M1509 ระบบจำนวนเชิงซ้อน2014
of 64
Transcript of M1509 ระบบจำนวนเชิงซ้อน2014
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1. 2
2. ni n !" ##. $
%. "&"' 1#
(. "")*"* 1(
+.
,'"- "'"" 1$/. "0 23
4. "0 21
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11. "5,"0 %3
12. ,'""- "' 67 %4
1#.
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1(. "8 +/
1+. ,'05 9 ++++ :; +$1/. "-&- 4+
14. "8,'05 4+
1$. "-
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1.
ก!"#$%&'% (#)ก*+ก,! %-#) 2 -
ก 2//2 +− 0 %- 2 - - / 13 2
ก 9/ 2 − 0 %- 2 - - / −3
"ก*+ก 45-% (#)4$ ก 1/2 + 0 %-*&&% (-ก'4$ !ก-78:ก$: ;'
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1 -
1. 9H −⋅− 2. F −⋅−
. C9 −⋅−
2. ni n !"
*7 1i i
2i 2D1E − −1
i 2i EiD E−1DEiD −i
Ci E 2i DE 2i D E−1DE −1D 1
'$%&ก! 5&กก,!4# (&L - ni %,!ก@7$!% ;
1i Fi 9i 1i 2i 6i 10i 1Ci i Hi 11i 1Fi Ci Gi 12i 16i
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$!! ; +#4$ ( n #)Kก nCi
1nCi + 2nCi +
nCi +
", nC2nC1nCnC iiii +++ +++
nC2nC1nCnC
iiii +++
⋅⋅⋅
*7 nDiE− ( n #)Kก
1DiE−
2DiE−
DiE−
CDiE−
$!! ; +#4$ ( n #)Kก nCDiE−
1nCDiE +−
2nCDiE +−
nCDiE +−
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2 -
1. 2i 2. 2Fi
. F6i C. 99i
F. 106i 6. 2FC9i
H. FGHH19F iii −+ G. CGGHF iCi +
9. F2120 iii +++ K 10. 2006FC iiiii +++++ K
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11. H29HF1 iiiii +++++ K 12. DiEDiDEiDEiE F222120 K
1.1H6F2F12
10099GGHH
iiii
iiii
+++
+++
1C. 2F10 DiEDiEDiE −+−+−
1F.19G2C2
DiEDiEDiEDiEDiE −++−+−+−+− K
16. 9H119HF iiiiiiii −++−+−+− K
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1H. ∑=
C0
10n
ni 1G. ∑=
F00
1M
MMi
19. ∑
=
1000
1M
Mi
M
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2. - 1G12GC iiii +++ ก!'$
ก. C
. -. 2
. 1
. - 11F912621F iiCiHi −−+ ก!'$
ก. F − Ci
. C − Fi-. F + Ci
. C + Fi
C.119
612
iFiH
ii6
+
+ %-ก!'$
ก. i2F
.2iF
-.2
iF−
.iF2−
F. Di2DEi2DEiHDEiHE CC2F19 −+++ %-ก!'$
ก. iH26 +
. iH100 −−
-. −C0
. −0
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6. 5 i 1− -
ก. 1
. −1-. i − 1
. −i − 1
H. - ∑ +=
20
1n1nE
ก. CE1 − iD. FE1 − iD
-. CE2 − 2iD
. FE2 − 2iD
#.
(N*#)#)% (ก$กก ?KLAMNFO nEA:FC
%&"$&!?,!ก@7 z $&
19G22 iiii ++++ K
niD
% (4' ก%&'% (#)",N*H %&"$& O
% ( z ∈ O -"$
9
-78! ; ,& $&ก! %&ก ,ก% (#)
$!"BN*
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&%&&= '=# EJ3 ID ( J #) ", I #) E4ก% (&ก! (-= !$! ก%&'=#-= !$!5#)!ก%ก+#)D "&4กK$%ก%&'=#"% ;'ก-7ก% (&ก!-&+ &ก $!! ; :&%&'=#$!% ;
5 z #) $&% ( z EJ3 ID 4$ z J + Ii ( i 1−
%&ก J #) z PQEzD
%&ก I #) N* z RSEzD%&ก Ii #) N*
F −+ iF + #)
9C −+ C + i #)
6 6 + 0i #)
G− iG0 + #)
5 z J + Ii - J I ! ; J ", I ก!8=&กK4$
1. J + Ii ( I ≠ 0 5=ก%&ก N*
2. 5 I 0 ", z J %&ก
. 5 J 0 ", z Ii %&ก N*"
C. 5 J 03 I 0 ", z 0 #)
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3 +% (ก$' #)$'$ E4$กก 1 $D N* N*"
1.
