គណះកមមករនពនិ្ឋនងិ · 2010. 8. 27. · GñksikSakñúgRKb;mCÆdæan edIm,IEktRmUvesovePAenH[kan;EtmansuRkitüPaB EfmeTot . CaTIbBa©b; eyIgxMJúGñkeroberog
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គណះកមម ករនពនឋ នង េរៀបេរៀង
េ ក លម ផលគន
េ ក ែសន ពសដឋ
គណះកមម កររតតពនតយបេចចកេទស
េ ក លម ឆន េ ក អង ស ង
េ ករស ទយ រ េ ក ទតយ េមង
េ ក នន សខ េ ក រពម សនតយ
គណះកមម កររតតពនយអកខ វរទឋ
េ ក លម មគកសរ
ករយកពយទរ រចនទពរ នង រកប
េ ក អង ស ង េ ក រព ម
កញញ ល គ ណ ក
រមភកថ
sYsþImitþGñksikSa CaTIemRtI ! ! esovePA smIkarGnuKmn_ EdleyIgxJMú)aneroberogenH rYmman KnøwHedaH RsaysmIkarGnuKmn_ nig lMhat;KMrU nig EpñkcugeRkayCaviBaØasaeRCIserIs. kMhusqÁgnana TaMgbec©keTs nig GkçraviruTæR)akdCaekItmanedayGectna BMuxaneLIy GaRs½yehtuenH eyIgxJMúrg;caMCanic©nUvmtiriHKn;EbbsßabnaBIsMNak; GñksikSakñúgRKb;mCÆdæan edIm,IEktRmUvesovePAenH[kan;EtmansuRkitüPaB EfmeTot . CaTIbBa©b; eyIgxMJúGñkeroberog sUmCUnBrdl;GñksikSaTaMgGs; TTYl)aneCaKC½ykñúgCIvit nig mansuxPaBl¥ . )at;dMbgéf¶TI 12 Ex ]sPa qñaM 2010 Gñkeroberog lwm plÁún Tel : 017 768 246
សមករអនគមន
- 1 -
េមេរៀនសេងខប
អនគមន ១-នយមនយ
E នង F ជសណមនទេទ ។
េបើទនកទនង f ពសណ E េទសណ F ភជ បធត
នមយៗៃនសណE េទនងធតែតមយគតៃនសណF
េនះទនកទនង f េ ថអនគមនពសណE េទ F
េគកនតសរេសរ FE:f →
)x(fyx =a
២-ែដនកនត នង សណរបភព
ែដនកនតៃនអនគមន f គជសណអេថរ x ែដល
ប ត លឲយអនគមន )x(fy = មនតៃមល ។
សណរបភពៃនអនគមន f ែដល ងេ យ I
គសណតៃមល )x(fy = ចេពះរគប x េនេលើែដនកនត
របស ។
សមករអនគមន
- 2 -
៣-អនគមនរចស
សនមតថេគមនអនគមន FE:f → ។
ចេពះអនគមន f េបើេគ ចបេងកើតអនគមន 1f − េ យ
ភជ បធតៃនសណ F េទសណ E វញ េនះេគថ 1f −
ជអនគមនរចសៃនអនគមន f ។
េដើមបរកអនគមន រចសៃនអនគមន f េគរតវ - ជនស )x(f េ យ y
- បតរ x ជ y រច y ជ x េគបនសមករមយរច
ទញរកតៃមលរបស y ។
- េបើសមករថមេនះមន ងy ជអនគមនៃន x េទេនះ
អនគមន f គម នអនគមនរចសេទ ។
- ជនស y េ យ )x(f 1− េគបនអនគមនរចស ។
េបើ f នង g ជអនគមនរចសគន េនះរកបរបស
ឆលះ គន េធៀបនងបនទ តពះទមយៃនអក កអរេ េន ។
សមករអនគមន
- 3 -
៤-អនគមនអចសបណងែសយល
អនគមនអចសបណងែសយល ជអនគមនកនត
េ យ xa)x(fy == ែដល IRx∈ នង a ជចននពត
វជជមន នងខសព 1 ។
រកបៃនអនគមនអចសបណងែសយល
2 3-1-2
2
3
0 1
1
x
y
)1a(,ay:)C( x >=
សមករអនគមន
- 4 -
ចេពះរគបចននពត 0a > នង 1a ≠ េគបន
1/ kxaa kx =⇔=
2/ )x(g)x(faa )x(g)x(f =⇔=
2 3 4-1-2
2
3
0 1
1
x
y
)1a0(,ay:)C( x <<=
សមករអនគមន
- 5 -
៥-អនគមនេ ករត
េបើេគមន xay = េនះ ylogx a= ែដល 0a,0y >> នង 1a ≠ ។ េគថ xa)x(f = មនអនគមនរចស xlog)x(f a
1 =− ។
ដចេនះ xlogy a= េ ថអនគមនេ ករតៃន x
មនេគល a ។
លកខណះៃនេ ករត
រគបចននពតវជជមន x នង y , 1a,0a ≠> េគមន
1/ ylogxlog)xy(log aaa +=
2/ ylogxlogyxlog aaa −=⎟⎠
⎞⎜⎝
⎛
3/ xlognxlog an
a =
4/ alog
1xlogx
a =
5/ 1aloga =
6/ 01loga =
7/ ba bloga =
សមករអនគមន
- 6 -
រកបៃនអនគមនេ ករត
2 3 4-1
2
-1
0 1
1
x
y
2 3 4-1
2
3
-1
0 1
1
x
y
1a,xlogy a >=
1a0,xlogy a <<=
សមករអនគមន
- 7 -
៦-អនគមន គ-េសស
េបើេគមនអនគមន )x(fy = មនែដនកនត D
- េគថ f ជអនគមនគលះរ ែត ⎩⎨⎧
=−∈−∈)x(f)x(f
Dx,Dx
- េគថ f ជអនគមនគលះរ ែត ⎩⎨⎧
−=−∈−∈
)x(f)x(fDx,Dx
៧-េដរេវ
េដរេវៃនអនគមន )x(f គជអនគមនកនតេ យ
h
)x(f)hx(flim)x('f0h
−+=
→ ។
របមនតរគះ
1/ 1nn u'nu'yuy −=⇒=
2/ u'vv'u'yuvy +=⇒=
3/ 2vu'vv'u'y
vuy −
=⇒=
4/ u2'u'yuy =⇒=
5/ 2v'v'y
v1y −=⇒=
សមករអនគមន
- 8 -
6/ uu e'u'yey =⇒=
7/ u'u'yulny =⇒=
8/ ucos'u'yusiny =⇒=
9/ usin'u'yucosy −=⇒=
10/ ucos
'u'yutany 2=⇒=
11/ usin
'u'yucoty 2−=⇒=
៨-អនគមនេកនើ -អនគមនចះ
-េគថ f ជអនគមនេកើនេលើចេនល ះ )b,a( លះរ ែត
រគប )b,a(x∈ េគមន 0)x('f > ។
-េគថ f ជអនគមនចះេលើចេនល ះ )b,a( លះរ ែត
រគប )b,a(x∈ េគមន 0)x('f < ។
-េគថ f ជអនគមនេថរេលើចេនល ះ )b,a( លះរ ែត
រគប )b,a(x∈ េគមន 0)x('f = ។
សមករអនគមន
- 9 -
៩-អនគមនខប
ឧបមថ f ជអនគមនមនែដនកនតD។
េគថ f ជអនគមនមនខប p លះរ ែត p ជចនន
វជជមនតចបផតែដល )x(f)px(f:Dx =+∈∀ ។
១០-ខបៃនអនគមនរតេកណមរត
-អនគមន )axsin(y = មនខប |a|
2p π=
-អនគមន )axcos(y = មនខប |a|
2p π=
-អនគមន )axtan(y = មនខប |a|
p π=
-អនគមន )axcot(y = មនខប |a|
p π=
១១-អនគមនទល
ឧបមថ f ជអនគមនមនែដនកនតD។ -េគថអនគមន f ជអនគមនទលេលើចនន M
េលើែដន D លះរ ែត M)x(f:Dx ≤∈∀ ។
-េគថអនគមន f ជអនគមនទលេរកមចនន m
សមករអនគមន
- 10 -
េលើែដន D លះរ ែត m)x(f:Dx ≤∈∀ ។
-េគថអនគមន f ជអនគមនទល េលើែដន D លះរ ែត M)x(fm:Dx ≤≤∈∀ ។
១២-អតបរម េធៀប នង អបបបរមេធៀបៃន អនគមន
- អនគមន f មនអតបរមេធៀបរតង 0x កល
⎩⎨⎧
<
=
0)x(''f0)x('f
0
0
- អនគមន f មនអបបបរមេធៀបរតង 0x កល
⎩⎨⎧
>
=
0)x(''f0)x('f
0
0
១៣-អនគមនប ត ក
ឧបមថេគមនអនគមនពរ f នង g ។
- េគកនត ងអនគមន f ប ត ក g េ យ
)]x(g[f)x(gf =o ។
- េគកនត ងអនគមន g ប ត ក f េ យ
)]x(f[g)x(fg =o ។
សមករអនគមន
- 11 -
រេបៀបេ ះរ យសមករអនគមន
១-រកអនគមន )x(f េ យ គ លទនកទនង )x(v)]x(u[f =
រេបៀបេ ះរ យ
េគមន )x(v)]x(u[f = (1) - ង )t(uxt)x(u 1−=⇒=
-ទនកទនង )1( ចសរេសរ
)]t(u[v)t(f 1−= -បតរ t េ យ x េគបន )]x(u[v)x(f 1−= ។
dUcenH )]x(u[V)x(f 1−= .
ឧទហរណ១
ចរកនតអនគមន )x(f េបើេគដងថ
1x6x8)1x2(f 3 +−=+ ចេពះរគប IRx∈
សមករអនគមន
- 12 -
ដេ ះរ យ
កនតអនគមន )x(f
េគមន )1(1x6x8)1x2(f 3 +−=+
ង 2
1txt1x2 −=⇒=+
ម (1) េគបន 1)2
1t(6)2
1t(8)t(f 3 +−
−−
=
3t3t
13t31t3t3t23
23
+−=
++−−+−=
ដចេនះ 3x3x)x(f 23 +−= ។
ឧទហរណ២
ចរកនតអនគមន )x(f េបើេគដងថ
2x1x2)
1x1x(f
−+
=+− ?
ដេ ះរ យ
កនតអនគមន )x(f
េគមន )1(2x1x2)
1x1x(f
−+
=+−
ង 1xt)1x(1x1xt −=+⇒
+−
=
សមករអនគមន
- 13 -
1t1tx
1tx)1t(1xtxt
−+
−=
−−=−−=+
ម (1) េគបន 2
1t1t
11t1t2
)t(f−
−+
−
+⎟⎠⎞
⎜⎝⎛
−+
−=
1t33t
2t21t1t2t2
−+
=+−−−−+−−
=
ដចេនះ 1x33x)x(f−+
= ។
ឧទហរណ៣
ចរកនតអនគមន )x(f េបើេគដងថ
2x2)2x2xx(f 2 +=+−+ ចេពះរគប IRx∈
ដេ ះរ យ
កនតអនគមន )x(f
េគមន )1(2x2)2x2xx(f 2 +=+−+ ង t2x2xx 2 =+−+
សមករអនគមន
- 14 -
)1t(22tx
2tx)1t(2
xxt2t2x2x
)xt(2x2x
2
2
222
22
−−
=
−=−
+−=+−
−=+−
ម )1( េគបន 1t
4t2t2])1t(2
2t[2)t(f22
−−+
=+−−
=
ដចេនះ 1x
4x2x)x(f2
−−+
= ។
ឧទហរណ៤
ចរកនតអនគមន )x(f េបើេគដងថ
1xx1x)
x1x(f 24
4
+−+
=+ ?
ដេ ះរ យ
កនតអនគមន )x(f
េគមន 1xx
1x)x1x(f 24
4
+−+
=+
សមករអនគមន
- 15 -
1
x1x
x1x
22
22
−+
+=
ង x1xt += ែដល 2
|x||1x||
x1x||t|
2
≥+
=+=
េគបន 2222
x12x)
x1x(t ++=+= ឬ 2t
x1x 2
22 −=+
េគបន 3t2t
12t2t)t(f 2
2
2
2
−−
=−−
−=
ដចេនះ 3x2x)x(f 2
2
−−
= ។
ឧទហរណ៥
ចរកនតអនគមន )x(f េបើេគដងថ
( )2xcosxsin)4
xcos(f −=⎥⎦⎤
⎢⎣⎡ π
− ?
ដេ ះរ យ
កនតអនគមន )x(f
េគមន ( )2xcosxsin)4
xcos(f −=⎥⎦⎤
⎢⎣⎡ π
−
សមករអនគមន
- 16 -
x2sin1
xcosxcosxsin2xsin 22
−=+−=
ង )4
xcos(t π−= ែដល 1t1 ≤≤−
េគបន )xcosx(sin22
4sinxsin
4cosxcost +=
π+
π=
ឬ 1t2x2sin)x2sin1(21)xcosx(sin
21t 222 −=⇒+=+=
េគបន 22 t22)1t2(1)t(f −=−−=
ដចេនះ 2x22)x(f −= ។
ឧទហរណ6
cUrkMnt;rkGnuKmn_ )x(fy = ebIeKdwgfa ³
1x1x)2x2xx(f 2
22
+−
=+−+ cMeBaHRKb;cMnYnBit x . ដេ ះរ យ
kMnt;rkGnuKmn_ )x(fy = ³ eKman )1(
1x1x)2x2xx(f 2
22
+−
=+−+ ebIeyIgtag t2x2xx 2 =+−+
សមករអនគមន
- 17 -
eyIg)an xt2x2x2 −=+−
1t,)1t(2
2tx
2t)1t(x2
2tx2tx2
xtx2t2x2x
)xt(2x2x
2
2
2
222
22
≠−−
=
−=−
−=−
+−=+−
−=+−
yktémø )1t(2
2tx2
−−
= CMnYskñúg )1( eyIg)an ³
8t8t)8t8t(t)t(f
8t8tt8t8t)t(f
4t8t44t4t4t8t44t4t
1])1t(2
2t[
1])1t(2
2t[)t(f
4
3
4
24
224
224
22
22
+−+−
=
+−+−
=
+−++−−+−+−
=+
−−
−−−
=
dUcenH 8x8x
)8x8x(x)x(f 4
3
+−+−
= .
សមករអនគមន
- 18 -
២-រកអនគមន )x(f េ យ គ លទនកទនង
)1()x(C)]x(v[f)x(B)]x(u[f)x(A =+ េដើមបរកអនគមនេនះេគរតវអនវតតនដចខងេរកម
-យក )t(v)x(u = រចទញរក )t(x ϕ= យកជនសកនង (1) េគបនសមករ
))t((C))]t((v[f))t((B)]t(v[f))t((A ϕ=ϕϕ+ϕ -េបើ )t(u))t((v =ϕ េនះេគបន
))t((C)]t(u[f))t((B)]t(v[f))t((A ϕ=ϕ+ϕ -បតរ t ជ x េគបនសមករ
)2())x((C)]x(u[f))x((B)]x(v[f))x((A ϕ=ϕ+ϕ - មសមករ (1) នង (2) េគបនរបពនឋសមករ ែដល ចេ ះរ យរក ))x(u(f ឬ ))x(v(f ។
-ឧបមថ )x(W))x(u(f = , េ យយក z)x(u =
េគទញ )z(ux 1−= េហើយ ))z(u(W)z(f 1−=
-បតរ z ជ x េគបន ))x(u(W)x(f 1−= ជអនគមន
ែដលរតវរក ។
សមករអនគមន
- 19 -
ឧទហរណ
ចរកនតអនគមន )x(f េបើេគដងថ
2x9x4)x21(fx)x21(f2 2 −−=++− ?
ដេ ះរ យ
កនតអនគមន )x(f
េគមន )1(2x9x4)x21(fx)x21(f2 2 −−=++− យក txt21x21 −=⇒+=− ជនសកនង (1) េគបន
2t9t4)t21(ft)t21(f2 2 −+=−−+ បតរ t ជ x េគបន
)2(2x9x4)x21(xf)x21(f2 2 −+=−−+ ម (1) នង (2) េគបនរបពនឋសមករ
⎩⎨⎧
−+=++−−
−−=++−
)2(2x9x4)x21(f2)x21(xf
)1(2x9x4)x21(xf)x21(f22
2
គណសមករ (1) នង 0x ≠ រចសមករ (2) នង 2
បនទ បមកេធវើផលបកអងគនងអងគេគបន 4x16xx4)x21(f)4x( 232 −+−=++
សមករអនគមន
- 20 -
េគទញ 4x
4x16xx4)x21(f 2
23
+−+−
=+
បនទ បពេធវើវធែចកពហធេគទទប ន
1x4)x21(f −=+ , ង 2
1txtx21 −=⇒=+
េគបន 3t21)2
1t(4)t(f −=−−
=
ដចេនះ 3x2)x(f −= ។
ឧទហរណ
eK[GnuKmn_ f kMnt´RKb´ }0;1{IRx −−∈ eday ½ 1x)
x1(f)x(f)1x2(x +=++ . rkGnuKmn_ )x(f rYc
cUrKNna )2009(f....)3(f)2(f)1(fS ++++= . ដេ ះរ យ
rkGnuKmn_ )x(f eKman )1(1x)
x1(f)x(f)1x2(x +=++
CMnYs x eday x1 kñúgsmIkar )1( eKán ½
សមករអនគមន
- 21 -
)2(2xxx)x(f
2xx)
x1(f
1x1)x(f)
x1(f)1
x2(
x1
22
++
=+
+
+=++
dksmIkar )1( nig )2( Gg:nwgGg:eKán ½
2x2x2)x(f
2x)1x(x2
2xxx1x)x(f
2xx)1x2(x
2
22
++
=++
++
−+=⎥⎦
⎤⎢⎣
⎡+
−+
eKTaján 1x
1x1
)1x(x1)x(f
+−=
+=
eKán [ ] ∑∑==
=−=⎟⎠⎞
⎜⎝⎛
+−==
2009
1k
2009
1k 20092008
200911
1k1
k1)k(fS
dUcen¼ 20092008)2009(f....)3(f)2(f)1(fS =++++= .
