|This article/section deals with mathematical concepts appropriate for a student in late high school or early university.|
The Gaussian integral is the integral:
It has a value of . The value is needed to normalize the Normal distribution.
First look at the double integral
So, the double integral is merely the square of the Gaussian integral.
Now, do the double integral in polar co-ordinates. − x2 − y2 = − (x2 + y2) = − r2 and dxdy = rdrdθ, so:
Substituting z = r2 into the integral,