How to compute the Inverse Laplace Transform of dirac-delta (i.e. $\mathcal{L^{-1}} \delta(x)$)?

If I go by the definition of ILT, I need to do something like $$\lim_{T\to\infty} \frac{1}{2 \pi i} \int_{\gamma - i T}^{\gamma + iT} \delta(s) e^{st} ds$$ where $\gamma$ is on real axis to the right of singularities of $\delta(s)$.

First of all, I am not sure if $x=0$ is to be considered as the singularity for $\delta(x)$, as $\delta(x)$ is a special type of function. Even if I do consider $x=0$ as the singularity, I am not sure how to compute the integral along an imaginary axis at $\gamma$ (say at $\gamma=1$), as I am not finding any definitions for $\delta(x)$ for complex $x$ when I search the literature.

  • $\begingroup$ There is no such thing as $\delta(s)$ in the context of Laplace transforms. Do you know that the inverse Laplace transform integral is the same as the (inverse) Fourier transform ? The (inverse) Fourier transform of $\delta(x)$ is $1/2\pi$. $\endgroup$ – reuns Jan 15 at 1:19

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