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Complex Analysis and Integral Transforms — Major (End Semester) 2023 question paper

MMMUT Mechanical Engineering previous year question paper for Complex Analysis and Integral Transforms (BSM-153), semester 2, Major (End Semester) 2023. All 28 questions are listed below, each with a written answer on Nexsus.

Paper details

  • Subject: Complex Analysis and Integral Transforms (BSM-153)
  • Branch: Mechanical Engineering
  • Semester: 2
  • Exam: Major (End Semester) 2023
  • Questions: 28

Questions asked in Complex Analysis and Integral Transforms Major (End Semester) 2023

  1. Q1. Attempt any five parts of the following.
  2. Q1(a). Find the analytic function f(z) = u + iv of which the real part u = e^x (x cos y - y sin y). [2 marks]
  3. Q1(b). What kind of singularities the following function have? (i) (x - sin z)/z^2 (ii) e^(1/2)/z^2. [2 marks]
  4. Q1(c). Show that an analytic function with constant modulus is constant. [2 marks]
  5. Q1(d). Determine the poles of the function f(z) = x^2/((x-1)^2(x+2)) and the residue at each pole. [2 marks]
  6. Q1(e). Find the Laplace inverse of the function x^2/(x^2 + 4a^2). [2 marks]
  7. Q1(f). Find Laplace transform of F(t) defined as F'(t) = { 1, 0 < t ≤ 1, t, 1 < t ≤ 2, 0, t ≥ 2 } [2 marks]
  8. Q1(g). Find the Z-transform of (a^k cos b k), k ≥ 0. [2 marks]
  9. Q2. Attempt any two parts of the following.
  10. Q2(a). Evaluate, using Cauchy's integral formula. [5 marks]
  11. Q2(a)(i). ∮(c cos z) dz around a rectangular with vertices 2 ± i, -2 ± i.
  12. Q2(a)(ii). ∮(c cos z) dz, where C is the circle |z| = 3.
  13. Q2(b). Show that the function f(z) = √(xy) is not analytic at the origin, although the Cauchy-Riemann equations are satisfied at the point. [5 marks]
  14. Q2(c). Expand ((x-2)(x+2))/((x+1)(x+4)) in Laurent series valid for (i) 1 < |z| < 4 (ii) |z| > 4 (iii) |z| < 1. [5 marks]
  15. Q3. Attempt any two parts of the following.
  16. Q3(a). Find the residue of f(z) = x^2/((x-1)^2(x-2)(x-3)) at its pole and hence evaluate ∮_C f(z) dz where C is the circle |z| = 2.5. [5 marks]
  17. Q3(b). Evaluate ∫_0^(∞) e^(-x(x-1)) dx = π e^(-x(x-1))/(2a). [5 marks]
  18. Q3(c). State and prove Taylor's Theorem. [5 marks]
  19. Q4. Attempt any two parts of the following.
  20. Q4(a). Solve the simultaneous equation dx/dt + 5x - 2y = t, dy/dt + 2x + y = 0 being given x(0) = 0 and y(0) = 0. [5 marks]
  21. Q4(b). Find Laplace transform of F(t) defined as [5 marks]
  22. Q4(b)(i). F(t) = { sin(t), 0 < t ≤ π/ω, 0, π/ω < t ≤ 2π/ω }
  23. Q4(b)(ii). Evaluate ∫_0^(π/ω) e^(-t) - π/ω dt.
  24. Q4(c). Find Inverse Laplace transform of functions: (i) ((x+2)^2)/((x^2+4x+8)^2) (ii) 1/((x^2+1)(x^2+9)). [5 marks]
  25. Q5. Attempt any two parts of the following.
  26. Q5(a). Find the Fourier transform of F(x) defined by F(x) = { 1, |x| < a, 0, |x| > a } and hence evaluate ∮(sin x)/x dx. [5 marks]
  27. Q5(b). Using finite Fourier transform, solve ∂u/∂t = ∂^2u/∂x^2 given u(0,t) = 0, u(4,t) = 0 and u(x,0) = 2x where 0 < x < 4, t > 0. [5 marks]
  28. Q5(c). Find the Z-transform of n^2a^n, and apply Z-transform to solve the difference equation: u_(n+2) + 6u_(n+1) + 9u_n = 2^n with u_0 = u_1 = 0. [5 marks]