Factorials!

Factorials

Factorials are simple.

$ 3! = 3 \times 2 \times 1 = 6 \textrm{ and } 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \textrm{ and so on} ... $

It can be calculated using some tricks too.
$ 6! = 6 \times ( 5 \times 4 \times 3 \times 2 \times 1 ) = 6 \times 5! = 6 \times 120 = 720 $

$ \frac{10!}{3! \times 8!} = \frac{10 \times 9 \times 8!}{3 \times 2 \times 1 \times 8!} = \frac{10 \times 9}{3 \times 2} = 15 $

But does the following make sense?
$ 0! = 1 \textrm{ and even more bizarre is } (1/2)! = (1/2) \sqrt{\pi} $
Well, how does that make sense?

Definition of Factorial

Note that the following integral identically equals to $n!$.
$ \int_0^\infty e^{-x} x^{n} dx = n! $

So, if we take this integral as the definition of factorial operation, then the weird results make sense.
$ 0! = \int_0^\infty e^{-x} x^{0} dx = \int_0^\infty e^{-x} dx = -e^{-x} \Big|_0^\infty = 1 $
$ (1/2)! = \int_0^\infty e^{-x} x^{1/2} dx = (1/2) \sqrt{\pi} $

With this definition, any real number's factorial can be calculated by simply integrating (well, the integral could be quite hard!).

Hmm... what about imaginary and complex numbers?

My Attempt at Complex Factorials

Coming soon...