# Is an inverse Laplace Transform always solvable?

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 $$\frac{P(s)}{Q(s)}$$ ?

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Do you mean $P$ and $Q$ are polynomials? –  S.B. Jul 3 '13 at 21:28
Yes you can.  –  Did Jul 3 '13 at 21:28
if degree of $P$ is less than degree of $Q$ then you decompose it to sum of $A/(s-a)^n$ and $(Bs+C)/(s^2+bs+c)^m$ and then you can get the inverse transforms. –  Maesumi Jul 3 '13 at 21:30

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.

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