Page 142 - Engineering Mathematics Workbook_Final

Complex Variables
→
38. The radius of convergence of 42. Define : f C C by
  1   n 2  0, if Re z = ( ) 0 or Im z = ( ) 0

  1+    f ( ) z = 

  n  − − − − − z is  , z otherwise
n
n= 0 n 3 . Then the set of points where f is

(a) e (b) 1/e analytic is

(c) 1 (d)  (a)  :Rez ( ) 0z  and Im ( )  0z 


z
( )  0
n
39. It is given that  a z converges at (b)  :Re z 
n
n= 0
z = 3 i + 4. Then the radius of (c)  :Re z  z ( ) 0 or Im z  ( )  0
convergence of the power series

 a z is (d)  :Im z 
z
( )  0
n
n= 0 n
43. Let S be the positively oriented circle


(a) 5 (b) 5
given by z − 3i = 2. Then the value
(c) <5 (d) > 5 dz
of  2 is
f
40. Let ( ) z be an analytic function. s z + 4
Then the value of  2 f e it ( ) (a) −  (b) 
( )cos t dt
0 2 2
equals − i i
(c) (d)
(a) 0 (b) 2 f ( ) 0 2 2

(c) 2 f 1 ( ) 0 (d)  f 1 ( ) 0 44. Let ( ) cos z − sin z for non-
f z =
z

1 zero z C and ( ) 0 = . Also, let
0
f
f
41. Let ( ) z = . Then the
z − 2 3z + 2 g z = 
( ) sinh z for z C .
1
coefficient of in the Laurent
z 3 (i) Then f(z) has a zero at z = 0 of
f
2
series expansion of ( ) z for z  order
is (a) 0 (b) 1
(a) 0 (b) 1 (c) 2 (d) greater than 2

(c) 3 (d) 5

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