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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. PART A Max. : 100 Marks (10 2=20 marks) Answer ALL questions. 01. 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. 02. State Cauchy’s integral formula for derivatives z 03. Develop Taylor’s series of f ( z ) about z = -1. ( z 1)( z 2) 04. Express z i / z i in the form of a+ib 1 05. Show that the Dirac delta function (ax) ( x) for a 0 . a 06. State convolution theorem. 07. Solve the differential equation T ’ + k 2 T 0 . x 2x , sin ...... in the interval 08. Obtain the orthonormalising constant for the series sin L L (L, L). 09. Evaluate 4 x2 3 d x using the knowledge of Gamma function. 0 10. Generate L3 (x) and L4(x) using Rodrigue’s formula for Laugerre polynomials. 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 halfrange Fourier sine series for the function f (x) = x ; 0 < x < 2. Use the (1) n 1 2 results to develop the series . 12 n 1 n 2 14. Verify that the system y11 + y 0 ; y1(0) = 0 and y (1) = 0 is a SturmLiouville System. Find the eigen values and eigen functions of the system and hence form a orthnormal set of functions. 15. (a) If f (x) = 1 k 0 1 Ak Pk ( x) obtain Parseval’s Identity 2 2 Ak k 0 2 k 1 2 { f ( x)} d x where Pk (x) stands for Legendre polynomials. / (b) Prove that H n (x) = 2n - 1 Hn (x) where Hn (x) stands for Hermite polynomials.(4+3.5) PART C (412.5=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. 2 d 17. (a) Evaluate using contour integration. 4 3 sin 0 ez d z (7+5.5) c z 2 16 c : z i 5 . 18. (a) The current i and the charge q in a series circuit containing an inductance L and dq di q E and i = capacitance c and emf E satisfy the equations L . Using dt c dt Laplace Transforms solve the equation and express i interms of circuit parameters. 1 (b) Find L1 2 , where L-1 stands for inverse Laplace transform. (3.5) (s a 2 ) 2 (b) Using suitable theorems evaluate 19. Solve the boundary value problem 2 y 2 y 2 . with Y (0, t) = 0; yx (L, t) = 0 a t2 x2 y (x, 0) = f (x) ; yt (x, 0) = 0 and y ( x, t ) M . 20. Solve Bessels differential equation using Froebenius power series method. + + + + +