 Weak modular form

Here a weak modular form is a holomorphic function on the complex upper half-plane satisfying the weight-$k$ transformation law, with no growth condition at any cusp of a modular group. This convention differs from weakly holomorphic modular forms, which are required to be meromorphic at the cusps. At level one, translation invariance gives a Laurent expansion on the punctured unit disc, possibly with infinitely many negative powers.