Page 141 - Engineering Mathematics Workbook_Final

Complex Variables


(b) f is conformal on z  1 (a) 0,   i (b) 0, ,2 i  i  

(c) f  kg for some real k (c) 0,   i , 2   i (d)   0

(d) f is one to one 2
i
35. The value of  exp (e − i ) d
 1  0
31. The principal value of log i    4    is equals
 
(a) 2 i  (b) 2
i
(a) i (b) (c)  (d) i
2
36. The sum of the residues at all the

i i cot z
f
(c) (d) poles of ( ) z = 2 , where a is
+
4 8 (z a )

a constant, (a  0, 1,  2,...... ) is
32. Consider the functions
2
f ( ) z = x + 2 iy and 1  1

2
(a)  −  cosec a
+
2
2
g ( ) z = x + y + ixy . At z = 0,  n=− (n a ) 2
(a) f is analytic but not g 1  1

2
(b) −   2 +  cosec a
+
(b) g is analytic but not f n=− (n a )
(c) both f and g are analytic 1  1 2

(c) −  −  cosec a
 n=− (n a ) 2
+
(d) neither f nor g is analytic

1 (d) 1  1 +  cosec a

2
33. The coefficient of in the expansion  ) 2
z n=− (n a+
 z 
of Log       , valid in z  1, is 37. Which of the following is not the real
 z − 1  part of an analytic function?
(a) -1 (b) 1 (a) x − y
2
2
1 1
(c) − (d) 1
2 2 (b)
1 x + 2 y 2
+
34. Let  be a simple closed curve in the
(c) cos coshx y
complex. Then the set of all possible

values of  dz is (d) x + x
2
 z (1 z− 2 ) x + y 2

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