Is the inverse of the Laplace transform unique?

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SUMMARY

The discussion centers on the injectivity of the Laplace transform, specifically whether the equality of two integrals, \(\int^{∞}_{0}e^{-st}f(t)dt = \int^{∞}_{0}e^{-st}g(t)dt\), implies that \(f(t) = g(t)\). BiP suggests that if \(\int_0^\infty e^{-st}(f(t)- g(t))dt= 0\), it raises the question of whether this condition guarantees \(f(t) = g(t)\). The consensus indicates that the injectivity of the Laplace transform is indeed a valid consideration in this context.

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Bipolarity
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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
 
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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)?
 

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