3g.) ;. u. & ... 1lI.~ul,lA-< a krN'lW'tt..,: (I) riiggwuyal"PP'>Iltokhan ("I...('u ("-uk,,,io..k k07"t'lit6.iMI All. A D.(>.) = Uf"lul,,,lt ki.Iigit"'>'OQ.l\lllnkll.1. intut IlIlithat6,,1O (lAArl CRAMER"""hAly. 274, nld.):';0 = 0 ..(.\) """, (x. ),~(>')k -=-1.. , 11.(lI. HI)M e&i jiltffiggWuy..k. A r(r.y: ..\) ~""llIO ...tIM'T"'t .. . h.. D{>') '! O .Ir~b-IIzi .. ~ldAAPXJ>licil clMllit.\Mt:II.""pcl! f (x ,II ;"\ ) vO/'L...u. ligyl\JWdror0:>.)JD(z . It. A)/(, ) Ity .1~ (4ry - ~) -I.'2 J.'_ ) d:E =18' I 1K(x,I IK, (f, lI) dt = O. Enni>lfoR"" d.l K 1(7 , , ) '" mindt-ll tov;,},bi4.r1l-fl-[.r+~1I-5~-~III]..\,. ),2 . Ad", k 1\. (.1 ,1/) m.'tmyi.i>g nullaVlOl rgyffil6; r(Z , lI:'\)I-A-pOllilt int"ll:J"~1rt \ lrt~eo; __ varMiggwnn)-d.)d,. '"~ lI"~_ I (I.r)dr.fK . (r ., ) = K (r.,) d,. -JK(I.t)K~_I{t. ,)dt";{sl = !IlZ) - A j K (y .r)t. (, )d,. "(11.2Ib)(11-2Oh)F..rre a ki>t in~n1rt .... h-vi-tI~k .. kilwtkdfaju in~1rtllrk~ ~ KIll .....j&Ji'rt8u. ~d>u.t'f ~ i:uttt:aIata""".ia (m..J.:>19. old.). Mi..... a:wnban .. baJ. oklali. ~a!ap- furw,,1A IWIllfUOlO".'ll POlltool. az llltrgrliJ jO"Pl'roxiIrIa..i6jahO'~ 1 llafO" ;"'ArnUalappontol k.. na w:lllol.MbII l)('YODui. allli 1I~1iu ~-uktr"Ut1sz.cr rni'I:ctct ""' a ""';',lllolMi idiit. FJ ..jAnla.tot< ..Jlcalma..ahb kvadr.. t,irafonll"J.&b~ 1wJ,iAbOI ""kinw.lk u'-),h K..:l ..2 --+( 1 - ),hK_}wt~n irhali>:(l1:l~)./ J(s)d.r "" QIUI(f) -L ~~J{r.J. -,J(l1.2~)=f .=I(x)d.cA2.(1l, 27a)integrAlt . Hc lynU'Sitsiik a>: int..,;Jand""t "lO' p(x) ponlllJulIllai. _Iy A2. J(x) fiiu;Wnyt toklwu:J inlA'l'pOWja:x. a1appon.p(r)L L,,(z )/ (x ,,), -,(x=Il l- 2m)( 11,27..)A ~.I!I K(r .. ).. .j) h.drlll.,'1rAfonmlu. D~ fiiQ lIZ r vallo>lOt6I.. A fii~'.. 'nY"I ~'fIWlltllrnil>lUl" ; < pont. &tt jrI7.i. ho,.y .. lrmtlrl\l'irAfnrmulM fl1gniMr vl!.lt.ozi:>. L-,&t"Az igy dLkkor l.. nJl""'~'W-Jl tl7-krlIt,,!vt.tP~ {t)=2 .In' d" [( /dt "-W].(11. 2Shj,-l..:lf"ju LEGE:' RE-p()!in()m (1 520, ()lr.'wntuumnu./l IIlllgOt biwnyoo ert..l..",l>cll;o! appro:cimAlj/l.. A ( I I .31/l ) .. If,,jul0 mag f1tsUakitA.""va1 kap juk: 1\(x.y) =1laa"= I,I, =0-O.S7j~ O,Sjj'~A, 10.2;;Ss(, )41/ inlu.n : 3 :..., "" 0.1127,I, = 0,:;,hAJ = 0,.1-44-4, = O.l3Oll7,I. = 0.