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A243308 Decimal expansion of h_3, a constant related to certain evaluations of the gamma function from elliptic integrals.

%I A243308 #14 Feb 16 2025 08:33:22
%S A243308 1,0,1,7,4,0,8,7,9,7,5,9,5,9,5,6,0,0,8,6,6,9,5,3,8,7,5,3,3,5,0,0,6,3,
%T A243308 4,2,5,9,9,5,2,5,6,9,1,8,5,4,5,4,1,1,8,9,9,9,1,5,0,5,4,2,3,7,5,3,5,2,
%U A243308 1,2,4,3,1,8,0,6,2,5,0,1,6,3,9,4,4,2,3,6,6,6,5,0,9,7,6,1,2,0,0,7,9,2,7
%N A243308 Decimal expansion of h_3, a constant related to certain evaluations of the gamma function from elliptic integrals.
%D A243308 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.5.4 Gamma function, p. 34.
%H A243308 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/GammaFunction.html">Gamma function</a>
%F A243308 4*K(4*sqrt(3)-7)/(sqrt(2+sqrt(3))*Pi), where K is the complete elliptic integral of the first kind.
%F A243308 3^(1/4)*GAMMA(1/3)^3/(2*2^(1/3)*Pi^2), where GAMMA is the Euler Gamma function.
%F A243308 GAMMA(1/6)^(3/2)/(2^(5/6)*sqrt(3)*Pi^(5/4)).
%e A243308 1.0174087975959560086695387533500634259952569...
%p A243308 Re(evalf(4*EllipticK(sqrt((4*sqrt(3)-7)))/(sqrt(2+sqrt(3))*Pi), 120)); # _Vaclav Kotesovec_, Apr 22 2015
%t A243308 RealDigits[4*EllipticK[4*Sqrt[3]-7]/(Sqrt[2+Sqrt[3]]*Pi), 10, 103] // First
%t A243308 RealDigits[1/ArithmeticGeometricMean[1, Sqrt[2 + Sqrt[3]]/2], 10, 103][[1]] (* _Jan Mangaldan_, Jan 06 2017 *)
%t A243308 RealDigits[2 EllipticK[(2 - Sqrt[3])/4]/Pi, 10, 103][[1]] (* _Jan Mangaldan_, Jan 06 2017 *)
%Y A243308 Cf. A073005, A175379, A175574.
%K A243308 nonn,cons,easy
%O A243308 1,4
%A A243308 _Jean-François Alcover_, Jun 03 2014