False theta functions closely resemble ordinary theta functions, however they
do not have the modular transformation properties that theta functions have. In
this paper, we find modular completions for false theta functions, which among
other things gives an efficient way to compute their obstruction to modularity.
This has potential applications for a variety of contexts where false and
partial theta series appear. To exemplify the utility of this derivation, we
discuss the details of its use on two cases. First, we derive a convergent
Rademacher-type exact formula for the number of unimodal sequences via the
Circle Method and extend earlier work on their asymptotic properties. Secondly,
we show how quantum modular properties of the limits of false theta functions
can be rederived directly from the modular completion of false theta functions
proposed in this paper.Comment: 20 page
