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>
More generally, one defines
- <math>\operatorname{Cl}_s(\theta) = \sum_{n=1}^\infty \frac{\sin(n\theta)}{n^s}<math>.
It is related to the polylogarithm by
- <math>\operatorname{Cl}_s(\theta)
= \Im (\operatorname{Li}_s(e^{i \theta}))<math>.
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>.
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.
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 0-8218-4532-2Template:Math-stub