![ClausensIntegral](https://cdn.statically.io/img/mathworld.wolfram.com/images/eps-svg/ClausensIntegral_1000.svg)
![ClausensIntegralReIm](https://cdn.statically.io/img/mathworld.wolfram.com/images/eps-svg/ClausensIntegralReIm_850.svg)
Clausen's integral, sometimes called the log sine integral (Borwein and Bailey 2003, p. 88) is the
case of the
Clausen function
where
is a dilogarithm.
Clausen's integral has the special value
![Cl_2(1/2pi)=K,](https://cdn.statically.io/img/mathworld.wolfram.com/images/equations/ClausensIntegral/NumberedEquation1.svg) |
(3)
|
where
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)](https://cdn.statically.io/img/mathworld.wolfram.com/images/equations/ClausensIntegral/NumberedEquation2.svg) |
(4)
|
where
,
![6K=2Cl_2(2beta)-3Cl_2(2beta-1/2pi)+Cl_2(2beta+1/2pi)](https://cdn.statically.io/img/mathworld.wolfram.com/images/equations/ClausensIntegral/NumberedEquation3.svg) |
(5)
|
where
,
and
![7/4sqrt(7)L_(-7)(2)=3Cl_2(gamma)-3Cl_2(2gamma)+Cl_2(3gamma)](https://cdn.statically.io/img/mathworld.wolfram.com/images/equations/ClausensIntegral/NumberedEquation4.svg) |
(6)
|
where
is a Dirichlet L-series and
(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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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.Referenced
on Wolfram|Alpha
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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