The Gamma function is an extension of the factorial function to real and complex arguments. Gamma (n) == (n - 1)! Hence, (-1)! == Gamma (0). But for non-positive integers, the Gamma function isn't analytic (it is elsewhere). http://mathworld.wolfram.com/GammaFunction.html.

But there are other extensions, like the Roman Factorial, which would give 1 as value for -1 http://mathworld.wolfram.com/RomanFactorial.html.

Abigail