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**

- Find the absolute value of
- Define harmonic function.
- Find the radius of convergence of the series .
- Using Cauchy integral formulas evaluatewhere is the unit circle .
- Define zero and poles of a function.
- Write Maclaurin series expansion of s.
- Define residue of a function.
- State Argument principle.
- Define isogonal mapping.
- Define critical point.

__PART – B__

**Answer any FIVE questions: ****(5**

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

__PART – C__

**Answer any TWO questions:**** (2 **

- (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.

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

(b) Using method of contour integration evaluate dx.

- (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.