

A301816


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


8



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
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OFFSET

0,1


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).


LINKS

Table of n, a(n) for n=0..86.
Iaroslav V. Blagouchine, A theorem for the closedform evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory, vol. 148, pp. 537592 and vol. 151, pp. 276277, 2015. arXiv version, arXiv:1401.3724 [math.NT], 2014.
Peter Luschny, Illustration of the real Stieltjes gamma function.


FORMULA

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


EXAMPLE

0.2754347245639200799552878777978068357987023238863074873733211475133063441...


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.


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.
Sequence in context: A246205 A340117 A160669 * A021367 A217102 A334851
Adjacent sequences: A301813 A301814 A301815 * A301817 A301818 A301819


KEYWORD

nonn,cons


AUTHOR

Peter Luschny, Apr 09 2018


STATUS

approved



