Complex Numbers

- i -

KNkmkarniBnnigeroberog

elak lwm pln nig elak Esn Bisid

KNkmkarRtYtBinit

elak lwm qun elak Gwug sMNag elak Tit em:g GkRsI Tuy rINa elak RBwm sunit elak pl bunqay elak elak nn; suxNa

GkRtYtBinitGkraviruT

elak lwm miKsir

karIkMuBTr GkrcnaRkb kBaa lI KuNaka elak lwm pln

- ii -

Garmfa esovePA sikSaKNitviTaedayxng Epk cMnYnkMupic EdlGksikSakMBugkan;sikSaenAkgdenH eyIgxJM)anxitxMRsavRCavcgRkgeLIg kgeKalbMNgTukCaksa sikSabEnmelIemeroncMnYnkMupicedayxng . enAkgesovePAenHrYmmanbICMBUkKW CMBUkTI1 emeronsegbPab;CamYy]TahrN_KMrU CMBUkTI2 lMhat;eRCIserIsnigdMeNaHRsay nig CMBUkTI3 CalMhat;Gnuvtn_ . esovePAenHminlhYseK hYsgenaHeT kMhusedayGectnaR)akdCaman GaRsyehtuenH eyIgxJMCaGkniBn nig eroberog rgcaMCanicnUvmtiriHKn;BIsMNak; GksikSakgRKb;mCdanedaykIrIkray edIm,IEklMGesovePAenH[kan;Etman suRkitPaBbEnmeTot . CaTIbBab;eyIgxJMsUmCUnBrcMeBaHGksikSaTaMgGs;CYbEtsuPmgl suxPaBl nigTTYlCyCMnkgkarsikSa nig muxrbrkargar RKb;eBlevla .

tdMbgfTI 28 mIna 2011 GkniBn lwm pln Tel : 017 768 246 www.mathtoday.wordpres.com

lwm pln nig Esn Bisid

- iii -

rkasiTiRKbyag

matikarerOg TMB&r CMBUkTI1 1-niymny 001 2-RbmaNviFIelIcMnYnkMupic 002 3-cMnYnkMupicqas; 008 4-Gnuvtn_cMnYnkMupickgdMeNaHRsaysmIkardWeRkTIBIr 010 5-cMnYnkMupickgbg; 011 6-m:UDlncMnYnkMupic 014 7-GaKuym:g;ncMnYnkMupic 017 8-TRmg;RtIekaNmaRtncMnYnkMupic 019

9-RbmaNviFIelIcMnYnkMupictamTRmg;RtIekaNmaRt 020 10-bMElgvilCMuvijKl;tRmuynbg;kMupic 024 11-bsTI n ncMnYnkMupictamTRmg;RtIekaNmaRt 026 12-TRmg;Giucs,:Ng;EslncMnYnkMupic 027 13-RbmaNviFIcMnYnkMupictamTRmg;Gics,:Ng;Esl 033 14-Gnuvtn_cMnYnkMupickgRtIekaNmaRt 034 15-Gnuvtn_cMnYnkMupickgsIVtcMnYnBit 037 16-Gnuvtn_cMnYnkMupickgFrNImaRt 039 17-bMElgcMnuckgbg;kMupic 051 18-)arIsg;nRbBncMNuckgbg;kMupic 054

- iv -

- v -

CMBUkTI2 kRmglMhat;eRCIserIs 055 EpkdMeNaHRsay 065 CMBUkTI3 lMhat;Gnuvtn_ 130

sikSaKNitviTaedayxng cMnYnkMupic

CMBUkTI1

cMnYnkMupic

1-niymny k> cMnYnnimit plKuNncMnYnBit c xusBIsUnnwg i ehAfacMnYnnim it . ehAfaktanim itEdl i 1i2 b 1i . ]TahrN_ ...,i3,

3i2,i5,i2 ehAfacMnYnnim it .

x> niymnycMnYnkMupic cMnYnkMupicCacMnYnEdlmanrag b.iaz Edl a nig b CacMnYnBit . eKtagsMNMuncMnYnkMupiceday . ehAfaEpkBitn a b.iaz EdlkMNt;tageday . a)zRe( ehAfaEpknim itn b b.iaz EdlkMNt;tageday . b)zIm( ]TahrN_1 i7,i5,i41,i34,i23,i21 ehAfacMnYnkMupic . ]TahrN_2 rkEpkBit nig Epknim itn i23z ? EpkBit nig Epknimitn i23z KW 2)zIm(;3)ZRe( .

eroberogeday lwm pln - TMBr1 -


2-RbmaNviFIelIcMnYnkMupic k> viFIbUkcMnYnkMupic ]bmafaeKmanBIrcMnYnkMupic b.iaz1 nig d.icz2 Edl CacMnYnBit . d,c,b,a eK)an )db(i)ca()idc()b.ia(zz 21 dUcenH )db.(i)ca(zz 21 . ]TahrN_ eK[cMnYnkMupic i23z1 nig i57z2 . KNna 21 zz eK)an )i5i2()73()i57()i23(zz 21 dUcenH . i34zz 21 x> viFIdkcMnYnkMupic ]bmafaeKmanBIrcMnYnkMupic b.iaz1 nig d.icz2 Edl CacMnYnBit . d,c,b,a eK)an )db(i)ca()idc()b.ia(zz 21 dUcenH )db.(i)ca(zz 21 . ]TahrN_ eK[cMnYnkMupic i23z1 nig i57z2 . KNna 21 zz eK)an )i5i2()73()i57()i23(zz 21 dUcenH . i710zz 21



K> viFIKuNcMnYnkMupic ]bmafaeKmanBIrcMnYnkMupic b.iaz1 nig d.icz2 Edl CacMnYnBit . d,c,b,a eK)an )d.ic)(b.ia(zz 21

)bcad(i)bdac(

bdibciadacbdiibciadac 2

dUcenH )bcad(i)bdac(zz 21 . ]TahrN_ eK[cMnYnkMupic i2z1 nig i31z2 . KNna 21 zz eK)an i55i3ii62)i31)(i2(z.z 221 dUcenH . i55z.z 21 X> viFIEckcMnYnkMupic ]bmafaeKmanBIrcMnYnkMupic b.iaz1 nig d.icz2 Edl CacMnYnBit . d,c,b,a

eK)an 2222

2

1dic

bdiibciadac)idc)(idc()idc)(iba(

idciba

zz

2222 dc)adbc(i)bdac(

dcbdibciadac

dUcenH 22222

1dcadbc.i

dcbdac

zz

.




i51z ]TahrN_ eK[cMnYnkMupic 1 nig i1z2 . KNna 2

1zz

eK)an 2

i4611

5i5i1)i1)(i1()i1)(i51(

i1i51

zz2

1

dUcenH i23zz2

1 . g> sVyKuNn i eKmansVyKuNn i dUcxageRkam

1ii.ii.ii

ii.ii

1ii.ii.ii

ii.ii

1)1()i(i

ii.ii

1i

ii

278

67

256

45

2224

23

2

1

CaTUeTA ii,1i,ii,1i 3n42n41n4n4 RKb; INn dUcenHcMnYn es InwgcMnYndEdlKW ni i,1,i nig 1 RKb; . INn


c> sVyKuNncMnYnkMupic eKmansVyKuNncMnYnkMupicdUcxageRkam

n

0k

kkknn

3342244

32233

222

i.ba)k,n(C)iba(

)ab4ba4(i)bba6a()iba(

)bba3(i)ab3a()iba(

ab2.iba)iba(

Edl )!kn(!k

!n)k,n(c . ]TahrN_ KNna nig . 32 )i2(,)i31( 4)i21( eK)an

i24716i3224i81)i21(

i112i6i128)i2(

i689i61)i31(

4

3

2

q> kMupicesIKa ]bmafa nig ibaz1 d.icz2 Edl CacMnYnBit . d,c,b,a

dbca

zz 21



dUcenHcMnYnkMupicBIres IKaluHRtaEtEpkBites IKa nig Epknim itesIKa . ]TahrN_ eK[ i432z1 nig i89z2 cUrkMNt;BIrcMnYnBit nig edIm,I[ 21 zz ? kMNt; nig

84

932i89i432

eKTaj)an . 3,2 C> KNnabskaerncMnYnkMupic ]bmafaeKmancMnYnkMupic b.iaz Edl CacMnYnBit b,a edIm,IKNnabskaern z eKRtUvGnuvtn_dUcxageRkam tag y.ixw Cabskaern b.iaz y,x CacMnYnBit eK)an zw2

iba)xy2(i)yx(

b.ia)y.ix(22

2

eKTaj)an

bxy2ayx 22

edaHRsayRbBnsmIkarenHeK)anKUcemIy )},();,{()y,x( 2211 dUcenH 222111 iw;iw .



]TahrN_1 KNnabskaerncMnYnkMupic i2021z tag y.ixw Cabskaern i2021z y,x CacMnYnBit eK)an zw2

i2021)xy2(i)yx(

i2021)y.ix(22

2

eKTaj)an

20xy221yx 22

bnab;BIedaHRsayRbBnsmIkarenHeK)anKUcemIy 2y,5x b 2y,5x dUcenH i25w;i25w 21 . ]TahrN_2 KNnabskaerncMnYnkMupic i2021z tag y.ixw Cabskaern i68z y,x CacMnYnBit eK)an zw2

i68)xy2(i)yx(

i68)y.ix(22

2

eKTaj)an

6xy28yx 22

bnab;BIedaHRsayeK)anKUcemIy 3y,1x b 3y,1x dUcenH i31w;i31w 21 .



3-cMnYnkMupicqas; k> niymny cMnYnkMupicqas;ncMnYnkMupic IRb;a,b.iaz KWCacMnYnkuMpicEdl kMNt;tageday b.iaz . ]TahrN_ cMnYnkMupicqas;n i34z KW i34z . x> lkN !> 2121 zz)zz( @> 2121 z.z)z.z( #>

2

1z

2

1zz

z

sRmaybBaak; tag nig ibaz1 idcz2 Edl CacMnynBit . d,c,b,a eK)an b.iaz1 nig d.icz2 man )db(i)ca(zz 21 enaH )db(i)ca(zz 21 ehIy )db(i)ca(idcibazz 21 dUcenH 2121 zz)zz( . eKman )bcad(i)bdac()idc)(iba(zz 21 eK)an )bcad(i)bdac(z.z 21 ehIy )bcad(i)idac()idc)(iba(z.z 21



dUcenH 2121 z.z)z.z( . eKman 2222

2

1dcadbc.i

dcbdac

)idc)(idc()idc)(iba(

idciba

zz

eK)an 22222

1dcadbc.i

dcbdac

zz

ehIy 22222

1dcadbc.i

dcbdac

)idc)(idc()idc)(iba(

idciba

zz

dUcenH 2

1z

2

1zz

z

. K> kenSamEpkBit nig EpknimitCaGnuKmn_n z nig z ]bmafaeKman enaH ibaz ibaz Edl CacMnYnBit . b;a eK)an a2ibaibazz enaH

2zza

ehIy ib2ibaibazz enaH i2zzb

dUcenH 2zz)zRe( nig

i2zz)zIm( .

-ebI enaH 0)zRe( zz naM[ z CacMnYnnim it . -ebI enaH 0)zIm( zz naM[ z CacMnYnBit .



4-Gnuvtn_cMnYnkMupickgdMeNaHRsaysmIkardWeRkTIBIr ]bmafaeKmansmIkardWeRkTIBIr 0cbzaz2 Edl CacMnYnBit . c,b,a,0a DIsRKImINg;nsmIkarKW ac4b2 -ebI smIkarmanbsBIepSgKaCacMnYnBitKW 0

a2bz;

a2bz 21

-ebI smIkarmanbsDbCacMnYnBitKW 0

a2bzz 21

-ebI smIkarmanbsBIepSgKaCacMnYnkMupicqas;KaKW 0

a2||ibz;

a2||ibz 21

]TahrN_ edaHRsaysmIkar 05z6z2 2 DIsRKImINg;nsmIkar 2i444036 eKTajbs

21.i

23

4i26z;

21.i

23

4i26z 21

dUcenHsmIkarmanbdBIrepSgKaCacMnYnkMupicqas;KaKW

21.i

23z;

21.i

23z 21 .



