KN³kmμkarniBnænigeroberog elak
lwm plÁún nig elak Esn Bisidæ KN³kmμkarRtYtBinitü
elak elak elak elak
lwm qun Titü em:g RBwm sunitü elak nn; suxNa
elak Gwug sMNag GñkRsI Tuy rINa elak pl b‘unqay
GñkRtYtBinitüGkçraviruTæ elak lwm miKÁsir
karIkMuBüÚTr½
GñkrcnaRkb
kBaØa lI KuNÑaka
elak lwm plÁún
-i-
GarmÖfa esovePA sikSaKNitviTüaedayxøÜnÉg Epñk cMnYnkMupøic EdlGñksikSakMBugkan;sikSaenAkñúgédenH eyIgxJMú)anxitxMRsavRCavcgRkgeLIg kñúgeKalbMNgTukCaÉksa sikSabEnßmelIemeroncMnYnkMupøicedayxøÜnÉg . enAkñúgesovePAenHrYmmanbICMBUkKW CMBkU TI1 emeronsegçbP©ab;CamYy]TahrN_KMrU CMBUkTI2 lMhat;eRCIserIsnigdMeNaHRsay nig CMBUkTI3 CalMhat;Gnuvtþn_ . esovePAenHminl¥hYseK hYsÉgenaHeT kMhusedayGectnaR)akdCaman GaRs½yehtuenH eyIgxJMúCaGñkniBnæ nig eroberog rgcaMCanic©nUvmtiriHKn;BIsMNak; GñksikSakñúgRKb;mCÄdæanedaykþIrIkray edIm,IEklMGesovePAenH[kan;Etman suRkitüPaBbEnßmeTot . CaTIbBa©b;eyIgxJMúsUmCUnBrcMeBaHGñksikSaTaMgGs;CYbEtsuPmgÁl suxPaBl¥ nigTTYlC½yCMn³kñúgkarsikSa nig muxrbrkargar RKb;eBlevla .
át´dMbgéf¶TI 28 mIna 2011
GñkniBnæ lwm plþún Tel : 017 768 246 www.mathtoday.wordpres.com - ii -
lwm plþún nig Esn Bisidæ
rkßasiT§iRKb´y¨ag - iii -
matikarerOg CMBUkTI1
1-niymn½y 2-RbmaNviFIelIcMnYnkMupøic 3-cMnYnkMupøicqøas; 4-Gnuvtþn_cMnYnkMupøickñúgdMeNaHRsaysmIkardWeRkTIBIr 5-cMnYnkMupøickñúgbøg; 6-m:UDúléncMnYnkMupøic 7-GaKuym:g;éncMnYnkMupøic 8-TRmg;RtIekaNmaRténcMnYnkMupøic 9-RbmaNviFIelIcMnYnkMupøictamTRmg;RtIekaNmaRt 10-bMElgvilCMuvijKl;tRmuyénbøg;kMupøic 11-b¤sTI n éncMnYnkMupøictamTRmg;RtIekaNmaRt 12-TRmg;Giucs,:ÚNg;EsüléncMnYnkMupøic 13-RbmaNviFIcMnYnkMupøictamTRmg;Gics,:ÚNg;Esül 14-Gnuvtþn_cMnYnkMupøickñúgRtIekaNmaRt 15-Gnuvtþn_cMnYnkMupøickñúgsIVútcMnYnBit 16-Gnuvtþn_cMnYnkMupøickñúgFrNImaRt 17-bMElgcMnuckñúgbøg;kMupøic 18-)arIsg;énRbBn½æcMNuckñúgbøg;kMupøic - iv -
TMB&r 001 002 008 010 011 014 017 019 020 024 026 027 033 034 037 039 051 054
CMBUkTI2 kRmglMhat;eRCIserIs EpñkdMeNaHRsay
055 065
CMBUkTI3 lMhat;Gnuvtþn_
130
-v-
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
CMBUkTI1
cMnYnkMupøic 1-niymn½y k> cMnYnnimμit plKuNéncMnYnBit c xusBIsUnünwg i ehAfacMnYnnimμit . i ehAfaÉktanimμitEdl i 2 1 b¤ i 1 . ]TahrN_ ³ 2i , 5i , 23i , 3i , ... ehAfacMnYnnimμit . x> niymn½ycMnYnkMupøic cMnYnkMupiøcCacMnYnEdlmanrag z a i.b Edl a nig b CacMnYnBit . eKtagsMNMuéncMnYnkMupøiceday ¢ . a ehAfaEpñkBitén z a i .b EdlkMNt;tageday Re( z ) a . b ehAfaEpñknimμitén z a i .b EdlkMNt;tageday Im( z ) b . ]TahrN_1 ³ 1 2i , 3 2i , 4 3i , 1 4i , 5i , 7i ehAfacMnYnkMupøic . ]TahrN_2³ rkEpñkBit nig Epñknimμitén z 3 2i ? EpñkBit nig Epñknimμitén z 3 2i KW Re(Z) 3 ; Im( z ) 2 . eroberogeday lwm plÁún
- TMBr½1 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
2-RbmaNviFIelIcMnYnkMupøic k> viFIbUkcMnYnkMupøic ]bmafaeKmanBIrcMnYnkMupøic z1 a i.b nig z 2 c i.d Edl a , b , c , d CacMnYnBit . eK)an z1 z 2 (a i.b) (c id) (a c) i(b d) dUcenH z1 z 2 (a c) i.(b d) . ]TahrN_ ³ eK[cMnYnkMupøic z1 3 2i nig z 2 7 5i . KNna z1 z 2 eK)an z1 z 2 ( 3 2i ) (7 5i ) (3 7) (2i 5i ) dUcenH z1 z 2 4 3i . x> viFIdkcMnYnkMupøic ]bmafaeKmanBIrcMnYnkMupøic z1 a i.b nig z 2 c i.d Edl a , b , c , d CacMnYnBit . eK)an z1 z 2 (a i.b) (c id) (a c) i(b d) dUcenH z1 z 2 (a c) i.(b d) . ]TahrN_ ³ eK[cMnYnkMupøic z1 3 2i nig z 2 7 5i . KNna z1 z 2 eK)an z1 z 2 ( 3 2i ) (7 5i ) (3 7) (2i 5i ) dUcenH z1 z 2 10 7i . eroberogeday lwm plÁún
- TMBr½2 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
K> viFIKuNcMnYnkMupøic ]bmafaeKmanBIrcMnYnkMupøic z1 a i.b nig z 2 c i.d Edl a , b , c , d CacMnYnBit . eK)an z1 z 2 (a i.b)(c i.d) ac iad ibc i 2bd ac iad ibc bd (ac bd ) i(ad bc )
dUcenH z1 z 2 (ac bd) i(ad bc) . ]TahrN_ ³ eK[cMnYnkMupøic z1 2 i nig z 2 1 3i . KNna z1 z 2 eK)an z1 .z 2 (2 i )(1 3i ) 2 6i i 3i 2 5 5i dUcenH z1 .z 2 5 5i . X> viFIEckcMnYnkMupøic ]bmafaeKmanBIrcMnYnkMupøic z1 a i.b nig z 2 c i.d Edl a , b , c , d CacMnYnBit . z1 a ib (a ib )(c id ) ac iad ibc i 2bd eK)an z c id (c id)(c id) c2 i 2d 2 2 ac iad ibc bd (ac bd ) i(bc ad ) 2 2 c d c2 d 2 dUcenH zz1 ac2 bd2 i. bc2 ad2 . c d c d 2
eroberogeday lwm plÁún
- TMBr½3 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
]TahrN_ ³ eK[cMnYnkMupøic z1 1 5i nig z 2 1 i . KNna zz1 eK)an zz1 115ii ((115ii)()(11ii)) 1 i151i 5 6 2 4i
2
2
dUcenH zz1 3 2i . 2
g> sV½yKuNén i eKmansV½yKuNén i dUcxageRkam ³ i1 i i 2 1 i 3 i 2 .i i i 4 (i 2 )2 ( 1)2 1 i 5 i 4 .i i i 6 i 5 .i i .i i 2 1 i 7 i 6 .i i i 8 i 7 .i i .i i 2 1 CaTUeTA i4n 1 , i4n 1 i , i4n 2 1 , i4n 3 i
RKb; n IN dUcenHcMnYn in esμInwgcMnYndEdl²KW i , 1 , i nig 1 RKb; n IN . eroberogeday lwm plÁún
- TMBr½4 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
c> sV½yKuNéncMnYnkMupøic eKmansV½yKuNéncMnYnkMupøicdUcxageRkam ³ (a ib )2 a 2 b 2 i .2ab (a ib )3 (a 3 3ab 2 ) i( 3a 2b b 3 ) (a ib )4 (a4 6a 2b 2 b4 ) i(4a 3b 4ab 3 ) n
(a ib ) C(n, k ) an k bk .i k n
k 0
Edl c(n, k ) k! (nn! k )! . ]TahrN_ ³ KNna (1 3i )2 , (2 i )3 nig (1 2i )4 . eK)an ³ (1 3i )2 1 6i 9 8 6i ( 2 i )3 8 12i 6 i 2 11i (1 2i )4 1 8i 24 32i 16 7 24i
q> kMupøicesμIKña ]bmafa z1 a ib nig z 2 c i.d Edl a, b, c, d CacMnYnBit . ac z1 z 2 bd
eroberogeday lwm plÁún
- TMBr½5 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
dUcenHcMnYnkMupøicBIresμIKñaluHRtaEtEpñkBitesμIKña nig EpñknimμitesμIKña . ]TahrN_ ³ eK[ z1 2 3 4i nig z 2 9 8i cUrkMNt;BIrcMnYnBit nig edIm,I[ z1 z 2 ? kMNt; nig ³
2 3 9 2 3 4i 9 8i 4 8 eKTaj)an 2 , 3 .
C> KNnab¤skaeréncMnYnkMupøic ]bmafaeKmancMnYnkMupøic z a i.b Edl a , b CacMnYnBit edIm,IKNnab¤skaerén z eKRtUvGnuvtþn_dUcxageRkam ³ tag w x i.y Cab¤skaerén z a i.b ¬ x , y CacMnYnBit¦ eK)an w 2 z ( x i .y )2 a i .b ( x 2 y 2 ) i( 2xy ) a ib x2 y 2 a eKTaj)an 2xy b
edaHRsayRbBn½æsmIkarenHeK)anKUcemøIy (x, y ) {(1 , 1 ); (2 , 2 )} dUcenH w1 1 i1 ; w 2 2 i 2 .
eroberogeday lwm plÁún
- TMBr½6 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
]TahrN_1 ³ KNnab¤skaeréncMnYnkMupøic z 21 20i tag w x i.y Cab¤skaerén z 21 20i ¬ x , y CacMnYnBit¦ eK)an w 2 z ( x i .y )2 21 20i ( x 2 y 2 ) i( 2xy ) 21 20i x 2 y 2 21 eKTaj)an 2xy 20
bnÞab;BIedaHRsayRbBn½æsmIkarenHeK)anKUcemøIy ³ x 5 , y 2 b¤ x 5 , y 2 dUcenH w1 5 2i ; w 2 5 2i . ]TahrN_2 ³ KNnab¤skaeréncMnYnkMupøic z 21 20i tag w x i.y Cab¤skaerén z 8 6i ¬ x , y CacMnYnBit¦ eK)an w 2 z ( x i .y )2 8 6i ( x 2 y 2 ) i( 2xy ) 8 6i x 2 y 2 8 eKTaj)an 2xy 6 bnÞab;BIedaHRsayeK)anKUcemøIy ³ x 1 , y 3
dUcenH w1 1 3i
; w 2 1 3i
eroberogeday lwm plÁún
b¤ x 1 , y 3
. - TMBr½7 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
3-cMnYnkMupøicqøas; k> niymn½y cMnYnkMupøicqøas;éncMnYnkMupøic z a i.b , a; b IR KWCacMnYnkuMpøicEdl kMNt;tageday z a i.b . ]TahrN_ ³ cMnYnkMupøicqøas;én z 4 3i KW z 4 3i . x> lkçN³ !> (z1 z 2 ) z1 z 2 @> (z1 .z 2 ) z1 .z 2 z1 z1 #> z z 2 2
sRmaybBa¢ak; tag z1 a ib nig z 2 c id Edl a , b , c , d CacMnynBit . eK)an z1 a i.b nig z 2 c i.d man z1 z 2 (a c) i(b d) enaH z1 z 2 (a c) i(b d) ehIy z1 z 2 a ib c id (a c) i(b d) dUcenH (z1 z 2 ) z1 z 2 . eKman z1z 2 (a ib)(c id) (ac bd) i(ad bc) eK)an z1 .z 2 (ac bd ) i(ad bc) ehIy z1 .z 2 (a ib)(c id) (ac id) i(ad bc) eroberogeday lwm plÁún
- TMBr½8 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
dUcenH (z1 .z 2 ) z1 .z 2 . ib (a ib )(c id ) ac bd bc ad i . 2 eKman zz1 ac id 2 2 2 (c id )(c id ) c d
2
c d
eK)an zz1 ac2 bd2 i. bc2 ad2 ehIy
c d 2 c d z1 a ib (a ib )(c id ) ac bd bc ad 2 i . z 2 c id (c id )(c id ) c d 2 c2 d 2
z1 z 1 dUcenH z z 2 2
.
K> kenSamEpñkBit nig EpñknimμitCaGnuKmn_én z nig z ]bmafaeKman z a ib enaH z a ib Edl a; b CacMnYnBit . eK)an z z a ib a ib 2a enaH a z 2 z ehIy z z a ib a ib 2ib enaH b z 2i z
dUcenH Re(z ) z 2 z nig Im( z ) z 2i z . -ebI Re(z ) 0 enaH z z naM[ z CacMnYnnimμit . -ebI Im( z ) 0 enaH z z naM[ z CacMnYnBit .
eroberogeday lwm plÁún
- TMBr½9 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
4-Gnuvtþn_cMnYnkMupøickñúgdMeNaHRsaysmIkardWeRkTIBIr ]bmafaeKmansmIkardWeRkTIBIr az 2 bz c 0 Edl a 0 , a, b , c CacMnYnBit . DIsRKImINg;énsmIkarKW b2 4ac -ebI 0 smIkarmanb¤sBIepSgKñaCacMnYnBitKW ³ z1
b b ; z2 2a 2a
-ebI 0 smIkarmanb¤sDúbCacMnYnBitKW z1 z 2 2ba -ebI 0 smIkarmanb¤sBIepSgKñaCacMnYnkMupøicqøas;KñaKW ³ z1
bi || bi || ; z2 2a 2a
]TahrN_ ³ edaHRsaysmIkar 2z 2 6z 5 0 DIsRKImINg;énsmIkar 36 40 4 4i 2 eKTajb¤s z1 6 4 2i 32 i. 12 ; z 2 6 4 2i 32 i. 12 dUcenHsmIkarmanb¤dBIrepSgKñaCacMnYnkMupøicqøas;KñaKW ³ 1 3 1 3 z1 i . ; z 2 i . . 2 2 2 2
eroberogeday lwm plÁún
- TMBr½10 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
5-cMnYnkMupøickñúgbøg; k> kartagcMnYnkMupøickñúgbøg; kñúgtRmuyGrtUNrem (xoy) eKGactagcMnYnkMupøic z a i.b ; a, b IR edaycMNuc M mYymankUGredaen (a, b) . eKfa M CacMnucrUbPaBéncMnYnkMupøic z a ib ehIy z ehAfaGaPicén cMnuc M(a, b) EdleKkMNt;sresr M(z ) . dUcKñaEdr eKk¾GactagcMnYnkMupøic z a i.b ; a, b IR edayviucTr½
u OM (a, b )
eKfa viucTr½
u
u
.
CavIucTr½rUbPaBéncMnYnkMupøic z a ib ehIy z ehAfaGaPicén
EdleKkMNt;sresr u (z ) . y b
M
o
a
eroberogeday lwm plÁún
x
- TMBr½11 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
x> viucTr½rUbPaBénplbUkcMnYnkMupøickñúgbøg;kMupøic kñúgtRmuyGrtUNrem (xoy) ]bmafaeKmancMnYnkMupøicBIr z1 nig z 2 ehIytag M1 nig M 2 CarUbPaBén z1 nig z 2 .
eK)an OM1 nig
OM 2
CaviucTr½rUbPaBén z1 nig z 2 .
y
M M1
M2
O
x
eKman z1 z 2 OM1 OM 2 OM dUcenHrUbPaBén z1 z 2 KWCaviucTr½Ggát;RTUgénRbelLÚRkam OM1MM 2 .
eroberogeday lwm plÁún
- TMBr½12 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
K> viucTr½rUbPaBénpldkcMnYnkMupøickñúgbøg;kMupøic kñúgtRmuyGrtUNrem (xoy) ]bmafaeKmancMnYnkMupøicBIr z1 nig z 2 ehIytag M1 nig M 2 CarUbPaBén z1 nig z 2 .
