A086054 Decimal expansion of Pi*log(2).

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

1,1

Comments

Madelung constant b2(2), negated.

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).

Links

Table of n, a(n) for n=1..105.

Eric Weisstein's World of Mathematics, Madelung Constants

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

Example

2.1775860903036021305006888982376139...

Mathematica

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

Crossrefs

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

Sequence in context: A072280 A217106 A329995 * A256392 A011134 A157240

Adjacent sequences: A086051 A086052 A086053 * A086055 A086056 A086057

Keyword

nonn,cons,easy

Author

Eric W. Weisstein, Jul 07 2003

Extensions

Corrected by Antti Ahti (antti.ahti(AT)tkk.fi), Nov 17 2004

More terms from Benoit Cloitre, May 21 2005

Status

approved