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Page 1
Edurev123
5. Cauchy's Residue Theorem
5.1 Using Cauchy's residue theorem, evaluate the integral
?? =? ?
?? ?? ?????? ?? ???????
(2013 : 15 Marks)
Solution:
Using trigonometry,
sin
4
??? ?=[
1
2
(1-cos?2?? ]
2
=
1
4
(1+cos
2
?2?? -2cos?2?? )
?=
1
8
[2+(1+cos?4?? )-2cos?2?? ]=
1
8
[cos?4?? -2cos?2?? +3]
?? ?=? ?
?? 0
?sin
4
??????? =
1
2
? ?
2?? 0
?sin
4
??????? ?=
1
2
? ?
2?? 0
?
1
8
[cos?4?? -2cos?2?? +3]????
?=
1
16
? ?
2?? 0
?[cos?4?? -2cos?2?? +3]????
?= Real part of
1
16
? ?
2?? 0
?(?? ??4?? -2?? ??2?? +3)????
Let ?? =?? ????
,???????????????????????????? =???? ????
???? ????? =
????
????
.
= R.P. of
1
16
???
?? (?? 4
-2?? 2
+3)????
????
where ?? is the unit circle.
???
?? ?? 4
-2?? 2
+3
?? ???? =2???? Sum of Residue of ?? (?? )
where ??????????????????????????????????????????????????????????? (?? )=
?? 4
-2?? 2
+3
??
Now ?? (?? ) has one pole at ?? =0 inside unit circle.
Residue at ?? =0=lim
?? ?0
????? (?? )
????????????????????????????????????????????????????????????????????=3
Page 2
Edurev123
5. Cauchy's Residue Theorem
5.1 Using Cauchy's residue theorem, evaluate the integral
?? =? ?
?? ?? ?????? ?? ???????
(2013 : 15 Marks)
Solution:
Using trigonometry,
sin
4
??? ?=[
1
2
(1-cos?2?? ]
2
=
1
4
(1+cos
2
?2?? -2cos?2?? )
?=
1
8
[2+(1+cos?4?? )-2cos?2?? ]=
1
8
[cos?4?? -2cos?2?? +3]
?? ?=? ?
?? 0
?sin
4
??????? =
1
2
? ?
2?? 0
?sin
4
??????? ?=
1
2
? ?
2?? 0
?
1
8
[cos?4?? -2cos?2?? +3]????
?=
1
16
? ?
2?? 0
?[cos?4?? -2cos?2?? +3]????
?= Real part of
1
16
? ?
2?? 0
?(?? ??4?? -2?? ??2?? +3)????
Let ?? =?? ????
,???????????????????????????? =???? ????
???? ????? =
????
????
.
= R.P. of
1
16
???
?? (?? 4
-2?? 2
+3)????
????
where ?? is the unit circle.
???
?? ?? 4
-2?? 2
+3
?? ???? =2???? Sum of Residue of ?? (?? )
where ??????????????????????????????????????????????????????????? (?? )=
?? 4
-2?? 2
+3
??
Now ?? (?? ) has one pole at ?? =0 inside unit circle.
Residue at ?? =0=lim
?? ?0
????? (?? )
????????????????????????????????????????????????????????????????????=3
?? ???? (?? )=6????
???????????????????????????????????????
1
16?? ??
?? ?? (?? )=
3
8
??
????????????????????????????????????????
0
?? ?sin
4
??????? = R.p
3
8
?? =
3?? 8
5.2 Evaluate the integral ?
?? ?? ?
????
(?? +
?? ?? ?????? ??? )
?? using residues.
(2014 : 20 Marks)
Solution:
Let
?? ?=? ?
?? 0
?
????
(1+
1
2
cos??? )
2
?=
1
2
? ?
2?? 0
?
4????
(2+cos??? )
2
?=? ?
2?? 0
?
2????
(2+cos??? )
2
.
Let the contour ' ?? ' be the unit circle |?? |=?? with centre at the origin.
Let ?? =?? ????
then cos??? ?=
1
2
(?? +
1
?? )
???? ?=???? ????
????
???? ?=
????
????
?? ?=?
0
2?? ?
2????
(2+cos??? )
2
=?
0
2?? ?
2????
???? (2+
?? 2
+1
2?? )
2
?=
1
?? ?
0
2?? ?
8?? (?? 2
+4?? +1)
2
????
????????????????????????????????????????????????? ?=
8
?? ?
?? ?
?? (?? 2
+4?? +1)
2
????
?=
8
?? ?
?? ??? (?? )????
where
?? (?? )=
?? (?? 2
+4?? +1)
2
(??)
