Integral from 0 to 1 of Complete Elliptic Integral of First Kind

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Theorem

Let $G$ denote Catalan's constant.

Then:

$2 G = \displaystyle \int_0^1 \map K k \rd k$

where $\map K k$ denotes the complete elliptic integral of the first kind:

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


Proof


Sources