Trying to solve this question:
Let $f(x)=e^{-x^2}$ be a Gaussian. Compute explicitly $(f*f)(x)$.
Using the definition of the convolution, and given the fact that the convolution of 2 Gaussians is another Gaussian, I got \begin{align*} (f*f)(x) &= \int_{-\infty}^{\infty}f(x-y)f(y)\,dy\\ &=\int_{-\infty}^{\infty}e^{-(x-y)^2}e^{-y^2}\,dy \end{align*} but I'm not sure how to proceed from here.
Any tips would be appreciated!
First, complete the square to get $-a((y+b)^2+cx^2)$, then you could take $e^{-acx^2}$ beyond the sign of the integral since integration goes over $y$ and change the integration variable to $(y+b)$. Finally, use the well-known formula for the Gaussian integral. As an answer, I've got $\sqrt{\frac{\pi}{2}}\cdot e^{-\frac{x^2}{2}}$