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Decimal expansion of Integral_{0<=x,y<=Pi/2} sqrt(1-cos^2(x)*cos^2(y)) dx dy.
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%I #24 May 03 2022 23:51:12

%S 2,0,7,7,6,8,1,4,6,0,0,2,8,1,5,8,2,0,5,7,8,3,1,2,0,5,5,6,7,8,5,5,2,9,

%T 0,1,2,8,0,3,7,7,9,0,5,7,6,2,4,7,8,2,4,1,0,0,6,3,5,0,3,8,0,4,8,2,2,6,

%U 3,5,5,3,1,4,6,3,2,0,3,8,4,6,3,3,0,1,6,0,0,0,0,9,6,9,2,1,9,0,7,5,2,3,4,0,5

%N Decimal expansion of Integral_{0<=x,y<=Pi/2} sqrt(1-cos^2(x)*cos^2(y)) dx dy.

%H Robert Ferréol, <a href="https://mathcurve.com/surfaces.gb/boheme/boheme.shtml">Bohemian dome</a>, Mathcurve.

%F Equals Sum _{n>=0} (Pi^2/(4*(2*n-1))*(binomial(2*n,n)/4^n)^3).

%F Equals (E(a) - K(a))^2 + E(a)^2 where a = 1/sqrt(2) and E (resp. K) is the complete elliptic integral of the second (resp. first) kind.

%e 2.0776814600281582057831205567855290128037790576247...

%p a:=1/sqrt(2):evalf((EllipticE(a)-EllipticK(a))^2+EllipticE(a)^2,50);

%t RealDigits[(EllipticE[1/2] - EllipticK[1/2])^2 + EllipticE[1/2]^2, 10, 105][[1]] (* _Amiram Eldar_, Mar 24 2022 *)

%Y Cf. A091670 ((1/Pi^2)*Integral_{0<=x,y<=Pi} 1/sqrt(1-cos^2(x)*cos^2(y)) dx dy).

%K cons,nonn

%O 1,1

%A _Robert FERREOL_, Mar 23 2022