JNTU
JNTU B.Tech II Year MATHEMATICS - IV Previous Question Paper (April/May-2023 - R18)
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 the limit of a complex function f(z).
Answer
A complex function f(z) is said to have a limit L as z approaches z₀, denoted by limz→z₀ f(z) = L, if for every given ε > 0, there exists a δ > 0 such that |f(z) - L| < ε whenever 0 < |z - z₀| < δ.
State Cauchy-Reimann equations in polar form and check the validity C-R equations for the function
Answer
The Cauchy-Riemann equations in polar form for a function f(r, θ) = u(r, θ) + iv(r, θ) are given by:
Given the function f(r, θ) = reiθ, we use Euler's formula eiθ = cos θ + i sin θ to express it as:
Here, u = r cos θ and v = r sin θ. We calculate the partial derivatives:
- ∂u/∂r = cos θ
- ∂u/∂θ = -r sin θ
- ∂v/∂r = sin θ
- ∂v/∂θ = r cos θ
Checking the equations:
- ∂u/∂r = cos θ and (1/r)(∂v/∂θ) = (1/r)(r cos θ) = cos θ. Thus, ∂u/∂r = (1/r)(∂v/∂θ) holds.
- ∂v/∂r = sin θ and -(1/r)(∂u/∂θ) = -(1/r)(-r sin θ) = sin θ. Thus, ∂v/∂r = -(1/r)(∂u/∂θ) holds.
Since both equations are satisfied, the Cauchy-Riemann equations hold for the given function.
Define the generalized Cauchy’s integral formula.
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Define an isolated singularity and essential singularity and give one example for each.
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Explain the process of evaluating an improper integral.
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Write the properties of the bilinear transformation.
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Write the Dirichlet’s conditions for the existance of Fourier transform.
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Define Fourier sine and cosine integrals.
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Classify the partial differential equation
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Find one independent solution of
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Part B
Show that the function f(z) = xy is not analytic at the origin even though CR equations are satisfied.
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Determine the analytic function whose real part is
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If f(z) is a regular function of z, prove that
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Prove that
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Evaluate
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Find Taylor's series expansion of
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Find the residue of
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By integrating around a unit circle, evaluate
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Define the bilinear transformation and hence find the bilinear transformation which maps the points z = 0, -i, -1 into the points w= i, 1, 0.
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Obtain the Fourier series for
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Find Fourier Sine transform and Fourier Cosine transform of
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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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An insulated rod OA of length l with insulated sides, has initial temperature
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