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

1,3

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

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

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