Is an inverse Laplace Transform always solvable?

Question

I just read on Wikipedia that if we got a certain Laplace Transform

$$\mathcal{L}\{f(t)\}= \frac{A}{s-\alpha_1} + \frac{B}{s-\alpha_2} + ... $$

can be solved like this:

$$f(t)= A e^{\alpha_1 t}+ Be^{\alpha_2 t}+ ...$$

My question now is: Given that we can always use partial fractions, can we solve every inverse Laplace Transform of the form $$\frac{P(s)}{Q(s)}$$ ?

Answer 1

Yes, but remember that:

• $\deg P$ can be greater than $\deg Q$,

• $Q$ can have multiple roots.

The first thing produces some delta function derivatives in the inverse, and the second means that exponentials $\times$ polynomials may also appear.