North Maharashtra University 2008 B.Sc Mathematics S.Y. - MTH – 221 Functions of a Complex Variable. -university question paper

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NORTH MAHARASHTRA UNIVERSITY, JALGAON

S.Y.B.Sc. Mathematics (Sem –II)
MTH – 221 . Functions of a Complex Variable.

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I) ques. of 2 Marks ;
10
1) State Cauchy’s Residue Theorem.
2) obtain all poles of f(z) = ( 1)( 9)
3 2
2
2
- +
+
z z
z
3) obtain the residues of f(z) a z =0,
Where, f(z) =
( )2 z z -1
ez
.
4) describe a rational function.
5) obtain the residues of f(z) = . ( 1)( 9)
3 2
2
2
- +
+
z z
z
6) obtain Zeros and poles of f(z) =
( )2 z z -1
ez
7) obtain all zeros and poles of f(z) =
( z ) ( z )
z z
1 4
2
2 2
2
+ +
-
8) Classify the poles of f(z) = ( 4)
1
z3 z +
9) Which of the poles of f(z) =
( 3z 1)( z 3)
1
+ +
Lies inside the circle z = 1.
7) Which of the poles of f(z) =
1
1
z two +
lies in the upper half of the z- plane.
8) obtain the poles of f(z) =
( z two a2 )( z two b2 )
1
+ +
which lie in the lower half of the complex
plane.
9) obtain all zeros and poles of f(z) =
( z )( z )
z
2 one two 4
2
+ +
and Classify them.
10) obtain all zeros and poles of
x2 +1
cos x
11) obtain all zeros and poles of
( x a )( x b )
x .sin x
2 two 2 2
3
+ +
III) ques. of 6 Marks :
1) State and prove Cauchy’s Residue Theorem.
2) Evaluate by Cauchy Residue Theorem : ? -
-
C
dz
z.( z )
z
1
5 two , where Cis the
11
Circle z = two taken Counter clockwise.
3)Evaluate: ( )( )dz
z z
z
C one 9
3 2
2
2
- +
+ ? by Cauchy’s Residue Theorem , where C is
i) the circle z - two = two ,
ii) the circle z = 4
4)Evaluate :
( )
dz
z z
ez
C
2 -1 ? , where C is the circle z = three traversed in positive direction,
5) Evaluate : dz
( z ) ( z )
z z
C one 4
2
2 2
2
+ +
- ? by Cauchy’s Residue Theorem , where Cis the
rectangle formed by the lines x= two 3
- -
+ , y = + .
6) Use Cauchy’s residue theorem to evaluate ?
= + 2
3 four z z .( z )
dz
7) Use Contour integration to evaluate ?
?
+
2
0 five three ?
?
cos
d
8) Evaluate : ?
?
+
2
0 five three ?
?
sin
d
9) Evaluate : ?
?
+
2
0
(cos two )2
d
?
?
10) Use method of contour integration to evaluate ?
?
+ 0
4 2
2
?
?
sin
d
11) Apply calculus of residues to evaluate ?
8
-8 x2 +1
dx
12) Evaluate by contour integration dx
x x
x x ?
8
-8 + +
- +
10 9
2
4 2
2
13) Evaluate :
( x a )( x b )
dx
2 two 2 2
0 + + ?
8
; where a >0,b>0
14) By Contour integration , evaluate dx
( x )( x )
x
2 one two 4
2
0 + + ?
8
15) Evaluate : dx
x
cos x ?
8
-8 two +1 by using Contour integration.
12
16) Evaluate by contour integration , dx
( x a )( x b )
x .sin x
2 two 2 2
3
0 + + ?
8
where a >0,b>0 .
17) Evaluate by Cauchy’s residues theorem dz
z
e
z
z
?
=
-
1
2
18) Evaluate by contour integration ?
?
+
2
0 five four ?
?
sin
d
19) Evaluate by Contour integration ?
?
+ 0 three two ?
?
cos
d
20) Evaluate ?
?
+ +
2
0 three two ? ?
?
cos sin
d
21) Evaluate : ?
?
d
cos
cos ?
?
-? five + 4
22) Evaluate : ?
8
-8 x4 +13x2 + 36
dx
23) Evaluate , ?
8
-8 x2 + x +1
dx
24) Evaluate : ?
8
+ 0
( x2 1)2
dx
25) Evaluate ; ?
8
- + 0
( x4 6x2 25 )
dx
26) Evaluate : ?
8
-8 +
dx
x
cos x
2 4
27) Evaluate : dx
x a
x sin x ?
8
-8 two + 2
28) Use Contour integration to prove that 2
1 two = ?
+ ?
?
-? ?
?
sin
d
29) Show that
2 2
2
0
2
0
2
a b sin a b
d
a b cos
d
-
?
=
+
=
+ ? ?
? ?
?
?
?
?
where a >0,b>0
30) Prove that dx
x a
cos mx ?
8
+ 0
2 two = 0
2
=
? e- ,m
a
ma and a >0
13
31) Prove that, 0
2 four two >
?
=
+
-
8
-8
? dx e sin a;a
x
x.sin ax a
I) Multipal option Questions;
1) The poles of f(z) = z two a2
ez
+
are .- - - - -
a) +2i ,b) 0,1 , c) +ai , d) None of these.
2) The poles of f(z) = ( two 1)3
1
z +
are
a) +3i , b) 2,3 ,c) +i ,d) None of these.
3) The sum of the residues at poles of f(z) = z two a2
ez
+
is
a) sin a,
a
1 b)
2
- one , c)
2
3 , d) None of these.
4) The sum of the residues of f(z) = ( two 1)3
1
z +
is ------
a) 0, b)1, c) -1,d) None of these.
5) The residue of f(z) = two 2 4
1
z z
z
-
+
at z = 0 is
a) one , b) 0 , c ) -1 , d ) None of these.
6) The sum of residues at its poles of f(z) = ( 1)2
1
z z -
is -- - - - -
a) one , b) 0 , c ) -1 , d ) None of these.
7) The simple poles of f(z) =
5 4
4
2
2
+ +
-
z z
z are
a) 1,4 b)-1,4 c) -1,-4 d) None of these.
8) For the function f(z) = ( 4)
3
2 2
2
+
+
z . z
z , the pole z=0 has order - - - - - -
a) 1, b)2, c)0 , d) None of these.

Earning:   Approval pending.