A212879 -id:A212879 - cp's OEIS frontend

A090986 Decimal expansion of Pi/sinh(Pi).

2, 7, 2, 0, 2, 9, 0, 5, 4, 9, 8, 2, 1, 3, 3, 1, 6, 2, 9, 5, 0, 2, 3, 6, 5, 8, 3, 6, 7, 2, 0, 3, 7, 5, 5, 5, 8, 4, 0, 7, 1, 8, 3, 6, 3, 4, 6, 0, 3, 1, 5, 9, 4, 9, 5, 0, 6, 8, 9, 6, 7, 8, 3, 8, 5, 6, 2, 4, 6, 1, 9, 1, 3, 6, 9, 4, 8, 7, 8, 8, 8, 1, 9, 1, 1, 5, 3, 1, 1, 7, 2, 1, 0, 6, 9, 3, 7, 6, 4, 4, 8, 6, 1, 0

Offset: 0

Benoit Cloitre, Feb 28 2004

Or, decimal expansion of Pi * csch(Pi).

			0.272029054982133162950236583672...

Jonathan M. Borwein, David H. Bailey, and Roland Girgensohn, "Two Products", Section 1.2 in Experimentation in Mathematics: Computational Paths to Discovery, Natick, MA: A. K. Peters, 2004, pp. 4-7.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Hyperbolic Cosecant.
Eric Weisstein's World of Mathematics, Infinite Product.

Cf. A112407, A144663, A144664, A144665, A144666, A144667, A144668, A144669, A212879.

SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)/Sinh(Pi(R)); // G. C. Greubel, Feb 02 2019

Re[N[Gamma[1+I]*Gamma[1-I], 104]] (* Vaclav Kotesovec, Dec 09 2015 *)
RealDigits[Pi/Sinh[Pi],10,120][[1]] (* Harvey P. Dale, May 16 2019 *)

default(realprecision, 100);  Pi/sinh(Pi) \\ G. C. Greubel, Feb 02 2019

numerical_approx(pi/sinh(pi), digits=100) # G. C. Greubel, Feb 02 2019

Equals Pi/sinh(Pi) = Product_{k>=1} k^2/(k^2+1).
Equals Pi * csch(Pi) = Product_{n >= 2} (n^2 - 1)/(n^2 + 1). - Jonathan Vos Post, Dec 07 2005
Equals Gamma(1+i)*Gamma(1-i), where i is the imaginary unit. - Vaclav Kotesovec, Dec 10 2015
Equals (1)(-i)*(1)_i where (n)_k denotes the rising factorial. - _Peter Luschny, May 06 2022
Equals 1 - 2*Sum_{n >= 1} (-1)^(n+1)/(n^2 + 1). - Peter Bala, Jan 01 2023
Equals A212879^2. - Amiram Eldar, Oct 25 2024

A212880 Decimal expansion of the negated argument of i!.

Original entry on oeis.org

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

Stanislav Sykora, May 29 2012

The value is in radians.

			0.30164032046753319788753165779...

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.

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

Cf. A001620 (gamma), A352619.

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

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

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

Original entry on oeis.org

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, 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, 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

Offset: 0

Stanislav Sykora, May 29 2012

Also the negated imaginary part of Gamma(i).

			0.498015668118356042713691117462198...

Stanislav Sykora, Mathematical Constants, Stan's Library, Vol. II.
Wikipedia, Particular values of the Gamma function

Cf. A212878 (-imag(i!)), A212879 (abs(i!)), A212880 (-arg(i!)), A090986.

RealDigits[Re[Gamma[I + 1]], 10, 105] (* T. D. Noe, May 29 2012 *)

PARI
```
real(I*gamma(I))
```

i! = gamma(1+i) = i*gamma(i).
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
Equals Integral_{x=0..+oo} exp(-x)*cos(log(x)) dx. - Jianing Song, Sep 27 2023
A212877^2 + A212878^2 = A090986 = Pi/sinh(Pi). - Vaclav Kotesovec, Dec 28 2023

A212878 Decimal expansion of the negated imaginary part of i!.

