According to Wikipedia, The formua for Laplace Transform of two functions is $$\mathcal{L} f(t) g(t) = \frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty} F(\sigma) G(s-\sigma) d\sigma$$
where, vertical line at $c$ is chosen to be in ROC of $F$
2 questions:
Why is the emphasis only on $F$ for choosing $c$? What about $G$? I couldn't find any derivation of this result on internet (including in the book that Wikipedia cites for this result). Either can someone please explain/derive this or point me to a derivation online?
If we have F=G and they have ROC of the form $Re(p) > 0$ (where $p$ is the complex variable of LT) $\color{blue}{AND}$ F=G has a branch point at $p=0$, after flipping the usual branch cut for $F$ (negative real axis) and similarly for $G(s-\sigma)$, I am left with a combined branch cut between $[0,Re(s)]$.In this case, can I choose $c$, such that $c > Re(s)$, such that this contour is on the right of the modified branch cut? Or am I not allowed to form this kind of branch cut as LT is supposed to be analytic in its ROC?