A212880 -id:A212880 - cp's OEIS frontend

A212879 Decimal expansion of the absolute value of i!.

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

Offset: 0

Stanislav Sykora, May 29 2012

Also absolute value of Gamma(i).

			0.5215640468649398411581802696...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A090986, A212877 (real(i!)), A212878 (-imag(i!)), A212880 (-arg(i!)).

N[Sqrt[Pi/Sinh[Pi]], 103] (* Vaclav Kotesovec, Dec 10 2015 *)
RealDigits[Abs[I!], 10, 120][[1]] (* Amiram Eldar, Oct 25 2024 *)

PARI
```
abs(gamma(I))
```

Equals abs(Gamma(i)), since i! = Gamma(1+i) = i*Gamma(i).
Equals sqrt(Pi/sinh(Pi)). - Vaclav Kotesovec, Dec 10 2015
Equals Product_{k>=1} k/sqrt(k^2+1) = sqrt(A090986). - Amiram Eldar, Oct 25 2024

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.

A352619 Decimal expansion of Sum_{k>=1} (-1)^(k+1) * zeta(2k+1)/(2k+1).

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Mar 24 2022

Is there a closed-form formula for this constant as for A352527?

			0.2755753444339996627189...

Bernard Candelpergher, Ramanujan Summation of Divergent Series, Springer, 2017, p. 35.

Cornel Ioan Vălean, Problema 327, La Gaceta de la Real Sociedad Matemática Española, Vol. 21, No. 2 (2018), pp. 331-343.

Cf. A001620, A212880, A352527.

Maple
```
evalf(gamma + argument(I!),100);
```

RealDigits[EulerGamma + Arg[Gamma[1 + I]], 10, 100][[1]] (* Amiram Eldar, Mar 24 2022 *)

Euler + arg(I*gamma(I)) \\ Michel Marcus, Mar 25 2022

Equals gamma + arg(i!) (see Vălean).
Equals A001620 - A212880.
Equals Sum_{k>=1} (1/k - arctan(1/k)). - Amiram Eldar, Jul 21 2022

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

A212879 Decimal expansion of the absolute value of i!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A352619 Decimal expansion of Sum_{k>=1} (-1)^(k+1) * zeta(2k+1)/(2k+1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica