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gamma_function [2016/10/26 15:38] nikolaj |
gamma_function [2016/10/26 16:10] nikolaj old revision restored (2015/12/16 09:59) |
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- | ===== Bayes algorithm ===== | + | ===== Gamma function ===== |
==== Function ==== | ==== Function ==== | ||
- | | @#FF9944: definition | @#FF9944: $\Gamma: (X\to {\mathbb R})\to X\to {\mathbb R}$ | | + | | @#FF9944: definition | @#FF9944: $\Gamma: \mathbb C\setminus\{-k\ |\ k\in\mathbb N\}\to \mathbb N$ | |
- | | @#FF9944: definition | @#FF9944: $bel(x) := N\$p_z(x|)\int_A p_u(x|x)bel'(x'){\mathrm d}$ | | + | | @#FF9944: definition | @#FF9944: $\Gamma(z) := \begin{cases} \int_0^\infty\ \ t^{z-1}\ \mathrm{e}^{-t}\ \mathrm d t & \mathrm{if}\ \mathrm{Re}(z)>0 \\\\ \frac{1}{z}\Gamma(z+1) & \mathrm{else} \end{cases}$ | |
----- | ----- | ||
=== Discussion === | === Discussion === | ||
- | $N*$ normaliz | + | $\Gamma(z)=\Pi(z-1)$ |
- | + | ||
=== Theorems === | === 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 === | === Reference === | ||
Wikipedia: [[http://en.wikipedia.org/wiki/Gamma_function|Gamma function]] | Wikipedia: [[http://en.wikipedia.org/wiki/Gamma_function|Gamma function]] | ||
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----- | ----- | ||
=== Context === | === Context === | ||
- | [[Function]] | + | [[Function integral on ℝⁿ]], [[Complex exponents with positive real bases]] |
+ | === Equivalent to === | ||
+ | [[Pi function]] | ||
=== Related === | === Related === | ||
[[Factorial function]] | [[Factorial function]] |