Fourier sine series over $[0,\pi]$ vs over $[-\pi,\pi]$?

Question

So from what I understand, on the interval $[-\pi,\pi]$, only even functions have cosine series and only odd functions have sine series. But then on $[0,\pi]$ functions have cosine and sine series? Why is this? And why does the same formula for finding the series work?

Answer 1

The trick is that for any function $f(x)$ on $[0,\pi]$ we can pretend that $f$ is an even or odd function on $[-\pi,\pi]$ by defining $f(x) := f(-x)$ or $f(x):=-f(-x)$ for $-\pi \leq x <0$. Then we just take the sine or cosine series for the extended version of $f$ on $[-\pi,\pi]$ and restrict it to $[0,\pi]$.