I've been reading a physics book about non-perturbative contributions to the path integral, and there is an equality I'm having difficulties to understand; I thought it might relate to an identity I do not know of Laplace transform.
$ S_{l}(\beta,g)=\sum_{n} \frac{\lambda^n}{2 \pi i n} \beta e^{-\beta}\int ds e^{-s\beta}[I_{l}(s)]^n$ (71)
It is easy to verify that $I_{l}(s)=I_{-l}(s)$. From equation (71) one now derives the result
$ S_{l}(\beta,g)=\sum_{N} e^{-\beta E_{NL}} $
with $E_{NL}=S_{N,l}+1$, in which $S_{NL}$ is the solution of the equation
$I_{l}(s)=\frac{1}{\lambda}$
Any suggestion is appreciated, and I hope my question is well placed. I did not include the expression for $I_{l}(s)$ because it does not seem fundamental to understand which relation he used, but I can obviously add it later if required.
EDIT: ok, I think I got it: he exchanges order of summation and integration, obatining the geometric serie, which has poles where that equation is satisfied. Is it right?