The Jacobi theta functions, like $$ \theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) , $$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the dependence of $\theta_3$ on its argument $z$. At a first look, the series is definitely convergent if $|q|<1$, with some vague hopes of not-horrible behaviour at the edge of the unit disk. However, as it turns out, not only does the series in the definition above diverge for $|q|>1$, but the theta function simply cannot be analytically continued past the edge of the unit disk, where it has a natural boundary.
I would like to understand exactly how the existence of this natural boundary, and the impossibility of analytical continuation beyond it, arises and how it can be rigorously proved. What is the cleanest, most flexible way of showing it's there?
(In particular, I would like to decide whether a similar series, $\sum_{n=-\infty}^\infty\exp(i(an^3+bn^2+cn))$, has a similar behaviour with respect to the equivalent parameter $q=e^{ib}$, so I would appreciate results which can be extended in that sort of direction.)