complex 123

8/7/2019 Complex 123

1/39

8/7/2019 Complex 123

2/39

GkcUlrYmRtYtBinitbeckeTselak lwm qunelak Esn BisidelakRs Tuy rINaelak Tit emgelak RBwm sunitelak pl bnqayGkRtYtBinitGkraviruTelak lwm miKsirkarIkMBT&rkBaa lI KuNakaGkniBn nig eroberogelak lwm pln nig elak Esn Bisid

8/7/2019 Complex 123

3/39

sikSaKNitviTaedayxng cMnYnkMupic

eroberogeday lwm pln - TMBr1 -

cMnYnkMupic

1-niymnyk> cMnYnnim it

plKuNncMnYnBitc xusBIsUnnwg i ehAfacMnYnnim it .

i ehAfaktanim

itEdl 1i2 b 1i .

]TahrN_ ...,i3,3

i2,i5,i2 ehAfacMnYnnim it .

x> niymnycMnYnkMupiccMnYnkMupicCacMnYnEdlmanrag b.iaz Edla nig b CacMnYnBit .

eKtagsMNMuncMnYnkMupiceday .a ehAfaEpkBitn b.iaz EdlkMNt;tageday a)zRe( .b ehAfaEpknim itn 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( .

8/7/2019 Complex 123

4/39



2-RbmaNviFIelIcMnYnkMupick> viFIbUkcMnYnkMupic

]bmafaeKmanBIrcMnYnkMupic b.iaz1 nig d.icz2 Edl d,c,b,a CacMnYnBit .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 d,c,b,a CacMnYnBit .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 .

8/7/2019 Complex 123

5/39



K> viFIKuNcMnYnkMupic]bmafaeKmanBIrcMnYnkMupic b.iaz1 nig d.icz2

Edl d,c,b,a CacMnYnBit .eK)an )d.ic)(b.ia(zz 21

)bcad(i)bdac(

bdibciadac

bdiibciadac2

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 d,c,b,a CacMnYnBit .

eK)an222

2

2

1

dic

bdiibciadac

)idc)(idc(

)idc)(iba(

idc

iba

z

z

2222dc

)adbc(i)bdac(

dc

bdibciadac

dUcenH

22222

1

dc

adbc.i

dc

bdac

z

z

.

8/7/2019 Complex 123

6/39



]TahrN_ eK[cMnYnkMupic i51z1 nig i1z2 . KNna2

1

z

z

eK)an 2i46

11

5i5i1

)i1)(i1(

)i1)(i51(

i1

i51

z

z

2

1

dUcenH i23

z

z

2

1 .g> sVyKuNn ieKmansVyKuNn 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 ni esInwgcMnYndEdlKW i,1,i nig 1 RKb; INn .

8/7/2019 Complex 123

7/39



c> sVyKuNncMnYnkMupiceKmansVyKuNncMnYnkMupicdUcxageRkam

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 32 )i2(,)i31( nig 4)i21( .eK)an

i24716i3224i81)i21(

i112i6i128)i2(

i689i61)i31(

4

3

2

q> kMupices IKa

]bmafa ibaz1 nig d.icz2 Edl d,c,b,a CacMnYnBit . db cazz 21

8/7/2019 Complex 123

8/39



dUcenHcMnYnkMupicBIres IKaluHRtaEtEpkBites IKa nig EpknimitesIKa .]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 b,a CacMnYnBitedIm,IKNnabskaern z eKRtUvGnuvtn_dUcxageRkam tag y.ixw Cabskaern b.iaz y,x CacMnYnBiteK)an zw

2 iba)xy2(i)yx(

b.ia)y.ix(

22

2

eKTaj)an

bxy2 ayx22

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

8/7/2019 Complex 123

9/39



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

i2021)xy2(i)yx(

i2021)y.ix(

22

2

eKTaj)an

20xy2 21yx22

bnab;BIedaHRsayRbBnsmIkarenHeK)anKUcemIy 2y,5x b 2y,5x

dUcenH i25w;i25w 21 .]TahrN_2 KNnabskaerncMnYnkMupic i2021z tag y.ixw Cabskaern i68z y,x CacMnYnBiteK)an zw2

i68)xy2(i)yx(

i68)y.ix(

22

2

eKTaj)an

6xy2

8yx 22

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

8/7/2019 Complex 123

10/39



3-cMnYnkMupicqas;k> niymny

cMnYnkMupicqas;ncMnYnkMupic IRb;a,b.iaz KWCacMnYnkuMpicEdlkMNt;tageday b.iaz .]TahrN_ cMnYnkMupicqas;n i34z KW i34z .x> lkN

