The complete elliptic integral of the second kind E is proportional to the circumference of the ellipse :
where a is the semi-major axis, and e is the eccentricity.
E may be defined as
or more compactly in terms of the incomplete integral of the second kind as
It can be expressed as a power series
which is equivalent to
In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as
The complete elliptic integral of the second kind can be most efficiently computed in terms of the arithmetic-geometric mean and its modification.
Read more about this topic: Elliptic Integral
Famous quotes containing the words complete, integral and/or kind:
“No man, said Birkin, cuts another mans throat unless he wants to cut it, and unless the other man wants it cutting. This is a complete truth. It takes two people to make a murder: a murderer and a murderee.... And a man who is murderable is a man who has in a profound if hidden lust desires to be murdered.”
—D.H. (David Herbert)
“Painting myself for others, I have painted my inward self with colors clearer than my original ones. I have no more made my book than my book has made mea book consubstantial with its author, concerned with my own self, an integral part of my life; not concerned with some third-hand, extraneous purpose, like all other books.”
—Michel de Montaigne (15331592)
“There are acacias, a graceful species amusingly devitalized by sentimentality, this kind drooping its leaves with the grace of a young widow bowed in controllable grief, this one obscuring them with a smooth silver as of placid tears. They please, like the minor French novelists of the eighteenth century, by suggesting a universe in which nothing cuts deep.”
—Rebecca West (18921983)