cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A086054 Decimal expansion of Pi*log(2).

Original entry on oeis.org

2, 1, 7, 7, 5, 8, 6, 0, 9, 0, 3, 0, 3, 6, 0, 2, 1, 3, 0, 5, 0, 0, 6, 8, 8, 8, 9, 8, 2, 3, 7, 6, 1, 3, 9, 4, 7, 3, 3, 8, 5, 8, 3, 7, 0, 0, 3, 6, 9, 2, 8, 6, 2, 9, 4, 3, 2, 5, 7, 9, 5, 2, 5, 3, 1, 9, 4, 3, 0, 8, 5, 4, 9, 1, 7, 6, 7, 4, 1, 9, 8, 6, 4, 3, 0, 3, 2, 8, 9, 6, 1, 6, 1, 0, 6, 6, 3, 0, 2, 5, 0, 5, 7, 6, 1
Offset: 1

Views

Author

Eric W. Weisstein, Jul 07 2003

Keywords

Comments

Madelung constant b2(2), negated.

Examples

			2.1775860903036021305006888982376139...
		

References

  • G. Boros and V. H. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, 2004 (equation 13.6.6).

Crossrefs

Cf. A000796 (Pi), A002162 (log(2)), A173623.

Programs

  • Mathematica
    RealDigits[Pi Log[2],10,120][[1]] (* Harvey P. Dale, Dec 31 2011 *)

Formula

Pi*log(2) = -(8/3)*int(log(x)/sqrt(1+4*x-4*x^2), x=0..1). - John M. Campbell, Feb 07 2012
Pi*log(2) = int((x/sin(x))^2, x=0..Pi/2) = int(log(x^2+1)/(x^2+1), x=0..infinity) = int(-log(cos(x)), x=-Pi/2..Pi/2) = int(arctan(1/x)^2, x=0..infinity). - Jean-François Alcover, May 30 2013
From Amiram Eldar, Jul 11 2020: (Start)
Equals Integral_{x=-1..1} arcsin(x) dx / x.
Equals Integral_{x=-Pi/2..Pi/2} x*cot(x) dx. (End)
Equals Integral_{x = 0..oo} log(x^2 + 4)/(x^2 + 4) dx. - Peter Bala, Jul 22 2022
Equals -Im(Polylog(2, 2)). - Mohammed Yaseen, Jul 03 2024

Extensions

Corrected by Antti Ahti (antti.ahti(AT)tkk.fi), Nov 17 2004
More terms from Benoit Cloitre, May 21 2005