Clausen function

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

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

1 General definition
2 Relation to polylogarithm
3 Kummer's relation
4 Relation to Dirichlet L-functions
5 Clausen function as a regularization (summability) method
6 Series acceleration
7 Special values
8 References

[edit] 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.

[edit] Relation to polylogarithm

It is related to the polylogarithm by

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

[edit] Kummer's relation

Ernst Kummer and Rogers give the relation

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

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

[edit] 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 $sin(n θ)$ 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. This allows relations between certain Dirichlet L-functions to be easily computed.

[edit] Clausen function as a regularization (summability) method

The Clausen function can also be viewed as a regularization method to give a meaning to the divergent Fourier series:

sin(θ) + 2sin(2θ) + 3sin(3θ) + ....

which can be taken to have the value $\operatorname{Cl}_{-1}(\theta)$ . By integrating, one may give a meaning to the series:

$\cos(\theta) + \cos(2\theta) + \cos(3 \theta)+..........= -\int d{\theta} \operatorname{Cl}_{-1}(\theta)$

The result can be generalized to every negative s by analytic continuation of the Clausen function. This regularization technique is similar to zeta function regularization in physics.

[edit] 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).

[edit] Special values

Some special values include

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

where K is Catalan's constant.

[edit] References

Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0-486-61272-4 : Section 27.8.
Leonard Lewin, (Ed.). Structural Properties of Polylogarithms (1991) American Mathematical Society, Providence, RI. ISBN 0-8218-4532-2
Jonathan M. Borwein, David M. Bradley, Richard E. Crandall (2000). "Computational Strategies for the Riemann Zeta Function". J. Comp. App. Math. 121: p.11.