Elliptic alpha functions relate the complete Elliptic Integrals of the First and
Second Kinds at Elliptic Integral Singular Values according to
(1) | |||
(2) | |||
(3) |
(4) | |||
(5) |
(6) |
(7) |
(8) |
J. Borwein has written an Algorithm which uses lattice basis reduction to provide algebraic values for .
See also Elliptic Integral of the First Kind, Elliptic Integral of the Second Kind, Elliptic Integral Singular Value, Elliptic Lambda Function
References
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.
Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. ``Ramanujan, Modular Equations, and Approximations
to Pi, or How to Compute One Billion Digits of Pi.'' Amer. Math. Monthly 96, 201-219, 1989.
Weisstein, E. W. ``Elliptic Singular Values.'' Mathematica notebook EllipticSingular.m.
© 1996-9 Eric W. Weisstein