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I. Hm bin phc
II. Chui phc
III. Tch phn ng phc
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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Tch phn ng phc
1. Tch phn ng phc
2. nh l Cauchy
Created and edited by: Nguyen Phuoc Bao Duy
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1. Tch phn ng phc
Created and edited by: Nguyen Phuoc Bao Duy
Cho f(z) l mt hm phc lin tc ti mi im trnmt ng cong n C trong mt
phng z v haiim u a v b:
Nu z = x + jy v f(z) = u(x,y) + jv(x,y):
01
( ) lim ( )k
n
k kC zk
f z dz f z z
( ) ( , ) ( , ) ( )C Cf z dz u x y jv x y dx jdy
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V d 4.01: Tnh tch phn phc dc theo ngC, ni hai im -1 + j v 5 + 3j
theo ng gp khcABD nh hnh:
Nu C l ng thng ni A v D th kt qu c gthay i khng?
p n: -4 + j.196/3
Created and edited by: Nguyen Phuoc Bao Duy
1. Tch phn ng phc2
Cz dz
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V d 4.02: Tnh tch phn ng phc
vi C l ng trn |z z0| = r (r l hng s)
p n:
Created and edited by: Nguyen Phuoc Bao Duy
10
1
( )nCdz
z z
1
0
2 ( 0)1
0 ( 0)( )nCj n
dznz z
1. Tch phn ng phc
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2. nh l Cauchy ng cong kn: l ng cong c im u v
im cui trng nhau, c th n hoc khng n:
Goursat sau ny chng minh c rng nh lCauchy ng cho c trng hp f(z)
khng lin tctrn C.
Created and edited by: Nguyen Phuoc Bao Duy
nh l Cauchy: Nu f(z) l hm gii tch vi o hmf(z) lin tc ti mi im bn
trong v trn ngcong kn n C, khi :
( ) 0C f z dz
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Min gii hn: l min c gii hn bi cc ngcong kn, c th n lin hoc a
lin:
Created and edited by: Nguyen Phuoc Bao Duy
Tch phn khng ph thucng i: Nu f(z) gii tchtrong mt min n linD,
khi tch phn caf(z) khng ph thuc vong ly tch phntrong D.
2 2
1 1
1 2
( ) ( )z z
z z
C C
f z dz f z dz
2. nh l Cauchy
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Created and edited by: Nguyen Phuoc Bao Duy
Nguyn l bin dng chu tuyn: Nu f(z) lmt hm giitch trong min n lin
D, v C1 l mt ng congkn trong D. Nu C1 c th bin dng (co gin) trthnh
ng cong kn C2 v qu trnh bin dngkhng vt ra khi min D, khi :
1 2
( ) ( )C Cf z dz f z dz
C1
C2
2. nh l Cauchy
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Created and edited by: Nguyen Phuoc Bao Duy
Nu f(z) c hu hn cc im bt thng z = zi (i =1,2,,n) bn trongmt ng
cong kn n C, v c nng con kn i bao quanh cc im bt thng ny(mi ng i ch
bao quanh duy nht mt im btthng zi), khi :
1 1 2
( ) ( ) ( ) ... ( )nC
f z dz f z dz f z dz f z dz
2. nh l Cauchy
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V d 4.03: Tnh tch phn
vi C l:
a. bt k ng cong kn no bao quanh z0b. bt k ng cong kn no khng bao
quanh z0.
V d 4.04: Tnh tch phn
vi C l
a. bt k ng cong kn no bao quanh hai im z =1 v z = -2j
b. bt k ng cong kn no bao quanh z = -2j nhngkhng bao quanh z =
1.
Created and edited by: Nguyen Phuoc Bao Duy
10
1
( )nCdz
z z
( 1)( 2 )Cz
dzz z j
2. nh l Cauchy
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V d 4.05: Tnh tch phn ng:
vi C l ng cong bao c ba im z = 1, - 2 and j.
Created and edited by: Nguyen Phuoc Bao Duy
Cng thc tch phn Cauchy: Cho f(z) l mt hm giitch bn trong v trn
bin mt ng cong kn nC. Nu z0 lmt im bt k bn trong C, th khi :
0
0
( )
01
0
( )2 . ( )
( ) 2. ( )
!( )
C
n
nC
f zdz j f z
z z
f z jdz f z
nz z
2( ) ( )
( 1)( 2)( )Cz
f z dz with f zz z z j
2. nh l Cauchy
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p n v d 4.05:
trong 1, 2 v 3 l cc ng cong kn ch baoquanh ln lt z = 1, - 2 v j.
Dng cng thc tchphn Cauchy ta c kt qu:
Created and edited by: Nguyen Phuoc Bao Duy
1 2
3 1 2 3
31 2
2 2
( 2)( ) ( 1)( )( )
1 22
( )( ) ( )( 1)( 2)
1 2
C
z z
z z j z z jf z dz dz dz
z zz
f zf z f zz zdz dz dz dz
z j z z z j
1 2 3( ) 2 (1) ( 2) ( ) 0C f z dz j f f f j
2. nh l Cauchy
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V d 4.06: Tnh tch phn ng
vi C l ng cong kn bao quanh z = 1.
p n: I = 12j
Created and edited by: Nguyen Phuoc Bao Duy
4
3( 1)Cz
I dzz
2. nh l Cauchy
