# The inverse Laplace transform of an entire function

A simple calculation shows that the Laplace transform of $f(t)=e^{-t^2/4}$ is the function $F(p)=\sqrt{\pi}e^{p^2}\operatorname{erfc}(p)$.
I would like to find the inverse Laplace transform of $F(p)$. However, $F(p)$ is an entire function for which $p=\infty$ is an essential singularity. This means that the usual methods based on contour deformation, residue calculation or series expansion don't seem to apply. Can anyone lend me a hand with this problem?

• Hi, the Fourier transform of $e^{-t^2/c}$ is $b e^{-a \omega ^2}$ and by analytic continuation $p = i \omega$ : the Laplace transform of $e^{-t^2/c}$ is $b e^{a p^2}$ and thus your $erfc(p)$ is wrong : $F(p) = \sqrt{\pi} e^{p^2}$. and so the problem is solved because the Gaussian is the Fourier (or Laplace) transform of itself ! – reuns Jul 18 '15 at 7:30
• The question deals with the unilateral Laplace transform and I am sorry I did not make that clear enough. In any event, the complementary error function has an asymptotic expansion (The Wolfram Functions Site) which allows for a rather simple inversion. I am grateful to @reuns anyway. – RLP Jan 11 '16 at 13:42