JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year MATHEMATICS-IV Previous Question Paper (August/September - 2022 - R16)
Answer any five questions. All questions carry equal marks.
Main
Show that f(z) = z̄ is not differentiable at z = 0.
Answer
f'(z) = lim_{Δz → 0} [f(z + Δz) - f(z)] / Δz
For z = 0, this becomes:
f'(0) = lim_{Δz → 0} [f(0 + Δz) - f(0)] / Δz = lim_{Δz → 0} [(Δz)̄ - 0] / Δz = lim_{Δz → 0} (Δz)̄ / Δz
Let Δz = Δx + iΔy. Then (Δz)̄ = Δx - iΔy. We examine the limit along different paths:
- Along the real axis (Δy = 0): lim_{Δx → 0} (Δx - i(0)) / Δx = lim_{Δx → 0} (Δx / Δx) = 1
- Along the imaginary axis (Δx = 0): lim_{Δy → 0} (0 - iΔy) / (iΔy) = lim_{Δy → 0} (-iΔy / iΔy) = -1
Find 'k' such that u(x, y) = 3x²y - y³ + kxy is Harmonic and find its Conjugate.
Answer
Given u = 3x²y - y³ + kxy:
∂u/∂x = 6xy + ky; ∂²u/∂x² = 6y
∂u/∂y = 3x² - 3y² + kx; ∂²u/∂y² = -6y
Substituting into Laplace's equation: 6y + (-6y) = 0. This holds for any value of k. Thus, the function is harmonic for all k ∈ ℜ.
To find the Harmonic Conjugate v(x, y), we use the Cauchy-Riemann equations:
1) ∂v/∂y = ∂u/∂x = 6xy + ky
2) ∂v/∂x = -∂u/∂y = -(3x² - 3y² + kx) = -3x² + 3y² - kx
Integrating (1) with respect to y:
v = 3xy² + (ky²/2) + f(x)
Differentiating with respect to x:
∂v/∂x = 3y² + f'(x)
Equating to (2): 3y² + f'(x) = -3x² + 3y² - kx ⇒ f'(x) = -3x² - kx
Integrating: f(x) = -x³ - (kx²/2) + C
Thus, the conjugate is: v(x, y) = 3xy² + (ky²/2) - x³ - (kx²/2) + C.
Check the continuity of f(z) = z²/|z| if z ≠ 0 and f(z) = 0 if z = 0.
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Find the analytic function f(z) whose real part u(x, y) = sin x cosh y.
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Evaluate ∮C (z - a)n dz = 0 if (n ≠ -1) where C is the circle |z - a| = r.
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Evaluate ∮C e-3z / (z + 2) dz where c is the circle |z| = 4 using Cauchy's integral formula.
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Find the Laurent's series of f(z) = z / ((z² - 1)(z² - 4)) for 1 < |z| < 2.
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Evaluate ∮C (1 + z) / (z(2 - z)) dz where C : |z| = 1 using Cauchy's residue theorem.
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Evaluate ∮0π 1 / (3 + 2 cos θ) dθ using Residue theorem.
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Evaluate ∮-∞∞ x² / ((x² + 1)(x² + 4)) dx using Residue theorem.
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Evaluate ∮0∞ (x sin 2x) / (16 + x²) dx using Residue theorem.
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Find the bilinear transformation which maps the points z = -1, i, 1 respectively onto w = i, 0, -i.
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Find the Fourier series of f(x) = {-1, -π < x < 0; 1, 0 < x < π}, f(x + 2π) = f(x).
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Find the Finite Fourier sine transform of f(x) = x/π in (0, π).
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A Rod of length 10 cm has its ends A and B kept at 50°C and 100°C until steady state conditions prevail. The temperature at A is then suddenly raised to 90°C and that at B is lowered to 60°C and the end temperatures are thereafter maintained. Find the temperature at a distance x from one end at a time.
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