៣-រកអនគមន )x(f នង )x(g េ យ គ លទនកទនង
⎩⎨⎧
=+=+
)x(C)]x(v[g)x(B)]x(u[f)x(A)x(C)]x(v[g)x(B)]x(u[f)x(A
22222
11111
ឧទហរណ cUrkMnt;rkGnuKmn_ )x(f nig )x(g ebIeKdwgfa ³ 2x)1x3(g2)1x2(f =++−
nig 1x2x2)2x6(g)3x4(f 2 ++−=−−− cMeBaHRKb; IRx∈ .
សមករអនគមន
- 22 -
ដេ ះរ យ
kMnt;rkGnuKmn_ )x(f nig )x(g ³ eKman )1(x)1x3(g2)1x2(f 2=++− nig )2(1x2x2)2x6(g)3x4(f 2 ++−=−−− eyIgtag 3t41x2 −=− naM[ 1t2x −= yk 1t2x −= CYskñúg ¬!¦ eK)an ³ [ ] [ ]
)3(1t4t4)2t6(g2)3t4(f
)1t2(1)1t2(3g21)1t2(2f2
2
+−=−+−
−=+−+−−
ebIeKyk tx = CYskñúg¬@¦ eK)an )4(1t2t2)2t6(g)3t4(f 2 ++−=−−−
tam ¬#¦ nig ¬$¦ eK)anRbB½næ
⎪⎩
⎪⎨⎧
++−=−−−
+−=−+−
)4(1t2t2)2t6(g)3t4(f
)3(1t4t4)2t6(g2)3t4(f2
2
dksmIkar¬#¦nig¬$¦eK)an t6t6)2t6(g3 2 −=− naM[ 2t2)2t6(g 2 −=−
yk 2t6x −= naM[ 6
2xt +=
សមករអនគមន
- 23 -
ehIy 18
)8x)(4x(2)6
2x(2)x(g 2 +−=−
+=
tam ¬$¦naM[ 1t2t2)2t2()3t4(f 22 ++−=−−− naM[ 1t2)3t4(f −=− yk 3t4x −= naM[
43xt +
= eKTaj
21x1)
43x(2)x(f +
=−+
=
dUcenH 18
)8x)(4x()x(g,2
1x)x(f +−=
+=
៤-សមករឌេផរងែសយលល បទមយ
k> niymn½y ³ smIkarDIeprg;EsüllIenEG‘rGUm:UEsnlMdab;TImYymanemKuN efrKWCasmIkarEdlmanTMrg;TUeTACa ( ) IRa,0ay'y:E ∈=− x> cMelIysmIkar³ smIkarDIeprg;EsüllMdab;TImYy ( ) IRa,0ay'y:E ∈=− mancMelIyTUeTACaGnuKmn_TMrg; ( ) axe.kxfy == Edl k CacMnYnBit .
សមករអនគមន
- 24 -
]TahrN_ 1³ cUredaHRsaysmIkarDIepr:g;EsülxageRkam ³ !> 0y2'y =− eday 2a = enaHsmIkarmancMelIyTUeTACaGnuKmn_
IRk,e.ky x2 ∈= . @> 0y3'y =+ eday 3a −= enaHsmIkarmancMelIyTUeTACaGnuKmn_
IRk,e.ky x3 ∈= − . #> 0y4'y =− edaydwgfa ( ) 30y = eday 4a = enaHsmIkarmancMelIyTUeTACaGnuKmn_
IRk,e.ky x4 ∈= eday ( ) 3e.k0y 0 == naM[ 3k = . dUcenHeK)an x4e.3y = CacMelIyrbs;smIkar . $> 6y3'y =− smIkarGacsresr ³
សមករអនគមន
- 25 -
( ) 02y3'y06y3'y=+−
=−− tag 2yz += naM[ 'y'z = eK)an 0z3'z =− naM[ IRk,e.kz x3 ∈= eday 2yz += eK)an x3e.k2y =+ naM[ x3e.k2y +−= CacMelIysmIkar. %> 0'y''y =− tag 'yz = naM[ ''y'z = smIkarGacsresr ³
0z'z =− naM[ IRk,e.kz x ∈= eday 'yz = eKTaj xe.k'y = naM[ )IRc,k(ce.kdx.e.ky xx ∈+== ∫ .
សមករអនគមន
- 26 -
៥-សមករឌេផរងែសយលល បទពរ
k> niymn½y³ smIkarDIepr:g;EsüllIenEG‘Gum:UEsnlMdab;TIBIrEdlman emKuNefrKWCasmIkarEdlmanTMrg;TUeTACa ( ) IRc,b,a,0a,0cy'by''ay:E ∈≠=++ . x> smIkarsMKal; ³ smIkarsMKal;rbs;smIkarDIepr:g;Esül
IRc,b,a,0a,0cy'by''ay ∈≠=++ CasmIkardWeRkTIBIrEdlmanrag 0cbrar 2 =++ . K> cMelIysmIkarDIepr:g;Esül ³ edIm,IrkcMelIysmIkarDIepr:g;Esül
IRc,b,a,0a,0cy'by''ay ∈≠=++ eKRtUvedaHRsaysmIkarsMKal; 0cbrar 2 =++ . -ebI 0ac4b2 >−=Δ enaHsmIkarsMKal;manb¤sBIr 1r nig
សមករអនគមន
- 27 -
2r kñúgkrNIenHsmIkarDIepr:g;EsülmancMelIyTUeTACaGnuKmn_
( ) xrxr 21 e.Be.Axfy +== Edl IRB,A ∈ . -ebI 0ac4b2 =−=Δ enaHsmIkarsMKal;manb¤sDub
021 ra2
brr =−== kñúgkrNIenHsmIkarDIepr:g;Esülman cMelIyTUeTACaGnuKmn_ ³
( ) ( ) xr0e.BAxxfy +== Edl IRB,A ∈ . -ebI 0ac4b2 <−=Δ enaHsmIkarsMKal;manrwsBIrCa cMnYnkMpøicqøas;KñaKW β+α= ir1 nig β−α= ir2 kñúgkrNIenHsmIkarDIepr:g;EsülmancMelIyTUeTACaGnuKmn_
( ) ( ) xe.xsinBxcosAxfy αβ+β== Edl IRB,A ∈ .
សមករអនគមន
- 28 -
]TahrN_ 1 cUredaHRsaysmIkarDIepr:g;EsüsxageRkam ³ !> 0y6'y5''y =+− mansmIkarsMKal; 06r5r 2 =+−
( ) ( ) ( ) 16.145 2 =−−=Δ manb¤s 3
215r,2
215r 21 =
+==
−=
cMelIysmIkar x3x2 e.Be.Ay += Edl IRB,A ∈ .
@> 0y4'y4''y =+− mansmIkarsMKal; 04r4r 2 =+−
( ) ( ) ( ) 04.144 2 =−−=Δ smIkamanb¤sDúb 2
a2brrr 021 =−===
cMelIysmIkarCaGnuKmn_ ( ) x2e.BAxy += Edl IRB,A ∈
សមករអនគមន
- 29 -
#> 0y13'y4''y =+− mansmIkarsMKal; 013r4r 2 =+−
( ) ( ) ( ) 22 i9913.12' =−=−−=Δ
smIkamanb¤sCacMnYnkMupøicqøas;KñaKW ⎢⎣
⎡−=+=
i32ri32r
2
1
eKTaj)an 3,2 =β=α dUcenH cMelIysmIkarCaGnuKmn_
( ) x2e.x3sinBx3cosAy += Edl IRB,A ∈ . ៦-េ ះរ យសមករឌេផរងែសយលែដលមន ង
)x(Eay'y:)E( =− ,
smIkarDIepr:g;EsülenHmancemøIyTUeTA pe yyy += Edl ey CacemøIyénsmIkar 0ay'y =− nig py CacemøIyBiessmYy énsmIkar )x(Eay'y =− .
សមករអនគមន
- 30 -
lMhat;KMrU eK[smIkarDIepr:g;Esül ( ) 6x10x4y4'y:E 2 −+−=− k-kMnt;cMnYnBit b,a nig c edIm,I[GnuKmn_ ( ) cbxaxxy 2
P ++= CacMelIyedayELkmYyrbs;smIkar ( )E . x-bgðajfaGnuKmn_ ( ) ( )xyxyy hP += CacMelIyTUeTArbs; ( )E enaHGnuKmn_ ( )xyh CacMelIyrbs;smIkarGUmU:Esn ( ) 0y4'y:'E =− . K-edaHRsaysmIkar ( )'E rYcTajrkcMelIyTUeTArbs;smIkar ( )E
dMeNaHRsay
kMnt;cMnYnBit b,a nig c
( ) 6x10x4y4'y:E 2 −+−=− edIm,I[GnuKmn_ ( ) cbxaxxy 2
P ++= CacMelIyedayELk
mYyrbs;smIkar ( )E luHRtEtGnuKmn_ ( ) ( )x'y,xy pp nig
សមករអនគមន
- 31 -
( )x''y p epÞógpÞat;nwgsmIkar ( )E .
eK)an ( ) ( ) ( ) 6x10x4xy4x'y:E 2pP −+−=−
eday ( )( )⎪⎩
⎪⎨⎧
+=
++=
bax2x'ycbxaxxy
p
2P
eK)an ( ) ( ) 6x10x4cbxax4bax2 22 −+−=++−+
naM[ ( ) ( ) 6x10x4c4bxa2b4ax4 22 −+−=−+−−−
eKTaj)an ⎪⎩
⎪⎨
⎧
−=−−=−
−=−
6c4b10a2b4
4a4 naM[
⎪⎩
⎪⎨
⎧
=−=
=
1c2b
1a
dUcenH 1c,2b,1a =−== nig ( ) ( )22
P 1x1x2xxy −=+−= .
x-karbgðaj
GnuKmn_ ( ) ( )xyxyy hP += CacMelIyrbs; ( )E
luHRtaGnuKmn_ 'y,y epÞógpÞat;smIkar
សមករអនគមន
- 32 -
edayeKman ( ) ( )x'yx'y'y hp += enaHeK)an ³ ( ) ( )[ ] ( ) ( )[ ]( ) ( )[ ] ( ) ( )[ ] ( )16x10x4xy4x'yxy4x'y
6x10x4xyxy4x'yx'y
2hhpp
2hphp
−+−=−+−
−+−=+−+
tamsRmayxagelIeKman
( ) ( ) ( )26x10x4xy4x'y 2pP −+−=−
¬ eRBaH ( )xyp CacMelIyrbs;smIkar ( )E ¦ .
tamTMnak;TMng ¬! ¦ nig ¬@¦ eKTaj)an ³
( ) ( )[ ] 6x10x4xy4x'y6x10x4 2hh
2 −+−=−+−+− naM[eKTaj)an ³
( ) ( ) 0xy4x'y hh =− TMnak;TMngenHbBa¢ak;faGnuKmn_ ( )xyh
CacMelIyrbs;smIkar ( ) 0y4'y:'E =− .
សមករអនគមន
- 33 -
K-edaHRsaysmIkar ( )'E ³ 0y4'y =−
eday 4a = dUcenHcMelIysmIkar ( )'E CaGnuKmn_
( ) IRk,e.kxy x4h ∈= .
TajrkcMelIyTUeTArbs;smIkar ( )E ³
tamsMrayxagelIcMelIysmIkar ( )E KWCaGnuKmn_TMrg;
( ) ( )xyxyy hp += edayeKman ( ) ( )2p 1xxy −=
nig ( ) x4h e.kxy =
dUcenH ( ) IRk,e.k1xy x42 ∈+−= CacMelIyrbs;smIkar ៧-រេបៀបេ ះរ យសមករឌេផរងែសយលែដលមន ង
)x(Ecy'by''ay:)E( =++ Edl 0a ≠
edIm,IedaHRsaysmIkarenHeKRtUvGnuvtþn_dUcxageRkam ³ -EsVgrkcemøIyBiessminGUm:UEsntageday py énsmIkar
សមករអនគមន
- 34 -
)x(Ecy'by''ay =++ Edl py manTRmg;dUc )x(E . -rkcemøIyTUeTAtageday hy énsmIkarlIenEG‘GUm:UEsn 0cy'by''ay =++ . -eK)ancemøIyTUeTAénsmIkar )E( KWCaGnuKmn_kMNt;eday hP yyy += . lMhat;KMrU eK[smIkarDIepr:g;Esül ( ) 34x24x4y4'y4''y:E 2 +−=+− k-kMnt;cMnYnBit b,a nig c edIm,I[GnuKmn_ ( ) cbxaxxy 2
P ++= CacMelIyedayELkmYyrbs;smIkar ( )E . x-bgðajfaGnuKmn_ ( ) ( )xyxyy hP += CacMelIyTUeTArbs; ( )E enaHGnuKmn_ ( )xyh CacMelIyrbs;smIkarGUmU:Esn ( ) 0y4'y4''y:'E =+− . K-edaHRsaysmIkar ( )'E rYcTajcMelIyTUeTArbs;smIkar ( )E . dMeNaHRsay kMnt;cMnYnBit b,a nig c
សមករអនគមន
- 35 -
( ) 34x24x4y4'y4''y:E 2 +−=+− edIm,I[GnuKmn_ ( ) cbxaxxy 2
P ++= CacMelIyedayELkmYy rbs;smIkar ( )E luHRtEtGnuKmn_ ( ) ( )x'y,xy pp nig ( )x''y p epÞógpÞat;nwgsmIkar ( )E .
eK)an ( ) ( ) ( ) ( ) 34x24x4xy4x'y4x''y:E 2pPp +−=+−
eday ( )( )( )⎪
⎩
⎪⎨
⎧
=
+=
++=
a2x''y
bax2x'ycbxaxxy
p
p
2P
eK)an
( ) ( ) ( ) 34x24x4cbxax4bax24a2 22 +−=++++−
( ) ( ) 34x24x4c4b4a2xa8b4ax4 22 +−=+−+−+
eKTaj)an ⎪⎩
⎪⎨
⎧
=+−−=−
=
34c4b4a224a8b4
4a4 naM[
⎪⎩
⎪⎨
⎧
=−=
=
4c4b
1a
សមករអនគមន
- 36 -
dUcenH 4c,4b,1a −=−== nig ( ) ( )22
P 2x4x4xxy −=+−= .
x-karbgðaj
GnuKmn_ ( ) ( )xyxyy hP += CacMelIyrbs; ( )E
luHRtaGnuKmn_ ''y,'y,y epÞógpÞat;smIkar
edayeKman ( ) ( )x'yx'y'y hp += nig ( ) ( )x''yx''y''y hp +=
enaHeK)an ³ [ ] [ ] [ ][ ] [ ] ( )134x24x4y4'y4''yy4'y4''y
34x24x4yy4'y'y4''y''y
2hhhppp
2hphphp
+−=+−++−
+−=+++−+
tamsRmayxagelIeKman
( ) ( ) ( ) ( )234x24x4xy4x'y4x''y 2pPp +−=+−
¬ eRBaH ( )xyp CacMelIyrbs;smIkar ( )E ¦ .
សមករអនគមន
- 37 -
tamTMnak;TMng ¬! ¦ nig ¬@¦ eKTaj)an ³
[ ] 34x24x4y4'y4''y34x24x4 2hhh
2 +−=+−++− naM[eKTaj)an ( ) ( ) ( ) 0xy4x'y4x''y hhh =+−
TMnak;TMngenHbBa¢ak;faGnuKmn_ ( )xyh CacMelIyrbs;
smIkar ( ) 0y4'y4''y:'E =+− .
K-edaHRsaysmIkar ( )'E ³ 0y4'y4''y =+−
smIkarsMKal; 04r4r 2 =+− 044', =−=Δ
naM[smIkarmanb¤sDúb 2rrr 021 ===
dUcenHcMelIysmIkar ( )'E CaGnuKmn_
( ) ( ) IRB,A,e.BAxxy x2h ∈+= .
TajrkcMelIyTUeTArbs;smIkar ( )E ³
សមករអនគមន
- 38 -
tamsMrayxagelIcMelIysmIkar ( )E KWCaGnuKmn_TMrg;
( ) ( )xyxyy hp += .
edayeKman ( ) ( )2p 2xxy −= nig ( ) ( ) x2
h e.BAxxy +=
dUcenH ( ) ( ) IRB,A,e.BAx2xy x22 ∈++−=
CacMelIyrbs;smIkar ( )E . ៨-រេបៀបេ ះរ យសមករឌេផរងែសយលែដលមន ង
)x(Qy)x(P'y:)E( =+ ែដល 0a ≠
edIm,IedaHRsaysmIkarenHeKRtUvGnuvtþn_dUcxageRkam ³ -KuNGgÁTaMgBIrénsmIkarnwg ∫ dx).x(P
e eK)an ³ )1(e)x(Qye)x(Pe'y
dx).x(Pdx).x(Pdx).x(P ∫∫∫ =+ -tagGnuKmn_ ∫=
dx).x(Peyz eK)an ³
)2(ye)x(Pe'y'zdx).x(Pdx).x(P ∫∫ +=
-tam )1( nig )2( eKTaj)an ³
សមករអនគមន
- 39 -
cdx.e)x(Qze)x(Q'zdx).x(Pdx).x(P
+=⇒= ∫∫∫
-Tajrk ∫−=dx).x(P
e.zy . lMhat;KMrU !> edaHRsaysmIkarDIepr:g;Esül 2x3ex4xy2'y −=+ KuNGgÁTaMgBIrénsmIkarnwg 2xxdx2
ee =∫ eK)an )1(x4yxe2e'y 3xx 22
=+ tagGnuKmn_ 2xyez = eK)an ³ )2(yxe2e'y'z
22 xx += tam )1( nig )2( eK)an ³ Cxdxx4zx4'z 433 +==⇒= ∫ eKTaj)an 22 x4x e)Cx(e.zy −− +== Edl C CacMnYnefrmYyNak¾)an .
