For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, and
$$F(x) = \sum_{n=0}^{+\infty} c_{n} \frac{(\mathrm{i}x)^{n}}{n!} \tag{S}$$
We admit that $\Gamma(x) \underset{x \rightarrow +\infty}{\sim} \sqrt{2\pi} x^{(x-1/2)} \mathrm{e}^{-x}$.
Investigate whether the series on the right-hand side of $(S)$ converges absolutely when $|x| = R$, where $R$ is the radius of convergence.