Page 152 - Engineering Mathematics Workbook_Final

Complex Variables

114. The value of c  e 2z dz where (a) 1 3 − 1 5 + 1 7 − ............
(z + ) 1 4 z z z
‘c’ is z = 3 is (b) 1 + 1 + 1 + ............
z 3 z 5 z 7

8 i  4 i  1 1 1
−
−
2
2
(a) e (b) e (c) + + + ............
3 3 z 2 z 4 z 6
8 i  1 − 1 + 1 −
−
1
(c) e (d) 0 (d) z 2 z 4 z 6 ............
3
118. The Taylor series expansion of
115. The value of  − 3z + 4 dz where f z =  is
( ) sin z about z =
z + 2 4z + 5 4
‘c’ is z = 1 is (a)
1       (z   − / ) 4 2 (z  − / ) 4 3  
1   1+  z −    + + + .... 
(a) 0 (b) 2    4  2! 3!  
10
(b)

4 1       (z   − / ) 4 2 (z  − / ) 4 3  
(c) (d) 1   1+  z −    − − − .... 
5 2    4  2! 3!  

z 3 z 5
116. The Laurent series expansion of (c) z − + − ......
1 3! 5!
f ( ) z = in the valid region
4z − z 2 (d) none
z  4 is 119. The taylor series expansion of

f z = − 1
( ) Tan z about z = 0 is


(a)  z n− 1 (b)  z n− 1 3 5
n i = 4 n+ 1 n= 0 4 n+ 1 z z
(a) z − + − .......
3 5
2n+
n+
1
2


(c)  z (d)  z 3 5
n= 0 4 n+ 2 n= 0 4 2n+ 2 z z
(b) z − − − .......
3 5
117. The Laurent series expansion of
1 z 3 z 5
f ( ) z = , in the region (c) z + + + .......
z (1 z+ 2 ) 3 5
z  1 is z 3 z 5
(d) z + + + .......
3 5

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