សមករអនគមន
- 40 -
@> edaHRsaysmIkarDIepr:g;Esül xsinx2
ex2xcosy'y −=+ KuNGgÁTaMgBIrénsmIkarnwg xsinxdxcos
ee =∫ eK)an )1(xe2excosye'y
2xxsinxsin =+ tagGnuKmn_ xsinyez = eK)an ³ )2(xecosye'y'z xsinxsin += tam )1( nig )2( eK)an ³ Cedxxe2zxe2'z
222 xxx +==⇒= ∫ eKTaj)an xsinxxsin e)Ce(e.zy
2 −− +== Edl C CacMnYnefrmYyNak¾)an . #> edaHRsaysmIkarDIepr:g;Esül xcosx 2
ex2siny'y −=− smIkarGacsresr xcosx 2
e.excosxsiny2'y −=− KuNGgÁTaMgBIrénsmIkarnwg xcosxdxsinxcos2 2
ee =∫− eK)an
សមករអនគមន
- 41 -
)1(exesinxcosy2e'y xxcosxcos 22=−
tagGnuKmn_ xcos2yez = eK)an ³
)2(xesinxcosy2e'y'z xcosxcos 22−=
tam )1( nig )2( eK)an ³ Cedxeze'z xxx +==⇒= ∫ eKTaj)an xcosxxcos 22
e)Ce(e.zy +== Edl C CacMnYnefrmYyNak¾)an . $> edaHRsaysmIkarDIepr:g;Esül xx32 2
e)1xx)(1x2(y)1x2('y −−+++=++ KuNGgÁTaMgBIrénsmIkarnwg xxdx)1x2( 2
ee ++=∫ eK)an
)1()1xx)(1x2(ey)1x2(e'y 32xxxx 22+++=++ ++
tagGnuKmn_ xx2yez += eK)an ³
)2(ye)1x2(e'y'z xxxx 22 ++ ++= tam )1( nig )2( eK)an ³
សមករអនគមន
- 42 -
C)1xx(
41dx)1xx)(1x2(z
)1xx)(1x2('z
4232
32
+++=+++=⇒
+++=
∫
eKTaj)an xx42xx 22e]C)1xx(
41[e.zy −−−− +++==
សមករអនគមន
- 43 -
kRmg viBaØasaKNitviTüaeRCIserIs
nigdMeNaHRsay
សមករអនគមន
- 44 -
viBaØasaKNitviTüaTI1 I-eK[ INn,i
31i
31A
nn
∈⎟⎠⎞
⎜⎝⎛ −−⎟
⎠⎞
⎜⎝⎛ += .
cUrbgðajfa 3
nsin.
)3(2
.iA n
1n π=
+
cMeBaHRKb; INn∈ .
II-cUrKNnalImIt
2x2x2....222limL
2xn −−+++++
=→
¬ man nb¤skaer ¦ III-eK[GaMgetRkal ∫ ++
=1
020 tt1
dtI nig ∫ ∈++
=1
02
n
n )INn(,dt.tt1
tI
k> cUrKNnatémøén 0I rYc Rsayfa )I( n CasIVútcuH. x> RsaybBa¢ak;fa
1n1III 2n1nn +
=++ ++ . K> Taj[)anfa 2n,
)1n(31
I)1n(3
1n ≥∀
−≤≤
+ .
TajrklImIt )In(lim nn +∞→.
សមករអនគមន
- 45 -
dMeNa¼Rsay I-bgðajfa
3nsin.
)3(2.iA n
1n π=
+
cMeBaHRKb; INn∈
eyIgman INn,i3
1i3
1Ann
∈⎟⎠⎞
⎜⎝⎛ −−⎟
⎠⎞
⎜⎝⎛ +=
tag
⎟⎠⎞
⎜⎝⎛ π
+π
=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
+=+=
3sin.i
3cos
32
23i
21
32
33.i1i
31Z
tamrUbmnþdWmr½eK)an ⎟⎠⎞
⎜⎝⎛ π
+π
=⎟⎠⎞
⎜⎝⎛ +=
3nsin.i
3ncos
)3(2i
31Z n
nnn
ehIy ⎟⎠⎞
⎜⎝⎛ π
−π
=3
nsin.i3
ncos)3(
2Z n
nn
eKTaj ⎟⎠⎞
⎜⎝⎛ π
−π
−⎟⎠⎞
⎜⎝⎛ π
+π
=3
nsin.i3
ncos)3(
23
nsin.i3
ncos)3(
2A n
n
n
n
សមករអនគមន
- 46 -
3
nsin.)3(
2.i
3nsin.i
3ncos
3nsin.i
3ncos
)3(2
n
1n
n
n
π=
⎟⎠⎞
⎜⎝⎛ π
+π
−π
+π
=
+
dUcenH 3
nsin.)3(
2.iA n
1n π=
+
.
II-KNnalImIt ³
2x
2x2....222limL
2xn −−+++++
=→
1n1nn
2x2xn
2xn
2xn
L41
41
LL
2x2....222
1lim
2x2x2....22
limL
2x2....222
12x
4x2.....222limL
2x2...222
2x2...2222x
2x2....222limL
−−
→→
→
→
=×=
++++++×
−−++++
=
++++++×
−−+++++
=
++++++
++++++×
−−+++++
=
tamTMnak;TMngenHbBa¢ak;fa *INn),L( n ∈ CasIVútFrNImaRt manersug
41q = .
សមករអនគមន
- 47 -
nigtYTImYy
41
)2x2)(2x(2xlim
)2x2)(2x(4x2lim
2x2x2limL
2x
2x2x1
=++−
−=
++−−+
=−−+
=
→
→→
tamrUbmnþ n
1n1n
1n 41
41.
41qLL =⎟
⎠⎞
⎜⎝⎛=×=
−− .
dUcenH n2xn 41
2x2x2....222limL =
−−+++++
=→
. III-k> KNnatémøén 0I rYc Rsayfa )I( n CasIVútcuH eyIg)an ∫∫
++=
++=
1
0 2
1
020
)t21(
43
dttt1
dtI
tag t21U += naM[ dtdU =
ehIycMeBaH [ ]1,0t ∈∀ enaH ⎥⎦⎤
⎢⎣⎡∈
23,
21U
eK)an 23
21
23
21 22
0 3U2
arctan3
2
U)23
(
dUI ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛=
+= ∫
សមករអនគមន
- 48 -
33633
23
1arctan3
23arctan3
2
π=⎟
⎠⎞
⎜⎝⎛ π
−π
=
−=
dUcenH 33tt1
dtI1
020
π=
++= ∫ .
mü:ageToteKman ∫ ++=
1
02
n
n dt.tt1
tI
nig dt.tt1
tI1
02
1n
1n ∫ ++=
+
+
cMeBaHRKb; [ ]1,0t ∈ eKman n1n tt ≤+ naM[ 2
n
2
1n
tt1t
tt1t
++≤
++
+
eKTaj dt.tt1
tdt.tt1
t1
0
1
02
n
2
1n
∫ ∫ ++≤
++
+
b¤ INn,II n1n ∈∀≤+ . dUcenH )I( n CasIVútcuH .
សមករអនគមន
- 49 -
x> RsaybBa¢ak;fa 1n
1III 2n1nn +=++ ++
eyIg)an
∫∫∫ +++
+++
++=++
++
++
1
02
2n1
02
1n1
02
n
2n1nn tt1dtt
tt1dtt
tt1dttIII
∫
∫∫
+=⎥⎦
⎤⎢⎣⎡
+==
++++
=++++
=
+
++
1
0
1
0
1nn
1
02
2n1
02
2n1nn
1n1t
1n1dtt
tt1dt).tt1(t
tt1dt)ttt(
dUcenH 1n
1III 2n1nn +=++ ++ .
K> Taj[)anfa 2n,)1n(3
1I
)1n(31
n ≥∀−
≤≤+
eyIgman )I( n CasIVútcuH . tamlkçN³énsIVútcuHeyIgman ³
n1n2nn2n1nn IIII3III ++≤≤++ −−++ eday
1n1III 2n1nn +
=++ ++ naM[
1n1
III n1n2n −=++ −−
eKTaj 1n
1I31n
1n −≤≤
+
សមករអនគមន
- 50 -
naM[ 2n,)1n(3
1I
)1n(31
n ≥∀−
≤≤+
. TajrklImIt )In(lim nn +∞→
man 2n,
)1n(31
I)1n(3
1n ≥∀
−≤≤
+
naM[ )1n(3
nnI
)1n(3n
n −≤≤
+
dUcenH ( )31
nIlim nn=
+∞→ .
សមករអនគមន
- 51 -
viBaØasaKNitviTüaTI2 I-eK[sIVúténcMnYnBit )U( n kMnt;eday ³ 3lnU0 = nig INn,)e1(lnU nU
1n ∈+=+ . cUrKNna nU CaGnuKmn_én n . II-edaHRsaysmIkarxageRkamkñúgsMNMukMupøic ³ 0)i32(2z)i71(z)i1(:)E( 2 =−−+−+
III-eK[GaMgetRkal ∫
π
=2
0
3nn dx.xcosxsinI Edl INn∈
k> cUrKNna nI CaGnuKmn_én n . x> cUrKNnaplbUk ( ) n210
n
0kkn I.........IIIIS ++++== ∑
=
CaGnuKmn_én n. rYcTajrktémøénlImIt nn
Slim+∞→
.
សមករអនគមន
- 52 -
IV-eK[cMnYnKt;viC¢man n . eKdwgfa n Ecknwg 7 [sMNl; % ehIy n Ecknwg 8 [sMNl; # . k> etIcMnYn n enaHEcknwg 56 [sMNl;b:unμan ? x> rkcMnYn n enaHedaydwgfa 5626n5616 << .
dMeNa¼Rsay I-KNna nU CaGnuKmn_én n eKman INn,)e1(lnU nU
1n ∈+=+ eKTaj n1n UU e1e +=+ b¤ 1ee n1n UU =−+ efr naM[ ( )nUe CasIVútnBVnþmanplsgrYm 1d = nigtYTImYy 3ee 3lnU0 == . eK)an n3e nU += naM[ )3nln(Un += . II-edaHRsaysmIkarkñúgsMNMukMupøic ³ 0)i32(2z)i71(z)i1(:)E( 2 =−−+−+ eyIgman )i32)(i1(8)i71( 2 −+++=Δ
សមករអនគមន
- 53 -
2)i31(i68
24i16i241649i141
+=+−=Δ
++−+−+=Δ
eKTajb¤s ⎢⎢⎢⎢
⎣
⎡
+=++
=++++
=
+=+
=+−−+
=
i23i1i51
)i1(2i31i71
z
i1i1
i2)i1(2
i31i71z
2
1
dUcenH i23z,i1z 21 +=+= . III-k> KNna nI CaGnuKmn_én n eyIgman
∫
∫π
π
=
=
2
0
2n
2
0
3nn
dx.xcos.xcosxsin
dx.xcosxsinI
tag xsinU = naM[ dx.xcosdU = cMeBaH ]
2,0[x π
∈ naM[ ]1,0[U∈ eyIg)an ³
សមករអនគមន
- 54 -
)3n)(1n(2
)3n)(1n(1n3n
3n1
1n1
U3n
1U
1n1
dU.UdU.UdU).U1(UI
1
0
3n1
0
1n
1
0
1
0
2nn1
0
2nn
++=
++−−+
=+
−+
=
⎥⎦⎤
⎢⎣⎡
+−⎥⎦
⎤⎢⎣⎡
+=
−=−=
++
+∫ ∫∫
dUcenH
)3n)(1n(2In ++
= .
x> KNnaplbUk ( ) n210
n
0kkn I.........IIIIS ++++== ∑
=
tamsRmayxagelIeyIgman ³
⎟⎠⎞
⎜⎝⎛
+−
++⎟
⎠⎞
⎜⎝⎛
+−
+=
+−
+=
++=
3n1
2n1
2n1
1n1
3n1
1n1
)3n)(1n(2In
eyIg)an ³
សមករអនគមន
- 55 -
)3n)(2n(2)8n3)(1n(
3n1
2n1
23
3n1
21
2n11
3k1
2k1
2k1
1k1
3k1
2k1
2k1
1k1S
n
0k
n
0k
n
0kn
++++
=
+−
+−=
⎟⎠⎞
⎜⎝⎛
+−+⎟
⎠⎞
⎜⎝⎛
+−=
⎟⎠⎞
⎜⎝⎛
+−
++⎟
⎠⎞
⎜⎝⎛
+−
+=
⎥⎦⎤
⎢⎣⎡
⎟⎠⎞
⎜⎝⎛
+−
++⎟
⎠⎞
⎜⎝⎛
+−
+=
∑ ∑
∑
= =
=
dUcenH )3n)(2n(2)8n3)(1n(Sn ++
++= .
KNnatémøénlImIt nnSlim
+∞→
eyIgman 3n
12n
123Sn +
−+
−= eyIg)an
23
3n1
2n1
23
lmSlimnnn
=⎟⎠⎞
⎜⎝⎛
+−
+−=
+∞→+∞→.
dUcenH 23Slim nn
=+∞→
.
សមករអនគមន
- 56 -
IV-k> etIcMnYn n enaHEcknwg 56 [sMNl;b:unμan ? tamsmμtikmμeKdwgfa n Ecknwg 7 [sMNl; % naM[man
INq1 ∈ Edl )1(5q7n 1 += ehIymüa:geTot n Ecknwg 8 [sMNl; # enaHnaM[man
INq2 ∈ Edl )2(3q8n 2 += tam ¬!¦ nig ¬@¦ eK)anRbBn½æ
⎩⎨⎧
+=+=
)2(3q8n)1(5q7n
2
1
b¤ ⎩⎨⎧
+=
+=
)4(21q56n7)3(40q56n8
2
1
dksmIkar ¬# ¦ nig ¬$¦ eK)an ³ 19)qq(56n 21 +−= tag INq,qqq 21 ∈−=
eK)an 19q56n += . TMnak;TMngenHmann½yfa cMnYn n enaHEcknwg 56 [sMNl; !( x> rkcMnYn n enaHedaydwgfa 5626n5616 << eKman 19q56n += naM[ 562619q565616 <+< b¤
565607q
565597
<<
សមករអនគមន
- 57 -
b¤ 567100q
565399 +<<+ naM[ 100q = .
cMeBaH 100q = eK)an 5619195600n =+= . dUcenHcMnYn n enaHKW 5619n = .
សមករអនគមន
- 58 -
viBaØasaKNitviTüaTI3 I- eK[sIVút 22n )1n(
1n1
1S+
++= Edl *INn∈ .
k-cMeBaHRKb; *INn∈ cUrbgðajfa 1n
1n1
1Sn +−+= .
x-KNnaplbUk
222222n )1n(1
n1
1.....31
21
121
11
1+
+++++++++=Σ
II- eK[GaMgetRkal 0a,ax
dx.xIa
033
n
n >+
= ∫
k> cUrkMnt;témørbs; n edIm,I[ nI minGaRs½ynwg a . x> KNna nI cMeBaHtémø n Edl)anrkeXIjxagelI . III-eK[cMnYnkMupøic i32Z ++= k> cUrbgðajfam:UDúl 26|Z| += x> sresr Z CaTRmg;RtIekaNmaRtrYcTajfa
426
12cos +
=π .
សមករអនគមន
- 59 -
IV-eKmanGnuKmn_ 3x4x
23x7y 2 +−−
= Edl xCacMnYnBitehIy 3x,1x ≠≠ k> cUrKNnaedrIev 'y nig ''y . x> edaHRsaysmIkar 0''y = .
dMeNa¼Rsay I- k>bgðajfa
1n1
n1
1Sn +−+=
eyIgman 22n )1n(1
n1
1S+
++= ¬ cMeBaHRKb; *INn∈ ¦
1n1
n11
)1n(n1)1n(n
)1n(n]1)1n(n[
)1n(n1)1n(n2)1n(n
)1n(nn1n2n)1n(n
)1n(nn)1n()1n(n
22
2
22
22
22
2222
22
2222
+−+=
+++
=
+++
=
+++++
=
++++++
=
+++++
=
សមករអនគមន
- 60 -
dUcenH 1n
1n11Sn +−+= .
x-KNnaplbUk ³ eK)an
∑=
=+
++++++=Σn
1kk2222n )S(
)1n(1
n11.....
21
111
∑= +
+=
+−+=⎟
⎠⎞
⎜⎝⎛
+−+=
n
1k 1n)2n(n
1n11n
1k1
k11
dUcenH 1n
)2n(nn +
+=Σ .
II-k> kMnt;témørbs; n edIm,I[ nI minGaRs½ynwg a eyIgman 0a,
axdx.xI
a
033
n
n >+
= ∫
eyIgtag t.ax = naM[ dt.adx = ehIycMeBaH [ ]a,0x∈ naM[ [ ]1,0t ∈ eK)an ∫ ∫ +
=+
= −1
0
1
03
n2n
33
n
n 1tdt.t
.aa)at(dt.a.)t.a(
I
tamTMnak;TMngenH edIm,I[ nI minGaRs½ynwg a luHRtaEt 02n =− b ¤ .
សមករអនគមន
- 61 -
dUcenH edIm,I[ nI minGaRs½ynwg aeKRtUv[ 2n = . x> KNna nI cMeBaHtémø n Edl)anrkeXIjxagelI cMeBaH 2n = eK)an ∫ +
=1
03
2
2 1xdxxI
tag 1xU 3 += naM[ dx.x3dU 2= ehIycMeBaH [ ]1,0x∈ naM[ [ ]2,1U∈ eyIg)an [ ] 2ln
31|U|ln
31
UdU
31I 2
1
2
12 === ∫ .
dUcenH 2ln31I2 = .
