I'm trying to show that that for a function $f(x)=\exp(-x^2)$, the fundamental solution of the heat equation reduces to a given solution $u(x,t)$.
But, I am stuck on the following part, that is showing:
$\int_{-\infty}^{\infty}e^{-s^2(1+4t)}e^{4xs\sqrt{t}}\,ds = e^{4tx^2\pi/(1+4t)}$$ \sqrt{\pi} \over\sqrt{1 + 4t} $
If this can be shown, the the rest of the problem is straightforward.
Any help would be much appreciated. Thanks!