The factorial function is defined as $$n!=\prod_{k=1}^{n}k$$ For $n>0$, $n$ being an integer. This definition can be extended to the complex numbers with positive real part by the Gamma function: $$\Gamma(n)=(n-1)!$$ This is the plot of the factorial function, in which the values of the factorial of non-integral numbers are evaluated by the gamma function: It can be seen that for the positive numbers, the factorial is always nonzero. Why is it so?
According to mathguy's comment, the factorial function is positive for all the complex numbers. Is this true? If yes, then why is it so?