# Questions tagged [gamma-function]

Questions on the gamma function $\Gamma(z)$ of Euler extending the usual factorial $n!$ for arbitrary argument, and related functions. The Gamma function is a specific way to extend the factorial function to other values using integrals.

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### Integral representation of $\frac{1}{\Gamma(z)}$

I am trying to find the integral representation of $\frac{1}{\Gamma(z)}$ in the real axis and cant seem to find it. I know that this must have a standard representation but still cant find it.
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### Showing that $\Gamma(s+1) = s\Gamma(s)$ and the Consequences of this Identity

For $\sigma \in \mathbb{R},$ define $\Omega_{\sigma} = \{ s\in \mathbb{C}\;|\; Re(s) > \sigma\}$. On $\Omega_0$, define $\Gamma(s)$ by \begin{align*} \Gamma(s) = \int_{0}^{\infty}x^{s-1}e^{-x}dx \...
I have read $[x^n] (1-x)^\alpha\sim \frac{n^{-\alpha-1}}{\Gamma(-\alpha)}$ as $n\rightarrow\infty$ where $[x^n]$ means coefficient of $x^n$ in what follows. But I have never seen a proof of this ...