$${\Gamma(z) = (n-1)!}$$
$${\Gamma(z) = \int_{0}^{\infty} x^{z-1} e^{-x} \ dx }$$
$${\Gamma(z) = \lim_\limits{n \to \infty} \left( \frac{n!\ n^z}{z (z+1) (z+1) \cdots (z + n)} \right)}$$
$${\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{n = 1}^{\infty} \left(1 + \frac{z}{n} \right)^{-1} e^{ \frac{z}{n} } }$$
$${\Gamma(z + 1) = z \Gamma(z)}$$
$${\Gamma(1-z) \Gamma(z) = \frac{\pi}{\sin(\pi z)} }$$
$${\Gamma\left(\frac{1}{2}-z\right) \Gamma\left(\frac{1}{2} + z\right) = \frac{\pi}{\cos(\pi z)} }$$
$${\Gamma(z) \Gamma\left(z + \frac{1}{2} \right) = 2^{1 - 2z} \sqrt{\pi} \ \Gamma(2z) }$$
$${\Gamma\left(z + \frac{1}{2} \right) = \frac{\sqrt{\pi} \ \Gamma(2z)}{2^{2z - 1} \Gamma(z)} }$$
$${\prod_{k = 0}^{n-1} \Gamma\left(z + \frac{k}{n} \right) = (2 \pi)^{\frac{n-1}{2}} n^{ \frac{1}{2} - nz } \Gamma(nz)}$$
$${\Gamma \left( \frac{1}{2} \right) = \sqrt{\pi}}$$