In probability theory the important Gaussian Distribution often leads to the integral .

In case you ever wondered where it comes from, here (informal but nice*) is a non-complex analysis demonstration.

We begin by considering the integral,

This is the integral over the entire plane .

Now there are two ways to get the entire plane.

Method 1:For we can create a rectangle . Now we make . That infinite rectangle will take the entire plane.

In other words, .

Now, the important realization is that this double integral has "seperated variables" meaning and we can write . Hence, (because it cannot be negative).

Method 2:We notice the expression in the double integral and try a polar coordinates substitution. If we let , we can let and . In other words, we draw a circle at the orgin with infinite radius, and this will take on values on the entire plane.

Thus, we end up with after a change to polar coordinates,

Thus, which implies .

And hence,

*)I am sure it can be made formal. But I do not know how do it because I am not familar with the theory of real multiple integration.