On the Euler Integral for the Positive and Negative Factorial
We reviewed the Euler integral for the factorial, Gauss' Pi function, Legendre's gamma function and beta function, and found that gamma function is defective in \Gamma(0) and \Gamma( x) because they are undefined or indefinable. And we came to a conclusion that the definition of a negative factorial, that covers the domain of the negative space, is needed to the Euler integral for the factorial, as well as the Euler $Y$ function and the Euler $Z$ function, that supersede Legendre's gamma function and beta function.