Clausen Function - Series Acceleration

Series Acceleration

A series acceleration for the Clausen function is given by

$\frac{\operatorname{Cl}_2(\theta)}{\theta} = 1-\log|\theta| + \sum_{n=1}^\infty \frac{\zeta(2n)}{n(2n+1)} \left(\frac{\theta}{2\pi}\right)^{2n}$

which holds for . Here, is the Riemann zeta function. A more rapidly convergent form is given by

$\frac{\operatorname{Cl}_2(\theta)}{\theta} = 3-\log\left -\frac{2\pi}{\theta} \log \left( \frac{2\pi+\theta}{2\pi-\theta}\right) +\sum_{n=1}^\infty \frac{\zeta(2n)-1}{n(2n+1)} \left(\frac{\theta}{2\pi}\right)^n$

Convergence is aided by the fact that approaches zero rapidly for large values of n. Both forms are obtainable through the types of resummation techniques used to obtain rational zeta series. (ref. Borwein, etal. 2000, below).

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