5-cMnYnkMupickgbg; k> kartagcMnYnkMupickgbg; kgtRmuyGrtUNrem )xoy( eKGactagcMnYnkMupic IRb,a;b.iaz edaycMNuc M mYymankUGredaen . )b,a( eKfa M CacMnucrUbPaBncMnYnkMupic ibaz ehIy ehAfaGaPicn z cMnuc EdleKkMNt;sresr . )b,a(M )z(M dUcKaEdr eKkGactagcMnYnkMupic IRb,a;b.iaz edayviucTr

. )b,a(OMu

eKfa CavIucTrrUbPaBncMnYnkMupic u ibaz ehIy ehAfaGaPicn z

viucTr EdleKkMNt;sresr . u )z(u

Mb

a o x

y



x> viucTrrUbPaBnplbUkcMnYnkMupickgbg;kMupic kgtRmuyGrtUNrem )xoy( ]bmafaeKmancMnYnkMupicBIr nig 1z 2z ehIytag nig CarUbPaBn nig . 1M 2M 1z 2z

eK)an nig CaviucTrrUbPaBn nig .

1OM

2OM 1z 2z

yM


eKman OMOMOMzz 2121

dUcenHrUbPaBn KWCaviucTrGgt;RTUgnRbelLRkam . 21 zz 21MMOM

2M

1M

x

O


K> viucTrrUbPaBnpldkcMnYnkMupickgbg;kMupic kgtRmuyGrtUNrem )xoy( ]bmafaeKmancMnYnkMupicBIr nig 1z 2z ehIytag nig CarUbPaBn nig . 1M 2M 1z 2z

eK)an nig CaviucTrrUbPaBn nig .

1OM

2OM 1z 2z


eK)an OM'OMOM)z(zzz 212121

dUcen rUbPaBn 21 zz KWCaviucTrGgt;RTUgnRbelLRkam . 21 'MMOM

y

'x

'

x

y 'M2

2M

M

1M

O


X> viucTrrUbPaBnplKuNcMnYnBit nwg cMnYnkMupickgbg;kMupic


y

'M

M

x O ]bmafa M nig CacMnucrUbPaBn nig 'M z )0(,z

rUbPaBn KW Edl z. 'OM OM'OM . 6-m:UDlncMnYnkMupic k> niymny kgtRmuyGrtUNrem xo )y( eKyk CarUbPaBn )b,a(M ibaz . rgVas; OM ehAfam:UDln ibaz . eKkMNt;tagm:UDln ibaz eday |z| b r EdlGacKNna)antam rUbmn 22 baOMr|z| .


y

M b |z|

P x o a kgRtIekaNEkg OMP eKman 222 OPMPOM eday bMP,aOP eK)an b 222 baOM 22 baOM . dUcenH 22 baOM|z| . x> lkN !> |z||z| @> 2|z|z.z #> |z|.|z||z.z| 2121 $>

|z||z|

zz

2

1

2

1 %> nn |z||z|



K> vismPaBRtIekaN RKb;cMnYnkMupic nig eKman 1z 2z |z||z||zz| 2121 sRmay tag nig iyxz1 ivuz2 eKman )vy(i)ux(zz 21 eK)an 2221 )vy()ux(|zz| nig 222221 vuyx|z||z| eday )yvxu(2vuyx|zz| 2222221 nig )vu)(yx(2vuyx|z||z| 22222222221 eK)an )yvxu()vu)(yx(2|zz||z||z| 2222221221 kenSam luHRtaEt 0|zz||z||z| 221221

0yuxyuv2vx

0vyxyuv2uxyvyuvxux

0)yvxu()vu)(yx(

2222

222222222222

22222

Bit 0)uyxv( 2 dUcenH . |z||z||zz| 2121



7-GaKuym:g;ncMnYnkMupic kgtRmuyGrtUNrem )xoy( eKyk CarUbPaBn . )b,a(M ibaz


mMuEdlpMeday ehAfaGaKuym:g;n )OM,Ox(

b.iaz . eKtag b )z(Arg CaGaKuym:g;n b.iaz . kgRtIekaNEkg OMP eKman b 2222 baOMr 22 bar RTwsIbTBItaKr

ra

OMOPcos nig

rb

OMMPsin .

edIm,IrkGaKuym:g;n b.iaz eKedaHRsaysmIkar

racos nig

rbsin eK)an Zk,k2)z(Arg .

P

y

Mb

x o a


]TahrN_ 1 rkGaKuym:g;ncMnYnkMupic i232z tamrUbmn 42)32(ba|z|r 2222

23

432

racos nig

21

42

rbsin

dUcenHGaKuym:g;n KW z Zk;k26

)z(Arg ]TahrN_ 2 rkGaKuym:g;ncMnYnkMupic 3i1z

tamrUbmn 23()1(ba|z|r 2)222

21

racos nig

23

rbsin

dUcenHGaKuym:g;n KW z Zk;k232)z(Arg

TahrN_ 3 rkGaKuym:g;ncMnYnkMupic 2i2z tamrUbmn 222ba|z|r 22

22

racos nig

22

rbsin

dUcenHGaKuym:g;n KW z Zk;k24

)z(Arg



8-TRmg;RtIekaNmaRtncMnYnkMupic cMnYnkMupic ehAfaTRmg;BICKNit . eKGacsresrTRmg;fImYy b.iaz eTotdUcxageRkam eKman 22 bar ehAfam:Dln b.iaz

racos nig

rbsin Edl ehAfaGaKuym:g;n . z

eK)an )sin.i(cosr)rb.i

ra(rb.iaz

dUcenH )sin.i(cosrz ehAfaTRmg;RtIekaNmaRtn . z ]TahrN_ 1 cUrsresr 3i1z CaragRtIekaNmaRt . eKman 2)3(1r 22 eK)an )

3sin.i

3(cos2)

23.i

21(2z .

]TahrN_ 2 cUrsresr i232z CaragRtIekaNmaRt . eKman 4)2()32(r 22 eK)an )

6sin.i

6cos(4)

21.i

23(4z .

)

65sini

65(cos4z

)]6

sin(.i)6

[cos(4z



9-RbmaNviFIelIcMnYnkMupictamTRmg;RtIekaNmaRt k> plKuNcMnYnkMupictamTRmg;RtIekaNmaRt RTwsIbT ]bmafaeKmancMnYnkMupic )sin.i(cosrz 11 nig )sini(cosrz 22 Edl 0r,0r 21 eK)an )]sin(i)[cos(rrz.z 2121 . sRmaybBaak; eK)an )sini(cosr).sin.i(cosrzz 2121 )cossincos(sini)sinsincos[(cosrrzz 2121 dUcenH )]sin(i)[cos(rrz.z 2121 . ]TahrN_ eK[cMnYnkMu[pic )

5sini

5(cos2z1

nig )152sini

152(cos3z2

KNna 21 z.z eK)an )]

152

5sin(i)

152

5[cos(6zz 21

)

3sin.i

3(cos6)

155sini

155(cos6

dUcenH )3

sin.i3

(cos6zz 21 .



x> plKuNEckcMnYnkMupictamTRmg;RtIekaNmaRt RTwsIbT ]bmafaeKmancMnYnkMupic )sin.i(cosrz 11 nig )sini(cosrz 22 Edl 0r,0r 21 eK)an )]sin(i)[cos(

rr

zz

2

1

2

1 . sRmaybBaak; eK)an

sinicossinicos.

rr

zz

2

1

2

1

)sin(i)cos(rr

sincos)cossincos(sini)sinsincos(cos.

rr

)sini)(cossini(cos)sini)(cossin.i(cos.

rr

2

1

222

1

2

1

dUcenH )]sin(i)[cos(rr

zz

2

1

2

1 . K> sVyKuNTI n ncMnYnkMupictamTRmg;RtIekaNmaRt RTwsIbT RKb;cMnYnBit nig eK)an 0r )nsin.in(cosr)]sin.i(cosr[z nnn Edl n CacMnYnKt;rWLaTIhV .



sRmaybBaak; ramrUbmn )]sin(i)[cos(rrz.z 2121 eK)anCabnbnab;dUcxageRkam

)3sin.i3(cosr

)]2sin(i)2[cos(r.rz.zz

)2sini2(cosr)]sin(i)[cos(r.rz.zz

3

223

22

]bmafa Bit )nsin.in(cosrz nn eK)an )]nsin(i)n[cos(r.rz.zz nn1n Bit ])1nsin(i)1n[cos(r 1n dUcenH . )nsin.in(cosrz nn X> rUbmndWmr eKman )nsin.in(cosr)]sin.i(cosr[z nnn eK)an )nsin.in(cosr)sin.i(cosr nnn dUcenH nsin.incos)sin.i(cos n ehAfarUbmndWmr .



]TahrN_ eK[cMnYnkMupic 94sin.i

94cosz

cUrsresr 32

z1zw CaragRtIekaNmaRt .

tamrUbmndWmreKman 98sini

98cos)

94sini

94(cosz 22

nig 34sini

34cos)

94sini

94(cosz 33

eK)an 34sin.i

34cos1

98sin.i

98cos

w

911sini

911cos)

92sini

92(cos

)]32

98sin(i)

32

98[cos(

32sini

32cos

98sin.i

98cos

.

32cos2

1

32cos

32sini2

32cos2

98sin.i

98cos

2

dUcenH

911sin.i

911cosw .



10-bMElgvilCMuvijKl;tRmuynbg;kMupic eKmancMnYnkMupic sin.icosw . ebI CarUbPaBn tambEmgvilpit O nig mMu enaH )'z('M )z(M eK)an z.)sini(cosz.w'z .

y


'M ]TahrN_ 1 enAkgbg;kMupic )xoy( eK[ M CacMnucrUbPaBn i3z . cUrkMNt; edaydwgfa CarUbPaBn M tambEmgvilpit O 'z )'z('MnigmMu

12 .

x O

M


ebI CarUbPaBn tambEmgvilpit O nigmMu )'z('M )z(M12 enaHeK)an

z)12

sini12

(cos'z

eday )6

sin.i6

(cos2)21.i

23(2i3z

eK)an )]612

sin(i)612

[cos(2'z dUcenH )

4sini

4(cos2'z .

]TahrN_ 2 enAkgbg;kMupic )xoy( eK[ M CacMnucrUbPaBn 3i1z . cUrkMNt; edaydwgfa CarUbPaBn M tambEmgvilpit O 'z )'z('MnigmMu

32 .

ebI CarUbPaBn tambEmgvilpit O nigmMu )'z('M )z(M32 enaHeK)an

z)32sini

32(cos'z

eday )]3

sin(.i)3

[cos(2)23.i

21(23i1z

eK)an )]33

2sin(i)33

2[cos(2'z dUcenH )

3sini

3(cos2'z .




tamTRmg;RtIekaNma t a aeKmancMnYnkMupic

11-bsTI n ncMnYnkMupic R ]bm f )sini(cosr z tag w )sini(cos CabsTI n enaH eK)an

eKTaj naM[

n z zw n)sini(cosr)nsinin(cosn

cos)ncos(rn

nk2

rn

sin)nsin( dUcenHbsTI n n z kMNt;eday

si.iw nk nk2n

nk2cosr

Edl n....,,2,1,0k 1 . KNnabsTIbIn ]TahrN_ 24.i24z

)4

eKman sini4

cos8)2

i2

(82z (224.i24 eKTaj

4,8r . tamrUbmnbsTI # n z kMNt;eday

2,1,0k,12k8sini

12k8cos 2wk

.

