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Gamma function

Definitions

Symbol: GammaFunction $\Gamma\!\left(z\right)$ Gamma function
Symbol: LogGamma $\log \Gamma\!\left(z\right)$ Logarithmic gamma function

Illustrations

Image: X-ray of $\Gamma\!\left(z\right)$ on $z \in \left[-5, 5\right] + \left[-5, 5\right] i$
Image: X-ray of $\log \Gamma\!\left(z\right)$ on $z \in \left[-5, 5\right] + \left[-5, 5\right] i$

Particular values

$\Gamma\!\left(n\right) = \left(n - 1\right)!$
$\Gamma\!\left(1\right) = 1$
$\Gamma\!\left(2\right) = 1$
$\Gamma\!\left(\frac{1}{2}\right) = \sqrt{\pi}$
$\Gamma\!\left(\frac{3}{2}\right) = \frac{\sqrt{\pi}}{2}$

Functional equations

$\Gamma\!\left(z + 1\right) = z \Gamma\!\left(z\right)$
$\Gamma\!\left(z\right) = \left(z - 1\right) \Gamma\!\left(z - 1\right)$
$\Gamma\!\left(z - 1\right) = \frac{\Gamma\!\left(z\right)}{z - 1}$
$\Gamma\!\left(z + n\right) = \left(z\right)_{n} \Gamma\!\left(z\right)$
$\Gamma\!\left(z\right) = \frac{\pi}{\sin\!\left(\pi z\right)} \frac{1}{\Gamma\!\left(1 - z\right)}$
$\Gamma\!\left(z\right) \Gamma\!\left(z + \frac{1}{2}\right) = {2}^{1 - 2 z} \sqrt{\pi} \Gamma\!\left(2 z\right)$
$\prod_{k=0}^{m - 1} \Gamma\!\left(z + \frac{k}{m}\right) = {\left(2 \pi\right)}^{\left( m - 1 \right) / 2} {m}^{1 / 2 - m z} \Gamma\!\left(m z\right)$
$\Gamma\!\left(z\right) = \exp\!\left(\log \Gamma\!\left(z\right)\right)$
$\log \Gamma\!\left(z + 1\right) = \log \Gamma\!\left(z\right) + \log\!\left(z\right)$

Integral representations

$\Gamma\!\left(z\right) = \int_{0}^{\infty} {t}^{z - 1} {e}^{-t} \, dt$

Series expansions

$\log \Gamma\!\left(1 + z\right) = -\gamma z + \sum_{k=2}^{\infty} \frac{\zeta\!\left(k\right)}{k} {\left(-z\right)}^{k}$
$\log \Gamma\!\left(z\right) = \left(z - \frac{1}{2}\right) \log\!\left(z\right) - z + \frac{\log\!\left(2 \pi\right)}{2} + \sum_{k=1}^{n - 1} \frac{B_{2 k}}{2 k \left(2 k - 1\right) {z}^{2 k - 1}} + R_{n}\!\left(z\right)$
Symbol: StirlingSeriesRemainder $R_{n}\!\left(z\right)$ Remainder term in the Stirling series for the logarithmic gamma function
$R_{n}\!\left(z\right) = \int_{0}^{\infty} \frac{B_{2 n} - B_{2 n}\!\left(t - \left\lfloor t \right\rfloor\right)}{2 n {\left(z + t\right)}^{2 n}} \, dt$
$\Gamma\!\left(z\right) = {\left(2 \pi\right)}^{1 / 2} {z}^{z - 1 / 2} {e}^{-z} \exp\!\left(\sum_{n=1}^{\infty} \left(z + n - \frac{1}{2}\right) \log\!\left(\frac{z + n}{z + n - 1}\right) - 1\right)$

Analytic properties

$\operatorname{HolomorphicDomain}\!\left(\Gamma\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C} \setminus \{0, -1, \ldots\}$
$\operatorname{Poles}\!\left(\Gamma\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \{0, -1, \ldots\}$
$\operatorname{EssentialSingularities}\!\left(\Gamma\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}$
$\operatorname{BranchPoints}\!\left(\Gamma\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{BranchCuts}\!\left(\Gamma\!\left(z\right), z, \mathbb{C}\right) = \left\{\right\}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \Gamma\!\left(z\right) = \left\{\right\}$

Complex parts

$\Gamma\!\left(\overline{z}\right) = \overline{\Gamma\!\left(z\right)}$

Bounds and inequalities

Related topics: Bounds and inequalities for the gamma function

$\Gamma\!\left(x\right) \lt {\left(2 \pi\right)}^{1 / 2} {x}^{x - 1 / 2} {e}^{-x} \exp\!\left(\frac{1}{12 x}\right)$
$\left|\Gamma\!\left(z\right)\right| \le {\left(2 \pi\right)}^{1 / 2} {\left|z\right|}^{x - 1 / 2} {e}^{-\pi \left|y\right| / 2} \exp\!\left(\frac{1}{6 \left|z\right|}\right)\; \text{ where } z = x + y i$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC