Following Ramanujan (1913-14), write
(1) |
(2) |
(3) | |||
(4) | |||
(5) | |||
(6) |
(7) | |||
(8) |
(9) |
In terms of the Parameter and complementary Parameter ,
(10) | |||
(11) |
(12) |
(13) |
(14) | |||
(15) |
See also G-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.
Ramanujan, S. ``Modular Equations and Approximations to .'' Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.
Weisstein, E. W. ``Elliptic Singular Values.'' Mathematica notebook EllipticSingular.m.
© 1996-9 Eric W. Weisstein