Page 161 - Engineering Mathematics Workbook_Final

Complex Variables

COMPLEX INTEGRATION USING
CAUCHY RESDUE THEOREM

i
169. Let  = e , then residue of
10
1
f ( ) z = at z = is
+
1 z 10
The value of I is
 
1 2 (a) − (b)
(a) i (b) i 10 10
2 3
i −  i  
3 4 (c) (d)
(c) i (d) i 5 5
4 5
sin z
170. The residue of ( ) z = at z = 0
f
[GATE-2017 EE SESSION-I] 8
z
COMPLEX INTEGRATION USING is
CAUCHY INTEGRAL 1
THEOREM (a) 0 (b) −
7!
168. Consider likely applicability of
Cauchy’s integral theorem to evaluate (c) 1 (d) none

the following integral counter 7!
clockwise around the unit circle C, [GATE]
I =  sec zdz z being a complex
,
−
c 171. If ( ) z = c + c z , then
1
f
variable. The of I will be 0 1
 1+ f ( ) z dz is given by
(a) I = 0 singularities set = 
unit z
circle
(b) I = 0 singularities set
+
  2n + 1   (a) 2 C (b) 2 (1 C )

=   , n = 0,1,2.... 1 0

  2  

+

(c) 2 jC (d) 2 j (1 C 0 )
1

(c) I = , singularities set [GATE-2009-EC]
2
 n
=  ; n = 0,1,2....  172. If C is a circle of radius r with centre
z , in the complex z-plane and if n is
0
(d) None of the above

[GATE-2005-CE]

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