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

0,1

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

Table of n, a(n) for n=0..88.

Eric Weisstein's World of Mathematics, Isoperimetric Quotient

Wikipedia, Elliptic integral

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

Cf. A091476, A363848, A363876.

Sequence in context: A094106 A277064 A276762 * A256609 A269892 A332437

Adjacent sequences: A363871 A363872 A363873 * A363875 A363876 A363877

Keyword

nonn,cons

Author

Tian Vlasic, Jun 25 2023

Status

approved