Prove that $\mathbb L^{-1}\{\mathbb p^\mathbb k\}=0$

Question

There is a question in my book at the end of which it is written that $\mathbb L^{-1}\{\mathbb p^\mathbb k\}=0$ for $\mathbb k$= 0,1,2,.....

But we know that $\mathbb L\{\mathbb 0\}$ = $\mathbb 0$ So here laplace transformation isn't one to one so how the inverse laplace transformation exists?

• How do I prove $\mathbb L^{-1}\{\mathbb p^\mathbb k\}=0$ ?

link

Original problem

Answer 1

We know $$\mathbb L[\delta(t)]=1\\\mathbb L[f'(t)]=s\mathbb L[f(t)](s)-f(0^+)$$ So $$\mathbb L^{-1}[s]=\mathbb L^{-1}\left[\mathbb L[\delta'(t)]-\delta(0^+)\right]=\delta'(t)$$ Similarly, $$\mathbb L^{-1}(s^k)=\delta^{(k)}(t)$$

However we use Laplace transforms on functions which are only non-zero for $t>0$, which is why we would then take $$\mathbb L^{-1}(s^k)=\left[\delta^{(k)}(t)\right]_{t>0}=0$$