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How do I estimate/bound the error in an inverse Laplace transform?

Suppose I have a Laplace transform, $$ F(s) = \int_0^{\infty}dx\ f(x)e^{-s x} \ . $$ I know that $$ F(s) \approx e^{A/(4s)} $$ (for $s$ real) where $A$ is very large, and I want to estimate $f(x) …