A242440 Decimal expansion of a constant related to a certain Sobolev isoperimetric inequality.

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

Offset: 0

Author

Jean-François Alcover, May 14 2014

Keywords

• nonn
• cons

Comments

Summarizing the definition: the supremum of the absolute value of a differentiable function f(x,y) is less than or equal to 0.318759... times the square root of the integral of the sum of squares of all partial derivatives of f.

Examples

			0.31875906098038662481197217247621254322535...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 222.

Programs

Maple

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

Mathematica

Sqrt[(EllipticF[Log[3]/2*I, 3/2]*I + EllipticK[-1/2])/(2*Pi*Sqrt[2])] // Re // RealDigits[#, 10, 100]& // First
RealDigits[Sqrt[(EllipticK[1/3] - EllipticF[ArcCot[Sqrt[2]], 1/3])/(2 Sqrt[3] Pi)], 10, 100][[1]] (* Jan Mangaldan, Jan 04 2017 *)

Formula

Equals sqrt( 1/(2*Pi) * Integral_{t >= 1} 1/(sqrt(t^2 + 2)*sqrt(t^2 + 3)) dt ).

sqrt((F(log(3)/2*i, 3/2)*i + K(-1/2))/(2*Pi*sqrt(2))), with i = sqrt(-1), F and K being the elliptic integrals.