A102886 - OEIS

%I #53 May 16 2024 13:17:20

%S 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,

%T 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,

%U 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

%N Decimal expansion of Serret's integral: Integral_{x=0..1} log(x+1)/(x^2+1) dx.

%C Named after the French mathematician Joseph-Alfred Serret (1819-1885). - _Amiram Eldar_, May 30 2021

%D Eric Billault et al, MPSI- Khôlles de Maths, Ellipses, 2012, exercice 11.10, pp. 252-264.

%D L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (94) on page 18.

%D I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 4.291.8.

%H Paul J. Nahin, <a href="https://doi.org/10.1007/978-3-030-43788-6">Inside interesting integrals</a>, Undergrad. Lecture Notes in Physics, Springer (2020), (2.2.4)

%H J.-A. Serret, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k16388p/f442n1.capture">Note sur l'intégrale Integral_{x=0..1} log(x+1)/(x^2+1) dx</a>, Journal de Mathématiques Pures et Appliquées, Vol. 9 (1844), page 436.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SerretsIntegral.html">Serret's Integral</a>.

%F Equals Integral_{x=0..1} arctan(x)/(x+1) dx. - _Jean-François Alcover_, Mar 25 2013

%F Equals Integral_{x=0..Pi/4} log(tan(x)+1) dx [see link J.-A. Serret and reference Billault]. - _Bernard Schott_, Apr 23 2020

%F 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

%F Equals -Integral_{x=0..1} x*arccos(x)*log(x) dx. - _Amiram Eldar_, May 30 2021

%e 0.27219826128795026631258611227970174341732296254616...

%t RealDigits[Pi*Log[2]/8, 10, 102][[1]] (* _Jean-François Alcover_, May 17 2013 *)

%o (PARI) Pi*log(2)/8 \\ _Michel Marcus_, Apr 23 2020

%o (PARI) intnum(x=0, 1, log(x+1)/(x^2+1)) \\ _Michel Marcus_, Apr 26 2020

%Y Cf. A086054 (Pi*log(2)).

%K nonn,cons,easy

%O 0,1

%A _Eric W. Weisstein_, Jan 15 2005