The One-Cycle Sawtooth Wave

Consider the one-cycle unit-amplitude sawtooth wave, given by the sine series

$$S_{\rm s} = \frac{2}{\pi} \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\, \sin(k\theta).$$

The corresponding FC series is then

$$S_{\rm c} = \frac{2}{\pi} \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\, \cos(k\theta),$$

and the complex power series $S_{z}$ is given by

$$S_{z} = \frac{2}{\pi} \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\, z^{k},$$

of which the two DP Fourier series above are the real and imaginary parts on the unit circle.