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%I #42 May 06 2022 07:32:01
%S 3,0,1,6,4,0,3,2,0,4,6,7,5,3,3,1,9,7,8,8,7,5,3,1,6,5,7,7,9,6,8,9,6,5,
%T 4,0,6,5,9,8,9,9,7,7,3,9,4,3,7,6,5,2,3,6,9,4,0,7,4,4,0,0,5,3,8,3,0,6,
%U 0,5,8,1,4,3,9,5,0,2,9,5,3,3,9,9,8,9,8,2,2,6,9,7,2,7,9,5,0,1,1,9,4,2,3,4,4
%N Decimal expansion of the negated argument of i!.
%C The value is in radians.
%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020, p. 5.
%H Mircea Ivan, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.118.07.653">Problem 11592</a>, The American Mathematical Monthly, Vol. 118, No. 7 (2011), p. 654; <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.120.07.660">Arggh! Eye Factorial ... Arg(i!)</a>, Solutions to problem 11592 by Nora Thornbe, Omran Kouba and Denis Constales, ibid., Vol. 120, No. 7 (2013), p. 662-664.
%H Cornel Ioan Vălean, <a href="https://gaceta.rsme.es/abrir.php?id=1450">Problema 327</a>, La Gaceta de la Real Sociedad Matemática Española, Vol. 21, No. 2 (2018), pp. 331-343.
%F Equals -arg(i*Gamma(i)), since i! = Gamma(1+i) = i*Gamma(i).
%F Equals lim_{n->infinity} ((Sum_{k=1..n} arctan(1/k)) - log(n)). - _Jean-François Alcover_, Aug 07 2014, after Steven Finch
%F Equals arctan(A212878/A212877). - _Vaclav Kotesovec_, Dec 10 2015
%F From _Amiram Eldar_, Jun 12 2021: (Start)
%F Equals 1 - Integral_{x=0..Pi/2} frac(cot(x)) dx, where frac(x) = x - floor(x) is the fractional part of x.
%F Equals gamma - Sum_{k>=1} (-1)^(k+1)*zeta(2*k+1)/(2*k+1) = A001620 - A352619.
%F Both formulae are from Vălean (2018). (End)
%F Equals log((Gamma(1-i)/Gamma(1+i))^(-i/2)). - _Vaclav Kotesovec_, Jun 12 2021
%e 0.30164032046753319788753165779...
%t RealDigits[-Arg[Gamma[1 + I]], 10, 105] // First (* _Jean-François Alcover_, Aug 07 2014 *)
%Y Cf. A212877 (real(i!)), A212878 (-imag(i!)), A212879 (abs(i!)).
%Y Cf. A001620 (gamma), A352619.
%K nonn,cons,easy
%O 0,1
%A _Stanislav Sykora_, May 29 2012
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