I'm seeing Laplace transforms for the first time, and I'm having trouble understanding the criteria for deciding when they exist.
I've read a few websites and books that seem to say that we only need a function $f(t)$ to be piecewise continuous for $t \geq 0$ and of exponential order for the Laplace transform to exist. However, the function $f(t) = t^{-1/2}$ is not even defined at $0$, but $\mathcal{L}[t^{-1/2}] = \frac{\sqrt{\pi}}{\sqrt{s}}$, and so it appears that these requirements can be relaxed a little. If we then consider $g(t) = t^{-2}$, apparently this transform does not exist. But it is definitely true that $ t^{-2} < e^t $ for large enough $t$, and so I don't see how $t^{-1/2}$ is any "better" than $t^{-2}$.