Trying to prove a property for the region of convergence of a Laplace transform

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I've been studying Schaum's outlines for Signals and Systems, and came across this:


image in question


And I understood the part until $$ \int_{t_{1}}^{t_{2}} |x(t)|e^{-\sigma_{0}}e^{-(\sigma_{1}-\sigma_{0})t}\,dt $$

Where did the $$ -\sigma_{0} -(\sigma_{1} - \sigma_{0})t $$ come from? Did the book add/divide the process? It didn't really explain what to do before that, so I might have overlooked something.

I got that the exponential with complex values is equal to one because of the absolute value, but after that, I get a bit lost by the addition of another exponential