I want to evaluate the following residue at its essential singularities $z \in I\pi\mathbb Z$,
$$\frac 1 {2\pi i} \oint dz \exp[i(-z+\coth z))].$$
I suppose what I can do is just perform Laurent expansion and correct the coefficient $a_{-1}$, but it just makes the situation worse because the integrand is composed function...
Is there any idea to help calculating such kind of residues?