Clausen,s Integral
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
29-7-2019
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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
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
(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.
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