The Shifted Square Wave

Consider the shifted unit-amplitude square wave, given by the cosine series

$$S_{\rm c} = \frac{4}{\pi} \sum_{j=0}^{\infty} \frac{(-1)^{j}}{k}\, \cos(k\theta),$$

where $k=2j+1$. The corresponding FC series is then

$$S_{\rm s} = \frac{4}{\pi} \sum_{j=0}^{\infty} \frac{(-1)^{j}}{k}\, \sin(k\theta),$$

where $k=2j+1$, and the complex power series $S_{z}$ is given by

$$S_{z} = \frac{4}{\pi} \sum_{j=0}^{\infty} \frac{(-1)^{j}}{k}\, z^{k},$$

where $k=2j+1$, of which the two DP Fourier series above are the real and imaginary parts on the unit circle.