In mathematics, the Weber modular functions are a family of three functions f, f1, and f2,[note 1] studied by Heinrich Martin Weber.
where τ is an element of the upper half-plane.
Then the Weber functions are These are also the definitions in Duke's paper "Continued Fractions and Modular Functions".
η ( τ )
is the Dedekind eta function and
2 π i τ
2 π i τ α
quotients immediately imply The transformation τ → –1/τ fixes f and exchanges f1 and f2.
So the 3-dimensional complex vector space with basis f, f1 and f2 is acted on by the group SL2(Z).
be the nome, The form of the infinite product has slightly changed.
But since the eta quotients remain the same, then
The utility of the second form is to show connections and consistent notation with the Ramanujan G- and g-functions and the Jacobi theta functions, both of which conventionally uses the nome.
π i τ
, define the Ramanujan G- and g-functions as The eta quotients make their connection to the first two Weber functions immediately apparent.
Then, Ramanujan found many relations between
which implies similar relations between
For example, his identity, leads to For many values of n, Ramanujan also tabulated
for even n. This automatically gives many explicit evaluations of
, which are some of the square-free discriminants with class number 2, and one can easily get
from these, as well as the more complicated examples found in Ramanujan's Notebooks.
The argument of the classical Jacobi theta functions is traditionally the nome
η ( τ )
η ( τ ) =
η ( τ + 1 )
, then they are just squares of the Weber functions
with even-subscript theta functions purposely listed first.
Using the well-known Jacobi identity with even subscripts on the LHS, therefore, The three roots of the cubic equation where j(τ) is the j-function are given by
Also, since, and using the definitions of the Weber functions in terms of the Jacobi theta functions, plus the fact that
η ( τ )
and have the same formulas in terms of the Dedekind eta function