New Microsoft Word Document 3 .doc - pdfMachine from ... · 3 n 2 n 3 n n1 1 k-eKtag ,nIN* 2U1 U2 V...
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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
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GarmÖkfa
esovePA BiPBsVúItcMnYnBitEdlGñksikßakMBugkan´enAkñúgéden¼
xMJúáTánxitxMRsavRCav nig niBnæeLIgkñúgeKalbMNgTukCaÉksar
RsavRCavsRmab´GñksikßaEdlmanbMNgcg´ec¼ cg´dwgGMBIsIVút
éncMnYnBit[kan´Etc,as´ . enAkñúgesovePA en¼ánRbmUlpþMúnUv
RbFanlMhat´sIVútcMnYnBitya¨geRcIn nigmanlkçN¼xusEbøk@Kña .
RbFanlMhat´nImYy@xMJúáTánxitxMeRCIserIsya¨gsRmitsRmaMgbMputRBm
TaMgeFIVdMeNa¼Rsayya¨gek,a¼k,ayEdlGac[Gñksikßagayyl´nigqab´
cgcaMGMBIviFIsaRsþeFIVdMeNa¼RsaylMhat´nImYy@ . buEnþeTa¼Caya¨gNa
k¾eday kgV¼xatbec©keTs Kruekaslü nigkMhusGkçraviruTæ Rákd
CaekItmaneLIgedayGectnaCaBMuxaneLIy . GaRs&yehtuen¼
xMJúáTCaGñkniBnæ rg´caMTTYlnUvmtiri¼Kn´EbbsSabnaBIsMNak´Gñksikßa
kñúgRKb´mCÄdæanedaykþIesamnsßrIkrayCanic©edIm,IEklMGesovePAen¼
[kan´EtmansuRkitüPaBEfmeTot .
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
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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© .
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BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 2 -
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 3 -
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 4 -
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 5 -
RbFanlMhat´
1-eK[ )u( n CasIVúténcMnYnBitkMnt´elI IN kMnt´eday
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 kMnt´eday
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 kMnt´eday ½
5u 0 nig 5nu2
1u n1n
( INn )
cUrKNna nu énsIVút )u( n CaGnuKmn_én n .
4- eK[sIVúténcMnYnBit )u( n kMnt´eday ½
10u 0 nig 1n2n3
2u
3
1u 2
n1n
( INn )
cUrKNna nu énsIVút )u( n CaGnuKmn_én n .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 6 -
5- eK[sIVúténcMnYnBit )u( n kMnt´eday ½
1u 0 nig nn1n 3)3n(u2u
( INn )
cUrKNna nu énsIVút )u( n CaGnuKmn_én n .
6- eK[sIVúténcMnYnBit )u( n kMnt´eday ½
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 kMnt´eday 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 kMnt´eday
)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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 7 -
9- eKmansIVút )u( n kMnt´eday
)1n(n
3).2n7(u
3
2u
3
11u
1n
n1n
1
k¿ tag n
3uv:*INn
n
nn . bgHajfa )v( n CasIVútFrNImaRt
x¿ KNna nv rYcTajrktémøén nu CaGnuKmn_én n .
10- eKmansIVút )u( n kMnt´eday
)1eln(u
0unu
1n
1
k¿ tag nun ev:*INn . rkRbePTénsIVút )v( n .
x¿ KNna nv rYcTajrktémøén nu CaGnuKmn_én n .
11- eKmansIVút )u( n kMnt´eday
n
nn u
uu
u
4
2
1
1
1
k¿ tag 2
1:*
n
nn u
uvINn . bgHajfa )v( n CasIVútFrNImaRt
x¿ KNna nv rYcTajrktémøén nu CaGnuKmn_én n .
12- eKmansIVút )u( n kMnt´eday
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
x¿ KNna nv rYcTajrktémøén nu CaGnuKmn_én n .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 8 -
13- eK[sIVúténcMnYnkMupøic )Z( n kMnt´eday 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 .
14- eKmansIVút )u( n kMnt´eday
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 9 -
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
17- eKmansIVút )u( n kMnt´eday
nnn nnuu
u
2).1(
12
2
1
1
cUrKNna nu CaGnuKmn_én n .
