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Elliptic Integral Singular Value--k_2


The second singular value k_2, corresponding to

 K^'(k_2)=sqrt(2)K(k_2),
(1)

is given by

k_2=tan(pi/8)
(2)
=sqrt(2)-1
(3)
k_2^'=sqrt(2)(sqrt(2)-1).
(4)

For this modulus,

 E(sqrt(2)-1)=1/4sqrt(pi/4)[(Gamma(1/8))/(Gamma(5/8))+(Gamma(5/8))/(Gamma(9/8))].
(5)

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Cite this as:

Weisstein, Eric W. "Elliptic Integral Singular Value--k_2." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipticIntegralSingularValuek2.html

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