Clausen Function

$\begin{figure}\begin{center}\BoxedEPSF{ClausenFunction.epsf}\end{center}\end{figure}$

Define

$\displaystyle S_n(x)$	$\textstyle \equiv$	$\displaystyle \sum_{k=1}^\infty {\sin(kx)\over k^n}$	(1)
$\displaystyle C_n(x)$	$\textstyle \equiv$	$\displaystyle \sum_{k=1}^\infty {\cos(kx)\over k^n},$	(2)

and write

$\begin{displaymath} \mathop{\it Cl}\nolimits _n(x)\equiv\cases{ S_n(x)=\sum_{k=... ... C_n(x)=\sum_{k=1}^\infty {\cos(kx)\over k^n} & $n$\ odd.\cr} \end{displaymath}$

(3)

Then the Clausen function $\mathop{\it Cl}\nolimits _n(x)$ can be given symbolically in terms of the Polylogarithm as

$\begin{displaymath} \mathop{\it Cl}\nolimits _n(x)=\cases{ {\textstyle{1\over 2... ...(e^{-ix})+\mathop{\rm Li}\nolimits _n(e^{ix})] & $n$\ odd.\cr} \end{displaymath}$

For

, the function takes on the special form

$\begin{displaymath} \mathop{\it Cl}\nolimits _1(x)=C_1(x)=-\ln\vert 2\sin({\textstyle{1\over 2}}x)\vert \end{displaymath}$

(4)

and for

, it becomes Clausen's Integral

$\begin{displaymath} \mathop{\it Cl}\nolimits _2(x)=S_2(x)=-\int_0^x \ln[2\sin({\textstyle{1\over 2}}t)]\,dt. \end{displaymath}$

(5)

The symbolic sums of opposite parity are summable symbolically, and the first few are given by

$\displaystyle C_2(x)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 6}}\pi^2-{\textstyle{1\over 2}}\pi x+{\textstyle{1\over 4}}x^2$	(6)
$\displaystyle C_4(x)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 90}}-{\textstyle{1\over 12}}\pi^2x^2+{\textstyle{1\over 12}}\pi x^3-{\textstyle{1\over 48}}x^4$	(7)
$\displaystyle S_1(x)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}(\pi-x)$	(8)
$\displaystyle S_3(x)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 6}}\pi^2 x-{\textstyle{1\over 4}}\pi x^2+{\textstyle{1\over 12}}x^3$	(9)
$\displaystyle S_5(x)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 90}}\pi^4 x-{\textstyle{1\over 36}}\pi^2 x^3+{\textstyle{1\over 48}}\pi x^4-{\textstyle{1\over 240}}x^5$	(10)

for $0\leq x\leq 2\pi$ (Abramowitz and Stegun 1972).

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Clausen's Integral and Related Summations'' §27.8 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1005-1006, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 783, 1985.

Clausen, R. ``Über die Zerlegung reeller gebrochener Funktionen.'' J. reine angew. Math. 8, 298-300, 1832.

Grosjean, C. C. ``Formulae Concerning the Computation of the Clausen Integral $\mathop{\it Cl}\nolimits _2(\alpha)$ .'' J. Comput. Appl. Math. 11, 331-342, 1984.

Jolley, L. B. W. Summation of Series. London: Chapman, 1925.

Wheelon, A. D. A Short Table of Summable Series. Report No. SM-14642. Santa Monica, CA: Douglas Aircraft Co., 1953.