Clausen function

Remarks

Equivalently, $\operatorname{Cl}_2$ is the sum of the uniformly convergent Fourier series $$\operatorname{Cl}_2(\theta) = \sum_{k=1}^{\infty} \frac{\sin k\theta}{k^2},$$ and $\operatorname{Cl}_2(\theta) = \operatorname{Im} \operatorname{Li}_2(e^{i\theta})$, where $\operatorname{Li}_2$ is the dilogarithm. The function is odd and $2\pi$-periodic, and it vanishes at every multiple of $\pi$.

Example: at $\theta = \pi/2$, $\sin(k\pi/2)$ is $0$ for even $k$ and $(-1)^{m}$ for $k = 2m + 1$, so $\operatorname{Cl}_2(\pi/2) = \sum_{m \ge 0} (-1)^m / (2m+1)^2 = G$, Catalan's constant. Its maximum on $\R$ is $\operatorname{Cl}_2(\pi/3) \approx 1.01494$.

Non-example: the series $\sum_{k \ge 1} \cos(k\theta)/k^2$ is not the Clausen function; it is the real part of $\operatorname{Li}_2(e^{i\theta})$, which equals $\frac{\pi^2}{6} - \frac{\pi\theta}{2} + \frac{\theta^2}{4}$ for $0 \le \theta \le 2\pi$.

Higher Clausen functions $\operatorname{Cl}_n$ are defined from $\operatorname{Li}_n(e^{i\theta})$ in a similar way.