eK)an OM1 nig
OM 2
CaviucTr½rUbPaBén z1 nig z 2 . y
M2 M1
x
O
x'
M
M2'
y'
eK)an z1 z 2 z1 ( z 2 ) OM1 OM'2 OM dUcen rUbPaBén z1 z 2 KWCaviucTr½Ggát;RTUgénRbelLÚRkam OM1MM'2 . eroberogeday lwm plÁún
- TMBr½13 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
X> viucTr½rUbPaBénplKuNcMnYnBit nwg cMnYnkMupøickñúgbøg;kMupøic ³ y M'
M
x
O
]bmafa M nig M' CacMnucrUbPaBén z nig z , ( 0)
rUbPaBén .z KW OM' Edl OM' OM . 6-m:UDúléncMnYnkMupøic k> niymn½y kñúgtRmuyGrtUNrem (xoy) eKyk M(a, b) CarUbPaBén z a ib . rgVas; OM ehAfam:UDúlén z a ib . eKkMNt;tagm:UDúlén z a ib eday | z | b¤ r EdlGacKNna)antam rUbmnþ | z | r OM a2 b2 . eroberogeday lwm plÁún
- TMBr½14 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
y M
b
|z| P o
a
x
kñúgRtIekaNEkg OMP eKman OM 2 MP 2 OP 2 eday OP a , MP b eK)an OM 2 a2 b2 b¤ OM a2 b2 . dUcenH | z | OM a2 b2 . x> lkçN³ !> | z || z | @> z.z | z |2 #> | z1 .z 2 || z1 | . | z 2 | $> zz1 || zz1 || 2 n
%> | z
2 n
|| z |
eroberogeday lwm plÁún
- TMBr½15 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
K> vismPaBRtIekaN RKb;cMnYnkMupøic z1 nig z 2 eKman | z1 z 2 | | z1 | | z 2 | sRmay ³ tag z1 x iy nig z 2 u iv eKman z1 z 2 (x u) i(y v) eK)an | z1 z 2 | (x u)2 (y v)2 nig | z1 | | z 2 | x2 y 2 u2 v 2 eday | z1 z 2 |2 x2 y 2 u2 v 2 2(xu yv ) nig | z1 | | z 2 |2 x2 y 2 u2 v 2 2 (x2 y 2 )(u2 v 2 ) eK)an ³
| z1 | | z 2 |2 | z1 z 2 |2 2 (x2 y 2 )(u2 v 2 ) (xu yv) kenSam | z1 | | z 2 |2 | z1 z 2 |2 0 luHRtaEt
( x 2 y 2 )(u 2 v 2 ) ( xu yv )2 0 x 2u 2 x 2 v 2 u 2y 2 v 2 y 2 x 2u 2 2xyuv y 2 v 2 0 x 2 v 2 2xyuv u 2y 2 0
Bit dUcenH | z1 z 2 | | z1 | | z 2 | . ( xv uy )2 0
eroberogeday lwm plÁún
- TMBr½16 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
7-GaKuym:g;éncMnYnkMupøic kñúgtRmuyGrtUNrem (xoy) eKyk M(a, b) CarUbPaBén z a ib . y M
b
o
P a
x
mMuEdlpÁúMeday ( Ox , OM ) ehAfaGaKuym:g;én z a i.b . eKtag b¤ Arg( z ) CaGaKuym:g;én z a i.b . kñúgRtIekaNEkg OMP eKman ³ r 2 OM 2 a 2 b 2 b¤ r a 2 b 2 ¬RTwsþIbTBItaKr½¦ OP a MP b cos nig sin . OM r OM r edIm,IrkGaKuym:g;én z a i.b eKedaHRsaysmIkar ³ a b cos nig sin eK)an Arg( z ) 2k , k Z . r r eroberogeday lwm plÁún
- TMBr½17 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
]TahrN_ 1 ³ rkGaKuym:g;éncMnYnkMupøic z 2 3 2i tamrUbmnþ r | z | a2 b2 (2 3 )2 22 4 a 2 3 3 b 2 1 ni g cos sin r 4 2 r 4 2
dUcenHGaKuym:g;én z KW Arg(z ) 6 2k ; k Z ]TahrN_ 2 ³ rkGaKuym:g;éncMnYnkMupøic z 1 i 3 tamrUbmnþ r | z | a2 b2 a 1 b 3 cos nig sin r 2 r 2
( 1)2 ( 3)2 2
dUcenHGaKuym:g;én z KW Arg(z ) 23 2k ; k Z TahrN_ 3 ³ rkGaKuym:g;éncMnYnkMupøic z 2 i 2 tamrUbmnþ r | z | a2 b2 2 2 2 a 2 b 2 ni g cos sin r 2 r 2
dUcenHGaKuym:g;én z KW Arg(z ) 4 2k ; k Z
eroberogeday lwm plÁún
- TMBr½18 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
8-TRmg;RtIekaNmaRténcMnYnkMupøic cMnYnkMupøic z a i.b ehAfaTRmg;BICKNit . eKGacsresrTRmg;fμImYy eTotdUcxageRkam ³ eKman r a2 b2 ehAfam:Dúlén z a i.b a b cos nig sin Edl ehAfaGaKuym:g;én z . r r eK)an z a i.b r( ar i. br ) r(cos i. sin ) dUcenH z r(cos i. sin ) ehAfaTRmg;RtIekaNmaRtén z . ]TahrN_ 1 ³ cUrsresr z 1 i 3 CaragRtIekaNmaRt . eKman r 12 ( 3 )2 2 eK)an z 2( 12 i. 23 ) 2(cos 3 i. sin 3 ) . ]TahrN_ 2 ³ cUrsresr z 2 3 2i CaragRtIekaNmaRt . eKman r (2 3 )2 (2)2 4 eK)an z 4( 23 i. 12 ) 4( cos 6 i. sin 6 ) . z 4[cos( ) i . sin( )] 6 6 5 5 z 4(cos i sin ) 6 6
eroberogeday lwm plÁún
- TMBr½19 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
9-RbmaNviFIelIcMnYnkMupøictamTRmg;RtIekaNmaRt k> plKuNcMnYnkMupøictamTRmg;RtIekaNmaRt RTwsþIbT ³ ]bmafaeKmancMnYnkMupøic z1 r1 (cos i. sin ) nig z 2 r2 (cos i sin ) Edl r1 0 , r2 0 eK)an z1 .z 2 r1r2[cos( ) i sin( )] . sRmaybBa¢ak; eK)an z1z 2 r1(cos i. sin ).r2 (cos i sin ) z1z2 r1r2[(cos cos sin sin) i(sin cos sin cos)
dUcenH z1 .z 2 r1r2[cos( ) i sin( )] . ]TahrN_ eK[cMnYnkMu[pøic 2 2 z1 2(cos i sin ) nig z 2 3(cos i sin ) 15 5 5 15 KNna z1 .z 2 eK)an z1z 2 6[cos( 5 215 ) i sin( 5 215 )]
5 5 i sin ) 6(cos i . sin ) 15 15 3 3 z1z 2 6(cos i . sin ) . 3 3 6(cos
dUcenH
eroberogeday lwm plÁún
- TMBr½20 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
x> plKuNEckcMnYnkMupøictamTRmg;RtIekaNmaRt RTwsþIbT ³ ]bmafaeKmancMnYnkMupøic z1 r1 (cos i. sin ) nig z 2 r2 (cos i sin ) Edl r1 0 , r2 0 eK)an zz1 rr1 [cos( ) i sin( )] . 2
2
sRmaybBa¢ak; i sin eK)an zz1 rr1 . cos cos i sin 2
2
r1 (cos i. sin )(cos i sin ) . r2 (cos i sin )(cos i sin )
r1 (cos cos sin sin ) i(sin cos sin cos ) . r2 cos 2 sin 2
r1 cos( ) i sin( ) r2
dUcenH zz1 rr1 [cos( ) i sin( )] . 2
2
K> sV½yKuNTI n éncMnYnkMupøictamTRmg;RtIekaNmaRt ³ RTwsþIbT ³ RKb;cMnYnBit nig r 0 eK)an ³ z n [r(cos i . sin )]n r n (cos n i . sin n )
Edl n CacMnYnKt;rWLaTIhV . eroberogeday lwm plÁún
- TMBr½21 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
sRmaybBa¢ak; ³ ramrUbmnþ z1 .z 2 r1r2[cos( ) i sin( )] eK)anCabnþbnÞab;dUcxageRkam ³ z 2 z .z r .r[cos( ) i sin( )] r 2 (cos 2 i sin 2 ) z 3 z 2 .z r 2 .r [cos( 2 ) i sin( 2 )] r 3 (cos 3 i . sin 3 ) ]bmafa zn rn (cos n i. sin n) Bit
eK)an zn1 zn .z rn .r[cos(n ) i sin(n )] r n 1[cos(n 1) i sin(n 1)] Bit dUcenH zn rn (cos n i. sin n) . X> rUbmnþdWmr½ eKman zn [r(cos i. sin )]n rn (cos n i. sin n) eK)an rn (cos i. sin )n rn (cos n i. sin n) dUcenH (cos i. sin )n cos n i. sin n ¬ehAfarUbmnþdWmr½ ¦ .
eroberogeday lwm plÁún
- TMBr½22 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
]TahrN_ ³ eK[cMnYnkMupøic z cos 49 i. sin 49 cUrsresr w
z2 1 z
3
CaragRtIekaNmaRt .
tamrUbmnþdWmr½eKman z 2 (cos 49 i sin 49 )2 cos 89 i sin 89 4 4 3 4 4 nig z (cos 9 i sin 9 ) cos 3 i sin 3 8 8 cos i . sin 9 9 eK)an w 4 4 1 cos i . sin 3 3 8 8 i . sin cos 9 9 2 2 2 2i sin cos 2 cos 2 3 3 3 8 8 i . sin cos 1 9 9 . 2 2 2 i sin cos 2 cos 3 3 3 8 2 8 2 [cos( ) i sin( )] 9 3 9 3 11 11 2 2 (cos i sin ) cos i sin 9 9 9 9 11 11 dUcenH w cos 9 i. sin 9 . 3
eroberogeday lwm plÁún
- TMBr½23 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
10-bMElgvilCMuvijKl;tRmuyénbøg;kMupøic eKmancMnYnkMupøic w cos i. sin . ebI M' (z' ) CarUbPaBén M(z ) tambEmøgvilp©it O nig mMu enaH eK)an z' w. z (cos i sin ) .z . y
M'
O
M x
]TahrN_ 1 ³ enAkñúgbøg;kMupøic (xoy) eK[ M CacMnucrUbPaBén z 3 i . cUrkMNt; z' edaydwgfa M' (z' ) CarUbPaBén M tambEmøgvilp©it O nigmMu 12 . eroberogeday lwm plÁún
- TMBr½24 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
ebI M' (z' ) CarUbPaBén M(z ) tambEmøgvilp©it O nigmMu 12 enaHeK)an
i sin ) z 12 12 3 1 eday z 3 i 2( 2 i. 2 ) 2(cos 6 i. sin 6 ) eK)an z' 2[cos(12 6 ) i sin(12 6 )] dUcenH z' 2(cos 4 i sin 4 ) . ]TahrN_ 2 ³ enAkñúgbøg;kMupøic (xoy) eK[ M CacMnucrUbPaBén z 1 i 3 z' (cos
.
cUrkMNt; z' edaydwgfa M' (z' ) CarUbPaBén M tambEmøgvilp©it O nigmMu 23 . ebI M' (z' ) CarUbPaBén M(z ) tambEmøgvilp©it O nigmMu
2 3
enaHeK)an
2 2 i sin ) z 3 3 eday z 1 i 3 2( 12 i. 23 ) 2[cos( 3 ) i. sin( 3 )] eK)an z' 2[cos( 23 3 ) i sin( 23 3 )] dUcenH z' 2(cos 3 i sin 3 ) . z' (cos
eroberogeday lwm plÁún
- TMBr½25 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
11-b¤sTI n éncMnYnkMupøictamTRmg;RtIekaNmaRt ]bmafaeKmancMnYnkMupøic z r(cos i sin ) tag w (cos i sin ) Cab¤sTI n én z enaH wn z eK)an n (cos n i sin n) r(cos i sin ) n r eKTaj cos(n) cos naM[ sin(n ) sin dUcenHb¤sTI n én z kMNt;eday ³
n r 2k n
2k 2k w k n r cos i . sin n n Edl k 0 , 1 , 2 , ...., n 1 .
]TahrN_ ³ KNnab¤sTIbIén z 4 2 i.4 2 eKman z 4 2 i.4 2 8( 22 i 22 ) 8(cos 4 i sin 4 )
eKTaj r 8 , 4 . tamrUbmnþb¤sTI # én z kMNt;eday ³
8k 8k w k 2cos i sin , k 0 , 1 , 2 . 12 12 dUcenH w 0 2(cos 12 i sin 12 ) ; w1 2(cos 34 i sin 34 ) nig w 2 2(cos 1712 i sin 1712 ) .
eroberogeday lwm plÁún
- TMBr½26 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
12-TRmg;Giucs,:ÚNg;EsüléncMnYnkMupøic k> rUbmnþGWEl (Euler's formula) cMeBaHRKb;cMnYnBit x eK)an eix cos x i sin x Edl e 2.71828... CaeKalelakarIteEB . rUbmnþGWElenHenAEtBitcMeBaH x CacMnYnkMupøick¾eday . sRmaybBa¢ak; -RsaybBa¢ak;edayeRbIedrIev ³ tagGnuKmn_ f ¬GacCaGnuKmn_kMupøic¦énGefr x kMNt;eday f (x)
cos x i sin x
eix ( sin x i cos x )eix ieix (cos x i sin x ) eK)an f ' (x) e2ix eix ( sin x i cos x i cos x i 2 sin x ) e 2ix sin x i cos x i cos x sin x 0 ix 0 ix e e naM[ f (x) CaGnuKmn_efrRKb; x . eK)an f (x) f (0) cos 0 0i sin 0 1 b¤ f (x) cos x ixi sin x 1 e e dUcenH eix cos x i sin x .
eroberogeday lwm plÁún
- TMBr½27 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
-RsaybBa¢ak;edayeRbIsmIkarDIepr:g;EsüllMdab;TImYy ³ eKtagGnuKmn_ g(x) cos x i. sin x eKman g' (x) sin x i cos x KuNGgÁTaMgBIrnwg i eK)an i.g' (x) i sin x csox eK)an g(x) ig' (x) 0 b¤ g' (x) i . g(x) 0 CasmIkarDIepr:g;EsüllMdab; I . eK)ancemøIyTUeTAénsmIkarenHKW g(x) k eix ebI x 0 enaH g(0) k Et g(0) cos 0 i. sin 0 1 enaH k 1 ehIyeK)an g(x) eix . dUcenH eix cos x i sin x . -RsaybBa¢ak;edayeRbIsmIkarDIepr:g;EsüllMdab;TIBIr ³ eRCIserIsGnuKmn_ h(x) eix eKman h' (x) i.eix nig h' ' (x) i 2eix eix eK)an h' ' (x) h(x) eix eix 0 CasmIkarDIepr:g;EsüllIenEG‘Gum:UEsn lMdab;TIBIr . smIkarDIepr:g;EsüllIenEG‘lMdab;BIrenHmancemøIylIenEG‘ÉkraCülIenEG‘cMnYnBIr EdlepÞógpÞat;vaKW h1(x) cos x nig h2 (x) sin x . bnSMlIenEG‘éncemøIy cMeBaHsmIkarDIepr:g;EsülGUm:UEsn k¾CacemøIymYypgEdr . eroberogeday lwm plÁún
- TMBr½28 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
dUcenHcemøIyTUeTAénsmIkarKW h(x) A cos x B sin x Edl A nig B CaBIrcMnYnefrEdlGacrk)antam h(0) A ei0 1 nig h' (0) B iei0 i eRBaH h' (x) A sin x B cos x . ehtuenHeK)an h(x) cos x i sin x . dUcenH eix cos x i sin x . -RsaybBa¢ak;edayeRbIesrIétlr½³ rUbmnþes‘rIétlr½cMeBaHGnuKmn_bI ex , cos x nig sin x KW ³ xn x x2 x3 ..... ..... e 1 n! 1! 2! 3! x
2n x 2 x4 x6 n x ..... ( 1) ...... cos x 1 ( 2n )! 2! 4! 6! 2n 1 x 3 x5 x7 n x sin x x .... ( 1) .... 3! 5! 7! ( 2n 1)!
edayCMnYs x eday ix kñúges‘TaMgbIenHeKTaj)an eix cos x i sin x . x> TRmg;Giucs,:ÚNg;Esül RKb;cMnYnkMupøic z a i.b Edl a , b CacMnYnBitGacsresrCaTRmg;mYyfμIeTot KW z r ei Edl r a2 b2 , cos ar , sin br . TRmg; z r ei ehAfaTRmg;Gics,:ÚNg;Esülén z a i.b . eroberogeday lwm plÁún
- TMBr½29 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
K> TMnak;TMngCamYyGnKmn_ u RtIekaNmaRt ³ cMeBaHRKb;cMnYnBit x eKman eix cos x i sin x (1) CMnYs x eday x eK)an e ix cos x i sin x (2) bUksmIkar (1) nig (2) eKTaj)an eix e ix 2 cos x
eix e ix eKTaj cos x 2 . dksmIkar (1) nig (2) eKTaj)an eix e ix 2i sin x
eix e ix eKTaj sin x 2i . eix e ix eix e ix dUcenH cos x 2 ; sin x 2i ¬ rUbmnþenHBitpgEdrcMeBaH x CacMnYnkMupøic ¦
.
X> TMnak;TMngCamYyGnuKmn_GIEBbUlik³
eix e ix eix e ix eKman cos x 2 ; sin x 2i CMnYs x eday i.x eK)an ³
ex e x ex e x cos(ix ) cosh x nig sin( ix ) i i sinh x 2 2 ex e x ex e x mü:ageToteKman cosh x 2 nig sinh x 2 eK)an ³ eix e ix eix e ix cosh(ix ) cos x , sinh( ix ) i sin x . 2 2
eroberogeday lwm plÁún
- TMBr½30 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
]TahrN_1 ³ sresr z 2 2i 3 CaTRmg;Giucs,:ÚNg;Esül ? eKman r 22 (2 3 )2 4 eK)an z 4( 12 i
3 ) 4(cos i sin ) 4 e 2 3 3 i
i
3
.
4
]TahrN_2³ sresr z 1 e CaragGics,:ÚNg;Esül ? eKman
z
i 1 e 4
i i e 8 (e 8
i8 dUcenH z 2 cos 8 .e ]TahrN_3³ KNna ii ?
i e 8)
eday cos 8 e
8
e 2
i
8
.
i eKman i cos 2 i. sin 2 e 2 i i eK)an ii e 2 e 2 . ]TahrN_4³ edaHRsaysmIkar cos x 2
kñúgsMNMukMupøic ?
eix e ix tamrUbmnþGWEleKman cos x 2 eix e ix eK)an 2 2 b¤ e2ix 4eix 1 0
eroberogeday lwm plÁún
i
tag t eix - TMBr½31 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
eK)ansmIkar t 2 4t 1 0 ' 4 1 3 eKTajb¤s t1 2 3 ; t 2 2 3 cMeBaH t 2 3 eK)an eix 2 3 enaH ix ln( 2 b¤ x i ln( 2 3 ) . cMeBaH t 2 3 eK)an eix 2 3 enaH ix ln( 2 b¤ x i ln( 2 3 ) . ]TahrN_5³ edaHRsaysmIkar sin x 3 kñúgsMNMukMupøic ? eix e ix tamrUbmnþGWEleKman sin x 2i eix e ix eK)an 2i 3 b¤ e2ix 6eix 1 0 eK)ansmIkar t 2 6t 1 0 , ' 9 1 10
3)
3)
tag t eix
manb¤s t1 3 10 , t 2 3 10 . -cMeBaH t 3 10 eK)an eix 3 10 enaH x i ln( 3 -cMeBaH t 3 10 (3 10 ) eK)an eix (3 10 ) (3 10 )ei ei ln( 3 10 ) eKTaj ix i ln( 3 10 ) b¤ x i ln( 3 10 ) . dUcenH x i ln( 3 10 ) , x i ln( 3 10 ) .
eroberogeday lwm plÁún
10 )
- TMBr½32 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
13-RbmaNviFIcMnYnkMupøictamTRmg;Gics,:ÚNg;Esül k> RbmaNviFIKuNcMnYnkMupøictamTRmg;Gics,:ÚNg;Esül eKmancMnYnkMupøic z r ei nig w ei Edl r 0 ; 0 ehIy , CacMnYnBit . eK)an z.w r. ei( ) . x> RbmaNviFIEckcMnYnkMupøictamTRmg;Gics,:ÚNg;Esül eKmancMnYnkMupøic z r ei nig w ei Edl r 0 ; 0 ehIy , CacMnYnBit . eK)an wz r ei( ) . K> sV½yKuNcMnYnkMupøictamTRmg;Gics,:ÚNg;Esül eKmancMnYnkMupøic z r ei cMeBaHRKb;cMnYnKt;rWLaTIhV n eK)an ³ n z n r ei r n ein . ]TahrN_ ³ eK[cMnYnkMupøic z 2e KNna z.w nig wz eK)an z.w
i( ) 6 e 6 12
eroberogeday lwm plÁún
6e
i
i
4
6
nig nig
w 3e
i
12
.
z 2 i ( 6 12 ) 2 i 12 e e w 3 3
- TMBr½33 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
14-Gnuvtþn_cMnYnkMupøickñúgRtIekaNmaRt k> rUbmnþmMuDúb cMeBaHRKb;cMnYnBit x eKman
eix e ix eix e ix cos x ; sin x 2 2i 2
2ix 2ix eix e ix e e 2 2 eK)an cos x 2 4 1 1 e2ix e i 2 x 1 1 2 cos x . cos 2x 2 2 2 2 2 eKTaj cos 2x 2 cos2 x 1 .
e2ix e 2ix eix e ix eix e ix ehIy sin 2x 2i 2. 2i . 2 dUcenH sin 2x 2 sin x cos x .
x> rUbmnþmMuRTIb 3
eix e ix 3 eKman cos x 2 e3ix e 3ix 3(eix e ix ) 8 2 cos 3x 6 cos x cos 3x 3 cos x 8 4 eKTaj)an cos 3x 4 cos3 x 3 cos x .
eroberogeday lwm plÁún
- TMBr½34 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic 3
eix e ix 3 ehIy sin x 2i (e 3ix e 3ix ) 3(eix e ix ) 8i 2i sin 3x 6i sin x 8i sin 3x 3 sin x 4 eKTaj)an sin 3x 3 sin x 4 sin3 x .
