Is the inverse of the Laplace transform unique?

Bipolarity · Apr 14, 2013

I've been wondering whether the Laplace transform is injective. Suppose I have that
[tex]\int^{∞}_{0}e^{-st}f(t)dt = \int^{∞}_{0}e^{-st}g(t)dt[/tex] for all s for which both integrals converge. Then is it true that [itex]f(t) = g(t)[/itex] ? If so, any hints on how I might prove it?

Thanks!

BiP

HallsofIvy · Apr 14, 2013

That would mean that
[tex]\int_0^\infty e^{-st}(f(t)- g(t))dt= 0[/tex]

Does that necessarily mean that f(t)= g(t)?

Is the inverse of the Laplace transform unique?

Discussion

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