Page 154 - Engineering Mathematics Workbook_Final

Complex Variables

( )
126. Let f z = e z be a complex (d) The function f has zeros of order
2 at 2 n , n =  
z 2018 1, 2,...
function. Then
[NET JUNE 2013]
(a) f(z) has a simple pole at z = 0
129. Consider the function
(b) f(z) has a pole at z = 0 of order
( z
2018 sin  / ) 2
f ( ) z = . Then f has
( ) z
(c) The residue of f(z) at z = 0 is sin 
1 poles at
2017! (a) all integers

(b) all even integers
(d) The residue of f(z) at z = 0 is
1 (c) all odd integers

2018! (d) all integers of the form 4k + 1,

Z
127. The value of the contour integral k 
2
→
  z − 1 dz is 130. Let f :C C be a holomorphic
z function and let u be the real part of f
(a) 0, when the curve is a and  be the imaginary of f. Then,
semicircle z = 2e i (0     ) for , x y  , f ( ' x  ) y 2 is equal
+
R
(b) − i , when the curve  is a to
)
 
semicircle z = 2e i (   (a) u + u (b) u + u
2
2
2
2
x
x
y
y
(c) − i , when the curve  is a (c) u + u (d) u − u
2
2
2
2
semicircle z = 2e i (   2 ) y y x y

131. Let f(z) be the meromorphic function
−
(d) 2i , when the curve  is a z
f
semicircle z = 2e i (0    2  ) given by ( ) z = . Then
−
(1 e z )sin z
128. Consider the function
)

−
f ( ) z = z 2 (1 cos z , z C . Which (a) z = 0 is a pole of order 2.

Z
of the following are correct? (b) For every k  , z = 2 ik is a
simple pole.
(a) The function f has zeros of order 2
=
k
at 0 (c) For every k  / z   0 , z  is a
(b) The function f has zeros of order simple pole.


1 at 2 n , n = 1, 2,..... (d) z  = 2 i is a pole.
 +
(c) The function f has zeros of order 4
at 0.

152