Mathematical Physics III — DU B.Sc. (Hons) Physics Previous Year Question Papers

4. Attempt four more questions from the rest. (a) Obtain cube roots of the given complex number z = -1 + i and locate them graphically. (c) If a function f(z) = u(x, y) + iv (x, y) is analytic then show that |Vul = |Vv| must hold. (d) Find the real constants a and b such that the given function f(z) = 3x - y + 5 + i(ax + by - 3) is (e) Find the Fourier transform of the function f(t) = 8(t). (f) Show that the Fourier transform where a is a constant and Fif(t)} = F(a). (8) Show that (x, y, z) = (x' +y' + 2)½ satisfies the Laplace's equation in 3-dimensions. Hence, prove that 1 + cos 72° + cos 144° + cos 216° + cos 288° = 0. (b) Show that the function (6) is not regular at the origin although the Cauchy- Riemann Conditions are satisfied at that point. (c) Expand the function (6) in Taylor series about z = 0 and hence find the "radius of convergence' of f(z). 3. (a) Show that (9) is harmonic in domain D not containing the origin. Find the function f(z) = u(x,y) + iv(x,y) that is analytic in the domain D. Also express the function f(z) in terms of z. (9) (b) Find the Fourier Integral transform of the function x = Plot f(x) and its Fourier transform F(a) for a = 3. (9) dz, where C encloses all the (b) Find the Fourier transform of the Gaussian distribution function (8) 5. (a) Using Contour integration, evaluate (9) (b) Using Contour integration, evaluate (9) (a) Using method of separation of variables, solve (b) Solve the one-dimensional time dependent heat diffusion equation to find the temperature distribution u(x, t) in a bar of length L with the following boundary and initial conditions: (12)