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A190573 Decimal expansion of Integral_{x = 0 to oo} Product_{m=1..oo} cos(x/m) dx.

%I A190573 #22 Feb 16 2025 08:33:14
%S A190573 7,8,5,3,8,0,5,5,7,2,9,8,6,3,2,8,7,3,4,9,2,5,8,3,0,1,1,4,6,7,3,3,2,5,
%T A190573 2,4,7,6,1
%N A190573 Decimal expansion of Integral_{x = 0 to oo} Product_{m=1..oo} cos(x/m) dx.
%D A190573 J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 101.
%D A190573 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.3, p. 429.
%H A190573 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InfiniteCosineProductIntegral.html">Infinite Cosine Product Integral</a>.
%e A190573 0.785380557298632873492583011467332524761...
%t A190573 (* This naive script is not suitable to get more than 10 digits. *)
%t A190573 digits = 10; m0 = 500; dm = 50; Clear[g];
%t A190573 f[n_Integer, m_] := NIntegrate[Product[Cos[x/k], {k, 1, m - 1}]*(((1 - x^2/(2*m^2))*Gamma[1 + m]^2)/(Gamma[1 + m - x/Sqrt[2]]*Gamma[1 + m + x/Sqrt[2]])), {x, n*Pi/2, (n + 1)*Pi/2}];
%t A190573 g[m_] := g[m] = NSum[f[n, m], {n, 0, Infinity}];
%t A190573 Print[g[m0]]; Print[g[m = m0 + dm]];
%t A190573 While[RealDigits[g[m], 10, digits][[1]] != RealDigits[g[m - dm], 10, digits][[1]], m = m + dm; Print[m, " ", RealDigits[g[m], 10, digits][[1]]]];
%t A190573 RealDigits[g[m], 10, digits][[1]] (* _Jean-François Alcover_, May 19 2016 *)
%K A190573 nonn,cons,more
%O A190573 0,1
%A A190573 _N. J. A. Sloane_, May 12 2011