I am revising Integral Transforms, and trying to find the Laplace Transform of the error function.
I have that $$erf(x)=\frac{2}{\sqrt \pi}\int_0^xexp(-t^2) dt$$
Thus I know the the Laplace transform will be the integral. $$\frac{2}{\sqrt \pi}\int_0^{\infty}e^{-{px}}\int_0^xexp(-t^2) dtdx$$
I was thinking of using integration by parts to solve this, integrating the $e^{-px}$ part of the equation, and differentiating the error function part? But I can't seem to get to the right answer doing this, which I know to be $$\frac{1}{p}exp(\frac{p^2}{4})(1-erf(\frac{p}{2})$$
Thanks in advance.