UPSC Exam > UPSC Notes > Mathematics Optional Notes for UPSC > Cauchy's Residue Theorem
Cauchy's Residue Theorem | Mathematics Optional Notes for UPSC PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
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 ?
-?? ?? ?
????
(?? 2
+?? 2
)
2
+?
?? ?
????
(?? 2
+?? 2
)
2
Now,
? ?
?? -?? ????
(?? 2
+?? 2
)
2
=? ?
?? ?? ????
(?? +???? )
2
(?? -???? )
2
Read More
|
387 videos|203 docs
|
Related Exams
About this Document
|
Dec 18, 2024 Last updated |
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
like and Cauchy's Residue Theorem Example, for UPSC 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Cauchy's Residue Theorem.
Introduction of Cauchy's Residue Theorem in English is available as part of our Mathematics Optional Notes for UPSC
for UPSC & Cauchy's Residue Theorem in Hindi for Mathematics Optional Notes for UPSC course.
Download more important topics related with notes, lectures and mock test series for UPSC
Exam by signing up for free. UPSC: Cauchy's Residue Theorem | Mathematics Optional Notes for UPSC
Description
Full syllabus notes, lecture & questions for Cauchy's Residue Theorem | Mathematics Optional Notes for UPSC - UPSC | Plus excerises question with solution to help you revise complete syllabus for Mathematics Optional Notes for UPSC | Best notes, free PDF download
Information about Cauchy's Residue Theorem
In this doc you can find the meaning of Cauchy's Residue Theorem defined & explained in the simplest way possible. Besides explaining types of
Cauchy's Residue Theorem theory, EduRev gives you an ample number of questions to practice Cauchy's Residue Theorem tests, examples and also practice UPSC
tests
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Exam
,shortcuts and tricks
,Viva Questions
,Cauchy's Residue Theorem | Mathematics Optional Notes for UPSC
,Objective type Questions
,Important questions
,mock tests for examination
,Cauchy's Residue Theorem | Mathematics Optional Notes for UPSC
,Semester Notes
,practice quizzes
,ppt
,video lectures
,MCQs
,Cauchy's Residue Theorem | Mathematics Optional Notes for UPSC
,Previous Year Questions with Solutions
,Extra Questions
,past year papers
,Free
,Summary
,study material
,Sample Paper
;
Additional Information about Cauchy's Residue Theorem for UPSC Preparation
Cauchy's Residue Theorem Free PDF Download
The Cauchy's Residue Theorem is an invaluable resource that delves deep into the core of the UPSC exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Cauchy's Residue Theorem now and kickstart your journey towards success in the UPSC exam.
Importance of Cauchy's Residue Theorem
The importance of Cauchy's Residue Theorem cannot be overstated, especially for UPSC aspirants.
This document holds the key to success in the UPSC exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Cauchy's Residue Theorem Notes
Cauchy's Residue Theorem Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Cauchy's Residue Theorem.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Cauchy's Residue Theorem Notes on EduRev are your ultimate resource for success.
Cauchy's Residue Theorem UPSC Questions
The "Cauchy's Residue Theorem 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.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Cauchy's Residue Theorem on the App
Students of UPSC can study Cauchy's Residue Theorem alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Cauchy's Residue Theorem,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Cauchy's Residue Theorem is prepared as per the latest UPSC syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup