In this article, I will be solving the infamous Gaussian integral shown below that is known for it’s surprising solution and rather creative method of solving. For those who are already familiar with this integral you may not learn much as I will just be going through the standard polar coordinates method, but for those who are not familiar, this may be an interesting article for you.
The Normal Distribution Curve
Before even attempting to solve this, I would like to just focus on the integrand for a second and graph it as it may seem familiar to some people:
Above is a graph of y = exp(-x²) which may remind you of the normal distribution curve (or bell curve), and that is because it actually is indeed the normal distribution curve, just not normalized so that the area under the curve is 1. The actual area under the curve is precisely what we will be calculating later in this article. In fact, the actual expression for the normal distribution curve is as follows:
Where μ is the mean and σ is the variance. If you look at this expression carefully you may notice that it does in fact have the…
