<s>[INST] I want to prove $k(x,t)=\frac{1}{\sqrt{4\pi t} } e^{\frac{-x^2}{4t}} $ I have this integral $$u(x,t)=\int _{-\infty}^{\infty} f(\eta)\left[\frac{1}{2\pi}\int _{-\infty}^{\infty}e^{iw(x-\eta)-w^2t}\ dw\right]\ d\eta=\int _{-\infty}^{\infty}k(x-\eta,t)f(\eta)\ d\eta$$
I want to prove $$k(x,t)=\frac{1}{\sqrt{4\pi t} }\ e^{\Large \frac{-x^2}{4t}} 
$$
Thanks for helping me out.
 [/INST] Completing square gives
\begin{align*}
 &= \int_{-\infty}^\infty \exp ( i\omega (x-\eta ) - \omega^2t)d\omega \\ 
 &= \int_{-\infty}^\infty \exp \left( - \left( i (x-\eta ) \cdot \frac{1}{2\sqrt t}\right )^2+2 \omega \sqrt{t} i (x-\eta ) \cdot \frac{1}{2\sqrt t} - \omega^2t  + \left( i (x-\eta ) \cdot \frac{1}{2\sqrt t}\right )^2\right)d\omega \\ 
 &= \exp\left(- \frac{(x-\eta)^2}{4t}  \right )\int_{-\infty}^\infty \exp \left(- \left(\omega \sqrt t -  \frac {i (x-\eta)} {2 \sqrt t} \right )^2 \right )d\omega\\ 
 &= \exp\left(- \frac{(x-\eta)^2}{4t} \right) \frac 1 {\sqrt t} \int_{-\infty}^\infty \exp \left(- \left(\omega \sqrt t - \frac {i (x-\eta)} {2 \sqrt t}  \right )^2 \right )d(\omega \sqrt t) \\ 
 &= \exp\left(- \frac{(x-\eta)^2}{4t} \right) \sqrt{\frac \pi t}
\end{align*}
</s>