A352615 Decimal expansion of Integral_{0<=x,y<=Pi/2} sqrt(1-cos^2(x)*cos^2(y)) dx dy.

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, 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, 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

Offset: 1

Author

Robert FERREOL, Mar 23 2022

Keywords

• cons
• nonn

Examples

			2.0776814600281582057831205567855290128037790576247...
		

Links

• Robert Ferréol, Bohemian dome, Mathcurve.

Crossrefs

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

Programs

Maple

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

Mathematica

RealDigits[(EllipticE[1/2] - EllipticK[1/2])^2 + EllipticE[1/2]^2, 10, 105][[1]] (* Amiram Eldar, Mar 24 2022 *)

Formula

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

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.