Hi, I have a few questions about Fourier series.

I understand the derivation of the formulas for working out the coefficients in your series, but I have a few queries about your bounds of integration, and whether or not you need to change these bounds depending on what domain you are working in

For example, when I calculate the fourier series for

over the domain of

and

.... I get the same as if my function was defined as

when

. And then

I believe these functions do NOT look the same.

Avoid my abuse of notation, but I only want to talk about bounds of integration....

When integrating we get

.

Now my professor has said that since the function is periodic we can 'shift' the bounds to make it easier, for example change it to

.

The thing I'm having difficulty with is that these integrals do not evaluate to the same thing. Take the example of

. Now drawing the function over the x-axis, I can see that the area under the curve(s) from -1/2 to 1/2 is the same as the area under the curve from 0 to 1. So I can see the motivation there. Am I stuffing up here because in the case when you consider in the domain of

that when you integrate from 0 to 1, if you look at the graph, it's actually the function

? Since you have shifted over, but you are still just integrating

which may cause some trouble...

If this isn't it... i'm really stuck... because I end up getting the same Fourier series when the function is initially defined in 2 different domains. And these functions do NOT look the same!

Any help would be appreciated, thanks!