Original entry on oeis.org

1, 5, 4, 9, 4, 9, 8, 2, 8, 3, 0, 1, 8, 1, 0, 6, 8, 5, 1, 2, 4, 9, 5, 5, 1, 3, 0, 4, 8, 3, 8, 8, 6, 6, 0, 5, 1, 9, 5, 8, 7, 9, 6, 5, 2, 0, 7, 9, 3, 2, 4, 9, 3, 0, 2, 6, 5, 8, 8, 0, 2, 7, 6, 7, 9, 8, 8, 6, 0, 8, 0, 1, 4, 9, 1, 1, 3, 8, 5, 3, 9, 0, 1, 2, 9, 5, 1, 3, 6, 6, 4, 7, 9, 4, 6, 3, 0, 7, 0, 7, 4, 9, 5, 9, 2

Offset: 0

Stanislav Sykora, May 29 2012

Also the negated real part of Gamma(i).

			0.15494982830181068512495513048...

Cf. A212877 (real(i!)), A212879 (abs(i!)), A212880 (-arg(i!)), A090986.

-Gamma[I] // Re // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, May 13 2013 *)

PARI
```
-imag(I*gamma(I))
```

i! = gamma(1+i) = i*gamma(i).
Equals -Integral_{x=0..+oo} exp(-x)*sin(log(x)) dx. - Jianing Song, Sep 27 2023
A212877^2 + A212878^2 = A090986 = Pi/sinh(Pi). - Vaclav Kotesovec, Dec 28 2023

A364355 Decimal expansion of the unique value of x such that Gamma(-x + i*sqrt(1-x^2)) is a real number and -1 < x < 1.

Original entry on oeis.org

5, 4, 1, 9, 7, 9, 8, 7, 1, 6, 9, 4, 8, 9, 0, 6, 0, 2, 4, 4, 3, 3, 2, 2, 7, 8, 7, 7, 9, 0, 9, 0, 4, 6, 8, 8, 0, 5, 5, 8, 2, 4, 2, 8, 0, 2, 9, 2, 7, 9, 3, 8, 4, 2, 7, 9, 5, 6, 1, 4, 5, 5, 1, 9, 4, 0, 0, 0, 0, 8, 1, 5, 8, 6, 3, 9, 1, 7, 2, 7, 4, 4, 0, 4, 6, 0, 2, 1, 5, 2, 1, 1, 5, 1, 5, 5, 5, 8, 8, 4, 8, 5, 5, 6, 6

Offset: 0

Artur Jasinski, Jul 20 2023

Gamma(-A364355 + i*sqrt(1-A364355^2)) = -0.6749332470449905963531... see A364356.

Also decimal expansion of the unique value of x in the range -1 < x < 1 for which the function Re(Gamma(-x + i*sqrt(1-x^2)))/abs(Gamma(-x + i*sqrt(1-x^2))) is minimized.

			x = 0.54197987169489060244332278779...

Cf. A090986, A212877, A212878, A212880, A212879, A364356.

RealDigits[x /. FindRoot[Im[Gamma[-x + I Sqrt[1 - x^2]]], {x, 0.5}, WorkingPrecision -> 106]][[1]]

A365317 Decimal expansion of real part of Gamma(exp(i*Pi/3)).

Original entry on oeis.org

3, 7, 9, 8, 0, 4, 8, 9, 1, 7, 9, 1, 3, 9, 0, 1, 5, 6, 9, 9, 1, 7, 9, 7, 0, 5, 1, 3, 4, 1, 6, 2, 3, 6, 2, 7, 6, 8, 9, 1, 2, 7, 0, 3, 5, 1, 9, 2, 7, 4, 2, 8, 5, 2, 0, 3, 5, 8, 9, 1, 9, 7, 6, 9, 7, 7, 8, 4, 6, 7, 4, 7, 8, 3, 5, 8, 6, 1, 4, 4, 5, 0, 8, 4, 6, 7, 1, 7, 8, 3, 0, 8, 3, 1, 8, 6, 3, 2, 2, 0, 9, 8, 7, 9, 0, 9

Offset: 0

Artur Jasinski, Sep 01 2023

For imaginary part of Gamma(exp(i*Pi/3)) see A365318.

For abs(Gamma(exp(i*Pi/3))) see A365319.

			0.37980489179139...

Juan Arias de Reyna and Jan van de Lune, 2013, On the exact location of the non-trivial zeros of Riemann's zeta function, arXiv:1305.3844 [math.NT], 2013, formula (4).
Eric Weisstein's World of Mathematics, Riemann-Siegel Functions.
Wikipedia, Riemann-Siegel theta function.

Cf. A212877, A212878, A212879, A212880, A365318, A365319.

RealDigits[Re[Gamma[Cos[Pi/3] + I Sin[Pi/3]]], 10, 106][[1]]
(* or *)
RealDigits[Sqrt[Pi/Cosh[Pi Sqrt[3]/2]] Cos[2 RiemannSiegelTheta[Sqrt[3]/2] + ArcTan[Tanh[Pi Sqrt[3]/4]] + Sqrt[3] Log[2 Pi]/2], 10, 106][[1]]

real(gamma(exp(I*Pi/3))) \\ Michel Marcus, Sep 01 2023

Equals sqrt(Pi*sech(Pi*sqrt(3)/2))*cos(2*theta(sqrt(3)/2)+(sqrt(3)/2)*log(2*Pi)+arctan(tanh(Pi*sqrt(3)/4))) where theta is Riemann-Siegel theta function.

A365318 Decimal expansion of negative imaginary part of Gamma(exp(i*Pi/3)).

Original entry on oeis.org

5, 1, 7, 2, 7, 9, 0, 9, 9, 4, 7, 4, 8, 4, 0, 1, 5, 1, 5, 9, 3, 3, 2, 3, 5, 0, 1, 7, 1, 5, 4, 1, 9, 0, 7, 2, 2, 1, 8, 4, 7, 0, 9, 0, 3, 3, 1, 4, 1, 7, 5, 9, 0, 8, 7, 9, 8, 3, 2, 3, 2, 2, 6, 4, 4, 9, 9, 0, 0, 3, 6, 0, 0, 3, 2, 7, 5, 1, 7, 7, 5, 8, 6, 8, 0, 1, 6, 4, 2, 2, 6, 3, 6, 1, 1, 6, 1, 1, 1, 0, 9, 6, 6, 0, 9, 2

Offset: 0

Artur Jasinski, Sep 01 2023

			0.51727909947484...
Gamma(cos(Pi/3) + I*sin(Pi/3)) = 0.37980489179139...-I*0.51727909947484...

Juan Arias de Reyna and Jan van de Lune, 2013, On the exact location of the non-trivial zeros of Riemann's zeta function, arXiv:1305.3844 [math.NT], 2013, formula (4).
Eric Weisstein's World of Mathematics, Riemann-Siegel Functions.
Wikipedia, Riemann-Siegel theta function.

Cf. A212877, A212878, A212879, A212880.

Cf. A365317 (real part), A365319 (abs).

RealDigits[-Im[Gamma[Cos[Pi/3] + I Sin[Pi/3]]], 10, 106][[1]]
(* or *)
RealDigits[Sqrt[Pi/Cosh[Pi Sqrt[3]/2]] Sin[2 RiemannSiegelTheta[Sqrt[3]/2] + ArcTan[Tanh[Pi Sqrt[3]/4]] + Sqrt[3] Log[2 Pi]/2], 10, 106][[1]]

-imag(gamma(exp(I*Pi/3))) \\ Michel Marcus, Sep 01 2023

Equals sqrt(Pi*sech(Pi*sqrt(3)/2))*sin(2*theta(sqrt(3)/2)+(sqrt(3)/2)*log(2*Pi)+arctan(tanh(Pi*sqrt(3)/4))) where theta is Riemann-Siegel theta function.

A364356 Decimal expansion of negative value of function Gamma(-A364355 + i*sqrt(1-A364355^2)).

Original entry on oeis.org

6, 7, 4, 9, 3, 3, 2, 4, 7, 0, 4, 4, 9, 9, 0, 5, 9, 6, 3, 5, 3, 1, 0, 0, 4, 4, 6, 9, 5, 4, 7, 2, 2, 1, 6, 4, 2, 5, 3, 7, 4, 9, 7, 5, 6, 2, 7, 7, 8, 7, 6, 6, 1, 1, 9, 2, 8, 7, 3, 0, 3, 2, 8, 9, 4, 1, 0, 6, 4, 8, 6, 5, 9, 1, 9, 3, 3, 5, 3, 9, 9, 3, 9, 1, 4, 4, 2, 1, 3, 1, 4, 1, 5, 6, 8, 0, 9, 1, 6, 2, 0, 6, 7, 9, 7

Offset: 0

Artur Jasinski, Aug 08 2023

Only for x = A364355 = 0.54197987169489060244332278779... the Gamma(-x + i*sqrt(1-x^2)) is a real number and -1 < x < 1 (for one case is an imaginary number see A364821 and for other values x is a complex number).

			Gamma(-A364355 + i*sqrt(1-A364355^2)) = -0.6749332470449905963531...

