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, a

_{0}is found by using n=0, in the formula for finding a

_{n}. b

_{n}is found by using sin(nx) instead of cos(nx) in the formula to find a

_{n}.

4. If the function f(x) is even, then which of the following is zero?

a) a_{n}

b) b_{n}

c) a_{0}

d) nothing is zero

View Answer

Explanation: Since b

_{n}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) a_{n}

b) b_{n}

c) a_{0}

d) everything is present

View Answer

Explanation: Since to find b

_{n}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 b

_{n}.

6. Find a_{0} 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 a_{0} 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 a_{n} if the function f(x) = x – x^{3}.

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 b_{n} if the function f(x) = x^{2}.

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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