Test: Fourier Series - Civil Engineering (CE) MCQ

The Fourier transform, 
CALCULATION:
We have;

Now take the derivative of ϕ(x, y) we have;

Again derivative we have;

Now putting these values in equation 1) we have;

Now, by applying Cauchy's differential equation we have;

Again solving we have;
The roots are written as;

Hence option 3) is the correct answer.

The Fourier series is the representation of non periodic signals in terms of complex exponentials, or equivalently in terms of sine and cosine waveform leads to Fourier series. In other words, Fourier series is a mathematical tool that allows representation of any periodic wave as a sum of harmonically related sinusoids.

f(x) = x³
find f(x) is even or odd
put x = -x
f(-x) = - x³
f(x) = -f(-x) hence it is odd function
for odd function, a_o = a_n = 0 
Fourier Series for odd function has only b_n term

Hence only sine terms are left in Fourier expansion of x³
Additional Information
Fourier Series

The Fourier series is the representation of non periodic signals in terms of complex exponentials or sine or cosine waveform. This was discovered by Jean Baptiste Joseph Fourier in 18th century.

Fourier Series is defined as

Calculation:
Given:
f(t) is an even periodic function, Since
f(-t) = f(t)
The constant term in the Fourier series is: a_o/2

Since the function is not continious, we need to break the integral according to interval
from -1 to 0, f(t) = 0,
from 0 to 1, f(t) = 4

Mistake Points
Avoid using the integral property of even function because the given function f(t) is not continuous. Though the function given is an even periodic function but it is not continuous

When the Dirichlet’s conditions are satisfied, then only for a signal, the fourier series exist. Fourier series is of two types- trigonometric series and exponential series.

Even if the Fourier series demands periodicity as the major necessity for its formation still it is not a part of Dirichlet’s condition. It is the basic necessity for Fourier series.