The question is to compute the Fourier Series of the function
$$h(x) = \sin\left(\frac{x}{4}\right)$$ on $[-\pi, \pi]$
My question is I know normally, we can use the following formula to compute $a_0, a_k, b_k$
$$a_0 = \frac{1}{L} \int_{-L}^{L}f(x)dx$$ $$a_k = \frac{1}{L} \int_{-L}^{L}f(x)\cos\left(\frac{k\pi x}{L}\right) dx$$ $$b_k = \frac{1}{L} \int_{-L}^{L}f(x)\sin\left(\frac{k\pi x}{L}\right) dx$$
But I noticed that for $\sin\left(\frac{x}{4}\right)$, the period $T = 2L = 8\pi$. If I use $L = 4\pi$ to plug into the formula, what is the meaning of $[-\pi, \pi]$? Could anyone explain for me?