Sine and cosine series

Remarks

The cosine series is the Fourier series of the even $2\pi$-periodic function equal to $f$ on $[0, \pi]$. The sine series is the Fourier series of the odd $2\pi$-periodic function equal to $f$ on $(0, \pi)$. Example: for $f(x) = x$ on $[0, \pi]$, integration by parts gives $b_n = \frac{2(-1)^{n+1}}{n}$, so the sine series is $2 \sum_{n \ge 1} \frac{(-1)^{n+1}}{n} \sin nx$. For the same $f$, $a_0 = \pi$, $a_n = 0$ for even $n \ge 2$ and $a_n = -\frac{4}{\pi n^2}$ for odd $n$. Non-example: $f(x) = 1/x$ on $(0, \pi]$ is not integrable on $[0, \pi]$, so its sine and cosine coefficients are not defined.