A363848 -id:A363848 - cp's OEIS frontend

A363874 Decimal expansion of the harmonic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

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

Tian Vlasic, Jun 25 2023

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.

			0.87892065082960412...

Eric Weisstein's World of Mathematics, Isoperimetric Quotient
Wikipedia, Elliptic integral

Cf. A091476, A363848, A363876.

First[RealDigits[Pi^2/(4 * NIntegrate[EllipticE[x^2]^2/Sqrt[1 - x^2], {x, 0, 1}, WorkingPrecision -> 100])]]

Pi^2/(4*intnum(x=0,1,(ellE(x)^2)/sqrt(1 - x^2))) \\ Hugo Pfoertner, Jun 25 2023

Equals Pi^2/(4*Integral_{x=0..1} (E(x)^2)/sqrt(1 - x^2) dx).

A363876 Decimal expansion of the geometric mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

Original entry on oeis.org

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

Offset: 0

Tian Vlasic, Jun 25 2023

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.

			0.916816923382168248...

Eric Weisstein's World of Mathematics, Isoperimetric Quotient
Wikipedia, Elliptic integral

Cf. A102753, A363848, A363874.

First[RealDigits[Pi^2/2*Exp[-1 - 2*NIntegrate[Log[EllipticE[x^2]], {x, 0, 1}, WorkingPrecision -> 100]]]]

Equals ((Pi^2)/2) * exp(-1-2*Integral_{x=0..1} log(E(x)) dx).

cp's OEIS Frontend

A363874 Decimal expansion of the harmonic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

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