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eroberogeday lwm pl:ún briBaØabR&tKNitviTüa nig BaNiC¢saRsþ

© 2008

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GñkshkarN_RtYtBinitübec©keTs

elak lwm qun

elak Esn Bisidæ

elak Titü em¨g

elakRsI Tuy rINa

elak RBwm suxnit

elak pl b�unqay

GñkrcnaRkb nig bec©keTskMBüÚT&r

kBaØa lI KuNÑaka

GñkRtYtBinitüGkçaraviruTæ

elak lwm miK:sir

© rkßasiTæi lwm pl:ún 2008


GarmÖkfa

esovePA BiPBsVúItcMnYnBitEdlGñksikßakMBugkanénAkñúgéden¼

xMJúáTánxitxMRsavRCav nig niBnæeLIgkñúgeKalbMNgTukCaÉksar

RsavRCavsRmab´GñksikßaEdlmanbMNgcgéc¼ cg´dwgGMBIsIVút

éncMnYnBit[kanÉtc,as´ . enAkñúgesovePA en¼ánRbmUlpþMúnUv

RbFanlMhat´sIVútcMnYnBitya¨geRcIn nigmanlkçN¼xusEbøk@Kña .

RbFanlMhatńImYy@xMJúáTánxitxMeRCIserIsya¨gsRmitsRmaMgbMputRBm

TaMgeFIVdMeNa¼Rsayya¨gek,a¼k,ayEdlGac[Gñksikßagayylńigqab´

cgcaMGMBIviFIsaRsþeFIVdMeNa¼RsaylMhatńImYy@ . buEnþeTa¼Caya¨gNa

k¾eday kgV¼xatbec©keTs Kruekaslü nigkMhusGkçraviruTæ Rákd

CaekItmaneLIgedayGectnaCaBMuxaneLIy . GaRs&yehtuen¼

xMJúáTCaGñkniBnæ rgćaMTTYlnUvmtiri¼KnÉbbsSabnaBIsMNak´Gñksikßa

kñúgRKb´mCÄdæanedaykþIesamnsßrIkrayCanic©edIm,IEklMGesovePAen¼

[kanÉtmansuRkitüPaBEfmeTot .

xMJúáTCaGñkniBnæsgÇwmfaesovePA BiPBsVúItcMnYnBit nwgnaMelakGñk

eq<a¼eTArkC&yCMn¼kñúgkarsikßa nig karRbLgRbECgnanaCaBMuxaneLIy

sUm[GñksikßaTaMgGs´mansuxPaBlðman®áCJaQøasév nigman

sMNaglðkñúgqakCIvit nig karsikßa !

át´dMbgéf¶TI 18 Ex kumÖ¼ qñaM 2008

GñkniBnæ nig RsavRCav lwm pl:ún


BiPBsIVúténcMnYnBit

© 2008 Lim Phalkun - 1 -

segçbemeronsIVútcMnYnBit

I> sIVúténcMnYnBit

1>sBaØaNsIVút

sIVúténcMnYnBitKWCaGnuKmn_elxEdlkMnt;BIsMNMu IN eTAsMNMu IR .

eKkMnt;sresrsIVútmYyeday )U( n b¤ INnn )U(

Edl )n(fUn .

2>GefrPaBénsIVút

k-sIVútekIn

eKfasIVút )U( n CasIVútekInelI IN kalNaRKb; INn eKman n1n UU

x-sIVútcuH

eKfasIVút )U( n CasIVútcuHelI IN kalNaRKb; INn eKman n1n UU

K-sIVútm:UNUtUn

eKfasIVút )U( n CasIVútm:UNUtUnkalNavaCasIVútekInCanic© b¤ CasIVút

cuHCanic© .




3>sIVútTal ;

k-sIVútTal;elI

eKfasIVút )U( n CasIVútTal;elIkalNamancMnYnBit M EdlbMeBjlkç½xN½Ð

MU:INn n .

x-sIVútTal;eRkam

eKfasIVút )U( n CasIVútTal;eRkamkalNamancMnYnBit m Edl

cMeBaH mU:INn n .

K-sIVútTal;

eKfasIVút )U( n CasIVútTal;kalNavaCasIVút Tal;elIpg nigTal;eRkampg

4>sIVútxYb

eKfasIVút )U( n CasIVútxYbEdlmanxYbesµI p kalNa

cMeBaH *INp,UU:INn npn

.

II> sIVútnBVnþ

1>niymn½y

sIVútnBVnþKWCasIVúténcMnYnBitEdlmantYnImYy² ¬eRkABItYTImYy¦esµInwgtYmun

bnÞab;bUknwgcMnYnefrmYy ehAfaplsgrYm . ebI )U( n CasIVútnBVnþman

plsgrYm d nigtYTImYy 0U enaHeK)an ³ INn,dUU n1n

.



2>tYTIn énsIVútnBVnþ ³

k> kñúgsMNMu IN tYTI n kMnt;eday d.nUU 0n .

x> kñúgsMNMu *IN tYTI n kMnt;eday d).1n(UU 1n .

3>plbUktYénsIVútnBVnþ

k> kñúgsMNMu IN plbUktYénsIVútkMnt;eday ³

2

)UU)(1n(U...UUUS n0

n210n

.

x> kñúgsMNMu *IN plbUktYénsIVútkMnt;eday ³

2

)UU(nU...UUUS 21

n321n

.

III> sIVútFrNImaRt

1>niymn½y

sIVútFrNImaRtKWCasIVúténcMnYnBitEdlmantYnImYy² ¬eRkABItYTImYy¦esµInwg

tYmunbnÞab;KuNnwgcMnYnefrmYyxusBIsUnü ehAfaersugénsIVútFrNImaRt.

ebI )U( n CasIVútFrNImaRt manersug 0q nigtYTImYy 0U enaHeK)an ³

INn,U.qU n1n .

2>tYTIn énsIVútFrNImaRt ³

k> kñúgsMNMu IN tYTI n kMnt;eday n

0n q.UU .

x> kñúgsMNMu *IN tYTI n kMnt;eday 1n1n q.UU

.



3>plbUktYénsIVútFrNImaRt

k> kñúgsMNMu IN plbUktYénsIVútkMnt;eday ³

1q,q1

q1.UU...UUUS

1n

0n210n

.

x> kñúgsMNMu *IN plbUktYénsIVútkMnt;eday ³

1q,q1

q1.UU...UUUS

n

1n321n

.

kMnt;sMKal; nig lkçN³

kñúgkarKNnaplbUk b¤ plKuNénsIVúteKGackMnt;sBaØamYy

sRmab;tagdUcxageRkam ³

-cMeBaHplbUk

n

okkn210n )U(U....UUUS .

-cMeBaHplKuN

n

0kkn210n )U(U......UUUP .

lkçN³énplbUk nig plKuNsVúIt ³

k>

n

0kk

n

0kk U.U. x>

n

0kk

n

0kk

n

0kkk VUVU

K>

n

0kk

1nn

0kk U.U. X>

n

0kk

n

0kk

n

0kkk VUV.U

g>

0V,

V

U

V

U n

0kkn

0kk

n

0kkn

0k k

k

.



RbFanlMhat´

1-eK[ )u( n CasIVúténcMnYnBitkMntélI IN kMntéday

TMnak´TMng ½

a

1au 0 nig 2uu 2

n1n Edl 1a .

cUrRsaybBa¢ak´fa n

n

2

2n

a

1au ?

2- eK[sIVúténcMnYnBit )( nu kMntéday

21

1u

2

1u

21u

n1n

0

k-tag 2 nn uv . bgHajfa )( nv CasIVútFrNImaRt .

x-KNna nv nig nu CaGnuKmn_én n .

3- eK[sIVúténcMnYnBit )u( n kMntéday ½

5u 0 nig 5nu2

1u n1n

( INn )

cUrKNna nu énsIVút )u( n CaGnuKmn_én n .


10u 0 nig 1n2n3

2u

3

1u 2

n1n

( INn )





1u 0 nig nn1n 3)3n(u2u

( INn )



1u 0 nig )u81u24

1(

8

1u nn1n

k¿ tag nn u81v . bgHajfa )2v(2

1v n1n

x¿ bgHajfasIVút 2vw nn CasIVútFrNImaRt rYcKNna

nw nig nv CaGnuKmn_én n .

K¿ TajrktY nu én n rYcKNnalImIt nn

ulim

.

7- eKmansIVúténcMnYnBit )u( n kMntéday ½

3u 0 nig n

2

n

3

n1n u3u3uu

tag 1uv nn .

bgHajfa 3

n1n vv rYcKNna nv nig nu CaGnuKmn_én n .

8- eKmansIVút )u( n kMntéday

)1n(n3

)1n2(2u

3

1u

2u

n1n

1

k¿ tag n

2uv:*INn nn . bgHajfa )v( n CasIVútFrNImaRt

x¿ KNna nv rYcTajrktémøén nu CaGnuKmn_én n .




)1n(n

3).2n7(u

3

2u

3

11u

1n

n1n

1

k¿ tag n

3uv:*INn

n

nn . bgHajfa )v( n CasIVútFrNImaRt



)1eln(u

0unu

1n

1

k¿ tag nun ev:*INn . rkRbePTénsIVút )v( n .



n

nn u

uu

u

4

2

1

1

1

k¿ tag 2

1:*

n

nn u

uvINn . bgHajfa )v( n CasIVútFrNImaRt



13123

129

4

2

3

1

0

nn

nnn

uu

uuu

u

k¿ tag 3

1:

n

nn u

uvINn . bgHajfa 3

1 nn vv




13- eK[sIVúténcMnYnkMupøic )Z( n kMntéday ½

INn;|Z|Z

2

1Z

2

3i1Z

nn1n

0

( |Z| n CamUDulén nZ ) .

snµtfa INn,)sin.i(cosZ nnnn

Edl IR;,0 nnn

k-cUrrkTMnak´TMngrvag n nig 1n ehIy n nig 1n .

x-rkRbePTénsIVút )( n rYcKNna n CaGnuKmn_én n .

K-cUrbgHajfa2

cos....2

cos2

coscos 1n2100n

rYcbBa¢ak´ n GnuKmn_én n .


nnn uuu

uu

3

2

3

43

4,1

12

10

k¿ tag nnn uuvINn 1: . bgHajfa )v( n CasIVútFrNImaRt

rYcKNna nv CaGnuKmn_én n .

x¿ KNna n210n v...vvvS CaGnuKmn_én n

rYcKNna nnS

lim .



K¿ Tajrktémøén nu nig nnu

lim .

15- k-KNnalImItxageRkam ½

2x

2x22.....222limL

2xn

( man n raDIkal´ )

x-tag n321n L....LLLS . cUrKNna nnSlim

16- eK[GnuKmn_ 2xx.....xx)x(f n

man n ¦skaer .

cUrKNnalImIt 2x

2)x(flimL n

2xn


nnn nnuu

u

2).1(

12

2

1

1

cUrKNna nu CaGnuKmn_én n .

18- eKmansIVút )u( n nig )( nv kMntéday

nnn

nnn

vuv

vuu

vu

4

3

4

14

1

4

3

1;3

1

1

00

k¿ tag nnn vuwINn : . bgHajfa )( nw CasIVútFrNImaRt

rYcKNna nw CaGnuKmn_én n .

x¿ bgHajfa nnn vuC CasIVútefrEdlRtUvkMnt´.

K¿ Tajrk nu nig nv CaGnuKmn_én n .



19- eK[sIVúténcMnYnBit INnn )U( eday ½

*INn,

5U6U2

2U6U5U

3U

n

3

n

2

n

3

n1n

1

k-eKtag *INn,1U2

2UV

n

nn

. cUrbgHajfa 3

n1n VV

x-eKyk *INn,VlnW nn .

cUrbgHajfa INn,n )W( CasIVútFrNImaRt rYcKNna nW

CaGnuKmn_én n .

K-cUrTajrk nV nig nU CaGnuKmn_énn rYcKNna nnUlim

.

20- eK[GnuKmn_ f kMntélI IR eday 7x9x3

6x6x)x(f

2

3

k-cUrKNnaedrIev )x('f rYcbgHajfaeKGacsresr 2 2

2 2

3(x 3x 2)f '(x)

(3x 9x 7)

.

x-tag nig Ca¦srbssmIkar 0)x('f .

eKyk )U(fU n1n

nig

n

nn U

UV cMeBa¼RKb´ *INn .

RsaybBa¢akfa 3

n1n VV rYcTajrk nV CaGnuKmn_énn nigx.

K-cUrKNna .........)x(ff........ff)x(F nn .



21-eK[sVúIténcMnYnBit )( nU kMntélI n eday½

10U nig aaUUINn nn sincos:

1 Edl

2a0

.

k¿ tag 2

aUV nn cot . cUrbgHajfa )( nV CasIVútFrNImaRtmYYy

x¿ KNnalImIt )....(lim nnVVV

10

nig nnU

lim .

22- eK[sIVúténcMnYnBit )( nU kMntéday ½

1010 UU ; nig nnn UaUUINn

cos:

122 Edl IRa .

k¿ tag INnUaiaUZ nnn

,)sin(cos1

.

cUrbgHajfa nn ZaiaZ )sin(cos 1

rYcTajrk nZ CaGnuKmn_nniga .

x¿ Tajrk nU CaGnuKmn_én n rYcTajrk naU

0lim .

23- eK[sIVút )u( n éncMnYnBitkMntélI *IN eday ½

)2n)(1n(n

1u n

.

k¿kMntćMnYnBit C,B,A Edl 2n

C

1n

B

n

A

)2n)(1n(n

1

x¿ tag n321n u....uuuS . KNna nnSlim

.

K¿ cMeBa¼RKb *INn eKtag

1n

nnn dx).x(guV Edl g

CaGnuKmn_kMntéday )2x)(1x(x

1)x(g



nigeKtag n321n V...VVV'S .

cUrbgHajfa dx).x(gS'S1n

1nn

ehIyTajrk n'S nig nn'Slim

24- eK[cMnYn

)n2(

444......444A

)1n(

222.....222B

nig

)n(

888......888C

cUrbgHajfa 7CBAS CakaerRákd .

25- eK[cMnYn

) elxn2(

111.....111A nig

) elxn(

444.....444B

cUrbgHajfa 1BA CakaerRákd .

26- eK[GnuKmn_ f kMnt´kñúgsMNMu IR eday

)3x3x(3

x)x(f

2

3

k-eKBinitüsIVút )U( n kMntéday )U(fU n1n

cMeBa¼RKb´ *INn .

cUrRsaybBa¢ak´TMnak´TMng 3

n1n U

31

U

31

.

x-eKtag ......)x(ff......ff)x(F nn

( GnuKmn_bNþaklMdab´ n )

cUrKNna )x(Fn .

K-snµtfa 7

3U1 nig nnn Uln)U3ln(V cMeBa¼RKb *INn



bgHajfa )V( n CasIVútFrNImaRtrYcKNna nV CaGnuKmn_én n

X-Tajrk nU CaGnuKmn_én n rYcbBa¢ak´témø nn

Ulim

.

27- eK[GnuKmn_ 2x)x(f 2 Edl IRx

k-eKyk )x(fU1 nig )U(fU n1n

cMeBa¼RKb´ *INn .

cUrbgHajfa .......)x(ff.......ffU nn .

x-RsayfaebI 2x eKán 2U n RKb´ *INn .

K-eKtag 4UUV 2

nnn RKb *INn nig 2x .

cMeBa¼RKb´ *INn cUrbgHajfa 2

n1n VV2 .

X-snµtfa 2lnVlnW nn cMeBa¼RKb´ *INn .

cUrrkRbePTénsIVút nW .

g-eRbIlTæplxagelIcUrTajrkGnuKmn_ ½

.........)x(ff........ff)x(F nn .

28- k-RbsinebI 1p cMeBa¼RKb *INn cUrbgHajfa ½

)1(np1)p1( n .

x-eK[ n321 a......,,a,a,a Ca ncMnYnminGviC¢man .

eKtag n

a....aaaA n321

n

nig n

n321n a......a.a.aG .

bgHajfaRbsinebI kk AG ehIy 0A k



eKán )p1(Aa.G 1k

k1k

k

k

Edl 1A

ap

k

1k .

K-edayeRbIlTæplxagelIcUrRsaybBa¢akfa ½

nnn aaaanaaaa ......... 321321 .

29- eK[

32

11

3

2

2

11

2

11

3

3

43

11

3

2

3

11

3

11

2

11

2

11

3

3

3

3

BI«TahrN_xagelIcUrrkrUbmnþTUeTAnigRsayrUbmnþena¼pg .

30- eK[sIVúténcMnYnBit )u( n kMntéday

INn;

3

2u

3

1u

4u

n1n

0

k¿ eKtag nn u1v . bgHajfa )v( n CasIVútFrNImaRtmYy .

x¿ KNnaplbUk n210n v....vvvS CaGnuKmn_én n

nig nnSlim

.

K¿ cUrKNna nv nig nu CaGnuKmn_én n nig

31- eK[sIVúténcMnYnBit )u( n kMntéday

INn;

2

3u

2

1u

2u

n1n

0

k¿ kMntćMnYnBit k edIm,I[ kuv nn CasIVútFrNImaRtmYy .

x¿ cUrKNna nu CaGnuKmn_én n nig nnulim

.




5u 0 nig 3n2u2

1u n1n

( INn )


33- f CaGnuKmn_kMntédayTMnakTMng ½

110 )()( ff nig INnnnfnfnf ,)()()( 12122

cUrkMnt´rk )(nf CaGnuKmn_én n .

34- eK[ 432 2cos2222,

2cos222,

2cos22

BI«TahrN_xagelIcUrrkrUbmnþTUeTAnigRsaybBaØakrUbmnþena¼pg

35- eK[sIVút nx kMntéday ½

3

4x1 nig ;....4;3;2n,nx22x)1n2( 1n

nn

cUrRsaytamvicaredaykMeNInfa ;...3;2;1n,1i2

)i;n(Cx

n

0in

36- cUrKNnaedrIevTI n énGnuKmn_ xsiney x

37- eK[sVúIténmYycMnYn n321 U........,,U,U,U EdlkMnt;eday ³

bnanU.....UUU 2n321 cMeBaHRKb; *INn

nig IRb,a,0a .

k> cUrRsaybBa¢ak;fasIVútenHCasIVútnBVnþ .

x> cUrKNnatYTImYy 1U nig plsgrYm d CaGnuKmn_én a nig b .



38- eK[sVúItnBVnþ n321 U........,,U,U,U EdlmanplsgrYm d

ehIy 0d eKBinitüsIVút )V( n mYykMnt;cMeBaHRKb; *INn eday

1a,0a,aV nUn .

k> cUrbgðajfa )V( n CasIVútFrNImaRt rYcKNna nV CaGnuKmn_én

d,U,a 1 nig n .

x> cUrRsayfa d

ndU

n321na1

a1.aV....VVVS 1

.

K> cUrKNnaplKuN n321n V.....VVVP CaGnuKmn_én 1U,a nig n .

39- eK[ )U( n CasIVútkMnt;eday baUU n1n cMeBaHRKb; *INn

nig IRb,a,0a .

k> cMeBaH 1a cUrrkRbePTénsIVút )U( n .

x> cMeBaH 1a eK]bmafa kVU nn cMeBaHRKb; *INn .

cUrkMnt;témø k edIm,I[ )V( n CasIVútFrNImaRt .

K> ]bmafa 1a . cUrKNna nU CaGnuKmn_én b,a nig n nigtY 1U .

40- eK[sIVúténcMnYnBit )a( n kMnt;eday

4a1 nig 2a

9a4a:*INn

n

n1n

k> cMeBaHRKb; *INn cUrbgðajfa 3a n .

x> eKyk 3a

1b

nn

. cUrrkRbePTénsIVút )b( n ?



K> cUrkMnt;rtY nb nig na CaGnuKmn_én n .

X> KNnaplbUk )b.a(S kk

n

1kn

.

41- eK[sIVút )U( n kMnt;eday

*INn,nU2U

5U

1nn

1

cUrKNna nU CaGnuKmn_én n rYcTajrklImIt

nn

n 2

Ulim .


INn,1nnU3

1U

1U

2n1n

0

cUrKNna nU CaGnuKmn_én n rYcTajrk

2n

n n

Ulim .


