Definition:Elliptic Integral of the First Kind/Complete

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Special Function

Definition 1

$\displaystyle \map K k = \int \limits_0^{\pi / 2} \frac {\d \phi} {\sqrt {1 - k^2 \sin^2 \phi} }$

is the complete elliptic integral of the first kind, and is a function of $k$, defined on the interval $0 < k < 1$.


Definition 2

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

is the complete elliptic integral of the first kind, and is a function of $k$, defined on the interval $0 < k < 1$.


Also see