III- k>bgðajfam:UDúl 26|Z| += eKman i32Z ++= eyIg)an 133441)32(|Z| 2 +++=++= 26)26(748|Z| 2 +=+=+= dUcenH 26|Z| += . x> sresr Z CaTRmg;RtIekaNmaRt
សមករអនគមន
- 62 -
eyIg)an )21
.i23
1(2i32Z ++=++=
)12
sin.i12
cos(12
cos4
)12
cos12
sini212
cos2(2
)6
sin.i6
cos1(2
2
π+
ππ=
ππ+
π=
π+
π+=
dUcenH )12
sin.i12
cos(12
cos4Z π+
ππ= .
Taj[)anfa 4
2612
cos +=
π eKman )
12sin.i
12cos(
12cos4Z π
+ππ
= naM[
12cos4|Z| π
= edayeKman 26|Z| += naM[ 26
12cos4 +=
π dUcenH
426
12cos +
=π .
IV-eKmanGnuKmn_ 3x4x
23x7y 2 +−−
= Edl xCacMnYnBitehIy 3x,1x ≠≠
សមករអនគមន
- 63 -
k> KNnaedrIev 'y nig ''y eyIgsn μt
3xB
1xA
3x4x23x7y 2 −
+−
=+−
−=
b¤ )3x)(1x(
)1x(B)3x(A3x4x
23x72 −−
−+−=
+−−
b¤ 3x4x
BA3x)BA(3x4x
23x722 +−
−−+=
+−−
eKTaj ⎩⎨⎧
−=−−=+
23BA37BA naM[ 1B,8A −==
ehtuenH 3x
11x
8y−
−−
= eyIg)an
22
2
22
22
22
)3x()1x(71x46x7
)2x()1x()1x()3x(8
)3x(1
)1x(8'y
−−−+−
=
−−−+−−
=
−+
−−=
nig 33
33
33 )3x()1x()1x(2)3x(16
)3x(2
)1x(16''y
−−−−−
=−
−−
= .
សមករអនគមន
- 64 -
dUcenH 33
32
22
2
)3x()1x()1x(2)3x(16''y,
)3x()1x(71x46x7'y
−−−−−
=−−−+−
=
x> edaHRsaysmIkar 0''y = eyIgman 0
)3x()1x()1x(2)3x(16''y 33
33
=−−
−−−=
naM[ 0)1x(2)3x(16 33 =−−− eKTajb¤s 5x = .
សមករអនគមន
- 65 -
viBaØasaKNitviTüaTI4 I-eKmanplbUk
000....1000n.......
10003
1002
101Sn ++++=
k> cUrRsayfa *INn,1x,
x1x
x.....xxx1x1
1 1nn32 ∈≠∀
−++++++=
−
+
rYcTaj[)anfa ³
2
n
21n2
)x1(x)]1n(nx[
)x1(1nx....x3x21
−+−
+−
=++++ − .
x> Tajrktémøén nS nig nnSlim
+∞→ .
II-eK[GnuKmn_ ³ 3m2x)5m(x)1m(mx
31)x(f 23
m −+−−+−= cUrkMnt;témø m edIm,I[GnuKmn_ )x(fm ekInCanic©elI IR . III-eK[sIVútGaMgetRkal ∫ −=
1
0
2nn dx.x1xI Edl INn∈
k> cUrrkTMnak;TMngrvag nI nig 2nI − x> KNnaplKuN 1n,I.IP 1nnn ≥∀= − CaGnuKmn_én n .
សមករអនគមន
- 66 -
K> cUrKNnaplbUk ( ) n21
n
1kkn P.........PPPS +++== ∑
=
CaGnuKmn_én n rYcTajrktémøénlImIt nnSlim
+∞→ .
X> rkrUbmnþKNna nI CaGnuKmn_én n . dMeNa¼Rsay
I- k>Rsayfa
x1xx.....xxx1
x11 1n
n32
−++++++=
−
+
eyIgman x1
xx1
1x1
x1x...xxx11n1n
n32
−−
−=
−−
=+++++++
dUcenH x1
xx....xx1x1
1 1nn2
−+++++=
−
+
.
müa:geTot )'x1
xx....xxx1()'x1
1(1n
n32
−++++++=
−
+
¬edrIevelIGgÁTaMgBIr¦
2
n1n2
2
2
1nn1n2
2
)x1(x]nx)1n[(nx........x3x21
)x1(1
)x1(x)'x1()x1(x)1n(nx....x3x21
)x1()'x1(
−−+
+++++=−
−−−−+
+++++=−−
−
−
+−
សមករអនគមន
- 67 -
dUcenH 2
n
21n2
)x1(x)]1n(nx[
)x1(1nx....x3x21
−+−
+−
=++++ −
x> Tajrktémøén nS nig nnSlim
+∞→
eKman
2
n
21n2
)x1(x)]1n(nx[
)x1(1nx....x3x21
−+−
+−
=++++ −
edayyk 101x = eK)an ³
n
n
1n
10.81)10n9(10
81100
8110
1)1nn101(100
81100
10n....
1003
1021
+−=
−−+=++++ −
b¤ nn 10.8110n9
8110
10n....
10003
1002
101 +
−=++++ .
dUcenH nn 10.8110n9
8110S +
−= nig 8110Slim nn
=+∞→
II-kMnt;témø m edIm,I[GnuKmn_ )x(fm ekInCanic©elI IR eKman 3m2x)5m(x)1m(mx
31)x(f 23
m −+−−+−= eK)an )5m(x)1m(2mx)x('f 2
m −−+−=
សមករអនគមន
- 68 -
edIm,I[GnuKmn_ )x(fm ekInCanic©elI IRluHRtaEtcMeBaHRKb; IRx∈ man 0)x('f > .
eK)an )1(0)5m(x)1m(2mx2 >−−+− vismIkar ¬!¦ BitCanic©cMeBaHRKb; IRx∈ kalNa
⎩⎨⎧
<Δ>=
0'0ma
eKman
1m3m2
m5m1m2m
)5m(m)1m('
2
22
2
+−=
−+++=
−++=Δ
0)1m2)(1m(' <−−=Δ naM[ 1m21
<< nig 0m > dUcenH 1m
21
<< . III-k> rkTMnak;TMngrvag nI nig 2nI − eyIgman ∫ −=
1
0
2nn dx.x1xI
nig dx.x1.xI1
0
22n2n ∫ −= −
−
សមករអនគមន
- 69 -
tag ⎪⎩
⎪⎨⎧
=
−=
dx.xdV
x1Un
2
naM[ ⎪⎪⎩
⎪⎪⎨
⎧
+==
−−=
+∫ 1nn
2
x1n
1dx.xV
dx.x1
xdU
eyIg)an ∫−+
+⎥⎦⎤
⎢⎣⎡ −
+=
++
1
02
2n1
0
21nn dx.
x1x
1n1x1.x
1n1I
n
1
02
1nn
1
0
1
0
2n2
n
n
1
02
2nn1
02
2nnn
n
I1n
1x1
xdx.x1n
1I
dx.x1x1n
1x1
dx.x1n
1I
dx.x1
)x1(xx1n
1dx.x1
)xx(x1n
1I
+−
−+=
−+
−−+
=
−
−−+
=−
−−+
=
∫
∫ ∫
∫∫
−
+
tag ⎪⎩
⎪⎨
⎧
−=
= −
2
1n
x1xdxdV
xU naM[
⎪⎩
⎪⎨⎧
−−=
−= −
2
2n
x1V
dx.x)1n(dU
eyIg)an [ ] n
1
0
22n1
021n
n I1n
1dx.x1x)1n(x1x
1n1
I+
−⎭⎬⎫
⎩⎨⎧
−−+−−+
= ∫ −−
eKTaj n2nn II)1n(I)1n( −−=+ − naM[ 2nn I2n1nI −+
−=
dUcenH TMnak;TMngrvag nI nig 2nI − KW
សមករអនគមន
- 70 -
2n,I2n1nI 2nn ≥∀
+−
= − . x> KNnaplKuN 1n,I.IP 1nnn ≥∀= − CaGnuKmn_én n eyIgman 1nnn I.IP −= naM[ n1n1n I.IP ++ = eday 2nn I
2n1nI −+
−= naM[ 1n1n I.
3nn
I −+ +=
eKTaj nn1n1n P3n
nI.I
3nn
P+
=+
= −+ naMeGay
3n2n.
2n1n.
1nn
3nn
PP
n
1n
++
++
+=
+=+ .
eyIg)an ∏∏−
=
−
=
+ ⎟⎠⎞
⎜⎝⎛
++
++
+=⎟⎟⎠
⎞⎜⎜⎝
⎛ )1n(
1k
)1n(
1k k
1k
3k2k
.2k1k
.1k
kP
P
2n3.
1n2.
n1
PP
2n1n....
54.
43
1nn....
43.
32
n1n....
32.
21
PP....
PP.
PP
3k2k
2k1k
1kk
PP
1
n
1n
n
2
3
1
2
)1n(
1k
)1n(
1k
)1n(
1k
)1n(
1k k
1k
++=
⎟⎠⎞
⎜⎝⎛
++
×⎟⎠⎞
⎜⎝⎛
+×⎟⎠⎞
⎜⎝⎛ −
=
⎟⎠⎞
⎜⎝⎛
++
×⎟⎠⎞
⎜⎝⎛
++
×⎟⎠⎞
⎜⎝⎛
+=⎟⎟⎠
⎞⎜⎜⎝
⎛
−
−
=
−
=
−
=
−
=
+ ∏ ∏∏∏
សមករអនគមន
- 71 -
naM[eKTaj 101n I.I.
)2n)(1n(n6
P.)2n)(1n(n
6P
++=
++=
¬ eRBaH 101 I.IP = ¦ eday ∫
π=⎥⎦
⎤⎢⎣⎡ +−=−=
1
0
1
0
220 4
xarcsin21x1
2xdx.x1I
nig 31)x1(
31dx.x1xI
1
0
1
0
23
221 =
⎥⎥⎦
⎤
⎢⎢⎣
⎡−−=−= ∫
eK)an )2n)(1n(n
1.
231
.4
.)2n)(1n(n
6Pn ++
π=
π++
=
dUcenH )2n)(1n(n
1.2
Pn ++π
= .
K> KNnaplbUk ( ) n21
n
1kkn P.........PPPS +++== ∑
=
eyIgman ³
⎥⎦
⎤⎢⎣
⎡++
−+
π=
++π
=
)2n)(1n(1
)1n(n1
4
)2n)(1n(n1.
2Pn
សមករអនគមន
- 72 -
eyIg)an ∑=
⎥⎦
⎤⎢⎣
⎡++
−+
π=
n
1kn )2k)(1k(
1)1k(k
14
S
)2n)(1n()3n(n.
8)2n)(1n(1
21
4
)2n)(1n(1
)1n(n1.....)
3.21
2.11(
4
+++π
=⎥⎦
⎤⎢⎣
⎡++
−π
=
⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛++
−+
++−π
=
dUcenH )2n)(1n(
)3n(n.8
Sn +++π
= nig 8
Slim nn
π=
+∞→ .
X> rkrUbmnþKNna nI CaGnuKmn_én n tamsRmayxagelIeyIgman 2n,I
2n1nI 2nn ≥∀
+−
= − -krNI 1p2n += ¬ cMnYness ¦ eyIg)an 1p21p2 I.
3p2p2
I −+ += b¤
3p2p2
II
1p2
1p2
+=
−
+
eKTaj 3p2
p2.....96.
74.
52
II
.....II.
II.
II
1p2
1p2
5
7
3
5
1
3
+=
−
+
3p2
p2.....96.
74.
52
II
1
1p2
+=+
naM[ 11p2 I.)3p2....(9.7.5
p2....6.4.2I
+=+ eday 3
1I1 =
សមករអនគមន
- 73 -
dUcenH 31.
)3p2....(9.7.5p2....6.4.2I 1p2 +
=+ . -krNI p2n = ¬ cMnYnKU ¦ eyIg)an 2p2p2 I.
2p21p2I −+
−= b¤
2p21p2
II
2p2
p2
+−
=−
eKTaj 2p21p2.....
85.
63.
41
II
.....II.
II.
II
2p2
p2
4
6
2
4
0
2
+−
=−
2p21p2.....
85.
63.
41
II
0
p2
+−
=
naM[ 01p2 I.)2p2....(8.6.4)1p2....(5.3.1
I+−
=+ eday 4
I0π
=
dUcenH 4
.)2p2....(8.6.4)1p2....(5.3.1I 1p2π
+−
=+ .
សមករអនគមន
- 74 -
viBaØasaKNitviTüaTI5 I-edaHRsaysmIkar 1y29x47 =+ kñúgsMNMucMnYnKt;rWLaTIhV II-eK[sIVúténcMnYnBit INnn )U( ∈ kMnt;eday ³
⎪⎩
⎪⎨⎧
∈∀−=
=
+ 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 . III-cUrKNnalImIt
xcosxsinxtanxsin2limA
4x −
−=
π→
IV- cUrbgðajfa ∫∫ −+=b
a
b
a
dx.)xba(fdx.)x(f
Gnuvtþn_ ³ cUrKNnaGaMgetRkal ∫
π
+=3
0
dx).xtan31ln(I
សមករអនគមន
- 75 -
dMeNa¼Rsay I-edaHRsaysmIkar 1y29x47 =+ kñúgsMNMucMnYnKt;rWLaTIhV eyIgman 1812947 +×= naM[ 294718 −= 1111829 +×= naM[ 47229)2947(29182911 −×=−−=−= 711118 +×= naM[ )47229()2947(11187 −×−−=−= b¤ 3292477 ×−×=
41711 +×= naM[ )329247()47229(7114 ×−×−−×=−=
b¤ 5294734 ×+×−= 1247 −×= naM[ )329247(2).529473(7241 ×−×−×+×−=−×= b¤ )13(29)8(471 ×+−×= eK)an )13(29)8(47y29x47 ×+−×=+
សមករអនគមន
- 76 -
)13y(29)8x(47 −−=+ naM[
⎩⎨⎧
∈∀=−−=+
Zq,q4713yq298x
dUcenH Zq,13q47y,8q29x ∈∀+=−−= . II-k> Rsayfa 2Un > cMeBaHRKb; INn∈ eyIgman 25U 0 >= Bit 23252UU 2
01 >=−=−= Bit eyIg]bmafavaBitdl;tYTI k KW 2Uk > Bit eyIgnwgRsayfavaBitdl;tYTI 1k + KW 2U 1k >+ Bit . eyIgman 2Uk > naM[ 4U 2
k > b¤ 22U 2k >−
eday 2UU 2k1k −=+ enaHeKTaj)an 2U 1k >+ Bit .
dUcenH 2Un > cMeBaHRKb; INn∈ . x> rkTMnak;TMngrvag 1nV + nig nV eyIgman 4UUV 2
nnn −+= naM[ 4UUV 2
1n1n1n −+= +++
សមករអនគមន
- 77 -
Et 2UU 2n1n −=+
eK)an 4)2U(2UV 22n
2n1n −−+−=+
)4U(U2U
44U4U2UV2
n2
n2
n
2n
4n
2n1n
−+−=
−+−+−=+
2)4UU(
24UU24U2
4UU2U
22nn
2nn
2n
2nn
2n
−+=
−+−=
−+−=
dUcenHeK)an 2n1n V
21V =+ CaTMnak;TMngEdlRtUvrk .
K> KNna nV rYcTajrk nU CaGnuKmn_én n tamsRmayxagelIeKman 2
n1n V21V =+
naM[ )V21(logVlog 2
n211n
21 =+
n211n
21 Vlog21Vlog +=+
សមករអនគមន
- 78 -
tag n21n VlogW = naM[ 1n
211n VlogW ++ =
eK)an n1n W21W +=+ b¤ )W1(2)W1( n1n +=+ + naM[ )W1( n+ CasIVútFrNImaRtmanersug 2q = nigtYTImYy 0W1+ Et 154554UUV 2
000 +=−+=−+= eRBaH 4UUV 2
nnn −+= . eK)an )
215(log)15(log1W1
21
210
+=++=+ .
tamrUbmnþ
n2
21
21
nn0n
)2
15(log
)2
15(log2q).W1(W1
+=
+=+=+
eday n21n VlogW =
enaH )2
V(logVlog1W1 n
21n
21n =+=+
សមករអនគមន
- 79 -
eKTaj n2
21
n
21 )
215(log)
2V(log +
=
naM[ n2
n 2152V ⎟⎟⎠
⎞⎜⎜⎝
⎛ +=
müa:geToteday 4UUV 2nnn −+=
naM[ 4U)UV( 2n
2nn −=−
b¤ 4UUUV2V 2n
2nnn
2n −=+−
eKTaj n
nn
2n
n V2V
21
V24V
U +=+
=
naM[ nn
n
n 22
2
2
n 215
215
215
12
15U ⎟⎟
⎠
⎞⎜⎜⎝
⎛ −+⎟⎟
⎠
⎞⎜⎜⎝
⎛ +=
⎟⎟⎠
⎞⎜⎜⎝
⎛ ++⎟⎟
⎠
⎞⎜⎜⎝
⎛ +=
dUcenH nnn 2
n
22
n 2152V,
215
215U ⎟⎟
⎠
⎞⎜⎜⎝
⎛ +=⎟⎟
⎠
⎞⎜⎜⎝
⎛ −+⎟⎟
⎠
⎞⎜⎜⎝
⎛ +=
សមករអនគមន
- 80 -
III-KNnalImIt xcosxsin
xtanxsin2limA4
x −−
=π
→
tag x4
z −π
= naM[ x4
z −π
= ebI 4
x π→ enaH 0z →
eK)an )z
4cos()z
4sin(
)z4
tan()z4
sin(2limA
0z−
π−−
π
−π
−−π
=→
22
21
])zsinz(cos
zzsin
zzsin
zzcos1
[lim2
1
zsin2zcoszsin1
zcoszsin1
zsinzcos
lim
zsin22zcos
22zsin
22zcos
22
ztan1ztan1)zsin
22zcos
22(2
lim
0z
0z
0z
−=−=
−−
−
=
−
+
−−−
=
−−−
+−
−−=
→
→
→
សមករអនគមន
- 81 -
dUcenH 22
xcosxsinxtanxsin2limA
4x
−=−−
=π
→
.
