Properties of Gamma Function

Theorem
Let $\N_0 = \N \cup \{ 0 \}$.

The gamma function has the following properties:

1. Euler's reflection formula:


 * $\displaystyle \Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)},\quad \forall z \notin -\N_0$

2. Legendre's duplication formula:


 * $\displaystyle \Gamma(z)\Gamma\left(z + \frac 12\right) = 2^{1-2z}\sqrt{\pi} \Gamma(2z),\quad \forall z \notin -\N_0$

3. For all $z$ with $|\arg(z)| < \pi - \epsilon$, $|z| > 1$,


 * $\displaystyle \frac{\Gamma'(z)}{\Gamma(z)} = \log z + \mathcal O\left( z^{-1} \right)$

where the implied constant depends on $\epsilon$.

4. $\Gamma(\overline{z}) = \overline{\Gamma(z)},\quad \forall z \notin -\N_0$