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

Answer ALL questions                                                                                                                 (10×2=20 )

  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

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

  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

Answer any TWO questions                                                                                                       (2×20=40)

  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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