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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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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
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The content of Contour integration is prepared as per the latest UPSC syllabus.
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