A363874 -id:A363874 - cp's OEIS frontend

A363848 Decimal expansion of the arithmetic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

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

Offset: 0

Tian Vlasic, Jun 24 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.

			0.933174653498462644...

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

Cf. A091476, A363874, A363876.

First[RealDigits[Pi^2/4 * NIntegrate[Sqrt[1-x^2]/EllipticE[x^2]^2, {x,0,1}, WorkingPrecision -> 100]]] (* Stefano Spezia, Jun 24 2023 *)

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

More terms from Stefano Spezia, Jun 24 2023

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

A363848 Decimal expansion of the arithmetic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions