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A363874
Decimal expansion of the harmonic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.
2
8, 7, 8, 9, 2, 0, 6, 5, 0, 8, 2, 9, 6, 0, 4, 1, 2, 4, 6, 2, 0, 2, 9, 7, 3, 2, 0, 0, 5, 3, 0, 7, 8, 4, 1, 6, 0, 2, 4, 9, 3, 3, 6, 4, 8, 6, 4, 2, 2, 9, 7, 7, 8, 0, 2, 0, 8, 9, 5, 7, 7, 3, 5, 2, 7, 1, 5, 0, 7, 2, 5, 3, 7, 1, 5, 9, 8, 8, 1, 9, 1, 8, 1, 8, 2, 8, 4, 3, 6
OFFSET
0,1
COMMENTS
The isoperimetric quotient of a curve is defined as Q = (4*Pi*A)/p^2, where A and p are the area and the perimeter of that curve respectively.
The isoperimetric quotient of an ellipse depends only on its eccentricity e in accordance to the formula Q = (Pi^2*sqrt(1-e^2))/(4*E(e)^2), where E() is the complete elliptic integral of the second kind.
LINKS
Eric Weisstein's World of Mathematics, Isoperimetric Quotient
FORMULA
Equals Pi^2/(4*Integral_{x=0..1} (E(x)^2)/sqrt(1 - x^2) dx).
EXAMPLE
0.87892065082960412...
MATHEMATICA
First[RealDigits[Pi^2/(4 * NIntegrate[EllipticE[x^2]^2/Sqrt[1 - x^2], {x, 0, 1}, WorkingPrecision -> 100])]]
PROG
(PARI) Pi^2/(4*intnum(x=0, 1, (ellE(x)^2)/sqrt(1 - x^2))) \\ Hugo Pfoertner, Jun 25 2023
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Tian Vlasic, Jun 25 2023
STATUS
approved