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:

f transforms like a modular form: for some integer k called the weight, for any elements of SL₂.
As a function of q=e^2πiτ, f is given by a Laurent series whose radius of convergence is 1.

Examples

The ring of level 1 modular forms is generated by the Eisenstein series E₄ and E₆ 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.