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A242440
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Decimal expansion of a constant related to a certain Sobolev isoperimetric inequality.
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2
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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
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OFFSET
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0,1
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COMMENTS
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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.
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 222.
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LINKS
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FORMULA
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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.
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EXAMPLE
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0.31875906098038662481197217247621254322535...
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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