## Clausen function

In mathematics, the Clausen function is defined by the following integral:

$\operatorname{Cl}_2(\theta) = - \int_0^\theta \log|2 \sin(t/2)| \,dt.$

It was introduced by Thomas Clausen ().

The Lobachevsky function Λ or Л is essentially the same function with a change of variable:

$\Lambda(\theta) = - \int_0^\theta \log|2 \sin(t)| \,dt = \operatorname{Cl}_2(2\theta)/2.$

though the name "Lobachevsky function" is not quite historically accurate, as Lobachevsky's formulas for hyperbolic volume used the slightly different function

$\int_0^\theta \log| \sec(t)| \,dt = \Lambda(\theta+\pi/2)+\theta\log 2$

## General definition

More generally, one defines

$\operatorname{Cl}_s(\theta) = \sum_{n=1}^\infty \frac{\sin(n\theta)}{n^s}$

which is valid for complex s with Re s >1. The definition may be extended to all of the complex plane through analytic continuation.

## Relation to polylogarithm

It is related to the polylogarithm by

$\operatorname{Cl}_s(\theta) = \Im (\operatorname{Li}_s(e^{i \theta}))$.

## Kummer's relation

Ernst Kummer and Rogers give the relation

$\operatorname{Li}_2(e^{i \theta}) = \zeta(2) - \theta(2\pi-\theta)/4 + i\operatorname{Cl}_2(\theta)$

valid for $0\leq \theta \leq 2\pi$.

## Relation to Dirichlet L-functions

For rational values of θ / π (that is, for θ / π = p / q for some integers p and q), the function sin(nθ) can be understood to represent a periodic orbit of an element in the cyclic group, and thus $\operatorname{Cl}_s(\theta)$ can be expressed as a simple sum involving the Hurwitz zeta function. This allows relations between certain Dirichlet L-functions to be easily computed.

## Series acceleration

A series acceleration for the Clausen function is given by

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

which holds for | θ | < 2π. Here, ζ(s) is the Riemann zeta function. A more rapidly convergent form is given by

$\frac{\operatorname{Cl}_2(\theta)}{\theta} = 3-\log\left[|\theta| \left(1-\frac{\theta^2}{4\pi^2}\right)\right] -\frac{2\pi}{\theta} \log \left( \frac{2\pi+\theta}{2\pi-\theta}\right) +\sum_{n=1}^\infty \frac{\zeta(2n)-1}{n(2n+1)} \left(\frac{\theta}{2\pi}\right)^n$

Convergence is aided by the fact that ζ(n) − 1 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).

## Special values

Some special values include

$\operatorname{Cl}_2\left(\frac{\pi}{2}\right)=K$

where K is Catalan's constant.