LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – APRIL 2012

MT 6603/MT 6600 – COMPLEX ANALYSIS

 

 

Date : 16-04-2012                   Dept. No.                                          Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

Answer all questions:                                                                                                                      (10

1. Find the absolute value of
2. Define harmonic function.
3. Find the radius of convergence of the series .
4. Using Cauchy integral formulas evaluatewhere  is the unit circle .
5. Define zero and poles of a function.
6. Write Maclaurin series expansion of s.
7. Define residue of a function.
8. State Argument principle.
9. Define isogonal mapping.
10. Define critical point.

PART – B

Answer any FIVE questions:                                                                              (5

11. Let f(z)= .  Show that f(z) satisfies CR equations at zero but not differential at .
12. Prove that is harmonic and find its Harmonic conjugate.
13. State and prove Liouvilles theorem and deduce Fundamental theorem of algebra.
14. Find the Taylors series to represent in .
15. Suppose is analytical in the region  and is not identically zero in .Show that the set of all zeros of  is isolated.
16. Use residue calculus to evaluate dz over  where  is the unit circle.
17. Show that any bilinear transformation can be expressed as a product of translation, rotation, magnification or contraction and inversion.
18. Find the bilinear transformation which maps the points  onto

PART – C

Answer any TWO questions:                                                                                               (2

19. (a) Derive CR equations in polar coordinates.

(b)   Find the real part of the analytic function whose imaginary part isand construct the analytic function.

20 .       (a) State and prove Cauchy integral formula.

(b)   Let F be an analytic inside and on a simple closed curve C. Let z be a point inside C. Show that f’(z) = dt.

21. (a) Expand in a Laurants series in (i) , ( ii)  .

(b) Using method of contour integration evaluate   dx.

22. (a) Show that any bilinear transformation which maps the unit circle = 1 onto  = 1 can be written in the form  where is real.

(b)State and prove Rouche’s theorem.

 

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