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 .