The tutorial that brought this assertion to me was: http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node2.html
"As the imaginary part $\omega=Im[s]$ of the complex variable $s=\sigma+j\omega$ has no effect in terms of the convergence, the ROC is determined solely by the real part $\sigma=Re[s]$."
This is talking about Laplace transform of a signal $x(t)$ w.r.t $s$, $L\{x(t)\}(s) = \int_{-\infty}^{+\infty}x(t)e^{-st}dt$.
The correctness of the assertion is not obvious to me. I tried several approaches to prove it as well as searching for a proof but failed. Could anyone show me a hint or guide me to a relevant reference?