How to derive the domain of Gamma function

Question

I was reading about the Gamma function, however, I have some trouble to figure out why the domain of

$$\Gamma(\alpha)= \int_0^{\infty} x^{\alpha -1}e^{-x}\,dx$$

How can I get $ \alpha > -1$?

Answer 1

$$ \int_0^\infty x^{\alpha-1} e^{-x} \,dx \le \int_0^1 x^{\alpha-1} \,dx + \int_1^\infty x^{\alpha-1} e^{-x}\,dx. $$ The first integral on the right converges if $\alpha>0.$ $$ \int_0^\infty x^{\alpha-1} e^{-x} \, dx \ge \frac 1 e \int_0^1 x^{\alpha-1} \,dx + \int_1^\infty x^{\alpha-1} e^{-x} \, dx. $$ The first integral on the right diverges to $+\infty$ if $\alpha<0.$

The second integral on the right can be shown to converge if $\alpha>0$ by going through the usual integration by parts.

By using the functional equation $\Gamma(\alpha+1) = \alpha\Gamma(\alpha),$ you can extend the domain to negative non-integers, but then it's not equal to the integral. By analytic continuation, you can extend it to complex numbers.