Mathworld page what you need is about half way down.
I am trying to expand f(x) = x into a real fourier series on the interval 0<=x<=2pi. The only examples I can find are on the interval -pi to pi, and when i try using the same method on my interval i come up with the answer zero, which is obviously wrong.
Where the coefficents are given by:
So we can see that
is the same thing as:
and now you can use your definition of a Fourier serier for a function of period on the interval , and because of the periodicity of a Fourier representation of a function this will give a series that is valid on any interval of length .
Also you are using too much cryptic notation that can be explained better in English, and the relegation of discontinuities to end points of the interval is unnecessary (in fact undesirable). The only caveat is that the series does not converge to whatever functional value you have assigned at jump discontinuities but to the mean of the limit from the right and left.
(If you are using Rudin as a reference, get another book you will never understand harmonic or any other branch of analysis if you start with Rudin, Rudin is for when you think you know a subject already)