A243267 -id:A243267 - cp's OEIS frontend

A243268 Decimal expansion of 1/2-G/Pi, the lowest limiting trough of a square wave Fourier series, where G is the Gibbs-Wilbraham constant. [negated].

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

Offset: 0

Jean-François Alcover, Jun 02 2014

			-0.08948987223608363511601442291245487...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.1 Gibbs-Wilbraham constant, p. 249.

Eric Weisstein's MathWorld, Wilbraham-Gibbs Constant

Cf. A036792, A036793, essentially the same as A243267.

G = SinIntegral[Pi]; RealDigits[1/2 - G/Pi, 10, 97] // First

A245535 Decimal expansion of the analog of the Gibbs-Wilbraham constant for L_1 trigonometric polynomial approximation.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 25 2014

			x0 = 1.376991769203938865765266614301624670814900061506257246...
g(x0) = 0.0657838882664480990565512180874704669499564803216...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.1 Gibbs-Wilbraham Constant, p. 249.

Eric Weisstein's MathWorld, Wilbraham-Gibbs Constant

Cf. A036792, A036793, A243267, A243268.

digits = 101; g[x_] := (PolyGamma[x/2] - PolyGamma[(x+1)/2] + 1/x)*Sin[Pi*x]/Pi; x0 = x /. FindRoot[g'[x] == 0, {x, 3/2}, WorkingPrecision -> digits+5]; RealDigits[g[x0], 10, digits] // First

Maximum g(x0) of the function g(x) = (psi(x/2) - psi((x+1)/2) + 1/x)*sin(Pi*x)/Pi, for x >= 1, where psi is the polygamma function.

A272335 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.

Original entry on oeis.org

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, 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, 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

Offset: 1

Jean-François Alcover, Apr 26 2016

			1.14272812693068128481021845956657111930110150452947023957171253...

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

R. P. Boyer and W. M. Y. Goh Generalized Gibbs phenomenon for Fourier partial sums and de la Vallée-Poussin sums, J. Appl. Math. Comput. 37 (2011) 421-442, p. 11.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2024; p. 33.

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

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

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

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

cp's OEIS Frontend

A243268 Decimal expansion of 1/2-G/Pi, the lowest limiting trough of a square wave Fourier series, where G is the Gibbs-Wilbraham constant. [negated].

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A245535 Decimal expansion of the analog of the Gibbs-Wilbraham constant for L_1 trigonometric polynomial approximation.

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A272335 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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