Working with a class of polynomials I've found this integral
$$A_n(x)=\frac{n!}{2\pi\,i}\int_C\frac{e^{x\,(e^t-1)}}{t^{n+1}}dt$$ where $C$ h is a closed circuit described in the positive sense surrounding the origin.
I wonder if it's possible find a upper bound of this integral (depending on $n$)
Using the residue theorem, we find that
$$\begin{align} A_n(x)&=e^{-x}\lim_{t\to 0}\frac{d^n}{dt^n}e^{xe^t}\\\\ &=e^{-x}\lim_{t\to 0}\left(\sum_{k=1}^n e^{xe^t}B_{n,k}(xe^t, xe^t, \dots, xe^t)\right)\\\\ &=\sum_{k=1}^n B_{n,k}(x,x, \dots, x)\\\\ &=B_n(x,x,\dots,x)\\\\ &=T_n(x)\\\\ &=\sum_{k=0}^n {n \brace k} x^k \end{align}$$
where $B_{n,k}(x_1,x_2,\dots, x_n)$ are the Bell polynomials, $B_n(x_1, x_2,\dots, x_n)$ is the $n$th complete exponential Bell polynomial, $T_n(x)$ is the Touchard polynomial, and ${n \brace k}$ are Stirling number of the second kind.