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

Function integral on ℝⁿ, Complex exponents with positive real bases

Equivalent to

Gamma function