See also Elliptic Alpha Function, Elliptic Integral Singular Value
References
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.
Weisstein, E. W. ``Elliptic Singular Values.'' Mathematica notebook EllipticSingular.m.