Hi, nerd blog! (This is a post that I wrote a long time ago and then never published. I figured it would be nice to publish it on March 14th.)
Today, we’re interested in the Gaussian integral
By a change of variables, we immediately see that is proportional to . But, what is the constant of proportionality? It’s actually nicer to ask a slightly different question: what is the unique value of such that . A quick numerical computation shows that . E.g., here’s some Mathematica code to find this value: .