A386735 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} {1/(x+y)}^2 dx dy, where {} denotes fractional part.
4, 0, 7, 1, 7, 0, 1, 2, 1, 1, 1, 4, 4, 0, 8, 6, 1, 1, 7, 4, 0, 0, 4, 8, 2, 0, 5, 1, 3, 6, 4, 0, 8, 4, 0, 6, 2, 7, 2, 8, 6, 5, 5, 7, 9, 0, 9, 6, 4, 2, 1, 9, 2, 8, 2, 0, 5, 7, 7, 3, 6, 4, 0, 9, 3, 6, 7, 3, 4, 9, 1, 6, 0, 5, 1, 0, 4, 0, 1, 7, 6, 5, 4, 0, 3, 7, 5, 1, 5, 9, 4, 0, 1, 9, 5, 5, 2, 1, 0, 2, 9, 1, 3, 6, 4
Offset: 0
Examples
0.40717012111440861174004820513640840627286557909642...
Links
- Ovidiu Furdui, Exotic fractional part integrals and Euler's constant, Analysis, Vol. 31 (2011), pp. 249-257.
- Huizeng Qin and Youmin Lu, Integrals of Fractional Parts and Some New Identities on Bernoulli Numbers, Int. J. Contemp. Math. Sciences, Vol. 6, No. 15 (2011), pp. 745-761. See eq. (3.1). Note that this equation has an error, 3/2 instead of 5/2.
Programs
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Mathematica
RealDigits[5/2 - Log[2] - EulerGamma - Pi^2/12, 10, 120][[1]]
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PARI
5/2 - log(2) - Euler - Pi^2/12
Formula
Equals 5/2 - log(2) - gamma - Pi^2/12.
For m >= 3, Integral_{x=0..1} Integral_{y=0..1} {1/(x+y)}^m dx dy = (2^(2-m) + m - 3)/((m-1)*(m-2)) + (m!/2) * Sum_{j>=1} ((j+1)!/(m+j)!) * (zeta(j+2) - 1).