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pi_function [2013/09/08 18:30] nikolaj |
pi_function [2015/01/12 18:19] nikolaj |
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| ===== Pi function ===== | ===== Pi function ===== | ||
| - | ==== Definition ==== | + | ==== Function ==== |
| - | | @#FFBB00: $\Pi: \mathbb C\setminus\{-k\ |\ k\in\mathbb N^*\}\to \mathbb N$ | | + | | @#FFBB00: definiendum | @#FFBB00: $\Pi: \mathbb C\setminus\{-k\ |\ k\in\mathbb N^*\}\to \mathbb N$ | |
| - | | @#FFBB00: $\Pi(z) := \begin{cases} \int_0^\infty\ \ t^{z}\ \mathrm{e}^{-t}\ \mathrm d t & \mathrm{if}\ \mathrm{Re}(z)>0 \\\\ \frac{1}{z+1}\Pi(z+1) & \mathrm{else} \end{cases}$ | | + | | @#FFBB00: definiendum | @#FFBB00: $\Pi(z) := \begin{cases} \int_0^\infty\ \ t^{z}\ \mathrm{e}^{-t}\ \mathrm d t & \mathrm{if}\ \mathrm{Re}(z)>0 \\\\ \frac{1}{z+1}\Pi(z+1) & \mathrm{else} \end{cases}$ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
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| === Theorems === | === Theorems === | ||
| ^ $n\in\mathbb N \implies \Pi(n)=n! $ ^ | ^ $n\in\mathbb N \implies \Pi(n)=n! $ ^ | ||
| - | ^ $\Pi(x)\cdot \Pi(x+1)=\frac{\pi}{\sin(\pi\ x)} $ ^ | + | ^ $\Pi(z)\cdot \Pi(-z)=\frac{\tau\ z/2}{\sin(\tau\ z/2)} $ ^ |
| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Gamma_function|Gamma function]] | Wikipedia: [[http://en.wikipedia.org/wiki/Gamma_function|Gamma function]] | ||
| ==== Parents ==== | ==== Parents ==== | ||
| - | === Requirements === | ||
| - | [[Function integral on ℝⁿ]], [[Complex exponents with positive real bases]] | ||
| === Context === | === Context === | ||
| + | [[Function integral on ℝⁿ]], [[Complex exponents with positive real bases]] | ||
| + | === Equivalent to === | ||
| [[Gamma function]] | [[Gamma function]] | ||
