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There is a very simple expression for the inverse of Fourier transform. What is the easiest known expression for the inverse Laplace transform?

Moreover, what is the easiest way to prove it?

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There is an integral expression in the Wikipedia article, en.wikipedia.org/wiki/Inverse_Laplace_transform –  Akhil Mathew Jul 28 '10 at 13:09

1 Answer 1

The Laplace transform can be simply interpreted as a Wick rotated Fourier transform of a function $f(t)$ which vanishes for $t<0$. Wick rotation means (in this case) changing the frequency $\omega$ of the Fourier transform into an imaginary parameter $s=-i\omega$ of the Laplace transform.

The reason for the inversion $$\mathcal{L}^{-1} \{F(s)\} = f(t) = \frac{1}{2\pi i}\lim_{T\to\infty}\int\limits_{\gamma-iT}^{\gamma+iT}e^{st}F(s)\,ds, \qquad s=\Re(\gamma)$$ to be a bit more complicated than the inverse Fourier transform (see wikipedia, to quote @Akhil) is this imaginary frequency which, if left purely imaginary, would lead to a non-convergent integral. The formula is still similar to the inverse Fourier transform. In fact I think (but have not verified) you could use it for that purpose to.

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