18- eKmansIVút )u( n nig )( nv kMnt´eday
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 10 -
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 kMnt´elI 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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 11 -
21-eK[sVúIténcMnYnBit )( nU kMnt´elI 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 kMnt´eday 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 éncMnYnBitkMnt´elI *IN eday ½
)2n)(1n(n
1u n
.
k¿kMnt´cMnYnBit 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_kMnt´eday )2x)(1x(x
1)x(g
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 12 -
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 kMnt´eday )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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 13 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 14 -
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 kMnt´eday
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 kMnt´eday
INn;
2
3u
2
1u
2u
n1n
0
k¿ kMnt´cMnYnBit k edIm,I[ kuv nn CasIVútFrNImaRtmYy .
x¿ cUrKNna nu CaGnuKmn_én n nig nnulim
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 15 -
32- eK[sIVúténcMnYnBit )u( n kMnt´eday ½
5u 0 nig 3n2u2
1u n1n
( INn )
cUrKNna nu énsIVút )u( n CaGnuKmn_én n .
33- f CaGnuKmn_kMnt´edayTMnakTMng 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 kMnt´eday 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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 16 -
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 ?
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 17 -
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 .
42- eK[sIVút )U( n kMnt;eday
INn,1nnU3
1U
1U
2n1n
0
cUrKNna nU CaGnuKmn_én n rYcTajrk
2n
n n
Ulim .
43- eK[sIVút )U( n kMnt;eday
*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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 18 -
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 .
x> cUrKNna nU CaGnuKmn_én n .
49- eK[sVúItnBVnþ )U( mantY ......17,13,9,5 nigsIVútnBVnþ )V(
mantY ,....25,19,13,7 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 19 -
k> etIkñúgcMeNam $( tYénsIVút U manb:unaµntYEdlCakaerR)akd ?
cUrkMnt;rktémøéntYTaMgenaH.
x> etIkñúgcMeNam $( tYénsIVút V manb:unaµntYEdCakaerR)akd ?
cUrkMnt;rktémøéntYTaMgenaH.
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
1kn321kn U....UUU)U(S .
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 20 -
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
k> cUrKNnaedrIev x'''f,)x(''f,)x('f nig xf )4( .
x> cUrbgðajfaedrIevTI n énGnuKmn_ f manTRmg; ³
x
nn
)n( e).xcosbxsina()x(f Edl na nig nb CasIVútcMnYnBit
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 21 -
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
x-eKyk *INn,VlnW nn .
cUrbgHajfa INn,n )W( CasIVútFrNImaRt rYcKNna nW
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 22 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 23 -
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 24 -
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 .
k> cUrbgðajfa )V( n CasIVútFrNImaRt rYcKNna nV CaGnuKmn_én
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 kMnt´cMeBa¼RKb´ *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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 25 -
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¨ra¨Em¨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
Edl m Caá¨ra¨Em¨Rt .
k-cUrkMnt´témørbs´ m edIm,I[smIkaren¼man¦sbIbeg;It
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 26 -
ánCasIVútnBVnþmYy .
x-cUreda¼RsaysmIkar )E( cMeBa¼témø m Edlánrk
eXIjxagelI .
72- eK[sIVúténcMnYnBit )U( n kMnt´edayTMnak´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 kMnt´eday ½
*INn,)U3)(U2)(U1(U1
4
1U
1U
nnnn1n
1
cUrKNna nU CaGnuKmn_én n .
74-eK[sIVúténcMnYnBit )U( n kMnt´elI INeday ½
INn,U2U
2U
n1n
0
k¿cUrKNna nU CaGnuKmn_én n .
x¿KNnaplKuN n210n U....UUUP .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 27 -
75-eK[sIVúténcMnYnBit )U( n kMnt´elI 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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 28 -
lMhatTI1
eK[ )u( n CasIVúténcMnYnBitkMnt´elI INkMnt´edayTMnak´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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 29 -
lMhatTI2
eK[sIVúténcMnYnBit )( nu kMnt´eday
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 30 -
lMhatTI3
eK[sIVúténcMnYnBit )u( n kMnt´eday ½
5u 0 nig 5nu2
1u n1n
( INn )
cUrKNna nu énsIVút )u( n CaGnuKmn_én n .