2. −C
.GH
C. F−
F. 12i
6. −Hi
H. i2
G. − Hi
9. i2 −
10. iC
21
+
11. iπ
12. i2G.0 −
1. 1.H
1C. i101
F1
+
1F. −1 + Gi
16. 1 − i
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N* N*"
1H. i
1G. −i
19. 1
20. 1 + i
4 +-",N*4#% ;
1. + Ci 2. C − Fi
. −C + Hi C. − − Gi
F. 1 6. −F
H. 2i G. −Hi
9. 0
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%. "&"'
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z J + Ii ",2
z T + Ui #) 2
1z 2z กK ( J T ", I U
5 - / ", V 4#% ;
1. / + i H + i 2. F + 9i F + Vi
. / + Gi H + Ci C. / + V + i + E/ − VDi
F. / + V − Ci 10 + E/ − VDi 6. V + /i V − /
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H. / − V + iV2 1 + Ci
G. E/ + 2VD + EC/ − VDi E2/ − 1D + EV − 6Di
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2. 5 E/ − VD + 2i 6 + E/ + VDi E/3 VD
ก. E13 2D
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. ก$ / ", V %-#)ก
- / − V กก i6//2
++ ViF6iV2
−+ -'$ก. −C
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กกก! 54$$&กก! ",N*กก!N*
ก,ก! ',ก! ",N*,ก!N* ,$!
5 1z EJ3 ID J + Ii
2z ET3 UD T + Ui
21 zz + EJ + T3 I + UD EJ + TD + EI + UDi
21zz − EJ − T3 I − UD EJ − TD + EI − UDi
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6 B,,!*W4#% ;
1. E2 + CiD + E + iD 2. EH + GiD + E2 − CiD
. EF + iD + E0 − iD C. E−2 + iD + EC + 2iD
F. E1 + iD + E2iD + E2 + HiD 6. E6 − iD + E− + 2iD + E1 + 6iD
H. EF + HiD − E + CiD G. EC − iD − E−2 + iD
9. E0 + iD − E1 − iD 10. EG − iD − EH + 0iD
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11. E0 − iD + EF + i2 D 12. − EC + 2iD
1. EF + iD + E2 − iD − E6 − iD 1C. Di2EDiFE −−−+
1F. 1 + 62F +− − 2 16. EG3 FD − E3 −2D
1H. EH3 D − E3 1D 1G. E23 −FD − E−F3 −2D
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7 ก$' E/ + ViD + E2 − iD −C + 2i ", - / ", V
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+. ,'"- "'""
5' O #) : (%กก - กก ' 1
z 32
z 3
z #)ก' O
1. O %!#X$กก! (- 5 1z ∈ O ", 2z ∈ O 4$ 1z + 2z ∈ O
2. O %!ก,!% (กก! (- 1z + 2z 2z + 1z
. O %!ก#,% (&ก,+ 4$กก
! (- E 1z + 2z D + z 1z + E 2z + z D
C. ก%ก,!ก@7กก
0 #)ก,!ก@7กก $&% ( 0 + 1z 1z + 0 1z
F. ก%กก5% 1z กก 1z - − 1z $&% ( 1z + E− 1z D E− 1z D + 1z 0
6.