IV- bgðajfa ∫∫ −+=b
a
b
a
dx.)xba(fdx.)x(f
tag dxdtxbat −=⇒−+= ebI btax =⇒= nig atbx =⇒= eyIg)an ∫ ∫ ∫∫ −+=−+−=−−+=b
a
a
b
b
a
a
b
dt).tba(fdt).tba(f)dt)(tba(fdx).x(f
eday ∫ ∫ −+=−+
b
a
b
a
dx).xba(fdt)tba(f
¬ lkçN³GaMgetRkalkMnt; ¦ dUcenH ∫∫ −+=
b
a
b
a
dx.)xba(fdx.)x(f .
KNnaGaMgetRkal ∫
π
+=3
0
dx).xtan31ln(I
tamrUbmnþxagelIeyIgGacsresr ³
សមករអនគមន
- 82 -
[ ]
∫∫
∫∫
∫∫
ππ
ππ
ππ
−π
=+−=
+−=⎟⎠⎞
⎜⎝⎛
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+−
+=⎥⎦⎤
⎢⎣⎡ −
π+=
3
0
3
0
3
0
3
0
3
0
3
0
I2ln3
2dx).xtan31ln(dx4lnI
dx.)xtan31ln(4lndx.xtan31
4lnI
dx.xtan31
xtan3.31lndx.)x
3tan(31lnI
2ln
32I2 π
= naM[ 2ln3
I π=
dUcenH 2ln3
dx).xtan31ln(I3
0
π=+= ∫
π
.
សមករអនគមន
- 83 -
viBaØasaKNitviTüaTI6 I-eK[sIVúténcMnYnkMupøic )Z( n kMnt;eday
⎪⎩
⎪⎨⎧
∈+−
++
=
=
+ INn,2
i21Z
2i1
Z
iZ
n1n
0
k> eKtag 1ZU:INn nn +=∈∀ . cUrbgðajfa n1n U
2i1
U+
=+ cMeBaHRKb; INn ∈ . x> cUrsresr nU CaragRtIekaNmaRt. Tajrk nZ CaGnuKmn_én n . II-cUrbgðajfa 3n22n3
n 73A ++ += Eckdac;nwg !! Canic©RKb; cMnYnKt;FmμCati n. III- eKeGayGaMgetRkal ³
∫+
=a)1n(
na2n xcos
dxI nig 0a,INn,xcos
dxJ *a)1n(
a2n >∈= ∫
+
.
k-bgðajfa nn321 JI....III =++++ . x-KNna nI nig nJ CaGnuKmn_én n .
សមករអនគមន
- 84 -
K-eRbIlTæplxagelIcUrbRgYmplbUk ³
( ) ( )a1ncosnacos1.........
a3cosa2cos1
a2cosacos1Sn +
+++=
IV-eK[ExSekag 2x
)1x(2y:)H(−−
= nigcMnuc )2;2(I . kMnt;rksmIkarrgVg; )C( manp©it I ehIyb:HnwgExSekag )H( .
dMeNa¼Rsay I-k> bgðajfa n1n U
2i1
U+
=+ cMeBaHRKb; INn ∈ eyIgman 1ZU:INn nn +=∈∀ naM[ 1ZU 1n1n += ++ eday
2i21
Z2i1
Z n1n+−
++
=+
eyIg)an 12
i21Z
2i1
U n1n ++−
++
=+
nn1n
n1n
U2i1
)1Z(2i1
U
2i1
Z2i1
U
+=+
+=
++
+=
+
+
dUcenH n1n U2i1U +
=+ .
សមករអនគមន
- 85 -
x> sresr nU CaragRtIekaNmaRtrYcTajrk nZ CaGnuKmn_én n eyIgman n1n U
2i1
U+
=+ naM[ )U( n CasIVútFrNImaRténcMnYnkMupøicmanersug
4sin.i
4cos
22
i22
2i1
qπ
+π
=+=+
= nigtY )
4sin.i
4(cos21i1ZU 00
π+
π=+=+=
tamrUbmnþ ³
1n
nn0n
)4
sin.i4
(cos2
)4
sin.i4
).(cos4
sin.i4
(cos2qUU
+π+
π=
π+
ππ+
π=×=
dUcenH ⎥⎦⎤
⎢⎣⎡ π+
+π+
=4
)1n(sin.i
4)1n(
cos2Un . müa:geTot 1ZU nn += naM[ 1]
4)1n(sin.i
4)1n([cos21UZ nn −
π++
π+=−=
dUcenH 4
)1n(sin2.i
4)1n(
cos21Znπ+
+⎥⎦⎤
⎢⎣⎡ π+
+−=
សមករអនគមន
- 86 -
II-bgðajfa 3n22n3n 73A ++ += Eckdac;nwg !!
eyIgman )11mod(533 ≡ naM[ )11mod(5.93 n2n3 =+ müa:geTot )11(mod572 ≡ naM[ )11(mod57 nn2 ≡ ehIy )11(mod273 ≡ eKTaj)an )11(mod5.27 n3n2 ≡+ eyIg)an
)11(mod05.115.25.973A nnn3n22n3 ≡=+≡+= ++ . dUcenH 3n22n3
n 73A ++ += Eckdac;nwg !! Canic©RKb; cMnYnKt;FmμCati n . III-k-bgðajfa nn321 JI....III =++++ eyIg)an
∫ ∫ ∫ ∫+ +
=+++=+++a2
a
a3
a2
a)1n(
na
a)1n(
a2222n21 xcos
dxxcos
dx...xcos
dxxcos
dxI...II
dUcenH nn321 JI....III =++++ .
សមករអនគមន
- 87 -
x-KNna nI nig nJ CaGnuKmn_én n eyIg)an ³
∫+ +
−+=⎥⎦
⎤⎢⎣
⎡==
a)1n(
na
a)1n(
na2n )natan(a)1ntan(xtan
xcosdxI
nig
∫+ +
−+=⎥⎦
⎤⎢⎣
⎡==
a)1n(
a
a)1n(
a2n atana)1ntan(xtan
xcosdxJ
dUcenH
a)1ncos(.acos)nasin(J,
a)1ncos()nacos(asinI nn +
=+
= . K-KNnaplbUk ( ) ( )a1ncosnacos
1.........
a3cosa2cos1
a2cosacos1
Sn ++++=
eyIg)an ³ ( ) ( )
[ ]∑ ∑= = +
===⎥⎦
⎤⎢⎣
⎡+
=
++++=
n
1k
n
1knk
n
a)1ncos(acosasin)nasin(J.
asin1I
asin1
a)1kcos(.kacos1
a1ncosnacos1.........
a3cosa2cos1
a2cosacos1S
សមករអនគមន
- 88 -
dUcenH ( ) ( ) a)1ncos(acosasin
nasina1ncosnacos
1...a2cosacos
1Sn +=
+++=
IV-kMnt;rksmIkarrgVg; )C( manp©it I ehIyb:HnwgExSekag )H( eKman
2x)1x(2y:)H(
−−
= nigcMnuc )2;2(I tamrUbmnþsmIkarrgVg; )C( manp©it )2;2(I sresr ³
222 R)2y()2x(:)C( =−+− Edl R CakaMénrgVg; . smIkarGab;sIuscMnucRbsBVrvag )H( nig )C( sresr ³
222
2
22
2
22
2
)2x(X,R)2x(
4)2x(
R2x
4x22x2)2x(
R22x
)1x(2)2x(
−==−
+−
=⎟⎠⎞
⎜⎝⎛
−+−−
+−
=⎥⎦⎤
⎢⎣⎡ −
−−
+−
2RX4X =+ naM[ )1(04XRX 22 =+−
edIm,I[rgVg; )C( b:HnwgExSekag )H( luHRtaEtsmIkar )1( man b¤sDúbeBalKWeKRtUv[ 016R4 =−=Δ naM[ 216R 4 ==
សមករអនគមន
- 89 -
smIkarrgVg; )C( Gacsresr ³
044y4y4x4x
4)2y()2x(22
22
=−+−++−
=−+−
dUcenH 04y4x4yx:)C( 22 =+−−+ .
សមករអនគមន
- 90 -
viBaØasaKNitviTüaTI7 I-cUrRsaybBa¢ak;facMnYn nnnn
n 124122462447E −−+= Eckdac;nwg 221Canic©cMeBaHRKb;cMnYnKt;Fm μCati n . II-eK]bmafa mS nig nS CaplbUk mtYdMbUg nig ntYdMbUg erogKñaénsVúItnBVnþmYyEdl )nm(;
nm
SS
2
2
n
m ≠= .
eKtag mU CatYTI m nig nU CatYTI n . cUrbgðajfa
1n21m2
UU
n
m
−−
=
III-eKmansIVút )n
1n21)....(n51)(
n31)(
n11(U 2222n
−++++=
cMeBaHRKb; *INn∈ . k> cMeBaHRKb; 0t ≥ cUrbgðajfa 1
t11t1 ≤+
≤−
x> Taj[)anfaRKb; 0x ≥ eKman x)x1ln(2xx
2
≤+≤− K> cUrrklImIt nn
Ulim+∞→
.
សមករអនគមន
- 91 -
dMeNa¼Rsay I- RsaybBa¢ak;facMnYn nE Eckdac;nwg 221 eyIgeXIjfa 1713221 ×= Edl 13 nig 17bzmrvagKña dUcenHedIm,IRsayfa cMnYn nE Eckdac;nwg 221eKRtUvRsay [eXIjfa cMnYn nE vaEckdac;nwg !#pg nig Eckdac;nwg !& pg. eKman nnnn
n 124122462447E −−+= )124462()122447( nnnn −+−= tamrUbmnþ )b....baa)(ba(ba 1n2n1nnn −−− +++−=− eyIg)an 11n q).124462(p).122447(E −+−= )q26p25(13q338p325 1111 +=+= Edl *INq,p 11 ∈ . TMnak;TMngenHbBa¢ak;fa cMnYn nE Eckdac;nwg !# . müa:geToteKman )122462()124447(E nnnn
n −+−=
*INq,p,)q20p19(17
q340p323q).122462(p).124447(
2222
22
22
∈+=+=
−+−=
សមករអនគមន
- 92 -
TMnak;TMngenHbBa¢ak;fa cMnYn nE Eckdac;nwg !& . dUcenHsrubesckþIeTAeyIg)an cMnYn nE Eckdac;nwg 221Canic© cMeBaHRKb;cMnYnKt;FmμCati n . II- bgðajfa
1n21m2
UU
n
m
−−
=
eyIgman 2
2
n
m
nm
SS
=
eday 2
)UU(nS;2
)UU(mS n1n
m1m
+=
+=
eK)an 2
2
n1
m1
nm
)UU(n2
2)UU(m
=+
×+
naM[ )1(nm
UUUU
n1
m1 =++
eday d).1n(UU,d).1m(UU 1n1m −+=−+= Edl dCaplsgrYménsIVút. yk d).1n(UU,d).1m(UU 1n1m −+=−+= CYskñúg ¬!¦ eK)an
nm
d)1n(U2d).1m(U2
1
1 =−+−+
សមករអនគមន
- 93 -
b¤ d)1n(mmU2d)1m(nnU2 11 −+=−+ eKTaj
2d
)nm(2)mmnnmn(d
)nm(2d)1n(md)1m(nU1 =
−+−−
=−
−−−=
eRBaH nm ≠ . eyIg)an
d.2
1n2d).1n(2dU
.d.2
1m2d)1m(2dU
n
m
−=−+=
−=−+=
eK)an 1n21m2
d.2
1n2
d.2
1m2
UU
n
m
−−
=−
−
= ebI 0d ≠ .
dUcenH 1n21m2
UU
n
m
−−
= .
III- k> cMeBaHRKb; 0t ≥ bgðajfa 1t1
1t1 ≤+
≤−
eKman 0t ≥ naM[ 1t1 ≥+ eKTaj)an )1(1t1
1≤
+
müa:geTot 0t2 ≥ naM[ 1t1 2 ≤− b¤ 1)t1)(t1( ≤+−
សមករអនគមន
- 94 -
eKTaj )2(t1
1t1+
≤−
dUcenHtam ¬!¦ nig ¬@¦ eK)an 1t1
1t1 ≤+
≤− .
x> Taj[)anfaRKb; 0x ≥ eKman x)x1ln(2xx
2
≤+≤−
tamsRmayxagelIeKman 1t1
1t1 ≤+
≤−
cMeBaH 0x ≥ eK)an ∫∫ ∫ ≤+
≤−x
0
x
0
x
0dt
t1dtdt).t1(
[ ] [ ] x0
x0
x
0
2
t)t1ln(2
tt ≤+≤⎥⎦
⎤⎢⎣
⎡−
x)1ln()x1ln(2xx
2
≤−+≤− eday 0)1ln( = . dUcenH x)x1ln(
2xx
2
≤+≤− . K> rklImIt nn
Ulim+∞→
∏
=⎟⎠⎞
⎜⎝⎛ −
+=−
++++=n
1k22222n n
1k21)n
1n21)....(n51)(
n31)(
n11(U
សមករអនគមន
- 95 -
tamsRmayxagelIeKman x)x1ln(2xx
2
≤+≤−
edayyk 2n1k2x −
= RKb; *INk,*INn ∈∈ eK)an ³
224
2
2 n1k2)
n1k21ln(
n2)1k2(
n1k2 −
≤−
+≤−
−−
∑∑∑===
⎟⎠⎞
⎜⎝⎛ −
≤⎥⎦⎤
⎢⎣⎡ −
+≤⎥⎦
⎤⎢⎣
⎡ −−
− n
1k2
n
1k2
n
1k4
2
2 n1k2)
n1k21ln(
n2)1k2(
n1k2
( )∑∏∑ ∑=== =
⎟⎠⎞
⎜⎝⎛ −
≤⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −
+≤⎥⎦
⎤⎢⎣
⎡ −−⎟
⎠⎞
⎜⎝⎛ − n
1k2
n
1k2
n
1k
n
1k4
2
2 n1k2
n1k21ln
n1k2
n1k2
eday ∑=
==−++++
=⎟⎠⎞
⎜⎝⎛ −n
1k2
2
22 1nn
n)1n2(.....531
n1k2
( )∑ ∑= =
+−=⎥⎦
⎤⎢⎣
⎡ −n
1k
n
1k
244
2
)1k4k4(n1
n1k2
333
444
n
1k
n
1k
n
1k44
24
n1
n)1n(2
n3)1n2)(1n(2
n.n1
2)1n(n4.
n1
6)1n2)(1n(n4.
n1
)1(n1)k4(
n1)k4(
n1
++
−++
=
++
−++
=
+−= ∑ ∑ ∑= = =
eKTaj 1Ulnn3
3)1n(6)1n2)(1n(21 n3 ≤≤
++−++−
eK)an 1Ulnlim nn=
+∞→ naM[ ...71828.2eUlim nn
==+∞→
.
សមករអនគមន
- 96 -
viBaØasaKNitviTüaTI8 I-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 . II-eK[GnuKmn_ xcos.e)x(f x= k> KNna )x('f rYcbgðajfa )
4xcos(e2)x('f x π+=
x> edayeFIVvicartamkMeNIncUrbgðafa )
4nxcos(.e2)x(f xn)n( π
+= III-eKmanGaMgetRkal
∫
π
=2
0
x dx.xcoseA nig ∫
π
=2
0
x dx.xsineB
សមករអនគមន
- 97 -
k> cUrbgðajfa 2
1e.i2
1ei1
1e 222)i1(
++
−=
+−
πππ+
x> KNna B.iA + rYcTajrktémøén A nig B .
dMeNa¼Rsay I-k>RsaybBa¢ak;fa n1n Z
2i3Z +
=+ eyIgman INn,U
2i3
UZ n1nn ∈∀−
−= + naM[ 1n2n1n U
2i3
UZ +++−
−= edayeKman n1n2n UU.3U −= ++ cMeBaHRKb; INn ∈ eK)an 1nn1n1n U
2i3
UU.3Z +++−
−−=
nn1n1n
n1nn1n1n
Z2
i3)U2
i3U(2
i3Z
)Ui3
2U(
2i3
UU2
i3Z
+=
−−
+=
+−
+=−
+=
++
+++
dUcenH INn,U2
i3UZ n1nn ∈∀
−−= + .
សមករអនគមន
- 98 -
x>kMnt;rkTMrg;RtIekaNmaRtén nZ edayeyIgman n1n Z
2i3Z +
=+ cMeBaHRKb; INn ∈ ¬tamsRmayxagelI ¦ naM[ )Z( n CasIVútFrNImaRténcMnYnkMupøicEdlmanersug
)6
sin.i6
(cos2
i3q
π+
π=
+=
nigtY 1U2
i3UZ 010 =−
−= ¬ eRBaH )1U,0U 10 == eK)an
6n
sin.i6
ncos)
6sin.i
6(cosqZZ nn
0nπ
+π
=π
+π
=×=
dUcenH 6
nsin.i
6n
cosZnπ
+π
= . K> TajrktY nU CaGnuKmn_én n rYcTajrktémø 2007U ³ eyIgman n1nn U
2i3
UZ−
−= + b¤ nn1nn U
2i
)U23
U(Z +−= + ¬ eRBaH IRUn ∈ ¦ eyIg)an nn1nn U
2i
)U23
U(Z −−= + ¬ cMnYnkMupøicqøas;én nZ ¦.
សមករអនគមន
- 99 -
eKTaj nnn U.iZZ =− naM[ i
ZZU nn
n−
= edayeyIgman
6n
sin.i6
ncosZn
π+
π=
nig 6
nsin.i
6n
cosZnπ
−π
= . eyIg)an
6n
sin2i
)6
nsin.i
6n
(cos)6
nsin.i
6n
(cosUn
π=
π−
π−
π+
π
=
dUcenH 6
nsin2Unπ
= . müa:geTotcMeBaH 2007n = eyIg)an ³
2)2
sin(2)3322
sin(2
2663sin2
62007sin2U2007
−=π
−=π+π
−=
π=
π=
.
dUcenH 2U2007 −= .
