How to get some insight in the following integral:

\begin{equation} \mathcal{I}(s)=\int_0^\infty x^{-x}e^{sx}\text{d} x \end{equation}

where $s$ is real (and the lower integration bound may be set to $a>0$)?

Alternatively, can we estimate this kind of series:

\begin{equation} \mathcal{S}(s)=\sum_{n=1}^\infty n^{-n}e^{sn} \end{equation} where $s$ is real?


The asymptotic behaviour of your series has been studied in great detail in the paper

G. H. Hardy, On the integral function $\Phi_{a,\alpha,\beta}(z)= \sum\frac{x^n}{(n+a)^{\alpha n+\beta}},$ Quarterly J. Math., 5 (1906) 37, 369-378. (Collected papers of G. H.Hardy, vol. IV, p. 128).

The Laplace integral can be studied by similar methods. On the positive ray an asymptotics can be obtained by Laplace method, and in the complex plane with the saddle point method.


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