&373,.~nullAtOi (1 1.1. Abra ).Al = 0.2;;1:'.=t, = 0.96214.K II.,o.ms.1.012111.1\72 '"' 1.2M03./ .. = -0.652;'1, K~ = 2.197'46 .( 11 .3.1)11:11 = XI I - 1.05191. K ,t = 1\'1 = 1.10517. 1\23 = K:r:z = 1.:>50838.A ,.."1. .':I"''';>:' In(XI ) = 0.93797 , .,:>(Z2 ) = O. ':(X3) = -0.931'97 1"I ,11. 1. Ahr...._{I 1.3-4)11.2 .4 .2. M a g-a p proximnclcA K( r. V) Ill"ll;ot h..lyt'-itj,lk a..(z)n.(r)tLr.\~'ktorokat, aklrora:tOlitouw ,.:dx) = r ... ,(x) == s . ~.., x) = I ..r, _O.$. rs = 11Lt.1Ol6013l 8ZAm1] X" . . X R ...u..I'poul,ot. k< . l"lIbaf;i~ny ez intgrll.I"!O"'IlI,t'l ~..v."b ,..;.k\"'n Il . IKlntokhan ('](-gi \.l' ki:E rn>ha.; intow,u,.gy.. nl.-t -v..akl WPgolc1"''''' ... ix) _ abol ~-(O) = O....-(O)'i} - n;;o,l. ...-(1) - 1 fbi. " peldAhan '" pon~'Olj.to.viU\ni ak."\IJuk." prt'>Mpohl1't.at.ilL.....k. S.. kklU i"k,,;.hh ajiLllb"ti> kiilOnf{o]. ,plill""-prol." ii"R\"pn~.'kha.',maJata. pI. 1I""..M>ezot,kpnllin{'i\.til< ;r) (t,i',(r)+III ..."I(r)...... .+szillt"l'voIhllnh""1l m,,;:n]cliL,fiiAAV('lly er....... n inr:""k'lik. ",kkor{ ! - n lx - ~'1hll~'(r ) " ( 11.39&) 'lO""'kt ~ly IDl'gOldAsl...1Lkkor -.ix) + .- (x) ... ..... rnego1dRM" ( I ] ,Y.la.) "'8Y"'nJ,1.lI..k. HOlO' .. honmg(on "'&Y"'nl.-t ~ Ult-~~t meghA.t..Vmh......uk, tl"kintljakWlatjuk.[infoftrio "IY"nJ,.t~,-.L K. ., cO .}~(z))(. '" 1. 2....)l'Zt(11.50)ofha urI .....~Ha -., "lO'!din> ortorlonnAlt fP~tk. kepc.=i:k"(1=1.2, .)J.:,(,) =JK(z.,)o;(z)drfiiggvklyPkM.. H.. "' ....(,) .. bomos"n ~ ~ lI"'8"ldAsa.. A1.a:I fmuAllf""""'.2.Algo ri t m usK(r.yl.,..)o{yldy=O.IlZ(11.:>lb)liZl\kJwr mcp.orozva f'Uf'l)'Pnloott-to,(z) fUggvi>rmy.-l. IlU\jd z-nnt Ln~jJ_ p.l'juk:(lL,k)Ml " bol1>Oln ..gymlpt l.f'ta:Z6~ ......(,) nqoIdaM.orto@t . 1.JllQI'l.-r _rinli diffx(1I.62a)fonnAIil< \citft",""tpjuk:f{Z)=/o.{7.rl"'- r (j =FR t; Plto LM-fN,1. 2....). II" a (1 1.61) '1{)'.. nletnek integral"l{Y''''i.iJNv1)'-,K..r..",,-ik", .c) - J(:I:) +-j-.r.AI--! ............nl....lill;,;(J-A(I~MJhn.).. fK(TY ,..