# Is the inverse of the Laplace transform unique?

1. Apr 14, 2013

### Bipolarity

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

Thanks!

BiP

2. Apr 14, 2013

### HallsofIvy

Staff Emeritus
That would mean that
$$\int_0^\infty e^{-st}(f(t)- g(t))dt= 0$$

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