## Gaussian Integral

A Gaussian function (named after one of the greatest mathematician, Carl Friedrich Gauss) is a function of the form:

We would have like to examine the following integral:

But we are facing a problem, what is the antiderivative function of ?

Well, You won’t be able to find one, at least not an elementary one. The integral of a Gaussian is the Gauss error function, but in this special case, there is a cool way to calculate this integral.

At first, Instead of looking at , lets take a look at . Meaning:

**Note:** In the second integral, I’ve changed the variable of integration to y. I can do that, because it’s a completely separated integral.

Now, notice that the following integral is equivalent to the following:

Well, That’s a bit weird. Now we have a double integral! We’ve made the problem much more complex! Or did we?

Well, yes, now we have a double integral, but there is a good reason for that. Now we can change our coordinate system to the polar coordinate system (Meaning: ) and remember to multiply by the Jacobian determinant! In our case, the Jacobian determinant is . So, we got:

But wait a minute, this looks familiar! Notice that:

Therefore: