I'm currently working on a problem which involves the following series: $$f(t)=\sum_{n=-\infty}^\infty a_ne^{-2\pi i\cdot pn\cdot\log(t)}$$ Where $p\in \mathbb R$. I'm interested in the short time asymptotics of this series (at $t\rightarrow 0^+$). Mainly - I'm interested to know if I can expand $f(t)$ as an asymptotic series in (possibly negative) powers of $t$. I'm actully not so interested in the entire series - I'm mostly interested in knowing if such an expansion exists, and if it does, to know the value of the free term, independent of $t$.
One idea I had in mind to deal with this, is to write each term as a Laurent series in $t$ (and then rearranging), but there's a problem with the $\log(t)$ inside the exponent (which has a branch point at $t=0$). This also makes me suspect that there can't be such an expansion, since if there were, then the function $f(t)$ would be meromorphic, but that's not true, due to the log-periodic oscillations of $f$ near $0$. But since I only need the series to exist for $t\rightarrow 0^+$, then maybe I can still find such a series for my specific problem (just not a Laurent series which converges in an annulus).
If anyone has an idea on how to do this, please let me know (you can assume that the series converges for $t>0$). The answer to my specific series will naturally depend on the coefficients $a_n$, but since I'm more interested in the general approach for series of this form (checking if an expansion exists and finding the free term), I'd like to hear your general ideas on how to approach such problems. Mainly - if there's a way to show that there can't be such an expansion - I'd also be happy to hear it.
Thanks in advance.