Talk:Laplace Transform of Sine

Domain of Laplace transform of sine
I think there is a typo in the statement: For $a\in\mathbb{C}$, the Laplace transform of $e^{at}$ is defined over $s\in\mathbb{C}$. In Proof 2 we use the Laplace Transform of Exponential, according to which

\[ \mathcal{L}\{e^{z t}\}(s) = \frac{1}{s-z}, \]

defined for $s\in\mathbb{C}$ with $\Re(s) > \Re(z)$.

In Proof 2 we use this result for $z = ia$ (with $a\in\mathbb{R}$), the Laplace transform is defined over $\Re(s) > \Re(ia) = 0$.