Clausen function

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

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

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General definition

More generally, one defines

<math>\operatorname{Cl}_s(\theta) = \sum_{n=1}^\infty \frac{\sin(n\theta)}{n^s}</math>

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

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Relation to polylogarithm

It is related to the polylogarithm by

<math>\operatorname{Cl}_s(\theta)

= \Im (\operatorname{Li}_s(e^{i \theta}))</math>.

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Kummer's relation

Ernst Kummer and Rogers give the relation

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

valid for <math>0\leq \theta \leq 2\pi</math>.

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Relation to Dirichlet L-functions

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

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Series acceleration

A series acceleration for the Clausen function is given by

<math>\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 </math>

which holds for <math>|\theta|<2\pi</math>. Here, <math>\zeta(s)</math> is the Riemann zeta function. A more rapidly convergent form is given by

<math>\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 </math>

Convergence is aided by the fact that <math>\zeta(n)-1</math> 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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Special values

Some special values include

<math>\operatorname{Cl}_2\left(\frac{\pi}{2}\right)=K</math>

where K is Catalan's constant.

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References

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