*INn,2)1n(UU

1Un

n1n

1

k> cUrKNna nU CaGnuKmn_én n .

x> KNnalImIt

n1n

n

n UU

Ulim

44- eK]bmafa mS nig nS CaplbUk m tYdMbUg nig n tYdMbUgerogKña

énsVúItnBVnþmYy Edl )nm(;n

m

S

S2

2

n

m .

eKtag mU CatYTI m nig nU CatYTI n .

cUrbgðajfa 1n2

1m2

U

U

n

m

?

45- eK[sIVúténcMnYnBit IN∈nn )U( kMnt;eday

INn,2UU

5U2

n1n

0

k> cUrRsayfa 2Un cMeBaHRKb; INn .



x> eKBinitüsIVút INnn )V( kMnt;edayTMnak;TMng 4UUV 2nnn .

cUrrkTMnak;TMngrvag 1nV nig nV .

K> cUrKNna nV rYcTajrk nU CaGnuKmn_én n .

46- eK[sIVútnBVnþ )U( mantY ........,12,7,2 nigsIVútnBVnþ )V(

mantY .....,8,5,2

etIkñúgcMeNam 151 tYénsIVútTaMgBIrenH manb:unµantYEdlmantémøesµIKña ?

47- eK[sIVúténcMnYnBit )U( n kMnt;eday ³

2U 0 nig RKb; 2

7n2U

2

1U:INn n1n

k> bgðajfaeKGackMnt;BIrcMnYnBit a nig b edIm,I[sIVút INn,)V( n

kMnt;edayTMnak;TMng banUV nn CasIVútFrNImaRt .

x> cUrKNna nU CaGnuKmn_én n .

48- eK[sIVúténcMnYnBit )U( n kMnt;eday 1U 0 nig RKb;

1n4nU2U3:INn 2n1n

k> bgðajfaeKGackMnt;bIcMnYnBit b,a nig c edIm,I[sIVút INn,)V( n

kMnt;edayTMnak;TMng cbnanUV 2nn CasIVútFrNImaRt .


49- eK[sVúItnBVnþ )U( mantY ......17,13,9,5 nigsIVútnBVnþ )V(

mantY ,....25,19,13,7 .



k> etIkñúgcMeNam $( tYénsIVút U manb:unaµntYEdlCakaerR)akd ?

cUrkMnt;rktémøéntYTaMgenaH.

x> etIkñúgcMeNam $( tYénsIVút V manb:unaµntYEdCakaerR)akd ?


K> etIkñúgcMeNam $(tYénsIVútTaMgBIrenHmanb:unµantYEdlmantémøesµIKña ?

50- eK[sIVút 22n )1n(

1

n

11S

Edl *INn .

k-cMeBaHRKb; *INn cUrbgðajfa 1n

1

n

11Sn

.

x-KNnaplbUk 222222n )1n(

1

n

11.....

3

1

2

11

2

1

1

11

51- eK[sIVút 1-n4

1n4U 2

2

n

+= Edl *INn .

k-cMeBaHRKb; *INn cUrbgðajfa 1n2

1

1n2

11U n

x-cUrKNnaplbUk

n

1kn321kn U....UUU)U(S .

52- eK[sIVút 22

n

)3

2n(

1

)3

1n(

11U

cMeBaHRKb; *INn .

cUrKNnaplbUk

n


53-eK[ *INnn ),U(

CasIVútnBVnþmYyEdlmantYTaMgGs;viC¢man

nigplsgrYm 0d . eKtag 2

1n

2

n

2n U

1

U

1

d

1S

.

cUrbgðajfa 1n1

n321 U

1

U

1

d

nS...SSS

.



54- eK[sIVúténcMnYnBit )x( n kMnt;eday ³

1x1 nig 1n2x

2xx

n

n1n

k-cUrRsayfacMeBaHRKb; *INn eKman 1n2x n .

x-KNna 1n2x

1.....

5x

1

3x

1S

n21

n

.

K> TajrklImIt

n

Slim n

n .

55- eKmanGnuKmn_ x2e).1x()x(f Edl x CacMnYnBit .

k> cUrKNnaedrIev x'''f,)x(''f,)x('f nig xf )4( .

x> cUrbgðajfaedrIevTI n énGnuKmn_ f manTRmg; x2

nn

)n( e).bxa()x(f

Edl na nig nb CasIVútcMnYnBitepÞógpÞat;TMnak;TMng ½

*INn,b2ab

a2a

nn1n

n1n

K> cUrkMnt; na nig nb CaGnuKmn_én n rYcTajrkkenSam )x(f )n( .

56- eKmanGnuKmn_ xsin.e)x(f x


x> cUrbgðajfaedrIevTI n énGnuKmn_ f manTRmg; ³

x

nn

)n( e).xcosbxsina()x(f Edl na nig nb CasIVútcMnYnBit



epÞógpÞat;TMnak;TMng

*INn,bab

baa

nn1n

nn1n

K> eKtag nnn b.iaz .

cUrbgðajfa n1n z).i1(z

rYcsresr nz CaTRmg;RtIekaNmaRt .

X> cUrkMnt; na nig nb CaGnuKmn_én n rYcTajrkkenSam )x(f )n( .

57- eK[GnuKmn_ 1x,x1

x)x(f

3 3

cUrKNna ].....])x(f[f[.....ff)x(Fn .

58-eK[sIVúténcMnYnBit INnn )U( eday ½

1U6U6

U3U9U7U 2

n3

n

n2

n3

n1n

nig 2U1 =

k-eKtag *INn,1U2

1UV

n

nn

. cUrbgHajfa 3

n1n VV



CaGnuKmn_én n .

K-cUrTajrk nV nig nU CaGnuKmn_én n rYcKNna nnUlim

.

59- eK[sIVúténcMnYnkMupøic )Z( n kMnt;eday ³

INn,2

i21Z

2

i1Z

iZ

n1n

0



k> eKtag 1ZU:INn nn .

cUrbgðajfa n1n U2

i1U

cMeBaHRKb; INn .

x> cUrsresr nU CaragRtIekaNmaRtrYcTajrk nZ CaGnuKmn_én n .

60- eK[sIVúténcMnYnBit )U( n kMnt;eday

INn,UU.3U

1U,0U

n1n2n

10

k> eKtag INn,U2

i3UZ n1nn

.

cUrRsaybBa¢ak;fa n1n Z2

i3Z

cMeBaHRKb; INn .

x> cUrkMnt;rkTMrg;RtIekaNmaRtén nZ .

K> rYcTajrktY nU CaGnuKmn_én n rYcTajrktémø 2007U .

61- eK[sIVúténcMnYnkMupøic )Z( n kMnt;edayTMnak;TMngxageRkam ³

1Z,0Z 10 nig n1n2n Z2

3i1Z

2

3i3Z

cMeBaHRKb; INn .

k> eKtag n1nn ZZU:INn . bgðajfa n1n U2

3i1U

x> KNna nU nig n210n U....UUUS CaGnuKmn_én n .

K> TajrkcMnYnkMupøic nZ .

62- eK[sIVúténcMnYnBit )U( n kMnt;eday ³

INn,

U2U81

U5U22U

1U

2

nn

2

nn1n

0



k> eKtag 1U

1U2V:INn

n

nn

. cUrbgðajfa 2

n1n VV

x> KNna nV nig nU CaGnuKmn_én n .

63- eK[sIVúténcMnYnBit )U( n kMnt;edayTMnak;TMng ³

2U 0 nig 2

nn

3

n1n U3U64

UU:INn

k> eKtag n

nn U2

UV:INn

. bgðajfa

3

n1n VV

x>KNna nV nig nU CaGnuKmn_én n .

64- eK[sIVúténcMnYnBit 123.....)1n(nUn

k> cUrKNnalImIt )nU(lim nn

.

x> eKtag )UU)(UU(

1V:INn

n

2

1n1n

2

2n

n

nig

n

1kkn VS .

KNna nn

Slim

.

65- eK[sIVúténcMnYnBit )U( n kMnt;eday4

nsin.2U

nn

Edl *INn

k>cUrbgðajfa 4

nsin

4

ncos

4

)1n(cos.2

x> Taj[)anfa 4

)1n(cos)2(

4

ncos)2(U 1nn

n

K> KNnaplbUk n321n U.........UUUS CaGnuKmn_én n

66- cMeBaHRKb; INn eKman nnn )i32(y.ix Edl IRy,x nn

k> cUrkMnt;rk nx nig ny CaGnuKmn_én n .



x> cUrbgðajfa n2nn1n1nn )13(2yxyx

nig 2n21n1nn1nn )13(2yyxx

67- eK[sVúItnBVnþ n321 U........,,U,U,U EdlmanplsgrYm d ehIy

0d . eKBinitüsIVút )V( n mYykMnt;cMeBaHRKb; *INn eday

1a,0a,aV nUn .


d,U,a 1 nig n .

x> cUrRsayfa d

ndU

n321n a1

a1.aV....VVVS 1

.

K> cUrKNnaplKuN n321n V.....VVVP CaGnuKmn_én 1U,a nig n

68- eK[GnuKmn_ 15x8x12

2x12x31)x(f

2

2

k-cUrrkEdnkMnt´énGnuKmn_ )x(f .

x-eKBinitüsIVút )U( n nig )V( n kMntćMeBa¼RKb´ *INn eday ½

)U(fU,)x(fU n1n1 nig

1U4

2UV

n

nn

.

cUrbgHajfa 2

n1n VV

rYcTaj[ánfa 1n2

1n VV

.

K-cUrKNna .........)x(ff........ff)x(F nn

rYcTajrk )1(Flim nn .



69- eK[GnuKmn_ 4

1x

2

1x.....xxx)x(f

k-cUrkMnt´bNþatémø INx edIm,I[GnuKmn_ )x(f

mantémøelxCacMnYnKt´ .

x-bgHajfacMeBa¼RKb´ *INp témø )1p(pf 22

CakaerRákd .

70-eK[smIkardWeRkTIbIGBaØatBit ½

01000m300m30mx)14m9m(mxx:)E( 23223

Edl m Caá¨raËm¨Rt .

k-cUrkMnt´témørbs´m edIm,I[smIkaren¼man¦sbIbeg;It

ánCasIVúFrNImaRtmYy .

x-cUreda¼RsaysmIkar )E( cMeBa¼témø m Edlánrk

eXIjxagelI .

71- eK[smIkarGBaØatBit ½

03m9m6m2x)4m3(2xm3x:)E( 2323


k-cUrkMnt´témørbs´ m edIm,I[smIkaren¼man¦sbIbeg;It



ánCasIVútnBVnþmYy .

x-cUreda¼RsaysmIkar )E( cMeBa¼témø m Edlánrk

eXIjxagelI .

72- eK[sIVúténcMnYnBit )U( n kMntédayTMnak´TMng ½

2

2U1 nig

2

U1U n

1n

cMeBa¼RKb´ *INn .

k-cUrbgHajfacMeBa¼RKb´ *INn eKman 1U2

2n .

x-cUrbgHajfa *INn,2

cosU1nn

rYcTajrklImIt nnUlim

.

73-eK[sIVúténcMnYnBit )U( n kMntéday ½

*INn,)U3)(U2)(U1(U1

4

1U

1U

nnnn1n

1

cUrKNna nU CaGnuKmn_én n .

74-eK[sIVúténcMnYnBit )U( n kMntélI INeday ½

INn,U2U

2U

n1n

0

k¿cUrKNna nU CaGnuKmn_én n .

x¿KNnaplKuN n210n U....UUUP .



75-eK[sIVúténcMnYnBit )U( n kMntélI IN eday ½

2

2U0 nig INn,

2

U11U

2

n1n

KNna nU CaGnuKmn_én n .

76-k¿cUrRsayfa )xtan3x3tan(8

1

xtan31

xtan2

3

x¿cUrKNnaplbUk

n

0k

k

2

k

3k

n

3

atan31

3

atan3

S .



lMhatTI1

eK[ )u( n CasIVúténcMnYnBitkMntélI INkMntédayTMnak´TMng

a

1au 0 nig 2uu 2

n1n

Edl 1a .

cUrRsaybBa¢ak´fa n

n

2

2

na

1au ?

dMeNa¼Rsay

RsaybBa¢ak´fa n

n

2

2

na

1au

eKman a

1au 0 Bit

2

222

01 a

1a2)

a

1a(2uu Bit

4

42

2

22

12 a

1a2)

a

1a(2uu Bit

«bmafavaBitdl´tYTI p KW p

p

2

2

pa

1au Bit

eyIgnwgRsayfavaBitdl´tYTI 1p KW 1p

1p

2

2

1pa

1au

Bit

eyIgman 2uu 2p1p Ettamkar«bma p

p

2

2p

a

1au

eyIgán 2)a

1a(u 2

2

21p p

p

2a

12au 1p

1p

2

21p

1p

1p

2

21p

a

1au

Bit. dUcen¼ n

n

2

2n

a

1au .



lMhatTI2

eK[sIVúténcMnYnBit )( nu kMntéday

21

1u

2

1u

21u

n1n

0

k-tag 2 nn uv . bgHajfa )( nv CasIVútFrNImaRt .

x-KNna nv nig nu CaGnuKmn_én n .

dMeNa¼Rsay

k-bgHajfa )( nv CasIVútFrNImaRt ½

eyIgman 2 nn uv

nnn

nn

nn

nn

vuv

uv

uv

uv

2

2)2(

2

1

2122

1

221

1

2

1

2

1

1

1

11

dUcen¼ )( nv CasIVútFrNImaRtmanersug 2

2q .

x-KNna nv nig nu CaGnuKmn_én n

tamrUbmnþ nn qvv 0 eday 1212200 uv

dUcen¼ n

nv

2

2 ehIy 2 nn uv naM[ nn vu 2

dUcen¼ nnu )

2

2(2 .



lMhatTI3

eK[sIVúténcMnYnBit )u( n kMntéday ½

5u 0 nig 5nu2

1u n1n

( INn )


dMeNa¼Rsay

KNna nu énsIVút )u( n CaGnuKmn_én n

eyIgtag banvu nn Edl IRb,a

eyIgán baanvu 1n1n

eday 5nu2

1u n1n

eKán 5n)banv(2

1baanv n1n

(*))b2

1a5(n)1a

2

1(v

2

1v n1n

ebI

0b2

1a5

01a2

1

naM[ 3b;2a

TMnak´TMng * eTACa n1n v2

1v

naM[ )v( n CasIVútFrNImaRtman

ersug2

1q tamrUbmnþ n

0n qvv eday 6n2vbanvu nnn

cMeBa¼ 0n eKán 6vu 00 naM[ 1656uv 00 .

eKán nnn 2

1

2

11v . dUcen¼ 6n2

2

1u

nn .



lMhatTI4


10u 0 nig 1n2n3

2u

3

1u 2

n1n

( INn )


dMeNa¼Rsay


eyIgtag cbnanvu 2nn Edl IRc,b,a

eyIgán cban)ba2(anvu 21n1n

eday 1n2n3

2u

3

1u 2

n1n

eKán

1n2n3

2)cbnanv(

3

1cban)ba2(anv 22

n2

1n

(*)c3

2ba1n)b

3

2a22(n)

3

2a

3

2(v

3

1v 2

n1n

ebI

0c3

2ba1

0b3

2a22

03

2a

3

2

naM[ 9c;6b;1a


1v

naM[ )v( n CasIVútFrNImaRtmanersug3

1q

tamrUbmnþ n0n qvv eday 9n6nvcbnanvu 2

n2

nn



cMeBa¼ 0n eKán 9vu 00 naM[ 19109uv 00

eKán nn 3

1v . dUcen¼ 2

nn )3n(3

1u .

lMhatTI5


1u 0 nig nn1n 3)3n(u2u

( INn )


dMeNa¼Rsay


eyIgtag nnn 3).ban(vu Edl IRb,a

eyIgán 1n1n1n 3)baan(vu

eday n

n1n 3)3n(u2u

eKán nnn

1n1n 3)3n(3)ban(v23)baan(v

(*)3.)ba33(n)a1(v2v nn1n

ebI

0ba33

01a naM[ 0b;1a

TMnak´TMng * eTACa n1n v2v

naM[ )v( n CasIVútFrNImaRtmanersug 2q



tamrUbmnþ n0n qvv eday n

nn

nn 3.nv3)ban(vu

cMeBa¼ 0n eKán 1vu 00 eKán nn 2v .

dUcen¼ nnn 3.n2u .

lMhatTI6


1u 0 nig )u81u24

1(

8

1u nn1n

k¿ tag nn u81v . bgHajfa )2v(2

1v n1n

x¿ bgHajfasIVút 2vw nn CasIVútFrNImaRt rYcKNna nw

nig nv CaGnuKmn_én n .

K¿ TajrktY nu én n rYcKNnalImIt nnulim

.

dMeNa¼Rsay

k¿ bgHajfa )2v(2

1v n1n

eKman nn u81v naM[ 1n1n u81v

Et )u81u24

1(

8

1u nn1n

eKánCabnþbnÞab´dUcxageRkam ½



nn2n

1n

nn1n

nn1n

v2

11

2

u811)

2

u811(v

4

u81

2

u81.21v

u81u24

11v

dUcen¼ )2v(2

1v n1n

.

x¿ bgHajfasIVút 2vw nn CasIVútFrNImaRt ½

eyIgman 2vw nn naM[ 2vw 1n1n

Et )2v(2

1v n1n

eKán nnn1n w2

1)2v(

2

12)2v(

2

1w

.

dUcen¼ )w( n CasIVútFrNImaRtmanersug 2

1q .

tamrUbmnþ n0n qww Et 12u812vw 000

dUcen¼ nn 2

1w nig nn 2

12v .

K¿ TajrktY nu énsIVút

eyIgman nn u81v naM[ 8

1vu

2

nn

3n21n

nn

2

n

n

2

1

2

1

8

38

14

1

2

1.44

8

1)2

12(

u



dUcen¼ 3n21nn 2

1

2

1

8

3u

.

KNnalImIt nn

ulim

eyIgán ]2

1

2

1

8

3[limulim

3n21nnn

n

eday 02

1

2

1lim

3n21nn

dUcen 8

3ulim n

n

.

lMhatTI7

eKmansIVúténcMnYnBit )u( n kMntéday ½

3u 0 nig n

2

n

3

n1n u3u3uu

tag 1uv nn .

bgHajfa 3

n1n vv

rYcKNna nv nig nu CaGnuKmn_én n .

dMeNa¼Rsay

k¿ bgHajfa 3

n1n vv

man 1uv nn

3

n

n

2

n

3

n

1n1n

)1u(

1u3u3u

1uv



dUcen¼ 3

n1n vv

KNna nv nig nu CaGnuKmn_én n ½

eyIgman 3

n1n vv

tag nn vlnw naM[ nn1n1n w3vln3vlnw

eKán )w( n CasIVútFrNImaRtmanersug 3q

nigtYdMbUg 2ln)1uln(vlnw 000

tamrUbmnþ )2ln(2ln3qwwn3nn

0n eday nn vlnw

dUcen¼ n3n 2v nig n3

n 21u .

lMhatTI8

eKmansIVút )u( n kMntéday

)1n(n3

)1n2(2u

3

1u

2u

n1n

1

k¿ tag n

2uv:*INn nn . bgHajfa )v( n CasIVútFrNImaRt


dMeNa¼Rsay

k¿ bgHajfa )v( n CasIVútFrNImaRt

eyIgman n

2uv nn



1n

2uv 1n1n

Et

)1n(n3

)1n2(2u

3

1u n1n

nn1n

n1n

n1n

n1n

v3

1)

n

2u(

3

1v

)1n(n3

)1n(2u

3

1v

)1n(n3

n62n4u

3

1v

1n

2

)1n(n3

)1n2(2u

3

1v

dUcen¼ )v( n CasIVútFrNImaRtmanersug 3

1q nigtY 42uv 11

x¿ KNna nv nig nu CaGnuKmn_én n

eyIgán 1n

1n1n 3

4q.vv

ehIy

n

2

3

4

n

2vu

1nnn

dUcen¼ n

2

3

4u;

3

4v

1nn1nn

.



lMhatTI9


)1n(n

3).2n7(u

3

2u

3

11u

1n

n1n

1

k¿ tag n

3uv:*INn

n

nn . bgHajfa )v( n CasIVútFrNImaRt


dMeNa¼Rsay


eyIgman n

3uv

n

nn

1n

3uv

1n

1n1n

Et )1n(n

3).2n7(u

3

2u

1n

n1n

n 1 n 1

n 1 n

n 1 n 1

n 1 n

n 1 n

n 1 n n n

2 (7n 2)3 3v u

3 n(n 1) n 1

2 (7n 2)3 n.3v u

3 n(n 1)

2 (7n 2 9n).3 2 3 2v u (u ) v

3 n(n 1) 3 n 3


2q nigtY

3

23

3

11v1

x¿ KNna nv nig nu CaGnuKmn_én n

eyIgán n

1n1n 3

2q.vv

ehIy n

3)

3

2(

n

3vu

nn

n

nn

dUcen¼ n

3

3

2u;

3

2v

nn

n

n

n

.



lMhatTI10


)1eln(u

0unu

1n

1

k¿ tag nun ev:*INn . rkRbePTénsIVút )v( n .


dMeNa¼Rsay

k¿ RbePTénsIVút )v( n ½

eyIgman nun ev

eyIgán )1ln(1

1

nu

n eun eev

1

1

1

1

nn

un

vv

ev n

TMnak´TMngen¼bBa¢ak´fa )( nv CasIVútnBVnþmanplsgrYm 1d .

x¿ KNna nv rYcTajrktémøén nu CaGnuKmn_én n

tamrUbmnþ dnvvn )1(1 eday 101

1 eev u nig 1d

eKán nnvn 1).1(1 .

ehIy nun ev naM[ )ln(ln nvu nn

dUcen¼ )ln(; nunv nn .



lMhatTI11


n

nn u

uu

u

4

2

1

1

1

k¿ tag 2

1:*

n

nn u

uvINn . bgHajfa )v( n CasIVútFrNImaRt


dMeNa¼Rsay


eyIgman 2

1

n

nn u

uv naM[

2

1

1

11

n

nn u

uv eday

n

nn u

uu

4

21

eKán nn

n

n

n

n

n

n

n

n vu

u

u

u

u

uu

u

v3

2

2

1.

