LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M. Sc. DEGREE EXAMINATION – PHYSICS

SECOND SEMESTER – NOVEMBER 2003

PH 2803 / PH 825 – MATHEMATICAL PHYSICS

 

15.11.2003                                                                                                  Max.   : 100 Marks

1.00 – 4.00

 

PART – A

 

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

 

1. Express x² + y² = 25 in zz^* and reⁱrepresentation.

 

2. State Liouville’s theorem.
3. Find Laurent Series of at z = 1 and name the Singularity.
4. Find the Jacobian of transformation of w = z².

 

5. Find L (F_Î (t)), where F_Î (t) represents Dirac delta function.

 

6. State parseval’s theorem.

 

7. Obtain the orthonormalizing constant for the set of functions given by ; n = 1, 2, 3  . . . .   in the interval –L to +L.

 

8. Solve the differential equation y¢ + k l² y = 0.

 

9. Write Laplace equation in spherical polar co-ordinates.

 

10. Using Rodrigue’s formula for Legendre polynomials, evaluate P₃(x).

 

PART – B

 

Answer any FOUR.                                                                                         (4 x 7.5 = 30)

 

11. Derive the necessary conditions for a function to be analytic.

 

12. Find the residues of f(z) = at its poles.

 

13. Expand f(x) = sin x, 0 < x < p in a fourier cosine series and hence prove that

 

 

(P.T.O)

-2-

 

 

 

 

14. Verify that the system y¢¢ + ly = 0; y¢ (0) = 0 and y(1) = 0 is a Sturm-Liouville system. Find the eigenvalues and eigenfunctions of the system. Prove that eigenfunctions are orthogonal.

 

15. Prove that L_n+1(x) = (2n + 1 – x)L_n(x) – n²  L_n-1(x) where L’s stand for Laugerre polynomials.

 

PART – C

 

Answer any FOUR.                                                                                       (4 x 12.5 = 50)

 

16. (i) Evaluate along

 

1. the parabola x = 2t, y = t² + 3

 

1. straight lines from (0, 3) to (2,3) and then from (2,3) to (2,4) and

 

7. a straight line from (0, 3) to (2, 4).    (7.5)

 

(ii) State and prove Poisson’s Integral formula for a circle.                                       (5)

 

17. Using contour Integration, evaluate for a>|b|.

 

18. An Inductor of 2 henrys, a resistor of 16 ohms and a capacitor of 0.02 farads are connected in series with an e.m.f E volts. Find the charge and current at any time t>0 if a) E =  300 V and   b) E = 100 sin 3t Volts

 

19. Generate Set of orthonormal functions from the sequence 1, x, x², x³ . . . . using Gram-Schmidt orthonormalization process.

 

20. Write Bessel’s differential equation and obtain the standard solution.

 

 

 

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