# 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.

In other words, we want to prove $$k(x-\eta,t)=\frac1{2\pi}\int_{-\infty}^\infty e^{\large i(x-\eta)\omega-t\omega^2}\ d\omega$$ or $$k(x,t)=\frac1{2\pi}\int_{-\infty}^\infty e^{\large ix\omega-t\omega^2}\ d\omega=\frac1{\pi}\int_{0}^\infty e^{\large -(t\omega^2+ix\omega)}\ d\omega$$ In general (click the link below for the complete proof) $$\int_{x=0}^\infty e^{-(ax^2+bx)}\,dx=\frac{1}{2}\sqrt{\frac{\pi}{a}}\exp\left(\frac{b^2}{4a}\right).$$ We have $a=t$ and $b=ix$, therefore \begin{align} k(x,t)&=\frac1{\pi}\frac{1}{2}\sqrt{\frac{\pi}{t}}\exp\left(\frac{(ix)^2}{4t}\right)=\large\color{blue}{\frac1{\sqrt{4\pi t}}\ e^{\Large-\frac{x^2}{4t}}}.\qquad\qquad\blacksquare \end{align}