3

2

63

22

24

2

14

2

1


2q .


tamrUbmnþ 11

nn qvv eday

3

2,

3

2

21

11

2

1

1

11

q

u

uv

dUcen¼ n

nv

3

2 .

müa¨geTot 2

1

n

nn u

uv naM[

nn

nn

n

n

n

nn v

vu

32

32

1)3

2(

1)3

2(2

1

12 1

.



lMhatTI12


13123

129

4

2

3

1

0

nn

nnn

uu

uuu

u

k¿ tag 3

1:

n

nn u

uvINn . bgHajfa 3

1 nn vv


dMeNa¼Rsay

k¿ bgHajfa 31 nn vv

eyIgman 3

1

n

nn u

uv naM[

3

1

1

11

n

nn u

uv

3

23

23

2

3

2

3

1 3

1

27279

133

313123

129

113123

129

n

n

nnn

nnn

nn

nn

nn

nn

n u

u

uuu

uuu

uu

uu

uu

uu

v

dUcen¼ 31 nn vv .


tag nn vw ln eKán nnnnn wvvvw 3ln3lnln 311

naM[ )( nw CasIVútFrNImaRtmanersug 3q nig 3lnln 00 vw

eKán )3ln(3ln3 30

nnnn qww Et nn vw ln eKán n

nv 33 .

ehIy 3

1

n

nn u

uv naM[

33

13

1

133

31

n

n

v

vu n .



lMhatTI13

eK[sIVúténcMnYnkMupøic )Z( n kMntéday

INn;|Z|Z

2

1Z

2

3i1Z

nn1n

0

( |Z| n CamUDulén nZ ) .

snµtfa INn,)sin.i(cosZ nnnn Edl IR;,0 nnn

k-cUrrkTMnak´TMngrvag n nig 1n ehIy n nig 1n .

x-rkRbePTénsIVút )( n rYcKNna n CaGnuKmn_én n .

K-cUrbgHajfa2

cos....2

cos2

coscos 1n2100n

rYbBa¢ak´ n

GnuKmn_én n .

dMeNa¼Rsay

k-rkTMnak´TMngrvag n nig 1n ehIy n nig 1n

eyIgman )sin.i(cosZ nnnn naM[ )sini(cosZ 1n1n1n1n

eday )|Z|Z(2

1Z nn1n

ehIy nn |Z|

eKán nnnn1n1n1n )sin.i(cos2

1)sini(cos

)

2sin.i

2(cos

2cos)sin.i(cos

)sin.icos1(2

1)sin.i(cos

nnnn1n1n1n

nnn1n1n1n

eKTaján 2

cos nn1n

nig

2n

1n



dUcen¼ 2

n1n

nig

2cos n

n1n

.

x-RbePTénsIVút )( n nig KNna n CaGnuKmn_én n ½

tamsRmayxagelIeyIgman n1n 2

1

naM[ n CasIVútFrNImaRt

manersugesµI 2

1q .

tamrUbmnþ n0n q

eday 3

sin.i3

cos2

3i1)sini(cosZ 0000

eKTaján 3

;1 00

dUcen¼ nn 2

1.

3

.

K-bgHajfa 2

cos....2

cos2

coscos n2100n

tamsRmayxagelIeKman 2

cos nn1n

¦

2cos n

n

1n

eKán

1nk

0k

1nk

0k

k

k

1k )2

cos(

2

cos.........2

cos.2

cos.cos 1n210

0

n

dUcen¼ 2

cos....2

cos2

coscos 1n2100n

.

müa¨geToteyIgman 2

cossin22

cos2

sin2sin n1n

nnn

( eRBa¼2

n1n

) .

eKTaj 1n

nn

sin

sin.

2

1

2cos



ehtuen¼ n

0n

n

1n

2

1

1

0nn sin

sin.

2

1

sin

sin.....

sin

sin.

sin

sin.

2

1

dUcen¼ )

2

1.

3sin(

1.

2

3

)2

1.

3sin

3sin

2

1

n

1n

n

nn

.

lMhatTI14


nnn uuu

uu

3

2

3

43

4,1

12

10

k¿ tag nnn uuvINn 1: . bgHajfa )v( n CasIVútFrNImaRt

rYcKNna nv CaGnuKmn_én n .

x¿ KNna nn vvvvS ...210 CaGnuKmn_én n

rYcKNna nnS

lim .


lim .

dMeNa¼Rsay

k¿ bgHajfa )v( n CasIVútFrNImaRt ½

eyIgman nnn uuv 1



nn

nnn

nnn

nnnn

nnn

vv

uuv

uuv

uuuv

uuv

3

1

)(3

13

1

3

13

1

3

4

1

11

11

111

121

dUcen¼ nv CasIVútFrNImaRtmanersug 3

1q nigtY

3

1010 uuv .

tamrUbmnþtYTI n énsIVútFrNImaRt nn qvv 0

dUcen¼ 1

3

1

n

nv .

x¿ KNna nn vvvvS ...210 CaGnuKmn_én n ½

tamrUbmnþ q

qvS

n

n

1

1 1

0

1

1

3

11

2

3

3

11

3

11

.1n

n

nS

dUcen¼

13

11

2

3nnS nig

2

3lim

nn

S .


lim

eyIgman nnn uuv 1



eyIgTaján

01

1

232

121

010

uuS

uuv

uuv

uuv

uuv

nn

nnn

eKán 10 nn Suu eday

13

11

2

3nnS nig 10 u

dUcen¼ 13.2

1

3

5

nnu nig

3

5lim

nn

u

lMhatTI15

k-KNnalImItxageRkam ½

2x

2x22.....222limL

2xn

( man n raDIkal´ )

x-tag n321n L....LLLS . cUrKNna nnSlim

dMeNa¼Rsay

KNnalImItxageRkam ½

eyIgman 2x

2x22.....222limL

2xn

eyIgán 4

1

2x2

1lim

)2x2)(2x(

4)x2(lim

2x

2x2limL

2x2x2x1



212x2x2

2x2x2

4

1L

4

1

2x

2x2lim

2x22

1limL

)2x22)(2x(

2x2lim

2x

2x22limL

eyIgsnµtfavaBitdl´lMdab´TI k KW kk 4

1L

eyIgnwgRsayfavaBitdl´lMdabTI )1k( KW 1k1k 4

1L

eyIgman 2x

2x2.....2222limL

2x1k

KuNPaKyknigPaKEbgnwg 2x2...2222)x(M

eKán )x(M).2x(

4x2....2222limL

2x1k

Bit

1kkkk1k

2x2x1k

4

1

4

1

4

1L

4

1L

)2(M

1L

2x

2x2....222lim

)x(M

1limL

dUcen¼ n2xn 4

1

2x

2x22.....222limL

x-KNna nnSlim

eKman n321n L....LLLS

)

4

11(

3

1

4

11

4

11

.4

1

4

1....

4

1

4

1

4

1

n

n

n32



kaNa n ena¼ 04

1n .

dUcen¼ 3

1Slim nn

.

lMhatTI16

eK[GnuKmn_ 2xx.....xx)x(f n man n ¦skaer .

cUrKNnalImIt 2x

2)x(flimL n

2xn

dMeNa¼Rsay

KNnalImIt 2x

2)x(flimL n

2xn

eyIgán ]2)x(f)[2x(

4])x(f[lim

2x

2)x(flimL

1n

21n

2x

1n

2x1n

eday ]2)x(f[)2x(42xx.....xxx4)]x(f[ n2

1n

eKán ]2)x(f)[2x(

]2)x(f[)2x(limL

1n

n

2x1n

n

1n1n

1n

n

2x1n

2x1n

2x1n

L.2)2(f

1

2)2(f

1L

2x

2)x(flim

2)x(f

1lim

2)x(f

1limL

edayeKman 2222....222)2(f 1n

eKTaj n1n L4

1

4

1L

¦ (*)1LL4 n1n

KuNGg:TaMgBIrén * nwg n4 eKán nn

n1n

1n 4L4L4



eyIgán

1nk

1l

k1nk

1kk

k1k

1k 4L4L4

3

14.44...444L4L4

1n1n32

1nn

eKTaj 1n1n

1n 4.3

1

3

1

4

LL

Et

4

1

2x

22xlimL

2x1

dUcen¼ 1nn 4.3

1

12

7L

.

lMhatTI17


nnn nnuu

u

2).1(

12

2

1

1

cUrKNna nu CaGnuKmn_én n .

dMeNa¼Rsay

KNna nu CaGnuKmn_én n

eyIgman nnn nn

uu2).1(

12 1

1

1122

)1(

122

2).1(

12

11

11

1

nnuu

nnuu

nnuu

nn

nn

nn

nn

nnn

nuu

kkuu

nn

nk

k

nk

kk

kk

k

1122

1

1122

1

)1(

1

)1(

11

1



eKTaj )1

5(2

1)2

11(

2

11 n

un

unnn

dUcen¼ nn n

nu

2.

15 .

lMhatTI18

eKmansIVút )u( n nig )( nv kMntéday

nnn

nnn

vuv

vuu

vu

4

3

4

14

1

4

3

1;3

1

1

00

k¿ tag nnn vuwINn : . bgHajfa )( nw CasIVútFrNImaRt

rYcKNna nw CaGnuKmn_én n .

x¿ bgHajfa nnn vuC CasIVútefrEdlRtUvkMnt´.

K¿ Tajrk nu nig nv CaGnuKmn_én n .

dMeNa¼Rsay

k¿ bgHajfa )( nw CasIVútFrNImaRt ½

eyIgman nnn vuw

nnnn

nnnnn

nnn

wvuw

vuvuw

vuw

2

1)(

2

14

3

4

1

4

1

4

3

1

1

111



dUcen¼ )( nw CasIVútFrNImaRtmanersug 2

1q

nigtY 2000 vuw .

tamrUbmnþ 10 2

1

2

12

nn

nn qww .

x¿ bgHajfa nnn vuC CasIVútefr ½

eyIgán 111 nnn vuC

nnnnnnnn CvuvuvuC 4

3

4

1

4

1

4

31

dUcen¼ nC CasIVútefrehIy 4000 vuCCn .

K¿ Tajrk nu nig nv CaGnuKmn_én n ½

tamsRmayxagelIeKTaján

)2(4

)1(2

11

nn

nnn

vu

vu

bUksmIkarBIren¼eKán 12

142

nnu naM[ nnu

2

12

dksmIkarBIren¼eKán 12

142

nnv naM[ nnv

2

12

dUcen¼ nnu

2

12 nig

nnv2

12 .



lMhatTI19

eK[sIVúténcMnYnBit INnn )U(eday

*, INn

UU

UUU

U

nn

nnn

562

265

3

3

23

1

1

k-eKtag *, INnU

UV

n

nn

12

2 . cUrbgHajfa 3

n1n VV

x-eKyk *INn,VlnW nn . cUrbgHajfa INn,n )W(

CasIVútFrNImaRtrYcKNna nW CaGnuKmn_én n .


.

dMeNa¼Rsay

k- bgHajfa ½ 3

n1n VV

eyIgman *, INnU

UV

n

nn

12

2

eyIgán 12

2

1

1

1

n

nn U

UV eday

562

2653

23

1

nn

nnn

UU

UUU

3

3

3

23

23

1

323

323

1

3

3

3

23

1

12

2

12

2

16128

8126

56241210

10124265

1562

2652

2562

265

n

n

n

n

nnn

nnnn

nnnn

nnnnn

nn

nn

nn

nn

n

U

U

U

U

UUU

UUUV

UUUU

UUUUV

UU

UU

UU

UU

V

)(

)(

)(

dUcen¼ 3

1 nn VV

.



x-bgHajfa INn,n )W( CasIVútFrNImaRt ½

eyIgman nn VlnW naM[ 1n1n VlnW

eday 3

n1n VV

eyIIIIIIIIIIIIIIIIIIIIgán nn

3

n1n W3Vln3VlnW

dUcen¼ INn,n )W( CasIVútFrNImaRtEdlmanersug 3q .

-KNna nW CaGnuKmn_én n ½

tamrUbmnþ 1n1n qWW

eday )ln()ln(lnln5

1

16

23

12

2

1

1

11

U

UVW nig 3q

eyIgán )(ln)()ln(5

133

5

1 11

nnnW .

K-Tajrk nV nig nU CaGnuKmn_én n ½

eyIgman nn VlnW eday )(ln5

13

1

nnW

eyIgán )(lnln5

13

1

nnV naM[

13

5

1

n

nV

müa¨geTot 12

2

n

nn U

UV naM[

25

152

5

121

5

12

21

21

1

1

1

3

3

3

3

n

n

n

n

n

nn V

VU

.

)(

)(

dUcen¼ 1

3

5

1

n

nV nig 25

1521

1

3

3

n

n

nU. .

eyIgman 25

1521

1

3

3

n

n

nU.

eyIgán nnlim U 2

.



lMhatTI20

eK[GnuKmn_ f kMntélI IR eday 7x9x3

6x6x)x(f

2

3

k-cUrKNnaedrIev )x('f rYcbgHajfaeKGacsresr ½

22

22

)7x9x3(

)2x3x(3)x('f

.

x-tag nig Ca¦srbssmIkar 0)x('f .

eKyk )U(fU n1n nig

n

nn U

UV cMeBa¼RKb´ *INn .

cUrRsaybBa¢ak´fa 3

n1n VV rYcTajrk nV CaGnuKmn_én

n nig x .

K-cUrKNna .........)x(ff........ff)x(F nn .

dMeNa¼Rsay

k-KNnaedrIev )x('f ½

22

322

)7x9x3(

)6x6x)(9x6()7x9x3)(6x3()x('f

22

234

22

234

22

3242234

)7x9x3(

)4x12x13x6x(3

)7x9x3(

12x36x39x18x3

)7x9x3(

54x54x9x36x36x642x54x18x21x27x9

dUcen¼ 22

234

)7x9x3(

)4x12x13x6x(3)x('f

.

bgHajfaeKGacsresr 22

22

)7x9x3(

)2x3x(3)x('f



tamrUbmnþ bc2ac2ab2cba)cba( 2222

eKán x12x4x64x9x)2x3x( 232422

¦ 4x12x13x6x)2x3x( 23422

dUcen¼ 22

22

)7x9x3(

)2x3x(3)x('f

.

x-RsaybBa¢ak´fa 3

n1n VV

ebI 0)x('f smmUl 0)2x3x( 22 naM[ 2

a

cx,1x 21

eKán 2,1 .

eday

n

nn U

UV cMeBa¼RKb´ *INn ena¼

2U

1UV

n

nn

eKán 2U

1UV

1n

1n1n

Et 7U9U3

6U6U)U(fU

n

2

n

n

3

nn1n

eyIgán 3

n

n

n

2

n

3

n

n

2

n

3

n

n

2

n

n

3

n

n

2

n

n

3

n

1n 2U

1U

8U12U6U

1U3U3U

27U9U3

6U6U

17U9U3

6U6U

V

dUcen¼ 3

n1n VV

.

- Tajrk nV CaGnuKmn_én n nig x

eyIgtag nn VlnW naM[ nn

3

n1n1n W3Vln3VlnVlnW

naM[ )W( n CasIVútFrNImaRtmanersug 3q nigtY 11 VlnW



eday 3

2

3

2

3

1

11 2x

1x

27x9x3

6x6x

17x9x3

6x6x

2U

1UV

eKTaj 3

1 2x

1xlnW

tamrUbmnþ n33

1n1n1n 2x

1xln

2x

1xln3qWW

Et nn VlnW eKTaj n3

n 2x

1xlnVln

dUcen¼ n3

n 2x

1xV

.

K-KNna .........)x(ff........ff)x(F nn

tag )x(fU1

)U(fU

)U(f.........)x(ff.........ffU

)U(f)x(fffU

)U(f)x(ffU

n1n

1nnn

23

12

eKTaj nnn U.........)x(ff........ff)x(F

tamsRmayxagelIeKman n3

n 2x

1xV

eday 2U

1UV

n

nn

naM[

1V

1V2U

n

nn



eKán

n3

33

3

3

n

)2x()1x(

)2x(1x2

12x

1x

12x

1x2

Un

nn

n

n

dUcen¼ nn

nn

33

33

n

)2x()1x(

)2x()1x(2)x(F

.

lMhatTI21

eK[sVúIténcMnYnBit )( nU kMntélI n eday½

10U nig aaUUINn nn sincos:

1 Edl

20

a .

k¿ tag 2

aUV nn cot . cUrbgHajfa )( nV CasIVútFrNImaRtmYYy

x¿ KNnalImIt )....(lim nnVVV

10

nig nnU

lim .

dMeNa¼Rsay

k¿ bgHajfa )( nV CasIVútFrNImaRtmYYy ½

man 2

aUV nn cot naM[

211

aUV nn cot

Et aaUU nn sincos

1

eKán 2

1

aaaUV nn cotsincos



aV

aa

Uaa

aU

aa

aaaU

a

aaaa

aU

aaaaaU

aaaaU

n

nn

n

n

n

n

cos

cos)cot(coscotcos

sincos;)sin(cotcos

)cos

sincossin(cotcos

)tancossin(cotcos

cotcossincos

22

2211

22

2

1

2

2

222

2

1222

22

2222

22

eday aVV nn cos1

naM[ )( nV CasIVútFrNImaRtmYymanersug

acos nig tY 2

12

00

aaUV cotcot .

x¿ KNnalImIt ½ )....(lim nnVVV

10

nig nnU

lim

eyIgman a

aa

q

qVVVV

nn

n cos

cos).cot(...

1

1

21

1

111

010

eyIgán

a

aaVVV

n

nn

n cos

cos)cot(lim)....(lim

1

1

21

1

10

eday 2

0

a ena¼ 10 acos nig 01

an

ncoslim

dUcen¼ a

a

VVV nn cos

cot)....(lim

1

21

10 .

müa¨geTot 2

aUV nn cot naM[

2

aVU nn cot

eday aa

qVV nnn cos)cot(

21

0



eKán 22

1a

aa

U nn cotcos)cot(

nig 222

1aa

aa

U n

nnncotcotcos)cot(limlim

eRBa¼ 0

an

ncoslim

dUcen¼ 2

aU nn

cotlim

.

lMhatTI22

eK[sIVúténcMnYnBit )( nU kMntéday ½

1010 UU ; nig nnn UaUUINn

cos:

122 Edl IRa .

k¿ tag INnUaiaUZ nnn

,)sin(cos1

.

cUrbgHajfa nn ZaiaZ )sin(cos 1

rYcTajrk nZ CaGnuKmn_

én n nig a .

x¿ Tajrk nU CaGnuKmn_én n rYcTajrk naU

0lim .

dMeNa¼Rsay

k¿ bgHajfa nn ZaiaZ )sin(cos 1

eyIgman nnn UaiaUZ )sin(cos 1

eyIgán 121

nnn UaiaUZ )sin(cos



n

nn

nn

nn

nnn

Uaia

UaiaUaiaaia

UUaia

UUaia

UaiaUaU

)sin(cos

)sin(cos)sin(cos

)sincos

)(sin(cos

)sin(cos

)sin(coscos

1

1

1

112

dUcen¼ nn ZaiaZ )sin(cos 1

.

