Recently, I saw this video showing a proof of the Gaussian integral, set to DNCE's Cake By The Ocean. It's pretty slick—I recommend watching with sound on. This inspired me to create a blogpost explaining what's being shown here.
We start with the Gaussian integral:
It's called the Gaussian integral because it's the integral of the Gaussian function, f(x)=e−x2. This function is pretty important, and shows up in a ton of different domains like physics, statistics, and signal processing. You're probably familiar with its graph:
This function has an interesting property: while the indefinite integral of f(x) cannot be written using elementary functions, the definite integral over (−∞,∞)can be solved analytically. This is what we will be exploring today.
The video starts by equating the integral to
This allows us to rewrite the expression as an integral of two variables:
∫−∞+∞e−x2 converges, so we can treat this as a case of an integral multiplied by a constant with respect to y. Thus, we have
e−y2 does not vary with respect to x, so we can move it into the inner integral, yielding
That x2+y2 tips us off that the next step will be to convert the integral to polar coordinates. To do so, we replace x and y with ρcosθ and ρsinθ, and our bounds of integration become ρ∈[0,∞) and θ∈[0,2π]