What is the inverse Laplace of a complete square?

Question

How do I find the inverse Laplace for something like this ?

$${8 \over ( s^2 + 16 )^2 }$$

I tried using partial fraction but it didn't help
any ideas on how to do it using differentiation or convolution theorem? or any other way !

Answer 1

By the residue theorem, the inverse Laplace transform of the above expression is

$$\left [\frac{d}{ds} \frac{8\,e^{s t}}{(s+4 i)^2} \right ]_{s=4 i} + \left [\frac{d}{ds} \frac{8\,e^{s t}}{(s-4 i)^2} \right ]_{s=-4 i}$$