K> rUbmnþplbUk nig pldkénmMuBIr eKman ei(a b ) eia .eib RKb;cMnYnBit a nig b eday eia .eib (cos a i sin a)(cos b i sin b) eiaeib (cosa cosb sina sinb) i(sina cosb sinb cosa)
ehIy ei(a b ) cos(a b) i sin(a b) dUcenH cos(a b) cos a cos b sin a sin b nig sin(a b) sin a cos b sin b cos a . mü:ageTot ei(ab) eia .e ib RKb;cMnYnBit a nig b eday eia .e ib (cos a i sin a)(cos b i sin b)
eiaeib (cosa cosb sina sinb) i(sina cosb sinb cosa)
ehIy ei(ab ) cos(a b) i sin(a b) eroberogeday lwm plÁún
- TMBr½35 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
dUcenH cos(a b) cos a cos b sin a sin b nig sin(a b) sin a cos b sin b cos a . X> rUbmnþbMElgBIplKuNeTAplbUk
eia e ia eib e ib cos a cos b . 2 2 ei ( a b ) ei ( a b ) e i ( a b ) e i ( a b ) 4 1 ei ( a b ) e i ( a b ) ei ( a b ) e i ( a b ) 2 2 2
1 cos(a b) cos(a b ) 2 dUcenH cos a cos b 12 cos(a b) cos(a b) eia e ia eib e ib sin a sin b . 2i 2i ei ( a b ) ei ( a b ) e i ( a b ) e i ( a b ) 4i 2 1 ei ( a b ) e i ( a b ) ei ( a b ) e i ( a b ) 2 2 2 1 cos(a b ) cos(a b ) 2 dUcenH sin a sin b 12 cos(a b) cos(a b)
eroberogeday lwm plÁún
- TMBr½36 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
15-Gnuvtþn_cMnYnkMupøickñúgsIVútcMnYnBit cMeBaHsIVúténcMnYnBit (an ) EdlepÞógpÞat;TMnak;TMngkMeNIn ³ an 2 pan 1 qan 0 Edl p , q CacMnYnBit smIkarsmÁal;rbs;sIVútenHKW z 2 pz q 0 kñúgkrNI p2 4q 0 smIkarmanb¤sBIrCacMnYnkMupøicqøas;Kña KW z1 nig z 2 . kñúgkrNIenHedIm,IKNna an eKRtUvGnuvtþn_dUcteTA ³ tagsIVútCMnYy zn an 1 z1an rYcRsayfa (zn ) CasIVútFrNImaRt éncMnYnkMupøicmYy. KNna zn rYcTajrk an . ]TahrN_ eKmansIVúténcMnYnBit (an ) kMNt;eday ³ a0 0 , a1 1 nig an 2 an 1 an Edl n 0 , 1 , 2 ,... cUrKNna an CaGnuKmn_én n smIkarsmÁal;énsIVút an 2 an1 an KW ³ r 2 r 1 b¤ r 2 r 1 0 ; 1 4 3 3i 2 manb¤s r1 1 2i 3 ; r2 1 2i 3 tagsIVútCMnYy zn an1 1 2i eK)an zn1 an 2 1 2i eroberogeday lwm plÁún
3
3
an
an 1
Et an 2 an1 an - TMBr½37 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
1 i 3 eK)an zn1 an1 an 2 an1 1 i 3 zn 1 an 1 an 2 1 i 3 2 zn 1 ( an 1 an ) 2 1 i 3 1 i 3 1 i 3 zn 1 (an 1 an ) 2 2 1 i 3 z n 1 an 2
eK)an (zn ) CasIVútFrNImaRt éncMnYnkMupøicmanersug q 1 2i 3 . tamrUbmnþ zn z 0 qn eday z0 a1 1 2i 3 a0 1 nig q 1 2i 3 cos 3 i sin 3 n n n eK)an zn (cos 3 i sin 3 ) cos 3 i sin 3 eday zn an1 1 2i 3 an (an1 a2n ) i 23 an eKTaj 23 an sin n3 dUcenH an 23 sin n3 .
eroberogeday lwm plÁún
- TMBr½38 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
16-Gnuvtþn_cMnYnkMupøickñúgFrNImaRt k> cm¶ayrvagBIrcMnuc y
B
A y1
O
A
x1
x2
x
eKmancMnYnkMupøicBIr z1 x1 i.y1 nig z 2 x2 i.y 2 . tag A nig B CacMnucrUbPaBén z1 nig z 2 kñúgbøg;kMupøic (xoy) . eK)an AB (x2 x1 )2 (y 2 y1 )2 ehIy z 2 z1 (x2 x1 ) i(y 2 y1 ) eK)an | z 2 z1 | (x2 x1 )2 (y 2 y1 )2 dUcenH AB | z 2 z1 | .
eroberogeday lwm plÁún
- TMBr½39 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
]TahrN_ 1 eK[BIrcMnuc A nig B manGahVikerogKña z1 1 2i nig z 2 2 6i cUrKNnacm¶ayrvagcMnuc A nig B eK)an AB | z 2 z1 | | ( 2 6i ) (1 2i ) |
| 3 4i | ( 3)2 42 5
dUcenH AB 5 . ]TahrN_2 eKmancMnYnkMupøic z1 2 a i nig z 2 3 i(6 a) Edl a IR kñúgbøg;kMupøic (xoy) eKtag A nig B CarUbPaBéncMnYnkMupøic z1 nig z 2 cUrkMNt;cMnYnBit a edIm,I[cm¶ayrvagcMnuc A nig B xøIbMput ? eK)an AB | z 2 z1 | | 3 i(6 a ) 2 a i |
| (1 a ) i(5 a ) | (1 a )2 (5 a )2 1 2a a 2 25 10a a 2 2a 2 12a 26 2(a 3)2 8
edIm,I[ AB xøIbMputluHRtaEt a 3 ehIy ABmin eroberogeday lwm plÁún
82 2
.
- TMBr½40 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
x> cMnucEckkñúgGgát;tampleFobmYy ³ kñúgbøg;kMupøic (xoy) eKtag A nig B CarUbPaBéncMnYnkMupøic z A nig zB yk P CacMnucmanGahVik z P CacMnucEckkñúgénGgát; AB tampleFob Edl 0 . y
B P A
y
O
ebI P CacMnucEckkñúgénGgát; AB tampleFob enaH
AP PB
eKman AP (z P z A ) nig PB (z B z P ) eK)an zp z A (z B z P ) dUcenH z P z A1 zB . eroberogeday lwm plÁún
- TMBr½41 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
krNI 1 enaHcMnuc P CacMnuckNþalénGgát; [AB] dUcenHGahVikéncMnuc P kNþalGgát; AB KW z P z A 2 zB . K> cMnucEckeRkAGgát;tampleFobmYy ³ kñúgbøg;kMupøic (xoy) eKtag A nig B CarUbPaBéncMnYnkMupøic z A nig zB yk Q CacMnucmanGahVik zQ CacMnucEckeRkAénGgát; AB tampleFob Edl 0 . y
Q
B A
y
O
ebI Q CacMnucEckeRkAénGgát; AB tampleFob enaH eKman
QA ( z A z Q )
eroberogeday lwm plÁún
nig
QA QB
QB ( z B z Q )
- TMBr½42 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
eK)an z A zQ (z B zQ ) dUcenH zQ z A1 zB . ]TahrN_ ³ eKmanBIrcMnuc A nig B CarUbPaBéncMnYnkMupøic z A 2 7i nig zB 1 i . 1 P CarUbPaBén z p CacMnucEckkñúgén AB tampleFob p 3 CarUbPaBén zQ CacMnucEckeRkAén AB tampleFob Q 23 cUrkMNt; z P nig zQ ? ebI P CarUbPaBén zp CacMnucEckkñúgén AB tampleFob p 13 Q
eK)an
1 1 2 7 i i 5 11 z A PzB 3 3 i zP 1 1 P 4 2 1 3
.
ebI Q CarUbPaBén zQ CacMnucEckeRkAén AB tampleFob Q 23 eK)an
z A QzB zQ 1 Q
2 2 i 3 3 8 19i 2 1 3
2 7i
dUcenH z P 54 112 i nig zQ 8 19i . eroberogeday lwm plÁún
- TMBr½43 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
K> pleFobRCug nig mMuénRtIekaNkñúgbøg; kñúgbøg;kMupøic (xoy) eKmanbIcMnuc A , B , C CarUbPaBéncMnYnkMupøic z A , z B , zC . C
y
C' B A
B'
x
O
nig OC' AC enaHeK)an OB' C' nig ABC CaRtIekaNb:unKña enaH BAC B' OC' .
sg;viucTr½
OB' AB
eKman OB'
AB ( z B z A )
eroberogeday lwm plÁún
nig
OC' AC ( z C z A )
- TMBr½44 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
eKman B'OC' XOC'XOB' b¤ BAC arg(zC z A ) arg(z B z A ) arg zzC zz A
B
A
| z z A | zC z A ehIypleFobRCug AC . C AB | z z | z z B
A
B
A
dUcenHebI A(z A ) , B(zB ) , C(zC ) begáIt)anCaRtIekaN ABC enaH z zA z zA . eK)an AC ni g C BAC arg C AB z z z z B
A
B
A
]TahrN_ ³ enAkñúgbøg;kMupøic ( XOY) eK[bIcMnuc A , B , C manGahVikerogKña ³ z A 2 i , z B 3 4i , z C 2 9i . cUrKNnapleFobRCug AC ni g BAC ? AB eKman zC z A 2 9i 2 i 4 8i nig zB z A 3 4i 2 i 1 3i eK)an zzC zz A 1438i i ((1438ii)()(1133ii)) 2 2i B
A
2 2 i ) 2 2 (cos i sin ) 2 2 4 4 AC 2 2 nig BAC . AB 4 2 2(
dUcenH
eroberogeday lwm plÁún
- TMBr½45 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
X> sMNMucMNuckñúgbøg;kMupøic kñúgbøg;kMupøicRbkbedaytRmuyGrtUNrm:al; (o, i , j ) eKtagcMnuc P(x; y ) CacMnucrUbPaBéncMnYnkMupøic z x i.y . sMNMuénrUbPaB P TaMgGs;EdlGacmaneTAtamlkçxNÐEdlRtUvbMeBjén z begáIt)anCasMNMuéncMnuc P kñúgbøg;kMupøic . ]TahrN_ 1 eK[cMnYnkMupøic z epÞógpÞat; (1 i )z (1 i ) z 4 . P CarUbPaBéncMnYnkMupøic z kñúgbøg;kMupøic ( xoy ) . rksMNMucMnuc P ? tag z x i.y enaH z x i.y eKman (1 i )z 2(1 i ) z 4 eK)an ³ y
(1 i)(x iy) (1 i)(x iy) 4 x iy ix y x iy ix y 4 2x 2y 4 y x 2
dUcenHsMNMucMnuc P KWCabnÞat; Edl mansmIkar ( ) : y x 2
eroberogeday lwm plÁún
1
0
1
x
( ) : y x 2
- TMBr½46 -
sikSaKNitviTüaedayxøÜnÉg
]TahrN_ 2 eK[cMnYnkMupøic z epÞógpÞat; | z 2 i | 3 . P CarUbPaBéncMnYnkMupøic z kñúgbøg;kMupøic ( xoy ) . rksMNMucMnuc P ? tag z x i.y CaGahVikéncMnuc P eKman | z 2 i | 3 eK)an | x iy 2 i | 3
cMnYnkMupøic
y
1
0
1
x
| ( x 2) i( y 1) | 3
( x 2)2 ( y 1)2 3
b¤ (x 2)2 (y 1)2 9 dUcenHsMMNMucMnuc P KWCargVg;p©it I(2 , 1) nig kaM R 3 . ]TahrN_ 3 eK[cMnYnkMupøic w zz 22 22ii ehIy P CacMnucrUbPaBén z kñúgbøg; (xoy) cUrkMNt;sMNMucMnuc P edIm,I[ w CacMnYnnimμitsuTæ ? edIm,I[ w CacMnYnnimμitsuTæluHRtaEt Re(w ) w 2 w 0 naM[ w w eday w zz 22 22ii nig w zz 22 22ii eroberogeday lwm plÁún
- TMBr½47 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
z 2 2i z 2 2i eK)an z 2 2i z 2 2i ebI z 2 2i ( z 2 2i )( z 2 2i ) ( z 2 2i )( z 2 2i ) z z ( 2 2i )z ( 2 2i )z 8 z .z ( 2 2i )z ( 2 2i )z 8
b¤ | z |2 8 tag z x iy Edl (x, y ) (2 , 2) eK)an x2 y 2 8 . dUcenHsMNMucMnuc P CargVg;p©it O kaM R 2 2 elIkElgEtcMnuc A(2,2) . 2z z 16
y
x2 y 2 8 1
0
eroberogeday lwm plÁún
1
x
- TMBr½48 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
]TahrN_ 4 eK[cMnYnkMupøic z ehIy P CacMnucrUbPaBén z kñúgbøg; (xoy) cUrkMNt;sMNMucMnuc P ebIeKdwgfa arg(z 2 2i ) 4 . tag z x iy enaHeK)an z 2 2i (x 2) i(y 2) eday arg(z 2 2i ) 4 enaH arg((x 2) i(y 2)) 4 eKTaj)an tan 4 yx 22 1 naM[ y x 4 dUcenHsMNMucMnuc P CabnÞat; (d) : y x 4 Edl xy 22 00 b¤ x 2 , y 2 .
]TahrN_ 5 enAkñúgbøg;kMupøic (xoy) eK[cMnuc P(z ) , Q(iz ) , R(2 2i ) cUrkMNt;sMNMucMnuc P edIm,I[bIcMnuc P , Q , R rt;Rtg;Kña ?
bIcMnuc P , Q , R rt;Rtg;KñaluHRtaEtRKb; t IR : PQ t QR b¤ zQ z P t(z R zQ ) . tag z x iy enaH zQ ix y eK)an ix y x iy t(2 2i ix y ) ( x y ) i( x y ) ( 2 y )t i( 2 x )t
eKTaj x y (2 y )t nig (x y ) (2 x)t eroberogeday lwm plÁún
- TMBr½49 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
eK)an xxyy 22 yx b¤ 2x x2 2y xy 2x xy 2y y 2 x 2 y 2 4x 4y 0 ( x 2)2 ( y 2)2 8
dUcenHsMNMucMnuc P CargVg;p©it I(2 , 2 ) mankaM R 2 2 . ]TahrN_ 6 enAkñúgbøg;kMupøic (xoy) eK[cMnuc P CarUbPaBén z EdlbMeBjlkçxNн z 2 2i 1 . cU r rksM N M u c M n u c P? zi 2 eK)an 2 | z 2 2i || z i | Edl z i b¤ 2 | z 2 2i |2 | z i |2 tag z x i.y eK)an ³ 2 | x iy 2 2i |2 | x iy i |2 2 | ( x 2) i( y 2) |2 | x i( y 1) |2 2[( x 2)2 ( y 2)2 ] x 2 ( y 1)2 2x 2 8x 8 2y 2 8y 8 x 2 y 2 2y 1 x 2 y 2 8x 10y 15 0 ( x 4)2 ( y 5)2 26
dUcenHsMNMucMnuc P CargVg;p©it I(4 , 5) nig kaM R eroberogeday lwm plÁún
26
. - TMBr½50 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
17-bMElgcMnuckñúgbøg;kMupøic k> bMElgkil kñúgbøg;kMupøic (xoy) eKmanBIrcMnuc M nig M' manGahVikerogKña z nig z' M' CarUbPaBén M tambMElgkilénviucTr½ v ( z v ) eK)an ³ z' z z v . y M' M
v
x
O
ebI M' CarUbPaBén M tambMElgkilénviucTr½
v ( z v ) enaH MM' v
eday MM'(z' z ) enaH z' z z v b¤ z' z z v . eroberogeday lwm plÁún
- TMBr½51 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
x> bMElgvilp©it kñúgbøg;kMupøic (xoy) eKmanBIrcMnuc M nig M' manGahVikerogKña z nig z' M' CarUbPaBén M tambMElgvilp©it ( z ) nigmMu eK)an ³ z' z ( z z )(cos i sin ) . Y
y
M'
M
X
x O
ebI M' CarUbPaBén M tambMElgvilp©it (z ) nigmMu enaHeK)an M M' r nig MM' Edl XM' nig XM . ehtuenH
z' z r .ei i e i z z r .e
Edl
dUcenH z' z (z z )(cos i sin ) . eroberogeday lwm plÁún
- TMBr½52 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
K> bMElgcaMgp©it kñúgbøg;kMupøic (xoy) eKmanBIrcMnuc M nig M' manGahVikerogKña z nig z' M' CarUbPaBén M tambMElgcaMgp©it ( z ) pleFob eK)an ³ z' z ( z z ) . y M' M
x O M'
CarUbPaBén M tambMElgcaMgp©it (z ) pleFob eK)an ³
M' M
dUcenH
eday
M'( z' z ) ; M ( z z )
z' z ( z z )
eroberogeday lwm plÁún
.
- TMBr½53 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
18-)arIsg;énRbBn½æcMNuckñúgbøg;kMupøic eK[ Ak , k 1k n CaRbBn½æmYyman n cMNucb:ugeder Edlman n
GahVik z Ak . ebI k 0 enaH)arIsg; G énRbBn½æmanGahVikmYy k 1 n
( k z A k )
kMNt;eday z G k 1 n
k
.
k 1
tamniymn½y ebI G Ca)arIsg;énRbBnæ½ Ak , k 1k n enaHeK)an ³ GAk OAk OG k GA k O k 1 n k . OA k k . OG O k 1
eday
n
eK)an b¤ b¤
OG . ( k ) k . OA k k 1 k 1
n
n
n
n
k 1
k 1
z G . ( k ) ( k z A k ) n
dUcenH
zG
( k z A k )
k 1
n
k
.
k 1
eroberogeday lwm plÁún
- TMBr½54 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
CMBUkTI2
kRmglMhat;eRCIserIsmandMeNaHRsay !> eKmancMnYnkMupøic z1 4 7i nig z 2 3 2i . k> cUrKNna z1 z 2 nig z1 z 2 . x> cUrKNna z1 z 2 nig zz1 . 2
@> eKman z1 1 2i nig z 2 3 i . cUrKNna U z 12 z 2 2 nig V z 13 z 2 3 ? #> cUrkMnt;cMnYnBit p nig q edIm,I[cMnYnkMupøic z 3 2i Cab¤srbs;smIkar z 2 ipz q 0 . $> kMnt;BIrcMnYnBit x nig y ebIeKdwgfa ³ ( 3 i )(1 ix ) (1 3i )( 3 2iy )
2(7 9i ) 1 i
.
%> cUrKNnab¤skaerén z 40 42i . ^> eK[cMnYnkMupøic a 2 3i ; b 3 i nig c 1 4i k-cUrsresr a 3 b 3 c 3 nig a b c CaTRmg;BICKNit . x-cUrepÞógpÞat;faeKGackMnt;cMnYnBit k edIm,I[ a 3 b 3 c 3 k . a b c eroberogeday lwm plÁún
- TMBr½55 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
&> eK[cMnYnkMupøic a 3 i ; u (x 1) i.(y 2) nig b 2 16i Edl x nig y CaBIrcMnYnBit . cUrkMnt;témø x nig y edIm,I[ au b 0 . *> eK[cMnYnkMupøic Z log 3 ( x 2 y ) i log 2 x log 2 y
i nig W 13 Edl x IR * ; y IR * . 1 2i k-cUrsresr W CaTRmg;BICKNit . x-kMnt; x nig y edIm,I[ Z W . (>eK[smIkar (E): z 2 az b 0 Edl a ;b IR cUrkMnt;témø a nig b edIm,I[cMnYnkMupøic z1 2 i 3 Cab¤srbs; smIkar (E) rYcTajrkb¤s z 2 mYYyeTotrbs;smIkar . etIGñkBinitüeXIjdUcemþccMeBaHcMnYnkMupøic z1 nig z 2 ? !0> eK[cMnYnkMupøic z Edlman z CacMnYnkMupøicqøas;rbs;va . edaHRsaysmIkar log 5 | z | z 7i z 3 i !!> eK[cMnYnkMupøicBIr ³ z 3x i ( 2x y ) nig W 1 y i [ 1 2 log5 ( x 3 ) ] Edl x nig y CacMnYnBit . kMnt; x nig y edIm,I[ W z .
eroberogeday lwm plÁún
- TMBr½56 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
!@> edaHRsaysmIkar 2z | z | 917ii Edl z CacMnYnkMupøic . !#> eK[cMnYnkMupøicBIr Z1 nig Z 2 Edl Z 2 0 . cUrRsaybBa¢ak;fa ZZ1 || ZZ1 || . 2
2
!$> eK[cMnYnkMupøicBIr Z1 nig Z 2 . cUrRsaybBa¢ak;fa | Z1 Z 2 | | Z1 | | Z 2 | . !%> eK[cMnYnkMupøicBIr Z1 nig Z 2 . k> cUrRsaybBa¢ak;fa | Z1 Z 2 | | Z1 | | Z 2 | x> Taj[)anfa (a c) 2 (b d) 2 a 2 b 2 cMeBaHRKb; a , b , c , d CacMnYnBit . !^> eK[cMnYnkMupøic ³ z1 6 2i. 2 nig z 2 1 i k>cUrsresr
z 1 , z 2 nig Z
z1 z2
c2 d2
CaragRtIekaNmaRt.
x>cUrsresr Z zz1 CaragBiCKNit. 2
K>Taj[)anfa cos 12
eroberogeday lwm plÁún
6 2 4
nig sin 12
6 2 . 4
- TMBr½57 -
sikSaKNitviTüaedayxøÜnÉg
!&> eK[cMnYnkMupøic z 2 2 i. 2 k-cUrsresr z 2 CaTMrg;BiCKNit . x-cUrsresr z 2 nig z CaTMrg;RtIekaNmaRt . K-TajrktMélR)akdén cos 8 nig sin 8 .
cMnYnkMupøic 2
.
!*> eK[cMnYnkMupøic Z cos 45 i. sin 45
cUrRsaybBa¢ak;fa (1 Z) 3 8 cos 3 25 (cos 65 i. sin 65 ) . !(> eK[cMnYnkMupøic Z 4 2(1 i ) k-cUrsresr Z CaTRmg;RtIekaNmaRt . x-KNnab¤sTI# én Z . @0> eKeGay f z z 3 2 3 i .z 2 41 i 3 .z 8i k>cUrbgðajfa z ¢ f z z 2i z 2 2 3z 4 x>edaHRsaysmIkar f z 0 kñúgsMNMukMupøic . @!> eK[cMnYnkMupøic z cos 47 i. sin 47 . cUrsresr (1 z )4 CaragRtIekaNmaRt .
eroberogeday lwm plÁún
- TMBr½58 -
sikSaKNitviTüaedayxøÜnÉg
@@> eK[cMnYnkMupøic k-cUrsresr z 1
z1 1 i
,z2
cMnYnkMupøic
nig
6 i. 2 z2 2
nig zz1 CaTMrg;RtIekaNmaRt . 2
x-cUrsresr zz1 CaTMrg;BiCKNit . 2
K-eRbIlTæplxagelIcUrTajrktMélR)akdén cos 12 nig sin 12 . @#> eK[sIVúténcMnYnkMupøic (Z n ) kMnt;eday 1 i 3 Z 0 2 1 Z n 1 2 Z n | Z n | ; n IN ¬ | Z n | Cam:UDulén Z n ¦.
snμtfa Z n n (cos n i. sin n ) , n IN Edl n 0 , n ; n IR . k-cUrrkTMnak;TMngrvag n nig n1 ehIy n nig n1 . x-rkRbePTénsIVút (n ) rYcKNna n CaGnuKmn_én n . K-cUrbgðajfa n 0 cos 0 cos 21 cos 22 .... cos n2 1 rYcbBa¢ak; n GnuKmn_én n . eroberogeday lwm plÁún
- TMBr½59 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
@$> eK[cMnYnkMupøic z x i.y Edl x nig y CaBIrcMnYnBit . cUrkMnt;témø x nig y ebIeKdwgfa ³ 10 ( 3 2i ) z (1 3i ) z ¬ z CacMnYnkMupøicqøas;én z ¦ . 2i @%> eK[ z ; z CacMnYnkMupøicEdl | z || z | r 0 . 1
2
1
2
2
2
z1 z 2 z1 z 2 1 2 2 2 r r z 1z 2 r z 1z 2
bgðajfa @^>eKyk z
1
; z 2 ; ....; z n
CacMnYnkMupøicEdlepÞógpÞat;TMnak;TMng
(k 1)z k 1 i(n k )z k 0 ; k 0 , 1 , 2 , ... , n 1
k-kMnt; z ebIeKdwgfa z z z .... z 2 x-cMeBaHtémø z Edl)ankMnt;xagelIcUrbgðajfa ³ 0
0
1
2
n
n
0
( 3n 1)n | z 0 | | z1 | | z 2 | .... | z n | n! 2
@&> eK[ z
2
1
2
2
CacMnYnkMupøicedaydwgfa ³ z z z z 0 .
; z2 ; z3
z 1 z 2 z 3 z 1z 2
cUrbgðajfa | z @*> cUrbgðajfa
1
2 3
3 1
|| z 2 || z 3 | 6z i 1 2 3iz
eroberogeday lwm plÁún
luHRtaEt
|z|
1 3
- TMBr½60 -
sikSaKNitviTüaedayxøÜnÉg
@(> eK[ z
1
cMnYnkMupøic
CacMnYnkMupøicEdlmanm:UDúlesμI 1 . 1 ) ( ) . z
; z 2 ; z 3 ; ....; z n
n Z (zk k 1
eKtag cUrbgðajfa 0 Z n #0> eK[cMnYnkMupøic z Edl | z | 1 . cUrbgðajfa ³ 2 |1 z | |1 z | 4 . n
k 1
k
2
2
#!> eK[cMnYnkMupøic Z (cos x 1 ) i.(sin x 1 ) cos x sin x Edl x CacMnYnBit. cUrkMnt´rkm¨UDulGb,brmaéncMnYnkMupøicen¼ ? #@> eK[ A 13 i 13 i , n IN . 2
2
2
n
cUrbgðajfa
A i.
( 3)
n
. sin
n 3
2 n 1 n
2
cMeBaHRKb; n IN .
##> kñúgbøg;kMupøic (O , i , j ) eK[bYncMnuc A , B , C , D
EdlmanGahVikerogKña nig Z 2 3i . cUrRsayfactuekaN ABCD carikkñúgrgVg;mYyEdleKnwgbBa¢ak;p©it nig kaMrbs;va . Z A 1 6i , Z B 4 5i , Z C 5 4i
eroberogeday lwm plÁún
D
- TMBr½61 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
#$> eK[ z nig z CaBIrcMnYnkMupøic . 1
2
cUrbgðajfa | 1 z z 1
| z 1 1 || z 2 1 | 2 | | z1 z 2 | 2
2
#%> eK[ z nig z CacMnYnkMupøicBIr . cUrRsayfa | z z | | z z | 2| z | | z | #^>eK[ z nig z CacMnYnkMupøicBIrEdl | z | | z | 1 nig z .z 1 . cUrRsayfa 1z z zz CacMnYnBitmYy . #&> eK[BhuFa P(x) (x sin a cos a) Edl n IN * cUrrksMNl´énviFIEckrvag P(x) nwg x 1 . #*> eK[cMnYnkMupøic z 2 cos i 2 sin 1
2
2
1
1
1
2
2
1
2
2
1
2
2
2
1
2
2
1
2
1 2
n
2
Edl IR . kñúgbøg;kMpøic (o , i , j ) eKehA M CacMnucrUbPaBén z . cUrkMNt;témøtUcbMput nig FMbMputén r OM ? #(> eK[cMnYnkMupøic z1 , z 2 , z 3 ehIyepÞógpÞat;TMnak;TMng ³ | z 1 || z 2 || z 3 | 1 nig
z 32 z12 z 22 1 0 z 2 z 3 z 1z 3 z 1z 2
cUrRsayfa | z1 z 2 z 3 | { 1 , 2 } . eroberogeday lwm plÁún
- TMBr½62 -
sikSaKNitviTüaedayxøÜnÉg
$0> eK[cMnYnkMupøic z1 nig
cMnYnkMupøic z2
Edl | z1 || z 2 | 1
cUrRsayfa | z1 1 | | z 2 1 | | z1z 2 1 | 2 $!> eK[sIVútcMnYnBit (an ) kMNt;eday ³
a 1 1 , a 2 1 a n 2 a n 1 a n , n 1 , 2 , 3 , ... 1 i 3 eKtagsIVútcMnYnkMupøic z n an1 an . 2 k> cUrRsayfa z n1 1 i 3 z n cMeBaHRKb; n 1 . 2 x> cUrdak; 1 i 3 CaTRmg;RtIekaNmaRtrYcTajrk z n CaGnuKmn_én n 2 K> TajrktYTUeTAénsIVút an . etI (an ) CasIVútxYbb¤eT ?
.
$@> eK[sIVúténcMnYnkMupøic ( z n ) kMNt;eday ³ 2 3i nig z n1 3 i z n 2 3 i 2 2 2 Edl n 1 , 2 , 3 , ... . z1
k> tag w n z n 1 . bgðajfa (w n ) CasIVútFrNImaRténcMnYnkMupøic rYcKNna w n CaGnuKmn_én n edaysresrlTæplCaTRmg;RtIekaNmaRt . eroberogeday lwm plÁún
- TMBr½63 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
x> Tajbgðajfa z n 2 cos n (cos n i sin n ) . 12 12 12 $#> eK[sIVúténcMnYnBit (u n ) nig ( v n ) kMNt;eday ³ un vn u n 1 2 nig un vn v n 1 2 k> eKBinitüsIVúténcMnYnkMupøic z n un i.v n
u1 v 1
2 2 2 2
Edl n 1
. cUrRsayfa (z n ) CasIVútFrNImaRténcMnYnkMupøic rYcKNna z n CaGnuKmn_én n edaysresrlTæplCaTRmg;RtIekaNmaRt . x> sMEdg un nig v n CaGnuKmn_én n . $$> eK[sIVúténcMnYnBit (un ) nig ( v n ) kMNt;eday ³ u 0 1 v0 3
u n 1 u n 2 v n 2 nig v n 1 2u n v n k> eKBinitüsIVúténcMnYnkMupøic z n un i.v n .
Edl n 0 2n
cUrRsayfa z n1 z n rYcTajfa z n z 0 . x> sMEdg un nig v n CaGnuKmn_én n . 2
eroberogeday lwm plÁún
- TMBr½64 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI1 eKmancMnYnkMupøic z1 4 7i nig z 2 3 2i . k> cUrKNna z1 z 2 nig z1 z 2 . x> cUrKNna z1 z 2 nig zz1 . 2
dMeNaHRsay k> KNna z1 z 2 nig z1 z 2 eyIg)an z1 z 2 (4 7i ) (3 2i ) 4 7i 3 2i 7 9i nig z1 z 2 (4 7i ) (3 2i ) 4 7i 3 2i 1 5i dUcenH z1 z 2 7 9i nig z1 z 2 1 5i . x> KNna z1 z 2 nig zz1
2
eyIg)an z1 z 2 (4 7i )(3 2i ) 12 8i 21i 14 2 29i nig z1 4 7i (4 7i)(3 2i) 12 8i 21i 14 26 13i 2 i z2
dUcenH
3 2i
( 3 2i )( 3 2i )
z1 z 2 2 29i
eroberogeday lwm plÁún
34
nig
z1 2i z2
13
.
- TMBr½65 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI2 eKman z1 1 2i nig z 2 3 i . cUrKNna U z 12 z 2 2 nig V z 13 z 2 3 ? dMeNaHRsay KNna U z 12 z 2 2 nig V z 13 z 2 3 eyIg)an ³
U 1 2i 2 3 i 2 1 4i 4 9 6i 1 5 10i
V 1 2i 3 3 i 3 1 6i 12 8i 27 27i 9 i 7 24i
dUcenH
U 5 10i
eroberogeday lwm plÁún
nig
V 7 24i
.
- TMBr½66 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI3 cUrkMnt;cMnYnBit p nig q edIm,I[cMnYnkMupøic z 3 2i Cab¤srbs;smIkar z 2 ipz q 0 . dMeNaHRsay kMnt;cMnYnBit p nig q ebI z 3 2i Cab¤srbs;smIkar z 2 ipz q 0 enaHvaRtUvepÞógpÞat;smIkar eK)an 3 2i 2 ip3 2i q 0
9 12i 4 3ip 2p q 0 5 2p q i.12 3p 0 eKTaj)an 5 2p q 0 naM[ q 2p 5 13 12 3p 0 p 4
dUcenH
p 4 , q 13
eroberogeday lwm plÁún
.
- TMBr½67 -
sikSaKNitviTüaedayxøÜnÉg
lMhat;TI4 kMnt;BIrcMnYnBit x nig y ebIeKdwgfa ³ ( 3 i )(1 ix ) (1 3i )( 3 2iy )
cMnYnkMupøic
2(7 9i ) 1 i
.
dMeNaHRsay kMnt;BIrcMnYnBit x nig y eKman (3 i )(1 ix) (1 3i )(3 2iy ) 2(17 i9i )
eK)an 3 3ix i x 3 2iy 9i 6y 2(7 9i )(1 i )
11 x 6y 6 3ix 2iy 8i 2(7 7i 9i 9) 2 x 6y 6 i.3x 2y 8 16 2i x 6y 6 16 eKTaj 3x 2y 8 2 b¤ x 6y 10 naM[ x 2 , y 2 . 3x 2y 10
dUcenH
x2 , y2
eroberogeday lwm plÁún
.
- TMBr½68 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI5 cUrKNnab¤skaerén z 40 42i . dMeNaHRsay tag W x i.y , x; y IR Cab¤skaerén z 40 42i eK)an W 2 z eday W 2 (x i.y ) 2 (x 2 y 2 ) 2ixy x 2 y 2 40 2 2 naM[ (x y ) 2ixy 40 42.i eKTaj 2xy 42
eday x y (x2 y 2 )2 4x2y 2 402 422 3364 eKTaj x 2 y 2 3364 58 . 2
2 2
x 2 y 2 58 1 eK)anRbBnæ½smIkar 2 2 x y 40 2 bUksmIkar ¬!¦ nig ¬@¦ eK)an 2x 2 98 naM[ x 7
dksmIkar¬!¦ nig ¬@¦ eK)an 2y 2 18 naM[ y 3 eday 2xy 42 0 naM[ x nig y mansBaØadUcKña naM[eKTaj)anKUcemøIy ³ x 7 , y 3 nig x 7 , y 3 . dUcenH W1 7 3i nig W2 7 3i .
eroberogeday lwm plÁún
- TMBr½69 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI6 eK[cMnYnkMupøic a 2 3i ; b 3 i nig c 1 4i k-cUrsresr a 3 b 3 c 3 nig a b c CaTRmg;BICKNit . x-cUrepÞógpÞat;faeKGackMnt;cMnYnBit k edIm,I[ a 3 b 3 c 3 k . a b c . dMeNaHRsay k¿sresr a 3 b 3 c 3 nig a b c CaTRmg´BICKNit ½ eyIgman a 3 (2 3i ) 3 8 36i 54 27i 46 9i b 3 ( 3 i ) 3 27 27i 9 i 18 26i c 3 (1 4i ) 3 1 12i 48 64i 47 52i
eyIgán a 3 b 3 c3 46 9i 18 26i 47 52i 111 87i
.
dUcen¼
a 3 b 3 c 3 111 87i
eyIgman
a b c ( 2 3i )( 3 i )(1 4i ) ( 6 2i 9i 3)(1 4i )
( 9 7i )(1 4i ) 9 36i 7i 28 37 29i
dUcen¼
a b c 37 29i
eroberogeday lwm plÁún
. - TMBr½70 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
x¿kMnt´cMnYnBit k eyIgman a 3 b 3 c 3 k .a b c eKTaj dUcen¼
a 3 b 3 c 3 111 87i k 3 a.b.c 37 29i k3 .
eroberogeday lwm plÁún
- TMBr½71 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI7 eK[cMnYnkMupøic a 3 i ; u (x 1) i.(y 2) nig b 2 16i Edl x nig y CaBIrcMnYnBit . cUrkMnt;témø x nig y edIm,I[ au b 0 . dMeNaHRsay kMnt;témøén x nig y eyIg)an au b 0 b a eday a 3 i ; u (x 1) i.(y 2) nig b 2 16i eK)an (x 1) i.(y 2) 2316i i 2(1 8i )( 3 i ) ( x 1) i( y 2) 32 i 2 2( 3 i 24i 8) ( x 1) i( y 2) 10 ( x 1) i( y 2) 1 5i eKTaj)an xy 12 51 naM[ x 2 ; y 3 u
dUcenH
x2 ; y3
eroberogeday lwm plÁún
. - TMBr½72 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI8 i eK[cMnYnkMupøic Z log 3 ( x 2 y ) i log 2 x log 2 y nig W 13 1 2i Edl x IR* ; y IR * . k-cUrsresr W CaTRmg;BICKNit . x-kMnt; x nig y edIm,I[ Z W . dMeNaHRsay k-sresr W CaTRmg;BICKNit i (12 i )(1 2i ) 12 24i i 2 eyIg)an W 12 2 5i 1 2i 1 4 5 dUcenH
W 2 5i
.
x-kMnt; x nig y edIm,I[ Z W
xy log ( )2 2 naM[ eyIg)an Z W smmUl 3 log 2 x log 2 y 5 eKTaj x ; y Cab¤sssmIkar u 2 12u 32 0 u 1 6 2 4 eday ' 36 32 4 eKTajb¤s u 6 2 8 2 dUcenH x 4 ; y 8 b¤ x 8 ; y 4 .
eroberogeday lwm plÁún
x y 12 x.y 32
- TMBr½73 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI9 eK[smIkar (E): z 2 az b 0 Edl a ;b IR cUrkMnt;témø a nig b edIm,I[cMnYnkMupøic z1 2 i 3 Cab¤srbs; smIkar (E) rYcTajrkb¤s z 2 mYYyeTotrbs;smIkar . etIGñkBinitüeXIjdUcemþccMeBaHcMnYnkMupøic z1 nig z 2 ? dMeNaHRsay kMnt;témø a nig b edIm,I[cMnYnkMupøic 2 i 3 Cab¤srbs;smIkarluHRtaEtvaepÞógpÞat;smIkar eK)an (2 i 3 )2 a(2 i 3 ) b 0 4 4i 3 3 2a ai 3 b 0 (1 2a b ) i(4 3 a 4 3 a 3 0 eKTaj)an b¤ 1 2a b 0
dUcenH
a 4 ; b 7
3) 0 a 4 b 7
.
