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Clausen's Integral

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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

Cl_2(theta)=-int_0^thetaln[2sin(1/2t)]dt
(1)
=i(1/6pi^2-1/4x^2)+x[ln(1-e^(ix))-ln2]-iLi_2(e^(ix))-xln[sin(1/2x)],
(2)

where Li_2(x) is a dilogarithm.

Clausen's integral has the special value

Cl_2(1/2pi)=K,
(3)

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

4Cl_2(1/3pi)=2Cl_2(2alpha)+Cl_2(pi+2alpha)-3Cl_2(5/3pi+2alpha)
(4)

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

6K=2Cl_2(2beta)-3Cl_2(2beta-1/2pi)+Cl_2(2beta+1/2pi)
(5)

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

7/4sqrt(7)L_(-7)(2)=3Cl_2(gamma)-3Cl_2(2gamma)+Cl_2(3gamma)
(6)

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

Cl_2(1/3pi)=sqrt(3)sum_(k=0)^(infty)[1/((6k+1)^2)+1/((6k+2)^2)-1/((6k+4)^2)-1/((6k+5)^2)]
(7)
=(sqrt(3))/9sum_(k=0)^(infty)((-1)^k)/(27^k)[(18)/((6k+1)^2)-(18)/((6k+2)^2)-(24)/((6k+3)^2)-6/((6k+4)^2)+2/((6k+5)^2)]
(8)

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

See also

Clausen Function, Lobachevsky's Function

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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 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. 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.

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Clausen's Integral

Cite this as:

Weisstein, Eric W. "Clausen's Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ClausensIntegral.html

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