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

Author

Eric W. Weisstein, Jul 07 2003

Keywords

• nonn
• cons
• easy

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).

Links

• Eric Weisstein's World of Mathematics, Madelung Constants

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