Given a constant $a\in\mathbb{C}$ and a function $f:\mathbb{R}\to\mathbb{C}$ with Laplace transform $F$, is there any chance to find the inverse Laplace transform of the function $g_1:(0,\infty)\to\mathbb{C}$ with $g_1(s)=\frac{F(s)}{sF(s)+a}$?
For sure, applying the Convolution Theorem for Laplace inversion, one may split $\mathcal{L}^{-1}(g_1)=f*\mathcal{L}^{-1}(g_2)$ where $g_2(s)=\frac{1}{sF(s)+a}$. But since I don`t know the inverse Laplace transform of $g_2$, this is perhaps unrewarding.
Thanks for help!