A364355 -id:A364355 - cp's OEIS frontend

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

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

A366345 Decimal expansion of y such that Gamma(A364821 + isqrt(1-A364821^2)) = -yi.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Oct 07 2023

x = A364821 = 0.149965974606491089853... is the only real number x such that Gamma(x + i*sqrt(1-x^2)) is an imaginary number and -1 < x < 1.

			Gamma(A364821 + i*sqrt(1-A364821^2)) = -i*0.5377003887835295919046...

Cf. A090986, A212877, A212878, A212880, A212879, A364355, A364356, A365317, A365318, A364821.

A366545 Decimal expansion of the value x for which the function Re(Gamma(-x + i*sqrt(1-x^2))) is minimized for -1 < x < 1.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Oct 12 2023

For Re(Gamma(-A366545 + i*sqrt(1-A366545^2))) = -0.930840199... see A366545.

			0.9565130903466545656...

Cf. A090986, A212877, A212878, A212880, A212879, A364355, A364356, A364821, A365317, A365318, A366345.

xmin = Re[x /. FindRoot[1/(2 Sqrt[1 - x^2]) I (Gamma[1 + x - I Sqrt[1 - x^2]] PolyGamma[0, x - I Sqrt[1 - x^2]] - Gamma[1 + x + I Sqrt[1 - x^2]] PolyGamma[0,
         x + I Sqrt[1 - x^2]]), {x, -0.98}, WorkingPrecision -> 110]];
 RealDigits[xmin, 10, 106][[1]]

cp's OEIS Frontend

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

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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.

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A366345 Decimal expansion of y such that Gamma(A364821 + isqrt(1-A364821^2)) = -yi.

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A366545 Decimal expansion of the value x for which the function Re(Gamma(-x + i*sqrt(1-x^2))) is minimized for -1 < x < 1.

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cp's OEIS Frontend

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

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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.

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A366345 Decimal expansion of y such that Gamma(A364821 + i*sqrt(1-A364821^2)) = -y*i.

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A366545 Decimal expansion of the value x for which the function Re(Gamma(-x + i*sqrt(1-x^2))) is minimized for -1 < x < 1.

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A366345 Decimal expansion of y such that Gamma(A364821 + isqrt(1-A364821^2)) = -yi.