KNnab¤smYYyeTotrbs;smIkar ³ ebI z1 ; z 2 Cab¤srbs;smIkarenaHtamRTwsþIbTEvüteK)an ³ z 1 z 2 a 4 eKTaj z 2 4 ( 2 i 3 ) 2 i 3 dUcenH z 2 2 i 3 . eyIgBinitüeXIjfacMnYnkMupøic z1 nig z 2 qøas;Kña . eroberogeday lwm plÁún
- TMBr½74 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI10 eK[cMnYnkMupøic z Edlman z CacMnYnkMupøicqøas;rbs;va . edaHRsaysmIkar log 5 | z | z 7i z 3 i dMeNaHRsay edaHRsaysmIkar log 5 | z | z 7 i z 3 i tag z x i.y naM[ z x iy Edl x ; y IR smIkarGacsresr ³
( x iy ) i( x iy ) 3i 7 x iy ix y log 5 x 2 y 2 3i 7 xy xy ( log 5 x 2 y 2 ) i 3i 7 7 x y x y 7 7 1 eKTaj)an x y b¤ 2 2 1 log x y 3 2 2 5 log x y 3 5 7 x y 7 b¤ 2 2 eKTajcemøIy x 3 , y 4 b¤ x 4 , y 3 . x y 25 dUcenH z1 3 4i ; z 2 4 3i CacemøIyrbs;smIkar . log 5 | x iy |
eroberogeday lwm plÁún
- TMBr½75 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI11 eK[cMnYnkMupøicBIr ³
nig W 1 y i [ 1 2 log5 ( x 3) ] Edl x nig y CacMnYnBit . kMnt; x nig y edIm,I[ W z . dMeNaHRsay kMnt; x nig y Re( W ) Re( z ) edIm,I[ W z luHRtaEt Im( W ) Im( z ) z 3x i ( 2x y )
(1) 1 y 3x eKTaj)an log 5 ( x 3) 2 x y ( 2) 1 2 tam (1) eKTaj y 3x 1 (3) ykCYskñúgsmIkar (2)
eK)an
1 2 log5 ( x 3 ) 2x 3x 1
2 log5 ( x 3) x
lkçx½NÐ x 3 0 b¤ x 3 tag t log 5 (x 3) naM[ x 3 5t b¤ x 5t 3 eK)an smIkar 2t 5t 3 b¤ 5t 2t 3 edayGgÁxageqVgénsmIkarCaGnuKmn_ekIn nig GgÁxagsþaMCaGnuKmn_efr enaHeyIg)ansmIkarmanb¤sEtmYyKt;KW t 1 . cMeBaH t 1 eK)an x 5 3 2 nig y 3(2) 1 7 dUcenH x 2 ; y 7 . eroberogeday lwm plÁún
- TMBr½76 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI12 edaHRsaysmIkar 2z | z | 917ii Edl z CacMnYnkMupøic . dMeNaHRsay edaHRsaysmIkar 9 7i 2z | z | tag z x i .y , x; y IR 1 i eK)an 2(x iy ) | x iy | ((917ii)()(11ii)) 2x 2iy x 2 y 2
9 9i 7 i 7 2
( 2x x 2 y 2 ) 2iy 1 8i
2x x 2 y 2 1 (1) eKTaj 2y 8 ( 2) tam (2) eKTaj y 4 ykeTACYskñúg (1) eK)an 2x x 2 16 1 2x 1 x 2 16
, x
1 2
( 2x 1) 2 x 2 16 4x 2 4x 1 x 2 16 3x 2 4x 15 0
; ' 4 45 7 2
eroberogeday lwm plÁún
- TMBr½77 -
sikSaKNitviTüaedayxøÜnÉg
eKTajb¤s x1 2 3 7 3 eK)an x 3 ; y 4 . dUcenH
z 3 4i
cMnYnkMupøic ; x2
27 5 1 3 3 2
¬minyk¦
CacemøIyrbs;smIkar .
eroberogeday lwm plÁún
- TMBr½78 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI13 eK[cMnYnkMupøicBIr Z1 nig Z 2 Edl Z 2 0 . cUrRsaybBa¢ak;fa ZZ1 || ZZ1 || . 2
dMeNaHRsay RsaybBa¢ak;fa
2
Z1 |Z | 1 Z2 | Z2 |
eyIgtag Z1 a i.b nig Z 2 c i . d Edl a , b , c , d CacMnYnBit . Z 1 a i .b (a i .b )(c i .d ) ac i .ad i .bc i 2 .bd eyIg)an 2 2 2 Z2
eK)an naM[
c i .d
(c i .d )(c i .d )
c i .d
Z 1 (ac bd ) i(bc ad ) ac bd bc ad i . Z2 c2 d2 c2 d2 c2 d2 2
Z1 ac bd bc ad 2 2 2 2 Z2 c d c d
2
(ac bd ) 2 (bc ad ) 2 c2 d2 a 2c 2 2abcd b 2d 2 b 2 c 2 2abcd a 2d 2 c2 d2 (a 2 c 2 a 2 d 2 ) ( b 2 c 2 b 2 d 2 ) c2 d2 a 2 ( c 2 d 2 ) b 2 (c 2 d 2 )
eroberogeday lwm plÁún
c d 2
2
(c 2 d 2 )(a 2 b 2 ) c2 d2
- TMBr½79 -
sikSaKNitviTüaedayxøÜnÉg
eKTaj
Z1 Z2
cMnYnkMupøic
(a 2 b 2 )(c2 d 2 ) (c d ) 2
2 2
a2 b2 c2 d 2
| Z | a2 b2 edayeKman 1 | Z 2 | c 2 d 2 Z1 |Z | 1 . dUecñH Z2 | Z2 |
eroberogeday lwm plÁún
- TMBr½80 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI14 eK[cMnYnkMupøicBIr Z1 nig Z 2 . cUrRsaybBa¢ak;fa | Z1 Z 2 | | Z1 | | Z 2 | . dMeNaHRsay RsaybBa¢ak;fa | Z1 Z 2 | | Z1 | | Z 2 | eyIgtag Z1 a i.b nig Z 2 c i . d Edl a , b , c , d CacMnYnBit . eyIgman Z1 Z 2 (a i.b)(c i.d) ac i.ad i.bc i 2 .bd naM[ Z1 Z 2 (ac bd) i.(ad bc) eK)an | Z1 Z 2 | (ac bd) 2 (ad bc) 2 a 2 c 2 2abcd b 2 d 2 a 2 d 2 2abcd b 2 c 2 (a 2 c 2 b 2 c 2 ) ( a 2 d 2 b 2 d 2 ) c 2 (a 2 b 2 ) d 2 (a 2 b 2 ) (a 2 b 2 )(c 2 d 2 )
eKTaj
| Z1 Z 2 | a 2 b 2 c 2 d 2
edayeKman dUcenH
| Z | a2 b2 1 | Z 2 | c 2 d 2 | Z 1 Z 2 | | Z 1 | | Z 2 |
eroberogeday lwm plÁún
.
- TMBr½81 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI15 eK[cMnYnkMupøicBIr Z1 nig Z 2 . k> cUrRsaybBa¢ak;fa | Z1 Z 2 | | Z1 | | Z 2 | x> Taj[)anfa (a c) 2 (b d) 2 a 2 b 2 c 2 d 2 cMeBaHRKb; a , b , c , d CacMnYnBit . dMeNaHRsay k> RsaybBa¢ak;fa | Z1 Z 2 | | Z1 | | Z 2 | kñúgbøg;kMupøic ( XOY) eyIgeRCIserIsviucTr½BIr U nig V manGahVikerogKña Z 1 nig Z 2 naM[viucTr½ U V manGahVik Z 1 Z 2 . tamlkçN³RCugrbs;RtIekaNeK)an U V | U V | | U | | V | eday ³
U
| U | | Z1 | , | V | | Z 2 |
| U V || Z1 Z 2 |
dUcenH
V
| Z1 Z 2 | | Z1 | | Z 2 |
eroberogeday lwm plÁún
- TMBr½82 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
x> Taj[)anfa (a c) 2 (b d) 2 a 2 b 2 c 2 d 2 eyIgtag Z1 a i.b nig Z 2 c i . d Edl a , b , c , d CacMnYnBit . man Z1 Z 2 (a c) i.(b d) naM[ | Z1 Z 2 | (a c) 2 (b d) 2 ehIy | Z1 | a 2 b 2 , | Z 2 | c 2 d 2 . tamsRmayxagelIeKman | Z1 Z 2 | | Z1 | | Z 2 | dUcenH
(a c ) 2 ( b d ) 2 a 2 b 2 c 2 d 2
eroberogeday lwm plÁún
.
- TMBr½83 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI16 eK[cMnYnkMupøic ³ z1
6 i. 2 ni g z2 1 i 2 k>cUrsresr z 1 , z 2 nig Z zz1 CaragRtIekaNmaRt. 2 x>cUrsresr Z zz1 CaragBiCKNit. 2 K>Taj[)anfa cos 12 6 4 2 nig sin 12 6 4 2 .
dMeNaHRsay z1 k>sresr z 1 , z 2 nig Z z CaragRtIekaNmaRt³ 2
3 6i 2 1 2 i . 2 cos i . sin 2 2 6 6 2 z 1 2 cos( ) i . sin( ) . 6 6
eKman z1 dUcenH
2 2 2 cos i . sin 2 i. 2 4 4 2 z 2 2 cos( ) i . sin( ) . 4 4
eKman z 2 1 i dUcenH
eroberogeday lwm plÁún
- TMBr½84 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
2 cos( ) i . sin( ) 6 4 6 4 2 2 Z cos i . sin . 12 12
eKman Z zz1 dUcenH
z1 x> sresr z CaragBiCKNit 2 eK)an Z 26(1i i )2 ( 26(1i i )(21)(1 i ) i ) Z
dUcenH
Z
6 2 6 2 i. 4 4
K> TajeGay)anfa cos 12 tamsRmayxagelIeKman ³ Z cos
i . sin 12 12
(1)
6 2 4
6i 6i 2 2 4
. nig sin 12
nig Z
6 2 4
6 2 6 2 i. 4 4
( 2)
edaypÞwm ¬!¦ nig ¬@¦ eKTaj)an ³ cos
12
6 2 4
eroberogeday lwm plÁún
nig
sin
12
6 2 4
.
- TMBr½85 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI17 eK[cMnYnkMupøic z 2 2 i. 2 2 . k-cUrsresr z 2 CaTMrg;BiCKNit . x-cUrsresr z 2 nig z CaTMrg;RtIekaNmaRt . K-TajrktMélR)akdén cos 8 nig sin 8 . dMeNaHRsay k-sresr z 2 CaTMrg;BiCKNit eyIg)an z 2 ( 2 2 i. 2 2 )2
2 2
2 2i
2 2
2 2 i. 2 2
2
2 2 2i . 4 2 2 2 2 2 2 2 .i
dUcenH
z 2 2 2 2 2 .i
x-sresr z 2 nig z CaTMrg;RtIekaNmaRt eKman z 2 2 2 2 2 .i dUcenH
2 2 4 cos i . sin 4 i. 2 4 4 2 z 2 4 cos i . sin nig z 2 cos i . sin 4 4 8 8
eroberogeday lwm plÁún
- TMBr½86 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
K-TajrktMélR)akdén cos 8 nig sin 8
tamsMrayxagelIeKman z 2 cos 8 i. sin 8 eday z 2 2 i. 2 2 eKTaj 2 cos 8 i. sin 8 2 2 i. 2
dUcenH
cos
2 2 8 2
eroberogeday lwm plÁún
nig
sin
2 2 8 8
2
.
- TMBr½87 -
sikSaKNitviTüaedayxøÜnÉg
lMhat;TI18 eK[cMnYnkMupøic Z cos 45 i. sin 45 cUrRsaybBa¢ak;fa dMeNaHRsay RsaybBa¢ak;fa ³
(1 Z ) 3 8 cos 3
cMnYnkMupøic
2 6 6 (cos i . sin ) 5 5 5
.
2 6 6 (cos i . sin ) 5 5 5 eyIgman 1 Z 1 cos 45 i. sin 45 tamrUbmnþ 1 cos 2 cos 2 2 nig sin 2 sin 2 cos 2 eyIg)an 1 Z 2 cos 2 25 2i sin 25 cos 25 2 2 2 1 Z 2 cos (cos i . sin ) b¤ 5 5 5 (1 Z ) 3 8 cos 3
tamrUbmnþdWmr½eyIg)an ³
3
2 2 2 (1 Z ) 3 2 cos (cos i . sin ) 5 5 5 2 6 6 (cos i . sin ) 8 cos 3 5 5 5 dUcenH (1 Z)3 8 cos 3 25 (cos 65 i. sin 65 )
eroberogeday lwm plÁún
. - TMBr½88 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI19 eK[cMnYnkMupøic Z 4 2(1 i ) k-cUrsresr Z CaTRmg;RtIekaNmaRt . x-KNnab¤sTI# én Z . dMeNaHRsay k-sresr Z CaTRmg;RtIekaNmaRt ³ eyIg)an Z 4 2(1 i ) 2 2 i . ) 8( cos i . sin ) 2 2 4 4 8 cos( ) i . sin( ) 4 4 3 3 8(cos i . sin ) 4 4 3 3 Z 8 (cos i . sin ) . 4 4 8 (
dUcenH
x-KNnab¤sTI# én Z eyIgtag Wk Cab¤sTI# én Z 8 (cos 34 i. sin 34 )
tamrUbmnþb¤sTI n : Wk n r cos( n2k ) i. sin( n2k )
eroberogeday lwm plÁún
- TMBr½89 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
3 3 2 k 2 k 4 4 3 eyIg)an Wk 8 cos ( 3 ) i. sin( 3 ) 3 8k 3 8k ) i . sin( 2 [cos( )] 12 12 -ebI k 0 : W0 2 (cos 4 i. sin 4 ) 11 11 -ebI k 1 : W1 2(cos 12 i sin 12 ) -ebI k 2 : W2 2 (cos 1912 i. sin 1912 ) .
eroberogeday lwm plÁún
- TMBr½90 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI20 eKeGay f z z 3 2 3 i .z 2 41 i 3 .z 8i k>cUrbgðajfa z ¢ f z z 2i z 2 2 3z 4 x>edaHRsaysmIkar f z 0 kñúgsMNMukMupøic . dMeNaHRsay k> bgðajfa z ¢ : f z z 2i z 2 2 3z 4 eyIgman f z z 2i .z 2 2 3z 4 edayBnøatGnuKmn_enHeyIg)an ³ f ( z ) z 3 2 3z 4z 2i( z 2 2 3z 4) z 3 2 3 .z 2 4z 2iz 2 4 3iz 8i z 3 2 3 i .z 2 41 i 3 .z 8i
dUcenH z ¢ f z z 2i z 2 2
3z 4
Bit
.
x> edaHRsaysmIkar ebI f z 0 naMeGay z 2i .z 2 2 3 .z 4 0 eKTajb¤s z 2i ehIy , z 2 2 3 .z 4 0 , ' 3 4 1 i 2 naMeGay z 1 3 i , z 2 3 i .
eroberogeday lwm plÁún
- TMBr½91 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI21 eK[cMnYnkMupøic z cos 47 i. sin 47 . cUrsresr (1 z )4 CaragRtIekaNmaRt . dMeNaHRsay sresr (1 z )4 CaragRtIekaNmaRt³ eKman z cos 47 i. sin 47 eK)an
4 4 i . sin 7 7 2 2 2 1 z 2 cos 2 2i . sin cos 7 7 7 2 2 2 2 cos (cos i . sin ) 7 7 7 1 z 1 cos
tamrUbmnþdWmr½eKGacsresr ³
4
2 2 2 (1 z ) 2 cos (cos i . sin ) 7 7 7 2 8 8 16 cos4 i . sin cos 7 7 7 4
dUcenH
(1 z )4 16 cos 4
eroberogeday lwm plÁún
2 8 8 i . sin cos 7 7 7
.
