cp's OEIS Frontend

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

%I A363874 #15 Sep 03 2023 10:19:20
%S A363874 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,
%T A363874 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,
%U A363874 7,2,5,3,7,1,5,9,8,8,1,9,1,8,1,8,2,8,4,3,6
%N A363874 Decimal expansion of the harmonic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.
%C A363874 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.
%C A363874 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.
%H A363874 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IsoperimetricQuotient.html">Isoperimetric Quotient</a>
%H A363874 Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Elliptic_integral">Elliptic integral</a>
%F A363874 Equals Pi^2/(4*Integral_{x=0..1} (E(x)^2)/sqrt(1 - x^2) dx).
%e A363874 0.87892065082960412...
%t A363874 First[RealDigits[Pi^2/(4 * NIntegrate[EllipticE[x^2]^2/Sqrt[1 - x^2], {x, 0, 1}, WorkingPrecision -> 100])]]
%o A363874 (PARI) Pi^2/(4*intnum(x=0,1,(ellE(x)^2)/sqrt(1 - x^2))) \\ _Hugo Pfoertner_, Jun 25 2023
%Y A363874 Cf. A091476, A363848, A363876.
%K A363874 nonn,cons
%O A363874 0,1
%A A363874 _Tian Vlasic_, Jun 25 2023