Weakly holomorphic modular form
In mathematics, a weakly holomorphic modular form is similar to a holomorphic modular form, except that it is allowed to have poles at cusps. Examples include modular functions and modular forms. Definition. To simplify notation this section does the level 1 case; the extension to higher levels is straightforward. A level 1 weakly holomorphic modular form is a function "f" on the upper half plane with the properties: Examples. The ring of level 1 modular forms is generated by the Eisenstein series "E" and "E" (which generate the ring of holomorphic modular forms) together with the  inverse 1/Δ of the modular discriminant. Any weakly holomorphic modular form of any level can be written as a quotient of two holomorphic modular forms. However, not every quotient of two holomorphic modular forms is a weakly holomorphic modular form, as it may have poles in the upper half plane.