system
Given the following post, respond as if you're commenting on Reddit:
human
Is there any Mathematical reason that (-0.5)!^2 = Pi, or is it just a coincidence?
gpt
It's not a coincidence. To extend the factorial function to non-natural numbers, you replace it with an integral. The factorial of x is then the integral of t^x e^-t dt from 0 to infinity, where t is a dummy variable.

For x=-1/2, this integral works out to the square root of pi. It is a similar integration to the Gaussian curve, which bounds an area of root-pi as well.