The Laplace transform is said to exist for a specific complex s if the magnitude of the...

The Laplace transform is said to exist for a specific complex s if the magnitude of the transform is finite – that is, if | X(s) | < ∞.

Show that a sufficient condition for the existence of the transform X(s) at s = is that

In other words show that x(t) exponentially weighted by is absolutely integrable. You will need to use the result that, for a complex function f(t),

Without rigorously proving eq.(P9.56-1), argue its plausibility.