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Date: 17-9-2019
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Define
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(2) |
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
(7) |
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(8) |
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(9) |
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(10) |
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(11) |
for (Abramowitz and Stegun 1972).
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.
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