LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

**B.Sc.** DEGREE EXAMINATION – **MATHEMATICS**

SIXTH SEMESTER – NOVEMBER 2012

# MT 6603/6600 – COMPLEX ANALYSIS

Date : 05/11/2012 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

__PART-A__

**Answer ALL questions (10×2=20 )**

- Show that the function is nowhere differentiable.
- When do we say that a function is harmonic.
- Find the radius of convergence of the series .
- State Cauchy Goursat theorem.
- Expand as a Taylor’s series about the point .
- Define meromorphic function with an example.
- Define residue of a function at a point.
- State argument principle.
- Define the cross ratio of a bilinear transformation.
- Define an isogonal mapping.

__PART-B__

**Answer any FIVE questions. (5×8=40) **

- Show that the function is discontinuous at given that when and .
- Find the analytic function of which the real part is .
- Evaluate along the closed curve containing paths and .
- State and prove Morera’s theorem.
- State and prove Maxmimum modulus principle.
- Find out the zeros and discuss the nature of the singularity of .

- State and prove Rouche’s theorem.
- Find the bilinear transformation which maps the points into the points

__PART C__

**Answer any TWO questions (2×20=40)**

- (a) Let be a function defined in a region such that and their first order partial derivatives are continuous in . If the first order partial derivatives of satisfy the Cauchy-Riemann equations at a point in D then show that f is differentiable at .

(b) Prove that every power series represents an analytic function inside its circle of convergence.

- (a) State and prove Cauchy’s integral formula.

(b) Expand in a Laurent’s series for (i) (ii)

(iii) .

- (a) State and prove Residue theorem.

(b) Using contour integration evaluate .

- (a) Let be analytic in a region and for .Prove that f is conformal at .

(b) Find the bilinear transformation which maps the unit circle onto the unit circle .

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