Now the poles of ?? (?? ) are given by
Page 3
Edurev123
5. Cauchy's Residue Theorem
5.1 Using Cauchy's residue theorem, evaluate the integral
?? =? ?
?? ?? ?????? ?? ???????
(2013 : 15 Marks)
Solution:
Using trigonometry,
sin
4
??? ?=[
1
2
(1-cos?2?? ]
2
=
1
4
(1+cos
2
?2?? -2cos?2?? )
?=
1
8
[2+(1+cos?4?? )-2cos?2?? ]=
1
8
[cos?4?? -2cos?2?? +3]
?? ?=? ?
?? 0
?sin
4
??????? =
1
2
? ?
2?? 0
?sin
4
??????? ?=
1
2
? ?
2?? 0
?
1
8
[cos?4?? -2cos?2?? +3]????
?=
1
16
? ?
2?? 0
?[cos?4?? -2cos?2?? +3]????
?= Real part of
1
16
? ?
2?? 0
?(?? ??4?? -2?? ??2?? +3)????
Let ?? =?? ????
,???????????????????????????? =???? ????
???? ????? =
????
????
.
= R.P. of
1
16
???
?? (?? 4
-2?? 2
+3)????
????
where ?? is the unit circle.
???
?? ?? 4
-2?? 2
+3
?? ???? =2???? Sum of Residue of ?? (?? )
where ??????????????????????????????????????????????????????????? (?? )=
?? 4
-2?? 2
+3
??
Now ?? (?? ) has one pole at ?? =0 inside unit circle.
Residue at ?? =0=lim
?? ?0
????? (?? )
????????????????????????????????????????????????????????????????????=3
?? ???? (?? )=6????
???????????????????????????????????????
1
16?? ??
?? ?? (?? )=
3
8
??
????????????????????????????????????????
0
?? ?sin
4
??????? = R.p
3
8
?? =
3?? 8
5.2 Evaluate the integral ?
?? ?? ?
????
(?? +
?? ?? ?????? ??? )
?? using residues.
(2014 : 20 Marks)
Solution:
Let
?? ?=? ?
?? 0
?
????
(1+
1
2
cos??? )
2
?=
1
2
? ?
2?? 0
?
4????
(2+cos??? )
2
?=? ?
2?? 0
?
2????
(2+cos??? )
2
.
Let the contour ' ?? ' be the unit circle |?? |=?? with centre at the origin.
Let ?? =?? ????
then cos??? ?=
1
2
(?? +
1
?? )
???? ?=???? ????
????
???? ?=
????
????
?? ?=?
0
2?? ?
2????
(2+cos??? )
2
=?
0
2?? ?
2????
???? (2+
?? 2
+1
2?? )
2
?=
1
?? ?
0
2?? ?
8?? (?? 2
+4?? +1)
2
????
????????????????????????????????????????????????? ?=
8
?? ?
?? ?
?? (?? 2
+4?? +1)
2
????
?=
8
?? ?
?? ??? (?? )????
where
?? (?? )=
?? (?? 2
+4?? +1)
2
(??)
Now the poles of ?? (?? ) are given by
(?? 2
+4?? +1)
2
=0??? =
-4±v16-4
2
=
-4±v12
2
=-2±v3 (twice)
??? (?? ) has poles of order at ?? =-2±v3 (twice)
Let ?? =-2+v3,?? =-2-v3
Clearly |?? |>1
Since |???? |=1?|?? |<1
Hence, the ?????? pole inside ?? is ?? =?? of order 2 .
????
?? ??? (?? )???? ?=??
??????
(?? 2
+4?? +1)
2
?=2???? ( residue at ?? =?? )
Now the residue at ?? =?? is
lim
?? ??? ?
?? ????
(?? -?? )
2
?? (?? 2
+4?? +1)
2
=lim
?? ??? ?
?? ????
?? (?? -?? )
2
=lim
?? ??? ?
-(?? +?? )
(?? -?? )
3
=
-(?? +?? )
(?? -?? )
3
=
-(-4)
(2v3)
3
=
(4)
8(3v3)
=
1
6v3
???????????????????????????????????????
?? ??? (?? )???? =2???? (
1
6v3
)=
2????
6v3
=
v3?? 3v3
?? from (i)
?????????????????????????????? ?
2?? 0
?
2????
(2+cos??? )
2
=
8
?? (
????
3v3
)=
8?? 3v3
??????????????????????????????????????????????????????????????? =? ?
?? 0
?
????
(1+
cos??? 2
)
2
=
8?? 3v3
5.3 State Cauchy's Residue theorem. Using it, evaluate the integral
?
