In my notes, it says that if $|f(t)| \leq e^{At}$ for some constant $A$, then the domain of the laplace transform of $f$ is $\{z\in\mathbb{C}:\mathrm{Re}(z)>\mathrm{Re}(A)\}$.
Find the domain of the Laplace transform of $\cos(\omega t)$ where $\omega\in\mathbb{R}$. So, since $|\cos(\omega t)| \leq 1 = e^{0t}$, then I would expect that the domain of the Laplace transform is the right-hand side of the complex plane.
However, I get a different result doing this:
$$\begin{align} \mathcal{L}(f(t)) :=& \int_0^\infty \cos (\omega t)e^{-zt}\ dt \\ =& \mathrm{Re}\left(\int_0^\infty e^{i\omega t}e^{-zt}\ dt\right) \\ =& \mathrm{Re}\left(\left[\frac{e^{-t(z-i\omega)}}{z-iw}\right]_{0}^\infty \right) \end{align}$$ which converges iff $\mathrm{Re}(z-iw)>0\iff \mathrm{Re}(z) > w$