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

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, 2, 4, 7, 6, 1

Offset: 0

Author

N. J. A. Sloane, May 12 2011

Keywords

• nonn
• cons
• more

Examples

			0.785380557298632873492583011467332524761...
		

References

• 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.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.3, p. 429.

Links

• Eric Weisstein's World of Mathematics, Infinite Cosine Product Integral.

Programs

Mathematica

(* This naive script is not suitable to get more than 10 digits. *)
digits = 10; m0 = 500; dm = 50; Clear[g];
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}];
g[m_] := g[m] = NSum[f[n, m], {n, 0, Infinity}];
Print[g[m0]]; Print[g[m = m0 + dm]];
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]]]];
RealDigits[g[m], 10, digits][[1]] (* Jean-François Alcover, May 19 2016 *)