I started to read about the history of the Gamma Function. There are three places I liked most,
- The early history of the factorial function (p. 239 - 243)
- Leonhard Euler's Integral: An Historical Profile of the Gamma Function
- Mathematical Thought from Ancient to Modern Times, Vol. 2 (p. 422 - 424)
To summarize very short what I have found out
Euler was able to make an exact formula of $n!$, for all $n\in \mathbb{Z}^+$ in his first letter to Goldbach (1729). He somehow determined $(n/2)!$ for all $n$ odd, and the domain of the exact formula of $n!$ was probably extended into $\mathbb{Q}^+$, as far as I understand. (*)
He made formula in terms of an integral, for all $n\in \mathbb{Z}^+$ in his second letter to Goldbach (1730). I assume it also works for all $n\in \mathbb{Q}^+$, since it is equivalent to the the first formula.
Using Legendres notation $\Gamma$, Euler found, in 1781, another nice representation that we are using today. A rewritten version is $$\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\,\mathrm{d}t \tag{1}$$ (which is defined for ... what, according to Euler?)
The formula $\Gamma(n+1)=n\Gamma(n)$ is known by Euler, so that he could determine $\Gamma(3/2)$ or similar. Hence it's for $n\in \mathbb{Q}^+$, see (*) above. Did he also know that it could be extended into $\mathbb{Q}\setminus \{0,-1,-2,\dots\}$?
We know well that $(1)$ is defined for $x\in \mathbb{R}^+$, in modern books. Did Euler know this? (I think he would define it for all $x\in \mathbb{Q}^+$). I mean, was he aware of it, though he didn't mention it, or was it something the later mathematicians added it to be more precise?
I am sorry if it's a long post. It was hard for me express what I had in my mind.