A364821 -id:A364821 - cp's OEIS frontend

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

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.

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]]

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

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

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

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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

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

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

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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.