Cf. A090986, A212877, A212878, A212880, A212879, A364355.

xmin=x /. FindRoot[Im[Gamma[-x + I Sqrt[1 - x^2]]], {x, 0.5}, WorkingPrecision -> 106];RealDigits[Re[-Gamma[-x + I Sqrt[1 - x^2]]/. x->xmin]][[1]]

A364821 Decimal expansion of the unique value of x such that Gamma(x + i*sqrt(1-x^2)) is an imaginary number and -1 < x < 1.

Original entry on oeis.org

1, 4, 9, 9, 6, 5, 9, 7, 4, 6, 0, 6, 4, 9, 1, 0, 8, 9, 8, 5, 3, 0, 9, 7, 0, 5, 3, 6, 6, 4, 1, 4, 5, 7, 3, 6, 6, 8, 7, 4, 1, 8, 4, 1, 0, 2, 3, 9, 9, 6, 9, 7, 4, 2, 9, 1, 1, 7, 8, 3, 1, 4, 7, 5, 5, 9, 8, 7, 2, 4, 7, 9, 7, 8, 9, 3, 9, 0, 2, 7, 0, 7, 3, 4, 1, 4, 6, 4, 6, 4, 2, 5, 3, 6, 5, 3, 0, 0, 8, 1, 0, 3, 6, 5, 8, 2

Offset: 0

Artur Jasinski, Oct 07 2023

Gamma(A364821 + i*sqrt(1-A364821^2)) = -i*0.5377003887835295919... see A366345.

Also decimal expansion of the unique value of x in the range -1 < x < 1 for which the function Im(Gamma(x + i*sqrt(1-x^2)))/abs(Gamma(x + i*sqrt(1-x^2))) is minimized.

			0.149965974606491...

Cf. A090986, A212877, A212878, A212880, A212879, A364355, A364356, A365317, A365318, A366345.

RealDigits[
  x /. FindRoot[Re[Gamma[x + I Sqrt[1 - x^2]]], {x, 0.15},
    WorkingPrecision -> 106]][[1]]

Last digit corrected by Vaclav Kotesovec, Sep 02 2025

A365319 Decimal expansion of abs(Gamma(exp(i*Pi/3))).

Original entry on oeis.org

6, 4, 1, 7, 3, 9, 3, 7, 2, 7, 8, 4, 7, 5, 5, 3, 2, 1, 5, 3, 8, 7, 3, 4, 3, 8, 8, 4, 1, 2, 2, 1, 4, 0, 3, 6, 1, 6, 8, 9, 2, 2, 9, 9, 1, 1, 6, 5, 3, 1, 6, 5, 9, 4, 0, 0, 8, 9, 4, 8, 4, 7, 6, 9, 3, 9, 8, 9, 0, 1, 3, 5, 5, 2, 9, 0, 3, 7, 4, 6, 4, 4, 2, 4, 7, 9, 5, 6, 1, 5, 3, 3, 8, 9, 0, 7, 4, 7, 1, 9, 8, 8, 9, 2, 5, 5

Offset: 0

Artur Jasinski, Sep 01 2023

Also abs(Gamma(exp(i*2*Pi/3))).
For real part of Gamma(exp(i*Pi/3)) see A365317.
For negative imaginary part of Gamma(exp(i*Pi/3)) see A365318.

			0.641739372784755...

Juan Arias de Reyna and Jan van de Lune, On the exact location of the non-trivial zeros of Riemann's zeta function, arXiv:1305.3844 [math.NT], 2013, formula (4).

Cf. A109219, A212877, A212878, A212879, A212880, A365317, A365318.

RealDigits[Abs[Gamma[Cos[Pi/3] + I Sin[Pi/3]]], 10, 106][[1]]
(* or *)
RealDigits[Sqrt[Pi/Cosh[Pi Sqrt[3]/2]], 10, 106][[1]]

abs(gamma(exp(I*Pi/3))) \\ Michel Marcus, Sep 01 2023

Equals sqrt(Pi/cosh(Pi*sqrt(3)/2)).

Equals 1/sqrt(3*A109219).

cp's OEIS Frontend

A090986 Decimal expansion of Pi/sinh(Pi).

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A212880 Decimal expansion of the negated argument of i!.

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

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A212878 Decimal expansion of the negated imaginary part of i!.

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A364355 Decimal expansion of the unique value of x such that Gamma(-x + i*sqrt(1-x^2)) is a real number and -1 < x < 1.

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A365317 Decimal expansion of real part of Gamma(exp(i*Pi/3)).

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A365318 Decimal expansion of negative imaginary part of Gamma(exp(i*Pi/3)).

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