Question

how to find Laplace transform of error function.

Answer 1

where
\[\Large erf(x)=\frac{2}{\sqrt{\pi}}\int\limits_{0}^{x}e^{-t^2}dt\]
\[\Large L[efr(x)] =?\]

Answer 2

\[\Large L[efr(t)] =\int_{0}^{\infty}\frac{2}{\sqrt{\pi}}\int\limits_{0}^{t}e^{-y^2}dy \ e^{-st} dt\\ \large =\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}\int\limits_{0}^{t}e^{-y^2}dy \ e^{-st} dt\\ \large =\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}\int\limits_{0}^{t}e^{-(y^2+st)}dy \ dt\\ \large\]we need to change \(dy \ dt\) to \(dt \ dy\) bounds of integrals will change also