The Triangular Wave

Consider the unit-amplitude triangular wave, given by the cosine series

$$S_{\rm c} = -\, \frac{8}{\pi^{2}} \sum_{j=0}^{\infty} \frac{1}{k^{2}}\, \cos(k\theta),$$

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

$$S_{\rm s} = -\, \frac{8}{\pi^{2}} \sum_{j=0}^{\infty} \frac{1}{k^{2}}\, \sin(k\theta),$$

where $k=2j+1$. Note that due to the factors of $1/k^{2}$ these series are already absolutely and uniformly convergent. The complex power series $S_{z}$ is given by

$$S_{z} = -\, \frac{8}{\pi^{2}} \sum_{j=0}^{\infty} \frac{1}{k^{2}}\, z^{k},$$

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