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Derivative of a factorial

What is ${\partial\over \partial x_i}(x_i !)$ where $x_i$ is a discrete variable? Do you consider $(x_i!)=(x_i)(x_i-1)...1$ and do product rule on each term, or something else? Thanks.
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Proof that $Γ'(1) = -γ$?

I know that $Γ'(1) = -γ$, but how does one prove this? Starting from the basics, we have that: $$Γ(x) = \int_0^\infty e^{-t} t^{x-1} dt$$ How do we differentiate this? How do we then find that ...
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Asymptotic approximation regarding the Gamma function $\Gamma$.

On the wikipedia page for Gamma function I saw an interesting formula $$\lim_{n\to \infty} \frac{\Gamma(n+\alpha)}{\Gamma(n)n^\alpha} = 1$$ for all $\alpha\in\Bbb C$. I couldn't find the source of ...
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Can there be only one extension to the factorial?

Usually, when someone says something like $\left(\frac12\right)!$, they are probably referring to the Gamma function, which extends the factorial to any value of $x$. The usual definition of the ...
Evaluate: $\lim\limits_{r \to \infty} \frac{\sqrt{r}}{e^{r}}\sum_{n=0}^{\infty}\frac{\Gamma{(n+3/2)}r^n}{(n!)^2}$
Evaluate: $$\lim\limits_{r \to \infty} \frac{\sqrt{r}}{e^{r}}\sum_{n=0}^{\infty}\frac{\Gamma{(n+3/2)}r^n}{(n!)^2}$$ My effort: \begin{aligned}\Gamma \left({\tfrac {1}{2}}+n\right)&={(2n)! \...