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).

1. a) Obtain the eigenvalues and eigenvectors of the following matrix A: b) Show that every eignenvlaue of a unitary matrix is of unit modulus. c) For the quadratic equation 2x2 + 4xy – y2 = 24, write down the matrix of coefficients and diagonalise it. Recast it in new variables and identify the conic section it represents. d) Define contravariant tensor and covariant tensors of rank two. Write the expression aij xi xj in a 3-D space. 2. a) Show that the function f (z) = z3 is analytic in the entire z-plane. b) Obtain the Taylor series representation of log (1 + z) about z = 0. c) Using the method of residues, evaluate the integral d) Evaluate the integral 3. a) Obtain the Fourier sine transform for the following function: b) Determine the Laplace transform of f (t) = t cosh(4t). c) Solve the initial value problem 4. a) Show that P1 (x) is orthogonal to [Pn (x)]2 on the interval (-1, 1). b) Show that c) Determine the first four coefficients of the expansion of the f(x) = 5x3 + x in a series of the form d) Using the generating function derive the relation