How to get the Laplace transformation of the function:
$$f(t) = \frac{a}{2\sqrt{\pi t}}\operatorname{erf}\left(\frac{-a}{4t}\right)$$
By definition of laplace transform,
$$\frac{a}{2\sqrt{\pi}}\int_0^∞\frac{1}{\sqrt{t}}e^{-st}\operatorname{erf}(\frac{-a}{4t})\,dt$$
Then substituting $\operatorname{erf}(\frac{-a}{4t})$ = $\int_0^{\frac{-a}{4t}}e^{-x^{2}}dx$...
It results into $\frac{a}{2\sqrt{\pi}}\int_0^\infty\frac{1}{\sqrt{t}}e^{-st}\frac{2}{\sqrt{\pi}}\int_0^{\frac{-a}{4t}}e^{-x^{2}}dxdt$ = $\frac{a}{\pi}\int_0^∞\frac{1}{\sqrt{t}}e^{-st}\int_0^{\frac{-a}{4t}}e^{-x^{2}}dxdt$
Then using change of integrals,
$$\frac{a}{\pi}\int_0^∞e^{-x^{2}}\int_{\frac{-a}{4x}}^∞\frac{1}{\sqrt{t}}e^{-st}dtdx$$
Then I am stuck on this part. Help would be appreciated. Thanks!