Gaussian integral over a union of intervals

Question

I am asking myself if there is any way of computing Gaussian integrals of the form $$\sum_{n=-\infty}^\infty \int_{2n}^{2n+1} e^{-t^2} \, dt,$$ i.e. integrating over all of $\mathbb R$ but leaving out "half" the intervals. Maybe there is no precise value but a way of expressing it in terms of other functions?

Answer 1

You are integrating over all intervals of the form $[2n,2n+1]$, i.e. intervals with even lower bound and odd upper bound. On the positive side of the real line, this includes intervals like $[0,1]$ and $[2,3]$, while on the negative side, it includes intervals like $[-2,-1]$ and $[-4,-3]$. Since the Gaussian $e^{-t^2}$ is an even function, we can take the absolute value of $t$, so we can integrate over $[1,2]$ instead of $[-2,-1]$, for instance. By doing this, we find that we are integrating over $[0,1]$, $[1,2]$, $[2,3]$, and so on. From this we conclude that $$\sum_{n=-\infty}^{\infty}\int_{2n}^{2n+1}e^{-t^2}\,\mathrm{d}t=\int_0^\infty e^{-t^2}\,\mathrm{d}t,$$ which is a well known integral.