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Page 1
Edurev123
6. Contour integration
6.1 If ?? ,?? ,?? are real numbers such that ?? ?? >?? ?? +?? ?? , show that :
? ?
?? ?? ?? ????
?? +?? ?????? ??? +?? ?????? ??? =
?? ?? v?? ?? -?? ?? -?? ??
(2009: 30 marks)
Solution:
Proof : Let
?? ?=?? ????
???? ?=???? ????
???? =???????? sin??? ?=
?? ????
-?? -????
2?? =
?? -?? -1
2?? sin??? ?=
?? ????
+?? -????
2
=
?? +?? -1
2
then,
Put values of ???? ,cos??? and sin??? in the L.H.S. and the curve becomes a circle : |?? |=1
Page 2
Edurev123
6. Contour integration
6.1 If ?? ,?? ,?? are real numbers such that ?? ?? >?? ?? +?? ?? , show that :
? ?
?? ?? ?? ????
?? +?? ?????? ??? +?? ?????? ??? =
?? ?? v?? ?? -?? ?? -?? ??
(2009: 30 marks)
Solution:
Proof : Let
?? ?=?? ????
???? ?=???? ????
???? =???????? sin??? ?=
?? ????
-?? -????
2?? =
?? -?? -1
2?? sin??? ?=
?? ????
+?? -????
2
=
?? +?? -1
2
then,
Put values of ???? ,cos??? and sin??? in the L.H.S. and the curve becomes a circle : |?? |=1
? ?
2?? 0
?
????
?? +?? cos??? +?? sin??? ?=???
?? ????
???? {?? +?? (
?? +?? -1
2
)+?? (
?? -?? -1
2?? )}
?=???
?? ????
????
2?? {2???? +???? (?? +?? -1
)+?? (?? -?? -1
)
?=???
?? 2??????
?? {2?????? +???? (?? 2
+1)+8(?? 2
-1)
?=???
?? 2????
?? 2
(?? +???? )+2?????? +???? -?? ?=???
?? 2????
(?? +?? ??){?? 2
+
2??????
?? +????
+
???? -?? ?? +????
}
=???
?? ?? (?? )????
where ?? is the circle of unit radius with centre at the origin
Now, poles of ?? (?? ) are given by
?? 2
+
2??????
?? +?? ?? +
?? ?? -?? ?? +?? ?? =0
?? =
-2????
?? +?? ?? ±
v
(
2????
?? +?? ?? )
2
-
4(?? ?? -?? )
(?? +?? ?? )
2
?? =
-2????
2(?? +?? ?? )
±
2
2v
?? 2
?? 2
-(?? ?? -?? )(?? +?? ?? )
(?? +?? ?? )
2
?? =
-???? ±v-?? 2
-{(?? ?? )
2
-?? 2
}
(?? +?? ?? )
?? =
-???? ±v-?? 2
+?? 2
+?? 2
?? +?? ?? ?? =
-???? ±v-1v?? 2
-?? 2
-?? 2
?? +?? ?? ?? =
-???? ±??v?? 2
-?? 2
-?? 2
?? +?? ?? ?
Now only
?? ?=
-???? +??v?? 2
-?? 2
-?? 2
?? +?? ?? lies inside the circle :|?? |=1
?=?? 0
(say)
Now, residue äi ?? =?? 0
Page 3
Edurev123
6. Contour integration
6.1 If ?? ,?? ,?? are real numbers such that ?? ?? >?? ?? +?? ?? , show that :
? ?
?? ?? ?? ????
?? +?? ?????? ??? +?? ?????? ??? =
?? ?? v?? ?? -?? ?? -?? ??
(2009: 30 marks)
Solution:
Proof : Let
?? ?=?? ????
???? ?=???? ????
???? =???????? sin??? ?=
?? ????
-?? -????
2?? =
?? -?? -1
2?? sin??? ?=
?? ????
+?? -????
2
=
?? +?? -1
2
then,
Put values of ???? ,cos??? and sin??? in the L.H.S. and the curve becomes a circle : |?? |=1
? ?
2?? 0
?
????
?? +?? cos??? +?? sin??? ?=???
?? ????
???? {?? +?? (
?? +?? -1
2
)+?? (
?? -?? -1
2?? )}
?=???
?? ????
????
2?? {2???? +???? (?? +?? -1
)+?? (?? -?? -1
)
?=???
?? 2??????
?? {2?????? +???? (?? 2
+1)+8(?? 2
-1)
?=???