- TMBr½92 -
sikSaKNitviTüaedayxøÜnÉg
lMhat;TI22 eK[cMnYnkMupøic z 1 1 i nig k-cUrsresr z 1
,z2
cMnYnkMupøic
z2
6 i. 2 2
nig zz1 CaTMrg;RtIekaNmaRt . 2
x-cUrsresr zz1 CaTMrg;BiCKNit . 2
K-eRbIlTæplxagelIcUrTajrktMélR)akdén cos 12 nig sin 12 . dMeNaHRsay k-sresr z 1 , z 2 nig zz1 CaTMrg;RtIekaNmaRt 2
eKman z 1 1 i eday r 12 12 2 eK)an z1 2 22 i. 22 2 cos 4 i. sin 4 .
eKman z 2 6 2i. 2 2 . 32 i. 2 3 1 z 2 2 i . 2 cos i . sin . 2 6 6 2 2 cos i . sin 4 cos i . sin eK)an zz1 4 . 12 12 2 2 cos i . sin 6 6
eroberogeday lwm plÁún
- TMBr½93 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
x-sresr zz1 CaTMrg;BiCKNit 2
1 i 21 i 6 i . 2 eK)an 6 i . 2 6 i . 2 6 i . 2 2 2 6 i . 2 i . 6 2 62 6 2 i. 6 2 4 dUcenH zz1 6 4 2 i. 6 4 2 . 2 K-TajrktMélR)akdén cos 12 nig sin 12 z1 z2
tamsMrayxagelIeKman ³
z1 z1 6 2 6 2 cos i . sin i. ni g z2 12 12 z2 4 4 eKTaj cos 12 i. sin 12 6 4 2 i. 6 4 2 dUcenH cos 12 6 4 2 nig sin 12 6 4 2 .
eroberogeday lwm plÁún
- TMBr½94 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI23 eK[sIVúténcMnYnkMupøic (Z n ) kMnt;eday
1 i 3 Z 0 2 1 Z n 1 2 Z n | Z n | ; n IN
¬ | Z n | Cam:UDulén Z n ¦. snμtfa Z n n (cos n i. sin n ) , n IN Edl n 0 , n ; n IR . k-cUrrkTMnak;TMngrvag n nig n1 ehIy n nig n1 . x-rkRbePTénsIVút (n ) rYcKNna n CaGnuKmn_én n . K-cUrbgðajfa n 0 cos 0 cos 21 cos 22 .... cos n2 1 rYcbBa¢ak; n GnuKmn_én n . dMeNaHRsay k-rkTMnak;TMngrvag n nig n1 ehIy n nig n1 eyIgman Z n n (cos n i. sin n ) naM[ Z n1 n1 (cos n1 i sin n1 ) eday Z n 1 12 ( Z n | Z n | ) ehIy | Z n | n eK)an
n 1 (cos n 1 i sin n 1 )
eroberogeday lwm plÁún
1 n (cos n i. sin n ) n 2
- TMBr½95 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
1 n 1 (cos n 1 i . sin n 1 ) n (1 cos n i . sin n ) 2 n 1 (cos n 1 i . sin n 1 ) n cos n (cos n i . sin n ) 2 2 2 eKTaján n 1 n cos 2n nig n 1 2n dUcenH n 1 2n nig n 1 n cos 2n .
x-RbePTénsIVút (n ) nig KNna n CaGnuKmn_én n tamsRmayxagelIeyIgman n 1 12 n naM[ n CasIVútFrNImaRtmanersug esμI q 12 . tamrUbmnþ n 0 q n 1 i Z (cos i sin ) eday 0 0 0 0 2 eKTaj)an 0 1 ; 0 3 dUcenH
n
1 . n 3 2
eroberogeday lwm plÁún
3
cos i . sin 3 3
.
- TMBr½96 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
K-bgðajfa n 0 cos 0 cos 21 cos 22 .... cos 2n tamsRmayxagelIeKman n 1 n cos 2n
b¤ eK)an
n 1 n cos n 2 k n 1 k k n 1 k 1 cos( ) 2 k 0 k k 0 n 1 2 n 1 cos 0 . cos . cos ......... cos 0 2 2 2
1 cos 2 .... cos n 1 . 2 2 2 mü:ageToteyIgman sin n 2 sin 2n cos 2n 2 sin n 1 cos 2n eKTaj cos 2n 12 . sinsinn n1 sin n 1 1 sin 0 sin 1 1 sin 0 . . ..... . ehtuenH n n sin sin n sin n 2 2 sin n 1 2 sin dUcenH n 1n 31 n 31 . 1 1 . 2 sin . ) 2 sin( . ) n 3 2 3 2n
dUcenH
n 0 cos 0 cos
eroberogeday lwm plÁún
- TMBr½97 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI24 eK[cMnYnkMupøic z x i.y Edl x nig y CaBIrcMnYnBit . cUrkMnt;témø x nig y ebIeKdwgfa ³ 10 ( 3 2i ) z (1 3i ) z ¬ z CacMnYnkMupøicqøas;én z ¦ . 2i dMeNaHRsay eKman (3 2i ) z (1 3i ) z 210 i eday z x i.y naM[ z x i.y eK)an (3 2i )(x iy ) (1 3i )(x iy ) 210 i 3x 3iy 2ix 2y x iy 3ix 3y
10( 2 i ) 5
(4x y ) i .( 5x 2y ) 4 2i
4x y 4 5x 2y 2 4 1 4 1 D 8 5 3 , Dx 82 6 5 2 2 2 4 4 Dy 8 20 12 5 2 Dy Dx 6 12 x 2 ,y 4 D 3 D 3
eKTaj)an eKman nig eK)an dUcenH
x 2 , y 4
eroberogeday lwm plÁún
. - TMBr½98 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI25
eK[ z ; z CacMnYnkMupøicEdl | z || z | r 0 . bgðajfa rz zzz rz zzz r1 1
2
1
2
2
1
2
2
2
1 2
1
2
2
1 2
2
dMeNaHRsay
bgðajfa ³ 2
2
z1 z 2 z1 z 2 1 2 2 2 r r z1z 2 r z 1z 2
tag z r(cos 2x i sin 2x) nig z Edl x IR ; y IR . eK)an ³ 1
2
r(co2y i sin 2y )
z1 z 2 r [ (cos 2x cos 2y ) i(sin 2x sin 2y ) ] r 2 z1z 2 r 2 r 2 [ cos( 2x 2y ) i . sin( 2x 2y ) ] 2 cos( x y ) cos( x y ) 2i sin( x y ) cos( x y ) r [ 2 cos 2 ( x y ) 2i sin( x y ) cos( x y ) ] 1 cos( x y ) . r cos( x y ) z1 z 2 1 sin( y x ) r 2 z1z 2 r sin( y x )
dUcKñaEdr eK)an ³
eroberogeday lwm plÁún
- TMBr½99 -
sikSaKNitviTüaedayxøÜnÉg 2
cMnYnkMupøic 2
z1 z 2 z1 z 2 1 cos 2 ( x y ) sin 2 ( y x ) 2 2 2 2 2 r z z r z z r cos ( x y ) sin ( y x ) 1 2 1 2 cos 2 ( x y ) 2 2 cos (x y ) 1 cos ( x y ) 2 cos ( x y ) sin 2 ( y x ) 2 2 sin ( x y ) sin (x y ) 1 2 sin ( y x ) cos 2 ( x y ) sin 2 ( y x) 2 2 cos ( x y ) sin (x y ) 1 2 2 cos ( x y ) sin ( y x )
eday
eRBaH
ehIy
eRBaH
dUcenH
2
2
z1 z 2 z1 z 2 1 2 2 2 r r z1z 2 r z 1z 2
eroberogeday lwm plÁún
.
- TMBr½100 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI26
eKyk z
; z 2 ; ....; z n
1
CacMnYnkMupøicEdlepÞógpÞat;TMnak;TMng
(k 1)z k 1 i(n k )z k 0 ; k 0 , 1 , 2 , ... , n 1
k-kMnt; z ebIeKdwgfa z z z .... z 2 x-cMeBaHtémø z Edl)ankMnt;xagelIcUrbgðajfa ³ 0
0
1
2
n
n
0
( 3n 1)n | z 0 | | z1 | | z 2 | .... | z n | n! 2
2
2
2
dMeNaHRsay
k-kMnt; z ebIeKdwgfa z z z .... z eKman (k 1)z i(n k )z 0 eK)an zz i. nk k1 0
0
k 1
1
2
n
2n
k
k 1
k ( p 1)
k 0
z k 1 p 1 n k i . z k 1 k k 0 zp n! i pCpn ; Cpn z0 p!(n p )!
eKTaj z
p
eday z
z1 z 2 .... z n 2
0
i p z 0 Cpn ; p 0 , 1 , 2 , ....
eroberogeday lwm plÁún
n
b¤ (z ) 2 n
p0
n
p
- TMBr½101 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
man (z ) z C i z (1 i) eK)an z (1 i) 2 eKTaj z (1 2 i) (1 i) dUcenH z (1 i) . x-cMeBaHtémø z Edl)ankMnt;xagelIcUrbgðajfa ³ n
n
p
p0
0
p0
n
n
p p n
0
n
0
n
n
0
n
n
0
0
( 3n 1)n | z 0 | | z1 | | z 2 | .... | z n | n! 2
2
2
2
edayGnuvtþn_vismPaB AM GM eyIg)an z 0 2 z1 2 ... z n 2 z 0
2
(C )
z 0 2 Cn2n 2n .
0 2 n
(C1n )2 .... (Cnn )2
( 2n )! n! n!
2n 2n( 2n 1)( 2n 2)....(n 1)) n! 2n 2n ( 2n 1) ( 2n 2) .... (n 1) n! n
n
( 3n 1)n n!
dUcenH
( 3n 1)n | z 0 | | z1 | | z 2 | .... | z n | n! 2
2
eroberogeday lwm plÁún
2
2
.
- TMBr½102 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI27
eK[ z
z1 z 2 z 3
CacMnYnkMupøicedaydwgfa ³ z z z z z z 0 .
cUrbgðajfa
| z1 || z 2 || z 3 |
1
; z2 ; z3
1 2
2 3
3 1
dMeNaHRsay
bgðajfa | z || z || z | eKman z z z z z z z z z 0 (1) z z z eKTaj z z z (z z ) 0 (2) yk (1) CMnYskñúg (2) eyIg)an ³ z z z 0 naM[ | z | | z | .| z | dUcKñaEdr | z | | z | . | z | nig | z | | z | . | z eyIg)an ³ 1
1
2
2
1
3
3
1 2
2
1 2
3 1
3
3
1
2
2
1 2
2 3
2
3
3
1
2
2
2
2
1
3
1
2
3
|
| z 1 |2 | z 2 |2 | z 3 |2 | z1 || z 2 | | z 2 || z 3 | | z 3 || z1 |
b¤ (| z | | z |) dUcenH | z || z
2
1
2
1
2
(| z 2 | | z 3 |)2 (| z 3 | | z1 |)2 0 || z 3 |
eroberogeday lwm plÁún
.
- TMBr½103 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI28
cUrbgðajfa
6z i 1 2 3iz
luHRtaEt
|z|
1 3
dMeNaHRsay
bgðajfa | z | 13 eKman 26z 3izi 1 lkçxNÐ 2 3iz 0 b¤ z 23i eK)an | 6z i || 2 3iz |
dUcenH
| 6z i |2 | 2 3iz |2 (6z i )(6 z i ) ( 2 3iz )( 2 3iz ) 36zz 6iz 6iz 1 4 6iz 6iz 9zz 27 zz 3 1 zz 9 1 | z |2 9 1 |z| 3
.
eroberogeday lwm plÁún
- TMBr½104 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI29
eK[ z ; z ; z ; ....; z CacMnYnkMupøicEdlmanm:UDúlesμI 1 . eKtag Z (z ) ( z1 ) . cUrbgðajfa 0 Z n 1
2
3
n
n
n
k
k 1
k 1
k
2
dMeNaHRsay
bgðajfa 0 Z n eday z ; z ; z ; ....; z CacMnYnkMupøicEdlmanm:UDúlesμI 1 enaHeKGactag z cos x i. sin x Edl x IR ; k 1 , 2 , 3 ,...., n eK)an Z (z ) ( z1 ) 2
1
2
3
n
k
k
k
k
n
n
k 1 n
k
k 1
k
n
cos xk i. sin xk cos xk i sin xk
k 1
k 1
2
2
n n cos xk sin xk 0 k 1 k 1
eK)an Z 0 müa:geTottamvismPaB Cauchy Schwartz 2
n n cos xk n cos 2 xk k 1 k 1
eroberogeday lwm plÁún
- TMBr½105 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
nig eKTaj Z n cos x n (sin 2
n n sin xk n sin 2 xk k 1 k 1 n
Zn
n
2
k
k 1 n
k 1
2
xk )
cos2 xk sin 2 xk
k 1
dUcenH
Z n.n n 2
. sMKal; ³ eKGacRsay Z n tammYyrebobeTotdUcxageRkam eday | z | 1 enaH z z1 RKb; k 1 ; 2 ;.....; n 0 Z n2
2
k
k
k
eK)an
n n 1 Z ( z k ) ( ) k 1 k 1 z k n n n n z k zk z k z k k 1 k 1 k 1 k 1
n
z k
k 1
2
2
n | zk | n2 k 1
eKTaj)an Z n . 2
eroberogeday lwm plÁún
- TMBr½106 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI30
eK[cMnYnkMupøic z Edl | z | 1 . cUrbgðajfa ³ 2 |1 z | |1 z | 4 . 2
dMeNaHRsay
bgðajfa 2 | 1 z | | 1 z | 4 tag z cos t i. sin t eK)an | 1 z | (1 cos t ) sin t 2 | sin 2t | ehIy | 1 z | (1 cos 2t ) sin 2t 2 | cos t | 2
2
2
2
2
2
t 2 | 1 2 sin 2 | 2 t t | 1 z | | 1 z 2 | 2 | sin | | 1 2 sin 2 | 2 2 t x sin ; 1 x 1 f 2
eK)an ehIytagGnuKmn_ edayyk kMnt;eday f (x) | x | | 1 2x | Edl 1 x 1 cMeBaH 1 x 1 eK)an 22 f (x) 2 dUcenH 2 | 1 z | | 1 z | 4 . 2
2
eroberogeday lwm plÁún
- TMBr½107 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI31 eK[cMnYnkMupøic Z (cos x 1 ) i.(sin cos x Edl x CacMnYnBit. cUrkMnt´rkm¨UDulGb,brmaéncMnYnkMupøicen¼ ? dMeNaHRsay rkm¨UDulGb,brmaéncMnYnkMupøic 2
2
x
1 ) sin 2 x
1 2 1 2 2 ) (sin x ) 2 2 cos x sin x 1 2 1 2 2 f ( x ) (cos 2 x ) (sin x ) 2 2 cos x sin x 1 1 4 sin x 2 cos 4 x 2 cos 4 x sin 4 x 1 1 ) 4 (cos 4 x sin 4 x ) ( 4 4 sin x cos x cos 4 x sin 4 x 4 4 4 (cos x sin x ) ( ) 4 4 sin x cos x
eyIgán tag
2
| Z | (cos 2 x
eroberogeday lwm plÁún
- TMBr½108 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
f ( x ) 4 (cos 4 x sin 4 x )(1
1 ) 4 4 sin x cos x
4 (cos 2 x sin 2 x ) 2 2 sin 2 x cos 2 x (1
16 ) sin 4 2x
1 16 4 (1 sin 2 2x )(1 ) 4 2 sin 2x
edayeKman
sin 2 2x 1
naM[
1 1 1 sin 2 2x 2 2
16 17 4 sin 2x eKTaj 4 (1 1 sin 2 2x)(1 164 ) 4 17 25 2 2 2 sin 2x eyIgán f ( x) 4 (1 1 sin 2 2x)(1 164 ) 25 2 2 sin 2x 5 5 2 eday | Z | f (x) eKTaján | Z | 2 2 dUcen¼m¨UDulGb,brmaén Z KW | Z |min 5 2 . 2
nig
1
eroberogeday lwm plÁún
- TMBr½109 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI32
eK[ cUrbgðajfa
n
1 1 A i i 3 3 A i.
2 n 1 n
( 3)
. sin
n 3
n
, n IN
.
cMeBaHRKb; n IN .
dMeNaHRsay
bgðajfa
A i.
2 n 1 ( 3)
n
. sin
n 3
cMeBaHRKb; n IN .
eyIgman , n IN tag Z 13 i 1 i3. 3 23 12 i 23 23 cos 3 i.sin 3 tamrUbmnþdWmr½eK)an Z 1 i 2 cos n3 i. sin n3 ( 3) 3 ehIy Z 2 cos n3 i. sin n3 ( 3) eKTaj A 2 cos n3 i. sin n3 2 cos n3 i. sin n3 n
1 1 i i A 3 3
n
n
n
n
n
n
n
n
n
n
( 3 )n
( 3 )n
n 1
n n n n 2 n i . sin i . . sin cos i . sin cos 3 3 3 3 3 ( 3 )n ( 3 )n 2n
dUcenH
A i.
2 n 1 ( 3)
n
eroberogeday lwm plÁún
. sin
n 3
.
- TMBr½110 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI33
kñúgbøg;kMupøic (O , i , j ) eK[bYncMnuc A , B , C , D EdlmanGahVikerogKña Z 1 6i , Z 4 5i , Z 5 4i nig Z 2 3i . cUrRsayfactuekaN ABCDcarikkñúgrgVg;mYyEdleKnwgbBa¢ak;p©it nig kaMrbs;va . A
B
D
C
dMeNaHRsay
RsayfactuekaN ABCDcarikkñúgrgVg; eyIgtag (c): x y ax by c 0 CasmIkarrgVg;carikeRkARtIekaN ABC . eyIg)an A (c) naM[ 1 6 a 6b c 0 b¤ a 6b c 37 (1) B (c) naM[ 4 5 4a 5b c 0 b¤ 4a 5b c 41 (2) C (c) naM[ ( 2) ( 3) 2a 3b c 0 b¤ 2a 3b c 13 (3) 2
2
2
2
2
2
2
eroberogeday lwm plÁún
2
- TMBr½111 -
sikSaKNitviTüaedayxøÜnÉg
eyIg)anRbB½næsmIkar
cMnYnkMupøic
a 6b c 37 4a 5b c 41 2a 3b c 13
bnÞab;BIedaHRsayRbB½næenHeK)ancemøIuy a 2 , b 2 , c 23 . smIkarrgVg;carikeRkARtIekaN ABC Gacsresr ³ (c) : x 2 y 2 2x 2y 23 0
b¤ (c) : (x 1) (y 1) 25 müa:geTotedayykkUGredaen D CYskñúgsmIkar 2
2
(c) :( 2 1) 2 ( 3 1) 2 25
vaepÞógpÞat;enaHnaM[ D (c) . edaybYncMnuc A , B , C , D sßitenAelIrgVg;mansmIkar (c) : ( x 1) ( y 1) 25 EtmYy enaHnaM[ctuekaN ABCDcarikkñúgrgVg; (c) manp©it I( 1 , 1 ) nig kaM R 5 . 2
2
eroberogeday lwm plÁún
- TMBr½112 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI34
eK[ z nig z CaBIrcMnYnkMupøic . cUrbgðajfa | 1 z z | | z z | 1
2
1 2
1
| z 1 1 || z 2 1 | 2
2
2
dMeNaHRsay
bgðajfa | 1 z z | | z z | tamvismPaBRtIekaNeK)an ³ 1 2
1
2
| z 1 1 || z 2 1 | 2
2
| 1 z 1z 2 | | z 1 z 2 | | 1 z 1z 2 z 1 z 2 |
nig | 1 z z | | z z | | 1 z z z z | eK)an | 1 z z | | z z | | (1 z z ) (z z ) | eday (1 z z ) (z z ) 1 z z z z (1 z eK)an | 1 z z | | z z | | (1 z )(1 z ) | 1 2
1
2
1 2
1
2
2
1 2
1
2
1 2
2
2
2
2
1
2
1
2
1
2
1
2
2
1 2
2
1 2
2
1
2
1
2
2
2
2 1
)(1 z 2 ) 2
2
2
| 1 z1z 2 | | z1 z 2 | 2 | 1 z12 || 1 z 2 2 | dUcenH | 1 z1z 2 | | z1 z 2 | 2 | (1 z1z 2 )2 (z1 z 2 )2 |
eroberogeday lwm plÁún
- TMBr½113 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI35 eK[ z nig z CacMnYnkMupøicBIr . cUrRsayfa | z z | | z z | 2| z | | z | dMeNaHRsay Rsayfa | z z | | z z | 2| z | | z | eKman | z z | (z z )( z z ) 1
2
2
1
2
2
1
2
1
2
2
2
2
1
2
1
2
2
2
2
1
2
2
1
2
1
2
1
2
| z 1 z 2 |2 z 1 z1 z 1 z 2 z1z 2 z 2 z 2
| z 1 z 2 |2 | z 1 |2 z 1 z 2 z1z 2 | z 2 |2 1
ehIy
| z 1 z 2 |2 ( z 1 z 2 )( z1 z 2 )
| z 1 z 2 |2 z 1 z1 z 1 z 2 z1z 2 z 2 z 2 | z 1 z 2 |2 | z 1 |2 z 1 z 2 z1z 2 | z 2 |2
bUkTMnak´TMng
1
nig
2
eKán ½
| z 1 z 2 |2 | z 1 z 2 |2 2 | z 1 |2 | z 2 |2
eroberogeday lwm plÁún
2
.
