Weber modular function
In mathematics, the Weber modular functions are a family of three functions f, f1, and f2, studied by Heinrich Martin Weber. Definition Let 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". The function is the Dedekind eta function and should be interpreted as . The descriptions as 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). Alternative infinite product Alternatively, let be the nome, The form of the infinite product has slightly changed. But since the eta quotients remain the same, then as long as the second uses the nome . 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 , define the Ramanujan G- and g-functions as The eta quotients make their connection to the first two Weber functions immediately apparent.