Page 145 - Engineering Mathematics Workbook_Final

Complex Variables

  The residue of ( ) z at its pole is
f
(a) − 3 (b) − 3
6 8 equal to 1. Then the value of  is

−
(c)  3 (d)  3 (a) -1 (b) 1
(c) 2 (d) 3
0
:
57. The straight lines L x = ,
1
L 2 : y = and L x + : y = 1 are 60. For the value of  obtained in above
0
g
problem, the function ( ) z is not
3
mapped by the transformation  = z conformal at a point
2
into the curves C , C and C  (1 3i )  (3 i + )
+
3
1
2
respectively. The angle of (a) 6 (b) 6
intersection between the curves at
 = 0 is (c) 2 (d) i 
3 2

(a) 0 (b) 61. The coefficient of (z  ) in the
2
−
4
Taylor series expansion of
  sin z

(c) (d)   if z 

z 
2 f ( ) z =  − around

=
58. Let  −  1 if z 
 2   is

   1 − ( − ) 2 + 4 dz = 4 1 − 1

   (z − ) 2 4 z    (a) 2 (b) 2
, where the close curve C is the 1 1
triangle having vertices at (c) (d) −
 − i i −   i i −  6 6
,
, i     , the integral f
  
 
 

 2   2  62. Let ( ) z be an entire function on C
being taken in anti-clockwise such that ( ) z  100log z for each
f
direction. Then one value of  is
f i =
2
z with z  . If ( ) 2i , then f(1)
(a) 1 i+ (b) 2 i+ must be
(c) 3 i+ (d) 4 i+ (a) 2
59. Consider the functions (b) 2i

z +  z (c) i
2
f ( ) z = 2 ,
(z + ) 1 (d) cannot be determined

  
g ( ) sinhz =   z −     ,   0.
 2 

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