- TMBr½114 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI36 eK[ z nig z CacMnYnkMupøicBIrEdl | z nig z .z 1 . cUrRsayfa 1z z zz CacMnYnBitmYy . dMeNaHRsay Rsayfa 1z z zz CacMnYnBitmYy 1
1
2
1
| | z2 | 1
2
1
2
1 2
1
2
1 2
tag
eday eKán eday
z1 z 2 1 z1z 2
ena¼
Z
z 1 .z1 | z 1 |2 1
ena¼
Z
z1 z 2 1 z1 .z 2 1 z1 z1
ehIydUcKña
z2
1 z2
1 1 z z1 z z2 Z 1 2 Z 1 1 z 2 z1 1 1 . z1 z 2 z z2 ZZ Z 1 1 z 1z 2
ena¼
eroberogeday lwm plÁún
CacMnYnBit .
- TMBr½115 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI37 eK[BhuFa P(x) (x sin a cos a) Edl n IN * cUrrksMNl´énviFIEckrvag P(x) nwg x 1 . dMeNaHRsay sMNl´énviFIEck tag R(x) CasMNl´énviFIEckrvag P(x) nwg x 1 -ebI n 1 ena¼ P(x) x sin a cos a dUcen¼ R(x) x sin a cos a CasMnl´énviFIEck . -ebI n 2 eKán P(x) (x 1)Q(x) R(x) Edl Q(x) CaplEck nwg R(x) Ax B eKán (x sin a cos a) (x 1)Q(x) Ax B ebI x i ena¼ (i sin a cos a) Ai B ¦ cos(na) i. sin(na) B i.A eKTaj A sin(na) nig B cos(na) dUcen¼ R(x) x sin(na) cos(na) . n
2
2
2
n
2
n
eroberogeday lwm plÁún
- TMBr½116 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI38
eK[cMnYnkMupøic z 2 cos i 2 sin Edl IR kñúgbøg;kMpøic (o , i , j ) eKehA M CacMnucrUbPaBén z . cUrkMNt;témøtUcbMput nig FMbMputén r OM ? dMeNaHRsay
kMNt;témøtUcbMput nig FMbMputén r OM eKman ³ OM 2 a 2 b 2 ( 2 cos ) 2 ( 2 sin ) 2 OM 2 4 cos sin
b¤ OM
4 cos sin
eK)an r 4 cos sin tamvismPaB Cauchy Schwarz eKman ³ | cos sin | 12 12 cos 2 sin 2 2
b¤ 2 cos sin 2 eKTaj 4 2 r 4 2 RKb; IR dUcenH rmin 4 2 nig rmax 4 2 .
eroberogeday lwm plÁún
- TMBr½117 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI39
eK[cMnYnkMupøic z1 , z 2 , z 3 ehIyepÞógpÞat;TMnak;TMng ³ | z 1 || z 2 || z 3 | 1 nig
z 32 z12 z 22 1 0 z 2 z 3 z 1z 3 z 1z 2
cUrRsayfa | z1 z 2 z 3 | { 1 , 2 } . dMeNaHRsay Rsayfa | z1 z 2 z 3 | { 1 , 2 } eKman
z 32 z12 z 22 1 0 z 2 z 3 z 1z 3 z 1z 2
eK)an z13 z 2 3 z 3 3 z1z 2 z 3 0 b¤ z13 z 2 3 z 3 3 3z1z 2 z 3 4z1z 2 z 3 tag z z1 z 2 z 3 eK)an ³ z 3 3z ( z 1 z 2 z 2 z 3 z 1 z 3 ) 4z 1 z 2 z 3 1 1 1 z z 1 z 2 z 3 3z ( ) 4 z1 z 2 z 3 3
z 3 z 1z 2 z 3 3z( z 1 z 2 z 3 ) 4
z 3 z 1 z 2 z 3 ( 3 z .z 4 ) z 1 z 2 z 3 3 | z | 2 4
eK)an | z |3 | z1z 2 z 3 (3 | z |2 4) | eroberogeday lwm plÁún
- TMBr½118 -
sikSaKNitviTüaedayxøÜnÉg
b¤ | z |3 | 3 | z |2 4 | -ebI 3 | z |2 4 0 b¤ | z | eK)an | z |3 3 | z |2 4
cMnYnkMupøic 2 3
| z |3 3 | z |2 4 0 (| z | 1)(| z | 2) 2 0 | z | 2 2 2 -ebI 3 | z | 4 0 b¤ | z | 3 eK)an | z |3 (3 | z |2 4) | z |3 3 | z |2 4 0 (| z | 1)(| z | 2) 2 0 | z | 1
dUcenH | z1 z 2 z 3 | { 1 , 2 } .
eroberogeday lwm plÁún
- TMBr½119 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI40
eK[cMnYnkMupøic z1 nig z 2 Edl | z1 || z 2 | 1 cUrRsayfa | z1 1 | | z 2 1 | | z1z 2 1 | 2 dMeNaHRsay
Rsayfa | z1 1 | | z 2 1 | | z1z 2 1 | 2 tamvismPaBRtIekaN | a | | b | | a b | eK)an ³ | z 2 1 | | z 1z 2 1 || z 2 1 z 1 z 2 1 |
| z 2 1 | | z 1z 2 1 || z 2 || 1 z 1 || 1 z 1 |
ehIy | z1 1 | | 1 z1 || (z1 1 1 z1 ) | 2 dUcenH | z1 1 | | z 2 1 | | z1z 2 1 | 2 .
eroberogeday lwm plÁún
- TMBr½120 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI41
eK[sIVútcMnYnBit (an ) kMNt;eday ³
a 1 1 , a 2 1 a n 2 a n 1 a n , n 1 , 2 , 3 , ... 1 i 3 eKtagsIVútcMnYnkMupøic z n an1 an . 2 k> cUrRsayfa z n1 1 i 3 z n cMeBaHRKb; n 1 . 2 x> cUrdak; 1 i 3 CaTRmg;RtIekaNmaRtrYcTajrk z n CaGnuKmn_ 2 én n .
K> TajrktYTUeTAénsIVút an . etI (an ) CasIVútxYbb¤eT ? dMeNaHRsay
k> Rsayfa z n1 1 i
3
zn
cMeBaHRKb; n 1
2 eKman z n an1 1 i 3 an 2 1 i 3 eK)an z n1 an 2 a n 1 2 eday an 2 an1 an
eroberogeday lwm plÁún
- TMBr½121 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
1 i 3 eK)an z n1 an1 an a n 1 2 1 i 3 a n 1 a n 2 1 i 3 2 (a n 1 an ) 2 1 i 3 1 i 3 1 i 3 (a n 1 an ) 2 2 1 i 3 zn . dUcenH z n1 2 x> dak; 1 i 3 CaTRmg;RtIekaNmaRt ³ 2 eK)an 1 i 3 1 i 3 cos i sin 2 2 2 3 3 Tajrk z n CaGnuKmn_én n ³
eday z n1 1 i
2
3
zn
enaH (z n ) CasIVútFrNImaRténcMnYn
cos i sin 2 3 3 1 i 3 1 i 3 z1 a 2 a1 cos i sin 2 2 3 3
kMupøicEdlmanersug q 1 i nigtY
eroberogeday lwm plÁún
3
- TMBr½122 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
tamrUbmnþ z n z1 qn1 (cos i sin )n 3
3
tamrUbmnþdWm½reK)an z n cos n i sin n . 3 3 K> TajrktYTUeTAénsIVút an eKman z n an1 1 i 3 an 2
eK)an z n (an1 an ) i
3 a n (1) 2 2 eday z n cos n i. sin n (2) 3 3 tamTMnak;TMng (1) & (2) eK)an 3 an sin n 2 3 dUcenH an 2 sin n . 3 3 2 ehIy (an ) CasIVútxYbEdlmanxYb p 6 . 3
eroberogeday lwm plÁún
- TMBr½123 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI42
eK[sIVúténcMnYnkMupøic (z n ) kMNt;eday ³ 2 3i 3i 2 3i z1 zn nig z n1 2 2 2 Edl n 1 , 2 , 3 , ... . k> tag w n z n 1 . bgðajfa (w n ) CasIVútFrNImaRténcMnYn kMupøic rYcKNna w n CaGnuKmn_én n edaysresrlTæplCaTRmg; RtIekaNmaRt . x> Tajbgðajfa z n 2 cos n (cos n i sin n ) . 12
dMeNaHRsay
12
12
k> bgðajfa (w n ) CasIVútFrNImaRténcMnYnkMupøic ³ eKman w n z n 1 eK)an w n1 z n1 1 3i 2 3i zn 1 2 2 3i 3i ( z n 1) wn 2 2
dUcenH (w n ) CasIVútFrNImaRténcMnYnkMupøic . eroberogeday lwm plÁún
- TMBr½124 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
KNna w n CaGnuKmn_én n ³ eK)an w n w1 qn1 eday w1 3 i cos i sin
2 6 6 3i nig q cos i sin 2 6 6 eK)an w n (cos i sin )n 6 6 dUcenH w n cos n i sin n ¬rUbmnþdWmr½¦ 6 6 n n n x> Tajbgðajfa z n 2 cos (cos i sin ) 12 12 12 eKman w n z n 1 enaH z n 1 w n n n z n 1 cos i sin 6 6 n n n 2 cos 2 2i . sin cos 12 12 12 n n n 2 cos (cos i sin ) 12 12 12 dUcenH z n 2 cos n (cos n i sin n ) . 12 12 12
eroberogeday lwm plÁún
- TMBr½125 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI43
eK[sIVúténcMnYnBit (un ) nig ( v n ) kMNt;eday ³ un vn u n 1 2 nig un vn v n 1 2 k> eKBinitüsIVúténcMnYnkMupøic z n un i.v n u1 v 1
2 2 2 2
Edl n 1
. cUrRsayfa (z n ) CasIVútFrNImaRténcMnYnkMupøic rYcKNna z n CaGnuKmn_én n edaysresrlTæplCaTRmg;RtIekaNmaRt . x> sMEdg un nig v n CaGnuKmn_én n . dMeNaHRsay
k> Rsayfa (z n ) CasIVútFrNImaRténcMnYnkMupøic ³ eKman z n u n i.v n eK)an z n1 un1 i.v n1
un vn u vn i. n 2 2 2i 2 2i 2 (u n iv n ) zn 2 1
dUcenH(z n ) CasIVútFrNImaRténcMnYnkMupøic . eroberogeday lwm plÁún
- TMBr½126 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
KNna z n CaGnuKmn_én n ³ eK)an z n z1 qn1 Et z1 u1 iv1 2 i
2 cos i . sin 2 2 4 4 2 2 nig q i cos i. sin 2 4 4 2 eK)an z n (cos i sin )n 4 4 dUcenH z n cos n i. sin n ¬rUbmnþdWm½r ¦ 4 4 x> sMEdg un nig v n CaGnuKmn_én n
eKman z n u n i.v n eday z n cos n i. sin n 4 n dUcenH un cos 4
eroberogeday lwm plÁún
4
nig
n v n sin 4
.
- TMBr½127 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
lMhat;TI44
eK[sIVúténcMnYnBit (un ) nig ( v n ) kMNt;eday ³ u 0 1 v0 3
u n 1 u n 2 v n 2 nig v n 1 2u n v n k> eKBinitüsIVúténcMnYnkMupøic z n un i.v n .
Edl n 0 2n
cUrRsayfa z n1 z n rYcTajfa z n z 0 . x> sMEdg un nig v n CaGnuKmn_én n . 2
dMeNaHRsay 2n
k>Rsayfa z n1 z n rYcTajfa z n z 0 ³ eKman z n un i.v n eK)an z n1 un1 iv n1 2
u n 2 v n 2 2iu n v n u n 2 2iu n v n (iv n ) 2 (u n iv n ) 2
dUcenH z n1 z n 2 .
eroberogeday lwm plÁún
- TMBr½128 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
müa:geTotebI n 0 enaH z1 z 0 2 ebI n 1 enaH z 2 z12 z 0 4 ebI n 2 enaH z 3 z 2 2 z 08 2k ]bmafavaBitdl;tYTI k KW z k z 0 2k 1 eyIgnwgRsayfavaBitdl;tYTI k 1 KW z k 1 z 0 2 2k eKman z k 1 z k Ettamkar]bma z k z 0 2k 2 2k 1 eK)an z k 1 (z 0 ) z 0 Bit . 2n dUcenH z n z 0 . x> sMEdg un nig v n CaGnuKmn_én n 2n eKman z n z 0 eday z 0 u 0 iv 0 1 i 3 2(cos i sin ) 3 3 2n 2n eK)an z n 2 (cos i sin ) 3 3 n n n 2 2 2 2 (cos i sin ) 3 3 2n 2n 2n 2n dUcenH un 2 cos ; v n 2 sin . 3 3
eroberogeday lwm plÁún
- TMBr½129 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
CMBUkTI3 lMhat;Gnuvtþn_ !> cUrsresrcMnYnkMupøicxageRkamCaTRmg;BICKNit a i.b ³ k> (2 5i )(3 i ) x> (1 i )(1 i ) K> (1 i )(1 2i )(1 3i ) X> (2 i )3 g> 4 1 3i c> 44 33ii q>
( 3 2i )(i 1) i 1
13 12i ( 2i 1)2 Q> 6i 8 2 i @> KNna (1 2i )3 nig (3i 4)4
2
C> 11 ii j> (i 2 31)(ii 3 2 )
#> eK[ x CacMnYnBit . kMNt; x edIm,I[ 13 ii 1x ii CacMnYnBit . $> eK[cMnYnkMupøic z1 3 2i nig z 2 2 4i . cUrkMNt;rUbPaBén z1 z 2 , z1 z 2 nig 2z 2 ? %> eK[ z1 2 i , z 2 1 2i nig z 2 1 3i k-KNna z13 z 23 z 33 nig z1 z 2 z 3 x-bgðajfa z13 z 23 z 33 3z1z 2z 3
eroberogeday lwm plÁún
- TMBr½130 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
^> eK[cMnYnkMupøicBIr z1 a i.b nig z 2 b i.a Edl a nig b CaBIrcMnYnBit . cUrkMnt;témørbs; a nig b edIm,I[ Z1 Z 2 7 5i . &> kMnt;cMnYnBit a nig b edIm,I[ (2 3i ) Cab¤sénsmIkar x 2 ax b 0 *> kMnt;cMnYnBit p nig q edIm,I[ 1 2i Carwsrbs;smIkar z 3 pz q 0 . (> k> cUrepÞógpÞat;fa 3 4i (2 i ) 2 x> edaHRsaysmIkar Z 2 (4 5i ) Z 3 11 i 0 . !0> k> cUrepÞógpÞat;fa 3 4i (1 2i )2 x> edaHRsaysmIkar Z 2 (2 i ) 2 Z 1 7i 0 . !!> eK[cMnYnkMupøic U 2 3i , V 3 2i nig W 1 5i . cUrepÞógpÞat;fa U 3 V 3 W 3 3U.V.W . !@> kMnt;BIrcMnYnBit x nig y ebIeKdwgfa ³ (1 3i )( x iy ) ( 2 i )( x iy ) 7 0 . !#> kMnt;BIrcMnYnBit x nig y ebIeKdwgfa (3 2i )x (1 3i )y 5(1123ii ) . z 2 2z 5
!$> eK[GnuKmn_ f (z ) 2 Edl z i . z 1 cUrKNna f (1 2i ) , f (1 i ) nig f (2 i ) .
eroberogeday lwm plÁún
- TMBr½131 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
!%> eK[ z 1 2i nig U a i.b . cUrkMnt;cMnYnBit a nig b edIm,I[ U 1 z z 2 z 3 z 4 . !^> k>eK[ z 1 i . cUrKNna z 2 . x> KNnaplbUk S 1 z z 2 z 3 ..... z 2006 . !&> eK[cMnYnkMupøic Z x i.y , x IR , y IR mankMupøicqøas;tageday Z . cUrkMnt;rktémø x nig y edIm,I[ (1 i ) Z (3 2i ) Z 2(19 i2i ) !*> eK[cMnYnkMupøic A 13 52ii , B 12 i6i nig C 816ii k> cUrsresr A , B , C CaTRmg;BICKNit . x> cUrepÞógpÞat;fa B 2 4A.C 4(2 i )2 . K> edaHRsaysmIkar A Z 2 B Z C 0 edaysresrb¤snImYy² CaTRmg;BICKNit . !(> eK[cMnYnkMupøic Z (x 2 y 2 ) 2ixy nig U 3 4i . k> cUrsresr U 3 CaTRmg;BICKNit . x> kMnt;cMnYnBit x nig y edIm,I[ Z U 3 . @0> k> cUrbgðajfa 77 36i (2 9i )2 x> edaHRsaysmIkar (2 i ) z 2 (8 i ) z 5(3 i ) 0 rYcsresrb¤snImYy²CaTRmg;BICKNit .
eroberogeday lwm plÁún
- TMBr½132 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
@!> cUrKNnab¤skaeréncMnYnkMupøic Z 45 28i . @@> cUrKNnab¤skaeréncMnYnkMupøic Z 40 42i . @#> eK[ Z 11 22 ii . cUrsresr Z CaTRmg;BICKNit rYcKNna Z 2 . @$> eK[cMnYnkMupøic z a b.i , a, b IR . cUrkMnt;témø a nig b edIm,I[ 16 z 2 23 z 2 i . @%> eK[smIkar (E) : z 3 (4 5i )z 2 5(1 3i ) 2(7 i ) 0 . k> kMnt;cMnYnBit b edIm,I[ z 0 bi Cab¤ssmIkar (E) . x> cUrsresrsmIkar (E) Carag (z z 0 )(z 2 pz q) 0 Edl p nig q CacMnYnkMupøicRtUvrk . K> edaHRsaysmIkar (E) kñúgsMNMukMupøic . @^> eK[bIcMnYnkMupøic ³ U i .x , V i .y , W i .z , , , , x, y , z IR 0 3 3 3 cUrbgðajfa U V W 3U.V.W luHRtaEt x y z 0 @&> eK[smIkar (E) : z 3 az 2 bz c 0 .
cUrrkcMnYnBit a, b, c edIm,I[ z 1 nig eroberogeday lwm plÁún
z 1 2i
.
