I need to find the domain of the Gamma function, that is to say all $z \in \mathbb{C}$, for which the integral:
$$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \mathrm dt$$
converges. I started by splitting up the integral into an integral running from $0$ to $1$ and another one from $1$ to $\infty$. I first tried to figure out for what $z \in \mathbb{C}$ the integral from $0$ to $1$ converges and I came to the conclusion, that $\Re(z) > 0$ is the condition.
The other integral, I believe, converges for every $z$, as the exponential function dominates the monomial eventually. So I concluded:
$$\exists \Gamma(z) \iff \Re(z) > 0$$
However, I just learned that this is wrong. I found out that the integral only diverges for non-positive integers. What did I do wrong or what is a better way to find the domain of the Gamma function?
Best Answer
The integral does converge iff $\Re z > 0$. However, it defines a function that can be analytically extended to the whole complex plane except the non-positive integers.
There are analogous integral representations for $\Gamma(z)$ which hold true for $\Re z < 0$. For instance, it is not difficult to show that for any $k\in\mathbb N$ $$\Gamma(z)=\int_{0}^{\infty} t^{z-1}\left(e^{-t}-1+t-\frac{t^2}{2!}+\dots+ (-1)^{k+1}\frac{t^k}{k!}\right)dt,$$ where $-k < \Re z < - k+1$ (the Cauchy–Saalschütz integral).