II- k> KNna )x('f rYcbgðajfa )4
xcos(e2)x('f x π+=
eyIgman xcose)x(f x=
សមករអនគមន
- 100 -
eyIg)an xsinexcose)x('f xx −=
)4
xcos(e2
)4
sinxsin4
cosx(cose2
)xsin22xcos
22(e2
)xsinx(cose
x
x
x
x
π+=
π−
π=
−=
−=
dUcenH )4
xcos(e2)x('f x π+= .
x> edayeFIVvicartamkMeNIncUrbgðajfa )
4nxcos(.e2)x(f xn)n( π
+= eyIgman )x(f)
4xcos(.e2)x('f )1(x =
π+= Bit
]bmafavaBitdl;tYTI k KW )4
kxcos(.e2)x(f xk)k( π+= Bit
eyIgnwgRsayfavaBitdl;tYTI 1k + KW ⎟
⎠⎞
⎜⎝⎛ π+
+=++
4)1k(xcose2)x(f x1k)1k(
eyIgman )')x(f()x(f )k()1k( =+
សមករអនគមន
- 101 -
⎥⎦⎤
⎢⎣⎡ π
+−π
+=
π+−
π+=
)4
kxsin()4
kxcos(e2
)4
kxsin(e2)4
kxcos(e2
xk
xkxk
eday ⎟⎠⎞
⎜⎝⎛ π+
+=π
+−π
+4
)1k(xcos2)4
kxsin()4
kxcos(
eK)an ⎟⎠⎞
⎜⎝⎛ π+
+=++
4)1k(xcose2)x(f x1k)1k( Bit .
dUcenH )4
nxcos(.e2)x(f xn)n( π+= .
III-k> bgðajfa 2
1e.i2
1ei1
1e 222)i1(
++
−=
+−
πππ+
tag i1i1
i11e.i
i11e.e
i11e
Z22
i22
)i1(
−−
×+−
=+−
=+−
=
ππππ+
2
1e.i2
1e2
i1ee.i 2222 ++
−=
+−+=
ππππ
dUcenH 2
1e.i2
1ei1
1e 222)i1(
++
−=
+−
πππ+
. x> KNna B.iA + rYcTajrktémøén A nig B
សមករអនគមន
- 102 -
eKman ∫
π
=2
0
x dx.xcoseA nig ∫
π
=2
0
x dx.xsineB
eyIg)an ∫ ∫
π π
+=+2
0
2
0
xx dx.xsine.ixdxcoseB.iA
( )
21e.i
21e
i11ee
i11
dx.edx.e.edx).xsin.ix(cose
dx.xsine.ixcose
222)i1(
2
0
x)i1(
2
0
2
0
2
0
x)i1(ixxx
2
0
xx
++
−=
+−
=⎥⎦⎤
⎢⎣⎡+
=
==+=
+=
πππ+π
+
π π π
+
π
∫ ∫ ∫
∫
dUcenH 2
1e.i2
1eB.iA22 +
+−
=+
ππ
ehIy 2
1eB,2
1eA22 +
=−
=
ππ
.
សមករអនគមន
- 103 -
viBaØasaKNitviTüaTI9 I-cUrRsayfa 0x ≥∀ eKman xx xe1ex ≤−≤ rYcTajfa 1
x1elim
x
0x=
−→
. II-eK[GaMgetRkal
dx.xcosxsin
xsinI2
0n 2n 2
n 2
n ∫
π
+=
nig dx.xcosxsin
xcosJ2
0n 2n 2
n 2
n ∫
π
+=
cUrbgðajfa nn JI = rYcTajrktémø nI nig nJ . III-eK[sIVúténcMnYnBit )x( n kMnt;eday ³ 1x1 = nig
1n2x2
xxn
n1n ++=−+
k-cUrRsayfacMeBaHRKb; *INn∈ eKman 1n2xn −= . x-KNna
1n2x1.....
5x1
3x1S
n21n ++
+++
++
=
K> TajrklImIt ⎥⎦⎤
⎢⎣⎡
+∞→ nSlim n
n .
សមករអនគមន
- 104 -
IV-eK[RtIekaN ABC mYymanRCugerogKñadUcxageRkam ³ cAB,bAC,aBC === .
]bmafa M CacMnucmYyenAkñúgRtIekaNenH ehIyeKtag y,x nig z CacMgayerogKñaBIcMnuc M eTARCug AC,BC nig AB énRtIekaN .cUrKNnatémøGb,brmaén 222 zyxT ++= CaGnuKmn_én c,b,a .
dMeNa¼Rsay I-Rsayfa 0x ≥∀ eKman xx xe1ex ≤−≤ tagGnuKmn_ x1e)x(f x −−= nig xx xe1e)x(g −−= cMeBaH 0x ≥∀ eK)an 1e)x('f x −= cMeBaHRKb; 0x ≥∀ eKman 01ex ≥− naM[ 01e)x('f x ≥−= naM[ )x(f CaGnuKmn_ekIncMeBaHRKb; 0x ≥∀ . tamlkçN³énGnuKmn_ekIncMeBaH 0x ≥ eK)an 0)0(f)x(f =≥ eKTaj )1(x1ex ≥−
សមករអនគមន
- 105 -
müa:geToteKman ³ 0xe)xee(e)x('g xxxx ≤−=+−= cMeBaH 0x ≥∀
naM[ )x(g CaGnuKmn_cuH 0x ≥∀ . tamlkçNHénGnuKmn_cuH cMeBaH 0x ≥ eK)an 0)0(g)x(g =≤ eKTaj)an )2(xe1e xx ≤− tam ¬!¦ nig ¬@¦ eyIgTaj)an xx xe1ex ≤−≤ cMeBaH 0x ≥∀ . Taj[)anfa 1
x1elim
x
0x=
−→
eKman xx xe1ex ≤−≤ naM[ xx
ex
1e1 ≤−
≤
eKTaj)an 1elimx
1elim1 x
0x
x
0x=≤
−≤
→→
dUcenH 1x
1elimx
0x=
−→
. II-bgðajfa nn JI = rYcTajrktémø nI nig nJ
eyIgman dx.xcosxsin
xsinI2
0n 2n 2
n 2
n ∫
π
+=
សមករអនគមន
- 106 -
nig dx.xcosxsin
xcosJ2
0n 2n 2
n 2
n ∫
π
+=
tag t2
x −π
= naM[ dtdx −= cMeBaH 0x = naM[
2t π= ehIy
2x π= naM[ 0t =
n
2
0n 2n 2
n 2
n
0
2n 2n 2
n 2
n
2
0n 2n 2
n 2
n
Jdt.tsintcos
tcosI
)dt()t
2(cos)t
2(sin
)t2
(sinI
\xcosxsin
xsinI
=+
=
−−
π+−
π
−π
=
+=
∫
∫
∫
π
π
π
dUcenH nn JI = .
2dxJ2I2JI
dx.xcosxsin
xcosdx.xcosxsin
xsinJ2I2JI
2
0nnnn
2
0n 2n 2
n 22
0n 2n 2
n 2
nnnn
π====+
++
+===+
∫
∫∫
π
ππ
សមករអនគមន
- 107 -
dUcenH 4
JI nnπ
== . III- k-RsayfacMeBaHRKb; *INn∈ ³ 1n2xn −= eKman 11.21x1 −== Bit ebI 1n = enaH
3x2xx
112 +=−
naM[ 313131
21x2 =−+=
++=
eK)an 12.23x2 −== Bit . ]bmafavaBitdl;tYTI k KW 1k2xk −= eyIgnwgRsayfavaBitdl;tYTI 1k + KW ³
1k21)1k(2x 1k +=−+=+ eyIgman
1k2x2
xxk
k1k ++=−+
naM[ 1k2x
2xx
kk1k +++=+
eday 1k2xk −= enaH
1k21k22
1k2x 1k ++−+−=+
សមករអនគមន
- 108 -
1k21k21k2)1k2()1k2(
)1k21k2(21k2x 1k
−−++−=
−−+−−+
+−=+
1k2x 1k +=+ Bit . dUcenH 1n2xn −= .
x-KNna 1n2x
1.....
5x1
3x1
Sn21
n ++++
++
+=
eyIg)an ∑=
⎟⎟⎠
⎞⎜⎜⎝
⎛++
=n
1k kn 1k2x
1S
eday 1n2x
2xx
nn1n ++=−+
eKTaj )xx(21
1n2x1
n1nn
−=++ +
eK)an )xx(21)xx(
21S 11n
n
1kk1kn −=⎥⎦⎤
⎢⎣⎡ −= +
=+∑
eday 1x1 = ehIy 1n2xn −= enaH 1n2x 1n +=+ dUcenH )11n2(
21Sn −+= .
សមករអនគមន
- 109 -
K> TajrklImIt ⎥⎦⎤
⎢⎣⎡
+∞→ nSlim n
n
eyIg)an ⎥⎦
⎤⎢⎣
⎡ −+=⎥⎦
⎤⎢⎣⎡
+∞→+∞→ n11n2
.21
limn
Slim
n
n
n
22
n1
n12lim
21
n=⎟⎟
⎠
⎞⎜⎜⎝
⎛−+=
+∞→
dUcenH 22
nSlim n
n=⎥⎦
⎤⎢⎣⎡
+∞→ .
IV-KNnatémøGb,brmaén 222 zyxT ++= eyIgtag S CaépÞRkLarbs;RtIekaN ABC eyIg)an AMCBMCAMB SSSS ++=
ax
21by
21cz
21S
AC.MP21
BC.MN21
AB.MQ21
S
++=
++=
B C
M
N
P
សមករអនគមន
- 110 -
eKTaj)an S2czbyax =++ tamvismPaBEb‘nUyIeyIg)an ³
T.cbaS2
zyx.cbaczbyax222
222222
++≤
++++≤++
eKTaj)an 222
2
cbaS4T++
≥ dUcenHtémøGb,rmaén 222 zyxT ++= KW ³
222
2
min cbaS4T++
= Edl )cp)(bp)(ap(pS −−−= nig
2cbap ++
= .
សមករអនគមន
- 111 -
viBaØasaKNitviTüaTI10 I-eKeGayGaMgetRkal ( ) *
1
0
n2n INn,dx.x.
1n21
I ∈−
= ∫ . k>KNna nI CaGnuKmn_én n . x>KNnaplbUk n321n I.........IIIS ++++= rYcTajrklImIt nn
Slim+∞→
. II-cUrkMnt;rkGnuKmn_ )x(fy = ebIeKdwgfa ³
1x1x)2x2xx(f 2
22
+−
=+−+ cMeBaHRKb;cMnYnBit x . III- eKmanBhuFa
2n,IR,)cossinx()x(P nn ≥∈θθ+θ=
cUrrksMNl;énviFIEckrvagBhuFa )x(Pn nig 1x2 + . IV- kñúgbøg;kMupøic )j,i,O(
→→ eK[bYncMnuc D,C,B,A manGahVikerogKña i45Z,i54Z,i61Z CBA +=+=+= nig i32ZD −−= .cUrRsayfactuekaN ABCDcarwkkñúgrgVg; EdleKnwgbBa¢ak;p©it nig kaM .
សមករអនគមន
- 112 -
dMeNa¼Rsay I- k-KNna nI CaGnuKmn_én n
*1
0
n2n INn,dx.)x(.
1n21I ∈−
= ∫
)1n2)(1n2(
1x1n2
11n2
1
dx.x1n2
1
1
0
1n2
1
0
n2
+−=⎥⎦
⎤⎢⎣⎡
+−=
−=
+
∫
dUcenH )1n2)(1n2(
1In +−= .
x-KNnaplbUk n321n I.........IIIS ++++= eyIgman ⎟
⎠⎞
⎜⎝⎛
+−
−=
+−=
1n21
1n21
21
)1n2)(1n2(1
In eyIg)an
)1n2
11n2
1(21.....)
71
51(
21)
51
31(
21)
311(
21Sn +
−−
++−+−+−=
1n2n
1n211n2.
21)
1n211(
21
+=
+−+
=+
− dUcenH
1n2nSn +
= .
សមករអនគមន
- 113 -
TajrklImIt nnSlim
+∞→
eyIg)an 21
1n2n
limSlimnnn
=+
=+∞→+∞→
. II-kMnt;rkGnuKmn_ )x(fy = ³ eKman )1(
1x1x
)2x2xx(f 2
22
+−
=+−+ ebIeyIgtag t2x2xx 2 =+−+ eyIg)an xt2x2x2 −=+−
1t,)1t(2
2tx
2t)1t(x2
2tx2tx2
xtx2t2x2x
)xt(2x2x
2
2
2
222
22
≠−−
=
−=−
−=−
+−=+−
−=+−
yktémø )1t(2
2tx2
−−
= CMnYskñúg )1( eyIg)an ³
សមករអនគមន
- 114 -
8t8t)8t8t(t)t(f
8t8tt8t8t)t(f
4t8t44t4t4t8t44t4t
1])1t(2
2t[
1])1t(2
2t[)t(f
4
3
4
24
224
224
22
22
+−+−
=
+−+−
=
+−++−−+−+−
=+
−−
−−−
=
dUcenH 8x8x
)8x8x(x)x(f
4
3
+−
+−= .
III-rksMNl;énviFIEckrvagBhuFa )x(Pn nig 1x2 + tag )x(R CaGnuKmn_sMNl;énviFIEckrvagBhuFa )x(Pn nig 1x2 + naM[GnuKmn_ )x(R RtUvmandWeRktUcCagb¤esμImYy . yk bax)x(R += Edl a nig b CacMnYnBitRtUvrk ehIy]bmafa )x(Q CaGnuKmn_plEck rvagBhuFa )x(Pn nig 1x2 + enaHeK)anTMnak;TMng ³
bax)1x).(x(Q)cossinx(
)x(R)1x).(x(Q)x(P2n
2n
+++=θ+θ
++=
សមករអនគមន
- 115 -
eyIgeRCIserIsyk ix = Edl 1i2 −= eyIg)an ³
bai)ncos()nsin(.ibai)1i)(i(Q)cossin.i( 2n
+=θ+θ+++=θ+θ
eRBaH )ncos()nsin(.i)cossin.i( n θ+θ=θ+θ ¬rUbmnþdWmr½ ¦ eKTaj)an )ncos(b,)nsin(a θ=θ= . dUcenH )ncos()nsin(.x)x(R θ+θ= . IV-RsayfactuekaN ABCDcarikkñúgrgVg; eyIgtag 0cbyaxyx:)c( 22 =++++ CasmIkarrgVg;carikeRkARtIekaN ABC eyIg)an )c(A∈ naM[ 0cb6a61 22 =++++ b¤ )1(37cb6a −=++ )c(B∈ naM[ 0cb5a454 22 =++++ b¤ )2(41cb5a4 −=++ )c(C∈ naM[ 0cb3a2)3()2( 22 =+−−−+− b¤ )3(13cb3a2 −=+−−
សមករអនគមន
- 116 -
eyIg)anRbB½næsmIkar ⎪⎩
⎪⎨
⎧
−=+−−−=++−=++
13cb3a241cb5a4
37cb6a
bnÞab;BIedaHRsayRbB½næenHeK)ancemøIuy 23c,2b,2a −=−=−= .
smIkarrgVg;carikeRkARtIekaN ABCGacsresr ³ 023y2x2yx:)c( 22 =−−−+ b¤ 25)1y()1x(:)c( 22 =−+− müa:geTotedayykkUGredaen DCYskñúgsmIkar
25)13()12(:)c( 22 =−−+−− vaepÞógpÞat;enaHnaM[ )c(D∈ . edaybYncMnuc D,C,B,A sßitenAelIrgVg;
25)1y()1x(:)c( 22 =−+− EtmYynaM[ctuekaNABCD carikkñúgrgVg; )c( manp©it )1,1(I nig kaM 5R = .
សមករអនគមន
- 117 -
viBaØasaKNitviTüaTI11 I-eK[GnuKmn_
7 7x20071x
)x(f+
=
cUrKNna [ ]].....])x(f[f[.....ff)x(Fn = . II-bgðajfacMnYn nnnn
n 610709743831A −+−= Eckdac;nwg !*&Canic© INn∈∀ III-eK[sIVúténcMnYnBit )U( n kMnt;eday ³
⎪⎩
⎪⎨
⎧
∈∀−++−
=
=
+ INn,U2U81U5U22
U
1U
2nn
2nn
1n
0
k>eKtag 1U1U2
V:INnn
nn +
−=∈∀ .
cUrbgðajfa 2n1n VV =+ .
x> KNna nV nig nU CaGnuKmn_én n . IV-eK[RtIekaN ABC mYyEkgRtg; A . cUrRsayfaeKmanTMnak;TMng ³ 222666 BCACAB3ACABBC =−− .
