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A102886
Decimal expansion of Serret's integral: Integral_{x=0..1} log(x+1)/(x^2+1) dx.
4
2, 7, 2, 1, 9, 8, 2, 6, 1, 2, 8, 7, 9, 5, 0, 2, 6, 6, 3, 1, 2, 5, 8, 6, 1, 1, 2, 2, 7, 9, 7, 0, 1, 7, 4, 3, 4, 1, 7, 3, 2, 2, 9, 6, 2, 5, 4, 6, 1, 6, 0, 7, 8, 6, 7, 9, 0, 7, 2, 4, 4, 0, 6, 6, 4, 9, 2, 8, 8, 5, 6, 8, 6, 4, 7, 0, 9, 2, 7, 4, 8, 3, 0, 3, 7, 9, 1, 1, 2, 0, 2, 0, 1, 3, 3, 2, 8, 7, 8, 1, 3, 2
OFFSET
0,1
COMMENTS
Named after the French mathematician Joseph-Alfred Serret (1819-1885). - Amiram Eldar, May 30 2021
REFERENCES
Eric Billault et al, MPSI- Khôlles de Maths, Ellipses, 2012, exercice 11.10, pp. 252-264.
L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (94) on page 18.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 4.291.8.
LINKS
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (2.2.4)
J.-A. Serret, Note sur l'intégrale Integral_{x=0..1} log(x+1)/(x^2+1) dx, Journal de Mathématiques Pures et Appliquées, Vol. 9 (1844), page 436.
Eric Weisstein's World of Mathematics, Serret's Integral.
FORMULA
Equals Integral_{x=0..1} arctan(x)/(x+1) dx. - Jean-François Alcover, Mar 25 2013
Equals Integral_{x=0..Pi/4} log(tan(x)+1) dx [see link J.-A. Serret and reference Billault]. - Bernard Schott, Apr 23 2020
Equals Pi*log(2)/8 = Sum_{n>0} (-1)^(n+1) * H(2n) / (2n+1) = H(2)/3 - H(4)/5 + H(6)/7 -... with H(n) = Sum_{j=1..n} 1/j the harmonic numbers. [Jolley]; improved by Bernard Schott, Apr 24 2020
Equals -Integral_{x=0..1} x*arccos(x)*log(x) dx. - Amiram Eldar, May 30 2021
EXAMPLE
0.27219826128795026631258611227970174341732296254616...
MATHEMATICA
RealDigits[Pi*Log[2]/8, 10, 102][[1]] (* Jean-François Alcover, May 17 2013 *)
PROG
(PARI) Pi*log(2)/8 \\ Michel Marcus, Apr 23 2020
(PARI) intnum(x=0, 1, log(x+1)/(x^2+1)) \\ Michel Marcus, Apr 26 2020
CROSSREFS
Cf. A086054 (Pi*log(2)).
Sequence in context: A095194 A254251 A095711 * A204382 A072981 A023399
KEYWORD
nonn,cons,easy
AUTHOR
Eric W. Weisstein, Jan 15 2005
STATUS
approved