Elliptic alpha functions relate the complete Elliptic Integrals of the First and
Second Kinds at Elliptic Integral Singular Values according to

(1) | |||

(2) | |||

(3) |

where is a Theta Function and

(4) | |||

(5) |

and is the Elliptic Lambda Function. The elliptic alpha function is related to the Elliptic Delta Function by

(6) |

(7) |

(8) |

J. Borwein has written an Algorithm which uses lattice basis reduction to provide algebraic values for .

**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

1999-05-25