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%I #5 Sep 15 2014 07:44:45
%S 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,
%T 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,
%U 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
%N Decimal expansion of the Landau-Kolmogorov constant C(5,1) for derivatives in L_2(0, infinity).
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 214.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Landau-KolmogorovConstants.html">Landau-Kolmogorov Constants</a>
%F 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)).
%e 2.702467331401968417841785516708665999600741467093925...
%t 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
%Y Cf. A244091, A245286, A245287.
%K nonn,cons
%O 1,1
%A _Jean-François Alcover_, Sep 15 2014
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