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Experimenting in Mathematica, I see that taking the Laplace transform of certain functions $f(t)\neq 0$ actually gives me a non-zero function $F(s)$. However, for these certain functions, taking the inverse Laplace transform of $F(s)$ results in an answer of zero. How can this be? I thought that since a function $f(t)$ has Laplace transform $F(s)$, then the inverse Laplace transform of $F(s)$ must be $f(t)$ ?

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To get a good answer, you probably need to be more specific. Do you have an example of such an $f$? –  mrf Feb 5 '13 at 12:20
    
What about a function of the form $f(t)=\left\{\begin{array}{cc}0 & \text{ if }t\geq 0 \\ 1 & \text{ if }t<0\end{array}\right.$ Is it true that $\mathcal{L}^{-1}\left\{F(s)\right\}=f(t)$? –  Jp McCarthy Feb 5 '13 at 12:28

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