Inverse Laplace Transform of $e^{c \cdot s^2}$ I am trying to find the Inverse Laplace Transform of the function
$$ F(s)=e^{c \cdot s^2} $$
where $c > 0$.
 A: I don't know if this will be of any use but note that (My original thought before I finished):
$$
\int e^{x^2}dx = \frac{\sqrt{\pi}}{2}\text{Erfi}[x]
$$
Now, if we proceed with your problem, we have
\begin{align}
\frac{1}{2\pi i}\lim_{u\to\infty}\int_{-iu}^{iu}\exp[cs^2 + st]ds &=
\frac{1}{2\pi i}\lim_{u\to\infty}\int_{-iu}^{iu}\exp\biggl[c\Bigl(s^2 + \frac{t}{c}s + \frac{t^2}{4c^2}\Bigr)-\frac{t^2}{4c}\biggr]ds\\
&= \frac{e^{-\frac{t^2}{4c}}}{2\pi i}\lim_{u\to\infty}\int_{-iu}^{iu}\exp\biggl[c\Bigl(s+\frac{t}{2c}\Bigr)^2\biggr]ds\\
&= \frac{e^{-\frac{t^2}{4c}}}{4 i\sqrt{c\pi}}\lim_{u\to\infty}\biggl[\text{Erfi}\Bigl(\frac{2iu+t}{2\sqrt{c}}\Bigr)-\text{Erfi}\Bigl(\frac{-2iu+t}{2\sqrt{c}}\Bigr)\biggr]\\
&= \frac{e^{-\frac{t^2}{4c}}}{4\sqrt{c\pi}}\lim_{u\to\infty}\biggl[\text{Erf}\Bigl(\frac{2u-it}{2\sqrt{c}}\Bigr)-\text{Erf}\Bigl(\frac{-2u-it}{2\sqrt{c}}\Bigr)\biggr]
\end{align}
For $t\geq 0$, $\text{Erf}[\infty - it] \to 1$ and $\text{Erf}[-\infty - it] \to -1$ so 
$$
\mathcal{L}^{-1}\{e^{cs^2}\}=\frac{e^{-\frac{t^2}{4c}}}{2\sqrt{c\pi}}
$$
