In mathematics, almost holomorphic modular forms, also called nearly holomorphic modular forms, are a generalization of modular forms that are polynomials in 1/Im(τ) with coefficients that are holomorphic functions of τ.
In terms of representation theory, modular forms correspond roughly to highest weight vectors of certain discrete series representations of SL2(R), while almost holomorphic or quasimodular forms correspond roughly to other (not necessarily highest weight) vectors of these representations.
A level 1 almost holomorphic modular form is a function f on the upper half plane with the properties: A level 1 quasimodular form is defined to be the constant term of an almost holomorphic modular form (considered as a polynomial in 1/Im(τ)).
Quasimodular forms can be interpreted as sections of certain jet bundles.
[2] For example, As the field generated by quasimodular forms of some level has transcendence degree 3 over C, this implies that any quasimodular form satisfies some nonlinear differential equation of order 3.
For example, the Eisenstein series E2 satisfies the Chazy equation (give or take a few constants).