• Support PF! Buy your school textbooks, materials and every day products Here!

Fourier series and the shifting property of Fourier transform

  • Thread starter MartynaJ
  • Start date
  • #1
5
1

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:

See above please
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:
  • Like
Likes Delta2

Answers and Replies

  • #2
1,459
874
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.
 
  • #3
5
1
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
 
  • #4
1,459
874
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?
 
  • Like
Likes Delta2
  • #5
5
1
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?
 
  • #6
1,459
874
I should have said integer values of uxL . Try a few ....

Revised!
 
  • #7
1,459
874
For instance, suppose u=(1/L)......
 

Related Threads on Fourier series and the shifting property of Fourier transform

  • Last Post
Replies
8
Views
978
  • Last Post
Replies
2
Views
1K
Replies
1
Views
1K
  • Last Post
Replies
4
Views
9K
Replies
5
Views
833
  • Last Post
Replies
2
Views
4K
Replies
0
Views
1K
  • Last Post
Replies
1
Views
960
Replies
2
Views
841
Replies
7
Views
1K
Top