MSc Physics (Electronics I) SEM I 2022 2023 Jan 2023 PHYSICS MATHEMATICAL METHODS Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate full marks

1. Q1 (a) Attempt any one: - 8 marks
   • (i) Derive Cauchy Riemann equations in cartesian form
   • b) Determine whether is analytic or not?
   • (ii) f(z) is analytic in a closed curve C, except at a finite number of poles within = (sum of residues) Hence Evaluate dz where c is the circle Izl = =
   • (b) Attempt any one: - 4
   • (i) Prove that u = x? — y* — 2xy — 2x + 3y is harmonic. Find a function v such that f(z) = u + iv is analytic Gi) Expand f(z) = Laurent series for
2. Q2 (a) Attempt any one: - 8 marks
   • (i) Find the inertia tensor about the origin for the mass distribution consisting of mass | at (0, 1, 1) and mass 2 at (1, —1,0). Find the principal moment of inertia and the principal axis
   • (ii) Find the characteristic frequencies and characteristic mode of vibration for the system of masses (m & 4m) and springs having spring constants (3k & k) as shown in the figure the motion is along a vertical line (Neglect mass of the spring.)b
   • (b) Attempt any one: - 4
   • (i) Using the Levi-Civita symbol, show that, A x (B x C) = B(A.C) — C(A.B)
   • (ii) Find the eigenvalues of the matrix {2 1 —2
3. Q3 (a) Attempt any one: - 8 marks
   • (ii) Solve the Bessel equation
   • (b) Attempt any one: - 4
   • (i) Using the generating function for Hermite polynomials
   • (ii) Using Green’s Theorem, Find the area of the region in the first quadrant bounded by the curves y =x, y= y=
4. Q4 (a) Attempt any one: - 8 marks
   • (i) State the expression of Fourier transform and its inverse in three-dimensional space. Find the Fourier transform of Yukawa potential
   • (ii) Solve the differential equation using Laplace transform
   • (b) Attempt any one: - 4
   • (i) Find the Laplace transform of f(t) = t sinhat
   • (ii) State and prove Fourier Convolution Theorem
5. Q5 Attempt any four: - 12 marks
   • (a) Determine the poles of the function and residue at the poles: f(z) =
   • (b) Use Cauchy’s integral formula to calculate
   • (c) What is the rank of the tensor ? Write its transformation equation
   • (d) For matrices M, C and D, show that M" = C where C! M C =D and D is
   • (e) Prove that = 1
   • (f) Find regular singular point of the differential equation
   • (g) Find the inverse Laplace transform of F(s) =

Read from the scan above, so a character or two may differ. The scan is the original.