-{y dyILIA . KonHl!lkib t tpusu Vohf'rra(i-Ie integralcgj..c ulctekII.."lIJ...nIM JIW'K"Jda,;;,.t"" I _mt~ '-sr VOLTERRAr.y, )_{Ior-,)0flok- lI1t,')(r.w.".""""'t'J'".,I,p = r(pf"rmAl' fplNlat"k" kOvrtkt.-z6k:110-,.. 1;'....... A kvw!rM"rnformubo ,w,,!'I>OIrtjaiHalr...vuna ..... bP!y7.Pt""lQ .c",>!. /Un, rr joOb okWBnAlr. ............. _ kiolMmil.lW\l. ..... .c _ ~ bdrm. N.U1 ~ az ~ f intl'ro1lJ.hlIlJOll ,~t k'.., ./".......iink Otzo"'" a ".c r alb.!",,",, IS. " Q "k"...I....1 ueform ,lAbtfi~.""rt lu-kJ"('''~"l ..A~.c fii~ .. {l16.:ij ~.!Wf'l.,.>I(P11641.)'"~..uuo.I ki""I!t;."7. k. ""'!P"4>{p ~lRl:poIil':OII r1) po ntok a p,) hclp.ctvektm okkal. rnely poutok>.t .. feliil..t panunNnvollltl" iultk hAlojAn mint ''''lk~pl)llt"kat k"p7lhetiink d . A ( 19 .2::.6) B Ih torhi'k 'Ullt100iAjarl\ a hf,.Urt.ati pout"kho>: h""li7.AtI' nddilnk "10' ft>lii!..t " t l\7.2. KOl>vert lllils A70 egyik "ziu nrp" rl' 7Rrrol ll mA.~i kra v...toatt.t'rp"t hrvjuk k",,,,rrl bb ....t.ml"t:tl~.962~rnAkn'>l itt rovid Attf>kinti",r ..dunk. T~'lik frI. bogy rp""""",",~ .....pont.nogyB-lllA K>I(i=1.2).(19,2T.k)I . OssZf'a 1:.'1 ..kkor E l N" ki'>zOfo kit("Yu. uiiwl a nonnaJW.1.A" wiatt a tiXpli numertkus hlbak19 .8 .2.1 . Bevezetes . h ihatlpusokA II7.iImitll,l;"']>!K1 ",,16 ""alIlolfllm, dvil,'!o: tlKYlUllOlo'.k" !llik, miut az B- 1(19.277,(l9.m)W>ntolllU.iun kivon.il>.a.A n u llak kitor!eM-llf'k .. lk.. r ii Wow LAttnk. bogy .. kril,k'\Il mrlwl..t kM.. ki_liUi!cgcg)'f'Jll6lch.... Ha l..""t.rzt lilY k..niljf,k d. bogy .. milw\.f'tRk""ItY "'" Ol>ZZuk a h.b,.r/",I/rl d",'klj..ladaUillak. EIU"'k lllt'gf,'lrJ,1 inw17. fdad"t..:t. "Il\ikor1( 19.211)I ~rl =Ix-11 :5o.s . 10-'10.(19.272)2.A n u m .. rik1l9 AZam it ftso k al apml1wl f't.. i Mind..n nnnc-riku.... ljMM mpu111.....\t'tf'k >IIlr"""'t4h,,1 tl'v6 nnn ~ h .. t~I""Vlhilil""'fl.il. A IwreItit"'i hiMk loJWi,. tt'r~, ......,. ...: 'VI1r. IlllAmltNl! vil4gitjllk~. ,\: A mbodroku t'g).. n l.. t gy okM: ar ... itT + r = 0 II. b..... c vNiJoo pgyiin.....ti>lr "" D tr' - 4tv ;::: 0 (........ KYOki>li::). Kritik,,,, >1>61 "ll.Ziik..c~ni."'l':bol agy''>lJ u(..-) ILZ a mq~"I,IM. IIm.. l.,. nem ill' y{rn) = Yo' pUlIt"" k"I.