Pi function
Function
| definiendum | $\Pi: \mathbb C\setminus\{-k\ |\ k\in\mathbb N^*\}\to \mathbb N$ |
| definiendum | $\Pi(z) := \begin{cases} \int_0^\infty\ \ t^{z}\ \mathrm{e}^{-t}\ \mathrm d t & \mathrm{if}\ \mathrm{Re}(z)>0 \\\\ \frac{1}{z+1}\Pi(z+1) & \mathrm{else} \end{cases}$ |
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: Gamma function
Parents
Context
Equivalent to
