A257963 Decimal expansion of the Integral_{x=0..1} arctan(arctanh(x))/x.
1, 0, 2, 5, 7, 6, 0, 5, 1, 0, 9, 3, 1, 3, 3, 0, 4, 5, 0, 3, 9, 8, 5, 4, 8, 6, 6, 0, 9, 6, 9, 5, 5, 2, 7, 9, 5, 3, 3, 4, 8, 7, 1, 8, 5, 6, 2, 1, 5, 0, 6, 9, 3, 9, 4, 2, 2, 3, 3, 8, 6, 8, 4, 4, 0, 1, 5, 8, 5, 1, 9, 2, 0, 8, 9, 9, 0, 7, 0, 9, 4, 2, 2, 2, 6, 7, 8, 7, 8, 7, 9, 1, 9, 7, 7, 9, 5, 3, 0, 7, 1, 3, 2, 9, 6
Offset: 1
Examples
= 1.02576051093133045039854866096955279533487185621506939422338684401585192089...
Links
- Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016.
- Iaroslav V. Blagouchine, Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal, Volume 35, Issue 1, pp. 21-110, 2014, DOI: 10.1007/s11139-013-9528-5 PDF file
Programs
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Maple
evalf(Pi*(log(GAMMA(1/Pi)) - log(GAMMA(1/2 + 1/Pi)) - log(Pi)/2),120); # Vaclav Kotesovec, May 17 2015
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Mathematica
nn = 111; RealDigits[ NIntegrate[ ArcTan[ ArcTanh[ x]]/x, {x, 0, 1}, AccuracyGoal -> nn, WorkingPrecision -> nn], 10, nn][[1]] (* or *) RealDigits[Pi (Log[Gamma[1/Pi]] - Log[Gamma[1/2 + 1/Pi]] - Log[Pi]/2), 10, 111][[1]] (* Robert G. Wilson v, May 14 2015 *)
Formula
The integral is equivalent to Pi*(log(Gamma(1/Pi)) - log(Gamma(1/2 + 1/Pi)) - log(Pi)/2), see page 82 of the second reference.
Comments