H

)43sini

43(cos2 dUcen w1;)12sini12(cos2w0

nig )

1217sini

1217(cos2w2

.



Mupic 's la)

12-TRmg;Giucs,:Ng;EslncMnYnk k> rUbmnGWEl (Euler formu cMeBaHRKb;cMnYnBit x eK)an xcoseix xsini

BitcMeBaH x CacMnYnkMupickeday .

tagGnuKmn_

Edl ...71828.2e CaeKalelakarIteEB . rUbmnGWElenHenAEt sRmaybBaak; -RsaybBaak;edayeRbIedrIev f GacCaGnuKmn_kMupicnGefr kMNt;eday x

ixexsinixcos)x(f

eK)an ix2ixix

e)xsinix(cosiee)xcosixsin()x('f

0

e0

e xsinxcosixcosixsin

e)xsinixcosixcosixsin(e

ixix

ix2

2ix

CaGnuKmn_efrRKb; . naM[ xf )x( eK)an 1

e0sini0cos)0(f)x( 1

exsinixcos)x(f ix

f 0 b

dUcenH . xsinixcoseix



eRbIsmIkarDIepr:g;Esl Mdab;TImYy nuKmn_

-RsaybBaak;eday l eKtagG xsin.ixcos)x(g eKman xsin)x('g xcosi

TaMgBIrnwg eK)an KuNGg (' i csoxxsini)xg.i eK)an 0)x('ig)x(g b 0)x(g.i)x(' g CasmIkarDIepr:g;EsllMdab; I . eK) IyTUeT renHKancem AnsmIka W

e ixek)x(g

ebI x naH k)0(0 g c Et 10sin.i0os)0(g enaH 1k ehIyeK)an .

RbIsmIkarDIe r:g;EsllMdab;TIBIr GnuKmn_

nig CasmIkarDIepr:g;EsllIenEGGum:UEsn

ix)x(g e dUcenH xsinixcoseix . -RsaybBaak;edaye p eRCIserIs e) ixx(h eKman ixx e ixe.i)x('h i2ei)x(''h eK)an x(''h 0ee)x(h) ixix lMdab;TIBIr . smIkarDIepr:g;EsllIenEGlMdab;BIrenHmancemIylIenEGkraClIenEGcMnYnBIr Edlepgpat;vaKW xcos)x(h1 nig xsin)x(h2 . bnSMlIenEGncemIy cMeBaHsmIkarDIepr:g;EslGUm:UEsn kCacemIymYypgEdr .


dUcenHcemIyTUeTAnsmIkarKW xcosA)x(h xsinB Edl 1 A nig CaBIrcMnYnefrEdlGacrk)antam

eRBaH B eA)0(h 0i

nig B)0('h )iie 0i xcosBxsinAx('h . eK)an ehtuenH si xinxcos)x(h .

rUbmnesrItlrcMeBaHGnuKmn_bI nig KW

dUcenH xsinixcoseix .-RsaybBaak;edayeRbIesrItlr

xcos,ex xsin

....)!1n2(

)1(....!7!!3

xxsin n xx5xx

......)!n2(

x)1(.....!6

x!4

x!2

x1xcos

.....!n

x.....!3

x!2

x!1x1e

1n2753

n2n

642

n32x

esTaMgbIenHeKTaj)an edayCMnYs x eday ix kg xsinixcoseix . g;Esl

Edl CacMnYnBitGacsresrCaTRmg;mYyfIeTot Edl

x> TRmg;Giucs,:N RKb;cMnYnkMupic az b.i b,a


KW rz ie rbs,

racos,bar 22 in .

TRmg; ehAfaTRmg;Gics,:Ng;Esln ierz b.iaz .



u aNmaRt K it

y eK)

K> TMnak;TMngCamYyGnKmn_RtIek cMeBaHR b;cMnYnB eKman eix x )1(xsinixcos CMnYs x eda x an e ix )2(xsinixcos bUksmIkar nig eKTaj)an )1( )2( xcos2ee ixix

2eex eKTaj cos .

ixix

ar nig eKTaj)an dksmIk )1( )2( xsini2ee ixix

i2eexsin

ixix eKTaj .

i2ee dUcenH xsin;

2eexcos

ixixixix . Mupic

gCamYyGnuKmn_GIEBbUlik

rUbmnenHBitpgEdrcMeBaH x CacMnYnkX> TMnak;TMn

eKman i2eexsin;

2exos ec

ixixixix CMnYs eday eK)an

x x.i

xcosh2

)ixcos( eexx

nig xsinhisin(

2eei)ix

xx

m:ageToteKman 2eexcosh

xx nig 2eexsinh

xx eK)an

xsini2ee)ixsinh(,e)ixcosh( xcos

2e ixixixix

.



_1 sresr]TahrN 3i22z CaTRmg;Giucs,:Ng;Esl ? 4)32(2r 22 eKman

3ie4)

eK)an 332

sini(cos4)23i1(4z .

]TahrN_2 sresr 4ie1z

CaragGics,:Ng;Esl ?

eKman eday 2ee

8cos

8i

8i )ee(ee1z 8i8i8i4i

dUcenH 8.8

cos2z ie

. _3 KNna ?

eKman

]TahrN ii

2ie

2sin.i

2cosi

2

i

2i

i eei eK)an

.

smIkar

]TahrN_4 edaHRsay 2xcos kgsMNMukMupic ?

nGWEleKman 2eetamrUbm xosc

ixix

)an 22ee ixix

eK b ixix2 01e4e tag ixet



eK)ansmIkar eKTajbs

01t4 t2 4' 31 32t;3 22t1

aHcMeB 32t eK)an enaH )32ln(ix ixe 32 b )3x . 2ln(i

aHcMeB 32t eK)an enaH )32ln(ix ixe 32 b )32ln(ix . ]TahrN_5 edaHRsaysmIkar 3xsin kgsMNMukMupic ?

i2eetamrUbmnGWEleKman xsin

ixix

eK)an 3i2ee ixix

b ixix2 01e6e tag ix

Ikar et

eK)ansm 10 19',01t6t2 manbs 103t,0 2 13t1 .-cMeBaH eK)an 103eix enaH )103ln(ix 103t -cMeBaH )103(103 t eK)an )103ln(iiix 1e ee)03()103( eKTaj )103ln(iix b )103ln(ix . dUcenH )103ln(ix,)103ln(ix .



Mn kMupic Rmg;Gics,:Ng;Esl

13-RbmaNviFIcMnYnkMupictamTRmg;Gics,:Ng;Esl k> RbmaNviFIKuNc Yn tamT eKmancMnYnkMupic ier nig ie z w Edl r ehIy CacMnYnBit . ,0;0 eK)an )(ie.rw.z . x> RbmaNviFIEckcMn k amTR gYn Mupict mg;Gics,:N ;Esl eKmancMnYnkMupic nig ie ierz w Edl r Iy 0;0 eh , CacMnYnBit . eK)an )(ie

w

. rz

K> sVyKuNcMnYnkMupictamTRmg;Gics,:Ng;Esl eKmancMnYnkMupic ierz cMeBaHRKb;cMnYnKt;rWLaTIhV n eK)an

innnin ererz . _ e Yn ic 6ie2z

nig 12ie3w

]TahrN K[cMn kMup .

KNna w.z nig wz

eK)an nig 12i)126(i e32e

32

wz

4i)126(i e6e6w.z



MupickgRtIekaNmaRt

cMeBaHRKb;cMnYnBit x eKman

14-Gnuvtn_cMnYnk k> rUbmnmMuDb

i2eexsin;

2eexcos

ixixixix

eK)an 4

2ee2eexcos

ix2ix22ixix2

21x2cos

21

21

2ee.

21x c os

x2iix22

eKTaj 1xcos2x2cos 2 .

2ee.

i2ee ehIy .2s

i2eex2in

ixixixixix2ix2 .

RTIb

eKman

dUcenH 2x2sin xcosxsin x> rUbmnmMu

3ixix3

2eexcos

4xcos3x3cos

8xcos6x3cos2

8)ee(3ee ixixix3ix3

eKTaj)an xcos3xcos4x3cos 3 .


ehIy 3ixix

3i2eexsin

4xsin3x3sin

i8xsini6x3sini2i8

)ee(3)ee( ixixix3ix3

eKTaj)an xsin4xsin3x3sin 3 . K> rUbmnplbUk nig pldknmMuBIr eKman RKb;cMnYnBit a nig b ibia)ba(i e.ee eday )bsinib)(cosasinia(cose.e ibia )acosbsinbcosa(sini)bsinasinbcosa(cosee ibia ehIy )basin(i)bacos(e )ba(i dUcenH bsinasinbcosacos)bacos( nig acosbsinbcosasin)basin( . m:ageTot ibia)ba(i e.ee RKb;cMnYnBit a nig b eday )bsinib)(cosasinia(cose.e ibia )acosbsinbcosa(sini)bsinasinbcosa(cosee ibia ehIy )basin(i)bacos(e )ba(i



dUcenH bsinasinbcosacos)bacos( nig acosbsinbcosasin)basin( . X> rUbmnbMElgBIplKuNeTAplbUk

2ee.

2eebcosacos

ibibiaia

)bacos()bacos(21

2ee

2ee

21

4eeee

)ba(i)ba(i)ba(i)ba(i


dUcenH )bacos()bacos(21bcosacos

i2ee.

i2eebsinasin

ibibiaia

)bacos()bacos(21

2ee

2ee

21

i4eeee


2


dUcenH )bacos()bacos(21bsinasin



15-Gnuvtn_cMnYnkMupickgsIVtcMnYnBit cMeBaHsIVtncMnYnBit Edlepgpat;TMnak;TMngkMeNIn )a( n 0qapaa n1n2n Edl CacMnYnBit q,p smIkarsmal;rbs;sIVtenHKW 0qpzz2 kgkrNI smIkarmanbsBIrCacMnYnkMupicqas;Ka 0q4p2 KW nig . kgkrNIenHedIm,IKNna eKRtUvGnuvtn_dUcteTA 1z 2z na tagsIVtCMnYy n11nn azaz rYcRsayfa CasIVtFrNImaRt )z( n ncMnYnkMupicmYy. KNna rYcTajrk . nz na ]TahrN_ eKmansIVtncMnYnBit ) kMNt;eday a( n 1a,0a 10 nig n1n2n aaa Edl ,...2,1,0n cUrKNna CaGnuKmn_n n na smIkarsmal;nsIVt n1n2n aaa KW 1rr2 b 22 i3341;01rr manbs

23i1r;

23i1r 21

tagsIVtCMnYy n1nn a23i1az

eK)an 1n2n1n a23i1az

Et n1n2n aaa



eK)an 1nn1n1n a23i1aaz

n1n

n1n1n

n1n1n

n1n1n

a2

3i1z

)a2

3i1a(2

3i1z

)a3i1

2a(2

3i1z

aa2

3i1z

eK)an ) CasIVtFrNImaRt ncMnYnkMupicmanersug z( n 23i1q .

tamrUbmn n0n qzz eday 1a

23i1az 010 nig 3sini3cos2

3i1q eK)an

3nsini

3ncos)

3sini

3(cosz nn

eday nn1nn1nn a23i)

2aa(a

23i1az

eKTaj 3nsina

23

n

dUcenH 3nsin

32an

.



