complex variables - madasmaths
TRANSCRIPT
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Created by T. Madas
Created by T. Madas
COMPLEX
VARIABLES
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Created by T. Madas
Created by T. Madas
CALCULATIONS
OF RESIDUES
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Created by T. Madas
Created by T. Madas
Question 1
( )2
sin zf z
z≡ , z ∈� .
Find the residue of the pole of ( )f z .
MM3-A , ( )0 1res z = =
Question 2
( ) 5ezf z z
−≡ , z ∈� .
Find the residue of the pole of ( )f z .
MM3-F , ( ) 1024
res z = =
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Created by T. Madas
Created by T. Madas
Question 3
( )2
2
2 1
2 1
z zf z
z z
+ +≡
− +, z ∈� .
Find the residue of the pole of ( )f z .
MM3-B , ( )1 4res z = =
Question 4
( )2
2 1
2
zf z
z z
+≡
− −, z ∈� .
Find the residue of each of the two poles of ( )f z .
MM3-E , ( ) 523
res z = = , ( ) 113
res z = − =
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Created by T. Madas
Created by T. Madas
Question 5
( )22 5 2
zf z
z z≡
− +, z ∈� .
Find the residue of each of the two poles of ( )f z .
MM3-J , ( )1 12 6
res z = = − , ( ) 223
res z = =
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Created by T. Madas
Created by T. Madas
Question 6
( )i
3
1 e z
f zz
−≡ , z ∈� .
a) Find the first four terms in the Laurent expansion of ( )f z and hence state the
residue of the pole of ( )f z .
b) Determine the residue of the pole of ( )f z by an alternative method
MM3-K , ( ) 102
res z = =
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Created by T. Madas
Created by T. Madas
Question 7
( )2
3 2
4
2 2
zf z
z z z
+≡
+ +, z ∈� .
Find the residue of each of the three poles of ( )f z .
( )0 2res z = = , ( ) ( )11 i 1 3i2
res z = − + = − + , ( ) ( )11 i 1 3i2
res z = − − = − +
Question 8
( )4
tan 3zf z
z≡ , z ∈� .
Find the residue of the pole of ( )f z .
MM3L , ( )0 9res z = =
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Created by T. Madas
Created by T. Madas
Question 9
( )( )( )
2
22
2
4 1
z zf z
z z
−≡
+ +, z ∈� .
Find the residue of each of the three poles of ( )f z .
( ) ( )12i 7 i25
res z = = + , ( ) ( )12i 7 i25
res z = − = − , ( ) 14125
res z = − = −
Question 10
( )1
e 1zf z ≡
−, z ∈� .
Find the residue of the pole of ( )f z , at the origin.
( )0 1res z = =
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Created by T. Madas
Created by T. Madas
Question 11
( )( )
223 10i 3
zf z
z z
≡
− −
, z ∈� .
Find the residue of each of the two poles of ( )f z .
( )5
3i256
res z = = , ( )51 i
3 256res z = = −
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Created by T. Madas
Created by T. Madas
Question 12
( )3
cot cothz zf z
z≡ , z ∈� .
Find the residue of the pole of ( )f z at 0z = .
MM3-D , ( ) 7045
res z = = −
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Created by T. Madas
Created by T. Madas
Question 13
( )6
5 4 3
1
2 5 2
zf z
z z z
+≡
− +, z ∈� .
Find the residue of each of the three poles of ( )f z .
( )651
2 24res z = = − , ( )
652
24res z = = , ( )
210
8res z = =
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Created by T. Madas
Created by T. Madas
Question 14
( )( ) ( )2
4
1 2i 6 i 1 2if z
z z≡
− + − +, z ∈� .
Find the residue of each of the two poles of ( )f z .
( )2 i ires z = − = , ( )( )1 2 i i5
res z = − = −
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Created by T. Madas
Created by T. Madas
Question 15
( )4
e
1
kzzf z
z≡
+, z ∈� , k ∈� , 0k > .
Show that the sum of the residues of the four poles of ( )f z , is
sin sinh2 2
k k
.
MM3-C , proof
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Created by T. Madas
Created by T. Madas
LAURENT
SERIES
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Created by T. Madas
Created by T. Madas
Question 1
Determine a Laurent series for
( )1
f zz
= ,
which is valid in the infinite annulus 1 1z − > .
MM3-A , ( )
( )
1
1
11
1
r
r
r
z z
∞+
=
− = −
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Created by T. Madas
Created by T. Madas
Question 2
Determine a Laurent series for
( )( )
2
3
e
1
z
f zz
=−
,
about its singularity, and hence state the residue of ( )f z about its singularity.
MM3-G , ( )( )
32
0
e 2 1
!
rr
r
zf z
r
∞−
=
− =
, ( ) 21 2eres z = =
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Created by T. Madas
Created by T. Madas
Question 3
Determine a Laurent series for
( ) ( )1
2 sin2
f z zz
= +
− ,
about 2z = .
MM3-H ,
( )( ) ( ) ( ) ( )
5 4 3 2
4 1 4 1 4... 1
25! 2 5! 2 3! 2 3! 2f z
zz z z z= + + − − + +
−− − − −
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Created by T. Madas
Created by T. Madas
Question 4
Determine a Laurent series for
( )2
1
1f z
z=
−,
which is valid in the punctured disc 0 1 2z< − < .
MM3-B , ( ) ( )
1
2 2
1
1 11
1 2
r r
r
r
z
z
∞+
+
=−
− − =
−
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Created by T. Madas
Created by T. Madas
Question 5
Determine a Laurent series for
( )1
4f z
z=
+,
which is valid for 4z > .
MM3-C , ( )
1
0
41
4
r
r
r
z z
∞
+
=
− =
+
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Created by T. Madas
Created by T. Madas
Question 6
Determine a Laurent series for
( )( )
5 3i
i
zf z
z z
+=
+,
which is valid in the annulus 1 i 2z< − < .
MM3-D , ( )
( )
( )
( )
( )1
0 0
3 i i5 3ii
i i 2i
r r
r r
r r
zz
z z z
∞ ∞
+
= =
− −+ = −
+ − −
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Created by T. Madas
Created by T. Madas
Question 7
Determine a Laurent series for
( )( )( )
1
1 2f z
z z=
− +,
which is valid in ...
a) ... the annulus 1 2 4z< − < .
b) ... in the region for which 2 4z − > .
MM3-E , ( ) ( ) ( ) ( )1
1 0
1 1 21 2 1
3 12 4
rr r r
r r
zf z z
∞ ∞
+ −
= =
− = − − − − ,
( )( ) ( )
( )1
0
1 4
3 2
r r
r
r
f zz
∞
+
=
− − −=
−
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Created by T. Madas
Created by T. Madas
VARIOUS
PROBLEMS
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Created by T. Madas
Created by T. Madas
Question 1
The complex number cos i sinz c a bθ θ= + + , 0 2θ π≤ < , traces a closed contour C ,
where a , b and c are positive real numbers with a c> .
By considering
1
C
dzz� ,
show that
( ) ( )
2
2 2
0
cos 2
cos sin
a cd
bc a b
πθ π
θθ θ
+=
+ + .
proof