# Fourier series and the shifting property of Fourier transform

## 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

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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.

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
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)......