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
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