Engineering Mathematics-II Unit 4 Notes AKTU (BAS203)

AKTU · BTECH · Semester 2 · Engineering Mathematics-II · Unit 4 · Notes

AKTU Engineering Mathematics-II (BAS203) Unit 4 notes for B.Tech Semester 2 – Complex Variable – Differentiation. Topics: Complex Variable Functions…

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Unit 4: Complex Variable – Differentiation – AKTU syllabus topics

Complex Variable Functions: Limit Continuity Differentiability
Analytic Functions: Cauchy-Riemann Equations, Harmonic Functions, Milne's Thompson Method
Conformal Mapping: Mobius Transformation

Q2d. Show that u = x⁴ − 6x²y² + y⁴ is harmonic function. Find complex function f(z) whose u is a real part. (10 marks, 2018-19)
Q6b. Find Mobius transformation that maps points z = 0, −i, 2i into the points w = 5i, ∞, −i/3 respectively. (10 marks, 2018-19)
Q2d. Find the harmonic conjugate of u = x³ − 3xy² + 3x² − 3y² and hence find the analytic function f(z) = u + iv. (7 marks, 2023-24)
Q6b. Find a bilinear transformation which maps the points i, −i, 1 of the z-plane into 0, 1, ∞ of the w-plane respectively. (10 marks, 2021-22)
Q6a. Let f(z) = x²y⁵(x + iy) / (x⁴ + y¹⁰) when z ≠ 0, f(z) = 0 when z = 0. Prove that Cauchy Riemann satisfies at z = 0 but function is not differentiable at z = 0. (10 marks, 2018-19)
Q2d. If f(z) = u + iv is analytic, and u − v = (e^y − cos x + sin x) / (cosh y − cos x), find f(z) such that f(π/2) = (3−i)/2. (7 marks, 2024-25)
Q6b. Define analytic function. Show that f(z) = |z|² is not analytic anywhere except at origin. (7 marks, 2023-24)
Q6a. Find the analytic function f(z) = u + iv given u + v = (x − y)(x² + 4xy + y²) using Milne's Thomson method. (7 marks, 2023-24)
Q6b. Find the bilinear transformation which maps the points z = 0, −1, i onto w = i, 0, ∞. Also, find the image of the unit circle |z| = 1. (7 marks, 2022-23)
Q1f. Show that the following function is harmonic: h(x, y) = x² + xy − y² (2 marks, 2024-25)

AKTU paper codes: BAS203, KAS203