Elliptic Integral Singular Value k2

The second Singular Value , corresponding to

$\begin{displaymath} K'(k_2)=\sqrt{2}\,K(k_2), \end{displaymath}$

(1)

is given by

$\displaystyle k_2$	$\textstyle =$	$\displaystyle \tan\left({\pi\over 8}\right)=\sqrt{2}-1,$	(2)
$\displaystyle k_2'$	$\textstyle =$	$\displaystyle \sqrt{2}\,(\sqrt{2}-1).$	(3)

For this modulus,

$\begin{displaymath} E(\sqrt{2}-1)={1\over 4}\sqrt{\pi\over 4}\left[{{\Gamma({\te... ...style{5\over 8}})\over\Gamma({\textstyle{9\over 8}})}}\right]. \end{displaymath}$

(4)