What are the Period and Fourier Series of the Function f(x)?

[hbomb]

Could someone please help me understanding this.
Let f(x) = 0, -2< x <0 and x, 0< x <2
f(x) repeats this pattern for all x

a) What is the period of f(x)?
b) Is f(x) even, odd, or neither?
c) Find the Fourier Series for f(x).

a) I found that the period is 2
b) odd
c) I'm not even sure I got close so I'm not even going to bother putting it up

 

[quasar987]

a) Do you know the definition of periodicity? A function f(x) is periodic of period L>0 if for all x we have f(x)=f(x+L). When we talk about the period of a function, we usually talk about the smallest such number L.

So according to your answer for a), the period is 2. But for instance, f(-1)=0 but f(-1+2)=1. So 2 isn't the period according to the definition above.

The definition is kind of abstract comparatively to how easy it is to find the period of most functions by just looking at their graph. By looking at the graph, the period is the length of the smallest interval such that the rest of the function is just a repetition(a "copy/paste") of the function in that interval.

b) Again, do you know what the definition of even and odd is or you just flipped a coin? f(x) is even if f(x)=f(-x) for all x. f(x) is odd if f(-x)=-f(x) for all x. So for instance, f(1)=1, but f(-1)=0. So f is not odd.

c) This is just a matter of calculating the two integrals for the coefficients and substituting the answers in the general form of the fouriers series.

 

[hbomb]

Well, I thought I knew what I was doing. My professor gave us a take home quiz that is due after Thanksgiving and he hasn't showed us how to do piecewise Fourier series. Possibly Monday he'll show us. I figure I could get a head start and learn how to do these problems.

 

[quasar987]

There is no difference btw piecewise and non-piecewise Fourier epansion. Both are simply about calculating the integrals giving the Fourier coefficients.

Take notice of the changes I made in post #2 concerning the definition of even and odd. I had written that f is even if f(x)=f(x) and odd if f(x)=f(-x) which is completely off.