JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II MATHEMATICS-IV Previous Question Paper (April/May - 2023 - R16)
Note: i) Question paper consists of Part A, Part B. ii) Part A is compulsory, which carries 25 marks. In Part A, Answer all questions. iii) 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 analytic function.
Answer
A complex function f(z) is said to be analytic at a point z0 if it is single-valued and possesses a unique derivative f'(z) at every point in some neighborhood of z0. A function analytic everywhere in a domain is called holomorphic in that domain.
Find the limit of the function.
Answer
The limit of a function f(z) as z approaches z0, denoted by limz→z0 f(z) = L, exists if for every ε > 0, there exists a δ > 0 such that |f(z) - L| < ε whenever 0 < |z - z0| < δ.
Determine the poles of the integral.
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Explain about isolated singularity and removable singularity.
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Write the cross ratio of four points.
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Find u and v values of the bilinear transformation.
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Define Fourier integral theorem.
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Define the periodic function.
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Write down the one-dimensional wave equation.
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Explain the method of separation of variables.
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Part B
Check the continuity of f(z).
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Discuss the analyticity of the function.
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Show that is harmonic and find its harmonic conjugate function.
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Using definition of derivative, find it, if it exists for the following function.
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Find the residue of at the given point.
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Evaluate the integral where C is the circle.
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Evaluate the complex integral where C is the circle.
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Find the Taylor’s series expansion for the function about the point.
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Using contour integration method prove the integral.
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Find the bilinear transformation such that the point z=i is the only fixed point and the point ‘1’ is mapped as ∞.
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Discuss fully the transformation, where c is a real number. What physical problem can we study with the help of this transformation?
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Evaluate the integral.
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Evaluate the Fourier series expansion of in the given interval.
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Find Fourier transform of the function.
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Find Fourier sine and cosine transform of the function in the interval.
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Find Fourier series expansion of the function.
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Solve the equation by method of separation of variables.
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Explain the possible solutions of one-dimensional heat equation.
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A tightly stretched string with fixed end points x=0 and x=l is initially in a position given. If it is released from rest position, find the displacement y at any time and any distance from the end x=0.
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