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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!

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    $\begingroup$ As done hundred times before somewhere on this site: complete the square and use standard gaussian integral afterwards $\endgroup$ – tired Nov 29 '16 at 19:01
  • $\begingroup$ @tired Sorry about that! And, also I had tried completing the square, but it made it more complicated i felt. After i did the square, how would it help make the integral easier to solve? Thanks $\endgroup$ – KidMe Nov 29 '16 at 19:03
  • $\begingroup$ @tired Don't mind, but your name suits after your comment! :-D $\endgroup$ – Qwerty Nov 29 '16 at 19:08
  • $\begingroup$ Hint: How does a change $x\rightarrow x+a$ changes the range of integration if it is initially $(-\infty,\infty)$ $\endgroup$ – tired Nov 29 '16 at 19:09
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    $\begingroup$ @KidMe correct. with this result anything should be clear i think $\endgroup$ – tired Nov 29 '16 at 19:22

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