?? ?
?? ?? +?? ?? (?? +?? )(?? -?? )
?? ???? ·{?? :|?? |=?? }
(2015 : 15 Marks)
Page 4
Edurev123
5. Cauchy's Residue Theorem
5.1 Using Cauchy's residue theorem, evaluate the integral
?? =? ?
?? ?? ?????? ?? ???????
(2013 : 15 Marks)
Solution:
Using trigonometry,
sin
4
??? ?=[
1
2
(1-cos?2?? ]
2
=
1
4
(1+cos
2
?2?? -2cos?2?? )
?=
1
8
[2+(1+cos?4?? )-2cos?2?? ]=
1
8
[cos?4?? -2cos?2?? +3]
?? ?=? ?
?? 0
?sin
4
??????? =
1
2
? ?
2?? 0
?sin
4
??????? ?=
1
2
? ?
2?? 0
?
1
8
[cos?4?? -2cos?2?? +3]????
?=
1
16
? ?
2?? 0
?[cos?4?? -2cos?2?? +3]????
?= Real part of
1
16
? ?
2?? 0
?(?? ??4?? -2?? ??2?? +3)????
Let ?? =?? ????
,???????????????????????????? =???? ????
???? ????? =
????
????
.
= R.P. of
1
16
???
?? (?? 4
-2?? 2
+3)????
????
where ?? is the unit circle.
???
?? ?? 4
-2?? 2
+3
?? ???? =2???? Sum of Residue of ?? (?? )
where ??????????????????????????????????????????????????????????? (?? )=
?? 4
-2?? 2
+3
??
Now ?? (?? ) has one pole at ?? =0 inside unit circle.
Residue at ?? =0=lim
?? ?0
????? (?? )
????????????????????????????????????????????????????????????????????=3
?? ???? (?? )=6????
???????????????????????????????????????
1
16?? ??
?? ?? (?? )=
3
8
??
????????????????????????????????????????
0
?? ?sin
4
??????? = R.p
3
8
?? =
3?? 8
5.2 Evaluate the integral ?
?? ?? ?
????
(?? +
?? ?? ?????? ??? )
?? using residues.
(2014 : 20 Marks)
Solution:
Let
?? ?=? ?
?? 0
?
????
(1+
1
2
cos??? )
2
?=
1
2
? ?
2?? 0
?
4????
(2+cos??? )
2
?=? ?
2?? 0
?
2????
(2+cos??? )
2
.
Let the contour ' ?? ' be the unit circle |?? |=?? with centre at the origin.
Let ?? =?? ????
then cos??? ?=
1
2
(?? +
1
?? )
???? ?=???? ????
????
???? ?=
????
????
?? ?=?
0
2?? ?
2????
(2+cos??? )
2
=?
0
2?? ?
2????
???? (2+
?? 2
+1
2?? )
2
?=
1
?? ?
0
2?? ?
8?? (?? 2
+4?? +1)
2
????
????????????????????????????????????????????????? ?=
8
?? ?
?? ?
?? (?? 2
+4?? +1)
2
????
?=
8
?? ?
?? ??? (?? )????
where
?? (?? )=
?? (?? 2
+4?? +1)
2
(??)
Now the poles of ?? (?? ) are given by
(?? 2
+4?? +1)
2
=0??? =
-4±v16-4
2
=
-4±v12
2
=-2±v3 (twice)
??? (?? ) has poles of order at ?? =-2±v3 (twice)
Let ?? =-2+v3,?? =-2-v3
Clearly |?? |>1
Since |???? |=1?|?? |<1
Hence, the ?????? pole inside ?? is ?? =?? of order 2 .
????
?? ??? (?? )???? ?=??
??????
(?? 2
+4?? +1)
2
?=2???? ( residue at ?? =?? )
Now the residue at ?? =?? is
lim
?? ??? ?
?? ????
(?? -?? )
2
?? (?? 2
+4?? +1)
2
=lim
?? ??? ?
?? ????
?? (?? -?? )
2
=lim
?? ??? ?
-(?? +?? )
(?? -?? )
3
=
-(?? +?? )
(?? -?? )
3
=
-(-4)
(2v3)
3
=
(4)
8(3v3)
=
1
6v3
???????????????????????????????????????
?? ??? (?? )???? =2???? (
1
6v3
)=
2????
6v3
=
v3?? 3v3
?? from (i)
?????????????????????????????? ?
2?? 0
?
2????
(2+cos??? )
2
=
8
?? (
????
3v3
)=
8?? 3v3
??????????????????????????????????????????????????????????????? =? ?