!> 2121zz)zz(

@> 2121 z.z)z.z( #>

2

1z

2

1

zz

z

sRmaybBaak;

tag ibaz1 nig idcz2 Edl d,c,b,a CacMnynBit .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

8/7/2019 Complex 123

11/39



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

2222

2

1

dc

adbc.i

dc

bdac

)idc)(idc(

)idc)(iba(

idc

iba

z

z

eK)an2222

2

1

dc

adbc.i

dc

bdac

z

z

ehIy2222

2

1

dc

adbc.i

dc

bdac

)idc)(idc(

)idc)(iba(

idc

iba

z

z

dUcenH 21z

2

1

zz

z

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

2

zza

ehIy ib2ibaibazz enaH

i2zzb

dUcenH2

zz)zRe(

nigi2

zz)zIm(

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

8/7/2019 Complex 123

12/39



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

a2

bz;

a2

bz 21

-ebI 0 smIkarmanbsDbCacMnYnBitKW

a2bzz 21

-ebI 0 smIkarmanbsBIepSgKaCacMnYnkMupicqas;KaKW a2

||ibz;

a2

||ibz 21

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

2

1.i

2

3

4

i26z;

2

1.i

2

3

4

i26z 21

dUcenHsmIkarmanbdBIrepSgKaCacMnYnkMupicqas;KaKW

2

1.i2

3z;2

1.i2

3z 21 .

8/7/2019 Complex 123

13/39



5-cMnYnkMupickgbg;k> kartagcMnYnkMupickgbg;

kgtRmuyGrtUNrem )xoy( eKGactagcMnYnkMupic IRb,a;b.iaz edaycMNucM mYymankUGredaen )b,a( .eKfa M CacMnucrUbPaBncMnYnkMupic ibaz ehIyz ehAfaGaPicncMnuc )b,a(M EdleKkMNt;sresr )z(M .

dUcKaEdr eKkGactagcMnYnkMupicIRb,a;b.iaz

edayviucTr)b,a(OMu .

eKfau CavIucTrrUbPaBncMnYnkMupic ibaz ehIyz ehAfaGaPicn

viucTr

u EdleKkMNt;sresr )z(u

.

M

y

xo a

b

8/7/2019 Complex 123

14/39



x> viucTrrUbPaBnplbUkcMnYnkMupickgbg;kMupickgtRmuyGrtUNrem )xoy( ]bmafaeKmancMnYnkMupicBIr 1z nig 2z

ehIytag 1M nig 2M CarUbPaBn 1z nig 2z .

eK)an

1OM nig

2OM CaviucTrrUbPaBn 1z nig 2z .

eKman OMOMOMzz 2121

dUcenHrUbPaBn 21 zz KWCaviucTrGgt;RTUgnRbelLRkam 21MMOM .

M

2M

1M

x

y

O

8/7/2019 Complex 123

15/39



K> viucTrrUbPaBnpldkcMnYnkMupickgbg;kMupickgtRmuyGrtUNrem )xoy( ]bmafaeKmancMnYnkMupicBIr 1z nig 2z

ehIytag 1M nig 2M CarUbPaBn 1z nig 2z .

eK)an

1OM nig

2OM CaviucTrrUbPaBn 1z nig 2z .

eK)an OM'OMOM)z(zzz 212121

dUcen rUbPaBn 21 zz KWCaviucTrGgt;RTUgnRbelLRkam 21 'MMOM .

y

'x x

'y

1M

'M2

2M

M

O

8/7/2019 Complex 123

16/39



X> viucTrrUbPaBnplKuNcMnYnBit nwg cMnYnkMupickgbg;kMupic

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

rUbPaBnz.

KW

'OM

Edl

OM'OM.6-m:UDlncMnYnkMupic

k> niymnykgtRmuyGrtUNrem )xoy( eKyk )b,a(M CarUbPaBn ibaz .rgVas; OM ehAfam:UDln ibaz .eKkMNt;tagm:UDln ibaz eday |z| b r EdlGacKNna)antamrUbmn 22 baOMr|z| .

