# Fourier series and the shifting property of Fourier transform

• MartynaJ
In summary, if ##f(x)=-f(x+\frac{L}{2})## and L is the period of the periodic function ##f(x)##, then the coefficient of the even term of its Fourier series is zero. This can be proven using the shifting property of the Fourier transform for certain integer values of u multiplied by L.
MartynaJ
Homework Statement
If ##f(x)=-f(x+L/2)##, where L is the period of the periodic function ##f(x)##, then the coefficient of the even term of its Fourier series is zero. Hint: we can use the shifting property of the Fourier transform.
Relevant Equations
Summary:: If ##f(x)=-f(x+L/2)##, where L is the period of the periodic function ##f(x)##, then the coefficient of the even term of its Fourier series is zero. Hint: we can use the shifting property of the Fourier transform.

So here's my attempt to this problem so far:

##f(x)=-f(x+\frac{L}{2})## Then using the shifting property of the Fourier Transform we get: ##F(u)=-F(u)e^{2\pi i u\frac{L}{2}}##

And a periodic function is a function in the form ##f(x)=f(x+L)##. Now using the shifting property of the Fourier Transform we get: ##F(u)=F(u)e^{2\pi i u L}##

Making these two functions equal, I get:
##-e^{2\pi i u\frac{L}{2}}=e^{2\pi i u L}##

Now I don't know what else to do to prove the question. Did I go wrong anywhere?
I also know that if the even terms of Fourier series is zero, this means that function is odd, i.e. ##f(-x)=-f(x)##

Last edited:
Delta2
The question asks about a Fourier Series (for periodic functions defined on interval L) not the Fourier Transform. Similar but different. Make sure you understand the distinction.

hutchphd said:
The question asks about a Fourier Series (for periodic functions defined on interval L) not the Fourier Transform. Similar but different. Make sure you understand the distinction.
Ya but it gives us the hint to use the shifting property of Fourier transform to solve the problem

Oh I see. Your constraint equations for F(u) is correct. Only for certain integer values of u can that be correct. What are they?

Delta2
hutchphd said:
Oh I see. Your constraint equations for F(u) is correct. Only for certain integer values of u can that be correct. What are they?
how can I find out?

I should have said integer values of uxL . Try a few ...

Revised!

For instance, suppose u=(1/L)...

## What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sinusoidal functions. It is used to decompose a complex function into simpler components for analysis and manipulation.

## What is the shifting property of Fourier transform?

The shifting property of Fourier transform states that shifting a function in the time domain results in a phase shift in the frequency domain. This means that a change in the time domain will affect the frequency components of a function.

## How is the shifting property applied in signal processing?

The shifting property is used in signal processing to manipulate signals in the frequency domain. By shifting a signal in the time domain, we can adjust the phase and amplitude of its frequency components, which can be useful for filtering, modulation, and other signal processing techniques.

## What is the difference between Fourier series and Fourier transform?

A Fourier series is used to represent periodic functions, while Fourier transform is used to represent non-periodic functions. Fourier series decomposes a function into a sum of sinusoidal functions, while Fourier transform decomposes a function into its frequency components.

## What are some practical applications of Fourier series and Fourier transform?

Fourier series and Fourier transform have many practical applications in various fields, including signal processing, image processing, data compression, and differential equations. They are also used in fields such as physics, engineering, and finance for analyzing and manipulating complex functions.

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