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 31 -
lMhatTI4
eK[sIVúténcMnYnBit )u( n kMnt´eday ½
10u 0 nig 1n2n3
2u
3
1u 2
n1n
( INn )
cUrKNna nu énsIVút )u( n CaGnuKmn_én n .
dMeNa¼Rsay
KNna nu énsIVút )u( n CaGnuKmn_én n
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
TMnak´TMng * eTACa n1n v3
1v
naM[ )v( n CasIVútFrNImaRtmanersug3
1q
tamrUbmnþ n0n qvv eday 9n6nvcbnanvu 2
n2
nn
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 32 -
cMeBa¼ 0n eKán 9vu 00 naM[ 19109uv 00
eKán nn 3
1v . dUcen¼ 2
nn )3n(3
1u .
lMhatTI5
eK[sIVúténcMnYnBit )u( n kMnt´eday ½
1u 0 nig nn1n 3)3n(u2u
( INn )
cUrKNna nu énsIVút )u( n CaGnuKmn_én n .
dMeNa¼Rsay
KNna nu énsIVút )u( n CaGnuKmn_én n
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 33 -
tamrUbmnþ n0n qvv eday n
nn
nn 3.nv3)ban(vu
cMeBa¼ 0n eKán 1vu 00 eKán nn 2v .
dUcen¼ nnn 3.n2u .
lMhatTI6
eK[sIVúténcMnYnBit )u( n kMnt´eday ½
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 34 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 35 -
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 kMnt´eday 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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 36 -
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 kMnt´eday
)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 .
dMeNa¼Rsay
k¿ bgHajfa )v( n CasIVútFrNImaRt
eyIgman n
2uv nn
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 37 -
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 38 -
lMhatTI9
eKmansIVút )u( n kMnt´eday
)1n(n
3).2n7(u
3
2u
3
11u
1n
n1n
1
k¿ tag n
3uv:*INn
n
nn . bgHajfa )v( n CasIVútFrNImaRt
x¿ KNna nv rYcTajrktémøén nu CaGnuKmn_én n .
dMeNa¼Rsay
k¿ bgHajfa )v( n CasIVútFrNImaRt
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
dUcen¼ )v( n CasIVútFrNImaRtmanersug 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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 39 -
lMhatTI10
eKmansIVút )u( n kMnt´eday
)1eln(u
0unu
1n
1
k¿ tag nun ev:*INn . rkRbePTénsIVút )v( n .
x¿ KNna nv rYcTajrktémøén nu CaGnuKmn_én 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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 40 -
lMhatTI11
eKmansIVút )u( n kMnt´eday
n
nn u
uu
u
4
2
1
1
1
k¿ tag 2
1:*
n
nn u
uvINn . bgHajfa )v( n CasIVútFrNImaRt
x¿ KNna nv rYcTajrktémøén nu CaGnuKmn_én n .
dMeNa¼Rsay
k¿ bgHajfa )v( n CasIVútFrNImaRt
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
dUcen¼ )v( n CasIVútFrNImaRtmanersug 3
2q .
x¿ KNna nv rYcTajrktémøén nu CaGnuKmn_én n
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 41 -
lMhatTI12
eKmansIVút )u( n kMnt´eday
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
x¿ KNna nv rYcTajrktémøén nu CaGnuKmn_én n .
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 .
x¿ KNna nv rYcTajrktémøén nu CaGnuKmn_én n
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 42 -
lMhatTI13
eK[sIVúténcMnYnkMupøic )Z( n kMnt´eday
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 43 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 44 -
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
eKmansIVút )u( n kMnt´eday
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 .
K¿ Tajrktémøén nu nig nnu
lim .
dMeNa¼Rsay
k¿ bgHajfa )v( n CasIVútFrNImaRt ½
eyIgman nnn uuv 1
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 45 -
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 .
K¿ Tajrktémøén nu nig nnu
lim
eyIgman nnn uuv 1
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 46 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 47 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 48 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 49 -
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
eKmansIVút )u( n kMnt´eday
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 50 -
eKTaj )1
5(2
1)2
11(
2
11 n
un
unnn
dUcen¼ nn n
nu
2.
15 .
lMhatTI18
eKmansIVút )u( n nig )( nv kMnt´eday
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 51 -
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 52 -
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 .
K-cUrTajrk nV nig nU CaGnuKmn_én n rYcKNna nnUlim
.
