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Is the inverse of the Laplace transform unique?

  1. Apr 14, 2013 #1
    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
     
  2. jcsd
  3. Apr 14, 2013 #2

    HallsofIvy

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    Science Advisor

    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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