(/\Z,, =kOvrllu.tik. T..h1i.t ,+1 ~udj .. !.-gf .lj..hh akkora.. mint 0 ~~u,lJe volt ..... igy "'" djM"" '" . 1u"",!l f'IAAk ~ '''''1''11 ..->hho'-k........hCCK.\""",rilPn f,olt,_ualh.w'>k.( 19 ,28 7h) 119.28i c)'1'(r) - !.(r. ' ),,(1').Ujo!lt....""tt ditf,"",llCiAl-v~~. .t1..td.:h ! I.r ,g ) .., all' probll..ookOril .)"ijtmWo)T fi"1ggvell)"'k ....... uhrutinok ..ljA.r4>ook fnrmilj'b"l1 A PASC Al. ADA. ALGOL 68 .... FORTRAN T7 nyrlvm. A vkW.t"" tartalolD; I . Knmpl...., aritmctika 14. S",jat.;rtl.lwk, ""jA.t...ktoruk 2. Pnlmomok lO'0khcl,..,i 15 . o..tenuiuAwluk 3 . Tr " n......... ndcns egyt'nlM:"k gyrt.ogouali7.Al""=ii. - ii, +2ii.' hill.5. IntrgralA>;6 . K,;- " ...... ~ diff......lldlil..!O'pnl,,(ok w .. i . Pan :i:.:!;" diff"r cndlil.. gy.. nl..t ,'k ~. Nllnll'ri kll.. difft'r" nd A.l...... Il. Int." Krill''K. t'ul..t .. k v18. Li"..... ri.. algrhta19. Statu.ztihU "d..t"k l'gy"?",,,, r..ld"l~U'~.AAA 20. Kun-d A o .,. n'i/;fl\III\!izi.ii, =~liv,'l.l: ,(2-h)'Yo.2+h(19.2x&)21. VCl,tl'> .A lJM mW L" iltnruk I'ul/otUlt rfw.b61 .oJl: L\tSL MATI! LIBRARY AlwAIl'" 1I11\1~nw.ikai rljariwlik,,,, !l7,AmitAAok ......u-ben. A Mathematiu hizonyoo.. ..!o5r,' 1I1l0ghatAr0znu r1_k IIVTint a V~I fiiggwny"bOl ~klist&1 Ad llgJoaniJK.v. mint gNilrus f'ljMMokMl. n. C'ZL'kb61az kti'b-kb61 w.tAruaa Illf1l a mindffikori IIlf'gnIdA~t. Mi..,,1 a r.'lluw.n4It pontolt..-duM IlWln....k F..xpon ~ciAli.. fijAAv;'Il~...k .. G1lOl1l\ll.-mggv'uy,.J,;Hiha-fiiggv"n)~'k8.9.Dr--l-fii!:R'....Y"k K..lvinftl~Pn~ k u.-I -rn~y k (tOrt ",n.hi) W.........tr __ ffI.. ..llil'tiklJ.'< fiiAAYfn....kIIProductISoin 10Sol\'1O"l'O"':tII..tdwl old IJlt1I: nUmIl)"'k""k lIZ ,..j"llIalmazra i1ll,,;~tl.,..'t. .'1.1. altalA""" Ill;urtjok,k. ""jilt""ktllliny". E~" kf'OXirn4riltil.'l~ I.artoma.nvok,, " szAmoI I\lJntt.. k. (-rnkiik mindig 0 Ct< 6. II,. nPk"t m..g"m ..ljiik, ak kor II Mathematin la."AAhhlln \11'0 '1111. ""kkni joh!> "ri"t .. mindfollkori.1', _ i eg""z "n"kPkrpU&)'IUlIlZt "" .......tmh:Iyt. ruiut A:iI uto""" I""kWww kapMijuk a kib6viU'1.IIItN;Ok- f..1M haUrin leivill az intf'grilciOo< inkrv.oJh>mOll todhbi r, pootobt ad!Iatunk u"'C. IDf'lY"'k a prohlPmAtileu.. WtoUl4.nyt "zt1kftik .... pontoMbbrw llZAmolnAk,-I.Di lJeN"ncialegyenif.tf'k n UlIlf'ri k ulI m egoldasa.1e1A.