16-Gnuvtn_cMnYnkMupickgFrNImaRt k> cmayrvagBIrcMnuc y


eKmancMnYnkMupicBIr 111 y.ixz nig 222 y.ixz . tag AnigBCacMnucrUbPaBn nig kgbg;kMupic 1z 2z )xoy( . eK)an 212212 )yy()xx(AB ehIy )yy(i)xx(zz 121212 eK)an 21221212 )yy()xx(|zz| dUcenH . |zz|AB 12

A

B

x

A

O 1x 2x

1y


]TahrN_ 1 eK[BIrcMnuc A nig manGahVikerogKa B i21z1 nig i62z2 cUrKNnacmayrvagcMnuc A nig B eK)an |zz|AB 12

54)3(|i43|

|)i21()i62(|22

dUcenH . 5AB ]TahrN_2 eKmancMnYnkMupic ia2z1 nig )a6(i3z2 Edl IRa kgbg;kMupic )xoy( eKtag nig CarUbPaBncMnYnkMupic nig A B 1z 2z cUrkMNt;cMnYnBit a edIm,I[cmayrvagcMnuc nig xIbMput ? A B eK)an |zz|AB 12

8)3a(2

26a12a2

aa1025aa21

)a5()a1(|)a5(i)a1(|

|ia2)a6(i3|

2

2

22

22

edIm,I[ AB xIbMputluHRtaEt 3a ehIy 228ABmin .



x> cMnucEckkgGgt;tampleFobmYy kgbg;kMupic )xoy( eKtag nig B CarUbPaBncMnYnkMupic nig A Az Bz yk P CacMnucmanGahVik CacMnucEckkgnGgt; Pz AB tampleFob Edl . 0

y


ebI P CacMnucEckkgnGgt; AB tampleFob enaH PBAP eKman nig )zz(AP AP

)zz(PB PB

eK)an )zz(zz PBAp dUcenH

1

zzz BAP .

A

BP

y O


krNI enaHcMnuc CacMnuckNalnGgt; 1 P ]AB[ dUcenHGahVikncMnuc P kNalGgt; AB KW

2zzz BAP

. K> cMnucEckeRkAGgt;tampleFobmYy kgbg;kMupic )xoy( eKtag nig B CarUbPaBncMnYnkMupic nig A Az Bz yk Q CacMnucmanGahVik CacMnucEckeRkAnGgt; Qz AB tampleFob Edl . 0

y


ebI Q CacMnucEckeRkAnGgt; AB tampleFob enaH QBQA eKman nig )zz(QA QA

)zz(QB QB

A

QB

y O


eK)an )zz(zz QBQA dUcenH

1

zzz BAQ . ]TahrN_ eKmanBIrcMnuc nig B CarUbPaBncMnYnkMupic A i72zA nig . i1zB CarUbPaBn CacMnucEckkgn P pz AB tampleFob 3

1p

CarUbPaBn CacMnucEckeRkAn Q Qz AB tampleFob 32

Q cUrkMNt; nig ? Pz Qz ebI P CarUbPaBn CacMnucEckkgn pz AB tampleFob 3

1p

eK)an i211

45

311

i31

31i72

1zzzP

BPAP

.

AB tampleFob 32

Q ebI Q CarUbPaBn CacMnucEckeRkAn Qz

eK)an i198321

i32

32i72

1zz

zQ

BQAQ

dUcenH i211

45zP nig i198zQ .



K> pleFobRCug nig mMunRtIekaNkgbg; kgbg;kMupic )xoy( eKmanbIcMnuc CarUbPaBncMnYnkMupic C,B,A . CBA z,z,z

C


y

sg;viucTr nig AB'OB AC'OC enaHeK)an nig 'C'OB

CaRtIekaNb:unKa enaH ABC 'OC'BBAC .

eKman nig )zz(AB'OB AB

)zz(AC'OC AC

O

A

B

'B

' C

x


eKman 'XOB'XOC'OC'B b

AB

ACABAC zz

zzarg)zzarg()zzarg(BAC

ehIypleFobRCug AB

AC

AB

ACzzzz

|zz||zz|

ABAC

. dUcenHebI begIt)anCaRtIekaN enaH )z(C,)z(B,)z(A CBA ABC

eK)an AB

ACzzzz

ABAC

nig

AB

ACzzzzargBAC .

]TahrN_ enAkgbg;kMupic eK[bIcMnuc manGahVikerogKa )XOY( C,B,A i92z,i43z,i2z CBA . cUrKNnapleFobRCug

ABAC nig BAC ?

eKman i84i2i92zz AC nig i31i2i43zz AB eK)an i22

)i31)(i31()i31)(i84(

i31i84

zzzzAB

AC

)4

sini4

(cos22)22i

22(22

dUcenH 22ABAC nig

4BAC .



X> sMNMucMNuckgbg;kMupic

kgbg;kMupicRbkbedaytRmuyGrtUNrm:al; eKtagcMnuc)j,i,o(

)y;x(P CacMnucrUbPaBncMnYnkMupic y.ixz . sMNMunrUbPaB P TaMgGs;EdlGacmaneTAtamlkxNEdlRtUvbMeBjn z begIt)anCasMNMuncMnuc P kgbg;kMupic . ]TahrN_ 1 eK[cMnYnkMupic z epgpat; 4z)i1(z)i1( . CarUbPaBncMnYnkMupic kgbg;kMupic P z )xoy( . rksMNMucMnuc ? P tag y.ixz enaH y.ixz


eKman 4z)i1(2z)i1( eK)an 4)iyx)(i1()iyx)(i1(

2xy4y2x24yixiyxyixiyx

dUcenHsMNMucMnuc P KWCabnat; Edl mansmIkar 2xy:)(

1

0 1 x

y

2xy:)(


]TahrN_ 2 eK[cMnYnkMupic z epgpat; 3|i2z| . CarUbPaBncMnYnkMupic kgbg;kMupic P z )xoy( . rksMNMucMnuc P ?

0 1

1

x

y

tag y.ixz CaGahVikncMnuc P eKman 3|i2z| eK)an 3|i2iyx|

3)1y()2x(

3|)1y(i)2x(|22

b 9)1y()2x( 22 dUcenHsMMNMucMnuc P KWCargVg;pit )1,2(I nig kaM 3R . ]TahrN_ 3 eK[cMnYnkMupic

i22zi22zw

ehIy P CacMnucrUbPaBn z kgbg; )xoy( cUrkMNt;sMNMucMnuc P edIm,I[ CacMnYnnim itsuT ? w edIm,I[w CacMnYnnim itsuTluHRtaEt 0

2ww)wRe(

naM[ ww eday

i22zi22zw

nig i22zi22zw



eK)an i22zi22z

i22zi22z

ebI i22z

8z)i22(z)i22(z.z8z)i22(z)i22(zz

)i22z)(i22z()i22z)(i22z(

16zz2 b | 8|z 2 tag iyx z Edl )2,2()y,x( eK)an x . 8y22 dUcenHsMNMucMnuc P CargVg;pit O kaM 22R elIkElgEtcMnuc )2,2(A .

0 1

1

x

y

8yx 22



]TahrN_ 4 eK[cMnYnkMupic z ehIy CacMnucrUbPaBn kgbg; P z )xoy( cUrkMNt;sMNMucMnuc P ebIeKdwgfa

4)i22zarg( .

tag iyxz enaHeK)an )2y(i)2x(i22z eday

4)i22zarg( enaH

4))2y(i)2xarg((

eKTaj)an 12x2y

4tan

naM[ 4xy dUcenHsMNMucMnuc P Cabnat; 4xy:)d( Edl b

02y02x

,2x y 2 . ]TahrN_ 5 enAkgbg;kMupic )xoy( eK[cMnuc )i22(R,)iz(Q,)z(P cUrkMNt;sMNMucMnuc P edIm,I[bIcMnuc R,Q,P rt;Rtg;Ka ?

bIcMnuc R,Q,P rt;Rtg;KaluHRtaEtRKb; QRtPQ:IRt

b . tag )zz(tzz QRPQ iyxz enaH yixzQ eK)an )yixi22(tiyxyix t)x2(it)y2()yx(i)yx( eKTaj t)y2(yx nig t)x2()yx(



eK)an x2y2

yxyx

b 22 yy2xyx2xyy2xx2

8)2y()2x(

0y4x4yx22

22

dUcenHsMNMucMnuc P CargVg;pit mankaM )2,2(I 22R . ]TahrN_ 6 enAkgbg;kMupic )xoy( eK[cMnuc CarUbPaBn EdlbMeBjlkxN P z

21

izi22z

. cUrrksMNMucMnuc P ? eK)an |iz||i22z|2 Edl iz b tag 22 |iz||i22z|2 y.ixz eK)an

26)5y()4x(

015y10x8yx

1y2yx8y8y28x8x2

)1y(x])2y()2x[(2

|)1y(ix||)2y(i)2x(|2

|iiyx||i22iyx|2

22

22

2222

2222

22

22

dUcenHsMNMucMnuc P CargVg;pit )5,4(I nig kaM 26R .



17-bMElgcMnuckgbg;kMupic k> bMElgkil kgbg;kMupic )xoy( eKmanBIrcMnuc nig manGahVikerogKa z nig M 'M 'z

CarUbPaBn M tambMElgkilnviucTr eK)an 'M )z(v v

. vzz'z

y


ebI CarUbPaBn M tambMElgkilnviucTr enaH 'M )z(v v v'MM

eday enaH )z'z('MM vzz'z b vzz'z .

'M

v

M

x

O


x> bMElgvilpit kgbg;kMupic )xoy( eKmanBIrcMnuc nig manGahVikerogKa z nig M 'M 'z CarUbPaBn M tambMElgvilpit 'M )z( nigmMu eK)an )sini)(coszz(z'z . Yy


ebI CarUbPaBn M tambMElgvilpit 'M )z( nigmMu enaHeK)an r'MM nig 'MM Edl nig 'MX MX . ehtuenH

ii

ie

e.re.r

zzz'z Edl

dUcenH )sini)(coszz(z'z .

O

M

'M

X

x


K> bMElgcaMgpit kgbg;kMupic )xoy( eKmanBIrcMnuc nig manGahVikerogKa z nig M 'M 'z CarUbPaBn M tambMElgcaMgpit 'M )z( pleFob eK)an . )zz(z'z


y

CarUbPaBn M tambMElgcaMgpit 'M )z( pleFob eK)an

eday M'M )zz(M;)z'z('M

dUcenH )zz(z'z .

'M

M

x

O


18-)arIsg;nRbBncMNuckgbg;kMupic eK[ nk1kk ,A CaRbBnmYyman n cMNucb:ugeder Edlman GahVik . ebI kAz

n

1kk 0 enaH)arIsg; G nRbBnmanGahVikmYy

kMNt;eday

n

1kk

n

1kkAk

G

)z(z .

tamniymny ebI GCa)arIsg;nRbBn nk1kk ,A enaHeK)an

OGAn

1kkk

eday OGOAGA kk

eK)an

OOG.OA.n

1kkkk

b

n

1kkk

n

1kk OA.)(.OG

b

n1k

kAkn

1kkG )z()(.z

dUcenH

n

1kk

n

1kkAk

G

)z(z .



CMBUkTI2

kRmglMhat;eRCIserIsmandMeNaHRsay !> eKmancMnYnkMupic nig i74z1 i23z2 . k> cUrKNna nig 21 zz 21 zz . x> cUrKNna nig 21 zz

2

1zz .

