Define
then the Clausen functions are defined by
|
(3)
|
sometimes also written as (Arfken 1985, p. 783).
Then the Clausen function can be given symbolically in terms of the polylogarithm
as
|
(4)
|
For ,
the function takes on the special form
|
(5)
|
and for ,
it becomes Clausen's integral
|
(6)
|
The symbolic sums of opposite parity are summable symbolically, and the first few are given by
for
(Abramowitz and Stegun 1972).
See also
Clausen's Integral,
Lobachevsky's
Function,
Polygamma Function,
Polylogarithm
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References
Abramowitz, M. and Stegun, I. 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, 1985.Borwein,
J. and Bailey, D. Mathematics
by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A
K Peters, pp. 89-90, 2003.Borwein, J.; Bailey, D.; and Girgensohn,
R. Experimentation
in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters,
p. 27, 2004.Borwein, J. M.; Broadhurst, D. J.; and Kamnitzer,
J. "Central Binomial Sums, Multiple Clausen Values and Zeta Functions."
Exp. Math. 10, 25-41, 2001.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 ." J. Comput. Appl. Math. 11,
331-342, 1984.Jolley, L. B. W. Summation
of Series. London: Chapman, 1925.Lewin, L. Dilogarithms
and Associated Functions. London: Macdonald, pp. 170-180, 1958.Lewin,
L. Polylogarithms
and Associated Functions. New York: North-Holland, 1981.Wheelon,
A. D. A
Short Table of Summable Series. Report No. SM-14642. Santa Monica, CA: Douglas
Aircraft Co., 1953.Referenced on Wolfram|Alpha
Clausen Function
Cite this as:
Weisstein, Eric W. "Clausen Function."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ClausenFunction.html
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