LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

SUPPLEMENTARY SEMESTER EXAMINATION – JUN 2006

M.Sc. DEGREE EXAMINATION

                                            PH 2803/PH 2900 – MATHEMATICAL PHYSICS

 

 

 

Date & Time : 27/062006/9.00 – 12.00         Dept. No.                                                       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 zz* representation for a circle.
2. State Cauchy’s integral formula for derivatives
3. Develop Taylor’s series of about z = -1.
4. Express in the form of a+ib
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 L₃ (x) and L₄(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. State and prove Cauchy’s residue theorem
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 Cosh y + 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
20. Solve Bessels differential equation using Froebenius power series method.

 

 

 

 

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