KNna nZ CaGnuKmn_én n nig a ½

eday nn ZaiaZ )sin(cos 1

naM[ )( nZ CasIVútFrNImaRténcMnYnkMupøic

Edlmanersug aiaq sincos nigtY 1010 UaiaUZ )sin(cos .

tamrUbmnþ )sin()cos()sin(cos nainaaiaqZZ nnn

0

dUcen¼ )sin(.)cos( nainaZn .

x¿ Tajrk nU CaGnuKmn_én n ½

eyIgman )()sin(cos 11 nnn UaiaUZ

nig )()sin(cos 21 nnn UaiaUZ

dksmIkar )(1 nig )(2 Gg:nwgGg:eKán ½

nnn UaiZZ sin2 naM[ ai

ZZU nn

n sin2

Edl 0asin .

eday )sin()cos( nainaZn nig )sin()cos( nainaZn



eKán a

na

ai

nainanainaUn sin

)sin(

sin

)sin()cos()sin()cos(

2

ehIy na

a

na

nan

a

naU

aana

sin)(

)sin(lim

sin

)sin(limlim

000

dUcen¼ nUna

0lim .

lMhatTI23

eK[sIVút )u( n éncMnYnBitkMntélI *IN eday )2n)(1n(n

1u n

k¿ kMntćMnYnBit C,B,A Edl 2n

C

1n

B

n

A

)2n)(1n(n

1

.

x¿ tag n321n u....uuuS . KNna nnSlim

.

K¿ cMeBa¼RKb *INn eKtag

1n

nnn dx).x(guV Edl g


1)x(g

nig n321n V...VVV'S .

cUrbgHajfa dx).x(gS'S1n

1nn

ehIyTajrk n'S nig nn

'Slim

.

dMeNa¼Rsay

k¿ kMntćMnYnBit C,B,A Edl 2n

C

1n

B

n

A

)2n)(1n(n

1

eyIgán n)1n(Cn)2n(B)2n)(1n(A1

smmUl A2n)CB2A3(n)CBA(1 2



eKTaj

0CB2A3

0CBA

1A2

naM[ 2

1C,1B,

2

1A .

x¿ KNna nnSlim

man

n

1kkn321n uu....uuuS

)2n(2

1

)1n(2

1

4

1

)2n(2

1

4

1

)1n(2

1

2

1

)2k(2

1

)1k(2

1

)1k(2

1

k2

1

)2k(2

1

1k

1

k2

1

n

1k

n

1k

n

1k

dUcen¼ 4

1

)2n(2

1

)1n(2

1

4

1limSlimnnn

.

K¿ bgHajfa dx).x(gS'S1n

1

nn

ehIyTajrk n'S nig nn'Slim

cMeBa¼RKb´ *INn eyIgman

1n

nnn dx).x(guV Edl g


1)x(g

nig n321n V...VVV'S

eyIgán

n

1k

n

1k

1n

nnnn dx).x(guV'S

n

k

n

nn dxxgdxxgdxxgu

1

2

1

3

2

1

).(...).().(

dUcen¼ dx).x(gS'S1n

1nn

.



müageToteKman )2x)(1x(x

1)x(g

manlMnaMdUc )2n)(1n(n

1u n

eKán )2x(2

1

1x

1

x2

1)x(g

eday )2n(2

1

)1n(2

1

4

1Sn

( tamsRmayxagelI )

eyIgán dx.)2x(2

1

1x

1

x2

1

)2n(2

1

)1n(2

1

4

1'S

1n

1n

))((ln

))((ln

lnln)ln()ln()ln())((

||ln||ln||ln))((

31

2

212

1

2

3

4

1

232

13

2

121

2

1

212

1

4

1

22

11

2

1

212

12

4

11

1

nn

n

nn

nnnnn

xxxnn

nnn

dUcen¼

)3n)(1n(

2nln

)2n)(1n(2

1

2

3ln

4

1'S n .

nig 2

3ln

4

1'Slim nn

.



lMhatTI24

eK[cMnYn

)2(

444......444n

A

)1(

222.....222

n

B nig

)(

888......888n

C

cUrbgHajfa 7 CBAS CakaerRákd .

dMeNa¼Rsay

bgHajfa 7 CBAS CakaerRákd

eyIgman )110(9

4444......444 2

)2(

n

n

A

)110(9

2222.....222 1

)1(

n

n

B

nig )110(9

8888......888

)(

n

n

C

eyIgán 7)110(9

8)110(

9

2)110(

9

4 12

nnnS

2

2

12

12

3

710.2

9

4910.2810.4

8

4910.810.210.49

63810.8210.2410.4

n

nn

nnn

nnn

S

S

S

S

dUcen¼ 7 CBAS CakaerRákd .



lMhatTI25

eK[cMnYn

) elxn

A2(

111.....111 nig

) elxn

B(

444.....444

cUrbgHajfa 1 BA CakaerRákd .

dMeNa¼Rsay

bgHajfa 1 BA CakaerRákd ½

eyIgman 9

110111.....111

2

2(

n

n

A

) elx

9

)110(4444.....444

(

n

n

B

) elx

eyIgán 19

)110(4

9

1101

2

nn

BA

2

2

2

3

2101

9

410.4101

9

9410.41101

n

nn

nn

BA

BA

BA

dUcen¼ 1 BA CakaerRákd .



lMhatTI26

eK[GnuKmn_ f kMnt´kñúgsMNMu IR eday )3x3x(3

x)x(f

2

3

k-eKBinitüsIVút )U( n kMntéday )U(fU n1n cMeBa¼RKb´ *INn

cUrRsaybBa¢ak´TMnak´TMng 3

n1n U

31

U

31

.

x-eKtag ......)x(ff......ff)x(F nn ( GnuKmn_bNþak´lMdab n )

cUrKNna )x(Fn .

K-eKsnµtfa 7

3U1 nig nnn Uln)U3ln(V cMeBa¼RKb *INn

cUrbgHajfa )V( n CasIVútFrNImaRt rYcKNna nV CaGnuKmn_

én n .

X-Tajrk nU CaGnuKmn_én n rYcbBa¢ak´témø nnUlim

.

dMeNa¼Rsay

k-RsaybBa¢ak´TMnak´TMng 3

n1n U

31

U

31

eyIgman )U(fU n1n eday

)3x3x(3

x)x(f

2

3

eyIgán )1()3U3U(3

UU

n

2

n

3

n1n



müa¨geTot 3)3U3U(3

U3U

n

2

n

3

n1n

2)3U3U(3

3U3U

)3U3U(3

27U27U9U3U

n2

n

3n

1n

n2

n

n2

n3

n1n

EckTMnak´TMng 2 nig 1 Gg:nigGg:eyIgán ½

3

n

3n

1n

1n

U

)3U(

U

3U

¦ 3

n1n U

31

U

31

.

x-KNna )x(Fn ½

eyIgman ......)x(ff......ff)x(F nn ( GnuKmn_bNþak´lMdab n )

cMeBa¼RKb´ *INn eyIgyk )3x3x(3

x)x(fU

2

3

1

)3U3U(3

UUfU

Uf......xff........ffU

Uf)x(fffU

)U(f)x(ffU

n

2

n

3

nn1n

1nn

23

12

dUcen¼eKTaj nn U........xff........ffxF


n1n U

31

U

31

eKán

n1n U

31ln3

U

31ln edaytag

n

n U

31lnT



eKán n1n T3T naM[ )T( n CasIVútFrNImaRtmanersug 3q

nigtY 3

3

2

1

1 x

3xln

x

27x27x91ln

)x(f

31ln

U

31lnT

tamrUbmnþ n33

1n1n

1n x

3xln

x

3xln.3qTT

Et

n

n U

31lnT eKTaj

n3

n x

3x

U

31

naM[

nn

n

n33

3

3n

x3x

x3

1x

3x

3U

.

dUcen¼

nn

n

33

3

n

x3x

x3........xff........ffxF

.

K-bgHajfa )V( n CasIVútFrNImaRt ½

eKman 3U1 nig nnn Uln)U3ln(V cMeBa¼RKb´ *INn

eyIgán

nn

nn U

31ln

U

U3lnV

naM[

1n

1n U

31lnV eday

3

n1n U

31

U

31

eKán n

n

3

n

1n V3U

31ln3

U

31lnV

dUcen¼ nV CasIVútFrNImaRtmanersug 3q



nigtY 8lnU

31lnV

1

1

.

-KNna nV CaGnuKmn_én n ½

tamrUbmnþ 2ln38ln.3qVV n1n1n

1n .

X-Tajrk nU CaGnuKmn_én n rYcbBa¢ak´témø nnUlim

½

eKman

n

n U

31lnV eday

n3n

n 2ln2ln3V

eKTaj n3

n

2U

31 naM[

12

3U n3n

.

dUcen¼ 12

3U n3n

nig 0Ulim nn

.

lMhatTI27

eK[GnuKmn_ 2x)x(f 2 Edl IRx

k-eKyk )x(fU1 nig )U(fU n1n cMeBa¼RKb´ *INn .

cUrbgHajfa .......)x(ff.......ffU nn .

x-RsayfaebI 2x eKán 2U n RKb´ *INn .

K-eKtag 4UUV 2

nnn RKb *INn nig 2x .

cMeBa¼RKb´ *INn cUrbgHajfa 2

n1n VV2 .

X-snµtfa 2lnVlnW nn cMeBa¼RKb´ *INn .



cUrrkRbePTénsIVút nW .

g-eRbIlTæplxagelIcUrTajrkGnuKmn_ ½

.........)x(ff........ff)x(F nn

dMeNa¼Rsay

k-bgHajfa .......)x(ff.......ffU nn

eyIgman )x(fU1 Bit ( tamsmµtikmµ )

)x(ff)U(fU 12 Bit ( eRBa¼ )U(fU n1n

)

)x(fff)U(fU 23 Bit

eyIgsnµtfavaBitdl´tYTI k KW .......)x(ff.......ffU kk Bit

eyIgnwgRsayfavaBitdl´tYTI 1k

KW .......)x(ff.......ffU 1k1k Bit

eyIgman .......xff.......ff......)x(ff........fffUfU 1kkk1k Bit

dUcen¼ .......)x(ff.......ffU nn .

x-RsayfaebI 2x eKán 2U n RKb´ *INn

eyIgman 2U)U(fU 2

nn1n

ebI 2x ena¼ 22x)x(fU 21 ¦ 2U1 Bit

eyIgsnµtfavaBitdl´tYTI k KW 2U k Bit



eyIgnwgRsayfavaBitdl´tYTI 1k KW 2U 1k

Bit

eyIgman 2UU 2

k1k

eday 2U k naM[ 4U 2

k ¦ 2242U 2

k

eKTaj 22UU 2

k1k

Bit .

dUcen¼ ebI 2x eKán 2U n .

K-bgHajfa 2

n1n VV2

eyIgman 4UUV 2

nnn

eyIgán 4UUV 2

1n1n1n

Et 2UU 2

n1n

2

n

22

nn1n

22

n

2

nn

2

n1n

2

nn

2

n1n

2

nn

2

n1n

2

n

4

n

2

n1n

22

n

2

n1n

V4UUV2

)4U(4UU2UV2

4UU2U2V2

4UUUV

U4UUV

4)2U(UV

dUcen¼ 2

n1n VV2

.

X-rkRbePTénsIVút nW .

eKman 2lnVlnW nn cMeBa¼RKb´ *INn



eKán 2lnVlnW 1n1n

eday 2

n1n VV2

¦ 2

VV

2

n1n

nn

2

n1n W22ln2Vln22ln

2

VlnW

dUcen¼ )W( n CasIVútFrNImaRtmanersug 2q .

g-rkGnuKmn_ .........)x(ff........ff)x(F nn

eday .......)x(ff.......ffU nn eKTaján nn U)x(F .

tamsRmayxagelIeyIgman )W( n CasIVútFrNImaRtmanersug

2q .

tamrUbmnþ 1

1n1n

1n W.2qWW

eday

2

Vln2lnVlnW 1

11

Et 4)x(f)x(f4UUV 22

111

2222242

1

2221

)4xx(2

14xx2xx4x2xV

4)2x(2xV

eKán 2

222

1 2

4xxln

4

)4xx(lnW

ehtuen¼ n2

22

21n

n 2

4xxln

2

4xxln.2W

eday

2

Vln2lnVlnW n

nn



eKTaj

n22

n

2

4xxln

2

Vln

¦ n2

2

n 2

4xx2V

müa¨geToteKman 4UUV 2

nnn

eKTaj n

n

n

2

nn V

2V

21

V24V

U

eday n2

2

n 2

4xx2V

eKán n

n

22

22

n

2

4xx

1

2

4xxU

n

n

n n

n

n n n

n

22

22

n 2 22 2

22

2 2 22 2 2

n 22 2

x x 42x x 4

U2 x x 4 x x 4

2 2

x x 42x x 4 x x 4 x x 4

U2 2 2x x 4

4

dUcen¼ nn 2

22

2

n 2

4xx

2

4xx)x(F

.



lMhatTI28

k-RbsinebI 1p cMeBa¼RKb´ *INn cUrbgHajfa ½

)1(np1)p1( n .

x-eK[ n321 a......,,a,a,a Ca ncMnYnminGviC¢man .

eKtag n

a....aaaA n321

n

nig n

n321n a......a.a.aG .

bgHajfaRbsinebI kk AG ehIy 0A k

eKán )p1(Aa.G 1k

k1k

k

k

Edl 1A

ap

k

1k .

K-edayeRbIlTæplxagelIcUrRsaybBa¢akfa ½

nnn aaaanaaaa ......... 321321 .

dMeNa¼Rsay

k-bgHajfa ½ )1(np1)p1( n

tamrUbmnþeTVFajÚtuneKman ½

nnn

22n

1n

0n

n pC.....pCpCC)p1( eday 2

n)1n(C,nC,1C 2

n1n

0n

eKán n2n p.....p2

n)1n(np1)p1(

edaycMeBa¼RKb 1p eKman 0p....p2

n)1n( n2

dUcen¼ )1(np1)p1( n .



x-bgHajfa )p1(Aa.G 1k

k1k

k

k

RbsinebI kk AG ena¼ 1kkk1k

kk aAaG

eday )p1(A)]1A

a(1[A)]Aa(A[AaA 1k

k

k

1k1kkk1kk

kk1k

kk

eRBa¼ 1A

ap

k

1k . dUcen¼ )p1(Aa.G 1k

k1k

k

k

.

K-RsaybBa¢ak´fa ½ nn321n321 a....aaana.....aaa

eyIgman 0aaa2a)aa( 22112

21

naM[ 2121 aa2aa Bit .

snµtfavaBitdl´tYTI k KW kk21k21 a.....aaka.....aa Bit

eyIgnwgRsayfavaBitdl´tYTI 1k KW k1k211k21 a.....aaka.....aa

tamkarsnµteyIgman kk21k21 a.....aaka.....aa

smmUl kk321

k321 a......aaak

a...aaa

¦ kk GA

Edl k

a....aaaA k321

k

nig k

k321k a......a.a.aG .

tamsRmayxagelIeKman )p1(Aa.G 1k

k1k

k

k

eday 1k

1k1kk211kkk Gaa.....a.aaG

eKTaj )p1(AG 1k

k1k1k

Edl 0A,11A

ap k

k

1k .

tamrUbmnþ )1( eKán 1k)1k

p1()p1(



naM[ 1k1k

k

1k

k )1k

p1(A)p1(A

.

eKTaj 1k1k

k

1k

1k )1k

p1(AG

¦ )1k

p1(AG k1k

eday1k

AaA)1

A

a(

1k

11A)

1k

p1(A k1k

k

k

1kkk

1k1kk21k1kkk A

1k

aa....aa

1k

AaAkA

eKTaj 1k

1k

k1k A)p1(AG

¦ k1k211k21 a.....aaka.....aa

Bit

dUcen¼ nn321n321 a....aaana.....aaa .



lMhatTI29

eK[

32

11

3

2

2

11

2

11

3

3

43

11

3

2

3

11

3

11

2

11

2

11

3

3

3

3

BI«TahrN_xagelIcUrrkrUbmnþTUeTAnigRsayrUbmnþena¼pg .

dMeNa¼Rsay

rkrUbmnþTUeTA½

eKman

32

11

3

2

2

11

2

11

3

3

nig

43

11

3

2

3

11

3

11

2

11

2

11

3

3

3

3

tam«TahrN_xagelIen¼eyIgánrUbmnþTUeTAdUcxageRkam ½

)1n(n

11

3

2

n

11

n

11

......

3

11

3

11

2

11

2

11

3

3

3

3

3

3

.

karRsaybBa¢ak´rUbmnþ ½

tag

n

2k3

3

3

3

3

3

3

3

n

k

11

k

11

n

11

n

11

......

3

11

3

11

2

11

2

11

P



eyIgman 1k)1k(

1)1k(k

1k

k

k

1k

1kk

1kk

1k

1k

1k

1k

k

11

k

11

2

2

3

3

3

3

eKán

n

2kn 1k)1k(

1)1k(k

1k

k

k

1kP

)1n(n

11

3

2

)1n(n

11

3

2

)1n(n

1)1n(n

3

212.1

1)1n(n.

1n

2.

n

1

1k)1k(

1)1k(k

1k

k

k

1k n

2k

n

2k

n

2k

dUcen¼

)1n(n

11

3

2

n

11

n

11

......

3

11

3

11

2

11

2

11

3

3

3

3

3

3

.

lMhatTI30

eK[sIVúténcMnYnBit )u( n kMntéday

INn;

3

2u

3

1u

4u

n1n

0

k¿ eKtag nn u1v . bgHajfa )v( n CasIVútFrNImaRtmYy .

x¿ KNnaplbUk n210n v....vvvS CaGnuKmn_én n

nig nnSlim

.

K¿ cUrKNna nv nig nu CaGnuKmn_én n nig



dMeNa¼Rsay

k¿ bgHajfa )v( n CasIVútFrNImaRtmYy ½

eyIgman nn u1v

nn1n

n1n

1n1n

v3

2)u1(

3

2v

3

2u

3

11v

u1v


1q nigtY 5u1v 00

x¿ KNnaplbUk nS CaGnuKmn_én n nig nnSlim

½

eyIgán q1

q1vv....vvvS

1n

0n210n

eday

3

1q

5v0

dUcen¼

1n

n 3

11

2

15S nig

2

15Slim nn

.

