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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