LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

M.Sc. DEGREE EXAMINATION – PHYSICS

SECOND SEMESTER – APRIL 2003

PH 2803 / PH 825  –  MATHEMATICAL PHYSICS

 

28.04.2003

1.00 – 4.00                                                                                                      Max : 100 Marks

                                                                PART – A                                       (10´ 2=20 marks)

      Answer ALL questions.

 

1. Starting from the general equation of a circle in the xy plane A (x² +y²) + Bx + Cy +D=0 arrive at the z z* representation for a circle.
2. State Liouville’s theorem.
3. Develop Laurent series of about z = -2.
4. Write the Jacobian of the transformation .
5. Show that the Dirac delta function .
6. State convolution theorem.
7. Solve the differential equation + .
8. Obtain the orthonormalising constant for the series in the interval     (-L, L).
9. Evaluate using the knowledge of Gamma function.
10. Generate L₂ (x) and L₃ (x) using Rodrigue’s formula for laugerre

 

 

 

                                                                PART – B                                      (4´ 7.5=30 marks)

      Answer any FOUR.

 

11. Obtain Cauchy Rieman equations from first principles of calculus of complex numbers.
12. Determine a function which maps the indicated region of w plane on to the upper half of the z – plane

v                                                                           y

w plane                                                                    z plane

p                              T

 

Q                      s         u                     p¹            Q¹                 S¹       T¹         _X

-b                     +b                                                 -1                 +1

13. Develop half-range Fourier sine series for the function f (x) = x ; 0 < x < 2. Use the results to develop the series .
14. Verify that the system y¹¹ + ; y¹(0) = 0 and y (1) = 0 is a Sturm-Liouville System. Find the eigen values and eigen functions of the system and hence form a orthnormal set of functions.
15. (a) If f (x) = obtain Parseval’s Identity
    where  P_k (x) stands for Legendre polynomials.

• Prove that  (x) = 2_{n – 1} H_n (x) where H_n (x) stands for Hermite polynomials.(4+3.5)

 

 

                                                               PART – C                                      (4´12.5=50 marks)

Answer any FOUR.

 

16. Show that u (x, y) = Sin x Coshy + 2 Cos x Sinhy + x² +4 xy – y² is harmonic Construct f (z) such that u  + iv is analytic.
17. (a)  Evaluate  using contour integration.

(b)  Using suitable theorems evaluate  c : .                                  (7+5.5)

18. (a) The current i and the charge q in a series circuit containing an inductance L and
    capacitance C and emf E satisfy the equations L  and i = . Using
    Laplace Transforms solve the equation and express i interms of circuit parameters.

• Find , where L^-1 stands for inverse Laplace transform.                 (3.5)

19. Solve the boundary value problem . with Y (0, t) = 0; y_x (L, t) = 0
    y (x, 0) = f (x) ;  y_t (x, 0)  = 0  and  and Interpret physically.
20. Solve Bessels differential equation using Froebenius power series method.

 

 

 

 

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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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