LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
SIXTH SEMESTER – APRIL 2008
MT 6600 - COMPLEX ANALYSIS
Date : 16/04/2008
Time : 9:00 - 12:00
Dept. No.
XZ 22
Max. : 100 Marks
SECTION – A
Answer ALL questions:
(10 x 2 = 20)
1. Prove that for any two complex numbers z1 and z 2 ,
z1  z2  z1  z 2 .
2. Verify Canchy – Riemann equations for the function
x 2  y 2  2ixy
3. Show that u  3x 2 y  2 x 2  y 3  2 y 2 is harmonic.
4. Define Möbius transformation.
5. State Cauchy – Goursat theorem.
6. State Cauchy’s Integral formula.
dz
7. Evaluate 
where C is the circle z  2  3 .
z 1
c
8. Find the Taylor’s series expansion sin z about z=0.
1
9. Obtain the Laurent’s series for
in z  3 .
z 3
z2
10. Find the residue of 2
at z  ai .
z  a2
SECTION – B
Answer any FIVE questions:
(5 x 8 = 40)
11. Prove that the function
x 2 y 5 ( x  iy )
f ( z) 
, if z  0
x 4  y10
 0,
if z  0
is not differentiable at the origin, but Canchy-Riemann equations are satisfied there.
12. Show that the function u( x, y)  x 3  3xy2 is harmonic and find the corresponding analytic
function.
2z  3
13. Show that the transformation w 
maps the circle x 2  y 2  4 x  0 onto a straight line
z4
4u  3  0 .
ez
dz if C is the positively oriented circle z  3.
14. Evaluate 
z2
c
15. State and prove Liouville’s theorem.
16. State and prove Residue theorem.
1
17. State and prove Taylor’s theorem.
z 2 1
18. Find the residues of 2
at its poles.
( z  1) 2
SECTION – C
Answer any TWO questions:
(2 x 20 = 40)
19. a) If f (z ) is a regular function of z , prove that
2
 2
2 
2
 2  2  f ( z )  4 f 1 ( z ) .
y 
 x
b) Find the bilinear transformation that maps the points z1  2, z2  i , z3  2 onto the points
w1  1, w2  i, w3  1 .
20. State and prove Cauchy’s theorem.
1
21. Expand
in the regions.
( z  1)( z  3)
i) z  1
ii) 1  z  3
iii) z  3
22. Using the method of Contour integration, prove that
2
1
2
i) 
do 
2  cos
3
0

ii)
x
0
dx

2
1
2
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2