 Zwegers proved this result in three different ways, by relating the mock theta functions to Hecke's theta functions of indefinite lattices of dimension 2, and to Appell–Lerch sums, and to meromorphic Jacobi forms.Zwegers's fundamental result shows that mock theta functions are the "holomorphic parts" of real analytic modular forms of weight 1/2.This allows one to extend many results about modular forms to mock theta functions.