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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?

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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.

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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.

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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 $$\mathbb{C} / \{n \in \mathbb{Z}, n \leq 0\}$$

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