I am getting a little confused when using the Laplace Transform. So, taking the laplace transform of 1 we get:
$\mathfrak L[1](p)= \int_0^\infty e^{-pt}dt=[-\frac1pe^{-pt}]_0^\infty=\frac1p$ when $\mathfrak R\mathfrak e[p]>0$
So what I am not understanding is the fact that we are not taking into account the imaginary component of p, and so if the real component of p is positive but the imaginary component is negative, won't the $e^{-pt}$ term not vanish to zero as t approaches $\infty$?