Can we plot the inverse Laplace transform of the function $(x+1/2)^{x-1/2}$?

Question

I understand that there is possibly no closed form, but can one somehow plot it using, say, Mathematica?

Answer 1

Yes, Mathematica can do this kind of thing. Here's a screenshot:

For completeness I'll paste the command I used:

Plot[Log[InverseLaplaceTransform[(x + 1/2)^(x - 1/2), x, t]], {t, -10,
   10}]

Answering followup questions from the comments:

1. To see why I wrapped it in Log before plotting, just look at the graph I got! If the log-scaled version is shooting off to 1000+ then the non-logscale version will grow obscenely fast and won't make a very pretty graph to look at.
2. It took 414 seconds on my laptop.
3. Actually I'm pretty sure it has an asymptote at $x=0$. You could test where the inverse transform is defined just by plugging in individual points like InverseLaplaceTransform[(x + 1/2)^(x - 1/2), x, 0.000001] and InverseLaplaceTransform[(x + 1/2)^(x - 1/2), x, -0.000001].