}] ~ (r ,. y,) polItoha km.dit6fflAA'"kIvt k"J-I, T"2 . P olinome gye n lete k nume ri k UII m .. goldlula A ~1"'THEMATICA "l'oliuumok gyi>kpit nll""'rik"""n mf'jl; tlldja batilfly~ni (1;'",,1101 1 ,,11] lIl....iT.AA. 1\.1>,,1 n /I ..,.illnfliwlkivant J>Ollt'lAAIiliti,t Aollj1'l1l "'I{. H" n i" ti-kd d IU'IO'juk.ll-kk" r gi'l'i ponu-Agg...:J "AAnlol, ~lillt, l ull J := NSolve [.l'" ~ 6KiiWwoill'-'"' dilferelK~wk \I"lQ' .WlcrenciAlpgyniPt-rrndK&t>rck llIUlK'rikUIIllll"p;oltLWwAIa Mal ' I>em~tiu lIZ rrcdnWn~..,kM O'gy Interpo h t ingFun cti Qn--PljA:rM fr~va1 Allitj.. d6.lgy I"he--l-ffi.,I VlOJik. ho,;y "'" int~~\" t i ('rti'k "k kd 1\ kovetkezd h..vitlt Irjuk rd .. IllwgAAPgy 0l!67'.?62 - U :.m I }. {r-> 0.1:'067262 + I ,J:':l92 I }. {r-> 1.07432}} .== y!Ol == O.x';OJ sea l OO. lt' [O) _97219.8. S: ,;" ..1ogi pd , h""~naiat,,973solUiI e ....r. kOzOUi 1.aI1.oiillwyban ..djll II ditff'T1"llriAlf'gyOP" tukt beI,-ctl.cllitjiik be, A 1Wwtkn.6 uUroo ~I t.OrthUk 8 Yi-gtil AZ ..rod""'ny a Ma,,", MjAt lM~pontoti~ r + ~kl'Tlil. 11& IllteI']XIhtlngFIUl IllteI']XIbtlDgFIUlCtiolll{O., -IO.} . l}10141 :"" PUUI.t rill f(iKKv"ny..k nUlll e ri kuH k iszamita.a A Maple imllt"INI uLAn rne~i t.,'ln' ~A1 A I>..~ild .... all " ...... lmr"y kilr"" " gy",kr"n ''ll}' ",' rhan j..l..nik meg . _ tl" g r>gy __ n}'f\np"rll.!,orr"I ..Ivit.AI""zt~ ..,1. ev a l f o p..rAtor Dlyen ki{"j,t.....,k """rnht{'k"i, mdy..k "11:,;"z"n altalan"", hd'l,ft"tt VI'lO''' fdhll"lI1131l> Altai d"filliAlt fliggV{'lly"k" I I",taIlJuoznllk. liZ .vu f(bf~)~n) (19 ,2'12)> fIIo1v e (f'q, x ) , _-\.074323739, 1,074323739A Maple """k II h t vale.. gyoktit ,,,It .. h'''g. A eomp ln ()I'"io~"l Ml'jnk n,,g .. komplex gyokiikct ill:> tsolv e(eq,.r.cOIIphx);1 ~. 