You can definitely extend the definition of factorial to the complex numbers using Euler's Gamma function, but the notation is different (indeed ! is reserved f

You can definitely extend the definition of factorial to the complex numbers using Euler's Gamma function, but the notation is different (indeed ! is reserved for integers). As far as the second question goes, that's taken to be the definition e^x := \sum_{n=0}^{\infty} (x^n)/(n!)