Page 147 - Engineering Mathematics Workbook_Final

Complex Variables

n
70. Let   a z be the Laurent series (d) Both f and g are analytic
n=−
n
1  n
expansion of ( ) z = 76. If  a (z − ) 2 is the Laurent
f
2z − 2 13z + 15 − n
3 a series of the function
in the annulus  z  5. Then 1 z + 4 z + 3 z 2
2 a f ( ) z = for z C /   2 .
2 (z − ) 2 3
is equal to
Then a equals.
−
2
i dz
71. The value of  dz =

−
4  z = 4 z cos z 77. For n Z define
1  ( i n i− )x
72. Let C = z  C : z =  2 . Then c = 2 −   e dx , then
n
1  z 7 cos   1    c 2 .
2 i  C   z 2     dz n Z n


73. Let (a) cosh (b) sinh 
)
3
2
+
2
3
u ( , x y = x + ax y bxy + 2y be (c) cosh 2 (d) sinh 2
)

a harmonic function and ( , x y its ( 2 / i  )
−
−
harmonic conjugate. If (0,0 ) 1= , 78. The principal value of ( ) 1 is
+
+
then a b  ( ) (a) e (b) e
1,1
2i
2
−
74. Let C be a simple positively oriented (c) e − 2i (d) e
2
circle of radius 2 centered at origin in
the complex plane. Then 79. In the Laurent series expansion of
  1

2 C    ze + 1/z tan + z 1   dz f ( ) z = valid for z − 1 1,
i    2 (z − 1 )(z − ) 3 2   ( z z − ) 1
  1
the coefficient of is
z − 1
f z
75. Let ( ) ( x= 2 + y 2 ) 2ixy+ and (a) -2 (b) -1
2
g ( ) 2z = xy + ( i x − y 2 ) for z C . (c) 0 (d) 1
Then in the complex plane C
80. Let :f C C→ be an entire function
(a) f is analytic but g is not analytic with f(0) = 1, f(1) = 2 and f’(0) = 0. If

there exists M > 0 such that
(b) g is analytic but f is not analytic
f 11 ( ) z  M for all z C , then f(2)
(c) neither f nor g is analytic
=

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