===== Pi function =====
==== Function ====
| @#FFBB00: definiendum | @#FFBB00: $\Pi: \mathbb C\setminus\{-k\ |\ k\in\mathbb N^*\}\to \mathbb N$ |
| @#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 ====
$\Pi(z)=\Gamma(z+1)$
=== Theorems ===
^ $n\in\mathbb N \implies \Pi(n)=n! $ ^
^ $\Pi(z)\cdot \Pi(-z)=\frac{\tau\ z/2}{\sin(\tau\ z/2)} $ ^
=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Gamma_function|Gamma function]]
==== Parents ====
=== Context ===
[[Function integral on ℝⁿ]], [[Complex exponents with positive real bases]]
=== Equivalent to ===
[[Gamma function]]