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:
“For which of you, intending to build a tower, does not first sit down and estimate the cost, to see whether he has enough to complete it?”
—Bible: New Testament, Luke 14:28.
“Make the most of your regrets; never smother your sorrow, but tend and cherish it till it come to have a separate and integral interest. To regret deeply is to live afresh.”
—Henry David Thoreau (1817–1862)
“A kind heart he hath. A woman would run through fire and water for such a kind heart.”
—William Shakespeare (1564–1616)