M

'M

O x

y

8/7/2019 Complex 123

17/39



kgRtIekaNEkg OMP eKman 222 OPMPOM eday bMP,aOP eK)an 222 baOM b 22 baOM .dUcenH 22 baOM|z| .

x> lkN!> |z||z| @> 2|z|z.z #> |z|.|z||z.z| 2121 $>

|z|

|z|

z

z

2

1

2

1 %> nn |z||z|

y

xo a

bM

P

|z|

8/7/2019 Complex 123

18/39



K> vismPaBRtIekaNRKb;cMnYnkMupic 1z nig 2z eKman |z||z||zz| 2121

sRmay tag iyxz1 nig ivuz2 eKman )vy(i)ux(zz 21 eK)an 2221 )vy()ux(|zz| nig

2222

21 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 0|zz||z||z| 221221 luHRtaEt

0yuxyuv2vx

0vyxyuv2uxyvyuvxux

0)yvxu()vu)(yx(

2222

222222222222

22222

0)uyxv(

2 BitdUcenH |z||z||zz| 2121 .

8/7/2019 Complex 123

19/39



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

mMuEdlpMeday )OM,Ox(

ehAfaGaKuym:g;n b.iaz .eKtag b )z(Arg CaGaKuym:g;n b.iaz .kgRtIekaNEkg OMP eKman

2222 baOMr b 22 bar RTwsIbTBItaKrr

a

OM

OPcos nig

r

b

OM

MPsin .

edIm,IrkGaKuym:g;n b.iaz eKedaHRsaysmIkar r

acos nig

r

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

P

y

xo a

b M

8/7/2019 Complex 123

20/39



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

23

432

racos nig

21

42

rbsin

dUcenHGaKuym:g;n z KW Zk;k26

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

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

1

r

acos nig

2

3

r

bsin


2)z(Arg

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

22 2

2

r

acos nig

2

2

r

bsin


)z(Arg

8/7/2019 Complex 123

21/39



8-TRmg;RtIekaNmaRtncMnYnkMupiccMnYnkMupic b.iaz ehAfaTRmg;BICKNit . eKGacsresrTRmg;f ImYyeTotdUcxageRkam eKman 22 bar ehAfam:Dln b.iaz

r

acos nig

r

bsin Edl ehAfaGaKuym:g;n z .

eK)an )sin.i(cosr)r

b.i

r

a(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)

2

3.i

2

1(2z

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

6sin.i

6cos(4)

2

1.i

2

3(4z

.

)6

5sini

6

5(cos4z

)]

6

sin(.i)

6

[cos(4z

8/7/2019 Complex 123

22/39



9-RbmaNviFIelIcMnYnkMupictamTRmg;RtIekaNmaRtk> 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 )15

2sini

15

2(cos3z2

KNna 21 z.z

eK)an )]15

2

5sin(i)

15

2

5[cos(6zz 21

)

3sin.i

3(cos6)

15

5sini

15

5(cos6

dUcenH )3

sin.i3

(cos6zz 21 .

8/7/2019 Complex 123

23/39



x> plKuNEckcMnYnkMupictamTRmg;RtIekaNmaRtRTwsIbT

]bmafaeKmancMnYnkMupic )sin.i(cosrz 11 nig )sini(cosrz 22 Edl 0r,0r 21 eK)an )]sin(i)[cos(

r

r

z

z

2

1

2

1 .sRmaybBaak;

eK)an

sinicossinicos.

rr

zz

2

1

2

1

)sin(i)cos(r

r

sincos

)cossincos(sini)sinsincos(cos.

r

r

)sini)(cossini(cos

)sini)(cossin.i(cos.

r

r

2

1

222

1

2

1

dUcenH )]sin(i)[cos(r

r

z

z

2

1

2

1 .K> sVyKuNTI n ncMnYnkMupictamTRmg;RtIekaNmaRt

RTwsIbT RKb;cMnYnBit nig 0r eK)an )nsin.in(cosr)]sin.i(cosr[z nnn

Edln CacMnYnKt;rWLaTIhV .

8/7/2019 Complex 123

24/39



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 )nsin.in(cosrz nn BiteK)an )]nsin(i)n[cos(r.rz.zz nn1n

])1nsin(i)1n[cos(r 1n BitdUcenH )nsin.in(cosrz nn .X> rUbmndWmreKman )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 .