1;I .tAhlAzat. AL f . oln parlUlCl!opci6icOlllpln aaxsols = PIrgyt'l O'IL kOlllpJ.'x RYiikOt (vlllU '10' po inom 0...:,..". komp t'X gyii k"t) hAl ftro".:z;t,~,"'" l'Z" h"IY"" ..rt>rlm"ny. Itt" harmadik I\rgnm"n'''Ill'' puult-'gr.. vnnatk"w ..el.. ~, &0< utol""" kWRlito ..ljoir.... bool,..; jelOl.....,.4,lq;;..L\hh 'l1O'0kiit mpglllltAlYl\I; (nli\k polinomegyeuletekre} 1I1 eq:= exp( -"""3) - 4 > : " h . ("'l . .r); -lll.IIjd t=ri6. Itt .. dil /~ lllRUUlCfI:" . v&lta. r (0.5): - {z(.5J = .5000000000,r(~)(.5) - .Ola4 II'\604721 akko pi'ldAul kiirhatjuk .. mrp;t! Mivt'l .. prim iti\' ruRg\-eov nnn iNut:rt. rl6orUir-OrIrjllk:> in t (up( - z ' 3). .r '" -2. 2); Ha n}(lOt~irjuk;t.,,-.'d.r>avalt (i.Dt {n p{ -r ~ 3 ) , z= -2..2), 15);Ilkkor ..l'O"drn"n}'lll 1\ 27 1.14 584 169 5583 iort" lwt kapjuk. Itt 10 M~pJe 11 bt":'I,i""tt k ine!itll ..ljArM r..th lW.nA1AAAvaJ 1[, jt>gy pon~al sziunolt Bi7.0 n)'Ol' ......tr kh..n no a m....w...r NJl readlib ( ' avaH /int ' ) : > ' IIYaH li nt' (up( -r~ 2), r-I(M)l) .11)llO, IO,_NCr ule):97620.1,B""rutdlt ..Mrnolo. .. !k. Mindkrt ~ r nIl"IiI"lIrouk .. konyvnhrwt. illrt_jr-Irn ~ ~itltgy pg)'f'nll'trpnd..".:rck algehrai m"l':"I,0 e Ikn.di ~ni. Hiivid idu 1I1.&D e&Y lij..,r ,,\Pnik 1Df'l!:: I O lit II - {I ..- ->-a+Sqrt'IIJ--Ib!}I(20.2b)(20.2lo't ..t. k Il kOvo1.~",," irja ki: .. 1 "2 _ II + 11 2 _ ill '17) I /2 ( _II _ (a 2 - -lb) l/~ (20.31,).\2. crl'gi'p"'" lllAt.-matikai rrndw.......k" ull!Ucrik"" IIlAt.eu1.II.tikA f..lNiatAillAk kM-... 6ti-TP ~ ..Ii,.tok pi-kLi.lll lIZa1Kchrai "KY"U1"'K'"IUII"""'m"l" ~wui ~.,. ....jM "p'rilciOlo rt'll''''''''''',I... \xillr-o -4.4222~}, {x-> -2.1421!5}. {J-> -0.9373!l7}, {.c-> 0.972291},{J-> 3.3t>8U2} , {J-> 5.172J7}}m..rik".. adatl\ink ""nnak. llUnll'rik,," lII ..go]'-IAMt lI:1,,r~tn"lIk. E"'rta:f,(20Ac) NSolve "t""lt""t """~""lj"k. AI20 .1 .2. Bevezetd peldak a legfoutosebb n lkalmezasi teruletekrol 20. 1.2. 1. Kep!(tkt'zelkKipl..tJ:l'ulbpll itt" l'1l: ag" h h ht ..l,",l,.."" m.. t .. m..tikai kifPj"".....k oldal "l/.Y'''l!:Y polinoen. IlSolv,,'''K.v..nl,l. viilt,,"". 1\ lO'kilkrt n jC'KY JlOll~ targy...Ja.4rl' ki",obb v_....ti-rii"k.A Mllple-lx'1l11lwvitd::> hohe(r"6 _ 2 . r"S _ 30 . z"4pri"",mUl. "tho+ 36 .c'3 + 190 . z"2 - 36 *.c - 150 '" O,z);(:.!{lAd)" lrbego'ipont""l'7-Amokat el6irt pont.-.