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%I #4 May 15 2016 18:00:31
%S 1,1,0,3,8,3,9,6,5,3,6,1,7,6,1,3
%N Decimal expansion of tau_2 (so named by S. Finch), the sum of squared eigenvalues of the Ruelle-Mayer linear operator G_2.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 2.17.1 Ruelle-Mayer Operators, p. 153.
%F Integral_{0..inf} Integral_{0..inf} J_1(2*sqrt(u*v)^2 / ((exp(u)-1) * (exp(v)-1)) du dv, where J_1 is the Bessel function of the first kind with parameter 1.
%e 1.103839653617613...
%t digits = 16; m0 = 100; dm = 5; Clear[f];
%t f[m_] := f[m] = NIntegrate[BesselJ[1, 2*Sqrt[u*v]]^2/((Exp[u]-1) * (Exp[v]-1)), {u, 0, m}, {v, 0, m - u}, MaxRecursion -> 30, WorkingPrecision -> digits + 10]; f[m = m0]; Print[m, " ", RealDigits[f[m], 10, digits][[1]]]; f[m = m0 + dm]; Print[m, " ", RealDigits[f[m], 10, digits][[1]]]; While[RealDigits[f[m], 10, digits][[1]] != RealDigits[f[m - dm], 10, digits][[1]], m = m + dm; Print[m, " ", RealDigits[f[m], 10, digits][[1]]]]; RealDigits[f[m], 10, digits][[1]]
%Y Cf. A038517, A242914.
%K nonn,cons,more
%O 1,4
%A _Jean-François Alcover_, May 15 2016
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