Is there an inverse Laplace Transform of $-a\cdot e^{-b\cdot x^c}$

Question

Is there an inverse Laplace Transform of $-a\cdot e^{-b\cdot x^c}$, where $a,b$ and $c$ are constants, in my case its: $a=-0.9898; b=0.3511; c=0.2553$

The property, that the function tends to zero for $x$ tending to $\infty$ is given, so that's why I think there has to be an inverse transformation. But I don't seem to find it.

Does anyone has an idea? Best wishes

Answer 1

Suggestion:

Use the Taylor expansion for the exponential function:

$$ e^{-b x^c} = \sum_{k=0}^{\infty} \frac{(-b x^c)^k}{k!}=\sum_{k=0}^{\infty} \frac{(-b)^k x^{ck}}{k!} $$

Now, invert each of the parcels in the sum and see if you get something that converges.