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Clausen Function



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

and write
\mathop{\it Cl}\nolimits _n(x)\equiv\cases{
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

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

For $n=1$, the function takes on the special form
\mathop{\it Cl}\nolimits _1(x)=C_1(x)=-\ln\vert 2\sin({\textstyle{1\over 2}}x)\vert
\end{displaymath} (4)

and for $n=2$, it becomes Clausen's Integral
\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).

See also Clausen's Integral, Polygamma Function, Polylogarithm


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.

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© 1996-9 Eric W. Weisstein