If I have a function f(x) defined on $[0,L)$, said to be periodic of period $L$ and such that $f(0)\neq0$, how should I get the Fourier coefficients? I'm hesitating between taking the even extension and integrating over $(-L/2,L/2)$ or same but integrating over $(-L,L)$ or even just integrating without taking the even extension.
Can you tell me? And maybe also tell me why if there is a real reason.
Thanks a lot!
You just need to integrate over one period, no matter where the limits are. So you could integrate over $[a,a+L)$ with any arbitrary $a$. Sometimes the integration can be simplified by choosing the limits in a clever way, but in principle the only important thing is to integrate over one full period. Continuity and function values do not matter.