We know that the power series of the exponential gives us $$\forall x \in \mathbb{R},\quad \mathrm{e}^x = \sum_{n=0}^{+\infty} \frac{x^n}{n!}.$$
We also know that the Euler's Gamma function $\Gamma$ is an extension of the factorial to complex numbers.
Since the integral can sometimes be considered as a continuous sum, I was wondering if the below function $f$ is a center of interest in Mathematics. If so, does it have a name? does it have special properties?
Let $f$ be the function defined on $\mathbb{R}$ by $$f:x\longmapsto \int_0^{+\infty} \frac{x^t}{\Gamma(t+1)}\,\mathrm{d}t.$$