K¿ KNna nv nig nu CaGnuKmn_én n nig nnulim

tamrUbmnþ n0n qvv eday

3

1q

5v0

dUcen¼ n

n 3

15v

.

müa¨geTot nn u1v naM[ nn v1u

dUcen¼ n

n 3

151u

nig 1ulim nn

.



lMhatTI31

eK[sIVúténcMnYnBit )u( n kMntéday

INn;

2

3u

2

1u

2u

n1n

0

k¿ kMntćMnYnBit k edIm,I[ kuv nn CasIVútFrNImaRtmYy .

x¿ cUrKNna nu CaGnuKmn_én n nig nnulim

.

dMeNa¼Rsay

k¿ kMntćMnYnBit k edIm,I[ )v( n CasIVútFrNImaRt ½

eyIgman kuv nn

)1(2

3k

2

1v

2

1v

2

3k

2

1)ku(

2

1v

k)2

3u

2

1(v

kuv

n1n

n1n

n1n

1n1n

tamTMnakTMng )1( edIm,I[ )v( n CasIVútFrNImaRtlu¼RtaEt

02

3k

2

1 dUcen¼ 3k .

x¿ KNna nu CaGnuKmn_én n nig nnulim

½

cMeBa¼ 3k eKman n1n v2

1v

naM[ )v( n CasIVútFrNImaRt

manersug 2

1q nigtY 532kuv 00 .



tamrUbmnþ n

n0n 2

15qvv

. eday kuv nn

naM[ nn vku

dUcen¼ n

n 2

153u

nig 3ulim nn

.

lMhatTI32


5u 0 nig 3n2u2

1u n1n

( INn )


dMeNa¼Rsay


eyIgtag banvu nn Edl IRb,a

eyIgán baanvu 1n1n

eday 3n2u2

1u n1n

eKán 3n2)banv(2

1baanv n1n

(*))b2

1a3(n)2a

2

1(v

2

1v n1n

ebI

0b2

1a3

02a2

1

naM[ 14b;4a


1v

naM[ )v( n CasIVútFrNImaRt



manersug2

1q tamrUbmnþ n

0n qvv

eday 14n4vbanvu nnn

cMeBa¼ 0n eKán 14vu 00 naM[ 914514uv 00

eKán nnn 2

9

2

19v .

dUcen¼ 14n42

9u

nn .

lMhatTI33

f CaGnuKmn_kMntédayTMnak´TMng ½

110 )()( ff nig INnnnfnfnf ,)()()( 12122

cUrkMnt´rk )(nf CaGnuKmn_én n .

dMeNa¼Rsay

kMnt´rk )(nf CaGnuKmn_én n ½

tag )()()( nfnfng 1

naM[ )()()( 121 nfnfng

121

121211

112121

nngng

nngnnfnfng

nfnnfnfng

)()(

)()()()(

)()()()(

eyIgán

)()(

)()()(1

0

1

0

121n

k

n

k

kkgkg

2125310 nngng )(...)()( Et 00 )(g



eKán 2nng )( eday )()()( nfnfng 1

eKTaj 21 nnfnf )()(

eyIgán

)()(

)()()(1

0

21

0

1n

k

n

k

kkfkf

6

1210

)()()()(

nnnfnf eday 10 )(f

dUcen¼ 6

6521

6

1211

2 ))(())(()(

nnnnnnnf .

lMhatTI34

eK[ 432 2cos2222,

2cos222,

2cos22

BI«TahrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBaØak´

rUbmnþena¼pg .

dMeNa¼Rsay

rkrUbmnþTUeTA ½

eKman 432 2cos2222,

2cos222,

2cos22

tamlMnaM«TahrN_eyIgGacTajrkrUbmnþTUeTAdUcxageRkam ½

1n

)n(2

cos22.........222

.

RsaybBaØakrUbmnþen¼ ½

eyIgtag

)n(

n 2......222A cMeBa¼RKb´ *INn



eyIgman 21 2cos22A

Bit

eyIg«bmafavaBitdl´tYTI p KW 1p

)p(

p 2cos22......222A

eyIgnwgRsayfavaBitdl´tYTI 1p KW 2p1p 2cos2A

Bit

eyIgman p1p A2A edaytamkar«bma 1pp 2

cos2A

eyIgán 2p2p

2

1p1p 2cos2

2cos4

2cos22A

Bit

dUcen¼ 1n

)n(2

cos22.........222

.

lMhatTI35

eK[sIVút nx kMntéday 3

4x1

nig ;....4;3;2n,nx22x)1n2( 1nn

n

cUrRsaytamvicaredaykMeNInfa ;...3;2;1n,1i2

)i;n(Cx

n

0in

dMeNa¼Rsay

RsaytamvicaredaykMeNInfa ;...3;2;1n,1i2

)i;n(Cx

n

0in

ebI 1n eKman 3

4

3

11

3

)1;1(C

1

)0;1(C

1i2

)i;1(Cx

1

0i1

Bit

( eRBa¼ 1!)11(!1

!1)1;1(C,1

)!01(!0

!1)0;1(C

)

«bmafavaBitdl´tYTI n KW

n

0in 1i2

)i;n(Cx Bit



eyIgnwgRsayfavaBitdl´tYTI )1n( KW

1n

0i1n 1i2

)i;1n(Cx Bit

eyIgman ;....4;3;2n,nx22x)1n2( 1nn

n

eKán 1n

n

n x1n2

n2

1n2

2x

naM[ n

1n

1n x3n2

2n2

3n2

2x

n

0i

1n

1n 1i2

)i;n(C

3n2

2n2

3n2

2x

eday )!in(!i

!n)i,n(C

ehIy )i,n(Ci1n

1n

)i1n()!in(!i

)1n(!n

)!i1n(!i

)!1n()i,1n(C

eKTaj )i;1n(C1n

i1n)i;n(C

eKán

n

0i

1n

1n )i;1n(C)1i2)(1n(

i1n

3n2

)1n(2

3n2

2x

)1()i;1n(C1i2

i1n

3n2

2

3n2

2x

n

0i

1n

1n

eyIgBInitü

n

0i

1n

0i 1)1n(2

)1n;1n(C

1i2

)i,1n(C

1i2

)i;1n(C

n

0i

1n

0i 3n2

1

1i2

)i,1n(C

1i2

)i;1n(C

( eRBa¼ 1)1n;1n(C )

n

0i

1n

0i 3n2

1)i;1n(C

)1i2(2

)3n2(

3n2

2

1i2

)i;1n(C

eday 2

1

1i2

i1n

)1i2(2

)i21()i1n(2

)1i2(2

3n2

n

0i

1n

0i 3n2

1)i;1n(C

2

1

1i2

i1n

3n2

2

1i2

)i;1n(C



3n2

1)i;1n(C

3n2

1)i;1n(C

1i2

i1n

3n2

2

1i2

)i;1n(C n

0i

n

0i

1n

0i

tamrUbmnþeTVFajÚtuneKman ½

1nn

0i

i1n

0i

i1n x)1n,1n(Cx)i;1n(Cx)i;1n(C)x1(

yk 1x eKán 1)i;1n(C2n

0i

1n

( eRBa¼ 1)1n;1n(C )

eKTaj 12)i;1n(C 1nn

0i

eKán 3n2

1)12(

3n2

1)i;1n(C

1i2

i1n

3n2

2

1i2

)i;1n(C 1nn

0i

1n

0i

)2(3n2

2)i;1n(C

1i2

i1n

3n2

2

1i2

)i;1n(C 1nn

0i

1n

0i

tam 1 nig 2 eKTaj

1n

0i1n 1i2

)i;1n(Cx Bit

dUcen¼

n

0in 1i2

)i;n(Cx .

lMhatTI36

cUrKNnaedrIevTI n énGnuKmn_ xsiney x

dMeNa¼Rsay

KNnaedrIevTI n ½

tag xeu nig xsinv

eyIgán x)n(xxx eu....;;e'''u;e''u,e'u

ehIy )2

xsin(xcos'v



)2

nxsin(v

)2

3xsin()

2

2xcos('''v

)2

2xsin()

2xcos(''v

)n(

tamrUbmnþT,ibniceKman

n

0p

p)pn(pn

)n( vuC)v.u(

eyIgán

n

0p

pn

xn

0p

xpn

)n( )2

pxsin(Ce)

2

pxsin(eCy

tag

n

0p

pn

n

0p

pn )

2

pxsin(CB;)

2

pxcos(CA

eKán

n

0p

pn

n

0p

pn )

2

pxsin(Ci)

2

pxcos(CB.iA

pn n i.p ix p 2

n np 0 p 0

p pC cos(x ) i.sin(x ) e . C e (1)

2 2

tamrUbmnþeTVFajÚtuneKán )2(eC)e1(n

0p

2

pi

pn

n2.i

tam 1 nig 2 eyIgTaján n

2.i

x.i e1eB.iA

eday 4.i

2i

e2)4

sin.i4

(cos2)2

2.i

2

2(2i1e1

eKán )

4

nx(i

n4

n.inx.i e)2(e2eB.iA

eKTaj

)4

nxsin()2(B;)

4

nxcos()2(A nn

.

dUcen¼ )4

nxsin(e)2(y xn)n( .



lMhatTI37

eK[sVúIténmYycMnYn n321 U........,,U,U,U EdlkMnt;eday ³

bnanU.....UUU 2n321 cMeBaHRKb; *INn nig IRb,a,0a .

k> cUrRsaybBa¢ak;fasIVútenHCasIVútnBVnþ .

x> cUrKNnatYTImYy 1U nig plsgrYm d CaGnuKmn_én a nig b .

dMeNa¼Rsay

k> tag bnanU....UUUS 2n321n

eyIg)an )1n(b)1n(aUSUU....UUUS 21nn1nn3211n

eKTaj bnanbbnaan2anS)1n(b)1n(aU 22n

21n

b¤ baan2U 1n

nig baan2baa2an2ba)1n(a2U n

eday a2)baan2()baan2(UU n1n efr

TMnak;TngenHbBa¢ak;fa )U( n CasIVútnBVnþ .

x>KNnatYTImYy 1U nig plsgrYm d CaGnuKmn_én a nig b

eyIgman baan2U n ¬ tamsRmayxagelI ¦

cMeBaH 1n eK)an babaa2U1 ehIy a2UUd n1n

dUcenH a2d,baU1 .



lMhatTI38

eK[sVúItnBVnþ n321 U........,,U,U,U EdlmanplsgrYm d ehIy 0d .

eKBinitüsIVút )V( n mYykMnt;cMeBaHRKb; *INn eday TMnak;TMng ³

1a,0a,aV nUn .


d,U,a 1 nig n .

x> cUrRsayfa d

ndU

n321n a1

a1.aV....VVVS 1

.

K> cUrKNnaplKuN n321n V.....VVVP CaGnuKmn_én 1U,a

nig n .

dMeNa¼Rsay

k> bgðajfa )V( n CasIVútFrNImaRt

eyIgman nU

n aV naM[ 1nU

1n aV

eyIg)an dUU

U

U

n

1n aaa

a

V

Vn1n

n

1n

efr ¬ eRBaH dUU n1n

¦

dUcenH )V( n CasIVútFrNImaRtmanersug daq .

KNna nV CaGnuKmn_én d,U,a 1 nig n

tamrUbmnþ 1n

1n qVV eday 1U

1 aV nig daq

eK)an d)1n(Ud)1n(U

n11 aa.aV

.



x>Rsayfa d

ndU

n321n a1

a1.aV....VVVS 1

tamrUbmnþ q1

q1.VV..........VVVS

n

1n321n

eday 1U

1 aV nig daq

dUcenH d

ndU

n321n a1

a1.aV....VVVS 1

.

K>KNnaplKuN n321n V.....VVVP CaGnuKmn_én 1U,a nig n

eyIg)an n321n V.....VVVP

2

)UU(nU....UUU

UUUU

n1n321

n321

aa

a.......a.a.a

dUcenH 2

)UU(n

n

n1

aP

.

lMhatTI39

eK[ )U( n CasIVútkMnt;eday baUU n1n

cMeBaHRKb; *INn

nig IRb,a,0a

k> cMeBaH 1a cUrrkRbePTénsIVút )U( n .

x> cMeBaH 1a eK]bmafa kVU nn cMeBaHRKb; *INn .

cUrkMnt;témø k edIm,I[ )V( n CasIVútFrNImaRt .

K> ]bmafa 1a . cUrKNna nU CaGnuKmn_én b,a nig n nigtY 1U .



dMeNa¼Rsay

k> cMeBaH 1a rkRbePTénsIVút )U( n

cMeBaH 1a eKmanTMnak;TMng bUU n1n

-ebI 0b enaH *INn,UU n1n naM[ )U( n CasIVútefr .

-ebI 0b enaH bUU n1n naM[ )U( n CasIVútnBVnþmanplsgrYm b

x> kMnt;témø k edIm,I[ )V( n CasIVútFrNImaRt

cMeBaH 1a eKman baUU n1n

eKman kVU nn naM[ kVU 1n1n eday baUU n1n

eK)an b)kV(akV n1n

b¤ bk)1a(aVV n1n

edIm,I[ )V( n CasIVútFrNImaRtluHRtaEt 0bk)1a( eKTaj)an

a1

b

1a

bk

.

K> KNna nU CaGnuKmn_én b,a nig nnigtY 1U

tamdMeNaHRsayxagelIeyIgeXIjfacMeBaH 1a eBlEdl a1

bk

enaHsIVút )V( nCasIVútFrNImaRtmanersug aq nigtY

a1

bUkUV 111

tamrUbmnþ 1n

1

1n

1n a).a1

bU(qVV

eday kVU nn

dUcenH a1

ba).

a1

bU(U 1n

1n

.



lMhatTI40

eK[sIVúténcMnYnBit )a( n kMnt;eday 4a 1

nig 2a

9a4a:*INn

n

n1n

k> cMeBaHRKb; *INn cUrbgðajfa 3a n .

x> eKyk 3a

1b

n

n

. cUrrkRbePTénsIVút )b( n ?

K> cUrkMnt;rtY nb nig na CaGnuKmn_én n .


n

1kn

.

dMeNa¼Rsay

k> cMeBaHRKb; *INn bgðajfa 3a n

eyIgman 34a1 Bit

32

7

24

916

2a

9a4a

1

12

Bit

]bmafavaBitdl;tYTI k KW 3a k Bit

eyIgnwgRsayfavaBitdl;tYTI 1k KW 3a 1k Bit .

eyIgman 3a k nig 2a

9a4a

k

k1k

eyIg)an 02a

3a3

2a

9a43a

k

k

k

k1k

eRBaH 3a k



eKTaj)an 3a 1k

Bit .

dUcenH cMeBaHRKb; *INn eK)an 3a n .

x>rkRbePTénsIVút )b( n

eKman 3a

1b

n

n

naM[ 3a

2a

32a

9a41

3a

1b

n

n

n

n1n

1n

eyIg)an 13a

3a

3a

1

3a

2abb

n

n

nn

nn1n

eday 1bb n1n

efrenaH )b( n CasIVútnBVnþmanplsgrYm 1d .

K>kMnt;rtY nb nig na CaGnuKmn_én n

tamrUbmnþ d).1n(bb 1n dUcenH nb n .

mü:ageToteday 3a

1b

n

n

naM[ 3b

1a

n

n

dUcenH 3n

1a n .

KNnalImIt

eyIgman 3n

1a n kalNa n enaH 0

n

1

dUcenH 3alim nn

.


n

1kn

eyIgman 3k

1a k nig kb k naM[ 1k3)3

k

1(kb.a kk



eyIg)an

n

1kn 2

)5n3(n

2

)1n34(n)1n3(........10741k3S

dUcenH 2

)5n3(nSn

.

lMhatTI41

eK[sIVút )U( n kMnt;eday

*INn,nU2U

5U

1nn

1

cUrKNna nU CaGnuKmn_én n rYcTajrklImIt

n

n

n 2

Ulim .

dMeNa¼Rsay

KNna nU CaGnuKmn_én n

eyIgman )1(nU2U 1nn

tag banVU nn naM[ b)1n(aVU 1n1n

Edl IRb,a

TMnak;TMng ¬!¦ Gacsresr nb)1n(aV2banV 1nn

)2(ba2n)1a(V2V

nb2a2an2V2banV

1nn

1nn

tamTMnak;TMng ¬@¦ ebI

0ba2

01a b¤ 2b,1a

enaHeK)an 1nn V2V

naM[ )V( n CasIVútFrNImaRtmanersug 2q .

tamrUbmnþtYTI n énsIVútFrNImaRteyIg)an 1n1n qVV

eday 2nVbanVU nnn eRBaH 2b,1a

naM[eKTaj 521VU 11 b¤ 2V1 .



eK)an n1n

n 222V Et 2nVU nn naM[ 2n2U n

n

dUcenHtYTI n énsIVútKW 2n2U n

n .

TajrklImIt

n

n

n 2

Ulim

eyIg)an 12

2n1lim

2

2n2lim

2

Ulim

nnn

n

nn

n

n

dUcenH 12

Ulim

n

n

n

.

lMhatTI42


INn,1nnU

3

1U

1U

2

n1n

0

cUrKNna nU CaGnuKmn_én n rYcTajrk

2

n

n n

Ulim .

dMeNa¼Rsay


eyIgman )1(1nnU3

1U 2

n1n

ebIsinCaeyIgbegáItsIVút )V( n mYyEdl kMnt;eday ³

INn,)2(cbnanUV 2

nn

ehIy c,b,a CabIcMnYnBit .

eyIg)an c)1n(b)1n(aUV 2

1n1n



)3()cba(n)ba2(anUV 2

1n1n

edayyk ¬!¦ CMnYskñúg ¬#¦ eK)anTMnak;TMng ³

)4()1cba(3n)1ba2(3n)1a(3UV3

)1cba(n)1ba2(n)1a(U3

1V

)cba(n)ba2(an1nnU3

1V

2

n1n

2

n1n

22

n1n

dksmIkar ¬$¦ nig ¬@¦ GgÁ nig GgÁeyIg)an ³

)5()3c2b3a3(n)3b2a6(n)3a2(VV3

cbnan)1cba(3n)1ba2(3n)1a(3VV32

n1n

22

n1n

ebI

03c2b3a3

03b2a6

03a2

naM[ 4

33c,6b,

2

3a

TMnak;TMng ¬%¦køayeTACa 0VV3 n1n

b¤ n1n V3

1V

TMnak;TMngenHbBa¢ak;fa )V( n CasIVútFrNImaRtmanersug 3

1q .

tamrUbmnþtYTI nénsIVútFrNImaRteyIg)an n

0n qVV .

cMeBaH 4

33c,6b,

2

3a eKman

4

33n6n

2

3UV 2

nn

ebI 0n eK)an 4

29

4

331

4

33UV 00 naM[ n

n )3

1(

4

29V .

edayehtufa 4

33n6n

2

3UV 2

nn naM[ 4

33n6n

2

3VU 2

nn

dUcenH 4

33n6n

2

3

3

1.

4

29U 2

nn .



TajrklImIt

2n

n n

Ulim


33n6n

2

3

3

1

4

29U 2

nn

eyIg)an 2

3

n4

33

n

6

2

3

3

1.

n4

29lim

n

Ulim

2n2n2n

n

dUcenH 2

3

n

Ulim

2n

n

.



lMhatTI43


*INn,2)1n(UU

1Un

n1n

1

k> cUrKNna nU CaGnuKmn_én n .

x> KNnalImIt

n1n

n

n UU

Ulim

dMeNa¼Rsay

k>KNna nU CaGnuKmn_én n

eyIgman nn1n 2).1n(UU naM[ n

n1n 2)1n(UU

eday n1nnn 2).1n(2.n2)].1n(n2[2)1n(

eKTaj n1nn1n 2)1n(2.nUU

eyIg)an

1nn1nn

3434

2323

212

2)2n(2)1n(UU

2.22.3UU

22.2UU

02UU

bUkTMnak;TMngenHeK)an n1n 2)1n(UU naM[ n

n 2).1n(1U

dUcenH nn 2).1n(1U .

x> KNnalImIt

n1n

n

n UU

Ulim

eyIgman nn 2).1n(1U nig n

n1n 2).1n(UU



eyIg)an 11n

1n

2).1n(

1lim

2).1n(

2).1n(1lim

UU

Ulim

nnn

n

nn1n

n

n

dUcenH 1UU

Ulim

n1n

n

n

.

lMhatTI44

eK]bmafa mS nig nS CaplbUk m tYdMbUg nig n tYdMbUgerogKña

énsVúItnBVnþmYy Edl )nm(;n

m

S

S2

2

n

m .

eKtag mU CatYTI m nig nU CatYTI n .

cUrbgðajfa 1n2

1m2

U

U

n

m

?

dMeNa¼Rsay

bgðajfa 1n2

1m2

U

U

n

m

?

eyIgman 2

2

n

m

n

m

S

S eday

2

)UU(nS;

2

)UU(mS n1

nm1

m

eK)an 2

2

n1

m1

n

m

)UU(n

2

2

)UU(m

naM[ )1(

n

m

UU

UU

n1

m1

eday d).1n(UU,d).1m(UU 1n1m

Edl d CaplsgrYménsIVút.

yk d).1n(UU,d).1m(UU 1n1m CYskñúg ¬!¦ eK)an

n

m

d)1n(U2

d).1m(U2

1

1

b¤ d)1n(mmU2d)1m(nnU2 11


© 2008 Lim Phalkun - 100 -

eKTaj 2

d

)nm(2

)mmnnmn(d

)nm(2

d)1n(md)1m(nU1

eRBaH nm .

eyIg)an d.2

1n2d).1n(

2

dU;.d.

2

1m2d)1m(

2

dU nm

eK)an 1n2

1m2

d.2

1n2

d.2

1m2

U

U

n

m

ebI 0d .

dUcenH 1n2

1m2

U

U

n

m

.

lMhatTI45

eK[sIVúténcMnYnBit IN∈nn )U( kMnt;eday

INn,2UU

5U2

n1n

0

k> cUrRsayfa 2Un cMeBaHRKb; INn .

x> eKBinitüsIVút INnn )V( kMnt;edayTMnak;TMng 4UUV 2nnn .

cUrrkTMnak;TMngrvag 1nV nig nV .

