  Ramanujan studied the analytic properties of many $q$-hypergeometric series.
Of those, mock theta functions have been particularly intriguing, and by work
of Zwegers, we now know how these curious $q$-series fit into the theory of
automorphic forms. The analytic theory of partial theta functions however,
which have $q$-expansions resembling modular theta functions, is not well
understood. Here we consider families of $q$-hypergeometric series which
converge in two disjoint domains. In one domain, we show that these series are
often equal to one another, and define mock theta functions, including the
classical mock theta functions of Ramanujan, as well as certain combinatorial
generating functions, as special cases. In the other domain, we prove that
these series are typically not equal to one another, but instead are related by
partial theta functions.
