LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

ZA 36

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – April 2009

MT 6603/MT 6600 – COMPLEX ANALYSIS

 

 

 

Date & Time: 18/04/2009 / 9:00 – 12:00        Dept. No.                                                          Max. : 100 Marks

 

 

PART – A

Answer ALL questions:                                                                               (10 x 2 = 20)  

1. Find the absolute value of .
2. Show that does not have a limit as .
3. When do you say that a sequence of function converges uniformly?
4. Using Cauchy integral formula evaluate where C is the circle .
5. Find zeros of the function .
6. When do you say that a point “a” is an isolated singularity of ?
7. Calculate the residue of .
8. State Cauchy’s residue theorem.
9. Define a bilinear transformation.
10. When do you say that a function is conformal at ?

PART – B

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

11. Illustrate by an example that C.R. equations are not sufficient for differentiability at a point.
12. State Liouville’s theorem and deduce the Fundamental theorem of algebra from it.
13. Let be analylic in a region D bounded by two concentric circles and on the boundary. Let be any point in D. Show that

.

14. State and prove Maximum modules theorem.
15. Expand as a power series in the region .
16. Evaluate where C is the square having vertices

.

 

 

17. Using contour integration evaluate .
18. Find the image of the circle under the map .

PART – C

Answer any TWO questions:                                                                       (2 x 20 = 40)

 

19. (a) Let nbe a function defined on the region D such that u and v and

their first order partial derivatives are continuous in D. If the first order partial derivatives of u and v satisfy the C.R. equations at , show that  is differentiable at

(b) Let be a given power series. Show that there exists a number R such that

such that

(i)  the series converges absolutely for every z with

(ii) if , the convergence uniform in .

(iii) if , the series diverges.                                                                        (10+10)

19. (a) State and prove Taylor’s theorem.

(b) Expand using Taylors series about the point .                                       (15+4)

20. (a) State and prove Argument theorem.

(b) State and prove Rouche’s theorem.

(c) Prove that .                                                                                      (7+6+7)

21. (a) Show that the transformation maps the circle into a straight

line given by .

(b) Find the bilinear transformation which maps the points respectively to

.

(c) Show that a bilinear transformation where maps the real axis into

itself if and only if a, b, c, d are real.                                                                          (6+6+8)

 

 

 

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