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 53 -
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 54 -
lMhatTI20
eK[GnuKmn_ f kMnt´elI 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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 55 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 56 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 57 -
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 kMnt´elI 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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 58 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 59 -
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 kMnt´eday 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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 60 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 61 -
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 éncMnYnBitkMnt´elI *IN eday )2n)(1n(n
1u n
k¿ kMnt´cMnYnBit 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_kMnt´eday )2x)(1x(x
1)x(g
nig n321n V...VVV'S .
cUrbgHajfa dx).x(gS'S1n
1nn
ehIyTajrk n'S nig nn
'Slim
.
dMeNa¼Rsay
k¿ kMnt´cMnYnBit 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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 62 -
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
CaGnuKmn_kMnt´eday )2x)(1x(x
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 63 -
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 64 -
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 65 -
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 66 -
lMhatTI26
eK[GnuKmn_ f kMnt´kñúgsMNMu IR eday )3x3x(3
x)x(f
2
3
k-eKBinitüsIVút )U( n kMnt´eday )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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 67 -
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
tamsRmayxagelIeKman 3
n1n U
31
U
31
eKán
n1n U
31ln3
U
31ln edaytag
n
n U
31lnT
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 68 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 69 -
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 70 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 71 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 72 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 73 -
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 74 -
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 75 -
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(
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 76 -
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 77 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 78 -
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 kMnt´eday
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 79 -
dMeNa¼Rsay
k¿ bgHajfa )v( n CasIVútFrNImaRtmYy ½
eyIgman nn u1v
nn1n
n1n
1n1n
v3
2)u1(
3
2v
3
2u
3
11v
u1v
dUcen¼ )v( n CasIVútFrNImaRtmanersug 3
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 80 -
lMhatTI31
eK[sIVúténcMnYnBit )u( n kMnt´eday
INn;
2
3u
2
1u
2u
n1n
0
k¿ kMnt´cMnYnBit k edIm,I[ kuv nn CasIVútFrNImaRtmYy .
x¿ cUrKNna nu CaGnuKmn_én n nig nnulim
.
dMeNa¼Rsay
k¿ kMnt´cMnYnBit 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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 81 -
tamrUbmnþ n
n0n 2
15qvv
. eday kuv nn
naM[ nn vku
dUcen¼ n
n 2
153u
nig 3ulim nn
.
lMhatTI32
eK[sIVúténcMnYnBit )u( n kMnt´eday ½
5u 0 nig 3n2u2
1u n1n
( INn )
cUrKNna nu énsIVút )u( n CaGnuKmn_én n .
dMeNa¼Rsay
KNna nu énsIVút )u( n CaGnuKmn_én n
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
TMnak´TMng * eTACa n1n v2
1v
naM[ )v( n CasIVútFrNImaRt
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 82 -
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_kMnt´edayTMnak´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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 83 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 84 -
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 kMnt´eday 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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 85 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 86 -
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 87 -
)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( .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 88 -
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 89 -
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 .
k> cUrbgðajfa )V( n CasIVútFrNImaRt rYcKNna nV CaGnuKmn_én
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 90 -
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 91 -
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 92 -
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 .
X> KNnaplbUk )b.a(S kk
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 93 -
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
.
X> KNnaplbUk )b.a(S kk
n
1kn
eyIgman 3k
1a k nig kb k naM[ 1k3)3
k
1(kb.a kk
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 94 -
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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 95 -
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
eK[sIVút )U( n kMnt;eday
INn,1nnU
3
1U
1U
2
n1n
0
cUrKNna nU CaGnuKmn_én n rYcTajrk
2
n
n n
Ulim .
dMeNa¼Rsay
KNna nU CaGnuKmn_én n
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 96 -
)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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 97 -
TajrklImIt
2n
n n
Ulim
tamsRmayxagelIeKman 4
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 98 -
lMhatTI43
eK[sIVút )U( n kMnt;eday
*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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 99 -
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
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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 .
BiPBsIVúténcMnYnBit
© 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 .
x> cUrKNna nU CaGnuKmn_én n .
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
BiPBsIVúténcMnYnBit
© 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 .
x> cUrKNna nU CaGnuKmn_én n .
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
BiPBsIVúténcMnYnBit
© 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 ¦ .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 107 -
lMhatTI49
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 ?
cUrkMnt;rktémøéntYTaMgenaH.
