%I #18 Sep 10 2024 04:30:01
%S 3,8,4,3,7,3,9,4,6,2,1,3,4,3,2,9,1,5,6,0,2,6,6,4,3,1,5,2,8,8,9,5,8,1,
%T 4,3,8,5,0,8,2,7,6,7,4,4,7,4,7,7,1,7,2,2,8,4,3,0,5,3,4,3,4,5,3,3,5,2,
%U 2,7,9,1,2,4,9,8,1,7,9,8,8,3,6,4,5,2,4,1
%N Decimal expansion of (log(2*Pi)+gamma)/(2*Pi).
%C gamma is A001620.
%D I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed., Academic Press, 2007, p. 656, 6.443.1.
%H A. Erdélyi, ed., <a href="https://ia804709.us.archive.org/0/items/in.ernet.dli.2015.141554/2015.141554.Tables-Of-Intergral-Transforms-Volume-2.pdf">Tables of Integral Transforms</a>, Vol. II, McGraw Hill, New York, 1954, p. 304, eq. (42).
%H Bakir Farhi, <a href="https://arxiv.org/abs/1312.7115">A curious formula related to the Euler Gamma function</a>, arXiv:1312.7115 [math.NT], 2013.
%H Fábio M. S. Lima, <a href="https://arxiv.org/abs/1906.04303">Closed-form expressions for Farhi's constant and related integrals and its generalization</a>, arXiv:1906.04303 [math.CA], 2019.
%H Niels Nielsen, <a href="https://archive.org/details/handbuchdertheor0000drni/page/203">Handbuch der Theorie der Gammafunktion</a>, Teubner, Leipzig, 1906, p. 203, eq. (5).
%H Jean-Christophe Pain, <a href="https://arxiv.org/abs/2408.14835">Expression of Farhi's integral in terms of known mathematical constants</a>, arXiv:2408.14835 [math.NT], 2024.
%H Michael Ian Shamos, <a href="https://citeseerx.ist.psu.edu/pdf/ae33a269baba5e8b1038e719fb3209e8a00abec5">Shamos's catalog of the real numbers</a>, 2011, p. 414.
%H Zurab Silagadze, <a href="https://mathoverflow.net/questions/158022/an-integral-related-to-the-euler-gamma-function">An integral related to the Euler gamma function</a>, MathOverflow, 2014.
%F Equals Integral_{x=0..1} sin(2*Pi*x) log(Gamma(x)) dx.
%e 0.38437394621343291560266431528895814385082767447477...
%p (log(2*Pi)+gamma)/2/Pi ; evalf(%) ;
%t RealDigits[(Log[2*Pi] + EulerGamma) / (2*Pi), 10, 120][[1]] (* _Amiram Eldar_, Aug 19 2024 *)
%o (PARI) (log(2*Pi) + Euler)/(2*Pi) \\ _Amiram Eldar_, Sep 09 2024
%Y Cf. A001620, A061444, A375366.
%K nonn,cons
%O 0,1
%A _R. J. Mathar_, Aug 13 2024