Clausen function

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

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

More generally, one defines

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

It is related to the polylogarithm by

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

Ernst Kummer and Rogers give the relation

$\operatorname{Li}_2(\exp(i \theta)) = \zeta(2) - \theta(2\pi-\theta) + i\operatorname{Cl}_2(\theta)$

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

For rational values of $θ / π$ (that is, for $θ / π = p / q$ for some integers p and q), the function $sin(n θ)$ 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.

References

Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . See section 27.8
Leonard Lewin, (Ed.). Structural Properties of Polylogarithms (1991) American Mathematical Society, Providence, RI. ISBN 0-8218-4532-2

Categories: Special functions

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