Elliptic Integral - Complete Elliptic Integral of The Second Kind

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.

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