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gamma_function [2015/12/16 09:59] nikolaj |
gamma_function [2016/10/26 15:46] nikolaj |
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| - | ===== Gamma function ===== | + | ===== Bayes algorithm ===== |
| ==== Function ==== | ==== Function ==== | ||
| - | | @#FF9944: definition | @#FF9944: $\Gamma: \mathbb C\setminus\{-k\ |\ k\in\mathbb N\}\to \mathbb N$ | | + | |
| - | | @#FF9944: definition | @#FF9944: $\Gamma(z) := \begin{cases} \int_0^\infty\ \ t^{z-1}\ \mathrm{e}^{-t}\ \mathrm d t & \mathrm{if}\ \mathrm{Re}(z)>0 \\\\ \frac{1}{z}\Gamma(z+1) & \mathrm{else} \end{cases}$ | | + | | @#FF9944: definition | @#FF9944: $\Gamma: (X\to {\mathbb R})\to X\to {\mathbb R}$ | |
| + | | @#FF9944: definition | @#FF9944: $bel(x) := N^*W_z(x)\int_A K_u(x,x')bel'(x'){\mathrm d}$ | | ||
| ----- | ----- | ||
| === Discussion === | === Discussion === | ||
| - | $\Gamma(z)=\Pi(z-1)$ | + | $N^*$ is supposed to be the normalization of the whole term on the right of it. |
| + | |||
| + | $K_u(x,x')$ 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 integral on ℝⁿ]], [[Complex exponents with positive real bases]] | + | [[Function]] |
| - | === Equivalent to === | + | |
| - | [[Pi function]] | + | |
| === Related === | === Related === | ||
| [[Factorial function]] | [[Factorial function]] | ||