8/7/2019 Complex 123

25/39



]TahrN_ eK[cMnYnkMupic9

4sin.i

9

4cosz

cUrsresr 3

2

z1

z

w CaragRtIekaNmaRt .tamrUbmndWmreKman

9

8sini

9

8cos)

9

4sini

9

4(cosz 22

nig

3

4sini

3

4cos)

9

4sini

9

4(cosz 33

eK)an3

4sin.i

3

4cos1

9

8sin.i

9

8cos

w

911sini

911cos)

92sini

92(cos

)]3

2

9

8sin(i)

3

2

9

8[cos(

3

2sini

3

2cos

9

8sin.i

9

8cos

.

3

2cos2

1

3

2cos

3

2sini2

3

2cos2

9

8sin.i

9

8cos

2

dUcenH9

11sin.i

9

11cosw

.

8/7/2019 Complex 123

26/39



10-bMElgvilCMuvijKl;tRmuynbg;kMupiceKmancMnYnkMupic sin.icosw .ebI )'z('M CarUbPaBn )z(M tambEmgvilpitO nig mMu enaHeK)an z.)sini(cosz.w'z .

]TahrN_ 1 enAkgbg;kMupic)xoy(

eK[M

CacMnucrUbPaBni3z

.cUrkMNt; 'z edaydwgfa )'z('M CarUbPaBn M tambEmgvilpitO

nigmMu12

.

x

y

O

M

'M

8/7/2019 Complex 123

27/39



ebI )'z('M CarUbPaBn )z(M tambEmgvilpitO nigmMu12

enaHeK)anz)

12sini

12(cos'z

eday )6

sin.i6

(cos2)2

1.i

2

3(2i3z

eK)an )]

612sin(i)

612[cos(2'z

dUcenH )

4

sini

4

(cos2'z .

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

nigmMu3

2 .ebI )'z('M CarUbPaBn )z(M tambEmgvilpitO nigmM

u 3

2 enaHeK)anz)

3

2sini

3

2(cos'z

eday )]

3sin(.i)

3[cos(2)

2

3.i

2

1(23i1z

eK)an )]

33

2sin(i)

33

2[cos(2'z

dUcenH )

3sini

3(cos2'z .

8/7/2019 Complex 123

28/39



11-bsTI n ncMnYnkMupictamTRmg;RtIekaNmaRt]bmafaeKmancMnYnkMupic )sini(cosrz

tag )sini(cosw CabsTI n n z enaH zwn eK)an )sini(cosr)nsinin(cosn eKTaj

sin)nsin(

cos)ncos(

rn

naM[

n

k2

rn

dUcenHbsTI n n z kMNt;eday

n k2sin.in k2cosrw nk

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

4sini

4(cos8)

2

2i

2

2(824.i24z

eKTaj

4,8r

. tamrUbmnbsTI # n z kMNt;eday 2,1,0k,

12

k8sini

12

k8cos2wk

.dUcenH )

4

3sini4

3(cos2w;)12

sini12

(cos2w 10

nig )12

17sini

12

17(cos2w2

.

8/7/2019 Complex 123

29/39



12-TRmg;Giucs,:Ng;EslncMnYnkMupick> rUbmnGWEl(Euler's formula)

cMeBaHRKb;cMnYnBitx eK)an xsinixcoseix Edl ...71828.2e CaeKalelakarIteEB .rUbmnGWElenHenAEtBitcMeBaH x CacMnYnkMupickeday .sRmaybBaak;

-RsaybBaak;edayeRbIedrIev tagGnuKmn_ fGacCaGnuKmn_kMupicnGefrx kMNt;eday

ixe

xsinixcos)x(f

eK)an

ix2

ixix

e

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

0e

0

e

xsinxcosixcosixsin

e

)xsinixcosixcosixsin(e

ixix

ix2

2ix

naM[ )x(f CaGnuKmn_efrRKb;x .

eK)an 1e 0sini0cos)0(f)x(f 0 b 1e xsinixcos)x(f ix dUcenH xsinixcoseix .

8/7/2019 Complex 123

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-RsaybBaak;edayeRbIsmIkarDIepr:g;EsllMdab;TImYy eKtagGnuKmn_ xsin.ixcos)x(g eKman xcosixsin)x('g KuNGgTaMgBIrnwg i eK)an csoxxsini)x('g.i eK)an 0)x('ig)x(g b 0)x(g.i)x('g

CasmIkarDIepr:g;EsllMdab;I .

eK)ancemIyTUeTAnsmIkarenHKWix

ek)x(g ebI 0x enaH k)0(g Et 10sin.i0cos)0(g

enaH 1k ehIyeK)an ixe)x(g .dUcenH xsinixcoseix .