-'lAAN h.....z.nAlhaljllk. A 1't'l1,Isz.'T"k AltlliAhnn ..lore urcghatercwtt 1"'1lll.ro;n.II.tA. Ai t.ol.... utM.ltM IL:I:t jP\nti. hosY tI.Z .. rffimfonyt \f'boogOpontool alakban v.ujuk.20.1.2.3. Grafikus abrazolasokA IcgtOOb IlZimit.6lpo.'ll ... lIll\A.en ttirthtik. Ha t...t;rkliinlr..07.P!' ....AZicY20 .1 .2.4 . ProgramozAs a ~za mft6ge-JWS kOTn.yezt>tbenMindo.vik ~ n lroboottW& \IlUl Am, bog lIpf'ci.iIi5 fclMIatok megoldM&ra ....j&t ~ Irjunk_ Ez ~t AI olyan~. ilomm ..1j4r~k Mh.ow-.nalAsat jeh-nti. mint. cikl......... koclt roIlut.allitAouk. peldaul 00. IF THE~. WHILE. FOR. .lMlWanwg~AdMk.operawruk azop/.t lJ.y"k rnindegyik r.."d"".kh..ly.......tRY irj.lw>lI)'ollllt"hh fiiAAvllly...k,t lu.l'llnk.6.Listllk es halmazok2. SzamokA mat ..matikai 1>rk. mrIy..kkrl. ~ k.SM.m tip"....fo"..jE...RaciooA!.iIo-mukurI nt 8gt1 r R.atioulValO..Kom pk-x....,,=kl tf"tlll6l.-g n.-il ...... l6i:tt.-.~tao 1I'@;;a1f1..amBevit..1 nnn ....w,....ee-pluInteger /Int eger &!aku ~hctnI..n tot1 pootl;" "fl:Y lrif..~ 1~~tAk ~1wt.~Ju..tllh>lwtl u,,1.aj.... Al\lf'W IY"I '-"""n ..lIIJr.hall "" "lluitdt f.ulfoI1ll ul.aolitM mlllAt n>Pg.A vaIOo.. """'"'" ...~Oponto!ut~t.llffiiri-I i.um:,.],.k. han . Ez"zunb!ui 1lUll.1t-n"""'-....n .. tolrs frA..,mod11llk aoak '-"KV c:gyszc"t~ilPn szinouiIn YltJ.to: t'Pover ja. bl TIM"I'" Po~rlb. RIlle: ... ('ir :> t-IIIr:>- t"gy "7.i>koz.Vi~""!l"ljuk 1II"1:" Rille ..... Set f,.upl k'f.di;d,~ kif..jt~;",,k.t! [,, 'KY" " a halol.-lali r kif..jl'7'''' rrt,pk.' p/'1 .IAlll (.gy valtO'w. ;. J0l.l>olnik. A Rule k"H"trukrill "l~wl "", I' KZt jcl,,"ti. 1",!lY a , kif..j"",;.,;hf.n ..Outl"l=r +(a+b)lo..,.z,.,. .IM"...I"I...."". kerfi].Inl- := 7 + yZ/. y _> a + b .IkivaJ.."ztja KZ ..],.(i ,,1..m,1. Hut[l] La s t [Ij kivAIAhZlja M \11"1>;lrrh"z ' '10' liOItit.1. A? "duu nt~',.Anrir .I"uukbUI "KY'''1li: lAhlAzlLla n = j ..... -tI'n; 1,,11:'1:= iabl e :BitlOlll i al [7, I, II. o. 7}llO " t (1:' 1 = {I, 7. 21 3.'). 3~. 21. 7. 1}Inl!:.!1 :-11[[3, -l] 0 1111121_ 631 APart [/' {i l. i2 .. J. {j L ]2 . .. }], u'r"''"lI.kl....uK7. I[[{il. i2, . . J. {j l. j 2, )IlUtI",ftfuo"lmnliHtat "lIil ..III, lI11wly 1\7. I J, ;2. ,..-,rs7i1m{, lil