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gamma_function [2016/10/26 15:46]
nikolaj
gamma_function [2016/10/26 16:10] (current)
nikolaj old revision restored (2015/12/16 09:59)
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-===== Bayes algorithm ​=====+===== Gamma function ​=====
 ==== Function ==== ==== Function ====
- +| @#FF9944: definition ​ | @#FF9944: $\Gamma: \mathbb C\setminus\{-k\ |\ k\in\mathbb ​N\}\to \mathbb ​N$ | 
-| @#FF9944: definition ​ | @#FF9944: $\Gamma: ​(X\to {\mathbb ​R})\to X\to {\mathbb ​R}$ | +| @#FF9944: definition ​ | @#FF9944: $\Gamma(z) := \begin{cases} \int_0^\infty\ \ t^{z-1}\ \mathrm{e}^{-t}\ \mathrm d t & \mathrm{if}\ \mathrm{Re}(z)>0 \\\\ \frac{1}{z}\Gamma(z+1\mathrm{else} \end{cases}$ |
-| @#FF9944: definition ​ | @#FF9944: $bel(x) := N^*W_z(x)\int_A K_u(x,x')bel'​(x'​){\mathrm ​d}$ |+
  
 ----- -----
 === Discussion === === Discussion ===
-$N^*$ is supposed to be the normalization of the whole term on the right of it. +$\Gamma(z)=\Pi(z-1)$
- +
-$K_u(x,x')$ ought to capture the propagation,​ possibly determined by actions $u$. +
- +
-$W_z$ ought to capture a redistribution of believe, due to some observation $z$+
- +
 === 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]]
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