A275373 - OEIS

%I #9 Jan 05 2017 03:27:14

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

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

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

%N Decimal expansion of the volume of the convex hull of two orthogonal disks in the "two-circle roller" configuration.

%H Steven R. Finch, <a href="http://arxiv.org/abs/1211.4514v3">Convex Hull of Two Orthogonal Disks</a>, arXiv:1211.4514v3 [math.MG] 2016 p. 15.

%F omega = arcsin(sqrt(sqrt(2) - 1)),

%F gamma =  (sqrt(2)-1)/2+E(omega, -1)+sqrt(2)(EllipticPi(-sqrt(2)-1,omega,-1) + EllipticPi(sqrt(2)-1,omega,-1)-F(omega,-1)),

%F VL = (8*gamma)/(3*sqrt(2)), where E is the complete elliptic integral of the second kind, EllipticPi is the incomplete elliptic integral of the third kind and F is the elliptic integral of the first kind.

%e 3.28181948744968941903219331047186568800423705235770280145947014746...

%t omega = ArcSin[Sqrt[Sqrt[2] - 1]];

%t gamma =  (Sqrt[2] - 1)/2 + EllipticE[omega, -1] + Sqrt[2] (EllipticPi[ -Sqrt[2] - 1, omega, -1] + EllipticPi[Sqrt[2] - 1, omega, -1] - EllipticF[omega, -1]);

%t VL = (8*gamma)/(3*Sqrt[2]);

%t RealDigits[VL, 10, 101][[1]]

%t RealDigits[2/3 (4 EllipticE[ArcTan[2^(1/4)], 1/2] - 2 Sqrt[2] EllipticF[ArcTan[2^(1/4)], 1/2] + 2 (Sqrt[2] + 1) EllipticPi[-1/Sqrt[2], ArcTan[2^(1/4)], 1/2] + 2 (Sqrt[2] - 1) EllipticPi[1/Sqrt[2], ArcTan[2^(1/4)], 1/2] + Sqrt[2] - 2), 10, 101][[1]] (* _Jan Mangaldan_, Jan 04 2017 *)

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Jul 25 2016