This set of Fourier Analysis Multiple Choice Questions & Answers (MCQs) focuses on “Fourier Series Expansions”.
1. Which of the following is not Dirichlet’s condition for the Fourier series expansion?
a) f(x) is periodic, single valued, finite
b) f(x) has finite number of discontinuities in only one period
c) f(x) has finite number of maxima and minima
d) f(x) is a periodic, single valued, finite
View Answer
Explanation: Dirichlet’s condition for Fourier series expansion is f(x) should be periodic, single valued and finite; f(x) should have finite number of discontinuities in one period and f(x) should have finite number of maxima and minima in a period.
2. At the point of discontinuity, sum of the series is equal to ___________
a) \(\frac{1}{2} [f(x+0) – f(x-0)] \)
b) \(\frac{1}{2} [f(x+0) + f(x-0)] \)
c) \(\frac{1}{4} [f(x+0) – f(x-0)] \)
d) \(\frac{1}{4} [f(x+0) + f(x-0)] \)
View Answer
Explanation: When there is a point of discontinuity, the value of the function at that point is found by taking the average of the limit of the function in the left hand side of the discontinuous point and right hand side of the discontinuous point. Hence the value of the function at that point of discontinuity is \(\frac{1}{2} [f(x+0) + f(x-0)] \).
3. What is the Fourier series expansion of the function f(x) in the interval (c, c+2π)?
a) \(\frac{a_0}{2}+∑_{n=1}^∞ a_n cos(nx) +∑_{n=1}^∞ b_n sin(nx) \)
b) \(a_0+∑_{n=1}^∞ a_n cos(nx) +∑_{n=1}^∞ b_n sin(nx) \)
c) \(\frac{a_0}{2}+∑_{n=0}^∞ a_n cos(nx) +∑_{n=0}^∞ b_n sin(nx) \)
d) \(a_0+∑_{n=0}^∞ a_n cos(nx) + ∑_{n=0}^∞ b_n sin(nx) \)
View Answer
Explanation: Fourier series expantion of the function f(x) in the interval (c, c+2π) is given by \(\frac{a_0}{2}+∑_{n=1}^∞ a_n cos(nx) +∑_{n=1}^∞ b_n sin(nx) \) where, a0 is found by using n=0, in the formula for finding an. bn is found by using sin(nx) instead of cos(nx) in the formula to find an.
4. If the function f(x) is even, then which of the following is zero?
a) an
b) bn
c) a0
d) nothing is zero
View Answer
Explanation: Since bn includes sin(nx) term which is an odd function, odd times even function is always odd. So, the integral gives zero as the result.
5. If the function f(x) is odd, then which of the only coefficient is present?
a) an
b) bn
c) a0
d) everything is present
View Answer
Explanation: Since to find bn we have sin(nx) and the function we have is also odd function, the product of odd function and another odd function yields even function, the only coefficient which exists is bn.
6. Find a0 of the function \(f(x) = \sqrt{\frac{1-cosx}{2}}.\)
a) \(\frac{4}{π} \)
b) \(\frac{2}{π} \)
c) \(\frac{π}{4} \)
d) \(\frac{π}{2} \)
View Answer
Explanation: \(f(x) = \sqrt{\frac{1-cosx}{2}} = sin(\frac{x}{2}) \)
\(a_0 = \frac{1}{π} \int_0^{2π} sin(\frac{x}{2})dx \)
\( = \frac{1}{π} (-cos(\frac{x}{2}))^2 \)
\( = \frac{(-2)}{ π (-2)} \)
\( = \frac{4}{π} \)
7. Find a0 of the function \(f(x) = \frac{1}{4} (π-x)^2.\)
a) \(\frac{π^2}{6} \)
b) \(\frac{π^2}{12} \)
c) \(5\frac{π^2}{6} \)
d) \(5\frac{π^2}{12} \)
View Answer
Explanation: \(f(x) = \frac{1}{4} (π-x)^2 \)
\(a_0 = \frac{1}{π} \int_0^{2π} \frac{1}{4} (π-x)^2 \)
= from 0 to 2 π
\( = \frac{1}{2} (\frac{1}{3} π^2 ) \)
\( = \frac{π^2}{6} \).
8. Find the sum of \(\frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} \) +……… using Fourier series expansion if f(x) = a when [0,π] and 2 π – x when [ π, 2 π].
a) \(\frac{π^2}{8} \)
b) \(\frac{π^2}{4} \)
c) \(\frac{π^2}{16} \)
d) \(\frac{π^2}{2} \)
View Answer
Explanation: \(a_0 = \frac{1}{π} (\int_0^π x dx)(\int_π^{2π} (2π-x) dx) \)
\( = \frac{1}{π} [π^2+4π^2-4 \frac{π^2}{2}-2π^2+\frac{π^2}{2}] \)
\( = π.\)
\(a_n = \frac{1}{π} (\int_0^π (xcos(nx)dx) )(\int_π^{2π}((2π-x)cos(nx))dx) \)
\( = \frac{-4}{πn^2} \) when n is odd and 0 when n is even
\( b_n = 0 \)
now, substituting x=0 in the given function and the Fourier series expansion, we get,
\(0 = \frac{π}{2}-\frac{4}{π} (\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} +….) \)
Therefore, \(\frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} +………= \frac{π^2}{8} \).
9. Find an if the function f(x) = x – x3.
a) finite value
b) infinite value
c) zero
d) can’t be found
View Answer
Explanation: Since the function is odd function, odd times even function (cos(nx)) is odd function. So in the given interval, the coefficient is zero.
10. Find bn if the function f(x) = x2.
a) finite value
b) infinite value
c) zero
d) can’t be found
View Answer
Explanation: Since the given function is even function, even times odd function(sin(nx)) gives odd function. Hence the coefficient in the given interval is zero.
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