@> eKman nig i21z1 i3z2 . cUrKNna nig ? 2221 zzU 3231 zzV #> cUrkMnt;cMnYnBit pnig q edIm,I[cMnYnkMupic i23z Cabsrbs;smIkar . 0qipzz2 $> kMnt;BIrcMnYnBit nig x y ebIeKdwgfa

i1)i97(2)iy23)(i31()ix1)(i3(

. %> cUrKNnabskaern i4240z . ^> eK[cMnYnkMupic i3b;i32a nig i41c k-cUrsresr nig 333 cba cba CaTRmg;BICKNit . x-cUrepgpat;faeKGackMnt;cMnYnBit k edIm,I[ cba.kcba 333



&> eK[cMnYnkMupic )2y.(i)1x(u;i3a nig i162b Edl nig x y CaBIrcMnYnBit . cUrkMnt;tm nig x y edIm,I[ 0bau . *> eK[cMnYnkMupic ylogxlogi)

2yx(logZ 223

nig i21i13W

Edl ** IRy;IRx . k-cUrsresr CaTRmg;BICKNit . Wx-kMnt; nig x y edIm,I[ . WZ (>eK[smIkar Edl 0bazz:)E( 2 IRb;a cUrkMnt;tm a nig b edIm,I[cMnYnkMupic 3i2z1 Cabsrbs; smIkar rYcTajrkbs mYYyeTotrbs;smIkar . )E( 2zetIGkBiniteXIjdUcemccMeBaHcMnYnkMupic nig ? 1z 2z!0> eK[cMnYnkMupic Edlman z z CacMnYnkMupicqas;rbs;va . edaHRsaysmIkar i3

7ziz|z|log5

!!> eK[cMnYnkMupicBIr )yx2(ix3z nig ]21[iy1W )3x(log5 Edl nig x y CacMnYnBit . kMnt; nig x y edIm,I[ . zW



!@> edaHRsaysmIkar i1i79|z|z2

Edl CacMnYnkMupic . z!#> eK[cMnYnkMupicBIr nig Edl 1Z 2Z 0Z2 . cUrRsaybBaak;fa

|Z||Z|

ZZ

2

1

2

1 . !$> eK[cMnYnkMupicBIr nig . 1Z 2Z cUrRsaybBaak;fa |Z||Z||ZZ| 2121 . !%> eK[cMnYnkMupicBIr nig . 1Z 2Zk> cUrRsaybBaak;fa |Z||Z||ZZ| 2121 x> Taj[)anfa 222222 dcba)db()ca( cMeBaHRKb; CacMnYnBit . d,c,b,a!^> eK[cMnYnkMupic

22.i6z1

nig i1z2

k>cUrsresr nig 21 z,z2

1zzZ CaragRtIekaNmaRt.

x>cUrsresr 2

1zzZ CaragBiCKNit.

K>Taj[)anfa 4

2612

cos nig 4

2612

sin .



!&> eK[cMnYnkMupic 22.i22z . k-cUrsresr CaTMrg;BiCKNit . 2zx-cUrsresr nig CaTMrg;RtIekaNmaRt . 2z zK-TajrktMlR)akdn

8cos nig

8sin .

!*> eK[cMnYnkMupic 54sin.i

54cosZ

cUrRsaybBaak;fa )56sin.i

56(cos

52cos8)Z1( 33 .

!(> eK[cMnYnkMupic )i1(24Z k-cUrsresr CaTRmg;RtIekaNmaRt . Zx-KNnabsTI# n . Z@0> eKeGay 8i.z3i14.zi32zzf 23 k>cUrbgajfa z 4z32z2izzf 2 x>edaHRsaysmIkar 0zf kgsMNMukMupic . @!> eK[cMnYnkMupic

74sin.i

74cosz .

cUrsresr CaragRtIekaNmaRt . 4)z1(



@@> eK[cMnYnkMupic nig i1z1 22.i6z2

k-cUrsresr nig 21 z,z2

1zz CaTMrg;RtIekaNmaRt .

x-cUrsresr 2

1zz CaTMrg;BiCKNit .

K-eRbIlTplxagelIcUrTajrktMlR)akdn 12

cos nig 12

sin . @#> eK[sIVtncMnYnkMupic kMnt;eday )Z( n

INn;|Z|Z21Z

23i1Z

nn1n

0

Cam:UDuln . |Z| n nZsntfa INn,)sin.i(cosZ nnnn Edl . IR;,0 nnn k-cUrrkTMnak;TMngrvag nig n 1n ehIy n nig 1n . x-rkRbePTnsIVt rYcKNna )( n n CaGnuKmn_n n . K-cUrbgajfa

2cos....

2cos

2coscos 1n2100n

rYcbBaak; GnuKmn_n n . n



@$> eK[cMnYnkMupic y.ixz Edl xnig y CaBIrcMnYnBit . cUrkMnt;tm nig x y ebIeKdwgfa

i210z)i31(z)i23( z CacMnYnkMupicqas;n . z

@%> eK[ CacMnYnkMupicEdl 21 z;z 0r|z||z| 21 .

bgajfa 22

212

212

212

21

r1

zzrzz

zzrzz

@^>eKyk CacMnYnkMupicEdlepgpat;TMnak;TMng n21 z....;;z;z1n,...,2,1,0k;0z)kn(iz)1k( 1 kk

k-kMnt; ebIeKdwgfa 0z nn210 2z....zzz x-cMeBaHtm Edl)ankMnt;xagelIcUrbgajfa 0z

!n)1n3(|z|....|z||z||z|n

2n

22

21

20

@&> eK[ CacMnYnkMupicedaydwgfa 321 z;z;z

0zzzzzzzzz 133221321 . cUrbgajfa |z||z||z| 321 @*> cUrbgajfa 1

iz32iz6

luHRtaEt 31|z|



@(> eK[ CacMnYnkMupicEdlmanm:UDles I 1 . n321 z....;;z;z;zeKtag

n

1k k

n

1kk )z

1()z(Z .

cUrbgajfa 2nZ0 #0> eK[cMnYnkMupic Edl z 1|z| . cUrbgajfa

4|z1||z1|2 2 . #!> eK[cMnYnkMupic )

xsin1x.(sini)

xcos1x(cosZ 2

22

2 Edl CacMnYnBit. xcUrkMntrkmUDulGb,brmancMnYnkMupicen ? #@> eK[ INn,i

31i

31A

nn

. cUrbgajfa

3nsin.

)3(2.iA n

1n cMeBaHRKb; n IN .

##> kgbg;kMupic eK[bYncMnuc )j,i,O( D,C,B,AEdlmanGahVikerogKa

i45Z,i54Z,i61Z CBA nig i32ZD . cUrRsayfactuekaN carikkgrgVg;mYyEdleKnwgbBaak;pit ABCD nig kaMrbs;va .



#$> eK[ nig CaBIrcMnYnkMupic . 1z 2zcUrbgajfa |1z||1z||zz||zz1| 22212121 #%> eK[ nig CacMnYnkMupicBIr . 1z 2zcUrRsayfa 2221221221 |z||z|2|zz||zz| #^>eK[ nig CacMnYnkMupicBIrEdl 1z 2z 1|z||z| 21 nig . 1z.z 21 cUrRsayfa

21

21

zz1zz

CacMnYnBitmYy .

#&> eK[BhuFa n)acosasinx()x(P Edl *INn cUrrksMNlnviFIEckrvag nwg )x(P 1x2 . #*> eK[cMnYnkMupic sin2icos2z Edl . IRkgbg;kMpic eKehA M CacMnucrUbPaBn z . )j,i,o(

cUrkMNt;tmtUcbMput nig FMbMputn OMr ? #(> eK[cMnYnkMupic ehIyepgpat;TMnak;TMng 321 z,z,z

1|z||z||z| 321 nig 01zzz

zzz

zzz

21

23

31

22

32

21

cUrRsayfa }2,1{|zzz| 321 .



$0> eK[cMnYnkMupic nig Edl 1z 2z 1|z||z| 21 cUrRsayfa 2|1zz||1z||1z| 2121 $!> eK[sIVtcMnYnBit kMNt;eday )a( n

...,3,2,1n,aaa1a,1a

n1n2n

21

eKtagsIVtcMnYnkMupic n1nn a23i1az .

k> cUrRsayfa n1n z23i1z cMeBaHRKb; n . 1

x> cUrdak; 2

3i1 CaTRmg;RtIekaNmaRtrYcTajrk CaGnuKmn_n n . nzK> TajrktYTUeTAnsIVt . etI CasIVtxYbbeT ? na )a( n$@> eK[sIVtncMnYnkMupic kMNt;eday )z( n

2i32z1

nig 2

i32z2

i3z n1n

Edl . ...,3,2,1n k> tag . 1zw nn bgajfa CasIVtFrNImaRtncMnYnkMupic )w( nrYcKNna CaGnuKmn_n n edaysresrlTplCaTRmg;RtIekaNmaRt . nw



x> Tajbgajfa )12nsini

12n(cos

12ncos2zn

. $#> eK[sIVtncMnYnBit nig kMNt;eday )u( n )v( n

22v

22u

1

1 nig

2vuv

2vuu

nn1n

nn1n

Edl 1n

k> eKBinitsIVtncMnYnkMupic nnn v.iuz . cUrRsayfa CasIVtFrNImaRtncMnYnkMupic rYcKNna )z( n nzCaGnuKmn_n n edaysresrlTplCaTRmg;RtIekaNmaRt . x> sMEdg nig CaGnuKmn_n . nu nv n$$> eK[sIVtncMnYnBit nig kMNt;eday )u( n )v( n

3v1u

0

0 nig Edl n

nn1n

2n

2n1n

vu2vvuu 0

k> eKBinitsIVtncMnYnkMupic nnn v.iuz . cUrRsayfa rYcTajfa . 2n1n zz

n20n zz

x> sMEdg nig CaGnuKmn_n . nu nv n



lMhat;TI1 eKmancMnYnkMupic nig i74z1 i23z2 . k> cUrKNna nig 21 zz 21 zz . x> cUrKNna z nig 21 z

2

1zz .

dMeNaHRsay k> KNna nig 21 zz 21 zz eyIg)an i97i23i74)i23()i74(zz 21 nig i51i23i74)i23()i74(zz 21 dUcenH i97zz 21 nig i51zz 21 .

x> KNna z nig 21 z2

1zz

eyIg)an i29214i21i812)i23)(i74(zz 21 nig i2

13i1326

4314i21i812

)i23)(i23()i23)(i74(

i23i74

zz2

1

dUcenH i292zz 21 nig i2zz2

1 .



lMhat;TI2 eKman nig i21z1 i3z2 . cUrKNna nig ? 2221 zzU 3231 zzV dMeNaHRsay KNna nig 2221 zzU 3231 zzV eyIg)an

i105

1i694i41i3i21U 22

i247

i9i2727i812i61i3i21V 33

dUcenH i105U nig i247V .



lMhat;TI3 cUrkMnt;cMnYnBit pnig q edIm,I[cMnYnkMupic i23z Cabsrbs;smIkar . 0qipzz2 dMeNaHRsay kMnt;cMnYnBit pnig q ebI Cabsrbs;smIkar enaHvaRtUvepgpat;smIkar i23z 0qipzz2

eK)an 0qi23ipi23 2 0p312.iqp25

0qp2ip34i129

eKTaj)an naM[

0p312

0qp25

4p135p2q

dUcenH 13q,4p .



lMhat;TI4 kMnt;BIrcMnYnBit x nig y ebIeKdwgfa

i1)i97(2)iy23)(i31()ix1)(i3(

. dMeNaHRsay kMnt;BIrcMnYnBit x nig y eKman

i1)i97(2)iy23)(i31()ix1)(i3(

eK)an

11)i1)(i97(2y6i9iy23xiix33

i2168y2x3.i6y6x 2

)9i9i77(2i8iy2ix36y6x

eKTaj

28y2x3166y6x

b naM[

10y2x310y6x

2y,2x . dUcenH . 2y,2x



lMhat;TI5 cUrKNnabskaern . i4240z dMeNaHRsay tag Cabskaern IRy;x,y.ixW i4240z eK)an eday zW2 ixy2)yx()y.ix(W 2222 naM[ eKTaj i.4240ixy2)yx( 22

42xy240yx 22

eday 33644240yx4)yx(yx 2222222222 eKTaj 583364yx 22 .

eK)anRbBnsmIkar

240yx

158yx22

22

bUksmIkar ! nig @ eK)an 98x2 2 naM[ 7x dksmIkar! nig @ eK)an naM[ 18y2 2 3y eday 042xy2 naM[ xnig ymansBaadUcKa naM[eKTaj)anKUcemIy

3y,7x nig 3y,7x . dUcenH nig i37W1 i37W2 .