?? 0
?
????
(1+
cos??? 2
)
2
=
8?? 3v3
5.3 State Cauchy's Residue theorem. Using it, evaluate the integral
?
?? ?
?? ?? +?? ?? (?? +?? )(?? -?? )
?? ???? ·{?? :|?? |=?? }
(2015 : 15 Marks)
Solution:
Cauchy's Residue theorem states that in a given region, integral of a function along the
closed curve is equal to 2???? times sum of its residues.
Given,
?? ?=??
?? ?
?? 2
+1
?? (?? +1)(?? -??)
2
???? ,?? :|?? |=2
?=??
?? ??? (?? )????
In the given region, |?? |=2,?? (?? ) is analytic everywhere except at ?? =0,-1,?? . ?? =0 is a
pole of order 1 .
?? =-1 is a pole of order 1 .
?? =?? is a pole of order 2 .
Now, Residue at ?? =0 :
lim
?? ?0
?
?? (?? 2
+1)
?? (?? +1)(?? -??)
2
?=
1+1
1(-??)
2
=
2
(-1)
?=-2
Residue at ?? =-1 :
lim
?? ?-1
?
(?? +1)(?? 2
+1)
?? (?? +1)(?? -1)
2
?=
1+?? -1
(-1)(-1-??)
2
=
-(1+?? -1
)
1-1+2?? ?=
(1+?? -1
)
2
??
Residue at ?? =?? :
Page 5
Edurev123
5. Cauchy's Residue Theorem
5.1 Using Cauchy's residue theorem, evaluate the integral
?? =? ?
?? ?? ?????? ?? ???????
(2013 : 15 Marks)
Solution:
Using trigonometry,
sin
4
??? ?=[
1
2
(1-cos?2?? ]
2
=
1
4
(1+cos
2
?2?? -2cos?2?? )
?=
1
8
[2+(1+cos?4?? )-2cos?2?? ]=
1
8
[cos?4?? -2cos?2?? +3]
?? ?=? ?
?? 0
?sin
4
??????? =
1
2
? ?
2?? 0
?sin
4
??????? ?=
1
2
? ?
2?? 0
?
1
8
[cos?4?? -2cos?2?? +3]????
?=
1
16
? ?
2?? 0
?[cos?4?? -2cos?2?? +3]????
?= Real part of
1
16
? ?
2?? 0
?(?? ??4?? -2?? ??2?? +3)????
Let ?? =?? ????
,???????????????????????????? =???? ????
???? ????? =
????
????
.
= R.P. of
1
16
???
?? (?? 4
-2?? 2
+3)????
????
where ?? is the unit circle.
???
?? ?? 4
-2?? 2
+3
?? ???? =2???? Sum of Residue of ?? (?? )
where ??????????????????????????????????????????????????????????? (?? )=
?? 4
-2?? 2
+3
??
Now ?? (?? ) has one pole at ?? =0 inside unit circle.
Residue at ?? =0=lim
?? ?0
????? (?? )
????????????????????????????????????????????????????????????????????=3
?? ???? (?? )=6????
???????????????????????????????????????
1
16?? ??
?? ?? (?? )=
3
8
??
????????????????????????????????????????
0
?? ?sin
4
??????? = R.p
3
8
?? =
3?? 8
5.2 Evaluate the integral ?
?? ?? ?
????
(?? +
?? ?? ?????? ??? )
?? using residues.
(2014 : 20 Marks)
Solution:
Let
?? ?=? ?
?? 0
?
????
(1+
1
2
cos??? )
2
?=
1
2
? ?
2?? 0
?
4????
(2+cos??? )
2
?=? ?
2?? 0
?
2????
(2+cos??? )
2
.
Let the contour ' ?? ' be the unit circle |?? |=?? with centre at the origin.
Let ?? =?? ????
then cos??? ?=
1
2
(?? +
1
?? )
???? ?=???? ????
????
???? ?=
????
????
?? ?=?
0
2?? ?
2????
(2+cos??? )
2
=?
0
2?? ?
2????
???? (2+
?? 2
+1
2?? )
2
?=
1
?? ?
0
2?? ?
8?? (?? 2
+4?? +1)
2
????
????????????????????????????????????????????????? ?=
8
?? ?
?? ?
?? (?? 2
+4?? +1)
2
????
?=
8
?? ?
?? ??? (?? )????
where
?? (?? )=
?? (?? 2
+4?? +1)
2
(??)
