Properties of Gamma Function

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Theorem

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


Gamma Difference Equation

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


Euler's Reflection Formula

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


Legendre's Duplication Formula

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

where $\N$ denotes the natural numbers.


Reciprocal times Derivative

For all $z \in \C$ such that $\cmod {\map \arg z} < \pi - \epsilon, \cmod z > 1$:

$\dfrac {\map {\Gamma'} z} {\map \Gamma z} = \ln z + \map {\mathcal O_\epsilon} {z^{-1} }$

where:

$\map {\mathcal O} {z^{-1} }$ denotes big-O notation
the implied constant depends on $\epsilon$.


Complex Conjugate

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


Gamma Function of One Half

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

Its decimal expansion starts:

$\map \Gamma {\dfrac 1 2} = 1 \cdotp 77245 \, 38509 \, 05516 \, 02729 \, 81674 \, 83341 \, 14518 \, 27975 \ldots$