Page 140 - Engineering Mathematics Workbook_Final

Complex Variables

(c) a 2n+ 1 = 0 for all n 26. In the Laurent series expansion of
1 1
0
0
(d) a  but a = f ( ) z = z − 1 − z − 2 valid in the
0
2
1
2
23. Let I I =  cot ( 2 ) z dz , where C is region z  , the coefficient of z 2
c (z i − ) is
the contour 4x + 2 y = 2 2 (counter
(a) -1 (b) 0
clock-wise). Then I is equal to
(c) 1 (d) 2
(a) 0
27. Let  = f ( ) z be the bilinear
−

(b) 2 i
transformation that maps -1, 0 and 1
  1  to -i, 1 and i respectively. Then
(c) 2 i    2 −     f (1 i − ) equals
 sinh   

+
−

2
2 i (a) 1 2i (b) 2i
(d) −
−
sinh  (c) 2 i + (d) 1 i
+
−
2
24. The real part of the principal value of 28. For the positively oriented unit circle,
−
4 4 i is 2Re ( ) z
 dz =
(a) 256 cos (ln 4) z = 1 z + 2

(b) 64 cos (ln 4) (a) 0 (b) i
(c) 16 cos (ln 4) (c) 2 i  (d) 4 i 

(d) 4 cos (ln 4) 29. The number of zeroes, counting
multiplicities, of the polynomial

5
3
2
25. If sin z =  a n (z − / ) 4 n , then a z + 3z + z + 1 inside the circle
6
n= 0 z = 2 is
equals
1 (a) 0 (b) 2
(a) 0 (b)
720 (c) 3 (d) 5

1 − 1 30. f = u + i and g  = i + be non-
(c) (d)
720 2 720 2 zero analytic functions on z  1.

Then it follows that

(a) ' 0f 

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