Cab¤ssmIkar (E) . - TMBr½133 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
@*> eKmansmIkar (E): z 2 3z 4 6i 0 k-kMnt;cMnYnBit b edIm,I[ z 0 b.i Cab¤srbs;smIkar (E) . x-cUredaHRsaysmIkar (E) kñúgsMNMukMupøic . @(> eKmansmIkar (E): z 2 5(1 i )z 3(4 3i ) 0 k-cUrepÞógpÞat;fa 48 14i (7 i ) 2 x-cUredaHRsaysmIkar (E) kñúgsMNMukMupøic . #0> eKmansmIkar (E) : z 2 (a i.b)z a 3 5i 0 Edl a , b IR . kMnt;témø a nig b edIm,I[ z1 3 2i Cab¤smYyrbs;smIkar (E) rYccUrkMnt;rkb¤s z 2 mYyeTotcMeBaHtémø a nig b Edl)anrkeXIj . #!> eKmansmIkar (E) : z 3 (5 i )z 2 (10 9i )z 2(1 8i ) 0 . k-kMnt;cMnYnBit b edIm,I[ z 0 b.i Cab¤srbs;smIkar (E) . x-cUrsresrsmIkar (E) Carag (z z 0 )(z 2 pz q) 0 Edl p nig q CacMnYnkMupøiucBIrEdleKRtUvrk . K-epÞógpÞat;fa 8 6i (1 3i ) 2 rYcedaHRsaysmIkar (E) kñúgsMNMukMupøic. #@> cUredaHRsaysmIkarxageRkamkñúgsMNMukMupøic ³ a / iz 2 ( 2 3i )z (1 5i ) 0 b / ( 2 i )z 2 5(1 i )z 2( 3 4i ) 0 c / (1 i )z 2 (1 7i )z 2( 2 3i ) 0
eroberogeday lwm plÁún
- TMBr½134 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
##> begáItsmIkardWeRkTIBIrEdlman nig Cab¤skñúgkrNInImYy²xageRkam ³ k> a / 2 11i , 2 11i b / 2 3i , 2 3i
x> a /
3 2i , 1 3i
b / 2 3i , 3 2i
K> a / 1 2i
, 3 2i
b/ 3 i , 1 i 3
#$> eKmancMnYnkMupøic 1 3i nig 1 2i . cUrsresrsmIkardWeRkTIBIrmYyEdlman Z1 2 2 nig Z 2 2 . 2 Cab¤s> #%> cUrkMnt;rkcMnYnkMupøicEdlmanm:UDúlesμI 8 ehIykaerrbs;vaCacMnYnnimμitsuTæ. #^> cUrkMnt;BIrcMnYnBit x nig y ebIeKdwgfa ³ 8 9i . (1 i )( x iy ) ( 3 2i )( x iy ) 1 2i #&> cUrkMnt;BIrcMnYnBit x nig y ebIeKdwgfa ³ 5 13i . ( 3 2i ) ( x y ) (1 2i )( x y ) 1 i #*> eKmansmIkar (E) : az 2 bz c 0 , a 0 , a, b, c IR . cUrbgðajfaebI z 0 Cab¤ssmIkar (E) enaH z 0 k¾Cab¤srbs; (E) Edr . eroberogeday lwm plÁún
- TMBr½135 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
1 i 5 2
1 i 5 #(> eKmancMnYnkMupøic nig 2 . k-cUrbegáItsmIkardWeRkTIBIrmYyEdlman nig Cab¤s .
x-eKtag S n n n cMeBaHRKb; n IN . cUrRsaybBa¢ak;TMnak;TMng S n 2 S n1 32 S n 0 ? 10
10
K-edaymin)ac;BnøatcUrKNna N 1 2i 5 1 2i 5 $0> eKmansmIkar (E) : z 2 ( )iz 0 , , IR *
.
cUrbgðajfasmIkar (E) b¤sBIrsuTæEtCacMnYnnimμitsuTæEdleKnwgbBa¢ak; . $!> eKmansmIkardWeRkTIBIr (E) : Az 2 Bz C 0 Edl A 0 ehIy A , B , C CacMnYnkMupøic . k> bgðajfaebI A C i.B enaHsmIkar (E) manb¤sBIrkMnt;eday ³ C z 1 i , z 2 i. . A x> bgðajfaebI A C i.B enaHsmIkar (E) manb¤sBIrkMnt;eday ³ C z 1 i , z 2 i. . A K> Gnuvtþn_ ³ cUredaHRsaysmIkarxageRkamkñúgsMNMukMupøic ³ a / (1 i )z 2 ( 2 3i )z 2 3i 0 b / (1 2i )z 2 (1 2i )z (1 i ) 0
eroberogeday lwm plÁún
- TMBr½136 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
$@> eK[cMnYnkMupøic Z1 1 2i nig Z 2 3 i . cUrkMnt;rkEpñkBit nigEpñknimμiténcMnYnkMupøic W ebIeKdwgfa W1 Z1
1
1 . Z2
$#> eKmancMnYnkMupøic nig Edl 3 2i nig . 5(1 i ) . cUrkMnt;EpñkBit nig Epñknimμitén Z 12 12 .
$$> eKmansmIkar (E) : i.Z 2 (2 3i ) Z 5 (1 i ) 0 k> kMnt;cMnYnBit a edIm,I[ Z1 a i Cab¤smYyrbs;smIkar (E) rYcKNna b¤s Z 2 mYyeTotrbs;smIkar . x> cUrkMnt;cMnYnBit p nig q edIm,I[ Zp Zq pZ q Z 2 . 1
2
1
2
$%> eKmancMnYnkMupøicBIr Z1 nig Z 2 Edl Z1 Z 2 3 i nig Z1 .Z 2 4 3i . k> cUrbgðajfa (Z1 ) 2 (Z 2 ) 2 0 . x> cUrkMnt;rkEpñkBit nig Epñknimμit én Z Z14 Z 2 4 . K> cUrepÞógpÞat;fa Z1 Z 2 2 4Z1 .Z 2 (Z1 Z 2 ) 2 rYckMnt;rk Z1 nig Z 2 . $^> eKmancMnYnkMupøic Z1 x 2 i xy nig Z 2 y 2 ixy Edl x, y IR . cUrkMnt; x nig y edIm,I[ Z1 Z 2 ( 2006 i 2007 ) 2 . eroberogeday lwm plÁún
- TMBr½137 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
$&> eK[cMnYnkMupøic ³ Z 1 (a b ) i (b c ) , Z 2 (b c ) i (c a )
nig Z 2 (c a) i(a b) Edl a , b , c CabIcMnYnBitxusKña. cUrbgðajfa Z1 3 Z 2 3 Z 3 3 3Z1 .Z 2 .Z 3 $*> cUrKNnab¤skaeréncMnYnkMupøicxageRkam ³ k> a / Z 48 14i b / Z 24 10i x> a / Z 40 42i b / Z 77 36i K> a / Z 55 48i b/ Z 1 i 6 $(> cUrsresrcMnYnkMupøicxageRkamCaTRmg;RtIekaNmaRt ³ k> a / Z 1 i 3 b / Z 2 3 2i x> a / Z 1 i b / Z 3 3i 2 K> a / Z 6 2i 2 X> a / Z 1 i3 g> a / Z 1 2 i
b / Z 2 2i b/Z 2 i 2 b/Z 2 3 i
%0> cUrsresrCaTRmg;RtIekaNmaRténcMnYnkMupøicxageRkam ³ k> a / Z 1 cos 49 i. sin 49 b / Z 1 sin 10 i. cos 10 eroberogeday lwm plÁún
- TMBr½138 -
sikSaKNitviTüaedayxøÜnÉg
x> a / Z sin 27 i(1 cos 27 ) K> a / Z
2 2 cos
b / Z 1 i tan
7
cMnYnkMupøic b / Z 1 cos
i . 2 2 cos 5 5
4 4 i sin 5 5
X> a / Z sin 10 i. cos 10
b / Z ( 3 2)(cos i sin ) 8 8 Z 2 2 2 2 cos i . 2 2 2 2 cos 4 4
g> . %!> cUredaHRsaysmIkarxageRkamrYcsresrb¤snImYy²CaragRtIekaNmaRt ³ b / 2z 2 2z 1 0 k> a / z 2 2z 4 0 x> a / z 2 3 .z 1 0 b / z 2 z 1 0 %@> cUrsresr Z 2 3 i. 2 3 CaragRtIekaNmaRt . %#> cUrsresr Z 2 2 i 2 2 CaragRtIekaNmaRt . %$> k-KNnatémøR)akdén sin 10 , cos 10 nig tan 10 . x-TajrkTRmg;RtIekaNmaRtén Z 1 i. 5 2 2 .
eroberogeday lwm plÁún
- TMBr½139 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
2i 6 %%> eK[cMnYnkMupøic z1 2 nig z 2 1 i k> cUrsresr Z zz1 CaTRmg; a b.i . 2 x> cUrsresr z 1 , z 2 nig Z zz1 CaTRmg;RtIekaNmaRt . 2 K> edayeRbIlTæplxagelIcUrTajrktémøR)akdén cos 12 nig sin 12 %^> eKmancMnYnkMupøic z1 1 i nig z 2 2 3 2i . 2 k> cUrsresr Z z 1 .z 2 CaTRmg;BICKNit .
.
x> cUrsresr z 1 , z 2 nig Z z 1 .z 2 CaTRmg;RtIekaNmaRt . K> edayeRbIlTæplxagelIcUrTajrktémøR)akdén cos 512 nig sin 512 . %&> eK[ z 1 i . cUrsresr z nig z 2007 CaTRmg;RtIekaNmaRt . %*> eK[ z 3 i . cUrsresr z nig z 2007 CaTRmg;RtIekaNmaRt . %(> eK[ z 12 i 23 . cUrsresr z nig z 2007 CaTRmg;RtIekaNmaRt . ^0> eK[cMnYnkMupøic Z 2 3 i k-cUrepÞógpÞat;fam:UDúl | Z | 6 2 . x-bgðajfa Z 2(1 cos6 i.sin6) rYcTajrkTRmg;RtIekaNmaRtén Z . K-Tajbgðajfa cos 12
eroberogeday lwm plÁún
6 2 4
. - TMBr½140 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
^!> eK[cMnYnkMupøic Z 2 1 i k-cUrepÞógpÞat;fam:UDúl | Z | 2 . 2 2 . x-bgðajfa Z 2(1 cos 4 i. sin 4 ) rYcTajrkTRmg;RtIekaNmaRtén Z . K-Tajbgðajfa cos 8 2 2 2 . ^@> eKmancMnYnkMupøic ³ 3i ni g Z 2 ( 3 1) i( 3 1) . Z1 2 k> cUrsresr U Z1 .Z 2 CaragBICKNit . x> cUrsresr U nig Z1 CaragRtIekaNmaRt rYcTajrkm:UDúl nigGaKuym:g; éncMnYnkMupøic Z 2 . K>cUrTajbgðajfa cos 12 6 4 2 nig sin 12 6 4 2 .
^#> eK[cMnYnkMupøic z 1 2i 3 .
cUrrkm:UDúl nigGaKuym:g;éncMnYnkMupøic U
eroberogeday lwm plÁún
z 2012 1 z
2
.
- TMBr½141 -
sikSaKNitviTüaedayxøÜnÉg
^$> eK[cMnYnkMupøic Z n
cMnYnkMupøic
n 2 n 1 i( 2n 1) (n 2 n 1) 2 ( 2n 1) 2
Edl n CacMnYnKt;FmμCati . k> bgðajfa Z n n 1 i n 11 i . x> KNna S n Z 0 Z1 Z 2 ..... Z n edaysresrlTæpl Edl)anCaragBICKNit . ^%> eKmansIVúténcMnYnkMupøic (Z n ) kMnt;eday ³ 1 i 3 1 i 3 3i 3 ni g Edl n IN . Z0 Z n 1 Zn 2 2 2 k> eKtag U n Z n 1 cMeBaHRKb; n IN . cUrbgðajfa U n1 1 2i 3 .U n , n IN x> cUrsresr U n CaTRmg;RtIekaNmaRt . K> cUrRsaybBa¢ak;fa ³ Z n 2 cos
(n 1) (n 1) (n 1) cos i . sin 6 6 6
X> cUrkMnt;TRmg;RtIekaNmaRtén Z n . ^^> ebI z CacMnYnkMupøicEdlepÞógpÞat;TMnak;TMng ³ z 2n (1 z ) n cMeBaHRKb; n IN enaHcUrRsaybBa¢ak; facMnYnkMupøic z nig 1 1z manm:UDúlesμIKña . eroberogeday lwm plÁún
- TMBr½142 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
^&> edaHRsaysmIkar z 3 z 3 i | z |2 1 i . ^*> eKmancMnYnkMupøic 1 2i 7 nig 1 2i 7 k> sresrsmIkardWeRkTIBIrEdlman , Cab¤s . x> tag n Z : S n n n . cUrRsayfa S n 2 S n1 2S n 0 ? ^(> eK[cMnYnkMupøic ³
Z 1 a1 i .b 1 Z 2 a 2 i .b 2 Z a i .b 3 3 3
Edl a1 , a 2 , a 3 , b1 , b 2 , b 3
CacMnYnBit . kñúglMhRbkbedaytMruyGrtUnrm:al; (o, i , j , k )
eK[RtIekaN ABC mYyEdl AB (a1 , a 2 , a 3 ) nig AC (b 1 , b 2 , b 3 ) . k>cUrkMnt;RbePTénRtIekaN ABC kalNaeKmanTMnak;TMng Z 12 Z 22 Z 23 0 . x> cUrRsayfaebI ABC CaRtIekaNEkgsm)aTkMBUl A
Z 16 Z 62 Z 63 enaHeK)an (Z1 .Z 2 .Z 3 ) 3 2
eroberogeday lwm plÁún
.
- TMBr½143 -
sikSaKNitviTüaedayxøÜnÉg
&0> eK[cMnYnBit x Edl x 2 k , k Z k> cUrRsaybBa¢ak;fa ³ 1 3 tan 2 x i .( 3 tan x tan 3 x )
x> eRbITMnak;TMngxagelIcUrTajbgðajfa ³ tan 3x
cMnYnkMupøic
cos 3x i sin 3x cos 3 x
3 tan x tan 3 x
1 3 tan 2 x K> cUrsresr tan( 3 x) nig tan( 3 x) CaGnuKmn_én tan x . 3x X> Taj[)anfa tan( 3 x) tan( 3 x) tan . tan x n g>cUrKNnaplKuN Pn [tan( 3 3n a). tan( 3 3n a)] k 0
.
&!> edaHRsaysmIkar | z | i.z 1 3i Edl z CacMnYnkMupøic . &@> edaHRsaysmIkar log 5 (z.z ) z 3 (1 2i ) . &#> edaHRsaysmIkar | z 1 i | iz 22 4i . &$> eK[sIVúténcMnYnkMupøic (Z n ) kMnt;eday ³ 1 Z 0 1 nig Z n 1 ( Z n | Z n | ) Edl n IN 2 KNna Z n edaysresrlTæpleRkamTRmg;RtIekaNmaRt .
eroberogeday lwm plÁún
- TMBr½144 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
i sin 9 CaTRmg;RtIekaNmaRt . &%> sresrcMnYnkMupøic z 49 4 cos i sin 9 9 i cos &^> sresrcMnYnkMupøic z 11 sin CaTRmg;RtIekaNmaRt . sin i cos &&> enAkñúgbøg;kMupøic (xoy) tamcMNuc M' (z' ) CarUbPaBén M(z 4 4i ) tambMElgvilp©it O nigmMu 6 . cUrkMMNt; z' ? cos
n n 2 2 (cos
n ) . 4 4 n n n n &(> bgðajfa ( 3 i ) 2 (cos 6 i. sin 6 ) . *0> eK[cMnYnkMupøic z cos 6 i. sin 6 k> edAcMnuc z , z 2 , z 3 , z 4 , z5 , z 6 , z7 elIrgVg;RtIekaNmaRt .
&*> bgðajfa (1 i )n
i sin
x> kMNt;témø 1 z z 2 z 3 z 4 .... z11 . *!> eKmancMnYnkMupøic 3 2i nig 4 5i EdlmanrUbPaBtagedaycMNucerogKña A nig B . k> KNnaRbEvg AB x> rkkUGredaencMNuckNþal AB .
eroberogeday lwm plÁún
- TMBr½145 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
*@> cMNuc A , B nig C CarUbPaBerogKñaéncMnYnkMupøic 2 i , 1 nig 3 2i . R)ab;RbePTénRtIekaN ABC . *#> cUrrk nig sg;sMNMucMNuc M manGahVik z EdlepÞógpÞat;lkçxNÐxageRkam ³ k> | z 1 i | 2 x> | 2z 3 2i | 4 K> | z i | 2 X> zz 1i 1 g> zz 1i 2 c> ziz21i 1 *$>k-cMeBaHRKb;cMnYnkMupøic z1 nig z 2 cUrbgðajfaeKmansmPaB ³ 2 ( | z1 |2 | z 2 |2 ) | z1 z 2 |2 | z1 z 2 |2
x-cUrbkRsaysmPaBenHtamEbbFrNImaRt . *%> cUrkMNt;cMnYnkMupøic z edaydwgfa | z | | 1z | | 1 z | . *^> cUredaHRsaysmIkarkñúgsMNMukMpøic z6 *&> eK[ e 1
2
i
4 7
1 i 3i
.
. cUrRsaybBa¢ak;fa ³
2 1
4
3 1
6
2
**> edaHRsaykñúgsMNMucMnYnkMupøic z6 8i rYcTajrktémø cos 12 nig sin 12 *(> eK[ z x i.y Edl x , y IR . bgðajfa | 2z | | x | | y | . eroberogeday lwm plÁún
- TMBr½146 -
sikSaKNitviTüaedayxøÜnÉg
cMnYnkMupøic
Éksareyag !> esovePAKNitviTüafñak;TI 11 kMritx esovePAKNitviTüafñak;TI 12 rbs;RksYgGb;rMyuvCn nigkILa ¬e)aHBum Mathématiques Géométrie (Terminales C et E) ( Genevieve HAYE , Bernard RANDÉ , Eric SERRA)
$> Complex Numbers from A to…Z ( Titu Andreescu , Dorin Andrica )
eroberogeday lwm plÁún
- TMBr½147 -
