Page 162 - Engineering Mathematics Workbook_Final

Complex Variables

a non-zero integer, then (a) the residue of z at z = 1 is
2
C  dz equals z − 1
(z − z 0 ) nt 1/2

2

(a) 2 n j (b) 0 (b) C  z dz = 0

nj 1 1

(c) (d) 2 n (c) C  dz = 1
2 2 z

[GATE-2015-EC-SET 3] (d) z (complex conjugate of z) is an
analytical function.
173. The value of  dz where C is
+
C ( 1 z 2 ) [GATE-2015-EC-SET 1]

i 176. In the following integral, the contour
the contour z − = 1 is C encloses the points 2 j and

2

− 2 j .
(a) 2 i  (b)  1 sin z
−
−
1
1
(c) tan z (d) i  tan z − 2 C  (z − 2 ) j 3 dz
[GATE-2007-EC]
The value of the integral is ………….
z
174. Given ( ) z = with z  , [GATE-2016]
a
X
−
(z a ) 2 dz
=
X
the residue of ( ) z z n− 1 at z a for 177. Evaluate  z sin z , where c is
n = 0 will be x + 2 y = 2 1

n
(a) a n− 1 (b) a (a) 1 (b) 2

n
(c) n a (d) n a n− 1 (c) 0 (d) -1
[ESE 2017 (EE)]
[GATE-2008 (EE)]
178. The countour C given below is on the
175. Let z = x + iy be a complex variable. complex plane z = x + jy, where
Consider that contour integration is j = − 1 .

performed along the unit circle in
anticlockwise direction. Which one of The value of the integral
the following statements is NOT 1 dz
TRUE?  C  z − 1 is ___________
2

[GATE 2018 (EC)]

160