Complete elliptic integral of the first kind

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The complete elliptic integral of the first kind K may be defined as

K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}

or

K(k) = \int_{0}^{1} \frac{dt}{\sqrt{(1-t^2)(1-k^2 t^2)}}.\!

It is a special case of the incomplete elliptic integral of the first kind:

K(k) = F(1;\,k) = F(\frac{\pi}{2}\,|\,k^2)\!

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}\!

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\}.\!

In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as

K(k) = \frac{\pi}{2} F \left(\frac{1}{2}, \frac{1}{2}; 1; k^2\right).\,\!

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.

[edit] Special values

K(0) = \frac 1 2 \pi\!
K(1) = \infty\!
K(\frac 1 2 \surd 2) = \frac 1 4 \pi^{-\frac 1 2} \Gamma(\frac 1 4)^2\!
K(\frac 1 4 \left(\surd 6-\surd 2\right)) = 2^{-\frac 7 3} 3^{\frac 1 4} \pi^{-1} \Gamma(\frac 1 3)^3\!
K(\frac 1 4 \left(\surd 6+\surd 2\right)) = 2^{-\frac 7 3} 3^{\frac 3 4} \pi^{-1} \Gamma(\frac 1 3)^3\!
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