Let $F(s) = \mathcal{L}(f(t))$ be the Laplace transform of $f(t)$. Then, the inverse Laplace transform is given by a path integral $$ f(t) = \frac{1}{2\pi i} \lim_{\omega\rightarrow\infty} \int_{\sigma - i \omega}^{\sigma+i\omega} e^{st} F(s) ds $$ where $\sigma$ is chosen to be greater than the real parts of all singularities of $F(s)$.
As I understand, the constant $\sigma$ can be chosen arbitrarily as far as it satisfies the above condition so that $F(s)$ remains bounded on the line $\Re(s)=\sigma$. Then, how is it that one will end with a unique result by solving this integral for different values of $\sigma$?