We will use a discrete analogue of the classical Laplace method to show that for infinitely many positive integers $n$, the main term of the asymptotic expansion of the scaled $q$-exponential $(-q^{-nt+1/2}u;q)_{\infty}$ could be expressed in $\theta$-function.
On A Limiting Relation Between Ramanujan's Entire Function $A_{q}(z)$ And $θ$-Function
Asymptotic expansions of several series and their application
What do sin$(x)$ and arcsinh$(x)$ have in Common?
![QR code for arXiv:math/0606781 [math.CA]](http://oss.arxitics.com/images/favicon.svg)