Date: Dec 16, 2012 1:17 AM
Author: thomasinventions@yahoo.com
Subject: Ellipse circumference in workable closed form
http://arithmeticgeometricmean.blogspot.com

8e^2(1-e^2)*d/dk(K(k=e^2))+2(1-e^2)*pi/(AGM(1,sqrt(1-e^2)))

The derivative of K is estimated with a small delta applied to the form involving the AGM:

[pi/(2*AGM(1,sqrt(1-(e^2+1E-26))))-pi/(2*AGM(1,sqrt(1-e^2)))]/[1E-26]

At least 14 digits accuracy for all eccentricities from zero up to the point of where the aspect ratio is 1 million to 1.

Thomas Blankenhorn