 Appell–Lerch sums, a generalization of Lambert series, were first studied by and .Watson studied the order 3 mock theta functions by expressing them in terms of Appell–Lerch sums, and Zwegers used them to show that mock theta functions are essentially mock modular forms.The Appell–Lerch series is

where 

and

The modified series

where 

and "y" = Im(τ) and

satisfies the following transformation properties

In other words, the modified Appell–Lerch series transforms like a modular form with respect to τ.