That's indeed a neat property! Re: "no good algorithm for inverse Laplace" -- there are certainly reasonably good numerical algorithms based on evaluating cont

That's indeed a neat property! Re: "no good algorithm for inverse Laplace" -- there are certainly reasonably good numerical algorithms based on evaluating contour integrals (see, e.g., https://en.wikipedia.org/wiki/Inverse_Laplace_transform https://en.wikipedia.org/wiki/Inverse_Laplace_transform). The inversion formulas are not usually taught in undergrad courses anymore (at least not in the US) because complex function theory has been largely taken out of undergrad engineering curricula, and even for math majors is very much optional.