K> etIkñúgcMeNam $(tYénsIVútTaMgBIrenHmanb:unµantYEdlmantémøesµIKña ?
dMeNa¼Rsay
k> etIkñúgcMeNam $( tYénsIVút U manb:unaµntYEdlCakaerR)akd ?
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
BiPBsIVúténcMnYnBit
© 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
x> etIkñúgcMeNam $( tYénsIVút V manb:unaµntYEdCakaerR)akd ?
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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 109 -
-cMeBaH 1r enaH 0111r 22 yk
-cMeBaH 2r enaH 3121r 22 ¬minyk¦
-cMeBaH 3r enaH 8131r 22 ¬minyk¦
-cMeBaH 4r enaH 15141r 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
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 110 -
2
48
228
220
28
172891)48(6V
131691)28(6V
111211)20(6V
7491)8(6V
K> etIkñúgcMeNam $(tYénsIVútTaMgBIrenHmanb:unµantYEdlmantémøesµIKña ?
-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 .
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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
1kn321kn U....UUU)U(S .
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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 113 -
lMhatTI52
eK[sIVút 22
n
)3
2n(
1
)3
1n(
11U
cMeBaHRKb; *INn .
cUrKNnaplbUk
n
1kn321kn U....UUU)U(S .
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
.
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 117 -
lMhatTI55
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( .
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
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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
k> cUrKNnaedrIev x'''f,)x(''f,)x('f nig xf )4( .
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
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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
)n( e).xcosbxsina()x(f
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
BiPBsIVúténcMnYnBit
© 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
)n( e).xcosbxsina()x(f
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
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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
x-eKyk *INn,VlnW nn .
cUrbgHajfa INn,n )W( CasIVútFrNImaRt rYcKNna nW
CaGnuKmn_én n .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 127 -
K-cUrTajrk nV nig nU CaGnuKmn_én n rYcKNna nnUlim
.
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
BiPBsIVúténcMnYnBit
© 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
.
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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 ¦
BiPBsIVúténcMnYnBit
© 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 .
BiPBsIVúténcMnYnBit
© 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
.
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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 .
BiPBsIVúténcMnYnBit
© 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
.
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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
.
BiPBsIVúténcMnYnBit
© 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
nig 2n21n1nn1nn )13(2yyxx
BiPBsIVúténcMnYnBit
© 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
nig 2n21n1nn1nn )13(2yyxx
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 .
BiPBsIVúténcMnYnBit
© 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 kMnt´cMeBa¼RKb´ *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
BiPBsIVúténcMnYnBit
© 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
.
BiPBsIVúténcMnYnBit
© 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
.
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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 .
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 147 -
lMhatTI70
eK[smIkardWeRkTIbIGBaØatBit ½
01000m300m30mx)14m9m(mxx:)E( 23223
Edl m Caá¨ra¨Em¨Rt .
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
BiPBsIVúténcMnYnBit
© 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
Edl m Caá¨ra¨Em¨Rt .
k-cUrkMnt´témørbs´ m edIm,I[smIkaren¼man¦sbIbeg;Itán
CasIVútnBVnþmYy .
x-cUreda¼RsaysmIkar )E( cMeBa¼témø m Edlán.
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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 kMnt´edayTMnak´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
BiPBsIVúténcMnYnBit
© 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
.
BiPBsIVúténcMnYnBit
© 2008 Lim Phalkun - 152 -
TajrklImIt nnUlim
eyIgán 10cos2
coslimUlim1nnnn
dUcen¼ 1Ulim nn
.
lMhatTI73
eK[sIVúténcMnYnBit )U( n kMnt´eday ½
*INn,)U3)(U2)(U1(U1
4
1U
1U
nnnn1n
1
cUrKNna nU CaGnuKmn_én n .
dMeNa¼Rsay
KNna nU CaGnuKmn_én n
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
BiPBsIVúténcMnYnBit
© 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 kMnt´elI 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
BiPBsIVúténcMnYnBit
© 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 kMnt´elI IN eday ½
2
2U0 nig INn,
2
U11U
2
n1n
KNna nU CaGnuKmn_én n .
dMeNa¼Rsay
KNna nU CaGnuKmn_én n ½
BiPBsIVúténcMnYnBit
© 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
BiPBsIVúténcMnYnBit
© 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
.
BiPBsIVúténcMnYnBit
© 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 .