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A102887
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Decimal expansion of Integral_{x=0..1} log(gamma(x))^2 dx.
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5
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1, 8, 6, 6, 3, 1, 7, 0, 8, 3, 7, 9, 3, 5, 6, 2, 0, 8, 0, 9, 9, 2, 9, 6, 7, 9, 3, 7, 9, 7, 8, 2, 8, 9, 7, 3, 9, 8, 0, 0, 4, 0, 4, 1, 8, 6, 7, 9, 5, 3, 3, 8, 8, 0, 9, 4, 0, 5, 5, 1, 4, 4, 9, 5, 9, 3, 0, 4, 0, 9, 6, 5, 9, 8, 4, 9, 0, 5, 6, 3, 0, 3, 4, 7, 5, 5, 2, 3, 9, 8, 6, 0, 2, 9, 2, 5, 7, 2, 5, 0, 8, 5
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OFFSET
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1,2
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COMMENTS
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Also equals (1/6)*log(2*Pi)^2 + 2*log(A)*log(2*Pi) - (1/6)*gamma*log(2*Pi) + Pi^2/48 + 2*gamma*log(A) + zeta''(2)/(2*Pi^2) (with A the Glaisher-Kinkelin constant). - Jean-François Alcover, Apr 29 2013
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REFERENCES
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George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 236.
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LINKS
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FORMULA
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Equals gamma^2/12 + Pi^2/48 + (gamma*log(2*Pi))/6 + log(2*Pi)^2/3 - ((gamma + log(2*Pi))*zeta'(2))/Pi^2 + zeta''(2)/(2*Pi^2).
Equals -Integral_{x=0..1, y=0..1} log(gamma(x*y))^2/log(x*y) dx dy. (Apply Theorem 1 or Theorem 2 in Glasser (2019).) - Petros Hadjicostas, Jun 30 2020
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EXAMPLE
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1.8663170837935620809929679379782897398...
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MATHEMATICA
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EulerGamma^2/12 + Pi^2/48 + (EulerGamma*Log[2*Pi])/6 + Log[2*Pi]^2/3 - ((EulerGamma + Log[2*Pi])*Zeta'[2])/Pi^2 + Zeta''[2]/(2*Pi^2)
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PROG
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(PARI) intnum(x=0, 1, log(gamma(x))^2) \\ Michel Marcus, Aug 27 2015
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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