K> cUrKNna nV rYcTajrk nU CaGnuKmn_én n .

dMeNa¼Rsay

k> Rsayfa 2Un cMeBaHRKb; INn

eyIgman 25U0 Bit


© 2008 Lim Phalkun - 101 -

23252UU 201 Bit

eyIg]bmafavaBitdl;tYTI k KW 2Uk Bit

eyIgnwgRsayfavaBitdl;tYTI 1k KW 2U 1k Bit .

eyIgman 2Uk naM[ 4U 2k b¤ 22U 2

k

eday 2UU 2k1k enaHeKTaj)an 2U 1k Bit .

dUcenH 2Un cMeBaHRKb; INn .

x> rkTMnak;TMngrvag 1nV nig nV

eyIgman 4UUV 2nnn naM[ 4UUV 2

1n1n1n

Et 2UU 2n1n eK)an 4)2U(2UV 22

n2

n1n

)4U(U2U44U4U2UV 2n

2n

2n

2n

4n

2n1n

2

)4UU(

2

4UU24U24UU2U

22nn

2nn

2n2

nn2n

dUcenHeK)an 2n1n V

2

1V CaTMnak;TMngEdlRtUvrk .

K> KNna nV rYcTajrk nU CaGnuKmn_én n

tamsRmayxagelIeKman 2n1n V

2

1V naM[ )V

2

1(logVlog 2

n

2

11n

2

1

n

2

11n

2

1 Vlog21Vlog tag n

2

1n VlogW naM[ 1n

2

11n VlogW

eK)an n1n W21W b¤ )W1(2)W1( n1n

naM[ )W1( n CasIVútFrNImaRtmanersug 2q nigtYTImYy 0W1


© 2008 Lim Phalkun - 102 -

Et 154554UUV 2000

eRBaH 4UUV 2nnn

eK)an )2

15(log)15(log1W1

2

1

2

10

.

tamrUbmnþ n2

2

1

2

1nn

0n )2

15(log)

2

15(log2q).W1(W1

eday n

2

1n VlogW enaH )2

V(logVlog1W1 n

2

1n

2

1n

eKTaj n2

2

1n

2

1 )2

15(log)

2

V(log

naM[

n2

n 2

152V

müa:geToteday 4UUV 2nnn naM[ 4U)UV( 2

n2

nn

b¤ 4UUUV2V 2n

2nnn

2n eKTaj

nn

n

2n

n V

2V

2

1

V2

4VU

naM[

nn

n

n 22

2

2

n 2

15

2

15

2

15

1

2

15U

.

dUcenH

nnn 2

n

22

n 2

152V,

2

15

2

15U


© 2008 Lim Phalkun - 103 -

lMhatTI46

eK[sIVútnBVnþ )U( mantY ........,12,7,2 nigsIVútnBVnþ )V( mantY .....,8,5,2

etIkñúgcMeNam 151 tYénsIVútTaMgBIrenH manb:unµantYEdlmantémøesµIKña ?

dMeNa¼Rsay

eyIgmantYTI p én sIVútnBVnþ )U( mantY ........,12,7,2

KW *INp,)1p(52U p

nigtYTI r énsIVút nBVnþ )V( mantY .....,8,5,2 KW *INr,)1r(32Vr

ebI rp VU enaHeK)an )1r(32)1p(52

b¤ )1()1r(3)1p(5

eday *INr,p enaHsmIkar )1( mancemøIykalNaman INn Edl ³

n51r nig n31p naM[eKTaj)an

1n5r

1n3p

eyIgsikSakñúgcMeNam 151 tYénsIVútTaMgBIrenHEdlmantémøesµIKña

mann½½yfa eKRtUv[ 151p1 nig 151r1 naM[eKTaj

1511n51

1511n31

b¤

30n0

50n0 naM[ 30n0 b¤ }30,....,3,2,1,0{n .

dUcenHeyIgsnñidæanfakñúgcMeNam 151 tYénsIVútTaMgBIrenHKWtYEdlman

témøesµIKñamancMnYn #! tY .


© 2008 Lim Phalkun - 104 -

lMhatTI47

eK[sIVúténcMnYnBit )U( n kMnt;eday ³

2U 0 nig RKb; 2

7n2U

2

1U:INn n1n

k> bgðajfaeKGackMnt;BIrcMnYnBit a nig b edIm,I[sIVút INn,)V( n

kMnt;edayTMnak;TMng banUV nn CasIVútFrNImaRt .


dMeNa¼Rsay

k> kMnt;BIrcMnYnBit a nig b edIm,I[sIVút INn,)V( n CasIVútFrNImaRt ³

eyIgman banUV nn naM[ baanUb)1n(aUV 1n1n1n

eday 2

7n2U

2

1U:INn n1n

eK)an baan2

7n2U

2

1V n1n Et banUV nn

b¤ banVU nn

naM[ baan2

7n2)banV(

2

1V n1n

)1()

2

7

2

ba(n)2a

2

1(V

2

1V

baan2

7n2b

2

1an

2

1V

2

1V

n1n

n1n

tamTMnak;TMng ¬!¦ edIm,I[ )V( n CasIVútFrNImaRtluHRtaEt

02

7

2

ba

02a2

1


© 2008 Lim Phalkun - 105 -

eKTaj)an 4a nig 1b . dUcenH 1b,4a .

x> KNna nU CaGnuKmn_én n .

tamsRmayxagelIcMeBaH 4a nig 1b enaHtam¬!¦naM[

n1n V2

1V

naM[ )V( n CasIVútFrNImaRtmanersug 2

1q nigtY 112bUV 00

eK)an n

n0n 2

1q.VV

.

dUcenH 1n42

1U

nn

lMhatTI48

eK[sIVúténcMnYnBit )U( n kMnt;eday ³

1U0 nig RKb; 1n4nU2U3:INn 2n1n

k> bgðajfaeKGackMnt;bIcMnYnBit b,a nig c edIm,I[sIVút INn,)V( n

kMnt;eday TMnak;TMng cbnanUV 2nn CasIVútFrNImaRt .


dMeNa¼Rsay

k> kMnt;bIcMnYnBit b,a nig c edIm,I[sIVút INn,)V( n CasIVútFrNImaRt

eyIgman cbnanUV 2nn naM[ cban)ba2(anUV 2

1n1n

eday 1n4nU2U3:INn 2n1n naM[

3

1

3

n4

3

nU

3

2U

2

n1n


© 2008 Lim Phalkun - 106 -

eK)an cban)ba2(an3

1

3

n4

3

nU

3

2V 2

2

n1n

Et cbnanUV 2nn naM[ cbnanVU 2

nn

eK)an cban)ba2(an3

1

3

n4

3

n)cbnanV(

3

2V 2

22

n1n

)1(3

1c

3

1ban)

3

4b

3

1a2(n)1a(

3

1V

3

2V 2

n1n

tamTMnak;TMng ¬!¦ edIm,I[ )V( n CasIVútFrNImaRtluHRtaEt

03

1c

3

1ba

03

4b

3

1a2

01a

naM[eKTaj)an 4c,2b,1a .

dUcenH 4c,2b,1a .

x> KNna nU CaGnuKmn_én n

cemøIy 4n2n)3

2(5U 2n

n ¬sisSedaHRsayxøÜnÉg ¦ .


© 2008 Lim Phalkun - 107 -

lMhatTI49

eK[sVúItnBVnþ )U( mantY ......17,13,9,5 nigsIVútnBVnþ )V(

mantY ,....25,19,13,7 .






dMeNa¼Rsay


tYTI p énsIVútnBVnþ )U( EdlmantYTImYy 5U1

nigplsgrYm 459d .

kMnt;eday 1p4)1p(45U p cMeBaHRKb; *INp .

cMeBaHRKb; 49p1 ]bmafa *INN,NU 2p

eK)an 2N1p4 naM[ )1(4

1Np

2

tamTMnak;TMng )1( edIm,I[ p CacMnYnKt;luHRtaEt k CacMnYness .

ebI *INr,1r2N ¬cMnYness ¦ enaH rr4

1)1r2(p 2

2


© 2008 Lim Phalkun - 108 -

Et 49p1 naM[ 49rr1 2 eday *INr eK)an 6r1

b¤ }6,5,4,3,2,1{r ehIy }42,30,20,12,6,2{p

dUcenHkñúgcMeNam $( tYénsIVút U man^tYEdlCakaerR)akd .

tYTaMgenaHKW 2

2 391)2(4U

242

230

220

212

26

131692)42(4U

111211)30(4U

9811)20(4U

7491)12(4U

5251)6(4U


tYTI k énsIVútnBVnþ )V( EdlmantYTImYy 7V1 nigplsgrYm 6713d

kMnt;eday 1k6)1k(67Vk cMeBaHRKb; *INk .

cMeBaHRKb; 49k1 ]bmafa *INt,tV 2k

eKTaj)an 2t1k6 naM[ )2(6

1tk

2

tam )2( edIm,I[ k CacMnYnKt;viC©manluHRtaEt 1t 2 Eckdac;nwg ^ .

eyIgyk rm6t Edl 5r5,r,*INm

eK)an 1r)mr2mr6(61rmr12m361)rm6(1t 222222

edIm,I[ 1t 2 Eckdac;nwg ^luHRta 1r 2

Eckdac;nwg ^ .

eday 5r5 enaHeK)an }5,4,3,2,1{r


© 2008 Lim Phalkun - 109 -

-cMeBaH 1r enaH 0111r 22 yk

-cMeBaH 2r enaH 3121r 22 ¬minyk¦



-cMeBaH 5r enaH 24151r 22 yk

dUcenHeK)an 5m6t,1m6t,1m6t,5m6t

eday 1'm61)1m(65m6,5'm65)1m(61m6

dUcenHtémø t EdlRtUvykmanEt 1m6t,5m6t

-ebI 5m6t eK)an 4m10m66

1)5m6(k 2

2

eday 49k1 naM[ 494m10m61 2 b¤ 3m2

kñúgkrNIenHeKTaj)antémø }28,8{k

-ebI 1m6t eK)an m2m66

1)1m6(k 2

2

eday 49k1 naM[ 49m2m61 2 b¤ 3m1

kñúgkrNIenHeKTaj)antémø }48,20,4{k

dUcenHeKTTYl)antémø }48,28,20,8,4{k

dUcenHkñúgcMeNam $( tYénsIVút V man%tYEdlCakaerR)akd .

tYTaMgenaHKW 2

4 5251)4(6V


© 2008 Lim Phalkun - 110 -

2

48

228

220

28

172891)48(6V

131691)28(6V

111211)20(6V

7491)8(6V


-tYTI p énsIVútnBVnþ )U( EdlmantYTImYy 5U1 nigplsgrYm 459d

kMnt;eday 1p4)1p(45U p cMeBaHRKb; *INp .

-tYTI k énsIVútnBVnþ )V( EdlmantYTImYy 7V1 nigplsgrYm

6713d kMnt;eday 1k6)1k(67Vk cMeBaHRKb; *INk .

cMeBaH 49p1 nig 49k1 ebI kp VU enaHeK)an ³

1k61p4 b¤ k3p2 naM[

*INr,r2k

r3p

eday 49p1 nig 49k1 eK)an

49r21

49r31 eKTaj 16r1

dUcenHkñúgcMeNam $(tYénsIVútTaMgBIrenHman16 tYEdlmantémøesµIKña .


© 2008 Lim Phalkun - 111 -

lMhatTI50

eK[sIVút 22n )1n(

1

n

11S

Edl *INn .

k-cMeBaHRKb; *INn cUrbgðajfa 1n

1

n

11Sn

.

x-KNnaplbUk 222222n )1n(

1

n

11.....

3

1

2

11

2

1

1

11

dMeNa¼Rsay

k-bgðajfa 1n

1

n

11Sn

eyIgman 22n )1n(

1

n

11S

¬ cMeBaHRKb; *INn ¦

1n

1

n

11

)1n(n

11

)1n(n

1)1n(n

)1n(n

]1)1n(n[

)1n(n

1)1n(n2)1n(n

)1n(n

n1n2n)1n(n

)1n(n

n)1n()1n(n

22

2

22

22

22

2222

22

2222

dUcenH 1n

1

n

11Sn

.

x-KNnaplbUk ³

eK)an

n

1kk222222n )S(

)1n(

1

n

11.....

3

1

2

11

2

1

1

11

n

1k

2

1n

)2n(n

)1n(n

1)1n(

1n

11n

1k

1

k

11


© 2008 Lim Phalkun - 112 -

lMhatTI51

eK[sIVút 1-n4

1n4U 2

2

n

+= Edl *INn .

k-cMeBaHRKb; *INn cUrbgðajfa 1n2

1

1n2

11U n

x-cUrKNnaplbUk

n


dMeNa¼Rsay

k-bgðajfa 1n2

1

1n2

11U n

eKman 1n4

1n4U

2

2

n

¬ cMeBaHRKb; *INn ¦

1n2

1

1n2

11

)1n2)(1n2(

)1n2()1n2(1

1n4

21

1n4

2)1n4(22

2

dUcenH 1n2

1

1n2

11U n

.

x-KNnaplbUk

n

1kn321kn U....UUU)U(S

eyIg)an

n

1kn 1k2

1

1k2

11S

1n2

)3n2(n

1n2

11n3n2

1n2

11n

)1n2

1

1n2

11(....)

7

1

5

11()

5

1

3

11()

3

1

1

11(

2

dUcenH 1n2

)3n2(nSn

.


© 2008 Lim Phalkun - 113 -

lMhatTI52

eK[sIVút 22

n

)3

2n(

1

)3

1n(

11U

cMeBaHRKb; *INn .

cUrKNnaplbUk

n


dMeNa¼Rsay

KNnaplbUk

n

1kn321kn U....UUU)U(S

22

n

)3

2n(

1

)3

1n(

11U

1n3

3

2n3

31

)2n3)(1n3(

)]2n3()1n3[(3)2n3)(1n3(

)2n3)(1n3(

9)2n3n9(

)2n3)(1n3(

]9)2n3n9[(

)2n3)(1n3(

81)2n3n9(18)2n3n9(

)2n3)(1n3(

]9)2n3n9(2[9)2n3n9(

)2n3)(1n3(

]5n6n18[9)2n3n9(

)2n3)(1n3(

]1n6n94n12n9[9)2n6n3n9(

)2n3)(1n3(

)1n3(9)2n3(9)1n3()2n3(

)2n3(

9

)1n3(

91

2

22222

222222

2222

2222

22

eyIg)an 1n3

n10n3

1n3

33n]

1k3

3

2k3

31[S

2n

1kn

dUcenH 1n3

)10n3(nSn

.


© 2008 Lim Phalkun - 114 -

lMhatTI53

eK[ *INnn ),U(

CasIVútnBVnþmYyEdlmantYTaMgGs;viC¢mannigplsgrYm

0d . eKtag 2

1n

2

n

2n U

1

U

1

d

1S

.

cUrbgðajfa 1n1

n321 U

1

U

1

d

nS...SSS

.

dMeNa¼Rsay

eyIgman 2

1n

2

n

2n U

1

U

1

d

1S

eday *INnn ),U(

CasIVútnBVnþmYyEdlmanplsgrYm 0d enaH

dUU n1n

. eyIgman 2

n2n

2n )dU(

1

U

1

d

1S

1nnnnnn

nn

nnnn

2nn

nn

22nn

nn

42nn

2n

2n

nn

2n

2n

2n

22n

2n

2n

2n

2

2n

2n

22n

2n

U

1

U

1

d

1

dU

1

U

1

d

1

)dU(U

U)dU(

d

1

)dU(U

d

d

1

)dU(U.d

d)dU(U

)dU(Ud

]d)dU(U[

)dU(U.d

dd).dU(U2)dU(U

)dU(U.d

]UddU2U[d)dU(U

)dU.(U.d

]U)dU[(d)dU(U

eyIg)an

n

1k 1n11nn

n

1kkn21 U

1

U

1

d

n

U

1

U

1

d

1)S(S...SS

dUcenH 1n1

n321 U

1

U

1

d

nS...SSS


© 2008 Lim Phalkun - 115 -

lMhatTI54

eK[sIVúténcMnYnBit )x( n kMnt;eday 1x1 nig 1n2x

2xx

n

n1n

k-cUrRsayfacMeBaHRKb; *INn eKman 1n2x n .

x-KNna 1n2x

1.....

5x

1

3x

1S

n21

n

.

K> TajrklImIt

n

Slim n

n .

dMeNa¼Rsay

k-RsayfacMeBaHRKb; *INn ³ 1n2x n

eKman 11.21x1 Bit

ebI 1n enaH 3x

2xx

1

12

naM[ 313131

21x 2

eK)an 12.23x 2 Bit .

]bmafavaBitdl;tYTI k KW 1k2x k

eyIgnwgRsayfavaBitdl;tYTI 1k KW 1k21)1k(2x 1k

eyIgman 1k2x

2xx

k

k1k

naM[ 1k2x

2xx

k

k1k

eday 1k2x k

enaH 1k21k2

21k2x 1k

1k21k21k2

)1k2()1k2(

)1k21k2(21k2


© 2008 Lim Phalkun - 116 -

1k2x 1k

Bit .

dUcenH 1n2x n .

x-KNna 1n2x

1.....

5x

1

3x

1S

n21

n

eyIg)an

n

1k k

n1k2x

1S eday

1n2x

2xx

n

n1n

eKTaj )xx(2

1

1n2x

1n1n

n

eK)an )xx(2

1)xx(

2

1S 11n

n

1kk1kn

eday 1x1 ehIy 1n2x n enaH 1n2x 1n

dUcenH )11n2(2

1Sn .

K> TajrklImIt

n

Slim n

n

eyIg)an

n

11n2.

2

1lim

n

Slim

n

n

n

2

2

n

1

n

12lim

2

1n

dUcenH 2

2

n

Slim n

n

.


© 2008 Lim Phalkun - 117 -

lMhatTI55

eKmanGnuKmn_ x2e).1x()x(f Edl x CacMnYnBit .


x> cUrbgðajfaedrIevTI n énGnuKmn_ f manTRmg; x2

nn

)n( e).bxa()x(f

Edl na nig nb CasIVútcMnYnBitepÞógpÞat;TMnak;TMng ½

*INn,b2ab

a2a

nn1n

n1n

K> cUrkMnt; na nig nb CaGnuKmn_én n rYcTajrkkenSam )x(f )n( .

dMeNa¼Rsay

k> KNnaedrIev x'''f,)x(''f,)x('f nig xf )4(

eKman x2e).1x()x(f

)1x()'e(e)'1x()x('f x2x2

x2x2x2x2 e)3x2()2x21(e)1x(e2e

dUcenH x2e)3x2()x('f

eK)an )3x2()'e(e)'3x2()x(''f x2x2

x2x2x2x2 e)8x4()6x42(e)3x2(e2e2

dUcenH x2e)8x4()x(''f


© 2008 Lim Phalkun - 118 -

eK)an )8x4()'e(e)'8x4()x('''f x2x2

x2x2x2x2 e).20x8()16x84(e)8x4(e2e4

dUcenH x2e)20x8()x('''f

eK)an )20x8()'e(e)'20x8()x(f x2x2)4(

x2x2x2x2 e)48x16()40x168(e)20x8(e2e8

dUcenH x2)4( e)48x16()x(f .

x>bgðajfaedrIevTI n énGnuKmn_ f manTRmg; x2

nn

)n( e).bxa()x(f

tamsRmayxagelIeyIgman ³

x2

11

x2 e)bxa(e)3x2()x('f Edl 3b,2a 11

x2

22

x2 e)bxa(e)8x4()x(''f Edl 8b,4a 22

x2

33

x2 e)bxa(e)20x8()x('''f Edl 20b,8a 33

x2

44

x2)4( e)bxa(e)48x16()x(f Edl 48b,16a 44

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

]bmafavaBitdl;edrIevlMdab;TI n KW ³ x2

nn

)n( e)bxa()x(f Bit

eyIgnwgRsay[eXIjfavaBitdl;edrIevlMdab;TI )1n( KW ³

x2

1n1n

)1n( e)bxa()x(f

eyIgman ))'x(f()x(f )n()1n( eday x2

nn

)n( e)bxa()x(f


© 2008 Lim Phalkun - 119 -

eK)an )bxa()'e(e)'bxa()x(f nn

x2x2

nn

)1n(

x2

nnn

)1n(

nnn

x2)1n(

nn

x2x2

n

)1n(

e)b2axa2()x(f

)b2xa2a(e)x(f

)bxa(e2ea)x(f

eKTaj x2

1n1n

)1n( e)bxa()x(f

Bit Edl n1n a2a

nig

nn1n b2ab

.

dUcenH edrIevTI n énGnuKmn_ f manTRmg; x2

nn

)n( e).bxa()x(f

Edl na nig nb CasIVútcMnYnBitepÞógpÞat;TMnak;TMng n1n a2a

nig nn1n b2ab

.

