In mathematics, the Weber modular functions are a family of three functions "f", "f"1, and "f"2, studied by Heinrich Martin Weber.
Definition.
Let formula_1 where "τ" is an element of the upper half-plane. Then the Weber functions are
formula_2
These are also the definitions in Duke's paper "Continued Fractions and Modular Functions". The function formula_3 is the Dedekind eta function and formula_4 should be interpreted as formula_5. The descriptions as formula_6 quotients immediately imply
formula_7
The transformation "τ" → –1/"τ" fixes "f" and exchanges "f"1 and "f"2. So the 3-dimensional complex vector space with basis "f", "f"1 and "f"2 is acted on by the group SL2(Z).
Alternative infinite product.
Alternatively, let formula_8 be the nome, 
formula_9
The form of the infinite product has slightly changed. But since the eta quotients remain the same, then formula_10 as long as the second uses the nome formula_8. 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.
Relation to the Ramanujan G and g functions.
Still employing the nome formula_8, define the Ramanujan G- and g-functions as
formula_13
The eta quotients make their connection to the first two Weber functions immediately apparent. In the nome, assume formula_14 Then,
formula_15
Ramanujan found many relations between formula_16 and formula_17 which implies similar relations between formula_18 and formula_19. For example, his identity,
formula_20
leads to
formula_21
For many values of "n", Ramanujan also tabulated formula_16 for odd "n", and formula_17 for even "n". This automatically gives many explicit evaluations of formula_18 and formula_19. For example, using formula_26, which are some of the square-free discriminants with class number 2,
formula_27
and one can easily get formula_28 from these, as well as the more complicated examples found in Ramanujan's Notebooks.
Relation to Jacobi theta functions.
The argument of the classical Jacobi theta functions is traditionally the nome formula_29 
formula_30
Dividing them by formula_3, and also noting that formula_32, then they are just squares of the Weber functions formula_33
formula_34
with even-subscript theta functions purposely listed first. Using the well-known Jacobi identity with even subscripts on the LHS,
formula_35
therefore,
formula_36
Relation to j-function.
The three roots of the cubic equation
formula_37
where "j"("τ") is the j-function are given by formula_38. Also, since,
formula_39
and using the definitions of the Weber functions in terms of the Jacobi theta functions, plus the fact that formula_40, then
formula_41
since formula_10 and have the same formulas in terms of the Dedekind eta function formula_3.
References.
Notes.
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