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

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

Author

Jean-François Alcover, Jul 25 2014

Keywords

• nonn
• cons

Examples

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

References

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

Links

• Eric Weisstein's MathWorld, Wilbraham-Gibbs Constant

Crossrefs

Cf. A036792, A036793, A243267, A243268.

Programs

Mathematica

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

Formula

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.