Laplace Transform of $y^n$

Question

When the Laplace transform of $y$ is denoted $Y(s)$, we have formulas for the derivatives of $y$ without actually knowing what $y$ is. Is there an explicit formula for $y^2$? More generally, $y^n$?

Answer 1

From this math stackexchange link, it seems there's no quick and easy way to do it. From the Wikipedia page on Laplace transform, it lists how to calculate $\mathcal{L}(f(t)g(t))$, in which case you could simply say $f(t)=g(t)$. But the formula deals with the convolution of two functions, namely

$$\mathcal{L}(f(t)g(t))=\frac{1}{2\pi i}\lim_{T\to\infty}\int_{c-iT}^{c+iT}F(\sigma)G(s-\sigma)d\,\sigma$$

And this integration is done along the vertical line $\textrm{Re}(\sigma)=c$, where $F(\sigma)$ and $G(s-\sigma)$ are the Laplace transform of $f(t)$ and $g(t)$, respectively. So it's clear that for an arbitrary function, taking the laplace of $f(t)^n$ is not an easy task.