Page 163 - Engineering Mathematics Workbook_Final

Complex Variables

179. What is the residue of the function 182. The contour integral  e dz with C
1
z
−
1 e 2z at its pole? C
z 4 as the counter clock wise unit circle
in the z-plane is equal to
4 4
(a) (b) −
3 3 (a) 0 (b) 2

2 2 1
(c) − (d) (c) 2 − (d) 
3 3
[GATE-2011 (IN)]
[ESE 2018 (COMMON PAPER)]
183. In the Laurent expansion of
LAURENT EXPANSION 1
f ( ) z = valid in the
180. The Taylor series expansion of f(z) = (z − 1 )(z − ) 2

sin z about z = is region 1 z  2 , the coefficient of
4
1
   2   3  is
   z −    z −     z 2
1       4     4   
(a) 1+     z −     +   +   + .....  
2   4  2! 3!  1
  (a) 0 (b)
    2

   2   3 
   z −    z −     (c) 1 (d) -1
1        4       4    
(b) 1+     z −     − − + .....  
2   4  2! 3!  [ESE 2018 (COMMON PAPER)]
 
     
1
184. The coefficient of in the laurent
z 3 z 5 z
(c) z − + − .......
3! 5!  z 
series expansion of log       valid
(d) none [GATE]  z − 1 
in z  1 is _____ [GATE]
sin z
181. For the function of a complex
z 3 185. In Laurent series expansion of

variable z, the point z = 0 is 1 1
f ( ) z = − valid in the
(a) a pole of order 3 z − 1 z − 2
1
2
(b) a pole of order 2 region z  , the coefficient of
z 2
(c) a pole of order 1 is _________. [GATE]

(d) not a singularity

[GATE-2007 (IN)]

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