cp's OEIS Frontend

A212877 Decimal expansion of the real part of i!, where i = sqrt(-1).

%I A212877 #35 Dec 28 2023 13:10:57
%S A212877 4,9,8,0,1,5,6,6,8,1,1,8,3,5,6,0,4,2,7,1,3,6,9,1,1,1,7,4,6,2,1,9,8,0,
%T A212877 9,1,9,5,2,9,6,2,9,6,7,5,8,7,6,5,0,0,9,2,8,9,2,6,4,2,9,5,4,9,9,8,4,5,
%U A212877 8,3,0,0,4,3,5,9,8,1,9,3,4,5,0,7,8,9,4,5,0,4,2,8,2,6,7,0,5,8,1,4,0,5,6,0,6
%N A212877 Decimal expansion of the real part of i!, where i = sqrt(-1).
%C A212877 Also the negated imaginary part of Gamma(i).
%H A212877 Stanislav Sykora, <a href="http://dx.doi.org/10.3247/SL2Math08.001">Mathematical Constants</a>, Stan's Library, Vol. II.
%H A212877 Wikipedia, <a href="http://en.wikipedia.org/wiki/Particular_values_of_the_Gamma_function#Imaginary_unit">Particular values of the Gamma function</a>
%F A212877 i! = gamma(1+i) = i*gamma(i).
%F A212877 Equals (1/2)*Integral_{x=-1/e..0} LambertW(x)*sin(log(-LambertW(x)))-LambertW(-1,x)*sin(log(-LambertW(-1,x))) dx. - _Gleb Koloskov_, Oct 01 2021
%F A212877 Equals Integral_{x=0..+oo} exp(-x)*cos(log(x)) dx. - _Jianing Song_, Sep 27 2023
%F A212877 A212877^2 + A212878^2 = A090986 = Pi/sinh(Pi). - _Vaclav Kotesovec_, Dec 28 2023
%e A212877 0.498015668118356042713691117462198...
%t A212877 RealDigits[Re[Gamma[I + 1]], 10, 105] (* _T. D. Noe_, May 29 2012 *)
%o A212877 (PARI) real(I*gamma(I))
%Y A212877 Cf. A212878 (-imag(i!)), A212879 (abs(i!)), A212880 (-arg(i!)), A090986.
%K A212877 nonn,cons,easy
%O A212877 0,1
%A A212877 _Stanislav Sykora_, May 29 2012