# Using Assumption with LaplaceInverseTransform

I want to compute the the Laplace inverse transform of the following

$$\bar f(s)=\frac{1}{s^2+a s+b}$$

which clearly depends on $a^2-4b$. When I use Mathematica, it just gives the answer for the case $a^2-4b>0$. I couldn't resolve the issue by using Assuming or Assumptions. I mean how can I get the answer for the cases $a^2-4b=0$ and $a^2-4b<0$.

$Assumptions = {a^2 - 4 b == 0}; f = 1/(s^2 + a s + b); InverseLaplaceTransform[f, s, t]  ## 1 Answer f = 1/(s^2 + a s + b); inv = InverseLaplaceTransform[f, s, t]  FullSimplify[inv, a^2 - 4 b > 0]  FullSimplify[inv, a^2 - 4 b < 0]  assumming a and b are Reals: FullSimplify[ComplexExpand@inv, a^2 - 4 b < 0]  Limit[inv, b -> a^2/4]  • (+1) Thanks for the attention. The result for$a^2-4b<0$is incorrect! :) Indeed, what you have obtained for$a^2-4b>0$and$a^2-4b<0\$ is the same – H. R. May 28 '18 at 13:39
• @H.R. - The result for a^2 - 4b < 0 is correct, it merely involves a complex argument to Sinh. If you want an explicitly real form, use Assuming[a^2 - 4 b < 0, inv // ComplexExpand[#, TargetFunctions -> {Re, Im}] & // FullSimplify] which evaluates to (2*Sin[(1/2)*Sqrt[-a^2 + 4*b]*t])/ (E^((a*t)/2)*Sqrt[-a^2 + 4*b]) – Bob Hanlon May 28 '18 at 13:52
• @H.R. The result is correct, o.k., for Realsdo this FullSimplify[ComplexExpand@inv, a^2 - 4 b < 0] – rmw May 28 '18 at 14:04
• Please add your last comment to the answer so that I can accept it. :) – H. R. May 28 '18 at 15:05