I am interested in Ecalle's generalization of the Borel-Laplace summation. I would like to see an explicit treatment of a summation of a transeries that include logarithmic terms.

The only example I have seen is the construction of Fatouc coordinates in the case of non-vanishing resiter, however this series has a single logarithm term that can just be subtracted.

If this is not possible, I would like to know if a closed form for the following integral can be found $$ I(z) = \int_{\Gamma_\theta} \frac{\log s}{2 \pi i s} e^{-z s} ds. $$ The contour of integration is shown in the following figure.

## A bit of context

We define the Laplace transform as $$L^{\theta}[f](z) = \int_{e^{i\theta}\mathbb{R}^+} f(s) e^{-z s} ds. $$

The classical Borel transform is defined as the formal inverse of Laplace transform. Since for any $\theta$ we have $L^\theta[s^n](z)=n!z^{-n-1}$ we define the Borel transform as $$B[z^{-n-1}](s) = \frac{s^n}{n!}.$$

Beacause the Borel transform introduces a factorial, it may happen that the Borel transform of a formal series can be a germ. If this germ can be extended towards infinity, we may be able to take its Laplace transform. This procedure is called Borel-Laplace summation. This procedure however can deal only with negative integer powers of $z$.

Ecalle generalized this, by defining the Laplace transform of a function $F$ by defining $$\mathcal{L}^{\theta}[F](z) = \int_{\Gamma_\theta} F(s) e^{-z s} ds, $$ with $\Gamma_\theta$ being the contour in the above figure.

If $f$ is entire, then we define $F(s)=f(s)\frac{\log s}{2\pi i}$ and we have $$ L^{\theta}[f](z) = \mathcal{L}^{\theta}[F](z). $$ In this setting the laplace transform of $\frac{\log s}{2 \pi i s}$ should the logarithm or something very close.