Clausen function

Graph of the Clausen function (red) and (green) ${\displaystyle \mathrm {Cl} _{2}(\theta )}$ ${\displaystyle \mathrm {Cl} _{4}(\theta )}$

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

${\displaystyle \operatorname {Cl} _{2}(\theta )=-\int _{0}^{\theta }\log |2\sin(t/2)|\,\mathrm {d} t.}$

general definition

More generally one defines for complex with : ${\displaystyle s}$ ${\displaystyle \operatorname {Re} (s)>1}$

${\displaystyle \operatorname {Cl} _{s}(\theta )=\sum _{n=1}^{\infty }{\frac {\sin(n\theta )}{n^{s}}}=\sin(\theta )+{\frac {\sin(2\theta )}{2^{s}}}+{\frac {\sin(3\theta )}{3^{s}}}+{\frac {\sin(4\theta )}{4^{s}}}+\cdots }$

This definition can be continued analytically on the entire complex level .

Relationship to the polylogarithm

The Clausen function is related to the polylogarithm :

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

Sorrow's relationship

Ernst Kummer and Rogers cite the following for valid relationships: ${\displaystyle 0\leq \theta \leq 2\pi }$

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

Relationship to the Dirichlet L functions

For rational values ​​of , the function can be understood as the periodic orbit of an element of a cyclic group . Consequently, it can be understood as a simple sum that includes the Hurwitz zeta function . This allows relationships between certain Dirichlet L functions to be calculated easily. ${\displaystyle \theta /\pi }$ ${\displaystyle \sin(n\theta )}$ ${\displaystyle \operatorname {Cl} _{s}(\theta )}$

The Clausen function as a regularization method

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

${\displaystyle \sin(\theta )+2\sin(2\theta )+3\sin(3\theta )+\dots }$

what can be denoted by. Integration gives: ${\displaystyle \operatorname {Cl} _{-1}(\theta )}$

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

This result can be generalized to all negative by analytical continuation . ${\displaystyle s}$

Series development

A series expansion for the Clausen function (for ) is ${\displaystyle |\theta |<2\pi }$

${\displaystyle {\frac {\operatorname {Cl} _{2}(\theta )}{\theta }}=1-\log |\theta |-\sum _{n=1}^{\infty }{\frac {\zeta (2n)}{n(2n+1)}}\left({\frac {\theta }{2\pi }}\right)^{2n}{\text{.}}}$

${\displaystyle \zeta (s)}$is the Riemann zeta function . A faster converging series is

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

The convergence is ensured by the fact that for large ones it quickly converges to 0. ${\displaystyle \zeta (n)-1}$ ${\displaystyle n}$

Special values

Some special values ​​are:

${\displaystyle \operatorname {Cl} _{2}\left({\frac {\pi }{2}}\right)=G}$,

where G is Catalan's constant .

More general:

${\displaystyle \operatorname {Cl} _{s}\left({\frac {\pi }{2}}\right)=\beta (s)}$

where is the Dirichlet beta function . ${\displaystyle \beta (s)}$

literature

• Leonard Lewin (Ed.): Structural Properties of Polylogarithms . American Mathematical Society, Providence (RI) 1991, ISBN 0-8218-4532-2 .
• Jonathan M. Borwein, David M. Bradley, Richard E. Crandall: Computational Strategies for the Riemann Zeta Function (PDF; 526 kB) . In: J. Comp. App. Math . 121, 2000, pp. 11.