?? 2????
?? 2
(?? +???? )+2?????? +???? -?? ?=???
?? 2????
(?? +?? ??){?? 2
+
2??????
?? +????
+
???? -?? ?? +????
}
=???
?? ?? (?? )????
where ?? is the circle of unit radius with centre at the origin
Now, poles of ?? (?? ) are given by
?? 2
+
2??????
?? +?? ?? +
?? ?? -?? ?? +?? ?? =0
?? =
-2????
?? +?? ?? ±
v
(
2????
?? +?? ?? )
2
-
4(?? ?? -?? )
(?? +?? ?? )
2
?? =
-2????
2(?? +?? ?? )
±
2
2v
?? 2
?? 2
-(?? ?? -?? )(?? +?? ?? )
(?? +?? ?? )
2
?? =
-???? ±v-?? 2
-{(?? ?? )
2
-?? 2
}
(?? +?? ?? )
?? =
-???? ±v-?? 2
+?? 2
+?? 2
?? +?? ?? ?? =
-???? ±v-1v?? 2
-?? 2
-?? 2
?? +?? ?? ?? =
-???? ±??v?? 2
-?? 2
-?? 2
?? +?? ?? ?
Now only
?? ?=
-???? +??v?? 2
-?? 2
-?? 2
?? +?? ?? lies inside the circle :|?? |=1
?=?? 0
(say)
Now, residue äi ?? =?? 0
?= lim
?? ??? 0
?
(?? -?? 0
)2
(?? +?? ?? )(?? 2
+
2??????
?? +?? ?? +
?? ?? -?? ?? +?? ?? )
?= lim
?? ??? 0
?
2(?? -?? 0
)
(?? +?? ?? )(?? -?? 0
)(?? -?? 1
)
( where ?? 1
=
-???? -??v?? 2
-?? 2
-?? 2
?? +?? ?? )
?=
2
(?? +?? ?? )(?? 0
-?? 1
)
?? 0
-?? 1
?=(
-???? +??v?? 2
-?? 2
-?? 2
?? +?? ?? )-(
-???? -??v?? 2
-?? 2
-?? 2
?? +?? ?? )
where,????????????????=
2?? v?? 2
-?? 2
-?? 2
?? +?? ?? =
1
?? v?? 2
-?? 2
-?? 2
Now,
? ?
2?? 0
?
????
?? +?? cos??? +?? sin??? ?=???
?? ?? (?? )????
?=2???? ( Residue at ?? =?? 0)
?=2???? ×
1
??v?? 2
-?? 2
-?? 2
?=
2?? v?? 2
-?? 2
-?? 2
. Proved.
=2???? ×
1
??v?? 2
-?? 2
-?? 2
? (By Cauchy's Residue Theorem)
6.2 Evaluate by Contour integration ?
?? ?? ?
????
(?? ?? -?? ?? )
?? /?? .
(2011: 15 marks)
Solution:
? ?
1
0
?
????
(?? 2
-?? 3
)
1/3
?=? ?
1
8
?
-
????
?? 2
(
1
?? 2
-
1
?? 3
)
1/3
?=? ?
8
1
?
????
?? (?? -1)
1/3
=? ?
8
0
?
?? -1/3
????
?? +1
=? ?
8
0
?
?? 2
3
-1
????
?? +1
Page 4
Edurev123
6. Contour integration
6.1 If ?? ,?? ,?? are real numbers such that ?? ?? >?? ?? +?? ?? , show that :
? ?
?? ?? ?? ????
?? +?? ?????? ??? +?? ?????? ??? =
?? ?? v?? ?? -?? ?? -?? ??
(2009: 30 marks)
Solution:
Proof : Let
?? ?=?? ????
???? ?=???? ????
???? =???????? sin??? ?=
?? ????
-?? -????
2?? =
?? -?? -1
2?? sin??? ?=
?? ????
+?? -????
2
=
?? +?? -1
2
then,
Put values of ???? ,cos??? and sin??? in the L.H.S. and the curve becomes a circle : |?? |=1
? ?
2?? 0
?
????
?? +?? cos??? +?? sin??? ?=???
?? ????
???? {?? +?? (
?? +?? -1
2
)+?? (
?? -?? -1
2?? )}
?=???
?? ????
????
2?? {2???? +???? (?? +?? -1
)+?? (?? -?? -1
)
?=???
?? 2??????
?? {2?????? +???? (?? 2
+1)+8(?? 2
-1)
?=???
?? 2????
