Page 144 - Engineering Mathematics Workbook_Final

Complex Variables
z  D, then which one of the 53. Which of the following is the

following statements is NOT correct? imaginary part of a possible value of
( )
i ?
f
(a) ( ) z =    f ( ) z    2 for all z  D
 z  1 
(b) f       = f ( ) z for all z  D (a)  (b) 2
 2  2

3
f z
(c) ( ) =    f ( ) z    3 for all z  D (c)  (d) 
8
4
→
(d) ( ) 1f 1 = 0 54. Let : f C C be analytic except for
a simple pole at z = 0 and let
( ) z
f
C
51. Let I =  dz , where : g C → be analytic. Then, the
C (z − 1 )(z − ) 2 Re ( ) ( )
 z  z z= 0  f z g z
f z = ( ) sin + cos and C is the value of is
2 2 Re f ( ) z
z=
0
curve z = 3 oriented anti-clockwise.
Then the value of I is (a) g(0)

(a) 4 i  (b) 0 (b) g’(0)

( )
(c) 2 i− (d) 4 i− (c) lim z f z
z→ 0

n
( ) ( )
52. Let  b z be the Laurent series (d) lim z f z g z
n
n=− z→ 0
1
expansion of the function , z
a sinh z 55. Let ( ) z = , z = x iy
+
f
−
0  z   . Then which one of the 8 z 3
following is correct?
( )
(i) Res f z is
1 7 z= 2
(a) b = 1, b = − , b =
−
0
2
6 2 360 1 1
(a) − (b)
1 7 8 8
(b) b = 1, b = − , b =
−
−
3
1
6 1 360 1 1
(c) − (d)
1 7 6 6
(c) b = 0, b = − , b =
−
0
2
6 2 360 56. The Cauchy principal value of
1 7   f ( ) x dx
(d) b = 1, b = − , b =
0 2 4 −
6 360

142