Domain of the laplace transform of $f(t) = e^{-at}$

Question

The question is:
Compute the laplace transform of $f(t) = e^{-at}$ and state the domain of the Laplace Transform where $a\in\mathbb{C}$.

I computed the Laplace Transform of $f$ as
$$ \mathcal{L}f(z) = \int_{0}^\infty f(t)e^{-zt}dt \\ = \int_0^\infty e^{-at}e^{-zt}dt \\ = \int_0^\infty e^{-(a+z)t}dt \\ = \frac{1}{a+z}. $$
I'm not sure what information to use to find the domain of this function... is it just $z \neq -a$?

Answer 1

Note that we have

$$I(L)=\int_0^L e^{-(a+z)t}\,dt=\frac{1-e^{-(a+z)L}}{a+z}$$

If $\text{Re}(a+z)>0$, then $\lim_{L\to \infty}I(L)=\frac1{a+z}$.

If $\text{Re}(a+z)\le 0$, then $\lim_{L\to \infty}I(L)$ fails to exist.