LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
SIXTH SEMESTER – APRIL 2011
MT 6603/MT 6600 - COMPLEX ANALYSIS
Date : 05-04-2011
Time : 9:00 - 12:00
Dept. No.
Max. : 100 Marks
PART – A
Answer ALL Questions
(10 x 2 = 20 marks)
1. Express the function f(z) = z3+z+1 in the form f(z) = u(x, y) + iv(x, y).
2. Show that the radius of convergence of the series
3. Find the modulus of
1 1.3 2 1.3.5 3
3
z
z 
z  ...is .
2 2.5
2.5.8
2
1  2i
.
1  (1  i) 2
4. Define removable singularity and pole for a function.
5. State Morera’s theorem.
6. State Cauchy’s Residue theorem.
7. Find the points where the mapping w 
1
is conformal.
Z
z2
8. Calculate the residues of
at z = 1, 2 and 3.
( z  1)( z  2)( z  3)
9. Find the Singular points and its nature for the function e1/z.
10. Find the fixed points of the transformation w =
1
.
z  2i
PART – B
Answer any FIVE questions
(5 x 8 = 40 marks)
11. If z1 and z2 are two complex numbers, show that
z1  z2  z1  z2  z1  z2 .
12. Given v(x,y) = x  6 x y  y find f(z) = u(x,y) + iv(x,y) such that f(z) is analytic.
4
2
2
4
13. State and Prove the fundamental theorem of algebra.
14. Obtain the Taylor’s or Laurent’s series for the function f(z) =
(ii) 1< z  2 .
1
when (i) Z 
(1  z 2 )( z  2)
15. Evaluate using Cauchy Residue theorem
z 1
dz where C is the circle z  1  i  2.
 2
C z  2z  4
ez
16. State Cauchy’s theorem and Cauchy’s integral formula. Evaluate  2
dz where C is
C z 4
positively oriented circle z  i  2 .
17. State and prove maximum modulus theorem.
18. Prove that the cross ratio is invariant under Mobius transformation.
PART – C
Answer any TWO questions
(2 x 20 = 40 marks)
 xy
if z  o
 2
2
19. a) Prove that for the function f ( z )   x  y
the Cauchy – Reimann equations
0
if z  0

are satisfied at z = 0, but f(z) is not differentiable at z = 0.
b) State and prove Cauchy – Hadamard’s theorem for radius of convergence.
(10 + 10)
20. a) If f(z) is analytic inside and on a simple closed curve C except for a finite number of
poles inside C and has no zero on C, Prove that
1 f '( z )
dz  N  P where N is the

2 i C f ( z )
number of zeros and P is the number of poles of f ( z ) inside C.
2
b) Using contour integration evaluate
d
 2  cos .
(10 + 10)
0
21. a) State and Prove Taylor’s theorem.
1 1 1
b) Show that 2  
z
4 4
 z 2
(1) (n  1) 

 when z  2  2.
n 1
 2 
n

n
(14+6)
22. a) Let f be analytic in a region D and f '( zo )  0 for zo  D. Prove that f is conformal at z o .
b) Show that by means of the inversion w 
w
1
the circle z  3  5 is mapped into the circle
2
3
5
 .
16 16
c) Find the general bilinear transformation which maps the unit circle z 1 onto w  1and
the points z  1to w  1and z  1to w  1.
$$$$$$$
(10+5+5)