Complete Elliptic Integral of the First Kind as Power Series

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Theorem

The complete elliptic integral of the first kind:

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

can be expressed as the power series:

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


Proof


Sources