JNTU
JNTU B.Tech II MATHEMATICS-IV Previous Question Paper (March - 2022 - R16)
Answer any five questions. All questions carry equal marks.
Main
Find the analytic function whose real part is e-x(x sin y - y cos y)
Answer
Given the real part u(x, y) = e-x(x sin y - y cos y). We use the Milne-Thomson method to find the analytic function f(z) = u + iv. First, compute the partial derivatives:
The derivative of the analytic function is f'(z) = ux(z, 0) - i uy(z, 0). Substituting y = 0:
Thus, f'(z) = 0 - i(e-z(z-1)) = i(z-1)e-z. Integrating with respect to z:
Find the value of ‘a’ if cos(ax)sinh(y) is harmonic.
Answer
A function φ(x, y) is harmonic if it satisfies Laplace's equation: φxx + φyy = 0. Given φ = cos(ax)sinh(y):
Substituting into Laplace's equation:
For this to hold for all x and y, we must have 1 - a2 = 0, which implies a2 = 1, so a = ±1.
Evaluate the integral of 1/(e^z - 1) over the contour |z|=2.
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Find the poles and the residues at each pole of (z^2+1)/(z^2-2z).
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Evaluate using the Residue theorem the integral of (z^2-2z)/(z^2+10z+9) over the circle |z|=2.
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Expand 1/((z-1)(z-2)) about:
|z| < 1
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1 < |z| < 2
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|z| > 2
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Find the image of 1 < x < 2 under the transformation w = ez.
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Find the bilinear mapping which maps the points z = 1, i, -1 into infinity, -i, 0.
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Obtain the Fourier series for the function f(x) = 0 in (-pi, 0) and x in (0, pi).
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Find the Fourier transform of f(x) = 1-|x| for |x|<1 and 0 for |x|>1, and hence evaluate the integral of sin^2(s)/s^2 from 0 to infinity.
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Obtain a cosine series for the function f(x) = x for 0 < x < pi/2 and f(x) = pi-x for pi/2 < x < pi.
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Find the Fourier sine transform of e-|x| and hence evaluate the integral of sin(mx)/(1+x^2) from 0 to infinity.
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Solve the boundary value problem utt = a2uxx, 0 < x < l, t > 0 with u(0,t)=0, u(l,t)=0, u(x,0)=0, and ut(x,0) = sin(3*pi*x/l).
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