5 1z 2z ", 1z + z 2z + z
H. 5 1z + z 2z + z ", 1z 2z
8 กก4#% ;
1. z + Ci 2. z F − Hi
. z −2 + 9i C. z F + 0i
F. z 0 + 2i
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. i ⋅ Fi C. −2i ⋅ i
F. E2 + 9iD i 6. E1F + CiD E−iD
H. 2i E1 − iD G. E + iD E + HiD
9. EF + 6iD E6i − FD 10. E−2 + FiD EC − iD
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11. E23 0D ⋅ E 2 3 0D 12. E1.23 0D ⋅ Eπ3 0D
1. E03 CD ⋅ E03 −1D 1C. E03 1D ⋅ E03 1D
1F. E1 + iD ⋅ E1 − iD 16. E2 − iD ⋅ E2 + iD
1H. E − iD ⋅ E + iD 1G. E + CiD ⋅ E − CiD
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19. E−23 D ⋅ E−23 − D 20. ET3 UD ⋅ ET3 −UD
21. E−/3 VD ⋅ E−/3 −VD 22. EJ3 ID ⋅ EJ3 − ID
2. DiCEDiE 2 −++ 2C. i21CD2HE 2 +−−
ก
1. -=7#)
2. N*"-=7N*"#)
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11 B,,!*W% (ก$'
1. EC + iD + E−1 + 2iD 2. EC + iDE−1 + 2iD
. YEC + iD + E−1 + 2iDZE1 + iD C. EC + iDE−1 + 2iD + EF − HiD
F. EC + iDE−1 + 2iD + E2 + iDE1 + iD 6. EC + iDE−1 + 2iDE1 + iD
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12 / ", V % ($-,ก!ก4#% ;
1. E/3 VD + EH3 −D EF3 2D 2. E2/3 VD 1 + E3 FD
. E/3 VD E63 −GD + EC3 2D C. E/3 2VD E03 1D E−23 HD
F. E/3 VD E03 6D E13 1D 6. DVi/Ei9 − Vi2F/i1C −−
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2. E1 − iDE2 + iDE + 2iD ก!'$
ก. 10 + 1i
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. 1 + 1i
. 2Di2EiC + ก!'$
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C. '$5=ก
ก. 5 1z + Fi ", 2z 1 + 2i ", 21 zz + C − Hi
. 5 1z + Fi ", 2z 1 + 2i ", 21 zz − −2 + i
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F. '$#)K
ก. E 3 0D + 0i i
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6. 5 / 2
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$. ,'"- "'"0
5' O #) : (%กก - ก-=7 ",' 1
z 32
z 3
z #)ก' O
1. O %!#X$ก-=7! (- 5 1z 3 2z #)ก' O ", 1z 2z ∈ O
2. %!ก,!% (ก-=7
1z 2z 2z 1z
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C. ก%ก,!ก@7ก-=7
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-=7E1D ! (-1z1
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H. 5 1z 2z ", 1z z 2z z
G. 5 1z 2z 1z z ", 1z ≠ 0 ", 2z z
9.
5 1z 2z 0 ", 1z 0 2z 0
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13 -1. ก$ 1z E23 FD ", 2z E13 D 21 zz + ", 21 zz ×
2. ก$ 1z EJ3 ID3 2z EC3 2D3 z EH3 FD ", Cz E13 2D
5 21 zz + z ", 1z × Cz
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14 -
1.iiC2
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2. - z ก DiFEi
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1C. ก$ก 21 zDi2CEzDiE ++− 1 + i
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F. z − Ci 6. z −2 − Fi
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9. z i2CH
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J1 +
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G. 5 J3 I #)'$L ", '$#)K
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24.2 ก ก!"#$ %#$&ก*+กก 2'ก$:!* (&'ก"&ก!#ก 'e@f%8@,3 e@f%!
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47 ก$' 2 − i ", 1 − i #)-ก UT/ I/J// 2C −−−+ 0 ( J3 I 3 T ", U #) J + I + T + U
48 B,ก",B,-=7กก
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−+ 0 -'$ก. "−23 23 −2i3 2i#
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50 z (ก$
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8/20/2019 M1509 2014
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52 ก$ 2z 2 − Ci - 1zz −+
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