គណះកមមករនពនិ្ឋនងិ · 2010. 8. 27. ·...

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 ។


- 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 ។


- 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 >=


- 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 <<=


- 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 =


- 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 <<=


- 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 −=⇒=


- 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 = ។


- 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


- 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 ។


- 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∈


- 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


2x1x2)

1x1x(f

−+

=+− ?

ដេ ះរ យ


េគមន )1(2x1x2)

1x1x(f

−+

=+−

ង 1xt)1x(1x1xt −=+⇒

+−

=


- 13 -

1t1tx

1tx)1t(1xtxt

−+

−=

−−=−−=+

ម (1) េគបន 2

1t1t

11t1t2

)t(f−

−+

−

+⎟⎠⎞

⎜⎝⎛

−+

−=

1t33t

2t21t1t2t2

−+

=+−−−−+−−

=

ដចេនះ 1x33x)x(f−+

= ។

ឧទហរណ៣


2x2)2x2xx(f 2 +=+−+ ចេពះរគប IRx∈

ដេ ះរ យ


េគមន )1(2x2)2x2xx(f 2 +=+−+ ង t2x2xx 2 =+−+


- 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


1xx1x)

x1x(f 24

4

+−+

=+ ?

ដេ ះរ យ


េគមន 1xx

1x)x1x(f 24

4

+−+

=+


- 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


( )2xcosxsin)4

xcos(f −=⎥⎦⎤

⎢⎣⎡ π

− ?

ដេ ះរ យ


េគមន ( )2xcosxsin)4

xcos(f −=⎥⎦⎤

⎢⎣⎡ π

−


- 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 =+−+


- 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

+−+−

= .


- 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 ។


- 19 -

ឧទហរណ


2x9x4)x21(fx)x21(f2 2 −−=++− ?

ដេ ះរ យ


េ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 −+−=++


- 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


- 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∈ .


- 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 +=


- 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 .


- 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


- 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 ∈+== ∫ .


- 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


- 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 ∈ .


- 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 ∈


- 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 =− .


- 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


- 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


- 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 =− .


- 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


- 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


- 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


- 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 ¦ .


- 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


- 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


- 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 .


- 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


- 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


- 42 -

C)1xx(

41dx)1xx)(1x2(z

)1xx)(1x2('z

4232

32

+++=+++=⇒

+++=

∫

eKTaj)an xx42xx 22e]C)1xx(

41[e.zy −−−− +++==


- 43 -

kRmg viBaØasaKNitviTüaeRCIserIs

nigdMeNaHRsay


- 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 +∞→.


- 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


- 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 = .


- 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 ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛=

+= ∫


- 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 .


- 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 −≤≤

+


- 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=

+∞→ .


- 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+∞→

.


- 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


- 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


- 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


- 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

=+∞→

.


- 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

<<


- 57 -

b¤ 567100q

565399 +<<+ naM[ 100q = .

cMeBaH 100q = eK)an 5619195600n =+= . dUcenHcMnYn n enaHKW 5619n = .


- 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 .


- 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

+−+=

+++

=

+++

=

+++++

=

++++++

=

+++++

=


- 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 ¤ .


- 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


- 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 ≠≠


- 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

−−−−−

=−

−−

= .


- 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 = .


- 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 .


- 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(

−−+

+++++=−

−−−−+

+++++=−−

−

−

+−


- 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 −−+−=


- 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 ∫ −= −

−


- 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


- 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

++=

⎟⎠⎞

⎜⎝⎛

++

×⎟⎠⎞

⎜⎝⎛

+×⎟⎠⎞

⎜⎝⎛ −

=

⎟⎠⎞

⎜⎝⎛

++

×⎟⎠⎞

⎜⎝⎛

++

×⎟⎠⎞

⎜⎝⎛

+=⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−

=

−

=

−

=

−

=

+ ∏ ∏∏∏


- 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


- 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 =


- 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

+−

=+ .


- 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


- 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 ×+−×=+


- 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 −+= +++


- 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 +=+


- 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 =+=+


- 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 ⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=


- 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

−=−=

−−

−

=

−

+

−−−

=

−−−

+−

−−=

→

→

→


- 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


- 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

.


- 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 .


- 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 +

=+ .


- 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+

+−=


- 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 =++++ .


- 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


- 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 ==


- 89 -

smIkarrgVg; )C( Gacsresr ³

044y4y4x4x

4)2y()2x(22

22

=−+−++−

=−+−

dUcenH 04y4x4yx:)C( 22 =+−−+ .


- 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+∞→

.


- 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

∈+=+=

−+−=


- 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 =−+−+


- 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( ≤+−


- 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


- 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

==+∞→

.


- 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


- 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 ∈∀

−−= + .


- 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 ¦.


- 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=


- 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( =+


- 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


- 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

.


- 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 .


- 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 ≥−


- 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

+=


- 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


- 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 ++−+−=+


- 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 −+= .


- 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


- 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 ++

= .


- 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 .


- 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 +

= .


- 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


- 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

++=


- 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 −=+−−


- 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 = .


- 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 =−− .


- 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


- 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 .


- 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 ∈ .


- 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 ==


- 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 =−− .


- 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||→→→→→→

++=++ .


- 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==⎟

⎠⎞

⎜⎝⎛ +++++

+∞→


- 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


- 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 =+−−+ .


- 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 = .


- 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


- 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 ≥−


- 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

==⎟⎠⎞

⎜⎝⎛ +++++

+∞→


- 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


- 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


- 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 ¦ .


- 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; .


- 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 .


- 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

−−

=++++= .


- 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


- 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 +=


- 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


- 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== ∫ .


- 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 .


- 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 −= .


- 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 =+−+ .


- 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 −=+ ++ .


- 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


- 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 −−=−=


- 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


- 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 −=+ ++

lkç½xNÐ½ ⎩⎨⎧

≥+∈

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

+−=

−+−−

=−+

=+


- 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 .


- 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


- 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


- 152 -

luHRtaEtsmIkar ( )E manb¤sBIrepSgKña eBalKWeKRtUv[ 0b4a2 >−=Δ eday 1b −= eKTaj)an 04a2 >+=Δ BitCanic©RKb;cMnYnBit a . dUcenH 1b,IRa −=∈ .


- 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 =


- 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 .


- 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

+∞→ .

គណះកមមករនពនិ្ឋនងិ · 2010. 8. 27. ·...

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Transcript of គណះកមមករនពនិ្ឋនងិ · 2010. 8. 27. ·...