What is the Laplace transform of this function?

Question

Let us say I have a function in the Laplace domain F(s) whose inverse Laplace transform is f(t). Is there some theorem/identity/derivation such that I can find the inverse Fourier transform of F(s)/(s-a)?

Answer 1

My hint from the comments:

Use the convolution theorem. Suppose $(f, F)$ and $(g, G)$ are transform pairs. Then $$ \mathcal{L} (f \star g) = \mathcal{L}(f) \mathcal{L}(g), $$ where $\star$ is convolution.

The crux:

Note that $$\mathcal{L} \left( e^{-at} \right) = \frac{1}{s-a}.$$ Now use the above.