Suppose we want to compute this integral using contour integration. $$\int_{-\infty}^{\,\infty}dt\;\frac{e^{-i(t+1/t)}}{t}$$
On first glance, I would suppose that there is a simple pole at $z=0$. Is it meaningful to then talk about the residue at $z=0$? Because then, we have $e^{-i\infty}$, which does not converge.
I have another thought, which is that maybe $z=0$ is not a simple pole, but I have to take into account $e^{-i(1/t)}$ as well? Which could potentially get messy.
Anyone has thoughts on this problem?
p.s. I am not sure whether this integral exists, rather, I am concerned with the problem relating to the residue.
$e^{-i/z} $ has an essential singularity at $z=0$