?? 2
(?? +???? )+2?????? +???? -?? ?=???
?? 2????
(?? +?? ??){?? 2
+
2??????
?? +????
+
???? -?? ?? +????
}
=???
?? ?? (?? )????
where ?? is the circle of unit radius with centre at the origin
Now, poles of ?? (?? ) are given by
?? 2
+
2??????
?? +?? ?? +
?? ?? -?? ?? +?? ?? =0
?? =
-2????
?? +?? ?? ±
v
(
2????
?? +?? ?? )
2
-
4(?? ?? -?? )
(?? +?? ?? )
2
?? =
-2????
2(?? +?? ?? )
±
2
2v
?? 2
?? 2
-(?? ?? -?? )(?? +?? ?? )
(?? +?? ?? )
2
?? =
-???? ±v-?? 2
-{(?? ?? )
2
-?? 2
}
(?? +?? ?? )
?? =
-???? ±v-?? 2
+?? 2
+?? 2
?? +?? ?? ?? =
-???? ±v-1v?? 2
-?? 2
-?? 2
?? +?? ?? ?? =
-???? ±??v?? 2
-?? 2
-?? 2
?? +?? ?? ?
Now only
?? ?=
-???? +??v?? 2
-?? 2
-?? 2
?? +?? ?? lies inside the circle :|?? |=1
?=?? 0
(say)
Now, residue äi ?? =?? 0
?= lim
?? ??? 0
?
(?? -?? 0
)2
(?? +?? ?? )(?? 2
+
2??????
?? +?? ?? +
?? ?? -?? ?? +?? ?? )
?= lim
?? ??? 0
?
2(?? -?? 0
)
(?? +?? ?? )(?? -?? 0
)(?? -?? 1
)
( where ?? 1
=
-???? -??v?? 2
-?? 2
-?? 2
?? +?? ?? )
?=
2
(?? +?? ?? )(?? 0
-?? 1
)
?? 0
-?? 1
?=(
-???? +??v?? 2
-?? 2
-?? 2
?? +?? ?? )-(
-???? -??v?? 2
-?? 2
-?? 2
?? +?? ?? )
where,????????????????=
2?? v?? 2
-?? 2
-?? 2
?? +?? ?? =
1
?? v?? 2
-?? 2
-?? 2
Now,
? ?
2?? 0
?
????
?? +?? cos??? +?? sin??? ?=???
?? ?? (?? )????
?=2???? ( Residue at ?? =?? 0)
?=2???? ×
1
??v?? 2
-?? 2
-?? 2
?=
2?? v?? 2
-?? 2
-?? 2
. Proved.
=2???? ×
1
??v?? 2
-?? 2
-?? 2
? (By Cauchy's Residue Theorem)
6.2 Evaluate by Contour integration ?
?? ?? ?
????
(?? ?? -?? ?? )
?? /?? .
(2011: 15 marks)
Solution:
? ?
1
0
?
????
(?? 2
-?? 3
)
1/3
?=? ?
1
8
?
-
????
?? 2
(
1
?? 2
-
1
?? 3
)
1/3
?=? ?
8
1
?
????
?? (?? -1)
1/3
=? ?
8
0
?
?? -1/3
????
?? +1
=? ?
8
0
?
?? 2
3
-1
????
?? +1
Let
??
?? ?? (?? )???? =??
?? ?? ?? -1
1-?? ????
where ?? is the contour consisting of a large semi-circle G,|?? |=?? in the upper half plane
indented at ?? =0, ?? =1.
?? 1
and ?? 2
are the semi-circles in the upper half plane with radii ?? 1
and ?? 2
and centres ?? =
0,?? =1 respectively. By Cauchy's residue theorem, we have
??
?? ??? (?? )???? =??
G
??? (?? )???? +? ?
-?? 1
-?? ??? (?? )???? +? ?
?? 1
??? (?? )???? +? ?
1-?? 2
?? 1
??? (?? )???? +? ?
?? 2
??? (?? )???? +? ?
?? 1+?? 2
??? (?? )???? =0(??)
(??? (?? ) has no pole inside ?? )
Now,
|??
G
??? (?? )???? |?=? ?
?? 0
?|
?? ?? -1
?? ?? (?? -1)?? 1-Re
????
·??Re
????
|???? =? ?
?? 0
?
?? ?? ?? -1
????
?=
?? ?? ?? ?? -1
?0 as ?? ?8 as ?? <1.
Since,
?lim
?? ?0
????? (?? )=lim
?? ?0
?
