A247373 Decimal expansion of the Landau-Kolmogorov constant C(5,1) for derivatives in L_2(0, infinity).

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

Offset

1,1

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 214.

Links

Table of n, a(n) for n=1..102.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants

Formula

C(5,1) = C(5,4) = sqrt(5)/(2^(4/5)*sqrt(c)), where c is the least positive root of f(c) = Pi^2/10, f(c) being integral_{0..infinity} (2*arctanh(x*sqrt(c/(1 + x^10))))/(x*sqrt(1 + x^10)).

Example

2.702467331401968417841785516708665999600741467093925...

Mathematica

digits = 102; f[c_?NumericQ] := NIntegrate[(2*ArcTanh[x*Sqrt[c/(1 + x^10)]])/(x*Sqrt[1 + x^10]), {x, 0, Infinity}, WorkingPrecision -> digits+5]; c0 = c /. FindRoot[f[c] == Pi^2/10, {c, 1/5}, WorkingPrecision -> digits+5]; C0[n_, 1] := (((n-1)^(1/n) + (n-1)^(-1+1/n))/c)^(1/2); RealDigits[C0[5, 1] /. c -> c0, 10, digits] // First

Crossrefs

Cf. A244091, A245286, A245287.

Sequence in context: A290789 A199292 A152779 * A021041 A245975 A188737

Adjacent sequences: A247370 A247371 A247372 * A247374 A247375 A247376

Keyword

nonn,cons

Author

Jean-François Alcover, Sep 15 2014

Status

approved