Complete elliptic integral of the second kind

From Wikipedia, the free encyclopedia

The complete elliptic integral of the second kind E may be defined as

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

or

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

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

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

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}\!

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\}.\!

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

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

[edit] Special values

E(0) = \frac 1 2 \pi\!
E(1) = 1\!
E(\frac 1 2 \surd 2) = \pi^{\frac 3 2} \Gamma(\frac 1 4)^{-2}+\frac 1 8 \pi^{-\frac 1 2} \Gamma(\frac 1 4)^2\!
E(\frac 1 4 \left(\surd 6-\surd 2\right)) = 2^{\frac 1 3} 3^{-\frac 3 4} \pi^2 \Gamma(\frac 1 3)^{-3} + 2^{-\frac {10} 3} 3^{\frac 3 4} \left(\surd 3 - 1\right) \pi^{-1} \Gamma(\frac 1 3)^3\!
E(\frac 1 4 \left(\surd 6+\surd 2\right)) = 2^{\frac 1 3} 3^{-\frac 1 4} \pi^2 \Gamma(\frac 1 3)^{-3} + 2^{-\frac {10} 3} 3^{\frac 1 4} \left(\surd 3 - 1\right) \pi^{-1} \Gamma(\frac 1 3)^3\!
Retrieved from "http://en.wikipedia.org../../../c/o/m/Complete_elliptic_integral_of_the_second_kind.html"