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

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, 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, 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

Offset: 1

Author

Jean-François Alcover, Jun 03 2014

Keywords

• nonn
• cons
• easy

Examples

			1.0174087975959560086695387533500634259952569...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.5.4 Gamma function, p. 34.

Links

• Eric Weisstein's MathWorld, Gamma function

Crossrefs

Cf. A073005, A175379, A175574.

Programs

Maple

Re(evalf(4*EllipticK(sqrt((4*sqrt(3)-7)))/(sqrt(2+sqrt(3))*Pi), 120)); # Vaclav Kotesovec, Apr 22 2015

Mathematica

RealDigits[4*EllipticK[4*Sqrt[3]-7]/(Sqrt[2+Sqrt[3]]*Pi), 10, 103] // First
RealDigits[1/ArithmeticGeometricMean[1, Sqrt[2 + Sqrt[3]]/2], 10, 103][[1]] (* Jan Mangaldan, Jan 06 2017 *)
RealDigits[2 EllipticK[(2 - Sqrt[3])/4]/Pi, 10, 103][[1]] (* Jan Mangaldan, Jan 06 2017 *)

Formula

4*K(4*sqrt(3)-7)/(sqrt(2+sqrt(3))*Pi), where K is the complete elliptic integral of the first kind.

3^(1/4)*GAMMA(1/3)^3/(2*2^(1/3)*Pi^2), where GAMMA is the Euler Gamma function.

GAMMA(1/6)^(3/2)/(2^(5/6)*sqrt(3)*Pi^(5/4)).