Page 143 - Engineering Mathematics Workbook_Final

Complex Variables

g ( ) z − 1
(ii) Then has a pole at z = 0 of (c) cos (1) (d) sin ( ) 1
zf ( ) z 2
order 48. Consider the function

(a) 1 (b) 2 f ( ) z = e iz
2
( z z + ) 1
(c) 3 (d) greater than 3
45. For the function (i) The residue of f at the isolated
 1  singular point in the upper half plane
f z =      , the point z z = +  : y   0 is
( ) sin
x iy C
 cos (1/ z ) 
= 0 is − 1 − 1
(a) (b)
(a) a removable singularity 2e e

(b) a pole (c) e (d) 1
2
(c) an essential singularity
(ii) The Cauchy Principal Value of
(d) a non-isolated singularity  sin xdx
the integral  2 is
15

n
f
46. Let ( ) z =  z for z C . If − ( x x + ) 1
n= 0 − −

+
−
−
C : z i − = 2 then  f ( ) z dz = (a) 2 (1 2e 1 ) (b) (1 e 1 )
C (z i − ) 15
)
+
−
+

(c) 2 (1 e (d) (1 e − 1 )
)
+
(a) 2 i  (1 15i
)
−
u
49. Let ( , x y = ) 2x (1 y for all real x
)
(b) 2 i (1 15i− ) and y. then a function ( , x y , so

)
(c) 4 i (1 15i+ ) that ( ) z = f u ( , x y + ) i ( , x y is
analytic, is
(d) 2 i
2
2
(a) x − 2 ( y − ) 1 (b) ( x − ) 1 − 2 y

n
47. Let  a n (z + ) 1 be the Laurent
2
n=− (c) ( x − ) 1 + 2 y (d) x + 2 ( y − ) 1
2
series expansion of
 z 
f
f ( ) sinz =       . Then a = 50. Let ( ) z be analytic o n
−
 z + 1  2 D = z C : z − 1 1 such that


(a) 1 (b) 0 f ( ) 1 = 1. If ( ) z = f z 2
f
( ) for all

141