?? ?? 1-?? =0,?? >0
?lim
?? 1
?0
?? ?
?? 1
??? (?? )???? =-??(?? -0)0=0
?lim
?? ?1
?(?? -1)
?lim
?? 2
?0
?? ?
?? 2
??? (?? )???? =lim
?? ?1
?(?? -1)
?? ?? -1
1-?? =-1
Hence, as ?? 1
?0,?? 2
?0,?? ?8, we have from (i)
Page 5
Edurev123
6. Contour integration
6.1 If ?? ,?? ,?? are real numbers such that ?? ?? >?? ?? +?? ?? , show that :
? ?
?? ?? ?? ????
?? +?? ?????? ??? +?? ?????? ??? =
?? ?? v?? ?? -?? ?? -?? ??
(2009: 30 marks)
Solution:
Proof : Let
?? ?=?? ????
???? ?=???? ????
???? =???????? sin??? ?=
?? ????
-?? -????
2?? =
?? -?? -1
2?? sin??? ?=
?? ????
+?? -????
2
=
?? +?? -1
2
then,
Put values of ???? ,cos??? and sin??? in the L.H.S. and the curve becomes a circle : |?? |=1
? ?
2?? 0
?
????
?? +?? cos??? +?? sin??? ?=???
?? ????
???? {?? +?? (
?? +?? -1
2
)+?? (
?? -?? -1
2?? )}
?=???
?? ????
????
2?? {2???? +???? (?? +?? -1
)+?? (?? -?? -1
)
?=???
?? 2??????
?? {2?????? +???? (?? 2
+1)+8(?? 2
-1)
?=???
?? 2????
?? 2
(?? +???? )+2?????? +???? -?? ?=???
?? 2????
(?? +?? ??){?? 2
+
2??????
?? +????
+
???? -?? ?? +????
}
=???
?? ?? (?? )????
where ?? is the circle of unit radius with centre at the origin
Now, poles of ?? (?? ) are given by
?? 2
+
2??????
?? +?? ?? +
?? ?? -?? ?? +?? ?? =0
?? =
-2????
?? +?? ?? ±
v
(
2????
?? +?? ?? )
2
-
4(?? ?? -?? )
(?? +?? ?? )
2
?? =
-2????
2(?? +?? ?? )
±
2
2v
?? 2
?? 2
-(?? ?? -?? )(?? +?? ?? )
(?? +?? ?? )
2
?? =
-???? ±v-?? 2
-{(?? ?? )
2
-?? 2
}
(?? +?? ?? )
?? =
-???? ±v-?? 2
+?? 2
+?? 2
?? +?? ?? ?? =
-???? ±v-1v?? 2
-?? 2
-?? 2
?? +?? ?? ?? =
-???? ±??v?? 2
-?? 2
-?? 2
?? +?? ?? ?
Now only
?? ?=
-???? +??v?? 2
-?? 2
-?? 2
?? +?? ?? lies inside the circle :|?? |=1
?=?? 0
(say)
Now, residue äi ?? =?? 0
?= lim
?? ??? 0
?
(?? -?? 0
)2
(?? +?? ?? )(?? 2
+
2??????
?? +?? ?? +
?? ?? -?? ?? +?? ?? )
?= lim
?? ??? 0
?
2(?? -?? 0
)
(?? +?? ?? )(?? -?? 0
)(?? -?? 1
)
( where ?? 1
=
-???? -??v?? 2
-?? 2
-?? 2
?? +?? ?? )
?=
2
(?? +?? ?? )(?? 0
-?? 1
)
?? 0
-?? 1
?=(
-???? +??v?? 2
-?? 2
-?? 2
?? +?? ?? )-(
-???? -??v?? 2
-?? 2
-?? 2
?? +?? ?? )
where,????????????????=
2?? v?? 2
-?? 2
-?? 2
?? +?? ?? =
1
?? v?? 2
-?? 2
-?? 2
Now,
? ?
2?? 0
?
????
?? +?? cos??? +?? sin??? ?=???
?? ?? (?? )????
?=2???? ( Residue at ?? =?? 0)
?=2???? ×
1
??v?? 2
-?? 2
-?? 2
?=
2?? v?? 2
-?? 2
-?? 2
. Proved.
=2???? ×
1
??v?? 2
-?? 2
-?? 2
? (By Cauchy's Residue Theorem)
6.2 Evaluate by Contour integration ?
?? ?? ?
????