lMhat;TI6 eK[cMnYnkMupic i3b;i32a nig i41c k-cUrsresr nig 333 cba cba CaTRmg;BICKNit . x-cUrepgpat;faeKGackMnt;cMnYnBit k edIm,I[ . cba.kcba 333 dMeNaHRsay ksresr nig 333 cba cba CaTRmgBICKNit eyIgman i946i2754i368)i32(a 33

i5247i6448i121)i41(c

i2618i9i2727)i3(b33

33

eyIgn

i87111

i5247i2618i946cba 333

dUcen i87111cba 333 .

eyIgman )i41)(i3)(i32(cba

i2937

28i7i369)i41)(i79()i41)(3i9i26(

dUcen i2937cba .



xkMntcMnYnBit k eyIgman cba.kcba 333 eKTaj 3

i2937i87111

c.b.acbak333

dUcen 3k .



lMhat;TI7 eK[cMnYnkMupic )2y.(i)1x(u;i3a nig i162b Edl nig x y CaBIrcMnYnBit . cUrkMnt;tm nig x y edIm,I[ 0bau . dMeNaHRsay kMnt;tmn x nig y eyIg)an 0bau

abu

eday )2y.(i)1x(u;i3a nig i162b eK)an

i3i162)2y.(i)1x(

i51)2y(i)1x(

10)8i24i3(2)2y(i)1x(

i3)i3)(i81(2)2y(i)1x( 22

eKTaj)an naM[

52y11x

;2x y 3

dUcenH 3y;2x .



lMhat;TI8 eK[cMnYnkMupic ylogxlogi)

2yx(logZ 223 nig i21

i13W

Edl . ** IRy;IRx k-cUrsresr CaTRmg;BICKNit . Wx-kMnt; nig x y edIm,I[ . WZ dMeNaHRsay k-sresr CaTRmg;BICKNit W eyIg)an i52

52ii2412

41)i21)(i12(

i21i12W

dUcenH i52W .

x-kMnt; nig x y edIm,I[ WZ

eyIg)an Z smmUl W

5ylogxlog

2)3yx(log

22

2 naM[

32y.x12yx

eKTaj y;x CabsssmIkar 032u12u2 eday eKTajbs 43236'

826u426u

2

1

dUcenH 8y;4x b 4y;8x .



lMhat;TI9 eK[smIkar Edl 0bazz:)E( 2 IRb;a cUrkMnt;tm a nig b edIm,I[cMnYnkMupic 3i2z1 Cabsrbs; smIkar rYcTajrkbs mYYyeTotrbs;smIkar . )E( 2zetIGkBiniteXIjdUcemccMeBaHcMnYnkMupic nig ? 1z 2zdMeNaHRsay kMnt;tm a nig b edIm,I[cMnYnkMupic 3i2 Cabsrbs;smIkarluHRtaEtvaepgpat;smIkar eK)an 0b)3i2(a)3i2( 2

0)3a34(i)ba21(0b3aia233i44

eKTaj)an

0ba2103a34 b

7b4a

dUcenH 7b;4a .

KNnabsmYYyeTotrbs;smIkar ebI Cabsrbs;smIkarenaHtamRTwsIbTEvteK)an 21 z;z

4azz 21 eKTaj 3i2)3i2(4z2 dUcenH 3i2z2 . eyIgBiniteXIjfacMnYnkMupic nig qas;Ka . 1z 2z



lMhat;TI10 eK[cMnYnkMupic z Edlman z CacMnYnkMupicqas;rbs;va . edaHRsaysmIkar i3

7ziz|z|log5

dMeNaHRsay edaHRsaysmIkar i3

7ziz|z|log5

tag y.ixz naM[ iyxz Edl IRy;x smIkarGacsresr

i37yxi)yxlog

7yx(

i37

yixiyxyxlog

i37

)iyx(i)iyx(|iyx|log

225

225

5

eKTaj)an

3yxlog7yx

17yx

225

b

3yxlog1

7yx22

5

4

b eKTajcemIy

25yx

7yx22


y,3x b ,4x y 3 .

dUcenH i34z;i43z 21 CacemIyrbs;smIkar .


lMhat;TI11 eK[cMnYnkMupicBIr

)yx2(ix3z nig ]21[iy1W )3x(log5 Edl nig x y CacMnYnBit . kMnt; x nig y edIm,I[ zW . dMeNaHRsay kMnt; nig x y edIm,I[ luHRtaEt zW

)zIm()WIm()zRe()WRe(

eKTaj)an

)2(yx221)1(x3y1

)3x(log5

tam eKTaj )1( )3(1x3y ykCYskgsmIkar )2(eK)an 1x3x221 )3x(log5 lkxN x2 )3x(log5 03x b 3x tag naM[ )3x(logt 5 t53x b 35x t eK)an smIkar b 352 tt 325 tt edayGgxageqVgnsmIkarCaGnuKmn_ekIn nig GgxagsaMCaGnuKmn_efr enaHeyIg)ansmIkarmanbsEtmYyKt;KW 1t . cMeBaH eK)an 1t 235x nig 71)2(3y dUcenH . 7y;2x



lMhat;TI12 edaHRsaysmIkar

i1i79|z|z2

Edl CacMnYnkMupic . zdMeNaHRsay edaHRsaysmIkar

i1i79|z|z2

tag IRy;x,y.ixz eK)an

)i1)(i1()i1)(i79(|iyx|)iyx(2

i81iy2)yxx2(

27i7i99yxiy2x2

22

22

eKTaj

)2(8y2)1(1yxx2 22

tam eKTaj )2( 4y ykeTACYskg eK)an )1(

22

22

22

2

2

7454';015x4x3

16x1x4x4

16x)1x2(21x,16x1x2

116xx2



eKTajbs 21

35

372x;3

372x 21 minyk

eK)an 4y;3x . dUcenH i43z CacemIyrbs;smIkar .



lMhat;TI13 eK[cMnYnkMupicBIr nig Edl 1Z 2Z 0Z2 . cUrRsaybBaak;fa

|Z||Z|

ZZ

2

1

2

1 . dMeNaHRsay

RsaybBaak;fa |Z||Z|

ZZ

2

1

2

1 eyIgtag nig b.iaZ1 d.icZ2 Edl CacMnYnBit . d,c,b,aeyIg)an

222

2

2

1

d.icbd.ibc.iad.iac

)d.ic)(d.ic()d.ic)(b.ia(

d.icb.ia

ZZ

eK)an 22222221

dcadbc.i

dcbdac

dc)adbc(i)bdac(

ZZ

naM[ 2

22

2

222

1

dcadbc

dcbdac

ZZ

22

2222

22

222222

22

22222222

22

22222222

22

22

dc)ba)(dc(

dc)dc(b)dc(a

dc)dbcb()daca(

dcdaabcd2cbdbabcd2ca

dc)adbc()bdac(



eKTaj 22

22

222

2222

2

1

dcba

)dc()dc)(ba(

ZZ

edayeKman

222

221

dc|Z|

ba|Z|

dUecH |Z||Z|

ZZ

2

1

2

1 .



lMhat;TI14 eK[cMnYnkMupicBIr nig . 1Z 2Z cUrRsaybBaak;fa |Z||Z||ZZ| 2121 . dMeNaHRsay RsaybBaak;fa |Z||Z||ZZ| 2121 eyIgtag nig b.iaZ1 d.icZ2 Edl CacMnYnBit . d,c,b,aeyIgman bd.ibc.iad.iac)d.ic)(b.ia(ZZ 221 naM[ )bcad.(i)bdac(ZZ 21 eK)an 2221 )bcad()bdac(|ZZ|

)dc)(ba()ba(d)ba(c

)dbda()cbca(

cbabcd2dadbabcd2ca

2222222222

22222222

22222222

eKTaj 222221 dcba|ZZ|

edayeKman

222

221

dc|Z|

ba|Z|

dUcenH |Z||Z||ZZ| 2121 .



lMhat;TI15 eK[cMnYnkMupicBIr nig . 1Z 2Zk> cUrRsaybBaak;fa |Z||Z||ZZ| 2121 x> Taj[)anfa 222222 dcba)db()ca( cMeBaHRKb; CacMnYnBit . d,c,b,adMeNaHRsay k> RsaybBaak;fa |Z||Z||ZZ| 2121 kgbg;kMupic eyIgeRCIserIsviucTrBIr nig manGahVikerogKa )XOY(

U

V

1Z nig naM[viucTr manGahVik 2Z VU 21 ZZ .

tamlkNRCugrbs;RtIekaNeK)an

eday |V||U||VU|


|ZZ||VU|

|Z||V|,|Z||U|

21

21

dUcenH |Z||Z||ZZ| 2121

U

V

VU


x> Taj[)anfa 222222 dcba)db()ca( eyIgtag nig b.iaZ1 d.icZ2 Edl CacMnYnBit . d,c,b,aman )db.(i)ca(ZZ 21 naM[ 2221 )db()ca(|ZZ| ehIy 222221 dc|Z|,ba|Z| . tamsRmayxagelIeKman |Z||Z||ZZ| 2121 dUcenH 222222 dcba)db()ca( .



lMhat;TI16 eK[cMnYnkMupic

22.i6z1

nig i1z2

k>cUrsresr nig 21 z,z2

1zzZ CaragRtIekaNmaRt.

x>cUrsresr 2

1zzZ CaragBiCKNit.

K>Taj[)anfa 4

2612

cos nig 4

2612

sin . dMeNaHRsay

k>sresr nig 21 z,z2

1zzZ CaragRtIekaNmaRt

eKman

6sin.i

6cos2

21.i

232

22i6z1

dUcenH

)6

sin(.i)6

cos(2z1 .

eKman

4sin.i

4cos2

22.i

222i1z2

dUcenH

)4

sin(.i)4

cos(2z2 .



eKman

)46

sin(.i)46

cos(22

zzZ2

1

dUcenH 12

sin.i12

cosZ .

x> sresr 2

1zzZ CaragBiCKNit

eK)an 4

22i6i6)i1)(i1(2)i1)(2i6(

)i1(22i6Z

dUcenH 4

26.i4

26Z .

K> TajeGay)anfa 4

2612

cos nig 4

2612

sin tamsRmayxagelIeKman

)1(12

sin.i12

cosZ nig )2(4

26.i4

26Z

edaypwm ! nig @ eKTaj)an

426

12cos nig

426

12sin .



lMhat;TI17 eK[cMnYnkMupic 22.i22z . k-cUrsresr CaTMrg;BiCKNit . 2zx-cUrsresr nig CaTMrg;RtIekaNmaRt . 2z zK-TajrktMlR)akdn

8cos nig

8sin .

dMeNaHRsay k-sresr CaTMrg;BiCKNit 2zeyIg)an 22 )22.i22(z

i.22222224.i222

22.i2222i22222

dUcenH i.2222z2

x-sresr nig CaTMrg;RtIekaNmaRt 2z zeKman i.2222z2

4sin.i

4cos4

22.i

224

dUcenH

4

sin.i4

cos4z2 nig

8

sin.i8

cos2z



K-TajrktMlR)akdn 8

cos nig 8

sin tamsMrayxagelIeKman

8sin.i

8cos2z

eday 22.i22z eKTaj 22.i22

8sin.i

8cos2

dUcenH 2

228

cos nig 8

228

sin .




