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A340350 Decimal expansion of Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + sin(x)^2*sin(y)^2) dy dx.

%I A340350 #7 Jan 07 2021 07:18:51
%S A340350 4,9,5,3,1,6,6,1,8,6,9,2,1,2,3,3,6,4,3,0,2,9,6,5,0,4,0,4,1,1,6,1,0,4,
%T A340350 7,5,8,8,7,1,7,8,8,4,1,7,6,7,9,7,4,5,1,8,2,4,6,4,7,4,5,9,3,4,1,1,2,3,
%U A340350 7,7,4,0,6,1,2,4,7,1,1,3,6,1,4,3,4,5,6,5,3,5,0,3,2,6,6,3,7,5,2,8,7,7,9,2,3,1
%N A340350 Decimal expansion of Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + sin(x)^2*sin(y)^2) dy dx.
%F A340350 Equals Pi * Integral_{x=0..Pi/2} log((1 + sqrt(1 + sin(x)^2))/2) dx.
%F A340350 Equals limit_{n->infinity} Pi^2 * (log(A340165(n)) / (2*n^2) - log(2)).
%F A340350 Equals limit_{n->infinity} Pi^2 * (log(A340167(n)) / (4*n^2) - log(2)).
%e A340350 0.49531661869212336430296504041161047588717884176797451824647459341123774...
%p A340350 evalf(Pi * Integrate(log((1 + sqrt(1 + sin(x)^2))/2), x = 0..Pi/2), 120);
%t A340350 RealDigits[N[Pi*Integrate[Log[(1 + Sqrt[1 + Sin[x]^2])/2], {x, 0, Pi/2}], 100]][[1]]
%o A340350 (PARI) Pi * intnum(x = 0, Pi/2, log((1 + sqrt(1 + sin(x)^2))/2))
%Y A340350 Cf. A097469, A340165, A340167, A340322, A340421, A340422.
%K A340350 nonn,cons
%O A340350 0,1
%A A340350 _Vaclav Kotesovec_, Jan 05 2021