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pi_function [2013/09/09 15:47] 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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Wikipedia: [[http://en.wikipedia.org/wiki/Gamma_function|Gamma function]] | Wikipedia: [[http://en.wikipedia.org/wiki/Gamma_function|Gamma function]] | ||
==== Parents ==== | ==== Parents ==== | ||
- | === Requirements === | + | === Context === |
[[Function integral on ℝⁿ]], [[Complex exponents with positive real bases]] | [[Function integral on ℝⁿ]], [[Complex exponents with positive real bases]] | ||
=== Equivalent to === | === Equivalent to === | ||
[[Gamma function]] | [[Gamma function]] |