The Gaussian Integral

The Gaussian or Probability Integral is an essential concept in mathematics particularly in the fields of probability theory, statistics and quantum mechanics.

The Gaussian integral, closely related to the erf function, is the integral of the one-dimensional Gaussian function over (-∞, ∞).

The Gaussian Integral can also be defined as the integral of the exponential of -x² over the entire real line.

It can be calculated using the trick of combining two one-dimensional Gaussians:

In this case, Since the variable in the integral is a dummy variable i.e, it integrates out in the end, we can rename from x to y.

When switching to polar coordinates we get:

Example

Prove the following:

Solution

Convert the integral into the polar coordinates (r, θ) where x² + y² = r², 

and dxdy = rdrdθ:

Evaluating the integrals:

Therefore: