Ramanujan g- and G-Functions

Following Ramanujan (1913-1914), write

(1)

(2)

These satisfy the equalities

			(3)
			(4)
			(5)
			(6)

 and  can be derived using the theory of modular functions and can always be expressed as roots of algebraic equations when  is rational. They are related to the Weber functions.

For simplicity, Ramanujan tabulated  for  even and  for  odd. However, (6) allows  and  to be solved for in terms of  and , giving

			(7)
			(8)

Using (◇) and the above two equations allows  to be computed in terms of  or 

(9)

In terms of the parameter  and complementary parameter ,

			(10)
			(11)

Here,

(12)

is the elliptic lambda function, which gives the value of  for which

(13)

Solving for  gives

			(14)
			(15)

Solving for  and  directly in terms of  then gives

			(16)
			(17)

Analytic values for small values of  can be found in Ramanujan (1913-1914) and Borwein and Borwein (1987), and have been compiled by Weisstein. Ramanujan (1913-1914) contains a typographical error labeling  as .

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.