LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

FG 31

M.Sc. DEGREE EXAMINATION – PHYSICS

SECOND SEMESTER – APRIL 2008

    PH 2900 / 2803 – MATHEMATICAL PHYSICS

 

 

 

Date : 26/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

Answer ALL questions.                                                        (10 x 2 = 20 marks)

 

1. State Cauchy’s integral formula for derivative.
2. Check whether is analytic or not.
3. Find the Taylor expansion for a function

1. State Laurent’s Theorem.
2. Give two examples of periodic functions.

Is periodic or not?

1. Find the Fourier transform of
2. Solve , where is a constant.
3. Are the set of functions are orthogonal in the limit . Expalin why?
4. Show that .
5. Using the knowledge of Gamma & Beta functions evaluate,

 

PART – B

Answer any FOUR questions.                                             (4 x 7.5 = 30 marks)

 

1. Derive Cauchy-Riemann equation in polar Co-ordinate System.
2. a) Using Cauchy’s Integral formula,

evaluate .

1. b) If is analytic in a closed region, show that .

1. Expand      in a series of sines with a period of 8.
2. Show that the eigen functions belonging to two different eigen values are orthogonal with respect to R(x) in (a, b)
3. Prove that

 

 

PART – C

Answer any FOUR questions.                                             (4 x 12.5 = 50 marks)

 

1. Derive Poisson’s Integral formula for circle.
2. Using Cauchy’s Residue theorem,

evaluate

1. a) Find the Fourier coefficients corresponding to the function

with the Period 10.

1. b) Prove that

1. Solve the boundary value problem,

1. Using the method of Frobenius, solve Laugerre’s differential equation.

 

 

 

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