54sin.i

54cosZ

cUrRsaybBaak;fa )56sin.i

56(cos

52cos8)Z1( 33 .

dMeNaHRsay RsaybBaak;fa

)56sin.i

56(cos

52cos8)Z1( 33

eyIgman 54sin.i

54cos1Z1

tamrUbmn 2

cos2cos1 2 nig 2

cos2

sin2sin eyIg)an

52cos

52sini2

52cos2Z1 2

b )52sin.i

52(cos

52cos2Z1

tamrUbmndWmreyIg)an

)56sin.i

56(cos

52cos8

)52sin.i

52(cos

52cos2)Z1(

3

33

dUcenH )56sin.i

56(cos

52cos8)Z1( 33 .



lMhat;TI19 eK[cMnYnkMupic )i1(24Z k-cUrsresr CaTRmg;RtIekaNmaRt . Zx-KNnabsTI# n . ZdMeNaHRsay k-sresr CaTRmg;RtIekaNmaRt ZeyIg)an )i1(24Z

)43sin.i

43(cos8

)4

sin(.i)4

cos(8

)4

sin.i4

cos(8)22.i

22(8

dUcenH )43sin.i

43(cos8Z .

x-KNnabsTI# n ZeyIgtag kW CabsTI# n )4

3sin.i43(cos8Z

tamrUbmnbsTI n :

)nk2sin(.i)

nk2cos(rW nk



eyIg)an

)

3

k243

sin(.i)3

k243

(cos8W 3k

)]12

k83sin(.i)12

k83[cos(2 -ebI )

4sin.i

4(cos2W:0k 0

-ebI )

1211sini

1211(cos2W:1k 1

-ebI )

1219sin.i

1219(cos2W:2k 2

.



lMhat;TI20 eKeGay 8i.z3i14.zi32zzf 23 k>cUrbgajfa z 4z32z2izzf 2 x>edaHRsaysmIkar 0zf kgsMNMukMupic . dMeNaHRsay k> bgajfa : z 4z32z2izzf 2 eyIgman 4z32z.i2zzf 2 edayBnatGnuKmn_enHeyIg)an )4z32z(i2z4z32z)z(f 23

Biti8z.3i14z.i32zi8iz34iz2z4z.32z

23

223

dUcenH z 4z32z2izzf 2 . x> edaHRsaysmIkar ebI 0zf naMeGay 04z.32z.i2z 2 eKTajbs ehIy i2z 2'2 i143,04z.32z, naMeGay i3z,i3z 21 .




74sin.i

74cosz .

cUrsresr CaragRtIekaNmaRt . 4)z1( dMeNaHRsay sresr CaragRtIekaNmaRt 4)z1( eKman

74sin.i

74cosz

eK)an 74sin.i

74cos1z1

)

72sin.i

72(cos

72cos2

72cos

72sin.i2

72cos2z1 2

tamrUbmndWmreKGacsresr

78sin.i

78cos

72cos16

)72sin.i

72(cos

72cos2)z1(

4

44

dUcenH

78sin.i

78cos

72cos16)z1( 44 .



lMhat;TI22 eK[cMnYnkMupic nig i1z1 2

2.i6z2

k-cUrsresr nig 21 z,z2

1zz CaTMrg;RtIekaNmaRt .

x-cUrsresr 2

1zz CaTMrg;BiCKNit .

K-eRbIlTplxagelIcUrTajrktMlR)akdn 12

cos nig 12

sin . dMeNaHRsay

k-sresr nig 21 z,z2

1zz CaTMrg;RtIekaNmaRt

eKman eday i1z1 211r 22 eK)an

4sin.i

4cos2

22.i

222z1 .

eKman 2

2.i3.22

2.i6z2

6sin.i

6cos2

21.i

232z2 .

eK)an 12

sin.i12

cos

6sin.i

6cos2

4sin.i

4cos2

zz2

1

.



x-sresr 2

1zz CaTMrg;BiCKNit

eK)an 2.i62.i6 2.i6i122

2.i6i1

zz2

1

426.i26

2626.i2.i62

dUcenH 4

26.i4

26zz2

1 .

K-TajrktMlR)akdn 12

cos nig 12

sin tamsMrayxagelIeKman

12sin.i

12cos

zz

2

1 nig 4

26.i4

26zz2

1

eKTaj 4

26.i4

2612

sin.i12

cos

dUcenH 4

2612

cos nig 4

2612

sin .



lMhat;TI23


)ZneK[sIVtncMnYnkMupic ( kMnt;eday

INn;|Z|Z21Z

23i1Z

nn1n

0

Cam:UDuln . |Z| n nZsntfa INn,)sin.i(cosZ nnnn Edl . IR;,0 nnn k-cUrrkTMnak;TMngrvag nig n 1n ehIy n nig 1n . x-rkRbePTnsIVt rYcKNna )( n n CaGnuKmn_n n . K-cUrbgajfa

2cos....

2cos

2coscos 1n2100n

rYcbBaak; GnuKmn_n n . ndMeNaHRsay k-rkTMnak;TMngrvag nig n 1n ehIy n nig 1n eyIgman )sin.i(cosZ nnnn naM[ )sini(cosZ 1n1n1n1n eday )|Z|Z(

21Z nn1n ehIy nn |Z|

eK)an nnnn1n1n1n )sin.i(cos21)sini(cos


)

2sin.i

2(cos

2cos)sin.i(cos

)sin.icos1(21)sin.i(cos

nnnn1n1n1n

nnn1n1n1n

eKTajn 2

cos nn1n nig 2

n1n

dUcenH

2n

1n nig 2cos

nn1n

. x-RbePTnsIVt nig KNna )( n n CaGnuKmn_n n tamsRmayxagelIeyIgman n1n 2

1 naM[ n CasIVtFrNImaRtmanersug esI

21q .

tamrUbmn n0n qeday

3sin.i

3cos

23i1)sini(cosZ 0000

eKTaj)an

3;1 00

dUcenH nn 21.

3 .



K-bgajfa 2

cos....2

cos2

coscos n2100n

tamsRmayxagelIeKman 2

cos nn1n

b 2

cos nn

1n

eK)an

1nk

0k

1nk

0k

k

k

1k )2

cos(

2

cos.........2

cos.2

cos.cos 1n2100

n

dUcenH 2

cos....2

cos2

coscos 1n2100n .

m:ageToteyIgman 2

cossin22

cos2

sin2sin n1nnnn

eKTaj 1n

nnsinsin.

21

2cos

ehtuenH n

0nn

1n

2

1

1

0nn sin

sin.

21

sinsin.....

sinsin.

sinsin.

21

dUcenH )

21.

3sin(

1.2

3

)21.

3sin

3sin

21

n1n

nnn

.



lMhat;TI24 eK[cMnYnkMupic y.ixz Edl xnig y CaBIrcMnYnBit . cUrkMnt;tm nig x y ebIeKdwgfa

i210z)i31(z)i23( z CacMnYnkMupicqas;n . z

dMeNaHRsay eKman

i210z)i31(z)i23(

eday y.ixz naM[ y.ixz eK)an

i210)iyx)(i31()iyx)(i23(

i24)y2x5.(i)yx4(

5)i2(10y3ix3iyxy2ix2iy3x3

eKTaj)an

2y2x54yx4

eKman 6282214

D,3582514

D x

nig 122082544

Dy

eK)an 4312

DD

y,236

DDx yx

dUcenH 4y,2x .



lMhat;TI25 eK[ CacMnYnkMupicEdl 21 z;z 0r|z||z| 21 .

bgajfa 22

212

212

212

21

r1

zzrzz

zzrzz

dMeNaHRsay bgajfa

2

2

212

212

212

21

r1

zzrzz

zzrzz

tag nig )x2sinix2(cosrz1 )y2siniy2co(rz2 Edl IRy;IRx . eK)an

)yxcos()yxcos(.

r1

])yxcos()yxsin(i2)yx(cos2[r)yxcos()yxsin(i2)yxcos()yxcos(2

])y2x2sin(.i)y2x2cos([rr])y2sinx2(sini)y2cosx2(cos[r

zzrzz

2

2221

221

dUcKaEdr )xysin()xysin(

r1

zzrzz

212

21

eK)an



)xy(sin)xy(sin

)yx(cos)yx(cos

r1

zzrzz

zzrzz

2

2

2

2

2

2

212

212

212

21

eday )yx(cos)yx(cos)yx(cos 2

2

2

eRBaH 1)yx(cos 2

ehIy )yx(sin)xy(sin)xy(sin 2

2

2

eRBaH 1)yx(sin 2

1)yx(sin)yx(cos)xy(sin)xy(sin

)yx(cos)yx(cos 22

2

2

2

2

dUcenH 22

212

212

212

21

r1

zzrzz

zzrzz

.



lMhat;TI26 eKyk CacMnYnkMupicEdlepgpat;TMnak;TMng n21 z....;;z;z

1n,...,2,1,0k;0z)kn(iz)1k( 1 kk k-kMnt; ebIeKdwgfa 0z nn210 2z....zzz x-cMeBaHtm Edl)ankMnt;xagelIcUrbgajfa 0z

!n)1n3(|z|....|z||z||z|n

2n

22

21

20

dMeNaHRsay k-kMnt; ebIeKdwgfa 0z nn210 2z....zzz eKman 0z)kn(iz)1k( k1k eK)an

1kkn.i

zz

k

1k

)!pn(!p

!nC;Cizz

1kkn.i

zz

pn

pn

p

0

p

1p

0k

)1p(

0k k

1k

eKTaj ....,2,1,0p;Cziz pn0pp eday b nn210 2z....zzz

n

0p

np 2)z(



man n0n

0p

n

0p

ppn0p )i1(ziCz)z(

eK)an nn0 2)i1(z eKTaj nn

n

0 )i1()i1(2z

dUcenH . n0 )i1(z x-cMeBaHtm Edl)ankMnt;xagelIcUrbgajfa 0z

!n)1n3(|z|....|z||z||z|n

2n

22

21

20

edayGnuvtn_vismPaB GMAM eyIg)an

2nn21n20n202n2120 )C(....)C()C(zz...zz

!n)1n3(

n)1n(....)2n2()1n2(n2

!n2

))1n)....(2n2)(1n2(n2!n

2

!n!n)!n2(.2Cz

n

nn

n

nnn2

20

dUcenH !n)1n3(|z|....|z||z||z|n

2n

22

21

20

.



lMhat;TI27 eK[ CacMnYnkMupicedaydwgfa 321 z;z;z

0zzzzzzzzz 133221321 . cUrbgajfa |z||z||z| 321 dMeNaHRsay bgajfa |z||z||z| 321 eKman 0zzzzzzzzz 133221321 eKTaj

)2(0)zz(zzz)1(zzz

21321

321

yk CMnYskg eyIg)an )1( )2(0zzz 2321 naM[ |z|.|z||z| 2123

dUcKaEdr nig |z|.|z||z| 3122 |z|.|z||z| 3221 eyIg)an

|z||z||z||z||z||z||z||z||z| 1332212

32

22

1 b 0|)z||z(||)z||z(||)z||z(| 213232221 dUcenH . |z||z||z| 321



lMhat;TI28 cUrbgajfa 1

iz32iz6

luHRtaEt 31|z|

dMeNaHRsay bgajfa

31|z|

eKman 1iz32iz6

lkxN 2 b 0iz3 3i2z

eK)an |iz32||iz6|

91|z|

91zz

3zz27zz9iz6zi641zi6iz6zz36

)zi32)(iz32()iz6)(iz6(|iz32||iz6|

2

22

dUcenH 31|z| .



lMhat;TI29 eK[ CacMnYnkMupicEdlmanm:UDles I . n321 z....;;z;z;z 1eKtag

n

1k k

n

1kk )z

1()z(Z .

