I What are the Laplace transforms of powers of y?

[space-time]

Let's say you have a function y(t). You know how derivatives of y have their own Laplace transforms? Well I was wondering if powers of y such as y^2 or y^3 have their own unique Laplace transforms as well. If so , how do you calculate them (because plugging them into the usual integral doesn't seem to work)?

 

[Ssnow]

On https://en.wikipedia.org/wiki/Laplace_transform, there is a table with properties of Laplace transform, one refer to the Laplace transform of $f(t)^{n}$ ...

 

[LCKurtz]

On https://en.wikipedia.org/wiki/Laplace_transform, there is a table with properties of Laplace transform, one refer to the Laplace transform of $f(t)^{n}$ ...

Are you sure? I don't see such a formula. Unless you are mistaking the one for $f^{(n)}(t)$ which is the n'th derivative. I have never seen a formula for $\mathcal L f(t)^n$ and I don't think there is a general one.

 

[Ssnow]

Are you sure?

Yes sorry I confuse the notation, I fact there isn't and explicit formula for this ...

 

[UnivMathProdigy]

On that same wikipedia page, there seems to be a way to do that if we consider the Laplace transform of the multiplication of functions; we just take the function f(t) and multiply it n times.

 

[JJacquelin]

Laplace transform is an efficient tool when linear operations are involves (sum, derivative, integral).
But it is not so, and generaly very complicated, when non-linear operations are involved (multiplication, division, power). Even in the simplest cases convolution is requiered, which is generaly arduous.