សមករអនគមន
- 118 -
V-eK[viucTr½bI →→→c,b,a EkgKñaBIr²kñúglMhéntMruyGrtUnrm:al;
cUrRsaybBa¢ak;fa ³ 222 ||c||||b||||a||||cba||
→→→→→→++=++ .
dMeNa¼Rsay I- KNna [ ]].....])x(f[f[.....ff)x(Fn = eyIgman
7 71x20071
x)x(f)x(F
+==
[ ] [ ][ ][ ] [ ]
[ ]7 7
1n
1n1nn
23
12
)x(F20071)x(F
)x(Ff)x(F
)x(Ff)x(fff)x(F)x(Ff)x(ff)x(F
−
−−
+==
−−−−−−−−−−−−−−−−
====
eyIg)an )x(F20071
)x(F)x(F 7
1n
71n7
n−
−
+=
naM[ 2007)x(F
1)x(F
)x(F20071)x(7F
17
1n7
1n
71n +=
+=
−−
−
eKTaj 2007)x(F
1)x(F
17
1n7n
=−−
efr
សមករអនគមន
- 119 -
naM[eKTaj)anfa ⎟⎟⎠
⎞⎜⎜⎝
⎛)x(F
17n
CasIVútnBVnþmanplsgrYm
2007d = nigtYTImYy 7
7
71
1 xx20071
)x(F1
U+
== .
tamrUbmnþ d).1n(UU 1n −+= eK)an 7
7
7
7
7n x
nx20071)1n(2007
xx20071
)x(F1 +
=−++
=
eKTaj 7 7
77
7
nnx20071
xnx20071
x)x(F
+=
+=
dUcenH [ ]7 7n
nx20071x
].....])x(f[f[.....ff)x(F+
==
II-bgðajfacMnYn nnnnn 610709743831A −+−=
Eckdac;nwg !*&Canic© INn∈∀ eyIgman 1711187 ×= . eday !! nig !& CacMnYnbfmrvagKñaenaHedIm,IRsayfacMnYn nA Eckdac;nwg !*& eKRKan;EtRsay[eXIjfavaEckdac;nwg !!pg ehIyvaEckdac;nig!&pg .
សមករអនគមន
- 120 -
tamrUbmnþ )b....baa)(ba(ba 1n2n1nnn −−− +++−=− eK)an )610709()743831(A nnnn
n −+−=
)q9q8(11q99q88q)610709(q)743831(
2121
21
+=+=
−+−=
tag INq,q,q9q8q 2121 ∈+= eK)an q11An = naM[ nA Eckdac;nwg !! . müa:geToteKman )709743()610831(A nnnn
n −−−=
)INp,p,p2p13p(,p17A)p2p13(17q34p221
p)709743(p)610831(
2121n
2121
21
∈−==−=−=
−−−=
TMnak;TMngenHbBa¢ak;fa nA Eckdac;nwg !& . dUcenH nnnn
n 610709743831A −+−= Eckdac;nwg !*& Canic© INn∈∀ . III-k >bgðajfa 2
n1n VV =+ eyIgman
1U1U2
V:INnn
nn +
−=∈∀ naM[
1U1U2V
1n
1n1n +
−=
+
++
eday 2nn
2nn
1n U2U81U5U22U
−++−
=+ cMeBaHRKb; INn ∈ .
សមករអនគមន
- 121 -
eK)an
2nn
2nn
2nn
2nn
2nn
2nn
2nn
2nn
1n
U2U81U5U22U2U81U10U44
1U2U81U5U22
1U2U81U5U222
V
−+++−+−−+−
=
+−++−
−⎟⎟⎠
⎞⎜⎜⎝
⎛−++−
=+
2n
2
n
n
n2n
n2n
n2n
n2n
1n
V1U1U2
)1U2U(3)1U4U4(3
3U6U33U12U12V
=⎟⎟⎠
⎞⎜⎜⎝
⎛+−
=
+++−
=+++−
=+
dUcenH 2n1n VV =+ .
x> KNna nV nig nU CaGnuKmn_én n edayeKman 2
n1n VV =+ eKTaj n1n Vln2Vln =+ naM[ )V(ln n CasVúItFrNImaRt manersug 2q = nigtYdMbUg )
21
ln()1U1U2
ln(Vln0
00 =
+−
= .
tamrUbmnþeK)an n2nn )
21
ln()21
ln(.2Vln ==
សមករអនគមន
- 122 -
naM[ nn
22
n )5.0(21
V =⎟⎠⎞
⎜⎝⎛=
ehIyeday 1U1U2V
n
nn +
−= naM[ n
n
2
2
n
nn
)5.0(2
)5.0(1V2V1
U−
+=
−+
=
dUcenH n
nn
2
2
n2
n)5.0(2
)5.0(1U,)5.0(V
−
+== .
IV-RsayfaeKmanTMnak;TMng 222666 BCACAB3ACABBC =−−
tamRTwsþIbTBItahÁr½kñúgRtIekaNEkg ABC eKman ³ )1(ACABBC 222 +=
elIkGgÁTaMgBIrén ¬!¦ CasV½yKuN # eK)an ³
)2()ACAB(AC.AB3ACABBC
ACAC.AB3AC.AB3ABBC
)ACAB(BC
2222666
6422466
3226
+=−−
+++=
+=
yk ¬!¦ CYskñúg ¬@ ¦eK)an 222666 BCACAB3ACABBC =−− .
សមករអនគមន
- 123 -
V-RsaybBa¢ak;fa 222 ||c||||b||||a||||cba||→→→→→→
++=++ eKman
)c.ac.bb.a(2)c()b()a()cba(||cba|| 22222→→→→→→→→→→→→→→→
+++++=++=++ eday →→→
c,b,a EkgKñaBIr²enaHeKman 0c.ac.bb.a ===
→→→→→→ eK)an 2222 ||c||||b||||a||||cba||
→→→→→→++=++
dUcenH 222 ||c||||b||||a||||cba||→→→→→→
++=++ .
សមករអនគមន
- 124 -
viBaØasaKNitviTüaTI12 I-eK[ExSekag
4x4x6x3y:)(
2
−+
=Γ nigcMnuc )3,1(I . cUrkMnt;smIkarrgVg; )C( manp©it I ehIyb:HnwgExSekag )(Γ II-RtIekaNEkgmYymanépÞRkLa 2cm6 ehIyRCugTaMgbI begáIt)anCasIVútnBVnþmYy. cUrkMnt;RCugTaMgbIrbs;RtIekaNenH . III-eKmansIVút )I( n kMnt;cMeBaHRKb; 1n ≥ eday ³ ∫ −=
1
0
xnn dx.e.)x1(.
!n1I
k-cUrKNnatY 1I . x-cUrbBa¢ak; 1nI + CaGnuKmn_én nI rYcTaj[)anfa ∑
=⎟⎠⎞
⎜⎝⎛−=
n
0pn !P
1eI K-cUrrklImIt nn
Ilim+∞→
rYcTajfa ³ 71828.2e
!n1....
!31
!21
!111lim
n==⎟
⎠⎞
⎜⎝⎛ +++++
+∞→
សមករអនគមន
- 125 -
IV-eK[bnÞat;BIrmansmIkarqøúHerogKñaxageRkam ³ ( )
32z
46y
32x:L1
+=
−=
−+
nig ( )12z
45y
93x:L2 −
−=
+=
− . k-cUrsresrsmIkarbnÞat;EkgrYm ( )Δ rvagbnÞat; ( )1L nig ( )2L x-KNnacMgayrvagbnÞat; ( )1L nig ( )2L .
dMeNa¼Rsay I-kMnt;smIkarrgVg; )C( manp©it I ehIyb:HnwgExSekag )(Γ eyIgman
4x4x6x3y:)(
2
−+
=Γ nigcMnuc )3,1(I tag R CakaMénrgVg; )C( EdlRtUvrk tamrUbmnþ 22
I2
I R)yy()xx(:)C( =−+− b¤ 222 R)3y()1x(:)C( =−+− smIkarGab;sIuscMnucRbsBVrvagrgVg;CamYyExSekagsresr ³
សមករអនគមន
- 126 -
[ ] 22
222
22
222
222
2
R)1x(16
9)1x(3)1x(
R)1x(16
)12x12x6x3()1x(
R)34x4x6x3()1x(
=−+−
+−
=−
+−++−
=−−+
+−
eyIgtag 1x,)1x(X 2 ≠∀−= eK)an ³
)1(081X)27R8(2X25
XR16)9X3(X16
RX16
)9X3(X
22
222
22
=+−−
=++
=+
+
edIm,I[rgVg; )C( b:HnwgExSekag )(Γ luHRtaEtsmIkar ¬!¦ manb¤¤¤¤¤¤¤¤¤¤¤¤¤sDúbeBalKWeKRtUv[ 02025)27R8(' 22 =−−=Δ naM[ 4527R8 2 =− naM[ 3R = . dUcenHsmIkarrgVg;sresr 9)3y()1x(:)C( 22 =−+− b¤eKGacsresrCaragTUeTA 01y6x2yx:)C( 22 =+−−+ .
សមករអនគមន
- 127 -
II-KNnargVas;RCugénRtIekaNEkg tamRTwsþIbTBItahÁr½kñúgRtIekaNEkg ABC eKman ³
)1(ACABBC 222 += eday BC,AC,AB CasIVútnBVnþenaHeKman ³
d2ABBC,dABAC +=+= Edl 0d > CaplsgrYménsIVút. tamTMnak;TMng ¬!¦ Gacsresr
222 )dAB(AB)d2AB( ++=+ b¤ 22222 dd.AB2ABABd4d.AB4AB +++=++
0)d3AB)(dAB(0d3AB.d2AB 22
=−+=−−
eKTaj)an dAB −= ¬minyk¦ nig d3AB = ¬yk ¦ eK)an d5BC,d4AC,d3AB === tag S CaépÞRkLarbs;RtIekaNeK)an
6d6AC.AB21S 2 === naM[ 1d = .
សមករអនគមន
- 128 -
dUcenH cm5BC,cm4AC,cm3AB === . III-k>cUrKNnatY 1I eKman ∫ ∫ −=−=
1
0
1
0
xx1 dx.e).x1(dx.e)x1(
!11I
tag ⎩⎨⎧
=
−=
dxedv
x1ux naM[
⎩⎨⎧
=
−=xev
dxdu
eK)an [ ] [ ] 2ee1)dx(ee)x1(I10
x1
0
x10
x −=+−=−−−= ∫
dUcenH 2eI −= . x-bBa¢ak; 1nI + CaGnuKmn_én nI eKman ∫ −=
1
0
xnn dx.e.)x1(.
!n1I
naM[ ∫ ++ −
+=
1
0
x1n1n dxe.)x1(.
)!1n(1I
tag ⎩⎨⎧
=
−= +
dxedv
)x1(ux
1n
naM[ ⎩⎨⎧
=
−+−=x
n
ev
)x1)(1n(du
eK)an [ ] ∫ −++
+−+
= ++
1
0
xn10
x1n1n dx.e)x1(
)!1n(1ne)x1(
)!1n(1I
∫ ++
−=−++
−=+
1
0n
xn1n I
)!1n(1
dx.e)x1(!n
1)!1n(
1I
សមករអនគមន
- 129 -
dUcenH )!1n(
1II n1n +−=+ .
Taj[)anfa ∑=
⎟⎠⎞
⎜⎝⎛−=
n
0pn !P
1eI
eKman )!1n(
1II n1n +−=+
cMeBaH !2
1II:1n 12 −== cMeBaH
!31II:2n 23 −==
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> cMeBaH
!n1
II:1nn 1nn −=−= − edayeFIVplbUkTMnak;TMngenHGgÁ nig GgÁ eK)an ³
!n1
....!3
1!2
1II 1n −−−−= eday
!11
!01
e2eI1 −−=−=
dUcenH ∑=
⎟⎟⎠
⎞⎜⎜⎝
⎛−=−−−−−=
n
0pn !p
1e!n
1....!2
1!1
1!0
1eI .
K-cUrrklImIt nnIlim
+∞→
cMeBaH [ ]1,0x∈ eKman ee1 x ≤≤ nig 0)x1( n ≥−
សមករអនគមន
- 130 -
eK)an nnxn )x1(e)x1(e)x1( −≤−≤− naM[ ∫∫∫ −≤−≤−
1
0
n1
0
xn1
0
n dx.)x1(!n
edx.e)x1(!n
1dx.)x1(!n
1
eday 1n
1)x1(1n
1dx.)x1(1
0
1
0
1nn
+=⎥⎦
⎤⎢⎣⎡ −
+−=−∫ +
eKTaj)an )1n(!n
eI)1n(!n
1n +≤≤
+ .
kalNa +∞→n naM[ 0)1n(!n
1→
+
dUcenH 0Ilim nn=
+∞→ .
Tajfa 71828.2e!n
1....!3
1!2
1!1
11limn
==⎟⎠⎞
⎜⎝⎛ +++++
+∞→
eKman ∑=
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
n
0pn !p
1eI naM[ n
n
0pIe
!p1
−=⎟⎟⎠
⎞⎜⎜⎝
⎛∑=
eK)an ( ) eIelim!p
1lim nn
n
0pn=−=⎟⎟
⎠
⎞⎜⎜⎝
⎛+∞→=+∞→
∑
eRBaH 0Ilim nn=
+∞→
dUcenH 71828.2e!n
1....!3
1!2
1!1
11limn
==⎟⎠⎞
⎜⎝⎛ +++++
+∞→
សមករអនគមន
- 131 -
IV-k>smIkarbnÞat;EkgrYm ( )Δ rvagbnÞat; ( )1L nig ( )2L tag ( ) ( )1AAA Lz,y,xA ∈ naM[kUGredaen A epÞógpÞat; smIkarbnÞat; ( )1L . eK)an p
32z
46y
32x AAA =
+=
−=
−+
naM[ ( )12p3z6p4y
2p3x
A
A
A
⎪⎩
⎪⎨
⎧
−=
+=
−−=
tag ( ) ( )2BBB Lz,y,xB ∈ naM[kUGredaen B epÞógpÞat; smIkarbnÞat; ( )2L . eK)an q
12z
45y
96x BBB =
−−
=+
=+
naM[ ( )22qz5q4y6q9x
B
B
B
⎪⎩
⎪⎨
⎧
+−=
−=
−=
ebI ( )AB CabnÞat;EkgrYmrvagbnÞat; ( )1L nig ( )2L
enaHeK)an ⎪⎩
⎪⎨
⎧
⊥
⊥⎯→⎯
⎯→⎯
2
1
UAB
UAB
r
r
naM[ ⎪⎩
⎪⎨
⎧
=
=⎯→⎯
⎯→⎯
0U.AB
0U.AB
2
1
r
r
សមករអនគមន
- 132 -
Edl 1Ur nig 2U
r CaviucTr½R)ab;TisbnÞat;( )1L nig ( )2L edayeKman )4p3q,11p4q4,4p3q9(AB +−−−−−+=⎯→⎯
nig )1,4,9(U,)3,4,3(U, 21 −−rr .
eK)an 0)4p3q()11p4q4(4)4p3q9(3U.AB 1 =+−−−−−+−+−=
⎯→⎯ r naM[ )3(020p34q14 =−−−
0)4p3q()11p4q4(4)4p3q9(9U.AB 2 =+−−−−−+−+=⎯→⎯ r
naM[ )4(084p14q98 =−+ . tam ¬#¦ nig ¬$¦ eK)anRbBn½æsmIkar ³
⎩⎨⎧
=−+=−−−084p14q98
020p34q14 naM[ ⎩⎨⎧
=−=1q
1p
yktémø 1p −= nig 1q = CYskñúgsmIkar ¬!¦ nig ¬@¦ eK)an )5,2,1(A − nig )1,1,3(B − . smIkarbnÞat; )AB( Gacsresr ³
សមករអនគមន
- 133 -
( )AB
A
AB
A
AB
A
zzzz
yyyy
xxxx:AB
−−
=−−
=−−
b¤ ( )6
5z32y
21x:AB +
=−−
=−
dUcenH ( )6
5z32y
21x: +
=−−
=−
Δ CabnÞat;EkgrYmEdlRtUvrk . x-KNnacMgayrvagbnÞat; ( )1L nig ( )2L eday A nig B CacMnucRbsBVénbnÞat;EkgrYmrvag ( )1L nig ( )2L enaHeK)an ³ ( ) ( )
( ) ( ) ( )2AB
2AB
2AB
21
zzyyxx
ABd)L(),L(d
−+−+−=
=
( ) 7)51()21()13()L(),L(d 22221 =++−−+−=
dUcenH ( ) 7)L(),L(d 21 = ¬ ÉktaRbEvg ¦ .
សមករអនគមន
- 134 -
viBaØasaKNitviTüaTI13 I- eK[sVúItnBVnþ n321 U........,,U,U,U EdlmanplsgrYm d ehIy 0d ≠ . eKBinitüsIVút )V( n mYykMnt;cMeBaHRKb; *INn∈ eday 1a,0a,aV nU
n ≠>= . k>bgðajfa )V( n CasIVútFrNImaRtrYcKNna nV CaGnuKmn_én d,U,a 1 nig n. x> cUrRsayfa d
ndU
n321n a1a1
.aV....VVVS 1
−−
=++++= K> cUrKNnaplKuN n321n V.....VVVP ××××= CaGnuKmn_én 1U,a nig n . II-eK[cMnYn INn,2007200722007A n20092008
n ∈+×+= . kMnt; n edIm,I[ nA CakaerR)akdéncMnYnKt; .
សមករអនគមន
- 135 -
III-eK[RtIekaN ABC mYyEkgRtg; A Edl 5BC = . M CacMnucmYyénRCug ]BC[ EdlmMu ∧∧
= MACMAB ehIy
7212
AM = . cUrKNnaRCug AB nig AC ? IV-eK[GnuKmn_ f kMnt;elI IR ehIyepÞogpÞat;TMnak;TMng³
( )( )
( )5432
32 x1x4x1x1
fx1
1xfx +=⎟
⎠⎞
⎜⎝⎛
+−
++
cUrKNnaGaMgetRkal³ ( )∫=1
0dx.xfI .
V- enAkñúgtMruyGrtUNrma:l;manTisedAviC¢man ⎟⎠⎞
⎜⎝⎛ →→→
k,j,i,0 manÉktþa cm1 enAelIGkS½ eK[BIrcMnuc )0,2,0(A − nig )1,2,1(B − . ( )P Cabøg;mansmIkar 04zy2x2 =+++ . cUrsresrsmIkarbøg; ( )Q kat;tamcMnuc A nig B ehIypÁúM CamYybøg)anmMuRsYcmYymantémø
4π
=θ .
