A negative, and a non integer factorial!

$n! = \Gamma(n+1) \\ (-1/2)! = \Gamma(1/2) \\ \Gamma(1/2) = \int_0^\infty x^{-1/2}e^{-x} dx$ Let $x = u^2.$ Then $\Gamma(1/2) = \int_0^\infty 2e^{-u^2} du \\ = \int_{-\infty}^{\infty} e^{-u^2} du$ This is a famous integral that eluded early attempts to solve, since there is no elementary antiderivative. The definite integral above, though, can be evaluated as $\boxed{\sqrt{\pi}.}$

By the way, even today, factorial is typically defined only for nonnegative integers, so I think the "undefined" answer should be replaced with a different answer so that people do not think you are referring to the classical definition $\prod_{k=1}^n k$ which is undefined for $n = -1/2.$

By definition, we have $n!=\Gamma(n+1)$ , the Gamma Function. Now $(-\frac{1}{2})!=\Gamma(\frac{1}{2})=\sqrt{\pi}$ by Euler's Reflection Formula, $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}$ .

Check the note in my profile😍✌️