# Loyola College B.Sc. Mathematics Nov 2012 Complex Analysis Question Paper PDF Download

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

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

PART-B

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

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

PART C

1. (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.

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

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

1. (a) State and prove Residue theorem.

(b) Using contour integration evaluate  .

1. (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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