K>kMnt; na nig nb CaGnuKmn_én n

tamsRmayxagelIeKman n1n a2a

nig nn1n b2ab

cMeBaHRKb; *INn .

tamTMnak;TMng n1n a2a

naM[ )a( n CasIVútFrNImaRtmanersug 2q

nigtY 2a1 .

tamrUbmnþ n1n1n

1n 22.2q.aa .

müa:geToteKman nn1n b2ab

naM[eKTaj n

nn1n 2ab2b

b¤ 1b2

1b

2

1n1n1nn

ebIeKtag

n1nn b2

1c

eK)an 1cc n1n

efr


© 2008 Lim Phalkun - 120 -

naM[ )c( n CasIVútnBVnþmanplsgrYm 2d nigtY 3bc 11

tamrUbmnþ 1n2)1n(23d)1n(cc 1n

eday n1nn b2

1c

naM[ 1n

n

1n

n 2).1n2(c.2b

dUcenH 1n

n

n

n 2).1n2(b,2a .

TajrkkenSam )x(f )n( ³

eKman x2

nn

)n( e).bxa()x(f eday n

n 2a nig 1n

n 2)1n2(b

eK)an x2nx21nn)n( e).2

1n2x(2e].2)1n2(x2[)x(f

dUcenH x2n)n( e).2

1n2x(2)x(f

.

lMhatTI56

eKmanGnuKmn_ xsin.e)x(f x


x> cUrbgðajfaedrIevTI n énGnuKmn_ f manTRmg;

x

nn

)n( e).xcosbxsina()x(f

Edl na nig nb CasIVútcMnYnBitepÞógpÞat;TMnak;TMng

*INn,bab

baa

nn1n

nn1n


© 2008 Lim Phalkun - 121 -

K> eKtag nnn b.iaz .

cUrbgðajfa n1n z).i1(z

rYcsresr nz CaTRmg;RtIekaNmaRt .

X> cUrkMnt; na nig nb CaGnuKmn_én n rYcTajrkkenSam )x(f )n( .

dMeNa¼Rsay

k> KNnaedrIev x'''f,)x(''f,)x('f nig xf )4(

eKman xsin.e)x(f x

eK)an xx e)'x(sinxsin)'e()x('f

)xcosx(sinee.xcosxsine xxx

dUcenH )xcosx(sine)x('f x .

eK)an xx e)'xcosx(sin)xcosx(sin)'e()x(''f

xcose2)xsinxcosxcosx(sine

e)xsinx(cos)xcosx(sinexx

xx

dUcenH xcose2)x(''f x .

eK)an xx e)'x(cos2xcos)'e(2)x('''f

)xsinx(cose2e.xsin2xcose2 xxx

dUcenH )xsinx(cose2)x('''f x .

eK)an xx)4( e)'xsinx(cos2)xsinx(cos)'e(2)x(f


© 2008 Lim Phalkun - 122 -

xsine4)xcosxsinxsinx(cose2

e)xcosxsin(2)xsinx(cose2xx

xx

dUcenH xsine4)x(f x)4( .

x> bgðajfaedrIevTI n énGnuKmn_ f manTRmg;

x

nn


tamsRmayxagelIeKman ³

x

11

xx e)xcosbxsina(e)xcosx(sin)xcosx(sine)x('f

Edl

1b

1a

1

1

x

22

xx e)xcosbxsina(e)xcos2xsin.0(xcose2)x(''f

Edl

2b

0a

2

2

x

33

xx e)xcosbxsina(e)xcos2xsin2()xsinx(cose2)x('''f

Edl

2b

2a

3

3

x

44

xx)4( e)xcosbxsina(e)xcos.0xsin4(xsine4)x(f

Edl

0b

4a

4

4

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

]bmafavaBitdl;edrIevlMdab;TI n KW xnn

)n( e).xcosbxsina()x(f Bit


© 2008 Lim Phalkun - 123 -

eyIgnwgRsayfavaBitdl;edrIevlMdab;TI )1n( KW x

1n1n)1n( e)xcosbxsina()x(f

Bit

eyIgman ))'x(f()x(f )n()1n(

)xcosbxsina()'e(e)'xcosbxsina( nnxx

nn

)xcosbxsina(ee)xsinbxcosa()x(f nn

xx

nn

)1n(

x

nnnn

xnnnn

e.xcos)ba(xsin)ba(

e)xcosbxsinaxsinbxcosa(

x

1n1n

)1n( e).xcosbxsina()x(f

Bit

¬eRBaH nn1n baa

nig nn1n bab

¦

dUcenH edrIevTI n énGnuKmn_ f manTRmg; x

nn


Edl na nig nb CasIVútcMnYnBitepÞógpÞat;TMnak;TMng

*INn,bab

baa

nn1n

nn1n

K> bgðajfa n1n z).i1(z

eKman nnn b.iaz naM[ 1n1n1n b.iaz

eday

*INn,bab

baa

nn1n

nn1n

eK)an )ba.(i)ba(z nnnn1n


© 2008 Lim Phalkun - 124 -

nnn

nn

nn

nnnn

z)i1()b.ia)(i1(

)bi1

i1a)(i1(

b)i1(a)i1(

b.ia.iba

dUcenH n1n z)i1(z

.

sresr nz CaTRmg;RtIekaNmaRt ³

edayeKman n1n z).i1(z

naM[ )z( n CasIVútFrNImaRténcMnYnkMupøic

Edlmanersug ³

)4

sin.i4

(cos2)2

2.i

2

2(2i1q

nigtY 111 ibaz

Et 1b,1a 11 naM[ )4

sin.i4

(cos2qi1z1

.

tamrUbmnþtYTI n énsIVútFrNImaRteKGacsresr ³

nn

1n

1n

1n

)4

sin.i4

(cos)2(

)4

sin.i4

(cos2).4

sin.i4

(cos2qzz

dUcenH )4

nsin.i

4

n(cos2z n

n

¬ tamrUbmnþdWmr½ ¦ .

X> kMnt; na nig nb CaGnuKmn_én n

eKman nnn b.iaz eday )4

nsin.i

4

n(cos2z n

n


© 2008 Lim Phalkun - 125 -

eKTaj )4

nsin.i

4

n(cos2b.ia n

nn

naM[

4

nsin.2b

4

ncos.2a

n

n

n

n

dUcenH 4

nsin.2b,

4

ncos.2a n

n

n

n

.

TajrkkenSam )x(f )n( ³

eKman x

nn

)n( e)xcosbxsina()x(f

eday4

nsin.2b,

4

ncos.2a n

nn

n

eK)an xnn)n( e).4

nsin.xcos2xsin.

4

ncos.2()x(f

dUcenH )4

nxsin(e.2)x(f xn)n( .

lMhatTI57

eK[GnuKmn_ 1x,x1

x)x(f

3 3

cUrKNna ].....])x(f[f[.....ff)x(Fn .

dMeNa¼Rsay

KNna ].....])x(f[f[.....ff)x(Fn

eyIgman 3 31

x1

x)x(f)x(F


© 2008 Lim Phalkun - 126 -

3 3

1n

1n1nn

23

12

)x(F1

)x(F)x(Ff)x(F

)x(Ff)x(fff)x(F

)x(Ff)x(ff)x(F

eyIg)an )x(F1

)x(F)x(F 3

1n

31n3

n

naM[ 1

)x(F

1

)x(F

)x(F1

)x(F

13

1n3

1n

31n

3n

eKTaj 1)x(F

1

)x(F

13

1n3n

efr naM[eKTaj)anfa

)x(F

13n

CasIVútnBVnþmanplsgrYm 1d nigtYTImYy 3

3

31

1 x

x1

)x(F

1U

.

tamrUbmnþ d).1n(UU 1n eK)an 3

3

3

3

3n x

nx11n

x

x1

)x(F

1

eKTaj 3 3

33

33

nnx1

x

nx1

x)x(F

dUcenH 3 3n

nx1

x].....])x(f[f[.....ff)x(F

.

lMhatTI58

eK[sIVúténcMnYnBit INnn )U( eday

1U6U6

U3U9U7U 2

n3

n

n2

n3

n1n

nig 2U1 = .

k-eKtag *INn,1U2

1UV

n

nn

. cUrbgHajfa 3

n1n VV



CaGnuKmn_én n .


© 2008 Lim Phalkun - 127 -


.

dMeNa¼Rsay

k- bgHajfa ½ 3

n1n VV

eyIgman *INn,1U2

1UV

n

nn

eyIgán 1U2

1UV

1n

1n1n

eday 1U6U6

U3U9U7U

2

n

3

n

n

2

n

3

n1n

3

n

n3

n

3n

n

2

n

3

n

n

2

n

3

n1n

2

n

3

nn

2

n

3

n

2

n

3

nn

2

n

3

n1n

2

n

3

n

n

2

n

3

n

2

n

3

n

n

2

n

3

n

1n

1U2

1U

)1U2(

)1U(

1U6U12U8

1U3U3UV

1U6U6U6U18U14

1U6U6U3U9U7V

1)1U6U6

U3U9U7(2

11U6U6

U3U9U7

V

dUcen¼ 3

n1n VV

.

x-bgHajfa INn,n )W( CasIVútFrNImaRt ½

eyIgman nn VlnW naM[ 1n1n VlnW

eday 3

n1n VV

eyIIIIIIIII II II II II II Igán nn

3

n1n W3Vln3VlnW

dUcen¼ INn,n )W( CasIVútFrNImaRtEdlmanersug 3q .

-KNna nW CaGnuKmn_én n ½


© 2008 Lim Phalkun - 128 -

tamrUbmnþ 1n1n qWW

eday )3

1ln()

14

12ln(

1U2

1UlnVlnW

1

111

nig 3q

eyIgán )3

1(ln3)3()

3

1ln(W nn

n .

K-Tajrk nV nig nU CaGnuKmn_én n ½

eyIgman nn VlnW eday )3

1(ln3W n

n

eyIgán )3

1(ln3Vln n

n naM[ n3

n 3

1V

müa¨geTot 1U2

1UV

n

nn

naM[

23

13

)3

1(21

)3

1(1

V21

V1U

n

n

n

n

3

3

3

3

n

nn

dUcen¼ n3

n 3

1V

nig

23

13U

n

n

3

3

n

.

-KNnalImIt ½ nnUlim

eyIgman 23

13U

n

n

3

3

n

eyIgán 1

3

21

3

11

lim23

13limUlim

n

n

n

n

3

3

n3

3

nnn

dUcen¼ 1Ulim nn

.


© 2008 Lim Phalkun - 129 -

lMhatTI59

eK[sIVúténcMnYnkMupøic )Z( n kMnt;eday

INn,2

i21Z

2

i1Z

iZ

n1n

0

k> eKtag 1ZU:INn nn .

cUrbgðajfa n1n U2

i1U

cMeBaHRKb; INn .

x> cUrsresr nU CaragRtIekaNmaRtrYcTajrk nZ CaGnuKmn_én n .

dMeNa¼Rsay

k> bgðajfa n1n U2

i1U

cMeBaHRKb; INn

eyIgman 1ZU:INn nn naM[ 1ZU 1n1n

eday2

i21Z

2

i1Z n1n

eyIg)an 12

i21Z

2

i1U n1n

nn1n

n1n

U2

i1)1Z(

2

i1U

2

i1Z

2

i1U

dUcenH n1n U2

i1U

.

x> sresr nU CaragRtIekaNmaRtrYcTajrk nZ CaGnuKmn_én n

eyIgman n1n U2

i1U

naM[ )U( n CasIVútFrNImaRténcMnYnkMupøicmanersug


© 2008 Lim Phalkun - 130 -

4sin.i

4cos

2

2i

2

2

2

i1q

nigtY )

4sin.i

4(cos21i1ZU 00

tamrUbmnþ

1nnn0n )

4sin.i

4(cos2)

4sin.i

4).(cos

4sin.i

4(cos2qUU

dUcenH

4

)1n(sin.i

4

)1n(cos2Un .

müa:geTot 1ZU nn naM[ 1]4

)1n(sin.i

4

)1n([cos21UZ nn

dUcenH 4

)1n(sin2.i

4

)1n(cos21Zn

.

lMhatTI60

eK[sIVúténcMnYnBit )U( n kMnt;eday

INn,UU.3U

1U,0U

n1n2n

10

k> eKtag INn,U2

i3UZ n1nn

.

cUrRsaybBa¢ak;fa n1n Z2

i3Z

cMeBaHRKb; INn .

x> cUrkMnt;rkTMrg;RtIekaNmaRtén nZ .

K> rYcTajrktY nU CaGnuKmn_én n rYcTajrktémø 2007U .

dMeNa¼Rsay

k> RsaybBa¢ak;fa n1n Z2

i3Z

eyIgman INn,U2

i3UZ n1nn

naM[ 1n2n1n U

2

i3UZ


© 2008 Lim Phalkun - 131 -

edayeKman n1n2n UU.3U cMeBaHRKb; INn

eK)an 1nn1n1n U2

i3UU.3Z

nn1n1n

n1nn1n1n

Z2

i3)U

2

i3U(

2

i3Z

)Ui3

2U(

2

i3UU

2

i3Z

dUcenH INn,U2

i3UZ n1nn

.

x>kMnt;rkTMrg;RtIekaNmaRtén nZ rYcTajrktY nU CaGnuKmn_én n

edayeyIgman n1n Z2

i3Z

cMeBaHRKb; INn ¬tamsRmayxagelI ¦

naM[ )Z( n CasIVútFrNImaRténcMnYnkMupøicEdlmanersug ³

)6

sin.i6

(cos2

i3q

nigtY 1U2

i3UZ 010

¬ eRBaH )1U,0U 10

eK)an 6

nsin.i

6

ncos)

6sin.i

6(cosqZZ nn

0n

dUcenH 6

nsin.i

6

ncosZn

.

K> TajrktY nU CaGnuKmn_én n rYcTajrktémø 2007U ³

eyIgman n1nn U2

i3UZ

b¤ nn1nn U2

i)U

2

3U(Z ¬ eRBaH IRU n ¦


© 2008 Lim Phalkun - 132 -

eyIg)an nn1nn U2

i)U

2

3U(Z ¬ cMnYnkMupøicqøas;én nZ ¦.

eKTaj nnn U.iZZ naM[ i

ZZU nn

n

edayeyIgman 6

nsin.i

6

ncosZn

nig

6

nsin.i

6

ncosZn

eyIg)an 6

nsin2

i

)6

nsin.i

6

n(cos)

6

nsin.i

6

n(cos

U n

.

dUcenH 6

nsin2Un

.

müa:geTotcMeBaH 2007n eyIg)an ³

2)2

sin(2)3322

sin(22

663sin2

6

2007sin2U 2007

.

dUcenH 2U2007 .

lMhatTI61

eK[sIVúténcMnYnkMupøic )Z( n kMnt;edayTMnak;TMngxageRkam ³

1Z,0Z 10 nig n1n2n Z2

3i1Z

2

3i3Z

cMeBaHRKb; INn .

k> eKtag n1nn ZZU:INn . bgðajfa n1n U2

3i1U

x> KNna nU nig n210n U....UUUS CaGnuKmn_én n .

K> TajrkcMnYnkMupøic nZ .


© 2008 Lim Phalkun - 133 -

dMeNa¼Rsay

k> bgðajfa n1n U2

3i1U

eyIgman n1nn ZZU naM[ 1n2n1n ZZU

edaycMeBaHRKb; INn man n1n2n Z2

3i1Z

2

3i3Z

eyIg)an 1nn1n1n ZZ2

3i1Z

2

3i3U

nn1n1n

n1n1n

n1n1n

U.2

3i1)ZZ(

2

3i1U

Z2

3i1Z

2

3i1U

Z2

3i1Z)1

2

3i3(U

dUcenH n1n U2

3i1U

.

x> KNna nU CaGnuKmn_én n

eyIgman n1n U2

3i1U

naM[ )U( n CasIVútFrNImaRt

éncMnYnkMupøicEdlmanersug 3sin.i

3cos

2

3.i

2

1

2

3i1q

nigtY 1ZZU 010 .

tamrUbmnþ 3

nsin.i

3

ncos)

3sin.i

3(cosqUU nn

0n

dUcenH 3

nsin.i

3

ncosUn

.


© 2008 Lim Phalkun - 134 -

KNna n210n U....UUUS CaGnuKmn_én n ³

eday )U( n CasIVútFrNImaRténcMnYnkMupøicenaHtamrUbmnþeyIg)an ³

q1

q1.US

1n

0n

eday 3sin.i

3cosq,1U0

eK)an 3

sin.i3

cos1

3

)1n(sin.i

3

)1n(cos1

3sin.i

3cos1

3sin.i

3cos1

S

1n

n

6

nsin.i

6

ncos

6

)1n(sin2

6sin.i

6cos)

2

1(2

6

)1n(sin.i

6

)1n(cos

6

)1n(sin2

i

i

6cos.i

6sin.

6sin2

6

)1n(cos.i

6

)1n(sin.

6

)1n(sin2

6cos

6sini2

6sin2

6

)1n(cos.

6

)1n(sin.i2

6

)1n(sin2

2

2

dUcenH

6

nsin.i

6

ncos

6

)1n(sin2Sn .

K> TajrkcMnYnkMupøic nZ

eyIgman n1nn ZZU:INn


© 2008 Lim Phalkun - 135 -

eyIg)an 01n

n

0k

n

0kk1kkn ZZZZUS

naM[ n01n SZZ

eday 0Z0 nig

6

nsin.i

6

ncos

6

)1n(sin2Sn

eK)an

6

nsin.i

6

ncos

6

)1n(sin2Z 1n

dUcenH

6

)1n(sin.i

6

)1n(cos

6

nsin2Zn .

lMhatTI62

eK[sIVúténcMnYnBit )U( n kMnt;eday

INn,U2U81

U5U22U

1U

2nn

2nn

1n

0

k> eKtag 1U

1U2V:INn

n

nn

. cUrbgðajfa 2

n1n VV

x> KNna nV nig nU CaGnuKmn_én n .

dMeNa¼Rsay

k >bgðajfa 2n1n VV

eyIgman 1U

1U2V:INn

n

nn

naM[ 1U

1U2V

1n

1n1n

eday 2nn

2nn

1nU2U81

U5U22U

cMeBaHRKb; INn


© 2008 Lim Phalkun - 136 -

eK)an 2nn

2nn

2nn

2nn

2nn

2nn

2nn

2nn

1nU2U81U5U22

U2U81U10U44

1U2U81

U5U22

1U2U81

U5U222

V

2n

2

n

n

n2n

n2n

n2n

n2n

1n V1U

1U2

)1U2U(3

)1U4U4(3

3U6U3

3U12U12V

Bit

x> KNna nV nig nU CaGnuKmn_én n

edayeKman 2n1n VV eKTaj n1n Vln2Vln naM[ )V(ln n

CasVúItFrNImaRtmanersug 2q nigtYdMbUg )2

1ln()

1U

1U2ln(Vln

0

00

.

tamrUbmnþeK)an n2n

n )2

1ln()

2

1ln(.2Vln naM[

n

n

22

n )5.0(2

1V

ehIyeday 1U

1U2V

n

nn

naM[ n

n

2

2

n

nn

)5.0(2

)5.0(1

V2

V1U

dUcenH n

n

n

2

2

n2

n)5.0(2

)5.0(1U,)5.0(V

.

lMhatTI63

eK[sIVúténcMnYnBit )U( n kMnt;edayTMnak;TMng ³

2U 0 nig 2nn

3n

1nU3U64

UU:INn

k> eKtag n

nn U2

UV:INn

. bgðajfa 3

n1n VV

x>KNna nV nig nU CaGnuKmn_én n .


