Complete elliptic integral of the second kind
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The complete elliptic integral of the second kind E may be defined as
or
It is a special case of the incomplete elliptic integral of the second kind:
It can be expressed as a power series
![E(k) = \frac{\pi}{2} \sum_{n=0}^{\infty} \left[\frac{(2n)!}{2^{2 n} n!^2}\right]^2 \frac{k^{2n}}{1-2 n}\!](../../../math/8/d/a/8da804c4cce41a5ffe9f83b23b5678b2.png)
which is
![E(k) = \frac{\pi}{2}\left\{1 - \left(\frac{1}{2}\right)^2 \frac{k^2}{1} - \left(\frac{1 \cdot 3}{2 \cdot 4}\right)^2 \frac{k^4}{3} - \cdots - \left[\frac{\left(2n - 1\right)!!}{\left(2n\right)!!}\right]^2 \frac{k^{2n}}{2 n-1} - \cdots \right\}.\!](../../../math/f/1/a/f1a9328c5213fdc7e9947b9bcc998913.png)
In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as
