I am pretty new to calculus, and I am trying to understand some basic rules to solve Fourier Series. I don't know how I deal with the region where a function is defined and its period. For instance:
$f(x)=x^{2}$ , on $0<x<\pi$
where $f(x)$ is an odd function and periodic with a period of $4\pi$. To find out its Fourier Sine Series, I know that I need to calculate the coefficient $B_{n}$ defined as:
$B_{n}=\frac{2}{T}\int_{\lambda}^{\lambda+T}f(t)sin(\frac{nt2\pi}{T})dt$
Where $T$ is the period and $\lambda$ is any $\lambda \epsilon \mathbb{R}$. What is the interval that I should integrate? Over the region $0$ and $\pi$? Over some interval with period of $4\pi$?
Thanks,
Uirá
I've posted a comment asking for a completion of the definition of the function $f$. However, your question about $\lambda$ can be answered generally without this information.
For the choice of $\lambda$ use whatever value makes the integral easy to evaluate. (Frequently, this means that any choice is as good as any other. This is because $\lambda$ doesn't appear in the integrand, so is only applied at the end via the Fundamental Theorem of Calculus.) Since the function is periodic, you may "stamp out" copies of the function between one choice of $\lambda$ and another. (You may need to saw off one end of the period and attach it to the other end, if the two choices of $\lambda$ don't differ by a whole period. But this won't change the value of the integral just the order that you go through the values of the integrand.) It is a good exercise to draw a picture of a periodic function with a few features per period and convince yourself that slightly increasing $\lambda$ is the same as slicing a little of the function from the left (end of $[\lambda, \lambda+T]$) and gluing it onto the right (end of that interval).