I need to calculate Fourier series of:
$$\sin(x)- \operatorname{IntegerPart}[\sin(x)]$$
This seems just a common sine function, with its value set to 0 at its max and mins, so the period is just the same as that of $\sin(x)$. But however I take it, it has at least 1 (2?) discontinuities inside it, and I don't know how to proceed.
My only guess comes from what I've read here:
If you have a removable discontinuity at a point, the Fourier series will converge to the limit of the function at the point.
Does it mean that Fourier series will just ignore that (removable) discontinuity, and hence my answer is $\sin(x)$?
Note: this is not homework, but I'm preparing for an exam and this exercise is from an old one.
When the sine is negative, then its integer part, as usually defined, is $-1$. So you have jump discontinuities as well as removable discontinuities. You'd have to add the integrals over $[-\pi,0]$ and $[0,\pi]$. The one removable discontinuity can be ignored; it doesn't affect the integral.