Page 137 - Engineering Mathematics Workbook_Final

Complex Variables

1. Let  denote the boundary of the e 2
square whose sides lie along x =  1 4. The value of  (z + ) 1 4 dz is
z =
and y =  1, where  is described in 2

the positive sense. Then the value of (a) 2 ie (b) 8 i  e
−
−

1
2
 z 2 dz is 3
 2z + 3 2 i 
−
2
(c) e (d) 0
i  3
(a) (b) 2 i 
4
5. The conjugate (also called
−
+
(c) 0 (d) 2 i  symmetric) point of 1 i with respect
to the circle z − 1 = 2 is
→
2. Let : f C C be given by
 ( ) 2 (a) 1 i − (b) 1 4i
+


f ( ) z =  z / z when Z  0; .
 + (d) 1 i −
−
 0 when Z = 0 (c) 1 2i

Then f 6. The harmonic conjugate of
)
2
2
u ( , x y = x − y + xy is
(a) is not continuous at Z = 0
(b) is differentiable but not analytic at (a) x − 2 y + 2 xy
Z = 0
(b) x − 2 y − 2 xy
(c) is analytic at Z = 0
1
(d) satisfied the Cauchy – Riemann (c) 2xy + ( y − 2 x 2 )
equations at Z = 0 2

3. The function (d) xy + 1 ( 2 y − 2 x 2 )
 ( ) 2 2 2


f ( ) z =  z / z if z  0
 7. The singularity of e sin z at z =  is
 0 if z = 0

(a) a pole
(a) satisfied the Cauchy – Riemann
equations at z = 0 (b) a removable singularity

(b) is not continuous at z = 0 (c) nonisolated essential singularity

(c) is differentiable at z = 0 (d) isolated essential singularity

(d) is analytic at z = 1

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