Complete elliptic integral of the first kind
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The complete elliptic integral of the first kind K may be defined as
or
It is a special case of the incomplete elliptic integral of the first kind:
It can be expressed as a power series
![K(k) = \frac{\pi}{2} \sum_{n=0}^{\infty} \left[\frac{(2n)!}{2^{2 n} n!^2}\right]^2 k^{2n}\!](../../../math/f/7/e/f7ef1dc81de81c7bddbbf5149e424e43.png)
which is
![K(k) = \frac{\pi}{2}\left\{1 + \left(\frac{1}{2}\right)^2 k^{2} + \left(\frac{1 \cdot 3}{2 \cdot 4}\right)^2 k^{4} + \cdots + \left[\frac{\left(2n - 1\right)!!}{\left(2n\right)!!}\right]^2 k^{2n} + \cdots \right\}.\!](../../../math/5/a/3/5a3f0d2d1e308865af0730f877bd5666.png)
In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as
The complete elliptic integral of the first kind is sometimes called the quarter period.
It can be computed in terms of the arithmetic-geometric mean.