សមករអនគមន
- 136 -
dMeNa¼Rsay I- k> bgðajfa )V( n CasIVútFrNImaRt eyIgman nU
n aV = naM[ 1nU1n aV +=+
eyIg)an dUUU
U
n
1n aaa
aV
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 1U1 aV = nig daq =
eK)an d)1n(Ud)1n(Un
11 aa.aV −+− == .
x>Rsayfa d
ndU
n321n a1a1
.aV....VVVS 1
−−
=++++=
tamrUbmnþ q1q1
.VV..........VVVSn
1n321n −−
=++++= eday 1U
1 aV = nig daq = dUcenH d
ndU
n321n a1a1
.aV....VVVS 1
−−
=++++= .
សមករអនគមន
- 137 -
K>KNnaplKuN n321n V.....VVVP ××××= eyIg)an n321n V.....VVVP ××××=
2
)UU(nU....UUU
UUUU
n1n321
n321
aa
a.......a.a.a+
++++ ==
=
dUcenH 2)UU(n
n
n1
aP+
= . II-kMnt; n edIm,I[ nA CakaerR)akd eyIgman INn,2007200722007A n20092008
n ∈+×+= )2007200721(2007 2008n2008 −+×+= edIm,I[ nA CakaerR)akdluHRtaEt 22008n 20072007 =− naM[ 2010n = . III- KNnaRCug AB nig AC tag yAC,xAB ==
eKman 0452
BACMACMAB ===
∧∧∧
tamRTwsþIbTsIunUskñúgRtIekaN ABM nig ACM eKman ³
សមករអនគមន
- 138 -
)1(45sin
BMBsin
AM0= nig )2(
45sinMC
CsinAM
0= bUkTMnak;TMng ¬!¦ nig ¬@¦ eK)an ³
)3(1235
21
.7
2125
Csin1
Bsin1
45sin.AMBC
Csin1
Bsin1
45sinBC
45sinMCBM
CsinAM
BsinAM
0
00
==+
=+
=+
=+
kñúgRtIekaNEkg ABC eKman 5y
BCACBsin ==
nig 5x
BCABCsin ==
TMnak;TMng ¬#¦ GacsresreTACa ³
1235
x5
y5
=+ naM[ )4(xy127yx =+
tamRTwsþIbTBItaKr½ kñúgRtIekaNEkg ABC eKman ³ 222 ACABBC += naM[ )5(yx25 22 +=
សមករអនគមន
- 139 -
eday xy2)yx(yx 222 −+=+ enaHtam¬%¦eK)an ³ )6(25xy2)yx( 2 =−+
tag 0yxS >+= nig 0y.xP >=
tam ¬%¦ nig ¬^¦ eK)anRbBn½æ ⎪⎩
⎪⎨⎧
=−
=
25p2S
P127S
2
naM[ 12P,7S == eK)an
⎩⎨⎧
×===+==+=4312xyP
437yxS
naM[ 4y,3x == b¤ 3y,4x == dUcenH 4AC,3AB == b¤ 3AC,4AB == . IV-KNnaGaMgetRkal³ ( )∫=
1
0
dx.xfI tag 3tx = naM[ dt.t3dx 2= nigcMeBaH [ ]1,0x∈ naM[ [ ]1,0t ∈ eK)an ∫ ∫==
1
0
1
0
23 dtt3).t(fdx).x(fI
naM[ ( )∫=1
0
32 1dt).t(ftI31
សមករអនគមន
- 140 -
müa:geTotebIeKtag t1t1x
+−
= naM[ 2)t1(dt2dx+
−=
cMeBaH [ ]1,0x∈ naM[ [ ]0,1t ∈ eK)an
∫ ∫ ∫ +−
+=
+−⎟
⎠⎞
⎜⎝⎛
+−
==1
0
0
1
1
022 dt).
t1t1(f
)t1(12)
)t1(dt2.(
t1t1fdx).x(fI
eKTaj)an ( )∫ +−
+=
1
02 2dt).
t1t1
(f)t1(
1I
21
bUkTMnak;TMng ¬!¦ nig ¬@¦ eK)an ³ dt.)
t1t1(f
)t1(1)t(ftI
21I
31 1
02
32∫ ⎥⎦
⎤⎢⎣
⎡+−
++=+
tamsmμtikmμeKman ³ ( )
( )( )543
232 x1x4
x1x1
fx1
1xfx +=⎟
⎠⎞
⎜⎝⎛
+−
++
eK)an 663
6164
)t1(61
dt.)t1(t4I65 1
0
641
0
543 =−
=⎥⎦⎤
⎢⎣⎡ +=+= ∫
dUcenH 563dx).x(fI
1
0== ∫ .
សមករអនគមន
- 141 -
V-sresrsmIkarbøg; ( )Q tag ( ) 0dczbyax:Q =+++ CasmIkarEdlRtUvrk. edaybøg; ( )Q kat;tamcMnuc A nig B enaHkUGredaencMnuc A nig B epÞógpÞat;nwgsmIkarbøg; ( )Q . eK)an
⎩⎨⎧
=++−+=++−+
0d)1(c)2(b)1(a0d)0(c)2(b)0(a
b¤ ⎩⎨⎧
=++−=+−
0dcb2a0db2
naM[eKTaj)an ⎪⎩
⎪⎨
⎧
−=
=
)2(ca
)1(2db
müa:geTotebIeyIgtag θ CamMupÁúMedaybøg; ( )P nig ( )Q enaHeK)an
QP
QP
n.nn.n
cos rr
rr
=θ
eday ( )1,2,2nPr nig )c,b,a(nQ
r CaviucTr½Nrm:al;énbøg; ( )P nig ( )Q .
សមករអនគមន
- 142 -
eK)an
22222222 cba3cb2a2
cba.122cb2a2cos
++
++=
++++
++=θ
eday 4π
=θ ¬bMrab;¦enaHeKTaj 22
cba3cb2a2
222=
++
++
naM[ )3()cba(9)cb2a2(2 2222 ++=++ yksmIkar ¬!¦ nig ¬@¦ CYskñúg ¬#¦ eK)an ³ )c
4dc(9)cdc2(2 2
222 ++=++−
)4
dc2(9)dc(22
22 +=+− 2222 d
49c18)dcd2c(2 +=+−
04
dcd4c16
04
dcd4c16
0d49c18d2cd4c2
22
22
2222
=++
=−−−
=−−+−
02dc4
2
=⎟⎠⎞
⎜⎝⎛ + naM[
8dc −= .
សមករអនគមន
- 143 -
tamTMnak;TMng ¬!¦ nig ¬@¦ eKTaj 2db = nig
8da = .
yktémø 2db,
8da == nig
8dc −= CYskñúgsmIkarbøg; ( )Q
eK)an ³ ( ) 0dz
8dy
2dx
8d:Q =+−+
smmUl ( ) 08zy4x:Q =+−+ . dUcenHsmIkarbøg; ( )Q EdlRtUvrkKW ³ ( ) 08zy4x:Q =+−+ .
សមករអនគមន
- 144 -
viBaØasaKNitviTüaTI14 I- eK[ )U( n CasIVútkMnt;eday baUU n1n +=+ cMeBaHRKb; *INn∈ nig IRb,a,0a ∈≠ . k> cMeBaH 1a = cUrrkRbePTénsIVút )U( n . x>cMeBaH 1a ≠ ]bmafa kVU nn += cMeBaHRKb; *INn∈ . cUrkMnt;témø k edIm,I[ )V( n CasIVútFrNImaRt . K> ]bmafa 1a ≠ . cUrKNna nU CaGnuKmn_én b,a nig nnigtY 1U . II-eK[ f CaGnuKmn_Cab;elI [ ]1,0 . cUrbgðajfa ∫∫
ππ π=
00dx).x(sinf
2dx).x(sinf.x ?
Gnuvtþn_³ cUrKNna ∫π
+=
02 xcos1dx.xsinxI .
III-edaHRsaysmIkar n34n
35CC 23
1n2
1n −=+ ++ .
សមករអនគមន
- 145 -
IV- eK[GnuKmn_ 1x
baxx)x(fy 2
2
+++
== manRkabtMnag ( )c kñúgtRmuyGrtUnremmYy. k-ExSekag ( )c kat;GkS½Gab;sIus ( )x0'x Rtg;cMnucman Gab;sIus α=x .bgðajfabnÞat;b:H ( )c Rtg; α=x man emKuNR)ab;Tis
1a2k 2 +α
+α= .
x-cUrkMnt;témø a nig b edIm,I[ExSekag ( )c kat;GkS½ Gab;sIus)anBIrcMnuc M nig N EdlbnÞat;b:H ( )c Rtg; M nig N EkgnwgKña .
dMeNa¼Rsay I-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 CasIVútnBVnþ
សមករអនគមន
- 146 -
manplsgrYm 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 +−+=+ tamTMnak;TMngenH edIm,I[ )V( n CasIVútFrNImaRtluHRtaEt
0bk)1a( =+− . eKTaj)an
a1b
1ab
k−
=−
−= . K> KNna nU CaGnuKmn_én b,a nig nnigtY 1U tamdMeNaHRsayxagelIeyIgeXIjfacMeBaH 1a ≠ eBlEdl
a1b
k−
= enaHsIVút )V( n CasIVútFrNImaRt manersug aq = nigtY
a1b
UkUV 111 −−=−=
សមករអនគមន
- 147 -
tamrUbmnþ 1n1
1n1n a).
a1b
U(qVV −−
−−=×=
eday kVU nn += dUcenH
a1b
a).a1
bU(U 1n
1n −+
−−= − .
II-bgðajfa ∫∫ππ π
=00
dx).x(sinf2
dx).x(sinf.x tag tx −π= naM[ dtdx −= nig cMeBaH [ ]π∈ ,0x naM[ [ ]0,t π∈ eK)an [ ]dt.)tsin(f).t(dx).x(sinf.x
0
0∫∫π
π
−π−π−=
∫ ∫∫∫π πππ
−π=−π=0 000
dt).t(sinf.tdt).t(sinfdt).t(sinf).t(dx).x(sinf.x
∫ ∫∫π ππ
−π=0 00
dx).x(sinf.xdx).x(sinfdx).x(sinf.x
naM[eKTaj)an ∫∫ππ π
=00
dx).x(sinf2
dx).x(sinf.x .
Gnuvtþn_³ KNna ∫π
+=
02 xcos1dx.xsinxI
eKman ∫ ∫ ∫π π π
−π
=−
=+
=0 0 0
222 xsin2dx.xsin
2xsin2dx.xsin.x
xcos1dx.xsin.xI
សមករអនគមន
- 148 -
tag xcosz = naM[ dx.xsindz −= ehIycMeBaH [ ]π∈ ,0x enaH [ ]1,1z −∈ eK)an [ ]∫
−
−π
=⎟⎠⎞
⎜⎝⎛ π
+ππ
=π
=+−π
=1
1
21
12 4442zarctan
2z1dz
2I
dUcenH 4xcos1
dx.xsinxI2
02
π=
+= ∫
π .
III-edaHRsaysmIkar n34n
35CC 23
1n2
1n −=+ ++
lkç½xNн ⎩⎨⎧
≥+∈
31nINn naM[ 2n ≥ .
eKman
( )( )
( ) ( )( )
( )2
1nn!1n.2
1nn.!1n!1n!.2
!1nC2
1n+
=−
+−=
−+
=+
( )( )
( ) ( ) ( )( )
( )( )6
1n1nn
!2n.61n.n.1n.!2n
!2n!.3!1nC3
1n
+−=
−+−−
=−+
=+
សមករអនគមន
- 149 -
smIkarGacsresr ³ ( ) ( )( )
n34
n35
61n1nn
21nn 2 −=
+−+
+ ( ) ( )
( )( ) 5n,2n,0n05n2nn
0n10n7n
0n8n10nnn3n3
n8n101nn1nn3
23
232
22
====−−
=+−
=+−−++
−=−++
naM[
eday 2n ≥ dUcenH 2n = b¤ 5n = . IV- bgðajfabnÞat;b:H ( )c Rtg; α=x manemKuNR)ab;Tis
1a2k 2 +α
+α=
eKman ( )1x
baxxxf 2
2
+++
= eK)an ( ) ( ) ( )
22
2222
)1x(baxx)'1x(1x)'baxx(x'f
++++−+++
=
22
22
)1x()baxx(x2)1x)(ax2(
+++−++
=
ebI k CaemKuNR)ab;Tisén bnÞat;b:H ( )c Rtg; α=x eK)an ( )α= 'fk .
សមករអនគមន
- 150 -
eK)an ( ) ( )1)1(
)ba.(2)1x(a2k 22
22
+α+α+αα−++α
=
müa:geTotExSekag ( )c kat;GkS½Gab;sIus ( )x0'x Rtg;cMnucman Gab;sIus α=x . eK)an ( ) 0
1baf 2
2
=+α+α+α
=α naM[ ( )20ba2 =+α+α yksmIkar ( )2 CYskñúg ( )1 eK)an
1a2
)1()1)(a2(k 222
2
+α+α
=+α
+α+α=
dUcenH 1a2k 2 +α
+α= .
x-kMnt;témø a nig b smIkarGab;sIuscMnucRbsBV M nig N rvagExSekag ( )c CamYyGkS½ ( )x0'x ³ ( ) 0
1xbaxxxf 2
2
=+++
= b¤ ( )E0baxx2 =++ tag 1k nig 2k CaemKuNR)ab;TisénbnÞat;b:H ( )c Rtg; M nig N eK)an ³
សមករអនគមន
- 151 -
( )1xax2x'fk 2
M
MM1 +
+== nig ( )
1xax2x'fk 2
N
NN2 +
+==
edIm,I[bnÞat;b:HenHEkgKñaluHRtaEt 1
1xax2.
1xax2k.k 2
N
N2M
M21 −=⎟⎟
⎠
⎞⎜⎜⎝
⎛++
⎟⎟⎠
⎞⎜⎜⎝
⎛++
=
naM[ )1x)(1x()ax2)(ax2( 2N
2MNM ++−=++
( ) ( )( )[ ]
( )301axx2)xx(a2xx)xx(
01xx2)xx(xxa)xx(a2xx4
01xxxxa)xx(a2xx4
0)1x)(1x(ax2.ax2
2NMNM
2N
2M
2NM
NM2
NM2N
2M
2NMNM
2N
2M
2N
2M
2NMNM
2N
2MNM
=+++++++
=+−++++++
=+++++++
=+++++
eday Mx nig Nx Cab¤ssmIkar ( )E enaHeKman
⎩⎨⎧
=
−=+
bxxaxx
NM
NM
TMnak;TMng ( )3 Gacsresr ³
( ) 1b01b1b2b
01ab2a2ba
22
2222
−==+=++
=+++−+
naM[
müa:geTotedIm,I[ ( )c kat;GkS½ ( )x0'x )anBIrcMnuc M nig N
សមករអនគមន
- 152 -
luHRtaEtsmIkar ( )E manb¤sBIrepSgKña eBalKWeKRtUv[ 0b4a2 >−=Δ eday 1b −= eKTaj)an 04a2 >+=Δ BitCanic©RKb;cMnYnBit a . dUcenH 1b,IRa −=∈ .
សមករអនគមន
- 153 -
lMhatGnuvtþn_ 1-cUrkMnt´rkGnuKmn_ )x(fy = ebIeKdwgfa ½ 1)0(f)0('f == nig x4x6)x('f
)1x2(2)x(''f
1x21 2
2 −=−
−−
2-cUrkMnt´rkGnuKmn_ )x(fy = ebIeKdwgfa ½ 2)0(f = nig 1x4x)x(f)x('f 22 +−= 3-cUrkMnt´rkGnuKmn_ )x(fy = ebIeKdwgfa ½ 4)1(f = nig 1x2x3x4)x('f)1x()x(fx2 232 +++=++ 4-cUrkMnt´rkGnuKmn_ )x(fy = ebIeKdwgfa ½ 2)1(f = nig x9x4)x(f2)x('fx 2 +=+ 5-eK[GnuKmn_ f kMnt´BIsMNMu IR eTA +*IR EdlcMeBa¼RKb´ IRy,x ∈ eKman )y(f)x(f)yx(f =+ nig 4)0('f =
សមករអនគមន
- 154 -
cUrkMnt´rkGnuKmn_ )x(f . 6-cUrkMnt´rkGnuKmn_ f manedrIevelI IR ebIeKdwgfa
IRy,IRx ∈∈∀ eKman )y(f)x(f2
yxf =⎟⎠⎞
⎜⎝⎛ + .
7-eK[GnuKmn_ x12x6x)x(f 23 +−= kMnt´elI IR k/ cUrkMnt´rkGnuKmn_ )x(f 1− CaGnuKmn_Rcas´ énGnuKmn_ )x(f . x/ cUrKNnaedrIevénGnuKmn_ )x(f nig )x(f 1− . 8-eKmanGnuKmn_ )x(f kMnt´elI IR EdlcMeBa¼Kb´ IRx∈ eKmanTMnak´TMng )x(fx2)x('f = ehIy 1)0(f = cUrkMnt´rkGnuKmn_ )x(f . 9-eK[GnuKmn_ f nig g manedrIevelI IR EdlcMeBa¼ RKb´ IRx∈ eKmanTMnak´TMng )x(g)x('g)x(f)x('f 22 = cUrrkTMnak´TMngrvag f nig g .
សមករអនគមន
- 155 -
10-eK[GnuKmn_ f kMnt´elI IR eday 2x1x)x(f ++= k/ bgHajfa f manedrIevelI IR nig )x(f)x('f.x12 2 =+ . x/ TajbBa¢ak´faedrIev ''f epÞogpÞat´ )x(f)x('xf4)x(''f)x1(4 2 =++ . 11-eK[GnuKmn_ xsinx)x(f −= nig xsin
6xx)x(g
3
−−= cMeBaH 0x > k/ cUrsikSaGefrPaBénGnuKmn_ )x(f nig )x(g . x/ RsaybBa¢ak;fa xxsin
6xx
3
≤≤− cMeBaH 0x > . K/ eKtag 3222n n
nsin.....n3sin
n2sin
n1sinS ++++= .
cUrrkkenSamGmén nS rYcTajrk nnSlim
+∞→ .
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