(?? ?? -?? ?? )
?? /?? .
(2011: 15 marks)
Solution:
? ?
1
0
?
????
(?? 2
-?? 3
)
1/3
?=? ?
1
8
?
-
????
?? 2
(
1
?? 2
-
1
?? 3
)
1/3
?=? ?
8
1
?
????
?? (?? -1)
1/3
=? ?
8
0
?
?? -1/3
????
?? +1
=? ?
8
0
?
?? 2
3
-1
????
?? +1
Let
??
?? ?? (?? )???? =??
?? ?? ?? -1
1-?? ????
where ?? is the contour consisting of a large semi-circle G,|?? |=?? in the upper half plane
indented at ?? =0, ?? =1.
?? 1
and ?? 2
are the semi-circles in the upper half plane with radii ?? 1
and ?? 2
and centres ?? =
0,?? =1 respectively. By Cauchy's residue theorem, we have
??
?? ??? (?? )???? =??
G
??? (?? )???? +? ?
-?? 1
-?? ??? (?? )???? +? ?
?? 1
??? (?? )???? +? ?
1-?? 2
?? 1
??? (?? )???? +? ?
?? 2
??? (?? )???? +? ?
?? 1+?? 2
??? (?? )???? =0(??)
(??? (?? ) has no pole inside ?? )
Now,
|??
G
??? (?? )???? |?=? ?
?? 0
?|
?? ?? -1
?? ?? (?? -1)?? 1-Re
????
·??Re
????
|???? =? ?
?? 0
?
?? ?? ?? -1
????
?=
?? ?? ?? ?? -1
?0 as ?? ?8 as ?? <1.
Since,
?lim
?? ?0
????? (?? )=lim
?? ?0
?
?? ?? 1-?? =0,?? >0
?lim
?? 1
?0
?? ?
?? 1
??? (?? )???? =-??(?? -0)0=0
?lim
?? ?1
?(?? -1)
?lim
?? 2
?0
?? ?
?? 2
??? (?? )???? =lim
?? ?1
?(?? -1)
?? ?? -1
1-?? =-1
Hence, as ?? 1
?0,?? 2
?0,?? ?8, we have from (i)
?? ?
0
-8
??? (?? )???? +? ?
1
0
??? (?? )???? +???? +? ?
8
1
??? (?? )???? =0
???? ?
0
-8
??? (?? )???? +? ?
8
0
??? (?? )???? =-????
???? ?
8
0
??? (-?? )???? +? ?
8
0
??? (?? )???? =-????
???? ?
8
0
?
(-?? )
?? -1
1+?? ???? +? ?
8
0
?
?? ?? -1
1-?? ???? =-????
???? ?
8
0
?
(-1)
?? -1
?? ?? -1
1+?? ???? +? ?
8
0
?
?? ?? -1
1-?? ???? =-????
???? ?
8
0
?
(?? ????
)
?? -1
?? ?? -1
1+?? ???? +? ?
8
0
?
?? ?? -1
1-?? ???? =-????
???? ?
8
0
?
?? -????
?? ?????? ?? ?? -1
1+?? ???? +? ?
8
0
?
?? ?? -1
1-?? ???? =-????
???-? ?
8
0
?
?? ?????? ?? ?? -1
1+?? ???? +? ?
8
0
?
?? ?? -1
1-?? ???? =-????
Equating imaginary parts, we have
-??
sin????? ?? ?? -1
1+?? ???? ?=-?? ????????????????????????????????????????????
?? ?? -1
1+?? ???? ?=
?? sin?????
Putting ?? =
2
3
, we have
??
?? -1/3
1+?? ???? =
?? sin?
2?? 3
=
?? sin?
2?? 3
=
?? v3/2
=
2?? v3
6.3 Evaluate by contour integration :
?? =? ?
2?? 0
????
1-2?? cos??? +?? 2
,?? 2
<1
Solution:
Given :
?? =? ?
2?? 0
????
1-2?? cos??? +?? 2
,?? 2
<1
Put ?? =?? ????
? ???? =???? ????
???? =????????
Then cos??? =
?? ????
+?? -????
2
=
1
2
(?? +
1
?? )
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Contour integration Notes
Contour integration Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Contour integration.
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Practice papers and question papers enable you to assess your progress effectively.
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Contour integration UPSC Questions
The "Contour integration UPSC Questions" guide is a valuable resource for all aspiring students preparing for the
UPSC exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
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The content of Contour integration is prepared as per the latest UPSC syllabus.
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