## Elliptic Lambda Function

The Group is the Subgroup of the Gamma Group with and Odd; and Even. The function

 (1)

where
 (2)

is a -Modular Function and are Theta Functions.

gives the value of the Modulus for which the complementary and normal complete Elliptic Integrals of the First Kind are related by

 (3)

It can be computed from
 (4)

where
 (5)

and is a Theta Function.

From the definition of the lambda function,

 (6)

For all rational , and are expressible in terms of a finite number of Gamma Functions (Selberg and Chowla 1967). is related to the Ramanujan g- and G-Functions by
 (7) (8)

Special values are

where

See also Elliptic Alpha Function, Elliptic Integral of the First Kind, Modulus (Elliptic Integral), Ramanujan g- and G-Functions, Theta Function

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987.

Bowman, F. Introduction to Elliptic Functions, with Applications. New York: Dover, pp. 75, 95, and 98, 1961.

Selberg, A. and Chowla, S. On Epstein's Zeta-Function.'' J. Reine. Angew. Math. 227, 86-110, 1967.

Watson, G. N. Some Singular Moduli (1).'' Quart. J. Math. 3, 81-98, 1932.