A212880 Decimal expansion of the negated argument of i!.

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, 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, 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

Offset: 0

Author

Stanislav Sykora, May 29 2012

Keywords

• nonn
• cons
• easy

Comments

The value is in radians.

Examples

			0.30164032046753319788753165779...
		

Links

• Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 5.
• Mircea Ivan, Problem 11592, The American Mathematical Monthly, Vol. 118, No. 7 (2011), p. 654; Arggh! Eye Factorial ... Arg(i!), Solutions to problem 11592 by Nora Thornbe, Omran Kouba and Denis Constales, ibid., Vol. 120, No. 7 (2013), p. 662-664.
• Cornel Ioan Vălean, Problema 327, La Gaceta de la Real Sociedad Matemática Española, Vol. 21, No. 2 (2018), pp. 331-343.

Crossrefs

Cf. A212877 (real(i!)), A212878 (-imag(i!)), A212879 (abs(i!)).

Cf. A001620 (gamma), A352619.

Programs

Mathematica

RealDigits[-Arg[Gamma[1 + I]], 10, 105] // First (* Jean-François Alcover, Aug 07 2014 *)

Formula

Equals -arg(i*Gamma(i)), since i! = Gamma(1+i) = i*Gamma(i).

Equals lim_{n->infinity} ((Sum_{k=1..n} arctan(1/k)) - log(n)). - Jean-François Alcover, Aug 07 2014, after Steven Finch

Equals arctan(A212878/A212877). - Vaclav Kotesovec, Dec 10 2015

From Amiram Eldar, Jun 12 2021: (Start)

Equals 1 - Integral_{x=0..Pi/2} frac(cot(x)) dx, where frac(x) = x - floor(x) is the fractional part of x.

Equals gamma - Sum_{k>=1} (-1)^(k+1)*zeta(2*k+1)/(2*k+1) = A001620 - A352619.

Both formulae are from Vălean (2018). (End)

Equals log((Gamma(1-i)/Gamma(1+i))^(-i/2)). - Vaclav Kotesovec, Jun 12 2021