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.
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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 E4 and E6 (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.