cUrbgajfa 2nZ0 dMeNaHRsay bgajfa 2nZ0 eday CacMnYnkMupicEdlmanm:UDles I n321 z....;;z;z;z 1enaHeKGactag kkk xsin.ixcosz Edl n,....,3,2,1k;IRxk eK)an

n

1k k

n

1kk )z

1()z(Z

0xsinxcos

xsinixcosxsin.ixcos

2n

1kk

2n

1kk

n

1kkk

n

1kkk

eK)an 0Z ma:geTottamvismPaB SchwartzCauchy

n

1kk

22n

1kk xcosnxcos



nig

n1k

k2

2n

1kk xsinnxsin

eKTaj )x(sinnxcosnZ n1k

k2

n

1kk

2

2

n

1kk

2k

2

nn.nZ

xsinxcosnZ

dUcenH . 2nZ0 sMKal; eKGacRsay tammYyrebobeTotdUcxageRkam 2nZ eday 1|z| k enaH

kk z

1z RKb;


n.....;;2;1 k

eK)an

n

1k k

n

1kk )z

1()z(Z

22n

1kk

2n

1kk

n

1kk

n

1kk

n

1kk

n

1kk

n|z|z

zzzz

eKTaj)an . 2nZ


lMhat;TI30 eK[cMnYnkMupic Edl . cUrbgajfa z 1|z|

4|z1||z1|2 2 . dMeNaHRsay bgajfa 4|z1||z1|2 2 tag tsin.itcosz eK)an |

2tsin|2tsin)tcos1(|z1| 22

ehIy |tcos|2t2sin)t2cos1(|z1| 222 |

2tsin21|2 2

eK)an

|2tsin21||

2tsin|2|z1||z1| 22

edayyk 1x1;2tsinx ehIytagGnuKmn_ f

kMnt;eday Edl |x21||x|)x(f 2 1x1 cMeBaH eK)an 1x1 2)x(f

22

dUcenH 4|z1||z1|2 2 .



lMhat;TI31 eK[cMnYnkMupic )

xsin1x.(sini)

xcos1x(cosZ 2

22

2 Edl CacMnYnBit. xcUrkMntrkmUDulGb,brmancMnYnkMupicen ? dMeNaHRsay rkmUDulGb,brmancMnYnkMupic

eyIgn 222222 )xsin1x(sin)

xcos1x(cos|Z|

tag 222222 )xsin1x(sin)

xcos1x(cos)x(f

)xcosxsinxsinxcos()xsinx(cos4

)xcos

1xsin

1()xsinx(cos4

xsin12xsin

xcos12xcos

44

4444

4444

44

44



)

x2sin161)(x2sin

211(4

)x2sin

161(xcosxsin2)xsinx(cos4

)xcosxsin

11)(xsinx(cos4)x(f

42

422222

4444

edayeKman naM[ 1x2sin 221x2sin

211 2

nig 17x2sin

161 4 eKTaj

225

2174)

x2sin161)(x2sin

211(4 4

2 eyIgn

225)

x2sin161)(x2sin

211(4)x(f 4

2

eday )x(f|Z| eKTajn 225

25|Z|

dUcenmUDulGb,brman KW Z225|Z| min .



lMhat;TI32 eK[ INn,i

31i

31A

nn

. cUrbgajfa

3nsin.

)3(2.iA n


dMeNaHRsay bgajfa

3nsin.

)3(2.iA n


eyIgman INn,i31i

31A

nn

tag

3sin.i

3cos

32

23i

21

32

33.i1i

31Z

tamrUbmndWmreK)an

3nsin.i

3ncos

)3(2i

31Z n

nnn

ehIy

3nsin.i

3ncos

)3(2Z nn

n

eKTaj

3nsin.i

3ncos

)3(2

3nsin.i

3ncos

)3(2A n

n

n

n

3nsin.

)3(2.i

3nsin.i

3ncos

3nsin.i

3ncos

)3(2

n

1n

n

n

dUcenH 3nsin.

)3(2.iA n

1n

.



lMhat;TI33 kgbg;kMupic eK[bYncMnuc )j,i,O( D,C,B,AEdlmanGahVikerogKa

i45Z,i54Z,i61Z CBA nig i32ZD . cUrRsayfactuekaN carikkgrgVg;mYyEdleKnwgbBaak;pit ABCD nig kaMrbs;va . dMeNaHRsay RsayfactuekaN carikkgrgVg; ABCDeyIgtag 0cbyaxyx:)c( 22 CasmIkarrgVg;carikeRkARtIekaN . ABCeyIg)an naM[ )c(A 0cb6a61 22 b )1(37cb6a naM[ )c(B 0cb5a454 22 b )2(41cb5a4 naM[ )c(C 0cb3a2)3()2( 22 b )3(13cb3a2



eyIg)anRbBnsmIkar

13cb3a241cb5a437cb6a

bnab;BIedaHRsayRbBnenHeK)ancemIuy 23c,2b,2a .

smIkarrgVg;carikeRkARtIekaN Gacsresr ABC 023y2x2yx:)c( 22 b 25)1y()1x(:)c( 22 ma:geTotedayykkUGredaen CYskgsmIkar D

25)13()12(:)c( 22 vaepgpat;enaHnaM[ . )c(DedaybYncMnuc sitenAelIrgVg;mansmIkar D,C,B,A

25)1y()1x(:)c( 22 EtmYy enaHnaM[ctuekaN carikkgrgVg; manpit ABCD )c( )1,1(Inig kaM 5R .



lMhat;TI34 eK[ nig CaBIrcMnYnkMupic . 1z 2zcUrbgajfa |1z||1z||zz||zz1| 22212121 dMeNaHRsay bgajfa |1z||1z||zz||zz1| 22212121 tamvismPaBRtIekaNeK)an |zzzz1||zz||zz1| 21212121 nig |zzzz1||zz||zz1| 21212121 eK)an |)zz()zz1(||zz||zz1| 22122122121 eday )z1)(z1(zzzz1)zz()zz1( 222122212221221221 eK)an |)z1)(z1(||zz||zz1| 222122121 |z1||z1||zz||zz1| 222 212121dUcenH |)zz()zz1(||zz||zz1| 22122122121



lMhat;TI35 eK[ nig CacMnYnkMupicBIr . 1z 2zcUrRsayfa 2221221221 |z||z|2|zz||zz| dMeNaHRsay Rsayfa 2221221221 |z||z|2|zz||zz| eKman )zz)(zz(|zz| 2121221 1|z|zzzz|z||zz|

zzzzzzzz|zz|2

221212

12

21

222121112

21

ehIy )zz)(zz(|zz| 2121221 2|z|zzzz|z||zz|

zzzzzzzz|zz|2

221212

12

21

222121112

21

bUkTMnakTMng nig eKn 1 2 2221221221 |z||z|2|zz||zz| .



lMhat;TI36 eK[ nig CacMnYnkMupicBIrEdl 1z 2z 1|z||z| 21 nig . 1z.z 21 cUrRsayfa

21

21

zz1zz

CacMnYnBitmYy .

dMeNaHRsay Rsayfa

21

21

zz1zz

CacMnYnBitmYy

tag 21

21

zz1zzZ

ena 21

21

z.z1zzZ

eday 1|z|z.z 111 2 ena 1

1 z1z ehIydUcKa

22 z

1z

eKn Z1zz

zz

z1.

z11

z1

z1

Z12

12

21

21

ena 21

21

zz1zzZ

CacMnYnBit . eday ZZ



lMhat;TI37 eK[BhuFa n)acosasinx()x(P Edl *INn cUrrksMNlnviFIEckrvag nwg )x(P 1x2 . dMeNaHRsay sMNlnviFIEck tag )x(R CasMNlnviFIEckrvag nwg )x(P 1x2 -ebI ena 1n acosasinx)x(P dUcen acosasinx)x(R CasMnlnviFIEck . -ebI eKn 2n )x(R)x(Q)1x()x(P 2 Edl CaplEck nwg )x(Q BAx)x(R eKn BAx)x(Q)1x()acosasinx( 2n ebI ena ix BAi)acosasini( n A.iB)nasin(.i)nacos( eKTaj nig )nasin(A )nacos(B dUcen )nacos()nasin(x)x(R .



lMhat;TI38 eK[cMnYnkMupic sin2icos2z Edl IRkgbg;kMpic eKehA M CacMnucrUbPaBn z . )j,i,o(

cUrkMNt;tmtUcbMput nig FMbMputn OMr ? dMeNaHRsay kMNt;tmtUcbMput nig FMbMputn OMr eKman

22222 )sin2()cos2(baOM sincos4OM2 b sincos4OM

eK)an sincos4r tamvismPaB SchwarzCauchy eKman

2sincos11|sincos| 2222 b 2sincos2 eKTaj 24r24 RKb; IR dUcenH 24rmin nig 24rmax .



lMhat;TI39 eK[cMnYnkMupic ehIyepgpat;TMnak;TMng 321 z,z,z


1|z||z||z| 321 nig 01zzz

zzz

zzz

21

23

31

22

32

21

cUrRsayfa }2,1{|zzz| 321 . dMeNaHRsay Rsayfa }2,1{|zzz| 321 eKman 01

zzz

zzz

zzz

21

23

31

22

32

21

eK)an 0zzzzzz 321333231 b 321321333231 zzz4zzz3zzz tag eK)an 321 zzzz

4|z|3zzz)4z.z3(zzzz 4)zzz(z3zzzz4)

z1

z1

z1(z3zzzz

zzz4)zzzzzz(z3z

2321321

3321321

3321

3213

3213132213

eK)an |)4|z|3(zzz||z| 23213


b |4|z|3||z| 23 -ebI b 04|z|3 2

32|z|

eK)an 4|z|3|z| 23

2|z|0)2|z)(|1|z(|

04|z|3|z|2

23

-ebI b 04|z|3 2 32|z|

eK)an )4|z|3(|z| 23

1|z|0)2|z)(|1|z(|

04|z|3|z|2

23

dUcenH . }2,1{|zzz| 321



lMhat;TI40 eK[cMnYnkMupic nig Edl 1z 2z 1|z||z| 21 cUrRsayfa 2|1zz||1z||1z| 2121 dMeNaHRsay Rsayfa 2|1zz||1z||1z| 2121 tamvismPaBRtIekaN |ba||b||a| eK)an

|z1||z1||z||1zz||1z||1zz1z||1zz||1z|

112212

212212

ehIy 2|)z11z(||z1||1z| 1111 dUcenH 2|1zz||1z||1z| 2121 .



lMhat;TI41 eK[sIVtcMnYnBit kMNt;eday )a( n

...,3,2,1n,aaa1a,1a

n1n2n

21

eKtagsIVtcMnYnkMupic n1nn a23i1az .

k> cUrRsayfa n1n z23i1z cMeBaHRKb; n . 1

x> cUrdak; 2

3i1 CaTRmg;RtIekaNmaRtrYcTajrk CaGnuKmn_ nz n n . K> TajrktYTUeTAnsIVt . etI CasIVtxYbbeT ? na )a( ndMeNaHRsay

k> Rsayfa n1n z23i1z cMeBaHRKb; n 1

eKman n1nn a23i1az

eK)an 1n2n1n a23i1az

eday n1n2n aaa



eK)an 1nn1n1n a23i1aaz

)a2

3i1a(2

3i1

)a3i1

2a(2

3i1

aa2

3i1

n1n

n1n

n1n

dUcenH n1n z23i1z .

x> dak; 2

3i1 CaTRmg;RtIekaNmaRt eK)an

3sini

3cos

23i

21

23i1

Tajrk CaGnuKmn_n n nz

eday n1n z23i1z enaH CasIVtFrNImaRtncMnYn )z( n

kMupicEdlmanersug 3

sini3

cos2

3i1q

nigtY 3

sini3

cos2

3i1a

23i1

az 121



tamrUbmn n1n1n )3sini3(cosqzz

tamrUbmndWmreK)an 3nsini

3ncoszn

. K> TajrktYTUeTAnsIVt na

eKman n1nn a23i1az

eK)an )1(a23i)

2aa(z nn1nn

eday )2(3nsin.i

3ncoszn

tamTMnak;TMng eK)an )2(&)1(

3nsina

23

n

dUcenH 3nsin

32an

. ehIy ( CasIVtxYbEdlmanxYb )an 6

3

2p .



lMhat;TI42 eK[sIVtncMnYnkMupic kMNt;eday )z( n

2i32z1

nig 2

i32z2

i3z n1n

Edl . ...,3,2,1n k> tag . b

Complex Numbers

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