-RsaybBaak;edayeRbIsmIkarDIepr:g;EsllMdab;TIBIr

eRCIserIsGnuKmn_ ixe)x(h eKman ixe.i)x('h nig ixix2 eei)x(''h eK)an 0ee)x(h)x(''h ixix CasmIkarDIepr:g;EsllIenEGGum:UEsnlMdab;TIBIr .

smIkarDIepr:g;EsllIenEGlMdab;BIrenHmancemIylIenEGkraClIenEGcMnYnBIrEdlepgpat;vaKW xcos)x(h1 nig xsin)x(h2 . bnSMlIenEGncemIycMeBaHsmIkarDIepr:g;EslGUm:UEsn kCacemIymYypgEdr .

8/7/2019 Complex 123

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dUcenHcemIyTUeTAnsmIkarKW xsinBxcosA)x(h EdlA nig B CaBIrcMnYnefrEdlGacrk)antam 1eA)0(h 0i nig iieB)0('h 0i eRBaH xcosBxsinA)x('h .ehtuenHeK)an xsinixcos)x(h .dUcenH xsinixcoseix .

-RsaybBaak;edayeRbIesrItlr

rUbmnesrItlrcMeBaHGnuKmn_bIxcos,e

x

nigxsin

KW

....)!1n2(

x

)1(....!7

x

!5

x

!3

x

xxsin

......)!n2(

x)1(.....

!6

x

!4

x

!2

x1xcos

.....!n

x.....

!3

x

!2

x

!1

x1e

1n2n

753

n2n

642

n32x

edayCMnYsx edayix kgesTaMgbIenHeKTaj)an xsinixcoseix .x> TRmg;Giucs,:Ng;Esl

RKb;cMnYnkMupic b.iaz Edl b,a CacMnYnBitGacsresrCaTRmg;mYyfIeTotKW

ierz Edl rb

sin,r

a

cos,bar22 .

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

8/7/2019 Complex 123

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K> TMnak;TMngCamYyGnuKmn_RtIekaNmaRt cMeBaHRKb;cMnYnBitx eKman )1(xsinixcoseix CMnYsx eday x eK)an )2(xsinixcose ix bUksmIkar )1( nig )2( eKTaj)an xcos2ee ixix eKTaj

2

eexcos

ixix .dksmIkar )1( nig )2( eKTaj)an xsini2ee ixix eKTaj

i2

eexsin

ixix .dUcenH

i2

eexsin;

2

eexcos

ixixixix . rUbmnenHBitpgEdrcMeBaH x CacMnYnkMupic

X> TMnak;TMngCamYyGnuKmn_GIEBbUlikeKman

i2

eexsin;

2

eexcos

ixixixix CMnYsx eday x.i eK)an

xcosh2

ee)ixcos(

xx nig xsinhi2

eei)ixsin(

xx m:ageToteKman

2

eexcosh

xx nig2

eexsinh

xx eK)an xsini

2

ee)ixsinh(,xcos

2

ee)ixcosh(

ixixixix .

8/7/2019 Complex 123

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]TahrN_1 sresr 3i22z CaTRmg;Giucs,:Ng;Esl ?eKman 4)32(2r 22 eK)an 3

i

e4)3

sini3

(cos4)2

3i

2

1(4z

.]TahrN_2 sresr 4

ie1z

CaragGics,:Ng;Esl ?

eKman )ee(ee1z8i

8i

8i

4i

eday 2

ee

8cos

8i

8i

dUcenH 8i

e.8

cos2z

.]TahrN_3 KNna ii ?

eKman2i

e2

sin.i2

cosi

eK)an 2i

2

i

i eei

.

]TahrN_4 edaHRsaysmIkar 2xcos kgsMNMukMupic ?tamrUbmnGWEleKman 2

ee

xcos

ixix eK)an 2

2

ee ixix b 01e4e ixix2 tag ixet

8/7/2019 Complex 123

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eK)ansmIkar 01t4t2 314' eKTajbs 32t;32t 21

cMeBaH 32t eK)an 32eix enaH )32ln(ix b )32ln(ix .cMeBaH 32t eK)an 32eix enaH )32ln(ix b )32ln(ix .]TahrN_5 edaHRsaysmIkar

3xsin kgsMNMukMupic ?

tamrUbmnGWEleKmani2

eexsin

ixix eK)an 3

i2

ee ixix b 01e6e ixix2 tag ixet eK)ansmIkar 1019',01t6t2 manbs 103t,103t 21 .-cMeBaH 103t eK)an 103eix enaH )103ln(ix -cMeBaH )103(103t eK)an )103ln(iiix ee)103()103(e eKTaj )103ln(iix b )103ln(ix .dUcenH )103ln(ix,)103ln(ix .

