Almost holomorphic modular form

In mathematics, almost holomorphic modular forms are a generalization of modular forms that are polynomials in 1/Im(τ) with coefficients that are holomorphic functions of τ. A quasimodular form is the holomorphic part of an almost holomorphic modular form. The operation of taking the holomorphic part gives an isomorphism between the spaces of almost holomorphic modular forms and quasimodular forms. The archetypal example of an almost homomorphic modular form is E₂(τ) – 3/πIm(τ), whose holomorphic part is the quasimodular Eisenstein series E₂(τ).

References

• Zagier, Don (2008), "Elliptic modular forms and their applications", The 1-2-3 of modular forms, Universitext, Berlin: Springer, pp. 1–103, doi:10.1007/978-3-540-74119-0, ISBN 978-3-540-74117-6, MR 2409678