The Two-Cycle Sawtooth Wave

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

$$S_{\rm s} = -\, \frac{4}{\pi} \sum_{j=1}^{\infty} \frac{1}{k}\, \sin(k\theta),$$

where $k=2j$. The corresponding FC series is then

$$S_{\rm c} = -\, \frac{4}{\pi} \sum_{j=1}^{\infty} \frac{1}{k}\, \cos(k\theta),$$

where $k=2j$, and the complex power series $S_{z}$ is given by

$$S_{z} = -\, \frac{4}{\pi} \sum_{j=1}^{\infty} \frac{1}{k}\, z^{k},$$

where $k=2j$, of which the two DP Fourier series above are the real and imaginary parts on the unit circle.