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
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.
Programs
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Maple
(log(2*Pi)+gamma)/2/Pi ; evalf(%) ;
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Mathematica
RealDigits[(Log[2*Pi] + EulerGamma) / (2*Pi), 10, 120][[1]] (* Amiram Eldar, Aug 19 2024 *)
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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.
Comments