Properties of Gamma Function

Theorem

The gamma function $\Gamma \left({z}\right)$ has the following properties:

$\map \Gamma {z + 1} = z \, \map \Gamma z$

$\forall z \notin \Z: \map \Gamma z \map \Gamma {1 - z} = \dfrac \pi {\map \sin {\pi z} }$

$\forall z \notin \set {-\dfrac n 2: n \in \N}: \map \Gamma z \map \Gamma {z + \dfrac 1 2} = 2^{1 - 2 z} \sqrt \pi \, \map \Gamma {2 z}$

where $\N$ denotes the natural numbers.

Let $z \in \C \setminus \Z_{\le 0}$.

Then:

$\ds \dfrac {\map {\Gamma'} z} {\map \Gamma z} = -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {z + n - 1} }$

where:

$\map \Gamma z$ denotes the Gamma function

$\map {\Gamma'} z$ denotes the derivative of the Gamma function

$\forall z \in \C \setminus \set {0, -1, -2, \ldots}: \map \Gamma {\overline z} = \overline {\map \Gamma z}$

$\map \Gamma {\dfrac 1 2} = \sqrt \pi$