A102886 Decimal expansion of Serret's integral: Integral_{x=0..1} log(x+1)/(x^2+1) dx.
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
Examples
0.27219826128795026631258611227970174341732296254616...
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.
- Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 308-309.
- Eric Weisstein's World of Mathematics, Serret's Integral.
Crossrefs
Cf. A086054 (Pi*log(2)).
Programs
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Mathematica
RealDigits[Pi*Log[2]/8, 10, 102][[1]] (* Jean-François Alcover, May 17 2013 *)
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PARI
Pi*log(2)/8 \\ Michel Marcus, Apr 23 2020
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PARI
intnum(x=0, 1, log(x+1)/(x^2+1)) \\ Michel Marcus, Apr 26 2020
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
Equals Integral_{x=0..log(2)} x/(e^x + 2*e^(-x) - 2) dx = -Integral_{x=0..Pi/2} log(sin(x))*sin(x)/sqrt(1+sin(x)^2) dx = Integral_{x=0..1} log((1 - x)/x)/(1 + x^2) dx = Integral_{x=0..Pi/4} x/((cos(x) + sin(x))*cos(x)) dx = Integral_{x=0..Pi/4} log(cot(x) - 1) dx (see Shamos). - Stefano Spezia, Nov 13 2024
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