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

0,1

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.

References

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

Links

Table of n, a(n) for n=0..99.

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.

Example

0.31875906098038662481197217247621254322535...

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 *)

Crossrefs

Sequence in context: A257142 A270861 A208656 * A245651 A007023 A176103

Adjacent sequences: A242437 A242438 A242439 * A242441 A242442 A242443

Keyword

nonn,cons

Author

Jean-François Alcover, May 14 2014

Status

approved