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 A257963 Decimal expansion of the integral_{x=0..1} arctan(arctanh(x))/x. 1
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS "The arctangent of the hyperbolic arctangent is analytic in the whole disk |x| < 1, and therefore, can be expanded into the MacLaurin series", see the first reference. 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 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. EXAMPLE = 1.02576051093133045039854866096955279533487185621506939422338684401585192089... MAPLE evalf(Pi*(log(GAMMA(1/Pi)) - log(GAMMA(1/2 + 1/Pi)) - log(Pi)/2), 120); # Vaclav Kotesovec, May 17 2015 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 *) CROSSREFS Cf. A257957, A257955, A257958, A257959, A155968, A049541, A000796. Sequence in context: A066035 A329283 A296168 * A167554 A202392 A290410 Adjacent sequences:  A257960 A257961 A257962 * A257964 A257965 A257966 KEYWORD nonn,cons AUTHOR Robert G. Wilson v, May 14 2015 STATUS approved

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Last modified May 14 19:53 EDT 2021. Contains 343903 sequences. (Running on oeis4.)