Now the poles of ?? (?? ) are given by
(?? 2
+4?? +1)
2
=0??? =
-4±v16-4
2
=
-4±v12
2
=-2±v3 (twice)
??? (?? ) has poles of order at ?? =-2±v3 (twice)
Let ?? =-2+v3,?? =-2-v3
Clearly |?? |>1
Since |???? |=1?|?? |<1
Hence, the ?????? pole inside ?? is ?? =?? of order 2 .
????
?? ??? (?? )???? ?=??
??????
(?? 2
+4?? +1)
2
?=2???? ( residue at ?? =?? )
Now the residue at ?? =?? is
lim
?? ??? ?
?? ????
(?? -?? )
2
?? (?? 2
+4?? +1)
2
=lim
?? ??? ?
?? ????
?? (?? -?? )
2
=lim
?? ??? ?
-(?? +?? )
(?? -?? )
3
=
-(?? +?? )
(?? -?? )
3
=
-(-4)
(2v3)
3
=
(4)
8(3v3)
=
1
6v3
???????????????????????????????????????
?? ??? (?? )???? =2???? (
1
6v3
)=
2????
6v3
=
v3?? 3v3
?? from (i)
?????????????????????????????? ?
2?? 0
?
2????
(2+cos??? )
2
=
8
?? (
????
3v3
)=
8?? 3v3
??????????????????????????????????????????????????????????????? =? ?
?? 0
?
????
(1+
cos??? 2
)
2
=
8?? 3v3
5.3 State Cauchy's Residue theorem. Using it, evaluate the integral
?
?? ?
?? ?? +?? ?? (?? +?? )(?? -?? )
?? ???? ·{?? :|?? |=?? }
(2015 : 15 Marks)
Solution:
Cauchy's Residue theorem states that in a given region, integral of a function along the
closed curve is equal to 2???? times sum of its residues.
Given,
?? ?=??
?? ?
?? 2
+1
?? (?? +1)(?? -??)
2
???? ,?? :|?? |=2
?=??
?? ??? (?? )????
In the given region, |?? |=2,?? (?? ) is analytic everywhere except at ?? =0,-1,?? . ?? =0 is a
pole of order 1 .
?? =-1 is a pole of order 1 .
?? =?? is a pole of order 2 .
Now, Residue at ?? =0 :
lim
?? ?0
?
?? (?? 2
+1)
?? (?? +1)(?? -??)
2
?=
1+1
1(-??)
2
=
2
(-1)
?=-2
Residue at ?? =-1 :
lim
?? ?-1
?
(?? +1)(?? 2
+1)
?? (?? +1)(?? -1)
2
?=
1+?? -1
(-1)(-1-??)
2
=
-(1+?? -1
)
1-1+2?? ?=
(1+?? -1
)
2
??
Residue at ?? =?? :
lim
?? ??? ?
?? ????
(?? -1)
2
?? 2
+1
?? (?? +1)(?? -1)
2
?=lim
?? ??? ?
?? ????
·
?? 2
+1
?? (?? +1)
?=lim
?? ??? ?{
-(?? ?? +1)
?? 2
(?? +1)
-
(?? ?? +1)
?? (?? +1)
2
+
?? ?? ?? (?? +1)
}
?=
+(1+?? ?? )
(+1)(1+??)
-
(1+?? ?? )
??(1+??)
2
+
?? ?? ??(?? +1)
?=
1+?? ?? 1+?? -
1+?? ?? 2
+
?? ?? ??(?? +1)
?=
1-?? 2
+
1+?? ?? 2
=
2-?? +?? ?? 2
??
?? ??? (?? )???? ?=2???? ( sum of residues)
?=2???? (-2+
(1+?? -1
)?? 2
+
2-?? +?? ?? 2
)
?=2???? (-2+
?? +???? -1
+2-?? +?? ?? 2
)
??
?? ??? (?? )???? ?=???? (?? ?? +???? -1
+2-4)
?=???? (?? ?? +
?? ?? -2)
??=???? (?? ?? +
?? ?? -2)
5.4 Show by applying residue theorem that
? ?
8
?? ????
(?? ?? +?? ?? )
?? =
?? ?? ?? ?? ,??? >??
(2018.: 15 Marks)
Solution:
Consider ?
-?? ?? ?
????
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Document Description: Cauchy's Residue Theorem for UPSC 2024 is part of Mathematics Optional Notes for UPSC preparation.
The notes and questions for Cauchy's Residue Theorem have been prepared according to the UPSC exam syllabus. Information about Cauchy's Residue Theorem covers topics
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Introduction of Cauchy's Residue Theorem in English is available as part of our Mathematics Optional Notes for UPSC
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Additional Information about Cauchy's Residue Theorem for UPSC Preparation
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