Finding the Fourier coefficient

Question

Compute the Fourier coefficient of

$$f(t) = \frac{t}{2\pi} - \left[\frac{t}{2\pi}\right]$$ where $[t]$ is the largest integer smaller than $t$.

Answer 1

$$ \eqalign{ & f(t) = {t \over {2\pi }} - \left\lfloor {{t \over {2\pi }}} \right\rfloor = \left\{ {{t \over {2\pi }}} \right\}\quad \Rightarrow \cr & \Rightarrow \quad \left\{ \matrix{ 0 \le f(t) < 1 \hfill \cr f(t + 2\pi ) = f(t) \hfill \cr} \right. \cr} $$

So $f(t)$ is a sawtooth wave of period $2 \pi$, avg $1/2$, amplitude $1/2$.

Answer 2

Note that $$f(t)=g(t)+{1\over 2}$$where $g(t)=g(-t)$. Therefore $$f(t)={1\over 2}+\sum_{n=1}^{\infty}a_n\sin nt$$ $$a_n={2\over \pi}\int_{0}^\pi {t\over 2\pi}\sin ntdt$$