The Group is the Subgroup of the Gamma Group with and
Odd; and Even. The function
(1) |
(2) |
gives the value of the Modulus
for which the complementary and normal complete Elliptic Integrals of
the First Kind are related by
(3) |
(4) |
(5) |
From the definition of the lambda function,
(6) |
(7) | |||
(8) |
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.
© 1996-9 Eric W. Weisstein