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Decimal expansion of a function approximation constant which is the analog of Gibbs's constant 2*G/Pi (A036793) for de la Vallée-Poussin sums.
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%I #21 Nov 18 2024 03:33:18

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

%T 1,1,9,3,0,1,1,0,1,5,0,4,5,2,9,4,7,0,2,3,9,5,7,1,7,1,2,5,3,0,9,9,2,9,

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

%N Decimal expansion of a function approximation constant which is the analog of Gibbs's constant 2*G/Pi (A036793) for de la Vallée-Poussin sums.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.1 Gibbs-Wilbraham Constant, p. 248.

%H R. P. Boyer and W. M. Y. Goh <a href="http://www.math.drexel.edu/~rboyer/papers/generalized_gibbs.pdf">Generalized Gibbs phenomenon for Fourier partial sums and de la Vallée-Poussin sums</a>, J. Appl. Math. Comput. 37 (2011) 421-442, p. 11.

%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020-2024; p. 33.

%F Equals (2/Pi)*Integral_{t=0..2*Pi/3} (cos(t) - cos(2*t))/t^2 dt.

%F Equals (2/Pi)*(2*Si(4*Pi/3) - Si(2*Pi/3)), where Si is the Sine integral function.

%e 1.14272812693068128481021845956657111930110150452947023957171253...

%t (2/Pi)(2 SinIntegral[4 Pi/3] - SinIntegral[2 Pi/3]) // N[#, 101]& // RealDigits // First

%Y Cf. A036792, A036793, A243267, A243268, A245535.

%K nonn,cons

%O 1,3

%A _Jean-François Alcover_, Apr 26 2016