Complete Elliptic Integral of the Second Kind as Power Series

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Theorem

The complete elliptic integral of the second kind:

$\displaystyle E \left({k}\right) = \int \limits_0^{\pi / 2} \sqrt{1 - k^2 \sin^2 \phi} \, \mathrm d \phi = \int \limits_0^1 \dfrac {\sqrt{1 - k^2 v^2} } {\sqrt{1 - v^2}} \, \mathrm d v$

can be expressed as the power series:

\(\displaystyle K \left({k}\right)\) \(=\) \(\displaystyle 2 + \dfrac \pi 2 \sum_{i \mathop \ge 0} \left({- \prod_{j \mathop = 1}^i \dfrac {2 j - 1} { 2 j} }\right)^2 \frac {k^{2 i} } {2 i - 1}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac \pi 2 \left({1 - \left({\dfrac 1 2}\right)^2 k^2 - \left({\dfrac {1 \cdot 3} {2 \cdot 4} }\right)^2 \frac {k^4} 3 - \left({\dfrac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} }\right)^2 \frac {k^6} 5 - \cdots}\right)\) $\quad$ $\quad$


Proof


Sources