JNTU
JNTU B.Tech II MATHEMATICS-IV Previous Question Paper (September/October - 2023 - R16)
Question paper consists of Part A, Part B. Part A is compulsory, which carries 25 marks. In Part A, answer all questions. In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
Part A
Define a regular function and give an example for a regular function.
Answer
A function f(z) is said to be regular (or analytic) at a point z₀ if it is single-valued and possesses a unique derivative at z₀ and at every point in some neighborhood of z₀. A function is regular in a domain if it is regular at every point in that domain.
An example of a regular function is the polynomial function f(z) = z², which is regular throughout the entire complex plane.
Define Harmonic and conjugate harmonic functions.
Answer
A function f(z) = u(x, y) + iv(x, y) is said to be harmonic in a domain if its real and imaginary parts, u and v, have continuous partial derivatives of the first and second order and satisfy Laplace's equation:
Two functions u(x, y) and v(x, y) are called conjugate harmonic functions if they satisfy the Cauchy-Riemann equations:
In this relationship, v is referred to as the harmonic conjugate of u, such that f(z) = u + iv is an analytic function.
State Cauchy’s integral formula.
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Define a pole of order 'm' and give an example for it.
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State fixed points or invariant points of a bilinear transformation.
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Explain the evaluation of the integral from -∞ to ∞ of f(x)dx using contour integration when poles of f(z) do not lie on the real axis.
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Define a periodic function.
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State Fourier integral theorem.
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Is the partial differential equation ∂²y/∂t² = c² ∂²y/∂x² parabolic? If not, what type of equation is it?
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State one dimensional heat equation.
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Part B
State and prove the necessary and sufficient conditions for a function w = f(z) to be analytic.
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Define an analytic function and hence find the analytic function whose imaginary part is e-x(x sin y - y cos y).
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Evaluate the integral of ez/(z+1)4 dz over the contour C where C is the circle |z| = 2.
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Expand f(z) = 2/(2z+1)3 about z = 0 as a Laurent’s series.
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Expand f(z) = 2/(2z+1)3 about z = 2 as a Laurent’s series.
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Evaluate by Cauchy’s residue theorem the integral of dθ / (5 + 4 sin θ) from -π to π.
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Find the bilinear transformation which maps the points 1, i, -1 in the z-plane on to the points i, 0, -i in the w-plane. Hence find the invariant points of this transformation.
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Write a note on the Fourier series of odd and even functions. Find the Fourier series expansion of the function f(x) = |cos x| in the interval [-π, π].
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Find the Fourier Transform of f(x) = 1 - x2 for |x| ≤ 1 and 0 for |x| > 1. Hence evaluate the integral from 0 to ∞ of (x cos x - sin x)/x3 cos(x/2) dx.
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Solve the equation ∂²u/∂x² = ∂u/∂y + 2u by the method of separation of variables, subject to the conditions u = 0 and ∂u/∂x = 1 + e-3y when x=0 for all values of y.
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Solve the one-dimensional wave equation by the method of separation of variables and hence find all the possible solutions.
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