A301816 Decimal expansion of the real Stieltjes gamma function at x = 1/2.

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

Offset: 0

Author

Peter Luschny, Apr 09 2018

Keywords

• nonn
• cons

Comments

Define the real Stieltjes gamma function (this is not a standard notion) as Sti(x) = -2*Pi*I(x+1)/(x+1) where I(x) = Integral_{-infinity..+infinity} log(1/2+i*z)^x/(exp(-Pi*z) + exp(Pi*z))^2 dz and i is the imaginary unit. We look here at the real part of Sti(x).

Examples

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Links

• Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory, vol. 148, pp. 537-592 and vol. 151, pp. 276-277, 2015. arXiv version, arXiv:1401.3724 [math.NT], 2014.
• Peter Luschny, Illustration of the real Stieltjes gamma function.

Crossrefs

Sti(0) = A001620 (Euler's constant gamma) (cf. A262235/A075266),

Sti(1/2) = A301816,

Sti(1) = A082633 (Stieltjes constant gamma_1) (cf. A262382/A262383),

Sti(3/2) = A301817,

Sti(2) = A086279 (Stieltjes constant gamma_2) (cf. A262384/A262385),

Sti(3) = A086280 (Stieltjes constant gamma_3) (cf. A262386/A262387),

Sti(4) = A086281, Sti(5) = A086282, Sti(6) = A183141, Sti(7) = A183167,

Sti(8) = A183206, Sti(9) = A184853, Sti(10) = A184854.

Programs

Maple

Sti := x -> (-4*Pi/(x + 1))*int(log(1/2 + I*z)^(x + 1)/(exp(-Pi*z) + exp(Pi*z))^2, z=0..64): Sti(1/2): Re(evalf(%, 100)); # Note that this is an approximation which needs a larger domain of integration and higher precision if used for more values than are in the Data section.

Formula

c = -Re((4/3)*Pi*Integral_{-oo..oo} log(1/2+i*z)^(3/2)/(exp(-Pi*z)+exp(Pi*z))^2 dz).