Is the Gamma function defined for complex numbers like so?

Question

I know $\Gamma (x)$ can be defined as...

$$\Gamma (x) = (x-1)!$$

And as...

$$\Gamma (x) = \int_{0}^{\infty} u^{x-1} e^{-u} \space du$$

But assuming we had a Complex variable $z$ and had the following...

$$(z-1)!$$

Would it be correct for me to say...

$$\Gamma (z) = (z-1)!$$

Or is that only defined for Natural values?

Answer 1

$\Gamma(z)$ is a function of a complex variable $z$ so that $\Gamma(n) = (n-1)!$ for $n = 1,2,...$, the positive integers. Just writing $\Gamma(z) = (z-1)!$ does not give a definition for $\Gamma(z)$. However, sometimes the notation $z!$ is used to denote the value of $\Gamma(z+1)$. If you define the symbol $z!$ to mean $\Gamma(z+1)$ then the statement $\Gamma(z) = (z-1)!$ is true. You can use the symbol $!$ this way if you want, but you should probably clarify your definitions beforehand.