A375367 Decimal expansion of (log(2*Pi)+gamma)/(2*Pi).

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, 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, 2, 7, 9, 1, 2, 4, 9, 8, 1, 7, 9, 8, 8, 3, 6, 4, 5, 2, 4, 1

Offset: 0

Author

R. J. Mathar, Aug 13 2024

Keywords

• nonn
• cons

Comments

gamma is A001620.

Examples

			0.38437394621343291560266431528895814385082767447477...
		

References

• I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed., Academic Press, 2007, p. 656, 6.443.1.

Links

• A. Erdélyi, ed., Tables of Integral Transforms, Vol. II, McGraw Hill, New York, 1954, p. 304, eq. (42).
• Bakir Farhi, A curious formula related to the Euler Gamma function, arXiv:1312.7115 [math.NT], 2013.
• Fábio M. S. Lima, Closed-form expressions for Farhi's constant and related integrals and its generalization, arXiv:1906.04303 [math.CA], 2019.
• Niels Nielsen, Handbuch der Theorie der Gammafunktion, Teubner, Leipzig, 1906, p. 203, eq. (5).
• Jean-Christophe Pain, Expression of Farhi's integral in terms of known mathematical constants, arXiv:2408.14835 [math.NT], 2024.
• Michael Ian Shamos, Shamos's catalog of the real numbers, 2011, p. 414.
• Zurab Silagadze, An integral related to the Euler gamma function, MathOverflow, 2014.

Crossrefs

Cf. A001620, A061444, A375366.

Programs

Maple

(log(2*Pi)+gamma)/2/Pi ; evalf(%) ;

Mathematica

RealDigits[(Log[2*Pi] + EulerGamma) / (2*Pi), 10, 120][[1]] (* Amiram Eldar, Aug 19 2024 *)

PARI

(log(2*Pi) + Euler)/(2*Pi) \\ Amiram Eldar, Sep 09 2024

Formula

Equals Integral_{x=0..1} sin(2*Pi*x) log(Gamma(x)) dx.