I have the following function:
$f(t)=\frac{A\exp(\omega_0t)}{(1/\omega_0+it)^2}$ where $A,\omega_0,t$ are real.
I want to calculate the inverse Laplace-Transformation:
$g(t)=\mathcal{L}^{-1}\left\{q(s)\right\}$
with $q(s)=1/(s+\mathcal{L}\{f(t)\}(s))$
Apparently this is not possible with the general approach.
Actually, I'm only interested in long/short time behaviour, so I really would be happy with an answer in this regime only.
I tried to find the poles of the function $q(s)$, but it's not possible with my knowledge. Visualizing the problem also leads to very chaotic results. The following picture shows a contourplot for $|\frac{1}{q(s)}|$ in order to find the poles of q(s). The x-axis is the real part, the y-axis the imaginary part of $s$. Values are $A=0.5$, $\omega_0=10$, and $\mathcal{L}\{f(t)\}(s)$ was calculated on the computer.
Is there any approach how to handle this problem?
Thanks already!