Download - Part 3-2
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I. Hm bin phc
II. Chui phc
III. Tch phn ng
IV. im bt thng, zeros v thng d
V. ng dng ca l thuyt thng ds
Part 3:
Hm bin phc
Created and edited by: Nguyen Phuoc Bao Duy
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Chui phc
1. nh ngha
2. Chui ly tha
3. Chui Taylor
4. Chui Laurent
Created and edited by: Nguyen Phuoc Bao Duy
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1. nh ngha
V d 2.01: Xt chui:
Chui trn hi t v 1/(1 z) khi |z| < 1, v phn kkhi |z| > 1.
Created and edited by: Nguyen Phuoc Bao Duy
Mt chui phc tng qut c dng:
trong fn(z) l mt hm phc theo bin z.
1 20
( ) ( ) ( ) ... ( ) ...n n
n
f z f z f z f z
2
0
11 ... ; | | 1
1n
n
z z z if zz
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Tiu chun DAlembert
V d 2.02: Tm min hi t (region of converge - ROC):
p n: ROC: Re{z} > 0.
Created and edited by: Nguyen Phuoc Bao Duy
Xt chui phc:
Tm gii hn:
L < 1: Chui hi t L > 1: Chui phn k L = 1: Cha th kt lun v tnh hi t ca chui.
1 20
1
( ) ( ) ( ) ... ( ) ...
( )lim
( )
n nn
n
nn
f z f z f z f z
f zL
f z
2
0
1 ...nz z z
n
e e e
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2. Chui ly tha
ROC ca chui ly tha:
Created and edited by: Nguyen Phuoc Bao Duy
Mt chui phc c dng
vi cc h s ai l hng s v z0 l mt im cnh trong mt phng phc, c gi l mt chui lytha quanh im z0.
2
0 0 1 0 2 00
( ) ( ) ( ) ...nn
n
a z z a a z z a z z
0
1
lim nn
n
az z R with R
a
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V d 2.03: Tm chui ly tha biu din hm 1/(z 3)trong 3 min sau:
a. |z| < 3
b. |z 2| < 1
c. |z| > 3
p n:
Created and edited by: Nguyen Phuoc Bao Duy
2
0
2
0
2 30
1 1 1. ... 3
3 3 3 3 9 27
1. 2 1 ( 2) ( 2) ... 2 1
3
1 1 3 1 3 9. ... 3
3
n
n
n
n
n
n
z z za zz
b z z z zz
c zz z z z z z
2. Chui ly tha
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V d 2.03 (tt):
Nhn xt: Chng ta c th chn bt k mt im z0trong mt phng phc v xc nh 1 chui ly tha hit v 1/(z 3) vi min hi t l mt hnh trn tm z0,bn knh R.
Created and edited by: Nguyen Phuoc Bao Duy
2. Chui ly tha
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Khai trin chui nh thc:
vi
Hai trng hp c bit thng s dng:
Created and edited by: Nguyen Phuoc Bao Duy
2( 1) ( 1)( 2)...( 1)(1 ) 1 ... ...2! !
k rk k k k k k rz kz z zr
1k and z
2
0
2
0
11 ... 1
1
11 ... ( ) 1
1
n
n
n
n
z z z zz
z z z zz
2. Chui ly tha
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3. Chui Taylor
ROC ca chui Taylor:
Created and edited by: Nguyen Phuoc Bao Duy
Mt chui ly tha c dng
trong f(z) l mt hm phc gii tch bn trongv trn bin mt ng cong kn C, c gi l khaitrin chui Taylor ca f(z) quanh im z0.
( )
0 00
1( ) ( )( )
!n n
n
f z f z z zn
0z z R
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V d 2.04: Xc nh khai trin chui Taylor ca hm
quanh im z = j, n s hng (z - j)4.
p n:
Created and edited by: Nguyen Phuoc Bao Duy
1( )
( 2 )f z
z z j
(1)
(2) (3) 2 4
(4)
1 1 1 1( )
( 2 ) 2 2
( ) 1; ( ) 0
( ) 2; ( ) 0 ( ) 1 ( ) ( ) ...
( ) 24
f zz z j j z j z
f j f j
f j f j f z z j z j
f j
3. Chui Taylor
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V d 2.05: Xc nh khai trin chui Maclaurin cacc hm s sau:
a. ez b. sinz c. cosz
p n:
Created and edited by: Nguyen Phuoc Bao Duy
Mt chui Taylor vi z0 = 0 c gi l chuiMaclaurin.
2 3
0
2 1 3 5
0
2 2 4
0
. 1 ...! 2 6
.sin ( 1)(2 1)! 6 120
.cos ( 1) 1 ...(2 )! 2 24
nz
n
nn
n
nn
n
z z za e z
n
z z zb z z
n
z z zc z
n
3. Chui Taylor
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Khai trin chui Taylor c th c xc nh thngqua php khai trin chui nh thc
Example 2.06: Tm khai trin Maclaurin ca cc hm sauv xc nh min hi t (ROC) tng ng:
p n:
Created and edited by: Nguyen Phuoc Bao Duy
1 2 2
3 4 2
1. ( ) . ( )
3 41 1
. ( ) . ( )1 3 2
za f z b f z
z zz
c f z d f zz z z
2
10
1 1 1 1 1. ( ) ...
3 3 3 3 3 9 271
3
ROC : 3
n
n
z z za f z
zz
z
3. Chui Taylor
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p n v d 2.06 (tt):
Created and edited by: Nguyen Phuoc Bao Duy
2 1
2 2 10
30
4 2 10
. ( ) 1 ; : 24 4
1. ( ) 1 2 ; : 1
1
1 1. ( ) 1 ; : 1
3 2 2
nn
nn
n
n
n
nn
z zb f z ROC z
z
zc f z z ROC z
z
d f z z ROC zz z
3. Chui Taylor
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V d 2.07: Tm khai trin Taylor ca cc hm sau quanhim z0 cho trc v xc nh min hi t (ROC) tngng:
Created and edited by: Nguyen Phuoc Bao Duy
02
0
0
1. ( ) ; 2
3 21
. ( ) ;( 2 )
. ( ) ; 4( 1)(2 )
a f z zz z
b f z z jz z j
zc f z z
z z
3. Chui Taylor
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4. Chui Laurent
ROC ca chui Laurent:
r1 c th bng 0
r2 c th bng
Xc nh chui Laurent bng
cch khai trin chui nh thc.
Created and edited by: Nguyen Phuoc Bao Duy
Nu f(z) gii tch trong hnh vnh khn gii hn bi2 ng trn C1 v C2 c bn knh ln lt l r1 v r2 (r1 3
c. 0 < |z + 1| < 2
d. |z| < 1
Created and edited by: Nguyen Phuoc Bao Duy
1( )
( 1)( 3)f z
z z
4. Chui Laurent