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Pune University - Sem - I : Methods of Mathematical Physics, April 2010

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Total No. of Questions : 7] P882 [Total No. of Pages : 3 [3722]-103 M. Sc. PHYSICS PHY UTN - 503 : Methods of Mathematical Physics (New Course) (2008 Pattern) Time : 3 Hours] [Max. Marks : 80 Instructions to the candidates : 1) Question No. 1 is compulsory. Attempt any four questions from the remaining. 2) Draw neat diagrams wherever necessary. 3) Figures to the right indicate full marks. 4) Use of logarithmic table and pocket calculator is allowed. Q1) Attempt any four of the following : [16] a) What are spherical harmonics? State the orthogonality condition satisfied by them. b) Let W be the subspace of R4 generated by {(1, 2, 5, 3), (2, 3, 1, 4), (3, 8, 3, 5)}. Find the basis and dimensions of W. c) Prove Schwartz inequality for vectors in an inner product space. d) State and prove Parseval s identity for Fourier series F(x). e) Find the Laplace transform of the function 2 = cos t f (t ) = 3 = 0 f) 2 t> , 3 2 ,t< 3 Show that the function f(z) = z2 is an analytic function of z. P.T.O. Q2) a) State Cauchy s integral formula and prove that f ( z0 ) = b) f (z) 1 z z0 dz . 2 i c Expand f(x) in the form of Fourier series where f(x) = 0 0<x<L L < x < 2L =1 Q3) a) x (0) = 4 d 2 + cos = 0 2 3 [8] Find the eigenvalues and the corresponding orthonormal eigenvectors of the given matrix. 1 0 1 A = 0 1 0 1 0 1 b) [8] Use the calculus of residue to prove 2 Q4) a) [8] Solve x (t ) + 4 x (t ) + 4 x (t ) = 4e 2t using Laplace transform Given : x(0) = 1 b) [8] [8] Using Gram-Schmidt orthogonalization procedure construct first three Legendre polynomials. un(x) = xn for 1 x 1 and n = 0, 1, 2 ......... and the density function w(x) = 1. [8] Q5) a) Obtain the orthogenality for Hermite polynomials 2 n x H n ( x) Hm ( x)e dx = 2 n ! mn [3722]-103 -2- [8] b) Let U & W be the subspaces of R4 generated by U {(1, 1, 0, 1), (1, 2, 3, 0), (2, 3, 3, 1)} and W {(1, 2, 2, 2), (2, 3, 2, 3), (1, 3, 4, 3)}. Obtain i) Q6) a) b) dim(U + W) ii) dim (U W ) 1 2s 2 4 Find L (s + 1) (s 2) (s 3) b) [8] Obtain an expression for the integral representation of Bessel function 1 J n ( x ) = cos(n x sin )d 0 Q7) a) [8] [8] Let V = R3. If W = {(a, b, c) : a 0}. then prove that W is not subspace of V. [4] Prove for Laguerre polynomial (1 + 2n x) Ln(x) = n2 Ln 1(x) + Ln+1(x). [4] c) Find the Fourier cosine transform of f(x) = e 2x + 4e 3x. [4] d) Expand 1 in a form of Taylor series for | z | < 1. ( z 1) ( z 2) rrrr [3722]-103 -3- [4]

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Additional Info : M.Sc. PHYSICS, PHY UTN - 503 : Methods of Mathematical Physics (New Course) (2008 Pattern), Pune University
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