IGNOU PHE-14 - Mathematical Methods in Physics-III

1. a) Determine the values of a, b, c when b) Verify the Cayley-Hamilton theorem of the matrix. and hence obtain. P-1. c) If Aij is an antisymmetric tensor and Bi is a vector, show that Aij Bi Bj = 0. d) What are the four conditions to be satisfied by the elements of a group? Show that the set of all complex numbers of unit magnitude u(1) ={z : |z|=1} forms a group. 2. a) Using the method of residues, evaluate the contour integral where C is defined by | z | < 4. b) Using the method of residues, evaluate the integral c) i) Show that the function f (z) = z3 is analytic in the entire z-plane. ii) Obtain the Taylor series expansion of cos2 z about z = 0. 3. a) Obtain the Fourier cosine transforms of the function: b) Calculate the inverse Laplace transform of the function: c) Solve the initial value problem using the method of Laplace transforms: y’’ - 2y’ - 3y = 0; y(0)= 1, y’(0) = 7 d) Calculate the Laplace transform of t n e at. 4. a) Show that b) Using the generating function for Legendre polynomials show that: c) Use Rodrigues’ formula for Laguerre polynomials to generate L4(x).