JNTU
JNTU B.Tech II MATHEMATICS-IV Previous Question Paper (November/December - 2018 - R16)
Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks.
PART A
State the necessary and sufficient conditions for a function f(z) to be analytic.
Answer
A function f(z) = u + iv is analytic in a domain D if and only if: 1) The partial derivatives ux, uy, vx, vy exist and are continuous in D, and 2) The Cauchy-Riemann equations ux = vy and uy = -vx are satisfied at every point in D.
Show that the function f(z) = xy + iy is everywhere continuous but is not analytic.
Answer
For f(z) = xy + iy, u = xy and v = y. Since u and v are polynomials in x and y, they are continuous everywhere, making f(z) continuous. Checking Cauchy-Riemann equations: ux = y, vy = 1, uy = x, vx = 0. Since ux ≠ vy (y ≠ 1) and uy ≠ -vx (x ≠ 0), the equations are not satisfied, so f(z) is not analytic.
Show that f(z) = 1/(1-ez) has a simple pole at z = 2πi.
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State Cauchy’s integral formula and use it to evaluate integral of (z2+4)/(z-3) dz where C is the circle |z| = 5.
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Find the fixed points of the mapping w = z + 2i.
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Find the residues at the poles of the function f(z) = (2z+1)/(z-1)2, C: |z| ≤ 4.
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If f(x) = x3 in [-π, π], find the Fourier coefficient bn.
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Find f(x) if its finite sine transform is given by fs(s) = (1+cos(sπ))/(sπ) where 0 < x < π, s = 1, 2, 3, …
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Classify the PDE: xuxx - uxy + yuyy = 1.
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Write the possible three solutions of the partial differential equation ∂2u/∂x2 = ∂2u/∂t2.
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PART B
Define analyticity of a function. Show that the function defined by f(z) = |xy| is not analytic at the origin although the C-R equations are satisfied at that point.
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Find the analytic function f(z) = u(r, θ) + iv(r, θ), when v(r, θ) = r2cos(2θ) - r cos(θ) + 2.
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Show that both the real and imaginary parts of an analytic function are harmonic.
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If f(z) = u + iv be an analytic function of z and if u - v = (x - y)(x2 + 4xy + y2) find f(z) in terms of z.
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State Cauchy integral theorem and use it to evaluate the integral e2z/((z-1)2(z-3)) dz where C is the circle |z| = 4.
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If Φ(a) = integral of (3z2+7z+1)/(z-a) dz, where C is the circle x2 + y2 = 4, find Φ(3), Φ'(1-i) and Φ''(1-i).
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Expand f(z) = 1/(z2-4z+3) in the region 1 < |z| < 3. Also name the series so obtained.
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Find the nature and location of the singularities of the function f(z) = e2z/(z-2)4 by finding its Laurent’s series expansion.
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State Residues theorem. Evaluate the integral by contour integration: integral from 0 to π of dθ/(13+5 cos θ).
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Find the residue of f(z) = z3/(z2-1) at z = ∞.
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Define bilinear transformation. Find the bilinear transformation which maps the points z = 1, i, -1 onto the points w = i, 0, -i and hence find the image of |z| < 1.
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Find the Fourier series for the function f(x) = πx/2 in 0 ≤ x ≤ 2.
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Find the Fourier transform of f(x) = 1 if |x| ≤ a, else 0. Hence prove that integral of sin2(ax)/x2 dx = πa/2.
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Develop f(x) = 2 for -2 < x < 0 and x for 0 < x < 2 in a series of sines and cosines and deduce the series for π2.
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Find the Fourier cosine transform of f(x) = e-x, x > 0.
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The ends A and B of a rod 20 cm long have the temperature at 30°C and 80°C until steady state conditions prevail. The temperature at the ends are suddenly changed to 40°C and 60°C respectively. Find the temperature distribution in the rod at time t.
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Write down one dimensional wave equation. A string is stretched and fastened to two points l cm apart. Motion is started by displacing the string in a sinusoidal arch of height y0 and then released from rest at time t = 0. Find the displacement at point x and at any time t.
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