Clausen's Integral

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

			(1)
			(2)

where  is a dilogarithm.

Clausen's integral has the special value

(3)

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

(4)

where ,

(5)

where , and

(6)

where  is a Dirichlet L-series and  (Borwein and Bailey 2003, pp. 89-90).

BBP-type formulas include

			(7)
			(8)

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

REFERENCES:

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 ." 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. http://crd.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf.

Borwein, 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.