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              Dept. No.                                                 Max. : 100 Marks

Time : 9:00 – 12:00

 

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
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 is conformal.
8. Calculate the residues of at z = 1, 2 and 3.
9. Find the Singular points and its nature for the function e^1/z.
10. Find the fixed points of the transformation w = .

PART – B

Answer any FIVE questions                                                                                  (5 x 8 = 40 marks)

11. If z₁ and z₂ are two complex numbers, show that
12. Given v(x,y) = find f(z) = u(x,y) + iv(x,y) such that f(z) is analytic.
13. State and Prove the fundamental theorem of algebra.
14. Obtain the Taylor’s or Laurent’s series for the function f(z) = when (i)

(ii) 1< .

15. Evaluate using Cauchy Residue theorem where C is the circle
16. State Cauchy’s theorem and Cauchy’s integral formula. Evaluate where C is positively oriented circle .
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)

19. a) Prove that for the function the Cauchy – Reimann equations

are satisfied at z = 0, but f(z) is not differentiable at z = 0.

 

1. 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  where N is the

number of zeros and P is the number of poles of inside C.

1. b) Using contour integration evaluate .                                              (10 + 10)
2. a) State and Prove Taylor’s theorem.
3. b) Show that when                                   (14+6)
4. a) Let f be analytic in a region D and for Prove that  is conformal at .
5. b) Show that by means of the inversion the circle is mapped into the circle

 

1. c) Find the general bilinear transformation which maps the unit circle onto and

the points                                                               (10+5+5)

 

 

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