===== Gamma function =====
==== Function ====
| @#FF9944: definition  | @#FF9944: $\Gamma: \mathbb C\setminus\{-k\ |\ k\in\mathbb N\}\to \mathbb N$ |
| @#FF9944: definition  | @#FF9944: $\Gamma(z) := \begin{cases} \int_0^\infty\ \ t^{z-1}\ \mathrm{e}^{-t}\ \mathrm d t & \mathrm{if}\ \mathrm{Re}(z)>0 \\\\ \frac{1}{z}\Gamma(z+1) & \mathrm{else} \end{cases}$ |

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=== Discussion ===
$\Gamma(z)=\Pi(z-1)$
=== Theorems ===
^ $n\in\mathbb N\land n\neq 0 \implies \Gamma(n)=(n-1)! $ ^

^ $\Gamma(z+1) = z\cdot\Gamma(z) $ ^
^ $\Gamma(z)\cdot\Gamma(1-z)  =\frac{\pi}{\sin(\pi\ z)} $ ^
^ $\Gamma(z)\cdot\Gamma(z+1/2)=2^{1-2z}\ \pi^{1/2}\ \Gamma(2z) $ ^
=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Gamma_function|Gamma function]]

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=== Context ===
[[Function integral on ℝⁿ]], [[Complex exponents with positive real bases]]
=== Equivalent to ===
[[Pi function]]
=== Related ===
[[Factorial function]]