Test: Fourier Series- 2 - Physics MCQ

The function is bounded, integrable and piece wise monotonic on  
Let us determine the Fourier coefficients



The correct answer is: - 1/n, for n even

The coefficient a₀ of the continuous Fourier series associated with the given function f(t) can be computed as


The correct answer is: 16/9

The coefficient a₁ of the continuous Fourier series associated with the above given function f(t) can be computed with k = 1 and T = 6 as following :


a₁ = –0.9119
The correct answer is: –0.9119

The function is piece wise monotonic, bounded and integrable on [-π, π]  Let us compute its Fourier coefficients

The function is continuous at all points of [-π, π] except 

which holds at all points with the exception of all discontinuities, 

At  the sum of the series

The correct answer is: 0

Since if we replace “t” by “–t”, then the function value remains the same!
The correct answer is: t²

Since the function’s value remains the same value after a period (or multiple periods) has passed!
The correct answer is: satisfies f(t + T) = f(t)

The coefficient  (corresponding to k = 1) can be expressed  as :

The coefficient b₁ of the continuous Fourier series associated with the above given function f(t) can be computed as

since  
and 
Hence 

b₁ = –0.7468
The coefficient a₁ of the continuous Fourier series associated with the above given function f(t) can be computed with k = 1 and T = 6 as following :


a₁ = –0.9119

The correct answer is: –0.4560 + 0.3734i

The substitution transforms the function into a periodic function with period . Moreover it is an even function.
∴ b_n = 0, n = 1, 2, 3,.....



The correct answer is:  for n even

Substitution of  transforms it into an odd periodic function on [-π, π],
so that the Fourier coefficients are
a_n = 0 for n = 0, 1, 2, 3


At x = 0, a point of continuity of the function, the sum of the series is zero, a fact which may be verified directly from series.
At x = l, the sum of series = 
The correct answer is: 0