© 2008 Lim Phalkun - 137 -

dMeNa¼Rsay

k> bgðajfa 3n1n VV

eyIgman n

nn U2

UV:INn

naM[ 1n

1n1n U2

UV

Et 2

nn

3n

1nU3U64

UU

eK)an 3n

3

n

n3nnn

3n

2nn

3n

2nn

3n

1n VU2

U

UU6U128

U

U3U64

U2

U3U64

U

V

dUcenH 3n1n VV .

x>KNna nV nig nU CaGnuKmn_én n

eyIgman 3n1n VV eKTaj n1n Vln3Vln

naM[ )V(ln n CasVúItFrNImaRtmanersug 3q nigtYdMbUg

)2

1ln()

U2

Uln(Vln

0

00

.

tamrUbmnþeK)an n3n

n )2

1ln()

2

1ln(.3Vln naM[

n

n

33

n )5.0(2

1V

ehIyeday n

nn U2

UV

naM[ n

n

3

3

n

nn

)5.0(1

)5.0(2

V1

V2U

dUcenH n

n

n

3

3

n3

n)5.0(1

)5.0.(2U,)5.0(V

.


© 2008 Lim Phalkun - 138 -

lMhatTI64

eK[sIVúténcMnYnBit 123.....)1n(nU n

k> cUrKNnalImIt )nU(lim nn

.

x> eKtag )UU)(UU(

1V:INn

n2

1n1n2

2nn

nig

n

1kkn VS .

KNna nn

Slim

.

dMeNa¼Rsay

KNnalImIt )nU(lim nn

eyIgman n123.......)1n(nnUn

b¤ n123......)1n(n

123....)1n(nU n

eyIg)an 2

1

n123......)1n(n

123.....)1n(lim)nU(lim

nn

n

dUcenH 2

1)nU(lim n

n

.

x> KNnalImIt nn

Slim

eKman 123.....)1n(nUn


© 2008 Lim Phalkun - 139 -

eK)an 123....n)1n(U 1n

b¤ n2

1n U1n123....n)1n(U

na[ )1n(UU n2

1n nig )2n(UU 1n2

2n

eK)an

n

1k

n

1kn 2n

1

2

1

2k

1

1k

1

)2k)(1k(

1S

dUcenH 2

1Slim n

n

.

lMhatTI65

eK[sIVúténcMnYnBit )U( n kMnt;eday 4

nsin.2U

nn

Edl *INn

k>cUrbgðajfa 4

nsin

4

ncos

4

)1n(cos.2

x> Taj[)anfa 4

)1n(cos)2(

4

ncos)2(U 1nn

n

K> KNnaplbUk n321n U.........UUUS CaGnuKmn_én n .

dMeNa¼Rsay

k>cUrbgðajfa 4

nsin

4

ncos

4

)1n(cos.2

tamrUbmnþ bsinasinbcosacos)bacos(

eyIg)an )4

sin4

nsini

4cos

4

n(cos2)

44

ncos(2

4

)1n(cos2

4

nsin

4

ncos)

4

nsin

2

2

4

ncos

2

2(2

4

)1n(cos2

dUcenH 4

nsin

4

ncos

4

)1n(cos.2

.


© 2008 Lim Phalkun - 140 -

x> Taj[)anfa 4

)1n(cos)2(

4

ncos)2(U 1nn

n

eyIgman 4

nsin

4

ncos

4

)1n(cos.2

naM[ 4

)1n(cos2

4

ncos

4

nsin

KuNGgÁTaMgBIrnwg n)2( eK)an ³

4

)1n(cos)2(

4

ncos)2(

4

nsin)2( 1nnn

dUcenH 4

)1n(cos)2(

4

ncos)2(U 1nn

n

.

K> KNnaplbUk n321n U.........UUUS CaGnuKmn_én n

eyIg)an

n

1kkn321n )U(U...........UUUS

4

)1n(cos)2(

4cos2

4

)1k(cos)2(

4

kcos)2( 1n

n

1k

1kk

dUcenH 4

)1n(cos)2(1S 1n

n

.

lMhatTI66

cMeBaHRKb; INn eKman nnn )i32(y.ix Edl IRy,x nn

k> cUrkMnt;rk nx nig ny CaGnuKmn_én n .

x> cUrbgðajfa n2nn1n1nn )13(2yxyx



© 2008 Lim Phalkun - 141 -

dMeNa¼Rsay

k> cUrkMnt;rk nx nig ny CaGnuKmn_én n

eKman nnn )i32(y.ix Edl IRy,x nn

tag )6

sin.i6

cos1(2)2

1.i

2

31(2i32Z

)12

sin.i12

cos(12

cos4)12

cos12

sin2.i.12

cos2(2Z 2

eyIg)an )12

nsin.i

12

ncos(

12cos4Zy.ix nnn

nn

dUcenH 12

nsin

12cos4y,

12

ncos

12cos4x nn

nnn

n

.

x>bgðajfa n2nn1n1nn )13(2yxyx


eyIgman nnn )i32(y.ix naM[ n

nn )i32(y.ix

eyIg)an 1nn1n1nnn )i32()i32()y.ix)(y.ix(

b¤ )i32(1)32()yxyx(i)yyxx( n2n1n1nn1nn1nn

eday 22 )13(23481)32( nig 2)13(2

132

eK)an

i)13(

2

1)13(2)yxyx(i)yyxx( 2n2n

n1n1nn1nn1nn

eKTaj 2n21n1nn1nn )13(2yyxx

nig n2nn1n1nn )13(2yxyx .


© 2008 Lim Phalkun - 142 -

lMhatTI67

eK[GnuKmn_ 15x8x12

2x12x31)x(f

2

2

k-cUrrkEdnkMnt´énGnuKmn_ )x(f .

x-eKBinitüsIVút )U( n nig )V( n kMntćMeBa¼RKb´ *INn

eday )U(fU,)x(fU n1n1 nig 1U4

2UV

n

nn

.

cUrbgHajfa 2

n1n VV rYcTaj[ánfa

1n2

1n VV

.

K-cUrKNna .........)x(ff........ff)x(F nn rYcTajrk )1(Flim nn

dMeNa¼Rsay

k-rkEdnkMnt´énGnuKmn_ )x(f ½

eKman 15x8x12

2x12x31)x(f

2

2

GnuKmn_en¼mann&ykalNa 015x8x12 2

ebI 015x8x12 2 19618016'

eKTaj¦s 6

5

12

144x,

2

3

12

144x 21

.

dUcen¼ }6

5,

2

3{IRD .

x-bgHajfa 2

n1n VV

eyIgman 1U4

2UV

n

nn

naM[

1U4

2UV

1n

1n1n


© 2008 Lim Phalkun - 143 -

eday 15U8U12

2U12U31)U(fU

n

2

n

n

2

nn1n

eKán 1)

15U8U12

2U12U31(4

215U8U12

2U12U31

V

n

2

n

n

2

n

n

2

n

n

2

n

1n

15U8U128U48U124

30U16U242U12U31V

n2

nn2

n

n2

nn2

n1n

2

n

n

n

2

n

n

2

n

n

2

n

n

2

n1n 1U4

2U

)1U8U16(7

)4U4U(7

7U56U112

28U28U7V

dUcen¼ 2

n1n VV .

-Taj[ánfa ½ 1n2

1n VV

eKman 2

n1n VV

ebI 1n eKán 2

12 VV Bit

ebI 2n eKán 22

1

2

23 VVV Bit

ebI 3n eKán 32

1

2

34 VVV Bit

snµtfavaBitdl´tYTI k KW 1k2

1k VV

eyIgnwgRsaybBa¢ak´favaBitdl´tYTI 1k KW k2

11k VV

eyIgman k1k 2

1

22

1

2

k1k VVVV

Bit .

dUcen¼ 1n2

1n VV

.


© 2008 Lim Phalkun - 144 -

K-KNna .........)x(ff........ff)x(F nn ½

tag )x(fU1

)U(fU

)U(f.........)x(ff.........ffU

)U(f)x(fffU

)U(f)x(ffU

n1n

1nnn

23

12

eKTaj nnn U.........)x(ff........ff)x(F

tamsRmayxagelIeKman 1U4

2UV

n

nn

naM[ 2U)1U4(V nnn

¦ 1V4

2VU

n

nn

eday

1n2

1n VV

nig 2

2

2

2

2

1

1

11x4

2x

1)15x8x12

2x12x31(4

215x8x12

2x12x31

2U4

2UV

eKán n1n 22

2n 1x4

2x)

1x4

2x(V

ehtuen¼ nn

nn

n

n

22

22

2

2

n

nn

)1x4()2x(4

)1x4(2)2x(

1)1x4

2x(4

2)1x4

2x(

1V4

2VU

dUcen¼ nn

nn

22

22

n

)1x4()2x(4

)1x4(2)2x()x(F

.


© 2008 Lim Phalkun - 145 -

-TajrklImIt ½ )1(Flim nn

cMeBa¼ 1x eKán n

n

2

2

n34

3.21)1(F

eyIgán 234

3.21lim)1(Flim n

n

2

2

nnn

.

dUcen¼ 2)1(Flim nn

.

lMhatTI69

eK[GnuKmn_ 4

1x

2

1x.....xxx)x(f

k-cUrkMnt´bNþatémø INx edIm,I[GnuKmn_

)x(f mantémøelxCacMnYnKt´ .

x-bgHajfacMeBa¼RKb´ *INp témø )1p(pf 22

CakaerRákd .

dMeNa¼Rsay

kMnt´témø INx edIm,I[GnuKmn_ )x(f mantémøelxCacMnYnKt´

eyIgman 2)4

1x

2

1()

4

1x(.

4

1x

2

1.2

4

1

4

1x

2

1x

eKTaján 4

1x

2

1)x(f .

smµtfamancMnYnKt´ *INk Edl k)x(f


© 2008 Lim Phalkun - 146 -

eKán k4

1x

2

1

¦ 4

1kk)

2

1k(

4

1x 22

eKTaj )1k(kx eday *INk

ena¼ IN)1k(kx

dUcen¼bNþatémø x EdlRtUrkena¼KW *INk,)1k(kx

x-bgHajfacMeBa¼RKb´ *INp témø )1p(pf 22 CakaerRákd

eyIgman 4

1x

2

1)x(f

eyIgán 4

1)1p(p

2

1)1p(pf 2222

2

2

24

p2

1p2

2

1

4

1p4p4

2

1

dUcen¼ 222 p)1p(pf CakaerRákd .


© 2008 Lim Phalkun - 147 -

lMhatTI70

eK[smIkardWeRkTIbIGBaØatBit ½

01000m300m30mx)14m9m(mxx:)E( 23223


k-cUrkMnt´témørbs´ m edIm,I[smIkaren¼man¦sbIbeg;Itán

CasIVúFrNImaRtmYy .

x-cUreda¼RsaysmIkar )E( cMeBa¼témø m Edlán

rkeXIjxagelI .

dMeNa¼Rsay

k-kMnt´témørbs´ m

tag 321 x,x,x Ca¦srbs´smIkar )E( .

tamRTwsþIbTEvüteyIgman ½

)3()10m(xxx

)2(14m9mxxxxxx

)1(mxxx

3

321

2

313221

321

ebIsmIkaren¼man¦sbI 321 x,x,x beg;ItánCasIVútFrNImaRt

ena¼eKman½ )4(xx.x 2231 .

tam )3( nig )4( eKán 332 )10m(x naM[ 10mx 2


© 2008 Lim Phalkun - 148 -

tam mx10mx:)1( 31 naM[ 10xx 31

tam 14m9mxx)xx(x:)2( 231312

14m014m

014m9m100m20m100m10

14m9m)10m()10)(10m(22

22

dUcen¼ 14m .

x-eda¼RsaysmIkar )E( cMeBa¼témø m Edlán

rkeXIjxagelI ½

cMeBa¼ 14m smIkarGacsresr

0)8x)(4x)(2x(64x56x14x 23

eKTaj¦s 8x,4x,2x 321 .

lMhatTI71

eK[smIkarGBaØatBit

03m9m6m2x)4m3(2xm3x:)E( 2323


k-cUrkMnt´témørbs´ m edIm,I[smIkaren¼man¦sbIbeg;Itán

CasIVútnBVnþmYy .

x-cUreda¼RsaysmIkar )E( cMeBa¼témø m Edlán.


© 2008 Lim Phalkun - 149 -

dMeNa¼Rsay

k-kMnt´témørbs´ m

tag 321 x,x,x Ca¦srbs´smIkar )E( .

tamRTwsþIbTEvüteyIgman ½

)3(3m9m6m2xxx

)2(8m6xxxxxx

)1(m3xxx

23321

313221

321

ebIsmIkaren¼man¦sbI 321 x,x,x beg;ItánCasIVútnBVnþ

ena¼ )4(x2xx 231

tam )1( nig )4( eKán m3x3 2 naM[ mx 2

tam 8m6xx)xx(x:)2( 31312

8m6xxm2 312

naM[ )5(8m6m2xx 231

tam )3( nig )5( eKán 3m9m6m2)8m6m2(m 232

¦ 3m .

x-eda¼RsaysmIkar )E(

cMeBa¼ 3m smIkarGacsresr ½

024x26x9x 23


© 2008 Lim Phalkun - 150 -

0)4x)(3x)(2x(

0)8x4x2x)(3x(

0)3x(8)3x(x6)3x(x

0)24x8()x18x6()x3x(

024x26x9x

2

2

223

23

eKTaj¦s 4x,3x,2x 321 .

lMhatTI72

eK[sIVúténcMnYnBit )U( n kMntédayTMnak´TMng ½

2

2U1 nig 2

U1U n

1n

cMeBa¼RKb´ *INn .

k-cUrbgHajfacMeBa¼RKb´ *INn eKman 1U2

2n .

x-cUrbgHajfa *INn,2

cosU1nn

rYcTajrklImIt nnUlim

dMeNa¼Rsay

k-bgHajfa 1U2

2n cMeBa¼RKb´ *INn

eKman 2

2U1 naM[ 1U

2

21 Bit

snµtfavaBitdl´tYTI k KW 1U2

2k Bit

eyIgnwgRsayfavaBitdl´tYTI 1k KW 1U2

21k Bit

eyIgman 1U2

2k


© 2008 Lim Phalkun - 151 -

12

U1

4

22

12

U1

4

22

2U12

21

k

k

k

eday 2

2

4

22

ena¼eKTaj 1

2

U1

2

2 k

¦ 1U2

21k Bit ( eRBa¼ 2

U1U k

1k

) .

dUcen¼ 1U2

2n cMeBa¼RKb´ *INn .

x-bgHajfa *INn,2

cosU1nn

eyIgman 111 2cos

4cos

2

2U

Bit .

snµtfavaBitdl´tYTI k KW 1kk 2cosU

Bit

eyIgnwgRsayfavaBitdl´tYTI 1k KW 2k1k 2cosU

Bit

eyIgman 2

U1U k

1k

Ettamkarsnµt 1kk 2cosU

eKán 2k

2k

2

1k

1k 2cos

22

cos2

22

cos1U

Bit .

dUcen¼ *INn,2

cosU1nn

.


© 2008 Lim Phalkun - 152 -

TajrklImIt nnUlim

eyIgán 10cos2

coslimUlim1nnnn

dUcen¼ 1Ulim nn

.

lMhatTI73

eK[sIVúténcMnYnBit )U( n kMntéday ½

*INn,)U3)(U2)(U1(U1

4

1U

1U

nnnn1n

1

cUrKNna nU CaGnuKmn_én n .

dMeNa¼Rsay


eyIgman )U3)(U2)(U1(U14

1U nnnn1n

)1()2

3U(

2

3U

2

3)

2

3U(

4

3U3U1U3U

4

1U

)1U3U(4

11)U3U(2)U3U(

4

1U

1)2U3U)(U3U(4

1U

1)2U)(1U()3U(U4

1U

2

n1n

2

nn

2

nn

2

n1n

2

n

2

nn

2

n

2

n

2

n1n

n

2

nn

2

n1n

nnnn1n


© 2008 Lim Phalkun - 153 -

tag )2

3Uln(V nn naM[ )2()

2

3Uln(V 1n1n

yk )1( CYskñúg )2( eKán nn

2

n1n V2)2

3Uln(2)

2

3Uln(V

naM[ )V( n CasIVútFrNImaRtmanersug 2q

nigtY 2

5ln)

2

3Uln(V 11

tamrUbmnþ 1n21n1n

1n )2

5ln(

2

5ln2q.VV

eday )

2

3Uln(V nn

eKTaján 1n2n )

2

5(

2

3U

¦ 1n2

n 2

5

2

3U

.

dUcen¼

1n2

n 2

5

2

3U

.

lMhatTI74

eK[sIVúténcMnYnBit )U( n kMntélI IN eday ½

INn,U2U

2U

n1n

0

k¿cUrKNna nU CaGnuKmn_én n .

x¿KNnaplKuN n210n U....UUUP .

dMeNa¼Rsay

KNna nU CaGnuKmn_én n ½

eyIgman 4

cos22U0


© 2008 Lim Phalkun - 154 -

8

cos24

cos22U2U 01

«bmafavaBitdl´tYTI p KW 2pp 2

cos2U

eyIgnwgRsayfavaBitdl´tYTI )1p( KW 3p1p 2

cos2U

Bit

eyIgman p1p U2U

Ettamkar«bma 2pp 2

cos2U

eyIgán 3p3p

2

2p1p 2cos2

2cos4

2cos22U

Bit

dUcen¼ 2nn 2cos2U

.

x¿ KNnaplKuN n210n U....UUUP

tamrUbmnþ acosasin2a2sin naM[ asin

a2sinacos2

2n2n

n

0k

n

0k

2k

1kn

0k2kkn

2sin

1

2sin

2sin

)

2sin

2sin

()2

cos2()U(P

lMhatTI75

eK[sIVúténcMnYnBit )U( n kMntélI IN eday ½

2

2U0 nig INn,

2

U11U

2

n1n

KNna nU CaGnuKmn_én n .

dMeNa¼Rsay

KNna nU CaGnuKmn_én n ½


© 2008 Lim Phalkun - 155 -

eyIgman 4

sin2

2U0

8

sin2

4sin11

2

U11U

22

01

«bmafavaBitdl´tYTI p KW 2pp 2

sinU

eyIgnwgRsayfavaBitdl´tYTI )1p( KW 3p1p 2

sinU

Bit

eyIgman 2

U11U

2

p

1p

Ettamkar«bma

2pp 2sinU

eyIgán 2

2sin11

U2p

2

1p

3p

3p

2

2p

2sin

22

sin2

22

cos1

Bit

dUcen¼ 2nn 2sinU

.

lMhatTI76

k¿cUrRsayfa )xtan3x3tan(8

1

xtan31

xtan2

3

x¿cUrKNnaplbUk

n

0k

k

2

k

3k

n

3

atan31

3

atan3

S

dMeNa¼Rsay

k¿ Rsayfa )xtan3x3tan(8

1

xtan31

xtan2

3


© 2008 Lim Phalkun - 156 -

tamrUbmnþ xtan31

xtanxtan3x3tan

2

3

eyIgán xtan31

xtan8xtan3

xtan31

xtanxtan3xtan3x3tan

2

3

2

3

dUcen¼ )xtan3x3tan(8

1

xtan31

xtan2

3

.

x¿KNnaplbUk

n

0k

k

2

k

3k

n

3

atan31

3

atan3

S

eyIgman )xtan3x3tan(8

1

xtan31

xtan2

3

edayyk k3

ax

eKán )3

atan3

3

a(tan

8

1

3

atan31

3

atan

k1k

k

2

k

3

eyIgán

n

1n

k

1kn

0k1k

kn 3

atan3a3tan

8

1)

3

atan3

3

atan3(

8

1S

dUcen¼ n

1n

n 3

atan

8

3

8

a3tanS

.


© 2008 Lim Phalkun - 157 -

lMhatTI77

k¿cUrRsayfa xtanx2tan2

1

xtan1

xtan2

3

x¿cUrKNnaplbUk

n

0kn

2

n

3k

n

2

atan1

2

atan2

S

dMeNa¼Rsay

k¿Rsayfa xtanx2tan2

1

xtan1

xtan2

3

tamrUbmnþ xtan1

xtan2x2tan

2

eyIgán xtan1

xtanxtan

xtan1

xtanxtanx2tan

2

12

3

2

dUcen¼ xtanx2tan2

1

xtan1

xtan2

3

.

x¿KNnaplbUk

n

0k

k

2

k

3k

n

2

atan1

2

atan2

S

eKman xtanx2tan2

1

xtan1

xtan2

3

yk k2

ax

eKán k1k

k

2

k

3

2

atan

2

atan

2

1

2

atan1

2

atan

eyIgán n

nn

0kk

k

1k

1kn 2

atan2a2tan

2

1

2

atan2

2

atan2S

dUcen¼ n

nn 2

atan2a2tan

2

1S .

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