Page 146 - Engineering Mathematics Workbook_Final

Complex Variables
+
−
63. Let C be the contour z = oriented initial point 1 2i and final point
2
+
in the anti-clockwise direction. The 1 2i. The value of C  1 2z dz =
+
+
3/z
value of the integral C  ze dz = 1 z
1 
+
(a) 3 i  (b) 5 i  (a) 4 − ln2 i
2 4
(c) 7 i (d) 9 i 1 
−
+
→
64. Let : f C / 3i C defined by (b) 4 + 2 ln2 i 4
z i −
f ( ) z = 1 
iz + 3 (c) 4 + ln2 i −
2 4

Which of the following statements 1 
+
about f is FALSE? (d) 4 − ln2 i
2 2
(a) f is formal on C/3i

68. If a C with a  1, then the value
(b) f maps circles in C/3i onto circles 2
−
in C of 1 a   dz dz where  is
 z a 2
+
(c) All the fixed points of f in the
region z  C :Im ( )  0z  the simple closed curve z = 1 taken
with the positive orientation is
(d) There is no straight line in C/3i

which is mapped onto a straight line 69. Let D = z  C : z   1 . Then there
in C by f. exists a non-constant analytic
function f on D such that for all
2
f
65. The function ( ) z = z + iz + 1 is n  2
differentiable at
  − 1 
(a) i (b) 1 (a) f   n   = 0
 
(c) -i (d) no point in C
  1  

66. The radius of convergence of the (b) f    n    = 0

power series  4 ( ) 1 n z− n 2n is
n= 0   1  
(c) f  1 −    = 0
( )  0 and let
67. Let  = z  C :Im z   n 
C be a smooth curve lying in  with   1 1  
(d) f  −    = 0
 2 n 

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