Clausen's Integral


Clausen's integral, sometimes called the log sine integral (Borwein and Bailey 2003, p. 88) is the n=2 case of the S_2 Clausen function


where Li_2(x) is a dilogarithm.

Clausen's integral has the special value


where K is Catalan's constant (Borwein and Bailey 2003, p. 89). Other identities include


where alpha=tan^(-1)(sqrt(3)/9),


where beta=tan^(-1)(1/3), and


where L_n(s) is a Dirichlet L-series and gamma=2tan^(-1)(sqrt(7)) (Borwein and Bailey 2003, pp. 89-90).

BBP-type formulas include


(Bailey 2000, Borwein and Bailey 2003, pp. 128-129).

See also

Clausen Function, Lobachevsky's Function

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Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1005-1006, 1972.Ashour, A. and Sabri, A. "Tabulation of the Function psi(theta)=sum_(n=1)^(infty)(sin(ntheta))/(n^2)." Math. Tables Aids Comp. 10, 54 and 57-65, 1956.Bailey, D. H. "A Compendium of BBP-Type Formulas for Mathematical Constants." 28 Nov 2000., J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 89-90, 2003.Clausen, R. "Über die Zerlegung reeller gebrochener Funktionen." J. reine angew. Math. 8, 298-300, 1832.Lewin, L. "Clausen's Integral." Ch. 4 in Dilogarithms and Associated Functions. London: Macdonald, pp. 91-105, 1958.Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, 1981.

Referenced on Wolfram|Alpha

Clausen's Integral

Cite this as:

Weisstein, Eric W. "Clausen's Integral." From MathWorld--A Wolfram Web Resource.

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