I am confused about the following two observations which seem contradictory:
It is stated that the region of convergence of the Laplace transform is a half space. That is $L(s)$ is defined for all $s$ with $Re(s)>c$, while undefined for all s with $Re(s)<c$.
In the discussion of the inverse laplace transform, it is stated that the integral$$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}e^{st}L(s)\mathrm{d}s$$can be evaluated by closing the "bromwich contour" (the line of integration) via a semicircle to the left of $c$ and then letting the radius tend to infinity. (Afterwards using residue theorem)
Now here is my problem: In order for 2.) to be feasible it is necessary that the integrand $e^{st}L(s)$ is defined on the whole complex plain (because if the radius of the semicircle becomes larger, then eventually every point - even those points with $Re(s)<<c$ will be enclosed by the circle.). But the Laplace transform L(s) is only defined for $Re(s)>c$. Hence, how is it possible to use 2.) when $L(s)$ is not defined in some parts of the complex plane? Can anyone shed some light on this?
$\endgroup$ 1 Reset to default
