What is the most convenient well-known expression for inverse of Laplace change?

There is a really straightforward expression for the inverse of Fourier change.

What is the most convenient well-known expression for the inverted Laplace change?

In addition, what is the most convenient means to confirm it?

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2019-05-07 13:35:01
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The Laplace change can be merely taken a Wick revolved Fourier change of a function $f(t)$ which disappears for $t<0$. Wick turning suggests (in this instance) transforming the regularity $\omega$ of the Fourier change right into a fictional parameter $s=-i\omega$ of the Laplace change.

The factor 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 little bit extra difficult than the inverted Fourier change (see wikipedia, to price estimate @Akhil) is this fictional regularity which, if left totally fictional, would certainly bring about a non - convergent indispensable. The formula is still comparable to the inverted Fourier change. Actually I assume (yet have actually not validated) you can utilize it for that objective to.

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2019-05-09 08:06:07
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