Weber Modular Function

The Weber modular functions are the three functions , , and defined on the upper half-plane by

			(1)
			(2)
			(3)

where is the Dedekind eta function (Schertz 2002, p. 326). Their values at imaginary quadratic arguments provide relatively simple defining polynomials for class fields (Yui and Zagier 1997, p. 1645).

Writing and , their infinite products are

			(4)
			(5)
			(6)

Here is the square of the nome used in the theory of Jacobi theta functions (Yui and Zagier 1997, p. 1646).

The functions satisfy

			(7)
			(8)

(Borwein and Borwein 1987, p. 69). They are related to the j-function by

(9)

The related functions and are defined by

			(10)
			(11)

and satisfy and (Atkin and Morain 1993; Schertz 2002, p. 326).

Letting be a root of unity, the transformation identities are

			(12)
			(13)
			(14)
			(15)
			(16)
			(17)

Thus the full modular group Gamma permutes the three functions up to roots of unity (Yui and Zagier 1997, pp. 1646-1647). They are also related to the Ramanujan g- and G-functions and the elliptic lambda function.

For a negative fundamental discriminant with , the value satisfies

(18)

where and are algebraic integers in the real subfield of the Hilbert class field of . When this imaginary quadratic field has class number one, and are ordinary integers. In particular, for the Heegner numbers listed below, is the unique real root of the corresponding minimal polynomial (Yui and Zagier 1997, pp. 1660-1661).

			minimal polynomial
11	1	1
19	0	-1
43	1	0
67	1	-1
163	3	2

The infinite product for gives the expansion

(19)

Consequently, if for positive , then

(20)

as . For , this explains the approximation of the Ramanujan constant by .

The value is also related to a continued fraction studied by Brillhart (Churchhouse and Muir 1969, p. 318; Stark 1971, p. 21). Its translate is the unique real root of

(21)

whose polynomial discriminant is . The continued fraction begins

(22)

Writing , its unusually large partial quotients include and (Stark 1971, p. 21). The continued fraction of has the same partial quotients from onward, with initial term 5 instead of 3.