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it is presentation on residue theoram for integrated msc students
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Contents
Terms
Introduction to residues
Residue theoram
Calculation of residues
Examples
Application
Bibliography
4
1
0
4
0
4
3
0
3
2
0
2
0
1
3
3 0
2
0 1 0 2 0
Introductio
n
1
0
4
0
4
3
0
3
2
0
2
0
1
3
3 0
2
0 1 0 2 0
C
n
n
C
n (^) n
1
1 0
0
0
We can use it to evaluate
integrals
Theoram
Let f ( z ) be an analytic function in a
closed curve C except at a finite
number of singular points within C,
then
k
i
z z
C
f z dz j f z
i 1
( ) 2 Res ( )
8
Proof of Residue Theorem
1
1
k
k
C C C C
f ( z ) dz f ( z ) dz f ( z ) dz f ( z ) dz
1 2
C
k
i
z z
C
i 1
Formula for finding the residue for a pole of
any order
0
m
m
0
2
0
2
0
1
0 1 0
0 ( 1 )
( 1 )
0 0
m
m
m
z z z^ z
^
m
m
m m m m
2
2 0
1
1 0
1
0 0 0 1 0
0
( 1 )^01
( 1 )
0
m
m
m
z z
Example
s
13
2 2 2 2
2
2 2
2
2 2
2
2 2
2
0
2 2
0
2
2
2
2
2
0
1 1 1
2
1 1 1
2
1
,
1 1 1
1
2
1 1
1 ( 1 )
( )
1 1 1 1
Res ( 0 ) Res ( )
.
( )
1
( )( )
1
Res ( ) lim
.
1
( )( )
1
Res ( 0 ) lim
| | 1 ,| | | | isin thecircle, isoutof the circle.
Wehave 3 simplepoles, 0 , 2 / 1 0 1 1.
( 2 / 1 )
1 1
2
1 1 1
1 1 / / 2
1 / / 2
, isrealand| | 1.
1 sin
sin
Example 3 :
a a
a
a a
a
i
ia
I
a a
a
a
a
i
a
a
z z z^ i
z z
z z
z
z z
f f z
z z z
z
z z z z z
z
f z z z
z z z z z z z
z
f z
z z z z z z
a
a
i
z z iz a z
dz
z z iz a
z
ia
dz
z az iz a
z
iz i
dz
a z z i
z z i
I
a a
a
d
I
z z
z
C C C
C
r= 1
z +
z
-
z 0
14
arctan.
1
Or
.
( )( )
1
2 Res ( ) 2 lim
on theupperhalf plane
1
1
2 Residues of
1
Example 4 :
2
2
2
x
x
dx
z i z i
i f i i z i
z
I i
x
dx
I
z i
.
2
'
( ) ( )
1
2 Res ( ) 2 lim
on the upper half plane
1
2 Residues of
, 0.
Example 5 :
2 2 3
2
2 2 2
2 2 2
z ai z ai a
i f ia i z ia
z a
I i
a
x a
dx
I
z ai
Conclusion
So the Residue allows us to evaluate
integrals of analytic functions f ( z )
over closed curves C when f ( z ) has
one singular point inside C.
NOTE-
The residue of a function at a
removable singularity is zero.
Bibliography
Higher engineering mathematics-
B.S.Agrewal (page no. 662-674)
Advanced engineering
mathematics- H.K.Das (page no.
515-532)
Mathmatical physics by Arfken
Wikipedia