# Inverse Laplace Transform of dirac-delta $\mathcal{L^{-1}} \delta(x)$

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.

• 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 $1/2\pi$. – reuns Jan 15 at 1:19