8/7/2019 Complex 123

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13-RbmaNviFIcMnYnkMupictamTRmg;Gics,:Ng;Eslk> RbmaNviFIKuNcMnYnkMupictamTRmg;Gics,:Ng;Esl

eKmancMnYnkMupic ierz nig iew Edl 0;0r ehIy , CacMnYnBit .eK)an )(ie.rw.z .x> RbmaNviFIEckcMnYnkMupictamTRmg;Gics,:Ng;Esl

eKmancMnYnkMupic

i

erz nig

i

ew Edl 0;0r ehIy , CacMnYnBit .eK)an )(ier

w

z .K> sVyKuNcMnYnkMupictamTRmg;Gics,:Ng;Esl

eKmancMnYnkMupic

i

erz cMeBaHRKb;cMnYnKt;rWLaTIhV n eK)an innnin ererz .]TahrN_ eK[cMnYnkMupic 6

i

e2z

nig 12ie3w .KNna w.z nig

w

z

eK)an 4i)

126(i

e6e6w.z nig 12i)126(i e

3

2e

3

2

w

z

8/7/2019 Complex 123

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14-Gnuvtn_cMnYnkMupickgRtIekaNmaRtk> rUbmnmMuDb

cMeBaHRKb;cMnYnBitx eKmani2

eexsin;

2

eexcos

ixixixix eK)an

4

2ee

2

eexcos

ix2ix22

ixix2

2

1x2cos

2

1

2

1

2

ee.

2

1xcos

x2iix22

eKTaj 1xcos2x2cos 2 .ehIy

2

ee.

i2

ee.2

i2

eex2sin

ixixixixix2ix2 dUcenH xcosxsin2x2sin .x> rUbmnmMuRTIbeKman

3ixix

3

2

eexcos

4xcos3x3cos

8xcos6x3cos2

8

)ee(3ee ixixix3ix3

eKTaj)an xcos3xcos4x3cos 3 .

8/7/2019 Complex 123

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ehIy3

ixix3

i2

eexsin

4

xsin3x3sin

i8

xsini6x3sini2

i8

)ee(3)ee( ixixix3ix3

eKTaj)an xsin4xsin3x3sin 3 .K> rUbmnplbUk nig pldknmMuBIreKman ibia)ba(i e.ee RKb;cMnYnBita 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( .m:ageTot ibia)ba(i e.ee RKb;cMnYnBita nig b eday )bsinib)(cosasinia(cose.e ibia

)acosbsinbcosa(sini)bsinasinbcosa(coseeibia

ehIy )basin(i)bacos(e )ba(i

8/7/2019 Complex 123

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

2

ee.

2

eebcosacos

ibibiaia

)bacos()bacos(2

1

2ee

2ee

21

4

eeee

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


dUcenH )bacos()bacos(2

1bcosacos

i2ee.

i2eebsinasin

ibibiaia

)bacos()bacos(2

1

2

ee

2

ee

2

1

i4

eeee


2


dUcenH )bacos()bacos(2

1bsinasin

8/7/2019 Complex 123

39/39


15-Gnuvtn_cMnYnkMupickgsIVtcMnYnBitcMeBaHsIVtncMnYnBit )a( n Edlepgpat;TMnak;TMngkMeNIn

0qapaa n1n2n Edl q,p CacMnYnBitsmIkarsmal;rbs;sIVtenHKW 0qpzz2 kgkrNI 0q4p2 smIkarmanbsBIrCacMnYnkMupicqas;KaKW 1z nig 2z . kgkrNIenHedIm,IKNna na eKRtUvGnuvtn_dUcteTA

tagsIVtCMnYy n11nnazaz rYcRsayfa )z( n CasIVtFrNImaRtncMnYnkMupicmYy.

KNna nz rYcTajrk na .]TahrN_ eKmansIVtncMnYnBit )a( n kMNt;eday

1a,0a 10 nig n1n2n aaa cUrKNna na CaGnuKmn_n n

16-Gnuvtn_cMnYnkMupickgFrNImaRtk> cmayrvagBIrcMnucx> cMnucEckGgt;tampleFobmYy

K> pleFobRCug nig mMunRtIekaNkgbg;X> sMNMucMNuckgbg;kMupic

complex 123

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