# What is the Domain of Gamma Function?

The gamma function is defined as $\Gamma(s)=\int_0^\infty t^{s-1}e^{-t}dt$ for s>0.

But then it says that

"The gamma function is defined for all complex numbers except the negative integers and zero."

So what is the domain of gamma function? Is it $\{x\in\mathbb R\,:\, x>0\}$ or all complex numbers except the negative integers and zero?

Well, first note that the integral that defines $\Gamma$ is only defined on the positive real numbers. If $p>0$, then, integration by parts yields the formula $\Gamma(p+1)=p\Gamma(p)$. Using this formula, one can extend the domain of definition inductively by setting first $\Gamma(p)=\frac{\Gamma(p+1)}{p}$ for $-1<p<0$ (note that the right hand side is well defined) and then proceed "backwards", that is for a positive integer $n$, set $\Gamma(p-n)=\frac{\Gamma(p-n+1)}{p-n+1}$ where $-1<p<0$. Since $\Gamma$ is not defined on $0$ (because there it is infinite) we cannot extend this formula in a nicely way to the negative integers and therefore, it is not defined there.
The gamma function is given by this integral for all positive $x$. Then there exists an analytic function with domain $\Bbb C\setminus\{0,-1,-2,\dots\}$ such that its restriction to positive axis coincides with the value of that integral. This continuation is unique and by convention we call it gamma function, too.
you said that The gamma function is defined for all complex numbers except the negative integers and zero yourself, im assuming this is true, and I would write that as $$\mathbb{C} / \{n \in \mathbb{Z}, n \leq 0\}$$